## Stabilization conditions for model predictive control of constrained PWA systems (2004)

Venue: | In Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, The Bahamas |

Citations: | 9 - 7 self |

### BibTeX

@INPROCEEDINGS{Lazar04stabilizationconditions,

author = {M. Lazar and Student Member and W. P. M. H. Heemels and S. Weil and A. Bemporad},

title = {Stabilization conditions for model predictive control of constrained PWA systems},

booktitle = {In Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, The Bahamas},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract — Model Predictive Control (MPC) has recently been applied to several relevant classes of hybrid systems with promising results. These developments generated an increasing interest towards issues such as stability and computational problems that arise in hybrid MPC. Stability aspects have been addressed only marginally. In this paper we present an extension of the terminal cost and constraint set method for guaranteeing stability in MPC to the class of constrained piecewise affine systems. Semidefinite programming is used to calculate the employed terminal weight matrix that ensures stability for quadratic cost based MPC. A procedure for computing a robust positively invariant set for piecewise linear systems is also developed. The implementation of the proposed method is illustrated by an example. Index Terms — Piecewise affine systems, Model predictive

### Citations

334 |
Constrained model predictive control: Stability and optimality
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(Show Context)
Citation Context ...n this paper we develop a priori stabilization conditions for quadratic cost based MPC of constrained PWA systems. The proposed method is an extension of the terminal cost and constraint set approach =-=[17]-=- for guaranteeing stability in linear or nonlinear MPC. The procedure for deriving the stabilization conditions is based on Lyapunov arguments, which yield, after non-trivial transformations, a set of... |

127 |
Nonlinear regulation: the piecewise linear approach
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(Show Context)
Citation Context ...and compact set in R n that contains the origin in its interior is called a C-set [4]. III. PROBLEM FORMULATION Consider the time-invariant discrete-time PWA system described by equations of the form =-=[20]-=-: xk+1 = Ajxk + Bjuk + fj when xk ∈ Ωj. (3) Here, xk ∈ X ⊆ R n is the state vector and uk ∈ U ⊆ R m is the control input vector at the discrete-time instant k ≥ 0. Aj ∈ R n×n , Bj ∈ R n×m , fj ∈ R n ,... |

81 | Robust constrained model predictive control using linear matrix inequalities
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(Show Context)
Citation Context ... Then (11) and P>0 are feasible if and only (18) is feasible. Proof: Given in [16].sRemark IV.3 An alternative solution to rewrite matrix inequalities of the form (11) as an LMI has been developed in =-=[14]-=- for uncertain linear systems, in the context of robust MPC. If the LMI (18) is feasible then, by Theorem IV.2, the terminal weight and the feedback gains are recovered as P = Z −1 and Kj := YjZ −1 fo... |

70 | Equivalence of hybrid dynamical models
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(Show Context)
Citation Context ...timal control and stabilization of hybrid systems in general and Piecewise Affine (PWA) systems in particular. This is motivated by the fact that PWA systems can model a broad class of hybrid systems =-=[11]-=-, [21]. Several results have been reported in this framework, e.g., see [6], [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to su... |

54 | Piecewise linear quadratic optimal control
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(Show Context)
Citation Context ... systems in particular. This is motivated by the fact that PWA systems can model a broad class of hybrid systems [11], [21]. Several results have been reported in this framework, e.g., see [6], [18], =-=[19]-=-, [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], [9], [13], [15]. However, al... |

51 | Interconnected automata and linear systems: A theoretical framework in discrete-time
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(Show Context)
Citation Context ...control and stabilization of hybrid systems in general and Piecewise Affine (PWA) systems in particular. This is motivated by the fact that PWA systems can model a broad class of hybrid systems [11], =-=[21]-=-. Several results have been reported in this framework, e.g., see [6], [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successf... |

41 | Optimal control of constrained piecewise affine discrete-time systems
- Mayne, Raković
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Citation Context ...g., see [6], [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], [9], =-=[13]-=-, [15]. However, all the implementations mentioned above faced two serious drawbacks. Firstly, the on-line computational load caused by the Mixed Integer Quadratic (or linear) Programming (MIQP) probl... |

39 |
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Citation Context ...is a convex set obtained as the intersection of a finite number of open and/or closed half-spaces. Moreover, a convex and compact set in R n that contains the origin in its interior is called a C-set =-=[4]-=-. III. PROBLEM FORMULATION Consider the time-invariant discrete-time PWA system described by equations of the form [20]: xk+1 = Ajxk + Bjuk + fj when xk ∈ Ωj. (3) Here, xk ∈ X ⊆ R n is the state vecto... |

35 | Robust Constraint Satisfaction: Invariant Sets and Predictive Control,PhDThesis
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Citation Context ...or the PWL system (20) the maximal positively invariant set in the sense of Definition II.1. Unfortunately, this set is not convex in general, but only a union (possibly infinite) of convex sets 4598 =-=[12]-=-. Only in the (rare) case when this set is a finite union of polyhedral sets, Problem 1 can be put in an MIQP form. However, this approach suffers from two other drawbacks. Firstly, it considerably co... |

25 | Optimal controllers for hybrid systems: Stability and piecewise linear explicit form
- Bemporad, Borrelli, et al.
- 2000
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Citation Context ...mework, e.g., see [6], [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in =-=[2]-=-, [9], [13], [15]. However, all the implementations mentioned above faced two serious drawbacks. Firstly, the on-line computational load caused by the Mixed Integer Quadratic (or linear) Programming (... |

19 | Optimal control of switched systems based on parameterization of the switching instants
- Xu, Antsaklis
- 2004
(Show Context)
Citation Context ...ms in particular. This is motivated by the fact that PWA systems can model a broad class of hybrid systems [11], [21]. Several results have been reported in this framework, e.g., see [6], [18], [19], =-=[22]-=- and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], [9], [13], [15]. However, all the ... |

18 |
Control of Systems Integrating Logic
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(Show Context)
Citation Context ...realtime implementation. Secondly, closed-loop stability is not guaranteed a priori. The first solution for guaranteeing stability of hybrid model based receding horizon control has been presented in =-=[1]-=- for Mixed Logical Dynamical (MLD) systems. This approach is based on enforcing a terminal state equality constraint. However, this method may require a long prediction horizon to guarantee feasibilit... |

16 |
Stability and stabilization of piecewise affine and hybrid systems: An LMI approach
- Mignone, Ferrari-Trecate, et al.
- 2000
(Show Context)
Citation Context ... (PWA) systems in particular. This is motivated by the fact that PWA systems can model a broad class of hybrid systems [11], [21]. Several results have been reported in this framework, e.g., see [6], =-=[18]-=-, [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], [9], [13], [15]. Howev... |

13 |
Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states
- Cwikel, Gutman
- 1986
(Show Context)
Citation Context ...ssume that P is the maximal positively invariant set for system (21) with arbitrary switching. Then we have that P is a positively invariant set for any linear system in (21) and then it follows from =-=[10]-=- that the convex hull of P is also a positively invariant set for any linear system in (21). Hence, the convex hull of P is a positively invariant set for system (21) in the sense of Definition II.2. ... |

8 |
An Efficient Algorithm for Computing the State Feedback Optimal Control Law for Discrete Time Hybrid Systems
- Borrelli, Baotić, et al.
- 2003
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Citation Context ...ffine (PWA) systems in particular. This is motivated by the fact that PWA systems can model a broad class of hybrid systems [11], [21]. Several results have been reported in this framework, e.g., see =-=[6]-=-, [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], [9], [13], [15].... |

8 |
Low complexity control of piecewise affine systems with stability guarantee
- Grieder, Kvasnica, et al.
- 2004
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Citation Context ...k, e.g., see [6], [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], =-=[9]-=-, [13], [15]. However, all the implementations mentioned above faced two serious drawbacks. Firstly, the on-line computational load caused by the Mixed Integer Quadratic (or linear) Programming (MIQP)... |

5 | The explicit solution of constrained LP-based receding horizon control
- Bemporad, Borrelli, et al.
- 2000
(Show Context)
Citation Context ...et) and by assuming positive invariance of the predefined target set. A notable exception is [2], where an extension of the results obtained for (linear) constrained LP-based receding horizon control =-=[3]-=- has been pursued. Unfortunately, this infinity norm based MPC approach did not yield conclusive stabilization conditions, but only a heuristic stabilization criterion. Another option is to determine ... |

3 |
Constrained stabilization with an assigned initial condition set
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- 1995
(Show Context)
Citation Context ...t (23) generated using (22) is a polyhedral set calculated in finite time. Due to the fact that this set is not the maximal positively invariant set for the PWL system (20), the backward procedure of =-=[5]-=- can be employed to enlarge P. However, this procedure must be implemented in the sense of Definition II.1 and then the new regions added to P must be such that the resulting set remains a polyhedron.... |

2 |
A semi-explicit MPC set-up for constrained piecewise affine systems
- Lazar, Heemels
- 2003
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Citation Context ...e [6], [18], [19], [22] and the references therein. Extension of Model Predictive Control (MPC) to this class of systems led to successful implementations such as the ones reported in [2], [9], [13], =-=[15]-=-. However, all the implementations mentioned above faced two serious drawbacks. Firstly, the on-line computational load caused by the Mixed Integer Quadratic (or linear) Programming (MIQP) problem pre... |

2 | Stabilizing receding horizon control of piecewise linear systems: An LMI approach
- Lazar, Heemels, et al.
- 2004
(Show Context)
Citation Context ...0 (AjZ + BjYj) 0 0 Z Theorem IV.2 Suppose that for j ∈ S0 the variables (P, Kj) and (Z, Yj) are related according to (17). Then (11) and P>0 are feasible if and only (18) is feasible. Proof: Given in =-=[16]-=-.sRemark IV.3 An alternative solution to rewrite matrix inequalities of the form (11) as an LMI has been developed in [14] for uncertain linear systems, in the context of robust MPC. If the LMI (18) i... |