During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

This article studies the robust and reliable $\mathscr {H}_{\infty }$ static output feedback (SOF) control for nonlinear systems with actuator faults in a descriptor system framework. The nonlinear plant is characterized by a discrete-time Takagi-Sugeno (T-S) fuzzy affine model with parameter uncertainties, and the Markov chain is utilized to describe the actuator-fault behaviors. Specifically, by adopting a state-output augmentation approach, the original system is firstly reformulated into the descriptor fuzzy affine system. Based upon a novel piecewise Markovian Lyapunov function (LF), the $\mathscr {H}_{\infty }$ performance analysis condition for the underlying system is then presented, and furthermore the robust and reliable SOF controller synthesis is carried out. It is shown that by invoking the redundancy properties induced by the descriptor formulation, combined with some convexifying techniques, the existence of the desired reliable controller can be explicitly determined by the solution of a convex optimization problem. Finally, simulation studies are applied to confirm the effectiveness of the developed method.

Reliable Output Feedback Control of Discrete-Time Fuzzy Affine Systems With Actuator Faults

In this paper, the problem of robust fault tolerant control for a class of singular systems subject to both time-varying state-dependent nonlinear perturbation and actuator saturation is investigated. A sufficient condition for the existence of a fixed-gain controller is first proposed which guarantees the regularity, impulse-free and stability of the closed-loop system under all possible faults. An optimization problem with LMI constraints is formulated to determine the largest contractively invariant ellipsoid. An adaptive fault tolerant controller is then developed to compensate for the failure effects on the system by estimating the fault and updating the design parameter matrices online. Both of these two controllers are in the form of a saturation avoidance feedback with the advantage of relatively small actuator capacities compared with the high gain counterpart. An example is included to illustrate the proposed procedures and their effectiveness.

Fault tolerant control for singular systems with actuator saturation and nonlinear perturbationFault tolerant control for singular systems with actuator saturation and nonlinear perturbation.

This paper deals with the quantized control problem for nonlinear semi-Markov jump systems subject to singular perturbation under a network-based framework. The nonlinearity of the system is well solved by applying Takagi–Sugeno (T-S) fuzzy theory. The semi-Markov jump process with the memory matrix of transition probability is introduced, for which the obtained results are more reasonable and less limiting. In addition, the packet dropouts governed by a Bernoulli variable and the signal quantization associated with a logarithmic quantizer are deeply studied. The major goal is to devise a fuzzy controller, which not only assures the mean-square $\bar { \sigma }$ -error stability of the corresponding system but also allows a higher upper bound of the singularly perturbed parameter. Sufficient conditions are developed to make sure that the applicable controller could be found. The further examination to demonstrate the feasibility of the presented method is given by designing a controller of a series DC motor model.

Network-Based Quantized Control for Fuzzy Singularly Perturbed Semi-Markov Jump Systems and its Application

In this paper, a class of discrete-time singularly perturbed switched systems $\left ({\text {SPSSs}}\right) $ with persistent dwell-time $\left ({\text {PDT}}\right)$ switching law is firstly proposed. Using the slow-state feedback control method, sufficient conditions to ensure the global uniform exponential stability of the closed-loop PDT SPSSs are derived. Based on the aforementioned conditions, the analyses of the extended dissipative performance of the closed-loop PDT SPSSs and a preferable decoupling method deriving the mode-dependent controller gains are given for the first time. Finally, the potential practicability of the method is verified by a purely numerical example and a tunnel diode circuit model, and a convex optimization method calculating the upper bound of the singular perturbation parameter is provided.

Extended Dissipative Control for Singularly Perturbed PDT Switched Systems and its Application

Stabilization bound of singularly perturbed systems subject to actuator saturation

In this paper, Lyapunov stability and strong passivity are defined for nonlinear descriptor systems. The new concepts facilitate the formulation of the relationship between the stability and passivity of nonlinear descriptor systems. A Lyapunov stability theorem which describes a sufficient condition for the systems to be globally asymptotically stable and of index one is derived. By the Lyapunov stability theorem, the connection between Lyapunov stability and strong passivity is established. Furthermore, strong passivity of feedback systems is discussed and two passivity theorems are given. Using the given passivity theorems, strongly absolute stability of Lur'e type descriptor systems is discussed. The obtained strongly absolute stability criterion is shown to be more general and less conservative than the existing methods. Finally, two examples are used to illustrate the advantages and effectiveness of the obtained methods.

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Lyapunov Stability and Strong Passivity Analysis for Nonlinear Descriptor Systems

Summary

This paper investigates the problems of controller design and stability analysis for singularly perturbed switched systems subject to actuator saturation. A set of well-defined sufficient conditions for the existence of state feedback controllers is proposed, under which the closed-loop system is locally asymptotically stable for arbitrary switching law as long as the singular perturbation parameter is sufficiently small. With the obtained controller, the estimation problem of stability bound and basin of attraction of the closed-loop system is reduced to solving a convex optimization problem. A numerical example and a hydraulic servo position system are used to illustrate the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

Controller design and analysis for singularly perturbed switched systems with actuator saturation

In this paper, sampled-data H∞ filtering problem is considered for Markovian jump singularly perturbed systems with time-varying delay and missing measurements. The sampled-data system is r...

Xiaoping Ma

Papers

Stabilization bound of singularly perturbed systems subject to actuator saturation

Lyapunov Stability and Strong Passivity Analysis for Nonlinear Descriptor Systems

Controller design and analysis for singularly perturbed switched systems with actuator saturation

Sampled-data H ∞ filtering for Markovian jump singularly perturbed systems with time-varying delay and missing measurements

Observer-based event-triggered control for singularly perturbed systems with saturating actuator