Control Theory Of Fractional Order Systems

Fractional order phenomena have been recognized in more and more science and engineering problems, which marks the progress of human beings’ cognition on the objective world. It also brings both opportunity and challenge when control and mod-ification of dynamic systems are expected to achieve higher goals. Control theory of fractional order systems is the foundation to stimulate the development of fractional technology, and is the key for being accepted and making favourable effect as one kind of solution for real problems. It is also a new fundamental science with engineering sig-nificance and wide applications but it is yet full of difficulties. This thesis is devoted to a creative study on it from easy to difficult and from shallow to deep, whereby theories of fractional order control systems can be established and improved.Firstly, robust stability of fractional order systems as a hot topic is studied. Three kinds of uncertainties are considered, which could directly impact the system stability. Robust stability condition for these uncertain fractional order systems are derived in term of LMI. Then stabilization controllers and less conservative LMI conditions are further proposed. It is a fact that H∞ norm is an important system specification that reflects system robust stability and the capability for disturbance rejection. Bounded real lemmas for fractional order systems are derived by utilizing the generalized KYP lemma for the first time. Then, H∞ controller synthesis of fractional order systems are given.The celebrated Routh test in stability theory is quite elegant and powerful but is only applicable for classical integer order systems. In this thesis, Routh-type test for fractional order systems is proposed for the first time. At the same time, singular cases arising in the Routh-type table are treated in a numerically efficient way. Furthermore, a general and complete Routh-type test for zero distribution of polynomials with com-mensurate fractional degrees and complex coefficients with respect to arbitrary sector regions in the Riemman surface is proposed. Besides, a simple graphical based test is given for more difficult polynomials with non-commensurate degrees.Existence of suitable Lyapunov functional for fractional order systems and its pos-sible structure are investigated since Lyapunov methodology plays a significant role in control system analysis and design. Inverse Lyapunov theorem of linear time invariant fractional order systems is proved for the first time. Lyapunov functional equation is derived and a systematic approach to construct a class of suitable Lyapunov functions is further proposed.Moreover, the generalized linear quadratic functional that characterizes the energy of fractional order control systems is derived and the LQR control problem is proposed to minimize that energy function. To solve this problem, an efficient mathematical tool called spacial product operation is created for the first time, which is efficient to ana-lyze the infinite dimensional state space equation of fractional order systems. Then the Bellman’s dynamic programming is applied to obtain the LQR control law for fractional order systems.Finally, numerical implementation problem of fractional order systems is consid-ered. Finite dimensional approximation method is given whereby the state space equa-tion of approximation for any general fractional order system can be obtained. The initialization problem is also addressed at the same time. Then the stability relationship between the true fractional order system and its fractional differential equation model is discussed.