Pole placement for linear discrete-time systems by periodic output feedbacks

Necessary and sufficient conditions for the existence of a solution to the pole placement problem by periodic output feedbacks for single-input single-output linear discrete-time systems are established. An algorithm for finding a sequence of output-feedback gains is presented and illustrated by a simple numerical example.     

Optimized robust control invariance for linear discrete-time systems: Theoretical foundations

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.     

Some issues of structure and control for linear time-invariant systems

The objective of this paper is to emphasise the role played by particular structures in the solution of some control problems. The so-called “structural approach” relies on various indicators of dynamical systems such as, for instance, finite and infinite zeros, kernel indices, …. The fundamental invariance properties of these structures under the action of some transformations groups (e.g. feedback) are at the origin of their key role. Structural solutions to “classical” control problems, such as disturbance rejection, model matching and non-interaction are now rather well known: zeros at infinity play a role in the existence of “proper” solutions, while finite (invariant) zeros allow for the characterisation of “fixed poles”, whose location in the complex plane gives answer to pole placement limitations (including stability). Among the recent contributions to this structural approach, a particular attention is here devoted to:

- “Partial” versions of some of these control problems: The control objective only concerns a finite number of (and not necessarily all) the first Markov parameters of the transfer function matrix of the controlled system (e.g. to be zero for disturbance rejection or model matching, to be diagonal for non-interaction). Some interesting new issues in the dual context of failure detection are also sketched.

- Generalised solutions: Based on proportional and derivative feedback laws, with new issues in the context of systems with variable internal structures, and also for systems with delays.

Geometric concepts, such as invariant and almost invariant subspaces, and algebraic counterparts, such as factorisations on some special rings, are intermediary tools which support the characterisations of those particular structures and which allow for a structural treatment of the considered control and/or observation problems.

The results are here presented without proof: references are given to previous published results (in most cases in books and journals which are easily available), and some simple examples are used to illustrate non-standard notions (among which systems with variable internal structure, and time domain left invertibility).

Most of the results here presented rely on long and intensive collaborations between the author and various colleagues.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Disturbance localization for left invertible linear periodic discrete-time systems

The disturbance localization problem for left invertible linear periodic discrete-time systems is solved using periodic state feedback controllers. The proposed technique is of algebraic nature and has the following two main characteristics: (i) It yields simple algebraic criteria for testing the solvability of the problem, as compared to known geometric criteria, which may not be so easy to check. (ii) It derives analytically the general expressions of all periodic controllers admissible for disturbance localization, as compared to known techniques, which lead to nonanalytic parametrizations of the admissible controllers via constructive procedures. Moreover, for the aforementioned class of periodic systems, the state feedback simultaneous disturbance localization and stabilization or pole placement problem is treated, and conditions for its solvability are established, on the basis of a decentralized control approach, that makes use of the equivalence between the above problem and the stabilization or pole placement problem of a general proper multichannel system by decentralized static output feedback.     

Robust stability and control of uncertain linear discrete-time periodic systems with time-delay

This paper deals with the problems of robust stability analysis and robust control of linear discrete-time periodic systems with a delayed state and subject to polytopic-type parameter uncertainty in the state-space matrices. A robust stability criterion independent of the time-delay length as well as a delay-dependent criterion is proposed, where the former applies to the case of a constant time-delay and the latter allows for a time-varying delay lying in a given interval. The developed robust stability criteria are based on affinely uncertainty-dependent Lyapunov–Krasovskii functionals and are given in terms of linear matrix inequalities. These stability conditions are then applied to solve the problems of robust stabilization and robust _H∞$H_{\infty }$  control via static periodic state feedback. Numerical examples illustrate the potentials of the proposed robust stability and control methods.     

Zero-error regulation of discrete-time linear periodic systems

In this paper the regulation problem for discrete-time linear periodic systems is investigated. A control structure is introduced which is shown to be capable of robustly zeroing the asymptotic error produced by periodically time-varying exogenous signals. The cases when the plant is time-invariant and/or the exogenous signals are constant are discussed.     

Adaptive quantized control for linear uncertain discrete-time systems

A direct adaptive control framework for linear uncertain systems for using communication channels is developed. Specifically, the control signals are to be quantized and sent over a communication channel to the actuator. The proposed framework is Lyapunov-based and guarantees partial asymptotic stability, that is, Lyapunov stability of the closed-loop system states and attraction with respect to the plant states. The quantizers are logarithmic and characterized by sector-bound conditions, with the conic sectors adjusted at each time instant by the adaptive controller, in conjunction with the system response. Furthermore, we extend the scheme to the case where the logarithmic quantizer has a deadzone around the origin so that only a finite number of quantization levels is required to achieve practical stability. Finally, a numerical example is provided to demonstrate the efficacy of the proposed approach.     

Regulator problem for linear systems with constraints on control and its increment or rate

This paper discusses the problem of constraints on both control and its rate or increment, for linear systems in state space form, in both the continuous and discrete-time domains. Necessary and sufficient conditions are derived for autonomous linear systems with constrained state increment or rate (for the continuous-time case), such that the system evolves respecting incremental or rate constraints. A pole assignment technique is then used to solve the inverse problem, giving stabilizing state feedback controllers that respect non-symmetrical constraints on both control and its increment or rate. An illustrative example shows the application of the method on the double integrator problem.     

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is (n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.     

Robust static output feedback control for linear discrete-time systems with time-varying uncertainties

This paper is concerned with the problem of designing robust static output feedback controllers for linear discrete-time systems with time-varying polytopic uncertainties. Sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities, and the results are extended to H₂ and H_∞ static output feedback controller designs. Numerical examples are given to illustrate the effectiveness of the proposed design methods.     

Periodically time-varying memory state-feedback controller synthesis for discrete-time linear systems

In this paper, we deal with discrete-time linear periodic/time-invariant systems with polytopic-type uncertainties and propose a new linear matrix inequality (LMI)-based method for robust state-feedback controller synthesis. In stark contrast with existing approaches that are confined to memoryless static controller synthesis, we explore dynamical controller synthesis and reveal a particular periodically time-varying memory state-feedback controller (PTVMSFC) structure that allows LMI-based synthesis. In the context of robust controller synthesis, we prove rigorously that the proposed design method encompasses the well-known extended-LMI-based static controller synthesis methods as particular cases. Through numerical experiments, we demonstrate that the suggested design method is indeed effective in achieving less conservative results, under both periodic and time-invariant settings. We finally derive a viable test to verify that the designed robust PTVMSFC is “exact” in the sense that it attains the best achievable robust performance. This exactness verification test works fine in practice, and we will show via a numerical example that exact robust control is indeed attained by designing PTVMSFCs, even for such a problem where the standard memoryless static state-feedback fails.     

Asymptotic stabilization of a class of continuous-time linear periodic systems

In this paper, the problem of the asymptotic stabilization of a class of continuous-time linear periodic systems is considered. First, under a simplifying assumption, necessary conditions, involving the concept of left eigenvector, are derived for the asymptotic stabilization of continuous-time linear periodic systems belonging to such a class. Then, under the same simplifying assumption, conditions are given under which a static state feedback control law can be designed, in an iterative manner, so that the closed-loop system is π-stable.     

Optimal design of linear control systems by an interactive optimization method

When designing linear control systems, one of the most difficult problems is that the designer almost has no theoretical basis for the determination of proper parameters in order to obtain a system with desired specifications. Poles and directions of eigenvectors in the pole assignment method or weighting matrices of the quadratic criterion function in the optimal regulator method are such parameters. The designer has to determine them by trial-and-error using computer simulation. The purpose of this paper is to propose an approach to helping determine proper parameters in linear control system design by the state space methods. In the case where the desired specifications are not given explicitly, the approach applies an interactive optimization method called the Interactive Simplex method to search the most suitable parameters directly in the parameter space. But, if the specifications are given explicitly, the design problem can be formulated as a multiobjective optimization problem. In this case, weights which indicate relative importance of different specifications are introduced and the Interactive Simplex method is applied in the weight space to indirectly find the most appropriate parameters. The approach is implemented as part of a CAD system. The designer has only to make pairwise comparisons of response curves which are shown on a graphics display terminal in order to obtain the most preferred control system. Two illustrative examples are demonstrated to indicate the efficiency of the approach.     

IQC-based -control of linear periodic systems

Stability analysis of linear periodically time-varying systems via integral quadratic constraints is extended to the problem of control design. A full-state feedback controller that satisfies exponential stability and -gain disturbance attenuation from an external disturbance to a controlled output is designed for linear systems with periodically time-dependent system matrices. The main result relies on dual forms of certain integral quadratic constraints. The solvability conditions for the problem are cast as a set of finite-dimensional linear matrix inequalities and thus, they are easily solvable. Moreover, the best possible disturbance attenuation level can be obtained as a convex problem.     

Robust output-feedback control of linear discrete-time systems

The problem of designing H_∞ dynamic output-feedback controllers for linear discrete-time systems with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states via the delayed inputs and outputs of the plant. The resulting model incorporates the uncertain parameters of the transfer function matrix in the state-space matrices. A recently developed linear parameter-dependent LMI approach to state-feedback H_∞ control of uncertain polytopic systems is then used to design a robust output-feedback controllers that are of order comparable to the one of the plant. These controllers ensure the stability and guarantee a prescribed performance level within the uncertainty polytope.     

Robust performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.     

An improved robust stability and robust stabilization method for linear discrete-time uncertain systems

This paper addresses the problems of the robust stability and robust stabilization of a discrete-time system with polytopic uncertainties.A new and simple method is presented to directly decouple the Lyapunov matrix and the system dynamic matrix.Combining this method with the parameter-dependent Lyapunov function approach yields new criteria that include some existing ones as special cases.A numerical example illustrates the improvement over the existing ones.     

On pole assignment of linear systems by gain output feedback

In this paper, the Geometric Approach is used to derive in a straightforward way a sufficient condition for pole assignability by gain output feedback. This result leads to a pole assignment procedure which reduces to solving a system of min (n - m, n - p) polynomial equations where n is the number of states, m the number of inputs, p the number of outputs. In the case where m + p > n, this system clearly appears to be linear. The degrees of freedom related to the pole assignment problem are expressed in terms of (right or left) eigenvectors.     

On state estimation and control in discrete-time systems of linear type

This note contains recurrence state estimation and control algorithms for general time-delay, discrete linear systems with the noises correlated on time intervals whose lengths change in time.