A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Trans. Automat. Contr., 45(10):1864--1876, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Space Agency. gives conditions for a particular class of linear time varying systems where the system matrix is a linear combination of a basis with respect to time varying coe#cients. 3] addresses observability and controllability for switched linear systems with known and periodic transitions. [1] proposes the notion of incremental observability of a hybrid system, which requires the solution of a mixed integer linear program in order to be tested. In this paper, we study the observability of a class of linear hybrid systems known as jump (or switched ) linear systems, i.e. systems ....

....from an unobservable state to an observable one, as we illustrate in the following example. Example 2 (Unique reconstruction from two unobservable linear systems) Consider a two dimensional linear hybrid system composed of the two linear systems x. 20) Let t 0 = 0, T = 2, x 0 = [0, 1] and assume that there is a single switch from system 1 to system 2 at time t 1 = 1. Then # = 2, 1, 2) O 2 (1) 2 (2) lies in the unobservable subspace of system 1. Also notice that the rank 2n condition is violated, since rank( O 2 (1) 2 (2) 2 4, thus Lemma ....

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A. Bemporad, G. Ferrari, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Transactions on Automatic Control, 45(10):1864--1876, October 2000.

....of a basis with respect to time varying coe#cients. 6] addresses the observability and controllability of switched linear systems with known and periodic transitions. 13] gives a condition for the observability of switched linear systems in terms of the existence of a discrete state trajectory. [3] proposes the notion of incremental observability for piecewise a#ne systems. Such a notion requires the solution of a mixed integer linear program in order to be tested. 16] derives necessary and su#cient conditions for the observability of discrete time jump linear systems. The conditions can ....

....2) yet it is possible to uniquely reconstruct the state trajectory from a particular output . Intuitively, this happens when the linear hybrid system switches from an unobservable state for system k to an observable one for system k # , as we illustrate in the following example. See [3] for additional examples in discrete time) Example 2 (Unique reconstruction from two unobservable linear systems) Consider a two dimensional linear hybrid system composed of the two linear systems x . 23) Let t 0 = 0, T = 2, x 0 = 0, 1] and assume that there is a single ....

A. Bemporad, G. Ferrari, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Transactions on Automatic Control, 45(10):1864--1876, October 2000.

....systems where the system matrix is a linear combination of a basis with time varying coe #cients. They relate to the conditions of [16] where the discrete state is controlled. 8] addresses observability and controllability for switched linear systems with known and periodic transitions. [2] proposes the notion of incremental observability of a hybrid system, which requires the solution of a mixed integer linear program in order to be tested. In a series of papers (see [7] and references therein) Krishnamurthy, Doucet et al. propose various forms of alternating minimization ....

A. Bemporad, G. Ferrari, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Transactions on Automatic Control, 45(10):1864--1876, October 2000.

....of a state space, a set of admissible controls, a family of controlled autonomous vector fields assigned to each discrete state, and collections of guards and reset maps. The main question investigated in the paper is the controllability of control hybrid systems. This issue has been addressed in [1,5,10,11]. In particular, in [11] a sufficient condition for controllabilityofhybrid systems is formulated in terms of so called arrival sets. In [1] the authors derive a necessary and sufficient algebraic condition for a certain subclass of piecewise affine hybrid systems. In [10] the notion of ....

....and reset maps. The main question investigated in the paper is the controllability of control hybrid systems. This issue has been addressed in [1,5,10,11] In particular, in [11] a sufficient condition for controllabilityofhybrid systems is formulated in terms of so called arrival sets. In [1], the authors derive a necessary and sufficient algebraic condition for a certain subclass of piecewise affine hybrid systems. In [10] the notion of controllability for hybrid systems is formalized by continuity of system functions. In this paper new sufficient conditions for the global ....

A.Bemporad, G. Ferrari-Trecate, and R. Morari. Observability and controllability of piecewise affine and hybrid systems. In Proceedings of the 38th IEEE Control Systems Society Conference on Decision and Control, pages 3966-- 3971, Phoenix, AZ., Phoenix, AZ, 1999.

.... model predictive control scheme which is able to stabilize MLD systems on desired reference trajectories while ful lling operating constraints is proposed in [BM99a] An appealing feature of MLD systems is their equivalence to the class of Piecewise Ane (PWA) systems, as proved constructively in [BFTM99] Veri cation: We have addressed the following reachability analysis problem of hybrid systems: For a set of initial states and a class of input signals, determine if a target set can be reached within a given time horizon, and, if so, the subset of initial conditions and input signals for which ....

.... the veri cation of an automotive system [BM99a, Tor99] and to the robust simulation of nonlinear electronic circuits [BGT00] The tool is also used for veri cation of safety properties against variations of the system parameters [BTM00b] stability analysis [BTM00a] and observability analysis [BFTM99] A Language for MLD Models: A declarative language for specifying hybrid systems has been developed which fully automatizes and optimizes the construction of MLD systems out of a high level description of the hybrid process. Several case studies have been modeled in such a language, called ....

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise ane and hybrid systems. In Proc. 38th IEEE Conf. on Decision and Control, Phoenix, AZ, 1999.

....consists of an automaton and, for each discrete state, of an a#ne system on a polyhedral set. A simple case of a polyhedral set is a simplex, the n dimensional generalization of the triangle in R 2 . The class of piecewise linear hybrid systems has been analyzed by several authors, see e.g. [2, 3]. Also the reachability of general hybrid systems has received considerable interest, see e.g. 1, 8, 9] A particular approach to the reachability problem was developed by the second co author in [16] In case of piecewise linear hybrid systems, this method requires the solution of a reachability ....

....T 3 Bu 2 # n T 3 (Av 2 a) so u 2 # 1 6 , with 1 # u 2 # 1. Hence u 2 # [ 1 6 , 1] Finally, u 3 has to satisfy (1) n T 1 Bu 3 n T 1 (Av 3 a) so u 3 3 2 , 2) n T 2 Bu 3 # n T 2 (Av 3 a) so u 3 # 1 2 , and 1 # u 3 # 1. So every u 3 # [ 1, 1 2 ] is a solution. To obtain an a#ne feedback, we fix the inputs at the vertices by choosing u 1 = 1 2 , u 2 = 0, and u 3 = 3 4 , and compute F = f 1 f 2 ) and g using formula (8) 0 1 0 1 1 1 1 1 1 1 1 A 0 f 1 f 2 g 1 A = 0 1 2 0 3 4 1 A . This yields the ....

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. In Proceedings of the 38th IEEE Conference on Decision and Control, pages 3966--3971, New York, 1999. IEEE Press.

....Appendix B. There are many definitions of hybrid systems, see [14] for an overview. Since then other definitions have been introduced that are more suitable for control theory, see [34, 55] The concept of a mixed logic dynamical (MLD) system has been introduced by A. Bemporad and M. Morari, see [9, 8]. It is established in [8] that the concept of a MLD system is a special case of a piecewise linear system. The definition of a piecewise linear hybrid system formulated above is a special case of the definition of a hybrid system introduced by the author in [55, 56] It differs from the other ....

....definitions of hybrid systems, see [14] for an overview. Since then other definitions have been introduced that are more suitable for control theory, see [34, 55] The concept of a mixed logic dynamical (MLD) system has been introduced by A. Bemporad and M. Morari, see [9, 8] It is established in [8] that the concept of a MLD system is a special case of a piecewise linear system. The definition of a piecewise linear hybrid system formulated above is a special case of the definition of a hybrid system introduced by the author in [55, 56] It differs from the other definitions mentioned ....

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise affine and hybrid systems. In Proceedings of the 38th IEEE Conference on Decision and Control, pages 3966--3971, New York, 1999. IEEE Press.

No context found.

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Transactions on Automatic Control, 2000. to appear.

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A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise ane and hybrid systems. IEEE Trans. Autom. Control, 45(10):1864{ 1876, 2000.

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A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise ane and hybrid systems. IEEE Trans. Autom. Control, 45(10):1864-1876, 2000. 11

....u(k) i ; 1) where i are convex polyhedra (i.e. given by a finite number of inequalities) in the input state space. The variables u(k) 2 R , x(k) 2 R and y(k) 2 R denote the input, state and output, respectively, at time k. PWA systems have been studied by several authors (see [2, 18, 22, 24, 28, 29, 31 33] and the references therein) as they form the simplest extension of linear systems that Dept. of Electrical Eng. Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, tel: 31 40 247 35 87, fax: 31 40 243 45 82, w.p.m.h.heemels tue.nl Systems and Control Eng. ....

....to allow a part of the inequalities in (2c) to be strict. On the other hand, from a numerical point of view this issue is not relevant. The equivalence of LC and MLD systems implies that all continuous PWA systems can be exactly written as LC systems as well (see also Corollary 2) 2 Proposition 5 [2] A completely well posed MLD system can be rewritten as a PWA system. Proposition 6 The classes of (constrained) MMPS and ELC systems coincide. Remark 3 As a consequence of the above result and Proposition 3 it is obvious that every LC system can be recast as an MMPS system. A more direct ....

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise affine and hybrid systems. IEEE Trans. Aut. Control, 45(10):1864--1876, 2000.

.... [9] and also hybrid systems in the Mixed Logic Dynamical (MLD) form [1] In particular, the MLD form is capable to model a large class of hybrid systems including linear hybrid dynamical systems, hybrid automata, some classes of discrete event systems, and systems with qualitative inputs outputs [1,3]. The algorithm to obtain the discrete time PWA representation of an MLD system and vice versa is reported in [3] In order to stress the importance of PWA systems it is worth recalling that in [2] the explicit form of Model Predictive Control (MPC) for linear constrained systems was derived and, ....

.... model a large class of hybrid systems including linear hybrid dynamical systems, hybrid automata, some classes of discrete event systems, and systems with qualitative inputs outputs [1,3] The algorithm to obtain the discrete time PWA representation of an MLD system and vice versa is reported in [3]. In order to stress the importance of PWA systems it is worth recalling that in [2] the explicit form of Model Predictive Control (MPC) for linear constrained systems was derived and, besides providing an algorithm for its computation, it was shown that the closed loop system has a PWA structure. ....

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Bemporad, A., Ferrari-Trecate, G., Morari, M.: Observability and Controllability of Piecewise A#ne and Hybrid Systems. IEEE Transactions on Automatic Control, 45(10), (2000), 1864--1876.

.... the piecewise afne state update mapping, system (1) is not well posed in general, as the state update function is twice (or more times) dened over common boundaries of sets C i (the boundaries will be also referred to as guardlines) This is a technical issue which can be avoided as in [9] In [10] the authors show that PWL systems are equivalent to the mixed logical dynamical (MLD) systems introduced in [4] These are hybrid systems de ned by the interaction of logic, nite state machines, and linear discrete time systems. The MLD form is based on the idea of transforming logic relations ....

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise afne and hybrid systems. IEEE Trans. Automatic Control, to appear. http://control.ethz.ch/.

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A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Trans. Automat. Contr., 45(10):1864--1876, 2000.

No context found.

A. Bemporad, G. Ferrari, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Transactions on Automatic Control, 45(10):1864--1876, October 2000.

No context found.

A. Bemporad, G. Ferrari, and M. Morari. Observability and controllability of piecewise affine and hybrid systems. IEEE Transactions on Automatic Control, 45(10):1864--1876, October 2000.

No context found.

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise a#ne and hybrid systems. IEEE Trans. Automat. Contr., 45(10):1864--1876, 2000.

No context found.

A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise ane and hybrid systems. In Proceedings of the 38th IEEE Conference on Decision and Control, pages 3966-3971, New York, 1999. IEEE Press.

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