On the computation of the transfer function matrix of singular systems

For singular systems, i.e. for systems of the form E[xdot] = Ax + Bu, with E singular, the problem of computing the transfer function matrix has been studied. An algorithm is developed which is similar to the corresponding algorithm proposed by Faddeev or Leverrier for regular systems. The present results involve the Drazin inverse and yield an expression for the transfer function matrix suitable for computer use.     

Delay-dependent H∞ filtering for discrete singular Markov jump systems with Wiener process and partly unknown transition probabilities

This paper is devoted to the investigation of the delay-dependent H_∞ filtering problem for a class of discrete-time singular Markov jump systems with Wiener process and partly unknown transition probabilities. The class of stochastic singular model under consideration is more general and covers the stochastic singular Markov jump time-varying delay systems with completely known and completely unknown transition probabilities as two special cases. Firstly, based on a stochastic Lyapunov–Krasovskii candidate function and an auxiliary vector function, by employing some appropriate free-weighting matrices, the discretized Jensen inequality and combining them with the structural characteristics of the filtering error system, a set of delay-dependent sufficient conditions are established, which ensure that the filtering error system is stochastically admissible. And then, a singular filter is designed such that the filtering error system is not only regular, causal and stochastically stable, but also satisfy a prescribed H_∞ performance for all time-varying delays no larger than a given upper bound. Furthermore, the sufficient conditions for the solvability of the H_∞ filtering problem are obtained in terms of a new type of Lyapunov–Krasovskii candidate function and a set of linear matrix inequalities. Finally, simulation examples are presented to illustrate the effectiveness of the proposed method in the paper.     

Observer-based control for singular nonhomogeneous Markov jump systems with packet losses

This paper is concerned with the observer-based H_∞ control for a class of singular Markov jump systems over a finite-time interval, where the transition probability (TP) is time-varying and is limited to a convex hull. Due to the limited capacity of network medium, packet losses are presented in the underlying systems. Firstly, using a stochastic Lyapunov functional, a sufficient condition on singular stochastic H_∞ finite-time boundedness for the corresponding closed-loop error systems is provided. Subsequently, a linear matrix inequality (LMI) condition on the existence of the H_∞ observer-based controller is developed from a new perspective. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed controller design method, wherein it is shown that the proposed method yields less conservative results than those in the literature.     

Dynamic output feedback H∞ control for fractional-order linear uncertain systems with actuator faults

This paper is concerned with the problem of robust fault-tolerant H_∞ dynamic output feedback control for fractional-order linear uncertain systems with the order satisfying 0 < α < 1 in the presence of actuator faults. A new linear matrix inequality (LMI) formulation corresponding to the H_∞ norm of fractional-order linear systems is proposed. Based on the new formulation and by introducing a new linearizing change of variables, sufficient conditions for robust fault-tolerant H_∞ dynamic output feedback controller designs are derived in term of LMIs. Furthermore, the proposed controller not only enables the system to keep robust stabilization, but also achieves a better H_∞ performance compared with the existing methods. Numerical examples are given to illustrate the design procedure and its effectiveness.     

H∞ control of singular time-delay systems via discretized Lyapunov functional

This paper addresses the problem of robust H_∞ control for uncertain continuous time singular systems with state delays. A new singular-type complete quadratic Lyapunov-Krasovskii functional (LKF) is introduced, which combines with the discretization LKF method to synthesis problems. An improved bounded real lemma (BRL) is presented to ensure the system to be regular, impulse free and stable with H_∞ performance condition. Based on the BRL, a memoryless state feedback controller is designed via linear matrix inequalities (LMIs), which greatly reduces the disturbance attenuation level. Numerical examples are given to illustrate improvements over some existing results.     

P-type iterative learning control algorithm for a class of linear singular impulsive systems

This paper is focused on the iterative learning control problem for linear singular impulsive systems. For the purpose of tracking the desired output trajectory, a P-type iterative learning control algorithm is investigated for such system. Based on the fundamental property of singular impulsive systems and the restricted equivalent transformation theory of singular systems, the convergence conditions of the tracking errors for the system are obtained in the sense of λ norm. Finally, the validation of the algorithm is confirmed by a numerical example.     

A useful identity in circuit theory

A useful identity expressing the derivative of an unknown variable x_k of X with respect to an entry in the coefficient matrix of a linear system AX = B is presented. If the derivatives of x_k with respect to each entry of A or their combinations are required, then the identity avoids the repeated solution of the linear equations, and may result in a symbolic solution provided A^-1 is known. A method to measure the sensitivity in an n-port linear system is proposed, and its relationships to Kron's method of tearing and Branin's formulae are also discussed.     

H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function

In this paper, the H_∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H_∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H_∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method.     

Design of fault diagnosis filters: A multi-objective approach

In this paper, fault detection and isolation (FDI) in linear uncertain dynamic systems is addressed. The main contributions consist of the formulation of the FDI problem as a filter-based multi-objective optimization problem and the study of the tools used for the solution. The design objectives are formulated in terms of H_∞, H_- and generalized H₂ performance specifications as well as regional constraints on filter poles. The problem is solved using linear matrix inequality (LMI) and generalized structured singular value (μ_g) techniques. Special design features are illustrated through a simulation example and experimental results from a controlled hydraulic process are provided to demonstrate the potential of the proposed procedure.     

An event-triggered asynchronous H∞ filtering for singular Markov jump systems with redundant channels

In this paper, the problem of asynchronous H_∞ filtering for singular Markov jump systems with redundant channels under the event-triggered scheme is studied. In order to save the resource of bandwidth limited network and improve quality of data transmission, we utilize event-triggered scheme and employ redundant channels. The redundant channels are modeled as two mutually independent Bernoulli distributed random variables. To formulate the asynchronization phenomena between the system modes and the filter modes, the hidden Markov model is proposed so that the filtering error system has become a singular hidden Markov jump system. The criterion of regular, causal and stochastically stable with a certain H_∞ performance for the filtering error system has been obtained. The co-design of asynchronous filter and the event-triggered scheme is proposed in terms of a group of feasible linear matrix inequalities. Two examples are given to show the effectiveness of the proposed method.     

Finite-time H∞ control for a class of discrete-time nonlinear singular systems

This paper investigates the finite-time control problems for a class of discrete-time nonlinear singular systems via state undecomposed method. Firstly, the finite-time stabilization problem is discussed for the system under state feedback, and a finite-time stabilization controller is obtained. Then, based on which, the finite-time H_∞ boundedness problem is studied for the system with exogenous disturbances. Finally, an example of population distribution model is presented to illustrate the validity of the proposed controller. Because there is no any constraint for singular matrix E in the paper, controllers can be designed for more discrete-time nonlinear singular systems.     

Mean-square exponential stability for a class of discrete-time nonlinear singular Markovian jump systems with time-varying delay

This paper investigates the problem of mean-square exponential stability for a class of discrete-time nonlinear singular Markovian jump systems with time-varying delay. The considered systems are with mode-dependent singular matrices _Er(k)$E_{r\left(k\right)}$. By using the free-weighting matrix method and the Lyapunov functional method, delay-dependent sufficient conditions which guarantee the considered systems to be mean-square exponentially stable are presented. Finally, some numerical examples are employed to demonstrate the effectiveness of the proposed methods.     

Topological interpretation of the inverse connection matrix A.sb

This paper discusses a new algebraic model of electrical networks by which a deeper understanding is achieved of the interrelation existing between a system of independent cut sets and the corresponding system of independent “contours”, in the sense of orthogonal network theory (1).The new model allows, by a redefinition of f-cut sets, to give an interpretation in terms of linear graph theory to the inverse connection matrix A^.s_b, i.e. to the transformation matrix between voltages in the orthogonal and the primitive reference frames, for a general choice of independent contours.Furthermore, a directly algebraic derivation of the voltage transformation is presented, recalling the properties of dual vector spaces and without invoking the principle of invariance of the instantaneous power.     

Study on stability and stabilizability of discrete-time mean-field stochastic systems

This paper is to study the mean square stabilizability and regional stability of discrete-time mean-field stochastic systems. Firstly, a necessary and sufficient condition is presented via the spectrum of linear operator to illustrate the stabilizability of discrete-time mean-field stochastic systems. B(0, γ)-stabilizability is introduced and transformed into solving linear matrix inequalities (LMIs). Secondly, B_M-stability is characterized, especially, the stabilities of circular region, sector region and annulus regions are discussed extensively. Finally, as applications, it is shown that B(0, γ₁; γ₂)-stability has close relationship with the decay rate of the system state response and the Lyapunov exponent.     

Inverse control of discrete-time multivariable systems

The inverse control of discrete-time single-input/single-output linear systems is extended to multivariable systems. The extension considered is far from being trivial when a multivariable system possesses interactions and multiple delays, leading to ill-conditioning or even singularity of the problem. A feedback control algorithm suggested is based on decoupling and compensating the plant by preceding it with the inverse controller. The inverse controller is derived by using the regularization of the pseudoinverse of the plant convolution matrix. In this way a regularized multivariable control with a minimum norm is provided. A numerical example of the inverse control of a distillation column is presented.     

Delay-dependent stochastic stability and H∞ analysis for time-delay systems with Markovian jumping parameters

This paper studies the problem of the stochastic stability and H_∞ disturbance attenuation for linear continuous-time time-delay systems that possess randomly Markovian jumping parameters. A delay-dependent sufficient condition on the stochastic stability with given H_∞ performance is proposed using the stochastic Lyapunov–Krasovskii stability theory. The conditions are formulated as a set of coupled linear matrix inequalities.     

Design of mixed H2-dissipative observers with stochastic resilience for discrete-time nonlinear systems

A linear matrix inequality based mixed H₂-dissipative type state observer design approach is presented for smooth discrete time nonlinear systems with finite energy disturbances. This observer is designed to maintain H₂ type estimation error performance together with either H_∞ or a passivity type disturbance reduction performance in case of randomly varying perturbations in its gain. A linear matrix inequality is used at each time instant to find the time-varying gain of the observer. Simulation studies are included to explore the performance in comparison to the extended Kalman filter and a previously proposed constant gain observer counterpart.