Restricted feedback control in discrete-time dynamical systems with memory.

We prove the existence of the chaotic behavior in dynamical systems generated by some class of time periodic nonautonomous equations on the plane. We use topological methods based on the Lefschetz Fixed Point Theorem and the Ważewski Retract Theorem.

Output stabilizability conditions for some classes of discrete-time linear and nonlinear control systems are formulated by using dynamic controllers. Methods for constructing static and dynamic controllers that provide a specified evaluation of the weighted damping level of external and initial disturbances are proposed for systems with controllable and observable outputs. The implementation of these methods with the use of static state feedback or complete order dynamic controllers is based on the solution of systems of linear matrix inequalities. The example of a controller synthesis for a two-mass mechanical system is given.

The problem of dynamic output feedback equivalence of nonlinear observable discrete-time control systems with outputs has been studied by Pawluszewics-Bartosiewics. In this paper we use those results to achieve equivalence of a nonlinear system to a linear one. In this way we obtain an algebraic criterion of dynamic output linearization for discrete-time systems. We recall the concept of anticipating output equivalence, where transformations of outputs and controls depend on future values, and the concept of dynamic output equivalence, depending on past and present controls and outputs values. In order to give the criterion of linearisation of a discrete-time control system we have to use the difference output algebra and the difference output universe associated with the given system.

This paper focuses on the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> state feedback control method for networked control systems and its application to the discrete-time direct-drive linear motion control system over a communication network with external disturbances and network-induced random time delay. The unavoidable network-induced random delay is modeled by a Markov chain with the uncertain transition probability. In this paper, the experimental linear motion control system is a double-sided linear switched reluctance machine (DLSRM). To tackle the external disturbances in the DLSRM, an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control strategy is designed by using Lyapunov stability theory and linear matrix inequalities technique. A mode-dependent state feedback H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> controller is designed for the closed-loop discrete-time direct-drive linear motion control system over a network to achieve a high-precision position control performance with antidisturbance capability. The experimental tests on the networked DLSRM system are presented to verify the effectiveness and practicability of the proposed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control method for industrial applications of networked control systems.

We establish conjugation notion in discrete-time systems, first introduced into the H ∞ control theory of continuous-time systems by Kimura (1989). In discrete-time systems, conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle. With the aid of J-lossless conjugation conjugation by a J.lossless system), it is shown that the parametrization of sub-optimal solutions of H ∞ model-matching problems is reduced to a Lyapunov-type equation. The parametrization of all solutions is given in an extremely simple way. It is further proved that the J-lossless conjugation of the H ∞ model-matching problem is a natural state-space representation of classical interpolation in discrete-time systems.

First, a function projective synchronization is defined in discrete-time dynamical systems, in which the drive and response state vectors evolve in a proportional scaling function matrix. Second, based on backstepping design with three controllers, a systematic, concrete and automatic scheme is developed to investigate the function projective synchronization of discrete-time chaotic systems. With the aid of symbolic-numeric computation, we use the proposed scheme to illustrate the function projective synchronization between the 2D Lorenz discrete-time system and the Fold discrete-time system, as well as between the 3D hyperchaotic Rössler discrete-time system and the Hénon-like map. Numeric simulations are used to verify the effectiveness of our scheme. By choosing different scaling functions, the interesting attractor figures of the drive and response systems are showed in a proportional scaling function.

In this paper, first, the design method of the discrete-time position control system of a body supported elastically is showed. In the design of the discrete-time position control system, the pulse transfer function of the body supported elastically and driven by a linear DC motor is derived by the z-transform, and the discrete-time control voltage of the linear DC motor is obtained by applying the inverse dynamics or the neuro-adaptive control in consideration of the delay time due to the computation time of the control digital computer. Then, from the experimental results, the usefulness of the present design method of the discrete-time position control system is confirmed, and the control performance of both control methods in regard to disturbance is compared. Furthermore, from the numerical simulation results, the effects of the sampling time on the control performance of the neuro-adaptive control is demonstrated.

In this paper we propose definitions of equivalence via stochastic bisimulation and of equivalence of stochastic external behavior for the class of discrete-time stochastic nonlinear control systems with possibly degenerate distributed disturbances. The notions are inspired by the analogue notions that exist for probabilistic chains and for behavioral systems, respectively. Connections between the two notions and also with stochastic reachability are discussed. For the subclass of discrete-time stochastic linear control systems with multivariate possibly degenerate normal distributed disturbances, geometric necessary and sufficient conditions for checking stochastic bisimulation equivalence and stochastic external behavior equivalence are derived. The proposed results nicely extend the theory of bisimulation equivalence and equivalence of external behavior known for non-deterministic control systems to a stochastic setting.

The paper focuses on unification of the study of continuous- and discrete-time nonlinear control systems using the mathematical tools of pseudo-linear algebra. The realization problem is studied through the language of differential forms. Necessary and sufficient conditions are given for the existence of the state space realization of the nonlinear input-output equation defined in terms of the pseudo-linear operator. The sufficiency part of the proof gives a constructive procedure (up to integration of the one-forms) for obtaining the observable state equations. The special cases for continuous- and discrete-time systems, described either in terms of the shift or difference operator, follow from the general result as the special cases. Moreover, the result accommodates also the systems described via q-shift or q-difference operators.

To estimate the growth rate of matrix products $A_{n}··· A_{1}$ with factors from some set of matrices $\mathscr{A}$, such numeric quantities as the joint spectral radius $ρ(\mathscr{A})$ and the lower spectral radius $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality $ρ(\mathscr{A})&lt;1$ serves as a criterion for the stability of a system, and the inequality $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})&lt;1 $ as a criterion for stabilizability.Given a set $\mathscr{A}$ of $N×M$ matrices and a set $\mathscr{B}$ of $M×N$ matrices. Then, for matrix products $A_{n}B_{n}··· A_{1}B_{1}$ with factors $A_{i}∈\mathscr{A}$ and $B_{i}∈\mathscr{B}$, we introduce the quantities $μ(\mathscr{A},\mathscr{B})$ and $η(\mathscr{A},\mathscr{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathscr{A},\mathscr{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}··· A_{1}B_{1}$ over all sets of matrices $A_{i}∈\mathscr{A}$ and the minimal growth rate over all sets of matrices $B_{i}∈\mathscr{B}$. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

Design of relay control for discrete time systems where the control is assumed to take values from a finite set of constant vectors is proposed in this paper. The stabilizability property of the plant is taken advantage of in designing a relay control which emulates locally the behaviour of a state feedback gain. A constructive method to compute the switching surfaces for the relay control and the domain of attraction is provided using linear matrix inequalities. The proposed relay feedback control design algorithm is shown to be applicable to linear parameter varying plants.

In this paper we consider the reaching law approach to the quasi-sliding mode control of discrete time systems. We analyze one of the most popular reaching laws which involves constant plus proportional decrease rate of the switching variable, and we propose a new version of this reaching law with the proportional part appropriately scaled. In our work the proportional part is scaled by a factor which depends on the current value of the switching variable. The reaching law introduced in this paper ensures quasi-sliding motion defined as remaining of the system representative point (state) in a band around the sliding hyperplane and crossing the hyperplane in each successive control period. The proposed reaching law guarantees faster convergence and offers better robustness with respect to external disturbances and modeling uncertainty than the previously presented reaching law characterized by the constant plus directly proportional decrease rate of the switching variable. Moreover, for a practically important class of time delay systems considered in the paper our reaching law ensures bounded control signal, which cannot be achieved with the reaching law proposed earlier.

In this paper we study the reaching law approach to the sliding mode control of discrete time linear dynamic systems. We propose a new non-switching type reaching law which determines the sliding variable decrease rate adjusted according to the current value of this variable. The reaching law introduced in this paper results in the quasi-sliding motion understood as remaining of the system state in a predefined vicinity of (band around) the sliding hyperplane. The reaching law does not cause chattering, i.e. it does not make the system state cross the hyperplane during the control process. The proposed reaching law guarantees fast convergence and offers good robustness with respect to modeling uncertainty and external disturbances.

The paper is devoted to the application of fractional calculus concepts to modeling, identification and control of discrete-time systems. Fractional difference equations (FAE) models are presented and their use in identification, state estimation and control context is discussed. The fractional difference state-space model is proposed for that purpose. For such a model stability conditions are given. A fractional Kalman filter (FKF) for this model is recalled. The proposed state-space model and fractional order difference equation are used in an identification procedure which produces very accurate results. Finally, the state-space model is used in closed-loop state feedback control form together with FKF as a state estimator. The latter is also given in an adaptive form together with FKF and a modification of recursive least squares (RLS) algorithm as a parameters identification procedure. All the algorithms presented were tested in simulations and the example results are given in the paper

This paper investigates the problem of Q–S synchronization for different dimensional chaotic dynamical systems in discrete-time. Based on two control laws and stability theory of dynamical systems in discrete-time, new synchronization schemes are derived. Numerical examples demonstrate the effectiveness and feasibility of the proposed control techniques.