Design of decentralised robust event-triggered non-fragile dissipative controllers for uncertain fractional-order interconnected systems

The study is concerned with a method of observer-based control for fractional-order uncertain linear systems with the fractional commensurate order a (1≤a<2) based on linear matrix inequality (LMI) approach. First, a sufficient condition for robust asymptotic stability of the observer-based fractional-order control systems is presented. Next, by using matrix's singular value decomposition and LMI techniques, the existence condition and method of designing a robust observer-based controller for such fractional-order control systems are derived. Unlike previous methods, the results are obtained in terms of LMIs, which can be easily obtained by Matlab's LMI toolbox. Finally, a numerical example demonstrates the validity of this approach.

ABSTRACTTheoretical results on robust passivity and feedback passification of a class of uncertain fractional-order (FO) linear systems are presented in the paper. The system under consideration is subject to time-varying norm-bounded parameter uncertainties in both the state and controlled output matrices. Firstly, some suitable notions of passivity and dissipativity for FO systems are proposed, and the relationship between passivity and stability is obtained. Then, a sufficient condition in the form of linear matrix inequality (LMI) for such system to be robustly passive is given. Based on this condition, the design method of state feedback controller is proposed when the states are available. Moreover, by using matrix singular value decomposition and LMI techniques, the existing condition and method of designing a robust observer-based passive controller for such systems are derived. Numerical simulations demonstrate the effectiveness of the theoretical formulation.

ABSTRACTThe decentralied robust H∞ control problem is considered for fractional-order interconnected systems subject to time-invariant norm-bounded uncertainties in the system, control input and interconnection matrices. First, sufficient conditions for designing a decentralised robust H∞ state feedback controller to guarantee that the closed-loop fractional-order interconnected systems are asymptotically stable with prescribed H∞ performance for all admissible uncertainties are given in terms of linear matrix inequalities. Second, a convex optimisation problem with linear matrix inequality constraints is formulated to design a decentralised robust H∞ state feedback controller with lower gains. Finally, one illustrative example is given to demonstrate the effectiveness of the proposed theoretical results.

LMI-based robust control of fractional-order uncertain linear systems

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0&lt;α&lt;2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0&lt;α&lt;1 and 1&lt;α&lt;2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

This paper concentrates on the study of the decentralized fuzzy control method for a class of fractional-order interconnected systems with unknown control directions. To overcome the difficulties caused by the multiple unknown control directions in fractional-order systems, a novel fractional-order Nussbaum function technique is proposed. This technique is much more general than those of existing works since it not only handles single/multiple unknown control directions but is also suitable for fractional/integer-order single/interconnected systems. Based on this technique, a new decentralized adaptive control method is proposed for fractional-order interconnected systems. Smooth functions are introduced to compensate for unknown interactions among subsystems adaptively. Furthermore, fuzzy logic systems are utilized to approximate unknown nonlinearities. It is proven that the designed controller can guarantee the boundedness of all signals in interconnected systems and the convergence of tracking errors. Two examples are given to show the validity of the proposed method.

Distributed control design with robustness to small time delays

Recent results by the authors have shown how to construct a class of structured controllers for large scale spatially interconnected systems via linear matrix inequalities. These controllers guarantee that the closed loop interconnected system is well-posed, stable and has H/sub /spl infin// norm less than unity. Of paramount importance in the control of interconnected systems is the requirement that the stability and performance of the controlled system be robust to arbitrarily small communication delays between subsystems; this amounts to a continuity property. In this paper, it is shown how to realize the structured controllers obtained from the linear matrix inequalities in order to ensure this continuity property for the closed loop system.

This paper proposes an algorithm for computation of Bode magnitude and phase responses for a large class of linear uncertain fractional-order systems using interval constraint propagation technique. It is first shown that the problem of finding the magnitude and phase of the uncertain fractional-order system over a frequency range can be formulated as a interval constraint satisfaction problem and then solved using branch and prune algorithm. The algorithm guarantees that the magnitude and phase responses are computed to prescribed accuracy. The other advantage of the method is that the magnitude and phase can be computed without any kind of integer-order approximation of the given fractional-order system. The proposed algorithm is demonstrated on three examples including a practical application of a gas turbine plant.

Design of fuzzy output feedback stabilization for uncertain fractional-order systems

This paper focuses on the linear matrix inequality (LMI) characterizations of fractional-order linear systems. Based on the generalized Kalman-Yakubovic-Popov (KYP) lemma, two bounded real lemmas of fractional-order linear systems are introduced with respect to two different norms respectively. Then an new bounded real lemma is proposed with more degrees of freedom. In terms of a set of LMIs then, it is generalized for a class of fractional-order uncertain linear systems with the convex polytopic uncertainties, which forms less conservative constraints on ℋ ∞ performance. Finally this result is demonstrated in a numerical example.

In this paper, the decentralized adaptive prescribed performance finite-time tracking control problem is investigated for a class of interconnected non-affine nonlinear systems with unmodeled dynamics, unknown control direction and unknown dead zone. A novel tracking control strategy combining Nussbaum-gain technique with finite-time control is presented. The designed controller can guarantee that tracking error of each subsystem is constrained by finite-time performance function and converges to the given neighborhood of the equilibrium point within a given arbitrarily settling time. Meanwhile, all the signals in the closed-loop interconnected system are bounded. In addition, the problem that the prescribed performance control method is difficult to realize decentralized control is solved by means of a special mathematical transformation. Specifically, the settling time is a design parameter which is independent of the system initial states in this method. The simulation results are presented to demonstrate the feasibility and superiority of the proposed control method.

A simple method for synchronization of uncertain fractional-order hyperchaotic systems is proposed in this paper. The method makes use of a unidirectional linear coupling approach due to its simple configuration and ease of implementation. To determine the coupling parameters, the synchronization error dynamics is first formulated as a fractional-order linear interval system. Then, the parameters are obtained by solving a linear matrix inequality (LMI) stability condition for stabilization of fractional-order linear interval systems. Thanks to the existence of an LMI solution, the convergence of the synchronization errors is guaranteed. The effectiveness of the proposed method is numerically illustrated by the uncertain fractional-order hyperchaotic Lorenz system.

The problem of designing robust observers for integer-order uncertain systems has been widely treated in the literature. Yet, the generalization of the existing results to the fractional-order framework represents a fertile area of research. In this context, this paper presents a novel robust observer design for a class of uncertain fractional-order linear systems. The main results of the paper are established by using the Lyapunov stability theory, the Caputo definition of fractional-order derivatives, and a convenient Linear Matrix Inequality (LMI) technique. In order to validate these theoretical results, a simulation third-order numerical example is treated in the end of the paper.

This paper proposes the static output feedback (SOF) controller design method for uncertain linear continuous-time systems based on the Linear Matrix Inequalities (LMIs) technique. In the present work, we consider norm-bounded parametric uncertainties. We show that the existence of the SOF for the closed-loop uncertain linear systems is cast as the feasibility of a set of Bilinear Matrix Inequalities (BMIs). Using some technical lemmas, the BMIs are transformed into LMIs, which are numerically exploitable. Thus, we present new LMI stability conditions for the SOF design using two different approaches. We show that our new LMI conditions are less restrictive compared with those available in the literature. A numerical example is illustrated in the end to show the superiority of our design method.