Bumpless transfer for discrete-time switched systems

Transcription

1 29 American Control Conference Hyatt Regency Riverfront, St. Lois, O, USA Jne 1-12, 29 WeB12.3 Bmpless transfer for discrete-time switched systems I. ALLOCI, L. HETEL, J. DAAFOUZ, ember, IEEE, C. IUNG, ember, IEEE, R. BONIDAL Abstract A bmpless transfer method for discrete-time switched linear systems is presented. It is based on an additional controller which is activated at the switching time for redcing the control discontinities. Dwell time conditions to garantee the stability of the closed-loop system are provided. Simlation tests on the Eisenhüttenstadt hot strip mill of Arcelorittal are shown. I. INTRODUCTION In practical control problems, several linear controllers are often sed to control the same non-linear plant, one for each operating point. This strategy avoids the non-linear control design, certainly more complicated. Nevertheless, switching among more controllers implies control discontinities ndesired transient behaviors. oreover, indstrial systems are often characterized by satrations de to the actators limits do not accept these discontinities. Conseqently, the dynamics performance can be largely modified the stability may not be garanteed anymore. Slope satrations of the control signal are well-nown for their destabilizing effect. The soltion of this problem is called bmpless transfer. A description of most poplar strategies for the bmpless transfer problem can been fond in [8], [9], [1] [13]. One of the first bmpless transfer schemes is proposed by [11] for non-linear plants. The idea consists in pre-setting the off-line controller state in order to redce the transient behavior at the switching time. In [18] [19], reslts of [11] are generalized for controllers which are not biproper have a minimm phase. A linear qadratic (LQ) optimization method is introdced to minimize the distance between the on-line the off-line controller otpt. In [5], the discontinity of the controller otpt is redced resetting the fast dynamics of the controller at the switching time. In [21], the desired transient behavior is obtained sing the L 2 anti-windp strctre [17]. Althogh the bmpless transfer problem has been widely stdied in literatre, only few articles address the switched systems framewor. In [1], a bmpless transfer soltion for continos-time switched systems the order of the controller is smaller than the order of the plant is given. The idea consists in forcing the otpt of the activated controller This wor has been spported by grants from "la région Lorraine, France" Arcelorittal aizières Research. I. alloci, L. Hetel, J. Daafoz C. Ing are the Centre de Recherche en Atomatiqe de Nancy, UR 739 CNRS - Nancy Université, ENSE, 2, Avene de la forêt de Haye Voevre-lès-Nancy, France. Corresponding athor: Jamal.Daafoz@ensem.inpl-nancy.fr. R. Bonidal is Arcelorittal aizières, R&D Indstrial Operations, BP 332, F aizières-lès-etz Cedex, France. to be eqal to the plant inpt at the switching time. An analogos strategy is proposed in [7] for continos-time LPV systems. However, as pointed ot in [2] [21], a constraint on the controller otpt does not garantee better performances of the plant otpt. In this article, a bmpless transfer control design for discrete-time switched systems is presented. We propose an additional controller which is activated at each switching time. The controller the plant otpt are forced to follow a desired profile for a given period of time. This minimizes the control discontinity garanteeing the plant otpt performances. The soltion is based on the LQ optimization theory, that has been introdced on the bmpless transfer framewor by [19]. Stability of the closed-loop system is garanteed by LI conditions [2] sing mltiple Lyapnov fnctions [3] the dwell time approach [15]. The article is organized as follows. In the next section, the problem is formlated. In section III, the optimization criterion to design the bmpless transfer controller (BT controller) is detailed. In section IV, stability conditions for the closed-loop system are investigated. In section V, simlation reslts of the Eisenhüttenstadt hot strip mill (Germany) are presented. Finally, a conclsion is given. II. PROBLE FORULATION Consider the discrete-time switched system { x +1 = A σ() x + B σ() y = C σ() x (1) x R n is the state, R r is the control signal, y R m is the otpt signal σ() : N Γ = {1,...,N} is the switching signal, which is assmed to be nnown a priori bt available in real time. The minimal interval of time between two switchings D j is assmed to be nown. oreover, the pair (A j,b j ) is spposed to be controllable the pair (A j,c j ) observable, j Γ. Frthermore, the state x is spposed to be available for feedbac the plant matrices (1) are assmed to be well-nown. A state-feedbac control law = K σ() x (2) which stabilizes the closed-loop system (1)-(2) is given. In order to redce the amplitde of the control signal discontinities, different strategies are possible. Here, a BT controller which is activated at the switching time for the /9/$ AACC 178

2 period of time τj < D j is proposed, that is: { K j x + bt = if τj K j x if > τj (3) K j x tj bt R r is the BT controller otpt Ξ = {1,...,τj } the nmber of times between the present time the last switching. For each mode j Γ, the closed-loop system (Fig. 1) can be written as { x +1 = (A j + B j K j )x + B j bt (4) y = C j x. enable 1 enable 2 bt K 1 bt K 2 Q 1 Q 2 ũ plant 1 ũ plant 2 x x C 1 C 2 spervisor y Fig. 1. Closed-loop system Γ = {1, 2} The BT controller Q j is designed in the next section. The signal ũ j is minimized sing a LQ criterion, ũ j represents the desired profile of the control signal. For simplicity reasons, a straight line is chosen as desired profile. Let t j be the switching time from the sbsystem i to the sbsystem j, (i,j) Γ Γ. We can define ũ j = ũ j + p j (5) ũ j = K i x tj 1 (6) is the control signal vale at the time before the switching p j determines the slope of the desired profile, i. e. p j = 1 (K j x tj K i x tj 1). (7) We obtain a vale of p which depends on the control signal discontinity (Fig. 2). In order to garantee the performances of the plant otpt y, also the difference between y the desired plant otpt ỹ is minimized. III. BUPLESS TRANSFER CONTROLLER DESIGN In this section, the method to design the BT controller is presented. In order to follow the desired profiles ũ ỹ, the problem is formlated as a classical LQ optimization problem. In the bmpless transfer framewor, this soltion K i x tj 1 t j p j t j + Fig. 2. evoltion BT controller switched on (thic line) switched off (thin line) has been proposed by [19], the difference between the on-line the off-line controller otpt is minimized before each switching. oreover, the difference between the on-line the off-line controller inpt is minimized. This strategy initializes the controller state then redces the transient behavior on the plant otpt. Since the antibmpless action consists in pre-setting the state of the offline controller before the switching, the method does not address control systems ot memory, sch as statefeedbac control laws. Another problem concerns the stability of the closed-loop system, which is not garanteed when arbitrary switchings occr. To solve these problems, for discrete-time switched systems we propose a BT controller which is activated at each switching time. Stability conditions for the closed-loop system are given in the next section. For each mode j Γ, the design of the BT controller is based on the minimization of the following qadratic cost fnction: J j = φ j + 1 T j f 2 φ j T j f T j f 1 = [z Wj z + z y W y j zy ], (8) z = ũ j (9) z y = y ỹ j (1) = 1 2 z T j f P j z T j f (11) Wj W y j are positive definite weighting matrices of appropriated dimensions. T j f = τ j + 1 is the terminal time P j is a positive semi-definite terminal weighting matrix. For simplicity reasons, we consider ỹ j =. The next theorem shows how to compte the signal bt. The proof, which is based on the Pontryagin s minimm principle [4], is similar to the one given in [19]. Theorem 1: Given the system (4), the qadratic cost fnction (8) the terminal time T j f, the BT controller which minimizes the signals z z y is given by x bt = Q j ũ j g j +1 (12) 179

3 Q j = (Ñj +1 Πj +1 A j K j ) (I + Ñj +1 Πj +1 B j) Ñj +1 Ñ j +1 = (W j ) 1 B j(i Π j +1 B j ) 1, (13) j Γ. The vales of Π j g j are given by the eqations Π j = A j(i Π j +1 B j ) 1 Π j +1 A j + C j (14) g j = A j(i Π j +1 B j ) 1 (g j +1 Πj +1 B jũ j ) (15) The bond condition is B j = B j (W j ) 1 B j C j = C jw y j C j. Π j T j f g j T j f = =. (16) Remar 1: In the finite horizon approach, the nowledge of all the ftre vales of ũ is reqired in order to solve (15) bacward in time. Then, in general this method cannot be applied to solve practical problems, as discssed in [19]. Nevertheless, from (5), in or case all the vales of ũ can be compted in the finite horizon Ξ. Only the nowledge of x tj 1 x tj is needed. Since these informations are available at each switching time, the method can always be sed. IV. STABILITY ANALYSIS In the previos sections, we assmed that the BT controller is switched on for times. As the original controller (2) has been designed ot taing into accont this fact, the stability is not garanteed anymore. Then, in this section, a stability condition for the closed-loop system (4) is given. For each mode j Γ, the closed-loop system x +1 = (A j + B j K j )x + B j bt (17) can be written in the eqivalent form v +1 = Y j ( )v (18) v = x x 1 ũ p is the agmented state the signals ũ p are defined in section II. The representation of a switched system an agmented state approach is jstified in [12]. We distingish two phases on the interval between two switchings: The bmpless transfer phase: the BT controller is on. We find H j +1 L j +1 I Y j ( ) τj=1 = K i Y j ( ) 2 τj = K j Ki Ā j +1 Ū j +1 I I I P j +1. In this case, the stability of (18) is not garanteed. The constrction of Y j is detailed in the appendix I. The recperation phase: the BT controller is off. We have A j + B j K j I Y j ( ) τj> = Y s j = Y s j is Schr constant j Γ.,... Given the minimal interval of time between two switchings D j, the vale of τj can always be redced in order to mae the system (18) stable. Let define the transition matrix between two switching times Z j (D j, ) = τj Y j ( ) τj τj =1 The following theorem checs stability. Y j s (D j 1). (19) the closed-loop system Theorem 2: Given D j, if there exist positive definite matrices P Yj = P Y j, P Zj = P Z j of appropriate dimensions scalars τj sch that the LIs Y s j P Yj Y s j P Yj (2) Z jp Zj Z j P Zj (21) Z jp Yj Z j P Zj (22) Y s j P Zi Y s j P Yj (23) are verified (i,j) Γ Γ, then the closed-loop system (18) is asymptotically stable. We give an idea of the proof. In the case of arbitrary switching law, a necessary condition for the asymptotic stability of the closed-loop system (18) is that each sbsystem is stable. Since for assmption Y s is Schr, from (19) there exists a vale of sch that also the sbsystem Z j is Schr, j Γ. In Fig. 3, the system evoltion is shown for Γ = {1,2}. For the closed-loop system stability, 18

4 s Y 1 Z 2 s Y 2 st st st 5.2 Z Fig. 3. System evoltion Fig. 4. j TABLE I BT CONTROLLER DATA D j W j I 1I 1I W y j I I I we also mst chec that switching among the sbsystems leads to a stable behavior. This is the meaning of the LIs (22) (23). Ths, the conditions of Theorem 2 garantee a decreasing trajectory after switching. This is eqivalent to say that the closed-loop system (18) is stable [16]. V. SIULATION RESULTS In this section, the strategy proposed in the previos sections is applied to the Eisenhüttenstadt hot strip mill (HS) of Arcelorittal. The rolling process consists in crshing a metal strip between two rolls in inverse rotation for obtaining a strip constant desired thicness. A HS is the association of several sts in a line, each st is constitted by a set of rolls. The lateral movement of the strip, reference to the mill axis, may indce a decrease of the prodct qality rolls damage. Then, in order to improve the reliability the process qality, this displacement mst be redced [6]. At the end of the treatment, the strip leaves the sts one after the others. Each time the strip leaves a st, the system dynamics changes. The HS is modeled as a switched system for sbsystems three switchings [14]. For each sbsystem j Γ, a different discrete-time state-feedbac control gain K j has been designed. Given the weighting matrices W, W y the minimal dwell time D j for a prodct of the HS database, the choice of the τj vales smmarized in Table I allow to find a soltion for the LI conditions of Theorem 2. This garantees the stability of the closedloop system. I denotes an identity matrix of appropriate dimension. Since the system never switches bac to the first y (mm) y (mm) y (mm) st st st Fig. 5. sbsystem, no BT controller is designed for j = 1. The signal bt is compted applying Theorem 1. Eqations (13) (14) can be solved off-line. At the opposite, to compte the eqation (15) we need to now the x tj 1 vale, t j is the switching time to the sbsystem j. Then, it can be compted only on-line, at the switching time. Since the otpt signal y corresponds to the displacement that has to be minimized, we choose ỹ =. Each st is controlled by a different control signal. For the simlated prodct, switchings occr at the instants = 129, = 1354 = In Fig. 4 it is shown the controller otpt for the last three sts. For each st, we propose a zoom of the zone corresponding to the biggest bmps on. The bold line shows the evoltion ot BT control as the thinnest line shows the evoltion when the BT controller is on. The evoltion of the strip displacement y is shown in Fig. 5. As expected, performances are better when the BT controller is on (thinnest line). In particlar, the strip displacement in the exit of the system (st 5 in Fig. 5) is redced from 4 to 8mm. VI. CONCLUSION In this article, a bmpless transfer method for discretetime switched systems has been proposed. The BT controller has been designed sing a LQ optimization method. The idea consists in forcing the controller otpt the plant y 181

5 otpt to follow a desired profile. This allows to avoid the control signal discontinities to improve the system performances. A LI criterion garanteeing the stability of the closed-loop system is proposed. Simlation tests on the Eisenhüttenstadt HS of Arcelorittal are provided. APPENDIX I CONSTRUCTION OF Y j Consider the switching from the sbsystem i to the sbsystem j. From Theorem 1, when the BT controller is on x bt = Q j ũ j (24) p j = 1 g j +1 ũ j = ũ j + p j (25) ũ j = K i x tj 1 (26) (K j x tj K i x tj 1). (27) The evoltion of the signal g j in (24) is given by (15), which can be rewritten as G j, = = τj η= g j = G j, ũ j + p j (28) τj η= ( τ j +1 η j ζ ζ=+1 ( τ j +1 η j ζ ζ=+1 ) ) j = A j(i Π j Bj ) 1. Π j +1 η Bj, Π j +1 η Bj (τ η) The closed form (28) allows to express (15) as a fnction of ũ j p j. Then, the system (17) becomes x +1 =Āj +1 x + ( B j +1 B jñj +1 Gj, +1 )ũ j+ ( Bj +1 B jñj +1 Gj,p +1 )p j Ā j = (I + B j Ñ j Π j )A j B j = B j (I + Ñj Π j B j ). When = 1, t j =. Then ũ j p j can be initialized as fnction of x x 1. Using (26) (27), we find H j +1 L j +1 I Y j ( ) τj=1 = K i K j Ki H j = Āj + 1 L j = ( B j B j Ñ j G j, )K i 1 ( B j B j Ñ j )K j ( B j B j Ñ j )K i. When 2 τj, ũ j p j remains constant. We have Ā j +1 Ū j +1 P j +1 Y j ( ) 2 τj τj = I I I Ū j = B j B j Ñ j G j, P j = Bj τj B j Ñ j. When the BT controller is off bt =, then A j + B j K j I Y j ( ) τj>τj = Yj s =.. is constant j Γ. REFERENCES [1] A.B. Arehart W.A. Wolovich. Bmpless switching controllers. In Conference on Decision Control, [2] S. Boyd, L.E. Ghaoi, E. Feron, V. Balarishnan. Linear atrix Ineqalities in system control theory. Society for Indstrial Applied athematics, [3] S. Branicy. ltiple Lyapnov fnction other analysis tools for switched hybrid systems. IEEE Transactions on Atomatic Control, 43(4): , [4] A. E. Bryson Y. C. Ho. Applied optimal control. John Wiley, [5] S.Y. Cheong.G. Safonov. Bmpless transfer for adaptive switching controls. In IFAC World Congress, 28. [6] J. Daafoz, R. Bonidal, C. Ing, P. Szczepansi, N. Namann, U. Koschac. New steering control at EKO Stahl finishing mill. In The Iron & Steel Technology Conference Exposition, 28. [7] K. Dong F. W. Online switching control design of LFT systems. In Conference on Decision Control, 26. [8] C. Edwards I. Postlethwaite. Anti-windp bmpless transfer schemes. Atomatica, 34(2):199 21, [9] S.F. Graebe A. Ahlen. The control hboo. CRC press, [1] R. Hans. Anti-windp bmpless transfer: A srvey. In 12th IACS world congress, [11] R. Hans,. Kinnaert, J. Henrotte. Conditioning techniqe, a general anti-windp bmpless transfer method. Atomatica, 23(6): , [12] L. Hetel, J. Daafoz, C. Ing. Eqivalence between the Lyapnv- Krasovsii fnctional approach for discrete delay systems the stability conditions for switched systems. In IFAC worshop on Time Delay Systems, 27. [13].V. Kothare, P.J. Campo,. orari, N. Nett. A nified framewor for stdy of anti-windp designs. Atomatica, 3(12): , [14] I. alloci, J. Daafoz, C. Ing, R. Bonidal, P. Szczepansi. Switched system modeling robst steering control of the tail end phase in a hot strip mill. To appear in Nonlinear Analysis: Hybrid Systems Applications,

6 [15] A.S. orse. Spervisory control of families of linear set-point controllers - part 1: Exact matching. IEEE Transactions on Atomatic Control, 41: , [16] P. Peleties R.A. DeCarlo. Asymptotic stability of m-switched systems sing Lyapnov-lie fnctions. In American Control Conference, [17] A. R. Teel N. Kapoor. The L 2 anti-windp problem: IIts definition soltion. In Eropean Control Conference, [18].C. Trner D.J. Waler. odified linear qadratic bmpless transfer. In American Control Conference, [19].C. Trner D.J. Waler. Linear qadratic bmpless transfer. Atomatica, 36: , 2. [2] L. Zaccarian A. R. Teel. A common framewor for anti-windp, bmpless transfer reliable designs. Atomatica, 38(1): , 22. [21] L. Zaccarian A. R. Teel. The L 2 (l 2 ) bmpless transfer problem for linear plants: Its definition soltion. Atomatica, 41(7): ,