The robust D-stability analysis of uncertain discrete-delay descriptor systems via genetic algorithms

The problem of assigning all poles of a closed-loop system in a specified disk by dynamic output feedback is considered for discrete systems with multiple non-commensurate time delays. This kind of pole assignment problem is called D-pole placement, and its relation to robustness properties is discussed. Thus, the robust D-pole placement is subsequently proposed and a robust criterion is derived to ensure D-stability of the discrete multiple time-delay systems subject to parametric uncertainties. Furthermore, with this robust criterion, the bounds can also be estimated for allowable parametric uncertainties that guarantee all the poles of the discrete uncertain multiple time-delay systems to remain inside the specified disk. Finally, an example is given to illustrate obtained results. >

The reachability and controllability of descriptor systems is studied. Since discrete time descriptor systems are subject to certain consistency conditions in order for a solution to exist, we consider the problem of constructing an admissible input that can drive the system in any reachable state (or the origin) and that also satisfies the consistency conditions. This is done by first decomposing the system into its two subsystems, via the Weierstrass decomposition. Furthermore, algebraic criteria regarding the reachability and controllability of a descriptor system that involve its causal and non-causal subsystems are formulated.

It is well known the descriptor systems have a wide application field. Usually it appear as a mathematical model of a real problem, mainly the model that involves the input output relationship. It is well known that a descriptor linear system has an unique solution if the pencil matrix of the system is regular. However, there are some systems that are not regular. Moreover, even though the system is regular the solution can contain the noncausal behavior. Therefore, it is necessary to normalize the descriptor system so as it has well behavior. In this paper, we propose a feedback to normalize a discrete descriptor system under disturbance. Furthermore, we establish a sufficient condition in order for the discrete descriptor system under disturbance can be normalized positively.

In this paper, we have investigated the stability and decaying rate testing problems of grey systems with time delay. Both continuous and discrete time-delay grey systems are discussed whose state matrices are interval matrices. In terms of the Gersgorin theorem, several new delay-independent criteria have been developed to preserve the system stability. By these conditions, new delay-dependent criteria are also presented in order to guarantee that the above-mentioned systems are stable with a specified decay rate. As far as we are aware, this paper seems to be the first to discuss robust stability problems for continuous and discrete time-delay grey systems. Finally, the applications of the results obtained have been illustrated by given numerical examples. We believe that the proposed schemes are applicable to robust control design.

A generalized pseudo-inverse approach to discrete-time finite horizon linear quadratic control of descriptor systems with final state constraint is considered. Necessary and sufficient conditions for existence of a unique optimal solution are provided. Both the test for existence of a solution, as well its computation, decompose hierarchically to problems of small size defined on subdivision of the time axis, thus rendering computational feasibility for problems too large to be handled directly.

Decoupled distributed Kalman fuser for descriptor systems

The problem of iterative learning control is considered for linear discrete-time systems with multiple time delays. In this paper, it is assumed that the initial condition for discrete time-delay systems is unknown at each iteration. For such a class of discrete time-delay systems, a method is presented whereby a class of P-type discrete iterative learning control schemes can be constructed. It is shown that under some given conditions, the constructed iterative learning controllers can guarantee the asymptotic convergence of the output error between the given desired output and the actual output through the iterative learning process. Finally, a numerical example is given to demonstrate the validity of the results.

Optimal D-stable control for discrete multiple time-delay systems with parametric uncertainties

It is well known that the modeling issues in Dynamical systems in chemical processes, networked control systems and automation systems are very complex and model may contain subsystems with different delays. Because of the mathematical complexity, the robust control of these types of multiple time delay system is a challenging one. The problem becomes more difficult and complex when the system is having time delays in both the denominator and the numerator. In this paper a new method is proposed for approximating general multiple time delay continuous-time transfer function system with multiple time delays in both the denominator and the numerator to a single input/output delay. Both Single Input Single Output(SISO) and Multi Input Multi Output(MIMO) cases have been considered. In this proposed methodology, the Approximated Generalized Time Moments (AGTM)/ Approximated Generalized Markov Parameter (AGMP) methods are used to transform the multiple time delay system into a single input/output delay system. A Linear Matrix Inequality (LMI) approach is developed for synthesizing robust H∞ control of the approximated time delay system. The efficacy of the proposed method is shown through illustrative examples. It is evident from the simulation results that the approximated system is stable with the proposed LMI H∞ controller.

By the modem time series analysis method, based on the autoregressive moving average (ARMA) innovation models and white noise estimation theory, using the optimal fusion rule weighted by diagonal matrices, a distributed descriptor Wiener state fuser is presented by weighting the local Wiener state estimators for the linear discrete stochastic descriptor systems with multisensor. It realizes a decoupled fusion estimation for state components. In order to compute the optimal weights, the formulas of computing the cross-covariances among local estimation errors are presented based on cross-covariances among the local innovation processes, input white noise, and measurement white noises. It can handle the fused filtering, smoothing, and prediction problems in a unified framework. Its accuracy is higher than that of each local estimator. A Monte Carlo simulation example shows its effectiveness and correctness.

A nonlinear observer is considered for a class of discrete nonlinear descriptor systems subject, to unknown inputs and faults. This class is partly characterized by globally Lipschitz nonlinearities, and a member system may be singular and possibly non-causal. The structure proposed for the observer makes it useful for both state estimation for feedback controls and residual generation for fault detection. Results on the existence of solutions are given and some useful bounds are derived which are important in the observer design. The proposed nonlinear observer is based on a transformed system and on the solution of a Riccati equation. The existence, convergence properties and robustness of the observer are investigated by the use of a quadratic Lyapunov function. A design algorithm is given and applied to the estimation of the states of a flexible joint robot.

A Constrained Optimization Approach for Robust D-Stability and Robust H 2 Performance Design

WAMS (Wide-Area Measurement Systems) involve all kinds of communicating channels. Different channels have different time delays and randomicity of time-delay. So closed-loop power systems ought to be modeled as nonlinear dynamic systems with multiple time delays and varying time delays. A method is proposed in this paper to design a controller that can adapt to feedback signals with multiple time delays and varying time delays. The method is based on the topological characteristics of stable time-delay regions, the LMIs (Linear Matrix Inequalities) method, GA (Genetic Algorithm) and the balanced truncation model-reduced algorithm. Numerical results form a tow-area four-machine power system demonstrate that the controller derived from the above method can damp low-frequency oscillations in the system for the feedback signals with multiple time delays and varying time delays.

Model reduction of discrete time-delay systems based on Charlier polynomials and high-order Krylov subspaces

The robust stability property of linear discrete time-delay singular systems with structured parametric uncertainties is considered. Under the assumptions that the nominal time-delay singular systems are regular, causal and stable, necessary and sufficient conditions based on μ-analysis and guardian map theory are obtained to guarantee the properties by transforming the problem into checking the nonsingularity of a class of uncertain matrices. A tight bound is obtained in terms of μ for keeping the discrete time-delay singular systems regular, causal and stable. The results can be easily extended to deal with robust D-stability problem. Two examples are given to show the feasibility of the approach.