SWITCHING CONTROL BASED ON MULTIPLE MODELS

Transcription

1 SWITCHING CONTROL BASED ON MULTIPLE MODELS A J Neves P Rocha T Mendonça Department of Mathematics, University of Aveiro Campo de Santiago, Aveiro, Portugal Fax: {ajneves,procha}@matuapt Department of Applied Mathematics Faculty of Sciences - University of Oporto Rua do Campo Alegre 687, Porto, Portugal Fax: tmendo@fcuppt Abstract: A switching control scheme based on a bank of multiple models is considered in order to solve the problem of tracking a constant reference The switching occurs among a finite family of candidate controllers associated to the models dynamics, according to a selection criterion based on the minimization of an identification error Assuming that the process to be controlled is described by one of the considered models (exact matching case) subject to a bounded disturbance, conditions under the model-controller bank are presented which guarantee the stability of the switching control system and the reference tracking objective Keywords: tracking, switching control, multiple models 1 INTRODUCTION In this paper the problem of tracking a constant reference by means of switching among linear controllers each of them associated to a model of the process dynamics is considered for the discretetime case The main goal is to present conditions for the transfer functions of the model-controller pairs that guarantee the reference tracking objective The general switching control scheme is illustrated in Figure 1 This scheme was considered in (Neves et al, 2002), but using switching among stabilizing controllers, in order to achieve the improvement of the controller response, for instance in situations it is impossible to obtain a single well tuned controller In this figure, ref denotes the reference and P is the process to be controlled K(σ) is a timevariant controller acting in the following way + ref e T σ K(σ) u Fig 1 Switching control scheme First, a model-controller bank B = {(M j, K j ) : j J = {1, 2,, N}} is constructed, such that, each model M j is a linear time-invariant model for the process dynamics and the corresponding controller K j is tuned to solve the tracking problem for M j Based on the values of u and y, the selection procedure S yields at each time instant k the index σ(k) J corresponding to the controller that should be active at that time This is achieved S P y

2 u through the minimization of a suitable function of the identification error e j := y y j, y j denotes the response of the model M j to the input u The signal σ() is known as the switching signal 2 SWITCHING STRATEGY The specification of the multiple model bank together with the corresponding controllers and a selection procedure S yields a concrete control strategy For the time being, as mentioned in the introduction, each admissible controller K j will be simply assumed to be linear and time invariant, with transfer function g j (z) and to be tuned to solve the reference tracking problem for the model M j The transfer function h j (z) associated with the model M j will be assumed to be strictly proper and to have no zeros for z = 1 This guarantees that the constant tracking problem is solvable for h j (z) The selection procedure S is depicted in Figure 2 and is implemented as follows Once a dwell time τ D IN and an instant T IN are fixed, switching is allowed to occur freely till the instant T From then on, switching is only allowed to occur at the instants: t i = T + i τ D, i IN 0 Let T := {0, 1,, T = t 0, t 1, t 2, } be the set of admissible switching instants At each instant k T, the controller K j to be chosen is the one associated with the model M j for which k f j (k) = e j (l) 2 is minimum l=0 M y + + e φ d y M 1 M N Fig 2 Selection procedure y 1 y N + + e N e 1 min j f j (k) = f(e σ ) Note that although the general switching scheme is similar to the one considered in (Morse, 1996; Morse, 1997), the control strategy is different from the one considered in that paper since another selection procedure is used Indeed, here the identification errors are directly computed from the S σ responses of the models to the input u as in (Morse, 1996; Morse, 1997) these errors are computed from the responses of certain estimators A different switching scheme combining fixed and adaptive models has been used in (Narendra and Balakrishnan, 1997) This approach known as multiple model switching and tuning (MMST) is used to adaptively control an unknown deterministic linear process or a linear process with a stochastic disturbance (Narendra et al, 2003) 3 ASSUMPTIONS ON THE MODEL-CONTROLLER BANK The model-controller bank is assumed to satisfy the following assumptions, analogous to the considered in (Neves et al, 2002): A1) The model transfer functions h j have a common pole z = α C The controller transfer functions g j have a common pole in z = 1 This assumption allows to define the modified models M j with transfer functions h j (z) := z α z 1 h j(z) and the associated modified controllers K j with transfer functions ḡ j (z) := z 1 z α g j(z) This corresponds to replace the pole in z = α in the models with a pole in z = 1 and the pole in z = 1 in the controllers with a pole in z = α Moreover, note that the output y j of the model M j can be written as with y j = h j v (1) v = z 1 z α u Regarding the modified controllers transfer functions, we assume that: A2) The transfer functions ḡ j are realized by the following minimal state-shared realizations denoted by Σ C j = (AC, B C, C C j, DC j ) x C (k + 1) = A C x C (k) + B C e T (k) (2) v(k) = Cj C x C (k) + Dj C e T (k) is the tracking error e T (k) = ref y(k)

3 4 SWITCHING UNDER EXACT MATCHING It is assumed that the process P to be controlled is nominally described by a model M that coincides with a model M i of the bank, with transfer function h = h i Moreover, the model output y is subject to a disturbance d, as shown in Fig 3, and the output y of the process P is given by with e = φ d y = y + e, (3) The transfer function φ(z) is supposed to be stable and proper y u h Fig 3 Process model d φ + + y Since the models M e M i coincide the corresponding responses y and y i to the input u are related by y = y i + ē i, (4) the error ē i results from possible differences between the initial states Substituting (4) in (3) we obtain y = y i + ē i + e (5) Now, let the transfer functions h j have minimal state space realizations Σ j = (A j, B j, C j ) Then, by (1), x j (k + 1) = A j x j (k) + B j v(k) (6) y j (k) = C j x j (k) As we have seen in (Neves et al, 2002) each system Σ i = (A i, B i, C i ) has an equilibrium state x e i corresponding to the solution (x i, v, y i ) (x e i, 0, ref) So, considering the shifted state x i (k) = x i (k) x e i (k), we obtain the state space equations x i (k + 1) = A i x i (k) + B i v(k), (7) e i (k) = C i x i (k) e T (k) i J\{i } Analogously defining x(k) = x i (k) x e i (k) and A = A i, B = B i, C = C i, the equations for the shifted state of the process model M i can be written as x(k + 1) = A x(k) + B v(k) (8) e T (k) = C x(k) e i (k), e i = y y i = ē i + φ d (9) In order to obtain one equation for the error ē i in (9) we consider the minimal realization (A, B, C ) for h = h i and define x, x i to be the states of the models M and M i, respectively Then, the outputs y and y i corresponding to the input u, are given by x (k + 1) = A x (k) + B u(k) and y (k) = C x (k) x i (k + 1) = A x i (k) + B u(k) y i (k) = C x i (k), thus, x (k + 1) x i (k + 1) = A (x (k) x i (k)) Therefore ē i (k) = C (x (k) x i (k)) ē i = C (A ) k ( x (0) x i (0) ) A3) The perturbation d has finite l 2 -norm A4) The transfer function h i is stable As noticed before the transfer function φ is stable and proper Together with assumption A3, this implies that φ d = e has finite l 2 -norm Further, from assumption A4 it follows that A is stable, so ē i has finite l 2 -norm and we conclude that also e i has finite l 2 -norm These considerations allow to derive the following lemma Lemma Assume that A3 and A4 hold There exists k IN such that, if j = arg min f j (k) for j k k, then e j has finite l 2 -norm Now, we introduce the following additional assumptions: This means that from a certain instant k on only controllers corresponding to models M j satisfying j J, with are selected J := { j J : e j 2 < } In order to investigate the performance of the closed-loop switching control system we start by taking σ(k) = j and use the tracking error equation obtained in (8) for e T The equations of the controller K j can be written as { x C (k + 1 = A C x C (k) B C C x(k) B C e i (k) v(k) = C C j xc (k) D C j C x(k) DC j e i (k) (10)

4 Substituting the last expression for v in equations (8) and (7), we obtain x(k + 1) = B C j C xc (k) + (A BDj C C) x(k) B Dj C e i (k) (11) e T (k) = C x(k) e i (k) and x i (k + 1) = A i x i (k) + B i Cj C xc (k) B i Dj C C x(k) B i Dj C e i (k) (12) e i (k) = C i x i (k) e T (k) for i J \{i } Without generality loss we simplify the notation assuming that i = N + 1 and J \{i } = {1,, N } Taking into account the expression for e T in (11), the errors e i associated to the models M i can be expressed in terms of the error e i as e i (k) = C i x i (k) + C x(k) + e i (k) (13) Thus, from (10), (11), (12) and (13), we obtain the state space equation X(k + 1) = F j X(k) + G j e i (k) (14) x C A C B C C 0 0 x BCj X = x 1 C A BDj C C 0 0, F j = B 1 Cj C B 1 Dj C C A 1 0, x N B N Cj C B N Dj C C 0 A N G j = B C BD C j B 1 D C j B N D C j and the output equations E(k) = H X(k) + L e i (k) e T (k) = C X(k) e i (k) e 1 0 C C 1 0 E =, H =, e N 0 C 0 C N 1 L = [ ] and C = 0 C (15) Therefore, when the controller K j is active the closed-loop transfer function from e i to E = [e 1 e N ] T, T j (z), is realized by (F j, G j, H, L) Taking (14) and (15) into account it is possible to show that, the transfer function γ i from e i to e T is given by γ i (z) = h i g j 1 = 1 Analogously the transfer function γ i from e i to e T can be written as 1 γ i (z) = 1 + h i g j Thus, the transfer function from e i to e i is equal to γ i = 1 + h i g j γ i and, consequently, we conclude that 1 + h 1 g j T j = 1 + h i 1 g j 1 + h i +1 g j 1 + h N g j (16) If we consider coprime factorizations h i = α i and β i g j = n j, T j can be written as d j with T j = β i := β i p j 1 β 1 p j i β i p j N β N p j i N +1 l=1 l i 1 = β i p j i β 1 p j 1 β N p j N (17) β l and p j l := β ld j + α l n j, (18) for l = 1,, N This representation allows to see that the MacMillan degree of T j, n(t j ), satisfyes the following inequality n(t j ) degree ( β i p j i ) (19)

5 It follows from (17) that if the N +1 polynomials β i p j i, i = 1,, N + 1, are coprime, then n(t j ) = degree ( β i p j i ) = degree(β 1 ) + + degree(β N +1) + degree(d j ) = n(h 1 ) + + n(h N +1) + n(g j ) = dim F j and (F j, G j, H, L) is a minimal realization of T j Thus, let the following assumption hold: A5) The polynomials β i p j i, i = 1, 2,, N +1 are coprime, for all j J Since it ensures the minimality of the realization (F j, G j, H, L), this assumption guarantees that the pair (H, F j ) is observable and, therefore, detectable Whence, there exists a matrix Γ j such that (F j Γ j H) is stable and the equations (14) and (15) can be written as X(k + 1) = (F j Γ j H) X(k) + (G j Γ j L) e i (k) + Γ j E(k) E(k) = H X(k) + L e i e T (k) = C X(k) e i (k), Now, the relationship between the errors e i, i = 1,, N + 1,, and the control error e T is given by the state space equations X(k + 1) = F j X(k) + G j Ẽ(k) (20) e T (k) = C X(k) + D Ẽ(k) F j := F j Γ j H, Gj := [ Γ j G j Γ j L ], Ẽ(k) := [ E T (k) e i (k) ] T and D := [ ] Taking into account that the switching signal σ variation is subject to a dwell time τ E we obtain the time-variant system described by the equations: X(k + 1) = F (k) X(k) + G(k) Ẽ(k) (21) e T (k) = C X(k) + D Ẽ(k) F (k) := F σ(k) and G(k) := Gσ(k) The matrices F (k) and G(k) assume a finite number of values since σ(k) J and further each matrix F (k) is stable Moreover, the imposition of a dwell time τ E from an instant T on guarantees that F (k) = F (t i ) and G(k) = G(ti ), (22) for k = t i,, t i+1 1, with t i = T + i τ E This together with the fact that Ẽ has finite l 2- norm allows to obtain the following result: Theorem With the previous notations assume that A1 to A5 hold Suppose that the time-variant system of the form (21) is subject to a dwell time τ E IN from an instant T IN on such that, (22) is verified for t i = T + iτ E with i IN 0 Then, there exists L IN such that, if τ E L, lim e T(k) = 0 k In other words, under our assumptions and provided that the dwell time is sufficiently large, our switching control scheme achieves the desired goal 5 CONCLUDING REMARKS In this paper the problem of tracking a constant reference by means of switching among linear controllers associated to a bank of multiple models has been considered We admit that the dynamics of the process to be controlled is described by one of the linear models in the bank (exact matching case), subject to a disturbance with finite l 2 -norm Using a switching strategy based on minimization of an identification error and considering certain assumptions on the model-controller bank, it is shown that the stability of the switching control scheme is guaranteed and the reference tracking objective is achieved Although these assumptions may be considered somewhat restrictive, in many applications PID controllers are used the conditions on the controllers are satisfied The same happens with the conditions on the models In fact, for instance in the modelling of certain biological processes (Mendonça and Lago, 1998) it is often possible to admit all models of the bank have a common pole, as it is required in assumption A1 6 ACKNOWLEDGEMENT The research of the authors has been (partially) supported by the R&D Unit Matemática e Aplicações (University of Aveiro, Portugal) through "Programa Operacional Ciência, Tecnologia e Inovação" (POCTI) of the Fundação para a Ciência e Tecnologia (FCT), co-financed by the European Union fund FEDER 7 REFERENCES Mendonça, T and P Lago (1998) PID control strategies for the automatic control of neuromuscular blockade Control Engineering Practice 6,

6 Morse, AS (1996) Supervisory control of families of linear set-point controllers -part 1: Exact matching IEEE Trans Automat Control 41(10), Morse, AS (1997) Supervisory control of families of linear set-point controllers -part 2: Robustness IEEE Trans Automat Control 42(11), Narendra, Kumpati S and J Balakrishnan (1997) Adaptive control using multiple models IEEE Trans Automat Control 42(2), Narendra, Kumpati S, Osvaldo A Driollet, Matthias Feiler and Koshy George (2003) Adaptive control using multiple models, switching and tuning International Journal of Adaptive Control and Signal Processing 17(2), Neves, A J, P Rocha and T Mendonça (2002) A switching control scheme for improving reference tracking In: Proceedings of CON- TROLO 2002, 5 t h Portuguese Conference on Automatic Control Aveiro, Portugal