Pole placement controllers for linear time-delay systems with commensurate point delays

This paper investigates the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement type controllers for single-input single-output linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled diophantine equations of polynomials. An extra complexity is incorporated to the above design to cancel extra unsuitable dynamics being generated when solving the above diophantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delaydependent) closed-loop spectrum. However, if the controller is simplified by deleting the above mentioned extra complexity, then the robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is maintained. In the current approach, the finite spectrum assignment is only considered as a particular case to the designer‘s choice of a (delay-dependent) arbitrary spectrum assignment objective.     

Stabilization methods for systems with two linear delays

In this paper we obtain stabilization methods for some systems with two linear delays. We present a numerical example which illustrates the effectiveness of one of stabilization methods.     

Robust adaptive control of linear time-delay systems with point time-varying delays via multiestimation

This paper presents an adaptive pole-placement based controller for continuous-time linear systems with unknown and eventually time-varying point delays under uncertainties consisting of unmodeled dynamics and eventual bounded disturbances. A multiestimation scheme is designed for improving the identification error performance and then to deal with possibly errors between the true basic delay compared to that used in regressor vector of measurements of the adaptive scheme and also to prevent the closed-loop system against potential instability. Each estimation scheme in the parallel disposal possesses a relative dead-zone which freezes the adaptation process for small sizes of the adaptation error compared to the estimated size of the absolute value of the contribution of the uncertainties to the filtered output versus time. All the estimation schemes run in parallel but only that which is currently in operation parameterizes the adaptive controller to generate the plant input at each time. A supervisor chooses the appropriate estimator in real time which respects a prescribed minimum residence time at each estimation algorithm in operation. That strategy is the main tool used to ensure the closed-loop stability under estimates switching. The relative dead-zone in the adaptation mechanism prevents the closed-loop system against potential instability caused by uncertainties.     

Stability criteria for linear systems with uncertain delays

New delay-independent and delay-dependent stability criteria for linear systems with multiple uncertain delays are established by using both the time-domain and the frequency-domain methods. The results are derived based on the established new preliminary lemmas and by using new-type stability theorems for general retarded dynamical systems and new analysis techniques in the time-domain and the frequency-domain. All the established stability criteria depend only on the eigenvalues related to the coefficient matrices of the systems and do not involve any free tuning parameters. In addition, some remarks are given to explain in detail the obtained results and to point out the limitations of the existing results in the literature.     

Structural constrained controllers for discrete-time linear systems

In this paper, we present an approach for designing structural constrained controllers for discrete-time linear systems, based on a new stabilizability property of the Riccati equation solution. First, the feedback stabilization problem under a general structural constraint is considered and a simple numerical procedure to solve it is presented. Special attention is given to the output feedback stabilization problem, for which sufficient conditions for the existence and convergence toward a stabilizing matrix are provided. Some examples are solved and comparisons with other methods available in the literature are made.This research has been developed with the financial support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Brasília, Brazil.     

Robust stabilization of linear systems with time-varying point delays via a delay-free dynamic controller

^** Email: wepdepam{at}lg.ehu.es This paper deals with the problem of robust closed-loop stabilizationagainst parametrical uncertainties of linear systems subjectto internal (i.e. in the state) and external (i.e. in the output),possibly time-varying and unbounded point delays of a boundedtime-derivative. The output-feedback linear stabilizing controlleris delay free and dynamic. It is assumed that the undelayedplant (i.e. the delay-free part of the plant) is stabilizableand detectable. The synthesis process of the stabilizing controllerinvolves three major actions. First, an augmented system isbuilt with the dynamic equations of both plant and controller.At this step, the controller structure is available but a particularstabilizing controller parametrization still remains undetermined.Subsequently, a Lyapunov matrix equation is ensured to be solvablefor the augmented closed-loop delay-free system so that sucha system is stable with a large stability abscissa related tothe amounts of uncertainties and delay contributions to thedynamics. At this stage, one takes the advantage that the augmentedsystem may be stabilized by an appropriate dynamic controllerof minimum order since the undelayed plant is stabilizable anddetectable. Finally, a complementary matrix equality is manipulatedto establish the closed-loop stability tolerance of the augmenteddelay system, related to that of the delay-free one, to thedelayed dynamics and parametrical uncertainties.     

A dynamic programming method for controlled linear systems with delays

A linear system with permanent delay is considered. A method of dynamic programming for constructing attainability sets and solving the problem of target control for the systems is used. The expressions for value functionals described by solutions to the corresponding Hamilton-Jacobi-Bellman equation are obtained. It is proved that these value functionals calculated by means of convex analysis satisfy the above equations. Strategies for synthesized control for the problem of hitting on the target set are given.     

Stabilization of some systems with two linear delays

We obtain methods for stabilizing systems with two linear delays and systems with two linear delays and one constant delay. We present a numerical example in which a stabilizing control is constructed.     

Singular linear systems of differential equations with delays

The differential equation Ax + Bx = Cx(t-l) + f is studied where A, B, C are square matrices. All matrices are allowed to be singular. Examples are given to show that not all initial functions are consistent and that for consistent initial conditions, solutions may be continuous for only a finite time period. Solutions are given explicitly by a recursion formula and consistent initial conditions are explicitly characterized     

Suboptimality and stability of linear distributed-parameter systems with finite-dimensional controllers

In order to implement feedback control for practical distributed-parameter systems (DPS), the resulting controllers must be finite-dimensional. The most natural approach to obtain such controllers is to make a finite-dimensional approximation, i.e., a reduced-order model, of the DPS and design the controller from this. In past work using perturbation theory, we have analyzed the stability of controllers synthesized this way, but used in the actual DPS; however, such techniques do not yield suboptimal performance results easily. In this paper, we present a modification of the above controller which allows us to more properly imbed the controller as part of the DPS. Using these modified controllers, we are able to show a bound on the suboptimality for an optimal quadratic DPS regulator implemented with a finite-dimensional control, as well as stability bounds. The suboptimality result may be regarded as the distributed-parameter version of the 1968 results of Bongiorno and Youla.This research was supported by the National Science Foundation under Grant No. ECS-80-16173 and by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0124. The author would like to thank the reviewer for many helpful suggestions.     

Stabilization of switched linear systems with mode-dependent time-varying delays

In this paper, we investigate the problem of stabilization via state feedback and/or state-based switching for switched linear systems with mode-dependent time-varying delays. By using the multiple Lyapunov functional method, we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and/or switching under time-varying delays with appropriate upper bounds. The main results are presented in terms of linear matrix inequalities (LMIs) which generalize some known results and can be easily tested by using the Matlab’s LMI Tool-box.     

Solution damping controllers for linear systems of the neutral type

For linear autonomous systems of the neutral type with commensurable delays in the state and control, we solve the problem of solution damping by a state feedback controller. In this case, the closed-loop system becomes a system of the neutral type with a finite spectrum. The present research is characterized by the fact that the original system does not necessarily have the property of complete controllability.     

Delay-dependent exponential stabilization for uncertain linear systems with interval non-differentiable time-varying delays

In this paper, the problem of exponential stabilization for a class of linear systems with time-varying delay is studied. The time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is not necessary to be differentiable. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Leibniz-Newton’s formula, new delay-dependent sufficient conditions for the exponential stabilization of the systems are first established in terms of LMIs. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.     

Methods for studying the asymptotic properties of systems with two linear delays

We study methods for the instability analysis of neutral-type systems with two delays linearly depending on the argument. In the case of instability of the neutral part of a system with constant coefficients, we suggest a stabilization algorithm. In addition, by using the Laplace transform, we obtain sufficient conditions for the instability of the solution of a given system.     

An explicit criterion for finite-time stability of linear nonautonomous systems with delays

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.     

Reduced-order controllers for unstable multivariable linear systems that can be balanced

This paper considers the problem of H_∞-stabilization of unstable multivariable linear systems. The major features of the approach are: (1) a reduced-order model is obtained using low-frequency balancing, the approximant will have the same number of unstable poles as the original system, (2) the controller design is accomplished by dynamic output feedback, and (3) sufficient conditions in the form of two algebraic Riccati equations and an upper bound explicitly characterize a H_∞-controller of lower dimensions. At the end, an illustrative example is given to show the simplicity of the procedure.     

Fractional order adaptive high-gain controllers for a class of linear systems

In this paper we show that a fractional adaptive controller based on high gain output feedback can always be found to stabilize any given linear, time-invariant, minimum phase, siso systems of relative degree one. We generalize the stability theorem of integer order controllers to the fractional order case, and we introduce a new tuning parameter for the performance behaviour of the controlled plant. A simulation example is given to illustrate the effectiveness of the proposed algorithm.     

Exponentially stabilizing finite-dimensional controllers for linear distributed parameter systems: Galerkin approximation of infinite dimensional controllers

The Galerkin method is presented as a way to develop finite-dimensional controllers for linear distributed parameter systems (DPS). The direct approach approximates the open-loop DPS and then generates the controller from this approximation; the indirect approach approximates the infinite-dimensional stabilizing controller. The indirect approach is shown to converge to the stable closed-loop system consisting of DPS and infinite-dimensional controller; conditions are presented on the behavior of the Galerkin method for the open-loop DPS which guarantee closed-loop stability for large enough finite-dimensional approximations.     

A sliding mode control for linear fractional systems with input and state delays

In this paper, a sliding mode control design for fractional order systems with input and state time-delay is proposed. First, we consider a fractional order system without delay for which a sliding surface is proposed based on fractional integration of the state. Then, a stabilizing switching controller is derived. Second, a fractional system with state delay is considered. Third, a strategy including a fractional state predictor input delay compensation is developed. The existence of the sliding mode and the stability of the proposed control design are discussed. Numerical examples are given to illustrate the theoretical developments.     

Regular linear systems governed by systems with state, input and output delays

^** Email: hadd{at}ucam.ac.ma^*** Email: idrissi{at}ucam.ac.ma In this paper, we give a new reformulation of linear systemswith delays in input, state and output. We show that these systemscan be written as a regular linear system without delays. Thetechnique used here is essentially based on the theory recentlydeveloped by Salamon and Weiss and the shift in semigroup properties.Our framework can be applied, in particular, when the delayoperators are given by Riemann–Stieltjes integrals.