Stability and Control of Fractional-order Systems with Delays and Periodic Coefficients

Abstract

This dissertation addresses various problems in numerical methods, stability, and control of fractional-order differential equations associated with delays and periodic coefficients. For this purpose, first, several fractional collocation differentiation matrices (FCDMs) are developed by the use of finite difference methods and spectral methods. It is shown that the methods of constructing spectral-FCDMs can be classified into two groups: direct and indirect methods. It is shown via an error analysis that use of the well-known property of fractional differentiation of polynomial bases applied to direct methods results in a limitation in the size of the obtained FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Moreover, an algorithm is presented to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo fractional derivatives. The proposed fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev-Gauss-Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. It is shown the proposed fractional differentiation matrices are highly efficient in solving fractional differential equations. Second, a general framework is presented for the numerical solution of multi-order fractional delay differential equations (FDDEs) in noncanonical form with irrational/ rational multiple delays by the use of a spectral collocation method. In contrast to the current numerical methods for solving fractional differential equations, the proposed framework can solve multi-order FDDEs in a noncanonical form with incommensurate orders. The framework can also enable the efficient solution of multi-order FDDEs with irrational multiple delays. Next, the framework is enhanced by the fractional Chebyshev collocation method in which a Chebyshev operation matrix is constructed for fractional differentiation. Spectral convergence and small computational time are two other advantages of the proposed framework enhanced by the fractional Chebyshev collocation method. In addition, the convergence, error estimates, and numerical stability of the proposed framework for solving FDDEs are studied. The advantages and computational implications of the proposed framework are discussed and verified in several numerical examples for practical engineering problems. Third, the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control are developed for a class of linear fractional order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This section of the dissertation is devoted to defining and approximating the monodromy operator for steady state solution of FPTD systems. It is shown that the monodromy operator cannot be realized in closed form for FPTD systems, and hence the short memory principle along with the fractional Chebyshev collocation (FCC) method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response. Fourth, a new method is proposed to identify unknown parameters of linear fractional order systems by discretization at the Gauss-Lobatto-Chebyshev collocation points. The proposed spectral parametric estimation method benefits from the spectral method exponential convergence feature and results in more accuracy in the estimated parameters and less computational time. Finally, a novel method is proposed to design optimal observer-based feedback control for linear fractional systems with constant or periodic coefficients based on the FCC method.