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A robust decentralized model reference adaptive control for non-minimum-phase interconnected systems
Automatica 35 (1999) 1499}1508
Technical Communique
A robust decentralized model reference adaptive control for non-minimum-phase interconnected systems夽 M. Makoudi* , L. Radouane LESSI, De& partement de Physique Faculte& des Sciences, B.P. 1796, 30000 Fe% s, Morocco Received 21 February 1997; revised 24 July 1998; received in "nal form 22 February 1999
Abstract In this paper, we present a decentralized model reference adaptive control for interconnected subsystems in the sense that no information exchange occurs between the subsystems. The approach is based on the interconnection output estimation using the polynomial series which o!ers a general solution for interconnected subsystems. The parameter estimation scheme is a combined adaptive data "ltering with a recursive least-squares algorithm with parameter projection and normalization. The problem of minimum phased subsystems is handled by an adaptive input}output data "ltering. Hence the zeros of each subsystem estimated model are relocated inside the unit circle. This estimated model which is minimum phased is then used for the control synthesis. It is shown that the stability conditions based on weak interconnections are relaxed. Also the robustness of the proposed adaptive control against unmodeled dynamics is stated. Finally, the results are illustrated by numerical examples. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive decentralized control; Model reference control; Polynomial approach; Interconnection estimation; Nonminimum phase; Data "ltering
1. Introduction In the control of Large-scale systems, one usually faces poor knowledge on the plant parameters and interconnections between subsystems. Thus the adaptive control technique in this case is an appropriate strategy to be employed. Moreover, if some subsystems distribute distantly, it is di$cult for a centralized controller to gather feedback signals from these subsystems. An e!ective way to handle this di$culty is to apply decentralized control strategies whereby each subsystem is controlled independently on the basis of its own performance criterion and locally available information. The majority of the results on Large-scale systems described in input/output form and decentralized control refer to systems that consist of
* Corresponding author. Tel.: 00212-5-642-389; fax: 00212-5642-5000; e-mail: [email protected]. 夽 The original version of this paper was presented at the IFAC international workshop on automation in the steel industry: current practice and future developments. Kyungjoo, Korea, 16}18 July 1997. This paper was recommended for publication in revised form by Editor Peter Dorato.
weakly coupled subsystems (Wen, 1992, 1994; Datta and Ioannou, 1991; Hejda et al., 1990; Ioannou, 1986; Ossman, 1989; Praly and Trulsson, 1986; Reed and Ioannou, 1988). The subsystems may be considered in isolation, and the controllers received for the isolated subsystems may be applied as decentralized controllers to the coupled overall system. However, the information which may be extracted from the interconnections are ignored. No attempt has been made to estimate them in order to improve the control performance and robustness. Here, we present a decentralized model reference adaptive control (DMRAC) for interconnected systems. The main idea is to predict the interconnection outputs acting on each subsystem to relax the hypothesis of weak interconnections. These predictions are used for the synthesis of the local control. The prediction method is based on expressing the interconnection outputs as a linear combination of a set of orthogonal known functions of a basis. Note that the polynomial approach has received much attention in mathematical modeling, identi"cation and control literature (Horng et al., 1986; Lee and Tsay, 1986; Dumont et al., 1985, 1986; James, 1994; Maciej, 1988; Greblicki, 1994; Wei, 1990; Heuberger et al., 1995). The parameter estimation scheme is a combined adaptive
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 5 4 - 0
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data "ltering with a recursive least-squares algorithm with parameter projection and normalization. The problem of minimum phase of the subsystems plant is handled by an adaptive input output data "ltering. Then the zeros of the estimated subsystem model are relocated inside the unit circle. The obtained minimum-phase models are used in the construction of the controllers. Relaxed constraints on the interconnections are necessary to ensure the global system boundedness. It is also shown that this scheme is robust with respect to unmodeled dynamics. The work is organised as follows: In Section 2, the system is de"ned and all assumptions are outlined. The parameter estimator is described in Section 3. In Section 4, global stability of the indirect adaptive controller is established. In Section 5, two numerical examples are given for illustration.
'* (t!1) is a regression vector and h* denotes a vector G G containing unknown parameters of the nominal subsystem model i.e. '*(t!1)2"[> (t!1)2> (t!ni); (t!di)2 G G G G ; (t!di!mi)], G h*2"[!a 2!a b 2 b ]. G G GLG G GKG Here, the objective is to derive a decentralized control law to stabilize the global system with no information exchange between the subsystems. It is assumed that for each subsystem i, the output < (t) is not available. Thus G we propose that each subsystem predicts these interconnection outputs in real time. The prediction method is based on expressing the interconnection outputs as a linear combination of a set of known functions of a basis.
2. Problem statement 2.1. Functional series modeling The system to be considered consists of N interconnected single-input}single-output (SISO) subsystems described in input/output form by A (q\)> (t)"B (q\); (t) G G G G , # C (q\)> (t!1)#m (t), GH H G H H$G i"1, 2 , N
(2) (3)
The polynomials C (q\) are the interconnection inputs GH of > (t) from the other subsystems > (t) with G H C (q\)"c #))) #c q\PGH, (4) GH GH GHPGH di is the time delay index and m (t) represents the unG modeled response of subsystem i. Subsystem (1) can be expressed as > (t)"'*(t!1)2h*#< (t), G G G where , < (t)" C (q\)> (t!1)#m (t), G GH H G H H$G
S(t)"
"q\BGB (q\); (t) G G , # C (q\)> (t!1)#m (t), (1) GH H G H H$G where > (t), ; (t) are the output and input, respectively, G G of the subsystem i. A (q\), B (q\) and C (q\) are G G GH scalar polynomials in the unit delay operator q\. A (q\)"1#a q\#)))#a q\LG, G G GLG B (q\)"b #)))#b q\KG, b O0. G G GKG G
Let ¹"+1,2, 2 , R, and let S(t)"+s (t), 2 , s (t),2 K be the set of linearly bounded independent sequences (functions of discrete-time t) on T. The elements of S will be further referred to as basis sequences or basis functions. Possible sets of functions that could be used include powers of time s (t)"tH\, j"1, 2 , m (which will H be referred to as the Legendre basis) or
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cos %( j!1)t if j is odd, sin %( j!1)t if j is even
(which is called the Fourier basis). Now we will use the functional modeling of the interconnections. We assume that for subsystem i, the unknown output < (t) may be expanded into a series. G < (t)"M2 (t)S(t)#= (t), (6) G G G where S(t)"[s (t), 2 , s (t)]2, M (t)3RK, K G = (t) is the residual between the output < (t) and the G G series which may be unbounded. M (t) is a time-varying G vector of unknown elements to be estimated. The vector M (t) is taken to be time varying, to represent more G precisely the interaction term < (t). This modelization is G more adequate than < (t)"M2S(t)#= (t) where M is G G G G a constant vector (see the appendix). By substitution of Eq. (6) into Eq. (1), we obtain A (q\)> (t)"B (q\); (t)#M2(t)S(t)#= (t). (7) G G G G G G Further assumptions concerning the plant are made: A1: An upper bound n* on the orders ni and mi is G known.
M. Makoudi, L. Radouane/Automatica 35 (1999) 1499}1508
A2: The time delays di, i"1, 2 , N are known. A3: There exists a known scalar o 3R> such that G #h*#4o . G G A4: There exists a "nite positive scalar k such that G "= (t)"4k Z (t). G G G Z (t)"p Z (t!1)#max [#' (t!1)#, Z ], G G G G G 0(p (1, Z '0, Z (0)'0. G G G Subsystem (7) can be given the following form: > (t)"'2 (t!1)h (t)#= (t). G G G G The augmented parameter and observation vectors are de"ned by h2 (t)"(!a 2 !a * b 2 b * M2 (t)), (8) G G GLG G GLG G '2(t!1)"(> (t!1)2 > (t!ni*) G G G ; (t!di)2 ; (t!di!ni*)S2(t)). (9) G G Thus the normalizing signal Z (t) depends also on G the interconnection terms through the vector (S(t)) in Eq. (9) and the vector M (t) in the augmented parameter G vector (8). Assumption A4 means that the unmodeled response = (t) (Eq. (7)) should be linearly bounded by the adaptive G input}output data activity measure Z (t), which depends G also on the vector S(t). We remark that the new bound k is less restrictive G then the classical one used in Wen (1992, 1994) and Praly (1986) which is given by "< (t)"4g Z (t), where < (t) is G G G G given by Eq. (5) and Z (t)"p Z (t!1)#max [#'* (t!1)#, Z ], G G G G G 0(p (1, Z '0, Z (0)'0, G G G '*(t!1)2"[> (t!1)2 > (t!ni*) G G G ; (t!di) 2 ; (t!di!ni*)]. G G In the present work, only the residual error = (t) is G constrained to satisfy assumption A4. This is due to the introduction of the term (S(t)) in the observation vector (Eq. (9)) and of the vector M (t) in the parameter vector G (Eq. (8)). The interconnections are thus taken into account by each subsystem. Then the normalizing signal Z (t) depends on local subsystem signals and implicitly G on the interconnection signals also. Therefore this new assumption makes the present work di!erent from the previous ones on the same subject. Assumption A4 is no more restrictive.
3. Parameter estimation We plan to design an adaptive DMRAC controller including adaptive data "ltering which should stabilise
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the subsystem and cause > (t) to track a bounded referG ence model > (t). GK 3.1. **Good data++ model The key issue to get a robust parameter estimation may be viewed as a &&good data'' model and a robust parameter adaptation algorithm with respect to bounded disturbances and time varying parameters. Here we de"ne a new good data model adapted to our problem. Let the % linear operator be de"ned by G % ( . ) (t!k)"aI(t)( . ) (t!k) (10) G G G G with 0(a (t)(1. G Given two asymptotically stable polynomials O (q\) G and P (q\) and an % operator, then operating on subG G system model (7) by the "lter % (O /P ) leads to G G G % A (q\)> (t)"% B (q\); (t) G G GD G G GD #% M2 (t)S (t)#= (t), G G D G % (> (t)#2#a * > (t!ni*)) G GD GLG GD "% (b ; (t!di)#2#b * ; (t!di!ni*)) G G GD GLG GD #% M2 (t)S (t)#= (t). G G D G As A (q\) and B (q\) are time-invariant polynomials, G G we obtain (% > (t)#2 #a * % > (t!ni*)) G GD GLG G GD "(b % ; (t!di)#2 #b * % ; (t!di!ni*)) G G GD GLG G GD #M2(t)S (t)#= (t) (11) G D G with = (t)"% = (t), G G GD where
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O (q\)( . ) (t)"P (q\)( . ) (t). G G G GD The proposed &&good data'' model (11) involves hence the following main design feature: the input}output signals are "ltered by % (O /P ). The "lter O /P , which G G G G G should be low-pass, can be used to reduce the &&highfrequency'' modes of the unmodeled dynamics. The % operator allows to remove the minimum phase design G assumption (Lemma 3). Note that when a (t)"1 (Eq. G (10)) the proposed "lter reduces to the classical one O /P G G considered in Giri et al. (1991) and in Sripada and Fisher (1987). Subsystem "ltered model (11) may be expressed in regression form as follows: > (t)"'2 (t!1, a )h (t)#= (t) GD GD G G G
(13)
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with the augmented parameter and observation vectors de"ned by h2(t)"(!a 2!a * b 2 b * M2(t)), G G GLG G GLG G '2 (t!1, a )"(a (t)> (t!1)2 aLG* (t)> (t!ni*) G GD GD G G GD ;aBG (t); (t!di) 2 aLG*>BG (t) G GD G ; (t!di!ni*)S2 (t)). GD D
"m (t)"(a (t)"m (t)" G G GD and
Lemma 1. Consider subsystem model (13) subject to assumption A4, then there exists a positive scalar k such G that "= (t)"4k Z (t) where k (k G G GD G G and
m (t)"*A (t)a (t)> (t!1)#*B (t)aBG (t); (t!di) G G G GD G G GD #a (t)F (t)#g (a (t)> (t!1) G GD G G GD #aBG (t); (t!di)), G GD then
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Z (t)"p Z (t!1)#max (#' (t!1)#, Z ). GD G GD GD G '2 (t!1)"(> (t!1)2 > (t!ni*) GD GD GD ; (t!di). 2 ; (t!di!ni*)S2 (t)). GD GD D
"< (t)"(a (t) G G
, (c #2 #c q\PGH)> (t!1)#m (t) GH GHPGH HD GD H H$G (a (t)"< (t)". G GD From assumption A4, we have
Thus the "lter % (O /P ) permits to relax the conG G G straints on the unmodeled dynamics.
"= (t)"4k Z (t) then "= (t)"4k Z (t), G G G GD G GD and "= (t)"4a (t)k Z (t)"k Z (t). G G G GD G GD As 0(a (t)(1, the bound k decreases, so large unG G modeled dynamics may be tolerated and classical weak interconnection assumption is relaxed.
Proof. Consider the "ltered subsystem model de"ned by
3.2. Robust parameter adaptation algorithm
% A (q\)> (t)"% B (q\); (t) G G GD G G GD #% M2(t)S (t)#= (t) G G D G where
Let hK (t) denote the estimate of h (t) at time t, using the G G least-squares parameter estimation with parameter projection, signal normalization and data "ltering
= (t)"% = (t)"< (t)!M2 (t)S (t), G G GD G G D , < (t)" % (c #2#c )> (t!1)#% m (t). G G GH GHPGH HD G GD H H$G Then , < (t)" (c a (t)#2#c aPGH (t)q\PGH) G GH G GHPGH G H H$G > (t!1)#m (t). HD G The unmodeled dynamics +m (t), may contain nonlineariG ties and bounded disturbances. We will consider without loss of generality, the following class: m (t)"*A (t)> (t!1)#*B (t); (t!di)#F (t) G G G G G G #g (> (t!1)#; (t!di)), G G G where #*A (t)#4i , #*B (t)#4i , G G G G where
#F (t)#4i , G G
m (t)"*A (t)> (t!1)#*B (t); (t!di)#F (t) GD G GD G GD GD #g (> (t!1)#; (t!di)), G GD GD
' (t!1, a )e (t) GD G GD hK (t)"I hK (t!1)# , G G Z (t)#'2 (t!1, a )' (t!1, a ) GD GD G GD G (15) where I represents the projection operator necessary to ensure #hK (t)#4o (note that if #hK (t)#4o then G G G G #hK * (t)#4o from assumption A2) e (t) is the prediction G G GD error de"ned as e (t)"> (t)!hK 2(t!1)' (t!1, a ), GD GD G GD G e (t)"> (t)!(!aL (t!1), 2 , !aL * (t!1), GD GD G GLG bK (t!1), 2 , bK * (t!1), M K 2(t!1))a G GLG G ;[a (t)> (t!1), 2 , aLG (t)> (t!ni*), G GD G GD aBG (t); (t!di), 2 , aBG>LG* (t) G GD G ; (t!di!ni*), S2 (t)]2, GD D e (t)"> (t)#aL (t!1)a (t)> (t!1)#2 GD GD G G GD #aL * (t!1)aLG* (t)> (t!ni*) GLG G GD K !b (t!1)aBG (t); (t!di)!2 G G GD !bK * (t!1)aBG>LG* (t); (t!di!ni*) GLG G GD !M K 2 (t!1)S (t). G D
(16)
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M. Makoudi, L. Radouane/Automatica 35 (1999) 1499}1508
The estimation error for subsystem model (17) can be written as e (t)"AK (t!1, q\)> (t)!BK (t!1, q\) GD G? GD G? ; (t!di)!M K 2 (t!1)S (t) GD G D "AK (t!1, q\)> (t)!BK (t!1, q\) G? GD G? ; (t)!M K 2 (t!1)S (t) GD G D where AK (t!1, q\) G? "1#aL (t!1)a (t)q\#2 G G #aL * (t!1)aLG* (t)q\LG*, BK (t!1, q\) G? GLG G "bK (t!1)aBG (t)#2 G G #bK * (t!1)aLG>BG (t)q\LG*. GLG G Thus at the present stage, the estimation error for subsystem i is de"ned in terms of the polynomials AK and G? BK , where BK is stable (Lemma 3). These polynomials G? G? will be used for the control synthesis (Eq. (19)). Lemma 2. Consider the subsystem model given in (13), subject to assumptions A1}A4 then the algorithm (15)}(16) has the following properties: (a) #hK (t)#4o , G G (b) ¹here exists a positive constant o such that for all GC (l, l)3N one has: J>J "e (t)" GD 4o #lk . GC G Z (t) > GD (c) ¹here exists a positive constant o such that for all GM (l, l)3N one has: J>J #hK (t)!hK (t!1)#4o #lk . G G GM G RJ> The proof of Lemma 2 may be carried out along the same lines as in Giri et al. (1991) and is then omitted. Lemma 3. Consider the ,ltered subsystem error model (18), subject to assumptions A1}A3, if "a (t)"(1/(ni*, G then BK (t!1, q\) is a stable polynomial. G? Proof. Given that LG* bK (t!1)(#hK (t!1)#(o G G I GI and 0(a (t)(1, we get G LG* bK (t!1) LG* bK (t!1) a (t) GI aI (t)(a (t) GI ( G o G G (t!1) G K K b (t!1) b (t!1) b K I G I G G
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BK (t!1, q\) is a stable polynomial if G? 1 a(t)/bK (t!1) o4 G G G ni* (see Zinober, 1983). Then a (t)( "bK (t!1)"/o (ni*. G G G 4. Decentralized adaptive control The proposed feedback control signal is generated from (Fig. 1) HK (t, q\)BK (t!1, q\); (t)#GK (t, q\)> (t) G G? GD G GD #QK (t#di, q\)S (t#di) G D "¹ (q\)> (t#di), (19) G GKD where > (t) is the given bounded output of the reference GK model, and O (qU)> (t)"P (qU)> (t). G GK G GKD ¹ (q\) is an arbitrary stable monic polynomial of G degree nti(ni*#di and HK (t, q\)"1#hK (t)q\#2 G G #hK (t)q\LFG, nhi(di GLFG GK (t, q\)"gL (t)#gL (t)q\#2 G G G #gL (t)q\LEG,ngi(ni* GLEG QK (t, q\)"qL 2 (t)#qL 2 (t)q\#2#qL 2 (t)q\LOG, G G G GLOG qL 3RK, j"0, 2 , nqi, nqi(di. GH The term QK (t#di, q\)S (t#di) is introduced beG D cause of the existence of the interconnections and is then used for their e!ect reduction. The estimation error is given by e (t)"> (t)!hK 2(t!1)' (t!1, a ) GD GD G GD G "AK (t!1, q\)> (t)!BK (t!1, q\); (t!di) G? GD G? GD !M K 2 (t!1)S (t). (20) G D Adding and subtracting q\BGBK (t!1, q\); (t) in G? GD Eq. (20) gives e (t)"AK (t!1, q\)> (t)!q\BGBK (t!1, q\) GD Ga GD G? ; (t)!M K 2 (t!1)S (t)#[q\BG . BK (t!1, q\) GD G D G? !BK (t!1, q\) . q\BG]; (t). (21) G? GD Operating on Eq. (21) by HK (t, q\) gives G HK (t, q\)AK (t!1, q\)> (t) G G? GD "q\BGHK (t, q\)BK (t!1, q\); (t) G G? GD #HK (t, q\)e (t)#HK (t, q\)M K 2 (t!1)S (t) G GD G G D #HK (t, q\)[q\BG . BK (t!1, q\) G G? !BK (t!1, q\) . q\BG]; (t) (22) G? GD
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Fig. 1 The proposed decentralized adaptive controller.
and in view of Eq. (20) then HK (t, q\)AK (t!1, q\)> (t) G G? GD "q\BG[¹ (q\)> (t#di)!QK (t#di, q\)S (t#di) G GKD G D !GK (t, q\)> (t)]#HK (t, q\)e (t) G GD G GD #HK (t, q\)M K 2 (t!1)S (t) G G D #HK (t, q\)[q\BG . BK (t!1, q\) G G? !BK (t!1, q\) . q\BG]. (23) G? HK (t, q\), GK (t, q\) and QK (t, q\) are determined by G G G solving the following equality:
Proof. We de"ne AK . BK and AK BK : AK . BK " aL (t)bK (t!i)q\G\HOBK . AK G H G H and AK BK " aL (t)bK (t)q\G\H"BK AK G H G H We also de"ne BM "BK (t!1, q\) and BM "BK (t#di!1, q\). From Eq. (18), we have
HK (t, q\)AK (t!1, q\)#q\BGGK (t, q\)"¹ (q\) (24) G G? G G QK (t, q\)"HK (t, q\)M K 2 (t!1). (25) G G G The closed-loop control system equation is given by
e (t#di)"AM > (t#di)!qBGBM ; (t) GD G? GD G? GD
¹ (q\)[> (t)!> (t)] G GD GKD "HK (t, q\)e (t)#HK (t, q\) G GD G ;[q\BG . BK (t!1, q\)!BK (t!1, q\).q\BG]; (t). G? G? GD (26)
HK e (t#di)"HK .AM > (t#di)!HK . qBG BM ; (t) G GD G G? GD G G? GD
The indirect adaptive control algorithm above is globally convergent as is stated by the following theorem. Theorem 1. Consider subsystem model (13), subject to assumptions A1}A4 in closed-loop with the adaptive control law (19). ¹hen the resulting closed-loop system is globally stable in the sense that (i) ; (t), > (t) are bounded for all time, G G (ii) if k is equal to zero, then we have: G lim (> (t)!> (t))"0 as tPR. G GK
Thus,
!M K 2 (t!1#di)S (t#di). G D
!HK MK 2 (t!1#di)S (t#di), G G D HK e (t#di)"HK AM > (t#di) G GD G G? GD #[HK .AM !HK AM ]> (t#di) G G? G G? GD M !HK .qBGB ; (t) G G? GD !HK MK 2 (t!1#di)S (t#di), G G D HK e (t#di)"[¹ !q\BGGK ]> (t#di) G GD G G GD #[HK . AM !HK AM ]> (t#di) G G? G G? GD M !HK . qBGB ; (t) G G? GD !HK M K 2 (t!1#di)S (t#di), G G D
(27)
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HK e (t#di)"¹ > (t#di)!GK > (t)!HK BM ; (t) G GD G GD G GD G G? GD
the adaptive decentralized control with the interconnection estimation:
!HK M K 2 (t!1#di)S (t#di) G G D
> (t)"!0.905> (t!1)#0.3> (t!2)#; (t!1)
#[HK .AM !HK AM ]> (t#di) G G? G G? GD #HK [BM !qBG . BM ]; (t). G G? G? GD
#1.5; (t!2)#0.35> (t!1)
(28)
#0.175> (t!2)#0.35> (t!1)
Or
#0.175> (t!2).
¹ > (t#di)#[HK .AM !HK AM ]> (t#di) G GD G G? G G? GD
> (t)"!0.805> (t!1)#0.1> (t!2)#; (t!1)
#H) [BM !qBGBM ]; (t) G G? G? GD
#1.5; (t!2)#0.35> (t!1)!0.175> (t!2)
"HK e (t#di)#¹ > (t#di). G GD G GKD
(29)
#0.35> (t!1)!0.175> (t!2).
Operating on Eq. (29) by AM gives G?
> (t)"!0.995> (t!1)#0.2> (t!2)#; (t!1)
AM .¹ > (t#di)#AM . [HK . AM !HK AM ]> (t#di) G? G GD G? G G? G G? GD #AM . HK .[BM !qBGBM ]; (t) G? G G? G? GD "AM . HK e (t#di)#AM ¹ > (t#di). G? G GD G? G GKD
!1.5; (t!2)!0.35> (t!1)!0.175> (t!2) !0.35> (t!1)!0.175> (t!2). Clearly, the subsystems plant are nonminimum phased. Suppose that we know o "4, o "5, o "6 (assump tion A3) and the reference signals > (t) i"1, 2, 3 are GK square waves with periods 100, 200, 300, respectively. The
(30)
Using Eq. (27) in Eq. (30), Eqs. (29)}(30) can be summarised as
¹ #[HK .AM !HK AM ]!HK [BM !qBGBM ] G G G? G G? G G? G? M M M M K K AM . [H . AM !H A ] ¹ B !AM . HK [B !qBGB ]#¹ [BM !qBGBM ] G? G G? G G? G G? G? G G? G? G G? G?
"
> (t#di) GD ; (t) GD
¹ HK 0 G > (t#di)# G e (t#di)# S (t#di). GKD GD K 2 AM ¹ AM .H !¹ !¹ M K (t!1#di) D G? G G? G G G R
Eq. (31) can be regarded as a linear time-varying system having inputs +e (t), and +> (t#di), and outputs GD GKD +; (t), and +> (t),. The terms in square brackets, for GD GD example, [HK .AM !HK AM ], and so on, arise due to the G G? G G? time-varying nature of the parameter estimates and a (t). G Since > (t), S (t#di) are bounded and following part GKD D iii) of Lemma 2, we can conclude that the model Eq. (31) is asymptotically time invariant and stable provided that ¹ and BK are both stable. Thus, from Eq. (31), G G? +; (t!di), and +> (t), are asymptotically bounded GD GD by +e (t),. If k is equal to zero, then +e (t), conGD G GD verges to zero and > (t) converges to > (t). Then GD GKD > (t) converges to > (t), which establishes the G GK theorem.
5. Numerical examples Example 1. To illustrate the performance and robustness of the proposed adaptive scheme, let us consider "rst the following interconnected subsystems, to which we apply
(31)
polynomials ¹ (q\), O (q\) and P (q\) are chosen as G G G ¹ (q\)"1#0.1q\!0.1q\, G O (q\)"1#0.2q\!0.1q\, G P (q\)"1!0.15q\#0.15q\ for i"1,2,3. G The unmodeled response (interconnections) are estimated using the Legendre series and the Fourier series truncated after three terms. We note that without adaptive input/output data "ltering and the interconnection compensation, the overall system is unstable. Figs 2-1} 2-3 show the system outputs. Example 2. Let Example 1, be a nominal system subject to unmodeled dynamics of di!erent classes as > (t)"(!0.905#0.1 sin(0.1t))> (t!1) #(0.3#0.1 cos(0.2t))> (t!2)#; (t!1) #1.5; (t!2)#0.35> (t!1)#0.175> (t!2)
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Fig. 2-1. (a) The "rst subsystem output > (t) using Legendre basis for Example 1. (b) The "rst subsystem output > (t)using Fourier basis for Example 1.
Fig. 2-2. (a) The second subsystem output > (t) using Legendre basis for Example 1. (b) The second subsystem output > (t) using Fourier basis for Example 1.
Fig. 2-3. (a) The third subsystem output > (t) using Legendre basis for example 1. (b) The third subsystem output > (t) using Fourier basis for Example 1.
#0.315> (t!1)#0.175> (t!2)
!0.175> (t!2)!0.35> (t!1)
#0.2 cos(2t)!0.2Y (t)
!0.175> (t!2)!0.2 cos(2t)!0.2(> (t!1)).
> (t)"(!0.805#0.1 sin(0.5t))> (t!1) #(0.1#0.1 cos(0.2t))> (t!2)#; (t!1) #1.5; (t!2)#0.35> (t!1) !0.175> (t!2)#0.35> (t!1) !0.175> (t!2)#0.2 sin(2t)!0.2> (t!1). > (t)"(!0.995#0.1 sin(0.7t))> (t!1) #(0.2#0.1 cos(0.5t))> (t!2) #; (t!1)!1.5; (t!2)!0.35> (t!1)
Under the same conditions of simulation as for Example 1, we obtain the subsystem behavior shown in Figs 3-1}3-3. As illustrated by the previous examples, by applying an adaptive input}output data "ltering and the interconnection compensation, the decentralized model reference adaptive control (DMRAC) scheme can be constructed even for nonminimum-phase subsystems in presence of unmodeled dynamics. Note that in the simulated examples, we have chosen deliberately the reference models > (t)(i"1, 2, 3) having di!erent periods to treat the GK generalized case when the interconnections are also through the local reference models > (t) (i"1, 2, 3). GK
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Fig. 3-1. (a) The "rst subsystem output > (t) using Legendre basis for Example 2. (b) The "rst subsystem output > (t) using Fourier basis for Example 2.
Fig. 3-2. (a) The second subsystem output > (t) using Legendre basis for Example 2. (b) The second subsystem output > (t)using Fourier basis for Example 2.
Fig. 3-3. (a) The third subsystem output > (t) using Legendre basis for Example 2. (b) The Third subsystem output > (t) using Fourier basis for Example 2.
6. Conclusion
Appendix
We have proposed a totally decentralized model reference adaptive control (DMRAC) for interconnected subsystems. The general approach is based on the interconnection output estimation using the polynomial series which o!ers a general solution for any class of interconnected subsystems described in input/output form. The parameter estimation scheme is a combined adaptive data "ltering with a recursive least-squares algorithm with parameter projection and normalization. The problem of minimum phase of the subsystems plant is handled by choosing the adaptive data "ltering parameters. Relaxed constraints on the interconnections and unmodeled dynamics are necessary to ensure the global system boundedness. The performance is illustrated by numerical examples.
General discrete orthogonal polynomials s (t) satisfy G the orthogonality property KG\ s (t)s (t)"d i, j"1, 2 , mi. G H GH R They satisfy the following recurrence relation: s (t)"u (t)s (t)#u s (t) G> G G G G\ i"1, 2 , mi, t"0, 2 , mi!1
(* )
u (t) and u are the recurrence coe$cients; their values G G depend on the particular discrete polynomials under consideration.
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We assume that for subsystem i, the unknown output < (t) may be expanded into a series of order mi as G < (t)"M2S(t)#= M (t) where S(t)"[s (t), 2 , s (t)]2. G G G KG Now consider the following series with expansion order l: S (t)"[s (t), 2 , st (t)]2 where l'mi. then < (t)"M 2S (t)#= (t) where "= (t)"("= M (t)" and the G G G G G parameter vectors M and M are assumed to be constant. G G Using the recurrence equation (*), we obtain s (t)"u (t)s (t)#u s (t) KG> KG KG KG KG\ and st (t)"ut (t)s (t)#ut s (t), KG KG\ then the expansion of < (t) into series of order l where G M is a constant vector of dimension l is equivalent to an G expansion of order mi where M is a time-varying vector G of dimension mi(l. This explains why a truncated series with time-varying parameters is more adequate to predict the interconnections with su$cient accuracy: < (t)"M2(t)S(t)#= (t). G G G References Datta, A., & Ioannou, P. (1991). Decentralized indirect adaptive control of interconnected systems. International Journal of Adaptive Control and Signal Processing, 5, 259}281. Dumont, G. A., Zervos, C., & Belanger, Pr. (1985). Automatic tuning of industrial PID controllers. American control conference, vol. 3, Boston (pp. 1573}1578). Dumont, G. A., & Zervos, C. (1986). Adaptive control based on orthonormal series representation. IFAC workshop on adaptive control and signal processing (pp. 371}376). Giri, F., M'saad, M. Dion, J. M., & Dugard, L. (1991). On the robustness of discrete time indirect adaptive (linear) controllers. Automatica, 27(1), 153}159.
Greblicki, W. (1994). Non parametric identi"cation of wiener systems by orthogonal series. IEEE ¹ransactions on Automatic Control, 39(1), 2077. Hejda, I., Vesely, V., & Murgas, J. (1990). Decentralized adaptive control of partially known systems. Problems of Control and Information ¹heory, 19, 31}45. Heuberger, P. S. C., Van Den Hof, P. M. J., & Bosgra, O. H. (1995). A generalized orthonormal basis for linear dynamical systems. IEEE ¹ransactions on Automatic Control, 40(3), 451. Horng, I. R., Chou, J. H., & Tsai, R. Y. (1986). Taylor series analysis of linear optimal control systems incorporating observers. International Journal of Control, 44, 1265}1272. Ioannou, P. (1986). Decentralized adaptive control of interconnected systems. IEEE Transactions on Automatic Control, 31, 291}298. James, L. (1994). Analysis on the Laguerre formula for approximating delay systems. IEEE ¹ransactions on Automatic Control, 39(7), 1517}1521. Lee, T. T., & Tsay, Y. F. (1986). Analysis and optimal control of discrete linear time-varying systems via discrete general orthogonal polynomials. International Journal of Control, 44, 1427}1436. Maciej, N. (1988). Functional series modelling approach to identi"cation of nonstationary stochastic systems. IEEE ¹ransactions on Automatic Control, 33(10), 955}961 Ossman, K. A. (1989). Indirect adaptive control for interconnected systems. IEEE ¹ransactions Automatic Control, 34, 908}910. Praly, L., & Trulsson, E. (1986). Decentralized indirect adaptive control APII, 20, 295}315. Reed, J. S., & Ioannou, P. A. (1988). Discrete-time decentralized adaptive control. Automatica, 24, 419}421. Sripada, N. R., & Fisher, D. G. (1987). Improved least squares identi"cation. International Journal Control, 46(6), 1889}1913. Wei, W. S. (1990). ¹ime series analysis. Copyright by Addison-Wesley Publishing Company. Wen, C. (1994). Decentralized adaptive regulation. IEEE. ¹ransactions on Automatic Control, 39, 2163}2166. Wen, C., & David, J. H. (1992). Global boundedness of discrete time adaptive control just using estimator projection. Automatica, 28, 1143}1157. Zinober, A. Si. (1983). Comments on &A simple criterion for stability of linear discrete systems'. International Journal of Control, 38, 1085.
Technical Communique
A robust decentralized model reference adaptive control for non-minimum-phase interconnected systems夽 M. Makoudi* , L. Radouane LESSI, De& partement de Physique Faculte& des Sciences, B.P. 1796, 30000 Fe% s, Morocco Received 21 February 1997; revised 24 July 1998; received in "nal form 22 February 1999
Abstract In this paper, we present a decentralized model reference adaptive control for interconnected subsystems in the sense that no information exchange occurs between the subsystems. The approach is based on the interconnection output estimation using the polynomial series which o!ers a general solution for interconnected subsystems. The parameter estimation scheme is a combined adaptive data "ltering with a recursive least-squares algorithm with parameter projection and normalization. The problem of minimum phased subsystems is handled by an adaptive input}output data "ltering. Hence the zeros of each subsystem estimated model are relocated inside the unit circle. This estimated model which is minimum phased is then used for the control synthesis. It is shown that the stability conditions based on weak interconnections are relaxed. Also the robustness of the proposed adaptive control against unmodeled dynamics is stated. Finally, the results are illustrated by numerical examples. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive decentralized control; Model reference control; Polynomial approach; Interconnection estimation; Nonminimum phase; Data "ltering
1. Introduction In the control of Large-scale systems, one usually faces poor knowledge on the plant parameters and interconnections between subsystems. Thus the adaptive control technique in this case is an appropriate strategy to be employed. Moreover, if some subsystems distribute distantly, it is di$cult for a centralized controller to gather feedback signals from these subsystems. An e!ective way to handle this di$culty is to apply decentralized control strategies whereby each subsystem is controlled independently on the basis of its own performance criterion and locally available information. The majority of the results on Large-scale systems described in input/output form and decentralized control refer to systems that consist of
* Corresponding author. Tel.: 00212-5-642-389; fax: 00212-5642-5000; e-mail: [email protected]. 夽 The original version of this paper was presented at the IFAC international workshop on automation in the steel industry: current practice and future developments. Kyungjoo, Korea, 16}18 July 1997. This paper was recommended for publication in revised form by Editor Peter Dorato.
weakly coupled subsystems (Wen, 1992, 1994; Datta and Ioannou, 1991; Hejda et al., 1990; Ioannou, 1986; Ossman, 1989; Praly and Trulsson, 1986; Reed and Ioannou, 1988). The subsystems may be considered in isolation, and the controllers received for the isolated subsystems may be applied as decentralized controllers to the coupled overall system. However, the information which may be extracted from the interconnections are ignored. No attempt has been made to estimate them in order to improve the control performance and robustness. Here, we present a decentralized model reference adaptive control (DMRAC) for interconnected systems. The main idea is to predict the interconnection outputs acting on each subsystem to relax the hypothesis of weak interconnections. These predictions are used for the synthesis of the local control. The prediction method is based on expressing the interconnection outputs as a linear combination of a set of orthogonal known functions of a basis. Note that the polynomial approach has received much attention in mathematical modeling, identi"cation and control literature (Horng et al., 1986; Lee and Tsay, 1986; Dumont et al., 1985, 1986; James, 1994; Maciej, 1988; Greblicki, 1994; Wei, 1990; Heuberger et al., 1995). The parameter estimation scheme is a combined adaptive
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 5 4 - 0
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data "ltering with a recursive least-squares algorithm with parameter projection and normalization. The problem of minimum phase of the subsystems plant is handled by an adaptive input output data "ltering. Then the zeros of the estimated subsystem model are relocated inside the unit circle. The obtained minimum-phase models are used in the construction of the controllers. Relaxed constraints on the interconnections are necessary to ensure the global system boundedness. It is also shown that this scheme is robust with respect to unmodeled dynamics. The work is organised as follows: In Section 2, the system is de"ned and all assumptions are outlined. The parameter estimator is described in Section 3. In Section 4, global stability of the indirect adaptive controller is established. In Section 5, two numerical examples are given for illustration.
'* (t!1) is a regression vector and h* denotes a vector G G containing unknown parameters of the nominal subsystem model i.e. '*(t!1)2"[> (t!1)2> (t!ni); (t!di)2 G G G G ; (t!di!mi)], G h*2"[!a 2!a b 2 b ]. G G GLG G GKG Here, the objective is to derive a decentralized control law to stabilize the global system with no information exchange between the subsystems. It is assumed that for each subsystem i, the output < (t) is not available. Thus G we propose that each subsystem predicts these interconnection outputs in real time. The prediction method is based on expressing the interconnection outputs as a linear combination of a set of known functions of a basis.
2. Problem statement 2.1. Functional series modeling The system to be considered consists of N interconnected single-input}single-output (SISO) subsystems described in input/output form by A (q\)> (t)"B (q\); (t) G G G G , # C (q\)> (t!1)#m (t), GH H G H H$G i"1, 2 , N
(2) (3)
The polynomials C (q\) are the interconnection inputs GH of > (t) from the other subsystems > (t) with G H C (q\)"c #))) #c q\PGH, (4) GH GH GHPGH di is the time delay index and m (t) represents the unG modeled response of subsystem i. Subsystem (1) can be expressed as > (t)"'*(t!1)2h*#< (t), G G G where , < (t)" C (q\)> (t!1)#m (t), G GH H G H H$G
S(t)"
"q\BGB (q\); (t) G G , # C (q\)> (t!1)#m (t), (1) GH H G H H$G where > (t), ; (t) are the output and input, respectively, G G of the subsystem i. A (q\), B (q\) and C (q\) are G G GH scalar polynomials in the unit delay operator q\. A (q\)"1#a q\#)))#a q\LG, G G GLG B (q\)"b #)))#b q\KG, b O0. G G GKG G
Let ¹"+1,2, 2 , R, and let S(t)"+s (t), 2 , s (t),2 K be the set of linearly bounded independent sequences (functions of discrete-time t) on T. The elements of S will be further referred to as basis sequences or basis functions. Possible sets of functions that could be used include powers of time s (t)"tH\, j"1, 2 , m (which will H be referred to as the Legendre basis) or
(5)
cos %( j!1)t if j is odd, sin %( j!1)t if j is even
(which is called the Fourier basis). Now we will use the functional modeling of the interconnections. We assume that for subsystem i, the unknown output < (t) may be expanded into a series. G < (t)"M2 (t)S(t)#= (t), (6) G G G where S(t)"[s (t), 2 , s (t)]2, M (t)3RK, K G = (t) is the residual between the output < (t) and the G G series which may be unbounded. M (t) is a time-varying G vector of unknown elements to be estimated. The vector M (t) is taken to be time varying, to represent more G precisely the interaction term < (t). This modelization is G more adequate than < (t)"M2S(t)#= (t) where M is G G G G a constant vector (see the appendix). By substitution of Eq. (6) into Eq. (1), we obtain A (q\)> (t)"B (q\); (t)#M2(t)S(t)#= (t). (7) G G G G G G Further assumptions concerning the plant are made: A1: An upper bound n* on the orders ni and mi is G known.
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A2: The time delays di, i"1, 2 , N are known. A3: There exists a known scalar o 3R> such that G #h*#4o . G G A4: There exists a "nite positive scalar k such that G "= (t)"4k Z (t). G G G Z (t)"p Z (t!1)#max [#' (t!1)#, Z ], G G G G G 0(p (1, Z '0, Z (0)'0. G G G Subsystem (7) can be given the following form: > (t)"'2 (t!1)h (t)#= (t). G G G G The augmented parameter and observation vectors are de"ned by h2 (t)"(!a 2 !a * b 2 b * M2 (t)), (8) G G GLG G GLG G '2(t!1)"(> (t!1)2 > (t!ni*) G G G ; (t!di)2 ; (t!di!ni*)S2(t)). (9) G G Thus the normalizing signal Z (t) depends also on G the interconnection terms through the vector (S(t)) in Eq. (9) and the vector M (t) in the augmented parameter G vector (8). Assumption A4 means that the unmodeled response = (t) (Eq. (7)) should be linearly bounded by the adaptive G input}output data activity measure Z (t), which depends G also on the vector S(t). We remark that the new bound k is less restrictive G then the classical one used in Wen (1992, 1994) and Praly (1986) which is given by "< (t)"4g Z (t), where < (t) is G G G G given by Eq. (5) and Z (t)"p Z (t!1)#max [#'* (t!1)#, Z ], G G G G G 0(p (1, Z '0, Z (0)'0, G G G '*(t!1)2"[> (t!1)2 > (t!ni*) G G G ; (t!di) 2 ; (t!di!ni*)]. G G In the present work, only the residual error = (t) is G constrained to satisfy assumption A4. This is due to the introduction of the term (S(t)) in the observation vector (Eq. (9)) and of the vector M (t) in the parameter vector G (Eq. (8)). The interconnections are thus taken into account by each subsystem. Then the normalizing signal Z (t) depends on local subsystem signals and implicitly G on the interconnection signals also. Therefore this new assumption makes the present work di!erent from the previous ones on the same subject. Assumption A4 is no more restrictive.
3. Parameter estimation We plan to design an adaptive DMRAC controller including adaptive data "ltering which should stabilise
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the subsystem and cause > (t) to track a bounded referG ence model > (t). GK 3.1. **Good data++ model The key issue to get a robust parameter estimation may be viewed as a &&good data'' model and a robust parameter adaptation algorithm with respect to bounded disturbances and time varying parameters. Here we de"ne a new good data model adapted to our problem. Let the % linear operator be de"ned by G % ( . ) (t!k)"aI(t)( . ) (t!k) (10) G G G G with 0(a (t)(1. G Given two asymptotically stable polynomials O (q\) G and P (q\) and an % operator, then operating on subG G system model (7) by the "lter % (O /P ) leads to G G G % A (q\)> (t)"% B (q\); (t) G G GD G G GD #% M2 (t)S (t)#= (t), G G D G % (> (t)#2#a * > (t!ni*)) G GD GLG GD "% (b ; (t!di)#2#b * ; (t!di!ni*)) G G GD GLG GD #% M2 (t)S (t)#= (t). G G D G As A (q\) and B (q\) are time-invariant polynomials, G G we obtain (% > (t)#2 #a * % > (t!ni*)) G GD GLG G GD "(b % ; (t!di)#2 #b * % ; (t!di!ni*)) G G GD GLG G GD #M2(t)S (t)#= (t) (11) G D G with = (t)"% = (t), G G GD where
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O (q\)( . ) (t)"P (q\)( . ) (t). G G G GD The proposed &&good data'' model (11) involves hence the following main design feature: the input}output signals are "ltered by % (O /P ). The "lter O /P , which G G G G G should be low-pass, can be used to reduce the &&highfrequency'' modes of the unmodeled dynamics. The % operator allows to remove the minimum phase design G assumption (Lemma 3). Note that when a (t)"1 (Eq. G (10)) the proposed "lter reduces to the classical one O /P G G considered in Giri et al. (1991) and in Sripada and Fisher (1987). Subsystem "ltered model (11) may be expressed in regression form as follows: > (t)"'2 (t!1, a )h (t)#= (t) GD GD G G G
(13)
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with the augmented parameter and observation vectors de"ned by h2(t)"(!a 2!a * b 2 b * M2(t)), G G GLG G GLG G '2 (t!1, a )"(a (t)> (t!1)2 aLG* (t)> (t!ni*) G GD GD G G GD ;aBG (t); (t!di) 2 aLG*>BG (t) G GD G ; (t!di!ni*)S2 (t)). GD D
"m (t)"(a (t)"m (t)" G G GD and
Lemma 1. Consider subsystem model (13) subject to assumption A4, then there exists a positive scalar k such G that "= (t)"4k Z (t) where k (k G G GD G G and
m (t)"*A (t)a (t)> (t!1)#*B (t)aBG (t); (t!di) G G G GD G G GD #a (t)F (t)#g (a (t)> (t!1) G GD G G GD #aBG (t); (t!di)), G GD then
(14)
Z (t)"p Z (t!1)#max (#' (t!1)#, Z ). GD G GD GD G '2 (t!1)"(> (t!1)2 > (t!ni*) GD GD GD ; (t!di). 2 ; (t!di!ni*)S2 (t)). GD GD D
"< (t)"(a (t) G G
, (c #2 #c q\PGH)> (t!1)#m (t) GH GHPGH HD GD H H$G (a (t)"< (t)". G GD From assumption A4, we have
Thus the "lter % (O /P ) permits to relax the conG G G straints on the unmodeled dynamics.
"= (t)"4k Z (t) then "= (t)"4k Z (t), G G G GD G GD and "= (t)"4a (t)k Z (t)"k Z (t). G G G GD G GD As 0(a (t)(1, the bound k decreases, so large unG G modeled dynamics may be tolerated and classical weak interconnection assumption is relaxed.
Proof. Consider the "ltered subsystem model de"ned by
3.2. Robust parameter adaptation algorithm
% A (q\)> (t)"% B (q\); (t) G G GD G G GD #% M2(t)S (t)#= (t) G G D G where
Let hK (t) denote the estimate of h (t) at time t, using the G G least-squares parameter estimation with parameter projection, signal normalization and data "ltering
= (t)"% = (t)"< (t)!M2 (t)S (t), G G GD G G D , < (t)" % (c #2#c )> (t!1)#% m (t). G G GH GHPGH HD G GD H H$G Then , < (t)" (c a (t)#2#c aPGH (t)q\PGH) G GH G GHPGH G H H$G > (t!1)#m (t). HD G The unmodeled dynamics +m (t), may contain nonlineariG ties and bounded disturbances. We will consider without loss of generality, the following class: m (t)"*A (t)> (t!1)#*B (t); (t!di)#F (t) G G G G G G #g (> (t!1)#; (t!di)), G G G where #*A (t)#4i , #*B (t)#4i , G G G G where
#F (t)#4i , G G
m (t)"*A (t)> (t!1)#*B (t); (t!di)#F (t) GD G GD G GD GD #g (> (t!1)#; (t!di)), G GD GD
' (t!1, a )e (t) GD G GD hK (t)"I hK (t!1)# , G G Z (t)#'2 (t!1, a )' (t!1, a ) GD GD G GD G (15) where I represents the projection operator necessary to ensure #hK (t)#4o (note that if #hK (t)#4o then G G G G #hK * (t)#4o from assumption A2) e (t) is the prediction G G GD error de"ned as e (t)"> (t)!hK 2(t!1)' (t!1, a ), GD GD G GD G e (t)"> (t)!(!aL (t!1), 2 , !aL * (t!1), GD GD G GLG bK (t!1), 2 , bK * (t!1), M K 2(t!1))a G GLG G ;[a (t)> (t!1), 2 , aLG (t)> (t!ni*), G GD G GD aBG (t); (t!di), 2 , aBG>LG* (t) G GD G ; (t!di!ni*), S2 (t)]2, GD D e (t)"> (t)#aL (t!1)a (t)> (t!1)#2 GD GD G G GD #aL * (t!1)aLG* (t)> (t!ni*) GLG G GD K !b (t!1)aBG (t); (t!di)!2 G G GD !bK * (t!1)aBG>LG* (t); (t!di!ni*) GLG G GD !M K 2 (t!1)S (t). G D
(16)
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The estimation error for subsystem model (17) can be written as e (t)"AK (t!1, q\)> (t)!BK (t!1, q\) GD G? GD G? ; (t!di)!M K 2 (t!1)S (t) GD G D "AK (t!1, q\)> (t)!BK (t!1, q\) G? GD G? ; (t)!M K 2 (t!1)S (t) GD G D where AK (t!1, q\) G? "1#aL (t!1)a (t)q\#2 G G #aL * (t!1)aLG* (t)q\LG*, BK (t!1, q\) G? GLG G "bK (t!1)aBG (t)#2 G G #bK * (t!1)aLG>BG (t)q\LG*. GLG G Thus at the present stage, the estimation error for subsystem i is de"ned in terms of the polynomials AK and G? BK , where BK is stable (Lemma 3). These polynomials G? G? will be used for the control synthesis (Eq. (19)). Lemma 2. Consider the subsystem model given in (13), subject to assumptions A1}A4 then the algorithm (15)}(16) has the following properties: (a) #hK (t)#4o , G G (b) ¹here exists a positive constant o such that for all GC (l, l)3N one has: J>J "e (t)" GD 4o #lk . GC G Z (t) > GD (c) ¹here exists a positive constant o such that for all GM (l, l)3N one has: J>J #hK (t)!hK (t!1)#4o #lk . G G GM G RJ> The proof of Lemma 2 may be carried out along the same lines as in Giri et al. (1991) and is then omitted. Lemma 3. Consider the ,ltered subsystem error model (18), subject to assumptions A1}A3, if "a (t)"(1/(ni*, G then BK (t!1, q\) is a stable polynomial. G? Proof. Given that LG* bK (t!1)(#hK (t!1)#(o G G I GI and 0(a (t)(1, we get G LG* bK (t!1) LG* bK (t!1) a (t) GI aI (t)(a (t) GI ( G o G G (t!1) G K K b (t!1) b (t!1) b K I G I G G
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BK (t!1, q\) is a stable polynomial if G? 1 a(t)/bK (t!1) o4 G G G ni* (see Zinober, 1983). Then a (t)( "bK (t!1)"/o (ni*. G G G 4. Decentralized adaptive control The proposed feedback control signal is generated from (Fig. 1) HK (t, q\)BK (t!1, q\); (t)#GK (t, q\)> (t) G G? GD G GD #QK (t#di, q\)S (t#di) G D "¹ (q\)> (t#di), (19) G GKD where > (t) is the given bounded output of the reference GK model, and O (qU)> (t)"P (qU)> (t). G GK G GKD ¹ (q\) is an arbitrary stable monic polynomial of G degree nti(ni*#di and HK (t, q\)"1#hK (t)q\#2 G G #hK (t)q\LFG, nhi(di GLFG GK (t, q\)"gL (t)#gL (t)q\#2 G G G #gL (t)q\LEG,ngi(ni* GLEG QK (t, q\)"qL 2 (t)#qL 2 (t)q\#2#qL 2 (t)q\LOG, G G G GLOG qL 3RK, j"0, 2 , nqi, nqi(di. GH The term QK (t#di, q\)S (t#di) is introduced beG D cause of the existence of the interconnections and is then used for their e!ect reduction. The estimation error is given by e (t)"> (t)!hK 2(t!1)' (t!1, a ) GD GD G GD G "AK (t!1, q\)> (t)!BK (t!1, q\); (t!di) G? GD G? GD !M K 2 (t!1)S (t). (20) G D Adding and subtracting q\BGBK (t!1, q\); (t) in G? GD Eq. (20) gives e (t)"AK (t!1, q\)> (t)!q\BGBK (t!1, q\) GD Ga GD G? ; (t)!M K 2 (t!1)S (t)#[q\BG . BK (t!1, q\) GD G D G? !BK (t!1, q\) . q\BG]; (t). (21) G? GD Operating on Eq. (21) by HK (t, q\) gives G HK (t, q\)AK (t!1, q\)> (t) G G? GD "q\BGHK (t, q\)BK (t!1, q\); (t) G G? GD #HK (t, q\)e (t)#HK (t, q\)M K 2 (t!1)S (t) G GD G G D #HK (t, q\)[q\BG . BK (t!1, q\) G G? !BK (t!1, q\) . q\BG]; (t) (22) G? GD
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Fig. 1 The proposed decentralized adaptive controller.
and in view of Eq. (20) then HK (t, q\)AK (t!1, q\)> (t) G G? GD "q\BG[¹ (q\)> (t#di)!QK (t#di, q\)S (t#di) G GKD G D !GK (t, q\)> (t)]#HK (t, q\)e (t) G GD G GD #HK (t, q\)M K 2 (t!1)S (t) G G D #HK (t, q\)[q\BG . BK (t!1, q\) G G? !BK (t!1, q\) . q\BG]. (23) G? HK (t, q\), GK (t, q\) and QK (t, q\) are determined by G G G solving the following equality:
Proof. We de"ne AK . BK and AK BK : AK . BK " aL (t)bK (t!i)q\G\HOBK . AK G H G H and AK BK " aL (t)bK (t)q\G\H"BK AK G H G H We also de"ne BM "BK (t!1, q\) and BM "BK (t#di!1, q\). From Eq. (18), we have
HK (t, q\)AK (t!1, q\)#q\BGGK (t, q\)"¹ (q\) (24) G G? G G QK (t, q\)"HK (t, q\)M K 2 (t!1). (25) G G G The closed-loop control system equation is given by
e (t#di)"AM > (t#di)!qBGBM ; (t) GD G? GD G? GD
¹ (q\)[> (t)!> (t)] G GD GKD "HK (t, q\)e (t)#HK (t, q\) G GD G ;[q\BG . BK (t!1, q\)!BK (t!1, q\).q\BG]; (t). G? G? GD (26)
HK e (t#di)"HK .AM > (t#di)!HK . qBG BM ; (t) G GD G G? GD G G? GD
The indirect adaptive control algorithm above is globally convergent as is stated by the following theorem. Theorem 1. Consider subsystem model (13), subject to assumptions A1}A4 in closed-loop with the adaptive control law (19). ¹hen the resulting closed-loop system is globally stable in the sense that (i) ; (t), > (t) are bounded for all time, G G (ii) if k is equal to zero, then we have: G lim (> (t)!> (t))"0 as tPR. G GK
Thus,
!M K 2 (t!1#di)S (t#di). G D
!HK MK 2 (t!1#di)S (t#di), G G D HK e (t#di)"HK AM > (t#di) G GD G G? GD #[HK .AM !HK AM ]> (t#di) G G? G G? GD M !HK .qBGB ; (t) G G? GD !HK MK 2 (t!1#di)S (t#di), G G D HK e (t#di)"[¹ !q\BGGK ]> (t#di) G GD G G GD #[HK . AM !HK AM ]> (t#di) G G? G G? GD M !HK . qBGB ; (t) G G? GD !HK M K 2 (t!1#di)S (t#di), G G D
(27)
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HK e (t#di)"¹ > (t#di)!GK > (t)!HK BM ; (t) G GD G GD G GD G G? GD
the adaptive decentralized control with the interconnection estimation:
!HK M K 2 (t!1#di)S (t#di) G G D
> (t)"!0.905> (t!1)#0.3> (t!2)#; (t!1)
#[HK .AM !HK AM ]> (t#di) G G? G G? GD #HK [BM !qBG . BM ]; (t). G G? G? GD
#1.5; (t!2)#0.35> (t!1)
(28)
#0.175> (t!2)#0.35> (t!1)
Or
#0.175> (t!2).
¹ > (t#di)#[HK .AM !HK AM ]> (t#di) G GD G G? G G? GD
> (t)"!0.805> (t!1)#0.1> (t!2)#; (t!1)
#H) [BM !qBGBM ]; (t) G G? G? GD
#1.5; (t!2)#0.35> (t!1)!0.175> (t!2)
"HK e (t#di)#¹ > (t#di). G GD G GKD
(29)
#0.35> (t!1)!0.175> (t!2).
Operating on Eq. (29) by AM gives G?
> (t)"!0.995> (t!1)#0.2> (t!2)#; (t!1)
AM .¹ > (t#di)#AM . [HK . AM !HK AM ]> (t#di) G? G GD G? G G? G G? GD #AM . HK .[BM !qBGBM ]; (t) G? G G? G? GD "AM . HK e (t#di)#AM ¹ > (t#di). G? G GD G? G GKD
!1.5; (t!2)!0.35> (t!1)!0.175> (t!2) !0.35> (t!1)!0.175> (t!2). Clearly, the subsystems plant are nonminimum phased. Suppose that we know o "4, o "5, o "6 (assump tion A3) and the reference signals > (t) i"1, 2, 3 are GK square waves with periods 100, 200, 300, respectively. The
(30)
Using Eq. (27) in Eq. (30), Eqs. (29)}(30) can be summarised as
¹ #[HK .AM !HK AM ]!HK [BM !qBGBM ] G G G? G G? G G? G? M M M M K K AM . [H . AM !H A ] ¹ B !AM . HK [B !qBGB ]#¹ [BM !qBGBM ] G? G G? G G? G G? G? G G? G? G G? G?
"
> (t#di) GD ; (t) GD
¹ HK 0 G > (t#di)# G e (t#di)# S (t#di). GKD GD K 2 AM ¹ AM .H !¹ !¹ M K (t!1#di) D G? G G? G G G R
Eq. (31) can be regarded as a linear time-varying system having inputs +e (t), and +> (t#di), and outputs GD GKD +; (t), and +> (t),. The terms in square brackets, for GD GD example, [HK .AM !HK AM ], and so on, arise due to the G G? G G? time-varying nature of the parameter estimates and a (t). G Since > (t), S (t#di) are bounded and following part GKD D iii) of Lemma 2, we can conclude that the model Eq. (31) is asymptotically time invariant and stable provided that ¹ and BK are both stable. Thus, from Eq. (31), G G? +; (t!di), and +> (t), are asymptotically bounded GD GD by +e (t),. If k is equal to zero, then +e (t), conGD G GD verges to zero and > (t) converges to > (t). Then GD GKD > (t) converges to > (t), which establishes the G GK theorem.
5. Numerical examples Example 1. To illustrate the performance and robustness of the proposed adaptive scheme, let us consider "rst the following interconnected subsystems, to which we apply
(31)
polynomials ¹ (q\), O (q\) and P (q\) are chosen as G G G ¹ (q\)"1#0.1q\!0.1q\, G O (q\)"1#0.2q\!0.1q\, G P (q\)"1!0.15q\#0.15q\ for i"1,2,3. G The unmodeled response (interconnections) are estimated using the Legendre series and the Fourier series truncated after three terms. We note that without adaptive input/output data "ltering and the interconnection compensation, the overall system is unstable. Figs 2-1} 2-3 show the system outputs. Example 2. Let Example 1, be a nominal system subject to unmodeled dynamics of di!erent classes as > (t)"(!0.905#0.1 sin(0.1t))> (t!1) #(0.3#0.1 cos(0.2t))> (t!2)#; (t!1) #1.5; (t!2)#0.35> (t!1)#0.175> (t!2)
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Fig. 2-1. (a) The "rst subsystem output > (t) using Legendre basis for Example 1. (b) The "rst subsystem output > (t)using Fourier basis for Example 1.
Fig. 2-2. (a) The second subsystem output > (t) using Legendre basis for Example 1. (b) The second subsystem output > (t) using Fourier basis for Example 1.
Fig. 2-3. (a) The third subsystem output > (t) using Legendre basis for example 1. (b) The third subsystem output > (t) using Fourier basis for Example 1.
#0.315> (t!1)#0.175> (t!2)
!0.175> (t!2)!0.35> (t!1)
#0.2 cos(2t)!0.2Y (t)
!0.175> (t!2)!0.2 cos(2t)!0.2(> (t!1)).
> (t)"(!0.805#0.1 sin(0.5t))> (t!1) #(0.1#0.1 cos(0.2t))> (t!2)#; (t!1) #1.5; (t!2)#0.35> (t!1) !0.175> (t!2)#0.35> (t!1) !0.175> (t!2)#0.2 sin(2t)!0.2> (t!1). > (t)"(!0.995#0.1 sin(0.7t))> (t!1) #(0.2#0.1 cos(0.5t))> (t!2) #; (t!1)!1.5; (t!2)!0.35> (t!1)
Under the same conditions of simulation as for Example 1, we obtain the subsystem behavior shown in Figs 3-1}3-3. As illustrated by the previous examples, by applying an adaptive input}output data "ltering and the interconnection compensation, the decentralized model reference adaptive control (DMRAC) scheme can be constructed even for nonminimum-phase subsystems in presence of unmodeled dynamics. Note that in the simulated examples, we have chosen deliberately the reference models > (t)(i"1, 2, 3) having di!erent periods to treat the GK generalized case when the interconnections are also through the local reference models > (t) (i"1, 2, 3). GK
M. Makoudi, L. Radouane/Automatica 35 (1999) 1499}1508
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Fig. 3-1. (a) The "rst subsystem output > (t) using Legendre basis for Example 2. (b) The "rst subsystem output > (t) using Fourier basis for Example 2.
Fig. 3-2. (a) The second subsystem output > (t) using Legendre basis for Example 2. (b) The second subsystem output > (t)using Fourier basis for Example 2.
Fig. 3-3. (a) The third subsystem output > (t) using Legendre basis for Example 2. (b) The Third subsystem output > (t) using Fourier basis for Example 2.
6. Conclusion
Appendix
We have proposed a totally decentralized model reference adaptive control (DMRAC) for interconnected subsystems. The general approach is based on the interconnection output estimation using the polynomial series which o!ers a general solution for any class of interconnected subsystems described in input/output form. The parameter estimation scheme is a combined adaptive data "ltering with a recursive least-squares algorithm with parameter projection and normalization. The problem of minimum phase of the subsystems plant is handled by choosing the adaptive data "ltering parameters. Relaxed constraints on the interconnections and unmodeled dynamics are necessary to ensure the global system boundedness. The performance is illustrated by numerical examples.
General discrete orthogonal polynomials s (t) satisfy G the orthogonality property KG\ s (t)s (t)"d i, j"1, 2 , mi. G H GH R They satisfy the following recurrence relation: s (t)"u (t)s (t)#u s (t) G> G G G G\ i"1, 2 , mi, t"0, 2 , mi!1
(* )
u (t) and u are the recurrence coe$cients; their values G G depend on the particular discrete polynomials under consideration.
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We assume that for subsystem i, the unknown output < (t) may be expanded into a series of order mi as G < (t)"M2S(t)#= M (t) where S(t)"[s (t), 2 , s (t)]2. G G G KG Now consider the following series with expansion order l: S (t)"[s (t), 2 , st (t)]2 where l'mi. then < (t)"M 2S (t)#= (t) where "= (t)"("= M (t)" and the G G G G G parameter vectors M and M are assumed to be constant. G G Using the recurrence equation (*), we obtain s (t)"u (t)s (t)#u s (t) KG> KG KG KG KG\ and st (t)"ut (t)s (t)#ut s (t), KG KG\ then the expansion of < (t) into series of order l where G M is a constant vector of dimension l is equivalent to an G expansion of order mi where M is a time-varying vector G of dimension mi(l. This explains why a truncated series with time-varying parameters is more adequate to predict the interconnections with su$cient accuracy: < (t)"M2(t)S(t)#= (t). G G G References Datta, A., & Ioannou, P. (1991). Decentralized indirect adaptive control of interconnected systems. International Journal of Adaptive Control and Signal Processing, 5, 259}281. Dumont, G. A., Zervos, C., & Belanger, Pr. (1985). Automatic tuning of industrial PID controllers. American control conference, vol. 3, Boston (pp. 1573}1578). Dumont, G. A., & Zervos, C. (1986). Adaptive control based on orthonormal series representation. IFAC workshop on adaptive control and signal processing (pp. 371}376). Giri, F., M'saad, M. Dion, J. M., & Dugard, L. (1991). On the robustness of discrete time indirect adaptive (linear) controllers. Automatica, 27(1), 153}159.
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