Slovak Society of Chemical Engineering Institute of Chemical and Environmental Engineering Slovak University of Technology in Bratislava PROCEEDINGS

Transcription

1 Slovak Society of Chemical Engineering Institute of Chemical and Environmental Engineering Slovak University of Technology in Bratislava PROCEEDINGS 37 th International Conference of Slovak Society of Chemical Engineering Hotel Hutník Tatranské Matliare, Slovakia May 24 28, 21 Editor: J. Markoš ISBN Bakošová, M., Vasičkaninová, A., Karšaiová, M.: Robust stabilization of a cstr with constrained input, Editor: Markoš, J., In Proceedings of the 37th International Conference of Slovak Society of Chemical Engineering, Tatranské Matliare, Slovakia, , 21.

2 ROBUST STABILIZATION OF A CSTR WITH CONSTRAINED INPUT 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, tel.: , monika.bakosova@stuba.sk Abstract: Hydrolysis of propylene oxide to propylene glycol in a CSTR was chosen as a controlled process. Considered reactor is the exothermic one with multiple steady states. Model uncertainties of the CSTR follow from the fact that there are two physical parameters, which values are known within some intervals, the reaction enthalpy and the pre-exponential factor. From the viewpoint of safety operation or in the case when the unstable steady-state coincides with the point that yields the maximum reaction rate at a prescribed temperature, it can be necessary to control CSTRs into the surroundings of the open-loop unstable steady-state. In this context, possibilities to stabilize the CSTR into its open-loop unstable stead-state were studied and the static output feedback approach was chosen for stabilization. The designed approach for robust stabilization of the CSTR is based on robust static feedback for linear systems with uncertainty and constrained control inputs. Necessary and sufficient conditions of positive invariance of polyhedral domains with respect to motion of uncertain systems are formulated. Conditions ensuring the asymptotic stability of uncertain linear systems with bounded control inputs are also given. The problem of robust static feedback controller calculation is solved by inversion method. The presented algorithm can be used for both, robust static state feedback and robust static output feedback controller design. The algorithm is simple and easy to use especially in the case of small number of vertices of uncertain parameters. The advantage of the procedure is that it can be used for P or PI controller design. Presented simulation results confirm that the CSTR can be successfully stabilized into its open-loop unstable steady-state using robust static output feedback in the presence of uncertainties as well as in the presence of control input boundaries. Keywords: CSTR, constrained input, uncertainty, static output feedback, robust control. 1 INTRODUCTION Chemical reactors are ones of the most important plants in chemical industry and exothermic continuous stirred tank reactors(cstrs) are very interesting systems from the control viewpoint because of their potential safety problems and the possibility of exotic behavior such as multiple steady states, see e.g. Molnár et al. [22]. Furthermore, operation of chemical reactors is corrupted by various uncertainties. Some of them arise from varying or not exactly known parameters, as e.g. reaction rate constants, reaction enthalpies, heat transfer coefficients, etc. Operating points of reactors change in other cases. All these uncertainties can cause poor performance or even instability of closed-loop control systems. Optimal control strategies, which are often used for reactor control design, can fail in the presence of uncertainties. Application of robust control approach is one way for overcoming all these problems, as it is shown e.g. in Alvarez-Ramirez and Femat [1999], Gerhard et al. [24], Bakošová etal.[25], Tlacuahuac etal.[25] andothers. Robustcontrol hasgrownasoneofthemostimportant areas in modern control design since works by Doyle and Stein [1981], Zames and Francis [1983] and many others. One of the solved problems is also the problem of robust static output feedback control (RSOFC), which has been till now an important open question in control engineering, see e.g. Iwasaki et al. [1994], Syrmos et al. [1997] and references therein. Technological processes in practice work usually with constraints on input variables. The most usual constraints are of saturation type and it means that there are limitations on the magnitude and/or derivative of control signals. So, the problem of control system design appears that is connected with stability and performance in the presence of input constraints. There are several approaches proposed in 113

3 the literature to solve this problem, e.g. the positive invariance concept[benzaouia et al., 21; Mesquine et al., 24b], the polynomial concept [Henrion et al., 21], predictive control techniques [Veselý, 29] and other approaches [Stoorvogel et al., 2]. From these approaches, the positive invariance approach is selected in this paper, because it gives simple methods to design static output-feedback controllers in the continuous-time case with symmetrical and nonsymmetrical constraints. 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 Blanchini [1999]. Several techniques have been proposed in the literature combining the techniques of robust and constrained control [Zhou et al., 1996; Mesquine et al., 24a]. This paper proposes a solution to this problem based on linearization technique and subsequent design of linear controllers with guaranteed robustness characteristics even in the presence of limitations on the control signals. Simulation results show the performance of such Pand PIcontrollers used for stabilization of acstr. 2 CONTROLLED CSTR Hydrolysis of propylene oxide to propylene glycol in a continuous stirred tank reactor (Molnár et al. [22]) was chosen asacontrolled process. Thereaction isas follows C 3 H 6 O + H 2 O C 3 H 8 O 2 (1) The reactor is fed with propylene oxide, methanol and water. Methanol is added to improve the solubility of propylene oxide in water. The excess of water provides higher selectivity to propylene glycol and eliminates the consecutive reactions of propylene oxide with propylene glycol. The reaction is of the first order with respect to propylene oxide as a key component. The dependence of the reaction rate constant on the temperature is described by the Arrhenius equation k = k e Ea RTr (2) where k is the pre-exponential factor, E a is the activation energy, R is the universal gas constant and T r isthetemperatureofthereactionmixture. Assumingidealmixinginthereactor, theconstant reacting volume and the same volumetric flow rates of the inlet and the outlet streams, the mass balance for any species j in thesystem is V r dc j dt = q r (c j c j ) + ν j rv r, (3) where V r is the reacting volume, c j is the molar concentration and c j is the feed molar concentration of the j-th component, q r is the volumetric flow rate of the reaction mixture, ν j is the stoichiometric coefficient of the j-th component, r = kc C3 H 6 O is the molar rate of the chemical reaction. It is assumed further that the specific heat capacities, densities and volumetric flow rates do not depend on temperature and composition, and also the heat of mixing and the mixing volume can be neglected. The simplified enthalpy balance of the reaction mixture used as a standard at reactor design (Ingham et al. [1994]) is V r ρ r C P r dt r dt = q rρ r C P r (T r T r ) UA (T r T c ) + rv r ( r H o ) (4) and the simplified enthalpy balance of the cooling medium is V c ρ c C P c dt c dt = q cρ c C P c (T c T c ) + UA (T r T c ) (5) where T is the temperature, ρ is the density, C P is the specific heat capacity, r H o is the reaction enthalpy, U is the overall heat transfer coefficient, A is the heat exchange area. The subscripts denote: the feed, c the cooling medium and r the reaction mixture. The values of constant parameters and steady-state inputs of the CSTR are summarized in Table

4 Table 1 Constant parameters and steady-state inputs of CSTR Variable Value Unit V r 2.47 m 3 V c 2 m 3 ρ r kgm 3 ρ c 998 kgm 3 C P r kjkg 1 K 1 C P c kjkg 1 K 1 AU 12 kjmin 1 K 1 E a /R 1183 K q r.72 m 3 min 1 q c.637 m 3 min 1 c C3 H 6 O,.824 kmol m 3 c C3 H 8 O 2, kmol m 3 T r K T c K Uncertainties of the over described reactor follow from the fact that there are two physical parameters in this reactor, which values are known within intervals (Table 2); the reaction enthalpy and the pre-exponential factor. The nominal values of these parameters are mean values of the intervals and they are used for deriving of the nominal model of the CSTR.Theminimum and the maximum values of the intervals are used for obtaining models, which create the vertex systems. Table 2 Uncertain parameters in CSTR Parameter Unit Minimal Value Maximal Value r H o kjkmol k min STEADY-STATE AND OPEN-LOOP ANALYSIS OF THE CSTR The steady state behavior of the CSTR with nominal values and also with all 4 combinations of minimum and maximum values of 2 uncertain parameters was studied at first. It can be stated that the reactor has always three steady states, two of them are stable and one is unstable. The situation for the nominal model is shown in Figure 1, where the curve Q GEN (red line) is the heat generated by the reactionandtheline Q OUT (blueline)istheheatwithdrawnfromthereactor. Thesteady-state operating points of the reactor are points, where the curve and the line intersect. The steady states are stable if the slope of the cooling line is higher then the slope of the heat generated curve. This condition is satisfied in the steady states at the temperatures T r = K and T r = K, and it is not satisfied in the steady state at T r = K. The steady-state behavior of the chemical reactor is similar for all vertex systems. From the viewpoint of safety operation or in the case when the unstable steady state coincides with the point that yields the maximum reaction rate at the prescribed temperature, it is necessary to control CSTRs into the prescribed open-loop unstable steady-state, see e.g. Pedersen and Jorgensen [1999], Antonelli and Astolfi [23], González and Alvarez [26], Salgado et al. [26], Salau et al. [26]. In this context, the open-loop behavior of the reactor in the surroundings of its unstable steady state was studied at first. The initial temperature of the reaction mixture was chosen T r () = K. 115

5 4 x (Q GEN,Q OUT )/(kj min 1 ) 2 1 T r /K T /K r Figure 1 Three steady states of CSTR Figure 2 Open-loop response of CSTR Simulation results obtained for the nominal system and 4 vertex systems are shown in Figure 2. They confirm that without feedback control, the temperature of the reaction mixture in the CSTR converges to the values characteristic either for the upper or the lower stable steady state. 4 ROBUST STATIC OUTPUT FEEDBACK CONTROLLER DESIGN FOR LINEAR UNCER- TAIN SYSTEM WITH CONSTRAINED INPUT 4.1 Problem statement model Consider the linear uncertain continuous-time system represented by the following state-space ẋ(t) = A (q A (t)) x(t) + B (q B (t)) u(t), x = x(t ) (6) y(t) = Cx(t) where x(t) R n is the state, u(t) Ω R m is the control input, y(t) R r is the controlled output. q A (t) Q A R p A q B (t) Q B R p B are vectors of uncertain parameters and Q A, Q B are compact convex sets including origin in their interiors. The control input is restricted by saturation to the following polyhedral set form Ω = {u(t) R m : u min u u max, u min, u max IntR m + } (7) Consider further that the controlled system (6) has the affine linear structure with A, B in the A(q(t)) = A + B(q(t)) = B + p A i=1 p B j=1 A i q Ai (t) (8) B j q Bj (t) (9) where q Ai (t), q Bj (t)represent i-th and j-th components of vectors 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, β l q l B, µ A k=1 µ B l=1 α k = 1, α k [, 1] (1) β l = 1, β l [, 1] (11) 116

6 Q B. Assume further that the pair (A(q A (t)), B(q B (t))) is controllable for every q A Q A and q B Thenominal system isgiven by ẋ(t) = A x(t) + B u(t), x = x(t ) (12) y(t) = Cx(t) The robust static output-feedback control problem is to find an output-feedback control law u(t) = Ky(t), K R m r, rank(k) = m (13) which stabilizes the nominal system(12) with respect to the control constraints(7) such that the feedback system is robustly stable against parametric uncertainty, e.a. the output feedback (13) stabilizes also the uncertain system (6). Stabilization of (12) means that the nominal closed loop ẋ(t) = (A + B KC)x(t) = (A + B F )x(t) = A CL x(t) (14) is stable, e.a. the eigenvalues of A CL have negative real parts and belong to the set D defined by (15) or (16). The application of the control law (7) divides the state space into two regions. The first region is the set of the linear closed-loop behaviour or D = {x(t) R n : u min KCx(t) u max, u min, u max IntR m + } (15) D = {x(t) R n : u min F x(t) u max, u min, u max IntR m + } (16) where F = KC. Theclosed loop in thefirst region is given ẋ(t) = (A(q A (t)) + B(q B (t))kc)x(t) = (A(q A (t)) + B(q B (t))f )x(t) = ( µb ) µ A = β l α k (A k + B l F ) x(t) = (17) = l=1 k=1 ( µb ) µ A β l α k A CL (q kl ) x(t) = A CL (q(t))x(t) l=1 k=1 where q(t) Q = (Q A Q B )and q kl denotes the vertices of Q, k = 1,..., µ A, l = 1,..., µ B. Thesecond region isthe set wherethe closed loop isnonlinear and given ẋ(t) = A(q A (t))x(t) + B(q B (t))sat(kcx(t)) = A(q A (t))x(t) + B(q B (t))sat(f x(t)) (18) It is necessary to emphasize [Mesquine et al., 24b] that positive invariance of domain D given by (15) or (16) with respect to motion of the closed-loop system (17) is the cornerstone to derive robust constrained regulators. 4.2 Robust stabilization with constrained input Definitions and theorems given in this section are taken from Mesquine et al. [24b]. Definition4.1 A set S R n ispositively invariant withrespect to motion of the system (6) if for every x(t ) S,the motion x(t, x(t )) S for all q(t) Q and all t > t. To obtain a solution of the stated problem, it is necessary to extend some results established without uncertainties to the case of uncertain systems. So, let us recall these results, which consist of positive invariance conditions and a definition of a nonquadratic Lyapunov function. The extension will be studied afterwards. Positive invariance conditions are formulated as follows. 117

7 Theorem 4.2 The domain (16) is positively invariant with respect to the system (14) if and only if there exist amatrix H R m m,such that F A + F B F = HF (19) HU (2) where H = [ ] [ ] H1 H 2 umax, U = H 2 H 1 u min { hii for i = j h + ij for i j, h + ij = Sup(h ij, ) { for i = j h ij for i j, h ij = Sup( h ij, ) (21) H 1 = (22) H 2 = (23) Now, the extension for the case of uncertain plants will be presented. This extension gives conditions for designing robust controllers in the presence of uncertainty and input limitations. The following lemma is fundamental to extend necessary and sufficient conditions for the positive invariance of the polyhedral domain. Let us suppose D p = { z R m : p 2 z p 1, p 1, p 2 IntR+ m } (24) with respect to motion of the autonomous uncertain system ż(t) = H(q(t))z(t) Lemma4.3 The set D p is positively invariant with respect to motion of system (25) if and only if it is positively invariant with respect to motion of system (25) at vertices q kl, k = 1,..., µ A, l = 1,..., µ B,of the set Q. The necessity and sufficiency of conditions formulated in the Lemma 4.3 are proved in Mesquine et al. [24b]. Let us further consider the nonautonomous uncertain system (6) with constrained control (7). The application of the output feedback control law(13) leads to the closed-loop system (17). Theorem 4.4 The subset D given by (16) is positively invariant with respect to motion of the uncertain system (17) with constrained control (7) if and only if there exist matrices H(q kl ) for k = 1,..., µ A, k = 1,..., µ B, such that F A CL (q kl ) = H(q kl )F (26) H(q kl )U (27) The necessity and sufficiency of conditions given in the Theorem 4.4 are proved in Mesquine et al. [24b]. The additional conditions to guaratee the asymptotic stability are formulated in Proposition 4.5. Proposition 4.5 F stabilizes asymptotically the uncertain linear system(17) with constrained control(7) if there exist matrices H(q kl ), k = 1,..., µ A, l = 1,..., µ B,such that (25) F A CL (q kl ) = H(q kl )F (28) H(q kl )U (29) A CL (q kl ) + A T CL(q kl ) isstable (3) Theproof of Proposition 4.5isagain given inmesquine et al. [24b]. 118

8 4.3 Algorithm The algebraic equation (19) F A + F B H = HF (31) plays a fundamental role in this design method. Two procedures can be used for solution of this equation, the direct and the inverse ones. The inverse procesure is used in this approach [Benzaouia, 1994]. The spectrum of A in (31) contains necessarily (at least) a symmetric set of n m non-null eigenvalues whenrank(f ) = m. Then assume σ(a ) Λ 2 = {λ j C : λ j, j = m + 1,..., n} (32) A ζ j = λ j ζ j, j = m + 1,..., n (33) where σ(a ) is the spectrum of A and ζ m+1,..., ζ n, are eigenvectors belonging to λ j and they are distinct. Let us givealso a diagonalizable matrix H R m m satisfying the following conditions σ(h) = Λ 1 = {λ i C : λ i, i = 1,..., m} (34) Hθ i = λ i θ i, i = 1,..., m (35) where θ 1,..., θ m are eigenvectors belonging to λ i and they are linearly independent. We further assume Λ 1 σ(a ) = (36) Theassigned spectrum to the matrix A + B F isthen given by Λ = Λ 1 Λ 2 (37) with A ξ k = λ k ξ k, k = 1,..., n (38) Thenecessary andsufficient condition of theexistence ofmatrix F that isthesolution of(31) isgivenin the following theorem. Theorem 4.6 For a given matrix H R m m satisfying (34) (36), there exists a real matrix F R m n of full rank as the unique solution of (31) if and only if ξ 1,..., ξ n are linearly independent and F = [θ 1,..., θ m, m+1,..., n ] [ξ 1,..., ξ n ] 1 (39) In Theorem 4.6, ξ i are eigenvectors associated to eigenvalues of A that are elements of Λ 2 and they are calculated as follows ξ i = (λ i I A ) 1 Bθ i, i = 1,..., m (4) To use the result of Theorem 4.6 for calculation F, the matrix H must be given satisfying following condition: σ(h) σ(a ) = Bθ i, i = 1,..., m (41) θ i, i = 1,..., m are linearly independent Algorithm 4.7 Thedataforthisproblemaresystem(6)withknownmatrices A i, i =,..., p A, B j, j =,..., p B,theconstraintsonthecontrol u min, u max andvertices q kl, k = 1,..., µ A, l = 1,..., µ B of the set of uncertainties Q. 119

9 Step 1: Choose a matrix H according to(41) and satisfying condition (2). Step 2: Use the inverse procesure to solve the equation (31), e.a. calculate F from (39). Step3: Use the direct procedure and compute H(q kl ) from the equation F A(q kl ) = H(q kl )F for k = 1,..., µ A, l = 1,..., µ B and verify condition (29). If condition (29) holds then go to the next step. Otherwise return to Step1and change thematrix H. Step4: Compute A CL (q kl ) for k = 1,..., µ A, l = 1,..., µ B, and verify condition (3). If condition (3) holds then use the feedback F as a solution of the robust constrained control problem. Otherwise return tostep 1and change the matrix H. 5 ROBUST STABILIZATION OF THE CSTR The approach described in Section 4 was applied to design of robust static output feedback controllers that asymptotically stabilize the reactor described in Sections 2 and 3. For the controller desing, it is considered that the reactor has one controlled output- the temperature of the reacting mixture T r [K] and two control inputs - the flow rate of the reacting mixture q r [m 3 s 1 ] and the flow rate of the cooling medium q c [m 3 s 1 ]). The control input boundaries were chosen as follows: q r.18 m 3 s 1, q c m 3 s 1. The matrices of the nominal linearized model at the operating point (open-loop unstable steady state) are A = , B = C = ( 1 ) (42) Theeigenvaluesof A are.299,.929,.26,.331,andtheyconfirmetheinstability ofthereactor at theoperating point. Thematricesinthelinearized uncertain model (6)of thereactor are q A1.1 A(q(t)) = q A q A q A3.138 (43) B(q(t)) = q B q B4, (44) The uncertainties are bounded as follows:.142 q A1.142, q A ,.157 q A3.157, q B , and.1285 q B Robust stabilization using P controller The task is to find a static output-feedback P controller K (13) that stabilizes the uncertain system with bounded inputs. The algorithm described in the previous section is used and the results are presented for the chosen matrix [ ] H = (45) Thecontroller F isobtained in theform [ ] F = (46) 12

10 and thestatic otput feedback P controller K is then [ ].343 K = The possibility to stabilize the CSTR using the designed static output-feedback controller K was verified by simulations. The controller stabilizes the CSTR with uncertainties and with constrained control as is shownin Fig. 3. Figs. 3 and 4 compare the stabilization of the reactor designed for the constrained and unconstrained case. The robust static output feedback controller for unconstrained case was obtained using LMI based approach described in Bakošová et al. [29]. The robust static output feedback controller found for the unconstrained inputs assures faster stabilization of the reactor for the nominal system and four vertex systems. The robust static output feedback controller designed for the constrained inputs leads to slowlier control responses and greater offsets. The control inputs for both, constrained and unconstrained cases are shown in Figs (47) T r /K 343 T r /K Figure3 Robuststaticoutput feedbackpstabilization of CSTR- constrained case Figure4 Robust staticoutputfeedback Pstabilization of CSTR- unconstrained case q r.8.6 q r Figure 5 Constrained input q r for robust P stabilization of CSTR Figure6 Unconstrained input q r forrobust P stabilization of CSTR 5.2 Robust stabilization using PI controller Now, the task is to find static output-feedback PI controller K = [K 1, K 2 ] that stabilizes the CSTR with uncertainties and with bounded inputs. The static output-feedback PI controller design problem is to find an output-feedback control law t u(t) = K 1 y(t) + K 2 y(τ)dτ (48) 121

11 q c.8 q c Figure 7 Constrained input q c for robust P stabilization of CSTR Figure8 Unconstrained input q c for robust P stabilization of CSTR with K 1, K 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 y(τ)dτ. The dynamics of the newly defined system can by described as follows ż 1 (t) = ẋ(t) = Az 1 (t) + Bu(t), x(t ) = x (49) ż 2 (t) = y(t) = Cz 1 (t) (5) or ż(t) = Az(t) + Bu(t) (51) where A = ( A C ) ( B, B = ). (52) Theoutput of the newly defined system canby described as follows where y(t) = Cz(t) C = ( C I ) (53) (54) with I R r r. So,the design of thestatic output feedback PIcontroller (48) istransformed to thedesign of the static output feedback Pcontroller K = [K 1 K 2 ] for thesystem (51) and (53). The algorithm described in Section 4 is used and the results are presented for the matrix H = [.15.25; 1. 5.]. The controller F is obtained in the form [ F = The static output feedback PI controller is then calulated and obtained as ] (55) K = [ ] (56) 122

12 The possibility to stabilize the CSTR using the designed static output-feedback controller K was verified by simulations. The controller stabilizes the CSTR with uncertainties and with constrained control as is shown in Fig. 9. Figs. 9 and 1 compare the stabilization of the reactor designed for the constrained and unconstrained case. The robust static output feedback controller for unconstrained case was obtained using LMI based approach described in Bakošová et al. [29]. The robust static output feedback controller found for the constrained inputs assures faster stabilization of the reactor for the nominal system and four vertex systems. The robust static output feedback controller designed for the unconstrained inputs leads to slowlier control responses. The control inputs for both, constrained and unconstrained cases are shown in Figs T r /K T r /K Figure 9 Robust static output feedback PI stabilization of CSTR- constrained case Figure 1 Robust static output feedback PI stabilization of CSTR- unconstrained case q r q r Figure 11 Constrained input q r for robust PI stabilization of CSTR Figure 12 Unconstrained input q r for robust PI stabilization of CSTR 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 not so easy in the case when the dimensions of the vector of uncertainty is high. The key role in the presented algorithm plays the choice of the matrix H. One possibility to obtain the reasonable choice of H is to use for its guess any controller found for unconstrained case with or withou uncertainty. 123

13 q c q c Figure 13 Constrained input q c for robust PI stabilization of CSTR Figure 14 Unconstrained input q c for robust PI stabilization of CSTR 6 ACKNOWLEDGMENTS The work has been supported by the Scientific Grant Agency of the Slovak Republic under grants 1/537/1, 1/71/9 and by the Slovak Research and Development Agency under the project APVV This support is very gratefully acknowledged. References ALVAREZ-RAMIREZ, J.; FEMAT, R Robust PI stabilization of a class of chemical reactors. Sytems and Control Letters, 38, ANTONELLI, R.; ASTOLFI, A. 23. Continuous stirred tank reactors: easy to stabilise? Automatica, 39, ISSN BAKOŠOVÁ, M.; PUNA, D.; DOSTÁL, P.; ZÁVACKÁ, J. 29. Robust stabilization of a chemical reactor. Chemical Papers, 63, 5, ISSN BAKOŠOVÁ, M.; PUNA, D.; MÉSZÁROS, A. 25. Robust controller design for a chemical reactor. In PUIGJANER, L.; ESPUNA, A., editors, European Symposium on Computer Aided Process Engineering - 15, Amsterdam: Elsevier. BENZAOUIA, A The resolution of equation XA+XBX=HX and the pole assignement problem. IEEE Trans. Automatic Control, 39, ISSN BENZAOUIA, A.; MESQUINE, F.; NAIB, M.; HMAMED, A. 21. Robust pole assignment in complex plane regions for linear uncertain constrained control systems. Int. J. Systems Science, 32, BLANCHINI, F Set invariance in control. Automatica, 35, 11, ISSN DOYLE, J. C.; STEIN, G Multivariable feedback design - concepts for a classical modern synthesis. IEEE Transactions on Automatic Control, 26, ISSN GERHARD, J.; MONNINGMANN, M.; MARQUARDT, W. 24. Robust stable nonlinear control and design of a CSTR in a large operating range. In Proceedings of the 7th International Symposium on Dynamics and Control of Process Systems, 92.pdf. Massachusetts, USA. GONZÁLEZ, P.; ALVAREZ, J. 26. Output-feedback control of continuous polymer reactors with continuous and discrete measurements. In International Symposium on Advanced Control of Chemical Processes, 135.pdf. Gramado. HENRION, D.; TARBOURIECH, S.; KUČERA, V. 21. Control of linear systems subject to input constraints: a polynomial approach. Automatica, 37, 4, ISSN INGHAM, J.; DUNN, I. J.; HEINZLE, E.; PŘENOSIL, J. E Chemical Engineering Dynamics. Weinheim: VCHVerlagsgesellschaft. IWASAKI, T.; SKELTON, R. E.; GEROMEL, J. C Linear quadratic suboptimal control with static output feedback. Systems Control Lett., 23,

14 MESQUINE, F.; BENLEMKADEM, A.; BENZAOUIA, A. 24a. Robust contrained regulator problem for linear uncertain systems. Journal of Dynamical and Control Systems, 1, 4, MESQUINE, F.; TADEO, F.; BENZAOUIA, A. 24b. Regulator problem for linear systems with constraints on control and its increment or rate. Automatica, 4, ISSN MOLNÁR, A.; MARKOŠ, J.; Ľ. JELEMENSKÝ 22. Accuracy of mathematical model with regard to safety analysis of chemical reactors. Chemical Papers, 56, ISSN PEDERSEN, K.; JORGENSEN, S. B Control of fold bifurcation application on chemostat around critical dilution rate. In Proceedings of the European Control Conference ECC 99, 128.pdf. Karlsruhe, Germany. SALAU, N. P. G.; SECCHI, A. R.; TRIERWEILER, J. O.; NEUMANN, G. A. 26. Multivariable control strategy based on bifurcation analysis of an industrial gas-phase polymerization reactor. In International Symposium on Advanced Control of Chemical Processes, 166.pdf. Gramado. SALGADO, J. D.; ALVAREZ, J.; MORENO, J. 26. Control of continuous reactors with nonmonotonic reaction rate. In International Symposium on Advanced Control of Chemical Processes, 183.pdf. Gramado. STOORVOGEL, A. A.; SABERI, A.; SANNUTI, P. 2. Performance with regulation constraints. Automatica, 36, 1, ISSN SYRMOS, V. L.; ABDALLAH, C. T.; DORATO, P.; GRIGORIADIS, K Static output feedback. a survey. Automatica, 33, ISSN TLACUAHUAC, A. F.; ALVAREZ, J.; GUERRA, E. S.; OAXACA, G. 25. Optimal transition and robust control design for exothermic continuous reactors. AIChE Journal, 51, VESELÝ, V. 29. Output stable model predictive control design with input constraints. JEE, 6, 6, ISSN ZAMES, G.; FRANCIS, B. A Feedback, minimax sensitivity and optimal robustness. IEEE Transactions on Automatic Control, 28, ISSN ZHOU, K.; DOYLE, J. C.; GLOVER, K Robust and Optimal Control. New Jersey: Prentice-Hall. 125