Control of linear time-varying systems by the gradient algorithm

Proceedings of the 36th

Conference on Decision & Control San Diego, California USA December 1997

FP02 4:50

Control

of

Linear Time-Varying Systems

by The Gradient

Algorithm

Min-Shin Chen

Department of Mechanical Engineering National Taiwan University

Taipei, Taipei, R.O.C. mschen@ccms.ntu.edu. tw

A b s t r a c t

In t.his paper, a new control design is proposed for a linear time-varying system which exhibits sustained but bounded state oscillations. The design is based on the gradient algorithm used in the adaptive parameter identification. The resultant observer-based state feed- back control guarantees exponential decay of the state oscillation given that the system is both uniformly con- trollable and uniformly observable. A unique feature of the proposed control design is that it requires neither information of t,ime derivatives of the time-varying pa- rameters, nor predict.ion of future information of the time-varying parameters.

1. Introduction

For linear time-varying systems, particularly for those with periodically time-varying system parame- ters. there have been a variety of control designs pro- posed. Examples can be found in [1,3,6,11]. For sys- tems with non-periodically time-varying parameters, the pole-placement like control in [5,12,13] provides a sound theoretical solution t o the control problem. However, their c,ontrol designs require time derivatives of the parameters up to the order of the system di- mension. Since it is impossible to have "clean" mea- surement of the time-varying parameters, the measure- ment noise will inevitably be amplified in the time dif- ferent,iation process, thus causing noticeable state os- cillations in the closed-loop response. Other control designs include the LQ optimal control [7] and the controllability-grammian-based control [4,9]. However, they suffer from the disadvantage that the control en- gineers must be able to predict how the time-varying parameters vary in the future in order to calculate the desired control input a t any time instant.

In this paper, a ntwi design is presented for the con- trol of a non-periodically time-varying system whose state exhibits sustainetl but bounded oscillations. In this new approach, one uses the open-loop state tran- sition matrix t o transfixm the system into one with

a zero system matrix, iind then utilizes the "gradient

algorithm" in adaptive parameter identification [8] to synthesize a stabilizing control. The same approach also leads to an observer design that asymptotically re- covers the system state from the system output when full state measurement is not available. The resultant observer-based state fel?dback control can be applied to systems with periodically or non-periodically time- varying parameters. Most importantly, it offers two advantages over other controls: (1) There is no need to predict future information of the time-varying parme- ters. The control design requires only past and present information of the parameters. (2) There is no need to take time derivatives of the time-varying parame- ters. This avoids the i~oise amplification problem in the pole-placement like control.

2. Problem Formulation

Consider a multivariable linear time-varying sys- tem

i ( t ) = A ( t ) ~ ( t )

+

B ( t ) ~ ( t ) , ~ ( 0 ) = 20 (1)

Y ( t ) = C ( t ) 4 t ) ,

where z ( t ) E R" is the system state vector, u ( t ) E R" is the control input, and y ( t ) E RP is the system out-

put. The system matriK A ( t ) E RnXn, the input ma- trix B ( t ) E RnXm, and the output matrix C ( t ) E Rpx" are time-varying matrices whose elements are bounded, piecewise continuous functions of time. Notice that

A ( t ) , B ( t ) and C ( t ) may be non-periodically time- varying. It is assumed t hat) the open-loop state trajec- tory z ( t ) exhibits sustained but bounded oscillations.

In other words, the state transition matrix @ ( t , t o ) E RnX" of the open-loop system remains bounded and nonzero for all t and t o . Therefore, there exist two

positive constants ml and m2 such that for all t and

t o 5

where ~i denotes the singular value of a matrix, and

the state transition matrix

[a]

@ ( t , t o ) is defined by

@ ( t o , t ) = W ( t , t o ) .

The objective of this paper is t o find a control to suppress the oscillatory behavior of the open-loop sys- tem under the condition that the only accessible signal in the system is the system output y ( t ) . In particular, an observer-based state feedback control

u ( t ) = -1i(t)i?(t), (4) is considered, where i ( t ) is an estimate of the state ~ ( t ) . For the existence of a stabilizing control as in E q . ( 4 ) , it is assumed that the system (1) is uniformly controllable as well as uniformly observable as defined below [9].

Definition 1 : The pair ( A ( t ) , B ( t ) ) is uniformly con- trollable if there exist A , /31 and

,&

E R+ such that for all t

>

0,

B11

L

Pc(t)

5

P 2 1 , (5) where P c ( t ) E

RnX"

is the controllability grammian defined by

A

q t

-

a,

T ) B ( T ) B T ( r ) @ y t -

a,

T)dT,

in which @ ( t , T ) is the state transition matrix defined in Eq.(3).

Definition 2 : The pair ( A ( t ) , C ( t ) ) is uniformly ob-

all t

>

0,

servable if there exist A , y1 a n d yz E R+ such that for

where Po(t) E R"'" is the observability grammian de- fined by

a

Po(t) =

LA

@ ( r , t - A)CT(T)C(T)@(T,t - A)&.

3. Preliminary

The control and observer designs in this paper will utilize the well-knwvn "gradient algorithm" orig- inally deveioped in the adaptive parameter identifica- tion problem. Consider the parameter estimation error dynamics of a gradient algorithm:

i ( t ) = -rw(t)zuT(t)z(t), z ( t ) E

R",

(7) where y is any positive constant, z ( t ) represents the pa- rameter estimation error, and w ( t ) E

R n X m

is usually called the "regressor". The follnwing theorem gives a sufficient condition on the exponential stability of the system (7).

Theorem 1 : If the regressor w(t) is "persistent ex- citing" in the sense that there exist positive constants A, a l , and a2 such that

1

a l l

5

L*

wT(T)uJ(T)dT

5

a21, vt

>

0, then the system (8) is exponentially stable, and

1l.(k.A)1l2

5

f k

11~(o)1121

k

=

.

,

where 0

<

p

<

1, and

ff1 p = 1 - 2 u y , U =

( 1

+

Y f f 2 4 q 2 . Proof: see proof of Theorem 2.5.1 in [lo].

4. State Feedback Control Design

In this Section it is temporarily assumed that the system state z ( t ) is accessible, and the goal is to find a state feedback control

u ( t ) = - - I i ( t ) z ( t ) (11) such that the closed-loop system (1) and (11) is ex- ponentially stable. The proposed control design starts with a coordinate transformation

z ( t ) = @ ( t , t o ) z ( t ) , (12) where z ( t ) is the new state coordinate, and the trans- formation matrix is exactly the open-loop state tran- sition matrix cP(t,to) in Eq.(3). Note that @ ( t , t o ) is always invertible by Theorem 4-2 in

[a].

Since @ ( t , t o ) is uniformly bounded as assumed in Eq.(2), the trans- formation ( 12) converts the stabilization problem of the

syst,eni state ~ ( t ) into that of the new state ~ ( t ) . Ac- cording t,o Eqs.( 1). ( 3 ) and ( l a ) , the governing equation of the ~ i t ' w state ~ ( t ) is given by

i ( t )

= @ ( t , t o ) - l B ( t ) u ( t ) . (13) Note carefully that in this new coordinate, the system matrix is identically zero. For a system with a zero syst,em matrix as in Eq.(13), Theorem 1 in the previous Section immediately suggests that the control u ( t ) can be chosen as

where 7 can be any positive constant. With this choice, the transformed closed-loop dynamics becomes

which has exact,ly the same structure as that in Eq.(7) w i t h @ - ' ( t . t o ) B ( t ) acting as the regressor.

The following lemma shows that the persistent excitation condition on the regressor @-'(t, to) B( t), which is required for the exponential stability of the system (15), is guaranteed by the uniform controllabil- i t y condition of the system (1).

Leiiiiiia 2 : If (.4(t), B ( t ) ) of the system ( 1 ) is uni-

f o r i d ) controllable as shown by Eq.(5), the observabil-

i t ) grammian P l ( t ) of the pair (0, BT(t)@-T(t,tO)) sat-

1sfics

P Z

t

@ - ' ( r , t " ) B ( r ) B T ( ~ ) @ - T ( r , t o ) d r 7 1 , (16)

LA

ml

where .3,'s are as in Eq.(5) and m,'s as in Eq.(2).

Proof Comparing the observability grammian PI ( t ) i n Eq.( 16) with the controllability grammian P c ( t ) of the system (1) in Eq.(5) shows that P l ( t ) = @(to,t

-

A)P,(t)@'(to, t - A). Hence, given any constant vector

t. one has

31 I[@T(tol t - A ) ~ l [ ~

5

zTp1(t)z

5

,&[I@'(to, t-A)zllz, due to Eq.(5). Since @(to,t - A) = @-l(t

-

A,to), it

follows from Eq.(3) that

End of proof.

The following theorern, which is the main result of this paper, shows that the proposed control (14) en- sures exponential stabihty of the closed-loop system.

Theorem 2 : Considcr the system (1) and the state feedback control (14). If the system (1) is uniformly controllable, and the open-loop state transition ma- trix satisfies the inequalities (2), the closed-loop system state z ( t ) converges to zero exponentially.

Proof : Based on Theorem 1, one concludes immedi- ately from Eqs.( 15) and (16) that the transformed state z ( t ) decays to zero exponentially in the sense that

ll.(~A)llz

I

Pk 11-(O)1IZ, k =

.

., where 0

<

p

<

1, and (17) m.:

P1

m; (m?

+

7P2&q2 l p = 1 - 2 a y , 6 =

in which Eq.(17) is obtained from Eq.(lO) with a1

=

,&/m; and a2 = p , / r i i T . Since Ilz(t)ll

5

m2llz(t)ll

due t o Eqs.(2) and (12),

z ( t )

also converges t o zero exponentially. End of proof.

5. Obslerver Design

In this section, it is assumed that the only accessi- ble signal in the system (1) is the system output y ( t ) . In order t o estimate the system state .c(t), a conven- tional Luenberger type observer is adopted:

i ( t )

=

A ( t ) i ( l ) -t B(t)u(t)

+

L(t)(y(t)

-

C ( t ) i ( t ) ) , i ( 0 ) = io, (18) where

i ( t )

E Rfl is an estimate of the system state ~ ( t ) ,

and L ( t )

E

R n x p is the observer feedback gain to be determined so that x( f ) approach z ( t ) exponentially. Denote the state estimrttion error by 2 =

i

-

z, and subtract Eq.( 18) from ]'hi.( 1) t o yield the state estima- tion error dynamics

i

= [ A ( t )

--

L (t) C( t)] 5. ₍₁₉₎ To design the observer feedback gain L ( t ) , the same coordinate transformation as in the control de- sign is applied t o the estimation error dynamics (19): Therefore,

5 ( t ) = @ ( t , to).i(t),

where i ( t ) defines the new coordinate, and @ ( t , t o ) is

as before the state transition matrix of A ( t ) . The gov-

erning equation of t ( t ) then becomes

t

= - @ - I ( t , t o ) L ( t ) C ( t ) @ ( t , t o ) Z .

Theorem 1 suggests that the observer feedback gain L ( t ) be chosen as

L ( t ) = I/ @ ( t , t o ) a r T ( t , t o ) G T ( t ) , v

>

0, (20)

where v can be any positive constant. The resultant estimation error dynamics becomes

i(t)

= --v @ ( t , to)CT(t)C(t)@(t, t o ) i ( t ) , (21)

which again has the same structure as the parameter estimation error dynamics of the gradient algorithm in Eq.(7) with now the regressor being QT(t,tO)CT(t).

The following lemma shows that this regressor a T ( t , t o ) C T ( t ) is persistently exciting if the system (1)

is unzformly observable.

Lemma 3 : If ( A ( t ) , C ( t ) ) of the system (1) is uni- formly observable as shown in Eq.(6), the observability grammian P2(t) of the pair (0, C(t)@(t,t,)) satisfies

?“I

L:

P2(t) =

t

@T(.,to)CT(.)C(7)@(.,to)~~

I

Y2m;L (22)

L-*

where yz’s are as in Eq.(6) and ma’s as in Eq.(2).

The proof of Lemma 3 duplicates that of Lemma 2 by noticing that Pz(t) = @ T ( t - A , t o ) P , ( t ) @ ( t - A , t o ) . With the persistent excitation condition (22), one can now conclude the exponential stability of the state es- tliriiation error dynamics (19) by quoting Theorem 1.

Theorem 3 : Consider the state estimation error dy- namics (19) and (20). If the system (1) is uniformly observable, and the open-loop state transition matrix satisfies the inequalities (a), the state estimation error

~ ( t )

-

z ( t ) converges to zero exponentially.

Proof : The proof follows exactly that of Theorem 2, and is omitted.

Finally, it is remarked that when the only accessi- ble signal is the system output y ( t ) , the state feedback control in Section 4 should be replaced by an observer- based state feedback control

u ( t ) = - K ( t ) x ( t ) , (23)

where Ii’(t) is as designed in Eq.(14), and 2 ( t ) is ob- tained from the observer (18). Under the uniform controllability and observabilit,y assumptions and by quoting Theorems 2, 3, and the well-known Separa- tion Property [2], it can be shown that the closed-loop system state under the control (23) converges t o zero exponentially.

6. Conclusions

This paper proposw a new control design for a class of linear time-vat ying syst,ems whose state ex- hibits sustained but bounded oscillations. The new approach utilizes the gradient algorithm, which is orig- inally used in the parameter identification problem, to find stabilizing control and observer feedback gains. It is worth mentioning that conventionally for a sys- tem with time-varying parameters, the observer feed- back gain depends on past and present information of the time-varying parameters, and the control feed- back gain, which is obtained as a dual result of the observer design, depends on futwe information of the time-varying parameters. However for the proposed design in this paper b0t.h the observer feedback gain and the control feedback gain depend only on past and present information of the time-varying parameters.

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