adaptive pole placement control - adaptive control

EE 264 SIST, ShanghaiTech

Adaptive Pole Placement Control

YW 11-1

Contents

Introduction

Motivating Examples

APPC scheme for general SISO plant via Polynomial approach

Adaptive Pole Placement Control 11-2

APPC = a pole placement control law + parameter adaptive law

The desired properties of the plant are expressed in terms of

desired pole locations to be placed by the controller

applicable to both minimum- and nonminimum-phase systems

includes direct and indirect two categories.

the use of adaptive laws with normalization and without

normalization does not extend to APPC.


Contents

Introduction

Motivating Examples



Scalar Example: Adaptive Regulation

Consider the scalar plant

y = ay + bu

where a and b 6= 0 are unknown constants, and the sign of b is

known.

The control objective: choose u so that the closed-loop pole is

placed at −am < 0, and y → 0 as t→∞.

The idea control law

u = −k∗y, k∗ = a+ amb




y = ay + bu


known.




u = −k∗y, k∗ = a+ amb



The closed-loop plant

y = −amy

achieves the control objective. Consider a, b unknown, use the

certainty equivalence approach to form the APPC as

u = −ky

with the estimate k is generated in two different ways:

Direct approach : ˙k = φ(y)

Indirect approach: k = a+am

b, b 6= 0




y = −amy



u = −ky




b, b 6= 0


Direct Adaptive Regulation

Add and subtract the term −amy in the plant equation to obtain

y = −amy + (a+ am) y + bu

= −amy + b (k∗y + u)

= −amy − bky

where k , k − k∗ is the parameter error. Equation relates the

parameter error term bky with the regulation error y and

motivates the Lyapunov function

V = y2

2 + k2|b|2γ





= −amy + b (k∗y + u)

= −amy − bky

where k , k − k∗ is the parameter error. Equation relates the

parameter error term bky with the regulation error y and

motivates the Lyapunov function

V = y2

2 + k2|b|2γ



time derivative of V is

V = −amy2 − bky2 + |b|γk ˙k = −amy2 − bky2 + bk

˙kγ

sgn(b)

The adaptive law

k = γy2 sgn(b)

leads to V = −amy2 ≤ 0 which implies

y, k, k ∈ L∞ and y ∈ L2.

y, u ∈ L∞; (Barbalat’s lemma) and y(t)→ 0 as t→∞

However, this does not guarantee that the closed-loop pole of the

plant is placed at −am even asymptotically with time.Adaptive Pole Placement Control 11-16




˙kγ

sgn(b)

The adaptive law

k = γy2 sgn(b)


y, k, k ∈ L∞ and y ∈ L2.




Indirect Adaptive Regulation



= −amy + (a− a+ bk)y + bu

= −amy − ay − bu+ bu = −amy − ay − bu

where we have using the algebraic equation

k = a+ am

b, u = ky

and a = a− a, b = b− b. This equation relates the regulation or

estimation error y with the parameter errors a, b.Adaptive Pole Placement Control 11-19




= −amy + (a− a+ bk)y + bu

= −amy − ay − bu+ bu = −amy − ay − bu

where we have using the algebraic equation

k = a+ am

b, u = ky

and a = a− a, b = b− b. This equation relates the regulation or

estimation error y with the parameter errors a, b.Adaptive Pole Placement Control 11-20


Motivates the Lyapunov candidate function

V = y2

2 + a2

2γ1+ b2

2γ2

for some γ1, γ2 > 0. The time derivative V is given by

V = −amy2 − ay2 − buy + a ˙aγ1

+ b ˙bγ2

Choosing˙a = ˙a = γ1y

2, ˙b = ˙b = γ2yu

we have

V = −amy2 ≤ 0Adaptive Pole Placement Control 11-21



V = y2

2 + a2

2γ1+ b2

2γ2



+ b ˙bγ2


2, ˙b = ˙b = γ2yu

we have



To guarantee b 6= 0, introduce a projection

˙a = γ1y2,

˙b =

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

0 otherwise

where b(0) is chosen so that b(0) sgn(b) ≥ b0. Furthermore, the

time derivative V satisfies

V =

−amy2 if |b| > b0 or if |b| = b0 and sgn(b)yu ≥ 0

−amy2 − byu if |b| = b0 and sgn(b)yu < 0



The projection introduces the extra term −byu

byu = byu− byu = (|b| − |b|) sgn(b)yu = (b0 − |b|) sgn(b)yu ≥ 0

Hence the projection can only make V more negative and

V ≤ −amy2 ∀t ≥ 0

always hold.

y, a, b ∈ L∞, y ∈ L2, and |b(t)| ≥ b0∀t ≥ 0, which implies that

k ∈ L∞.

y ∈ L∞, and y(t)→ 0 as t→∞Adaptive Pole Placement Control 11-26





V ≤ −amy2 ∀t ≥ 0

always hold.


k ∈ L∞.


Scalar example : Adaptive Tracking

Consider the plant

y = −ay + bu

with a, b are unknown constants, but the sign of b 6= 0 is known.

Control objective:

i) closed-loop pole is at −am and u, y ∈ L∞ii) y(t) tracks the reference signal ym(t) = c, ∀t ≥ 0, where c 6= 0

is a known constant.



Consider the plant

y = −ay + bu

with a, b are unknown constants, but the sign of b 6= 0 is known.

Control objective:

i) closed-loop pole is at −am and u, y ∈ L∞ii) y(t) tracks the reference signal ym(t) = c, ∀t ≥ 0, where c 6= 0

is a known constant.



Define the tracking error e = y − c satisfies

e = ae+ ac+ bu

Then, the tracking objective becomes e(t) = y(t)− ym → 0 as

t→∞ exponentially fast. If a, b, c are known, we can choose an

idea control law

u = −k∗1e− k∗2, k∗1 = a+ amb

, k∗2 = ac

b

leading to

e = −ame


Direct Adaptive Tracking

The CE control law

u = −k1e− k2

As before, we rewrite the error equation as

e = −ame+ b (u+ k∗1e+ k∗2)

= −ame− b(k1e+ k2)

where k1 = k1 − k∗1, k2 = k2 − k∗2,which relates the tracking error

e with the parameter errors k1, k2 and motivates the Lyapunov

function

V = e2

2 + k21|b|2γ1

+ k22|b|2γ2



The CE control law

u = −k1e− k2

As before, we rewrite the error equation as

e = −ame+ b (u+ k∗1e+ k∗2)

= −ame− b(k1e+ k2)

where k1 = k1 − k∗1, k2 = k2 − k∗2,which relates the tracking error

e with the parameter errors k1, k2 and motivates the Lyapunov

function

V = e2

2 + k21|b|2γ1

+ k22|b|2γ2



The time derivative V is forced to satisfy

V = −ame2

by choosing

k1 = γ1e2 sgn(b), k2 = γ2e sgn(b)

The APPC scheme may be written as

u = −k1e− γ2 sgn(b)∫ t

0e(τ)dτ

k1 = γ1e2 sgn(b)

which is referred to as the direct adaptive proportional plus

integral (API) controller.Adaptive Pole Placement Control 11-35


The time derivative V is forced to satisfy

V = −ame2

by choosing

k1 = γ1e2 sgn(b), k2 = γ2e sgn(b)

The APPC scheme may be written as

u = −k1e− γ2 sgn(b)∫ t

0e(τ)dτ

k1 = γ1e2 sgn(b)

which is referred to as the direct adaptive proportional plus

integral (API) controller.Adaptive Pole Placement Control 11-36

Indirect Adaptive Tracking

The CE control law

u = −k1e− k2

with

k1 = a+ am

b, k2 = ac

b

provided b 6= 0. Again we rewrite the error equation as

e = −ame+ (a+ am) e+ ac+ bu

= −ame+ ame+ ay + bu

= −ame− ay − bu+ ay + bu

= −ame− ay − bu



The CE control law

u = −k1e− k2

with

k1 = a+ am

b, k2 = ac

b








Leading to an adaptive law

˙a = γ1ey,

˙b =

γ2eu if |b| > b0 or if |b| = b0 and sgn(b)eu ≥ 0

0 otherwise

where b(0) satisfies b(0) sgn(b) ≥ b0. One can verify the

boundedness of and convergence of the tracking error by

considering the Lyapunov function

V = e2

2 + a2

2γ1+ b2

2γ2


Contents

Introduction

Motivating Examples



PPC

Consider the SISO LTI 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)

yp is required to follow a certain class of reference signals ym

Remark: In general, by assigning the closed-loop poles to those of

A∗(s), we can guarantee closed-loop stability and convergence of

the plant output yp to zero, provided that there is no external

input.Adaptive Pole Placement Control 11-43

PPC






polynomial A∗(s)






PPC

Assumptions

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

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

P3. The reference signal ym ∈ L∞ is assumed to satisfy

Qm(s)ym = 0

where Qm(s), known as the internal model of ym, is a known

monic polynomial of degree q with all roots in <[s] ≤ 0 and

with no repeated roots on the jω -axis. The internal model

Qm(s) is assumed to be coprime with Zp(s).Adaptive Pole Placement Control 11-47

PPC

Assumptions

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

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


Qm(s)ym = 0

where Qm(s), known as the internal model of ym, is a known

monic polynomial of degree q with all roots in <[s] ≤ 0 and

with no repeated roots on the jω -axis. The internal model

Qm(s) is assumed to be coprime with Zp(s).Adaptive Pole Placement Control 11-48

PPC

Assumptions


Qm(s)ym = 0

and Qm(s) is assumed to be coprime with Zp(s).

Example

if ym(t) = c+ sin(3t), where c is any constant, then

Qm(s) = s(s2 + 9

)and, according to P3, Zp(s) should not have s

or s2 + 9 as a factor.


Indirect APPC via Polynomial Approach

The design of the APPC scheme is based on three steps:

1. a control law is developed that can meet the control objective

in the known parameter case. i.e. PPC

2. an adaptive law is designed to estimate the plant parameters

online. The estimated plant parameters are then used to

calculate the controller parameters at each time t

3. the APPC is formed by replacing the controller parameters in

Step 1 with their online estimates.


APPC via polynomial approach

Step 1. PPC for known parameters. We consider the control law

up = − P (s)Qm(s)L(s) (yp − ym)

where P (s), L(s) are polynomials (with L(s) monic) of degree

q + n− 1, n− 1, respectively, chosen to satisfy the polynomial

equation

L(s)Qm(s)Rp(s) + P (s)Zp(s) = A∗(s)

Assumptions P2 and P3 guarantee that Qm(s)Rp(s) and Zp(s)

are coprime, which guarantees hat L(s), P (s) exist and are unique.



Step 1. PPC for known parameters. We consider the control law


where P (s), L(s) are polynomials (with L(s) monic) of degree

q + n− 1, n− 1, respectively, chosen to satisfy the polynomial

equation

L(s)Qm(s)Rp(s) + P (s)Zp(s) = A∗(s)

Assumptions P2 and P3 guarantee that Qm(s)Rp(s) and Zp(s)

are coprime, which guarantees hat L(s), P (s) exist and are unique.




yp = ZpP

A∗ym

and the control law

up = RpP

A∗ym

if ym ∈ L∞ and ZpPA∗ ,

RpPA∗ are proper with stable poles, it follows

that yp, up ∈ L∞.

The tracking error e1 = yp − ym can be shown to satisfy

e1 = −LRpA∗

Qmym = −LRpA∗

[0]

it is easily to see e1 → 0 as t→∞.Adaptive Pole Placement Control 11-53



yp = ZpP

A∗ym

and the control law

up = RpP

A∗ym





e1 = −LRpA∗

Qmym = −LRpA∗

[0]



Since L(s) is not necessarily Hurwitz, the control law


may have unstable poles. An alternative realization is

Λ(s)up = Λ(s)up −Qm(s)L(s)up − P (s) (yp − ym)

where Λ(s) is any monic Hurwitz polynomial of degree n+ q − 1.

Filtering each side with 1Λ(s) , we obtain

up = Λ− LQmΛ up −

P

Λ (yp − ym)

The control law is implemented using 2(n+ q − 1) integrators to

realize the proper stable transfer functions Λ−LQm

Λ , PΛ .Adaptive Pole Placement Control 11-56









P

Λ (yp − ym)





Step 2. Estimation of plant polynomials The following PM can be

derived:

z = θ∗>p φ

where

z = sn

Λp(s)yp, θ∗p =[θ∗>b , θ∗>a

]>, φ =

[α>n−1(s)

Λp(s) up,−α>n−1(s)

Λp(s) yp

]>αn−1(s) =

[sn−1, . . . , s, 1

]>, θ∗a = [an−1, . . . , a0]> , θ∗b = [bn−1, . . . , b0]> ,

and Λp(s) is a monic Hurwitz polynomial.

Note that, if the degree of Zp(s) is less than n− 1, then some of

the first elements of θ∗b will be equal to zero.Adaptive Pole Placement Control 11-59


Step 2. Estimation of plant polynomials The following PM can be

derived:

z = θ∗>p φ

where

z = sn

Λp(s)yp, θ∗p =[θ∗>b , θ∗>a

]>, φ =

[α>n−1(s)

Λp(s) up,−α>n−1(s)

Λp(s) yp

]>αn−1(s) =

[sn−1, . . . , s, 1

]>, θ∗a = [an−1, . . . , a0]> , θ∗b = [bn−1, . . . , b0]> ,

and Λp(s) is a monic Hurwitz polynomial.

Note that, if the degree of Zp(s) is less than n− 1, then some of

the first elements of θ∗b will be equal to zero.Adaptive Pole Placement Control 11-60


A wide class adaptive laws can be designed to estimate θ∗p. For

example, the gradient algorithm

θp = Γεφ, ε =z − θTp φm2s

, m2s = 1 + φTφ

where Γ = Γ> > 0 is the adaptive gain and

θp =[bn−1, . . . , b0, an−1, . . . , a0

]Tare the estimated plant parameters form the plant polynomial

Rp(s, t) = sn + an−1(t)sn−1 + · · ·+ a1(t)s+ a0(t),

Zp(s, t) = bn−1(t)sn−1 + · · ·+ b1(t)s+ b0(t)



Step 3. Adaptive control law

up =(Λ(s)− L(s, t)Qm(s)

) 1Λ(s)up − P (s, t) 1

Λ(s) (yp − ym)

with online estimates L(s, t), P (s, t) calculated at each frozen

time t using the polynomial equation

L(s, t) ·Qm(s) · Rp(s, t) + P (s, t) · Zp(s, t) = A∗(s)

Remark: stabilizability problem in indirect APPCThe existence and uniqueness of L(s, t), P (s, t) is guaranteed, provided

that Rp(s, t)Qm(s), Zp(s, t) are coprime at each frozen time tm which

means that at certain points in time the solution L(s, t) P (s, t) may not

exist.Adaptive Pole Placement Control 11-63


Step 3. Adaptive control law

up =(Λ(s)− L(s, t)Qm(s)

) 1Λ(s)up − P (s, t) 1

Λ(s) (yp − ym)

with online estimates L(s, t), P (s, t) calculated at each frozen

time t using the polynomial equation

L(s, t) ·Qm(s) · Rp(s, t) + P (s, t) · Zp(s, t) = A∗(s)

Remark: stabilizability problem in indirect APPCThe existence and uniqueness of L(s, t), P (s, t) is guaranteed, provided

that Rp(s, t)Qm(s), Zp(s, t) are coprime at each frozen time tm which

means that at certain points in time the solution L(s, t) P (s, t) may not

exist.Adaptive Pole Placement Control 11-64


Theorem: Assume that the estimated plant polynomials

RpQm, Zp are strongly coprime at each time t. Then all the

signals in the closed loop are uniformly bounded, and the tracking

error converges to zero asymptotically with time.

Remark: By strongly coprime we mean that the polynomials do

not have roots that are close to becoming common. For example,

the polynomials s+ 2 + ε, s+ 2, where s is arbitrarily small, are

not strongly coprime.



Example: Consider the plant

yp = b

s+ aup

where a and b are constants. The desired roots

A∗(s) = (s+ 1)2

and the reference signal ym = 1.


adaptive pole placement control - adaptive control

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