State feedback controller for a class of MIMO non triangular systems

Transcription

1 State feedbac controller for a class of MIMO non triangular systems M Farza, M M Saad, O Gehan, E Pigeon and S Hajji GREYC UMR 67 CNRS, Université de Caen, ENSICAEN 6 Boulevard Maréchal Juin, 45 Caen Cedex, France {mfarza,msaad,gehan,pigeon}@greycensicaenfr Unité de Recherche UCA, ENIS, Département de Génie électrique, BP W, 338 Sfax, Tunisia hjjisfin@yahoofr Abstract: This paper presents a controller design for a class of MIMO nonlinear systems involving some uncertainties The latter is particularly composed by cascade subsystems and each subsystem is associated to a subset of the system outputs and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems The main contribution consists in extending the available control results to allow more interconnections between the subsystems Of fundamental interest, it is shown that the underlying tracing error exponentially converges to zero in the absence of uncertainties, and can be made as small as desired by properly specifying the control design parameter in the presence of uncertainties Keywords: Nonlinear systems, MIMO systems, Tracing, State feedbac high gain control INTRODUCTION In this paper, one aims at addressing the output tracing problem for MIMO uniformly observable systems which dynamicalbehaviorcanbedescribedbythefollowingstate representation: where { xt = Axtφut,xtButνt yt = Cxt x denotes the state of the system and is composed as follows x x x x = IRn with x x = IRn x q x λ where x i = x i, x i,p IR p with x i,j IR for =,,q, i =,,λ, j =,,p with q q n = p λ = n; p and λ = = u and y respectively denotes the input and the output of the system u u u = u q, y = y y y q IRp where u = u, u,p,y = y, y,p =,,q with u,i,y,i IR and hence q p = p = IR p for The matrices A and C are respectively given by I p A = diaga,,a q, A = I p and C = diagc,,c q, C = [I p ] 3 B = diagb,,b q, B T = [ I p ] 4 φu,x denotes the nonlinear function field which is composed as follows φ u,x φ u,x φu,x =, φ u,x = φ q u,x φ u,x φ u,x φ λ u,x with φ u,x IR n and each function φ i u,x IR p is differentiable with respect to x and assumes the following structural dependence on the state variables for i λ : φ iu,x=φ ix =φ ix,x,,x,x,x,,x i 5 Copyright by the International Federation of Automatic Control IFAC 97

2 for i = λ : φ λ u,x = φ λ u,,u,x,x,,x q 6 νt is an unnown function that can be approximated more or less precisely by a nown function that shall be denoted by ˆνt If no approximation is available, one taes ˆνt = The function νt as well as ˆνt assumes the following structure: ν t ν t νt = IRn,ν t = ν q t ν t ν t ν λ t IRn where the functions νi t IRp are given by for =,,q and i =,,λ νi = ν t for =,,q and i = λ For =,,q, let ε = ˆν ν 7 Then one assumes that the ε s, are unnown bounded functions, ie β > ; t ; ε t β 8 The control problem to be addressed consists in an admissible asymptotic tracing of an output reference trajectory y y that will be noted y = IRp where y IR p for y q =,,q is assumed to be smooth enough, ie lim t yt yt α 9 where α is a non negative scalar with is equal to zero if lim νt ˆνt = and henceforth β = t The problems of observation and control of nonlinear systems have received a particular attention throughout the last four decades Considerable efforts were dedicated to the analysis of the structural properties to understand better the concepts of controllability and of observability of nonlinear systems Hammouri and Farza [3], Rajamani [998], Isidori [995], Gauthier and Kupa [994], Fliess and Kupa [983] Several control and observer design methods were developed thans to the available techniques, namely feedbac linearisation, flatness, high gain, variable structure, sliding modes and bacstepping Farza et al [5], Agrawal and Sira-Ramirez [4], Fliess et al [999], Sepulchre et al [997], Fliess et al [995], Isidori [995], Krstič et al [995] The main difference between these contributions lies in the design model, and henceforth the considered class of systems, and the nature of stability and performance results A particular attention has been devoted to the design of state feedbac control laws incorporating an observer satisfying the separation principle requirements as in the case of linear systems Mahmoud and Khalil [996] Inthispaper,oneproposesastatefeedbaccontrollerfora class of non triangular MIMO nonlinear systems described by The following features are worth to be mentioned The last equation of each subsystem, where the input intervenes, may depend on the whole state The system cannot be henceforth considered as a cascade of subsystems that can be sequentially controlled This is the main issue in the control design Theabsenceofzerosintheconsideredclassofsystems is particularly motivated by pedagogical purposes as oneseestoemphasizetherationalbehindthecontrol design method Nevertheless, the proposed method can be straightforwardly extended to minimum phase systems see for instance Praly [3] For the sae of space limitation, only state feedbac control is considered in this paper Nevertheless, as the system is uniformly observable, the proposed state feedbac control law can be combined with an appropriate high gain observer to get an output state feedbac controller as in Hajji et al [8] A high gain state observer for a class of systems similar to has been proposed in Farza et al [] Notice that the authors in Liu et al [999] considered a stabilization problem for a class of non uniformly observable systems that has not a triangular form The proposed method cannot be extended to tacle neither the output feedbac stabilization, nor the tracing problem The state feedbac control design under consideration is particularly suggested from the high gain observer design bearing in mind the control and observation duality Of particular interest, the controller gain involves a well defined design function which provides a unified framewor for the high gain control design, namely several versions of sliding mode controllers are obtained by considering particular expressions of the design function Furthermore, it isshownthatanintegralactioncanbesimplyincorporated into the control design to carry out a robust compensation of step lie disturbances This paper is organized as follows The reference model is introduced in section with an explicit procedure for the determination of its state and input Some useful preliminaries,notationsanddefinitionsaregiveninsection 3 with a technical lemma which is crucial for the controller design Section 4 is devoted to the state feedbac control design with a full convergence analysis of the tracing error The possibility to incorporate an integral action into the control design is shown in section 5 Simulation results are given in section 6 in order to show the effectiveness of the proposed control design method REFERENCE MODEL System can be written under the following more developed form as follows { x = A x φ u,xb u ν y = C x =,,q Taing into account the class of systems, it is possible x x to determine the system state trajectory x = x q 98

3 IR n and the system input sequence u IR p with x IR n and u IR p, =,,q, corresponding to the output trajectory yt This allows to define an admissible reference model as follows { x = A x φ u, xb u ˆν y = C x for =,,q where ˆν is the available approximation of ν The determination of the components of the reference model state, x as well as those of its input, u is described in the next subsection Determination of the state and input of the reference model One shall firstly detail the determination of x i, i =,,λ Indeed, one has: x = y x i = x i φ i x,, x i for i =,,λ 3 By assuming that the reference trajectory y is smooth enough, one can recursively determine the components of x from y and its λ first time derivatives, ie y i = di y dt i, i =,,λ, as follows: x i = βi y, y,, y i, i =,,λ 4 where the functions βi are defined as follows: β y = y for i =,,λ : i βi βi = y j j= y j φ i x,, x i i βi = j= φ i y j y j β y,β y, y,,βi y,, y i Now, one can proceed in a similar manner in order to determine recursively x i, for =,,q and i =,,λ, as follows: x i =β i y,, y i l= λ l,, 5 y,, y i λ, y,, y i =,,q and i =,,λ 6 where the functions β i are defined as follows: for =,,q : β = y for i =,,λ : i βi βi = l= j= y j l i y j βi l j= y j φ i x,, x, x,, x i i β i i = y j l= j= y j l l j= φ i y j βi y j y j β,,β λ,,β λ,β,,β i 7 Once all the components of the reference model state x are determined, one can compute the components of the reference model input using system as follows: u = x λ φ λ x ˆν t and for =,,q u = x λ φ λ u,, u, x ˆν t 8 The tracing problem 9 can be hence turned to a state trajectory tracing problem defined by lim et α where et = xt xt 9 t Such a problem can be interpreted as a regulation problem for the tracing error system obtained from the system and model reference state representations and, respectively, namely e = A e φ u,x φ u, x B u u ν ˆν y = y y = C e for =,,q 3 SOME DEFINITIONS AND NOTATIONS For =,,q, let Δ ρ be the diagonal matrix defined by: Δ ρ = diag I p, ρ δ I p,, ρ I δ λ p where ρ > is a real number and one defines δ which indicates the power of ρ as follows: q δ = q λ i 3 for =,,q i= δ q = One also defines for =,,q and i =,,λ, the following sequence of scalar numbers: σ i =σ i δ with σ = λ δ λ δ η 3 where < η is an arbitrarily small non negative number One can chec that σ i for =,,q and i =,,λ Similarly to the Δ s, one defines for =,,q the diagonal matrices Λ s as follows: Λ ρ = ρ σ Δ ρ 4 Noticethat,accordingtothedefinitionoftheσi sgivenby 3, one has σλ = λ δ η and therefore one has σλ σλ l l =η l for,l =,,q5 This means that whatever is the difference between λ and λ l, the difference between σ λ and σ l λ l which are the powers of ρ on the last rows of Λ ρ and Λ l ρ, respectively can be made as small as desired by choosing η small enough very close to zero 99

4 Now, taing into account the structures of Λ, Δ ρ and A, respectively given by 4, and, one can show that the following identities hold: Λ ρa Λ ρ = Δ ρa Δ ρ = ρδ A ρ σ C Λ ρ = C Δ ρ = C 6 Two other set of matrices that shall be used in the controller design are S and P, =,,q which are defined as follows Indeed, let consider the following algebraic Lyapunov equation S A T S S A = C T C 7 where A and C are defined in and 4, respectively It has been shown that such this equation has a unique solution S that is symmetric positive definite and one has Gauthier et al [99] S CT = Cλ I p,,c λ λ I p T 8 where Cλ i λ! = i!λ i! for i =,,λ Similarly, consider the following algebraic Riccatti equation: P A T P P A = P B B T P 9 Again, equation 9 admits a unique solution P that is symmetric positive definite and it can be expressed as follows: P = T S T with T = I p I p I p, 3 where S is the solution of 7 This can be checed by multiplying the left and right sides of equation 7 by T S and S T, respectively and by using the facts that T A T = A T, T T = I and B = T C T As a result, one has: B T P = C S T = C λ λ I p,,c λ I p 3 In the sequel, one shall denote by λ min and λ max the smallest and largest eigenvalues of, respectively One also defines the following bloc diagonal matrix: P = diagp,p,,p q 3 Before ending this section, one shall give a technical lemma, given in [Faral] needed in the proof of our main result This lemma allows to provide a sequence of reals that reflects in some sense the interconnections between the blocs nonlinearities Lemma 3 Let χ,i l,j = if φ i x l u,x 33 j otherwise for,l =,,q, i =,,λ and j =,,λ l Then, the sequence of real numbers σi defined by 3 are such that if χ,i l,j = then σl j σ i δ l δ η q 34 4 STATE FEEDBACK CONTROL As it was early mentioned, the proposed state feedbac controldesignisparticularlysuggestedbythedualityfrom the high gain observer design proposed in Farza et al [] The underlying state feedbac control law is then given by v e = K ρ λ δ B T P Δ ρe u x = u v e 35 where e = x x, ρ > is the controller design parameter, Δ ρ and P are given by and 9, respectively and finally K : IR p IR p is a bounded design function satisfying the following property ξ Ω one has ξ T K ξ ξt ξ 36 where Ω is an arbitrarily fixed compact subset of IR p Manyfunctionssatisfyingcondition36aregiveninFarza et al [5] Before stating the fundamental result of the paper, one needs the following technical assumption, usually adopted when using high gain approaches, namely Assumption φu,x is a globally Lipschitz nonlinear function with respect to x uniformly in u One can then states the following fundamental result Theorem 4 Assume that system satisfies assumption, then ρ > ; ρ > ρ ; λ ρ > ; μ ρ > ; α ρ > such that for {,,q} x t x t λ ρ e μρt x x α ρ β where β is the upper bound of ν ˆν Moreover, λ ρ is polynomial in ρ, lim μ ρ = and lim α ρ = ρ ρ Setch of the proof of Theorem 4 Set the tracing error et = xt xt and let e t be the th subcomponent of et One has: e =A e φ u,x φ u, x B K ρ λ δ B T P Δ ρe B ε 37 where ε = ˆν ν as defined by 7 For =,,q, set e = Λ ρe 38 where Λ ρ is given by 4 From equation 37 and using identities 6, one gets e =ρ δ A e Λ ρφ u,x φ u, x Λ ρb ε ρ σ ρ σ λ λ δ B K ρ σ λ δ B T δ P e Set V e = e T P e 39 q where P is given by 9 and let V e = V e be the candidate Lyapunov function =

5 Proceeding as in Farza et al [], one can show that for =,,q, one has V e t exp ρ n γμ P ρ η q t V e q λ max Pβ n γμ P ρ η q ρ λ δ which yields to 4 e t ρ λ δ η γμp μ P e ρ n ρ η q t e qμ P 4 n γμ P ρ η q ρ ηβ where γ and μ P are positive real constants This ends the proof of the theorem It is worth noticing that the tracing error converges exponentially to zero in the absence of uncertainties and can be made as small as desired in the presence of uncertainties Indeed, the tracing error remains in a ball with a radius proportional to which can be ρ η made arbitrarily small for high values of λ Nevertheless, one notices that this radius is also proportional to the estimation error on ν, namely ˆνt νt Thus, a good estimate ˆνt of νt also allows to obtain an ultimate bound for the tracing error which is relatively small without requiring high values for ρ 5 INTEGRAL ACTION One can easily incorporate an integral action, into the proposed state feedbac control design for performance enhancement purposes, by simply introducing suitable state variables as follows e = e where e is the integral of the error output associated to the bloc The state feedbac gain is then determined from the control design model e = A e φ u,x φ u, x B u u ˆν ν y = C e = e 4 with =,,q and e e = p I e, A = p p, B = B C = C p, φ p u,x = φ u,x Indeed, the control design model structure 4 is similar to that of the error system and hence the underlying state feedbac control design is the same The state feedbac control law incorporating an integral action is then given by v e = K ρ λ δ BT P Δ ρ e 43 u e = u v e with Δ ρ = diag I p, ρ δ I p,, ρ λ I p 44 δ where P is the unique symmetric positive definite matrix solution of the following Riccatti algebraic equation P P A A T P = P B BT P 45 and K is a function satisfying condition 36 6 EXAMPLE In the following, simulation results are given to show the effectiveness of the proposed controller design using the followingmimononlinearsystemwhichisunderform: x = x x s x = x 3 x x 3 = x x 3 x x, x, x ν 46, y = x x, = x, x x, s, x, = x, x s, x, = x x, x u u, ν, 47 x, = x x 3, x u 3, ν, = y, y, T = x = x, x, T y For the output y, the desired output reference sequence, ie y, is generated from a third order generator with unitary static gain and a trapezoidal input sequence, while the desired output reference sequences for the outputs y and y 3 are y, = sint and y, = cost Simulation have been carried out by considering the disturbances ν,ν, and ν, which are zero until t = s at which they tae the following expressions: ν = sin5t, ν, = 3sint and ν, = 3sin5t Moreover, additive step lie disturbances, s, and s,, have been applied on the dynamics of x, and x, at times t = s and t = 35s, respectively and their respective amplitudes are s, = 3, s, = 3 For the first input, a step lie disturbance, s, with an amplitude equal to 5 has been applied on its dynamics between times t = 3s and t = 6s No estimates for the unmodelled disturbances have been considered, ie these estimates have been taen equal to zero The expression of all the design functions K is K ξ = tanhξ and the value of the design parameter ρ has been fixed to The resulting input/output performances are shown in figure where the outputs are particularly displayed with their desired reference trajectories The obtained results show the good performance of the proposed controller A zoom on the tracing errors is shown in figure to emphasize the tracing performance Notice that the tracing errors go to zero in the absence of unmodelled disturbances In the presenceofsuchdisturbancesfromt = s,thetracing errors lie in a ball with a relatively small radius These performances confirm the theoretical results developed in the previous sections 7 CONCLUSION A unified high gain state feedbac control design framewor has been developed to address an admissible tracing

6 Input u Input u, Input u, Output y Output y, Output y, Time s Time s Fig Input/output performances e e e 3 5 x 3 Errors 5 x 3 x Fig Zoom on the tracing errors problem for a class of controllable nonlinear systems The unifying feature is provided through a suitable design functionthatallowstorediscoverallthosewellnownhigh gain control methods, namely the sliding modes control The effectiveness of the proposed feedbac control method has been emphasized throughout an illustrative example in simulation Notice that an output feedbac control can be derived by combininganappropriateobserverwiththeproposedstate feedbac as the system under consideration is observable for any input This shall be presented in a forthcoming wor REFERENCES S K Agrawal and H Sira-Ramirez Differentially Flat Systems Taylor and Francis Group, 4 M Farza, M M Saad, and M Seher A set of observers for a class of nonlinear systems In Proc of the IFAC World Congress, Prague, Czech Republic, 5 MFarza,MM Saad,MTrii,andTMaatoug Highgain observer for a class of non-triangular systems Systms & Control Letters, 6:7 35, M Fliess and I Kupa A finiteness criterion for nonlinear input-output differential systems SIAM Journal on Control and Optimisation, :7 78, 983 M Fliess, J Levine, P Martin, and P Rouchon Flatness and defect of nonlinear systems : introductory theory and examples Int J of Control, 6:37 36, 995 M Fliess, J Levine, P Martin, and P Rouchon A lie-baclund approach to equivalence and flatness of nonlinear systems IEEE Trans Auto Control, AC-44: 9 937, 999 JP Gauthier and IAK Kupa Observability and observers for nonlinear systems SIAM J Control Optim, 3: , 994 JP Gauthier, H Hammouri, and S Othman A simple observer for nonlinear systems - application to bioreactors IEEE Trans on Aut Control, 37:875 88, 99 SHajji,MFarza,MM Saad,andMKamoun Observerbased output feedbac controller for a class of nonlinear systems In In Proc of the 7th IFAC World Congress, Seoul, Korea, 8 H Hammouri and M Farza Nonlinear observers for locally uniformly observable systems ESAIM J on Control, Optimisation and Calculus of Variations, 9: , 3 A Isidori Nonlinear Control Systems Springer, 3rd edition, 995 M Krstič, I Kanellaopoulos, and P Kootovič Nonlinear and Adaptive Control Design John Wiley and Sons, 995 X Liu, K Zhou, and G Gu Global stabilzation of MIMOminimum-phasenonlinearsystemswithoutstrict triangularconditions Systms & Control Letters,36:359 36, 999 N A Mahmoud and H Khalil Asymptotic regulation of minimum phase nonlinear systems using output feedbac IEEE Trans Auto Control, AC-4:4 4, 996 H Nijmeijer Observability of a class of nonlinear systems : a geometric approach Ricerche di Automatica, :5 68, 98 L Praly Asymptotic stabilization via output feedbac for lower triangular systems with output dependent incremental rate IEEE Transactions on Automatic Control, 486:3 8, 3 R Rajamani Observers for Lipschitz Nonlinear Systems IEEE Transactions on Automatic Control, 433:397 4, 998 RSepulchre,MJanovič,andPKootovič Constructive Nonlinear Control Springer, 997 S Seshagiri and H Khalil Robust output feedbac regulation of minimum-phase nonlinear systems using conditional integrators Automatica, 4:43 54, 996