- Email: [email protected]
or time varying delay
Automatica 35 (1999) 1417}1426
Brief Paper
Robust decentralized adaptive control for non-minimum phase systems with unknown and/or time varying delay夽 M. Makoudi*, L. Radouane LESSI FaculteH des Sciences, DeH partement de Physique, B.P. 1796, Atlas, 30000, Morocco Received 28 July 1997; revised 15 October 1998; received in "nal form 8 February 1999
Abstract This paper presents a decentralized model reference adaptive control (DMRAC) for interconnected subsystems with unknown and/or time-varying time delay. The decentralization 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 signal normalization. A &&good data'' model is de"ned by an adaptive "ltering of the input and output signals. The obtained model permits to deal with non-minimum phased subsystems with unknown or time-varying dead time and at the same time to relax the hypothesis of weak interconnections for decentralized control. The performance of the studied scheme is illustrated by numerical examples. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Large-scale systems; Decentralized adaptive control; Non-minimum phase systems; Time-varying delay; Good data model; Interconnection prediction
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$cult for a centralized controller to gather feedback signals from the subsystems. Also, the design and implementation of the centralized controller may be complicated. An e!ective way to handle this di$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 decentralized control of large-scale systems described in input/output form refer to systems that consist of weakly coupled subsystems (Wen and Hill, 1992; Datta and Ioannou, 1991;
***** * Corresponding author. Tel.: (212)5-642389; fax: (212)5-642500; e-mail: [email protected]. 夽 This papeer was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor G. Dumont under the direction of Editor C.C. Hang.
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 local controllers received for the isolated subsystems may be applied as decentralized controllers to the overall system. However, the information which may be extracted from the interconnections are generally ignored and considered as perturbations (Ossman, 1989; Praly and Trulsson, 1986). No attempt has been made to estimate them in order to improve the control performance and robustness. In this paper, we present a decentralized model reference adaptive control (DMRAC) for interconnected subsystems. 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 modelling, identi"cation and control literature (Heuberger et al., 1995; Lee and Tsay, 1986; Niedzwecki, 1988; Greblicki, 1994; James, 1994; Wei, 1990; Zervos et al., 1985, 1988; Zervos and Dumont, 1988). The parameter estimation
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 2 - 7
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scheme is a combined adaptive data "ltering with a recursive least-squares algorithm with parameter projection and signal normalization. The problem of minimum phase is handled by adjusting the data "ltering parameter at each instant. Then all the subsystem zeros are relocated inside the unit circle. It is demonstrated that the input/output data "ltering permits also to solve the di$cult problem of model reference adaptive control in the presence of unknown time-delay variations, where only the upper limit on the time delay must be known. Then, relaxed constraints on the interconnections are necessary to ensure the global system boundedness. The scheme robustness with respect to unmodelled dynamics is also simultaneously improved. The paper is organized as follows: In Section 2, we present the system description. A robust DMRAC for non-minimum phase subsystems with varying time delay is presented in Section 3. In Section 4, the robustness of the DMRAC is established without a priori information about the exact invariant time delay. Numerical examples are "nally given for illustration.
2. Problem statement 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!di) G G G G , # C (q\)> (t!1)#m (t), (1) GH H G H HOG where > (t), ; (t) are the output and input, respectively, of G G the subsystem i. A (q\), B (q\) and C (q\) are scalar G G GH polynomials in the unit delay operator q\. A (q\)"1#a q\#2#a q\LG, G G GLG
(2)
B (q\)"b #2#b q\KG, b O0. (3) G G GKG G The polynomials C (q\) are the interconnection inputs GH of > (t) from the other subsystems > (t) with G H C (q\)"c #2#c q\PGH. (4) GH GH GHPGH The time-delay index di (di51), is unknown and/or time varying. m (t) represents the unmodelled response of G subsystem i. Subsystem (1) can be expressed as > (t)"'*(t!1)2h*#< (t), G G G G where , < (t)" C (q\)> (t!1)#m (t). GH H G G H O H G
(5)
'*(t!1)2 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) G G G ; (t!di)2; (t!di!mi)], G G h*2"[!a 2!a b 2b ]. G G GLG G GKG In this paper, the objective is to derive a decentralized control law to stabilize the global system with no information exchange among the subsystems. It is assumed that for subsystem i, the output < (t) is not available. G Thus, we propose that each subsystem predicts its 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. 2.1. Functional series modelling Let ¹"+1, 2,2, a, and let S(t)"+s (t),2, s (t),2 be K the set of linearly bounded independent sequences (functions of discrete-time t) on ¹. The elements of S will be further referred to basis functions. Possible sets of functions that could be used include powers of time s (t)"tH\, j"1,2, m (which will be referred to as the H Legendre basis) or cosine functions
S(t)"
cos n( j!1)t if j is odd sin n( j!1)t
if j is even.
(which is called the Fourier basis). Now, we will use the functional modelling of the interconnections. We assume that for subsystem i, the unknown output < (t) may be expanded into a series as G < (t)"M2 (t)S(t)#= (t), (6) G G G where S(t)"[s (t),2, s (t)]2, s (t) are known functions of K G a basis, M (t)3RK. 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 (1), we obtain A (q\)> (t)"B (q\); (t!di)#M2(t)S(t)#= (t). (7) G G G G G G Further assumptions concerning the plant are made: A1: An upper bound ni* on the orders ni, mi and rij, i, j"1, 2,2, N is known. A2: There exists a known scalar o 3R> such that G #h*#4o . G G
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
A3: 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)"#"; (t!1)"#"S(t)",Z ], G G G G G G 0(p (1, Z '0, Z (0)'0. G G G In this paper, we plan to design a decentralized adaptive control which should stabilize the global system and cause > (t) to track a bounded reference model > (t). G GK A major drawback with model reference adaptive control methods is that they require a priori knowledge of the dead time of the process transfer function which must be minimum phased. In many industrial processes, the time-delay di is either unknown accurately or even time varying.
3. DMRAC in presence of varying time delay
(8) by the "lter % (O /P ) leads to G G G % A (q\)> (t)"% B (q\); (t)#% M2(t)S (t) G G G G G G G G #% = (t) G G with
1419
(14)
O (q\)( . ) (t)"P (q\)( . ) (t). G G G The plant "ltered model (14) may be expressed in regression form as follows: > (t)"'2 (t!1, a )h (t)#= (t) (15) G G G G G '2 (t!1, a ) G G "(a (t)> (t!1)2a (t)LG*> (t!ni*)a (t); (t!1)2 G G G G G G a (t)LG*>BGK?V; (t!ni*!dimax)S2(t)), G G = (t)"% = (t). G G G
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. Let the % linG ear operator be de"ned as (Makoudi and Radouane, 1995, 1997)
Remarks. (1) The plant "ltered model (14) hence involves the main design features. The input}output signals are "ltered by % (O /P ). The "lter (O /P ), which should G G G G G be low-pass, is used to reduce the high-frequency modes of the unmodelled dynamics. The new &&good data'' model involves in spite of the low-pass "ltering, an exponential adaptive "ltering (% ). The % operator allows to remove G G the minimum phase assumption while improving the robustness of the DMRAC (Lemma 3). This is equivalent to generating a minimum phase estimated model for each subsystem. Note that when a (t)"1 the proposed "lter G reduces to the classical one O /P considered in Giri et al. G G (1991) and Sripada and Fisher (1987). (2) Note that in the previous works dealing with small interconnections (see, for example, Wen and Hill, 1992; Datta and Ioannou, 1991; Hejda et al., 1990; Praly and Trulsson, 1986), the classical assumption made for Eq. (5) is "< (t)"4k Z (t) where Z (t) depends only on the locally G G G G available information. k is required to be su$ciently G small for stability purpose. In our work, only the residual error = (t) (Eq. (8)) is constrained to satisfy assumption G A3. This is due to the introduction of the term (S(t)) in the observation vector and of the vector M (t) in the paraG meter vector (Eq. (11)). The interconnections are thus taken into account by each subsystem. Then the normalizing signal Z (t) depends on local subsystem signals G and implicitly on the interconnection signals also. Therefore, this new assumption makes the present work di!erent from the previous ones on the same subject. Assumption A3 is no more restrictive.
% > (t!k)"aI(t)> (t!k), (12) G G G G % ; (t!k)"aI (t); (t!k) (13) G G G G with 0(a (t)(1 ∀t, where a (t) will be adjusted; this G G corresponds to an adaptive exponential "ltering. Given two asymptotically stable polynomials O (q\) G and P (q\), and a % operator, then operating on plant G G
Lemma 1. Consider subsystem (7) subject to A3, then for subsystem model (14), there exists a scalar k '0 such that G "= (t)"4"k Z (t), G G G Z (t)"p Z (t!1)#max("> (t!1)"#"; (t!1)" G G G G G #"S (t)", Z ), 0(p (1, Z '0, Z (0)'0. G G G GD
If the time-delay di is time-varying i.e. dimin4di(t)4 dimax where dimin51 and only dimax is known, the normal approach in literature (Dumont, 1982; Vodel and Edgar, 1980) is to model subsystem (7) by A (q\)> (t)"B (q\); (t)#M2(t)S(t)#= (t) G G G G G G with
(8)
q\LG*\BGK?V. B (q\)"b q\#2#b * G G GLG >BGK?V Subsystem (8) can be expressed as
(9)
> (t)"'2(t!1)h (t)#= (t), G G G G where
(10)
'2(t!1)"(> (t!1)2> (t!ni*) G G G ; (t!1)2; (t!dimax!ni*)S2(t)), G G
h2 (t)"(!a 2!a * b 2b * M2 (t)). G G GLG G GLG >BGK?V G (11) Compared to model (7), this corresponds to an extension of the B-polynomial with (dimax-1) extra parameters. 3.1. Adaptive **good data++ model
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¹hus, the ,lter % (O /P ) permits to reduce the e+ects of the G G G unmodeled dynamics.
projection, signal normalization and data "ltering (Wen and David, 1992).
Proof. Consider the "ltered subsystem model de"ned by:
' (t!1, a )e (t) G G G hK (t)"I hK (t!1)# , G G Z (t)#'2 (t!1, a )' (t!1, a ) G G G G G
% A (q\)> (t)"% B (q\); (t)#% M2 (t)S (t)#= (t), G G G G G G G G G where = (t)"% = (t)"< (t)!M2(t)S (t), G G G G G , < (t)" % (c #2#c )> (t!1)#% m (t). G G GH GHPGH H G G H HOG Then , < (t)" (c a (t)#2#c aPGH(t)q\PGH)> (t!1)#m (t). G GH G GHPGH G H G H O H G The unmodelled dynamics +m (t), may contain nonG linearities and bounded disturbances. We will consider, without loss of generality, the following class: m (t)"*A (t)> (t!1)#*B (t); (t!1)#F (t) G G G G G G #g (> (t!1)#; (t!1)), G G G where
(16) where I represents the projection operator necessary to ensure #hK (t)#4o for all t if #hK (0)#4o . e (t) is the G G G G G prediction error de"ned as e (t)"> (t)!hK 2(t!1)' (t!1, a ). G G G G G
(17)
Some useful properties of estimator (16), (17) are now given in Lemma 2. Lemma 2. Consider the system model given in (8), subject to assumptions A1}A3, then the algorithm (16)}(17) 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)" G 4o #lk . GC G Z (t) RJ> G
#*A (t)#4i , #*B (t)#4i , #F (t)#4i , G G G G G G where
(c) ¹here exists a positive constant o such that for all GM (l, l)3N one has
m (t)"*A (t)> (t!1)#*B (t); (t!1)#F (t) G G G G G G #g (> (t!1)#; (t!1)) G G G and m (t)"*A (t)a (t)> (t!1)#*B (t)a (t); (t!1) G G G G G G G #a (t)F (t)#g (a (t)> (t!1)#a (t); (t!1)), G G G G G G G then "m (t)"(a (t)"m (t)" G G G and
, "< (t)"(a (t) (c #2#c q\PGH)> (t!1)#m (t) GH GHPGH H G G G H O H G (a (t)"< (t)". G G From assumption A4, we have "= (t)"4k Z (t) then G G G "= (t)"4k Z (t) and "= (t)"4a (t)k Z (t)"k Z (t). As G G G G G G G G G 0(a (t)(1, the bound k decreases, so large unmodelG G led dynamics may be tolerated and the classical weak interconnection assumption is relaxed.
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. Given hK (t), we form AK (t, q\), BK (t, q\) from the G G? G? coe$cients as AK (t, q\)"1#aL (t)a (t)q\#2#aL * (t)a (t)LG*q\LG*, G? G G GLG G (18) BK (t, q\)"a (t)bK (t)#2#aLG*>BGK?V(t) G? G G G (t)q\LG*\BGK?V>. bK * GLG >BGK?V
(19)
Lemma 3. Consider the system (8), subject to assumptions A1 and A2, if "a (t)"("bK (t)"/(o (ni*#dimax!1) ∀t, G G G then BK (t, q\) is a stable polynomial. G?
3.2. Robust parameter adaptation algorithm
Proof. BK (t, q\) is a stable polynomial if G?
Let hK (t) denote the estimate of h (t) at time t using G G the least-squares parameter estimation with parameter
LG*>BGK?V bK (t) 1 GI aI\(t)4 G ni*#dimax!1 K (t) I b G
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
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(see Zinober, 1983), where ni*#dimax is the order of BK (t, q\). As "a (t)"41, we get G? G * LG*>BGK?V bK (t) LG >BGK?V bK (t) GI aI\ (t)4a(t) GI G bK (t) G bK (t) G GR I I 1 4 ni*#dimax!1 and
GK (t, q\) is determined by solving the following equality: G
LG*>BGK?V bK (t)(#hK (t)#(o YGI G G I then, the condition "a (t)"("bK (t)"/(o (ni*#dimax!1) G G G assures the stability of BK (t, q\). G?
(27)
3.3. Decentralized adaptive control The proposed feedback control signal is generated from BK (t!1, q\); (t)#GK (t, q\)> (t)#M K 2 (t)S (t#1) G? G G G G "¹ (q\)> (t#1). (20) G GK > (t) is the given bounded output of the reference model GK satisfying O (q\)> (t)"P (q\)> (t). (21) G GK G GK ¹ (q\) is an arbitrary stable monic polynomial of degree G nti, and GK (t, q\)"gL (t)#2#gL (t)q\LEG, (22) G G GLEG where nti(ni*#1 and ngi(ni*. The term MK 2(t)S (t#1) is introduced because of the G existence of the interconnections and is then used for their e!ect reduction. The estimation error of the subsystem i is given by e (t)"> (t)!hK 2 (t!1)' (t!1, a ) G G G G G "AK (t!1, q\)> (t)!BK (t!1, q\); (t!1) G? G G? G !MK 2(t!1)S (t). (23) G Adding and subtracting q\. BK (t!1, q\); (t) in G? G Eq. (23) gives AK (t!1, q\)> (t)"q\BK (t!1, q\); (t)#MK 2(t)S (t) G? G G? G G #e (t)#[BK (t!1, q\). q\ G G? !q\. BK (t!1, q\)]; (t) (24) G? G and in view of Eq. (20) AK (t!1, q\)> (t)"q\(¹ (q\)> (t#1) G? G G GK !M K 2 (t)S (t#1)!GK (t, q\)> (t)) G G G #e (t)#M K 2 (t!1)S (t) G G #[BK (t!1, q\) . q\ G? !q\BK (t!1, q\]; (t). (25) G? G
AK (t!1, q\)#q\GK (t, q\)"¹ (q\). G? G G
(26)
The closed-loop control system equation is then ¹ (q\)[> (t)!> (t)]"e (t)#(BK (t!1, q\) . q\ G G GK G G? !q\. BK (t!1,q\)); (t). G? G
The indirect adaptive control algorithm above is globally convergent, as shown below. Theorem 1. Consider the decentralized adaptive control system consisting of the plant (8), subject to assumptions A1}A3 in closed-loop with the adaptive control law (20). 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 lim(> (t)!> (t))"0 as G G GK tPR Proof. We de"ne AK . BK and AK BK AK . BK " aL (t)bK (t!i)q\G\HOBK . AK , G H G H
(28)
AK BK " aL (t)bK (t)q\G\H"BK AK . G H G H
(29)
We also de"ne BM "BK (t!1, q\).
(30)
From (17), we have K 2(t)S (t#1), e (t#1)"AK > (t#1)!BK ; (t)!M G? G G G G? G
(31)
e (t#1)"¹> (t#1)!GK > (t)!BM ; (t) G G G G G? G !M K 2(t)S (t#1)#[AK !AM ]> (t#1) G G? G? G ![BK !BM ]; (t) G? G? G
(32)
or ¹> (t#1)#[AK !AM ]> (t#1)![BK !BM ]; (t) G G? G? G G? G? G "e (t#1)#¹ > (t#1). G G GK
(33)
Operating on (33) by AK gives G? ¹AK > (t#1)#AK .[AK !AM ]> (t#1) G? G G? G? G? G !AK . [BK !BM ]; (t)"AK e (t#1)#¹ AK > (t#1). G? G? G? G G? G G G? GK (34)
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Using (31) in (34), then (33) and (34) can be summarized as
¹ #[AK !AM ] ![BK !BM ] G G? G? G? G? AK . [AK !AM ]#[AK . ¹ !AM ¹ ] ¹ BK !AK . [BK !BM ]!¹ [BK !BM ] G? G? G? G? G G? G G G? G? G? G? G G? G?
> (t#1) ¹ 1 0 GD G > (t#1)# " e (t#1)# S (t#1). GK G ; (t) ¹ AK AK !¹ !¹ M K 2 GD G G? G? G G G
Eq. (35) can be regarded as a linear time varying system having inputs +e (t#1), and +> (t#1), and outputs G GK +; (t), and +> (t),. The terms in square brackets, for G G example, [AK !AM ], and so on, arise due to the timeG? G? varying nature of the parameter estimates and a (t). Since G > (t), S (t) are bounded and following part (iii) of GK Lemma 2, we can conclude that model (35) is asymptotically time invariant and stable provided that ¹ and G BK are both stable. Thus, from (35), +; (t),and +> (t), G? G G are asymptotically bounded by +e (t),. If k is equal to G G zero, +e (t), converges to zero and > (t) converges to G G > (t). Then > (t) converges to > (t). This establishes the GK G GK theorem.
4. DMRAC in presence of unknown time delay In this section, we consider the non-minimum phase subsystem described by A (q\)> (t)"B (q\); (t!di)#M2(t)S(t)#= (t), (36) G G G G G G where the time-delay di is constant and unknown. Eq. (36) can expressed as A (q\)> (t)"b ; (t!1)#2#b ; (t!di)#2 G G G G GBG G #b * ; (t!ni*)#M2(t)S(t)#= (t) GLG G G G or A (q\)> (t)"B (q\); (t)#M2(t)S(t)#= (t), G G G G G G where
(37)
B (q\)"b q\#2#b * q\LG*. (38) G G GLG According to De Keyser (1986), the following condition on the coe$cients b holds: GH "b "(0.2"b " for j"0,2, di!1, (39) GH GBG Then, operating on the plant model (37) by the "lter P (O /P ) leads to G G G P A (q\)> (t)"P B (q\); (t)#P M2 (t)S (t)#= (t). G G G G G G G G G (40) In regression form we have > (t)"'2 (t!1, a )h (t)#= (t), G G G G G '2 (t!1, a )"(a (t)> (t!1)2a (t)LG*> (t!ni*) G G G G G G a (t); (t!1)2a (t)LG* ; (t!ni*)S (t)). G G G G
(35)
The unknown process parameters h (t) are estimated G using the algorithms (16) and (17). Given hK (t),we form BK (t, q\) as G G? K BK (t, q\)"a (t)b (t)#2#a (t)LG* bK * (t)q\LG*. (41) GLG G? G G G From Lemma 3, BK (t, q\) is a stable polynomial if G? "a (t)"("bK (t)"/o (ni*!1. G G G After "ltering, given that 0(a (t)(1, condition (39) G becomes a (t)bK (t)'a (t)bK (t)'2'aLG* (t)bK * (t). G G G G G GLG Then the subsystem model (36) has the same behaviour as model (40) with time-delay di"1. The proposed feedback control signal is generated from BK (t!1, q\); (t)#GK (t, q\)> (t)#M K 2 (t)S (t#1) G? G G G G "¹ (q\)> (t#1). G GK The indirect model reference adaptive control is globally convergent in the sense that ; (t), > (t) are bounded for G G all time and > (t) converges to > (t) (Theorem 1). G GK 5. Numerical examples Two examples that illustrate our results are given in this section. In the "rst example, we consider three interconnected non-minimum phase subsystems with timevarying delay di(t)3[dimin, dimax]. The second example is used to demonstrate the performance of the DMRAC for non-minimum phase subsystems without a priori information about the constant time delay. Example 1. To illustrate the performance and robustness of the proposed adaptive scheme, let us consider "rst the following interconnected subsystems with time-varying delay, to which we apply the adaptive decentralized control with the interconnection estimation. A (q\)> (t)"B (q\); (t!d1(t))#C (q\)> (t) #C (q\)> (t)#m (t), A (q\)> (t)"B (q\); (t!d2(t))#C (q\)> (t) #C (q\)> (t)#m (t),
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
A (q\)> (t)"B (q\); (t!d3(t))#C (q\)> (t) #C (q\)> (t)#m (t) with A (q\)"1#0.905q\!0.3q\#0.2q\, B (q\)"1#1.5q\ C (q\) "!0.5q\!0.7q\"C (q\), 14d1(t)4, m (t)"0.1> (t!4)#0.01 cos(2t); (t!1) #0.01 sin(2t); (t!2) A (q\)"1#0.805q\!0.1q\#0.02q\ (q\) "1#1.5q\ C (q\)"!0.5q\!0.7q\ "C (q\), 24d2(t)4, m (t)"0.01> (t!4)#0.01 cos(2t); (t!1) #0.01 sin(2t); (t!2), A (q\)"1#0.995q\!0.2q\#0.1q\, B (q\)"1!1.5q\ C (q\) "!0.5q\#0.7q\"C (q\), 24d3(t)5, m (t)"0.05> (t!4)#0.01 cos(2t); (t!1) #0.01 sin(2t); (t!2). Clearly, the subsystems are non-minimum phased. Suppose that we know o "4, o "5, o "6 and the refer ence signals > (t) are square waves with periods 300, 200 GK and 100, respectively. The data "ltering parameters are chosen as "a (t)"""bK (t)"/4o i"1, 2, 3 and S(t) is the G G G Legendre series S(t)"(1, t, t, t,2). The low-pass "lter (O /P ) is such that G G O (q\)"1#0.1q\!0.1q\ P (q\) G G "1!0.2q\#0.2q\ for i"1, 2, 3. Figs. 1(1), 1(3) and 1(5) show the system output. Figs. 1(2), 1(4) and 1(6) show the time-delay variations. The interconnections are estimated using the Legendre series truncated after three terms. We note that without input/output data "ltering, and the interconnection compensation, the overall system is unstable (see Figs. 1(7)}1(9). Now, we consider again the "rst example with added noise on the subsystem outputs. To each output, a Gaussia noise with mean"3 and variance"5 (Fig. 1(10)) is added. Under the same conditions of simulation as before, and for a low-pass "lter O (q\)"1#0.01q\! G 0.01q\, P (q\)"1!0.2q\#0.2q\ for i"1, 2, 3, G the system behaviour is given by Figs. 1(11), 1(12) and 1(13). This illustrates the robustness of the proposed approach against noise and unmodelled dynamics.
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Example 2. To show the performance of the DMRAC without a priori information about the exact time-delay di, we consider the same system as in Example 1 with unknown time-delay di (di51). The system is described by A (q\)> (t)"B (q\); (t!d1)#C (q\)> (t) #C (q\)> (t)#m (t), A (q\)> (t)"B (q\); (t!d2)#C (q\)> (t) #C (q\)> (t)#m (t), A (q\)> (t)"B (q\); (t!d3)#C (q\)> (t) #C (q\)> (t)#m (t), where B (q\)"b #b q\, d1"4, B (q\)"b #b q\, d2"3, B (q\)"b #b q\ and d3"2. Under the same conditions of simulation, we obtain the subsystem behaviour shown in Figs. 2(1)}2(3). As illustrated by the previous examples, by applying an adaptive input}output data "ltering, the DMRAC scheme can be constructed even for non-minimum phase subsystems with unknown and/or time-varying delay in the presence of unmodelled dynamics. Simulations show excellent performance and robustness of the DMRAC against unmodelled dynamics and time delay variations. In all cases, excellent tracking was achieved.
6. Conclusion In this paper, we have proposed a decentralized model reference adaptive control for interconnected non minimum phase subsystems with time-varying delay. The approach is based on the interconnection output estimation using the polynomial series. The parameter estimation scheme is a combined data "ltering with a recursive least-squares algorithm with projection and signal normalization. The problem of minimum phase of the subsystems is handled by an adaptive data "ltering. It is shown that the adaptive input/output data "ltering also permits to solve the di$cult problem of model reference adaptive control in the presence of unknown time delay variations.
Appendix General discrete orthogonal polynomials s (t) satisfy G the orthogonality property K\ s (t)s (t)"d , i, j"1,2, m. G H GH R
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They satisfy the following recurrence relation: s (t)"u (t)s (t)#u s (t), i"1,2, m, G> G G G G\ t"0,2, m!1.
(* )
u (t) and u are the recurrence coe$cients; their values G G depend on the particular discrete polynomials under consideration. We assume that for subsystem i, the unknown output < (t) may be expanded into a series of G
Fig. 1. (1) The "rst subsystem output for Example 1; (2) the "rst subsystem time-delay d1(t) variation for Example 1; (3) the second subsystem output > (t) for example 1; (4) Tte second subsystem time-delay d2(t) variation for Example 1; (5) the third subsystem output > (t) for Example 1; (6) the third subsystem time-delay d3(t) variation for Example 1; (7) the "rst subsystem output > (t) for Example 1 without input/output data "ltering, and the interconnection compensation; (8) the second subsystem output > (t) for Example 1 without input/output data "ltering, and the interconnection compensation; (9) the third subsystem output > (t) for Example 1 without input/output data "ltering, and the interconnection compensation; (10) the Gaussia noise with mean"3 and variance"5 which is added to each subsystem output for Example 1; (11) the "rst subsystem output > (t) for example1for which the noise (Fig. 1(10)) is added; (12) the second subsystem output > (t) for Example 1 for which the noise (Fig. 1(10)) is added; (13) the third subsystem output > (t) for Example 1 for which the noise (Fig. 1(10)) is added.
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
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Fig. 1. Continued.
Fig. 2. (1) The "rst subsystem output > (t) for Example 2 with unknown time delay; (2) the second subsystem output > (t) for Example 2 with unknown time delay; (3) the third subsystem output > (t) for Example 2 with unknown time delay.
order m as < (t)"M2S(t)#= M (t) where S(t)"[s (t),2, s (t)]2. G G G K Now, consider the following series with expansion order t : S (t)"[s (t),2, st (t)]2, where t'm. then < (t)"M 2S (t)#= (t), G G G where "= (t)"("= M (t)" and the parameter vectors M and G G G M are assumed to be constant. G Using the recurrence equation (*), we obtain s (t)"u (t)s (t)#u s (t) K> K K K K\
and st (t)"ut (t)s (t)#ut s (t). K K\ Then the expansion of < (t) into a series of order where G M is a constant vector of dimension t , is equivalent to an G expansion of order m where M is a time-varying vector G of dimension m(t . 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
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References Datta, A., & Ioannou, P. (1991). Decentralized indirect adaptive control of interconnected systems. Internationl Journal of Adaptive Control and Signal Processing, 5, 259}281. De Keyser, B.M.C. (1986). Adaptive dead time estimation, Second IFAC workshop on adaptive control and signal processing, Lund, Sweden (pp. 209}213). Dumont, G. A. (1982). Adaptive dead time compensation. Proceedings of the 6th IFAC Symposium on identi,cation and system parameter estimation, Arlington, VA (pp. 397}406). 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., Paul, S. C., Van den Hof, M. J., & Bosgra, O. H. (1995). A generalized orthonormal basis for linear dynamical systems. IEEE ¹ransactions on Automatic Control, 40, 451}465. Ioannou P. (1986). Decentralized adaptive control of interconnected systems. IEEE ¹ransactions on Automatic Control, 31, 291}298. James, L. (1994). Analysis on the Laguerre formula 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 CON¹RO¸, 44, 1427}1436. Makoudi, M., & Radouane, L. (1995). Robust model reference adaptive control for Non-minimum phase systems. IFAC =orkshop motion control, Munich Germany, October 9}11. Makoudi, M., & Radouane, L. (1997). A robust decentralized model reference adaptive control for non minimum phase systems. IFAC international workshop on automation in the steel industry: Current practice and future developments, Kyungjoo, Korea, July 16}18. Also in Automatica, 35(8) 1499}1508. Niedzwiecki, M. (1988). Functional series modeling approach to identi"cation of non stationary stochastic systems. IEEE ¹ransactions on Automatic Control, 33, 955}961. Praly, L., & Trulsson, E. (1986). Decentralized indirect adaptive control. APII, 20, 295}315. Sripada, N. R., & Fisher, D. G. (1987). Improved least squares identi"cation. International Journmal of Control, 46(6) 1889}1913. Vodel, E. F., & Edgar, T. F. (1980). An adaptive dead-time compensator for process control. Proceedings of the ISA/80, Houston, TX, paper TPS-E.
Wei, W. S. (1990). ¹ime series analysis. Copyright by Addison-Wesley Publishing Company, Reading, MA. Wen, C., & David, J. H. (1992). Global boundedness of discrete time adaptive control just using estimator projection. Automatica, 28, 1143}1157. Zinober, A. S. (1983). Comments on &A simple criterion for stability of linear discrete systems'. International Journal of Control, 38, 1085. Zervos, C. C., Belanger, P. R., & Dumont, G. A. (1985). On PID controller tuning using orthonormal series identi"cation. IFAC workshop on adaptive control of chemical processes, Frankfut/ Main, October 21}22 (pp. 13}18). Zervos, C. C. (1988). Adaptive control based on orthonormal series representation. Ph.D. Dissertation, University of British Columbia, Vancouver. Zervos, C. C., & Dumont, G. A. (1988). Deterministic adaptive control based on Laguerre series representation. International Journal of Control, 48, 2333}2359.
Meryem Makoudi was born in Fez, Morocco on 11 October 1963. She graduated from the University sidi Mohamed Ben Abdellah, Faculty of Science, Fez, and received the &&The`se de 3` cycle'' degree in Automatic Control and Informatics in 1992. Since 1993, she has been a Professor of informatics at the faculty of legal, economical and social sciences, Fez. Her current research interests are in Large-scale systems, Decentralized control, Adaptive control, Nonlinear systems, Electrical networks, Irrigation canals and Industrial Applications.
Larbi Radouane was born in Sefrou, Morocco in 1953. He graduated from the University Mohamed V, Faculty of Science Rabat, and received the &&The`se de 3` Cycle'' and &&The`se d'Etat'' degrees in 1980 and 1985, respectively. Since 1985, he has been a Professor of Automatic control and signal processing at the University sidi Mohamed ben Abdellah, Faculty of Science, Fez where he is presently Directeur de recherche and Head of the Laboratoire d'Electronique, Signaux Syste`mes et d'Informatique (LESSI). His current research interests are in Large scale systems, Decentralized Control, Adaptive control, Signal and Image processing, and Industrial Applications.
Brief Paper
Robust decentralized adaptive control for non-minimum phase systems with unknown and/or time varying delay夽 M. Makoudi*, L. Radouane LESSI FaculteH des Sciences, DeH partement de Physique, B.P. 1796, Atlas, 30000, Morocco Received 28 July 1997; revised 15 October 1998; received in "nal form 8 February 1999
Abstract This paper presents a decentralized model reference adaptive control (DMRAC) for interconnected subsystems with unknown and/or time-varying time delay. The decentralization 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 signal normalization. A &&good data'' model is de"ned by an adaptive "ltering of the input and output signals. The obtained model permits to deal with non-minimum phased subsystems with unknown or time-varying dead time and at the same time to relax the hypothesis of weak interconnections for decentralized control. The performance of the studied scheme is illustrated by numerical examples. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Large-scale systems; Decentralized adaptive control; Non-minimum phase systems; Time-varying delay; Good data model; Interconnection prediction
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 the subsystems. Also, the design and implementation of the centralized controller may be complicated. 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 decentralized control of large-scale systems described in input/output form refer to systems that consist of weakly coupled subsystems (Wen and Hill, 1992; Datta and Ioannou, 1991;
***** * Corresponding author. Tel.: (212)5-642389; fax: (212)5-642500; e-mail: [email protected]. 夽 This papeer was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor G. Dumont under the direction of Editor C.C. Hang.
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 local controllers received for the isolated subsystems may be applied as decentralized controllers to the overall system. However, the information which may be extracted from the interconnections are generally ignored and considered as perturbations (Ossman, 1989; Praly and Trulsson, 1986). No attempt has been made to estimate them in order to improve the control performance and robustness. In this paper, we present a decentralized model reference adaptive control (DMRAC) for interconnected subsystems. 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 modelling, identi"cation and control literature (Heuberger et al., 1995; Lee and Tsay, 1986; Niedzwecki, 1988; Greblicki, 1994; James, 1994; Wei, 1990; Zervos et al., 1985, 1988; Zervos and Dumont, 1988). The parameter estimation
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 2 - 7
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scheme is a combined adaptive data "ltering with a recursive least-squares algorithm with parameter projection and signal normalization. The problem of minimum phase is handled by adjusting the data "ltering parameter at each instant. Then all the subsystem zeros are relocated inside the unit circle. It is demonstrated that the input/output data "ltering permits also to solve the di$cult problem of model reference adaptive control in the presence of unknown time-delay variations, where only the upper limit on the time delay must be known. Then, relaxed constraints on the interconnections are necessary to ensure the global system boundedness. The scheme robustness with respect to unmodelled dynamics is also simultaneously improved. The paper is organized as follows: In Section 2, we present the system description. A robust DMRAC for non-minimum phase subsystems with varying time delay is presented in Section 3. In Section 4, the robustness of the DMRAC is established without a priori information about the exact invariant time delay. Numerical examples are "nally given for illustration.
2. Problem statement 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!di) G G G G , # C (q\)> (t!1)#m (t), (1) GH H G H HOG where > (t), ; (t) are the output and input, respectively, of G G the subsystem i. A (q\), B (q\) and C (q\) are scalar G G GH polynomials in the unit delay operator q\. A (q\)"1#a q\#2#a q\LG, G G GLG
(2)
B (q\)"b #2#b q\KG, b O0. (3) G G GKG G The polynomials C (q\) are the interconnection inputs GH of > (t) from the other subsystems > (t) with G H C (q\)"c #2#c q\PGH. (4) GH GH GHPGH The time-delay index di (di51), is unknown and/or time varying. m (t) represents the unmodelled response of G subsystem i. Subsystem (1) can be expressed as > (t)"'*(t!1)2h*#< (t), G G G G where , < (t)" C (q\)> (t!1)#m (t). GH H G G H O H G
(5)
'*(t!1)2 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) G G G ; (t!di)2; (t!di!mi)], G G h*2"[!a 2!a b 2b ]. G G GLG G GKG In this paper, the objective is to derive a decentralized control law to stabilize the global system with no information exchange among the subsystems. It is assumed that for subsystem i, the output < (t) is not available. G Thus, we propose that each subsystem predicts its 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. 2.1. Functional series modelling Let ¹"+1, 2,2, a, and let S(t)"+s (t),2, s (t),2 be K the set of linearly bounded independent sequences (functions of discrete-time t) on ¹. The elements of S will be further referred to basis functions. Possible sets of functions that could be used include powers of time s (t)"tH\, j"1,2, m (which will be referred to as the H Legendre basis) or cosine functions
S(t)"
cos n( j!1)t if j is odd sin n( j!1)t
if j is even.
(which is called the Fourier basis). Now, we will use the functional modelling of the interconnections. We assume that for subsystem i, the unknown output < (t) may be expanded into a series as G < (t)"M2 (t)S(t)#= (t), (6) G G G where S(t)"[s (t),2, s (t)]2, s (t) are known functions of K G a basis, M (t)3RK. 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 (1), we obtain A (q\)> (t)"B (q\); (t!di)#M2(t)S(t)#= (t). (7) G G G G G G Further assumptions concerning the plant are made: A1: An upper bound ni* on the orders ni, mi and rij, i, j"1, 2,2, N is known. A2: There exists a known scalar o 3R> such that G #h*#4o . G G
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A3: 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)"#"; (t!1)"#"S(t)",Z ], G G G G G G 0(p (1, Z '0, Z (0)'0. G G G In this paper, we plan to design a decentralized adaptive control which should stabilize the global system and cause > (t) to track a bounded reference model > (t). G GK A major drawback with model reference adaptive control methods is that they require a priori knowledge of the dead time of the process transfer function which must be minimum phased. In many industrial processes, the time-delay di is either unknown accurately or even time varying.
3. DMRAC in presence of varying time delay
(8) by the "lter % (O /P ) leads to G G G % A (q\)> (t)"% B (q\); (t)#% M2(t)S (t) G G G G G G G G #% = (t) G G with
1419
(14)
O (q\)( . ) (t)"P (q\)( . ) (t). G G G The plant "ltered model (14) may be expressed in regression form as follows: > (t)"'2 (t!1, a )h (t)#= (t) (15) G G G G G '2 (t!1, a ) G G "(a (t)> (t!1)2a (t)LG*> (t!ni*)a (t); (t!1)2 G G G G G G a (t)LG*>BGK?V; (t!ni*!dimax)S2(t)), G G = (t)"% = (t). G G G
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. Let the % linG ear operator be de"ned as (Makoudi and Radouane, 1995, 1997)
Remarks. (1) The plant "ltered model (14) hence involves the main design features. The input}output signals are "ltered by % (O /P ). The "lter (O /P ), which should G G G G G be low-pass, is used to reduce the high-frequency modes of the unmodelled dynamics. The new &&good data'' model involves in spite of the low-pass "ltering, an exponential adaptive "ltering (% ). The % operator allows to remove G G the minimum phase assumption while improving the robustness of the DMRAC (Lemma 3). This is equivalent to generating a minimum phase estimated model for each subsystem. Note that when a (t)"1 the proposed "lter G reduces to the classical one O /P considered in Giri et al. G G (1991) and Sripada and Fisher (1987). (2) Note that in the previous works dealing with small interconnections (see, for example, Wen and Hill, 1992; Datta and Ioannou, 1991; Hejda et al., 1990; Praly and Trulsson, 1986), the classical assumption made for Eq. (5) is "< (t)"4k Z (t) where Z (t) depends only on the locally G G G G available information. k is required to be su$ciently G small for stability purpose. In our work, only the residual error = (t) (Eq. (8)) is constrained to satisfy assumption G A3. This is due to the introduction of the term (S(t)) in the observation vector and of the vector M (t) in the paraG meter vector (Eq. (11)). The interconnections are thus taken into account by each subsystem. Then the normalizing signal Z (t) depends on local subsystem signals G and implicitly on the interconnection signals also. Therefore, this new assumption makes the present work di!erent from the previous ones on the same subject. Assumption A3 is no more restrictive.
% > (t!k)"aI(t)> (t!k), (12) G G G G % ; (t!k)"aI (t); (t!k) (13) G G G G with 0(a (t)(1 ∀t, where a (t) will be adjusted; this G G corresponds to an adaptive exponential "ltering. Given two asymptotically stable polynomials O (q\) G and P (q\), and a % operator, then operating on plant G G
Lemma 1. Consider subsystem (7) subject to A3, then for subsystem model (14), there exists a scalar k '0 such that G "= (t)"4"k Z (t), G G G Z (t)"p Z (t!1)#max("> (t!1)"#"; (t!1)" G G G G G #"S (t)", Z ), 0(p (1, Z '0, Z (0)'0. G G G GD
If the time-delay di is time-varying i.e. dimin4di(t)4 dimax where dimin51 and only dimax is known, the normal approach in literature (Dumont, 1982; Vodel and Edgar, 1980) is to model subsystem (7) by A (q\)> (t)"B (q\); (t)#M2(t)S(t)#= (t) G G G G G G with
(8)
q\LG*\BGK?V. B (q\)"b q\#2#b * G G GLG >BGK?V Subsystem (8) can be expressed as
(9)
> (t)"'2(t!1)h (t)#= (t), G G G G where
(10)
'2(t!1)"(> (t!1)2> (t!ni*) G G G ; (t!1)2; (t!dimax!ni*)S2(t)), G G
h2 (t)"(!a 2!a * b 2b * M2 (t)). G G GLG G GLG >BGK?V G (11) Compared to model (7), this corresponds to an extension of the B-polynomial with (dimax-1) extra parameters. 3.1. Adaptive **good data++ model
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¹hus, the ,lter % (O /P ) permits to reduce the e+ects of the G G G unmodeled dynamics.
projection, signal normalization and data "ltering (Wen and David, 1992).
Proof. Consider the "ltered subsystem model de"ned by:
' (t!1, a )e (t) G G G hK (t)"I hK (t!1)# , G G Z (t)#'2 (t!1, a )' (t!1, a ) G G G G G
% A (q\)> (t)"% B (q\); (t)#% M2 (t)S (t)#= (t), G G G G G G G G G where = (t)"% = (t)"< (t)!M2(t)S (t), G G G G G , < (t)" % (c #2#c )> (t!1)#% m (t). G G GH GHPGH H G G H HOG Then , < (t)" (c a (t)#2#c aPGH(t)q\PGH)> (t!1)#m (t). G GH G GHPGH G H G H O H G The unmodelled dynamics +m (t), may contain nonG linearities and bounded disturbances. We will consider, without loss of generality, the following class: m (t)"*A (t)> (t!1)#*B (t); (t!1)#F (t) G G G G G G #g (> (t!1)#; (t!1)), G G G where
(16) where I represents the projection operator necessary to ensure #hK (t)#4o for all t if #hK (0)#4o . e (t) is the G G G G G prediction error de"ned as e (t)"> (t)!hK 2(t!1)' (t!1, a ). G G G G G
(17)
Some useful properties of estimator (16), (17) are now given in Lemma 2. Lemma 2. Consider the system model given in (8), subject to assumptions A1}A3, then the algorithm (16)}(17) 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)" G 4o #lk . GC G Z (t) RJ> G
#*A (t)#4i , #*B (t)#4i , #F (t)#4i , G G G G G G where
(c) ¹here exists a positive constant o such that for all GM (l, l)3N one has
m (t)"*A (t)> (t!1)#*B (t); (t!1)#F (t) G G G G G G #g (> (t!1)#; (t!1)) G G G and m (t)"*A (t)a (t)> (t!1)#*B (t)a (t); (t!1) G G G G G G G #a (t)F (t)#g (a (t)> (t!1)#a (t); (t!1)), G G G G G G G then "m (t)"(a (t)"m (t)" G G G and
, "< (t)"(a (t) (c #2#c q\PGH)> (t!1)#m (t) GH GHPGH H G G G H O H G (a (t)"< (t)". G G From assumption A4, we have "= (t)"4k Z (t) then G G G "= (t)"4k Z (t) and "= (t)"4a (t)k Z (t)"k Z (t). As G G G G G G G G G 0(a (t)(1, the bound k decreases, so large unmodelG G led dynamics may be tolerated and the classical weak interconnection assumption is relaxed.
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. Given hK (t), we form AK (t, q\), BK (t, q\) from the G G? G? coe$cients as AK (t, q\)"1#aL (t)a (t)q\#2#aL * (t)a (t)LG*q\LG*, G? G G GLG G (18) BK (t, q\)"a (t)bK (t)#2#aLG*>BGK?V(t) G? G G G (t)q\LG*\BGK?V>. bK * GLG >BGK?V
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Lemma 3. Consider the system (8), subject to assumptions A1 and A2, if "a (t)"("bK (t)"/(o (ni*#dimax!1) ∀t, G G G then BK (t, q\) is a stable polynomial. G?
3.2. Robust parameter adaptation algorithm
Proof. BK (t, q\) is a stable polynomial if G?
Let hK (t) denote the estimate of h (t) at time t using G G the least-squares parameter estimation with parameter
LG*>BGK?V bK (t) 1 GI aI\(t)4 G ni*#dimax!1 K (t) I b G
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
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(see Zinober, 1983), where ni*#dimax is the order of BK (t, q\). As "a (t)"41, we get G? G * LG*>BGK?V bK (t) LG >BGK?V bK (t) GI aI\ (t)4a(t) GI G bK (t) G bK (t) G GR I I 1 4 ni*#dimax!1 and
GK (t, q\) is determined by solving the following equality: G
LG*>BGK?V bK (t)(#hK (t)#(o YGI G G I then, the condition "a (t)"("bK (t)"/(o (ni*#dimax!1) G G G assures the stability of BK (t, q\). G?
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3.3. Decentralized adaptive control The proposed feedback control signal is generated from BK (t!1, q\); (t)#GK (t, q\)> (t)#M K 2 (t)S (t#1) G? G G G G "¹ (q\)> (t#1). (20) G GK > (t) is the given bounded output of the reference model GK satisfying O (q\)> (t)"P (q\)> (t). (21) G GK G GK ¹ (q\) is an arbitrary stable monic polynomial of degree G nti, and GK (t, q\)"gL (t)#2#gL (t)q\LEG, (22) G G GLEG where nti(ni*#1 and ngi(ni*. The term MK 2(t)S (t#1) is introduced because of the G existence of the interconnections and is then used for their e!ect reduction. The estimation error of the subsystem i is given by e (t)"> (t)!hK 2 (t!1)' (t!1, a ) G G G G G "AK (t!1, q\)> (t)!BK (t!1, q\); (t!1) G? G G? G !MK 2(t!1)S (t). (23) G Adding and subtracting q\. BK (t!1, q\); (t) in G? G Eq. (23) gives AK (t!1, q\)> (t)"q\BK (t!1, q\); (t)#MK 2(t)S (t) G? G G? G G #e (t)#[BK (t!1, q\). q\ G G? !q\. BK (t!1, q\)]; (t) (24) G? G and in view of Eq. (20) AK (t!1, q\)> (t)"q\(¹ (q\)> (t#1) G? G G GK !M K 2 (t)S (t#1)!GK (t, q\)> (t)) G G G #e (t)#M K 2 (t!1)S (t) G G #[BK (t!1, q\) . q\ G? !q\BK (t!1, q\]; (t). (25) G? G
AK (t!1, q\)#q\GK (t, q\)"¹ (q\). G? G G
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The closed-loop control system equation is then ¹ (q\)[> (t)!> (t)]"e (t)#(BK (t!1, q\) . q\ G G GK G G? !q\. BK (t!1,q\)); (t). G? G
The indirect adaptive control algorithm above is globally convergent, as shown below. Theorem 1. Consider the decentralized adaptive control system consisting of the plant (8), subject to assumptions A1}A3 in closed-loop with the adaptive control law (20). 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 lim(> (t)!> (t))"0 as G G GK tPR Proof. We de"ne AK . BK and AK BK AK . BK " aL (t)bK (t!i)q\G\HOBK . AK , G H G H
(28)
AK BK " aL (t)bK (t)q\G\H"BK AK . G H G H
(29)
We also de"ne BM "BK (t!1, q\).
(30)
From (17), we have K 2(t)S (t#1), e (t#1)"AK > (t#1)!BK ; (t)!M G? G G G G? G
(31)
e (t#1)"¹> (t#1)!GK > (t)!BM ; (t) G G G G G? G !M K 2(t)S (t#1)#[AK !AM ]> (t#1) G G? G? G ![BK !BM ]; (t) G? G? G
(32)
or ¹> (t#1)#[AK !AM ]> (t#1)![BK !BM ]; (t) G G? G? G G? G? G "e (t#1)#¹ > (t#1). G G GK
(33)
Operating on (33) by AK gives G? ¹AK > (t#1)#AK .[AK !AM ]> (t#1) G? G G? G? G? G !AK . [BK !BM ]; (t)"AK e (t#1)#¹ AK > (t#1). G? G? G? G G? G G G? GK (34)
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Using (31) in (34), then (33) and (34) can be summarized as
¹ #[AK !AM ] ![BK !BM ] G G? G? G? G? AK . [AK !AM ]#[AK . ¹ !AM ¹ ] ¹ BK !AK . [BK !BM ]!¹ [BK !BM ] G? G? G? G? G G? G G G? G? G? G? G G? G?
> (t#1) ¹ 1 0 GD G > (t#1)# " e (t#1)# S (t#1). GK G ; (t) ¹ AK AK !¹ !¹ M K 2 GD G G? G? G G G
Eq. (35) can be regarded as a linear time varying system having inputs +e (t#1), and +> (t#1), and outputs G GK +; (t), and +> (t),. The terms in square brackets, for G G example, [AK !AM ], and so on, arise due to the timeG? G? varying nature of the parameter estimates and a (t). Since G > (t), S (t) are bounded and following part (iii) of GK Lemma 2, we can conclude that model (35) is asymptotically time invariant and stable provided that ¹ and G BK are both stable. Thus, from (35), +; (t),and +> (t), G? G G are asymptotically bounded by +e (t),. If k is equal to G G zero, +e (t), converges to zero and > (t) converges to G G > (t). Then > (t) converges to > (t). This establishes the GK G GK theorem.
4. DMRAC in presence of unknown time delay In this section, we consider the non-minimum phase subsystem described by A (q\)> (t)"B (q\); (t!di)#M2(t)S(t)#= (t), (36) G G G G G G where the time-delay di is constant and unknown. Eq. (36) can expressed as A (q\)> (t)"b ; (t!1)#2#b ; (t!di)#2 G G G G GBG G #b * ; (t!ni*)#M2(t)S(t)#= (t) GLG G G G or A (q\)> (t)"B (q\); (t)#M2(t)S(t)#= (t), G G G G G G where
(37)
B (q\)"b q\#2#b * q\LG*. (38) G G GLG According to De Keyser (1986), the following condition on the coe$cients b holds: GH "b "(0.2"b " for j"0,2, di!1, (39) GH GBG Then, operating on the plant model (37) by the "lter P (O /P ) leads to G G G P A (q\)> (t)"P B (q\); (t)#P M2 (t)S (t)#= (t). G G G G G G G G G (40) In regression form we have > (t)"'2 (t!1, a )h (t)#= (t), G G G G G '2 (t!1, a )"(a (t)> (t!1)2a (t)LG*> (t!ni*) G G G G G G a (t); (t!1)2a (t)LG* ; (t!ni*)S (t)). G G G G
(35)
The unknown process parameters h (t) are estimated G using the algorithms (16) and (17). Given hK (t),we form BK (t, q\) as G G? K BK (t, q\)"a (t)b (t)#2#a (t)LG* bK * (t)q\LG*. (41) GLG G? G G G From Lemma 3, BK (t, q\) is a stable polynomial if G? "a (t)"("bK (t)"/o (ni*!1. G G G After "ltering, given that 0(a (t)(1, condition (39) G becomes a (t)bK (t)'a (t)bK (t)'2'aLG* (t)bK * (t). G G G G G GLG Then the subsystem model (36) has the same behaviour as model (40) with time-delay di"1. The proposed feedback control signal is generated from BK (t!1, q\); (t)#GK (t, q\)> (t)#M K 2 (t)S (t#1) G? G G G G "¹ (q\)> (t#1). G GK The indirect model reference adaptive control is globally convergent in the sense that ; (t), > (t) are bounded for G G all time and > (t) converges to > (t) (Theorem 1). G GK 5. Numerical examples Two examples that illustrate our results are given in this section. In the "rst example, we consider three interconnected non-minimum phase subsystems with timevarying delay di(t)3[dimin, dimax]. The second example is used to demonstrate the performance of the DMRAC for non-minimum phase subsystems without a priori information about the constant time delay. Example 1. To illustrate the performance and robustness of the proposed adaptive scheme, let us consider "rst the following interconnected subsystems with time-varying delay, to which we apply the adaptive decentralized control with the interconnection estimation. A (q\)> (t)"B (q\); (t!d1(t))#C (q\)> (t) #C (q\)> (t)#m (t), A (q\)> (t)"B (q\); (t!d2(t))#C (q\)> (t) #C (q\)> (t)#m (t),
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
A (q\)> (t)"B (q\); (t!d3(t))#C (q\)> (t) #C (q\)> (t)#m (t) with A (q\)"1#0.905q\!0.3q\#0.2q\, B (q\)"1#1.5q\ C (q\) "!0.5q\!0.7q\"C (q\), 14d1(t)4, m (t)"0.1> (t!4)#0.01 cos(2t); (t!1) #0.01 sin(2t); (t!2) A (q\)"1#0.805q\!0.1q\#0.02q\ (q\) "1#1.5q\ C (q\)"!0.5q\!0.7q\ "C (q\), 24d2(t)4, m (t)"0.01> (t!4)#0.01 cos(2t); (t!1) #0.01 sin(2t); (t!2), A (q\)"1#0.995q\!0.2q\#0.1q\, B (q\)"1!1.5q\ C (q\) "!0.5q\#0.7q\"C (q\), 24d3(t)5, m (t)"0.05> (t!4)#0.01 cos(2t); (t!1) #0.01 sin(2t); (t!2). Clearly, the subsystems are non-minimum phased. Suppose that we know o "4, o "5, o "6 and the refer ence signals > (t) are square waves with periods 300, 200 GK and 100, respectively. The data "ltering parameters are chosen as "a (t)"""bK (t)"/4o i"1, 2, 3 and S(t) is the G G G Legendre series S(t)"(1, t, t, t,2). The low-pass "lter (O /P ) is such that G G O (q\)"1#0.1q\!0.1q\ P (q\) G G "1!0.2q\#0.2q\ for i"1, 2, 3. Figs. 1(1), 1(3) and 1(5) show the system output. Figs. 1(2), 1(4) and 1(6) show the time-delay variations. The interconnections are estimated using the Legendre series truncated after three terms. We note that without input/output data "ltering, and the interconnection compensation, the overall system is unstable (see Figs. 1(7)}1(9). Now, we consider again the "rst example with added noise on the subsystem outputs. To each output, a Gaussia noise with mean"3 and variance"5 (Fig. 1(10)) is added. Under the same conditions of simulation as before, and for a low-pass "lter O (q\)"1#0.01q\! G 0.01q\, P (q\)"1!0.2q\#0.2q\ for i"1, 2, 3, G the system behaviour is given by Figs. 1(11), 1(12) and 1(13). This illustrates the robustness of the proposed approach against noise and unmodelled dynamics.
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Example 2. To show the performance of the DMRAC without a priori information about the exact time-delay di, we consider the same system as in Example 1 with unknown time-delay di (di51). The system is described by A (q\)> (t)"B (q\); (t!d1)#C (q\)> (t) #C (q\)> (t)#m (t), A (q\)> (t)"B (q\); (t!d2)#C (q\)> (t) #C (q\)> (t)#m (t), A (q\)> (t)"B (q\); (t!d3)#C (q\)> (t) #C (q\)> (t)#m (t), where B (q\)"b #b q\, d1"4, B (q\)"b #b q\, d2"3, B (q\)"b #b q\ and d3"2. Under the same conditions of simulation, we obtain the subsystem behaviour shown in Figs. 2(1)}2(3). As illustrated by the previous examples, by applying an adaptive input}output data "ltering, the DMRAC scheme can be constructed even for non-minimum phase subsystems with unknown and/or time-varying delay in the presence of unmodelled dynamics. Simulations show excellent performance and robustness of the DMRAC against unmodelled dynamics and time delay variations. In all cases, excellent tracking was achieved.
6. Conclusion In this paper, we have proposed a decentralized model reference adaptive control for interconnected non minimum phase subsystems with time-varying delay. The approach is based on the interconnection output estimation using the polynomial series. The parameter estimation scheme is a combined data "ltering with a recursive least-squares algorithm with projection and signal normalization. The problem of minimum phase of the subsystems is handled by an adaptive data "ltering. It is shown that the adaptive input/output data "ltering also permits to solve the di$cult problem of model reference adaptive control in the presence of unknown time delay variations.
Appendix General discrete orthogonal polynomials s (t) satisfy G the orthogonality property K\ s (t)s (t)"d , i, j"1,2, m. G H GH R
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They satisfy the following recurrence relation: s (t)"u (t)s (t)#u s (t), i"1,2, m, G> G G G G\ t"0,2, m!1.
(* )
u (t) and u are the recurrence coe$cients; their values G G depend on the particular discrete polynomials under consideration. We assume that for subsystem i, the unknown output < (t) may be expanded into a series of G
Fig. 1. (1) The "rst subsystem output for Example 1; (2) the "rst subsystem time-delay d1(t) variation for Example 1; (3) the second subsystem output > (t) for example 1; (4) Tte second subsystem time-delay d2(t) variation for Example 1; (5) the third subsystem output > (t) for Example 1; (6) the third subsystem time-delay d3(t) variation for Example 1; (7) the "rst subsystem output > (t) for Example 1 without input/output data "ltering, and the interconnection compensation; (8) the second subsystem output > (t) for Example 1 without input/output data "ltering, and the interconnection compensation; (9) the third subsystem output > (t) for Example 1 without input/output data "ltering, and the interconnection compensation; (10) the Gaussia noise with mean"3 and variance"5 which is added to each subsystem output for Example 1; (11) the "rst subsystem output > (t) for example1for which the noise (Fig. 1(10)) is added; (12) the second subsystem output > (t) for Example 1 for which the noise (Fig. 1(10)) is added; (13) the third subsystem output > (t) for Example 1 for which the noise (Fig. 1(10)) is added.
M. Makoudi, L. Radouane/Automatica 35 (1999) 1417}1426
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Fig. 1. Continued.
Fig. 2. (1) The "rst subsystem output > (t) for Example 2 with unknown time delay; (2) the second subsystem output > (t) for Example 2 with unknown time delay; (3) the third subsystem output > (t) for Example 2 with unknown time delay.
order m as < (t)"M2S(t)#= M (t) where S(t)"[s (t),2, s (t)]2. G G G K Now, consider the following series with expansion order t : S (t)"[s (t),2, st (t)]2, where t'm. then < (t)"M 2S (t)#= (t), G G G where "= (t)"("= M (t)" and the parameter vectors M and G G G M are assumed to be constant. G Using the recurrence equation (*), we obtain s (t)"u (t)s (t)#u s (t) K> K K K K\
and st (t)"ut (t)s (t)#ut s (t). K K\ Then the expansion of < (t) into a series of order where G M is a constant vector of dimension t , is equivalent to an G expansion of order m where M is a time-varying vector G of dimension m(t . 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
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References Datta, A., & Ioannou, P. (1991). Decentralized indirect adaptive control of interconnected systems. Internationl Journal of Adaptive Control and Signal Processing, 5, 259}281. De Keyser, B.M.C. (1986). Adaptive dead time estimation, Second IFAC workshop on adaptive control and signal processing, Lund, Sweden (pp. 209}213). Dumont, G. A. (1982). Adaptive dead time compensation. Proceedings of the 6th IFAC Symposium on identi,cation and system parameter estimation, Arlington, VA (pp. 397}406). 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., Paul, S. C., Van den Hof, M. J., & Bosgra, O. H. (1995). A generalized orthonormal basis for linear dynamical systems. IEEE ¹ransactions on Automatic Control, 40, 451}465. Ioannou P. (1986). Decentralized adaptive control of interconnected systems. IEEE ¹ransactions on Automatic Control, 31, 291}298. James, L. (1994). Analysis on the Laguerre formula 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 CON¹RO¸, 44, 1427}1436. Makoudi, M., & Radouane, L. (1995). Robust model reference adaptive control for Non-minimum phase systems. IFAC =orkshop motion control, Munich Germany, October 9}11. Makoudi, M., & Radouane, L. (1997). A robust decentralized model reference adaptive control for non minimum phase systems. IFAC international workshop on automation in the steel industry: Current practice and future developments, Kyungjoo, Korea, July 16}18. Also in Automatica, 35(8) 1499}1508. Niedzwiecki, M. (1988). Functional series modeling approach to identi"cation of non stationary stochastic systems. IEEE ¹ransactions on Automatic Control, 33, 955}961. Praly, L., & Trulsson, E. (1986). Decentralized indirect adaptive control. APII, 20, 295}315. Sripada, N. R., & Fisher, D. G. (1987). Improved least squares identi"cation. International Journmal of Control, 46(6) 1889}1913. Vodel, E. F., & Edgar, T. F. (1980). An adaptive dead-time compensator for process control. Proceedings of the ISA/80, Houston, TX, paper TPS-E.
Wei, W. S. (1990). ¹ime series analysis. Copyright by Addison-Wesley Publishing Company, Reading, MA. Wen, C., & David, J. H. (1992). Global boundedness of discrete time adaptive control just using estimator projection. Automatica, 28, 1143}1157. Zinober, A. S. (1983). Comments on &A simple criterion for stability of linear discrete systems'. International Journal of Control, 38, 1085. Zervos, C. C., Belanger, P. R., & Dumont, G. A. (1985). On PID controller tuning using orthonormal series identi"cation. IFAC workshop on adaptive control of chemical processes, Frankfut/ Main, October 21}22 (pp. 13}18). Zervos, C. C. (1988). Adaptive control based on orthonormal series representation. Ph.D. Dissertation, University of British Columbia, Vancouver. Zervos, C. C., & Dumont, G. A. (1988). Deterministic adaptive control based on Laguerre series representation. International Journal of Control, 48, 2333}2359.
Meryem Makoudi was born in Fez, Morocco on 11 October 1963. She graduated from the University sidi Mohamed Ben Abdellah, Faculty of Science, Fez, and received the &&The`se de 3` cycle'' degree in Automatic Control and Informatics in 1992. Since 1993, she has been a Professor of informatics at the faculty of legal, economical and social sciences, Fez. Her current research interests are in Large-scale systems, Decentralized control, Adaptive control, Nonlinear systems, Electrical networks, Irrigation canals and Industrial Applications.
Larbi Radouane was born in Sefrou, Morocco in 1953. He graduated from the University Mohamed V, Faculty of Science Rabat, and received the &&The`se de 3` Cycle'' and &&The`se d'Etat'' degrees in 1980 and 1985, respectively. Since 1985, he has been a Professor of Automatic control and signal processing at the University sidi Mohamed ben Abdellah, Faculty of Science, Fez where he is presently Directeur de recherche and Head of the Laboratoire d'Electronique, Signaux Syste`mes et d'Informatique (LESSI). His current research interests are in Large scale systems, Decentralized Control, Adaptive control, Signal and Image processing, and Industrial Applications.