lecture 07 eailc

7/22/2019 Lecture 07 EAILC

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Lecture 7: Deterministic Self-Tuning Regulators

Feedback Control Design for Nominal Plant Model

via Pole Placement

Indirect Self-Tuning Regulators

Direct Self-Tuning Regulators

c Leonid Freidovich. A ril 22, 2010. Elements of Iterative Learnin and Ada tive Control: Lecture 7 . 1/20


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Consider a single input single output (SISO) system

y(t) + a1y(t 1) + + any(t n)

= b0u(t d0) + + + bmu(td0m)



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y(t) + a1y(t 1) + + any(t n)

= b0u(t d0) + + + bmu(td0m)It can be re written in the regression form

y(t) = a1y(t 1) any(t n) + b0u(t d0) +

+ b1u(td01) + + bmu(td0m)

= (t 1)T0



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y(t) + a1y(t 1) + + any(t n)

= b0u(t d0) + + + bmu(td0m)It can be re written in the regression form

y(t) = a1y(t 1) any(t n) + b0u(t d0) +

+ b1u(td01) + + bmu(td0m)

= (t 1)T0

where 0 =

a1, . . . , an, b0, . . . , bm

T

(t 1) = y(t 1), . . . ,y(t n),u(t d0), . . . , u(td0m)

T



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RLS Algorithm (discrete time)

Given the data

y(t), (t 1)N

t0defined by the model

y(t) = (t 1)T

0

, t = t0, t0+ 1, . . . , N



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RLS Algorithm (discrete time)

Given the data

y(t), (t 1)N

t0defined by the model

y(t) = (t 1)T

0

, t = t0, t0+ 1, . . . , N

With the initial guess (t0)and a measure of our trust in thisguess: P(t0) > 0, we can use the RLS algorithm:

(t) = (t 1)+ K(t)

y(t) (t 1)T(t 1)

K(t) = P(t)(t 1) = P(t 1) (t 1) + (t 1)TP(t 1)(t 1)

P(t) =

1

P(t 1)

P(t 1) (t 1) (t 1)T P(t 1)

+ (t 1)T P(t 1) (t 1)



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RLS Algorithm (continuous time)

Given the data

y(), ()t

t=t0defined by the model

y() = ()T0, [t0, t]

obtained from A(p) y() = B(p) u() using filtered signals

yf() = Hf(p) y()anduf() = Hf(p) u()as follows

y() = pn yf(), ()T =

pn1 yf(), . . . , p

m1 uf()



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RLS Algorithm (continuous time)

Given the data

y(), ()t

t=t0defined by the model

y() = ()T0, [t0, t]

obtained from A(p) y() = B(p) u() using filtered signals

yf() = Hf(p) y()anduf() = Hf(p) u()as follows

y() = pn yf(), ()T =

pn1 yf(), . . . , p

m1 uf()

With the initial guess (t0)and a measure of our trust in this

guess: P(t0) > 0, we can use the RLS algorithm:

(t) = P(t) (t)

y(t) (t)T(t)

P(t) = P(t) P(t)

(t) (t)T

P(t)



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Algorithm using RLS and MD Pole Placement

Off-line Parameters: Given polynomialsBm, Am, Ao



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Step 1: Estimate the coefficients of AandB = B+ B, i.e. 0,

using the Recursive Least Squares algorithm.



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Step 2: Apply the Minimum Degree Pole Placement algorithm

(deg A=deg Am, deg B=deg Bm, deg Ao=deg Adeg B+1, Bm=BBpm)

R= Rp B+, T = Ao Bpm, S, R p : A Rp+B

S= Ao Am

with estimates forAandB taken from the previous Step.



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S= Ao Am


Step 3: Compute the feedback control law

R u(t) = Tuc(t) S y(t)



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S= Ao Am


Step 3: Compute the feedback control law

R u(t) = Tuc(t) S y(t)

Repeat Steps 1, 2, 3 (until the performance is satisfactory).



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Example 3.4 (Recursive Modification of Example 3.1)

Given a continuous time system

y+ y = u,



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y+ y = u,

its pulse transfer function with sampling periodh= 0.5sec is

G(q) = b1q+ b2

q2 + a1q+ a2=

0.1065q+ 0.0902

q2 1.6065q+ 0.6065



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y+ y = u,

its pulse transfer function with sampling periodh= 0.5sec is

G(q) = b1q+ b2

q2 + a1q+ a2=

0.1065q+ 0.0902

q2 1.6065q+ 0.6065

The task is to synthesize a 2-degree-of-freedom controller suchthat the complementary sensitivity transfer function is

Bm(q)Am(q)

= 0.1761qq2 1.3205q 0.4966

We will take uc(t) = 1 for t 0 except for

uc(t) = 1 for 15 t < 30, uc(t) = 0 for 50 t 70



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We have found the controller via MD pole placement

R(q)u(t) = T(q)uc(t) S(q)y(t)

with

R(q) = q+ 0.8467

S(q) = 2.6852 q 1.0321

T(q) = 1.6531 q

that stabilizes the discrete plant. Will it stabilize the

continuous-time system as well?



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We have found the controller via MD pole placement


with

R(q) = q+ 0.8467

S(q) = 2.6852 q 1.0321

T(q) = 1.6531 q

that stabilizes the discrete plant. Will it stabilize the

continuous-time system as well?

Let us design the adaptive controller which would recover theperformance for the case when parameters of the plant - the

polynomialsA(q)andB(q) are not known!



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0 10 20 30 40 50 60 70 80 902

1.5

1

0.5

0

0.5

1

1.5

2

time

output

The response of the closed-loop system with adaptive controller



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0 10 20 30 40 50 60 70 80 905

4

3

2

1

0

1

2

3

4

5

time

control

The control signal



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0 5 10 15 20 25 30 35 402

1

0

1

a1

0 5 10 15 20 25 30 35 400.5

0

0.5

1

a2

0 5 10 15 20 25 30 35 400.1

0.15

0.2

0.25

b1

0 5 10 15 20 25 30 35 400

0.2

0.4

b2

time (sec)

Adaptation of parameters defining polynomials A(q)andB(q)



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Lecture 6: Deterministic Self-Tuning Regulators

Feedback Control Design for Nominal Plant Model

via Pole Placement





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The previous algorithm consists of two steps

Step 1: Computing estimates for the polynomialsA(q),B(q)



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The previous algorithm consists of

Step 1: Computing estimates A(q), B(q)forA(q),B(q)

Step 2: Computing polynomials defining the controller

R(q), T(q), S(q)

based on A(q), B(q).



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R(q), T(q), S(q)


Such procedure has potential drawbacks like

Redundant step in design: why to know A(q)andB(q)?



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R(q), T(q), S(q)


Such procedure has potential drawbacks like


Control design methods (pole placement, LQR, . . . ) are

sensitive to mistakes in A(q), B(q). Is there a way to avoidsuch poorly conditioned numerical computations?



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R(q), T(q), S(q)


The procedure has potential drawbacks like


Control design methods (pole placement, LQR, . . . ) are

sensitive to mistakes in A(q), B(q). Is there a way to avoidsuch poorly conditioned numerical computations? E.g.:

A(q) = q(q 1), B(q) = q+



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The idea of another approach comes from the observation thatfor the plant

A(q) y(t) = B+(q) B(q) u(t)

and for the target system (the model to follow)

Am(q) y(t) = Bmp(q) B(q) uc(t)

the Diophantine equation

Ao(q) Am(q)= A(q) Rp(q)+ B(q) S(q)

can be seen as a regression to estimate coefficients of RandS



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A(q) y(t) = B+(q) B(q) u(t)






Ao(q)Am(q)y(t) = A(q)Rp(q)+ B(q)S(q) y(t)



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A(q) y(t) = B+(q) B(q) u(t)






Ao(q)Am(q)y(t) = A(q)Rp(q)y(t) + B(q)S(q)y(t)



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A(q) y(t) = B+(q) B(q) u(t)






Ao(q)Am(q)y(t) = Rp(q)A(q)y(t) + B(q)S(q)y(t)



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A(q) y(t) = B+(q) B(q) u(t)






Ao(q)Am(q)y(t) = Rp(q)B(q)u(t) + B(q)S(q)y(t)



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A(q) y(t) = B+(q) B(q) u(t)






Ao(q)Am(q)y(t) = Rp(q)B+(q)B(q)u(t)+B(q)S(q)y(t)



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A(q) y(t) = B+(q) B(q) u(t)






Ao(q)Am(q)y(t) = B(q) R(q) u(t) + S(q)y(t)



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Direct Self-Tuning for Minimum-Phase Systems

Suppose thatB(q)is stable and can be canceled, i.e.

A(q)y(t) = B(q)u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)

= B+(q)B(q)u(t) = B+(q)b0u(t)



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A(q)y(t) = B(q)u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)

= B+(q)B(q)u(t) = B+(q)b0u(t)

then Diophantine equation

Ao(q)Am(q)= A(q)Rp(q)+ B

(q)S(q)



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A(q)y(t) = B(q)u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)

= B+(q)B(q)u(t) = B+(q)b0u(t)



(q)S(q)

leads to the regression

Ao(q)Am(q)y(t) = b0

Rp(q)u(t) + S(q)y(t)



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A(q)y(t) = B(q)u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)

= B+(q)B(q)u(t) = B+(q)b0u(t)



(q)S(q)

leads to the regression

Ao(q)Am(q)y(t) = b0

Rp(q)u(t) + S(q)y(t)

which is in the form (t) = (t)T with

Ao(q)Am(q)y(t) = (t), (t)T = b0

R(q)u(t) + S(q)y(t)



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In the formula

(t) = b0R(q)u(t) + b0S(q)y(t) = (t)T

the vector of regressors is

(t) = [u(t), . . . , u(t l), y(t), . . . , y(t l)]T

and the vector of parameters is

= [b0r0, . . . , b0rl, b0s0, . . . , b0sl]T



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In the formula

(t) = b0R(q)u(t) + b0S(q)y(t) = (t)T

the vector of regressors is

(t) = [u(t), . . . , u(t l), y(t), . . . , y(t l)]T

and the vector of parameters is

= [b0r0, . . . , b0rl, b0s0, . . . , b0sl]T

Supposeis estimated by , then

R(q)= ql +2

1

ql1 + . . . , S(q)= ql +l+2

1

ql1 + . . .


Di S lf T i f Mi i Ph S


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Some modification to the regression model

Ao(q)Am(q)y(t) = (t) = b0R(q)u(t)+b0S(q)y(t) = (t)T

can be made if we introduce the signals

uf(t) = 1

Ao(q)Am(q)u(t), yf(t) =

1

Ao(q)Am(q)y(t)


Di t S lf T i f Mi i Ph S t


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Some modification to the regression model

Ao(q)Am(q)y(t) = (t) = b0R(q)u(t)+b0S(q)y(t) = (t)T

can be made if we introduce the signals

uf(t) = 1

Ao(q)Am(q)u(t), yf(t) =

1

Ao(q)Am(q)y(t)

Then the regression model is

y(t) = b0R(q)uf(t) + b0S(q)yf(t)

= R(q)uf(t) + S(q)yf(t)

= (t)T


Di t S lf T i R l t S


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Direct Self-Tuning Regulators: Summary

Given polynomialsAm(q),Bm(q),Ao(q)

the relative degreed0 of the plant (deg Ao = d0 1)


Direct Self Tuning Regulators: Summary


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Step 1: Estimate coefficients of polynomialsR(q)andS(q)

from the regression model

y(t) = b0R(q)uf(t) + b0S(q)yf(t) = (t)T




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Step 2: Compute the control signal by


where forBm(q) = qd0Am(1) (minimal delay and unit static gain):

T(q)= Ao(q)Am(1)




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where forBm(q) = qd0Am(1) (minimal delay and unit static gain):

T(q)= Ao(q)Am(1)

Repeat Steps 1 and 2


Direct ST Reg for Non Minimum Phase Systems


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Direct ST Reg. for Non-Minimum-Phase Systems

IfB(q)is not stable or cannot be canceled, i.e.

A(q) y(t) = B(q) u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)

= B+(q)B(q)u(t), B(q) const


Direct ST Reg for Non-Minimum-Phase Systems


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Direct ST Reg. for Non-Minimum-Phase Systems


A(q) y(t) = B(q) u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)



Ao(q)Am(q)= A(q)R(q)+ B

(q)S(q)




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Direct ST Reg. for Non Minimum Phase Systems


A(q) y(t) = B(q) u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)




(q)S(q)leads to the regression

Ao(q)Am(q)y(t) = B(q) R(q)u(t) + S(q)y(t)




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Direct ST Reg. for Non Minimum Phase Systems


A(q) y(t) = B(q) u(t)

= b0u(td0) + b1u(td01) + + bmu(td0m)




(q)S(q)leads to the regression

Ao(q)Am(q)y(t) = B(q) R(q)u(t) + S(q)y(t)

which is nonlinear in parameters and has unstable factor

(t) = B

(q)

R(q)u(t) + S(q)y(t)

= (t 1)T




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Given

polynomialsAm(q),Bm(q),Ao(q)

the relative degreed0 of the plant




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Given



Step 1: Estimate coefficients of R(q)and S(q)from

y(t) = R(q)uf(t) + S(q)yf(t) = (t)T

Cancel possible common factors of R(q)and S(q)to computeR(q)andS(q)




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g g y

Given







R(q)u(t) = T(q)uc(t) S(q)y(t)where

T(q)= Ao(q)Bmp(q)




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g g y

Given







R(q)u(t) = T(q)uc(t) S(q)y(t)where

T(q)= Ao(q)Bmp(q)

Repeat Steps 1 and 2


Next Lecture / Assignments:


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g

Next meeting (April 26, 13:00-15:00, in A205Tekn).

Homework problems: Consider the process:

G(s) = 1

s(s + a), whereais an unknown parameter. Assume

that the desired closed-loop system is

Gm(s) = 2

s2 + 2s + 2. Construct

continuous-time indirect algorithm,

discrete-time indirect algorithm, continuous-time direct algorithm,

discrete-time direct algorithm.


lecture 07 eailc

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