Stabilization via output static feedback in discrete-time linear systems

This paper is concerned with the use of output static feedback in linear time-invariant finite dimensional systems. We consider here the use of an extra relaxation constraint as a mean to obtain efficiently computable (sufficient) conditions for the synthesis of output static stabilizing controllers. We present novel necessary and sufficient conditions for output static feedback stabilizability, that are computable in terms of the solvability of a concave programming problem. We add an extra relaxation constraint to the above conditions, to define a new concept: output static stabilization in the relaxed sense. This (conservative) stabilization concept is fully characterized in terms of the LQR problem. The dependence of the above stabilization concept on the state-space representation of a system is analyzed. We show that for a particular class of plants, the stabilization problem in the above sense can be cast as a convex programming problem. We fully characterize a particular class of plants that are stabilizable in the above sense. A consequence of this characterization is somehow surprising: the identification of a class of plants stabilize via output static feedback, for which stabilizing feedback matrices can be (almost) expressed in analytic closed form. A result which is indeed a generalization of a well-known fact for SISO systems.

Interval observer design for unknown input estimation of linear time-invariant discrete-time systems

Delay-dependent H∞ control stabilization of linear systems with time-varying delays

It is shown that a 2-D linear time-invariant discrete-time system could be bounded-input bounded-output (BIBO) unstable even if the 2-D system, which is implemented by finite word length, is free of overflow oscillations with zero input, according to a previous criterion for the absence of overflow oscillations in 2-D discrete-time systems. An improved criterion is presented to ensure both stability properties.

On Discrete-Time Linear Systems with Almost-Periodic Kernels

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.

Optimal performance of discrete-time control systems based on network-induced delay

The paper presents the algorithm for robustness estimation of large deviations in free motion of linear discrete-time systems. A tracking discrete-time systems with the modal control law are discussed, where the modal control law is designed taking into account the value of delay and the deviation. Two cases of discrete-time systems with control signal delay are considered in the paper. The first case relates to predictable delay of control signal, and the second one relates to unpredictable delay of control signal. It is assumed that parameters of the system are linearly dependent on the uncertainties. The problem is solved with the state space approach and the sensitivity theory methods. An upper bound estimation of trajectory deviations for discrete-time systems is obtained. The estimation contains the condition number of the eigenvectors matrix of the system state matrix. Therefore, sensitivity functions of singular values of the eigenvectors matrix are used to calculate the robustness estimation of the deviations. Based on the obtained equations, an algorithm for the robustness estimation of large deviations in linear discrete-time systems with parametric uncertainties is proposed. The proposed method is compared with an alternative LMI based approach for the peak estimation problem. Advantages and disadvantages of both approaches are illustrated with examples.KeywordsDiscrete-time systemDelayLarge deviationsPeak effectSensitivity analysisUpper boundsRobustnessParametric uncertainty

Simultaneous global external and internal stabilization of linear time-invariant discrete-time systems subject to actuator saturation

OUTPUT FEEDBACK DESIGN BY COUPLED LYAPUNOV-LIKE EQUATIONS

The main idea of the theory of discrete-time single-input single-output linear time-invariant systems (LTI-system) is that every such system has an input-output map that can be represented by the familiar convolution. Such a system is called system of convolution type. It was recently discovered by Boyd and Sandberg, that not all LTI-systems are systems of convolution type. Since not all LTI-systems are of convolution type it would be interesting to characterize the systems of convolution type. In this paper such a characterization is given with the help of the approximation behaviour of the system with respect to FIR-systems. A necessary and sufficient condition is given, such that the system is a system of convolution type. Some applications for the filter design are also described.

In this paper, the finite-horizon near optimal adaptive regulation of linear discrete-time systems with unknown system dynamics is presented in a forward-in-time manner by using adaptive dynamic programming and Q-learning. An adaptive estimator (AE) is introduced to relax the requirement of system dynamics, and it is tuned by using Q-learning. The time-varying solution to the Bellman equation in adaptive dynamic programming is handled by utilizing a time-dependent basis function, while the terminal constraint is incorporated as part of the update law of the AE. The Kalman gain is obtained by using the AE parameters, while the control input is calculated by using AE and the system state vector. Next, to relax the need for state availability, an adaptive observer is proposed so that the linear quadratic regulator design uses the reconstructed states and outputs. For the time-invariant linear discrete-time systems, the closed-loop dynamics becomes non-autonomous and involved but verified by using standard Lyapunov and geometric sequence theory. Effectiveness of the proposed approach is verified by using simulation results. The proposed linear quadratic regulator design for the uncertain linear system requires an initial admissible control input and yields a forward-in-time and online solution without needing value and/or policy iterations. Copyright © 2014 John Wiley & Sons, Ltd.

This paper considers a subclass of hybrid systems, which is the discrete-time linear switching systems whose changes in the dynamics are determined by exogenous, uncontrollable events, and resets are present. In the presence of uncertain parameter and non-symmetrical constrained control, based on the results of nominal linear switching systems, a necessary and sufficient condition to test the robustness of a state feedback controller is given. The candidate state feedback controller for this test is that stabilizing nominal discrete-time linear switching systems. Also, the method in this paper can be extended to the case which has state constraints. At last, a numerical example is presented to illustrate the technique.

Inverse systems of linear systemsThe concept of inverse systems for standard and positive linear systems is introduced. Necessary and sufficient conditions for the existence of the positive inverse system for continuous-time and discrete-time linear systems are established. It is shown that: 1) The inverse system of continuous-time linear system is asymptotically stable if and only if the standard system is asymptotically stable. 2) The inverse system of discrete-time linear system is asymptotically stable if and only if the standard system is unstable. 3) The inverse system of continuous-time and discrete-time linear systems are reachable if and only if the standard systems are reachable. The considerations are illustrated by numerical examples.

In this paper, a new finite-time stability (FTS) concept, which is defined as positive FTS (PFTS), is introduced into discrete-time linear systems. Differently from previous FTS-related papers, the initial state as well as the state trajectory is required to be in the non-negative orthant of the Euclidean space. Some test criteria are established for the PFTS of the unforced system. Then, a sufficient condition is proposed for the design of a state feedback controller such that the closed-loop system is positively finite-time stable. This condition is provided in terms of a series of linear matrix inequalities (LMIs) with some equality constraints. Moreover, the requirement of non-negativity of the controller is considered. Finally, two examples are presented to illustrate the developed theory.