Adaptive Control of Non-Minimum Phase Systems with Independent Tracking and Regulation Specifications

Copyright © IFAC Software for Computer Control Madrid. Spain 1982

ADAPTIVE CONTROL OF NON-MINIMUM PHASE SYSTEMS WITH INDEPENDENT TRACKING AND REGULATION SPECIFICATIONS R. Lozano L. Centro de Investigaci6n y de Estudios Avanzados del IPN, Departamento de Ingenieria Electrica, Ap. Postal 14-740, 07000 Mexico Abstract . Loca l convergence for adaptive contro l algorithms applied to v,eneral dis crete , deterministic , linear , time-invariant systems has been established in (1) . However, this algorithm does not allow to obtain an independent behaviour in tracking and regulation . In this technical note , the adaptive control algorithm of (1) is mo dified in order to be able to specify independently the tracking and regulation objec tives . The performances of this new adaptive control structure are evaluated by si mulations. Keywords . Adaptive control, Discrete time systems , Optimal control , Parameter Est imation, Stability , Tracking systems . INTRODUCTION Goodwin and Sin presented in (1 ) an adaptive control algorithm for discrete non -minimum phase systems in which the closed loop poles are arbitrarily assigned in the limit and the system input and output remain bounded for all time. However their scheme does not allow to specify independently the tracking and regulation objectives . This paper presents a modified version of the adaptive control algorithm in (1) that in sures independent behaviour in trackin~ and regulation as preconized in (2) and (3) for minimum phase systems . The advantages of this adaptive control scheme over that in (1) are discussed . 11 CONTROL OF KNOWN PARAMETERS LINEAR

PLANTS . Consider a SISO (single input-single output) linear time - invariant plant described by: -1 -d -1 A(q ) y(t)= q B(q ) u(t) + wet) (2.1)

f 0; b

m

-d -

B( z

-1

)= z

-d

B( z

-1

if B( l)= 0

)j

B

t > 0

(2 . 6)

where N':(q- 1) is a given asymptot~callY stable polynomial (A*(O)= 1) and E(q-1) is a finite dimensional polynomial (to be defined later) . It readily follows that both tracking and regulation objectives will be achieved if the following equation holds. -1

(2 . 7)

Since A(q ) and B(q ) are relatively prime polynom i als , there exist unique polynomials E(q71) and r(q - 1) both of order (r-l), r= max(n , m+d) , and E(O) = 1 such that : -1 -1 -d -1 -1 -1 A(q ) E(q )+q B(q )r(q )= A*(q ) (2.8) See appendi x A in (1). Using the above identity and Eq . (2 . 1), it follows that N':(q - 1) (y(t+d)-B(q-1)y'·:(t))::. B(q-1 ) (E(q-1)u(t) + r ( q- 1) yet) _ -1

A":( q B) y;:(t) ) + E(q - 1 ) wet)

f 0 ( 2 . 3)

1) The control should be such that in tracking (w(t)= 0) , the transfer function from a reference sequence y*(t) to the plant output yet) must be equal to: z

A*(q-1) y(t +d)= E(q - 1) wet)

-1

B(q- 1 )= b o + b 1 q -1 + . .. + b m q - m o The control objectives are:

2) The control should be such that in regu lation (y*(t)= 0) the transfer function from the disturbance wet) to the plant output yet ) must be asymptotically stable, i .e . :

A":(q-1) (y(t+d)-B(q-1)y(t))= E( q-1) wet)

where yet) , u(t) denote the system output and input respectively , wet) is a bounded perturbation and A(q-1) and B(q- 1) are relatively prime scalar polynomial in the unit delay ope rator q-1, d > 1, represents J pure time delay . Thus A(q-1)= 1 + a q - 1 + ... + an q - n ; an f 0 1 (2.2)

b

whi ch avoids an unbounded control input by preserving the plant zeroes . The B-coefficient is introduced to insure a zero steady state tracking error (y(t) - y*(t)) whenever possible.

( 2 . 4)

(2 . 9)

Thus, if the input is generated by the control law: E( q

-1

)u(t)

A*(q - 1) y*(t)

B

-

r(q

-1

)y(t)

(2.10)

The objectives in Eq. (2.7) are reached . Also from Eqs . (2.1), (2 . 8) and (2.1 0 ), we obtain: _1 1 A* (q- 1) y*(q - 1) A*(q -)u(t)= A( q- )

(2.5)

B

where [ B=

- r(q

B(1) otherwise

(2.6)

SFC- J *

285

-1

) wet)

( 2 . 11)

R. Lozano L.

286

It is now clear from Eqs . (2.7) and (2.11) that yet) and u(t) are both bounded sequences . Note thai if A*(q-1)= 1 i.e. all the closed l oop regulation poles are placed at the origin (see Eq . (2.7)1 and if w(t) = 0 , then yet) will exactly track the sequence q- d B( q-l) y,', ( t) for t > d . For A* ( q-l) f 1, the error (y(t) - q-d B(q- l) y"'(t)) will converge to zero only asymptotically, but the control input as well as the plant output will be in general less abrupt . For comparison purposes , let's rewrite the control law used in (1): (2.12) -d -1 premultiplying the above equation by q B(q ) and using Eqs . (2 . 1) and ( 2 . S) it follows that: -1 -d -1 -1 A*( q ) y(t)= q B( q ) F(q ) y*(t) +

A

- F(t,q

-1

) wet)

( 2 .13)

The control law in Eq . (2 . 10) presents three main advantages over the control law in Eq. ( 2 . 12): 1) The regulation and tracking object ive s are independently specified . It can be seen from Eq . ( 2 .1 3) that when the contro ller_ 1 in Eq. (2.1 2 ) is used , the zeroes of A",( ) are simultaneous ly the poles in tracking and in regulation . 2 ) The zeroes of the transfer function from y*(t) to yet) are exclusively those of the plant (z -d B(z-l »). Note that when the control law in Eq . (2.12) is used the zeroes of the controller (F(z - l») are also zeroes of the transfer function Y(z)/Y*(z) . 3 ) The steady state tracking error (y ( t) y*(t» is cancelled in an easier way . III ADAPTIVE CONTROL ALGORITHM . We will consider in this section w(t)= 0 for simplici ty purposes , then Eq . ( 2.1) can be written in the form: y(t) =
T

(3.1)

8

whe re :

-1

A

A(t,q

(3.2)

r!=

(-a 1 ,·

··,

( 3 . 3)

Sr )

-a r , Sl"' "

A

)E(t,q

A

S(t) =

{1

i, . .. , d-l and i >

( 3 .4)

m+d .

8(t)= 8(t - 1) +

Ft
F
= F t+l t

t

T

F

F

t

1 +
and the adaptive control law:


t F

o

> 0

(3.6)

A

) + B(t,q

-1

A

) F(t,q

-1

1 )= A*(q- )

it B(t,l) = 0

(3.9)

otherwise

B( t ,1 )

The symbols in Eqs. (3.5), ( 3 . S) and (3.9) mean: T

A

A

A

A

"-

GCt) = (-a (tl. .... - ar(t). b (t) ..... b (t») 1 1 r ( 3. 10) A

A(t , q

-1

)= 1 + a (t) q 1

-1

+ ... + ar(t) q- r (3.11) (3.12 )

We can now establish the following lemma, whose proof follows exactly the proof of theorem 3 .1 in (1) and will be omitted . Lemma 3.1: Subject to the following assumptions: r = max (n, m+d) is known A(q-l) and q - d B(q -1 ) are relatively prime

A. l A.2

A

A. 3

A

-

f(1) is within 118(1)

811 < g as in

lemma 3 . 2 in (1). Then the algor i thm (3.5) to (3.9) leads to: i) u(t) and yet) bounded 1

A_l

ii) lim {A*(q- ) (y(t+d) - B(t , q t+oo where

~(t,

q-l) =

~(t,

) y*(t»)= 0 (3.13)

q -l) / Set)

( 3 .1 4)

IV SI MULATION RESULTS Consider the plant in Eq. (2.1) (example 5.4 in (1) with: -1 -d -1 -1 -2 q B(q )= A(q ) = 1-2.5q + 2 . 5q - q

-2

for t <

(4.1)

T

A(q- l) = 1-3q - 2 + 2q - 3 ; q - d B(q -1 )= q

Consider the following least squares adaptive control algorithm:

-1

) are solution of

A

and: r= max(n,m+d); a = 0 for i > n; Si= 0 for i= i

(3.7) A_1

-1

( 3 . S) and:


) yet)

where E(t,q ) and F(t,q the identity:

E( q- l) u(t)= F( q-l) (y*(t) - yet»)

+ E( q

-1

A

-1

( O. 5 - O. Sq

-1

) for t >

T

(4. 2 )

The regulation behaviour will be given by: -1 -1 3 A*(q ) = (1 - 0 . 4q ) (4.3) The reference sequence will be generated by: -1 -2 -3 (1+1.9q - 1. 23q + 0 . 265q ) y*(t) = = (0.2Sq

-2

+ 0 . 22q

-3

) u*(t)

(4,4)

where u'" (t) is a unit pulse train which period equals to 100 steps . A forgetting factor will be introduced in Eq . (3,6) such that the trace of Fk remains cons tant as in (2) and ( 3) with Fo = 10.1 6 , This is a slig~tly different algorithm but it also has the properties required to establish lemma

Adaptive Control of Non-Minimum Phase Systems

287

3.1.

The adaptive control scheme will be used in tracking and in regulation. In both cases, the initial controller will be designed as in section 2 using the initial values in (4.1). Figure 4.1a shows the tracking performance of the adaptive controller when the plant parameters are abruptly changed at time T= 30. The solid line represents the plant output and the dashed line represents the compensated referencE sequence By'" (t) • Figure 4.1b shows the regulation behaviour (y*(t)= 0) of the adaptive controller when the plant parameters are abruptly changed at time T= 1 and wet) is a periodic impulse perturbation w(t)= 2 for t= 0,60, 120, ••• Note that the regulation behaviour be observed in Fig. 4.1a from step In this case , the plant output has tered by a plant parameters change regulated around y*(t)= 1 for 30 <

can also 30 to 50. been aland it t < 50.

Acknowledgments The author wishes to thank Dr. I. D. LANDAU and Dr. L. DUGARD for the numerous discussions on the subject.

OUTPUT

4.

3.

2.

8.

2.

(2) 1. D. LANDAU and R. LOZANO (1981). "Unification of discrete time-explicit model reference adaptive control designs", Automatica, 17. NQ 4 pp 595-611. (3) R. LOZANO and 1. D. LANDAU, (1981). "Redesign of explicit and implicit discrete time model reference adaptive control scheme s ". Int. Journal of Control Vol. 33 , NQ 2 , 247-268. (4) R. KUMAR, J. B. MOORE, (1981). "Minimum variance control harnessed for non-m~n~­ mum-phase plants". Proc. of the 8th IFAC Congress . (5) R. LOZANO, (1981). "Adaptive control with forgetting factor". Proc. of the 8th IFAC. World Congress. To appear in Automatica.

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REFERENCES

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(1) G. C. GOODWIN and K. S. SIN (1981). "Adaptive Control of non-minimum-phase systems". IEEE Trans. on Auto. Control,

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