Robust Static Output Feedback Stabilization of an Exothermic Chemical Reactor with Input Constraints

Transcription

1 Robust Static Output Feedback Stabilization of an Exothermic Chemical Reactor with Input Constraints MONIKA BAKOŠOVÁ, ANNA VASIČKANINOVÁ, MÁRIA KARŠAIOVÁ Slovak University of Technology in Bratislava, Faculty of Chemical and Food Technology Institute of Information Engineering, Automation and Mathematics Radlinského 9, Bratislava SLOVAKIA Abstract: Robust static feedback stabilization of linear systems with uncertainties and bounded control inputs is studied. Necessary and sufficient conditions of positive invariance of polyhedral domains with respect to motion of uncertain linear systems are formulated together with conditions ensuring the asymptotic stability of uncertain linear systems with bounded control inputs. The problem of robust static feedback controller calculation is solved by the inversion method. The approach has several advantages. It enables to include control input boundaries into the controller design. It can be used for both, robust static state feedback and robust static output feedback controller design. The designed controllers can be P or PI ones. The designed approach is verified by simulation results obtained for an exothermic continuous stirred tank reactor with two uncertain parameters. The reactor is open-loop unstable. The possibility to stabilize the reactor using static output feedback P or PI controllers in the presence of boundaries on control inputs is confirmed. Key Words: constrained input, uncertainty, robust control, static output feedback, continuous stirred tank reactor 1 Introduction Technological processes usually work with constraints on input variables. The most usual constraints are ofsaturation type and it means that there are limitations on magnitude of control signals and/or magnitude of their changes. So, it is necessary to solve the problem of controller design taking these limitation into account. There are several approaches proposed in the literature to solve the problem of controller design with the control input boundaries, e.g. the positive invariance concept [1, 2], the polynomial concept [3], predictive control techniques [4] and other approaches [5]. From these approaches, the positive invariance approach is selected in this paper, because it gives simple methods to design static outputfeedback controllers with control input constraints in the continuous-time case. The concept of positive invariance is involved in many control problems as constraints, robustness, pole assignment and optimization. Overview of positively invariant sets is given e.g. in[6]. Operation of technological processes is corrupted by different uncertainties. Some of them arise from varying or not exactly known parameters, as e.g. densities, specific heat capacities, reaction rate constants, reaction enthalpies, heat transfer coefficients, etc. The other reason is using linearization techniques for modelling or control system design. The uncertainty is caused by the difference between the real behaviour of the system and thebehaviour of the plant model. Various techniques have been proposed in the literature combining the techniques of robust and constrained control [7, 8]. In this paper, a solution of robust constrained control problem for linear continuous-time systems is described. Robustness conditions for static feedback controllers with constrained control inputs are formulated and the algorithm for obtaining robustly stabilizing P or PI controllers is given. The second part of the paper is devoted to the application of the robust constrained control design to an exothermic continuous stirred tank reactor (CSTR). CSTRs are ones of the most important plants in process industry and exothermic reactors are very interesting systems from the control viewpoint because of their potential safety problems and the possibility of exotic behaviour such as multiple steady states, see e.g. [9]. During the last years, several approaches have been proposed to solve the stabilization problem of chemical reactors with uncertainties, see e.g. [10], [11], [12], [13], [15] and others. This paper proposes a solution of this problem based on linearization technique and subsequent design of robust static output feedback controllers in the presence of boundaries on control inputs. Simula- ISSN: ISBN:

2 tion results confirm possibility to stabilize the CSTR with uncertainties using static output feedback P or PI controllers in the presence of boundaries on control inputs. 2 Robust static output feedback controller design for linear uncertain system 2.1 Problem statement Consider a linear uncertain continuous-time system represented by the following state-space model ẋ(t) = A(q A (t)) x(t) + B (q B (t)) u(t) x 0 = x(t 0 ) (1) y(t) = Cx(t) where x(t) R n is the state, u(t) Ω R m is the control input and y(t) R r is the controlled output. Vectors q A (t) Q A R p A, q B (t) Q B R p B represent uncertain parameters and Q A,Q B are compact convex sets that include origin in their interiors. The control input is restricted by saturation to the following polyhedral set Ω = {u(t) R m : u min u u max, u min, u max Int(R+ m )} (2) Consider further that the controlled system(1) has the affinelinear structure with A,B inthe form A(q(t)) = A 0 + B(q(t)) = B 0 + p A i=1 p B j=1 A i q Ai (t) (3) B j q Bj (t) (4) where q Ai (t), q Bj (t) represent i-th and j-th components ofvectors q A (t),q B (t), respectively. Convexity and compactness of the sets Q A,Q B imply that there exist µ A vertices q k A, k = 1,...,µ A and µ B vertices q l B, l = 1,...,µ B such that q A = q B = µ A k=1 µ B l=1 α k q k A, µ A k=1 α k = 1, α k [0,1] (5) µ B β l qb l, β l = 1, β l [0,1] (6) l=1 Assume further that the pair (A(q A (t)),b(q B (t))) is controllable for every q A Q A and q B Q B. Thenominal system isgiven by ẋ(t) = A 0 x(t) + B 0 u(t), x 0 = x(t 0 ) (7) y(t) = Cx(t) The robust static output feedback control problem isto findan output feedback control law u(t) = Fy(t), F R m r, rank(f) = m (8) or an state feedback control law u(t) = Kx(t), K R m n, rank(k) = n (9) which stabilize the uncertain system (1) with respect to the control constraints (2) such that the closed-loop system is robustly stable in the presence of parametric uncertainty. The nominal system (7) is stabilized as well. Stabilization of(7) means that the nominal closed loop ẋ(t) = (A 0 + B 0 FC)x(t) = (A 0 + B 0 K)x(t) = A CL0 x(t) (10) is stable, e.a. the eigenvalues of A CL0 have negative real parts. Application of the control law(2) divides the state spaceintotworegions. Thefirstregionisthesetofthe linear closed-loop behaviour D 1 = {x(t) R n : u min FCx(t) u max or u min Kx(t) u max, (11) u min,u max Int(R m + )} The uncertain closed loop is given in this region ẋ(t) = (A(q A (t)) + B(q B (t))fc) x(t) = = (A(q A (t)) + B(q B (t))k)x(t) = = A CL (q(t))x(t) (12) where q(t) Q = (Q A Q B ) and q kl, k = 1,...,µ A, l = 1,...,µ B, are vertices of Q. The second region is the set of the nonlinear closed-loop behaviour D 2 = {x(t) R n : u = sat(fcx(t)) = sat(kx(t))} (13) and the uncertain closed loop is given ẋ(t) = A(q A (t))x(t) + B(q B (t))sat(fcx(t)) = = A(q A (t))x(t) + B(q B (t))sat(kx(t)) (14) ISSN: ISBN:

3 2.2 Robust P stabilization with constained input The cornerstone for deriving robust constrained P controllers F satisfying (8) is the positive invariance of domain D 1 (11) with respect to motion of the closedloop system (12) [2]. Definitions and theorems given in this section are taken from [2] and the proofs of them can be found also there. Definition 1 A set S R n is positively invariant with respect to motion of the system (1) if for every x(t 0 ) S, the motion x(t,x(t 0 )) S for all q(t) Q and all t > t 0. Theorem 2 The domain D 1 (11) is positively invariant with respect to the system (10) if and only if there exist amatrix H R m m, such that KA 0 + KB 0 K = HK (15) HU 0 (16) where K = FC and [ ] [ ] H1 H H = 2 umax, U = (17) H 2 H 1 u min { hii for i = j H 1 = h + ij for i j, h + ij = Sup(h ij,0) (18) { 0 for i = j H 2 = h ij for i j, h ij = Sup( h ij,0) (19) Consider the polyhedral domain D p = { x(t) R n : p 2 x p 1, p 1,p 2 Int(R n + )} (20) and the autonomous uncertain system ẋ(t) = A(q(t))x(t), x 0 = x(t 0 ) (21) Lemma3 Theset D p (20)ispositively invariant with respecttomotionofsystem(21)ifandonlyifitispositively invariant with respect to motion of system (21) at vertices q kl, k = 1,...,µ A, l = 1,...,µ B of the set Q. Theorem 4 Thesubset D 1 (11)ispositivelyinvariant with respect to motion of the uncertain system (12) with constrained control (2) if and only if there exist matrices H(q kl ) for k = 1,...,µ A, k = 1,...,µ B such that KA CL (q kl ) = H(q kl )K (22) H(q kl )U 0 (23) Theconditions statedinlemma3nadtheorem4 are necessary and sufficiet [2]. The additional conditions to guaratee the asymptotic stability are formulated in Proposition 5. Proposition 5 K stabilizes asymptotically the uncertain linear system (12) with constrained control (2) if there exist matrices H(q kl ), k = 1,...,µ A, l = 1,...,µ B such that KA CL (q kl ) = H(q kl )K (24) H(q kl )U 0 (25) A CL (q kl ) + A T CL (qkl ) is stable (26) 2.3 Robust PI stabilization with constained input Forthesystem(1),itisnecessarytofindastaticoutput feedback t u(t) = F 1 y(t) + F 2 y(τ)dτ (27) with F 1,F 2 R m r. Define a new state z(t) = [z1 T(t),zT 2 (t)]t, where z 1 (t) = x(t)and z 2 (t) = t 0 y(τ)dτ. Thedynamicsof the newly defined system can by described as follows ż 1 (t) = ẋ(t) = A(q A (t)) z 1 (t) + B (q B (t)) u(t) z 1 (t 0 ) = x(t 0 ) = x 0 (28) ż 2 (t) = y(t) = Cz 1 (t) or ż(t) = A(q A (t)) z(t) + B (q B (t)) u(t) (29) y(t) = Cz(t) (30) where ( ) A 0 A =,B = C 0 ( B 0 0 ),C = ( C 0 0 I ) (31) with I R r r. So, the design of the robust static output feedback PI controller (27) can transformed to the design of the static output feedback P controller F = [F 1 F 2 ], F R m 2r, for thesystem (29), (30). 2.4 Design procedure The algebraic equation (15) representing the condition for positive invariance of (10) plays a fundamental role in the design procedure for robust constrained controller. Two procedures can be used for solution of this equation, the direct and the inverse ones. The inverse procesure is used in this approach [14]. ISSN: ISBN:

4 Thespectrum of A 0 in15contains necessarily (at least) a symmetric set of n m non-null eigenvalues when rank(k) = m. Assume σ(a 0 ) Λ 2 = {λ j C : λ j 0, j = m + 1,...,n} (32) A 0 ζ j = λ j ζ j, j = m + 1,...,n (33) where ζ m+1,...,ζ n are eigenvectors belonging to λ j and they are distinct. Give also a diagonalizable matrix H R m m satisfying thefollowing conditions σ(h) = Λ 1 = {λ i C : λ i 0, i = 1,...,m} (34) Hθ i = λ i θ i, i = 1,...,m (35) where θ 1,...,θ m areeigenvectorsbelongingto λ i and they are linearly independent. Assume further Λ 1 σ(a 0 ) = 0 (36) The assigned spectrum to the matrix A CL0 = A 0 + B 0 K isgiven by with Λ = Λ 1 Λ 2 (37) A CL0 ξ k = λ k ξ k, k = 1,...,n (38) The necessary and sufficient condition of the existence of matrix K, which is the solution of (15), is given in the following theorem [14]. Theorem 6 For a given matrix H R m m satisfying (34) and (36), there exists a real matrix K R m n of full rank as the unique solution of (15) if and only if ξ 1,...,ξ n are linearly independent and K = [θ 1,...,θ m,0 m+1,...,0 n ][ξ 1,...,ξ n ] 1 (39) In Theorem 6, ξ i are eigenvectors associated to the eigenvalues of A CL0 and they are calculated as follows ξ i = (λ i I A 0 ) 1 B 0 θ i, i = 1,...,m (40) Touse the result oftheorem 6for calculation K, the matrix H satisfying following condition must be given: σ(h) σ(a 0 ) = 0 Bθ i 0, i = 1,...,m θ i, i = 1,...,m, are linearly independent (41) Algorithm The input data for the problem of robust constrained controller design are system (1) with known matrices A i,i = 0,...,p A, B j,j = 0,...,p B, the constraints on the control input u min,u max and vertices q kl,k = 1,...,µ A,l = 1,...,µ B of the set of uncertainties Q. Step1: Chooseamatrix H accordingto(41)andsatisfying condition (16). Step 2: Use the inverse procesure and calculate K from (39). Step3: Usethedirectprocedureandcompute H(q kl ) from (24) and verify condition (25). If condition (25) holds then go to the next step. Otherwisereturn tostep 1and change the matrix H. Step4: Compute A CL (q kl ) for k = 1,...,µ A,l = 1,...,µ B and verify condition (26). If condition (26) holds then calculate F from K = FC anduse F asarobust staticoutput feedback controller that robustly stabilizes uncertain closedloop system with constrained control input. OtherwisereturntoStep1andchange thematrix H. Thealgorithmcanbeusedforboth,PorPIrobust controller design. 3 Application to the CSTR The approach described in the theoretical section was applied to design of robust static output feedback controller that asymptotically stabilizes the reactor with uncertainties and bounded control input. The reactor is used for production of propyleneglycol from propyleneoxide and is detaily described in [15]. Two uncertain parameters are considered in this reactor, e.a. pre-exponential factor k in the reaction rate constant and reaction enthalpy ( r H). Their values can lie in following intervals: k < , >min 1, ( r H) < , > kj kmol 1. It is considered further that the reactor has one controlled output - the temperature of the reacting mixture T r [K] and two control inputs - the volumetric flow rates of the reacting mixture q r [m 3 min 1 ] and the cooling medium q c [m 3 min 1 ]. For the controller design, the linearized model of the reactor is used. The matrices in the descriptionoftheuncertainlinearmodel(1)ofthereactorare ISSN: ISBN:

5 A(q(t)) = q A q A q A q A B(q(t)) = C = ( q B q B Theuncertaintiesareboundedasfollows: q A , q A , q A , q B , and q B The control input boundaries are chosen according to (2) as follows: u [m 3 min 1 ], u [m 3 min 1 ]. The rector at the operating point is open-loop unstable and this fact is confirmed by the eigenvalues of A 0, which are ,0.0898, , The open loop behaviour of the nominal model of the CSTRanditsfourvertexsystemsobtained for 4combinations of uncertain parameters is shown in Fig. 1. ) The controller K = [ ] (44) which stabilizes asymptotically the nominal system with constrained control is calculated using (39). Then the robustness of this controller with respect to the parametric uncertainties is studied and the conditions (25) and (26) are verified. Both of them are also fulfilled. Therefore the controller K stabilizes asymptotically the uncertain system with constrained control (2). Then, the static output feedback controller is calulated and obtained as F = [ ] T. Fig. 2 shows the stabilization of the reactor using static output feedback P controller F = [ ] T. Thecontroller isabletostabilize the nominal system as well as all vertex systems. The nominal system and four vertex systems for simulation purposes are represented by the original nonlinear model of the reactor obtained for nominal values of uncertain parameters and by the nonlinear models of the reactor obtained for all combinations of minimum and maximum values of uncertain parameters. Designed P controller robustly stabilizes the CSTR to the open-loop unstable operaiting point, but using of the P controller leads to the offsets in the control responses of vertex systems T r /K T r /K t/min t/min Figure 1: Open-loop behaviour of CSTR The task is to find static output-feedback P controller F that stabilizes the uncertain system with bounded inputs. The algorithm described in the previous section is used and the results are presented for the matrix [ ] H =. (42) Chosen H satisfies conditions (41) and (16): H 0 U = [ (43) ] T Figure 2: Robust static output feedback stabilization of CSTR using P controller Similarly, the static output-feedback PI controller F that stabilizes the uncertain system with bounded inputs is found. The algorithm described in the previous section is used and the results are presented for the matrix [ ] H =. (45) The obtained static output feedback PI controller is [ ] F =. (46) ISSN: ISBN:

6 Fig. 3 shows the PI stabilization of the reactor using designed robust static output feedback PI controller for the constrained case. The controller is able to stabilize the nominal systems as well as four vertex systems of the original nonlinear model into the openloop unstable operating point without offsets. T r /K t/min Figure 3: Robust static output feedback stabilization of CSTR using PI controller 4 Conclusion The paper presents approach to the design of static feedback controllers for linear systems with uncertainty and constrained inputs. Given algorithm is simple and easy to use especially in the case of small number of vertices of uncertain parameters. Checking and assuring necessary and sufficient conditions for asymptotic stabilization of uncertain system with bounded inputs is notso easy inthe case whenthe dimensionsofthevectorofuncertainty ishigh. Thekey role in the presented algorithm plays the choice of the matrix H. The advantage of the algorithm is that it can be used for both, the P or PI robust static output feedbacck controller design. Acknowledgements: The authors gratefully acknowledge the support by the Scientific Grant Agency of the Slovak Republic under the grants 1/0537/10, 1/0071/09 and the Slovak Research and Development Agency under the project VV References: [1] A. Benzaouia, F. Mesquine, M. Naib and A. Hmamed, Robust pole assignment in complex plane regions for linear uncertain constrained control systems, Int. J. Systems Science 32, 2001, pp [2] F. Mesquine, F. Tadeo and A. Benzaouia, Regulator problem for linear systems with constraints on control and its increment or rate, Automatica 40, 2004, pp [3] D. Henrion, S. Tarbouriech and V. Kučera, Control of linear systems subject to input constraints: a polynomial approach, Automatica 37, 2001, [4] V. Veselý, Output stable model predictive control design with input constraints, JEE 60, 2009, pp [5] A. A. Stoorvogel, A. Saberi and P. Sannuti, Performance with regulation constraints, Automatica 36, 2000, pp [6] F. Blanchini, Set invariance in control, Automatica 35, 1999, pp [7] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, New Jersey 1996 [8] F. Mesquine, A. Benlemkadem and A. Benzaouia, Robust contrained regulator problem for linear uncertain systems, Journal of Dynamical and Control Systems 10, 2004, [9] A. Molnár, J. Markoš and Ľ. Jelemenský, Accuracy of mathematical model with regard to safety analysis of chemical reactors, Chemical Papers 56, 2002, pp [10] J. Alvarez-Ramirez and R. Femat, Robust PI stabilization of a class of chemical reactors, Sytems and Control Letters 38, 1999, pp [11] J. Gerhard, M. Mönningmann and W. Marquardt, Robust stable nonlinear control and design of acstrin alarge operating range, Proc. 7th Int. Symp. Dynamics and Control of Process Systems, Massachusetts, USA, 2004, 92.pdf. [12] M. Bakošová, D. Puna and A. Mészáros, Robust Controller Design for a Chemical Reactor. In: L. Puigjaner and A. Espuna(Eds.), European Symposium on Computer Aided Process Engineering - 15, Elsevier, Amsterdam, 2005, pp , [13] A. F. Tlacuahuac, J. Alvarez, E. S. Guerra and G. Oaxaca, Optimal transition and robust control design for exothermic continuous reactors, AIChE Journal 51, 2005, [14] A. Benzaouia, The resolution of equation XA+XBX=HX and the pole assignement problem, IEEE Trans. Automatic Control 39, 1994, [15] M. Bakošová, D. Puna, P. Dostál and J. Závacká, Robust stabilization of a chemical reactor, Chemical Papers 63, 2009, ISSN: ISBN: