## Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems (1998)

Venue: | IEEE Transactions on Automatic Control |

Citations: | 191 - 4 self |

### BibTeX

@ARTICLE{Johansson98computationof,

author = {Mikael Johansson and Anders Rantzer},

title = {Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems},

journal = {IEEE Transactions on Automatic Control},

year = {1998},

volume = {43},

pages = {555--559}

}

### Years of Citing Articles

### OpenURL

### Abstract

. This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach. Keywords. Piecewise linear systems, Lyapunov stability, linear matrix inequalities. 1. Introduction Construction of Lyapunov functions is one of the most fundamental problems in systems theory. The most direct application is stability analysis, but analogous problems appear more or less implicitly also in performance analysis, controller synthesis and system identification. Consequently, methods for constructing Lyapunov functions for general nonlinear systems is of great theoretical and practical interest. The objective of this paper is to develop a uniform and compu...

### Citations

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Citation Context ...n but not from the circle criterion. The failure of the circle criterion comes as no surprise, as the circle criterion relies on the existence of a common Lyapunov function on the form V (x) = x T Px =-=[11]-=-, which we know does not exist. The standard proof of the Popov criteria, on the other hand, uses the Lyapunov function V (x) = x T Px+ 2j R Cx 0 OE(oe) doe. For the simple switched system, this funct... |

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Citation Context ... defined for all ts0. It is however possible to extend the trajectory definition further, using the concept sliding mode. Such definitions have been suggested by, for example, Filippov [16] and Utkin =-=[17]-=-. For piecewise affine systems, the approach of Filippov suggests the following definition of system trajectories. We say that a continuous piecewise C 1 function x(t) 2 [ i2I X i is a trajectory of t... |

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Filippov, Differential Equations with Discontinuous Righthand Sides
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Citation Context ...ng trajectories defined for all ts0. It is however possible to extend the trajectory definition further, using the concept sliding mode. Such definitions have been suggested by, for example, Filippov =-=[16]-=- and Utkin [17]. For piecewise affine systems, the approach of Filippov suggests the following definition of system trajectories. We say that a continuous piecewise C 1 function x(t) 2 [ i2I X i is a ... |

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Citation Context ...x = 2 X i , the inequality (7) may be simpler to satisfy than the Lyapunov inequalities in Proposition 1. In some cases, the S-procedure is a non-conservative way to account for quadratic constraints =-=[13]. Ma-=-trices for use in the S-procedure can be constructed from the system description as follows. Let U i be a matrix with non-negative entries and let �� E i satisfy (2). Then, it follows that x T E T... |

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Citation Context ...yapunov function should be continuous across cell boundaries. The dependence on the discrete state may even be discontinuous, if the value of the Lyapunov function decreases at the switching instants =-=[15]-=-. If the switching conditions for the discrete state can be expressed as affine equalities in the continuous state, it is possible to formulate the search for this type of Lyapunov functions as an LMI... |

71 | Piecewise linear quadratic optimal control
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Citation Context ...ty that also covers polytopic Lyapunov functions, see [1], [2] and the references therein. The technique presented here has been extended to treat performance analysis and optimal control problems in =-=[3]-=-, and smooth nonlinear systems in [4]. This paper is based on the report [5]. Similar work has also been reported in [6]. 2. Model Representation We consider analysis of piecewise affine systems of th... |

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Citation Context ... Systems with switching vector fields require special attention when it comes to modeling and simulation. All simulations in this paper are performed in Omsim with proper treatment of discrete events =-=[7]-=-. 3. Quadratic Stability In some cases, it is possible to prove stability of piecewise linear systems using a globally quadratic Lyapunov function V (x) = x T Px. The computations are usually based on... |

21 |
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Citation Context ...lysis of a class of hybrid systems. The use of piecewise quadratic Lyapunov functions appears to be a powerful extension of quadratic stability that also covers polytopic Lyapunov functions, see [1], =-=[2]-=- and the references therein. The technique presented here has been extended to treat performance analysis and optimal control problems in [3], and smooth nonlinear systems in [4]. This paper is based ... |

16 |
Robust stability analysis and controller design with quadratic Lyapunov functions
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Citation Context ...r analysis of a class of hybrid systems. The use of piecewise quadratic Lyapunov functions appears to be a powerful extension of quadratic stability that also covers polytopic Lyapunov functions, see =-=[1]-=-, [2] and the references therein. The technique presented here has been extended to treat performance analysis and optimal control problems in [3], and smooth nonlinear systems in [4]. This paper is b... |

5 |
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Citation Context ...has been extended to treat performance analysis and optimal control problems in [3], and smooth nonlinear systems in [4]. This paper is based on the report [5]. Similar work has also been reported in =-=[6]-=-. 2. Model Representation We consider analysis of piecewise affine systems of the formsx(t) = A i x(t) + a i ; for x(t) 2 X i (1) Here, fX i g i2I ` R n is a partition of the state space into a number... |

3 | Modelling, analysis & synthesis of hybrid systems - Pettersson, Lennartson - 1996 |

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System Modeling, Identification and Control using Fuzzy Logic
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Citation Context ...nds to zero exponentially. Quadratic stability of a family of linear system is attractive, since stability follows independently of cell partition and for a large class of switching schemes, see e.g. =-=[8]-=-. Furthermore, the conditions of Proposition 1 are linear matrix inequalities in P , and verifying these conditions amounts to solving a convex optimization problem for which efficient software is pub... |

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Ghaoui, LMITOOL: An Interface to Solve
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Citation Context ... conditions of Proposition 1 are linear matrix inequalities in P , and verifying these conditions amounts to solving a convex optimization problem for which efficient software is publically available =-=[9, 10]-=-. Current software is capable of treating several hundreds of variables in a matter of seconds. It can sometimes be of interest to verify that no common solution P in Proposition 1 exists. This verifi... |

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Citation Context ...cell X i . Consequently, it is sufficient to require that x T (A T i P + PA i )x ! 0 for x 2 X i : (6) One way to introduce piecewise analysis in the stability conditions is the so-called S-procedure =-=[12]-=-. We thus construct matrices S i such that x T S i xs0 for x 2 X i , and obtain the following relaxed conditions for quadratic stability A T i P + PA i + S i ! 0: (7) The intuition behind the method i... |

1 | Die absolute Stabilit ��t von Regelsystemen., chapter - Aiserman, Gantmacher - 1965 |

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