adaptive pole placement control ii - adaptive control

EE 264 SIST, ShanghaiTech

Adaptive Pole Placement Control II

YW 12-1

Contents

Adaptive Observer

APPC for general SISO plant via ss approach

Adaptive Linear Quadratic Control

Modification for Solving Stability Issue

Adaptive Pole Placement Control II 12-2

Adaptive Luenberger observer

Consider the LTI SISO plant

x = Ax+Bu, x(0) = x0

y = C>x

In the case A,B,C are known, the Luenberger observer is in the

form of

˙x = Ax+Bu+K(y − y), x(0) = x0

y = C>x

where K is chosen such that A−KC> is Hurwitz, guarantees

that x→ x exponentially fast. The existence of K is ensured by

the observability of pair (A,C>)Adaptive Pole Placement Control II 12-3


Consider the LTI SISO plant

x = Ax+Bu, x(0) = x0

y = C>x

In the case A,B,C are known, the Luenberger observer is in the

form of

˙x = Ax+Bu+K(y − y), x(0) = x0

y = C>x

where K is chosen such that A−KC> is Hurwitz, guarantees

that x→ x exponentially fast. The existence of K is ensured by

the observability of pair (A,C>)Adaptive Pole Placement Control II 12-4


Idea: (A,B,C)→ G(s)→ G(s)→ (A, B, C)

mapping of the 2n estimated parameters of G(s) to the n2 + 2n

parameters of A,B,C is not uniqueunless (A,B,C) is in a

observer canonical form, i.e., the plant is represented as

xo =[−ap |

In−10

]xo + bρu

y = [1, 0, . . . , 0]xo

where ap = [an−1, an−2, . . . , a0]> and bp = [bn−1, bn−2, . . . , b0]>

are the coefficients of the transfer function

G(s) = y(s)u(s) = bn−1s

n−1 + bn−2sn−2 + · · ·+ b0s

sn + an−1sn−1 + an−2sn−2 + · · ·+ a0s



Idea: (A,B,C)→ G(s)→ G(s)→ (A, B, C)




xo =[−ap |

In−10

]xo + bρu

y = [1, 0, . . . , 0]xo



G(s) = y(s)u(s) = bn−1s

n−1 + bn−2sn−2 + · · ·+ b0s

sn + an−1sn−1 + an−2sn−2 + · · ·+ a0s



Then the adaptive observer is given by

˙x = A(t)x+ bp(t)u+K(t)(y − y), x(0) = x0

y = [1, 0, . . . , 0]x

where x is the estimate of xo and

A(t) =[−ap(t) |

In−10

], K(t) = a∗ − ap(t)

ap(t) and bp(t) are the estimates of the vectors ap and bp,

respectively. a∗ ∈ Rn is chosen so that

A∗ =[−a∗|In−1

0

]is a Hurwitz matrix.Adaptive Pole Placement Control II 12-8


Then the adaptive observer is given by

˙x = A(t)x+ bp(t)u+K(t)(y − y), x(0) = x0

y = [1, 0, . . . , 0]x

where x is the estimate of xo and

A(t) =[−ap(t) |

In−10

], K(t) = a∗ − ap(t)

ap(t) and bp(t) are the estimates of the vectors ap and bp,

respectively. a∗ ∈ Rn is chosen so that

A∗ =[−a∗|In−1

0

]is a Hurwitz matrix.Adaptive Pole Placement Control II 12-9


Theorem: The adaptive Luenberger observer with gradient-based

algorithm guarantees the following properties:

(i) If choose u ∈ L∞ and A is a stable matrix, all signals are

bounded.

(ii) Furthermore, if choose u is sufficiently rich of order 2n, then

the state observation error |x− xo| and the parameter estimation

error θ converge to zero exponentially fast.

Brief Proof. (i) The observer equation may be written as

˙x = A∗x+ bp(t)u+(A(t)−A∗

)xo

(ii) The state observation error x = x− xo satisfies

˙x = A∗x+ bpu− apyAdaptive Pole Placement Control II 12-10


Theorem: The adaptive Luenberger observer with gradient-based

algorithm guarantees the following properties:

(i) If choose u ∈ L∞ and A is a stable matrix, all signals are

bounded.

(ii) Furthermore, if choose u is sufficiently rich of order 2n, then

the state observation error |x− xo| and the parameter estimation

error θ converge to zero exponentially fast.

Brief Proof. (i) The observer equation may be written as

˙x = A∗x+ bp(t)u+(A(t)−A∗

)xo

(ii) The state observation error x = x− xo satisfies

˙x = A∗x+ bpu− apyAdaptive Pole Placement Control II 12-11

Contents

Adaptive Observer





APPC via state space approach

The plant

yp = Gp(s)up, Gp(s) = Zp(s)Rp(s)

where Gp(s) is proper and Rp(s) is a monic polynomial.

The control objective is to choose up so that the closed-loop

poles are assigned to those of a monic Hurwitz polynomial A∗(s)

and yp → ym

and Assumptions are the same as before

P1. Rp(s) is a monic polynomial whose degree n is known.

P2. Zp(s), Rp(s) are coprime and degree (Zp) < n.

P3. Qm(s)ym = 0 and Qm(s) is assumed to be coprime with Zp(s).Adaptive Pole Placement Control II 12-13


Step 1. PPC for known parameters We start by considering

the expression

e1 = Zp(s)Rp(s)

up − ym

for the tracking error. Filtering each side of with Qm(s)Q1(s) , where

Q1(s) is an arbitrary monic Hurwitz polynomial of degree q, and

using Qm(s)ym = 0, we obtain

e1 = ZpQ1RpQm

up, up = QmQ1

up

In this way, have converted the tracking problem into the

regulation problem of choosing up to regulate e1 to zero.Adaptive Pole Placement Control II 12-14



the expression

e1 = Zp(s)Rp(s)

up − ym




e1 = ZpQ1RpQm

up, up = QmQ1

up




Let (A,B,C) be a state-space realization of error equation in the

observer canonical form, i.e.

e = Ae+Bup, e1 = C>e

where A =[−θ∗1 |

In+q−10

], B = θ∗2, C = [1, 0, . . . , 0]> and

θ∗1, θ∗2 ∈ Rn+q are the coefficient vectors of the polynomials

Rp(s)Qm(s)− sn+q and Zp(s)Q1(s), respectively.

Note, because RpQm, Zp are coprime, any possible zero-pole

cancellation in (6.51) between Q1(s) and Rp(s)Qm(s) will occur

in R[s] < 0 due to Q1(s) being Hurwitz, which implies that

(A,B) is always stabilizable.Adaptive Pole Placement Control II 12-17


Let (A,B,C) be a state-space realization of error equation in the

observer canonical form, i.e.

e = Ae+Bup, e1 = C>e

where A =[−θ∗1 |

In+q−10

], B = θ∗2, C = [1, 0, . . . , 0]> and

θ∗1, θ∗2 ∈ Rn+q are the coefficient vectors of the polynomials

Rp(s)Qm(s)− sn+q and Zp(s)Q1(s), respectively.

Note, because RpQm, Zp are coprime, any possible zero-pole

cancellation in (6.51) between Q1(s) and Rp(s)Qm(s) will occur

in R[s] < 0 due to Q1(s) being Hurwitz, which implies that

(A,B) is always stabilizable.Adaptive Pole Placement Control II 12-18


Consider the feedback control law

up = −Kce, up = Q1Qm

up

where e is the state of the full-order Luenberger observer

˙e = Ae+Bup −Ko

(C>e− e1

)and Kc and Ko are solutions to the polynomial equations

det (sI −A+BKc) = A∗c(s)

det(sI −A+KoC

>)

= A∗o(s)

where A∗c and A∗o are given monic Hurwitz polynomials of degree

n+ q.Adaptive Pole Placement Control II 12-19


The design of A∗c and A∗o polynomials

the roots of A∗c(s) = 0 represent the desired pole locations of

the transfer function of the closed-loop plant

the roots of A∗o(s) are equal to the poles of the observer

dynamics

The existence of Kc and Ko

The existence of Kc follows from the controllability of (A,B)

The existence of Ko follows from the observability of (A,C).



Theorem: If Assumptions P1-P3 hold, Consider the system

yp = Gp(s)up, Gp(s) = Zp(s)Rp(s)

where Gp(s) is proper and Rp(s) is a monic polynomial. The

control law

up = Q1Qm

up, up = −Kce

˙e = Ae+Bup −Ko

(C>e− e1

)guarantees that all signals in the closed-loop plant are bounded

and e1 converges to zero exponentially fast.



Step 2. Estimation of plant parameters The adaptive law for

estimating the plant parameters is given by, for instance

θp = Γεφ

ε = z − θ>φm2s

, m2s = 1 + φ>φ

z, φ and θp =[θ>a , θ

>b

]>are as defined same as ones given in

polynomial approach.



Step 3. Adaptive control law Using the CE approach, the

adaptive control law is given by

˙e = Ae+ Bup − Ko

(c>e− e1

)up = −Kce, up = Q1(s)

Qm(s) up

where

A(t) =[−θ1(t) | In+q−1

0

], B(t) = θ2(t)

θ1(t) and θ2(t) are the coefficient vectors of the polynomials

Rp(s, t)Qm(s)− sn+q =(sn + θ>a (t)αn−1(s)

)Qm(s)− sn+q

Zp(s, t)Q1(s) = θ>b (t)αn−1(s)Q1(s)

respectively.Adaptive Pole Placement Control II 12-24


and

Ko(t) = α∗0 − θ1(t)

α∗0 is the coefficient vector of A∗0(s); and Kc(t) is calculated at

each time t by solving the polynomial equation

det(sI − A+ BKc

)= A∗c(s)

Note, the stabilizability problem arises in adaptive control law,

where for the calculation of Kc to be possible the pair

(A(t), B(t)) has to be controllable at each time t and for

implementation purposes strongly controllable.Adaptive Pole Placement Control II 12-25


and

Ko(t) = α∗0 − θ1(t)

α∗0 is the coefficient vector of A∗0(s); and Kc(t) is calculated at

each time t by solving the polynomial equation

det(sI − A+ BKc

)= A∗c(s)

Note, the stabilizability problem arises in adaptive control law,

where for the calculation of Kc to be possible the pair

(A(t), B(t)) has to be controllable at each time t and for

implementation purposes strongly controllable.Adaptive Pole Placement Control II 12-26


Theorem Assume that the polynomials Zp, RpQm are strongly

coprime at each time t. Then all the signals in the closed-loop

APPC scheme via state-space approach are uniformly bounded,

and the tracking error e1 → 0 asymptotically .

Example We consider the same scalar plant

yp = b

s+ aup

where a and b are unknown constants with b 6= 0. The input up is

to be chosen so that the poles of the closed-loop plant are placed

at the roots of A∗(s) = (s+ 1)2 = 0, and yp tracks the reference

signal ym = 1.Adaptive Pole Placement Control II 12-27


Theorem Assume that the polynomials Zp, RpQm are strongly

coprime at each time t. Then all the signals in the closed-loop

APPC scheme via state-space approach are uniformly bounded,

and the tracking error e1 → 0 asymptotically .

Example We consider the same scalar plant

yp = b

s+ aup

where a and b are unknown constants with b 6= 0. The input up is

to be chosen so that the poles of the closed-loop plant are placed

at the roots of A∗(s) = (s+ 1)2 = 0, and yp tracks the reference

signal ym = 1.Adaptive Pole Placement Control II 12-28

Contents

Adaptive Observer





LQC

Idea: using an optimization technique to achieve our tracking or

regulation objective by minimizing a certain cost function that

reflects the performance of the closed-loop system.

Unify the regulation or tracking problem of the system as

e = Ae+Bup

e1 = C>e

where up = Q1(s)Qm(s) up, and up is to be chosen so that e ∈ L∞ and

e1 → 0 as t→∞.


LQC

The desired up to meet this objective is chosen as the one that

minimizes the quadratic cost

J =∫ ∞

0

(e2

1(t) + λu2p(t)

)dt

where λ > 0, a weighting coefficient to be designed, penalizes the

level of the control input signal. The optimum control input upthat minimizes J is

up = −Kce, Kc = λ−1B>P

where P = P> satisfies the Riccati Equation

A>P + PA− PBλ−1B>P + CC> = 0Adaptive Pole Placement Control II 12-32

LQC

The desired up to meet this objective is chosen as the one that

minimizes the quadratic cost

J =∫ ∞

0

(e2

1(t) + λu2p(t)

)dt

where λ > 0, a weighting coefficient to be designed, penalizes the

level of the control input signal. The optimum control input upthat minimizes J is



A>P + PA− PBλ−1B>P + CC> = 0Adaptive Pole Placement Control II 12-33

LQC



A>P + PA− PBλ−1B>P + CC> = 0

Remark:

1. With λ > 0 and finite, the LQC guarantees A−BKc is

Hurwitz, e, e1 → 0 exponentially fast, and up ∈ L∞2. The location of the eigenvalues of A−BKc depends on the

particular choice of λ. In general, there is NO guarantee that

one can find a λ so that the closed-loop poles are equal to the

roots of the desired polynomial A∗(s)Adaptive Pole Placement Control II 12-34

LQC

Consider the state e may not be available for measurement, we

need to use


where e is the state of the observer equation

˙e = Ae+Bup −Ko

(C>e− e1

)where Ko are solutions to the polynomial equations

det(sI −A+KoC

>)

= A∗o(s)

where A∗o is given monic Hurwitz polynomials of degree n+ q.


LQC

Theorem The LQ control law

up = Q1(s)Qm(s) up

with


guarantees that all the eigenvalues of A−BKc are in <[s] < 0,

all signals in the closed-loop plant are bounded, and e1(t)→ 0 as

t→∞ exponentially fast.


ALQC

Use the CE approach to form the adaptive control law

˙e = Ae+ Bup − Ko

(c>e− e1

)up = Q1(s)

Qm(s) up, up = −Kce, Kc = λ−1B>P

where A, B, Ko are generated using the gradient-based adaptive

law (see parameter estimation schemes) and P (t) is calculated by

solving the Riccati equation

A>(t)P (t) + P (t)A(t)− P (t)B(t)λ−1B>(t)P (t) + CC> = 0


ALQC

Theorem: Assume that the polynomials Rp(s, t)Qm(s) and

Zp(s, t) are strongly coprime at each time t. Then the ALQC

scheme guarantees that all signals in the closed-loop plant are

bounded and the tracking error e1 converges to zero as t→∞.Remark 1: the ALQC scheme depends on the solvability of the

algebraic Riccati equation. For the solution P (t) = P>(t) > 0 to exist,

the pair (A, B) has to be stabilizable at each time t. A sufficient

condition for (A, B) to be stabilizable is that the polynomials

Rp(s, t)Qm(s) and Zp(s, t) are coprime at each time t.

Remark 2: For P (t) to be uniformly bounded, however, we will require

Rp(s, t)Qm(s) and Zp(s, t) to be strongly coprime at each time t.Adaptive Pole Placement Control II 12-38

ALQC











Contents

Adaptive Observer





Loss of stabilizability

Recall scalar example

y = −ay + bu

with indirect APPC law

u = −ky, k = a+ am

b

The system loss the stabilization, when b→ 0. The solution is a

projection operation

˙b =

γ2yu if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0

0 otherwise


Modification for ensure stabilizability

a) Parameter Projection Methods: exists a convex subset C0 of the

parameter space that is assumed to have the following properties:

(i) The unknown plant parameter vector θ∗p ∈ C0.(ii) Every member θp of C0 has a corresponding level ofstabilizability greater than ε∗ for some known constant ε∗ > 0.Advantage: The projection based on gradient method is simple and does

not alter the usual properties of the adaptive law that are used in the

stability analysis.

Disadvantage: This approach relies on the rather strong assumption that

the set C0 is known. No procedure has been proposed for constructing

such a set C0 for a general class of plants.







stability analysis.





Projection operation

The gradient algorithm with projection is computed by applying

the gradient method to the following minimization problem with

constraints:

minimize J(θ)

s.t. θ ∈ S

where S is a convex subset of Rn with smooth boundary almost

everywhere. Assume that S is given by

S = {θ ∈ Rn | g(θ) ≤ 0}

where g : Rn → R is a smooth function.Adaptive Pole Placement Control II 12-47

Projection operation

The adaptive laws based on the gradient method can be modified

to guarantee that θ ∈ S by solving the constrained optimization

problem given above to obtain

θ = Pr(−Γ∇J) =

−Γ∇J if θ ∈ S0 or θ ∈ δ(S)

and − (Γ∇J)>∇g ≤ 0

−Γ∇J + Γ ∇g∇g>

∇g>Γ∇gΓ∇J otherwise

where δ(S) = {θ ∈ Rn | g(θ) = 0} and S0 = {θ ∈ Rn | g(θ) < 0}

denote the boundary and the interior, respectively, of S. E.g.

gradient algorithm based on the instantaneous cost function:

∇J = −εφAdaptive Pole Placement Control II 12-48


Theorem The gradient adaptive laws and the LS adaptive laws

with the projection modifications retain all the properties that are

established in the absence of projection and in addition guarantee

that θ(t) ∈ S∀t ≥ 0, provided θ(0) ∈ S and θ∗ ∈ S.

Other modification methods:

b) Heuristics methods, such as re-initialization, ignore the

undesired value

c) Correction Approach

d) Persistent Excitation Approach

e) Switching Methods ...Adaptive Pole Placement Control II 12-49


Theorem The gradient adaptive laws and the LS adaptive laws

with the projection modifications retain all the properties that are

established in the absence of projection and in addition guarantee

that θ(t) ∈ S∀t ≥ 0, provided θ(0) ∈ S and θ∗ ∈ S.

Other modification methods:

b) Heuristics methods, such as re-initialization, ignore the

undesired value

c) Correction Approach

d) Persistent Excitation Approach

e) Switching Methods ...Adaptive Pole Placement Control II 12-50

adaptive pole placement control ii - adaptive control

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