New Results on Input/Output Variable Structure ...

1993 American Control Conference ( San Francisco, CA )

1

New Results on Input/Output Variable Structure Model-Reference Adaptive Control (I/O VS-MRAC): design and stability analysis Liu Hsu

Department of Electrical Eng. COPPE/UFRJ - CP 68504 21.945/970 - Rio de Janeiro - Brazil

Aldayr D. de Araujo

Department of Electrical Eng. UFRN 59.072/970 - Natal - Brazil

ABSTRACT: The I/O VS-MRAC redesign and stability analysis presented earlier for the case of plants with relative degree n  2 (ACC'92), is generalized to the case of arbitrary n . The redesign is based on the explicit consideration of input disturbances which, in addition to actual external disturbances, may include the disturbance originated by the uncertainty of the plant high frequency gain (HFG). This leads to considerably simpler and less restrictive stability analysis, as well as to more eective controller design for external disturbance rejection. It is shown that the overall error system is globally exponentially stable with respect to some small residual set, in agreement with the remarkable performance exhibited by the proposed controller. Simpli ed I/O VS-MRAC schemes are also discussed.

1 Introduction

The acronym I/O VS-MRAC designates the class of variable structure (VS) controllers which require only input/output measurements to be implemented, rst proposed in [9], for n =1, then extended in [7], for arbitrary n . The main interest of the I/O VS-MRAC relies on its remarkable stability and performance robustness properties [1, 3, 5, 6, 7, 8, 9], similar to those exhibited by VS control systems based on full state measurement to implement the appropriate sliding modes [14]. Recent related works were presented in [5, 6, 13]. A variant of the I/O VS-MRAC was proposed in [15] and the multivariable extension was considered in [2]. The stability analysis of the case n = 1 is rather simple [9]. However, when n  2 is much more involved [1, 8]. Based on the key observation that the case of plants with known HFG is drastically simpler to analyze, in [10] was proposed a redesign of the I/O VS-MRAC where the uncertainty of the HFG is incorporated as an input disturbance to be rejected by means of an appropriate redesign of the relay modulation functions relative to some `nominal' plant with known HFG. Stability analysis, taking into account the eect of the averaging lters used to approximately obtain some equivalent controls [14], resulted much simpli ed and some restrictive assumptions made in earlier works [1, 8] were removed. Details, however, were presented only for the case n =2: This paper deals with the general case n  1 and con rms that the approach of [10] is well suited for generalization. In fact this paper is the rst to present a complete stability analysis for the general case. The analysis also suggests simpli ed implementations of the I/O VS-MRAC.

Fernando Lizarralde

Department of Electrical Eng. COPPE/UFRJ - CP 68504 21.945/970 - Rio de Janeiro - Brazil

model reference having input r and output ym characterized by the rational transfer function M (s)= Km NDmm ((ss)) 1 . The basic objective of model following is to nd a control law u(t) such that the output error e0 := y ? ym tends to zero asymptotically, for arbitrary initial conditions and arbitrary piece-wise continuous uniformly bounded reference signals r(t). The usual assumptions for MRAC are made [12, p.183].

2.1 The controller structure

As in the standard MRAC [12] the following input and output State Variable Filters (SVF) are used

v_1 = v1 + gf u; v_2 = v2 + gf y

(1)

where v1 ; v2 2 c (i =0; : : : ; N ), for some constant c> 0, is valid 8t  t0 , t0 nite, then there exists some nite time ts  t0 , such that, e0i (t) = 0, 8t  ts ; (d) if an external unknown bounded disturbance d(t), with jd(t)j  d 8t (d constant), acts at the plant input, all signals in the system remain bounded and the tracking error is ultimately bounded by kd d, kd > 0; (e) if in (d) one also has j i (t)j > cd (i = 0; : : : ; N ) 8t > t0 , t0 nite, c being an appropriate positive constant, then, the property (b) holds. In practice (Ui )0eq cannot be perfectly measured, but can be approximately implemented by ltering the high frequency components of Ui through a low-pass lter with suciently small time constants. In this paper the stability analysis of the actual I/O VS-MRAC system, where (Ui )0eq is replaced by (Ui )av = Fi?+11 (i+1 s)Ui is presented for the case n  1 generalizing (in detail) the previous results of [1, 8, 10].

Lemma 1 Consider the input/output relationship e00 (t) = H (s)[u + d(t) + (t)] (8) where H (s) is SPR and d(t) is l:i:. If u is given by the discontinuous feedback law u = ?f (t)sgn(e00), where f is l:i: and satis es f (t) jd(t)j 8t, then e00 (t) tends, at least exponentially to zero as t ! 1. Moreover, if f (t) satis es f (t) jd(t)j+ 8t, with arbitrary small positive constant , then e00 (t) becomes identically zero after some nite time tr  0.

Proof: see [9, 10]. Lemma 2 Consider the input/output relationship

e0i (t) = L?i 1 (s)[u + d(t) + (t)] + i (t) (9) where Li (s)= s+ (> 0); d(t) is l:i: and i (t) is an absolutely continuous function (8t). If u is given by the discontinuous feedback law u = ?f (t)sgn(e0i ), where f is l:i. and f (t)  jd(t)j 8t, then the signals e0i (t) and e^i (t) := e0i (t) ? (t) are

2.2 A formulation for the uncertainty on Kp

Following [10], if Kp = Kn +
k , where Kn is some nominal value of Kp and
k is the uncertainty on Kp , one can write y = Kn[Np (s)=Dp (s)](u + du ) (6) du =  u (7) Thus, the uncertainty on the HFG can be formulated as an input disturbance du . As evidenced in previous works [1, 8] external input disturbance rejection can be achieved by adequately redesign the modulation functions fi . The diculty

bounded by







je^i (t)j ; e0i (t)  C0 e?t + e?t  Re?t +2 supj i (t)j (10) t

where C0  0 is any constant satisfying C0  je^i (0)j. Proof: see [10]. Lemma 3 Consider the input/output relationship

z = W (s)d 2

(11)

Let 0 be a positive constant satisfying 0 < 0 < minj jRe(pj )j (pj are the poles of W (s)), and d^(t) be a signal satisfying d^ = s + 0 d (12) 0 where d(t) is an instantaneous upper bound of d(t), i.e., jd(t)j  d(t) 8t. Then there exists a positive constant c1 such that (13) jz (t)j  c1 d^(t) + exp where \exp" is a term which depends on the initial conditions and decays exponentially to zero with rate 0 : Proof: see [11].

Remark 1 In [7] the averaging lters were believed to be necessarily of order 2 or higher. This paper shows that this is not true { it is in fact sucient to use rst order lters. The essential tool that has led to the new result is Lemma 2, which gives a bound for the solution of error equation (9) where i is an \output disturbance".

In order to obtain less conservative (i.e., smaller) modulation functions, the following identity [12, p.320] is used: Wd (s) = 1 ? G^ 1 (s), where G^ 1 (s) = ^uT (sI ? )?1 gf , (^uT = [^1 :::^n?1 ],  and gf from (1)), i.e., G^ 1 u is the contribution of the input lter to the ideal matching control. Then, the equation of the auxiliary error e0N can be rearranged as (d = du +de; du =  u):   e0N = L?N1 ?UN ? U^ + Un ? (G^ 1 u ? kn?1 r) ? Wd de + N (23) 4 Stability Analysis where k = K =K , k = K =K ,  = (k ? kn )=kn and  = This section considers the problem of redesign and stability 1 +  = kn kn?1 n(notem that  >p0). m analysis of the I/O VS-MRAC algorithm (Fig. 1) for a plant Bounds for the auxiliary errors with \known" HFG Kn (6), under the action of a general input 4.2 In what follows, \exp" denotes exponentially decaying terms disturbance d(t) = du(t) + de(t), where du is given by (7) and (which include the eect of initial conditions of some state rede is uniformly bounded 8t. The term du allows the case of alization of the operators in (20){(22) and L?i 1 in (18)) and all unknown HFG to be embedded in the case of known HFG. The redesign is centered on the derivation of conditions to be K s denote positive constants. satis ed by the modulation functions. Sucient conditions for Theorem 1 Considering the auxiliary errors given by (15){ such functions are derived in order to guarantee stability and (18), with the relay modulation functions satisfying the followdisturbance rejection. ing conditions (d = du + de) Noting that, whatever k , known or unknown, there exists   T   ? 1 U = ! ! such that u = U + k r ? Wd de gives the perfect fi  L?i+11 ;N (U^ ? Un ? Wd d) ; i = 0; :::;N ? 1 (24) model following control law, it is reasonable to assume that fN  ?1 (U^ + G^ 1 u) ? Un ? ?1 kn?1 r ? ?1 Wd de (25) juj  K3 j !j + Kred (14) then, the auxiliary error e00 tends to zero, at least, exponenwhere K3 and Kred are adequate positive constants. This as- tially, and the errors e0i (i = 1; : : : ; N ) are bounded by sumption is consistent with the modulation function fN (di 0 ei (t)  2i;max KiC (t) + exp (26) rectly related with u) obtained later on in Section 5. It also assures that all signals u(t); d(t); i and i belong to L1e . j i (t)j  i;max Kei C (t) + exp (27) 4.1 Error equations with averaging lters As already mentioned, the implementation of the equivalent where (0 = 0, i = 1; : : : ; N )  0 0  controls is made through averaging lters. Thus, the auxiliary i;max = j=0max [j =(j+1 : : : i )] ; i;max =max i ; i;max ;:;i ? 1 error equations are expressed by (28)   e00 = kn ML ?U0 ? L?1 (U^ ? Un ? Wd d) (15) C (t) = M C1 (t) + j Wd j C2 (t) (29) C1 (t) = supt j !(t)j , C2 (t) = supt jd(t)j, M  j ! ? !n j e0i=1;:::;N = Fi?1 (Ui?1 ) ? L?i 1 (Ui ) (16) with and L1 operator norm. where U^ +kn?1 r = ^!T ! +kn?1r is the ideal control law that leads N ( s ) to perfect model following by the plant Kn Dpp (s) (6) with d  0. Proof: By Lemma 1, we have that the error e00 (15) converges to zero, at least, exponentially, if the signal f0 satis es (24). The dynamic system governing e00 can be represented as On the other hand, considering (20), the following inequality   x_ e = Ac xe +kn b0c ?U0 ? L?1 (U^ ? Un ? Wd d) ; e00 = hTc xe (17) holds (i = 1; : : : ; N ) After some algebraic manipulations of (15) and (16) we obtain (F1;i ? 1) supj ui (t) ? expj i  F L C (t) = i Ki0C (t) (30) (i = 1; : : : ; N ) i 1;i i;N t   {z } | e0i = L?i 1 ?Ui ? L?i+11 ;N (U^ ? Un ? Wd d) + i (18) O(1) where Li;j = Li Li+1 : : : Lj (Li;j = 1 if j < i), Fi;j is de ned in Note that all terms are absolutely continuous and thus their similar way and exponentially decaying components can be incorporated in the  (t) term of Lemma 2. Thus, since ui (t) is bounded by 0 i=1;:::;N = ui ? ei ? ei; e1  0 (19) (30) then, by Lemma 2, if condition (24) holds, the error e01 is bounded by ui = (F1;i ? 1)F1?;i1 L?i;N1 (U^ ? Un ? Wd d) (20) 0 e1 (t)  21;max K1 C (t) + exp; K1 = K10 (31) i?1 X ei = Lj;i?1 Fj?+11 ;i e0j = Li?1 Fi?1 ( e;i?1 + e0i?1 ) (21) Then, from (21): j=1 0 sup j e2 ? expj  20 ;max K2"C (t) (32) ei = (kn MF1;i Li;N )?1 e00 (22) t

3

 (t) + EXP , with C (t) In what follows, the proof will be completed by induction. (39) is bounded by j e(t)j  N;max KC First, we suppose that the two following inequalities hold for de ned by expression (18). From e = e ? kn bc e^N , we further obtain the inequalities some i  3



e0i?1 (t)  2 i?1;max Ki?1 C (t) + exp

(33)

0 Ki"C (t) sup j ei ? expj  i;max

(34)

t

j e(t)j  N;max KeC (t) + EXP je0 (t)j  N;max K0 C (t) + EXP

(40) (41)

From the relation ! = !m + e, where !mT = [vmT 1 ym vmT 2 ] is

We then show that they are also valid for i + 1. Indeed, from the regressor vector corresponding to the reference model and (21), (33) and (34) one has

is a constant matrix, it follows that 0 ? 1 " sup j e;i+1 ? expj  i+1 Li Fi+1 [i;max Ki j !j  Km + K j ej (42) t

|

{z

O(1)

}

From (40) and (42) we obtain (C1 (t) = supt j !(t)j ) +2Ki i;max =i+1 ]C (t) C1 (t)  N;max K1 C (t) + K2R(0) + Km  i0+1;max Ki"+1 C (t) (35)

(43)

Thus, by Lemma 2, condition (24) or (25), inequalities (30) On the other hand, from (7) and considering an external and (35), we have that (27) is valid and bounded disturbance, we can obtain



e0i (t)  2Ki i;max C (t) + exp

sup jd(t)j = C2 (t)   sup ju(t)j + Ked

(36)

t

t

(44)

Since (33)(34) imply (35)(36) and (31)(32) hold, the proof by Since by assumption (14), we have induction is completed. 4.3 Error system stability sup ju(t)j  K3 C1 (t) + Kred (45) t Consider the error system (2) for the nominal plant with u = ?UN + Un + kn?1 r (see Fig. 1) and U^ = ^T !, i.e., then, from (44) and (45), C2 (t) can be expressed by ^ e_ = Ac e + kn bc [?UN ? U + Un + Wd d] (37) 0 (46) C2 (t)  K4 C1 (t) + Kred   ^ note that k and U have been replaced by kn and U , respec- and using (43) in (46), we conclude that C2 also has a bound tively. From (18) an explicit expression for (?UN ? U^ + Un ) can be of the type (43) and consequently obtained and then from (37), with e^N := e0N ? N , we have C (t)  (K5 + K6 R(0))(1 ? N;max K7 )?1 (47)   e_ = Ac e + kn bc e^_ N + N e^N (38) Now, from (40) and (47) one has Thus, making the variable transformation e := e ? kn bc e^N , we obtain e_ = Ac e + kn (Ac + IN )bc e^N (39) In order to fully account for the initial conditions we will consider the full state vector z of the system composed by subsystems (37)(17)(18)(19), which can be described by the states e(t), xe (t), e0i (t) and x i (t) (i = 1; : : : ; N ), i.e., z T = [eT xTe e01 : : : e0N xT 1 : : : xT N ]. For de niteness, the state x i is that of a minimal realization of the operator i?1 (F1;i ? 1)F1?;i1 L?i;N1 with input i [U^ ? Un ? Wd d] and output ui :

j ej  N;max K8 R(0) + O(N;max ) + EXP

(48)

Then, from inequalities (26), (30), (47), (48) and the exponential stability of (17) and since that the initial time is irrelevant in deriving the above expressions, we can write for arbitrary t  t0  0

R(t)  (max K9 + K10 e?a(t?t0 ) )R(t0 ) + O(N;max ) (49) where max = maxi=1;:::;N i;max . For max small enough, i.e., for the time constants of the averaging lters satisfying the following relationships: i small enough and j =(j+1 : : : i ) small enough (j = 1; : : : ; i ? 1; i = 2; : : : ; N ); the equation (49) implies that R(t) converges exponentially to a residual value of order N;max , independent of R(t0 ). Consequently, the same conclusion holds for the output error e0 (t).

Remark 2 Note?1that in the ?error equations considered above, the ?1 ?1 1

operators Wd , L in (17), Li+1;N in (18), Li;N in (20), Li?1 Fi in (21) and (MF1;i Li;N )?1 in (22) can be considered as zero-state response operators which do not contribute with additional state variables. Indeed, note that (37) in fact represents (2) which does not involve operator Wd . Similarly, (17) can also be expressed without L?1 . Thus, e and xe can represent the states of (37) and (17), respectively. On the other hand, the initial conditions associated with operators L?i 1 and Fi?1 in (16) (rewritten as (18)) can be In this section, explicit formulas for the modulation functions taken into account by the states e0i and x i , respectively. are derived. It will be supposed that an instantaneous upper

5 The relay modulation functions

bound of jde (t)j denoted de (t) is known, i.e., jde(t)j  de (t) 8t. For convenience we de ne the norm of z as R(z ) = j ej + j xej + First note that U  = !T ! = (U^ + G^ 1 u)= and 2n ? 2nn = 0 i=1 [jei j + j x i j ]. In order to simplify notation we use R(t) ?1 kn?1 . Then, expressions (24)(25) can be rearranged as instead R(z (t)) and denote by \EXP" any term of the form KR(0)e?at, where K and a are positive constants. fi=0;:::;N ?1  L?i+11 ;N [(U  ? Un ) ? (u ? Un ) ? Wd de] (50) Now, since e0N and e^N are bounded by (26), (see (27)), and Ac fN  (U  ? Un ) ? (2n ? 2nn )r ? ?1 Wd de (51) is exponentially stable, we conclude that the error e(t) given by PN

4

fN = KuN u^ + KyN y^ + Krd

and these inequalities are satis ed if fi=0;:N ?1  T ji j + i ? nT i + L?i+11 ;N Wd de (52) fN  T j!j + 2n jrj + ?1 jWd dej (53) where 0 <      and  jj. Now, for the implementation of the modulation functions (52){ (53) it is necessary to nd an upper bounds for the signals zi=1;:::;N de ned by zi=0;:::;N ?1 = L?i+11 ;N Wd de = L?i+11 ;N (1 ? G^ 1 )de (54) zN = ?1 Wd de = ?1 (1 ? G^ 1 )de (55) The transfer function G^ 1 (s) is unknown, however, its poles are known and xed by design as the roots of det(sI ? ) = 0. Then, the required bounds can be established with the Lemma 3. Making use of these results and including the exponential term of Lemma 3 in the term (t) of Lemmas 1 and 2, the modulation functions (52){(53) can be implemented as fi=0;:::;N ?1 = T ji j +  i ? nT i + ci d^ei (t) (56) fN = T j!j + 2n jrj + cN d^eN (t) + ?1 de(t) (57) where   , ci=0;:::;N being appropriate positive constants and d^ei = s+ i ide , i = minj jRe(pi;j )j; d^N = s+

NN de , N = mink jRe(pN;k )j; where pi;j and pN;k are poles of L?i+11 ;N (1?G^ 1 ) and G^ 1 respectively. Remark 3 The modulation functions (56){(57), when de  0 (no

(63)

Version C (Relay I/O VS-MRAC [1]): If only local stabil-

ity is required, drastic simpli cation can be obtained by using constant modulation functions. It should be stressed that the simpli cations above are obtained either at the expense of larger modulation functions Versions A, B, or loss of global stability in the Version C. Large modulation functions may induce numerical problems, excessive control action, etc.

6 Conclusions

In this paper, earlier results about design and stability analysis of the I/O VS-MRAC, taking into account the eect of the averaging lters, were generalized to the case of arbitrary n . Global exponential stability of some small residual set in the error space was obtained. The present analysis reveals that the averaging lters can be chosen of order one, contrary to what was believed originally. Moreover, the new design can also be used for more eective external disturbance rejection. Simpli ed I/O VS-MRAC versions which do not require the knowledge of the plant order (only of n ) have also been presented.

References

[1] A. Araujo and L. Hsu, \Further developments in variable structure adaptive control based only on I/O data," in Proc. 11th IFAC World Congress, vol. 4, (Tallinn), pp. 293{298, 1990. [2] C. J. Chien, A. C. Wu, L. C. Fu, and K. C. Sun, \A robust model reference using VS design: The general case for multivariable plants," in Proc. ACC, (Chicago), pp. 2730{2734, 1992. [3] R. R. Costa and L. Hsu, \Robustness of VS-MRAC with respect to unmodeled dynamics and external disturbance," Int. J. Adaptive Contr. Signal Process., vol. 6, pp. 19{33, 1992. [4] A. Filippov, \Dierential equations with discontinuous righthand side," Amer. Math. Soc. Trans., vol. 42, no. 2, 1964. [5] L. C. Fu, \New approachto robust MRAC for a class of plants," Int. J. Contr., vol. 53, no. 6, pp. 1359{1375, 1991. [6] L. C. Fu, \A new robust MRAC using VS design for relative degree 2 plants," Automatica, vol. 28, no. 5, pp. 911{926, 1992. [7] L. Hsu, \Variable structure model reference adaptive control using only I/O measurement: The general case," IEEE Trans. Aut. Contr., vol. 35, no. 11, pp. 1238{1243, 1990. [8] L. Hsu, A. D. Araujo, and R. R. Costa, \On the design of VSMRAC systems using I/O data," in Proc. Int. Workshop on VS systems and their applications, (Sarajevo), 1990. [9] L. Hsu and R. R. Costa, \Variable structure model reference adaptive control using only input and output measurement: Part I," Int. J. Contr., vol. 49, no. 2, pp. 399{416, 1989. [10] L. Hsu and F. Lizarralde, \Redesign and stability analysis of I/O VS-MRAC systems," in Proc. ACC, (Chicago), pp. 2725{ 2729, 1992. [11] P. Ioannou and K. Tsakalis, \A robust direct adaptive controller," IEEE Trans. Aut. Contr., vol. 31, no. 11, pp. 1033{ 1043, 1986. [12] K. Narendra and A. Annaswamy, Stable Adaptive Systems. Prentice Hall, 1989. [13] K. Narendra and Bovskovic, \A combined direct-indirect and variable structure method for robust adaptive control," IEEE Trans. Aut. Contr., vol. 37, no. 2, pp. 262{268, 1992. [14] V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems. MIR, 1978. [15] A. C. Wu, L. C. Fu, and C. F. Hsu, \Robust MRAC for plant with arbitrary relative degree using variable structure design," in Proc. ACC, (Chicago), pp. 2735{2739, 1992.

external disturbance), are essentially the same (slightly less conservative) as the former ones obtained in [8].

5.1 Simpli ed modulation function

In recent versions of the I/O VS-MRAC, [15, 2] used simpli ed relay modulation functions of the type fi=0;:::;N (t) = Ki1 m(t) + Ki2 (58) m_ (t) = ? m(t) + Kuy (ju(t)j + jy(t)j + 1) (59) with positive K 0s and . This simpli cation can be justi ed from Lemmas 2 and 3. On the other hand, since i and i are generated by stable linear lters with inputs u(t) and y(t), and u(t), respectively, Lemma 3 can be used to generate more easily implementable modulation functions, in the sense that simpler ltering devices are required. Following this streamline of ideas, some alternative simpli ed modulation functions are presented below. Here, note that the knowledge of the plant order is not needed, only of n . Version A: Introduce the signals u^(t), y^(t) and d^(t) as u^(t)= G (s) ju(t)j, y^(t)= G (s) jy(t)j and d^(t)= G (s)de(t), where G (s) = s+1  . Then, as can be easily shown from Lemma 3, with appropriate positive constants  and K 0s, the following modulation functions satisfy (52){(53) fi=0;:::;N ?1 = Kui u^ + Kyiy^ + Kyi0 jyj + Kdi d^ (60) 0 de (61) fN = KuN u^ + KyN y^ + 2n jrj + KdN d^ + KdN Version B: Constant upper bounds for jr(t)j and jde(t)j can also be used so that (60){(61) can be replaced by fi=0;:::;N ?1 = Kui u^ + Kyi y^ + Kyi0 jyj + Kdi (62) 5