Conditions for stabilizability of linear switched systems (original) (raw)

Linear Quadratic State Feedback Design for Switched Linear Systems with Polytopic Uncertaities

In this paper, we consider the design of stable linear switched systems for polytopic uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop switched systems can stabilize unstable open loop systems or stable open loop systems but in which there is no solution for a common Lyapunov matrix. For continuous time switched linear systems, we show that if there exists solution in an associated Riccati equation for the closed loop systems sharing one common Lyapunov matrix, the switched linear systems are stable. For the discrete time switched systems, we derive an LMI to calculate a common Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop linear quadratic state feedback regulators guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence. Keywords: Continuous time linear switched system, discrete time switched linear systems, linear quadratic state feedbac...

On Switched Control Design of Linear Time-Invariant Systems with Polytopic Uncertainties

Mathematical Problems in Engineering, 2013

This paper proposes a new switched control design method for some classes of linear time-invariant systems with polytopic uncertainties. This method uses a quadratic Lyapunov function to design the feedback controller gains based on linear matrix inequalities (LMIs). The controller gain is chosen by a switching law that returns the smallest value of the time derivative of the Lyapunov function. The proposed methodology offers less conservative alternative than the well-known controller for uncertain systems with only one state feedback gain. The control design of a magnetic levitator illustrates the procedure.

Stability Analysis of Discrete-time Switched Linear Systems with Parametric Uncertainties

International Journal of Industrial Electronics, Control and Optimization (IECO), 2019

This paper considers the stability problem of discrete-time switched linear systems in the presence of parametric uncertainties. This type of uncertainty is sometimes called structured uncertainty because of its known structure. However, some of the parameters in the system are uncertain. From a practical viewpoint, it is important to guarantee the robust stability of uncertain switched systems. Therefore, based on the structure of the uncertainty matrix and the common Lyapunov function for the nominal switched system, sufficient conditions for robust exponential stability of the discrete-time uncertain switched system (under any switching signal) are derived. These sufficient conditions are formulated in terms of matrix inequalities and using fixed values for some parameters, they will be solved via LMI techniques and based on numerical methods. Moreover, a procedure is proposed to determine the maximum admissible bounds of the uncertain parameters to guarantee the exponential stability of the uncertain switched system. Finally, numerical examples are provided to verify the proposed theoretical results.