FROM OPEN-LOOP TRAJECTORIES TO STABILIZING STATE FEEDBACK -APPLICATION TO A CSTR-

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1 FROM OPEN-LOOP TRAJECTORIES TO STABILIZING STATE FEEDBACK -APPLICATION TO A CSTR- Nicolas Marchand Mazen Alamir Laboratoire d Automatique de Grenoble UMR 5528, CNRS-INPG-UJF, ENSIEG BP 46, 3842 Saint Martin d Hères Cedex, France Abstract: In this paper, a new state feedback is presented for the stabilization of a nonlinear systems. It is based on the generation of open-loop trajectories used to design a stabilizing feedback. A general theoretical background based on Lyapunov theory is given. The application of the feedback to a Continuous Stirred Tank Reactor, known to exhibit unstable steady states, is provided. Résumé: Dans cet article, un retour d état basé sur la génération de trajectoires en boucle ouverte est proposé dans le but de stabiliser des systèmes non linéaires. Un fond théorique assorti d une preuve basée sur la théorie des fonctions de Lyapunov ainsi que l application de cette méthode un réacteur chimique connu pour exhiber des etats stationnaires instables sont proposés. Copyright c 1998 IFAC Keywords: Nonlinear control, Lyapunov function, Global stability, Open-loop, State feedback 1. INTRODUCTION There exist only few methods for designing stabilizing control laws for nonlinear systems when no special structure of the system is assumed. Like for the linear case, where the open-loop properties are used to design the feedback, some of these methods are based on the generation of open-loop trajectories. One of those control strategies based on the generation of an open-loop trajectory, that enables at least conceptually to stabilize a very wide class of nonlinear system is the optimal receding horizon control (Keerthi and Gilbert, 1987; Mayne and Michalska, 199; Alamir and Bornard, Unfortunately, the open-loop trajectory is the solution of the minimization of a cost function over a non convex set which is in most cases computationally infeasible or at least very costly. M.A. Zarkh (Zarkh, 1995 proposed the use of the open-loop control with a revising feedback in order to avoid frequent computations of the control. Open-loop trajectories are often used in the robotic framework where steering a system from an initial point to a desired one has always been a major subject of study. The work of Teel et al. (Teel et al., 1995 can be mentioned here as an example of steering open-loop controls used for the design of a stabilizing closed loop controller for chained form systems. Open-loop trajectories take also an important place in path generation and obstacle avoidance (see for ex. (Kawaji and Matsunaga, In the framework of differentially flat systems (Fliess et al., 1992; Fliess et al., 1995; Fliess et al., 1996, open-loop trajectories are widely used. Indeed, open-loop trajectories can be parametrized by the trajectory of what is called the flat output, avoiding the integration of the system equations. This enables the design of a stabilizing state feedback by tracking the error with respect to an a priori defined open-loop trajectory satisfying a stable asymptotic behaviour. In this paper, we propose a new way for designing a stabilizing state feedback based on the generation of

2 steering open-loop trajectories. Sufficient conditions on the open-loop trajectory generator are given such that the proposed feedback asymptotically stabilizes the given nonlinear system. A link to the infinite horizon theory is established. The paper is organized as follows. In section 2, the system under consideration and the proposed stabilizing state feedback are exposed. The section ends with remarks on the feedback and its relation with existing fields of nonlinear control. Section 3 gives the proof of the asymptotic stability of the closed loop system. The last section is devoted to the application of the method to a continuous stirred tank reactor. 2. PROBLEM STATEMENT AND SOLUTION METHOD The connection between open-loop controls, and therefore open-loop trajectories, and closed loop stability has always been an important point in the literature (necessary conditions for the stabilization (Brockett et al., 1983, theory of optimal receding horizon (Mayne and Michalska, 199, etc. Our aim in this paper is to give conditions on open-loop trajectories in order to build a stabilizing feedback based on them. This work can be tied up with the receding horizon where minimization of a cost function gives an open-loop control used to define the closed loop control that ensures the global asymptotic stability of the system. In this section, the system under consideration is first presented, and the proposed stabilizing feedback follows. 2.1 System under consideration In this paper, the following time invariant nonlinear system will be considered: ẋ(t = f(x(t, u(t (1 where x(t R n is the state vector and u(t R m the control at time t. x will denote the initial state. We shall suppose that the system verifies the following assumption: Assumption 2.1. f : (x, u f(x, u is continuously differentiable on R n R m. We shall now give some notations. 2.2 notations L 2 will denote the space of time functions square integrable on [, + [. For any z R n, z 2 shall denote z T z. For any z R n, z i shall denote the i th component of the vector z. x(t; t ; x ; u will denote the solution at instant t of (1 when it exists, under the control u(. and starting from the state x at instant t. Following Hahn (Hahn, 1967, a function Φ : R + R + will be said of class K if Φ is continuous, Φ( = and Φ is monotonically increasing. 2.3 Trajectory generator C x will denote the set of open-loop control such that if u C x, lim x(t; ; x ; u =. An admissible t open-loop control will denote a control in C x. The corresponding trajectory will be called admissible open-loop trajectory and the set of admissible openloop trajectories will be denoted by T x. A trajectory generator will denote a function T defined as follows: T : R n R R n (2 (x, t T (x, t such that x R n, T (x,. T x. u T (x,. C x will denote the corresponding admissible open-loop control. A trajectory generator is simply a function that gives an admissible trajectory for the system (1 from an initial position. Definition 2.1. An admissible generator will denote a trajectory generator satisfying: (1 t 1, t 2 R +, t 1 t 2 and x R n, T (T (x, t 1, t 2 = T (x, t 1 + t 2 (2 x R n, T (x,. L 2 (3 T is locally K-lipschitz in every neighbourhood of x uniformly w.r.t. the time t. 2.4 Remarks The existence for any x R n of an open-loop trajectory T (x,. and therefore of a control u T that steers the system to zero implies the global asymptotic controllability of the system (1 at the origin. Note that assumptions 2.1 and definition 2.1 can be tied up with some classical assumptions of the optimal receding horizon theory (see for instance the work of D.Q. Mayne and H. Michalska (Mayne and Michalska, 199 and their proof of the stability of the finite receding horizon with terminal constraints. If the uniqueness of the optimal solution ˆx(x,. is supposed and its continuous differentiability with respect to x, the choice of T (x,. := ˆx(x,. in an infinite horizon scheme verifies definition 2.1. In that case, (1 is nothing else than the Bellman s optimality principle and the

3 L 2 property is implied by the existence of a finite cost. Recently a link between asymptotic controllability and feedback (eventually discontinuous stabilizability has been established by Clarke (Clarke et al., The design of the proposed feedback is based on the knowledge of a control Lyapunov function. This can be linked in our case to the knowledge of an admissible generator in the sense of definition 2.1. The L 2 property is only a technical trick that allows us to define a Lyapunov function by the equation (3. However, one can extend the results to non-l 2 trajectories using Massera s lemma (Massera, 1949 that proves the existence of a function transforming a non-l 2 trajectory in a L 2 one. 3. CLOSED-LOOP STABILITY Theorem 3.1. Suppose the system under consideration satisfies assumption 2.1 and that there exists an admissible generator T, then the closed loop system ẋ = f(x, u T (x, is globally asymptotically stable in the Lyapunov sense. Proof : Let us define the function V as follows: V (x = + T (x, τ 2 dτ (3 Starting from position x at time t, dx OL (resp. dx CL will denote the evolution of the state system on the open-loop (resp. closed loop trajectory during the time dt. One has: dv (x CL = V (x + dx CL V (x = V (x + dx CL dt Let us define I(x by: I(x := V (x + dx CL + = + T (x,τ+dt 2 dτ T (x,τ 2 + dτ T (x,τ 2 dτ dt T (x,τ 2 dτ dt + T (x,τ+dt 2 dτ Since T (x, τ + dt = T (x + dx OL, τ (2.1 (1, it follows that I(x satisfies: with: J (x I(x J (x (4 + J (x:= T (x+dx CL,τ T (x+dx OL,τ 2 dτ Since T is L 2, there exists t (x > such that: J (x 2 t (x T (x+dx CL,τ T (x+dx OL,τ 2 dτ T (., t is locally K-lipschitz in a neighbourhood N(x of x, uniformly w.r.t. t on [, t (x]: J (x 2t (xk(x dx CL dx OL 2 (5 In N(x, dx OL = f(x, u T (x, dt + O(dt 2 and dx CL = f(x, u T (x, dt + O(dt 2 (with of course O(dt 2 depending also on x that will be omitted since it remains constant in this proof. It follows that: dx CL dx OL 2 = O(dt 4 hence: and: dv (x CL = dt (5 J (x = O(dt 4 (4 I(x = O(dt 4 T (x,τ 2 dτ + O(dt 4 dv (x CL lim = T (x, 2 = x 2 dt dt V verifies the following properties: V (x φ( x with φ of class K. dv (x CL ψ( x with ψ of class K. dt (The choice of ψ is obvious and one should take + for instance φ( x = min y = x T (y, t 2 dt. Its continuity is ensured by the one of T. φ( = and φ strictly increasing are consequences of (1 of definition 2.1 The zero equilibrium point is therefore globally asymptotically stable under the feedback defined previously. Remark 3.1. The application of theorem 3.1 on an autonomous system ẋ = f(x gives a well known result in the stability theory literature (Hahn, 1967; Rouche et al., 1977: for system whose motion depends continuously on the initial conditions, the global uniform attractivity implies the global asymptotic stability in the Lyapunov sense. Indeed, property 3 of definition 2.1 is implied by the differentiability of f (Rouche and Mawhin, Property 1 is a consequence of the uniqueness of the solution of the differential equation. The L 2 property coupled with the differentiability of the motion with respect to the initial conditions gives the global uniform attractivity of the origin (see chapter V of (Hahn, Since the trajectory x(.; ; x depends continuously on x, it implies the global asymptotic stability of the origin. Theorem 3.1 gives explicitly a Lyapunov function for those systems. 4. APPLICATION TO A EXOTHERMIC CONTINUOUS STIRRED TANK REACTOR In this section, the application on a Continuous Stirred Tank Reactor of the state feedback exposed in the previous section is presented. The dynamic and the static behaviour of the CSTR will first be

4 presented. The design of the feedback and the proof of its stability are then given. The section ends with simulations. 4.1 Dynamics of the CSTR We consider the exothermic first-order reaction in a CSTR (Continuous Stirred Tank Reactor. This is a continuously fed process. The reaction takes place in a jacket that is immersed in a heat bath. The state equations are as follows: ( γx2 ẋ 1 =D(1 x 1 exp γ+x 2 ( γx2 ẋ 2 =BD(1 x 1 exp γ+x 2 x 1 (6 x 2 +β(x J x 2 (7 ẋ J =δ(u x J (8 where x 1 is the reactant concentration, x 2 (considered as the output is the dimensionless reactor temperature and x J is the dimensionless jacket fluid temperature. x J is the manipulated variable. In order to take care of the time lag in this control input that adds a difficulty to the control problem, equation (8 was added. The control variable is u and the parameter δ enables to handle the rate of heat transfer in the jacket. x = (x 1, x 2, x J will denote the initial conditions of the state vector x = (x 1, x 2, x J. The parameters used are: γ = 2, D =.72, B = 8, β =.3 and δ = 3. The classical exothermic continuous stirred tank reactor (x 1 and x 2 as state variables and x J as control is known to exhibit complex static and dynamic behaviour, although only a two state model is used. Adding a third state to account for the cooling jacket temperature dynamics renders the problem even more complicated. Further details on this CSTR model can be found in (Buescher and Baum, 1995 who used learning strategies on neural network to drive the system from a stable steady state to an unstable one, or in (Uppal et al., 1974 and the references therein. 4.2 Steady state behaviour In this subsection, the steady state behaviour of the CSTR will be analysed. It is easily verified that, given the final value x f 2 of the reactor temperature, steady state values of the remaining states are: x f 1 = DΦ(xf DΦ(x f 2 [ ] (9 u f = x f J = 1 (1 + βx f 2 β DΦ(x f 2 (1 where: Φ(x f 2 = exp ( γx f 2 γ + x f 2 (11 One feature of the exothermic CSTR which makes it difficult to control is that it has unstable equilibrium points. Figure 1 shows the greatest real part of the eigenvalues of the transition matrix and the possible asymptotic value of the output for a given input. It pointed out that steady states values of x f 2 in the interval [2, 3.5] are unstable and that, if no feedback is used and for an identical asymptotic value of the control, the output of the system will jump from the desired but unstable final output to an undesired stable one (see also figure 2. max(real(eig(df/dx x2f Fig. 1. Evolution of the real part of the eigenvalues of f x with respect to xf Control law Recall that our aim is to bring the CSTR from a given initial state x := (x 1, x 2, x J to a final equilibrium point corresponding to x f 2. An overlook on the system equations shows that as long as differentiable evolutions are considered, the trajectory of the state vector x := (x 1, x 2, x J and the input u are uniquely determined by the evolution of x 1. Recall T(. is a solution of the differential system equations (6-8. Denoting by T i (. the i th component of the vector T (., we shall restrict our search to the trajectories T (. of the following form: T 1 (t = Ψ(ta = a + N a i exp( σ i t (12 i=1 with: i, j i > j > σ i > σ j > Ψ(t = (1, exp( σ 1 t,..., exp( σ N t is a functional basis and a = (a,..., a N T as the coordinates of T 1 (. in it. The design of the control law can be broken up in the following steps:

5 step 1: [Compute a(x(t, x f 2 ] The initial condition x(t and the desired final value of reactor temperature x f 2 gives us a linear system of equations in the unknown a. The choice of N = 3 ensures that the solution is unique. It is given by: a = σ 1... σ N σ σn 2 with: ẍ 1(t= ẋ 1(t [ 1+D exp( γx2 (t γ+x 2 (t 1 ] + x 1 (t DΦ(x f 2 1+DΦ(x f 2 ẋ 1 (t ẍ 1 (t [ x 2(t+BD(1 x 1(t exp( γx2 (t + γ+x 2 (t ] ( Dγ β(x J (t x 2 (1 x 2(t 1 (t γx2 (t (γ+x 2 (t 2 exp γ+x 2 (t For arbitrary values of (σ 1, σ 2, σ 3, the initial conditions x(t and the desired final value of reactor temperature x f 2 entirely and uniquely determine T 1 (x(t, x f 2, t for t [, + [. In the following, we shall denote T (. := T (x(t, x f 2,. and T i(. := T i (x(t, x f 2,. with i = 1, 2 or 3. step 2: [Compute u T (x(t, x f 2,.] Once T 1 (x(t, x f 2,. is known, T 2(., T 3 (. and the corresponding control u T (. can be expressed as follows: determined by T 1 (., T (x(t, x f 2,. also satisfies 1 of definition 2.1. T 1 (. x f 1 is L 2. A limited expansion of T 2 (. when t gives: T 2(t a γ ln( D(1 a a γ ln( D(1 a + γ 2a 1 a a γ ln( D(1 a exp( σ 1t (13 Inverting relation (9, one can deduce from (13 that T 2 (. x f 2 is also L 2. Equation (7 indicates that T 3 (. x f J is also L 2. T (x, x f 2,. is therefore L 2. T is obviously differentiable with respect to x. Since T is an admissible generator, theorem 3.1 implies that the feedback u T (x(t, x f 2, asymptotically stabilizes the CSTR system around the desired steady state output x f 2. It is worth noting that the proposed method is based on the knowledge of the state vector as every state feedback and that problem of the design of a state observer remains. 4.5 Simulation T 2(x(t,x f γ ln Φ( t 2, t= γ ln Φ( t T 3(x(t,x f 2, t= 1 β ( T 2( t+(β+1t 2( t B( T 1( t+t 1( t T(x(,3,. with: Φ( t= u T (x(t,x f 2 Ṫ3 ( t, t= δ +T 3( t Φ( t= Ṫ1 ( t+t 1 ( t D(1 T 1 ( t Φ( t= T 1 ( t(1 T 1 ( t+ṫ 1 2 ( t+ṫ1( t D(1 T 1 ( t 2 T 2( t= Φ( tφ( t Φ 2 ( t Φ 2 ( t d 3 T 1 dt 3 ( t(1 T 1 ( t+ṫ1( t T 1 ( t+ T 1 ( t D(1 T 1 ( t 2 + 2Ṫ1 ( t ( T 1 ( t(1 T 1 ( t+ṫ 2 1 ( t+ṫ1 ( t D(1 T 1 ( t 3 T Φ( tγ 2 2( t= Φ( t(γ ln Φ( t 2 γ 2 (γ ln Φ( t 2 + Φ( t 2γ 2 Φ( t (γ ln Φ( t 3 T 3( t= 1 β ( T 2( t+(β+1 T 2( t B( T 1( t+ T 1( t step 3: [Apply u T (x(t, x f 2, ] The control law is given by applying at each time t the control u T (x(t, x f 2,. The choice of (σ 1, σ 2, σ 3 remains free and can be used to specify the evolution of the state during the transient x2 1.5 x1 Integrated open loop trajectory Fig. 2. Comparison between T (x(, 3,. and the numerical integration of the open-loop system Figure 2 shows the trajectory T (x(, x f 2,. generated at time t = with x f 2 = 3 and the behaviour of the system when the corresponding open-loop control u T (x(,. is applied. It underlines the instability of steady states x f 2 in [2, 3.5]: the open-loop system jumps (due to numerical integration errors from the desired steady state x f 2 = 3 to a stable one xf Figure 3 shows the closed loop trajectory of the state of the system. The closed loop stability of an openloop unstable steady state is achieved. 4.4 Stability of the closed loop Let us now verify that the above defined T is an admissible generator: It is easy to check that T 1 (. verifies 1 of definition 2.1. Since T 2 (. and T 3 (. are uniquely 5. CONCLUSION The design of nonlinear stabilizing feedback remains a quite difficult issue. This paper gives conditions on open-loop trajectory generators in order to use them for the design of a nonlinear feedback. The proof of

6 x2f Fig. 3. Closed loop trajectory the stabilizing property of the proposed state feedback is given. The construction of such a feedback is proposed on a continuous stirred tank reactor. The stabilization result given here has its own theoretical interest since it answers an old question. Indeed, the use of open-loop path trajectory in order to construct a stabilizing feedback has always been an intuitive approach used in various control fields (flat systems, receding horizon with sampling time. However, in our knowledge, this has never gave rise to an explicit state feedback law. It is worth noting that the possibility to derive an analytic state feedback depends on strong triangular - like structural condition (satisfied by the CSTR. However, an approximate numerical version of admissible generators can be proposed (?. The result of this paper can then be used in order to choose the parameters of the underlying implementation (uniqueness and regularity of the solution,.... x2 REFERENCES Alamir, M. and G. Bornard (1994. On the stability of receding horizon control of nonlinear discrete time systems. Systems & Control Letters 23, Alamir, M. and G. Bornard (1996. Globally stabilizing discontinuous feedback law for a class of nonlinear systems. European Journal of Control 2, Brockett, R. W., R. S. Millman and H. S. Susmann (1983. Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory. Birkhäuser. Boston-Basel-Stuttgart. Buescher, K. L. and C. C. Baum (1995. A twotimescale approach to nonlinear model predictive control. In: Proc. American Control Conference. Evanston, USA. pp Clarke, F. H., Y. S. Ledyaev, E. D. Sontag and A. Subbotin (1997. Asymptotic controllability implies feedback stabilization. IEEE Trans. on Automatic Control 42(1, x1 xj Fliess, M., J. Levine, Ph. Martin and P. Rouchon (1992. On differentially flat nonlinear systems. Comptes Rendus des Sances de l Acadmie des Sciences 315, Série I. Fliess, M., J. Levine, Ph. Martin and P. Rouchon (1995. Flatness and defect of nonlinear systems: introductory theory and examples. Int. Journal of Control 61(6, Fliess, M., J. Levine, Ph. Martin and P. Rouchon (1996. Flat System: Theory and Practice. Summer School. Laboratoire d Automatique de Grenoble. Hahn, W. (1967. Stability of motion. Springer Verlag. Berlin-Heidelberg. Kawaji, S. and N. Matsunaga (1992. Path generation using piecewise polynomials and its application to obstacle avoidance. In: Proc. IEEE Conf. on Decision and Control. pp Keerthi, S. S. and E. G. Gilbert (1987. Moving horizon approximation for a general class of optimal nonlinear infinite horizon discrete time systems. In: Information sciences and systems. proc. 2th annual conference, Princeton university. pp Massera, J. L. (1949. On Liapounoff s conditions of stability. Annals of Mathematics 5(3, Mayne, D. Q. and H. Michalska (199. Receding horizon control of nonlinear systems. IEEE Trans. on Automatic Control 35(7, Rouche, N. and J. Mawhin (1973. Equations différentielles ordinaires. Chap. V, p Masson et Cie. Rouche, N., P. Habets and M. Laloy (1977. Stability theory by Liapunov s direct method. Springer Verlag. Berlin-Heidelberg. Teel, A. R., R. M. Murray and G. C. Walsh (1995. Non-holonomic control systems: from steering to stabilization with sinusoids. Int. Journal of Control 62(4, Uppal, A., W. H. Ray and A. B. Poore (1974. On the dynamic behaviour of continuous stirred tank reactors. Chemical Engineering Science 29, Zarkh, M. A. (1995. Optimal control with revising feedback. In: Proceedings of the 3rd European Control Conference, Rome, Italy.