Workshop Future Directions in Systems and Control Theory · A Framework for Control, State Estimation, and Verification of Hybrid Systems Manfred Morari Alberto Bemporad Giancarlo

Workshop

Future Directions in Systems and Control Theory Thursday, June 24

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A Framework for Control, State Estimation, and

Verification of Hybrid Systems

Manfred Morari

Alberto Bemporad Giancarlo Ferrari-Trecate

Domenico Mignone Fabio Danilo Torrisi

Automatic Control Laboratory

Swiss Federal Institute of Technology (ETH)

[email protected]

Technical Reports: http://www.aut.ee.ethz.ch

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Premise

The discussed problems are important but

inherently difficult

All useful techniques must involve significant

off-line and/or on-line computation

Results

New System Type:

Mixed Logical Dynamical (MLD) System

Many practical problems can be represented

in MLD form

Control, estimation, and verification

require solution of Mixed-Integer Linear (or

Quadratic) Programs (MILP, MIQP) for

which efficient techniques are becoming

available

■■■■■ £Uf/*nam»hchm I

Example: Verification of an Automotive Active Leveler

System

• Controller

f>OTl f<OTh f<OTh I f>ITl f = 1 -{-as

■h

f>OTh

• For a given set of initial conditions, variation of

parameters, and road conditions:

(VP#1) Find worst performance

(VP#2) Verify that height is within desired limits

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Example: Control of a

Hydroelectric Power Plant Water \Q\

reference level

Level sensor

Objective:

• Maximize power generation

Manipulated variables:

• Turbine ON / OFF

• Stepper motors (—1,0,1)

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Hydroelectric Power Plant

Constraints

• on actuators (flaps, gates)

• on level

Targets ( = end point constraints)

• level

• actuators (flaps, gates)

Preference hierarchy / heuristics

1. Flaps rather than gates

2. Turbine ON > Tmin

3. Stepper motor ON > Tmi

4. Transitions: turbines / flaps / gates

5. Stepper motor OFF < T„

mm

max

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Example: Three Tank System

Nominal behaviour:

• Liquid level in tank 1 controlled by pump 1

• Liquid level in tank 3 controlled by switching

valve Vi

Faults:

01: Leak in tank 1

</>2: Valve Vi blocked closed

Maintaining nominal operating condition

requires fault detection & control

reconfiguration

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Example: Gas Supply System (Kawasaki Steel)

Joint electric 'SL, power plant

-> Process under consideration

Heavy oil >

Maintain hold-up

Maximize use of gas for combustion

Limitations on gas flows, heat produced, gas

volume in holders

Complex rules for boiler switching

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Premise





Results

New System Type:



in MLD form



Quadratic) Programs (MILP, MIQP) for


available

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General Class: MLD Systems

General formulation of Mixed Logical-Dynamic

(MLD) systems

x(t+l) = Ax(t) + BlU(t) + B2S(t) + B3z(t)

y(t) = Cx(i) + DlU(t) + D28(t) + D3z(t)

E2S(t) + E3z(t) < Eît) + E4x(t) + E5

State

x = xc

x£ , xc G Mnc, xt G {0, l}n*

Output :

y , Vc e M?c, yi G {0, l]Pi

Input :

u = uc , uc G Rmc, ut G {0, l}m*

Auxiliary binary variables : S G {0, l}rz

Auxiliary continuous variables : z G Wc

Zürich I

Assumptions Recall: x(jb+l) = Ax(t) + BlU(t) + B2S(t) + B3z(t)

y(t) = Cx(t) + DlU(t) + D25(t) + D3z(t)

E25(t) + E3z(t) < ElU{t) + E4x(t) + E5

Well-Posedness :

{x(t), n(t)} -* {z(t + l)} single valued

{x(t), u(t)} -> {y(t)} single valued

This allows to define trajectories in x-space

Physical Constraints :

G Rnc+mc : Fxc + Guc<H C± xc

uc

C is a bounded polyhedral set (not restrictive

in practice)

This allows to define upper and lower bounds

M, m

Note: Physical constraints are included in

the inequality

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Propositional Calculus and

Linear Integer Programming

Truth table

Xi x2 i^> X\ X± V X2 Xt A X2 Xi —)■ X2 Xx ++ X2 F F T T

F T F T

T T F F

F T T T

F F F T

T T F T

T F F T

I ! Propositional logic •£>■ Integer linear inequalities

I6{F,T} & 6 € { 0, 1 }

xxvx2 is equivalent to 5i + 52>l

XXAX2 is equivalent to *1 +*2 > 2

<^J Xi is equivalent to <5i<0

X\ —■> X% is equivalent to Si - s2 < 0

x1<+x2 is equivalent to $! - 52 = 0

Logic Facts Involving

Continuous Variables

[fix) < 0] A X f(x)-ö<-l+m(l-ö)

[fix) < 0] V X fix) < MS

~ [fix) < o] fix) > e

[/(x)<0] -»■ X -fix) + e < (e - m)6

[/O)<0] o X \ fix) < M(l - 5) 1 fix) > e+im-e)8 \

• M, m = upper-, lower-bounds on f(x)

• e = small tolerance (e.g. machine precision)

• Xe{F, T}, xeW1, 5 G {0,1}

if f(x) linear => Mixed-Integer Linear Inequalities

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Product of variables

53 = 5i52 ( -81 + S3 < 0 ) -S2 + ö3 < 0 ( 5i + 52 - 53 < 1

y = öf(x)

' y < MÖ y > mö

* V < f(x)-m(l-6) . y > /(x)-M(l-5)

M, m = upper-, lower-bounds on f(x)

e = small tolerance (e.g. machine precision)

if f(x) linear^ Mixed-Integer Linear Inequalities

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An Example

5(0

10 timer

20

System:

x(f +1)

y(t)

= 0.8

= [10]x(t)

cosa(t) -sina(t) sina(t) COSa(t) x(t) + u{t)

a(t) z

• Rewrite as x(t + l) = [I I\z(t)

' 10 " -10-e

-M -M M M M M -M -M

0 0 0 0

[1 0]x(t) > 0 -| if [1 0]x(t) < 0 - if 3 "

m +

ro 01 0 " 1 °1 "10' 0 0 0 -1 0 —e r 0 0 0 0 -i 0 0 0 0 0 i 0 0 M 0 -j 0 0 M

0 0 z(t)< [0 1]'

- [o 1/ [oir

u(*) + -Ai ®(t) + M M

0 i A2 0 0 0

-i 0 - [o 1]'

0 -A2

I 0 N

0 0 0 -I A/ 0 0 1 0 1 0 0 -1 L ° [l

The origin is globally asymptotically stable

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System Theory

for MLD Systems

Well- Posed n ess

Stability

Reachability/Controllability

Observability/Reconstructibility

Optimal Control

State Estimation

Observability Test for Autonomous MLD Systems

DEFINITION of incremental observability:

Vxlt x2 G X(0)

T-l X) ||y(*, *l) " tf(*> *2)lloo > Hl*l - a2||l t=0

GOAL : compute minimum number

of output measurements necessary to

distinguish two initial states x±9x2 G X(0)

TEST :

0. Fix wm-m as small as desired (w < tymin=practical indistinguishability)

1. Fix Tmax=maximum number of measurements (T > Tmax=practical indistinguishability)

2. T<- 1 3. MinimizeX^llyÖÔ-K^^lloo-tüminlki-^lli

4. If minimum < 0, increase T.

5. If T > Tmax, STOP (system unobservable)

6. Go to 3

Mixed-Integer Linear Program

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Observability Index: Example

xi x2

■(t + 1) =

y(t) — Xl(t),

f

J

1.1 o" 0 1

xi X2

(

0 0.9 xi

{ 0.9 0 X2

iy(t^,)-y(t,yi

) if € < X!(t) < 1, €> 0

(t) otherwise

e < zi(0) < 1

rp A log | log 1.1 + 1

e T 0.9 3 0.4 11 0.1 26 0.01 50

0.001 74 0 oo

Observability index T does not depend on

state dimension n

T can be arbitrarily large (also T = oo)

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System Theory

for MLD Systems

Well-Posed ness

Stability



Optimal Control

State Estimation

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Model Predictive Control

Reference

^ r

Optimizer 1 u(t) 1 . fe. Plant f y(t)

—►

i

Observer

X{ 't)

MODEL : a model of the plant is needed to

predict the future behaviour of the plant

PREDICTIVE : optimization is based on the

predicted future evolution of the plant

CONTROL : control complex constrained

multivariable plants

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Optimization Problem

future

-u 0 O P P O

Predicted outputs y(t+k\t)

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Manipulated w(t+fc) Inputs

output horizon

Compute the optimal sequence of manipulated

inputs which minimizes

• tracking error = output - reference

• subject to constraints on inputs and outputs

On-line optimization => Receding Horizon

DMAHUMIA* " Zürich

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Receding Horizon

t+11+2 t+l+m t+l+p

• Optimize at time t (new measurements)

• Only apply the first optimal move u(t)

• Repeat the whole optimization at time t+1

Advantage of on-line optimization:

FEEDBACK!

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Receding Horizon - Examples

Chess Game

Investment plans

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Predictive Controller • On-line optimization problem (MIQP)

T-l

min J(t£-\*(t)) 4 £>(*) - tiell^ + ||S(fc|t) - S«|& ô" } fc=0

Ve\\Q5

subj. to <

X(T\t) = Xe

xQz + l\t) = Ax(k\t) + B!v(k)+

B25(k\t) + B3z(k\t)

y(k\t) = Cx(k\t) + D!v(k) + D25(k\t) + D3z(k\t)

E25(k\i) + E3z(k\t) < ElV(k) + EAx(h\t) + E5

According to a receding horizon philosophy,

set u(t) = t£(0)

Repeat everything at time t + 1

Feasibility of MIQP ©t = 0 => closed-loop

stability

Global optimum not needed !

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Closed-Loop Example jc-space y(t),r(t)

0.5

*2 0

■OS

-i o^ •

j y^NJ/.v __j

-1 -.y^ I -*^<^1 — ^.

0.2

0

■02.

-0.4

-0.6

-0.8

-1

-0.5 05

*1

u(t)

. U» I j — i • »-

.1 J_J__i_J_-l-i f--l- f ^ I

1 I I'll1 f'l I ' I -4—i—r~*" i—Vr ~ 7—A—i r i—

\ /* I1/ \'i ' \ f'l 111 1 1/ _Jki__JL I \ J ' 1 j

' 'l - ' ■'. ^ .'.'•'■'•! -v.

0 20 40 60 80 100 120

timer

0.5

-0.5

1.5

1

05

0

-05

0 20 40 60 80 100 120

time/

5(0

, , , , ,

- in i ■ 1- —■ i ' |i i,.——m

1 1 1 1- (

0 20 40 60 80 100 120

timer

x{t +1)

y(t)

= 0.8

«(*) =

cosa(t) — sina(t) sin a(t) cosa(t)

[1 0]x(t) § if [1 0]x(t) > 0

-§ if [1 0]a;(t) < 0

*(«) + «(*)

Computational Aspects

The on-line optimization problem is a

Mixed-Integer Quadratic Program (MIQP)

No need to reach global optimum (see proof

of the theorem)

Available methods: Generalized Benders'

Decom position, Outer Approxi mation,

LP/QP based branch and bound, and

Branch and Bound.

Branch & Bound is the most effective

General purpose B&B MIQP solvers

available, for both dense and sparse

problems (e.g. Fletcher-Leyffer's)

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System Theory

for MLD Systems

We 11-Posed n ess

Stability



Optimal Control

State Estimation

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Fault Detection and State Estimation

for Hybrid Systems

• Use Moving Horizon Estimation for MLD

Systems (dual of MPC)

• Mixed logic dynamic fault (MLDF) form

(Bemporad, Mignone, Morari, 1998):

x(t + 1) =Ax(t) + BlU(t) + B2S(t) + B3z(t)

+ B6ct>(t) + m y(t) =Cx(t) + DlU(t) + D2S(t) + Dzz(t) +

+ D6<j>{t) + C« E2S(t) + E3z(t) < ElU(t) + E4x(t) +E5 + E6<f>(t)

Faults: <j>(t) e {0,1}^

Disturbances: £(t) e Rn, C(t) e Rp

• At each time t the estimates </>(t), x(t) are obtained by minimizing an MIQP over a horizon extending backwards in time

0

Y, HfKfc|t)-v(fc)l& + *'(c(*l*)>W).

5(fc|t),£(fc|i),$(fc|t),2(fc|t))

k=t-T+\

Zürich HMAHMJ

Three Tank System: Fault Detection - I

0.6

0.5

0.4

^" 0.2

0.1

0

-0.1

0.1

1-0.1

(D

-0.3

-0.4

measured output y{t)

, 1 1

i 1 1

0.6 estimated output y(tlt-k)

0.4

8 0.2

20 40 60 time

state estimation error

80

I Y . iW l^.t ( -Jr

i_ _-/C 1 '

i i '

-0.2

1

0.8

J).6

*0.4

0.2

0

I i 1

■ 1 1

20 40 60 time

The estimated fault ${t-1 It)

20 40 time

60 80 20 40 time

60

</>]_: leak in tank 1 at time t = 38

80

\ »

l i 1

I I l

I I ■ I I I

j i i

80

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Three Tank System: Fault Detection - II

measured output estimated output

100

state estimation error 0.3

02

0.1

1-0-1 ,

-02

-0.3

-0.4

I I I I

I I I I

*< I 1 1 1 1

-r\ r—i i / i i i i

J L L J ' i t i i

i i i i 20 40 60

time 80 100

20 40 60 80 100 The estimated fault ^ (t-1 It)

1

1 I

i The estimated fault ♦2<t—11t)

80

■--

T 1 1 1 1 1

—1 1 i 1 I 1

1 1 I

I I

1 i 1

20 40 time

60 80

fa: leak in tank 1 for 20s < t < 60s

fa: valve V\ blocked for t > 40s

Three Tank System: Fault Detection - III

measured output estimated output

20 40 60 80 100 time

state estimation error

40 60 80 100 time

20 40 60 80 100

Estimated fault <>-, (t—11t)

«s- ~|- i i

—I 0 Estimated fault 4>2(t-1lt) 80

0! leak in tank 1 for 20s < t < 60s

</>2 valve Vi blocked for t > 40s

+ logic constraint [hi < hv] => <j>2 — 0

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Research Program

Modeling Language and Compiler

— Express models and specs with "standard vocabulary

— Translate into MILP/MIQP

Verification/Controllability/Reachability

— Use tools from Polyhedral Computation

— New case studies

State Estimation/Fault detection

— Dual of Model Predictive Control

— Convergence properties of estimator

Optimization Algorithms

— Sparse QP solver

— New Branch-and-Bound strategies

Model Reduction/Approximation

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HYSDEL (HYbrid Systems DEscription Language)

Describe Hybrid Systems in a compact way

— Propositional Logic

— Dynamics

— Constraints

Automatically generate MLD models

MLD model is not unique in terms of the

number of auxiliary variables => optimize

model: # binary variables = min.!

Reduction of model complexity for models

containing purely logic relations

(Truth Tables => Convex Hull)

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HYSDEL Example

Automotive Active Leveler X Description of Tariables and constants

state f,h,xll,zl2; input d,dc,dev;

const OTh, 0T1, ITh, IT1; const Ml,H2in3,H4Imi,m2,n3,n4; const e; const T8,cbar,eTbar,eats,cmax>cmin,eTmazIe'vnin;

X Variable tjpea

real f ,h,zl.z2,z3,d,dc.dev; logic dl.d2.d3.d4.d5,d6,d7,d8.d9,dl0.dll.di2.dl3.di4;

X Relations

dl = -Cf-ITh. <= 0. Hi, «d, e>; d2 = {f-m <= 0, M2, «2, e}-; d3 = ti-OTh >= 0, M3, «3, e>: d4 = Cf-0T1 >= 0, H4, »4, e>;

d5 - "ill fc d3; X Should be accepted also: d5-(l-xll)kd3, d5=(l-ili)*d3 dg = ill ft "dl; d7 - zl2 ft d2; d8 - "rl2 ft "d4; d9 - d5 I d6; dlO • d7 I d8; dll - "ill ft xl2; dl2 = ill ft -il2; dl3 = Td9 ft 'dlO ft di4); di4 = xli | xl2;

zi = dll * (cbar + dc) {cnax,cniii,e>; z2 = dl2 * (evbax -I- der) {eTmax,tnniiLn.,B}; z3 = (eats » f + <1 - eats) » h)*dl3 {10*0111,10*0Tl,e};

X Other constraints

must ill + xl2 <= 1; Bust ~(d9 fc dlO); aust ""(dll ft dl2);

X update

update f = z3; update n = h + Ts * (d + zl + z2); update ill = d9; update zl2 = dlO;

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General Transformation of Truth Tables

into Linear Integer Inequalities

T:

XI x2 a • • xn— 1 Xn - = F(xi,.. • ? xn—l)

0 0 0 1 0 0 1 0

a a a

a a

a a a

1 1 1 1

A5<£, i G {0, If

Theorem:

T/?e polytope P = {5 : AS < B} is the convex

hull of the rows of the truth table T

P = conv(T)

(Bemporad and Mignone) □

Every logic proposition can be translated into

linear integer inequalities

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Truth Tables => Linear Integer Inequalities

Example: "AND"

fa fa fa

0 0 0 0 1 0 1 0 0 1 1 1

(1,1,1)

*;>(u,o)

- 8i (0,0,0) (1,0,0)

conv([8MäM°M}]) = -fa + fa < o

S : -fa + fa < 0 fa -f- fa — fa < 1

Algorithm to compute convex hull: lrs (David Avis, McGill Univ.) ftp://mutt.cs.mcgill.cs/pub/c

Other algorithms: qhull, chD, Hull, Porto, cdd.

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Research Program

Modeling Language and Compiler

— Express models and specs with "standard" vocabulary


Verification/Controllability/Reachability



State Estimation/Fault detection



Optimization Algorithms



Model Reduction/Approximation

.1 Zurich

Formal Verification of Hybrid Systems

• MLD Model:

x(t + l) = Ax(t) + B!w(f) + B25(t) + B3z(t) y(t) = Cx(t) + Dtw(t) + D2S(t) + D3z(t)

E25(t) + E3z(t) < E!w(t) + E4x(t) + E5

• Verification Problem VP #1: Find max range for y(t) vt > o , w(t) e w , x(o) e x(o)

• Verification Problem VP #2: Vw e W and Vz(0) e X(0) verify that x(t) e Xs (set of safe states)

• Simple solution (VP #1): Solve VT > 0

max y(T) = Cx(T) + DlW(T) + D26(T) + D3z(T)

{x(o) e x(o) w{t) e W, 0<t<T

x(t + l) = Ax(t) + BlW(t) + B25(t) + B3z{t) E2S(t) + E3z{t) < ElU(t) + E4x(t) + E5

Similarly for VP #2: feasibility test with x(T) g Xs

IMPRACTICAL ! (even for polyhedral sets W, Xs, X(0), because of integer constraints)

Zürich

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Verification Algorithm RMK: when S(t) = const , system behaves linearly.

IDEA: hypothetically partition Rn< = UÜiG (N < 2r0 where xc(t) e C% <=> 6(t) = Si

tion Ak ̂ orith m:

—>

i tr

t=

=4^

~'5r

t=3. th t=2/ { x)

t=l/[-

d= «]

t=0 \

I8S H?] " LI J

6= 111 H!l

max{u;(fc)^(fc)}t=o)a,c(o)j<5(t)} [ ^t) ]

subj. to <

x(k + 1) = Az(fc) + jBiiü(fc) + B2S(k) + J33s(fc) £2<SO0 + E3z(k) < Exwik) + £4a;(fc) + E5

w(k) eW, fc = 0,... ,t 6(k)=6i9 fc = 0,.-. ,t —1, <5(t)G{0,l}'

[ z(0) G Af(0) T/

Reachable set implicitly defined by linear inequalities (Hp: W, #(0), #s polytopes)

Example: Verification of an Automotive Active Leveler

System

Controller

f<on

f<ITl

ggs&ggHHft

f>OTl f<OTh f<OTh / f>ITl f 1 + as

■h

f>ITh f>OTh

• For a given set of initial conditions, variation of

parameters, and road conditions:

(VP#1) Find worst performance

(VP#2) Verify that height is within desired limits

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Verification of an Automotive Active Leveler - II

• System can be transformed into MLD form

Continuous states 2 Logic states 2 Disturbance inputs 3 Boolean auxiliary variables 14 Continuous auxiliary variables 3

• Results and computational times:

-44.54 < h(t) < 25.00 4 m 25 m

Pentium II 400 MHz SPARCstation 20

(M-code)

— No use of fast iterative reach-set projection

— Use of rectangular approximation of initial sets

Using HyTech (Stauner et al., 1997):

-47 < h(i) < 27 1 62 m | SPARCstation 20 ]

Analytical solution (Elia and Brandin, 1998):

43 < h{t) < 23 by-hand

Note: exact limits in discrete-time ^ continuous-time

Research Program

• Modeling Language and Compiler

— Express models and specs with "standard" vocabulary


• Verification/Controllability/Reachability



• State Estimation/Fault detection



• Optimization Algorithms



• Model Reduction/Approximation

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Branch &i Bound for Optimal Control

(and Fault Detection) of Hybrid Systems (A.Bemporad, D. Mignone, M.Morari)

Key Idea:

1. Optimal sequences { 5(0), 5(1), ... , 5(T) }

have FEW switches

example: [S(t) = 1] <+ [x(t) > 0]

2. Order the sequences by number of switches:

0000, 1111; 0001, 0011; 0010, 0110; ...

mm m II. Zürich MklM

Branch & Bound for Hybrid Systems

5(0) = o 8(0) = 1

5(1) = o 8(1) = 1

First explore combinations of 6 with

low number of switches ("outside first"

strategy)

Assumption: global optimum is reached

early =^ large subtrees are fathomed.

If not enough available computation time

look only for sequences having a number of

switches < Kmax

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Three Tank System: Control -I

^x^ssaa^

Number of QPs in receding horizon control using

different tree exploring startegies

2000

& ~ 1600

O E1200

i . ~... ,-~-.-. ~ r«,-.

ddpthHrvt bnHudfli tint outside first

•ff-rxTrifr

VvV 20 time 30 40

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Three Tank System: Control -II

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Trajectories using suboptimal MIQP solutions:

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REFERENCES

BOOK

1. J.W. Polderman and JCVV, Introduction to Mathematical Systems The- ory: A Behavioral Approach, Springer-Verlag, New York, 1998.

GENERAL ARTICLES

2. JCVV, Paradigms and puzzles in the theory of dynamical systems, IEEE Transactions on Automatic Control, Volume 36, pages 259-294, 1991.

3. JCW, Models for dynamics, Dynamics Reported, Volume 2, pages 171-

269, 1989.

4. JCVV, From time series to linear system , Part I. Finite dimensional linear time invariant systems, Part II. Exact modelling, Part III. Approxi-

mate modelling, Automatica, Volume 22, pages 561-580, pages 675-694, 1986; Volume 23, pages 87-115, 1987.

5. JCW, The behavioral approach to systems and control, European Jour- nal of Control, Volume 2, pages 250-259, 1996.

CONTROL

6. JCW and H.L. Trentelman, #oo control in a behavioral context: The full information case , IEEE Transactions on Automatic Control, Vol- ume 44, pages 521-536, 1999.

7. JCW, On interconnections, control, and feedback, IEEE Transactions on Automatic Control, Volume 42, pages 326-339, 1997.

8. JCW, Generic eigenvalue assignability by real memoryless output feedback made simple, Communications, Computation, Control and Signal Processing, (Eds. A. Paulraj, V. Roychowdhury, and CD. Schaper),

Kluwer, pages 343-354, 1997.

9. JCW and H.L. Trentelman, On quadratic differential forms, SIAM Journal on Control and Optimization, Volume 36, pages 1702-1749,

1998.

10. JCW, Path integrals and stability, Mathematical Control Theory (Eds. J. Baillieul and J.C. Willems), Springer Verlag, pages 1-32, 1998.

-47-

RECENT PROGRESS IN

FEEDBACK PASSIVATION DESIGNS

Petar V. Kokotovic

Center for Control Engineering and Computation

University of California, Santa Barbara

ACTIVATION OF CONCEPTS

Over the last ten years several descriptive

concepts have been activated into constructive

design tools such as

• Lyapunov functions -* Control Lyapunov

functions (CLF)

• Total stability -> Input/state stability (ISS)

• Passivity —* Passivation

• Optimality —> Inverse optimality

leading to design procedures like

• Cascade passivation

• Backstepping: robust, adaptive, stochastic

• Forwarding

• Passivation using LMI's

FOUNDATIONS:

absolute stability: Lurie (1951),

passivity: Popov (1960),

PR Lemma: Yakubovich (1962), Kaiman (1963),

passivity and small gain: Zames (1966),

inverse optimality: Kaiman (1964), Moylan

and Anderson (1973),

dissipative systems: Willems (1972), Hill and

Moylan (1976),

conicity and margins: Safonov and Athans

(1977), Safonov (1980).

DESIGN TOOLS AND PROCEDURES:

Lyapunov design: Meilakhs (1976,1978),

Jurdjevic and Quinn (1978), Corless and

Leitmann (1981), Qu (1998),

control Lyapunov functions: Artstein (1983),

Sontag (1989),

backstepping: Tsinias (1989), Byrnes and

Isidori (1989),

ISS and ISS small gain: Sontag (1989), Jiang,

Teel and Praly (1994), Teel (1996),

passivation: PK and Sussmann (1989), Byrnes,

Isidori and Willems (1991),

adaptive backstepping: Kanellakopoulos, PK

and Morse (1991), Krstic, Kanellakopoulos and

PK (1992),

robust backstepping: Freeman and PK (1992),

Marino and Tomei (1993),

ISS and Hoo backstepping: Jiang, Teel and

Praly (1994), Pan and Ba§ar (1998), Krstic and

Li (1998),

stochastic backstepping: Krstic and Deng

(1998), Pan and Ba§ar (1999),

forwarding: Teel (1992), Mäzene and Praly

(1996), Jankovic, Sepulchre and PK (1996).

In this lecture

• Locally optimal backstepping design

Ezal, Pan and PK (1997)

• Passivation redesign

Arcak, Seron, Braslavsky and PK (1998)

• Observer-based feedback passivation

Larsen and PK (1999)

• Passive observer design

Arcak and PK (1999)

• Example: Ship steering

Arcak, Fossen and PK (1999)

CLF Construction

Backstepping Lemma If V(X) is a CLF for

X = F(X) + G(X)u

and

L(F+G«)V = LFV(X) + LGV(X) <x(X) < -a(X)

then, V+(X,x)=V(X) + (x-a(X))2

is a CLF for

X = F(X) + G(X)x

x = f(X,x) + g(X,x)u g(X,x)^0.

Example X = X + x

x = u.

V(X) = X2 , ct(X) = -X3 - X

V+(X,x)=X2 + (x + X + X3)2.

Xo = Fo(Xo) + G0(X0)xi

X2 = f2(Xy^>X2) + 92[X,'X>1>X2)'X>3 (SF)

Xn = fTt(X,x) + gn(X,x)u.

gt(X,.-.,Xi)^0

L(F0 + G0ao)Vo(Xo) < —CTo(Xo)

Step 1. Set X = X0,x

backstepping lemma:

Step i. (i = 2, • • • ,n)

xi, V(X)=Vo(X0), apply

V+^VTCX,^).

X =

Xo

xt_i

, x = xt -* V+ =: Vt(X,xi,--- ,xt).

Step n Vr^{X)X^,^'^ ,xn)

Locally Optimal Backstepping Design

K. Ezal et al. - CDC 97 Best Student Paper

X = Ax + f(x) + G\{x)'W -h 02{x)u (strict-feedback class)

Robust backstepping:

Vi := Vi-i + öiZ?

Zi+i := Xi+i-ai(xu...,Xi)

dcti

dxj (0) = a u

Globally Inverse Optimal: u = -r 1(x)5nzn

/•CO

J = [q(x) + r(x)u2 - i2w'w] dt Jo

r(x), q(x) — a posteriori constructed!

Desire Local Hoo -Optimality: i = Ax + ßiiü + B2u

Requires r(0) = Ä, fe*(0) = Q, £» = <?x(0)

A > 0, Q = Q' > 0 - a priori specified

Solution: Cholesky factorization

PA + ÄP + P (7"2#i£i - B2R~lB'2) P + Q = 0

P = L'

o

0

<$n_

L, L

1 0

-aii 1

-Otn-1,1 — CXri-l.ri-l

0'

Result: Locally H«, -optimal, globally inverse optimal

Stability region of an H«, design

»1 = x\ -f- x2 + w

X2 = U

Linearized system:

X\ = X2 + w

X2 = u

Cost functional: /»OO

J = j [a£ + x2 + u2 --y2!!/2] <ft

Desired attenuation: 7 = 5 > 7* = 1.27

Optimal linear control:

t* = -B'Px = -I.O6X1 - 1.78s<

0.8

0.6

0.4

0.2

»? 0

-0.2

-0.4

-0.6

-0.8

EXP0N.- STABLE

.» ^

\

-J » L.

0.2 0.4 0.6 VST —. L-i %

0.8

Global Asymptotic Stability of Nonlinear Locally Optimal Design (w = 0)

Linear Nonlinear

Passivation Redesign Arcak, Seron, Braslavsky and PK (1998)

H o Nominal plant

x = o(x) + r(x)v

A Input unmo deled dynamics

v = HA(£,U).

A is 0-GAS and 3 pos.def. TT(£.) :

TT(£,) < —Au2 H- vu, A > 0 (excess of passivity)

This includes stable linear A(s) with

Re{A(jcu)}> A.

Find pos.def. V(x) such that

UV(x) < (LrVM)2 VxÔ.

Then. u = -kLrV(x) k>x GAS,

Proof: Shortage vs. excess of passivity

Shortage of passivity 1 /k: kV = kUV + kLrVv < k(LrV)2 + kLrVv

Excess of passivity A: it < —Au + vu

u(X)y =kv(x) + n(y => u<o vxô. X = 0=^>u=0=>£, = 0is the largest invariant

set by 0-GAS of A.

When does such V(x) exist?

Iff 3 CLF V(x) such that

r UV(x) limsup 2 < +00,

Construction of V(x)-

Find positive 9(-) such that

(i) L°V(x) <9(V(x))> w (LrV(X))2

(ii) lirat-ôo J0 9(s)ds = +00

Then.

(L)

Lemma When Ho is strict feedback (SF), our

backstepping CLF Vn(X,xi, • • • ,xn) constructed

from Vo(Xo) with

TLF0+G0CC0Ô(XO) < — o"o(Xo)

satisfies (L) if Vo and <JQ have pos.def. Hessians.

Passivation Redesign Example

X = Xs + x

X = V

12(s + 35)(s + 20) v = u = A(s)u.

7(s + 40)(s + 30)

Excess of passivity Re{A(jtu)} > A = 1.

V0(X)=X2, cc0(X) = -X-X3, a0(X)=2X;

3i2 V1(X,x)=X/ + (x + X + Xi)

UVl(X,x) <3 + 9vf(X)X)

(LrV^X.x))2 2 4

Choose e(V1) = | + |V2, k=i = 1,

u = _icLrV = -eiVOLpV, = -2Q(V^[x+X+X3).

Feedback Passivation (FPR) Design

In design we use state feedback

u(x) = oe(x) + ß(x)ü

to make H passive for the input-output pair

r>*

A

y u ß(X) H =►-

/ i X

a(-)

This is possible if

• H is relative degree one:

u appears in the y-equation,

• H is (weakly) minimum phase:

the "zero dynamics" which remain when the

output is kept at zero, tj(t) = 0, are stable.

•

Observer-based FPR Design Larsen and PK (1999)

x = fo(^,0) + g(x)-y2)Ui

E, = A£, + Bu

yi = C]£, y2 = C2£,

Design an asymptotically stable observer

AjM + MA0 = -I, A0 := A + LC2, M > 0.

• Design a state feedback gain K to stabilize the linear part of the system with A^ := A — BK Hurwitz and (Ak) B, Ci) passive by solving

A^P + PAk = -Q<0 P>0

PB = C}.

• Design a stabilizing u from the CLF

W(x, £,, e) = V(x) + -lJ?l + -eTMe. A. \x

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x = Ax-iKx)+Y(y,u), y = Cx>

Observer:

* = A*-1|>(«) + y(y,u) + L(y - CS)

Observer gain matrix L to be designed for

M*) =

^TL(XTL)

I|H(X-L): nondecreasing (ND)

(ND) =$> a [i|)i(xi) -ikfe - cr)] > 0 (sector)

Error dynamics:

e = (A-LC)e-(oMx)-iK*))

Indicator matrix K locates the nonlinearities:

K = diag(ki,--- ,kn)

/

K = 0 if \|H(XO=0

1 otherwise

Note: PK = K does not imply P = I

z := Ke

4>(t,z):=il)(x)-il»(*)

zTcj)(t,z) > 0 (sector)

e = (A-LC)e-K<t>(t,z)

z = Ke.

Observer Design

v e = (A-LC)e + Kv

z = Ke

Find P = PT > 0 and L such that

,T (A-LC)'P+P(A-LC)<0 and PK = K (PR)

Then, whenever e(t) exists, it satisfies

|e(t)|<k|e(0)|.

Proof: PR and sector properties.

Control Law Design

x = Ax-iMx)+Y(y>u), y = Cxy

Find u = oc[y, it) to guarantee

x(t)| <max{ ß(|x(0)|,t), 9 (supT€[0>t)|e(T)|) }. (ISS)

Main Result

If (PR) and (ISS) a,re satisfied, then (CL) is GS.

If, in addition, (K, A — LC) is detectable and,

i|Vs &re either strictly increasing or identically

zero, then (CL) is GAS.

Proof:

(PR) implies |e(t)| < k|e(0)| and, with (ISS),

proves GS. From (PR) and Barbalat's lemma

z(t) = Ke(t) -> 0.

(CL) is autonomous and z = 0 dynamics are

e = (A — LC)e. Detectability implies e(t) —> 0

and GAS follows from ISS.

Feasibility

(PR) is an LMI in LTP and P = PT > 0:

(A - LC)TP + P(A - LC) PK - K

KP-K 0 <0.

Feasibility depends on K, the location of I|H(*I)/

Example y = x-\

XT = X2

x2 = -iM*2)+YCy)+u.

ih(x-,) = 0 ->K = 0 0

0 1

(PR) satisfied with P = I, L = [ 1 1 ].

Evaluate V for V = \ e] + \ e2 along

e-| = —ei + e2

e2 = -ei -(iMx2)-iM£2)).

V = -e 2

i e2(^2(x2)-^2(x2-e2)) < -^

Example: Ship Steering Arcak, Fossen and PK (1999)

f] = ][r\)y

Mrv = —D(-v)-v — g(r|) + T.

D(-v)-v = D0-v +

5iCvi)

Do=Dj>0,

Sehe)

6i(-)'s are ND,

Observer:

ft = J(ri)^ + (ri-ft),

M^ = -D(^-g(Ti)+T + JT(ti)(Ti--ft)

V(TI,-V) = -fjTfi + -yrMrv,

V 1 1 yn fj — T-V'DO-V-'V [5(-v) -6(-v -~.T --v)]

1_T- 1-T„ - < -jT[ T| — --V D0-V.

Observer Based Control Design

V

xi

X3

= ^1

X2 +Xi

X3 -X2(l +x|) +U

2u.

A =

0 1 0

0 -1 1

0 0 0

M*) = 0

X-

0

C = [1 0 0]

^K =

T("y>u) = u

2u

0 0 0

0 1 0

0 0 0

(PR) satisfied with L = [ 3 3 1 ] T

ISS Control Law Design by Backstepping:

V — Xi

*i — X2 + x1

*2 — *3~ x2(l + x2) +u

X3 — 2u.

Relative degree two + ISS zero dynamics

—> two steps of observer backstepping:

iStep i. xi = x2 +xf

ai (xi) = —xf — xi — riixi , n-| > 0.

xi = —xi — TLi xi + e2 + (£2 — tti)

5tep jg. Z2 '•= %2 — oc-[ {x-[) = X2 — oci (xi) + e2

z2 =u+q(x1)x)--—e2 OXi

u= a2(x1)£) =-q(xi,Ä) -XT -z2-n2 (fff J 2:2

Proof of ISS:

XT —x^ — TLIXT -fz2 + e2

Z2 = —Xi — Z2 — Tl2 3ai \

9XT y z2-

3ai

3xi e2

ISS Lyapunov function: V(xi, zz) =x\+z\

—> (*i >*2) subsystem is ISS with input e2-

ISS of zero dynamics:

r) := x3 - 2x2

f| = — 2r| — 2x2 + 2xf (ZD)

Cascade of ISS systems is ISS:

e2 (x-|,x2)

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Documents

Workshop Future Directions in Systems and Control Theory · A Framework for Control, State Estimation, and Verification of Hybrid Systems Manfred Morari Alberto Bemporad Giancarlo