LMIs in Control/Click here to continue/Observer synthesis/Switched Observer with State Jumps

Switched Systems Observer synthesis for switched linear systems results in switched observers with state jumps. The System where , , and is the index function in discrete state given by deciding which one of the linear vector fields is active at a certain time instant. The Data The matrices are system matrices of appropriate dimensions and are known. The unknown variables of the observer synthesis LMI are and . The Problem Formulation Given a State-space representation of a system given as above. The dynamics of the continuous time observer is defined as: where is the state estimate of the vector field , is the observer gains, is the index function, and is the output of the mode location observer. The observer is divided into two parts, the mode location observer estimating the active dynamics and the continuous-time observer estimating the continuous state of the switched system. The estimated state jumps will be updated according to where is the set of times when the mode location observer switches mode, which are the times when changes value. The LMI: The following are equivalent: (a)There exists and such that where and the states of the hybrid observer is updated according to (b) If for some then Conclusion: Using multiple Lyapunov functions and properly updating the continuous estimated states when the mode changes occur, an observer can be synthesized by solving a linear matrix inequality problem above. External Links A list of references documenting and validating the LMI. S. Pettersson and B. Lennartson. Hybrid system stability and robustness verification using linear matrix inequalities. International Journal of Control, 75(16-17):1335–1355, 2002. Stefan Pettersson. Designing switched observers for switched systems using multiple Lyapunov functions and dwell-time switching. IFAC Proceedings Volumes, 39(5):18–23, 2006. 2nd IFAC Conference on Analysis and Design of Hybrid Systems. Stefan Pettersson. Switched state jump observers for switched systems. IFAC Proceedings Volumes, 38(1):127–132, 2005. 16th IFAC World Congress. Return to Main Page: