Multidimensional Linear Discrete System Stability Analysis Using Single Square Matrix

The anisotropic norm of the linear discrete time invariant system is the measure of the system output sensitivity to a random input with the mean anisotropy bounded by some nonnegative value a. The mean anisotropy characterizes the degree of predictability of the stochastic signal. The anisotropic norm of a system is the induced norm with the extreme cases of H2 norm and H∞ norm, respectively, under a → 0 and a → ∞. The lower bound of the anisotropic norm of the linear discrete time invariant system was established in the present paper using the methods of linear algebra.

The authors present an algorithm for order formulation of linear time invariant discrete systems using a new version of evolutionary computing technique namely, modified particle swarm optimization (MPSO). In the modified PSO, the movement of a particle is governed by three behaviors namely, inertia, cognitive and social. The cognitive behavior helps the particle to remember its previous visited best position. This paper proposes to split the cognitive behavior into two sections. This modification helps the model order formulation with minimum integral squared error. The results are shown in the form of unit step and unit impulse time response curves and compared with Genetic algorithm and general PSO based optimization techniques. The proposed method is illustrated through numerical example from literature.

: This report considers a feedback control law for linear time-varying and time invariant discrete systems based on a receding horizon concept applied to a minimum energy problem with fixed terminal constraints. The control law is shown to be asymptotically stable and to result in a new method for stabilizing linear time-varying systems as well as extending some well known methods for stabilizing time invariant systems. In particular, the stabilizing gains of the feedback control law are obtained from the solution to a discrete Riccati equation over an arbitrary finite time interval, which is relatively easy to compute. The gain matrix reduces to a constant matrix for linear time invariant systems. Some stability results will turn out to be special cases of these results. The results parallel those for linear continuous time systems, although the technical details are tedious and more involved.

In this paper we consider the problem of stabilization of a single input linear discrete time invariant system where the control can take values from a countable set. The problem definition includes the design of a quantization scheme that will produce the coarsest such set while guaranteeing stability. This problem is cast as a static combinatorial resource allocation or coverage problem where in the control values are viewed as resources which have to cover the state space. A relaxed notion of stability is used to represent the coverage cost function. The resulting quantizer on applying Deterministic Annealing algorithm is logarithmic and explicitly gives the partition of the state space.

On Lower Bound of Anisotropic Norm

Another approach for generating reduced order discrete time system using composite method is being proposed. The proposed method is validated by considering a Linear Time Invariant Discrete System (LTIDS) having single input single output(SISO) of higher order being represented in frequency domain. Eigen Spectrum Analysis and Least Squares moment matching methods form the proposed composite method to obtain the reduced numerator and denominator polynomials. The suitability of the proposed method is judged by comparing the step, impulse responses of the original and reduced systems. The performances of the systems are measured in terms of Steady State Error (SSE). It is seen from the numerical examples that the results are quite competitive as compared to other methods thus justifying the applicability of the proposed method.

— In this paper, an effective procedure to determine the reduced order model of higher order linear time invariant discrete systems is discussed. A new procedure has been proposed for evaluating Time moments of the original high order system. Numerator and denominator polynomials of reduced order model are obtained by considering first few redefined time moments of the original high order system. The proposed method has been verified using numerical examples. Keywords- order reduction; eigen values; large scale discrete systems; modeling. I.

The problem of intermittent fault (IF) detection is studied for a class of linear discrete-time stochastic systems with unknown disturbance under the influence of a class of networked phenomena by using the moving horizon estimation (MHE). Aiming to normalise the transmission of data in network systems, the Round-Robin (RR) protocol is introduced for linear discrete-time stochastic systems, so the system is converted to a linear periodic system. By using the lifting technique and superposition principle, the linear periodic discrete-time system is transformed into a linear discrete time-invariant system. An MHE strategy is developed to cope with unknown disturbance by introducing provisional estimates of external unknown disturbances. By utilizing the moving horizon estimator, a scalar residual decoupled from the unknown disturbance is designed. Finally, the effectiveness of the proposed method is verified by a simulation example.

There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

Pole assignment for linear systems by periodic output feedback

This article shows a synthesis of an adaptive controller for an unknown discrete linear time invariant system with a delayed and lossy feedback. The controller identifies system parameters by an intermittent gradient estimator, and provides a tracking control by using the identified parameters.

Some problems associated with the disturbed matrix Lyapunov equation for linear time-invariant discrete systems

This paper introduces the notion of absolutely distinguishable discrete dynamic systems, with particular applicability to linear time-invariant (LTI) systems. The motivation for this novel type of distinguishability stems, in particular, from the stability and performance requirements of worst-case adaptive control methodologies. The main results presented herein show that, in most practical cases, a persistence of excitation type of condition and a minimum number of iterations are required to properly distinguish two dynamic systems. We also demonstrate that the former constraint can be written as a lower bound on the intensity of the exogenous disturbances. The applicability of the developed theory is illustrated with a set of examples.

Many different types of adaptive or universal controllers", capable of dealing with a very broad range of linear time-invariant systems, have been proposed. One class of such controllers uses switching between a finite, or at least countable, number of fixed controllers until stability is detected. Such controllers are very attractive from a theoretical viewpoint, in providing stabilization and asymptotic performance for a broad class of plants. However, such controllers are also known to have very poor transient properties, due to the long time required to search for a stabilizing feedback. The key contribution of this paper is to introduce the "method of localization" which can greatly improve the speed of the search. The method is described for linear time-invariant discrete time systems of known nominal order, with disturbances and noise present. Analysis and simulations demonstrate the potential for greatly improved transient performance.

It is well known that the Kalman filter is globally optimal for linear time-invariant systems excited with white Gaussian noise along the process and measurement channels. However, its application to practical situations is often suboptimal because the noise covariance matrices typically are not accurately known. This paper presents a new covariance matching Kalman filter (CMKF) algorithm which estimates the measurement (R) or the process (Q) noise covariance matrix in addition to the state. The convergence of the filter for estimating Q or R matrices respectively is proved for discrete stochastic linear time-invariant systems under relatively mild conditions of observability.