vss2020/Homepage-Summer-School/... · 2019. 4. 13. · Outline 1 Introduction 2 Generalized Super Twisting Algorithm (GSTA) A Quadratic Strong Lyapunov Function for the GSTA Convergence

Homogeneity Based Design of Sliding ModeControllers

Jaime A. Moreno

Universidad Nacional Autonoma de MexicoElectrica y Computacion, Instituto de Ingenierıa,

04510 Mexico D.F., Mexico, [email protected]

8th-12th April 2019, Rio de Janeiro, BrasilEECI International Graduate School in Control 2019, M13

www.eeci-igsc.euInternational Summer School on Sliding Mode Control -

Variable Structure Systems

Course Overview

Generalized Super-Twisting Algorithm

HOSM Control and Homogeneity

Lyapunov-Based Design of Higher-Order Sliding Mode(HOSM) Controllers

HOSM Differentiation/Observation: A Lyapunov Approach

Construction of Lyapunov Functions using GeneralizedForms

Continuous HOSM Controllers

Homogeneity Based SMC Jaime A. Moreno UNAM 2

Course Overview








Course Overview








Course Overview








Course Overview








Course Overview








Part I



Outline

1 Introduction

2 Generalized Super Twisting Algorithm (GSTA)A Quadratic Strong Lyapunov Function for the GSTAConvergence TimeGSTA with perturbations: LMI

3 The GSTA with Variable (Adaptive) Gains

4 Uniformity of\ the GSTA with Constant GainsA Non Quadratic Strong Lyapunov Function for the GSTAUniformity in the ConvergenceAn Alternative Robust Lyapunov Function

5 Output Feedback with Twisting and GSTA: A LyapunovApproach

6 Conclusions


Overview

1 Introduction





6 Conclusions


Second Order Sliding Modes

Super-Twisting Algorithm (STA)[Levant, 1993]

x1 = −k1 |x1|12 sign (x1) + x2

x2 = −k2 sign (x1) ,

Used for control to attenuate chattering,Used for exact differentiaton and observation.

Twisting Algorithm [Emel’yanov, Levant, 1985].

x1 = x2

x2 = −k1 sign (x1)− k2 sign (x2) , k1 > k2

Prescribed Convergence Law (Terminal Sliding ModeController)[Levant, 1993; Man et al., 1994]

x1 = x2

x2 = −α sign(x2 + β |x1|

12 sign (x1)

),


Quasi-Continuous Controller [Levant, 2006]

x1 = x2

x2 = −αx2+β|x1|12 sign(x1)

|x2|+β|x1|12

,

Many others: Sub-Optimal Controller, etc. [Bartolini,Ferrara, Usai, 1998]


Arbitrary Order Sliding Modes

“Arbitrary Order Robust Exact Differentiator” [Levant,2003].

x1 = −k1 |x1|n−1n sign (x1) + x2

x2 = −k2 |x1|n−2n sign (x1) + x3 ,

...xn = −kn sign (x1) ,

Used for exact differentiaton and observation.

Nested Sliding Mode Controllers [Levant, 2003].

x1 = x2

x2 = x3

x3 = −α sign

(x3 + 2

(|x2|3 + |x1|2

) 16

sign(x2 + |x1|

23 sign (x1)

)),

for n = 3 and recursively for n > 3


Quasi-Continuous Controller [Levant, 2006]

x1 = x2

x2 = x3

x3 = −αx3+2

(|x2|+|x1|

23

)− 12(x2+|x1|

23 sign(x1)

)|x3|+2

(|x2|+|x1|

23

) 12

,

Many Others are possible


Analysis of HOSM Algorithms

Homogeneity

Definition

dilation: κ > 0, (positive) weights (m1, m2, · · · , mn)dκ : (x1, x2, · · · , xn)→ (κm1x1, κ

m2x2, · · · , κmnxn)homogeneous vector field (or inclusion) of degree q

F (x) = κ−qd−1κ F (dκx)

local stability = global stability (as for linear systems)If q < 0 Finite-Time stabilityRobustnessBut: It does not assure stability.

Contraction analysis using geometry to certify stability


Lyapunov analysis of HOSM I

Weak (non strict) Lyapunov Functions:

V (x) > 0 , V (x) ≤ 0 , ∀x 6= 0

They have been used for some SOSM algorithms [Orlov,2005; ....]

Super Twisting Algorithm

x1 = −k1 |x1|12 sign (x1) + x2 , V (x) = k2 |x1|+ 1

2x2

2

x2 = −k2 sign (x1) V (x) = −k1k2 |x1|1/2

Twisting Algorithm

x1 = x2 V (x) = k1 |x1|+ 12x2

2

x2 = −k1 sign (x1)− k2 sign (x2) , V (x) = −k2 |x2|

They assure stability (using generalized Lasalle’s Lemma)


Lyapunov analysis of HOSM II

They cannot assure robustness (under perturbations), andno convergence time estimation

Strong (strict) Lyapunov Functions:

V (x) > 0 , V (x) < 0 , ∀x 6= 0

It assures (robust) stabilityIf V (x) ≤ −γV p (x) , 0 ≤ p < 1, Finite-Time stability andConvergence Time Estimation

Recently some results for SOSM algorithms have been obtained:


Lyapunov analysis of HOSM III

Construction using Zubov’s Theorem [Polyakov andPoznyak, 2009, 2011]: for Twisting and Super TwistingAlgorithms, Suboptimal Controller, Terminal SMController.

Plus: systematic and general method; valid for system withperturbations; tight convergence time estimation.Difficulties: Solution of a PDE or PD Inequality; “complex”Lyapunov Fs

Quadratic Lyapunov functions for GeneralizedSupertwisting Algorithms [Moreno and Osorio, 2008;Moreno, 2009,...]

Some other results [Orlov, ....]


Why Strong Lyapunov funct. for HOSM?

Advantages of Lyapunov functions:

Simple analysis tool

It allows the estimation of convergence time

Robustness analysis tool

It does not rely on homogeneity

Several Extensions possible:

Uniform algorithms: convergence time independent of i.c.Variable (adaptive) gainsMultivariable (?)

Design tool: Control Lyapunov Functions

Addition of extra terms to improve performance

Analysis and design of interconnected systems

Some requirements to a Lyapunov method

Systematic and efficient“Simple” Lyapunov functions (recall quadratic LF forLinear Systems)


Overview

1 Introduction





6 Conclusions


Generalized Super Twisting Algorithm(GSTA)

x1 = −k1 (t)φ1 (x1) + x2 + ρ1 (t, x)x2 = −k2 (t)φ2 (x1) + ρ2 (t, x) ,

φ1 (x1) = µ1|x1|p sign (x1) + µ2|x1|q sign (x1) ,

φ2 (x1) = µ21p|x1|2p−1 sign (x1) + µ1µ2 (p+ q) |x1|p+q−1 sign (x1) +

+ µ22q|x1|2q−1 sign (x1) ,

µ1, µ2 ≥ 0 constant. Non homogeneous.

q ≥ 1 ≥ p ≥ 12 are real numbers.

φ1 (x1), φ2 (x1) monotone increasing continuous if p > 12 .

If p = 12 φ2 (x1) discontinuous at x1 = 0.

Trajectories in the sense of Filippov


Particular cases with constant gains k1, k2:

A linear Algorithm: (µ1, µ2, p, q) = (1, 0, 1, 1).

x1 = −k1x1 + x2

x2 = −k2x1 .

The classical Super-Twisting Algorithm (STA) [Levant,93]: (µ1, µ2, p, q) =

(1, 0, 1

2 , q).

x1 = −k1 |x1|12 sign (x1) + x2

x2 = −12k2 sign (x1) .

φ2 (x1) is discontinuous.

A Homogeneous Algorithm: For p ≥ 12

x1 = −k1 |x1|p sign (x1) + x2

x2 = −k2p |x1|2p−1 sign (x1) .

Generalized Super-Twisting Algorithm (GSTA) [Moreno2009]: p = 1

2 , q = 1.


Stability Properties: dependent on p, q.

Low order term High order Stability Type

p = 1 q = 1ExponentialRobust

Non UniformPractical

12 < p < 1 q = 1

Finite-TimeRobust


p = 12 q = 1

Finite-TimeExact


p = 1 q > 1ExponentialRobust

UniformPractical

12 < p < 1 q > 1

Finite-TimeRobust

UniformPractical

p = 12 q > 1

Finite-TimeExact

UniformPractical

Exact: Convergence to the origin under bounded perturbations.Homogeneity Based SMC Jaime A. Moreno UNAM 18

Outline

1 Introduction





6 Conclusions


Quadratic Lyapunov Function

Family of quadratic strong Lyapunov Functions:

VQ(x) = ζTPζ , P = P T > 0 ,

Algebraic Lyapunov Equation (ALE):

ATP + PA = −Q

ζ =

[φ1 (x1)x2

], A =

[−k1 1−k2 0

].

Time derivative of Lyapunov Function:

VQ(x) = φ′1 (x1) ζT(ATP + PA

)ζ = −φ′1 (x1) ζTQζ


Figure : The Lyapunov function VQ(x), p = 1/2, µ2 = 0.


Stability of GSTA without Perturbations

Theorem

If µ1 > 0, k1 , k2 are constant. The statements are equivalent:

(i) x = 0 is asymptotically stable.

(ii) A Hurwitz.

(iii) k1 > 0 , k2 > 0.

(iv) ∀Q = QT > 0, ALE has a unique solution P = P T > 0.

VQ(x) is a global, strong Lyapunov function.

VQ ≤ −γ1 (Q,µ1)V3p−1

2p

Q (x)− γ2 (Q,µ2) |x1|q−1 VQ (x) ,

γ1 (Q,µ1) , µ1p

1 pλmin Qλ

1−p2p

min Pλmax P

, γ2 (Q,µ2) , µ2qλmin Qλmax P

Remarkable: Stability of GSTA ⇔ Stability of ξ = Aξ.Homogeneity Based SMC Jaime A. Moreno UNAM 22

Outline

1 Introduction





6 Conclusions


Convergence Time

Proposition

Suppose k1 > 0 , k2 > 0, µ1 > 0.• If 1

2 ≤ p < 1: Finite-Time Convergence

T (x0) =

2p

(1−p)γ1(Q,µ1)V1−p2p

Q (x0) µ2 = 0 or q > 1

2p(1−p)γ2(Q,µ2) ln

(1 + γ2(Q,µ2)

γ1(Q,µ1)V1−p2p

Q (x0)

)µ2 > 0 and q = 1

• If p = 1: exponential convergence.

For Design: T depends on the gains!


Figure : Convergence time for p = 1/2, q = 1: µ2 = 0 (blue) andµ2 > 0 (green).


Outline

1 Introduction





6 Conclusions


GSTA with perturbations: LMI

x1 = −k1φ1 (x1) + x2

x2 = −k2φ2 (x1) + ρ2 (t, x) .

|ρ2 (t, x)| ≤ δ(pµ2

1 |x1|2p−1 + (p+ q)µ1µ2 |x1|p+q−1 + qµ22 |x1|2q−1

)The robust stability analysis can be performed through the LMI[

ATP + PA+ εP + δ2CTC PBBTP −1

]≤ 0 ,

A =

[−k1 1−k2 0

], C =

[1 0

], B =

[01

].

Globally, robustly stable. VQ (x) = ζTPζ Lyapunovfunction.For 1 > p ≥ 1

2 , µ1 > 0 Finite-Time convergenceFor p = 1, µ1 > 0 Exponential convergence.For p = 1

2 Exact Stability.Homogeneity Based SMC Jaime A. Moreno UNAM 27

Frequency Domain Interpretation: TheCircle Criterium

Classical Circle Criterium: LMI is satisfied ⇔ Nyquist diagramof G (s) = C (sI −A)−1B is in circle with radius δ, ⇔

maxω|G (jω)| < 1

δ, G (s) =

1

s2 + k1s+ k2.

Figure : Nyquist Diagramm of G(jω).


maxω|G (jω)|2 =

1k2

2if k2 − 1

2k21 < 0

1k2

1(k2− 14k2

1)if k2 − 1

2k21 > 0

.

Figure : Robust Stability Region for the Gains k1, k2


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Time (sec). p=3/2

Unp

ertu

rbed

x1,

x2

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Time (sec). p=1

Unp

ertu

rbed

x1,

x2

0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec). p=3/4

Unp

ertu

rbed

x1,

x2

0 0.5 1 1.5 2 2.5 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec). p=1/2

Unp

ertu

rbed

x1,

x2

x1x2

x1x2

x1x2

x1x2

Figure : States of the GSOA without perturbation,p =

(32 , 1 , 3

4 ,12

), µ1 = 1, µ2 = 0. k1 = 3, k2 = 2


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Time (sec). p=3/2

Per

turb

ed x

1, x

2

0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec). p=1

Per

turb

ed x

1, x

2

0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec). p=3/4

Per

turb

ed x

1, x

2

0 0.5 1 1.5 2 2.5 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec). p=1/2

Per

turb

ed x

1, x

2

x1x2

x1x2

x1x2

x1x2

Figure : States of the GSOA with a persistent perturbationρ2(t, x) = 1.9 sin(10t), p =

(32 , 1 , 3

4 ,12

), µ1 = 1, µ2 = 0.


Practical Stability: for large perturbations

|ρ1| ≤ δ1 , |ρ2| ≤ δ2 ,

1 if µ2 = 0, p = 12 , δ2 <

µ21λminQ4λmaxP then

VQ ≤ −κ(12µ

21λmin Q − 2δ2λmax P

)λ

12max P

V12

Q (x) ,

∀ ‖ζ‖2 >µ21δ1λmax P

2 (1− κ)(12µ

21λmin Q − 2δ2λmax P

) ,⇒ Practical Stability (ISS), if δ2 is small enough.

2 If 12 < p ≤ 1 then for any 0 < κ < 1

VQ ≤ −κpµ

1p

1 λmin Q

λ3p−12p

max PV

3p−12p

Q (x)−κµ2λmin Qλmax P

VQ (x) , ∀ ‖ζ‖2 /∈ D

D is a compact set, containing the origin ⇒ Practical Stability(ISS)


0 2 4 6 8 10 12

0

2

4

6

8

10

Time (sec). p=3/2

Per

turb

ed x

1, x

2

0 5 10 150

5

10

15

Time (sec). p=1

Per

turb

ed x

1, x

2

0 5 10 150

5

10

15

20

25

30

35

Time (sec). p=3/4

Per

turb

ed x

1, x

2

0 5 10 150

100

200

300

400

500

Time (sec). p=1/2

Per

turb

ed x

1, x

2

x1x2

x1x2

x1x2

x1x2

Figure : States of the Super-Twisting Algorithm with a constantperturbation ρ2(t, x) = 10, p =

(32 , 1 , 3

4 ,12

), µ1 = 1, µ2 = 0,

k1 = 3, k2 = 2.


Overview

1 Introduction





6 Conclusions


Variable Gain GSTA

x1 = −k1 (t, x)φ1 (x1) + x2 + ρ1 (t, x)x2 = −k2 (t, x)φ2 (x1) + ρ2 (t, x) ,

|ρ1 (t, x)| ≤ g1 (t, x) |φ1 (x1)||ρ2 (t, x)| ≤ g2 (t, x) |φ2 (x1)|

where g1 (t, x) ≥ 0, g2 (t, x) ≥ 0 are known continuous functions.If β > 0, ε > 0, δ > 0 arbitrary constants,

k1(·) =δ +1

β

1

4ε[2εg1 + g2]2 + 2εg2 + ε+ [2ε+ g1]

(β + 4ε2

)

k2(·) =β + 4ε2 + 2εk1(·)

Then

x = 0 is robustly GAS.Homogeneity Based SMC Jaime A. Moreno UNAM 35

VQ (x) = ζTPζ constant, strong, robust Lyapunov function.

For 1 > p ≥ 12 , µ1 > 0 Finite-Time Convergence

T (x0) =

2p

(1−p)γ1V

1−p2p

Q (x0) if µ2 = 0 or q > 1

2p(1−p)γ2

ln

(1 + γ2

γ1V

1−p2p

Q (x0)

)if µ2 > 0 and q = 1

γ1, γ2 some constants.

For p = 1, µ1 > 0 Exponential Convergence.

If p = 12 Exact Stability.

Useful for Adaptive Control [Shtessel 2011]

Proof: It is possible to construct a constant, robust Lyapunovfunction V (x) = ζTPζ with A(t)TP + PA(t) < 0, andζT =

[φ1 (x1) , x2

].


Overview

1 Introduction





6 Conclusions


Outline

1 Introduction





6 Conclusions


An Alternative Strong Lyapunov Function

Theorem

Suppose k1 > 0 , k2 > 0 constant, α = k2δ, β = 1, δ > 0.

VN (x) = α |φ1 (x1)|2−β |φ1 (x1)|1q sign (x1) |x2|

2q−1q sign (x2)+δx2

2 ,

is a global, strong Lyapunov function for δ sufficiently large.

VN ≤ −1

qνmin

(1

4δmax 1, k2

) 3q−12q

V3q−1

2q

N (x) ,

νmin , minx1∈R

pµ1 |x1|p−q + qµ2(µ1 |x1|p−q + µ2

) q−1q

.

For q ≥ 1, µ2 > 0 global asymptotic stability of the origin isassured.


Outline

1 Introduction





6 Conclusions


Lyapunov function and Unif. Convergence

W (x) = VQ (x) + VN (x) strong Lyapunov func. for GSOA.Uniform Convergence: When the convergence time has anupper bound independent of the initial condition!

101

102

103

104

0

2

4

6

8

10

12

14

16

norm of the initial condition ||x(0)|| (logaritmic scale)

Converg

ence T

ime T

NSOSMO

GSTA with linear term

STO

Figure : Convergence time when the initial condition grows.


Finite-Time and Uniform Convergence forp < 1 < q

Proposition

Suppose k1 > 0 , k2 > 0, µ1, µ2 > 0 and 12 ≤ p < 1 < q. Then

Finite-Time and Uniform convergence

T (x0) =2q

(q − 1)κ2

(1

µq−12q

− 1

Wq−12q (x0)

)+

2p

(1− p)κ1µ

1−p2p ,

where W (x) = VQ (x) + VN (x), µ, κ1, κ2 are some constants.

T (x0) ≤ Tmax =2q

(q − 1)κ2

(κ2

κ1

) q−pp(q−1)

+2p

(1− p)κ1

(κ1

κ2

) q(1−p)q−p

i.e. any trajectory converges to x = 0 in a time smaller thanTmax.


Outline

1 Introduction





6 Conclusions


Robust stability for perturbations in sector

Proposition

Assume k1 > g1 , k2 >(

4q2

3q−1

) q2q−1

g3q−12q−1

2 are constant. VN (x) is

a global, robust, strong Lyapunov function.

VN ≤ −2ψminνmin

(1

4δmax 1, k2

) 3q−12q

V3q−1

2q

N (x) ,

with appropriate νmin, ψmin.

For q > 1, µ2 > 0 GAS.

For µ1, µ2 > 0, 12 ≤ p < 1 < q Finite-Time and uniform

stability.

For p = 12 exact stability.

Using W (x): Practical Stability for every q ≥ 1.


Overview

1 Introduction





6 Conclusions


OF Twisting Controller

Plant with relative degree 2

x1 = x2

x2 = a (t, ξ) + b (t, ξ)u ,

a, b unknown scalar functions.

0 < bm ≤ |b (t, ξ)| ≤ bM , |a (t, ξ)| ≤ a ,

Discontinuous Observer

˙x1 = −l1φ1 (x1 − x1) + x2˙x2 = −l2φ2 (x1 − x1) + 1

2 (bm + bM )u .

Output feedback

u = −k1 sign (x1)− k2 sign (x2)


LF for State Feedback: Twisting Controller

Closed Loop with State Feedback (we assume b (t, ξ) > 0)

x1 = x2

x2 = −b (t, ξ) k1 sign (x1)− b (t, ξ) k2 sign (x2) + a (t, ξ) .

(Lipschitz) Continuous (strict) Lyapunov Function

V (x) =

(π1 |x1|+

1

2x2

2

) 32

+ π2x1x2

π2 > 0 , p > 0 , π1 = p+

(1 +

bMbm

)a+

2

32

12π2 .

V (x) ≤ −γV23 (x) .


Under the conditions for the gains

k2 +p

bM> k1 −

a

bm> k2 >

1

2

(1 +

bMbm

)a

bm+

232

3

π2

bm+

1

2

(1

bm− 1

bM

)p > 0

x = 0 is a robust, finite-time, globally stable point.Convergence-Time estimation

T (x0) ≤ 3

γV

13 (x0) .


OF Closed Loop Stability

Estimation errors e1 = x1 − x1 and e2 = x2 − x2. Closed loopdynamics

Π :

x1 = x2

x2 = −b (t, ξ) k1 sign (x1)− b (t, ξ) k2 sign (x2) + a (t, ξ) + χ (t, ξ, x2, e2) ,

Ξ :

e1 = −l1φ1 (e1) + e2

e2 = −l2φ2 (e1) + ρ2 (t, ξ, u) ,

where

χ (t, ξ, x2, e2) , b (t, ξ) k2 [sign (x2 (t))− sign (x2 (t) + e2 (t))] ,

ρ2 (t, ξ, u) , −a (t, ξ) +

(1

2(bm + bM )− b (t, ξ)

)u .


Bounds of the perturbations

|χ (t, ξ, x2, e2)| ≤ 2bMk2 ,

|ρ2 (t, ξ, u)| ≤ 1

2g(

1 + 3µ |e1|12 + 2µ2 |e1|

),

g , 2a+ (bM − bm) (k1 + k2) .

If (l1, l2) and (k1, k2) are appropiately designed, then CL isfinite time stable.


What about LF for HOSM?

Second order robust exact differentiator

x1 = −k1 |x1|23 sign (x1) + x2

x2 = −k2 |x1|13 sign (x1) + x3 ,

x3 = −k3 sign (x1) + δ (t) ,

Lyapunov Function

V (x) = γ1 |x1|43 − γ12 |x1|

23 sign (x1)x2 + γ2 |x2|2 + γ13x1x3

−γ23x2 |x3|2 sign (x3) + γ3 |x3|4

V (x) = −(

4

3γ1k1 − γ12k2 + γ13k3

)|x1|+ 2

(2

3γ1 − γ2k2 +

1

3γ12k1

)|x1|

13 sign (x1)x2

− 2

3γ12|x2|2

|x1|13

− (γ12 + γ13k1) |x1|23 sign (x1)x3 + γ23k2 |x1|

13 sign (x1) |x3|2 sign (x3)

− 4γ3k3 sign (x1)x33 + (2γ2 + γ13 + 2γ23k3 sign (x1x3))x2x3 − γ23 |x3|3 .


Ongoing Work

Systematic method to construct Lyapunov functions forSOSM and HOSM: Twisting Algorithm, Terminal(Prescribed Time) Controller. [Tonametl Sanchez]

Lyapunov-Based Adaptive SM Control: [Shtessel, Moreno,Fridman]

Uniformization of HOSM algorithms [Emmanuel Cruz]

SOSM for PDEs: Supertwisting Algorithm [RamonMiranda]

Robust and Robust Parameter Estimation [Eder Guzman]

Robust observation and control for Chemical andBioprocesses [A. Vargas, E. Rocha, A. Vande Wouwer, J.Alvarez...]


Overview

1 Introduction





6 Conclusions


Conclusions

Developing a Lyapunov-based methodology for HOSMseems to be important for:

To have quantitative analysis and design tools for HOSMTo integrate the HOSM to the standard Nonlinear Methods:interconections and combinations

We are at the beginning of this development: much work isneeded

To find Lyapunov functions is good but to find asystematic an efficient way of construction is better.

Thanks to colleagues and students: L. Fridman, M. Osorio,M.T. Angulo, E. Cruz-Zavala, E. Guzman, R. Miranda, T.Sanchez, ....


Part II



Outline

7 Preliminaries

8 HOSM Control Problemρ = 1, First Order Sliding Mode (FOSM) Control Problemρ = 2, Second Order Sliding Mode (SOSM) Control

9 HomogeneityClassical HomogeneityWeighted HomogeneityWeighted Homogeneity for systems with inputs(perturbations)Weighted Homogeneity and Precision under perturbationsHomogeneous Approximation/DominationExample: ”Danger” of Non Homogeneous Controllers

10 Homogeneous Design of HOSM (Levant 2005)

11 Plaidoyer for a Lyapunov-Based Framework for HOSM


Overview

7 Preliminaries






Differential Inclusions (DI)

Consider a dynamical system

x = f(x) .

We know that

If f(x) discontinuous according to Filippov we obtain a DI

x ∈ F (x) , F (x) ⊂ Rn .

If f(x) is uncertain, i.e. ‖f(x)‖ ≤ f+ we can write

x ∈[−f+, f+

]⇒ x ∈ F (x) .

In case of discontinuity or/and uncertainty we obtain aDifferential Inclusion from a Differential Equation


Filippov Differential Inclusions (DI)

x ∈ F (x)

is Filippov DI if ∀x ∈ Rn, the set-valued function F (x) ⊂ Rn is

not empty;

compact;

convex;

upper-semicontinuous, i.e.

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

wheredist(x, A) = inf |x− a||a ∈ A .

A solution x(t) is an absolutely continuous functionsatisfying the DI almost everywhere.

Filippov DI have the usual properties, except foruniqueness.



x ∈ F (x)


not empty;

compact;

convex;


limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0






x ∈ F (x)


not empty;

compact;

convex;


limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0






x ∈ F (x)


not empty;

compact;

convex;


limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0






x ∈ F (x)


not empty;

compact;

convex;


limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0






x ∈ F (x)


not empty;

compact;

convex;


limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0





Higher Order Sliding Mode (HOSM)

Consider a Filippov DI x ∈ F (x), with a smooth outputfunction σ = σ(x). If

1 The total time derivatives σ, σ, . . . , σ(r−1) are continuousfunctions of x

2 The setσ = σ = . . . = σ(r−1) = 0 (1)

is a nonempty integral set (i.e., consists of Filippovtrajectories)

3 The Filippov set of admissible velocities at the r-slidingpoints (??) contains more than one vector

the motion on the set (??) is said to exist in an r -sliding(rth-order sliding) mode. The set (??) is called the r-slidingset. The nonautonomous case is reduced to the one consideredabove by introducing the fictitious equation t = 1.


Overview

7 Preliminaries






HOSM Control

SISO smooth, uncertain system

z = f (t, z) + g (t, z)u, σ = h (t, z) ,

z ∈ Rn, u ∈ R, σ ∈ R: sliding variable.

f (t, z) and g (t, z) and n uncertain.

Control objective: to reach and keep σ ≡ 0 in finite time.

Relative Degree ρ w.r.t. σ is well defined, known andconstant.

Reduced (Zero) Dynamics asymptotically stable (byappropriate selection of σ).


The basic DI

Defining x = (x1, ..., xρ)T = (σ, σ, ..., σ(ρ−1))T , σ(i) = di

dtih (z, t)

The regular form

∑T :

xi = xi+1, i = 1, ..., ρ− 1,xρ = w (t, z) + b (t, z)u, x0 = x (0) ,

ζ = φ(ζ, x) ζ0 = ζ(0) ,

0 < Km ≤ b (t, z) ≤ KM , |w (t, z)| ≤ C .

Reduced Dynamics Asymptotically stable:

ζ = φ(ζ, 0) ζ0 = ζ(0) ,

The basic Differential Inclusion (DI)∑DI :

xi = xi+1, i = 1, ..., ρ− 1,xρ ∈ [−C, C] + [Km, KM ]u .


The Basic Problems

Bounded memoryless feedback controller

u = ϑρ(x1, x2, · · · , xρ) ,

Render x1 = x2 = · · · = xρ = 0 finite-time stable.

Motion on the set x = 0 is ρth-order sliding mode.

ϑρ necessarily discontinuos at x = 0 for robustness [−C, C].

Problem 1

How to design an appropriate control law ϑr ?

Problem 2

How to estimate in finite time the required derivativesx = (x1, ..., xρ)

T = (σ, σ, ..., σ(ρ−1))T ?


Outline

7 Preliminaries






ρ = 1, First Order Sliding Mode (FOSM)Control

Introduced in the mid 60’s: Fantastic results, maturetheory, lots of applications,... Utkin, Emelianov, .....Edwards, Spurgeon, Sira-Ramirez, Loukianov, Zinober, ....

σ ∈ [−C, C] + [Km, KM ]u .

u = ϑ1 = −k sign (σ), k > CKm

.

Robust (=insensitive) control, simple realization, finitetime convergence to the sliding manifold,...

Lyapunov design using V (σ) = 12σ

2.

No derivative estimation required.

Mature theory including Multivariable case, adaptation,design of sliding surfaces, ...


Sliding modes

Mathematical Aspects ISliding Mode Equations (cont).

A.F. Filippov, Application of the theory of differential equations with discontinuous

right-hand sides to non-linear problems of automatic control, Proceedings of 1st IFAC

Congress in Moscow, 1960, Butterworths, London, 1961.

grad sxn

x1

s(x)=0

fsm

!f"#$

%

&''

!

(

0)( if

0)( if )( ),(

xsf

xsfxfxfx!

Convex

Hullfsm belongs to convex hullYu.I. Neimark, Note on A. Filippov’s paper,

1st IFAC Congress.

dx/dt=Ax+bu+dv,

u=-sign(s), v=-sign(s),

s=cx

Nonuniqueness !?(f


Continuous vs. Discontinuous Control: Afirst order plant

Consider a plant

σ = α+ u, σ(0) = 1

where α ∈ (−1, 1) is a perturbation.Continuous (linear) Control

σ = α− kσ, k > 0, σ(0) = 1

Comments:

RHS of DE continuous (linear).

If α = 0 exponential (asymptotic) convergence to σ = 0.

If α 6= 0 practical convergence.


Discontinuous Control

σ = α− sign(σ), σ(0) = 1

with α ∈ (−1, 1).

σ > 0⇒ σ =< 0

σ < 0⇒ σ => 0

y σ(t) ≡ 0,∀t ≥ T .

Comments:

¿0 = α− sign(0)?

RHS of DE isdiscontinuous.

After arriving at σ = 0,sliding on σ ≡ 0.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−1

−0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.4

0.6

0.8

1

1.2

σuu

Finite-Time convergence.

Differential Inclusion.

σ ∈ [−α, α]− sign(σ)Homogeneity Based SMC Jaime A. Moreno UNAM 69

0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

t

σ

Lineal

Discontínuo

0 5 10 15 20−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

t

u

Lineal

Discontínuo

Notice the chattering = infinite switching of the controlvariable!


Outline

7 Preliminaries






ρ = 2, Second Order SM (SOSM) Control

Introduced in the mid 80’s: Wonderful results, maturegeometric theory, lots of applications,... Levant, Fridman,Bartolini, Ferrara, Shtessel, Usai, Feng, Man, Yu, Furuta,Spurgeon, Orlov, Perruquetti, Barbot, Floquet, Defoort, ....

∑DI :

x1 = x2,x2 ∈ [−C, C] + [Km, KM ]u .

Some controllers (see Fridman 2011):

Twisting Controller (Emelyanov, Korovin, Levant 1986).ϑ2(x) = −k1sign(x1)− k2sign(x2).Super-Twisting Algorithm (Levant 1993) (as differentiatorLevant 1998).The Sub-Optimal Algorithm (Bartolini, Ferrara, Usai 1997).Terminal Sliding Mode Control (Man, Paplinski, Wu, Yu(1994, 1997..)).


A second order plant

x1 = x2

x2 = φ (x1, x2) + uy = x1

φ: Perturbation/uncertainty.Question: Can we just feedback the output (as for FO case)?Two alternative output controllers:

Continuous (linear) output controller (Homogeneous TimeInvariant (HTI))

x1 = x2

x2 = −k1y

Discontinuous output controller (HTI)

x1 = x2

x2 = −k1sign(y)


0 5 10 15 20 25 30 35 40−3

−2

−1

0

1

2

3

t

x

Linear controller

x1

x2

0 5 10 15 20 25 30 35 40−5

−4

−3

−2

−1

0

1

2

3

4

5

t

x

Discontinuous controller

x1

x2

Both Oscillate!It is impossible to stabilize a double (or triple etc) integrator bystatic output feedback!


State feedback

State feedback controllers:

Continuous (linear) state feedback controller (HTI)

x1 = x2

x2 = −k1x1 − k2x2

Exponential ConvergenceRobust, but Sensitive to perturbations: Practical stability.

Continuous HTI state feedback controller

x1 = x2

x2 = −k1 dx1c13 − k2 dx2c

12

d·cρ = | · |ρ sign(·)Finite Time ConvergenceRobust, but Sensitive to perturbations: Practical stability.


First Order Sliding Mode Controller

x1 = x2

x2 = −k2sign(x2 + k1x1)

Rewrite as a first order system with a stable (first order) zerodynamics: with σ = x2 + k1x1 sliding variable

x1 = −k1x1 + σσ = −k2sign(σ)


Behavior with perturbation

0 10 20 30 40−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

x

Linear controller

x1

x2

0 10 20 30 40

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

t

x


x1

x2

Linear controller stabilize exponentially and is notinsensitive to perturbation

SM control also stabilizes exponentially but is insensitive toperturbation!


The Twisting Controller

A discontinuous HTI controller able to obtain finite-timeconvergence and insensitivity to perturbations:

x1 = x2

x2 = −k1sign(x1)− k2sign(x2)

0 10 20 30 40−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

x

Linear controller

x1

x2

0 10 20 30 40−3

−2

−1

0

1

2

3

t

x


x1

x2


ρ = 2, SOSM Control....

Robust (=insensitive) control, finite time convergence tothe sliding manifold,...

No Lyapunov design or analysis.

Analysis and Design is very geometric: Beautiful butdifficult to extend to ρ > 2.

Solution: Homogeneity!


Overview

7 Preliminaries






Outline

7 Preliminaries






Classical Homogeneity

Classical Homogeneity for functions. (Euler, Zubov,Hahn,...)

Let n, m be positive integers. A mapping f : Rn → Rm ishomogeneous with degree δ ∈ R iff ∀λ > 0 :

f(λx) = λδf(x).

Some examples:

Linear function: Let A ∈ Rm×n then f(x) = Ax ishomogeneous of degree δ = 1, since

f(λx) = A(λx) = λ(Ax) = λf(x) .

f(x1, x2) =x3

1+x32

x21+x2

2is continuous and homogeneous of degree

δ = 1 (but not linear!), since

f(λx1, λx2) =(λx1)3 + (λx2)3

(λx1)2 + (λx2)2=λ3(x3

1 + x32)

λ2(x21 + x2

2)= λf(x1, x2) .





f(λx) = λδf(x).

Some examples:


f(λx) = A(λx) = λ(Ax) = λf(x) .

f(x1, x2) =x3

1+x32

x21+x2



f(λx1, λx2) =(λx1)3 + (λx2)3

(λx1)2 + (λx2)2=λ3(x3

1 + x32)

λ2(x21 + x2

2)= λf(x1, x2) .





f(λx) = λδf(x).

Some examples:


f(λx) = A(λx) = λ(Ax) = λf(x) .

f(x1, x2) =x3

1+x32

x21+x2



f(λx1, λx2) =(λx1)3 + (λx2)3

(λx1)2 + (λx2)2=λ3(x3

1 + x32)

λ2(x21 + x2

2)= λf(x1, x2) .


f(x1, x2) =

dx1c12 +dx2c

12

x1+x2if x1 + x2 6= 0

0 otherwise

is discontinuous and homogeneous of degree δ = −12 , since

f(λx1, λx2) =dλx1c

12 + dλx2c

12

λx1 + λx2= λ−

12 f(λx1, λx2)

Quadratic Form: if x ∈ Rn and P ∈ Rn×n, q(x) = xTPx ishomogeneous of degree δ = 2, since

q(λx) = (λx)TP (λx) = λ2xTPx = λ2q(x) .

Classical Form = homogeneous polynomial: if x ∈ Rn, e.g.

p(x) = α1x1x2x3 + α2x1x3x5 + α3x21x5 + α4x

32 + · · ·

is homogeneous of degree δ = 3.


f(x1, x2) =

dx1c12 +dx2c

12

x1+x2if x1 + x2 6= 0

0 otherwise



12 + dλx2c

12

λx1 + λx2= λ−

12 f(λx1, λx2)





32 + · · ·



f(x1, x2) =

dx1c12 +dx2c

12

x1+x2if x1 + x2 6= 0

0 otherwise



12 + dλx2c

12

λx1 + λx2= λ−

12 f(λx1, λx2)





32 + · · ·




Classical Homogeneity for vector fields. (Zubov, Hahn,...)

Let n be a positive integer. A vector field f : Rn → Rn ishomogeneous with degree δ ∈ R iff ∀λ > 0 :

f(λx) = λδf(x).

Associated with the vector field f(x) is the DifferentialEquation x = f(x), and it has a flow (solution) ϕ(t, x).

Homogeneity of vector field ⇒ Homogeneity of Flow

If ∀λ > 0 :

f(λx) = λδf(x)⇒ ϕ(t, λx) = λϕ(λδ−1t, x)



Classical Homogeneity for vector fields. (Zubov, Hahn,...)

Let n be a positive integer. A vector field f : Rn → Rn ishomogeneous with degree δ ∈ R iff ∀λ > 0 :

f(λx) = λδf(x).

Associated with the vector field f(x) is the DifferentialEquation x = f(x), and it has a flow (solution) ϕ(t, x).


If ∀λ > 0 :

f(λx) = λδf(x)⇒ ϕ(t, λx) = λϕ(λδ−1t, x)


Some examples:

Linear system: Let A ∈ Rn×n then x = f(x) = Ax ishomogeneous of degree δ = 1 and the flow is ϕ(t, x) = eAtx

ϕ(t, λx) = eAt(λx) = λeAtx = λϕ(λ0t, x) .

If x is scalar. System x = −sign(x) is homogeneous withdegree δ = 0, since

f(λx) = sign(λx) = sign(x) = λ0f(x) .

The flow is

ϕ(t, x) =

sign(x)(| x | −t) if 0 ≤ t ≤| x |0 if t >| x |

and it is homogeneous

ϕ(t, λx) =

sign(λx)(| λx | −t) if 0 ≤ t ≤| λx |0 if t >| λx |

= λϕ(t

λ, x) .


Some examples:

Linear system: Let A ∈ Rn×n then x = f(x) = Ax ishomogeneous of degree δ = 1 and the flow is ϕ(t, x) = eAtx

ϕ(t, λx) = eAt(λx) = λeAtx = λϕ(λ0t, x) .

If x is scalar. System x = −sign(x) is homogeneous withdegree δ = 0, since

f(λx) = sign(λx) = sign(x) = λ0f(x) .

The flow is

ϕ(t, x) =

sign(x)(| x | −t) if 0 ≤ t ≤| x |0 if t >| x |

and it is homogeneous

ϕ(t, λx) =

sign(λx)(| λx | −t) if 0 ≤ t ≤| λx |0 if t >| λx |

= λϕ(t

λ, x) .


Outline

7 Preliminaries






Weighted Homogeneity or(quasi-homogeneity) (Zubov, Hermes)

A generalized weight is a vector r = (r1, · · · , rn), withri > 0.

A dilation is the action of the group R+ \ 0 on Rn givenby

Λr : R+ \ 0 × Rn → Rn

(λ, x) → diagλrix

we will denote this for simplicity as Λrx , Λr(λ, x), λ > 0.

Weighted Homogeneity for functions. (Zubov, Hermes...)

Let n, m be positive integers. A mapping f : Rn → Rm isr-homogeneous with degree δ ∈ R iff ∀λ > 0 :

f(Λrx) = λδf(x).





Λr : R+ \ 0 × Rn → Rn





f(Λrx) = λδf(x).





Λr : R+ \ 0 × Rn → Rn





f(Λrx) = λδf(x).


Some remarks

Function f(x1, x2) = x1 + x22 is not homogeneous, but it is

(2, 1)-homogeneous of degree δ = 2 since,

f(λx) = λx1 + (λx2)2 = λx1 + λ2x22 6= λδf(x), ∀λ 6= 1 .

f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .

Classical Homogeneity = Weighted Homogeneity withr = (r1, · · · , rn) = (1, · · · , 1).Values on the unit sphere define an r-Homogeneousfunction.If f(x) is r-homogeneous of degree δ then it is(αr)-homogeneous of degree (αδ) for any α > 0Euler’s Theorem: Let V : Rn → R be differentiable. V isr-homogeneous of degree δ if and only if

n∑i=1

rixi∂V

∂xi(x) = δV (x) .


Some remarks




f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .


n∑i=1

rixi∂V

∂xi(x) = δV (x) .


Some remarks




f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .


n∑i=1

rixi∂V

∂xi(x) = δV (x) .


Some remarks




f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .


n∑i=1

rixi∂V

∂xi(x) = δV (x) .


Some remarks




f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .


n∑i=1

rixi∂V

∂xi(x) = δV (x) .


Weighted Homogeneity

Weighted Homogeneity for (set-valued) vector fields.(Zubov, Hermes, Levant, ...)

Let n be a positive integer. A vector field f : Rn → Rn (aset-valued vector field f : Rn ⇒ Rn) is r-homogeneous withdegree δ ∈ R iff ∀λ > 0 :

f(Λrx) = λδΛrf(x).

Associated with the (set-valued) vector field f(x) is theDifferential Equation x = f(x) (DI x ∈ f(x)), and it has a flow(solution) ϕ(t, x).

r-Homogeneity of vector field ⇒ r-Homogeneity of Flow

If ∀λ > 0 :

f(Λrx) = λδΛrf(x)⇒ ϕ(t, Λrx) = Λrϕ(λδt, x)


Weighted Homogeneity

Weighted Homogeneity for (set-valued) vector fields.(Zubov, Hermes, Levant, ...)

Let n be a positive integer. A vector field f : Rn → Rn (aset-valued vector field f : Rn ⇒ Rn) is r-homogeneous withdegree δ ∈ R iff ∀λ > 0 :

f(Λrx) = λδΛrf(x).

Associated with the (set-valued) vector field f(x) is theDifferential Equation x = f(x) (DI x ∈ f(x)), and it has a flow(solution) ϕ(t, x).

r-Homogeneity of vector field ⇒ r-Homogeneity of Flow

If ∀λ > 0 :

f(Λrx) = λδΛrf(x)⇒ ϕ(t, Λrx) = Λrϕ(λδt, x)


Some examples:

Super-Twisting (ST) Algorithm:

x1 = −k1 dx1c12 + x2

x2 ∈ −k2 dx1c0 + [−1, 1] .

is (2, 1)-homogeneous of degree −1, since

−k1

⌈λ2x1

⌋ 12 + λx2 = λ2−1(−k1 dx1c

12 + x2)

−k2

⌈λ2x1

⌋0+ [−1, 1] = λ1−1(−k2 dx1c0 + [−1, 1]) .


Twisting Algorithm:

x1 = x2

x2 ∈ −k1 dx1c0 − k2 dx2c0 + [−1, 1] .

is (2, 1)-homogeneous of degree −1, since

λx2 = λ2−1(x2)

− k1

⌈λ2x1

⌋0 − k2 dλx2c0 + [−1, 1] =

λ1−1(−k1 dx1c0 − k2 dx2c0 + [−1, 1]) .


Two important examples:

Levant’s arbitrary order differentiator:

xi = −ki dx1 − f(t)cn−in + xi+1

...

xn ∈ −kn dx1 − f(t)c0 .

is (n, n− 1, · · · , 1)-homogeneous of degree −1.

Homogeneous Controller of a chain of integrators:

x1 = x2

...

xn = −n∑i=1

ki dxicαi

is r-homogeneous of degree δ ∈ [−1, 0] with

ri = 1 + (i− n)δ, αi =1 + δ

1 + (i− n)δ.


Two important examples:

Levant’s arbitrary order differentiator:

xi = −ki dx1 − f(t)cn−in + xi+1

...

xn ∈ −kn dx1 − f(t)c0 .

is (n, n− 1, · · · , 1)-homogeneous of degree −1.

Homogeneous Controller of a chain of integrators:

x1 = x2

...

xn = −n∑i=1

ki dxicαi

is r-homogeneous of degree δ ∈ [−1, 0] with

ri = 1 + (i− n)δ, αi =1 + δ

1 + (i− n)δ.


Dynamic interpretation of r-homogeneity

System x = f(x) is r-homogeneous of degree δ, i.e.f(Λrx) = λδΛrf(x).

State Transformation z = Λrx

z = Λrx = Λrf(x) = λ−δf(Λrx)

and thereforedz

d(λδt)= f(z)

System x = f(x) is invariant under the transformation

Gλ : (t, x) 7→ (λ−δt, Λrx) .






and thereforedz

d(λδt)= f(z)


Gλ : (t, x) 7→ (λ−δt, Λrx) .






and thereforedz

d(λδt)= f(z)


Gλ : (t, x) 7→ (λ−δt, Λrx) .


Properties of Homogeneous Systems

Zubov, Hahn, Hermes, Kawski, Rosier, Aeyels, Sepulchre,Grune, Praly, Perruquetti, Efimov, Polyakov,..... Levant,Orlov, Bernuau et al. 2013

If x = 0 Locally Attractive (LA) ⇔ GloballyAsymptotically Stable (GAS)

local contraction ⇒ global contraction ⇒ globalasymptotic stability

If x = 0 GAS and δ < 0 ⇔ x = 0 Finite Time Stable

If x = 0 GAS and δ = 0 ⇔ x = 0 Exponentially Stable (e.g.LTI systems)

If x = 0 GAS and δ > 0 ⇔ x = 0 Asymptotically Stable

If x = 0 GAS ⇔ It exists a Homogeneous LyapunovFunction
























































Outline

7 Preliminaries






Weighted Homogeneity with inputs

Consider a System with inputs u ∈ Rm

x = f(x, u) .

State and input weight vectors r = (r1, · · · , rn), ri > 0,ρ = (ρ1, · · · , ρm), ρi > 0

State and input dilations Λr and Λρ.

Weighted Homogeneity for (set-valued) vector fields withinputs.

A vector field f : Rn × Rm → Rn (a set-valued vector fieldf : Rn × Rm ⇒ Rn) is (r, ρ)-homogeneous with degree δ ∈ R iff∀λ > 0 :

f(Λrx, Λρu) = λδΛrf(x, u).




x = f(x, u) .









x = f(x, u) .







Associated with the (set-valued) vector field f(x, u) is theDifferential Equation x = f(x, u) (DI x ∈ f(x, u)), and it has aflow (solution) ϕ(t, x, u).


If f is homogeneous then ∀λ > 0 :

ϕ(t, Λrx, Λρu(λδ·)) = Λrϕ(λδt, x, u(·))


Dynamic interpretation of homogeneity

System x = f(x, u) is homogeneous of degree δ, i.e.f(Λrx, Λρu) = λδΛrf(x, u).

State and input Transformation z = Λrx, w = Λρu

z = Λrx = Λrf(x, u) = λ−δf(Λrx, Λρu)

and thereforedz

d(λδt)= f(z, w)


Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .

Internal stability ⇒ external stability (iISS, ISS) [Bernuauet al. 2013]






and thereforedz

d(λδt)= f(z, w)


Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .







and thereforedz

d(λδt)= f(z, w)


Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .







and thereforedz

d(λδt)= f(z, w)


Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .



Outline

7 Preliminaries






Weighted Homogeneity and Precision

Consider a System with a scalar and constant input u ∈ R

x = f(x, u) .

so thatϕ(t, Λrx, λ

ρu) = Λrϕ(λδt, x, u) .

Suppose that asymptotically or after a finite time for some u0

limt→∞|ϕi(t, x, u0)| = |ϕ∞i(u0)| ≤ ai .

Therefore (using λ = ( uu0)

1ρ )

|ϕ∞i(u)| = |ϕ∞i(λρu0)| = λri |ϕ∞i(u0)| ≤ νiuriρ ,

with νi = aiu− riρ

0 .


Outline

7 Preliminaries






Homogeneous Approximation

First Lyapunov’s Theorem: x = 0 is LAS for x = f(x) if

x = 0 is AS for linearized system x = Ax, where A = ∂f(0)∂x .

This is not true for Taylor Approximations. Example fromBacciotti & Rosier, 2005:

x1 = −x31 + x3

2 ,

x2 = −x1 + x52 .

x = 0 is GAS for approximation of order 3 (black), but it isnot AS with perturbation (red) of higher order 5!But true for homogeneous approximations (Not unique!)

x1 = −x31 + x2 ,

x2 = −x51 + x2

2 .

x = 0 is GAS for homogeneous approximationr1 = 1, r2 = 3, δ = 2 (black), and it is still AS withperturbation (red), which is homogeneous of degree 3.


Homogeneous Domination

Typical control (similar for observation) problem:

x1 = x2+f1(x1) ,

x2 = x3+f2(x1, x2) ,

...

xn = u+ fn(x) .

Homogeneous Approximation (Black):

x1 = x2 ,...

xn = u = φ(x) .

It is homogeneous of degree δ = −1, 0, +1 and weightsr = (r1 + δ, r1 + 2δ, · · · , r1 + nδ): NOT UNIQUE! ⇒ Fixed byselection of the control law φ(x) ⇒ Domination of other termsfi.


Outline

7 Preliminaries






Alternative Integral + state feedback controllers for System

x1 = x2

x2 = u+ ρ (t) ,

Linear Integral + state feedback controller (Homogeneous)

u = −k1x1 − k2x2 + x3

x3 = −k3x1

Linear state feedback + Discontinuous Integral controller(Not Homogeneous)

u = −k1x1 − k2x2 + x3

x3 = −k3sign(x1)

Discontinuous I-Controller (Extended Super-Twisting)(Homogeneous)

u = −k1|x1|13 sign(x1)− k2|x2|

12 sign(x2) + x3

x3 = −k3sign(x1)


Controller without perturbation

0 10 20 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)

Linear Integral controller

x

1

x2

u

0 10 20 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)

Linear+Discontinuous Integrator

x

1

x2

u

0 10 20 30−7

−6

−5

−4

−3

−2

−1

0

1

2

3

t(x

,u)

Super−Twisting controller

x

1

x2

u


Controller with perturbation

0 5 10 15 20 25 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)


x

1

x2

u

0 5 10 15 20 25 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)


x

1

x2

u

0 5 10 15 20 25 30−7

−6

−5

−4

−3

−2

−1

0

1

2

3

t

(x,u

)


x

1

x2

u


Overview

7 Preliminaries






Homogeneous Design of HOSM [?]

∑DI :


u = ϑr(x1, x2, · · · , xρ) ,

ϑr homogeneous of degree 0 (discontinuous at x = 0) withweights rs = (ρ, ρ− 1, ..., 1)

ϑr(ερx1, ε

ρ−1x2, . . . , εxρ)

= ϑr (x1, x2, . . . , xρ) ∀ε > 0

Local boundedness ⇒ global boundedness

System∑

DI is homogeneous of degree −1 with weights rs

Local contractive ⇔ Global, uniformly Finite-Time stabilityRobustness of stability ⇒ Accuracy with respect tohomogeneous perturbations |xi| ≤ γiτρ+1−i = O

(τρ+1−i),

γi constants.


Concrete Homogeneous HOSM Controllers

Notation: dxcp = |x|psign (x).

Nested Sliding Controllers (NSC)

u2L = −k2

⌈x2 + β1 dx1c

12

⌋0,

u3L = −k3

⌈x3 + β2(|x2|3 + |x1|2)

16

⌈x2 + β1 dx1c

23

⌋0⌋0

,

Quasi-Continuous Sliding Controllers (QCSC)

u2C = −k2(x2+β1dx1c1/2)|x2|+β1|x1|1/2

u3C = −k3x3+β2(|x2|+β1|x1|3/2)

−1/2(x2+β1dx1c3/2)

|x3|+β2(|x2|+β1|x1|3/2)1/2 .


Scaling the gains: again homogeneity!

Example: 2nd Order Nested Controller

x1 = x2

x2 = −k2

⌈x2 + β1 dx1c

12

⌋0

Linear change of coordinates: z = λ2x, 0 < λ ∈ R

z1 = λ2x2 = z2

z2 = −λ2k2

⌈z2

λ2+ β1

⌈ z1

λ2

⌋ 12

⌋0

= −λ2k2

⌈z2

λ+ β1 dz1c

12

⌋0

Preserves stability!


Example: 3rd Order Nested Controller

x1 = x2, x2 = x3,

x3 = −k3

⌈x3 + β2(|x2|3 + |x1|2)

16

⌈x2 + β1 dx1c

23

⌋0⌋0

z = λ3x, 0 < λ ∈ R⇒ Preserves stability!

z1 = z2, z2 = z3,

z3 = −λ3k3

⌈z3

λ3+ β2

(∣∣∣ z2

λ3

∣∣∣3 +∣∣∣ z1

λ3

∣∣∣2) 16⌈z2

λ3+ β1

⌈ z1

λ3

⌋ 23

⌋0⌋0

= −λ3k3

⌈z3

λ2+ β2

(∣∣∣z2

λ

∣∣∣3 + |z1|2) 1

6 ⌈z2

λ+ β1 dz1c

23

⌋0⌋0


Balance

Advantages:

Beautiful and powerful theory: more qualitative thanquantitative

Simple: local = global, convergence = finite-time = robust

scaling the gains (for nested controllers) for convergenceacceleration

u = λρϑr

(x1,

x2

λ, . . . ,

xρλρ−1

), λ > 1


Balance....

Limitations:

Beyond homogeneity unclear how to design ϑr!

It does not provide quantitative results, e.g.

Stabilizing GainsConvergence Time estimationAccuracy GainsPerformance quantities

Behavior with respect to non homogeneous perturbations

Behavior under interconnection

Design for performance

Due to Limitations we require other methods, e.g. Lyapunov(but not exclusively)


Overview

7 Preliminaries






Motivation

Lyapunov based methods

Standard methods in nonlinear control theory

Design for robustness, optimality, performance, etc.

Gain tuning methods

Lyapunov Function for Internal Stability and Lyapunov-likeFunctions for External Stability (e.g. ISS, iISS,...)

LF for design: Control Lyapunov Functions (CLF)

Interconnection analysis is possible

Robustness analysis: noise, uncertainties, perturbations

Objective:

A Lyapunov Based framework for HOSM

Belief: Combination of Homogeneity and LF ⇒ powerful tool!


Main Problem: Construction of LF

Existence of LF for f(x) Filippov Differential Inclusion

It exists a smooth p.d. V (x) [Clarke et al. 1998]

f(x) homogeneous ⇒ V (x) homogeneous [Nakamura et al.2002, Bernuau et al. 2014]

There are basically two issues:

What is the form or structure of the LF?

How to decide if V (x) and W (x) are positive definite(p.d.)?

There are many works on these general topics. But there arefew for HOSM algorithms and taking advantage of thehomogeneity properties.


State of the art

Orlov, 2005 Weak LF for Twisting Algorithm. MechanicalEnergy.

Moreno and Osorio, 2008 Strong non smooth LF forSuper–Twisting. No method.

Polyakov and Poznyak, 2009 Strong non smooth LF forTwisting and ST. Zubov’s Method.

Santiesteban et al., 2010 Strong LF for Twisting with linearterms. No method.

Polyakov and Poznyak, 2012 Strong LF for Twisting, Terminaland Suboptimal. Zubov’s Method.

Sanchez and Moreno, 2012 Strong non smooth LF for Twisting,Terminal and a sign controller. TrajectoryIntegration method.


Some attempts (in our group)

First steps: the quadratic form approach

Young’s inequality and extensions

Trajectory integration

Reduction method

Generalized Forms approach

Homogenous Control Lyapunov Functions

etc.

I will not talk about Lille’s Group Implicit Lyapunov Functions(ILF) approach (Polyakov, Efimov, Perruquetti,...)!


Part III

Lyapunov-Based Design of

Higher-Order Sliding Mode (HOSM)

Controllers


Outline

12 Control Lyapunov Functions and Families of HOSM CLFs

13 HOSM Controllers

14 HOSM Controllers: Some Examples

15 Gain Calculation

16 Simulation Example


Robust Stabilization Problem (E. Cruz)

Perturbed nonlinear system

x ∈ F (x) + g (x) ξ (x)u ,

x ∈ Rn, u ∈ Rg(x) known vector field

F (x) set-vector field, ξ multivalued ⇒ Uncertainties.

Assumptions

0 < Km ≤ ξ (x) ≤ KM

F, g are r-homogeneous of degree l.

[?]


Overview


13 HOSM Controllers


15 Gain Calculation



Control Lyapunov Function (CLF)

(Homogeneous) CLF

V (x) ∈ C1, p.d., r-homogeneous of degree m > −l

∂V (x)

∂xg(x) = 0⇒ sup

v∈F (x)

LvV < 0, ∀x ∈ Rn \ 0 .

r-homogeneous of degree 0 (discontinuous) Controllers

u = −kϕ1 (x) = −k⌈Lg(x)V (x)

⌋0,

u = −kϕ2 (x) = −kLg(x)V (x)

‖x‖l+mr,p

,

If k ≥ k∗, x = 0 is GAS, and if l < 0 it is Finite-Time Stable.

‖x‖r,p =(|x1|

pr1 + ·+ |xn|

prn

) 1p

, p ≥ max ri, homogeneous norm .


HOSM design based on CLF

The Basic Uncertain System∑DI :


The Design is reduced to find a CLF.

By a Back-Stepping-like procedure construct a CLF

Define

xTi = [x1, · · · , xi] ,

r = (r1, r2, ..., rρ) = (ρ, ρ− 1, ..., 1) ,

αρ ≥ αρ−1 ≥ · · · ≥ α1 ≥ ρ ,

m ≥ ri + αi−1 .


Family of CLFs

CLF r-homogeneous of degree m: ∀γi > 0, ∃ki > 0

V (x) = γρWρ (xρ) + · · ·+ γiWi (xi) + · · ·+ γ1ρ

m|x1|

mρ

Wi (xi) =rim|xi|

mri − dνi−1c

m−riri xi +

(1− ri

m

)|νi−1|

mri ,

νi (xi) = −ki dσicri+1αi , σ1 = dx1c

α1r1

σi(xi) = dxicαiri − dνi−1c

αiri = dxic

αiri + k

αirii−1 dσi−1c

αiαi−1 ,

V (x) is a continuously differentiable and r-homogeneous CLF ofdegree m.


Overview


13 HOSM Controllers


15 Gain Calculation



HOSM Controllers

HOSM Discontinuous Controller

Discontinuity at σρ(x) = 0

uD (x) = −kρ dσρ (x)c0 , kρ 0 ,

HOSM Quasi-Continuous Controller

Discontinuity only at x = 0

uQ (x) = −kρσρ (x)

M(x), kρ 0 ,

M (x) is any continuous r-homogeneous positive definitefunction of degree αρ.

ρ-th order sliding mode x = 0 is established in Finite-Time.


Convergence Time Estimation


T (x0) ≤ mτρV1mρ (x0) ,

where τρ is a function of the gains (k1, ..., kρ), Km and C.


Variable-Gain HOSM Controller

If C = C + Θ (t, z), with constant C and time-varyingΘ (t, z) ≥ 0 known.

Variable-Gain Controller

The Discontinuous Variable-Gain HOSM Controller

uD (x) = − (K (t, z) + kρ) dσρ (x)c0 , kρ 0 ,

and the Quasi-Continuous Variable-Gain HOSM Controller

uQ (x) = − (K (t, z) + kρ)σρ (x)

M(x), kρ 0 ,

stabilize the origin x = 0 in Finite-Time if kρ 0 andKmK (t, z) ≥ Θ (t, z).


Overview


13 HOSM Controllers


15 Gain Calculation



Discontinuous Nested HOSM Controllers

For αρ ≥ · · · ≥ α1 ≥ ρ, the discontinuous Controllers of ordersρ = 2, 3, 4 are given by

u2D = −k2

⌈dx2cα2 + kα2

1 dx1cα22

⌋0

u3D = −k3

⌈dx3cα3 + kα3

2

⌈dx2c

α22 + k

α22

1 dx1cα23

⌋α3α2

⌋0

u4D = −k4

dx4cα4+kα43

dx3cα32 +k

α32

2

⌈dx2c

α23 +k

α23

1 dx1cα24

⌋α3α2

α4α3

0

and are, in general, of the type of the Nested Sliding Controllers(Levant 2005).


Discontinuous Relay Polynomial HOSMControllers

Cruz-Zavala & Moreno (2014, 2016) [?, ?, ?]; Ding, Levant & Li(2015,2016) [?, ?].For αρ = αρ−1 = · · · = α1 = α ≥ ρ discontinuous ”relaypolynomial” controllers

u2R = −k2sign(dx2cα + k1 dx1c

α2

),

u3R = −k3sign(dx3cα + k2 dx2c

α2 + k1 dx1c

α3

)u4R = −k4sign

(dx4cα + k3 dx3c

α2 + k2 dx2c

α3 + k1 dx1c

α4

)where for ρ = 2, k1 = kα1 ; for ρ = 3, k1 = kα2 k

α21 , k2 = kα2 ; and

for general ρ, ki =∏ρ−1j=i k

αρ−jj .


Quasi-Continuous Nested or RelayPolynomial HOSM Controllers

For arbitrary βi > 0

u2Q = −k2dx2cα2 + kα2

1 dx1cα22

|x2|α2 + β1 |x1|α22

,

u3Q = −k3

dx3cα3 + kα32

⌈dx2c

α22 + k

α22

1 dx1cα23

⌋α3α2

|x3|α3 + β2 |x2|α32 + β1 |x1|

α33

u4Q = −k4

dx4cα4+kα43

dx3cα32 +k

α32

2

⌈dx2c

α23 +k

α23

1 dx1cα24

⌋α3α2

α4α3

|x4|α4 + β3 |x3|α42 + β2 |x2|

α43 + β1 |x1|

α44


Overview


13 HOSM Controllers


15 Gain Calculation



Numerical Gain Calculation

ki, for i = 1, · · · , ρ− 1, selected to render V (x) a CLF,

kρ to obtain V < 0.

Fix m, ρ, αi and γi. Set k1 > 0, for i = 2, · · · , ρ

ki > maxxi∈Si

Φi (xi) =: Gi (k1, · · · , ki−1) , (2)

Maximization feasible since1 Φ is r-homogeneous of degree 0:achieves all its values on the

homogeneous unit sphere Si = xi ∈ Ri : ‖xi‖r,p = 1, and2 Φ is upper-semicontinuous ⇒ it has a maximum on Si.

Parametrization

ki = µikρ

ρ−(i−1)

1 , kρ >1Km

(µρkρ1 + C) , (3)

for some positive constants µi independent of k1.

Parametrization can be used for all controllers, but kρdifferent for discontinuous and quasi-continuous controllers.


Analytical Gain calculation

It is possible (but cumbersome) to provide for any order ananalytical estimation of the values of the gains using classicalinequalities.

The simplest case with ρ = 3, u3D, u3R and u3Q

For any values of α3 ≥ α2 ≥ r1 = 3, m ≥ r2 + α2, γ1 > 0,0 < η < 1 and k1 > 0,

k2 >r22

m−2r2α2

m− 1

(m− r1)2α2−r2α2

m− 1

m−r1r2

(γ1 + m−r2

r1kmr21

)m−1r2

(ηγ1k1)m−r1r2

.


Remarks

By homogeneity the gain scaling with any L ≥ 1

kT = (k1, · · · , kρ)→ kTL = (L1ρk1, · · · , L

1ρ+1−iki, · · · , Lkρ)

preserves the stability for any αj .

Convergence will be accelerated for L > 1, or the size ofthe allowable perturbation C will be incremented to LC.

The gains obtained by means of the LF can be very largefor practical applications, so that a simulation-based gaindesign is eventually necessary (see [?]).

The gain design problem is an important and unexploredone.


Overview


13 HOSM Controllers


15 Gain Calculation



Example

Kinematic model of a car [?]

z1 = v cos (z3) , z2 = v sin (z3) , z3 =(vL

)tan (z4) , z4 = u,

z1, z2: cartesian coordinates of the rear-axle middle point,

z3: the orientation angle,

z4: the steering angle, (Actual control)

v: the longitudinal velocity (v = 10 m.s−1),

L: distance between the two axles (L = 5 m), and

u: the control input.

u is used as a new control input in order to avoid discontinuitieson z4.


Control Objective

Control Task: steer the car from a given initial position tothe trajectory z2ref = 10 sin (0.05z1) + 5 in finite time.

Turn on the controllers after Observer converged (0.5 sec.).

Sliding variable σ = z2 − z2ref, relative degree ρ = 3, Model

x1 = x2, x2 = x3, x3 = φ(·) + γ(·)u,

where x =[σ σ σ

]T.

Simulations: Euler’s method, sampling time τ = 0.0005.

Bounds: |φ| ≤ C0 = 49.63, Km = 6.38 ≤ γ ≤ KM = 46.77.

Controllers: (i) Levant’s Discontinuous Controller (L3)with β1 = 1, β2 = 2 and k3 = 20; (ii) Levant’sQuasi-Continuous Controller (Q3) with β1 = 1, β2 = 2 andk3 = 24.5; and (iii) Proposed Discontinuous Controller(E3) u3D = −k3dσ3c0 with k1 = 1, k2 = 1.5, k3 = 20.


Simulations

0 10 20 30−5

0

5

10

15

20

Time [s]

z2[m

]

(a)

0 10 20 30−5

0

5

10

15

20(d)

Time [s]

z2[m

]

0 10 20 30−5

0

5

10

15

20(g)

Time [s]

z2[m

]

0 10 20 30

−10

−5

0

5

10

(b)

Time [s]

x1,

x2,x

3

0 10 20 30

−10

−5

0

5

10

(e)

Time [s]x

1,

x2,x

3

0 10 20 30

−10

−5

0

5

10

(h)

Time [s]

x1,

x2,x

3

0 10 20 30−0.4

−0.2

0

0.2

0.4

(c)

Time [s]

z4[r

ad

]

0 10 20 30−0.4

−0.2

0

0.2

0.4

(f)

Time [s]

z4[r

ad

]

0 10 20 30−0.4

−0.2

0

0.2

0.4

(i)

Time [s]

z4[r

ad

]

Figure : Left column: Levant’s Discontinuous Controller (L3), MiddleColumn: Levant’s Quasi-Continuous Controller (Q3); Right Column:Proposed Discontinuous Controller (E3).


Simulations

9 11 13 15−0.35

−0.25

−0.15

−0.05

0.05

0.15

0.25

0.35

Time [s]

Accura

cy x

3

(a)

17 21 25 29 33 35−0.35

−0.25

−0.15

−0.05

0.05

0.15

0.25

0.35(b)

Time [s]

Accura

cy x

3

9 11 13 15−0.35

−0.25

−0.15

−0.05

0.05

0.15

0.25

0.35(c)

Time [s]

Accura

cy x

3

10 20 30−0.35

−0.25

−0.15

−0.05

0.05

0.15

0.25

0.35(d)

Time [s]A

ccura

cy x

3

Figure : Accuracy: (a) with (L3); (b) with (Q3); (c) with (E3); (d)with (Q3) (k3 = 70).

Advantage of (E3): combines fast convergence rate of (L3)with smooth transient response of (Q3).


Resume

New: Methodological approach to design HOSM controllersusing CLF.

Different alternatives to find CLFs: Back-stepping,Polynomial methods, ...

It can be extended to design controllers with Fixed-Timeconvergence.

Drawback: Calculation of gains ki needs maximization of0-degree homogeneous functions.


Part IV

HOSM Differentiation/Observation: A

Lyapunov Approach


Outline

17 Basic Observation Problem

18 Super-Twisting Observer

19 Generalized Super-Twisting Observers

20 Lyapunov Approach for Second-Order Sliding ModesStability AnalysisGSTA with perturbations: ARI

21 Example: Reaction rate estimation in Bioreactors

22 The arbitrary order HOSM DifferentiatorDifferentiation ProblemSome known DifferentiatorsA family of Homogeneous DifferentiatorsExamples

23 Conclusions


Overview







23 Conclusions


Basic Observation Problem

Variations of the observation problem: with unknown inputs,practical observers, robust observers, stochastic framework todeal with noises, ....


An important Property: Observability

Consider a nonlinear system without inputs, x ∈ Rn, y ∈ R

x (t) = f (x (t)) , x (t0) = x0

y (t) = h (x (t))

Differentiating the output

y (t) = h (x (t))

y (t) =d

dth (x (t)) =

∂h (x)

∂xx (t) =

∂h (x)

∂xf (x) := Lfh (x)

y (t) =∂Lfh (x)

∂xx (t) =

∂Lfh (x)

∂xf (x) := L2

fh (x)

...

y(n−1) (t) =∂Ln−2

f h (x)

∂xx (t) =

∂Ln−2f h (x)

∂xf (x) := Ln−1

f h (x)

where Lkfh (x) are Lie’s derivatives of h along f .


Evaluating at t = 0y (0)y (0)y (0)...

y(k) (0)

=

h (x0)Lfh (x0)L2fh (x0)

...Lkfh (x0)

:= On (x0)

On (x): Observability map

Theorem

If On (x) is injective (invertible) → The NL system isobservable.


Observability Form

In the coordinates of the output and its derivatives

z = On (x) , x = O−1n (z)

the system takes the (observability) form

z1 = z2

z2 = z3...

zn = φ (z1, z2, . . . , zn)y = z1

So we can consider a system in this form as a basic structure.


A Simple Observer and its Properties

Plant: x1 = x2 , x2 = w(t)Observer: ˙x1 = −l1 (x1 − x1) + x2 , ˙x2 = −l2 (x1 − x1)Estimation Error: e1 = x1 − x1, e2 = x2 − x2

e1 = −l1e1 + e2 , e2 = −l2e1 − w (t)

Figure : Linear Plant with an unknown input and a Linear Observer.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer Linear Observer

Figure : Behavior of Plant and the Linear Observer without unknowninput.


0 20 40 600

20

40

60

80

100

120

140

Time (sec)S

tate

x1

0 20 40 600.5

1

1.5

2

2.5

3

3.5

4

Time (sec)

Sta

tet

x2

0 20 40 60−2

−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2


Figure : Behavior of Plant and the Linear Observer with unknowninput.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2


Figure : Behavior of Plant and the Linear Observer without UI withlarge initial conditions.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2


Figure : Behavior of Plant and the Linear Observer without UI withvery large initial conditions.


Recapitulation.

Linear Observer for Linear Plant

If no unknown inputs/Uncertainties: it convergesexponentially fast.

If there are unknown inputs/Uncertainties: no convergence.At best bounded error.

Convergence time depends on the initial conditions of theobserver

Is it possible to alleviate these drawbacks?


Recapitulation.







Recapitulation.







Recapitulation.







Sliding Mode Observer (SMO)

Figure : Linear Plant with an unknown input and a SM Observer.Homogeneity Based SMC Jaime A. Moreno UNAM 158

0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear ObserverLinear Observer

Nonlinear Observer

Figure : Behavior of Plant and the SM Observer without unknowninput.


0 20 40 600

20

40

60

80

100

120

140

Time (sec)

Sta

te x

10 20 40 60

0.5

1

1.5

2

2.5

3

3.5

4

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear ObserverLinear Observer

Nonlinear Observer

Figure : Behavior of Plant and the SM Observer with unknown input.


Recapitulation.

Sliding Mode Observer for Linear Plant

If no unknown inputs/Uncertainties: e1 converges in finitetime, and e2 converges exponentially fast.

If there are unknown inputs/Uncertainties: no convergence.At best bounded error. Only e1 converges in finite time!


It is not the solution we expected! None of the objectiveshas been achieved!


Recapitulation.







Recapitulation.







Recapitulation.







Overview







23 Conclusions


Super-Twisting Algorithm (STA)

Plant:x1 = x2 ,x2 = w(t)

Observer:˙x1 = −l1 |e1|

12 sign (e1) + x2 ,

˙x2 = −l2 sign (e1)

Estimation Error: e1 = x1 − x1, e2 = x2 − x2

e1 = −l1 |e1|12 sign (e1) + e2

e2 = −l2 sign (e1)− w (t) ,

Solutions in the sense of Filippov.


Figure : Linear Plant with an unknown input and a SOSM Observer.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withoutunknown input.


0 20 40 600

20

40

60

80

100

120

140

Time (sec)S

tate

x1

0 20 40 600.5

1

1.5

2

2.5

3

3.5

4

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withunknown input.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer without UIwith large initial conditions.


Recapitulation.

Super-Twisting Observer for Linear Plant

If no unknown inputs/Uncertainties: e1 and e2 converge infinite-time!

If there are unknown inputs/Uncertainties: e1 and e2

converge in finite-time! Observer is insensitive toperturbation/uncertainty!

Convergence time depends on the initial conditions of theobserver. This objective is not achieved!


Recapitulation.







Recapitulation.







Overview







23 Conclusions


Generalized Super-Twisting Algorithm(GSTA)

Plant:x1 = x2 ,x2 = w(t)

Observer:˙x1 = −l1φ1 (e1) + x2 ,˙x2 = −l2φ2 (e1)

Estimation Error: e1 = x1 − x1, e2 = x2 − x2

e1 = −l1φ1 (e1) + e2

e2 = −l2φ2 (e1)− w (t) ,


φ1 (e1) = µ1 |e1|12 sign (e1) + µ2 |e1|

32 sign (e1) , µ1 , µ2 ≥ 0 ,

φ2 (e1) =µ2

1

2sign (e1) + 2µ1µ2e1 +

3

2µ2

2 |e1|2 sign (e1) ,Homogeneity Based SMC Jaime A. Moreno UNAM 170

Figure : Linear Plant with an unknown input and a Non LinearObserver.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withoutunknown input and large initial conditions.


0 20 40 600

10

20

30

40

50

60

Time (sec)S

tate

x1

0 20 40 600

0.5

1

1.5

2

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withoutunknown input and very large initial conditions.


0 20 40 600

20

40

60

80

100

120

140

Time (sec)S

tate

x1

0 20 40 600.5

1

1.5

2

2.5

3

3.5

4

Time (sec)

Sta

tet

x2

0 20 40 60−1

0

1

2

3

4

Time (sec)

Estim

atio

n e

rro

r e

1

0 20 40 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Estim

atio

n e

rro

r e

2

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer with UI withlarge initial conditions.


Effect: Convergence time independent ofI.C.

101

102

103

104

0

2

4

6

8

10

12

14

16

norm of the initial condition ||x(0)|| (logaritmic scale)

Co

nve

rge

nce

Tim

e T

NSOSMO

GSTA with linear term

STO

Figure : Convergence time when the initial condition grows.


Recapitulation.

Generalized Super-Twisting Observer for Linear Plant




Convergence time is independent of the initial conditions ofthe observer!.

All objectives were achieved!

How to proof these properties?


Recapitulation.









Recapitulation.









Recapitulation.









Recapitulation.









What have we achieved?

An algorithm

Robust: it converges despite of unknowninputs/uncertainties

Exact: it converges in finite-time

The convergence time can be preassigned for any arbitraryinitial condition.

But there is no free lunch!

It is useful for

Observation

Estimation of perturbations/uncertainties

Control: Nonlinear PI-Control

in practice?

Some Generalizations are available but Still a lot is missing


Overview







23 Conclusions


Lyapunov functions:

1 We propose a Family of strong Lyapunov functions, thatare Quadratic-like

2 This family allows the estimation of convergence time,

3 It allows to study the robustness of the algorithm todifferent kinds of perturbations,

4 All results are obtained in a Linear-Like framework, knownfrom classical control,

5 The analysis can be obtained in the same manner for alinear algorithm, the classical ST algorithm and acombination of both algorithms (GSTA), that is nonhomogeneous.


Generalized STA

x1 = −k1φ1 (x1) + x2

x2 = −k2φ2 (x1) ,(4)


φ1 (e1) = µ1 |e1|12 sign (e1) + µ2 |e1|q sign (e1) , µ1 , µ2 ≥ 0 , q ≥ 1 ,

φ2 (e1) =µ2

1

2sign (e1) +

(q +

1

2

)µ1µ2 |e1|q−

12 sign (e1) +

+ qµ22 |e1|2q−1 sign (e1) ,

Standard STA: µ1 = 1, µ2 = 0

Linear Algorithm: µ1 = 0, µ2 > 0, q = 1.

GSTA: µ1 = 1, µ2 > 0, q > 1.


Outline







23 Conclusions


Quadratic-like Lyapunov Functions

System can be written as:

ζ = φ′1 (x1)Aζ , ζ =

[φ1 (x1)x2

], A =

[−k1 1−k2 0

].

Family of strong Lyapunov Functions:

V (x) = ζTPζ , P = P T > 0 .

Time derivative of Lyapunov Function:

V (x) = φ′1 (x1) ζT(ATP + PA

)ζ = −φ′1 (x1) ζTQζ

Algebraic Lyapunov Equation (ALE):

ATP + PA = −Q


Figure : The Lyapunov function.


Lyapunov Function

Proposition

If A Hurwitz then x = 0 Finite-Time stable (if µ1 = 1) andfor every Q = QT > 0, V (x) = ζTPζ is a global, strongLyapunov function, with P = P T > 0 solution of the ALE,and

V ≤ −γ1 (Q,µ1)V12 (x)− γ2 (Q,µ2)V (x) ,

where

γ1 (Q,µ1) , µ1λmin Qλ

12minP

2λmax P, γ2 (Q,µ2) , µ2

λmin Qλmax P

If A is not Hurwitz then x = 0 unstable.


Convergence Time

Proposition

If k1 > 0 , k2 > 0, and µ2 ≥ 0 a trajectory of the GSTA startingat x0 ∈ R2 converges to the origin in finite time if µ1 = 1, andit reaches that point at most after a time

T =

2

γ1(Q,µ1)V12 (x0) if µ2 = 0

2γ2(Q,µ2) ln

(γ2(Q,µ2)γ1(Q,µ1)V

12 (x0) + 1

)if µ2 > 0

,

When µ1 = 0 the convergence is exponential.

For Design: T depends on the gains!


Outline







23 Conclusions


GSTA with perturbations: ARI

GSTA with time-varying and/or nonlinear perturbations

x1 = −k1φ1 (x1) + x2

x2 = −k2φ2 (x1) + ρ (t, x) .

Assume2 |ρ (t, x)| ≤ δ

Analysis: The construction of Robust Lyapunov Functions canbe done with the classical method of solving an AlgebraicRicatti Inequality (ARI), or equivalently, solving the LMI[

ATP + PA+ εP + δ2CTC PBBTP −1

]≤ 0 ,

where

A =

[−k1 1−k2 0

], C =

[1 0

], B =

[01

].


Overview







23 Conclusions


General Model of a Bioreactor

ξ = Kϕ(ξ, t)−Dξ −Q(ξ) + F

Notationξ ∈ Rn State vector,concentrations

ϕ ∈ Rq Vector of reaction rates

K ∈ Rn×q Matrix of yield coefficients

D Dilution rate, D = FV

F Vector of supply rates

Q Removal rates components

Main Issue: ϕ(ξ, t) uncertain


Observers and estimators

Main approaches (state-of-the-art)

Detailed reaction rates ϕ

Extended Kalman Observer

High Gain Observer

Extended LuenbergerObserver

Linear Observers

Partial or no knowledge ofreaction rates

Interval observers

Dissipative approach

Asymptotic Observer

High Gain Observer

GSTO(generalized super-twisting observer)


Objective

Reaction rate estimation using GSTO´s


Generalized STA

x1 = −k1φ1 (x1) + x2 + ρ1 (t, x)x2 = −k2φ2 (x1) + ρ2 (t, x) ,

(5)

φ1 (x1) = m1|x1|12 sign (x1) +m2 |x1|q sign (x1) , m1 , m2 ≥ 0 , q >

1

2

φ2 (x1) =m2

1

2sign (x1) +

2q + 1

2m1m2 |x1|

2q−12 sign (x1) +

+m22q |x1|2q−1 sign (x1) ,

Standard STA: m1 = 1, m2 = 0

Linear Algorithm: m1 = 0, m2 > 0, q = 1In particular High Gain Observer (HGO)

GSTA: m1 = 1, m2 > 0, q > 12


Generalized STA


(5)


1

2

φ2 (x1) =m2

1

2sign (x1) +

2q + 1

2m1m2 |x1|

2q−12 sign (x1) +

+m22q |x1|2q−1 sign (x1) ,



GSTA: m1 = 1, m2 > 0, q > 12


Generalized STA


(5)


1

2

φ2 (x1) =m2

1

2sign (x1) +

2q + 1

2m1m2 |x1|

2q−12 sign (x1) +

+m22q |x1|2q−1 sign (x1) ,



GSTA: m1 = 1, m2 > 0, q > 12


Basic Idea: A Simple Microbial growth

Considering the simple bioprocess model1

X = µX −DXS = −kµX +D(Sin − S)

(6)

X and S are concentrations of biomass and substrate, µ is thespecific growth rate, k yield coefficient, D dilution rate (controlinput), Sin substrate concentration in the input. For simulationpurposes

µ =µmaxS

ks + S

where µmax is the maximal value in the specific growth rate, ksis the saturation constant.

1M. Farza, M. Nadri, H. Hammouri, Nonlinear observation of specificgrowth rate in aerobic fermentation process


Specific growth rate: GSTO

Considering that ϕ = µX, then the reduced model

X = ϕ−DXϕ = δ(t)

(7)

where ϕ is the reaction rate. The objective is estimate µ, formake this consider

˙X = ϕ−DX − l1φ1(X)˙ϕ = −l2φ2(X)

Ω1 (8)

Thus, an estimation of µ is given by

µΩ1 =ϕ

X


Specific growth rate: finite time parameterestimator

In this case, the model is

X = µX −DXµ = δµ(t)

(9)

But, in this case we apply an estimator to reconstruct thevariant time parameter

Super-Twisting VariableEstimator (STVE)

˙X = µX −DX − l1φ1(X)

˙µ = −Xl2φ2(X)

Ω2

Avoid the singularity of

µΩ1 =ϕ

X


Simulation

Model

Initial ConditionsX(0) = 1.41(g/l)S(0) = 2.17(g/l)Parametersk = 2, ks = 5.0(g/l)µmax = 0.33(h−1), Sin = 5(g/l)Dilution rate D

Observers Ω1,Ω2Initial conditions and parametersX(0) = 1.41(g/L)µ(0) = 0.15(h−1)S(0) = 4(g/L)

q = 1, l1 = 2, l2 = 1Ω1 m1 = 0.2,m2 = 1Ω2 m1 = 0.3,m2 = 1


Specific growth rate, estimation with the proposed observers


Observer Ω2 and HGO

ΩHG

˙X = µX1 −DX − 2θ(X)˙µ = − θ2

X(X), θ = 3

Despite noise in X

Enhanced velocity and noiserejection in the Ω2


Non measurable state S

Recall, the original model

X = µX −DXS = −kµX +D(Sin − S)

An AO is designed, definingZ = kX + S, such that

˙Z = −DZ +DSinS = Z − kX algebraic

and the error is given by

˙Z = −D(Z − Z)

With µ provided by ESTV, we have

˙S = −kµX +D(Sin − S)

the error dynamic is

˙S = −kµX −DS


Estimation of S despite noise

Because in the ESTV-case the dynamic of S is included


Overview







23 Conclusions


Outline







23 Conclusions


Differentiation

Signal f (t) is a Lebesgue-measurable function on [0,∞).

f(t) = f0(t) + v(t): unknown

f0(t), unknown base signal, n-times differentiable,|f0|(n)(t) ≤ L, L known|v(t)| ≤ η uniformly bounded noise signal.

Using: ς1 = f0 (t) , . . . , ςi+1 = f(i)0 (t) , di

dtif0 (t), i = 1, ..., n,

state representation of the base signal

ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v

Differentiator = Observer with (bounded) Unknown Input


Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Differentiation





dtif0 (t), i = 1, ..., n,


ςi = ςi+1 , i = 1, · · · , n− 1,

ςn = f(n)0 (t)

y = ς1 + v



Outline







23 Conclusions


The Linear or High Gain Differentiator

xi = −ki1

εi(x1 − f) + xi+1 ,

... i = 1, · · · , n− 1

xn = −kn1

εn(x1 − f) ,

Smooth differentiator

Detailed analysis possible using linear methods: Vasiljevicand Khalil (2008)

Quadratic Lyapunov Function

Gain optimization


Trade off → Optimization


Levant’s Robust and Exact Differentiator

xi = −kiλin dx1 − fc

n−in + xi+1 ,

... i = 1, · · · , n− 1

xn = −knλ dx1 − fc0 ,

bzep = |z|psign(z)

Levant 1998 (2nd order), 2003 (arbitrary order)

Discontinuous: Filippov’s Differential Inclusion

In the absence of noise it converges exactly in finite time.

Basic for Higher Order Sliding Modes. Extensions: J.P.Barbot, Fridman, ....

Convergence proof: Geometry and Homogeneity

It provides Qualitative properties. No gain design method.




n−in + xi+1 ,

... i = 1, · · · , n− 1


bzep = |z|psign(z)










n−in + xi+1 ,

... i = 1, · · · , n− 1


bzep = |z|psign(z)










n−in + xi+1 ,

... i = 1, · · · , n− 1


bzep = |z|psign(z)










n−in + xi+1 ,

... i = 1, · · · , n− 1


bzep = |z|psign(z)










n−in + xi+1 ,

... i = 1, · · · , n− 1


bzep = |z|psign(z)









No Lyapunov Function available for arbitrary order.

For order n = 2:

Polyakov and Poznyak (2009).Moreno and Osorio (2008,2012) a non-smooth Lyapunovfunction

V (e) = [de1c12 , e2]P [de1c

12 , e2]T

It provides necessary and sufficient conditions.Detailed analysis and (gain) design possible.

For order n = 3:

Moreno (2012). Non smooth Lyapunov Function.It provides sufficient conditions.Analysis and (gain) design possible. Nonlinear inequalitiesto solve.Sanchez et al. (2015,2016): smooth LF. Use of SOS-likemethods.




For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:





For order n = 2:


V (e) = [de1c12 , e2]P [de1c

12 , e2]T


For order n = 3:



Outline







23 Conclusions


Homogeneous Differentiators


ri+1r1 + xi+1 ,

... i = 1, · · · , n− 1

xn = −knλ dx1 − fcrn+1r1 ,

0 < ri+1 = ri + d , i = 1, . . . , n , rn = 1, −1 ≤ d ≤ 0

For d = 0: Linear (HG) Differentiator (Khalil andCoauthors).

For d = −1: Levant’s Differentiator. (1998, 2003,...).

For −1 < d ≤ 0 Differentiator is continuous

For −1 = d Differentiator is discontinuous, i.e. (DifferentialInclusion).




ri+1r1 + xi+1 ,

... i = 1, · · · , n− 1


0 < ri+1 = ri + d , i = 1, . . . , n , rn = 1, −1 ≤ d ≤ 0








ri+1r1 + xi+1 ,

... i = 1, · · · , n− 1


0 < ri+1 = ri + d , i = 1, . . . , n , rn = 1, −1 ≤ d ≤ 0








ri+1r1 + xi+1 ,

... i = 1, · · · , n− 1


0 < ri+1 = ri + d , i = 1, . . . , n , rn = 1, −1 ≤ d ≤ 0






The differentiation error

Differentiation error ei , xi − f (i−1)0

ei = −kiλin de1 + vc

ri+1r1 + ei+1 ,

... i = 1, · · · , n− 1

en = −knλ de1 + vcrn+1r1 − f (n)(t) ,

If f (n)(t) ≡ 0 and −1 < d ≤ 0 homogeneous withhomogeneity degree d and weights r = [r1, · · · , rn].

If f (n)(t) ∈ [−L, L] and d = −1 is a homogeneous DI withhomogeneity degree d = −1 and weightsr = [n, n− 1, · · · , 1].

Family parametrized by degree of homogeneity −1 ≤ d ≤ 0





ri+1r1 + ei+1 ,

... i = 1, · · · , n− 1









ri+1r1 + ei+1 ,

... i = 1, · · · , n− 1






Properties of homogeneous differentiators

Properties

Set λ = 1. ∃ki(L, d) such that as t→∞ the differentiationerror ei

For v (t) ≡ 0 and polynomial signals, Snp =f (n) (t) ≡ 0

For d = 0: converges exponentially,For −1 ≤ d < 0: converges in finite time.

For v (t) ≡ 0 and n-Lipschitz signals SnL =∣∣f (n) (t)

∣∣ ≤ LFor d = −1: converges in finite time (if kn > L),For −1 < d ≤ 0: Ultimately Uniformly Bounded.

For a uniformly bounded noise (|v (t)| ≤ η) and n-Lipschitz

signals SnL =∣∣f (n) (t)

∣∣ ≤ L,

for −1 ≤ d ≤ 0, ei is Ultimately Uniformly Bounded.

The same holds for any λ ≥ 1.



Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,





Properties








∣∣ ≤ L,




Remarks

For d = 0 this is well known. Lyapunov and Frequencyanalysis is possible.

For −ε < d < 0 Perruquetti et al. (2008) show convergencefor Hurwitz gains.

For continuous cases −1 < d ≤ 0 there exist smoothLyapunov functions: Yang and Lin (2004), Qian and Lin(2005), Andrieu et al. (2006,2008,2009),...

We extend the approach to the discontinuous case d = −1.

Analysis, design and comparison for the whole family.

For n = 2 the quadratic LF can be applied for −1 ≤ d < 1(Moreno 2009, 2011, 2013).


Remarks








Remarks








Remarks








Remarks








Remarks








The Lyapunov Function I

Fix p ≥ r1 + r2 = 2− (2n− 3) d > 1 and define

zi =eiki−1

, ki =kiki−1

, k0 = 1 , δ (t) = −f(n) (t)

kn−1.

Zi (zi, zi+1) =rip|zi|

pri − zi dzi+1c

p−riri+1 +

(p− rip

)|zi+1|

pri+1 ,

Zi are continuously differentiable, positive semidefinite and

Zi (zi, zi+1) = 0 if and only if dzicpri = dzi+1c

pri+1 .


The Lyapunov Function II

Lyapunov Function

For every p ≥ 2− (2n− 3) d > 1 and βi > 0 each differentiatorof the family −1 ≤ d ≤ 0 admits a strong, proper, smooth andr−homogeneous of degree p Lyapunov function of the form

V (z) =

n−1∑j=1

βjZj (zj , zj+1) + βn1

p|zn|p

βi > 0 , i = 1, · · · , n .

V (z) is positive definite and (due to homogeneity) radiallyunbounded.

For the linear case (d = 0, p = 2) V is a quadratic form.


Idea of the Proof

The basic idea (similar to Yang and Lin 2004, Qian andLin 2005, Andrieu et al. 2006, 2008, ...) is to reducestepwise the observer and showing convergence for asmaller observer.

For the discontinuous case an issue: V is discontinuous.Properties of continuous homogeneous functions are notvalid. Two ways out

The derivative satisfies

V ≤ −α2Vp+dp + α3LV

p−1p +

n∑i=1

kiµiVp−rip |η|

ri+1r1 ,

for some µi > 0.

Using standard arguments: differentiation error is ISS withrespect to the noise η (t) and fn0 (t).


Idea of the Proof





p−1p +

n∑i=1

kiµiVp−rip |η|

ri+1r1 ,

for some µi > 0.



Idea of the Proof





p−1p +

n∑i=1

kiµiVp−rip |η|

ri+1r1 ,

for some µi > 0.



Idea of the Proof





p−1p +

n∑i=1

kiµiVp−rip |η|

ri+1r1 ,

for some µi > 0.



Some observations

Noise influence:For d = −1: ∃µi(ki), s.t.

|ei| = |xi − f i−10 | ≤ µi|η|n−i+1

n

For d = 0: ∃µi0(ki), s.t.

|ei| = |xi − f i−10 | ≤ µi0|η|

Gain selection: From down upwards (kn, kn−1, · · · , k1)independent of the order. Calculation by maximization of ahomogeneous function.

Parameter λ > 1 accelerates convergence and increases theallowed bound L, but it also increases the noise effect.


V ≤ −κV (z)p−1p , κ > 0 , , T (z0) ≤ p

κV

1p (z0) .


Some observations


|ei| = |xi − f i−10 | ≤ µi|η|n−i+1

n


|ei| = |xi − f i−10 | ≤ µi0|η|




V ≤ −κV (z)p−1p , κ > 0 , , T (z0) ≤ p

κV

1p (z0) .


Some observations


|ei| = |xi − f i−10 | ≤ µi|η|n−i+1

n


|ei| = |xi − f i−10 | ≤ µi0|η|




V ≤ −κV (z)p−1p , κ > 0 , , T (z0) ≤ p

κV

1p (z0) .


Some observations


|ei| = |xi − f i−10 | ≤ µi|η|n−i+1

n


|ei| = |xi − f i−10 | ≤ µi0|η|




V ≤ −κV (z)p−1p , κ > 0 , , T (z0) ≤ p

κV

1p (z0) .


Some observations


|ei| = |xi − f i−10 | ≤ µi|η|n−i+1

n


|ei| = |xi − f i−10 | ≤ µi0|η|




V ≤ −κV (z)p−1p , κ > 0 , , T (z0) ≤ p

κV

1p (z0) .


Some observations


|ei| = |xi − f i−10 | ≤ µi|η|n−i+1

n


|ei| = |xi − f i−10 | ≤ µi0|η|




V ≤ −κV (z)p−1p , κ > 0 , , T (z0) ≤ p

κV

1p (z0) .


Outline







23 Conclusions


First Order differentiator (n = 2)

x1 = −k1 dx1 − fc1

1−d + x2

x2 = −k2 dx1 − fc1+d1−d ,

x2(t) ≈ f (1)0 .

Homogeneous of degree is −1 ≤ d ≤ 0, weightsr1 = 1− d, r2 = 1, and r3 = 1 + d.

For d = −1 Levant’s robust and exact differentiator, ford = 0 linear differentiator.


First Order differentiator (n = 2) contd. I

Lyapunov Function is

V (z1, z2) =1− d2− d

|z1|2−d1−d − z1z2 +

(β + 1

2− d

)|z2|2−d ,

Derivative V

V = −k1 |σ1|2 + k2 (1 + β) s1 dz1c1+d1−d − k2β |z1|

21−d

σ1 =(dz1c

11−d − z2

), s1 =

(z1 − dz2c1−d

)


First Order differentiator (n = 2) contd. II

Required value of k1 satisfies

k1

k2

=k2

1

k2> ω2 , max

z∈R2g2 (z1, z2) ,

g2 (z1, z2) ,

(s1 − β dz2c1−d

)dz1c

1+d1−d

|σ1|2.

g2 (z1, z2) homogeneous of degree zero, upper semicontinuousand has a maximum, achieved on the homogeneous sphere.


Second Order differentiator (n = 3)

x1 = −k1 dx1 − fc1−d1−2d + x2

x2 = −k2 dx1 − fc1

1−2d + x3

x3 = −k3 dx1 − fc1+d1−2d

x2 (t) ≈ f (1) (t) and x3 (t) ≈ f (2) (t).

V (z) = Z1 (z1, z2) + β2Z2 (z2, z3) +β3

p|z3|p

k22

k1k3> ω23 ,

k1k2

k3> ω13 ,


Simulations

Signal f0 (t) = 0.5 sin (0.5t) + 0.5 cos (t),

Bounded derivative∣∣∣f (3)

0 (t)∣∣∣ ≤ 1.

Simulations for linear d = 0, homogeneous d = −0.5 andLevant’s d = −1 differentiators.

Noise v (t) = ε sin (ωt), ε = 0.001, and ω = 1000.

Gains k1 = 3, k2 = 1.5√

3, k3 = 1.1.

Euler-method with step size τ = 3× 10−4.


Simulations



0 (t)∣∣∣ ≤ 1.



Gains k1 = 3, k2 = 1.5√

3, k3 = 1.1.



Simulations



0 (t)∣∣∣ ≤ 1.



Gains k1 = 3, k2 = 1.5√

3, k3 = 1.1.



Simulations



0 (t)∣∣∣ ≤ 1.



Gains k1 = 3, k2 = 1.5√

3, k3 = 1.1.



Simulations



0 (t)∣∣∣ ≤ 1.



Gains k1 = 3, k2 = 1.5√

3, k3 = 1.1.



Simulations



0 (t)∣∣∣ ≤ 1.



Gains k1 = 3, k2 = 1.5√

3, k3 = 1.1.



Simulations: No noise

0 10 20 30

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

e1

d=0

d=−0.5

d=−1

0 10 20 30−1

−0.5

0

0.5

1

1.5

2

t

e2

f0(t)=0.5sin(0.5t)+0.5cos(t), noise=0

d=0

d=−0.5

d=−1

0 10 20 30−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

e3

d=0

d=−0.5

d=−1


Simulations: Noisy measurement

0 10 20 30

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

e1

d=0

d=−0.5

d=−1

0 10 20 30−1

−0.5

0

0.5

1

1.5

2

t

e2

f0(t)=0.5sin(0.5t)+0.5cos(t), noise=0.001sin(1000t)

d=0

d=−0.5

d=−1

0 10 20 30−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

e3

d=0

d=−0.5

d=−1


Overview







23 Conclusions


Conclusions

A family of Homogeneous continuous and discontinuousdifferentiators is proposed.Unified family of differentiable LFs is given.It allows to

Gain calculation (also use of SoS-like methods).But the set of stabilizing gains is not covered!Coefficients for the noise and High derivative effect can becalculated (conservative!).Comparison is possible (future work).Convergence time estimation.

The discontinuous differentiator is the only capable ofexactness.It brings together homogeneous continuous anddiscontinuous observation.Extension to nonlinear observers in observability(triangular) form is possible (Bernard, Praly, AndrieuNOLCOS2016).


Conclusions





Conclusions





Conclusions





Conclusions





Conclusions





Conclusions





Conclusions





Conclusions





Conclusions





Conclusions





Part V

Construction of Lyapunov Functions

using Generalized Forms


Outline

24 The idea of the Method

25 Generalized forms properties

26 Positive definiteness of classic and generalized forms

27 Lyapunov function design

28 Examples

29 Example: The arbitrary order HOSM Differentiator


Overview





28 Examples



Generalized Forms (GF) method [Sanchezand Moreno 2014, 2016]

Basic Idea:

Transform a PDE

∂V (x)

∂xf(x) = −W (x)

⇒ Algebraic equation + Positive Definiteness

Motivation: Lyapunov functions for LTI systems

System: x = Ax , x ∈ Rn ,LF Candidate: V (x) = xTPx ,

LF Derivative: −V = W (x) = xTQx ,

Algebraic Lyapunov Equation: PA+ATP = −Q .


Classic forms:

Homogeneous polynomial of degree m ∈ Z≥0

f(x) =

M∑j=1

αj

n∏i=1

xρi,ji , ρi,j ∈ Z≥0 ,

n∑i=1

ρi,j = m

Finite M ∈ Z>0, αj ∈ R, x ∈ Rn.

Example

f(x) = 2x41 + 3x3

1x2 + 5x21x

22 − 6x1x

32 + 8x4

2 .


Generalized forms (GF):

Homogeneous function of degree m ∈ R≥0 with weights r

f(x) =

N∑j=1

αj

n∏i=1

υi,j(xi, ρi,j) , ρi,j ∈ R≥0 ,

n∑i=1

riρi,j = m

υi,j(xi, ρi,j) = |xi|ρi,j , dxicρi,j

Finite N ∈ Z>0, αj ∈ R, x ∈ Rn.

Example

f(x) = κ1|x1|5π2 + κ2 dx1c

π2 |x2|

4π3 , κi ∈ R .

m = 5π, r = [2, 3]>.

Classic forms ⊂ GFs


Motivational example

Homogeneous polynomial system (κ = 2, r = [1, 3]>)

Σ : x1 = −x31 + x2 , x2 = −x5

1 ,

Weak Lyapunov function [Bacciotti & Rosier, 2005]

V (x) = 16x

61 + 1

2x22 , V = −x8

1 ,

Theorem [Sanchez, 2016]

For Σ, there is no strict LF in the class of homogeneouspolynomials of any degree m for any weights r.

Strict Lyapunov function for Σ

V (x) = α1x61 − α12x1 dx2c

53 + α2x

22 , (GF!)


GF systems: HOSM

Twisting algorithm [Levant, 1993]

x1 = x2 , x2 = −k1 dx1c0 − k2 dx2c0 ,

Super–Twisting algorithm [Levant, 1993]

x1 = −k1 dx1c12 + x2 , x2 = −k2 dx1c0 ,

CTA [Torres et al., 2013]

x1 = x2

x2 = −k1 dx1c13 − k2 dx2c

12 + x3

x3 = −k3 dx1c0 − k4 dx2c0

d · cρ = sign(·)| · |ρ


More GF systems

Continuous homogeneous systems (Finite time)

x1 = x2 , x2 = −k1 dx1c12 − k2 dx2c

23 , κ = −1 , r = [3, 2]>

Polynomial homogeneous systems

x1 = −x31 + x2 , x2 = −x5

1 , κ = 2 , r = [1, 3]>

Linear systems

x = Ax , x ∈ Rn , κ = 0 , r = [1, . . . , 1]>


Overview





28 Examples



GFs properties

Theorem1 Sums of GFs (degree m, weights r) are GFs of degree m.

2 Fa and Fb GF of degree ma and mb, with weights r, FaFb isa GF of degree ma +mb.

Theorem1 A GF is differentiable almost everywhere (coordinate

hyperplanes).

2 A continuous GF is differentiable everywhere if itsexponents ρi,j 6= 0 are such that

ρi,j ≥ 1 , if υi(xi, ρi,j) = dxicρi,jρi,j > 1 , if υi(xi, ρi,j) = |xi|ρi,j

, ∀i, j .

3 Partial derivatives of a differentiable GF are GFs.


GFs properties

Corollary

x = f(x), x ∈ Rn, is a GF system of degree κ with r.

V : Rn → R is a GF of degree m with weights r.

⇒ W (x) = −∇V (x) · f(x) is GF of degree m = m+ κ.

Structure for positive definiteness

V (x, α) =n∑i=1

αi|xi|mri +

q∑j=1

αj

n∏i=1

υi,j(xi, ρi,j),

W (x, β) =n∑i=1

βi|xi|mri +

q∑j=1

βj

n∏i=1

υi,j(xi, ρi,j) ,


Polynomial characterization

Commensurable exponents: ρi/ρj ∈ Q

Isomorphism: dγ : Pn → Dγdγ(y) = [σ1y

µ11 , . . . , σny

µnn ]>, µi ∈ Q>0

Hyperoctants: Dγ ⊂ Rn. σi = sign(xi), x ∈ DγPn = z ∈ Rn | zi > 0, i = 1, 2, . . . , n.

Lemma

If f : Rn → R is GF of degree m with weights r and rationalexponents, then there exist µi ∈ Q>0 such that everyfDγ dγ : Pn → R is a form.

Associated forms of a GF f

f(x) : f1(y), . . . , f2n(y) , fi : Pn → R


Polynomial characterization

Isomorphism:

y xd1(y)

y y

y x

xx

d2(y)

d3(y) d4(y)

Lemma

A GF f : Rn → R is positive definite if its associated forms arepositive definite.


Example

GF

V (x) = |x1|53 + x1x2 + |x2|

52 , m = 5 , r = [3, 2]>

Isomorphism

dγ(z) = [σ1z31 , σ2z

22 ]>

Vγ = V dγ : P → RD1 = x1 ≥ 0, x2 ≥ 0, V1(z) = z5

1 + z31z

22 + z5

2

D2 = x1 ≤ 0, x2 ≥ 0, V2(z) = z51 − z3

1z22 + z5

2

D3 = x1 ≤ 0, x2 ≤ 0, V3(z) = z51 + z3

1z22 + z5

2

D4 = x1 ≥ 0, x2 ≤ 0, V4(z) = z51 − z3

1z22 + z5

2

V (x) : V1(z), V2(z), V3(z), V4(z)Homogeneity Based SMC Jaime A. Moreno UNAM 242

Overview





28 Examples



Polya’s Theorem

Theorem (Polya, 1928)

The (classic) Form F : Pn \ 0 → R, is positive if and only ifthere exists p0 ∈ N such that for all p ≥ p0, the coefficients ofthe form

G(z) = (z1 + z2 + · · ·+ zn)pF (z), ∀z ∈ Pn \ 0,

are positive.

Example

V (z) = α1z31 − α2z

21z2 + α3z

32 , αi > 0 ,

Gp(z) = (z1 + z2)p V (z) ,


Polya’s Theorem

p = 1

G1(z) = α1z41 + (α1 − α2)z3

1z2−α2z21z

22 + α3z1z

32 + α3z

42 .

p = 2

G2(z) = α1z51 + (2α1 − α2)z4

1z2 + (α3 − α2)z21z

32

+(α1 − 2α2)z31z

22 + 2α3z1z

42 + α3z

52 .

Inequalities

α1 > 0, 2α1 − α2 ≥ 0, α3 − α2 ≥ 0, α1 − 2α2 ≥ 0, α3 > 0 ,


Polya’s Theorem

System of linear inequalities Avα 0

Av =

1 0 0 2 0 10 1 0 −1 −1 −20 0 1 0 1 0

> , α = [α1 α2 α3]> .

Polyhedral cone

C = α ∈ Rd : Aα 0

Minkowski-Weyl

C = α = Bγ : 0 γ ∈ Rq a1

a2

a3

A: Faces, B: EdgesSoftware: Skeleton [Zolotykh, 2012]


SOS representation

Sum of Squares (SOS) representation: Hilbert’s 17th prob.

The (classic) Form F of degree 2q is positive semi-definite if

F (z) =

N∑i=1

(fi)2 .

Example

F (z) = z21 + 2z1z2 + z2

2 = (z1 + z2)2 ,

Example

F (z) =z61 − 2z4

1z2z3 + z21z

42 + z2

1z22z

23 − 2z1z

22z

33 + z6

3

=(z31 − z1z2z3)2 + (z3

3 − z1z22)2 .


SOS representation

SOS-Quadratic form [Choi et al., 1995]

F (z) : SOS ⇐⇒ F (z) = y(z)TPy(z) ,

F = F (z;α) , LMI problem: P (α) ≥ 0

Software: SOSTOOLS [Prajna et al., 2002-2005] .

Positive definiteness

F (z) = F (z)− εn∑i=1

zmi , ε ∈ R>0


Positive definite GFs

Let F = F (x;α) be a GF and Fi its associated forms

Polya’s Theorem

Gp(z;α) = (z1 + z2 + · · ·+ zn)pFi(z;α) ,

Linear inequalities: Aiα 0 ,

SOS representation

Fi(z;α) = Fi(z;α)− εn∑i=1

zmi , ε ∈ R>0

LMIs: Pi(α) ≥ 0 .

Adequate isomorphism.


Comments

Polya

Necessary and Sufficient condition for positivedefiniteness

Leads to Linear Inequalities (not LMIs)

The complete solution for a given power p can becompletely characterized

Available Software (e.g., Skeleton)

SOS

A Sufficient condition for positive definiteness

Leads to LMIs

Available Software (e.g., SOSTOOLS)

Allows to include objective functions (optimization)


Overview





28 Examples



GF Lyapunov function

GF system:

x = f(x; k)

GF LFcandidate:

V (x;α)

Derivative:

W (x;β) ,

Associated forms

Vi(z;α), Wi(z;β)

β bilinear

β = β(α, k) , β = M(k)α , β = M(α)k


Algorithm. I

Given a GF system x = f(x; k) of degree κ with weights r,

Step 1 Chose some terms υij(xi, ρi) in

V (x, α) =n∑i=1

αi|xi|mri +

q∑j=1

αj

n∏i=1

υi,j(xi, ρi,j)

Step 2 Take the derivative of V along the trajectories ofthe system and obtain

W (x, β) =

n∑i=1

βi|xi|mri +

q∑j=1

βj

n∏i=1

υi,j(xi, ρi,j)


Algorithm. II

Step 3 Considering

Homogeneity:∑n

i=1 riρi,j = m

Differentiability:

m > maxiriρi,j ≥ 1 , for υi(xi, ρi,j) = dxicρi,jρi,j > 1 , for υi(xi, ρi,j) = |xi|ρi,j

restrict the exponents ρi,j and the signs of αj suchthat the coefficients βi can be strictly positive. Ifnot, go back to Step 1 and increase q or changeυij(xi, ρi).

Step 4 Set m and ρi,j .

Step 5 Chose µi in: dγ(y) = [σ1yµ11 , . . . , σny

µnn ]>


Algorithm. III

Step 6 Compute the associated forms

V1, . . . , V2n , W1, . . . ,W2n

Step 7 Find α and k for positive definiteness of Vi, Wi

Solving Polya’s inequalities

Finding SOS representation

Bilinear problem!


Analysis (k given):

Polya’s procedure

AViα 0, AWiβ 0 , β = M(k)α

Solve for α the system of linear inequalities:

AViα 0, AWiM(k)α 0

SOS procedure

Define the forms:

Vj(y) = Vj(y)− εn∑i=1

yδi , Wj(y) = Wj(y)− εn∑i=1

yδi ,

δ, δ, degrees of Vi, Wi. Solve for α the system of LMIs:

PVi(α) ≥ 0, PWi(α) ≥ 0


Design:

Polya’s procedure

AViα 0, AWiβ 0 , β = M(α)k

Solve for α the system AViα 0 and choose an α∗

Solve for k the system AWiM(α∗)k 0

SOS procedure

PVi(α) ≥ 0, PWi(α, k) ≥ 0

Solve for α the LMIs PVi(α) ≥ 0 and choose an α∗

Solve for k the LMIs PWi(α∗, k) ≥ 0


Overview





28 Examples



Super–Twisting I

Super–Twisting algorithm [Levant, 1993]

x1 = −k1 dx1c12 + x2 , x2 = −k2 dx1c0 ,

Homogeneous of degree κ = −1 with weights r = [2, 1]>.

Lyapunov function candidate

V (x) = α1|x1|m2 + α2 dx1cρ1 dx2cρ2 + α3|x2|m.

Homogeneity ρ2 = m− 2ρ1.Necessary conditions for positive definiteness: α1, α3 > 0.Differentiability: m > 2, ρ1 ≥ 1 and ρ2 = m− 2ρ1 ≥ 1.

Choosing m = 3

V (x) = α1|x1|32 + α2 dx1cρ1 dx2c3−2ρ1 + α3|x2|3.


Super–Twisting II

V = −W (x)

W (x) =3α1k12 |x1| − 3α1

2 dx1c12 x2 + 3α3k2 dx2c2 dx1c0 +

α2k2(3− 2ρ1)|x1|ρ1 |x2|2−2ρ1 + α2k2ρ1 dx1cρ1− 12 dx2c3−2ρ1

−α2ρ1|x1|ρ1−1|x2|2−2ρ1 .

ρ1 = 1 and −α2 = α2 > 0.

LF Candidate

V (x) = α1|x1|32 − α2x1x2 + α3|x2|3 ,

W (x) = β1|x1| − β2 dx1c12 x2 + β3|x2|2 + β4 dx1c0 |x2|2 ,


Super–Twisting III

Coefficients of the Derivative

β1 = 32α1k1 − α2k2, β2 = 3

2α1 + α2k1, β3 = α2, β4 = 3α3k2

Note: βi is linear in αj and linear in kj but not in both.

LF conditions

Find αi, ki so that V > 0 and W > 0.

Isomorphism

dγ(z) = [σ1z21 , σ2z2]>


Super–Twisting IV

Vγ = V dγ : P → RD1 = x1 ≥ 0, x2 ≥ 0, V1(z) = α1z

31 − α2z

21z2 + α3z

22

D2 = x1 ≤ 0, x2 ≥ 0, V2(z) = α1z31 + α2z

21z2 + α3z

22

D3 = x1 ≤ 0, x2 ≤ 0, V3(z) = α1z31 + α2z

21z2 + α3z

22

D4 = x1 ≥ 0, x2 ≤ 0, V4(z) = α1z31 − α2z

21z2 + α3z

22

Wγ = W dγ : P → RD1 = x1 ≥ 0, x2 ≥ 0, W1(z) = β1z

21 − β2z1z2 + (β3 + β4)z2

2

D2 = x1 ≤ 0, x2 ≥ 0, W2(z) = β1z21 + β2z1z2 + (β3− β4)z2

2

D3 = x1 ≤ 0, x2 ≤ 0, W3(z) = β1z21 + β2z1z2 + (β3− β4)z2

2

D4 = x1 ≥ 0, x2 ≤ 0, W4(z) = β1z21 − β2z1z2 + (β3 + β4)z2

2


Super–Twisting V

x1x2 < 0: z1 ≥ 0, z2 ≥ 0

V (z) = α1z31 + α2z

21z2 + α3z

32 ,

W (z) = β1z21 + β2z1z2 + (β3 − β4)z2

2 .

β3 > β4.

x1x2 ≥ 0: z1 ≥ 0, z2 ≥ 0

V (z) = α1z31 − α2z

21z2 + α3z

32 ,

W (z) = β1z21 − β2z1z2 + (β3 + β4)z2

2 .

Just these forms must be analysed!(V was analysed in the example of Polya’s theorem)


Super-Twisting, Polya’s procedure

Fix α = [2.1, 1, 1.1]>,

G2(z) = (z1 + z2)pW (z) ⇒ Awβ 0 ,

Double description

AwM(α)[1 k>]> > 0 ⇔ k = Bwγ ,

γ ∈ Rq, γi > 0,∑q

i=1 γi = 1, q is the number of columns of Bw.

Solution for p = 6Bw =

[3.788 2.325 3.0190.303 0.303 0.257

],

For example, with γ = (1/3)[1, 1, 1]>

k1 = 3.04, k2 = 0.28


Super-Twisting, SOS procedure I

Change of variables: z1 > 0, z2 > 0, y1, y2 ∈ R

(z1, z2) 7→ (y21, y

22).

Classical Forms of even degree: y ∈ R2

V (y) = α1y61 − α2y

41y

22 + α3y

62 ,

W (y) = β1y41 − β2y

21y

22 + (β3 + β4)y4

2 .

SOS ⇒ LMI [Parrilo, 2000]

V (y) = V (y)− ε(y61 + y6

2) > 0 , ε > 0 .

V (y) = ψT (y)Qvψ(y) , ψ(y) = [y31, y

21y2, y1y

22, y

32]T .


Super-Twisting, SOS procedure II

Qv =

α1 0 −λ1 00 2λ1α1α0 − α2 0 −λ2

−λ1 0 α0 − 2λ2α2α3 00 −λ2 0 α3

> 0 .


Overview





28 Examples



HOSM Differentiator

f(t) = f0(t) + ν(t),∣∣∣f (n)

0 (t)∣∣∣ ≤ L

Levant’s Differentiator

xi = −ki dx1 − fcn−in + xi+1 , i = 1, · · · , n− 1

xn = −kn dx1 − fc0 .

Dynamics of the Differentiation error: zi =xi−f

(i−1)0

ki−1

zi = −ki(dz1 + νc

n−in − zi+1

), ki =

kiki−1

,

zn = −kn dz1 + νc0 − f(n)0 (t)

kn−1.

Homogeneous: degree d = −1, weights r = (n, n− 1, · · · , 1).


Generalized Form as Lyapunov Function

LF: for p ≥ 2n− 1 and any βi > 0

V (z) =

n−1∑j=1

βjZj (zj , zj+1) + βn1

p|zn|p , βi > 0

Zi (zi, zi+1) = n+1−ip|zi|

pn+1−i +

− zi dzi+1cp−n−1+in−i +

(p−n−1+i

p

)|zi+1|

pn−i .


V ≤ −κV (z)p−1p , κ > 0

T (z0) ≤ p

κV

1p (z0) .


Gains calculation by SOS

Gain Calculation using SOS and p = 2n− 1

n k1 k2 k3 k4 L2 2.12 1.02 – – 13 3.01 4.95 1.03 – 14 5.81 17.75 15.45 1.02 1


Summary

Pros

Provides a computable way to calculate LFs for a fairlygeneral class of homogeneous systems.

It can be extended to non homogeneous systems.

The algebraic problem to solve is a system of linearinequalities (Polya) or an LMI (SOS). It is linear in thecoefficients of the LF candidate and in the gains.

Cons

Restricted to ”polynomial” systems

Course of high p for Polya and SOS.


Summary

Pros

Provides a computable way to calculate LFs for a fairlygeneral class of homogeneous systems.

It can be extended to non homogeneous systems.

The algebraic problem to solve is a system of linearinequalities (Polya) or an LMI (SOS). It is linear in thecoefficients of the LF candidate and in the gains.

Cons

Restricted to ”polynomial” systems

Course of high p for Polya and SOS.


Overview





28 Examples



References I

A. Bacciotti and L. Rosier.

Liapunov functions and stability in control theory.

Communications and Control Engineering. Springer, Berlin, 2ndedition, 2005.

Emmanuel Bernuau, Denis Efimov, Wilfrid Perruquetti, andAndrey Polyakov.

On homogeneity and its application in sliding mode control.

Journal of the Franklin Institute, 351(4):1866–1901, 2014.

Special Issue on 2010-2012 Advances in Variable StructureSystems and Sliding Mode Algorithms.

M. D. Choi, T. Y. Lam, and B. Reznick.

Sum of squares of real polynomials.

In Proceedings of Symposia in Pure mathematics, volume 58,pages 103–126. American Mathematical Society, 1995.


References II

E. Cruz-Zavala and J. A. Moreno.

Levant’s arbitrary order exact differentiator: A Lyapunovapproach.

IEEE Transactions on Automatic Control., Submitted,Submitted 2016.

Shihong Ding, Arie Levant, and Shihua Li.

Simple homogeneous sliding–mode controller.

Automatica, 67:22 – 32, 2016.

H. Hermes.

Homogeneus coordinates and continuous asymptotically stabilizingfeedback controls, in, Differential Equations, Stability and Control(S. Elaydi, ed.), volume 127 of Lecture Notes in Pure and AppliedMath., pages 249–260.

Marcel Dekker, Inc., NY, 1991.


References III

M. Hestenes.

Calculus of Variations and Optimal Control Theory.

John Wiley and Sons, 1966.

Y. Hong.

H∞ control, stabilization, and input–output stability of nonlinearsystems with homogeneous properties.

Automatica, 37(6):819–829, 2001.

A. Levant.

Sliding order and sliding accuracy in sliding mode control.

International Journal of Control, 58(6):1247–1263, 1993.

Arie Levant.

Homogeneity approach to high-order sliding mode design.

Automatica, 41(5):823–830, 2005.


References IV

J. A. Moreno and M. Osorio.

A Lyapunov approach to second–order sliding mode controllersand observers.

In Decision and Control, 2008. CDC 2008. 47th IEEEConference on, pages 2856–2861, Dec 2008.

Y. Orlov.

Finite time stability and robust control synthesis of uncertainswitched systems.

SIAM Journal on Control and Optimization, 43(4):1253–1271,2004.


References V

P. A. Parrilo.

Structured Semidefinite Programs and Semialgebraic GeometryMethods in Robustness and Optimization.

PhD thesis, California Institute of Technology, Pasadena,California, 2000.

G. Polya.

Uber positive Darstellung von Polynomen.

Vierteljahrschrift Naturforschenden Ges, 73:141–145, 1928.

A. Polyakov and A. Poznyak.

Unified Lyapunov function for a finite–time stability analysis ofrelay second–order sliding mode control systems.

IMA Journal of Mathematical Control and Information,29(4):529–550, 2012.


References VI

Andrei Polyakov and Alex Poznyak.

Lyapunov function design for finite-time convergence analysis:“Twisting” controller for second–order sliding mode realization.

Automatica, 45(2):444–448, 2009.

S. Prajna, A. Papachristodoulou, and P. A. Parrilo.

SOSTOOLS: Sum of squares optimization toolbox for MATLAB,2002–2005.

Available from www.cds.caltech.edu/sostools andwww.mit.edu/∼parrilo/sostools.

T. Sanchez and J. A. Moreno.

Construction of Lyapunov Functions for a Class of Higher OrderSliding Modes algorithms.

In 51st IEEE Conference on Decision and Control (CDC), pages6454–6459, 2012.


References VII


A constructive Lyapunov function design method for a class ofhomogeneous systems.

In IEEE 53rd Annual Conference on Decision and Control(CDC), pages 5500–5505, 2014.

Tonametl Sanchez.

Construction of Lyapunov functions for continuous anddiscontinuous homogeneous systems.

PhD thesis, Universidad Nacional Autonoma de Mexico, MexicoCity, 2016.

http://132.248.9.195/ptd2016/junio/408055776/Index.html.


References VIII

R. Santiesteban, L. Fridman, and J. A. Moreno.

Finite–time convergence analysis for “Twisting” controller via astrict Lyapunov function.

11th International Workshop on Variable Structure Systems(VSS), Mexico City, Mexico, June 2010.

T. SA¡nchez and J. A. Moreno.

Recent trends in Sliding Mode Control, chapter Construction ofLyapunov functions for High Order Sliding Modes.

Institution of Engineering and Technology, 2016.

V. Torres-Gonzalez, L.M. Fridman, and J.A. Moreno.

Continuous Twisting Algorithm.

In Decision and Control (CDC), 2015 IEEE 54th AnnualConference on, pages 5397–5401, Dec 2015.


References IX

N. Y. Zolotykh.

New modification of the Double Description Method forconstructing the skeleton of a polyhedral cone.

Computational Mathematics and Mathematical Physics,52(1):146–156, 2012.

V. I. Zubov.

Methods of A. M. Lyapunov and their applications.

Groningen: P. Noordho: Limited, 1964.


Overview

30 Homogeneity

31 Proofs


Homogeneity (Scaling property) I

Classic

g(εx) = εmg(x)

Weighted

f (εr1x1, . . . , εrnxn) = εmf(x)

Example

Consider the function f : R2 → R given by

f(x1, x2) = κ1|x1|5π2 + κ2|x1|

π2 |x2|

4π3 , κi ∈ R .

This function is homogeneous of degree m = 5π with theweights r = [2, 3]>. Moreover, with p = 1/π, it is homogeneousof degree m = 5 with the weights r = [2/π, 3/π]>.


Homogeneity (Scaling property) II

Definition (see, e.g., Bacciotti and Rosier 2005 )

Let Λrε be the square diagonal matrix given by

Λrε = diag(εr1 , . . . , εrn), where r = [r1, . . . , rn]>, ri ∈ R>0, and

ε ∈ R>0. The components of r are called the weights of thecoordinates. Thus, a function f : Rn → R is homogeneous ofdegree m ∈ R (with the weights r) iff (Λr

εx) = εmf(x), ∀x ∈ Rn, ∀ε ∈ R>0.

Lemma (see, e.g., Hong 2001 )

Let f : Rn → R be homogeneous of degree m ∈ R withr = [r1, . . . , rn]>, ri ∈ R>0, then f is also homogeneous ofdegree pm ∈ R with r = [pr1, . . . , prn]>, for any p ∈ R>0.


Homogeneity (Scaling property) III

Definition

Given a vector of weights r = [r1, . . . , rn]>, a homogeneousnorm is a map x 7→ ‖x‖r,q, where for any q ≥ 1

‖x‖r,q =

(n∑i=1

|xi|qri

) 1q

, ∀x ∈ Rn .

The set Sr,q = x ∈ Rn : ‖x‖r,q = 1 is the correspondinghomogeneous unit sphere.


Homogeneity (Scaling property) IV

Definition (see, e.g., Orlov 2005, Levant 2005 )

The vector field f : Rn → Rn, f = [f1(x), . . . , fn(x)]>, ishomogeneous of degree k ∈ R (with the weights r) iffi (Λr

εx) = εk+rifi(x), i = 1, 2, . . . , n, ∀x ∈ Rn, ∀ε > 0.

A dynamical system x = f(x), x ∈ Rn, is said to behomogeneous of degree k if f is homogeneous of degree k.

A vector-set field F ⊂ Rn is called homogeneous of degreek ∈ R if the identity F (Λr

εx) = εkΛrεF (x) holds for all

x ∈ Rn and any ε ∈ R>0 for some vector of weights r. Thisis equivalent to the invariance of the differential inclusionx ∈ F (x) with respect to the combined time-coordinatetransformation Gε : (t, x) 7→ (ε−kt,Λr

εx).

A differential inclusion is said to be homogeneous if itsvector-set field F is homogeneous.


Homogeneity (Scaling property) V

Theorem (see, e.g., Bernuau 2014 )

Consider the homogeneous differential inclusion x ∈ F (x) ofdegree k with the weights r. Let φ(t; t0, x0) denote a system’ssolution with initial condition x0, at the time t0, thus

φ(t; t0,Λrεx) = Λr

εφ(εkt; εkt0, x) .


Overview

30 Homogeneity

31 Proofs


Proof of Theorem (polynomial system) I

The system is homogeneous of degree κ = 2l with weightsr = [l, 3l]>, for any l ∈ R>0. Consider the following polynomialfunction

V (x) = α1xp11 + α2x

p22 +

N∑i=1

βixq1i1 xq2i2 , p1, p2, q1i, q2i ∈ Z>0 ,

p1, p2 ∈ Z>0, q1i, q2i ∈ Z≥0, for some N ∈ Z>0. Note that all the(positive definite) homogeneous polynomials of degree m withweights r = [l, 3l]> are described by V if α1, α2 > 0 and

lp1 = m, 3lp2 = m, lq1i + 3lq2i = m. (10)

In order to have V positive definite it is necessary thatα1, α2 > 0. Moreover p1 and p2 must be even. From (??),


Proof of Theorem (polynomial system) II

p2 = m/3l. The derivative of V along the trajectories of thesystem is

V = −p1α1k1xp1+21 + p1α1x

p1−11 x2 − p2α2k2x

51xp2−12 +

+

N∑i=1

βi

(−q1ik1x

q1i+21 xq2i2 + q1ix

q1i−11 xq2i+1

2 − q2ik2xq1i+51 xq2i−1

2

).

The first term of V is negative definite in x1 but it is necessaryto have a negative definite term in x2. Note that the only wayto obtain it is from the term βiq1ix

q1i−11 xq2i+1

2 if βi < 0 andq1i = 1 for some i. However, from (??)q2i = (m− lq1i)/3l = (m/3l)− (q1i/3), since m/3l = p2 andq1i = 1 we have that q2i = p2 − 1/3, therefore q2i cannot be aninteger and this concludes the proof.


Proof of Lemma (Isomorphism) I

Lemma

If f : Rn → R is a GF of degree m with the weightsr = [r1, . . . , rn]>, and its exponents are rational numbers, thenthere exist µi ∈ Q>0 such that each fDγ dγ : P → R is a classicform restricted to P.

From the hypothesis of the lemma,

f(x) =

N∑j=1

αj

n∏i=1

υi,j(xi, ρi,j) ,

n∑i=1

riρi,j = m, ρi,j ∈ Q≥0 .

Denote with fγ to the function given by fγ(y) = (fDγ dγ)(y),and note that υi,j(σiy

µii , ρi,j) = υi,j(σiyi, µiρi,j). Thus

fγ(y) =

N∑j=1

αj

n∏i=1

υi,j(σiyi, µiρi,j) .


Proof of Lemma (Isomorphism) II

Denote with LCDi the least common denominator of all theexponents of the variable xi in f , i.e., of all ρi,j with i fixed.Define µi = ciLCDi, ci ∈ Z>0. Hence, it is clear that all theproducts µiρi,j are integer numbers, and therefore, all theexponents in each fγ are integers. Note that, for a fixed i, σi isconstant in each Dγ , and recall that y ∈ P. Hence, there areonly two cases for the functions υi,j ,

1 υi,j(σiyi, µiρi,j) = dσiyicµiρi,j = σiyµiρi,ji ,

2 υi,j(σiyi, µiρi,j) = |σiyi|µiρi,j = yµiρi,ji .


Proof of Lemma (Isomorphism) III

Therefore each fγ is polynomial restricted to P. Now, supposethat for some j and j′ in f , ρa,j , ρb,j′ 6= 0 for somea, b ∈ 1, 2, . . . , n, moreover, ρi,j = ρi,j′ = 0 for all i 6= a, b.Hence, a necessary condition for homogeneity of f israρa,j = rbρb,j′ = m, this implies that ρa,j/ρb,j′ = rb/ra. Also, anecessary condition to make fγ homogeneous with weightsr = [1, . . . , 1]> is µaρa,j = µbρb,j′ . Thus,

µaµb

=ρb,j′

ρa,j=rarb⇔ µa

ra=µbrb.

Since a, b are arbitrary, the relation µa/ra = µb/rb, for any pair(a, b) ∈ 1, . . . , n × 1, . . . , n, is a necessary condition to make


Proof of Lemma (Isomorphism) IV

fγ homogeneous. Define the constant µ = µi/ri. Since f ishomogeneous of degree m, then, for any j,

m =

n∑i=1

riρi,j =

n∑i=1

µ

µriρi,j =

1

µ

n∑i=1

µiρi,j ⇔n∑i=1

µiρi,j = µm .

This last equality shows that fγ is homogeneous of degree µmwith the weights r = [1, . . . , 1]>. Therefore, by choosingµi = ciLCDi, ci ∈ Z>0 satisfying µi/ri is constant for any i, thefunction fγ is a classical form restricted to P.


Polya’s inequalities I

Polya’s Theorem provides a linear system of inequalitiesAα [0]. The construction of A is done considering the numberof monomials of a form with degree m and n variables:Nmn = (m+n−1)!

m!(n−1)! . The exponents in each monomial can be

expressed as the vector q[j], j = 1, 2, . . . , Nmn such that the sum

of all its components is m, namely, q[j] =[q

[j]1 q

[j]2 . . . q

[j]n

],∑n

i=1 q[j]i = m. Applying Polya’s procedure with a given p on a

form F a new form G of degree p+m is obtained, therefore thisG contains Np+m

n coefficients and monomials, the exponents ofthis form can be organized in the Np+m

n vectors


Polya’s inequalities II

s[k] =[s

[k]1 s

[k]2 . . . s

[k]n

],∑n

i=1 s[k]i = p+m. For each p, A is

given by A =[Ak,j

],

Ak,j =

p!

(sk1−qj1)!...(skn−q

jn)!, if sn ≥ qn

0, other case.


Part VII



Outline

32 Continuous HOSM Controllers33 Motivation34 Preview: Relative degree r = 1: Super-Twisting35 r = 2 Continuous Terminal Sliding Mode Controller36 r = 2 Continuous Twisting Controller37 Continuous Integral Controller (PID)38 Discontinuous Integral Controller: State Feedback

The I-ControllerLyapunov FunctionCaveat: Lack of Homogeneity

39 Discontinuous Integral Controller: Output Feedback40 Conclusions41 Discontinuous Integral Controller: Arbitrary Order

Example: Magnetic Suspension System42 Adaptive Continuous Twisting Algorithm43 General Conclusions and Open Problems


Overview






Overview






Discontinuous HOSM Controller

Perturbed second-order plant

x1 = x2

x2 = u+ µ (t) ,

Discontinuous controller (SOSM), e.g. Twisting controller,rejects bounded perturbation,

strong chattering,

Precision|x1| ≤ ν1τ

2, |x2| ≤ ν2τ .

Chattering reduction requires continuous control signal.


Chattering Attenuation: Standard

x1 = x2

x2 = x3 , u+ µ (t)

x3 = u+ µ (t)

u = k3ϑ2 (x1, x2, x3)

Properties

Levant 2003

Continuous control signal u(t) ⇒ chattering attenuation

Rejects Lipschitz continuous (possibly unbounded)perturbation,

Precision |x1| ≤ ν1τ3, |x2| ≤ ν2τ

2, |x3| ≤ ν3τ

Drawback: It requires (x1, x2) and x2!


Overview






Preview: Relative degree r = 1, theSuper-Twisting

System:x1 = u+ ρ(x, t)

Discontinuous Control:

u = −k sign(x1)

Discontinuous Integral Control (Super-Twisting):

u = −k1|x1|12 sign(x1) + z

z = −k2 sign(x1)

Closed Loop System:

x1 = −k1|x1|12 sign(x1) + x2

x2 = −k2 sign(x1) + ρ(x, t)





u = −k sign(x1)


u = −k1|x1|12 sign(x1) + z

z = −k2 sign(x1)

Closed Loop System:

x1 = −k1|x1|12 sign(x1) + x2

x2 = −k2 sign(x1) + ρ(x, t)





u = −k sign(x1)


u = −k1|x1|12 sign(x1) + z

z = −k2 sign(x1)

Closed Loop System:

x1 = −k1|x1|12 sign(x1) + x2

x2 = −k2 sign(x1) + ρ(x, t)





u = −k sign(x1)


u = −k1|x1|12 sign(x1) + z

z = −k2 sign(x1)

Closed Loop System:

x1 = −k1|x1|12 sign(x1) + x2

x2 = −k2 sign(x1) + ρ(x, t)


Super-Twisting

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Tiempo [s]

x1

Figure : State Trajectory with ρ(t) = 0.5 sin(t) + 0.25 sin(2t)

Robust stabilization in finite time

Continuous control signal


Super-Twisting

0 1 2 3 4 5 6 7 8 9 10

−4

−3

−2

−1

0

1

Tiempo [s]

u

Figure : Super-Twisting Control Signal


Overview






Continuous Terminal Sliding ModeController

u = −k1L23 dφL (x1, x2)c

13 + z

z = −k2L dφL (x1, x2)c0

φL (x1, x2) = x1 +α

L12

dx2c32 , ki > 0, L > 0 .

Stability proof: Lyapunov function

V (x) = β |x1|53 + x1x2 +

2

5α |x2|

52 − 1

k31

x2x33 + γ |x3|5 ,

x3 , z + µ


Properties

Kamal, Moreno, Chalanga, Bandyopadhyay, Fridman(2016).


It rejects Lipschitz continuous (possibly unbounded)perturbation,

Precision

|x1| ≤ ν1τ3, |x2| ≤ ν2τ

2, |x3| ≤ ν3τ

Advantage: It only requires (x1, x2) and not x2!

Estimation of the perturbation: z(t)→ µ(t).


Gain calculation by function maximization

Set 1 2 3 4

k1 4.4 4.5 7.5 16

k2 2.5 2 2 7

α 20 28.7 7.7 1

∆ 1 1 1 1

Table : Sets of gain values obtained by maximization for L = 1.


Phaseportrait: Sliding-like behavior

0 1 2 3 4 5−8

−7

−6

−5

−4

−3

−2

−1

0

Phase portrait of x1 and x

2

x1

x2

x

1 vs. x

2

φ = 0

Figure : Phaseportrait


Phaseportrait: Twisting-like behavior

−1 0 1 2 3 4 5−6

−5

−4

−3

−2

−1

0

1

2

Phase portrait of x1 and x

2

x1

x2

x

1 vs. x

2

φ = 0

Figure : Phaseportrait


Overview






Continuous Twisting Algorithm

u(x) = −k1dx1c13 − k2dx2c

12 + z

z = −k3dx1c0 − k4dx2c0,

Stability proof: Lyapunov function

V (x) = α1|x1|53 +α2x1x2+α3|x2|

52 +α4x1dx3c2−α5x2x

33+α6|x3|5 .

x3 , z + µ


Properties

Torres, Sanchez, Fridman, Moreno (2015).


It rejects Lipschitz continuous (possibly unbounded)perturbation,

Precision

|x1| ≤ ν1τ3, |x2| ≤ ν2τ

2, |x3| ≤ ν3τ

Advantage: It only requires (x1, x2) and not x2!

Estimation of the perturbation: z(t)→ µ(t).

Gain calculation using Polya’s Theorem.

Convergence Time estimation

Tc ≤5

γV

15 (x(0)) ,


Virtues of Continuous HOSM

Continuous control signal ⇒ chattering attenuation.

Extension to arbitrary order

xi = xi+1, i = 1, ..., ρ− 1,

xρ = −k1φ (x) + z + µ (t)

z = −k2dφ (x)c0

Rejects Lipschitz (possibly unbounded) continuousperturbations versus bounded perturbations of HOSM.

Requires only x and not xρ.

Lyapunov approach also extended for arbitrary ordersystems.

Interesting approach from (Chitour, Harmouche,Laghrouche).


Overview






Continuous Integral Controller (PID)

System

x1 = x2

x2 = u+ ρ (t) ,

PID-Controller (e.g. linear)

u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,

k1,2 (x1, x2) continuous, kI (x3) continuous/discontinuous.

Constant perturbations/references ⇒ Asymptoticconvergence and insensitive to perturbation!

Arbitrary perturbations/ref ⇒ Practical convergence.

Estimation of ρ(t) is not required for implementation.

More general: Internal Model Principle based controller.



System

x1 = x2

x2 = u+ ρ (t) ,


u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,








System

x1 = x2

x2 = u+ ρ (t) ,


u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,








System

x1 = x2

x2 = u+ ρ (t) ,


u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,








System

x1 = x2

x2 = u+ ρ (t) ,


u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,








System

x1 = x2

x2 = u+ ρ (t) ,


u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,








System

x1 = x2

x2 = u+ ρ (t) ,


u = −k1 (x1, x2) + kI (x3)

x3 = −k2 (x1, x2) ,







Proposed Solution

Combine Integral Action and Discontinuous Control.

k1 (x1, x2) and kI (x3) continuous, k2 (x1, x2) discontinuous.

Insensitive to any Lipschitz perturbation (i.e. withbounded derivative).

No estimation of the perturbation ρ(t) required forimplementation.

Continuous control signal ⇒ Chattering reduction.

For simplicity (!?) we add Homogeneity.


Proposed Solution








Proposed Solution








Proposed Solution








Proposed Solution








Proposed Solution








Overview






Outline






Continuous and Homogeneous StateFeedback Controller

u = −k1 dx1c13 − k2 dx2c

12

Closed Loop System:

x1 = x2

x2 = −k1 dx1c13 − k2 dx2c

12 + ρ (t) ,

Lyapunov Function:

V (x1, x2, x3) = γ1 |x1|53 + γ12x1x2 + |x2|

52 ,

Sensitive to perturbations.


Homogeneous Integral + State FeedbackController

Discontinuous Integral Controller (k1, k2, k3 > 0, k4 ∈ R)

u = −k1 dx1c13 − k2 dx2c

12 + z

z = −k3

⌈x1 + k4 dx2c

32

⌋0

Closed Loop System:

x1 = x2

x2 = −k1 dx1c13 − k2 dx2c

12 + z + ρ (t) ,

z = −k3

⌈x1 + k4 dx2c

32

⌋0


Homogeneous Integral + State FeedbackController

Discontinuous Integral Controller (k1, k2, k3 > 0, k4 ∈ R, L > 0)

u = −k1L23 dx1c

13 − k2L

12 dx2c

12 + z

z = −k3L⌈x1 + k4L

− 32 dx2c

32

⌋0

Closed Loop System:

x1 = x2

x2 = −k1L23 dx1c

13 − k2L

12 dx2c

12 + z + ρ (t) ,

z = −k3L⌈x1 + k4L

− 32 dx2c

32

⌋0

L > 0 scaling gain:If ρ(t) = 0: Stability for L = 1 ⇒ Stability for any L > 0.


Remarks

In contrast to the continuous Integral Controller:It tracks exactly, in finite time and robustlyarbitrary references with bounded r(t)despite arbitrary (time) Lipschitzperturbations/uncertainties, i.e. ‖ρ(t)‖ ≤ ∆, ∆ constantwithout an Internal Model.

For implementation: r(t) and r(t) are required but not r(t).

Define x3 = z + ρ. After convergence ⇒ x(t) = 0 ⇒z(t) = −ρ(t): Integral action estimates the perturbation!

Control signal is continuous ⇒ Chattering attenuation.Gain selection:

Set k1, k2 so that state feedback stable and well-behavedwithout perturbation.Select k4 = 0, k4 > 0, k4 < 0.Select k3 small to assure stability.Select L sufficiently large to compensate theperturbations/uncertainties.


Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Remarks







Related controllers

A similar algorithm is the ”Continuous TwistingAlgorithm”.The proof is based on a Generalized Formstechnique.

u = −k1 dx1c13 − k2 dx2c

12 + z

z = −k3 dx1c0 − k4 dx2c0

The ”High-Order Super Twisting”

u = −k1

⌈x2 + k2 dx1c

23

⌋ 12

+ z

z = −k3

⌈x2 + k2 dx1c

23

⌋0


Related controllers

A similar algorithm is the ”Continuous TwistingAlgorithm”.The proof is based on a Generalized Formstechnique.

u = −k1 dx1c13 − k2 dx2c

12 + z

z = −k3 dx1c0 − k4 dx2c0

The ”High-Order Super Twisting”

u = −k1

⌈x2 + k2 dx1c

23

⌋ 12

+ z

z = −k3

⌈x2 + k2 dx1c

23

⌋0


Outline






Homogeneous and smooth LyapunovFunction

Homogeneous and smooth Lyapunov Function (L = 1)

V1 (x1, x2; x3) = k1

(3

4|x1|

43 − x1

x3

k1+

1

4

∣∣∣∣x3

k1

∣∣∣∣4)

+1

2x2

2 ,

V (x1, x2, x3) = γV54

1 (x1, x2; x3) +

(x1 −

(x3

k1

)3)x2 +

µ

5|x3|5 .

Its derivative is given by

V (x1, x2, x3) ∈ −W1 (x1, x2; x3)− k3W2 (x1, x2; x3)


The Lyapunov function fulfills following differentialinequality

V (x) ≤ −κV45 (x) ,

for some κ > 0 depending on the gains and ∆.

It implies robust finite time stability.

Convergence time estimation:

T (x0) ≤ 5

κV

15 (x0) .



V (x) ≤ −κV45 (x) ,




T (x0) ≤ 5

κV

15 (x0) .



V (x) ≤ −κV45 (x) ,




T (x0) ≤ 5

κV

15 (x0) .


Outline






Caveat

Alternative Integral + state feedback controllers:

Linear Integral + state feedback controller (Homogeneous)

u = −k1x1 − k2x2 + x3

x3 = −k3x1

Linear state feedback + Discontinuous Integral controller(Not Homogeneous)

u = −k1x1 − k2x2 + x3

x3 = −k3sign(x1)

Discontinuous I-Controller (Extended Super-Twisting)(Homogeneous)

u = −k1|x1|13 sign(x1)− k2|x2|

12 sign(x2) + x3

x3 = −k3sign(x1)


Controller without perturbation

0 10 20 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)


x

1

x2

u

0 10 20 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)


x

1

x2

u

0 10 20 30−7

−6

−5

−4

−3

−2

−1

0

1

2

3

t(x

,u)


x

1

x2

u


Controller with perturbation

0 5 10 15 20 25 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)


x

1

x2

u

0 5 10 15 20 25 30−10

−8

−6

−4

−2

0

2

4

t

(x,u

)


x

1

x2

u

0 5 10 15 20 25 30−7

−6

−5

−4

−3

−2

−1

0

1

2

3

t

(x,u

)


x

1

x2

u


Remarks

Linear stabilizes exponentially and is not insensitive toperturbation

Linear + Discontinuous Integrator causes oscillations(Harmonic Balance). This is structural and for any n > 2.Eliminated by Homogeneity.

Extended ST: Convergence in finite time and insensitive toperturbations.


Overview






Homogeneous Output Feedback Controller

Homogeneous State Feedback Controller + HomogeneousObserver

˙x1 = −l1 dx1 − x1c23 + x2

˙x2 = −l2 dx1 − x1c13 − k1 dx1c

13 − k2 dx2c

12

u = −k1 dx1c13 − k2 dx2c

12 .

Homogeneous Integral + Output Feedback Controller

˙x1 = −l1 dx1 − x1c23 + x2

˙x2 = −l2 dx1 − x1c13 − k1 dx1c

13 − k2 dx2c

12

u = −k1 dx1c13 − k2 dx2c

12 + z

z = −k3

⌈x1 + k4 dx2c

32

⌋0,


Simulations

We have implemented three controllers:

A State Feedback (SF) controller with discontinuousintegral term, with gains k1 = 2, k2 = 5, k3 = 0.5, k4 = 0,and initial value of the integrator z (0) = 0.

An Output Feedback (OF) controller with discontinuous

integral term, with controller gains k1 = 2λ23 , k2 = 5λ

12 ,

k3 = 0.5λ, k4 = 0, λ = 3, observer gains l1 = 2L,l2 = 1.1L2, L = 4, observer initial conditions x1 (0) = 0,x2 (0) = 0, and initial value of the integrator z (0) = 0.

A Twisting controller, given by u = −k1 dx1c0 − k2 dx2c0,with gains k1 = 1.2, k2 = 0.6.

Perturbation ρ (t) = 0.4 sin (t)


Simulations

0 5 10 15 20 25 30−2

−1

0

1

2

3

4

time

Po

sitio

n

OF

Obs

SF

Twisting


Simulations

0 5 10 15 20 25 30−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time

Ve

locity

OF

Obs

SF

Twisting


Simulations

0 5 10 15 20 25 30−5

−4

−3

−2

−1

0

1

2

3

time

Inte

gra

tor

Sta

te

OF

Perturbation

SF


Simulations

0 5 10 15 20 25 30−20

−15

−10

−5

0

5

10

15

time

Co

ntr

ol

OF Control

SF Control

Twisting Control


Overview






Conclusions

The Discontinuous Integral Controller

tracks exactly, in finite time and robustlyarbitrary references with bounded r(t)despite arbitrary (time) Lipschitzperturbations/uncertainties, i.e. ‖ρ(t)‖ ≤ ∆, ∆ constantwithout an Internal Model.

Separate design of State Feedback and Observer;

Neither continuous Observer nor continuous StateFeedback Controller are insensitive to perturbations;

Insensitivity against perturbations is achieved bydiscontinuous Integral Control;

For implementation: r(t) is required but not r(t) and r(t).

Design is Lyapunov-Based.

Generalization to arbitrary order possible.


Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Overview






Discontinuous Integral Controller

Homogeneous Discontinuous Integral control∑T :

xi = xi+1, i = 1, ..., ρ− 1,xρ = u+ w (t) , x0 = x (0) ,

|w (t, z)| ≤ C .

u = k1ϑ1(x1, x2, · · · , xρ) + xρ+1 ,

xρ+1 = k2ϑ2(x1, x2, · · · , xρ) ,

ϑ1(·) homogeneous,

ϑ2(·) homogeneous of degree 0 (discontinuous!),


Block Diagram of Discontinuous I-Control

ϑ2(x)∫

Plant

ϑ1(x) Differentiator

υ xρ+1 u y σ−

r(t)w(t)

x


Virtues of Discontinuous Integral Control

Continuous control signal u(t).

Rejects/Tracks Lipschitz perturbations/references versusconstant signals for Continuous I-Control.


Increased precision.

For ρ = 1: Super-Twisting!

For ρ = 2 different versions: Zamora et al. 2013, Kamal etal. 2015 and 2016, Torres et al. 2015, ...

Recently, smooth LF for arbitrary ρ for two possible cases:

ϑ2(x1, x2, · · · , xρ): k2 not restricted.ϑ2(x1): k2 small.

Output feedback: uses a continuous observer!





































































































Outline






Magnetic Suspension System

Figure : ECP Model 730: Magnetic Suspension System


Magnetic Suspension System

x1 = x2

x2 = − kmx2 −

aL0

2m

x23

(a+ x1)2+ g

x3 =1

L(x1)

(−Rx3 + aL0

x2x3

(a+ x1)2+ u

)L(x1) = L1 +

aL0

a+ x1

x1 = y ∈ R+: position of the disc,

x2 = y ∈ R: velocity,

x3 = Ic: current in the coil,

u = V : voltage.


Discontinuous I-Controller

Control Objective: Position Tracking errore1 = y − r(t) ≡ 0 after finite time.

Tracking Error Dynamics

e1 =e2

e2 =e3

e3 =− k3λ− d

4+2d

⌈de3c

44+2d + k

44+2d

2 λ− 4d

(4+d)(4+2d) de2c4

4+d +

k4

4+2d

2 k4

4+d

1 λ− 12d

(4+d)(4+2d) e1

⌋ 4+3d4

+ z + w(t),

z =− kIλ de1c4+4d

4 .

Homogeneity degree: d ∈ [−1, 0]


Euler’s integration method of fixed-step, sampling time1× 10−4[s].

Gains: k3 = 21, k2 = 7, k1 = 3, kI = 2

d = 0: Lineal controller. λ = 100.

d = −0.5. Continuous Nonlinear I-Controller. λ = 2

d = −1: Discontinuous I-controller. λ = 2


Experiment 1: Position Tracking


Experiment 1: Tracking error


Experiment 1: Velocity


Experiment 1: Current


Experiment 1: Control Signal


Exp. 2: Position with varying mass


Experiment 2: Regulation error

l


Experiment 2: Control Signal


Overview






Problem statement

Uncertain double integrator

x1 = x2

x2 = u+ ∆(t) (11)

where x1, x2 ∈ R are the states, u ∈ R is the control input and∆(t) is the perturbation.

Assumptions:

The states x1, x2 are measurable.

∆(t) Lipschitz continuous and bounded

|∆(t)| ≤ δp , (12)

with δp ≥ 0 unknown.


Adaptive Continuous Twisting Algorithm(ACTA) I

Continuous Twisting with adaptive gain L(t)

u = −L23 (t)k1bx1e

13 − L

12 (t)k2bx2e

12 + η

η = −L(t)(k3bx1e0 + k4bx2e0

), (13)

L(t) =

`, if Te(t) 6= 0 or ||x(t)|| 6= 0

0, if Te(t) = 0 and ||x(t)|| = 0(14)

where x = (x1, x2) and ` > 0 is a positive constant.Function Te(t) represents a timer with behavior given by

Te(t) =

ti + τ − t if ti ≤ t ≤ ti + τ

0 if t > ti + τ


Adaptive Continuous Twisting Algorithm(ACTA) II

where τ > 0 is a constant dwell time. The times ti are definedas the instants when ||x(t)|| changes from zero to a non zerovalue. For i = 0, t0 = 0, and for i > 0 ti are defined as theinstants such that

∃ ε− > 0, ∀µ ∈ (0, ε−) ||x(ti − µ)|| = 0

∃ ε+ > 0, ∀µ ∈ (0, ε+) ||x(ti + µ)|| 6= 0 .

The idea of this adaptation law is to let the adaptive gain growuntil ||x(t) = 0||, but for at least a fixed time τ . Every time ti,when x deviates from zero due to an increase of the size of theperturbation, the gain will grow again for at least a time τ untilit becomes zero again. This process is repeated until ||x(t)||remains in zero for all future times. It is possible to show that


Adaptive Continuous Twisting Algorithm(ACTA) III

this happens in finite time. Off course, n practice the idealcondition x = 0 will be replaced by x belonging to a smallneighborhood of zero.


Closed Loop with ACTA

x1 = x2

x2 = −L23 (t)k1bx1e

13 − L

12 (t)k2bx2e

12 + x3

x3 = −L(t)(k3bx1e0 + k4bx2e0

)+ ∆(t) (15)

Main Result ACTA

Suppose: gains k1, k2, k3, and k4 stabilize unperturbed CTAand L(0) > 0. Then

x(t) ≡ 0 for t ≥ T .

L(t) is bounded.


Idea of Proof I

Change of variables:

z1 =x1

L(t)3q+1, z2 =

x2

L(t)2q+1, z3 =

x3

L(t)q+1,

L(t) > 0∀t ≥ 0, 0 < q ∈ R is to be selected.

System (??) in the new coordinates

z1 = − (3q + 1)L(t)

L(t)z1 +

z2

Lq(t)

z2 = − (2q + 1)L(t)

L(t)z2 −

k1

Lq(t)bz1e

13 − k2

Lq(t)bz2e

12 +

z3

Lq(t)

z3 = − (q + 1)L(t)

L(t)z3 −

k3bz1e0

Lq(t)− k4bz2e0

Lq(t)+

∆(t)

Lq+1(t).


Idea of Proof II

Lyapunov function candidate

V (z) = α1|z1|53 +α2z1z2+α3|z2|

52 +α4z1bz3e2−α5z2z

33+α6|z3|5.

Derivative of the Lyapunov Function candidate

V = − 1

Lq(t)

(1− U(z)

W (z)

∆(t)

L(t)

)W (z)− 5q

L(t)

L(t)H(z)

where W (z) > 0, H(z) > 0 for large q.

If L(t) grows V < 0.


Simulation Example I

x1 = x2

x2 = u+ ∆(t)

u = −L23 (t)k1bx1e

13 − L

12 (t)k2bx2e

12 + η

η = −L(t)(k3bx1e0 + k4bx2e0

)Parameters: ` = 15, τ = 1sGains

k1 = 0.96746, k2 = 1.40724, k3 = 0.00844, k4 = 0.004601 .


Simulation Example II

0 50 100 150 200 2500

500

1000

1500

2000

2500

3000Adaptive gain L(t) and perturbation ∆(t)

seconds

Gain L(t)

Perurtbation ∆(t)

Figure : Adaptive gain L and perturbation ∆(t)


Simulation Example III

0 50 100 150 200 250−5

0

5

10

15

20Euclidean norm of vector x

seconds

50 100 150 200

0

0.02

0.04

0.06

28 30 32 340

0.01

0.02

52 54 56 580

0.02

0.04

Figure : Euclidean norm of state vector x(t)


Simulation Example IV

0 50 100 150 200 250−2000

−1500

−1000

−500

0

500

1000

1500

2000Control signal and perturbation ∆(t)

seconds

Control signalPerturbation

Figure : Control signal


Simulation Example V

0 50 100 150 200 2500

200

400

600

800

1000

1200

1400

1600

1800

2000Control signal with opposite sign and perturbation ∆(t)

seconds

30.5 31 31.5

35

35.005

52 54 56

94.995

95

95.005

−uPerturbation

Figure : Control signal with opposite sign and perturbation


Overview






Conclusions

1 Development of a Lyapunov based approach to HOSM andhomogeneous control is an important task.

2 We require constructive methods to efficiently design controllersand observers for this class of systems.

3 We have provided some possible approaches. Each has itsstrengths and its weaknesses.

4 Still a lot of work has to be done

5 Other interesting approaches: Implicite Lyapunov Functions(ILF) by Lille Group!


Open Problems

1 Gain (and Structure) Design for Performance.

2 Performance comparison of HOSM Controllers with othercontrollers (e.g. FOSM).

3 Is there a family of LF for HOSM providing necessary andsufficient stability conditions ? Towards a more systematicLyapunov Design.

4 (Truly) multivariable HOSM controllers and Observers.Some results for ST from Ch. Edwards,...

5 Adaptive Algorithms. Important results from Y. Shtessel,F. Plestan, ...

6 Parameter estimation...

7 Implementation of HOSMs, Discretization methods,...


Thank you! Gracias!


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Communications and Control Engineering. Springer, Berlin, 2ndedition, 2005.

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Simple homogeneous sliding–mode controller.

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Automatica, 37(6):819–829, 2001.

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vss2020/Homepage-Summer-School/... · 2019. 4. 13. · Outline 1 Introduction 2 Generalized Super Twisting Algorithm (GSTA) A Quadratic Strong Lyapunov Function for the GSTA Convergence