Linearization of discrete-time systems by exogenous dynamic feedback

It is shown that a discrete-time system may be linearizable by exogenous dynamic feedback, even if it cannot be linearized by endogenous feedback. This property is completely unexpected and constitutes a fundamental difference with respect to the continuous-time case. The notion of exogenous linearizing output is introduced. It is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability. Necessary and sufficient conditions for the existence of an exogenous linearizing output are provided. The results of the paper are obtained using transformal operator matrices. The properties of such operators are studied. The theory is applied to the exact discrete-time model of a mobile robot, showing that the above-mentioned property concerns not only academic examples, but also physical systems.     

On H∞-control of discrete-time nonlinear systems

Following recent works on continuous-time nonlinear H_∞-control, where connections with game theory and passivity conditions have been set, the present paper studies the corresponding problem for discrete-time systems. The paper describes sufficient conditions for the existence and the construction of a feedback law which imposes a prescribed level of disturbance attenuation with internal stability. Both cases of state feedback and measurement feedback are considered.     

The effect of sampling on linear equivalence and feedback linearization

We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback. For n = 2, we show, under the assumption of completeness of ad_FG, that if the discretized system is lineariable by state coordinate change and feedback, then the continuous time affine complete analytic system is linearizable by state coordinate change only. Also, we suggest a method of proof when n ≥ 3.     

On condition/event systems with discrete state realizations

This paper introduces condition/event (C/E) systems as a class of continuous-time discrete event dynamic systems (DEDS) with two types of discrete-valued input and output signals:condition signals andevent signals. In applications such as discrete control, C/E systems provide an intuitive continuous-time modeling framework amenable to block diagram representation. In this paper we consider C/E systems with discrete state realizations, and study the relationship between continuous-time C/E systems and untimed models of their sequential inputoutput behavior called C/E languages. We show that C/E systems with discrete state realizations are necessarilytime-change invariant (Theorem 3.1), which means the ensemble of admissible continuous-time input-output behaviors is completely characterized by the C/E language for the system (Theorem 4.1). It is also shown that deterministic C/E systems with discrete state realizations are necessarily discrete-time (clocked) systems (Corollary 3.1), and that finite discrete state realizations exist for a C/E system only if its related C/E language has a finite state generator (Theorem 4.2). Finally, we develop equivalent discrete-state realizations for C/E systems resulting from cascade and feedback interconnections. The paper concludes with a discussion of several directions for future research.Please direct correspondence concerning this paper to B.H. Krogh at the above address.     

Adaptive control of linearizable systems

The authors give some initial results on the adaptive control of minimum-phase nonlinear systems which are exactly input-output linearizable by state feedback. Parameter adaptation is used as a technique to make robust the exact cancellation of nonlinear terms, which is called for in the linearization technique. The application of the adaptive technique to control of robot manipulators is discussed. Only the continuous-time case is considered; extensions to the discrete-time and sampled-data cases are not obvious     

Discretization and event triggered digital output feedback control of LPV systems

This paper investigates the problem of discretization and digital output feedback control design for continuous-time linear parameter-varying (LPV) systems subject to a time-varying networked-induced delay. The proposed discretization procedure converts a continuous-time LPV system into an equivalent discrete-time LPV system based on an extension of the Taylor series expansion and using an event-based sampling. The scheduling parameters are continuously measured and modeled as piecewise constant. A new transmission of the measured output to the controller is triggered by significant changes in the parameters, yielding time-varying transmission intervals. The obtained discretized model has matrices with polynomial dependence on the time-varying parameters and an additive norm-bounded term representing the discretization residual error. A two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the LPV model. The conditions can also be used to provide robust (i.e., independent of the scheduling parameter) static output feedback controllers. The viability of the proposed design method is illustrated through numerical examples.     

Solution to the inverse regulator problem for discrete-time systems

A complete solution is given for the inverse problem of LQ optimal regulators for the discrete-time multi-input case. The Kalman equation for discrete-time systems is derived to characterize the optimal regulators. It is then shown that this equation holds for some quadratic weighting if and only if the feedback satisfies a condition corresponding to the circle criterion for continuous-time systems. The optimality criterion given here consists of this condition and an additional condition to ensure detectability. An example is given to illustrate the results.     

Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems

A continuous-time system cannot be recovered solely from its uniformly sampled discrete-time model through the zero-order hold discretization or step-invariant transformation, but our studies indicate that it can be recovered uniquely from its non-uniformly sampled discrete-time model. In this paper, we discuss some related issues of non-uniformly sampled systems, including model derivation, controllability and observability, computation of single-rate models with different sampling periods, reconstruction of continuous-time systems, and parameter identification of non-uniformly sampled discrete-time systems. A numerical example is also given for illustration.     

Distributed robust control of linear multi-agent systems with parameter uncertainties

This article considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighbouring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the H _∞ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust H _∞ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.     

A computational method for simultaneous LQ optimal control designvia piecewise constant output feedback

The paper is concerned with simultaneous linear-quadratic (LQ) optimal control design for a set of LTI systems via piecewise constant output feedback. First, the discrete-time simultaneous LQ optimal control design problem is reduced to solving a set of coupled matrix inequalities and an iterative LMI algorithm is presented to compute the feedback gain. Then, simultaneous stabilization and simultaneous LQ optimal control design of a set of LTI continuous-time systems are considered via periodic piecewise constant feedback gain. It is shown that the design of a periodic piecewise constant feedback gain simultaneously minimizing a set of given continuous-time performance indexes can be reduced to that of a constant feedback gain minimizing a set of equivalent discrete-time performance indexes. Explicit formulas for computing the equivalent discrete-time systems and performance indexes are derived. Examples are used to demonstrate the effectiveness of the proposed method.     

Quadratic stabilizability of switched linear systems with polytopic uncertainties

In this paper, we consider quadratic stabilizability via state feedback for both continuous-time and discrete-time switched linear systems that are composed of polytopic uncertain subsystems. By state feedback, we mean that the switchings among subsystems are dependent on system states. For continuous-time switched linear systems, we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices, then the switched system is quadratically stabilizable via state feedback. For discrete-time switched linear systems, we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non-negative scalars and a common positive definite matrix. For both continuous-time and discrete-time switched systems, we propose the switching rules by using the obtained common positive definite matrix.     

Normal forms for multi‐input flat systems of minimal differential weight

We present normal forms for nonlinear control systems that are the closest to static feedback linearizable ones, that is, for systems that become feedback linearizable via the simplest dynamic feedback, which is the one‐fold prolongation of a suitably chosen control. They form a particular class of flat systems, namely those of differential weight n + m + 1, where n is the number of states and m is the number of controls. We also show that the dynamic feedback may create singularities in the control space depending on the state and we discuss them. We also address the issue of the normalization of the system only versus that of the system together with a flat output. Finally, we illustrate our results by several examples.     

Orbital feedback linearization for multi‐input control systems

A nonlinear control system is said to be orbital feedback linearizable if there exist an invertible static feedback and a change of time scale (depending on the state) which transform the system into a linear system. We give geometric necessary and sufficient conditions describing multi‐input control‐affine systems that are orbital feedback linearizable out of equilibria and in the case of equal controllability indices. We also describe a construction of the time rescaling needed to orbitally linearize the system. Moreover, we analyze close relations between orbital feedback linearizable control‐affine systems and control‐linear systems that are feedback equivalent to a multi‐chained form comparing geometric structures corresponding to both problems. We illustrate our results by two examples, one being a rigid bar moving in .Copyright © 2014 John Wiley & Sons, Ltd.     

Stability analysis of sampled-data systems via open-/closed-loop characteristic polynomials contraposition

In this paper, a class of sampled-data systems with double feedbacks are scrutinized for internal/external stability through the open-/closed-loop characteristic polynomials contraposition. A sampled-data system with double feedbacks consists of a continuous-time plant and two feedback control loops: a discrete-time one for digital control objectives with sampler and zero-order hold synchronized, and a continuous-time one for analog control performances. Contraposition stability criteria are worked out by exploiting what we call the contraposition return difference relationship in between the open- and closed-loop characteristic polynomials defined via the lifted model. The criteria for asymptotical stability are necessary and sufficient, independent of open-loop poles distribution of the lifted model and locus orientation specification. Moreover, the criteria for L_p-stability are sufficient, while retaining the technical merits of the criteria for asymptotical stability. All criteria present stability conditions in the standard discrete-time sense that are implementable either graphically with loci plotting, or numerically without loci plotting. Examples are included to illustrate the results.     

On zero-order hold equivalents of distributed parameter systems

Various connections are established between linear time-invariant distributed parameter continuous-time systems and their zero-order hold discrete-time equivalents. These connections are established in both the time and frequency domains. The time-domain connections relate various growth constants and norm bounds of the continuous-time systems considered to those of their zero-order hold discrete-time equivalents. The frequency-domain connection provides an upper bound on the difference between the frequency response of a continuous-time system and that of its zero-order hold discrete-time equivalent     

Stabilization of nonlinear discrete-time systems using statedetection

Some recent results concerning stabilization of continuous-time systems using state detection are extended to the case of discrete-time nonlinear systems. It is shown that if a discrete-time locally detectable system can be stabilized by a state feedback law, then it can also be stabilized by a feedback law that depends on the output of a weak detector     

Linearization of discrete-time systems

We characterize the equivalence of single-input single-output discrete-time nonlinear systems to linear ones, via a state-coordinate change and with or without feedback. Four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not; the interdependence of these problems is analyzed. An important feature that distinguishes these discrete-time problems from the corresponding problem in continuous-time is that the state-coordinate transformation is here directly computable as a higher composition of the system and output maps. Finally, certain connections are made with the continuous-time case.     

A multirate digital control via a discrete-time observer for non-linear singularly perturbed continuous-time systems

Based on Euler's methodology, we present a new discretization scheme of two-time-scale non-linear continuous-time systems. This scheme of the discretization consists in using two periods, slow and fast, which are determined with respect to slow and fast variables rate respectively, and allows us to define and then implement a multirate digital control for such systems to cope with this multirate measurements on-line. To this end, two reduced order observers are constructed for the slow and fast subsystems by referring to the knowledge of the left-inversion of the state-to-measurements map problem.     

A perceptron network for functional identification and control ofnonlinear systems

Tracking control of a general class of nonlinear systems using a perceptron neural network (PNN) is presented. The basic structure of the PNN and its training law are first derived. A novel discrete-time control strategy is introduced that employs the PNN for direct online estimation of the required feedforward control input. The developed controller can be applied to both discrete- and continuous-time plants. Unlike most of the existing direct adaptive or learning schemes, the nonlinear plant is not assumed to be feedback linearizable. The stability of the neural controller under ideal conditions and its robust stability to inexact modeling information are rigorously analyzed.