Initial Condition Problems Of Fractional Order Systems

Among the plentiful areas of modern science and technology,researches of systems and controls have greatly improved people's daily life and productivity.Meanwhile,the introduction of fractional calculus has also helped to inject a new vitality into this old subject.Compared with the conventional integer order case,while the studies of fractional order systems have indeed opened a door for understanding natural world as well as creating value,the singularities have also brought extra difficulties for us.A good case in point here is the initial condition problems.It is true that these parts of work might be challenging,but they are also the key points which we have to face and the basis which leads fractional order system theories to practical applications.Therefore,this dissertation will start from the nature of fractional order systems and investigate the initial condition problems profoundly.Firstly,by introducing a named aberration phenomenon,the complexity and sig-nificance of initial condition problems of fractional order systems are demonstrated by this dissertation.Furthenrmore,in order to reveal the nature of this odd phenomenon,the infinite dimensional property and long memory property are emphasized.As a re-sult,the relationship between pseudo state-space model under Riemann-Liouville and Caputo derivative and exact state-space model is figured out.Also,the concepts of pre-initial process and initialization function are introduced.All of these work provide basis and foundation for the follow-up studies.Secondly,the fractional numerical realization problems are also investigated by this dissertation.For the part of fractional derivative,a general approach for calculat-ing fractional calculus under different types of derivatives is presented.A fractional order tracking differentiator is proposed.From the consideration of initial conditions,the effect of pre-initial process on computation of fractional derivatives is also figured out.For the part of response of fractional order systems,a numerical method which is applicable to any general fractional order system is given.In addition,An explicit scheme is proposed for the case of nonzero initial conditions.Furthermore,it is noticed that the rational approximation of fractional order sys-tems brings new ideas for discussing these problems under integer order frameworks.This dissertation also starts from system identification and figures out approximation methods for both of fractional order operators and systems by vector fitting method.A direct scheme is presented to meet the requirement of low order.Meanwhile,a strat-egy for allocating the exact initial states is proposed to keep the equivalence of initial conditions between the original system and approximation one.Finally,the estimation of nonzero initial conditions is also investigated in this dis-sertation.Starting from the infinite dimensional property,a least square based approach is given to estimate the exact initial states and track the exact output.A state observer is designed to obtain the estimation of exact initial states.In addition,based on the long memory property,an effective method is also proposed to fit the initialization function which is another description on initial conditions.