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of Proceedings the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 WeA01.3 A New Local Control Strategy for Control of Discrete-Time Piecewise Af ne Systems Thomas Erhard Hodrus, Michael Buchholz, Volker Krebs Institut f r Regelungs und Steuerungssysteme, u Universit t Karlsruhe (TH), Germany a phone: +49 721 / 608 2462 fax: +49 721 / 608 2707 e mail: {hodrus,krebs}@irs.uni karlsruhe.de Abstract In this paper we propose an enhanced local control strategy for the control of discrete-time piecewise af ne systems on full-dimensional polytopes. The control strategy is divided into a local control and a supervisory control problem. The local control problem is to reach and cross one selected facet of a polytope ensuring that the next sample of the time-discrete trajectory is picked up in the adjacent polytope. The procedure is based on conditions, given as inequalities, for the discrete-time gradient of the system, evaluated in the vertices of the polytopes. Solving an optimization problem with respect to the inequality conditions, a performance index is minimized. The performance index represents a minimal retention period of the trajectory in the polytope or the minimal quadratic sum of the input signal. The control law is then obtained by a simple matrix inversion. The supervisory control problem is to nd a suitable combination of polytopes and local control strategies that transfers the trajectory to the polytope that contains the operating point. I. INTRODUCTION The piecewise linear approach for nonlinear systems was proposed in [1]. This approach has been generalized and nally led to hybrid systems theory. In [2] the hybrid phenomena were formulated. After this milestone a lot of research has been done, especially in modeling of hybrid systems. The analysis of this class of systems and the synthesis of controllers are still sophisticated, even nowadays. There are two different trends in literature for the design of controllers for piecewise af ne systems that represent a sub class of hybrid systems. In [3] and [4] continuous-time controllers and in [5], [6] and [7] discrete-time controllers are designed. Discrete-time af ne systems can be obtained from measurements of technical processes by identi cation of the hybrid system [8], [9] or by theoretical modeling [10], [11]. In the theoretical modeling approach, given continuous piecewise af ne models are discretized, calculating the discrete-time af ne model in each domain from the respective continuous model. As already presented in [12], the control strategy consists of a local part and a supervisory part. In this paper, a new approach for the local control strategy for the control of discrete-time piecewise af ne systems will be presented. In section II some basic notions as polytope, facet, af ne system, and af ne control law are given. Using these terms, the problem of dynamical systems restricted to a polytope is presented. The simplex as a special polytope is introduced. The aim of the local controller design is formulated. Suf cient conditions for the discrete-time gradient in the vertices of the polytope are given in section III. In section IV, both, the local control part and the supervisory control part of the strategy, are presented. At the beginning, the considered state space is partitioned into a simplex structure. Each simplex has a corresponding vertex in a graph. The local control strategy completes the graph by adding selected edges. After the local control is explored, the completed graph structure is used for supervisory control. The task of the supervisory control is to nd an optimal path through this graph structure. The local and the supervisory control strategy are calculated off-line. The online effort is to determine the active simplex and therefore, the active control law, solving a linear search problem. This approach is able to track very fast dynamics because of the reduced on-line effort. Finally, the strategy is applied to a two-tank system in section V. The two-tank system can be modeled as a piecewise af ne system with discrete and continuous input signals. A performance index will determine which piecewise af ne dynamic assigned to a combination of discrete inputs is the best. The plant in the example is taken from [12]. The start-state vector and the end-state vector are equal, too. A comparison with the results in [12] clari es the advantages of the new method in a short discussion. The contribution is nished with a conclusion in section VI. II. PROBLEM STATEMENT The considered state space x RN , N N is bounded on the set . Assuming there are M points v 1 , ..., v M , with M N + 1, in state space RN , such that there exists no hyperplane of RN , containing all these M points. The full-dimensional polytope P is de ned as the convex hull of v 1 , ..., v M . If a point v i , i = 1, ..., M cannot be written as convex combination of the points 0-7803-9568-9/05/$20.00 2005 IEEE 4181 v 1 , ..., v i 1 , v i+1 , ..., v M , it is called a vertex of the polytope P. The polytope is completely characterized by its set of vertices. A full-dimensional polytope with M = N + 1 is called a full-dimensional simplex. The intersection of a nite number of half spaces can also describe a polytope. If an integer K N + 1, non-zero vectors n1 , ..., nK RN , and scalars 1 , ..., K R exists, such that P = {x RN | i = 1, ..., K : nT x i } i (1) R,i and ( i + H i u0,i ) = R,i equal to an autonomous discrete-time af ne system xk+1 = R,i xk + R,i . (6) is valid, then (1) is the implicit description of a polytope. The intersection of a full-dimensional polytope P with one of its supporting hyperplanes Fi = {x RN | nT x = i } P i (2) is called a facet Fi of P , if the dimension of the intersection is equal to N 1. By convention the vector ni , that is the normal vector of the facet Fi , is of unit length and points out of the polytope. In [4] a method is presented to partition full-dimensional polytopes into full-dimensional simplices by using e.g. the Delaunay-triangulation. Therefore, in the sequel, the fulldimensional simplex P in RN with N + 1 facets is used. Using numbered vertices of a simplex, the facets can be named with the number of that vertex which is not a vertex of the facet. On each full-dimensional simplex Pi a discrete-time af ne system (3) xk+1 = i xk + H i uk + i is considered, with i RN N , H i RN m and i RN . The state x RN is assumed to be contained in the simplex Pi . The input signal u takes values from the bounded set U Rm of continuous inputs. The inside P i of the polytope P is P i = P \ P where P denotes the hull of the polytope P . The union of the simplices Pii with Pii Pji = i = j forms the subset = Pi = (Pii Pi ) of the state space within which the piecewise af ne system is de nded. The aim of the local controller design is to nd discretetime af ne control laws uk = Ri xk + u0,i (4) If there are discrete inputs uD to the system, uD is assumed to be an element of a bounded (integer) set, see details in [3] or section V, a nite number of discretetime piecewise af ne systems can be determined for each possible combination of the elements of the nite discrete input vector uD . Let each combination be numbered serially in the variable xD . For each combination xD the same continuous state space with the same partitioning into simplices is considered. There are max(xD ) dynamics de ned for one simplex. If the trajectory enters a simplex or is within the start simplex, this is a necessary condition that the discrete input uD is changed by the supervisory control strategy. In the next section, suf cient conditions for feedback control are formulated. III. SUFFICIENT CONDITIONS FOR FEEDBACK CONTROL TO A FACET In [4] a method was introduced for continuous-time systems that uses conditions of the gradient x of the system. The gradient is evaluated at the vertices of a simplex by projecting the gradient on the normal vectors of the bounding hyperplanes. This results into a linear inequality system for the input u(vi ) in these vertices v i . de ned on the simplex Pi , such that the trajectories of the controlled system xk+1 = ( i H i Ri )xk + ( i + H i u0,i ), (5) Fig. 1. Admissible range for the gradient vector starting at any possible point in Pi will stay in the simplex Pi (details in [12]), or enter one of the N + 1 adjacent simplices through one speci c facet of the simplex (focused in this paper). To obtain an unique solution, if the trajectory hits the common facet Fi , while leaving simplex Pi and entering simplex Pj , the dynamic of Pj is assumed to be dominant. The controlled system in equation (5) is with ( i +H i Ri ) = First, the continuous inequalities are presented. The index e represents the number of the facet Fe where the trajectories should leave the simplex. The vertex v e is the only vertex that is not part of the facet Fe . For illustrating the following equations, a two dimensional problem is given in gure 1, where e = 3. 4182 v i V \ {v e } : nT e x(v i , u(v i )) > 0 (7a) F v i V \ {v e } : Fj F \ {Fe , Fi } : nT j x(v i , u(v i )) 0 (7b) F Fj F \ {Fe } : n+1 j=1 j=e The ranges of the direction of the differences in the vertices of the facet Fe are narrowed to the neighbor simplex. In gure 2 the narrowed ranges are sketched. nT j x(v e , u(v e )) 0 (7c) F nT j x(v e , u(v e )) < 0 (7d) F The rst system of inequalities (7a) requires that the projection of the gradient on the normal vector of the facet Fe in the vertices point out of the simplex P . The second system of inequalities (7b) requires that the projection of the gradient on the normal vector of all other facets point into simplex P . E.g. in vertex v 1 in gure 1, the projection of the gradient on the normal vector of F2 has to point into the simplex P . The admissible range of the directions of the gradients in the vertices are marked as dotted lines. The third condition (7c) forces the gradient in vertex v e to point into the simplex P . The last condition (7d) forces the gradient not to vanish in the vertex v e . In the next step these four conditions (7a-7d) from [4] are modi ed for discrete-time systems. The gradient x has to be expressed as a difference xk+1 xk . With xk+1 = xk + H uk + the expressions above modify to the following equations, I n is the unity matrix of the dimension n: v i V \{v e } : nT e ( I n ) v i + H u(v i )+ > 0 (8a) F v i V \ {v e } : Fj F \ {Fe , Fi } : nT j ( I n )v i + Hu(v i ) + 0 (8b) F Fj F \ {Fe } : nT j ( I n )v e + Hu(v e ) + 0 (8c) F n+1 j=1 j=e Fig. 2. Range narrowed to the neighbor simplex by additional conditions If the trajectory crosses the facet Fe , the dynamic of the neighbor simplex P is valid for the rest of the sampling time. If the dynamic in simplex P and the dynamic in simplex P are equal, the next sample is guaranteed to be picked up in the neighbor simplex P by satisfying the conditions (8a8d,9). In general equal dynamics will not be the case. This means that the trajectory could evolve that way, without the further additional conditions (10,11), that the next sample is picked up outside the neighbor simplex P . If these two conditions are added, the next sample is picked up in the neighbor simplex, even if the dynamics in simplex P and simplex P are different: v i V \ {v e } : Fj F \ {Fe } : nT j I n v i + H u(v i ) + 0 (10) F v i V \ {v e } : Fe : I nT Fe n nT j ( I n )v e + Hu(v e ) + < 0 (8d) F As illustrated in gure 1 the next sample is not necessarily picked up in the neighbor simplex P , if the conditions (8a8d) are met. I.e. a trajectory starts in xk in simplex P and in the next sample xk+1 the neighbor simplex P is overleaped, although the conditions from [4] are met. Therefore, the discrete-time versions (8a-8d) of the continuous conditions (7a-7d) are only necessary to solve the problem. By adding the conditions (9- 11) the conditions turn out to be suf cient. The additional conditions v i V \ {v e } : Fj F \ {Fe } : T nFj ( I n ) v i + H u(v i ) + 0 (9) keep the direction of the difference with respect to the dynamic in simplex P in the neighbor simplex P . The vertices of the neighbor simplex are v i V. v i + H u(v i ) + 0 (11) condition The (10) is the correspondency to condition (9) with the dynamic , H and , valid in the neighbor simplex P . Condition (11) forces the trajectory to enter the neighbor simplex P . There will be no, one or an in nite number of solutions that satisfy these conditions. In the last case the degree of freedom can be used to perform an optimization. This will be discussed in the next section in detail. IV. CONTROL STRATEGY Given a discrete-time piecewise af ne system in state space , partitioned into simplices, the relation between the numbered simplices can be stored in a graph structure. Each simplex Pi is represented by a vertex in this graph. The control strategy consists of two parts. 4183 First, the local control strategy will be investigated. Using the conditions presented in section III, the local strategy tries to nd a discrete-time piecewise af ne control law, ensuring that every trajectory, starting in simplex Pi will leave the simplex through a speci c facet Fj , j 1, ..., N + 1 of the simplex Pi and the sample picked up next is within the adjacent simplex P . The simplex Pi and P share the same facet Fj . If a valid control law is found, an oriented edge is added to the graph, connecting simplex Pi with P . This edge is used to store the two determined parameters R and u0 of the control law (4). If discrete inputs uD exist, there is a third parameter xD . To determine these parameters the best possibility will be chosen minimizing a performance index in such a way that the retention time in the simplex P is minimized or the control energy is minimized. It is assumed that a change in xD is only allowed, if the rst sample is picked up after the trajectory crossed a facet. A. Local control The local control problem, as mentioned in section II, is to enter the adjacent simplex P through a de ned facet Fe if the N state x R is not contained in the nal simplex PF . The sequence of simplices is determined by the supervisory control. The control law that solves the problem, consists of two parameters R and u0 . The aim of this procedure is to nd a matrix R and the vector u0 , such that constraints in the input signal u are satis ed. For each facet of each simplex, the local control algorithm tries to nd a control law (4). If there are discrete inputs uD , the calculations are done for each xD . The effort for exploring all simplices is then multiplied with max(xD ). A point x in the simplex can be written as a convex combination of the vertices x= 1 After merging these n + 1 u(v 1 )T . . = . u(v T ) n+1 equations, the matrix equation vT 1 1 T . . R . . T . . u0 vT 1 n+1 =(T 1 )T B is obtained. In [4] it is shown, that the matrix T B is always invertible for a simplex. Therefore the unique solution u(v 1 )T RT . . = TT B . uT 0 T u(v n+1 ) is found. As already mentioned in section III, the suf cient conditions (8a-8d,9-11) can result in an in nite number of solutions. To obtain an appropriate control law, the inputs ui (v i ) have to be xed to a de ned value. This degree of freedom can be used to nd the best solution u (v i )T with respect to the chosen performance index J(v i ) := J(v i , u(v i )) = nT (( I n )v i + Hu(v i ) + ) e in each vertex v i . The optimization problem results in the max-min-Problem max min J(v i ) v i V that can easily transformed into a min-max-Problem min max J(v i ) v i V . v1 + . . . + n+1 v n+1 , i [0, 1] , Min-max-problems are standard optimization problems, here a so called constrained min-max-problem has to be solved due to the inequality constraints. A solver for this problem is e.g. fminimax, which is a part of the MATLAB OPTIMIZATION TOOLBOX. The stabilizing control in the nal simplex PF can be determined with the methods presented in [12]. B. Supervisory control i.e. for the inputs u(x) = 1 u(v 1 ) + . . . + n+1 u(v n+1 ) is valid [4]. If inputs u(v i )(i = 1, . . . , n + 1) are found that satisfy the inequalities and the input limitations an af ne feedback law can be determined. Choosing the inputs in the vertices that way that they ful ll the inequality conditions (8a-8d,9-11), then the feedback law is u(v i ) = R v i + u0 , in u(v i )T = vT i 1 RT uT 0 , i = 1, . . . , n + 1. i = 1, . . . , n + 1. The local control provides a graph with several weighted edges. The weight represent the performance index J(u ). The supervisory control reduces to a problem to nd an allowed path in this graph. From a start vertex to the end vertex the shortest path with the smallest sum of the weights of the used edges has to be found by the supervisory control strategy. This is also a standard problem and can be solved with Dijkstra s algorithm [13]. V. EXAMPLE The system under investigation is the two-tank system shown in gure 3. For each tank there is an in ux. The left tank is labeled with 1 and the right tank with 3, and the in uxes are labeled with q1 and q3 , respectively. In table I, the discrete states xD as combinations of the discrete input vector uD = (V 1, V 3, V 13u) are given. A Rewriting this law, by transposing the equations, results 4184 Fig. 3. Two-tank system TABLE I COMBINATIONS OF THE VALVES AND DISCRETE STATE xD xD V1 V3 V13u 1 0 0 0 2 0 0 1 3 0 1 0 4 0 1 1 5 1 0 0 6 1 0 1 7 1 1 0 8 1 1 1 Fig. 4. The trajectory of the two-tank system, starting in x0 = (3, 20)[cm] moving to xF = (53, 40)[cm]. TABLE II TRAJECTORY FROM x0 TO xF SEQUENTIALLY MOVES OVER THE valve is open if the value of the binary variable is one, e.g. V 1 = 1. The dynamic for the two-tank system for xD = 8 is 1 (t) = a1 2g A1 3 (t) = SIMPLICES Pi 4 52 1 10 27 3 5 42 2 11 32 2 6 15 1 xF 30 7 sign( 1 (t) a13 2g | 1 (t) A1 q1,max uq,1 (t) 1 (t) + A1 3 (t)) 3 (t)| Step i xD Step i xD x0 43 7 7 37 4 2 44 8 8 34 2 3 51 2 9 33 1 For simplex 44 , details are given: 8 = 44 0.9743 0.0044 , 8 = 44 0.0079 0.9831 0.6412 0.0014 and H 8 = 44 0.0026 0.6441 0.1686 0.2591 a13 2g | 1 (t) sign( 1 (t) 3 (t)) A3 q3,max a3 2g 3 (t) + uq,3 (t). A3 A3 T 3 (t)| The input u(t) = (uq,1 (t), uq,3 (t)) is added linearly to the nonlinear state equation. The state vector is x(t) = ( 1 (t), 3 (t))T . The root function in the state equation is approximated with discrete-time af ne systems valid on corresponding simplices for the controller design, simulation later on is done on the nonlinear system. In gure 4, the partitioned state space , x = {R2 |0 x1 = 1 60[cm], 0 x2 = 3 60[cm]}, into simplices is given. The simplices are numbered. The trajectory moves from the initial state x0 = (3, 20)[cm] to nal state xF = (53, 40)[cm] crossing the simplices composed in table II. The information which valves are closed or opened, is given by xD marked with big digits in these twelve used simplices in gure 4. The upper index denotes the xD to indicate the necessary combination of the valves. The two parameters of a control law are given: R8 44 51 = 0.0624 0.0126 0.0711 0.0042 , u8 0,44 51 = 0.3752 0.6272 The lower index denotes, where the trajectories start and where they will lead to, e.g. 44 51 means the control law is valid in simplex 44 and will force the trajectory to enter simplex 51 . There was also a control law found for R8 44 50 , but no that considers the limitation of law was found for R8 44 43 the input vector u. Other possible control laws were found for R1 44 50 , 3 R44 50 , R3 , R5 , R5 , R6 , R6 44 51 44 50 44 43 44 50 44 51 , , R7 and R7 . R7 44 50 44 51 44 43 The performance index in this example was de ned to describe the retention time of the trajectory in the investigated 4185 simplex. This is the time the trajectory needs to leave the simplex through the facet Fe , if it starts in the vertex v e . The control law R8 44 51 has the performance index 8 J44 51 = 19.32. This performance index is smaller than 3 7 the performance index J44 51 = J44 51 = 32.84 of the 3 7 control law R44 51 and R44 51 , and it is smaller than 6 the performance index J44 51 = 46.46 of the control law 6 R44 51 . Therefore, the supervisory control chose a path through the graph the way, that simplex 44 has to be left through the facet that is adjacent to simplex 51 , using valve position 8 . The in uxes q1 and q3 represent the continuous inputs u of the system. The needed continuous inputs u are guaranteed to stay within their upper bounds cu and lower bounds cl cl u = R x + u0 cu with cl = 0 0 and cu = 1 1 . To compare the results in this paper with the results in [12], gure 4 is investigated. The difference is that the method in [12] forces the trajectory to move on a straight line. In simplex 44 it is obvious that this is not the case in this paper. Also remember that the simulation is done with the nonlinear system, not with the af ne one. If the sampling period is too large, than it might happen that the neighbor simplex P is overleaped by the trajectory. A found controller has to be checked about that. If this problem occurs, it indicates that the sampling time or the locations of the vertices is unbalanced. VI. CONCLUSION A new approach for the design of controllers for piecewise af ne hybrid systems was presented. Based on a discretetime model, the design of the controller is divided into a local control and a supervisory control. The supervisory control is performed in a graph, done by path planning, the local control is done by solving an optimization problem, ful lling inequality conditions. The resulting control law is an af ne feedback law, valid on the corresponding simplex. Linear search algorithms reduce the on-line computational effort for the control action, determining which control law is active. For that reason, the presented method is suitable even for more complex industrial applications. Future work will focus on nding af ne control laws for a general polytope structure and will investigate systems with less inputs than states. REFERENCES [1] E. D. Sontag, Nonlinear regulation: The piecewise linear approach, IEEE Transactions on Automatic Control, vol. 26, pp. 346 358, 1981. [2] M. Branicky, Studies in hybrid systems: Modeling, analysis, and control, Ph.D. dissertation, MIT, 1995. [3] G. M. Nenninger, Modellbildung und Analyse hybrider dynamischer Systeme als Grundlage f r den Entwurf hybrider Steuerungen, in u Fortschritt-Berichte VDI. D sseldorf: VDI Verlag GmbH, 2001, vol. u 902. [4] L. Habets and J. van Schuppen, A control problem for af ne dynamical systems on a full-dimesnional polytope, Automatica, no. 40, pp. 21 35, 2004. [5] A. Bemporad and M. Morari, Control of systems integrating logic, dynamics, and contraints, Automatica, vol. 35, pp. 407 427, 1999. [6] F. Borrelli, Discrete time constrained optimal control, Ph.D. dissertation, Eidgen ssische Technische Hochschule Z rich, Oktober 2002. o u [7] J. Imura, Optimal control of sampled-data piecewise af ne system, Automatica, vol. 40, pp. 661 669, 2004. [8] E. M nz and V. Krebs, Identi cation of hybrid systems using a u priori knowledge, in Proceedings of the IFAC World Congress, CD, Barcelona, Spain, 2002. [9] G. Ferrari-Trecate, Identi cation of piecewise af ne and hybrid systems, in Proceedings of the ACC, CD, Arlington, USA, 2001. [10] T. Hodrus and E. M nz, Hybride Ph nomene in zeitdiskreter Darstelu a lung, at-Automatisierungstechnik, vol. 51, no. 12, pp. 574 582, 2003. [11] T. E. Hodrus, M. Schwarz, and V. Krebs, A time discretization method for a class of hybrid systems, in Proceedings of NOLCOS Symposium on Nonlinear Control Systems, 2004. [12] T. E. Hodrus, M. Buchholz, and V. Krebs, Control of discrete-time piecewise af ne systems, in Proceedings of IFAC World Congress, 2005. [13] G. L. Nemhauser, K. Rinnooy, and M. J. Todd, Optimization, ser. Handbooks in operations research and management science. Amsterdam: Elsevier Science Publishers B.V., 1989, vol. 1. In gure 5, the two states, i.e. the levels in tank 1 and 3, the in uxes to tank 1 and 3 and the binary state of the discrete inputs for the valves are plotted over time. Fig. 5. The level of the two-tank system and the binary state of the valves. For non-zero values the valves V 1, V 2 and V 13u are open. If there is a function value for V 1, the valve V 1 is open. There is no steady state error. The calculation on a 1.5 GHz Athlon for the local control was done in about 10 minutes, using MATLAB R12.1 4186

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