Nonlinear and Adaptive Control

Zhengtao Ding

Control Systems CentreSchool of Electrical and Electronic Engineering

University of Manchester, P O Box 88Manchester M60 1QD, United Kingdom

January 24, 2007

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Aims

To introduce techniques for analysis and design of nonlinear and adaptivecontrol systems.

Learning Outcomes

On completion of the course, students should be able to

• Demonstrate a knowledge of effects of nonlinearities on the operation ofcontrol systems.

• Show an understanding of methods for reducing nonlinear effects in controlsystems.

• Apply phase-plane method to second-order nonlinear systems.

• Obtain locally linearized models for nonlinear dynamic systems.

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• Use the describing function method for predicting nonlinear feedback sys-tems behaviour.

• Investigate the stability of nonlinear systems by using Lyapunov functions.

• Use backstepping technique for control design of a special class of non-linear systems.

• Design and analysis adaptive control of uncertain linear systems.

• Understand robustness of adaptive schemes.

• Design adaptive control for special class of uncertain nonlinear systems.

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1. Introduction to Nonlinear Systems

1.1 Nonlinear functions and nonlinearities

A function f : R 7→ R is linear, if for any x, y, k ∈ R,

f (kx) = kf (x) (1)

f (x + y) = f (x) + f (y) (2)

A function does not satisfy these conditions is a nonlinear functions. Similarconditions can be imposed for linear maps between Rm and Rn for anypositive integers m and n.

Linear dynamic systems

x = Ax + Bu (3)

y = Cx + Du (4)

for state x ∈ Rn, u ∈ Rm, and u ∈ Rp, and matrices are in suitabledimensions.

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Single-Input-Single-Output (SISO) systems with input saturation

x = Ax + Bσ(u) (5)

y = Cx (6)

where σ : R 7→ R is a saturation function defined as

σ(u) =

−1 for u < −1u otherwise1 for u > 1

(7)

The saturation function σ is a nonlinear function and this system is a non-linear system.

A general nonlinear system is often described by

x = f (x, u) (8)

y = h(x, u) (9)

where functions f and h are often assumed to be smooth enough (ie, havingcontinuous derivatives to certain orders).

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System with unknown parameters

x = ax + u (10)

where a is an unknown parameter. How to design a control system to ensurethe stability of the system? If a range a− < a < a+ is known, then we candesign a control law as

u = −cx− a+x (11)

with c > 0, which results the closed-loop system

x = −cx + (a− a+)x (12)

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Adaptive control can be used in the case of complete unknown a.

u = −cx− ax (13)˙a = x2 (14)

If we let a = a− a, the closed-loop system is described by

x = −cx + ax (15)˙a = −x2 (16)

This adaptive system is nonlinear, even though the original uncertain systemis linear. This adaptive system is stable, but how to show it?

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1.2 Common nonlinearities in control systems

• Nonlinear components, such springs, resistors etc.

• Dead zone and saturation in actuators

• Switching devices, relay and hysteresis relay

• Backlash

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1.3 Common nonlinear systems behaviours

• Dynamics (of linearized systems) depends on operating point (hence theinput).

• There may be more than one equilibrium point.

• Limit cycles (persistent oscillations) may occur, and also chaotic motion.

• For a pure sinusoidal input at a fixed frequency, the output waveforms isgenerally distorted, and the output amplitude is a nonlinear function ofthe input amplitude.

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1.4 Compensating nonlinearities and nonlinear control design

• Gain Scheduling switching controller parameters in different operatingregions.

• Feedback Linearisation By using nonlinear control laws.

•Nonlinear Function Approximations using fuzzy logic and neural net-works.

• Adaptive Control such self-tuning control model reference adaptive con-trol, and nonlinear adaptive control.

• Bang-bang Control Switching between operating limits (time-optimalcontrol).

• Variable Structure Control, sliding mode control.

• Backstepping Design design control input in an iterative way by movingdown the different virtual control until the control input is design.

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• Forwarding Control an iterative design working in a way opposite tobackstepping.

• Semiglobal Control Design achieving stability in arbitrarily large re-gions, often by using high gain control.

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2. State Space Models

2.1 Nonlinear systems and linearisation around an operating point

Consider

x = f (x, u) (17)

y = h(x, u) (18)

An operating point at (xe, ue) is taken with x = xe and u = ue beingconstants such that f (xe, ue) = 0. A linerized model around the operationpoint can then be obtained. Let

x = x− xe (19)

u = u− ue (20)

y = h(x, u)− h(xe, ue) (21)

then the linearized model is given by

˙x = Ax + Bu (22)12

y = Cx + Du (23)

where

ai,j =∂fi

∂xj(xe, ue) (24)

bi,j =∂fi

∂uj(xe, ue) (25)

ci,j =∂hi

∂xj(xe, ue) (26)

di,j =∂hi

∂uj(xe, ue) (27)

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2.2 Some special forms

Strict feedback from

x1 = x2 + φ1(x1)

x2 = x3 + φ2(x1, x2)...

xn−1 = xn + φn−1(x1, x2, . . . , xn−1)

xn = u + φn(x1, x2, . . . , xn) (28)

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Output feedback from

x1 = x2 + φ1(y)

x2 = x3 + φ2(y)...

xr = xr+1 + φr(y) + bru...

xn = φn(y) + bnu

y = x1 (29)

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Normal from

z = f0(z, ξ1, . . . , ξr)

ξ1 = ξ2

ξ2 = ξ3...

ξr = fr(z, ξ1, . . . , ξr) + bru (30)

with z ∈ Rn−r.Special normal from

z = f0(z, ξ1)

ξ1 = ξ2

ξ2 = ξ3...

ξr = fr(z, ξ1, . . . , ξr) + bru (31)

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2.3 Autonomous systems

Autonomous systems: a dynamic system which does not explicitly de-pend on time. Often it can be expressed as

x = f (x) (32)

For an autonomous system, the set of all the trajectories provides a com-plete geometrical representation of the dynamic behaviour. This is oftenreferred to as the phase portrait, especially for second order systems in theformat:

x1 = x2

x2 = φ(x1, x2) (33)

Singular point: the point xe such that f (xe) = 0.

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2.4 Second order systems

Also referred to as phase plane.

x1 = f1(x1, x2)

x2 = f2(x1, x2) (34)

Classification of singular points (based of eigenvalues)

λ1 < 0, λ2 < 0 Stable Nodeλ1 > 0, λ2 > 0 Unstable Nodeλ1 < 0, λ2 > 0 Saddle Pointλ1,2 = µ± iν(µ < 0) Stable Focusλ1,2 = µ± iν(µ > 0) Unstable Focusλ1,2 = ±iν(ν > 0) Centre

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Slope of trajectory

dx2

dx1=

f1(x1, x2)

f2(x1, x2)(35)

An Isocline is a curve on which f1(x1,x2)f2(x1,x2)

is constant.

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Example 2.1 The swing equation of a synchronous machine

Hδ = Pm − Pe sin δ (36)

where H is the inertia, δ, the rotor angle, Pm, mechanical power, Pe themaximum electrical power generated.

The state space model is obtained by let x1 = δ and x2 = δ as

x1 = x2

x2 =u− Pe sin x1

Hy = Pe sin x1 (37)

where u = Pm as the input.

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Singular points

x1e = arcsin(ue/Pe) or π − arcsin(ue/Pe)

x2e = 0 (38)

Linearised model

A =

0 1

−Pe cos x1eH 0

B =

0

− 1H

C = [ Pe cos x1e 0 ]

Stable or unstable?

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2.5 Limit cycles

A special trajectory which takes the form of a closed curve, and is intrinsicto the system, and not imposed from outside.

Limits cycles are periodic solutions.A limit cycle is stable is nearby trajectories converge to it asymptotically,

unstable is move away.A general condition for the existence of limit cycles in the phase plane is

the Poincare-Bendixson theorem:If a trajectory of the second-order autonomous system remains in a finite

region, then one of the following is true:(a) the trajectory goes to an equilibrium point(b) the trajectory tends to an asymptotically stable limit cycle(c) the trajectory is itself a stable limit cycle

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Example 2.2 The Van der Pol oscillatorThis is one of the best known models in nonlinear systems theory, originally

developed to describe the operation of an electronic oscillator, which dependson the existence of a region with effective negative resistance.

x1 = x2

x2 = −x1 + ε(x2 − x32) (39)

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2.6 Strange attractors and chaos

For high order nonlinear systems, there are more complicated features if thetrajectories remain in a bounded region.Positive limit set of a trajectory the set of all the points for which thetrajectory converge to.

Strange limit sets are those limit sets which may or may not be asymp-totically attractive to the neighbouring trajectories. The trajectories theycontain may be locally divergent from each other, within the attracting set.Such struectures are associated with the quasi-random behaviour of solutionscalled chaos.

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Example 2.3 The Lorenz attractorThis is one the most widely studied examples of strange behavour in or-

dinary differential equations, which is originated from studies of turbulentconvection by Lorenz. The equation is in the form

x1 = σ(x2 − x1)

x2 = (1 + λ− x3)x1 − x2

x3 = x1x2 − bx3 (40)

where σ, λ and b are positive constants.

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3. Describing Functions

3.1 Describing function fundamentals

If a nonlinear system contains only one nonlinear component and the otherpart is linear, then this kind of systems may be analyzed using describingfunctions. Some genuinely nonlinear systems can also be arranged in theform of with only one nonlinear element.

The basic assumption for describing function analysis are:

• there is only a single nonlinear component

• the nonlinear component is not time-varying

• corresponding to a sinusoidal input x = sin(ωt), only the fundamentalcomponent has to be considered

• the nonlinearity is odd

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Basic definitionFor a nonlinear component described by single-value nonlinear function f (),

its output w(t) = f (A sin(ωt)) to a sinusoidal input A sin(ωt) is a periodicalfunction, although it may not be sinusoidal in general. A periodical functioncan be expanded in Fourier series

w(t) =a0

2+

∞∑n=1

[an cos(nωt) + bn sin(nωt)] (41)

where

a0 =1

π

∫ π−π w(t)d(ωt)

an =1

π

∫ π−π w(t) cos(nωt)d(ωt)

bn =1

π

∫ π−π w(t) sin(nωt)d(ωt)

Deu to the assumption that f is an odd function, we have a0 = 0, and the

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fundamental component is given by

w1 = a1 cos(ωt) + b1 sin(ωt) = M sin(ωt + φ) (42)

where

M(A, ω) =√√√√a2

1 + b21

φ(A, ω) = arctan(a1/b1)

In complex expression, we have w1 = Mej(ωt+φ) = (b1 + ja1)ejωt.

Describing function is defined, similar to frequency response, as the com-plex ratio of the fundamental component of the nonlinear element againstthe input by

N(A, ω) =Mejωt+φ

Aejωt =b1 + ja1

A(43)

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Example 3.1 Describing function of hardening springThe characteristics of a hardening spring are given by

w = x + x3/2 (44)

Given the input A sin(ωt), the output is

w = A sin(ωt) +A3

2sin3(ωt) (45)

Since w is an odd function, we have a1 = 0. The coefficient b1 is given by

b1 =1

π

∫ π−π[A sin(ωt) +

A3

2sin3(ωt)] sin(ωt)d(ωt)

= A +3

8A3 (46)

Therefore the describing function is

N(A, ω) = N(A) = 1 +3

8A2 (47)

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3.3 Describing functions for common nonlinear components

Saturation A saturation function is described by

f (x) =

kx for |x| < asign(x)ka otherwise

(48)

The output to the input A sin(ωt), for A > a, is symmetric over quarters ofa period, and in the first quarter,

w1(x) =

kA sin(ωt) 0 ≤ ωt ≤ γka γ < ωt ≤ π/2

(49)

where γ = sin−1(a/A). The function is odd, hence we have a1 = 0, andthe symmetry of w1(t) implies that

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b1 =4

π

∫ π/40 w1 sin(ωt)d(ωt)

=4

π

∫ γ0 kA sin2(ωt)d(ωt) +

4

π

∫ π/2γ ka sin(ωt)d(ωt)

=2kA

π[γ +

a

A

√√√√√√√√1− a2

A2] (50)

Therefore the describing function is given by

N(A) =b1

A=

2k

π

sin−1 a

A+

a

A

√√√√√√√√1− a2

A2

(51)

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Ideal relay The output from the ideal relay (sign function) is give by, withM > 0,

w1(x) =

−M −π ≤ ωt < 0M 0 ≤ ωt < π

(52)

It is again an odd function, hence we have a1 = 0. The coefficient b1 isgiven by

b1 =∫ π/20 M sin(ωt)(.ωt) =

4M

π(53)

and therefore the describing function is given by

N(A) =4M

πA(54)

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3.4 Describing function analysis of nonlinear systems

Extended Nyquist criterion: Consider a linear feedback system with for-ward transfer function G(s) and feedback control gain H(s). The charac-teristic equation of this system is given by

G(s)H(s) + 1 = 0 or G(s)H(s) = −1 (55)

The Nyquist criterion can then be analyze the stability of the system basedon the Nyquist plot the open-loop transfer function G(s)H(s) with the en-circlement of the point (−1, 0). An extension to this criterion for the caseof the forward transfer function is KG(s), characteristic equation is

KG(s)H(s) + 1 = 0 or G(s)H(s) = −1/K (56)

The same argument used in the derivation of Nyquist can be applied again,with (−1, 0) being replace by (1/K, 0). Note that K can be a complexnumber.

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Existence of limit cyclesConsider a unit feedback system with N(A, ω) and G(s) in the forward

path. The equations describing this system can be put as

x = −y

w = N(A, ω)x

y = G(jω)w

with y as the output. From the above equations, it can be obtained that

y = G(jω)N(A, ω)(−y) (57)

and it can be arranged as

[G(jω)N(A, ω) + 1]y = 0 (58)

If there exists a limit cycle, then y 6= 0, which implies that

G(jω)N(A, ω) + 1 = 0 (59)

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or

G(jω) = −1/N(A, ω) (60)

Therefore the amplitude A and frequency ω of the limit cycle must satisfythe above equation. The equation 60 is difficult to solve in general. Graphicsolutions can be found by plotting G(jω) and −1/N(A, ω) to see if theyintersect each other. The intersection points are the solutions. It is easy toplot −1/N if it is independent of the frequency.Stability of limit cycles Discussions based on the extended Nyquist cri-terion.

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Example 3.1 Consider a linear transfer function G(s) = Ks(s+1)(s+2) and

an ideal relay with M = 1 in a closed loop. Determine if there exists a limitcycle. If so, determine the amplitude and frequency of the limit cycle.

(ω =√

2, A = 23π)

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4. Stability Theory

4.1 Definitions

Consider the system

x = f (x) (61)

with x = 0 as an equilibrium point.Lyapunov Stability The equilibrium point x = 0 is said to be stable if forany R > 0, there exists r > 0 such that if ‖x(0)‖ < r, then ‖x(t)‖ < Rfor all t ≥ 0. Otherwise the equilibrium is unstable.

Critically stable linear system is Lyapunov stable.

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Asymptotic Stability An equilibrium point 0 is asymptotically stable if itis stable (Lyapunov) and if in addition there exists some r > 0 such that‖x(0)‖ < r implies that limt→∞ x(t) = 0.

Linear systems with poles in the left side of complex plane are asymptoti-cally stable.

Consider x = −x3

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Exponential Stability An equilibrium point 0 is exponentially stable if thereexist two positive real numbers α and λ such that

∀t > 0, ‖x(t)‖ < α‖x(0)‖e−λt (62)

in some ball Br around the origin.For linear systems, asymptotic stability implies exponential stability.

Global Stability If asymptotic (exponential) stability holds for any initialstate, the equilibrium point is said to be globally asymptotically (exponen-tially) stable.

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4.2 Linearization and local stability

Lyapunov’s linearisation method

• If the linearized system is strictly stable (ie, with system’s poles in theopen left-half complex plane), then the equilibrium point is asymptoticallystable for the actually nonlinear system.

• If the linearized system is unstable (ie, with system’s poles in the openright-half complex plane), then the equilibrium point is unstable.

• If the linearized system is marginally stable (with poles on the imaginaryaxis), then the stability of the original system cannot be concluded usingthe linearised model.

Examples

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4.3 Lyapunov’s direct method

Positive definite functions A scalar function V (x) is said to be locallypositive definite if V (0) = 0 and in a ball Br, x 6= 0 implies V (x) > 0. Ifthe above properties hold for the entire space, the V (x) is said to be globallypositive definite.

Lyapunov function If in a ball BR, the function V (x) is positive definiteand has continuous partial derivatives, and if its time derivative along anystate trajectory of system (61) is negative semidefinite, ie,

V (x) ≤ 0 (63)

then V (x) is a Lyapunov function.

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Lyapunov Theorem for Local Stability If in a ball BR, there exists ascalar function V (x) with continuous first order derivatives such that

• V (x) is positive definite (locally in BR)

• V (x) is negative semidefinite (locally in BR)

then the equilibrium point 0 is stable. Furthermore, if V (x) is locally negativedefinite in BR, then the stability is asymptotic.

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Example 4.1 S simple pendulum is described by

θ + θ + sin θ = 0 (64)

Consider the scalar function

V (x) = (1− cos θ) +θ2

2(65)

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Lyapunov Theorem for Global Stability Assume that there exists ascalar function V (x) with continuous first order derivatives such that

• V (x) is positive definite

• V (x) is negative definite

• V (x) →∞ as ‖x‖ → ∞then the equilibrium at the origin is asymptotically stable.

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4.4 Lyapunov analysis of linear-time-invariant systems

Consider LTI system x = Ax. If a Lyapunov function candidate is give by

V (x) = xTPx (66)

then direct evaluation gives

V = xTATPx + xTPAx := −xTQx (67)

where

ATP + PA = −Q (68)

If Q is positive definite, then the system is asymptotically stable.

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Theorem of stability of LTI systems A necessary and sufficient conditionfor a LTI system x = Ax to be strictly stable is that, for any positivedefinite matrix Q, the unique solution P of Lyapunov equation (68) is positivedefinite.Proof: Briefly, choose a positive definite Q, and define

P =∫ ∞0 exp(AT t)Q exp(At)dt (69)

and note

−Q =∫ ∞0 d[exp(AT t)Q exp(At)] (70)

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5. Advanced Stability Theory

5.1 Positive Real Systems

Consider a SISO dynamic system

x = Ax + bu

y = cTx (71)

and its transfer function matrix is given by

G(s) = cT (sI − A)−1b (72)

Positive Reals Systems A system described in (75) is said to be positivereal if

<(G(s)) ≥ 0 for all <(s) ≥ 0 (73)

It is strictly positive real if G(s− ε) is positive real for some ε > 0.Examples: G(s) = 1

s+λ with λ > 0.

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Theorem 5.1 A transfer function G(s) is strictly positive real (SPR) if andonly if

• G(s) is a strictly stable transfer function

• the real part of G(s) is strictly positive along the jω axis, ie.,

∀ω ≥ 0,<[G(jω)] > 0 (74)

Some necessary conditions for SPR

• G(s) is strictly stable

• The Nyquist plot of G(jω) lies entirely in the right half of complex plane

• G(s) has a relative degree of 0 or 1

• G(s) is strictly minimum phase

Examples: G1 = s−1s2+as+b

, G2 = s+1s2−s+1

, G3 = 1s2+as+b

, G4 = s+1s2+s+1

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Theorem 5.2 A transfer function G(s) is positive real (PR) if and only if

• G(s) is a stable transfer function

• The poles of G(s) on the jω axis are simple and associated residues arereal and non-negative

• <[G(jω)] ≥ 0 for any ω such that jω is not a pole of G(s)

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Kalman-Yakubovic Lemma Consider a controllable LTI system

x = Ax + bu

y = cTx (75)

Its transfer function matrix is given by

G(s) = cT (sI − A)−1b (76)

is strictly positive real if and only if there exist positive definite matrices Pand Q such that

ATP + PA = −Q (77)

Pb = c (78)

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5.2 Stability of feedback system

Consider the closed-loop system

x = Ax + bu

y = cTx

u = −F (y)y (79)

If cT (sI −A)−1b is strictly positive real, then what is the condition on f (y)for the stability if the closed-loop system?

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From K-Y lemma, there exist positive definite P and Q. Define V =xTPx, its derivative is given by

V = −xTQx + 2xTPBu

= −xTQx− 2xTPBF (y)y

= −xTQx− 2xT cF (y)y

= −xTQx− 2F (y)y2 (80)

Therefore if F (y) > 0 then system is exponentially stable. The condition isonly a sufficient condition.

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5.3 Circle Criterion

Consider the case α < F (y) < β. Under what condition of cT (sI −A)−1b,the closed-loop system (79) is stable?

Consider the transform

F =F − α

β − F(81)

and obviously we have F > 0. How to use this transform for analysis ofsystems stability?

Consider the characteristic equation of (79). We can write

G(s)F + 1 = 0 (82)

which gives

G(s)(F − α) = −αG− 1 (83)

G(s)(β − F ) = βG + 1 (84)

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With (83) divided by (84), we can obtain1 + βG

1 + αG· F − α

β − F+ 1 = 0 (85)

Let

G :=1 + βG

1 + αG(86)

Based on the analysis of feedback stability of SPR system, we can concludethat the closed-loop system is stable if G is SPR.

What is the condition of G if G is SPR? From (86), it can be obtainedthat

G =G− 1

β − αG(87)

For the case of 0 < α < β, G is SPR is equivalent to that Nyquist plot ofG does not encircle the circle centered as (−1

2(1/α + 1/β), 0) with radius of12(1/α− 1/β). That is why it is referred as circle criterion.

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6. Feedback Linearization

6.1 Introduction

Consider the following nonlinear systems

x = f (x) + g(x)u

y = h(x) (88)

Two Examples:

x1 = x2 + y3

x2 = y2 + u

y = x1 (89)

x1 = x2 + y3 + u

x2 = y2 + u

y = x1 (90)

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6.2 Input-Output Linearization

Lie Derivative A Lie derivative of a function h(x) along the vector fieldf (x) is defined as

Lfh(x) =∂h(x)

∂xf (x) (91)

Notation

Lkfh(x) = Lf (Lk−1

f h(x)) =∂Lk−1

f h(x)

∂xf (x) (92)

Relative Degree A system has relative degree ρ at a point x if the followingconditions are satisfied:

LgLkfh(x) = 0 for k = 0, . . . , ρ− 2 (93)

LgLρ−1f h(x) 6= 0 (94)

Examples. Use (89) and linear systems to demonstrate the concepts.

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It can be obtained that

y(k) = Lkfh(x) for k = 0, . . . , ρ− 1 (95)

y(ρ) = Lρfh(x) + LgL

ρ−1f h(x)u (96)

If we define the input as

u =1

LgLρ−1f h(x)

[−Lρfh(x) + v] (97)

which results at

y(ρ) = v (98)

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Therefore the system is input-output linearized by defining

ξi = Li−1f h(x) for i = 1, . . . , ρ (99)

and the linearized part of the system is described by

ξ1 = ξ2 (100)... (101)

ξρ−1 = ξρ (102)

ξρ = v (103)

The other part of system dynamics are characterized by the zero dynamics.Examples

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6.3 Full State Linearization

Consider

x = f (x) + g(x)u (104)

where x ∈ rn, and u ∈ R. The full state linearization can be achieved ifthere exists a function h(x) such that the relative degree regrading h(x) asthe output is n, the dimension of the system.Lie Bracket For two vector fields f (x) and g(x), their Lie Bracket is definedas

[f, g](x) =∂g

∂xf (x)− ∂f

∂xg(x) (105)

Notations:

ad0fg(x) = g(x) (106)

ad1fg(x) = [f, g](x) (107)

adkfg(x) = [f, adk−1

f g](x) (108)

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Example

f (x) =

x2

− sin x1 − x2

, g(x) =

0x1

(109)

Example

f (x) = Ax, g(x) = b (110)

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Distribution The collection of vector spaces

∆ = span{f1(x), . . . , fn(x)} (111)

The dimension of distribution is defined as

dim(∆(x)) = rank[f1(x), . . . , fn(x)] (112)

Involutive A distribution is involutive if

g1 ∈ ∆ and g2 ∈ ∆ ⇒ [g1, g2] ∈ ∆

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Example Let ∆ = span{f1, f2} where

f1(x) =

2x210

, f2(x) =

10x2

(113)

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Theorem 6.1 The system (104) is feedback linearizable if and only if

• the matrix [g(x), adfg(x), . . . , adn−1f g(x)] has full rank;

• the distribution span{g(x), adfg(x), . . . , adn−2f g(x)} is involutive.

63

Example

x1 = a sin x2

x2 = −x21 + u

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7. Backstepping Design

7.1 Integrator Backstepping

Consider

x = f (x) + g(x)ξ

ξ = u (114)

with x ∈ Rn and ξ ∈ R. There exist a function φ(x) with φ(0) = 0 and apositive definite function V (x) such that

∂V

∂x[f (x) + g(x)φ(x)] ≤ −W (x) (115)

where W (x) is positive definite.Condition (115) implies that the system x = f (x)+g(x)φ(x) is asymptot-

ically stable. How to design the control input so that the system is stable?

65

Consider

x = f (x) + g(x)φ(x) + g(x)(ξ − φ(x)) (116)

Let z = ξ − φ(x). It is easy to obtain that

x = f (x) + g(x)φ(x) + g(x)z (117)

z = u− φ = u− ∂φ

∂x(f (x) + g(x)ξ) (118)

Consider a Lyapunov function candidate

Vc(x, ξ) = V (x) +1

2z2 (119)

Its derivative is given by

Vc =∂V

∂x[f (x) + g(x)φ(x)] +

∂V

∂xg(x)z

+z[u− ∂φ

∂x(f (x) + g(x)ξ)] (120)

66

Let

u = −cz − ∂V

∂xg(x) +

∂φ

∂x(f (x) + g(x)ξ) (121)

with c > 0 which results at

Vc = −W (x)− cz2 (122)

Theorem 7.1 For a system described in (114), the control design given in(121) ensures the global asymptotic stability of the closed-loop system.

67

Example 7.1

x1 = x21 + x2

x2 = u (123)

Example 7.2

x1 = −x1 + x21x2

x2 = u (124)

Example 7.3

x1 = x21 − x1x2

x2 = u (125)

68

7.2 Iterative Backstepping

Consider a nonlinear system in the strict feedback form

x1 = x2 + φ1(x1)

x2 = x3 + φ2(x1, x2)...

xn−1 = xn + φn−1(x1, x2, . . . , xn−1)

xn = u + φn(x1, x2, . . . , xn) (126)

Backstepping technique introduced in the previous section can be appliediteratively. Define the notations

z1 = x1 (127)

zi = xi − αi−1(x1, . . . , xi−1), for i = 2, . . . , n (128)

where αi−1 for i = 2, . . . , n are stabilizing functions obtained in the iterativebeackstepping design.

69

Step 1:

z1 = [x2 − α1] + α1 + φ1(x1)

= z2 + α1 + φ1(x1) (129)

Let

α1 = −c1z1 − φ1(x1) (130)

The resultant dynamics of z1 is

z1 = −c1z1 + z2 (131)

70

Step 2: The dynamics of z2 is

z2 = x2 − α1

= x3 + φ2(x1, x2)−∂α1

∂x1(x2 + φ1(x1))

= z3 + α2 + φ2(x1, x2)−∂α1

∂x1(x2 + φ1(x1)) (132)

Design α2 as

α2 = −z1 − c2z2 − φ2(x1, x2) +∂α1

∂x1(x2 + φ1(x1)) (133)

The resultant dynamics of z2 is given by

z2 = −z1 − c2z2 + z3 (134)

71

Step i : For 2 < i < n, the dynamics of zi is given by

zi = xi − αi−1(x1, . . . , xi−1)

= xi+1 + φi(x1, . . . , xi)

−i−1∑j=1

∂αi−1

∂xj(xj+1 + φj(x1, . . . , xj))

= zi+1 + αi + φi(x1, . . . , xi)

−i−1∑j=1

∂αi−1

∂xj(xj+1 + φj(x1, . . . , xj)) (135)

Design αi as

αi = −zi−1 − cizi − φi(x1, . . . , xi)

+i−1∑j=1

∂αi−1

∂xj(xj+1 + φj(x1, . . . , xj)) (136)

The resultant dynamics of zi is given by

zi = −zi−1 − cizi + zi+1 (137)

72

Step n: At the final step, we have

zn = xn − αn−1(x1, . . . , xn−1)

= u + φn(x1, . . . , xn)

−n−1∑j=1

∂αn−1

∂xj(xj+1 + φj(x1, . . . , xj)) (138)

Design the final control input as

u = −zn−1 − cnzn − φn(x1, . . . , xn)

+n−1∑j=1

∂αn−1

∂xj(xj+1 + φj(x1, . . . , xj)) (139)

The resultant dynamics of zn is given by

zn = −zn−1 − cnzn (140)

73

Stability Analysis Define a Lyapunov function candidate

V =1

2

n∑i=1

z2i (141)

It is easy to obtain that

V = z1[−c1z1 + z2] + z2[−z1 − c2z2 + z3] + . . .

+zn−1[−zn−2 − cn−1zn−1 + zn] + zn[−zn−1 − cnz2]

= − n∑i=1

ciz2i (142)

Therefore the closed loop system under the proposed control is asymptoticallystable.Theorem 7.2 For a system in the strict feedback form (126), the controlinput (139) renders the closed-loop system asymptotically stable.

74

Example 7.4

x1 = x2 + x21

x2 = x3 + x1x2

x3 = u + x23 (143)

Example 7.5

x1 = −x1 + x21x2

x2 = x3

x3 = u (144)

75

8. Adaptive Control I

8.1 Introduction

The basic objective of adaptive control is to maintain consistent performanceof a system in the presence of uncertainty or unknown variation in plantparameters.MRAC Model Reference Adaptive Control consists of reference model whichproduce the desired output, and difference between the plant output andthe reference output is then used to adjust the control parameter and thecontrol input directly. MRAC is often in continuous-time domain, and fordeterministic plants.STC Self-Tuning Control estimates systems parameters and then computethe control input from the estimated parameters. STC is often in discrete-time and and for stochastic plants.

We will focus on MRAC in this course.

76

Design of adaptive controllers Compared with the conventional con-trol design, adaptive control is more involved, with the need to design theadaptation law. Adaptive control design usually involves the following threesteps:

• Choose a control law containing variable parameters

• choose an adaptation law for adjusting those parameters

• analysis the convergence properties of the resulting control system

77

8.2 Adaptive control of first-order systems

Consider a first order system

y + apy = bpu (145)

The output y is to follow the output of the reference model

ym + amym = bmr (146)

The reference model is stable, ie., am > 0. The signal r is the referenceinput. The design objective is to make the tracking error e = y−ym convergeto 0.

78

Model Reference Control Rearrange the system model as

y + amy = bp[u−ap − am

bpy] (147)

and therefor the we obtain

e + ame = bp[u−ap − am

bpy − bm

bpr]

:= bp[u− ayy − arr] (148)

where ay =ap−am

bp, ar = bm

bp.

If all the parameters are known, the control law is designed as

u = arr + ayy (149)

79

Adaptive control law With parameters unknown, let ar and ay denotetheir estimates of ar and ay. The control law under the certainty equivalenceprinciple is given by

u = arr + ayy (150)

Adaptive laws The estimates are updated by

˙ar = −sign(bp)γrer (151)˙ay = −sign(bp)γyey (152)

were γr and γy are positive real design parameters, and are often referred toas adaptive gains.

The closed loop system under the adaptive control law is given by

e + ame = bp[−ayy − arr] (153)

where ar = ar − ar and ay = ay − ay.

80

Stability analysis Consider the Lyapunov function candidate

V =1

2e2 +

|bp|2γr

a2r +

|bp|2γy

a2y (154)

Its derivative along the trajectory (153) and adaptive laws (151) and (151)is given by

V = −ame2 + ar[|bp| ˙ar − ebparr] + ay[|bp| ˙ay − ebpayy]

= −ame2 (155)

Therefore the system is Lyapunov stable with all the variables e, ar and ay

are all bounded. Furthermore, we can show that

∫ ∞o e2(t)dt =

V (0)− V (∞)

am< ∞ (156)

Therefore we have established that e ∈ L2∩L∞ and e ∈ L∞. To concludethe stability analysis, we need Babalat’s Lemma.

81

Barbalat’s Lemma If a function f (t) is uniformly continuous for t ∈[0,∞), and ∫∞

0 f (t)dt exists, then limt→∞ f (t) = 0.Since e and e are bounded , e2 is uniformly continuous. Therefore we

can conclude from Barbalat’s Lemma that limt→∞ e2(t) = 0 and hencelimt→∞ e(t) = 0.

82

Simulation study of first order system

Gp =2

s− 1(157)

Gm =1

s + 2(158)

83

9. Adaptive Control II

9.1 Model reference control of high order systems

Consider a nth-order system

y(s) = kpZp(s)

Rp(s)u(s) (159)

where kp is the high frequency gain, Zp and Rp are monic polynomials withorders of n−ρ and n respectively with ρ as the relative degree. The referencemodel is given by

y(s)m = kmZm(s)

Rm(s)r(s) (160)

where km > 0 and Zm and Rm are monic Hurwitz polynomials.

84

MRC of systems with ρ = 1 Following a similar manipulation to the firstorder system, we have

y(s)Rm(s) = kpZp(s)u(s)− (Rp(s)−Rm(s))y(s)

= kpZm(s)[Zp(s)

Zm(s)u(s) +

Rm(s)−Rp(s)

Zm(s)y(s)]

= kpZm(s)[u(s)− θT1 α(s)

Zm(s)u(s)− θT

2 α(s)

Zm(s)y(s)− θ3y(s)](161)

where θ1 ∈ Rn−1, θ2 ∈ Rn−1, θ3 ∈ R and

α(s) = [sn−2, . . . , 1]T (162)

Hence, we have

y = kpZm(s)

Rm(s)[u(s)− θT

1 α(s)

Zm(s)u(s)− θT

2 α(s)

Zm(s)y(s)− θ3y(s)] (163)

85

and

e1 = kpZm(s)

Rm(s)[u(s)− θT

1 α(s)

Zm(s)u(s)− θT

2 α(s)

Zm(s)y(s)− θ3y(s)− θ4r](164)

where e1 = y− ym and θ4 = kmkp

. The control input for the model reference

control is given by

u =θT1 α(s)

Zm(s)u +

θT2 α(s)

Zm(s)y + θ3y + θ4r

= θTω (165)

whereθT = [θT

1 , θT4 , θ3, θ4]

ω = [ωT1 , ωT

2 , y, r]T

ω1 =α(s)

Zm(s)u; ω2 =

α(s)

Zm(s)y

86

Example 9.1 Design MRC for the system

y(s) =s + 1

s2 − 2s + 1u (166)

with the reference model

ym(s) =s + 3

s2 + 2s + 3r (167)

87

MRC of systems with ρ > 1 For a system with ρ > 1, the input in thesame format as (165) can be obtained. The only difference is that Zm isof order n − ρ < n − 1. In this case, we let P (s) be a monic polynomialwith order ρ− 1 so that Zm(s)P (s) is of order n− 1. We adopt a slightlydifferent approach from the case of ρ = 1. Consider the identity

y =Zm

Rm[RmP

ZmPy]

=Zm

Rm[QRp + ∆1

ZmPy]

= kpZm

Rm[QZpu + k−1

p ∆1y

ZmP]

= kpZm

Rm[u +

∆2

ZmPu +

k−1p ∆1

ZmPy] (168)

where Q is a monic polynomial of order ρ − 1, ∆1 and ∆2 are polynomials

88

of order n− 1 and n− 2 respectively. Therefore we can write

y = kpZm

Rm[u− θT

1 α(s)

ZmPu− θT

2 α(s)

ZmPy − θ3y] (169)

and

e1 = kpZm

Rm[u− θT

1 α(s)

ZmPu− θT

2 α(s)

ZmPy − θ3y − θ4r] (170)

Hence the control input is designed as

u =θT1 α(s)

Zm(s)P (s)u +

θT2 α(s)

Zm(s)P (s)y + θ3y + θ4r]

= θTω (171)

with the same format as (165) but

ω1 =α(s)

Zm(s)P (s)u; ω2 =

α(s)

Zm(s)P (s)y

89

Example 9.2 Design MRC for the system

y(s) =1

s2 − 2s + 1u (172)

with the reference model

ym(s) =1

s2 + 2s + 3r (173)

90

9.2 Adaptive Control (ρ = 1)

Basic Assumptions Adaptive control design in this course assumes

• the known system order n,

• the known relative degree ρ

• the minimum phase of the plant,

• the known sign of the high frequency gain sign[kp].

91

We choose the reference model as SPR. If the parameters are unknown,the control law based on the certainty equivalence principle is given as

u = θTω (174)

where θ is the estimate of θ. The adaptive law is designed as

˙θ = −sign[kp]Γe1ω (175)

where Γ ∈ R2n, the adaptive gain, is a positive definite matrix, and ω isgenerated in the same way as in the MRC case.

92

Stability Analysis In this can we can write

e1 = kpZm

Rm(θTω − θTω)

= kmZm

Rm[− kp

kmθTω] (176)

where θ = θ − θ. Furthermore, we can put in the state space form as

e = Ame + bm[− kp

kmθTω]

e1 = cTme (177)

where {Am, bm, cm} is a minimum state space realization of kmZm(s)Rm(s), ie,

cTm(sI − Am)−1bm = kmZm(s)

Rm(s)(178)

93

Since {Am, bm, cm} is SPR, there exist positive definite matrices P and Qsuch that

ATmP + PAm = −Q (179)

Pbm = cm (180)

Define a Lyapunov function candidate as

V =1

2eTPe +

1

2| kp

km|θTΓ−1θ (181)

Its derivative is given by

V = −1

2eTQe + eTPbm[− kp

kmθTω] + | kp

km|θTΓ−1 ˙θ

= −1

2eTQe + e1[−

kp

kmθTω] + | kp

km|θTΓ−1 ˙θ

= −1

2eTQe (182)

94

We can now conclude the boundedness of e and θ. Furthermore it can beshown that e ∈ L2 and e1 ∈ L∞. Therefore from Barbalat’s lemma we havelimt→∞ e1(t) = 0. The boundedness of other system state variables can beestablished from the minimum phase property of the system.

95

9.3 Adaptive Control with ρ > 1

For systems with higher relative degrees, the certainty equivalence principlecan still be use to design the control, ie,

u = θTω (183)

with ω be the same as in (171). However the adaptive law is more involvedas the reference model is no longer SPR. If L(s) is a polynomial of orderρ− 1 which makes km

ZmLRm

SPR, then the error model is given by

e1 = kmZmL

Rm[k(uf − θTφ)] (184)

where k =kpkm

, uf = 1L(s)u and φ = 1

L(s)ω. An auxiliary error is constructedas

ε = e1 − kmZmL

Rm[k(uf − θTφ)]− km

ZmL

Rm[εn2

s] (185)

96

where k is an estimate of k, n2s = φTφ+u2

f . The adaptive laws are designedas

˙θ = −sign[bp]Γεφ (186)˙k = γε(uf − θTφ) (187)

97

10. Robust Issues in Adaptive Control

10.1 Introduction

Adaptive control and stability analysis have been carried under the conditionthat there are only parameter uncertainty in the system. However, manytypes of non-parametric uncertainties do exist in practice. These include

• high-frequency unmodelled dynamics, such as actuator dynamics or struc-tural vibrations

• low-frequency unmodelled dynamics, such as Coulomb frictions.

• measurement noise

• computation roundoff error and sampling delay

Such non-parametric uncertainties will affect the performance of adaptivecontrol systems when they are applied to control practical systems. Theymay cause instability. Let us consider a simple example.

98

Consider the system output is described by

y = θω (188)

The adaptive law˙θ = γεω (189)

ε = y − θω (190)

will render the convergence of the estimate θ by taking

V =1

2γθ2 (191)

as a Lyapunov function candidate and the analysis

V = −θ(y − θω)ω

= −θ2ω2 (192)

Now, if the signal is corrupted by some unknown bounded disturbance d(t),ie.,

y = θω + d(t) (193)99

The same adaptive will have a problem. In this case,

V = −θ(y − θω)ω

= −θ2ω2 − θdω

= −θ2ω2

2− 1

2(θω + d)2 +

d2

2(194)

From the above analysis, we cannot conclude the boundedness of θ even wehave ω bounded. In fact, if we take θ = 2, γ = 1 and ω = (1 + t)1/2 ∈ L∞and let

d(t) = (1 + t)−1/4(5

4− 2(1 + t)−1/4) (195)

It can then be obtained that

y(t) =5

4(1 + t)−1/4,→ 0 as t →∞ (196)

˙θ =

5

4(1 + t)−3/4 − θ(1 + t)−1 (197)

100

which has a solution

θ = (1 + t)1/4 →∞ as t →∞ (198)

In this example, we have observed that adaptive law designed for disturbance-free system fails to remain bounded even the disturbance is bounded andconverges to zero as t tends to infinity.

In this session, we conside the simple model

y = θω + d(t) (199)

with d as bounded disturbance. A number of robust adaptive laws will beintroduced for this simple model. In the following we keep use ε = y − θω

and V = θ2

2γ . Once the students understand the basic ideas, they can extendthe robust adaptive laws to adaptive control.

101

10.2 Dead-zone Modification

The adaptive law is modefied as

˙θ =

γεω |ε| > g0 |ε| ≤ g

(200)

where g is a constant satisfying g > |d(t)| for all t. For ε > g, we have

V = −θεω

= −(θω − θω)ε

= −(y − d(t)− θω)ε

= −(ε− d(t))ε

< 0 (201)

Therefore we have

V =

< 0 |ε| > g0 |ε| ≤ g

(202)

and we can conclude that V is bounded.102

10.3 σ-Modification

The adaptive law is modefied as

˙θ = γεω − γσθ (203)

where σ is a positive real constant. In this case, we have

V = −(ε− d(t))ε + σθθ

= −ε2 + d(t)ε− σθ2 + σθθ

≤ −ε2

2+

d20

2− σ

θ2

2+ σ

θ2

2

≤ −σγV +d2

0

2+ σ

θ2

2(204)

where d0 ≥ |d(t)|, ∀t ≥ 0. To establish the boundedness of V , we need astandard result, which is a special case of the comparison lemma.

103

Comparison Lemma Let f, V : [0,∞) 7→ R. Then

V ≤ −αV + f, ∀t ≥ t0 ≥ 0 (205)

implies that

V (t) ≤ e−α(t−t0)V (t0) +∫ tt0 e−α(t−τ )f (τ )dτ, ∀t ≥ t0 ≥ 0 (206)

for any finite constant α.

104

Applying the comparison lemma to (204), we have

V (t) ≤ e−σγt + V (0)∫ t0 e−σγ(t−τ )[

d20

2+ σ

θ2

2]dτ, (207)

V (∞) ≤ 1

σγ[d2

0

2+ σ

θ2

2] (208)

Therefore we can conclude that V ∈ L∞.

105

10.4 Robust Adaptive Control

The robust adaptive laws introduced can be applied to various adaptive con-trol schemes. We demonstrate the application a robust adaptive to MRACwith ρ = 1. We start directly from the error model

e = Ame + bm[−kθTω + d(t)]

e1 = cTme (209)

where k = kp/km and d(t) is a bounded disturbance with bound d0, whichrepresents the non-parametric uncertainty in the system. As discussed earlier,we need a robust adaptive law to deal with the bounded disturbances. If wetake σ-modification, then the adaptive law is

˙θ = −sign[kp]Γe1ω − σΓθ (210)

This adaptive law will ensure the boundedness of the variables.

106

Stability Analysis Let

V =1

2eTPe +

1

2|k|θTΓ−1θ (211)

Its derivative is given by

V = −1

2eTQe + e1[−kθTω + d] + |k|θTΓ−1 ˙θ

= −1

2λmin(Q)‖e‖2 + e1d + |k|σθT θ

= −1

2λmin(Q)‖e‖2 + |e1d| − |k|σ‖θ‖2 + |k|σθTθ (212)

Note

|e1d| ≤1

4λmin(Q)‖e‖2 +

d20

λmin(Q)

|θTθ| ≤ 1

2‖θ‖2 +

1

2‖θ‖2 (213)

107

Hence we have

V ≤ −1

4λmin(Q)‖e‖2 − |k|σ

2‖θ‖2 +

d20

λmin(Q)+|k|σ

2‖θ‖2

≤ −αV +d2

0

λmin(Q)+|k|σ

2‖θ‖2 (214)

where α is a positive real and

α =min{1

2λmin(Q), |k|σ}max{λmaxP, |k|

λmin(Γ)}(215)

Therefore, we can conclude the boundedness of V from the comparisonlemma, which further implies the boundedness of the tracking error e1 andthe estimate θ.

108

11. Adaptive Control of Nonlinear Systems

11.1 First Order Nonlinear Systems

Consider a first order nonlinear system described by

y = u + φT (y)θ (216)

where φ : R 7→ Rp is a smooth nonlinear function, and θ ∈ Rp is a unknowvector of constant parameters. For this system, adaptive control law can bedesigned as

u = −cy − φT (y)θ (217)˙θ = Γyφ(y) (218)

where c is a positive real constant, and Γ is a positive definite gain matrix.

109

The closed-loop dynamics is given by

y = −cy + φT (y)θ (219)

with the usual notation θ = θ − θ.Stability Analysis Let

V =1

2y2 +

1

2θTΓ−1θ (220)

and its derivative is obtained as

V = −cy2 (221)

which ensures the boundedness of y and θ. We can show limt→∞ y(t) = 0in the usual way.

110

11.2 Adaptive Backstepping

Consider a second order nonlinear system

x1 = x2 + φT1 (x1)θ

x2 = u + φT2 (x1, x2)θ (222)

where φ1 : R 7→ R, φ2 : R2 7→ Rp are smooth nonlinear functions, andθ ∈ Rp is an unknow constant parameters. From the previous section, weknow that if x2 = −c1x1 − φT

1 (x1)θ, then the first part of the system isstable. Hence if we set

α = −c1x1 − φT1 (x1)θ (223)

then backstepping control design can be used to designed the control inputu in a similar way as for the systems without unknown parameters.

111

For the adaptive backstepping design, we define

z1 = x1 (224)

z2 = x2 − α (225)

We have the dynamics for z1 as

z1 = z2 + α + φT1 (x1)θ

= −c1z1 + z2 + φT1 (x1)θ (226)

Consider the dynamics for z2

z2 = x2 − α(x1, θ)

= u + φT2 (x1, x2)θ −

∂α

∂x1x1 −

∂α

∂θ

˙θ

= u + [φ2(x1, x2)−∂α

∂x1φ1(x1)]

Tθ − ∂α

∂x1x2 −

∂α

∂θ

˙θ (227)

Notice that the adaptive law designed based on the dynamics of z1 will notwork because the dynamics of z2 involves the unknown parameters θ as well.

112

We leave the adaptive law to the Lyapunov function based analysis later.From the dynamics of z2, we design the control input as

u = −z1 − c2z2 − [φ2(x1, x2)−∂α

∂x1φ1(x1)]

T θ +∂α

∂x1x2 +

∂α

∂θ

˙θ (228)

The resultant dynamics of z2 is given by

z2 = −z1 − c2z2 + [φ2(x1, x2)−∂α

∂x1φ1(x1)]

T θ (229)

Consider a Lyapunov function candidate

V =1

2[z2

1 + z22 + θTΓ−1θ] (230)

113

We then have

V = z1[−c1z1 + z2 + φT1 θ]

+z2[−z1 − c2z2 + [φ2 −∂α

∂x1φ1]

T θ]

− ˙θT

Γ−1θ

= −c1z21 − c2z

22 + [z1φ1 + z2(φ2 −

∂α

∂x1φ1)− Γ−1 ˙

θ]T θ (231)

From the above analysis, we decide the adaptive law as

˙θ = Γ[z1φ1 + z2(φ2 −

∂α

∂x1φ1)] (232)

and we have

V = −c1z21 − c2z

22 (233)

Therefore we have the boundedness of z1, z1 and θ and limt→∞ zi = 0 fori = 1, 2.

114

11.3 Adaptive Control of Strict Feedback Systems

Consider a nonlinear system in the strict feedback form

xi = xi+1 + φTi (x1, x2, . . . , xi)θ for i = 1, . . . , n− 1

xn = u + φTn (x1, x2, . . . , xn)θ (234)

Adaptive control design can be carried using backstepping iteratively in nsteps. The main difficulty is in design of adaptive law, as the seam unknownparameter appears in every step. The initial method used multiple estiamtesfor θ. Later the tuning function method was proposed to solve the multipleestimate problem. For more details please refer to (Krstic et al, 1995)

115

Example 11.1 Design an adaptive control law for

x1 = x2 + (ex1 − 1)θ

x2 = u (235)

Example 11.2 Design an adaptive control law for

x1 = x2 + x31θ + x2

1

x2 = (1 + x21)u + x2

1θ (236)

116