Stability Analysis And Controller Design Of Fractional Order Positive Systems

It has been proved by the modern physics that there are many fractional phenomena in practical engineering and science.Compared with traditional integer order state equation,the fractional order state equation can better describe physical systems in reality.At present,frac-tional order calculus is widely applied to image processing,signal processing,fractal theory and other fields.Fractional order system has become a research hotspot in the field of control science.In addition,positive systems are an important branch of the systems theory.Due to the widespread applications in science and technology,positive systems have been of great interest to many researchers.The nonnegativity restriction on variables results in the fact that positive systems are defined on cones and not on linear space,which implies that many results for gen-eral systems are not available for positive systems.A great number of novel and straightforward theoretical results have been reported by capturing the nonnegativity characteristic of variables.Based on the previous results on fractional order systems and positive systems,this dis-sertation is concerned with diagonal stability,controller design and H? control for fractional order positive systems with the fractional order ? satisfying 0<?<1.The main contributions of this dissertation are summarized as follows:Firstly,the diagonal stability for fractional order positive systems is investigated.For inte-ger order positive systems,diagonal stability is a classic result.By diagonal stability,we imply the existence of a positive definite diagonal matrix for the Lyapunov linear matrix inequalities(LMls).Based on this result,we propose an extension of diagonal stability for Metzler matrix.Conditions of the stability for fractional order positive systems with diagonal solutions is also proved.The diagonal solutions of the LMIs for fractional order positive systems involve a di-agonal positive definite matrix and a skew-symmetric anti-diagonal matrix,and the result fits to the properties of fractional order systems.Secondly,the design method of feedback controller for fractional order positive systems is studied.The result on diagonal stability for fractional order positive systems can be used to design the state feedback controller such that closed-loop fractional order systems are stable and positive.At the end of the chapter,numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods.Finally,H? control problem for fractional order positive systems is studied.The necessary and sufficient LMIs condition for the H? performance of fractional positive systems is given,and the existence of diagonal matrix solution is proved.The sufficient and necessary condition for designing the state feedback controller is given,which makes the closed-loop system is positive and stable and satisfies the H? performance.At the end of the chapter,numerical ex-amples are provided to demonstrate the effectiveness and applicability of the proposed methods.