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Section: New Results

Stability analysis of Piece-Wise Affine Systems

Participants : Giorgio Valmorbida, Leonardo Broering Groff, Joao Manoel Gomes Da Silva Jr [Universidade Federal do Rio Grande do Sul] .

Piece-wise affine systems appear when linear dynamics are defined in different partitions of the state space. This type of system naturally appears whenever actuators have different stages or saturate or whenever non-linear control laws are obtained as the solution to a parameterised optimization problem as, for instance for systems with feedback laws based on the so-called explicit Model Predictive Control. Even though the dynamics is simple to describe, the stability analysis, performance assessment and robustness analysis are difficult to perform since, due to the often used explicit representation, the Lyapunov stability and dissipation tests are often described in terms of a number of inequalities that increase exponentially on the number of sets in the partition. Moreover regional stability and uncertainties corresponding to modification on the partition are difficult to study in this scenario.

To overcome these difficulties we have proposed an implicit representation for this class of systems in terms of ramp functions. The main advantage of such a representation lies on the fact that the ramp function can be exactly characterized in terms of linear inequalities and a quadratic equation, namely a linear complementarity condition. Thanks to the characterization of the ramp function and the implicit description of the PWA system the verification of Lyapunov inequalities related to piecewise quadratic functions can be cast as a pair of linear matrix inequalities.

The obtained formulation opens several possibilities to study the class of piecewise affine systems and their robustness properties. Indeed the fact that some partitions are uncertain is more easily coped with the proposed approach as they are described as parametric uncertainties of the implicit representation. Also, systems of larger dimension can be studied.

The stability analysis of the particular subclass of systems given by asymmetric saturation can also be performed with discontinuous Lyapunov functions for discrete-time systems.