Adaptive Control For Uncertain Fractional Order Systems

With the advancement of engineering technology, high requirement on control has been put forward in more and more fields. As a generalization of integer order calculus (IOC), fractional order calculus (FOC) can describe a considerable amount of complex systems more precisely and concisely, which thereby reduces the robustness require-ment of the controller. In addition, the degree of freedom in controller design increas-es and the control quality is improved along with the introduction of FOC. However, the practical systems cannot be described exactly by the model due to environmental changes or/and component aging over working time. As a result, study on fractional order control for uncertain fractional order systems (FOS) is of great theoretical and practical significance.The adaptive approach has achieved fruitful results on control uncertain integer order systems (IOS). Yet, when it extends to the fractional order case, there are still many difficulties and challenges what we have to face. FOC can be regarded as an natural but nontrivial extension of IOC. Nevertheless, the system has changed in nature from integer order case to fractional order case. For instance, the eigenfunctions are single-valued for the former case but multi-valued for the latter case; the state space descriptions for IOS are often finite dimensional while FOS have an infinite dimen-sional state space expression. All these make the revelent theoretical system change in essential, and surely hold back the development of adaptive control for FOS. Conse-quently, based on the essential characteristic of FOS, this dissertation will utilize the indirect Lyapunov method, to study on fractional order adaptive control, and provide an effective solution for fractional order control problem of uncertain FOS.Firstly, the fractional order direct model reference adaptive control (FO-DMRAC) scheme is modified. On the one hand, for the single-input single-output (SISO) case with order0<α<1, to reduce the dependency of parameter estimation has on the tracking error and to avoid undesirable jump transient, the prediction error is added to the parameter adjusting law which results in a modified control scheme. And further extension of the FO-DMRAC approach to the case of1<α<2is promoted. On the other hand, for the multi-input multi-output (MIMO) case, the right gain matrix is adopted, and then a novel control scheme for plants with arbitrary relative degree is present, which does not require a stringent symmetry assumption related with the plant high frequency gain matrix. With the aid of continuous frequency distribution model (CFDM) and indirect Lyapunov method, the stability of the closed-loop control system, the asymptotic convergence of output tracking error and parameter estimation error are established.Secondly, to obtain better convergence characteristics and control performance, this dissertation first proposes and thoroughly analyzes the fractional order indirect model reference adaptive control (FO-IMRAC) scheme. This part begins with system parameter estimation. Considering the cases with and without constraints, parame-ter estimation approaches are developed respectively. On the basis of this result, FO-IMRAC controller design methods are given for both SISO univariate and multivariate FOS, including, selection of reference model, construction of controller structure and calculation of controller parameter, etc.Afterwards, as the two aforementioned methods can only be applied to linear para-metric models, this dissertation develops the fractional order adaptive backstepping control technique which can be applied to nonlinear fractional order systems. When all states are measurable, the given plant can be converted into the normalized para-metric strict-feedback form by introducing appropriate transformations of coordinates. Then, new error variables are constructed to design fractional order adaptive backstep-ping state feedback controller (FO-ABSFC). When only partial states are measurable, a state estimator is introduced firstly. Then new error variables and error system are constructed. The problem of proving the stability of the closed-loop control system under asymptotic estimation can be solved via a novel Lyapunov function. Finally, the general design process and realization method of fractional order adaptive backstep-ping output feedback controller (FO-ABOFC) are given with the help of the proposed fractional order tracking differentiator (FOTD).Furthermore, numerical approximation for fractional order integral operators is researched simultaneously in two ways:the optimal performance and the lowest or-der, based on the concept of system identification skillfully. In the former case, the Oustaloup recursive approximation approach is not a rigorous variable poles one, s-ince the approximation model cannot reach the situation with complex poles. What is more, the approach results from inaccurate amplitude frequency characteristic, so this dissertation uses the vector fitting approach to realize the rational approximation of true sense of variable poles. In the latter case, by designing a fixed pole approximation ap-proach, we transform the corresponding problem into a linear least square problem. In consideration of the characteristics of pure integrator, the dissertation gives the method of selecting an optimal initial pole. According to the proposed fractional order integral operator approximation, the approximation of FOS is developed. By pointing out the relationship between the fractional order pseudo state space model and its frequency distribution model, we discuss and resolve the system response problem with non-zero initial condition. The aforementioned related work provides an effective solution to validate the proposed fractional order adaptive control scheme in this dissertation.