Continuous Piecewise Linear Control for Nonlinear Systems: The Parallel Model Technique

Transcription

1 Continuous Piecewise Linear Control for Nonlinear Systems: The Parallel Model Technique Andrés G. García Universidad Nacional del Sur Dto. de Ing. Eléctrica y de Computadoras Instituto de Investigación en Ing. Eléctrica (IIIE, UNS-CONICET) Av. Alem 1253, B. Blanca Argentina Osvaldo E. Agamennoni Universidad Nacional del Sur Dto. de Ing. Eléctrica y de Computadoras Instituto de Investigación en Ing. Eléctrica (IIIE, UNS-CONICET) Av. Alem 1253, B. Blanca Argentina Abstract: This paper describes a systematic technique for obtaining controllers for Nonlinear Systems using a Continuous Piecewise Linear Approximation (CPWL) of the given Nonlinear vector field. The method proposes the use of a CPWL approximation of the Nonlinear System and then a theory is developed to show that the stabilization of the CPWL aproximation ODE yields stability for the Nonlinear ODE. An example is presented in order to show the capabilities of this idea but also the practical applicability. Finally some conclusions and future work are depicted. Key Words: Nonlinear ODE s, Continuous Piecewise Linear ODE s, Control Systems. 1 Introduction Nonlinear Controller design is a very involved issue into the control community. While for linear systems there exists a wide range of methodologies from analysis to readily implementable strategies (see [1], [2] and [4]), for Nonlinear dynamics only a few methodologies are known to be effective and usually encounter in practice hard problems regarding time consuming or demanding tremendous amounts of capabilities from computer resources (see [6],[7] and [8]). Since Control problems are not more than a parameterized Initial Value Problem (IVP) for Ordinary Differential Equations (ODE s), for designing control strategies, is natural to resort primarily to techniques able to solve -or approximate- such a problems. One methodology which has shown to be effective either from numerical issues or theoretical analysis is the use of Continuous Piecewise Linear Approximations (CPWL), this idea was early studied by Sacks in a qualitative fashion (see [14]) and later extended to approximate solutions by Girard, De Feo, Storace and Johansson (see [13], [19] and [11]). Is worth noticing that two main research streams are related to Continuous Piecewise Linear (CPWL) ODE s: Dynamic Systems which are written as CPWL ODE s. As approximation to Nonlinear Continuous ODE s. First item was pioneered studied and defined by Chua in the early s 70 s (see [16] and [17]). This work gave origin to a Canonical Representation for a CPWL Basis able to represent any CPWL vector field developed by Julián (see [9]). Using this basis a reliably toolbox written in Matlab code were developed in order to approximate a given vector field with some desired degree of error (see [10]), on the other hand with a different procedure than a CPWL basis the Phd thesis of Johansson also leads a Matlab toolbox (see [11]). Second item was conducted by Storace and De Feo who made several numerical experiments to investigate topological properties of Nonlinear ODE s of low dimensions. However, this work is not proving in rigor that the properties of the given the Nonlinear ODE and the CPWL approximation are shared by both systems, they only present extensive simulations and Continuation numerical packages to show this idea (see [15]). One attempt to overcome this inconvenient providing a way to decide if both (Nonlinear ODe and its CPWL approximation) share properties, are the works in [13] and [18] where Dynamic Error Bounds are derived. The former paper proved that the trajectories of the Nonlinear ODE and the trajectories of the approximation CPWL differs in norm around the order of the grid size -see [9] for a precise definition of gride size, while the second paper proved that the dynamics of 39 ISSN

2 the error bound introduced in that paper is the same as the CPWL approximation ODE. As is evident, the result in [18] when applied to a system with a stable CPWL approximation will lead the conclusion that the trajectories of both systems are stable and this precisely the topic of the present paper, the concept of Parallel Model, that means, given a Nonlinear control system to the type x(t = f(x) + B u and a CPWL approximation of f(x), f CP W L (x), then a CPWL controller u CP W L (x) stabilizing x CP W L (t) = f CP W L (x CP W L )+B u CP W L is also stabilizing x(t = f(x) + B u. A remarkable point here is the fast implementation of CPWL functions with microelectronics suggests that the future implementation of electronic controller could be appealing via CPWL vector fields which approximate any nonlinear dynamical system with some degree of accuracy, see [20] This paper is organized as follows: Section 2 introduces precise definitions for the kind of Nonlinear systems considered in this paper and the goal addressed,section 2.1 presents the formulation of an error bound able to ensure stability for both systems, Section 3 provides an analysis of the asymptotic properties of the approximate and real systems, Section 4 shows how the developed theory works in practice applicability in a real case and finally Section 5 depict some conclusions and future directions for research. 2 ABOUT THE PROBLEM CON- SIDERED A Nonlinear Control System is a set of ODEs to the form Ẋ(t) = f(x, u) where X Rn and with X R m. In this context, one important aim is to determine a suitable controller u endowing the system with some desired properties (stability, asymptotic stability, robustness, performance, etc). As is very well known this aim is very difficult in practice for general cases with any nonlinear control system as exposed above, being available only a few partial results for special cases. The idea in what follows is to produce an approximation of the Nonlinear Vector Field with a CPWL one, moving the task of designing the controller to the side of the approximate vector field and keeping track some error bound. Notice that the design procedure for the obtained CPWL ODE can be carried out with the available techniques for designing of controllers for Piecewise Linear ODE s in general as the one in [11] or [24]. However, the theory that it will be developed in the present paper will give support to the techniques in [11]. In this paper we aim to develop a controller u(t) for the following class of Nonlinear Systems: X(t) = f(x) + B u(t), (1) where X R n, f : R n R n is a smooth vector field and B is a constant matrix B R nm. The point is to approximate the Nonlinear vector field f by using a CPWL one as follows: X CP W L = f CP W L (X CP W L ) + B u(t), (2) where X are the approximate trajectories to X in (1). Incidentally notice that the same controller u(t) is applied to both systems, moreover, if we define the control vector u(t) to be Continuous Piecewise Linear 1 with the same partition as for f(x), we have: Ẋ = f(x) + B u CP W L X CP W L = f CP W L (X CP W L ) + B u CP W L u CP W L = A i u X CP W L + B i u (3) where the notation A i and B i was introduced in [18] indicating the matrices forming the linear approximation into the simplex i th. This is the reason why this technique will be called Parallel Model, since we need the trajectories of the CPWL model in order to be applied into u CP W L to effectively control Ẋ = f(x) + B u CP W L 2.1 CPWL Approximations for Nonlinear Control Systems The class of Nonlinear control systems defined in (1) can be described in a formal way as follows: Ẋ(t) = f(x) + B u CP W L f(x) = [f 1 (X) f n (X)] T f i U R n R i = 1,..., n X = [X 1 X n ] T, where T stands for transpose. Let us consider U divided in r simplices using a boundary configuration H as described in [9] with the CPWL approximation f CP W L in the simplex i th to the given vector field f denoted by: f i CP W L (X) = [f i 1CP W L (X) f i ncp W L (X)]T f i CP W L (X) = Ai X + B i 1 u(t) could be Piecewise Linear Discontinuous as discussed in [11], pp ISSN

3 In what follows we use the idea presented in [9],pp. 100, where the maximum norm was used in order to characterize the size of the error: f 1 (X) f i 1CP W L (X) ε 1 f 2 (X) f i 2CP W L (X) ε 2 i = 1,..., r.. f n (X) fncp i W L (X) ε n (4) Then we are looking for an error bound for the trajectories such that: X X i h i (ε), i = 1,..., r, (5) where h i : R n R + is a continuous function, ε = [ε 1... ε n ] T, X i is the approximate trajectory to X(t) running into the i th simplex and. stand for the maximum norm. In this way, the approximating CPWL control system will be: Ẋ i = f i CP W L(X i ) + B (A (i) u X i + B (i) u ) where X i = [X i 1 Xi n] T is the state vector of the CPWL approximation ODE while X(t) are the trajectories of the Nonlinear ODE. 2. Next subsection develop Error Bounds in the sense of (5) which share the dynamics of the CPWL approximation ODE. 2.2 Obtaining an Error Bound for the Approximate Trajectories Taking into account (4), is possible to obtain an error bound in the spirit of (5) as follows: f j (X) f i jcp W L (X) ε j, j = 1,..., n, i = 1,..., r, (6) Defining the Error for the Trajectories for the i th simplex as follows: E i (t) = X X i (t). (7) Adding and subtracting B u CP W L, u CP W L = A i u X i + Bu i and considering the notation introduced in previous section: f CP W L (X) = A i X + B i, then equation (6) becomes: 2 This notation is providing a formal statement of equation (2) Ėi j (Ai + B A i u) E i ε j, j = 1,..., n, i = 1,..., r. (8) Here we wrote E i = [E1 i Ei n] T. Notice that (8) is in fact a differential inequality in Ej i for each j = 1,..., n, however this equation can be recast in a matrix way as follows: ε Ė i (A i + B A i u) E i ε, i = 1,..., r, (9) where ε = [ε 1 ε n ] T. The last step will be then to solve the Matrix Inequality in equation (9) and for that a Theorem by Coppel will be used -see [21], pp ): Theorem 1 Let y(x) be a solution of the scalar ODE dy(x) dx = f(x, y) where f(x, y) is continuous. If u(x) is continuous and satisfies u(a) y(a) and du(x) dx f(x, u) on [a, b], then: Similarly, if s(a) y(a) and ds(x) dx then s(x) y(x) on [a, b]. u(x) y(x) (10) f(x, s) on [a, b], Then the Error Bounds in equation (9) can be restated now: E 1 (t) E(t) E 2 (t) Ė 1 (t) = (A i + B A i u) E i ε Ė 2 (t) = (A i + B A i u) E i + ε j = 1,.., n (11) It is worth noticing that the stability issues of the Error Bound in (11) are shared by the CPWL ODE X CP W L = f CP W L (X CP W L ) + B u CP W L (X CP W L ). On the other hand, the point missing it is to provide a way to determine when a CPWL system is stable and for that we have the following Theorem 3 : Theorem 2 Given an Continuous CPWL ODE: Ẋ = A i X+B i for a Domain Ω with only equilibrium points in the interior of the simplices 4 with all the matrices A i stable (negative definite) and such that the trajectories remains inside Ω for all x Ω, then the ODE is stable in the sense that the equilibrium points are attractive for any initial condition x(0) Ω. 3 The proof is given in the appendix. 4 That means to exclude the possibility for equilibrium points in the frontiers 41 ISSN

4 3 CPWL AND NONLINEAR SYS- TEMS SHARE PROPERTIES As depicted in previous section, for systems which all the matrices A i are stable for any simplex in the CPWL approximation ODE, then the Error Bounds in equation (11) also go to an equilibrium point showing that the Nonlinear ODE is following the CPWL approxiamtion when the same CPWL controller is used in both systems. Here arises the important remark that for a sufficient amount of simplices (ensuring low values of ε), the properties imposed to the CPWL approximating system are also Shared by the Nonlinear given system. In this way, we can design a controller to track some reference using the CPWL system, regulate some given outputs, check stability, design optimal controllers, etc. One of the most important properties in a Nonlinear control system is the concept of controllability (observability), where a controller u is required in order to drive the system from an initial point x(0) to a point x(t ) for some time T (see [23], pp. 511). In this way, is useful to introduce the concept of quasicontrollability: Definition 3 (ρ-quasi-controllability): A Nonlinear system to the form Ẋ = f(x) + g(x) u is called ε- Quasi-Controllable if there exists a controller u, such that the trajectories are close to the point X T in ε when t goes to infinity: X(T ) = X T X(0) = X 0 ρ lim t X(t) X T ρ ρ > 0 (12) Theorem 4 : Given a Nonlinear system Ẋ = f(x) + B u and its CPWL approximation: Ẋ i = f i P W L (Xi ) + B u(t), then the Nonlinear system is ρ-quasi-controllable. Proof: According to the definition in (12), we only have to realize that the quantity ρ is not more than equilibrium of the Error Bound: (A i + B A i u) 1 ε obtained in equation (11), for some simplex I where the point X(T ) belongs. Is interesting to notice that the definition of Quasi-Controllability becomes the classic Controllability definition when the number of simplices tend to infinity. In this way this concept is telling how possible is to drive a non-linear system close to some desired point using a controller designed in the basis of the CPWL approximation. 4 EXAMPLE: Academic Example A simpel example taken from [11] is the inverted pendulum: ẋ1 (t) = x 2 ẋ 2 (t) = 0.1 x 2 + sin(x 1 ) + u CP W L (13) In this way and using 10 simplices in each coordinate with a range of [ 1, 1] in the x coordinate and [ 10, 10] for y to produce a CPWL of the Drift [x 2, 0.1 x 2 + sin(x 1 )], it is possible to apply the stabilization technique depicted in [11], pp. 102, to design a control law give by u CP W L = [ ] [x, y]. With this controller we, the maximum error is of 0.03 in absolute value, so both CPWL and Nonlinear systems are stabilized by the same controller. 5 Conclusion A methodology for designing controllers for some a class of Nonlinear Control systems was presented. In this methodology an approximation of the Nonlinear Drift of the given Control system is approximated with a CPWL one. Using the CPWL approximation ODE as a Parallel Control system, the design of a controller with any desired property (stability, optimal strategy, etc) for the Nonlinear system is then translated into the design for the CPWL one. It was shown that the same properties assigned to the CPWL system are shared by the Nonlinear system when the same controller is applied to both dynamics. An example showed an application of the theory developed, specially the idea of Quasi-controllability introduced in section 3. This concept is pointing into the direction of proving strong properties using the CPWL control system, let s say it would be interesting to show that the controllability of CPWL and Nonlinear systems are equivalent. 6 Appendix Proof of Theorem 2 Then is possible to define an auxiliary ODE as follows: Ai X + B Z = i = Ẋ, X V ertices Z(T vertex ), Otherwise then: (14) 42 ISSN

5 Ż = A i, X V ertices 0, Otherwise where T vertex are the instants where the trajectories of the CPWL ODE is crossing from one simplex to the next one. Notice that this definition is preventing problems whit the discontinuities of Ż at the vertices. In this way, we have: Ż = A i Z Ż = 1 2 (Z Z) x Consider next the following integral: Ẋ dz = In other words: t 0 Ẋ Ż dt = (15) = 1 X(t) 2 (Z Z) dx X(0) } X } d(z Z) Ẋ dz = Z(t) 2 2 In order to solve this integral, the continuity of the variable Z(t) in the borders or frontiers in virtue of definition (14) is invoked. Then: = 1 2 Ẋ dz = m 1 i=1 m i=1 X(Ti+1 ) + X(T i ) X(t) X(T m 1 ) Ti+1 T i Ẋ Ż dt = (Z Z) dx + X (Z Z) dx. X where. 2 is the l2-norm (see [28] for a further reading) and T i, i = 1,.., m are the crossing times or the times where the trajectories cross from one simples to the next one. On the other hand the integral in equation (15) also means: t 0 t Ẋ Ż dt = Ẋ A i Ẋ dt (16) If we require ẋ A i Ẋ < 0, then this equation leads: 0 ẋ dz < 0 Z(t) 2 2 < 0 Notice that only for the very special case of Piecewise Quadratic functions as Ẋ Ai Ẋ, the requirement Ẋ A i Ẋ < 0 is equivalent to say that A i is asymptotically stable or negative definite (see [27]). Focusing the aim of stability we need to consider the integral in equation (16) whit t tending to infinity, which yields: i=1 Ti+1 T i Ẋ A i ẋ dt = finite It is well known that a necessary condition for a convergence of a an infinite sum is the limit at infinity of the general term tending to zero, in this case this leads: lim i Ti+1 T i Ẋ A i ẋ dt = 0 (17) This only leaves two possibilities: Ẋ A i ẋ 0 as t lim i (T i T i+1 ) = 0 The first case is our desired objective since Ẋ A i ẋ 0 means ẋ 0, however the second possibility requires a special care, it is indicating that the equilibriums are allocated in the frontiers leaving the possibility of instabilities even when the dynamics of the individuals simplices are stable as reported in [26] and [25]. Acknowledgements: The research was supported by CIC, ONICET, Universidad Nacional del Sur and Agencia Nacional de Promoción Científica y Técnica. References: [1] A. Bemporad, F. Borrelli and M.Morari, Model Predictive Control Based on Linear Programming-The Explicit Solution, IEEE Transactions on Automatic Control. 47, 2004, pp [2] A. Bemporad, M. Morari and V.D. Pistikopoulos, The Explicit Solution of Model Predictive Control Via Multiparametric Quadratic Programming, Proceedings of the American Control Conference, Chicago. Illinois, 2000, pp ISSN

6 [3] A. Bemporad, M. Morari and V.D. Pistikopoulos, The Explicit Linear Quadratic Regulator for Constrained Systems, Automatica, 38, 2002, pp [4] F. Borrelli, A. Bemporad and M. Morari,Geometric Algorithm for Multiparametric Linear Programming, Journal of Optimization theory and applications, 118, 2003, pp [5] A. Bemporad, F. Borrelli, and M. Morari, Optimal Controllers for Hybrid Systems: Stability and Piecewise Linear Explicit Form, Proceedings 39th IEEE Conference on Decision and Control, Sydney, 2000, pp [6] I. Rodrigues and J. How, Automated Control Design for a Piecewise-Affine Approximation of a class of Nonlinear Systems, Proceeding of the American Control Conference, Arlington, 2001, pp [7] L. Rodrigues and S. Boyd, Piecewise-Affine State Feedback Using Convex Optimization, Proceeding of the American Control Conference, Boston, Massachusetts, 2004, pp [8] Z. Wan and M.V. Kothare, Efficient Scheduled Stabilizing Output Feedback Model Predictive Control for Constrained Nonlinear Systems, IEEE Transactions on Automatic Control, 49, 2004, pp [9] P.M. Julián, A High Level Canonical Piecewise Linear Representation: Theory and Applications, Phd. Thesis, 1999, Universidad Nacional del Sur. [10] P.M. Julián, A Toolbox for Piecewise Linear Approximations of Multidimensional Functions, personales/pjulian/cpwl.htm. [11] M. Johansson, Piecewise Linear Control Systems, Springer Verlag, [12] O. De Feo and M. Storace, PWL identification of dynamical systems: some examples, Proceeding of ISCAS, Vancouver, Canada, 2004, pp [13] A. Girard, Approximate Solutions Of ODEs Using Piecewise Linear Vector Fields, Proceedings of Computer Algebra in Scientific Computing, Yalta, Ukraine, 2002, pp [14] E. Sacks, Automatic Qualitative Analysis of Dynamic Systems Using Picewise Linear Approximations, Artificial Intelligence., 3, 1990, pp [15] M. Storace and O. De Feo, Piecewise-LInear Approxiamtions of Nonlinear Dynamical Systems, IEEE Transcations on Circuits and Systems-I:Regular Papers, 51, 2004, [16] L.O. Chua, Efficient computer algorithms for piecewise-linear analysis of resistive nonlinear networks, IEEE Transactions on Circuit Theory, 1, 18, 1971, pp [17] L.O. Chua, S.M. Kang, Section-wise piecewiselinear functions: canonical representation, properties and applications, Proceedings of the IEEE, 6, 65, 1977, pp [18] A. Garcia, S. Biagiola, L. Castro, J. Figueraoa and O. Agamennoni, Approximate Solutions for Non-Linear Autonomous ODEs On the Basis of PWL Approximation Theory, Applied Analysis and Differential Equations- Proceedings of ICAADE, Iasi, Romania, [19] O. De Feo and M. Storace, PWL Identification of Dynamical sysems: some examples, Proceeding of ISCAS, Vancouver, Canada, 2004, pp [20] M. Parodi, M. Storace and P.M. Julián, Synthesis of multiport resistors with piecewise-linear characteristics: A mixedsignal architecture, Journal on Circuit Theory and Applications, 33, 2005, pp [21] W.A. Coppel, Disconjugacy, Lecture Notes in Math., 220, Springer Verlag, Berlin, 1971 [22] G. Inalhan, M. Tillerson and J. How, Relative dynamics and control of spacecrafts formations in eccentric orbits, Journal of Guidance, Control, and Dynamics, 25, 2002, pp [23] S. Sastry, Nonlinear Systems: Analysis, Stabiliy and Control, Springer Verlag, New York, [24] D. Liberzon and S. Morse, Basic Problems in Stability and Design of Switchied Systems, IEE Control Systems Magazine, 9, 19, 1999, pp [25] V. Carmona, E. Freire, E. Ponce, F. Torres and J. Ros, Some Recent Results for Continous Switched Linear Systems, 12th International Power Electronics and Motion Control Conference, [26] V. Carmona, E. Freire, E. Ponce and F. Torres, nstability in the Simplest Class of Continuous Switched Linear Systems with Stable Components, 16th IFAC World Congress. Praga, Chec Republic, [27] H.G. Kwatny and G.L. Blankenship, Nonlinear Control and Analytical Mechanics, [28] R.A. Horn and C.R. Johnson, Matrix Analysis, ISSN