Piecewise Aggregation And Optimized Design Of Switched Linear Systems

A switched system is an especial class of hybrid systems. It consists of severalsubsystems and a rule that orchestrates the switching between them. For theory andapplications reasons, the switched system has been attacting more and more interests ofinvestigators. It has been used widely in many industry fields and has some potentialdevelopments. Common Lyapunov functions, multiple Lyapunov functions, quadraticLyapunov functions and switched Lyapunov functions are important tools for stability,stabilization and performance optimization of switched systems.If the switched system is asymptotically stabilizable, then there is a switched Lyapunovfunction. It is a significative work to search a proper switched Lyapunov function and designa state-feedback switching law based on the switched Lyapunov function. If a discrete-timeswitched linear system is asymptotically stabilizable, then there is a nonconvex minimumquadratic Lyapunov function. Uing the switched Lyapunov function based on cost function,we can design an optimal state-feedback switching law.When a continuous-time switched linear system is asymptotically stabilizable, severalpiecewise contractive switching pathsθ_i and contractive regionsΩ_ican be found, and apiecewise state-feedback switching law called by∧_i=1^kθ_i^Ω_ican be designed to stabilize thesystem. To optimize the infinite horizon cost function of continuous-time switched systems, inthis work we do the following jobs.Firstly, we investigate a gradient optimization condition on finite interval and methodsbased on Hamilton functions, and present a conjugate gradient algorithm with armijo steps tosearch the optimal piecewise switching pathsθ_i^o. To ensure that the optimal piecewiseswitching paths are also contractive, we present a sufficient condition about restrictedeigenvalues of cost function.Secondly, an aggregated system derived from optimal piecewise switching paths is adiscrete-time switched system. A nonconvex minimum quadratic Lyapunov functionV_k（x）=min{x^TPx:P∈Z_k}can be found by a switched Riccati mapping. We can outline apruning procedure for removing the redundant elements from setsZ_kand obtaining theequivalent subsets with smaller cardinalities. An optimal piecewise state-feedback switchinglaw called by∧_i=1^ko（θ_i^o）^Ω_iocan be designed by these optimal pathsθ_i^oand contractive regionsΩ oiderived from V_k（x）, where k^o is the number of optimal contractive regions.Thirdly, the optimal state-feedback switching law∧_i=1^ko（θ_i^o）^Ω_ioguides an optimal costswitching of the aggregated system and a sub-optimal switching of the correspondingcontinuous-time switched system. To attain a smaller cost error, we use a biggerk^o, but therunning time of the procedure is longer. Another synchronous state-feedback switching law isused to overcome these demerits.The switched Lyapunov functions and piecewise contractive switching paths are analyzedand computed. The performance optimization of switched linear systems is extended formfinite interval to infinite horizon. To break the computational bottleneck, the procedures bringa switched Lyapunov function and a state-feedback switching law, and avoid the complicatedtheoretical analysis. Some simulated examples are shown, and the experiment data validatesthe optimal idea based on piecewise contractive switching path and aggregated system, andcorresponding conclusions.