Aufomaticu, Vol. 33, No. 4, pp. 53?.-550, 1997 @I 1997 Elsevicr Science Ltd. All rights reserved

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PIE SOOO5-1098(96)0018%3 ala5-1098/97 s17.oo+o.al

Continuous I/O Robust Control of SISO Time-varying Systems

ZHIHUA QU,* EDWARD W. KAMENS and JOHN F. DORSEY 5

Based on the approach of model reference control, an input-output design is presented to generate continuous robust controlfor systems with fast time- varying parameters and uncertainties.

Key Words-Continuous control; output feedback; robust control; robust stability; time-varying systems; SISO; nonlinear systems; model reference control; uncertain dynamic systems; nonlinear control.

Abstract-In this paper output tracking problem of time- varying systems is investigated. A system under consideration contains time-varying parameters which may change very fast and may be subjected to high-order nonlinear disturbances. A robust controller requiring only input-output measure- ment is proposed. The control guarantees at least stability of uniform ultimate boundedness which can be made ar- bitrarily close to asymptotic stability through choosing a design constant. The resulting control is continuous, uniformly bounded, and designed by a recursive mapping procedure. The only information of unknown systems required by the robust control approach are the bounding function on the magnitude of nonlinear disturbances and the bounds on the parameters and their derivatives, which represent the dis- tinct feature of the proposed result. That is, time-varying parameters are not restricted to be either slow time-varying or of known structure. al997 Elsevier Science Ltd.

1. INTRODUCTION

Among different analysis and control design tech- niques, Lyapunov’s second method has its domi- nant role since it can be used for general nonlinear and time-varying systems. If the plant under con- sideration has fast time-varying parameters and is subject to unknown and possibly nonlinear distur- bances, a robust control designed using the Lya- punov’s direct method will be the natural choice. The robust control design characterized by pioneer work (Corless and Leitmann, 198 1; Gutman, 1979) is usually carried out in terms of a state space model

Received 23 December 1993; revised 13 October 1995; revised 2 July 1996; received in final form 17 September 1996. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hassan Khalil under the direction of Editor Tamer Basar. Corresponding author Professor Zhihua Qu. Tel. +I 407 823 5976; Fax + 1 407 823 5835; E-mail [emailprotected].

* Department of Electrical and Computer Engineering, Uni- versity of Central Florida, Orlando, FL 32816, U.S.A.

5 School of Electrical Engineering, Georgia Institute of Tech- nology, Atlanta, GA 30332, U.S.A.

533

and often requires certain structural conditions on uncertainties, the matching conditions. Although the robust control approach can be widely applied by its nature, it has two major limitations, First, the matching conditions (Corless and Leitmann, 198 1; Gutman, 1979) are more or less required, even though recent progress has been reported in Qu and Dorsey (199 1), Qu ( 1992), Qu and Dawson (1995), Qu and Dawson (1994), Qu (1993b) and the references cited therein. Second, most robust con- trol schemes require full-state feedback except for some special cases shown in Dawson et al. (1992) and the references contained therein, and a result on state tracking may not necessarily imply output tracking since the output matrix may contain un- known parameters. For linear time-invariant sys- tems, asymptotic output tracking results have been reportedin Schmitendorf and Barmish (1986,1987) in which robust control requires state feedback and impose the restrictions that the disturbance is con- stant and that the reference signal is generated from unit step function by finite integration or differen- tiation. For nonlinear systems, several results on output stabilization have been reported. If the sys- tem is output linearizable by input injection and is minimum phase and if the nonlinearities are known and parameters are constant, it has been shown in Marino and Tomei (1993) that there exists a glob- ally stable output-feedback adaptive control. For systems satisfying the generalized matching con- ditions (Qu, 1993b), it was shown in Khalil and Esfandiari (1993) that semi-global stability can be guaranteed under an observer-based robust con- trol. For systems with input-to-state stable inverse dynamics, sufficient conditions were derived in Tee1 and Praly (I 993) under which stability of bounded- ness or asymptotic stability can be ensured under an observer-based output-feedback controller. Like the state-feedback results, observer-based controls

534 Z. Qu et al.

require certain structural properties. The approach to overcome these limitations is

to develop a robust control design procedure based on input-output dynamic descriptions rather than state space models. Such an idea was pioneered in Qu et al. (1994) in which a procedure for design- ing robust control, called model reference robust control, was introduced. The design procedure of Qu et af. (1994) integrates model reference control scheme into robust control methodology and has been proven in Qu et al. (1994) Qu (1993a,c) to be effective for time-invariant systems with unknown parameters, unmodeled dynamics, weakly nonmin- imum phase dynamics, and nonlinear uncertainty and disturbance. The intention of this paper is to extend the idea to time-varying systems. The exten- sion is shown to solve the long-standing problem, the output tracking problem of fast time-varying systems under nonlinear uncertainties and distur- bances.

The idea of model reference control (MRC) has been widely studied in the literature of linear con- trol theory, especially adaptive control of linear time-invariant systems (Landau, 1979; Narendra and Annaswamy, 1989; Sastry and Bodson, 1989; Slotine, 1991). In the past decade, extensive re- search effort has been made in order to extend model reference adaptive control (MRAC) to time- varying systems. It is only recently that some suc- cess has been achieved using the MRAC approach. The most recent and important result is given in Tsakalis and Ioannou (1989a) in which linear time-varying systems are stabilized under MRAC if the parameters are either slow time-varying or are unknown constant multiples of known fast- changing time functions. Although this result pro- vides a MRC structure which guarantees exact model matching for general linear time-varying systems with known parameters, it (and its later result (Tsakalis and Ioannou, 1989b)) fails to re- move the major assumption of parameters being slowly time varying. Also, it has been well docu- mented that MRAC has performance degradation under bounded, additive disturbances and even instability in the presence of unmodeled dynamics. Furthermore, if there exists any nonlinearity, sta- bility results based on the MRAC method will, in general, be local since it uses the certainty equiv- alence principle. All these facts illustrate the need of finding a better alternative for controlling fast time-varying systems.

There has been some work on combining model reference control with other control design meth- ods, such as variable structure control (VSC). It has been shown in Chien and Fu (1992) that a dis- continuous VSC can guarantee asymptotic stabil- ity when the relative degree of the plant is one. The

use of discontinuous controls makes difficult the extension of the result to systems of high relative degree, introduces the possibility of intensive chat- tering, and cannot guarantee existence of classical solution.

In this paper we consider analysis and design of I/O robust control for time-varying systems. We first investigate how to combine MRC of time-varying systems with robust control design of Qu et al. (1994). Although the MRC structure in Tsakalis and Ioannou (1989a) achieves exact model match- ing, it will be shown in this paper that the structure in Tsakalis and Ioannou (1989a) is not adequate for robust control design of Qu et al. (1994). This leads us to introduce a new MRC structure which will be integrated into a recursive mapping proce- dure to generate a robust controller. The controller is shown to have several advantages over MRAC and VSC schemes. The advantages include continu- ity of control law, conceptual simplicity, easier im- plementation, guaranteed robust stability and per- formance under nonlinear disturbances, capable of handling arbitrarily fast time-varying parameters, and straightforward stability proof. These proper- ties indicate that the proposed robust control gives for the first time a complete solution to output tracking of fast time-varying systems.

This paper is organized as follows: Section 2 con- tains formulation of the output tracking problem. Perfect tracking of linear time-varying systems un- der perfect knowledge is discussed in Section 3, in which the existing MRC structure is briefly reviewed and a new MRC structure is then proposed. In Sec- tion 4, examples are discussed to show intuitively why robust control along the line of Qu et al. (1994) works better for time-varying systems than the ex- isting techniques. In Section 5, an input-output ro- bust controller and its design procedure is devel- oped.

2. PROBLEM FORMULATION

To simplify our discussion, we shall restrict our attention in this paper to single-input single-output systems whose dynamics is described by the follow- ing input-output differential equation:

A,($, t)[y(t)l = b,o(t&h, t)[u(t) + d(y, t)l

5 &Jw$J(s, t)[u(t) + d(y, t)l, (1)

where A, (s, t) and BP (s, t) are linear time-varying (LTV), manic polynomial differential operators (PDOs for which rules and properties are given in Solodov (1966)) given by

A,(s, t) = J” + a,~(tW’-’ + . . . + a,,,(t),

B,,(s, t) = P + b,&)d”-’ + . . . + b,,&),

Robust control of time-varying systems 535

m I n - 1, the symbol s denotes the differential op- erator d/dt, and d(y, t) denotes lumped sum of non- linear time-varying uncertainties or disturbances that can be bounded by a function of the system output.

With respect to the model of TV plants, we make the following assumptions:

Assumption 1. The indices n and m are constant and exactly known, and the LTV PDOs A,, (s, t 1 and B,,(s, t) are right coprime (Tsakalis and Ioannou,

1989a). Moreover, the sign of k,,(t) 2 b,,o(t) is fixed and assumed without loss of any generality to be positive.

Assumption 2. The parameters of the system, api and b,Jt), 1 I i I n and 0 I j I m are continu- ously differentiable up to (max(5n - 3m - 5,6n - 3m - 7))th order. The parameters may not be known and may drift arbitrarily fast, but the values of the parameters and their derivatives of neces- sarily high order belong to known compact sets in ‘x. For mathematical convenience, compact sets can be enlarged to be polygons, that is, there are

known constants +,i, api, $,j, spj, gik, &ik, gjk,

and z,,+ such that, for all 1 I i I n, 0 I j I m, 1 5 k I max(5n - 3m - 5,6n - 3m - 7}, and for all t I 0

where & > 0. The upper and lower bounds do not have to be constant.

Assumption 3. The disturbance d(y, t) is bounded in size as, for all (y, t)

Mb, t) II 5 P(Y, t),

where II - II denotes the standard Euclidean norm (Stewart, 1973), and function p(y, t) is known and well-defined in the sense that it is uniformly bounded with respect to time and locally uniformly bounded with respect to y and/or its stable filtered versions.

Assumption 4. In the presence of disturbance d(y, t), the system (1) under a continuous, bounded control has a classical solution.

Assumption 5. The PDO B,,(s, t) is exponentially stable with rate no larger than -YE for some JJB > 0 (Tsakalis and Ioannou, 1989a).

Remark 1. In Assumption 3, admissible uncertain- ties or disturbances are those that, although they

may depend on the state of the system, can be bounded by a function depending only on the sys- tem output. For example, consider

d(y, t) = d,,(t) + e-dl(r--T)(‘--T)

Xb.h(-r)y(T) + ddT)y2(T)

+codd4(~)j(~)lldt +ds(t) sin[J2(t)]y’(t),

where Idi(t)l I 1 for i = 0,2,3,4, 5 and dl (t) 2 2. Although the uncertainty is a function of y, p and j;, it is admissible since its bounding function can be chosen to be

f

p(y, t) = 1 + I

e++[l + [y(T)1

10

+y2Wdt + ly(t)l".

In this paper, we do not consider the case that the bounding function depends on, in addition to the output, the state of the system. It is possible to apply the proposed method to achieve semi-global stabil- ity results by incorporating into the robust control framework an observer to estimate the state of the system using the input-output data.

Remark 2. The bounding function p(x, t) is usually obtained by taking the Euclidean norm. Since it is not unique, it follows from the inequality 2 I Q) IT I I 1 + (p2T that p(y, t) can be assumed without loss of any generality to be differentiable at least once with respect to its arguments.

The control objective is to find a continuous- feedback controller to guarantee output tracking under nonlinear uncertainties, while the controller requires only input and output measurements. The performance for output tracking is required to be, at least, uniformly ultimately bounded stability, which can be made arbitrarily close to asymptotic stability or exponential attraction. Because the system under consideration is fast time varying and contains non- linear uncertainties, the prevailing approach is ro- bust control methodology. In Qu et al. (1994), a ro- bust controller has been designed for time-invariant and some simple time-varying systems. We intend to extend the previous discussions of robust control design in Qu et al. (1994) to general time-varying systems in the form of (1).

In order to specify the transient performance of output tracking, the model-reference con- trol (MRC) formulation is integrated into the Lyapunov-based robust control design. That is, robust control is to make the output of system (1) track the output of a reference model under any

536 2. Qu et al.

given uniformly bounded reference signal r(t), in the presence of significant uncertainty d(y, t) and under fast drifting of system parameters. The refer- ence model is chosen to be a time-invariant linear system described by the transfer function

Ym(s) = b &.Pm + bmlPm-l + - - . + b,,,,,,m

R(s) Pm + a,ls”m-l+ - - . + am,

!ik - Bmh) i G ts) “A,(S) m ’

where r(t) represents a continuous and uniformly bounded reference input signal. It is assumed that both &(s) and B,,,(s) be coprime and Hurwitz polynomials (Kailath, 1980). It is well-known from linear system theory that perfect tracking is feasible only if n, - m, r n - m. Assume that n, - m,,, = n-m does not lose any generality since, if otherwise, Gm(s) in (2) can always be obtained by multiplying the reference signal by a proper and stable transfer function. In the following discussion, it is assumed for simplicity that n, = n - m and m, = 0, that is, Bm(s) = 1.

In the next section, we begin our robust control design with model matching under perfect knowl- edge. The existing results on model reference con- trol of known, linear time-varying systems will first be reviewed, that is, the nonstandard structure pro- posed in Tsakalis and Ioannou (1989a). Then, a modification on the nonstandard controller struc- ture used in Tsakalis and Ioannou (1989a) will be made in order to obtain a new controller structure. The new controller structure will be used in Section 5 to give a complete solution to robust control of time-varying systems.

3. MRC OF LTV SYSTEMS WITH PERFECT KNOWLEDGE

In this section, it is assumed that perfect knowl- edge of the LTV plant be available and that there be no uncertainty (that is, d(y, t) = 0 for all y and t). This is based on the intuition that the MRRC problem for unknown systems is not solvable un- less the MRC problem is first solved under perfect knowledge. Under perfect knowledge, it has been shown that the standard MRC structure in Landau (1979), Narendra and Annaswamy (1989), Sastry and Bodson (1989), Slotine (199 1) is sufficient for linear time-invariant systems. It was recently real- ized in Tsakalis and Ioannou (1987), and the refer- ences cited therein, that the standard MRC struc- ture is not adequate for LTV systems. It has been shown in Tsakalis and Ioannou (1989a) that a non- standard MRC structure, shown in Fig. 1, is ca- pable of achieving an exact matching between the closed-loop I/O description and the desired trans- fer function.

The difference between the standard MRC struc- ture and the MRC structure in Fig. 1 is that aux- iliary signals employed are defined differently. As shown in Fig. 1, the two auxiliary signals w1 w2 are defined by

and

31 = AOWl + &(t)u, $2 = AOW2 + Q,(t)y, (3)

where WI, w2 E %(“-I) are auxiliary sub-state vec- tors, A0 E %(n-‘)x(n-l) is a constant stable ma-

trix with det[sZ - A01 4 P(s). The output matrix Bo E XQ(n-i) is a constant vector such that the pair (Ao, Bo) is observable. The MRC law proposed in Tsakalis and Ioannou (1987) is given by

u(t) = k(t)r(t) + Q,(t)y + B;w, + B;w2, (4)

where k(t), 80(t) E % together with &(t), e,(t) E #(“-l) are time-varying parameters in the con- troller. The standard MRC structure is defined similarly except that 81(t) and &(t) in (3) are exchanged with their corresponding BO in (4), respectively.

The result in Tsakalis and Ioannou (1989a) on the problem of exact I/O operator matching under the MRC system in Fig. 1 can be summarized by the following lemma.

Lemma 1. (Tsakalis and Ioannou, 1989a) Con- sider the system represented by (1) whose pa- rameters are known and satisfy Assumptions 1, 2, and 5. Then, there exist bounded parameters e,*(t), Of(t), O:(t), k*(t) for control (4) so that the closed-loop system defined by Fig. 1 is internally stable and its I/O operator from r(t) to y is the same as the transfer function Gm(s) of the refer- ence model (2). Moreover, the control parameters are the solutions of the following PDO equations:

Nl (s, t) = Z’(s) - Wh t)B,b, t),(5)

N3h t)k,-‘(tM,h t)

-Nzh t) = J’(sMtM,(s)lk,, (6)

where adj(R) denotes the adjoint matrix of an in- vertible matrix R, and

N1 (s, t) = B;fadj(sZ - Ao)& (t),

iv2b, t) = Bladj(sZ - A0W2(t) + P(s)80(t).

The step-by-step procedure of solving control parameters is the following: let k*(t) = km/k,,(t); solve for manic PDOs A$(s, t) and nT3(s, t) from equation (6); determine manic PDO N1 (s, t) from equation (5); find control parameters from the definitions of NI (s, t) and N2(4 t). It was shown in Tsakalis and Ioannou (1989a) that the parame- ters in (5) and (6) are differentiated at most up to (max{2n - 2,3n - 4)) times.

Robust control of time-varying systems 537

Fig. 1. The MRC structure for known LTV systems proposed by Tsakalis and Ioannou (1989a).

Although the structure in Fig. 1 gives a complete solution to the MRC problem of LTV systems un- der perfect knowledge, it will be shown in Section 5 that the structure is not adequate for designing output-feedback robust control if the relative de- gree between A(s, t) and B(s, t) is greater than one. Such a difficulty is caused by the fact that PDOs are generally not commutative. Inspired by the MRC structure in Fig. 1 which is improved for LTV sys- tems from the standard MRC structure, we propose a new MRC structure, shown in Fig. 2, in which the auxiliary signals are redefined to be

1 tii = AOWl + 0((s)& (r)-u

0((s) ’ (7)

where (x(s) is any manic Hurwitz polynomial of order p with 0 I p I n - m (it will be shown in Section 5 that the index p should be chosen to be n - m - l), WI, wz E li(fl-1+2P) are modified aux- iliary state vectors, A0 E li(n-‘+2”)x(n-‘+2p) with det[sl- ,401 = P(s) for a Hurwitz polynomial P(s) of order (n - 1 + 2p), and Be E #(n-1+2p) is chosen such that (Ao, Bo) is observable. Similar to the dis- cussions in Tsakalis and Ioannou (1989a), it can be shown that the modified MRC structure in Fig. 2 also represents a solution to the exact matching problem of I/O description between the closed-loop plant and the reference system. Such a result is summarized by the following lemma whose proof is conceptually the same as that of Lemma 1 and therefore omitted.

Lemma 2. Consider the system represented by (1) whose parameters are known and satisfy Assump- tions 1, 2, and 5. Let o((s) be an arbitrary but given Hurwitz polynomial of s. In the closed sys- tem shown in Fig. 2, the control is

+@w, + lgw2, (9)

where w](t) and we are auxiliary signals given in (7) and (8), and k(t), 80(t) E R, B,(t), e,(t) E R(n-1+2p), are the time-varying parameters in the controller. Then, there exist bounded parameters e,*(t), Q,* (t), Q,*(t), k* (t) so that the closed-loop system under control (9) is internally stable and its I/O operator for r(t) - y is the same as the transfer function G,,,(S) of the reference model (2). More- over, the parameters in the controller are the solu- tions of the following PDO equations:

iv1 6, t) = P(s)&) - N3h t)BJx t)a(s), (10)

N3h Ok,-‘(t)A,(S, t)oc(s) - N2(s, t)

= P(s)a(s)k(t)A,(s)/k,~, (11)

where

NI (s, t) = @adj(sl- Ao)oc(s)BI (0,

N(s, t) = @adj(sl- A&(s)B2(t)

+mhmeOw.

The parameters of the controller in Fig. 2 can be determined by the same procedure as that follow- ing Lemma 1. Using the analysis in Tsakalis and Ioannou (1989a), one can show that (10) and (11) may contain the time derivatives of the parameters of, at most, up to (max{ 2n + 3p - 2,3n + 3p - 4)) order. It is obvious that the new structure in Fig. 2 includes that in Fig. 1 as a special case with p = 0.

As will become clear in the subsequent discus- sions, the MRC structure in Fig. 2 is suitable to output-feedback robust control design. The main idea in arriving at the new MRC structure is that, using the new structure, we can define the filtered

538 Z. Qu et al.

Fig. 2. A new MRC structure for known LTV systems for time-varying systems.

versions of the auxiliary signals, the output, and the input as

1 yq = -

1

a(s) w1t E2 = -

a(s) w2,

1 UC-u

1

odd ’ El = -

a(s) w’J

and these filtered signals are related by (7) and (8), that is

$1 = A(-Ji& + 81 (t)ii, e* = A0il72 + &(t)jT

The relationship that WI depends on a filtered ver- sion of ti is particularly important since it guaran- tees that an output-feedback robust control can be designed, that it is bounded and well-defined, and that it ensures internal stability of the overal non- linear system. Note that the MRC structure in Fig. 2 requires higher-order time derivatives of the plant parameters than those in Fig. 1. Also, auxiliary sig- nals ~1 and w2 are of higher dimension. These ex- tensions are necessary and they do not cause any more complication in robust control design (in Sec- tion 5) than the MRRC for time-invariant systems. This is because only the bounds on the system pa- rameters are needed and because the vector auxil- iary signals can also be bounded in Euclidean norm by some new scalar (thus dimension-independent) auxiliary signals.

The discussions in this paper are based on the I./O operator (PDO. or transfer function) method which inherently assumes zero initial conditions for all internal states of the system. This treatment is valid despite the fact that the plant is time-varying and that the overall system is nonlinear (due to the presence of d (y, t) and to the design of a nonlinear robust controller). This conclusion can be seen by the following two observations:

First, as in the LTI case (Landau, 1979; Naren- dra and Annaswamy, 1989; Sastry and Godson,

1989; Slotine, 199 l), cancellation of stable PDOs is validated by the following lemma proven for time- varying systems in Tsakalis and Ioannou (1989a):

Lemma 3. (Tsakalis and Ioannou, 1989a) Let US, t) be an exponential stable PDO with rate -BL < 0. Consider the following two systems with I/O pairs (y,, u) and (~2, U) satisfying:

Us, t)y1 = J% t)u; y7. = US, tlz, us, t)z = 24,

where x2 is an intermediate variable. Then, the dif- ference between y2 - u and y1 - u decays exponen- tially to zero with rate at least -fin.

Second, although the overall system is nonlinear, the nonlinearities are isolated as inputs u(t) and d(y, t) into the linear part, and they depend not on the internal state of the system but only on the in- put and output data. As has been shown in Qu et al. (1994), Qu (1993a) and later in Section 5, the same perfect control, to be denoted by U * , can be added and subtracted, and this manipulation of adding zero can map the linear part into a globally stable linear part with remaining inputs u( t 1, - U * , and d (y, t). These inputs are again isolated away from the stable linear part. As a result, nonzero initial conditions in the linear part (neglected when I/O operator is used) only contribute to the system out- put and the solution of the state is an additive term that exponentially decays to zero. It has been shown in Qu et al. (1994), Qu (1993a) that, in a Lyapunov argument, the same type of stability result can be concluded whether or not there is an exponentially decaying, additive term in the solution of the state. Therefore, we need only develop robust control and robust stability results in the following sections for zero initial conditions.

For notational briefness in the subsequent dis- cussions, let

Robust control of time-varying systems

e*(t) = [ k* 0) 0,*(t) Es:’ WIT Fe; (tHT]T,

539

As will be shown later, 0((s) = 1 should be cho- sen for this system (since n - m - 1 = 0). It fol- lows from (9), (lo), and (11) that the following con- trol is a perfect nominal tracking control (that is, lim,,, (y, - y) = 0 under u*):

w* 0) = [r(t) y(t) [$(t)lT [w;(t)lTIT, (12)

where the asterisk is used to denote the solution(s) under perfect knowledge, 8* E (112n+4p is the vector containing all control parameters, and w* E #2n+4p the vector of all signals including reference, auxil- iary and output signals. Therefore, the control laws (4) and (9) can be written in a compact form as

u(t) = u*(e*(t),~*(tn (13)

It is obvious that control (13) is (9) and, if p = 0, it reduces to (4). Then, it follows from the above discussions that, whenever U* (8* (t), w*(t)) is known, the control u(t) = U*(8* (t), w*(t)) guar- antees y - y, as t - OQ for any r(t) under the condition that d(y, t) = 0. The question now is what robust control should be used if the plant is unknown and/or if d(y, t) # 0. The solution to this question will be provided in Section 5, which is based on the new MRC structure in Fig. 2. Also in Section 5, we shall show why the MRC structure in Fig. 1 is not sufficient for Lyapunov-based robust control design.

Before proceeding with mathematical develop- ment of the MRRC for time-varying systems, we choose to devote the next section to intuitive ex- planations through two simple examples on what forms of robust control laws are to be used and why these robust controllers work.

4. WHY DOES ROBUST CONTROL WORK?

Let us introduce an important notation for sub- sequent analysis. Let IpI or llpll denote the magni- tude of p, depending whether p is a scalar or vec- tor. Let II lpll I represent an upper (known) bound on the magnitude of p whenever p is unknown.

To illustrate the basic idea of how the proposed input-output robust control works, consider the following two examples. They are second-order sys- tems of relative degree one and two, respectively. Again, for simplicity, we assume that d(y, t) be zero in this section.

Example 1. A second-order LTV system

j; + al (t)j + az(t)y = ti + 61 (t)u, (14)

where al(t), a*(t), 61 (t) > 0 are time-varying pa- rameters. The control objective is to make the plant output y track the output of the reference model

J&1 + 2ym = r,

where r is any given bounded and continuous refer- ence input. Note that the reference model is strictly positive real (Narendra and Annaswamy, 1989).

u*(t) = r(t) + e,*y+ w1 + w2,

where

til = -wl + e+, ti2 = -w2 + e;y,

and 0; are given by

e:(t) = i -b,(t), e&t) = al(t) - 3,

e;(t) = a2(t) - 2 - e;(t) - b,*(t).

Since the perfect knowledge of the system parame- ters is not available, we cannot let u = u* (t). How- ever, the actual control u can always be rewritten as

U= (u- e,*y- w1 - w2) +e,*y+wl+w2

=[u+ r(t)-u*]+ ely+ wI+ w2.

Since u*(t) is the perfect control and since the term [u + r - u* ] can be viewed as a dummy reference input, the system dynamics can be mapped into

3 + 2y = u + r(t) - u*(t).

Defining the error signal e(t) = y,,(t) - y(t) yields

t+2e= -u+u*(t). (15)

Uncertainty u* (t) can easily be bounded (since 07 can be bounded using the bounds on the system pa- rameters), and it satisfies the matching conditions (Corless and Leitmann, 1981). So, robust control u can be designed to compensate for u* (t) .

As will be shown later, the proposed MRRC for this example has the following form:

em Illu*U)llI u(t) = lel . II,u*~t~lll + Ee-Bt . lllu*(t)llL (16)

where E > 0, fl 1 0 are design constants. As will be shown in the next section, the above control can compensate for uncertainty u*(t) and is continu- ous and bounded. It is worth pointing out that the bounding function II Iu* (t) ]I ( on the size of u* (t) is a function of w1 which is a filtered version of u. This means that control (16) depends implicitly on itself and, in the presence of the self dependence, a com- plete stability proof must show explicitly bounded- ness of control (16).

As was shown in Qu et al. (1994) the control (16) with /3 > 0 has the capability of implicitly learning uncertainty, and consequently the resulting perfor- mance is exponential stability. It should be noted that control (16) can theoretically compensate for

540 Z. Qu et al.

any bounded uncertainty. Therefore, the smooth- ness of the control depends on the uncertainty be- ing compensated for. For the same reason the con- trol may be sensitive to noises and become chatter- ing. By trading-off tracking accuracy, the problem of chattering can be eliminated by simply choosing fi = 0 and E not too small.

To illustrate how a robust control in the form of (16) compensates for time-varying functions, we choose to simulate system (15) under (16) (simula- tion of MRRC for system (14) will be done in the next section

d . Figure 3 shows the simulation using

SIMNON c in which the control parameters are E = 1, j? = 0.1, and the bounding function is

Illu*(t)llI = 1 +2leWI

+ e-(‘+[ le(T) 1 + IU(T) l]dT.

10

The ‘uncertainty’ is chosen to be

u* 0) = 0.5 sin t + e(t) cos2t

+ 1 e-w) I

[e(T) COS(3-r)

to

+ Sin(3T)u(T)]dT.

Example 2. Consider LTV system

j + al (t)j + az(t)y = u,

where al (t ) and a2 (t) are time-varying parameters. The plant output y is required to track the output of the reference model

j;, + 3jm + 2y,, = r.

It will be shown later that polynomial 0((s) should be of first order (since n - m - 1 = 1) in order to find a well-defined I/O robust control for this example. However, to reveal basic ideas, we shall consider here a(s) = 1. It follows from (9), (lo), and (11) that the perfect nominal tracking control is

u*(t) = r(t) + O,*y + WI + ~2,

where

ti, = -WI + e:zJ, ti2 = -w2 + e:y,

and 0: are given by

02 (t) = al(t) - 3,

eo* (t) = -5 + al(t) + cil (t) + a2(t) - 0: (th(t),

e: (1) = az(t) + Li2(t) - er(t)a2(t)

-e,* (t) - b,*(t) - 2.

Without perfect knowledge of the system, we can- not set u = u* (t). However, by employing the same

technique of rewriting the actual control as used in Example 1, we can map the system into

j + 3j + 2y = u + r(t) - u*(t),

and then into

Z+3i+2e= -u+u*(t). (17)

As before, uncertainty u*(r) can be bounded by a function of y, WI and ~2.

Unlike Example 1, an output-feedback robust control cannot be designed directly for error dy- namics (17). This is because, although the uncer- tainty is matched (Corless and Leitmann, 198 l), the reference model as well as error system (17) is not strictly positive real. That is, a robust control de- signed using the standard state space robust control theory will require measurement of both y and j. To get around this problem, define

1 U=-u s+ 1.5 .

Using the intermediate variable ii, the error system can be rewritten to be of two cascaded subsystems as

t+3t+2e= -t- 1.5ii+u*(t),

ti=-1.5u+u,

both of which are strictly positive real. Using the strictly positive real property, output-feedback ro- bust control can be designed for both of the two subsystems, and the cascaded structure can be used to generate the overall control u( t ) in a similar fash- ion to those in Qu and Dawson (1995, 1994), Qu (1993b). The trade-off is that the uncertainty u* (t ) is no longer matched; instead it satisfies the gener- alized matching conditions (Qu, 1993b).

Asymptotic or exponential stability is not achiev- able in general for systems with uncertainties satis- fying only the generalized matching conditions and of relative degree greater than one. For the con- ditions under which exponential stability can be achieved, refer to the discussions in Qu (1993b), Qu and Dawson (1995) for general nonlinear uncer- tain systems. As shown in Qu and Dawson (1995, 1994), Qu (1993b), the loss of the matching condi- tions for the uncertainty often implies the loss of asymptotic stability or exponential stability. An in- tuitive explanation is the following: if Z would be freely chosen, the intermediate control variable ‘ii could be designed in a similar form as that in (16). As explained in Example 1, a robust control in the form of (16) can compensate any continuous un- certainty. There is no limitation on the uncertainty u*(t) except its size bounding function )I Iu* (t) II I. Inside any given size bounding function, the uncer- tainty may vary continuously but arbitrarily fast;

Robust control of time-varying systems

Fig. 3. Illustration of robust control performance.

this implies that a robust control in the form of (16) has the capability to change arbitrarily fast as well in order to match up with the change in the uncertainty. Consequently, the time derivative of il cannot be bounded a priori; since u = t + 1.5% ex- ponential stability may not be achieved under any bounded control for plants of high relative degree. For this reason, exponential stability for plants of relative degree greater than one will not be pursued in this paper. The control objective for these sys- tems is to make the tracking error smaller than any given (nonzero) accuracy requirement.

The explicit form of output-feedback robust con- trol and the simulation results for this example will be presented in the next section.

With the basic ideas of robust control illustrated by the above examples, we can present mathemat- ical development of the MRRC design. Note that the MRRC design for time-varying systems is con- ceptually identical to that for time-invariant sys- tems, which shows the superior capability of the MRRC scheme.

5. I/O ROBUST TRACKING CONTROL OF TV SYSTEMS

When the parameters of the plant are unknown, we propose to follow the basic idea exposed in Sec- tion 4 and design a robust control which guarantees stability and performance even under nonlinear un- certainty d(y, t). We shall investigate robust control design based on Lyapunov’s direct method and on the new MRC structure in Fig. 2, from which insuf- ficiency of the MRC structure in Fig. 1 for robust control design becomes obvious.

541

Due to the existence of uncertainty d(y, t), the total input to the plant is

U(t) = u(t) + d(y, t).

Thus, the auxiliary signals have to be modified to be

ti; = Aow;” + or(s# (t) --& + d(y, t)l, (18)

Ii; = Aow; + a(s)ez* (1) -&)y, (19)

where Aa and BO are the same as those in (7). Note that wr depends on uncertainty, therefore cannot be calculated, and should be bounded.

Despite the lack of perfect knowledge of the plant, it follows from (13) that, no matter what control ~0) is to be designed, the total input to the plant can be rewritten as

U(t) = U*(e* (t), w* 0)) + [u(t) + d(y, t)

-U*(e*(t), w*(t))1

= &)k* (t) 1 1

-r(t) + a((s)0,* (1) -y 0((s) (Y(s)

+B;w: + B;w,*

+Mt) + d(y, t) - u*(e* (r), w*(t))]

= cuwt* (t) & 1 r(t)

+ [ cds)k* 0) -& 1

-1 [u(t) + d(y, f)

- u*(e*(t), w*(t))] + ds)gw--&~ 1 +B;w;c + B;w;.

(20)

542 Z. Qu et al.

Since ideal parameter and signal vectors 8* (t) and w* (t) are determined based on Lemma 2 and since the term

I [ 1 1 -1

r(t) + a(s)k* (t) - 0((s)

[u(t) + d(y, t)

- u*(e*(t), w*m]

can be viewed as a dummy ‘total reference input’, the plant output under control (20) can be rewritten as

a(s)k*(t)-$ I -1

[u(t)

- u*(e*(t),w*w) +d(y,t)l , 1

(21)

where k* (t) = k,,,/k,,(t), and a(s) will soon becho- sen to be a manic polynomial of (n-m - 1) th order. Let us define the output tracking error e(t) to be

e(t) = y,(t) -v(t),

where y,n(t) = G,(s)r(t). Note that

aWk*W-& I[

1 1 ’ a(s)-- = k*(t) ~4s) 1 1.

It then follows from Lemma 3 and the discussions following that the dynamics of the output tracking error e(t) can be rewritten to be

e(t) = Gm(s) [

1 1 o(b)--

k* 0) o(b) I x[-u(t) + u*te*w, w*(t)) - d(y,

1 = 77,” (s) -

[

1

k*(t) ---u(t) + a@, u, t)

0((s)

= l&(s) j-& [-ii(t) + ;I& u. t,] I

where

(22)

Z(Y> u, t) = --+‘(e*(r). w*(t)) - d(y, t)l.

The transfer function &(s) in (22) is defined by

B,(s) &Cd = G,,(s)a(s) A kmo, m

(23)

where a(s) is chosen from now on to be a manic and Hurwitz polynomial in s of degree n-m - 1. In order to design a robust controller requiring only input and output data, it is necessary to have c,,,(s) be a strictly positive real (SPR) transfer function. The conditions for a transfer function of relative degree one to be SPR can be found on page 64 of Narendra and Annaswamy (1989). If n - m - 1 = 0, it is assumed that Gm(s) be SPR. If the relative degree of G,n(s) is larger than one, it is assumed that a(s) can be chosen such that ??,,, and is SPR.

This assumption is made based on the well-known fact that, if c,,,(s) is stable, minimum phase and of relative degree one but not SPR, a SPR transfer function can be generated for the reference model with a(s) by filtering the reference input.

The term a(~, U, t) in (22) stands for the total uncertainty in the closed-loop system, and can be bounded as follows: assuming w: (0) = w;(O) = r(0) = 0 without loss of any generality

lacy, u, t) 1 = Ik* W-&(l) + e~w--&

+h B&J: 0)

1 +-

a(s) E$w2*(t) - -&d(y, t)

I IIIk*(t)III * IW)l + IIle,*wllI * IFWI

+ ’ hdt - T) - IIle:(T)II I 1 kc’(T) IdT

+ ’ hdt - T) - IIle;(T)II I to

1 IY(T) IdT

+ [h&t-T)+hl(t-T) I IO

Xiiiqwiii - P~),T)~T

b II I&, u. t) II I,

where h, (t 1, h, (t ) 1 0 are stable impulse responses defined as follows. Function h,(t) has almost the same expression as the inverse Laplace transform of l/a(s) except that all trigonometric functions are removed and that some (or all) of its coefficients are made positive in order to ensure h,(t) 2 0. Func- tion h,(t) is defined in the same way as h,(s) but is based on llB;feAo’ (I. Function hi(t) 2 0 is the con- volution of h, (t ) and h,( t ) . All above convolutions can be implemented easily by defining several scalar auxiliary signals.

As will be shown later, if n - m - 1 > 0, the bounding function 11 la(y, U, t) II I is required to be differentiable. In this case, we can choose

Ill&u,t)llI = 1 + $IIlk*(t)ll12 - IM12 L

+~llIe;(T)ll12 .i%) t

+ ho(f - T)iiie:(T)IlI I to

+ * hdt - T) - iiie,*(T)~~ I 10

Robust control of time-varying systems 543

+ s [h,(t - T) + hl(t - T)

X 111 e;r (T) IIll . P(J’(T). T)dT.

By definition, h,(t), h,(t), and hi(t) are dif- ferentiable. It is worth emphasizing here that II la(v, U, t) II I only contains a filtered version of ii and that the time derivative of II @(y, U, t) 11 I does not contain u(t) but rather ii and its filtered ver- sion. This can be seen by noting that

$jh,,(t - T) - IIIe:(T)III . bdT)ldT lo

= ’ h20 - T) * IIle;(T)III . lU(-r)ldT, I lo

(24)

where h2(t) is the inverse Laplace transform of the product of s and the Laplace transform of h,(t). By induction, one can see that, after taking time derivative i times (no matter whether the operation of developing bounding function(s) takes place be- tween taking time derivatives), the resulting expres-

sion will depend on the filtered versions of &u( t ). Thus, so long as the number of differentiations is kept to no more than n - m - 1, there will be no direct dependence on u(t) but only on its filtered versions. It is this dependence that allows us to de- sign a well-posed nonlinear robust control and to prove boundedness of the robust control and inter- nal stability of the overall system.

Based on the error dynamics (22) we can proceed with our robust control design. The following anal- ysis is an extension of the results in Qu et al. (1994) to fast TV systems. We shall discuss separately two disjoint and complementary cases: n - m - 1 = 0 and n - m - 1 > 0. If n - m - 1 = 0, the PDOs A(s, t) and B(s, t) of the plant have relative degree one. In this case, o((s) = 1, and a simple design of robust control is given by the following theorem whose proof is included in Appendix A.

Theorem 1. Consider the system represented by (1) which satisfies Assumptions 1-5. Let the robust control be

ii(t) = /-de, y, 24 t)

Z+O (Me, y, w t) I + Et+) dy, u, t), (25)

where fl 2 0 and E > 0 are design constants, &,,s and b,,s > 0 are upper and lower bounds on b,,o(t) (that is, on k,,(t)) as defined in Assumption 3, and

g(JJ, U, t) = 2&711111aJ u, t)lll, Pky, u, t) = e(t)g(y, 24, t).

Then, the output tracking error e(t) of the plant converges to zero exponentially if fl > 0 or to a hyper-ball whose radius is a class X function (Khalil, 1991) of E if /I = 0. Moreover, the control is continuous and uniformly bounded (with respect to any given initial condition of the plant), and the overall system is internally stable.

The main advantage of control ii(t) is that it works for fast TV systems in which parameter drift- ing can be arbitrarily large but bounded. As shown in Example 1, there is a trade-off between tracking accuracy (exponential stability) and smoothness of the control in the presence of noises and computer round-off error.

Example 3. (Continuation of Example 1.) To ver- ify the theoretical analysis, we make the following choices for simulation purposes:

h(t) = 2 - sin(lOt), al(t) = cos(l5t),

a201 = sin(l5t), d(y, t) = sin(t) + ycos(t) - 0.5y2,

and zero initial conditions. The reference input is chosen to be r(t) = sin(2t). The robust control is implemented using

p(y,t) = l+y2, E=4, fl= 1,

II Ial I = 2 + Y2 + 2lyl

+ f e-w7)

I

[[U(T) I + )y(T) I + 2 + 2y2(T)ldT.

10

The simulation results are shown in Fig. 4 and they are consistent with the theoretical results. As dis- cussed in Section 4, the simulation also shows that, after the tracking error is forced to be sufficiently small, the robust control starts to respond to the round-off errors existing in numerical calculations using a computer.

It should be noted that the choice of bounding function 11 Iall I is not unique. It follows from the mean value theorem that, for some s E (to, t)

I

I e-(f-T)ly(T) Idr = ly(s) I - / e--(‘-r)dT

10 10

5 sup ly(-r)l. IOSTSf

Therefore, bounding function 111~11 I can be chosen (through enlargement) to be

II Ial I = 4 + Y2 + 2lYl + sup [IYWI + 2y2w1 I(J53Sl

544 2. Qu

If the control is implemented digitally, the convo- lution of u(t) can also be eliminated as for that of y(t). This means that, by redefining the bounding functions properly, explicit computation of auxil- iary signals is not needed for the proposed robust control.

The result in Theorem 1 does not apply to sys- tems with relative degree n-m > 1 since the control (25) guaranteeing exponential stability or stability of uniform ultimate boundedness under nonlinear uncertainties has to be implemented as

u(t) = oc(s)ii(t), (26)

which requires time derivatives of ii(t) and there- fore measurement of output derivatives of orders up to n - m - 1. The way to remove measurement of output derivatives is to redesign the robust con- troller as suggested in Section 4; that is, first make the uncertainty be generalized matched and then apply the recursive design procedure.

Tothisend,letI =n-m-I > Oandletzibethe variables of the new state in the controllable canoni- cal realization of the transfer function 1 /a(s). That is, if a(s) = s’ + o(Is’-’ + . . . + o(/, we define for i= 1,. . ., 1- 1 that ZI = Ii and ii = zi+l . The ac- tual robust control (26) can then be represented in terms of the state space equation as

il = z2,

ii = Zi+l, Vi=2;..,I-1,

ii = -0[1z1 - * * * - a/z1 + u(t),

u=z1. (27)

Using the new state variables zi, the state space re- alization (A.l) (in the proof of Theorem 1 in Ap- pendix A) can be rewritten as

f, = Ax, + B [-z, + &, u, t,] /k*(t),

e(t) = cx,. (29

Furthermore, it follows from definitions of zi and (22) that

- 1 i = sG,(s)~*(~) - [-u(t) + Z(y, 24, t)] 9

and,forvi= l;..,l-1

(29)

The system consisting of (27) and (28) satisfies the generalized matching conditions (Qu, 1993b), and thus the recursive design procedure can be applied. To yield an input-output robust controller, the rela- tions in (29) on time derivatives of the state variables are used to eliminate the first-order time derivatives

et al.

of the fictitious controllers required in the process of a recursive design.

With state space equations (27) (28) and (29) in hand, we can proceed with input-output robust design for TV systems of high relative degree. The design procedure is a straightforward application of Lyapunov’s second method, and the robust control is generated by the following recursive, nonlinear mappings:

v1 = ccl(e,y,u,t)t~l(e,y,u,~)t

25, (p:(+ y, u, f) + 6:) glk, .x f.4 0,

v2 = Vl - Zl

+~2(e,y,u,t)t~z(e,y,u,t)t

2 (j&e, y, u, 0 + ES) g2(e, y, u, t),

if1>2

Vi = Vi-2 - Zi-2 + Vi-1 - Zi-1

+Pi(e,J f4 t)lPi(e,_Y, U, t)l 2 (p:(e. y. u, 1) + 4)

gi(S Y, U, t),

VI+1 = o(lZl + . . -+oL~zl+v/_l-z/_l+v/-z~

+2(l~,+l(e,y,u,1)1+9+1) gr+lky, u, I),

(30)

wherei=3,.=*,l,Ej>O,j= 1;.*,1+1,are constants, and

gl(e,y,u,t) =2i;,0ttt~(y,u,t)ttt,

PI (e, y, u, t) = 4th (e, y, u, t),

gz(e,y.u,t)=$o+ tet2+2tttiltll.

P2kys u, t) = (VI - zlkz(e, y, u, th

gi(~,y,u,t)=2llI~i-lttt,

cli(e,y, 4 t) = (vi-i -zi-lki(e,y, 24, t),

gr+~~e,y,~,~~=~lll~~ltI~

ko+lky, u, f) = (VI - zr)gr+i(e,y, u, t).

It should be noted that every step in the recur- sive mapping basically involves finding a bounding function of 1 iit for obtaining vi+i. It starts with bounding l(y, u, t), which has been done in the de- velopment preceding Theorem 1. Note again that 11 \afy, u, t) 111 is a function not directly of zi = ti but only through a stable filter. The term J&l can be bounded by first developing bounds for the first- order partial derivatives of vi with respect to its vari- ables, and then by determining the bounds for the first-order time derivatives of its variables using the relations in (29). That is, 11 liill I can be found in a similar way to that of 11 la(y, u, t) II 1, differentiability can be guaranteed by properly choosing the bound- ing functions, and II I iti II I is an implicit function of u only through a filtered version of zi_l (the same chain of dependence has been used and discussed in more detail in Qu et al. (1994)). This implicit de-

Robust control of time-varying systems 545

Fig. 4. Simulation results of Example 1.

pendence is instrumental in the proof of Theorem 2 (to be stated below) as it was in the proof of The- orem 1.

We are now in a position to state the main result of this paper. The proof can be found in Appendix A.

Theorem 2. Consider the system represented by (1) which satisfies Assumptions l-5. Then, under the robust control u(t) given by

u(t) = V+1, (31)

where v/+1 is defined by the outcome of the map- ping procedure (30), the plant output tracking error e(t) is globally and uniformly ultimately bounded. More specifically, the magnitude of the output tracking error e(t) is bounded from above by an exponentially decaying time function, and, as time approaches infinity, becomes no larger than a design parameter E (whose value depends on the value of Ciz: Ei through a fixed class K func- tion). Furthermore, the robust control u(t) and all internal variables of the system are all uniformly continuous and globally, uniformly bounded.

The idea of designing robust control using a non-

linear mapping such as (30) was originally proposed in Qu and Dawson (1995, 1994), Qu (1993b). It is worth mentioning that designing control through a mapping procedure was also reported in adaptive control theory, for example, the backstepping pro- cedure reported in Kanellakppoulos et al. (1991) and the references therein. Although both mapping procedures are conceptually similar, the mapping (30) is somewhat more general mathematically since it is capable of handling the class of unknown but bounded nonlinear functions of the state and since it yields a stable controller u which is a nonlinear function of the filtered versions of itself.

Example 4. (Continuation of Example 2.) To ver- ify the theoretical analysis, we make the following choices for simulation purpose:

q(t) = cos(l5t), az(t) = sin(l5t),

d(y, t) = cos(t) + 0.5cos(y) + sin(t)y2,

and zero initial conditions. The reference input is chosen to be r(t) = sin(2t).

As explained in the preceding discussions, the ro- bust control design procedure is conceptually the same as that for linear time-invariant systems except

546 Z. Qu et al.

that MRC structure has to be modified for time- varying systems. Since the modified MRC structure leads to a lumped total uncertainty which will be bounded for robust control design, the most con- vincing way to show robustness and effectiveness of MRRC is to apply the MRRC result explic- itly selected for time-invariant systems to fast time- varying systems. Based on this reasoning, we ap- ply here the same robust controller selected in Qu et al. (1994) for a class of relative-degree-two time- invariant systems to the time-varying system. That is, the robust control u is implemented as follows:

U = o(lZ1 + vr - Zl

cl2k y, 4 t) +2(l~2k,Y9t4t)l +E2)

g2k.x 24 t),

v1 Cll(e,y,u,t)l~l(e,y,u,t)l

=

2 (pfky, 24 t) + 4)

glk, y, 24 0,

where p(y, t) = 2 +fl, o(1 = 10, ~1 = ~2 = 5.0 and the expressions of gr (e, y, U, t) and gz(e, y, u, t) can be found in Qu et al. (1994). The simulation results are shown in Fig. 5 in which the plots are consistent with theoretical results.

Before concluding the discussions in this section, let us look at why the MRC structure proposed by Tsakalis and Ioannou (1987) and shown in Fig. 1 is not directly applicable to Lyapunov-based robust control design. Note that any control u(t) can be rewritten as

u(t) = u,* (e* (t), w*(t))

+[u(t) - q+w*w, w*w)l, (32)

where U;” (0*, w*(t)) is given by (13) with p = 0 and obtained from Lemma 1 (and Fig. 1) if perfect knowledge of the plant is available. Since the vectors of ideal parameters and signals 0* (t) and w* 0) satisfy Lemma 1, the plant output under control (32) must be

1

1 y(t) = G(s) r(t) + k+.(tl -[u(t)

- qw*w, w*(t)) + d(y, t)l . (33) 1

It then follows from the definition of the output tracking error that

1 e(t) = G(s) k* (t) --I.-u(t)

+v:w*w, w*(t)) - d(y, t)l

= Gnw ---&+t) + ZY, u, t) , 1 (34)

where

ZY, u, t )

= -$&)ru~‘e*(t). w*(t)) - d(y, t)l.

As before, we can design an input-output robust controller if and only if the following two condi- tions hold: (1) the transfer function Em(s) is SPR; (2) the variable substitution E = K(t)&&u(t), for some k’ ( t ) bounded away from zero, can be used to generate a recursive mapping for the actual con- trol variable u(t); and (3) the recursive mapping is well-defined. The first condition is easy to satisfy. It is easy to verify that a nonlinear mapping similar to (30) exists if and only if i or &Z can be cal- culated by properly choosing known or unknown k’(t). Since the two PDOs $I and & do not commute, the second condition is satisfied if and only if one of the following conditions holds:

(i) k*(t) is time-invariant, which implies that k,,(t) = kp is a constant.

(ii) n - m = 1, in which case (x(s) = 1. (iii) kp(t) = k+(t), where c(t) is a known time

function. It follows from the discussions up to (24) that, if the MRC structure in Fig. 1 is used, Z in (24) and other bounding functions have to be replaced by U. This makes the recursive mapping not well-defined since the bounding functions in the latter steps of the recursive mapping will depend explicitly on the control input or even its derivatives. We then know that, if the MRC structure in Fig. 1 is used, the third condition does not even satisfy for case (iii). In summary, the MRC structure in Fig. 1 fails, after being combined with Lyapunov-based robust con- trol design, to be applicable to the general case of a plant of high relative degree with unknown or time-varying gain kp (t ) .

6. CONCLUSION

In this paper we consider the input-output ro- bust control problem of fast time-varying systems with possibly high-order nonlinear uncertainties. Inspired by the MRC structure for known LTV sys- tems proposed in Tsakalis and Ioannou (1987) we first propose a new MRC structure which is a minor modification of the existing one. We then integrate the new MRC structure into the recursive robust control design methodology in Qu (1993b), Qu et al. (1994) to generate a robust controller. The resulting control has several significant features: the control is continuous and uniformly bounded, guarantees existence of a classical solution for the system, re- quires only input-output measurement and ensures stability of uniform ultimate boundedness, which can be made arbitrarily close to asymptotic stabil- ity. The most fundamental difference between this result and the existing ones (most of them using an

Robust control of time-varying systems 547

Fig. 5. Simulation of Example 2.

adaptive control approach) is that a time-varying input-output model is used, the system may contain any uncertainty bounded by a high-order nonlinear function of system output and/or its filtered ver- sions, and the system parameters are not required to be either slow time-varying or fast time-varying of known structure (Tsakalis and Ioannou, 1989a). Another feature of MRRC is that the resulting ro- bust controller is simpler than the existing adaptive control schemes since its implementation does not involve any adaptation law and since explicit cal- culation of auxiliary signals can be avoided by em- ploying the mean value theorem (as shown in the second part of Example 1). The theoretical analy- sis as well as robust control design in this paper is merely a straightforward application of Lyapunov argument. Acknowledgements-Professor Zhihua Qu was supported in part by U.S. National Science Foundation under grant MSS- 9110034.

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APPENDIX A. PROOFS

Proof of Theorem 1. Suppose that the triple {A, R, C] is a minimal realization of the transfer function ??&); that is, the realization

xc = Ax,. +B [-ii(t) +&m,t)]/k*(t).

e(t) = Cx,, (A.1)

is controllable and observable and yields C(sl-A) -I B = -if&, (s). Since E,,,(s) is SPR, it follows from the Kalman-Yakubovich Lemma in Khalil (1991) that there exist symmetric positive definite (s.p.d.) matrices P and Q such that

ATPiPA= -Q, PR = CT.

Define V to be the Lyapunov function as

I’, = k,,,x;fPx,.

Taking its time derivative along the trajectory of the plant yields

km ri, = k,,,x;(PA + ATP)x, + 2x;fPRr k (t) [;s(Y# u. t) - ii(t)]

= -k,,,xTQxp + 2e(t)k,(t) [a(~, u, t) - R(t)]

I -kmxrQxf + 21e(t)l . Epo . [a(y, u, t) 1 - Zkr(t)e(t)u(t)

5 -k,x;fQxr + [ le(t)lg(v. u. t) - Xr(t)e(t)u(t)]

= -kmxTQxr + I&y, u, t)I - k& Me,xu,t)12 bfl Me, y, u, t) I + i3-B’

5 -kdQxe + Me, y, u, 0 I - Me,y, u, t)12

ItiCe, y. u, 0 I + ee+

= -k,,,x;Qx, + It&y, u, t)l

Me, y, u, t) I + ee-pr ee@

LinCQ) 5 --V, + ee-e’.

h,,(P)

Now, let us define

s,(t) = Vr + h,V, - eeeBf,

where h, 0 A,,,l”( Q)/h-(P). It follows from the inequality on I$ that s,(t) 5 0. Solving the above first-order differential equation yields

,

V,(t) = e-“r(‘-‘o)V,(to) + e-A’(r-T)[sr(t) + ce-pT]dr

i0

I

se -“r”-‘o)Vr(t~) f E I

e -h&-r) e-BTdT

10

Ep[e-8(r-rai _ e-h,(t-fo)]

=e -wfoty,(to) + if h, f B, .s(t - to)e-e’

if &. = B.

Therefore, if fi > 0, V, converges to zero exponentially, and so does the state x,; consequently, the output tracking error e(r) converges to zero exponentially. If g = 0, the ultimate bound on V,(m) is e/h, which is proportional to E; consequently, the state x, and tracking error e are uniformly ultimately bounded.

Although x, and e are shown to be bounded, internal sta- bility of the overall system cannot be established until u = ii is

Robust control of time-varying systems 549

shown to be uniformly bounded. The boundedness of u cannot be claimed directly from y being bounded since the relationship

(A.3

contains a self-dependence of u in terms of a filtered version of u. Fortunately, one can show that u is bounded in three steps as follows:

(I) Note that lll~(y,u,r)llI depends not directly on u but through a first-order filter (represented by h, (2)). It fol- lows from (A.2) that, if some filtered version of u through a first-order filter is bounded, function 1) I& u, t) II 1 is bounded and in turn so is the control u. Therefore, to conclude boundedness of u, we need only to show that some filtered version of u through a first-order filter is bounded.

(2) Note that the PDO B,(s, r) is exponentially stable. Recall from system dynamics that, for any constant c > 0

--&u(r) = c,;’ (3, l)A,,(S, t)y(t) - d(Y, 0. (A.3)

where C,(s, t) e k,,(r)B,,(s. r)(s + c) is also exponentially stable and of the same order as A,, b, 1).

(3) The right-hand side of (A.3) is bounded if y is bounded. It follows from the previous analysis that y is bounded. Therefore, we can conclude sequentially that Au, then 1) @(y, u, t) )I 1, and finally u are bounded.

This concludes our proof.

Proof of Theorem 2. Choose Lyapunov function V to be

V = Vp + i(Zi - vi)* = kmx;fPxe + i(Zi - Vi)2, i=l i=l

where V, is the same as that defined in the proof of Theorem 1. Taking its time derivative along the trajectory of the system given by (27) and (28) yields

P = k,x:(PA + ATP)x, + 2x;PB $)[-a +;i(Y,uA]

I

+ 12(Z, - Vj)(ii - ii) i=l

= -k,xTQxr + Ze(t)k,,(t) [-VI + $(y, u, r)]

-2e(t)k,,(t) [ZI - VI 1

I-l

+ C2(z; - Y;)czj+] - dj) i=l

+2(z) - by)(&) - o(IZ/ - . . . - ct/Zl - G/)

= -kmx;fQxv + 2e(t)k,(t) [-VI + ;i(y, u, t)] + 2(z1 - ~1)

x(V2 - k,,(r)e(t) - C)

I-I

+ 12(Zi - Vj)(Vi+l + Zi-l - Vi-1 - tii) i=2

+2(z/ - V/)(U - DllZ/ - . . * - (x/z1 + Z[-1 - V/-l - G/).

It follows that

Wt)k,,(t) [-VI + si(y, u, t)] 5 ICCI~Y, u, t)l

If4(e,y,u,t)13

-Iplk,.Wt)l2+E:

114 (e, Y, u, t) 16:

= I/.oky,u,t)l*+E~

5 El.

since, if IPt(e,y,u,t)l >_ et

and if IPI (e, y. I(. t) I 5 EI

IPI (CY, w C) IEI 25

lbf~(e,y, u. t)k~

I~I (e, y. u. t) I2 + El 4

= ICclk.y,u,~)l < ,,

El -

By repeating the above derivation, we can show that, for i E {2,. . ..I-- II

2(z1 - VI) (~2 - k,,(t)e(t) - $1) 5 -2(z1 - ~1)~ + ~2.

2(Zi - Vi)(Vi+l + Z/-l - Vi-1 - ii) 5 -2(Zj- Vi)2 + Ci+l,

2(Z/ - vr)(u(t) - fflZ/ - . - . - a/z1 + Z/-I - v,-, - i/j

5 -2(Z/ - v,)2 + E/+1.

Therefore, we have

I /+I

P 5 -k,X;TQxc - 2 x(Zi - vi)* + 1 pi 5 AV + h&ce2. i=l j=l

where C is the output matrix defined in (A.l)

n = minkJti,, (Q). 21 maxIk,,,h,,(P), 1)’

A = max(k,&n(P), 11,

Now, let

S(l) = P + AV - A?&.

It follows that s(t) L 0. Solving the above first-order differential

equation yields

V(t) = e- “(‘-‘~)V(to) + e+(‘+(s(t) + h@*)dT

Ill

-c e-h(r-tO)V(~O) + Ace2 - e-x(‘-T)AdT

10

=e -Akto) V(l()) + &$(, _ e-A(w))

- hc2. ast--

Therefore, V is uniformly ultimately bounded. Therefore, the state xr and the output tracking error e(r) are globally and uniformly ultimately bounded. The convergence of V(r) is ob- viously exponential, and so is the convergence of e(t). More- over, it is easy to show that

lim sup Ilxrl12 5 ce* ‘-Oa +>_,

lim sup (IelI 5 E. ‘-- T,,

The above argument also shows that zi - vi for i = I, . . . , I are all bounded. To show the system is internally stable, we must prove that zr, vi, and u are all bounded. This can be done inductively as follows. First, note from (31) and (30) that

IVI I 5 gl ky, u. t),

and that gt (e, y, II. t) is an implicit function u only through zt passing a first-order, stable filter. Second, note that, for constant c>o

-&ZI U) = I

(s + c) a(s) u(r) = C;’ (3, t)A,>(s. t)j4r) - d(y, t),

550 Z. Qu et al.

where C&, t) 4 k,(r)B,(s. t)(s + c)cr(s) is exponentially stable vi - .ri is uniformly bounded and that vi is bounded in terms and of the same order as A&t). We know from y being of gi(e, Y. U, t), Zj and vj with j < i, one can argue inductively uniformly bounded that gt (e, J u, t) and in turn vt and zt are that the intermediate control and state variables vi(t) and Zi, uniformly bounded. Now, recall that gi(e, 3 u, i) is an implicit j= I,... ,I, and consequently the actual control u(r) = v~+r function of +I and its filtered versions. Using the fact that are all globally and uniformly bounded.