Systems & Control Letters 18 (1992) 245-252 North-Holland

245

Rejection of persistent bounded disturbances: Nonlinear controllers M.A.

Dahleh

*

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

J.S. S h a m m a Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, USA Received 4 May 1991 Revised 6 December 1991

Abstract: This paper considers nonlinear time-varying (NLTV) compensation for linear time-invariant (LTI) plants subject to persistent bounded disturbances. It is shown using two different approaches that using NLTV compensation instead of LTI compensation does not improve the optimal rejection of persistent bounded disturbances. The first approach is to derive a bound on the achievable performance over all stabilizing NLTV controllers. Using results from t' Loptimal control, it follows that in some special cases this bound can be achieved by LTI compensation. This approach involves the introduction of an operator analogous to the Hankel operator in ~ - o p t i m a l control and is of independent interest. The second approach is to assume the NLTV controller is sufficiently smooth to admit a time-varying linearization. This time-varying linearization is then used to construct an LTI controller which achieves the same performance as the original NLTV controller. These results extend previous work by the authors regarding linear time-varying compensation.

Keywords: Disturbance rejection; nonlinear time-varying compensation; /l-optimal control; Hankel operators; linearization.

Notation L T I .'= l i n e a r t i m e - i n v a r i a n t . L T V := l i n e a r t i m e - v a r y i n g . N L T V := n o n l i n e a r t i m e - v a r y i n g .

t'~ := { f = ( . . . ,

f(-

1), f(0), f(1), f ( 2 ) . . . . ): II f II~ := s u p . I f(n)[ < oo}.

e~(Z+) := {f~e~: f(n) = 0,

V n < 0}. [ 1 := { f E tim: II f n l : = E n [ f ( n ) [ < ~}. c o = { f ~ e ~ Ilimk_~ ± ~ x ( k ) = 0}.

II Z II := s u p ~ t ~ f . 0 II Tf I1~/11 f I1~. = f ( n ) if n < 0, a n d 0 o t h e r w i s e .

II_f(n)

1. Problem statement I n this p a p e r , w e c o n s i d e r t h e u s e o f N L T V c o m p e n s a t i o n to a c h i e v e o p t i m a l d i s t u r b a n c e r e j e c t i o n w i t h L T I p l a n t s . T h i s p r o b l e m h a s b e e n c o n s i d e r e d in [1,5,7,9,12]. I n [5,7,9], it was s h o w n t h a t N L T V c o m p e n s a t i o n d o e s n o t i m p r o v e t h e o p t i m a l r e j e c t i o n o f f i n i t e - e n e r g y (i.e., £ 2) d i s t u r b a n c e s . I n [12], it

Correspondence to: Prof. J. Shamma, Dept. of Electrical Engineering, 4-174 EE/Computer Sci. Building, University of Minnesota, 200 Union Str. Minneapolis, MN 55455, USA. * Supported by the Center for Intelligent Control Systems under the Army Research Office grant DAAL03-86-K-0171 and by the Wright Patterson Air Force Base under grant F33615-90-c-3608 0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

M.A. Dahleh, J.S. Shamma / Rejection of persistent bounded disturbances

246

was shown that LTV compensation does not improve the optimal rejection of persistent bounded (i.e., 1 =) disturbances. This was extended to continuous-time systems in [1]. Possible advantages of NLTV control are discussed in [8] and references contained therein. The results of [1,12] hold for LTV compensation only. In this paper, we consider nonlinear compensation with persistent bounded disturbances. It is shown using two approaches that NLTV compensation again does not improve the optimal disturbance rejection of / = disturbances. The first approach involves the introduction of an operator analogous to the Hankel operator in ~ - o p t i m a l control which is of independent interest. This operator leads to a bound on the achievable performance which can be achieved by LTI compensation. The second approach uses a linearization of the nonlinear controller to construct an LTI controller which achieves the same performance as the original NLTV controller. In the discussion that follows, familiarity with the disturbance rejection problem framework and related notions of stabilization, causality, and well-posedness is assumed (cf. [6,15]). In particular, unless otherwise specified, all operators are norm-bounded causal mappings over signals with support ( - ~, ~). To set up the problem, let Tzw(K) denote the closed-loop mapping from the exogenous disturbances, w, to the regulated variables, z, as a function of the controller K. Let J T V denote all norm-bounded causal NLTV operators on t ~. Let _%Fvv denote the subset of J T V which are linear. Similarly, let S~w denote the subset of -~xv which are time-invariant. The A transform of an element H ~.Z,
inf

ILT1 - T2QT3 II,

Q ~ ~YTV

/XTV := inf{ II T~w(K)I1: K is any stabilizing LTV controller} =

inf

I I T , - T 2 Q ~11,

g c ~vv

]'£TI : : inf{ II T=w(K)II: K is any stabilizing LTI controller}

=

inf

IIT~ - T2QT3 II.

Here, T1,2, 3 E,.~TI are discrete-time multiple-input/multiple-output systems determined by the discrete-time LTI plant and disturbance rejection problem under consideration. In the remainder of this paper, any extra assumptions on T~,2,3 will be introduced as needed. The following theorem concerns LTV compensation: T h e o r e m 1.1 [12]. /'£TV: itETl"

In this paper, we show that under certain conditions /-/'NL= //'TI"

2. Main results 2.1. A Hankel-like operator

In this section, a Hankel-like operator (cf. [6]) is defined for general operators on 1 =. This Hankel-like operator leads to a lower bound on the achievable performance over NLTV compensators. In the special case where T2 = I, it is shown that this bound may be achieved by an LTI compensator. In this section, T1,2.3 are assumed to be single-input/single-output with the exception of Example 2.2.

M.A. Dahleh,J.S. Shamma / Rejectionof persistentboundeddisturbances

247

Let a 1. . . . . a n, b 1. . . . . bm be the zeros of the transfer functions of T2 and T3 respectively inside in the open unit disc. For simplicity, assume they are real and distinct. The forthcoming analysis still goes through in the general case. Define the functions

Uai(k)=a]-k

Vk<0, j=l

Vbj(k)=bj -k V k < O , j = l

. . . . . n, . . . . . m.

Let = span{ Ua,},

~ ' = span{ vbj} .

and T are subspaces inside / ~ supported on the nonpositive integers. Given any operator H on / ~, define a Hankel-like operator as follows:

F H : ~ + ~ - - ~ { ~, u + v - - * H _ ( u * H e + e * H v ) where H _ denotes the projection on the nonpositive integers, * denotes convolution, and e denotes the unit pulse at the origin, e(k)=

1, 0,

k=0, k~:O.

In the case of an LTI operator H, the operator F , has a simple representation. Let u + v ~ ~ + •. Then

U+U= ~aiUai+ ~fliUb~ i-1 i-I and m

II_(H(u +v)) = ~aiI4(ailua + i=1

]~_~iltt(bi)ub. i-I

The norm of F . is defined as

II FHII =

II r . f II

sup

In the case where H is LTI, this norm can be computed exactly via solving a linear programming problem. This is captured in the following proposition. Proposition 2.1.

Let H be an LTI operator with A-transform I4. Then m

II FH II = max ~ a,fl

subject to n

aiI~(ai) + Y'~ flitQ(bi)

i=1

i=1

~

aiaki + i=1

fliaki <1,

Vk >_O.

i=1

Proof. By direct computation,

II FH II =

max a,/3

max ~ OliI~(ai)a7 k + ~ fliI~(bi)bZ k k<0_

i=1

i=1

M.A. Dahleh, J.S. Shamma / Rejection of persistent bounded disturbances

248

subject to

i:~l a i a ~ k + i - l ~ i b f k

<1,

Vk
It remains to be shown that the function to be maximized achieves the m a x i m u m at k = O. To prove this, assume it achieves the m a x i m u m at k = k * . Let ~i=c~ia~ -k*, 13i=[3ibi k*. T h e n &i, /3i are feasible solutions which gives the same value at k = O. [] T h e following t h e o r e m establishes the connection b e t w e e n { ~-optimal control and the above Hankellike operator. Theorem

2.1 [3].

ttZTi =

IIFT, II

It is interesting to notice that when Q is LTI, one can easily show that II FT, II is a lower bound for/-£TI as follows. Let ci9 = T~ - T2QT 3 with impulse response ~b • t 1. T h e n I1'/'11 = IIT~ -T2QT3[[ >

sup

IIH_(u*ch*e) +H_(e*6*t.')ll~=

lIFT, II.

uc~,c~'7/.llu+t. I1~_<1 This lower bound is not valid in general if Q • ~ T v . This poses a serious p r o b l e m in proving the general result we desire. In the special case where T 2 = I (and hence ~" = 0), the lower b o u n d is valid and the desired result can be proved. O f course this includes the case where T~-I is stable. Theorem

2.2. If T 2 = I then /tZNL ~LbT1.

P r o o f . For all v • •

:

with I[ v II ~ < 1,

II TI - QT3 l[ > II II (T~ - OT3)t, II ~ = II H_Tlv - FI_QII_T3v I[= = II II_Ztt' I1~. T h e above is true since ~ ( b i) = 0. Hence, II T~ - QT~ II -> II FT, II = ~T1However, the lower b o u n d is achieved by an L T I Q.

[]

While the conditions of T h e o r e m 2.2 are not the most general, there are in fact some interesting p r o b l e m s in which T 2 has a stable inverse. Below are a few examples. E x a m p l e 2.1. Weighted input-sensitivity minimization for a stable plant. T h e m a p from the reference input to the input of the plant, with a controller in the feedback loop, is given by S i = W1(I+KP)-LW2. Incorporating the p a r a m e t r i z a t i o n of all stabilizing controllers, S i is given by

S i = W,( I - Q P ) W 2 Both W~ and W 2 are assumed to have a stable inverse. T h e result above implies that nonlinear controllers will not offer any advantage in {1 tracking p r o b l e m s with stable plants. T h e parallel result for output sensitivity is still open. E x a m p l e 2.2. Robust stability with coprime factor perturbations. T h e r e is an important reason for considering this example. Even though the result we presented earlier is for the square case, it is still valid for the non-square case, i.e for the case where T 3 is a row vector. In this example we will sketch the p r o o f of this result in this special problem. T h e general non-square case follows in the same way.

M.A. Dahleh, J.S. Shamma / Rejection of persistent bounded disturbances

249

Define the following class of plants:

( N + A 2 ) ( M + A 1 ) - ' and IIA, II <

~2= { P r e =

1}

with Z~i being / ~ bounded LTV operators and P0 = NM-1 satisfying the Bezout identity

(~?

-N

A sufficient condition for robustly stabilizing the above family with any NLTV controller is given by inf

[l[I?

0I+Q[A7

h/llll_
Q ~Wrv

This condition is also necessary if the controllers are restricted to be linear, possibly time-varying [2,12]. The necessity of this condition for N L T V controllers is as follows. First, the underlying notion of stability is finite-gain stability over c o rather than t'~. Second, the operator Q is restricted to be continuous and have pointwise fading-memory [13]. We note that the construction in [2,12,13] leads to a construction of admissible LTV A i such that either of the following conditions occurs. The first condition is that the plant (N + Ae)(M + A1)- ~ has an internal cancellation. That is, the operators M + A~ and N + A 2 are no longer coprime. This corresponds to an admissible plant which is not stabilizable. The second is that the admissible plant (N + Aa)(M + A1) -1 is stabilizable, but not using the particular Q with the property I1[12 0] + Q[A7 341 II > 1. It turns out that above infimization is achieved via a linear time-invariant Q. Define the subspace (inside g = × / =) as follows:

~/=(v=(_MN)x,x~c°,x(k)=O

V k > 1}.

Then for any Q ~ XTV and v ~ 7f, it is true that

I1([ o]

i xH >_ ix
Equivalently,

-

v ~

II v II

which was shown in [2] to be achieved by an LTI Q. The generalization to arbitrary T3 follows in a similar fashion. So far, there does not exist a general result that proves or disproves the general case where T2 does not have a stable inverse. In the sequel, a smaller lower bound on II T1 - T2QT3 II is furnished. However, it is not evident that there exists a causal Q that achieves the bound. Theorem 2.3. Let T3 = I. Then

I[rT, u Ill /.LNL ~_~

sup

Proof. By direct computation,

IITI- T2QII >>_II(TI- T2Q)flI~ Y f ~ e ~ ( Z + ) , l l f l l ~ _ < l >_ l l u * ( Z l - T2O)fll~, t l u l l l < l > IIII_(u*(Zl-Z2O)f)ll~ = l l l I _ ( U * Z l f ) ll~, u ~ .

25O

MA. Dahleh, J.S. Shamma / Rejection of persistent bounded disturbances

This leads to

II T~ - T2Q II >-

sup

sup

II I I _ ( u * T l f ) I1~ =

Ilulll_
sup II FT, U II ,. Ilulll_
[]

The interpretation of this bound is as follows. Fix any f~lo~(Z+), then there exist a Q f such that

II(T,-T2Qf)flI~=

IlII_(u,T,f)lloo.

sup Ilu IIl_
F h i s Qf however may very well be a non-causal function of f , and hence does not qualify as a candidate solution for the original problem. A consequence of this theorem is that in the case of a fixed input minimization [4], nonlinear time varying compensation does not improve the performance. This is clear from the fact that the above lower bound is valid for each f regardless of Q and can be achieved with Q time-invariant [4].

2.2. Linearization In this section, we show that the use smooth N L T V compensation instead of LTI compensation does not improve the achievable rejection of persistent bounded disturbances. The systems Ti,2,3 are now assumed to be multi-input/multi-output. The smoothness condition in this context is in terms of the compensation being linear&able. The following definition is adapted from [15, Chapter 7].

Definition 2.1. An operator H ~ "/]/'TV is linearizable if there exists a linear operator H L ~"~TV such that lim

sup

il n f - H L f II

=0.

f~0 In this case, H L is called the linearization of H. The main result of this section is as follows: Theorem 2.4. Let tXNL be defined as in (1) with the infimization being over all Q E,/~TV which are linearizable. Then tXT1 = IZNL. Proof. Let Q E./~TV be linearizable, and let lIT 1 - T 2 Q T 3 11 =/z. We will show that there exists a 0 ~-~TI such that

II T,

-

T2QT 3

II ~ ~.

Towards this end, let QL denote the linearization of Q. Then from Definition 2.1, given any e > 0, there exists an a > 0 such that

sup f~o

I[ T2QT3f - T2QL T 3 f II

M.A. Dahleh, J.S. Shamma / Rejection of persistent bounded disturbances

251

Then /x >__ sup

I[(T1 - T2QT3)f [I

ll/llo_<~

=

sup Ill I1~_<~ f=/=0

>_

sup

Ilfil~

II(Z, - T2QL T3) f - ( T2QT3 - T2QL T3) f II II f I1= II(T~ - T2QLT3)f [I

-

sup

f=/=0

[I( T2QT3 - T2QL T3) f II

Ill I1=

f=/=0

> 1[T1 - T2QLT3 II - e . Since e is arbitrary, it follows that ~.2:TI such that 1[ T1 - T20T3 II -< ~ .

II T a - T2QLT3 II ~ ~. U p o n applying T h e o r e m 1.1, there exists a

[]

The idea in the proof of T h e o r e m 2.4 is first to show that L T V compensation gives the same performance as linearizable N L T V compensation. We then use the results from [12] to show that LTI compensation gives the same performance as linearizable N L T V compensation.

3. Concluding remarks Even for the problem of disturbance rejection, nonlinear controllers can offer some advantage as seen in the following example. Let z denote the unit delay operator. Let T 3 = I, and let

T 1 - TzQ=

z2

1 Q. 1

Then for any w ~ f=,

w(n) ( ( T 1 - T2Q)w)(n ) = w ( n - 1) w(n - 2)

-

(Qw)(n) (Qw)(n) (Qw)(n)

Given this structure, an optimal Q may be constructed as follows. Define wU(n) := m a x ( w ( n ) , w ( n - 1), w(n - 2 ) ) ,

w E ( n ) :-----m i n ( w ( n ) , w ( n - - 1), w ( n - - 2 ) ) .

Then set

(QNLW)(n) = I ( w U ( n ) + w L ( n ) ) . It can be shown that this selection of QNL leads to l[ T1 - TzQNL I1 = 1. This choice of Q is nonlinear. However, the same norm can be achieved by using the linear Q = 0. Nevertheless, the compensator QNL achieves better performance in the sense that signal-by-signal, the response using QNL is smaller than using Q = 0. That the two choices lead to the same norm means there exists a signal such that the responses are the same size. Note that the choice of QNL is not differentiable. Thus, the performance is not characterized by the small signal behavior.

252

M.A. Dahleh, J.S. Shamma / Rejection of persistent bounded disturbances

A c o m m e n t is in o r d e r r e g a r d i n g the use of i n d u c e d n o r m s to assess t h e p e r f o r m a n c e o f n o n l i n e a r f e e d b a c k systems. F o r l i n e a r i z a b l e systems, the overall p e r f o r m a n c e is at best t h e ' s m a l l - s i g n a l ' p e r f o r mance. Thus, it s e e m s n a t u r a l that l i n e a r c o n t r o l l e r s w o u l d p e r f o r m as well as l i n e a r i z a b l e n o n l i n e a r controllers. It turns out that the use of i n d u c e d n o r m s to assess p e r f o r m a n c e m a y be too restrictive in the p r e s e n c e of n o n l i n e a r c o m p e n s a t i o n . T h e r e a s o n is t h a t this d e f i n i t i o n r e q u i r e s the ratio of the e r r o r - n o r m to the d i s t u r b a n c e - n o r m to be small w i t h o u t r e g a r d to the size of the m a g n i t u d e o f the disturbances. M o r e precisely, it m a y be that the r e g u l a t e d v a r i a b l e is small while t h e ratio o f e r r o r - n o r m to d i s t u r b a n c e - n o r m is large. This l e a d s to q u e s t i o n i n g t h e utility of i n d u c e d n o r m s to quantify p e r f o r m a n c e in n o n l i n e a r systems. O n e a l t e r n a t i v e is to c o n s i d e r the worst case p e r f o r m a n c e over a given class o f d i s t u r b a n c e s . F o r example, let ~¢" d e n o t e s o m e b o u n d e d set of d i s t u r b a n c e s . T h e n d e f i n e the p e r f o r m a n c e m e a s u r e

:= s u p II T~ww II. wETf

Such a p e r f o r m a n c e objective is p a r t i c u l a r l y w e l l - s u i t e d to n o n l i n e a r systems. It avoids using i n d u c e d n o r m s a n d a d d r e s s e s directly the d e s i r e d goal o f k e e p i n g r e g u l a t e d v a r i a b l e s small. F u r t h e r m o r e , it allows the class of d i s t u r b a n c e s to be d e f i n e d as desired. F o r e x a m p l e , o n e m a y d e f i n e 7 f as --={w~t~:

l[wl[ < c , and ~ l w ( n )

la < c 2 ) .

n

This d e f i n i t i o n allows b o t h a m a g n i t u d e a n d e n e r g y b o u n d on the d i s t u r b a n c e s o f interest. Such n o t i o n s o f p e r f o r m a n c e have b e e n c o n s i d e r e d in [10,11].

Acknowledgements T h e a u t h o r s t h a n k Paul M i d d l e t o n for suggesting t h e p r e c e e d i n g QNL e x a m p l e .

References [1] H. Chapellat, M. Dahleh and S.P. Bhattacharyya, Optimal disturbance rejection for periodic systems, Technical Report No. 89-019, TAMU, College Station, TX (1989). [2] M.A. Dahleh, BIBO stability robustness for coprime factor perturbations, to appear. [3] M.A. Dahleh and J.B. Pearson, Jr., t LOptimal feedback controllers for MIMO discrete-time systems, IEEE Trans. Automat. Control 32 (4) (1987) 314-322. [4] M.A. Dahleh and J.B. Pearson, Minimization of a regulated response to a fixed input, IEEE Trans. Automat. Control 33 (1988) 924-930. [5] A. Feintuch and B.A. Francis, Uniformly optimal control of linear time-varying systems, Systems Control Lett. 5 (1985) 67-71. [6] B.A. Francis, A Course in ~ - O p t i m a l Control Theory (Springer-Verlag, New York, 1987). [7] P.P. Khargonekar and K.R. Poolla, Uniformly optimal control of Linear time-varying plants: Nonlinear time-varying controllers, Systems Control Left. 5 (1986) 303-308. [8] P.P. Khargonekar and K.R. Poolla, Robust control of Linear time-invariant plants using switching and nonlinear feedback, Proceedings of the 28th IEEE Conference on Decision and Control (1989) 2205-2207. [9] P.P. Khargonekar, K.R. Poolla and A. Tannenbaum, Robust control of linear time-invariant plants by periodic compensation, IEEE Trans. Automat. Control 30 (1985) 1088-1096. [10] K. Poolla and J.S. Shamma, Asymptotic performance through adaptive robust control, Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HA (Dec. 1990). [11] J.S. Shamma, Performance limitations in sensitivity reduction for nonlinear plants, Systems Control Lett. 17 (1991) 43-47. [12] J.S. Shamma and M.A. Dahleh, Time-varying vs. time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization, IEEE Trans. Automat. Control 36 (7) (July 1991) 838-847. [13] J.S. Shamma and R. Zhao, Fading memory and necessity of the small gain theorem, 30th IEEE Conference on Decision and Control (Dec. 1991). [14] M.S. Verma, Coprime factorizational representations and stability of nonlinear feedback systems, Internat. J. Control 48 (1988) 897-918. [15] J.C. Willems, The Analysis of Feedback Systems (MIT Press, Cambridge, MA, 1971).