Optimal control of systems with delayed observation sharing patterns via input–output methods

In this paper we present an input–output point of view of certain optimal control problems with constraints on the processing of the measurement data. In particular, we consider norm minimization optimal control problems under the so-called one-step delay observation sharing pattern. We present a Youla parametrization approach that leads to their solution by converting them to nonstandard, yet convex, model matching problems. This conversion is always possible whenever the part of the plant that relates controls to measurements possesses the same structure in its feedthrough term with the one imposed by the observation pattern on the feedthrough term of the controller, i.e., (block-)diagonal. When that is not the case, it amounts to the so-called non-classical information pattern problems. For the case, using loop-shifting ideas, a simple sufficient condition is given under which the problem can be still converted to a convex, model matching problem. We also demonstrate that there are several nontrivial classes of problems satisfying this condition. Finally, we extend these ideas to the case of a N-step delay observation sharing pattern.     

Relationship between poles and zeros of input–output and chain-scattering systems

A system is frequently represented by transfer functions in an input–output characterization. However, such a system (under mild assumptions) can also be represented by transfer functions in a port characterization, frequently referred to as a chain-scattering representation. Due to its cascade properties, the chain-scattering representation is used throughout many fields of engineering. This paper studies the relationship between poles and zeros of input–output and chain-scattering representations of the same system.     

Delay-dependent stability of discrete-time interconnected systems

This paper studies the delay-dependent stability problem of discrete-time interconnected systems with timevarying delays. By using vector Lyapunov function approach and linear matrix inequalities (LMIs), new stability conditions are derived. These results proposed in this paper are all at subsystems level. After comparing with the existing results, it is shown that these conditions are less conservative. A numerical example is presented to demonstrate the effectiveness of the results.     

A delay-partitioning approach to the stability analysis of discrete-time systems

This paper revisits the problem of stability analysis for linear discrete-time systems with time-varying delay in the state. By utilizing the delay partitioning idea, new stability criteria are proposed in terms of linear matrix inequalities (LMIs). These conditions are developed based on a novel Lyapunov functional. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size. We have also established that the conservatism of the conditions is a non-increasing function of the number of partitions. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.     

Input-output-to-state stability for discrete-time systems

This paper presents Lyapunov characterizations of uniform output-to-state stability and uniform input-output-to-state stability (IOSS) (with respect to disturbances) for discrete-time (DT) nonlinear systems. We show the equivalence of the following three properties for DT systems: uniform IOSS, existence of a smooth Lyapunov function for uniform IOSS, and existence of a (state-)norm estimator. This equivalence result is a DT counterpart of Krichman et al. [2001. Input-output-to-state stability. SIAM Journal on Control and Optimization 39, 1874-1928. Theorem 2.4] and a generalization of Jiang and Wang [2002. A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49-58. Theorem 1.1] and Jiand and Wang [2001. Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857-869. Theorem 1, 1⇔4].     

Stabilisation of discrete-time systems via Schur stability region

In this paper, we consider the stabilisation problem of discrete-time systems by affine compensator. Using the known properties of the Schur stability region of monic polynomials, we give conditions under which stabilising controllers exist. A necessary and sufficient condition is obtained when the difference of the order of the system and the number of stabilising parameters is one. In the general case, two solution algorithms: division-elimination and least-square minimisation algorithms are suggested.     

On robustness of constrained discrete-time systems to state measurement errors

In this note we show that robustness with respect to additive disturbances implies robustness with respect to state measurement errors and additive disturbances for a class of discrete-time closed-loop nonlinear systems. The main result is formulated in terms of input-to-state stability and includes the possible presence of input and state constraints. Moreover, the state feedback controllers are allowed to be discontinuous and set-valued and thus the result also applies to model predictive control laws.     

On the Lyapunov stability of discrete-time processes modeled by difference inclusions

The subject of this note is the stability in the sense of Lyapunov of difference inclusions. Connection with the convergence of algorithms is also discussed.     

A new LMI condition for robust stability of discrete-time uncertain systems

In this paper, a new robust stability condition is derived for uncertain discrete-time linear systems with convex polytopic uncertainties. The condition is expressed in terms of a set of linear matrix inequalities (LMIs) involving only the vertices of the polytope domain. It enables us to determine robust stability of uncertain systems easily by solving some LMIs. A rigorous proof is given to show that an interesting result appeared recently is a special case of the proposed condition. Numerical examples also demonstrate the merit of the present condition in the aspect of conservativeness over other results in the literature.     

On bounded input–bounded output stability of a class of nonlinear retarded systems

Nonlinear retarded systems are considered. Explicit bounded input–bounded output stability conditions are derived. Copyright © 2000 John Wiley & Sons, Ltd.     

Improved stability criterion and its applications in delayed controller design for discrete-time systems

This paper is concerned with the problem of delay-dependent stability analysis for discrete-time systems with interval-like time-varying delays and the problem of stabilization for discrete-time linear systems via time-delayed controllers. The first problem is solved by applying a novel Lyapunov functional, and an improved delay-dependent stability criterion is obtained in terms of a linear matrix inequality. Based on this, a sufficient condition for the solvability of the second problem is presented. The reduced conservatism of the proposed stability result is shown through a numerical example, while the applicability of the time-delayed controller design method is demonstrated by an inverted pendulum system.     

Observers for discrete-time nonlinear systems

The problem of observers for discrete-time nonlinear systems has been considered and a simple, easy-to-implement algorithm is given whose convergence properties are guaranteed for autonomous and forced systems. Some numerical examples show the effectiveness of the proposed observer.     

Input-to-state stability for discrete-time time-varying systems with applications to robust stabilization of systems in power form

Input-to-state stability (ISS) of a parameterized family of discrete-time time-varying nonlinear systems is investigated. A converse Lyapunov theorem for such systems is developed. We consider parameterized families of discrete-time systems and concentrate on a semiglobal practical type of stability that naturally arises when an approximate discrete-time model is used to design a controller for a sampled-data system. An application of our main result to time-varying periodic systems is presented, and this is used to solve a robust stabilization problem, namely to design a control law for systems in power form yielding semiglobal practical ISS (SP-ISS).     

Input–output decoupling of nonlinear systems by static measurement feedback

This paper gives necessary and sufficient conditions for solvability of the strong input–output decoupling problem by static measurement feedback for nonlinear control systems.     

On robust stability of discrete-time adaptive nonlinear control

We consider adaptive control of discrete-time nonlinear systems with a single unknown parameter in this paper. We demonstrate that the necessary and sufficient condition for the existence of a feedback stabilizer that is robust to bounded noise is that the nonlinear growth rate of the system dynamics is less than 4. This result further confirms the conclusion of Guo [On critical stability of discrete-time adaptive nonlinear control, IEEE Trans. Automat. Control 42 (1997) 1488–1499] which addresses unbounded noise in a stochastic setting. Also in our worst-case approach, we find that much simpler adaptive stabilizers can be constructed when the nonlinear growth rate is less than 4.     

Stability analysis for discrete-time switched time-delay systems

The stability analysis problem is studied in this paper for a class of discrete-time switched time-delay systems. By using a newly constructed Lyapunov functional and the average dwell time scheme, a delay-dependent sufficient condition is derived for the considered system to be exponentially stable. The obtained results provide a solution to one of the basic problems in discrete-time switched time-delay systems, that is, to find a switching signal for which the switched time-delay system is exponentially stable. Two illustrative examples are given to demonstrate the effectiveness of the proposed results.     

Anti-windup design with guaranteed regions of stability for discrete-time linear systems

The purpose of this paper is to study the determination of stability regions for discrete-time linear systems with saturating controls through anti-windup schemes. Considering that a linear dynamic output feedback has been designed to stabilize the linear discrete-time system (without saturation), a method is proposed for designing an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system in the presence of saturation. It is shown that the closed-loop system obtained from the controller plus the anti-windup gain can be locally modeled by a linear system with a deadzone nonlinearity. Then, based on the use of a new sector condition and quadratic Lyapunov functions, stability conditions in an LMI form are stated. These conditions are then considered in a convex optimization problem in order to compute an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system. Moreover, considering asymptotically stable open-loop systems, it is shown that the conditions can be slightly modified in order to determine an anti-windup gain that ensures global stability. An extension of the proposed results to the case of dynamic anti-windup synthesis is also presented in the paper.     

Stability of leaderless discrete-time multi-agent systems

The paper presents a result which relates connectedness of the interaction graphs in multi-agent discrete-time systems with the capability for global convergence to a common equilibrium of the system. In particular, we extend previously known results by Bertsekas and Tsitsiklis and by Moreau, by including the possibility of arbitrary bounded time delays in the communication channels and relaxing the convexity of the allowed regions for the state transition map of each agent.