Stabilization of linear time-invariant interval systems viaconstant state feedback control

This paper investigates the stabilization problem of linear uncertain systems via constant state feedback control. The systems under consideration contain time-invariant uncertain parameters whose values are unknown but bounded in given compact sets and are thus called interval systems. The criterion for the asymptotic stability of the closed-loop system, obtained when a state feedback control is applied, is that all the eigenvalues of the resulting system matrix are in the strict left half of the complex plane. First, the author shows that to insure an interval system stabilizable, some entries of the system matrices must be sign invariant. More precisely, the number of the least-required, sign-invariant entries in system matrices is equal to the system order. Then, the author studies the stabilizability of a set of interval systems called standard systems which contain sufficient numbers of sign-invariant entries in proper locations. After dividing all standard systems into some subsets by the uncertainty locations, the author then derives necessary and sufficient conditions under which every system in a subset is stabilizable, regardless of its parameter varying bounds. The conditions show that all uncertain entries in system matrices should form a particular geometrical pattern called a “generalized antisymmetric stepwise configuration”. For an interval system satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example. The result is further generalized for nonstandard systems via linear transformation     

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.     

Stabilization of perturbed systems via linear optimal regulator

Linear quadratic state feedback regulators make the resulting closed-loop systems stable enough, i.e. they realize robust stabilization. Many attempts at robust stabilization using linear quadratic regulators have been reported. One of the major trends of formulating uncertainty in systems is to express perturbed parameters as the sum of two terms, i.e. nominal values and the deviation from them. In this paper, it is assumed that the upper and lower bounds for each uncertain parameter can somehow be estimated. This enables us to dispense with nominal values. The main aim is to contrive a robust feedback stabilization law for systems with parameters falling into certain ranges via a linear quadratic regulator based only upon information on their bounds. The systems under consideration are therefore those having interval system matrices (in which each element has the above-mentioned two-sided bounds). A certain feedback law is a stabilizing law for a system with an interval system matrix if and only if the same feedback law remains so for systems with system matrices whose entries are all possible combinations of the endpoints of their variation range.     

A multistage reduction technique for feedback stabilizing distributed time-lag systems

A direct pole relocation theory is advanced for linear time-invariant systems with distributed delays in both state and control variables. The principal tools of the theory include (i) the finite cardinality of the unstable spectrum; (ii) a set of matrices, each of which is a left zero of the system characteristic quasi-polynomial matrix; and (iii) a linear transformation which reduces the delay system to a sufficiently high-order delay-free system whose spectrum contains the delay system unstable spectrum. It is shown that if the delay system is spectrally stabilizable, then it shares a common feedback stabilizing control law with its delay-free counterpart. This point of contact with a delay-free system permits the determination of the control law using well-established ordinary system methods. The workability of the approach hinges on the ability to partition the unstable spectrum (augmented with additional poles from the stable spectrum, if necessary) into N symmetric sets. When this partition is impossible, a spectral controllability invariance theorem facilitates resolution of the problem.     

不确定离散广义系统的D稳定鲁棒控制

研究了具有圆盘区域极点约束的一类不确定离散广义系统的鲁棒控制问题.首先,研究了控制输入项不含扰动的不确定离散广义系统,提出了广义二次D镇定的概念,基于矩阵不等式和广义Riccati方程,给出了一种广义二次D镇定器的设计方法,所得到的结论能够实现研究目标;然后,讨论了控制输入项含有扰动的不确定离散广义系统,在一定的假设条件下,给出了期望状态反馈增益阵的存在条件及其解析表达式.最后,用数值示例说明所给方法的有效性及可行性.     

Discrete two-time-scale systems

In this paper reduced-order modelling and control analysis of linear, discrete-time systems having dominant and non-dominant modes are presented. Decoupling of modes is achieved using an explicitly invertible linear transformation. A matrix norm condition is derived, the satisfaction of which enables approximate expressions for the block-diagonalizing matrices, eigenvalue distribution and state trajectories to be obtained. Design of stabilizing feedback controllers is developed and it is shown that two gain matrices are needed for separate assignment of dominant and non-dominant eigenvalues. The theoretical analysis is illustrated by control system examples.     

Robust control through piecewise Lyapunov functions for discrete time-varying uncertain systems

This paper is concerned with the robust stabilization by state feedback of a linear discrete-time system with time-varying uncertain parameters. An optimization problem involving a set of linear matrix inequalities and scaling parameters provides both the robust feedback gain and the piecewise Lyapunov function used to ensure the closed-loop stability. In the case of linear time-varying systems involving the convex combination of two matrices, only two scaling parameters constrained into the interval [0,?1] are needed, allowing a simple numerical solution as illustrated by means of examples.     

An LMI approach to stabilization of linear discrete-time periodic systems

The problem of stabilization of linear discrete-time periodic systems is considered. LMI based conditions for stabilization via static periodic state feedback as well as via static periodic output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient whereas for output feedback the result is only sufficient as it depends on the particular state-space representation used to describe the system. The problem of quadratic stabilization in the presence of either norm-bounded or polytopic parameter uncertainty is also treated. As an application of the output feedback stabilization technique, we consider the problem of designing a stabilizing (respectively, quadratically stabilizing) static periodic output feedback controller for linear time-invariant discrete-time systems which are not stabilizable (respectively, quadratically stabilizable) by static constant output feedback.     

Semi-global stabilization of discrete-time linear systems with position and rate-limited actuators

It is shown via explicit construction of feedback laws that, if a discrete-time linear system is asymptotically null controllable with bounded controls, then, when subject to both actuator position and rate saturation, it is semi-globally stabilizable by linear state feedback. If, in addition, the system is also detectable, then it is semi-globally stabilizable via linear output feedback.     

Stabilization of non-linear systems†

The problem of stabilizing a discrete-time non-linear system is considered. For a rather large class of common stabilizable non-linear systems, a procedure leading to the stabilization of a given non-linear system Σ belonging to that class is derived. In this procedure, a pair of compensators is constructed, consisting of a precompensator and an output feedback compensator, which, when connected in closed loop around the system Σ, yield a closed-loop system that is internally stable for bounded input sequences. The procedure allows the construction of infinitely many different pairs of such compensators, thus facilitating the choice of a convenient one.     

Gain margin and Lyapunov analysis of discrete-time network synchronization via Riccati design

This paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete-time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single-input controllable system is analyzed. The solution matrix entries are represented using unknown closed-loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi-input and single-input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete-time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.     

An efficient LMI approach for the quadratic stabilisation of a class of linear,uncertain, time-varying systems

The purpose of this article is to provide a numerically efficient method for the quadratic stabilisation of a class of linear, discrete-time, uncertain, time-varying systems. The considered class of systems is characterised by an interval time-varying (ITV) matrix and constant sensor and actuator matrices. It is required to find a linear time-invariant (LTI) static output feedback controller yielding a quadratically stable closed-loop system independently of the parameter variation rate. The solvability conditions are stated in terms of linear matrix inequalities (LMIs). The set of LMIs includes the stability conditions for the feedback connection of a unique suitably defined extreme plant with an LTI output controller and the positivity of a closed-loop extremal matrix. A consequent noticeable feature of the article is that the total number of LMIs is independent of the number of uncertain parameters. This greatly enhances the numerical efficiency of the design procedure.     

An eigenstructure assignment approach for constrained linear continuous-time singular systems

This paper establishes necessary and sufficient algebraic conditions for positive invariance of convex polyhedra with respect to some linear continuous-time singular systems. They can be considered as an extension for linear singular systems of the classical positive invariance relations for regular linear systems. For a stabilizable and impulse controllable singular system with constrained inputs, a stabilizing state feedback control guaranteeing the closed-loop positive invariance of some polyhedral sets determined from the feedback matrix is studied. An analysis of the closed-loop positive invariance relations is thus presented in terms of eigenstructure and stability properties. An eigenstructure assignment technique is proposed depending on the number of stable finite poles of the singular system.     

Proper,denominator assigning feedback compensators equivalent to state variable feedback

For linear, time-invariant, stabilizable multivariable systems, we examine the problem of the existence and computation of proper denominator assigning and internally stabilizing feedback compensators, which give rise to a closed-loop system, whose transfer function matrix is equal to one that can be obtained by the action of state variable feedback. We establish a sufficient condition for the solution of this problem for the class of systems with the same number of inputs and outputs and non-singular transfer function matrix with all its zeros located in the open left half of the complex plane.     

A behavioral approach to the control of discrete linear repetitive processes

This paper formulates the theory of linear discrete time repetitive processes in the setting of behavioral systems theory. A behavioral, latent variable model for repetitive processes is developed and for the physically defined inputs and outputs as manifest variables, a kernel representation of their behavior is determined. Conditions for external stability and controllability of the behavior are then obtained. A sufficient condition for stabilizability is also developed for the behavior and it is shown under a mild restriction that, whenever the repetitive system is stabilizable, a regular constant output feedback stabilizing controller exists. Next, a notion of eigenvalues is defined for the repetitive process under an action of a closed-loop controller. It is then shown how under controllability of the original repetitive process, an arbitrary assignment of eigenvalues for the closed-loop response can be achieved by a constant gain output feedback controller under the above restriction. These results on the existence of constant gain output feedback controllers are among the most striking properties enjoyed by repetitive systems, discovered in this paper. Results of this paper utilize the behavioral model of the repetitive process which is an analogue of the 1D equivalent model of the dynamics studied in earlier work on these processes.     

Stabilization of exponentially unstable discrete-time linear systems by truncated predictor feedback

Predictor state feedback solves the problem of stabilizing a discrete-time linear system with input delay by predicting the future state with the solution of the state equation and thus rendering the closed-loop system free of delay. The solution of the state equation contains a term that is the convolution of the past control input with the state transition matrix. Thus, the implementation of the resulting predictor state feedback law involves iterative calculation of the control signal. A truncated predictor feedback law results when the convolution term in the state prediction is discarded. When the feedback gain is constructed from the solution of a certain parameterized Lyapunov equation, the truncated predictor feedback law has been shown to achieve asymptotic stabilization of a system that is not exponentially unstable in the presence of an arbitrarily large delay by tuning the value of the parameter small enough. In this paper, we extend this result to exponentially unstable systems. Stability analysis leads to a bound on the delay and a range of the values of the parameter for which the closed-loop system is asymptotically stable as long as the delay is within the bound. The corresponding output feedback result is also derived.     

Stabilisation of discrete-time bilinear systems by constant control inputs

ABSTRACT

In this paper, the stabilisation problem of discrete-time bilinear systems by using constant control inputs is considered. It is shown that the spectrum of a matrix derived from the system matrices plays a key role in the stabilisation design. Sufficient conditions for the systems to be stabilizable by constant control inputs are presented in the cases when the derived matrix has no complex eigenvalue as well as in the cases when it has complex eigenvalues. Particularly, it is proved that, if the derived matrix has only nonzero real eigenvalues, then the systems can always be stabilised by constant control inputs. Finally, simulations are given to demonstrate the obtained results.     

Stability of linear systems with periodic feedback and sampled-data systems

A linear multivariable time-invariant system with three periodic state feedback strategies—dynamic, static, and of the instantaneous state—is considered. For each of these feedback strategies the stability question of an appropriate closed-loop system in relation to its dependence upon the repetition frequency is examined. The proposed approach to the question is based on exponential matrix representations of the dynamics of the considered closed-loop systems. It is shown that at sufficiently large values of the repetition frequency the stability question may be decided on the basis of the stability of a certain matrix which is common for all the considered feedback strategies. It is also shown that peculiar to discrete-time control, the dead-beat stability phenomenon may appear only for a dynamic or static feedback strategy and it may be achieved when the repetition frequency is small enough. The proposed approach results in some new characterization of the stability property for multivariable sampled-data systems.     

Dynamic Output Feedback for Discrete-Time Systems Under Amplitude and Rate Actuator Constraints

This work proposes a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. The nonlinear effects introduced by the saturations in the closed-loop system are taken into account by using a generalized sector condition, which leads to theoretical conditions for solving the problem directly in the form of linear matrix inequalities.