Completely Distributed State Estimation for Jointly Observable Uncertain Linear Systems

A stabilization algorithm for a class of uncertain linear systems

This paper introduces a notion of observability for a class of uncertain linear systems with structured uncertainty. In the uncertain systems under consideration, the uncertainty is described by averaged integral quadratic constraints. In order to define a notion of observability for uncertain linear systems, the paper introduces a robust observability function which extends the usual definition of the observability Gramian to the case of uncertain systems. Using this observability function, a corresponding unobservable cone is defined, and an uncertain system is said to be robustly observable if this cone contains only the origin. The paper presents an algorithm for finding the robust observability function and corresponding unobservable cone. This algorithm involves solving a parameterized Riccati differential equation.

This paper investigates the robust output tracking problem for a class of large‐scale linear uncertain systems with interactions and time delays. Time delays exist in both states and controls. Using the Riccati equation, a procedure for determining decentralized linear control laws is presented such that the closed‐loop system asymptotically tracks the reference output and rejects any constant but unknown disturbances. The main feature of this approach is that the uncertain systems may contain time delays in both states and controls as well as in interactions between subsystems. A numerical example is included to show the results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

A minimax optimal control strategy for partially observable uncertain quasi-Hamiltonian systems

This note is concerned with the problem of stabilizing an uncertain linear system via state feedback control. An uncertain system which admits a stabilizing state feedback control and some associated quadratic Lyapunov function is said to be quadratically stabilizable. In a number of recent papers, conditions are given under which quadratic stabilizability via nonlinear control implies quadratic stabilizability via linear control. These papers restrict the manner in which the uncertain parameters are permitted to enter structurally into the state equation in order to establish this result. This note presents an example which shows that this implication is not true for more general uncertain linear systems. To this end, we describe an uncertain linear system which is quadratically stabilizable via nonlinear control but not quadratically stabilizable via linear control.

We consider the guaranteed cost control problem of a class of uncertain linear discrete-time systems. The uncertain systems under consideration depend on norm-bounded time-varying uncertain parameters. Results on the design of robust state feedback guaranteed cost controllers are presented.

Based on the stabilizability of a nominal system (i.e. a system in the absence of uncertainty), by making use of the Lyapunov stability criterion and combining with the algebraic Riccati equation, a new approach for designing a robust linear state feedback controller for uncertain linear dynamical systems is presented. Using this approach, the BIBO stability of uncertain linear dynamical systems is also discussed, Some analytical methods and the Bellman-Gronwall inequality are employed to investigate the robust stabilization conditions on the feedback controller. The main features of this approach are that no matching condition about uncertainty is needed and the uncertain systems can be asymptotically stabilized. An example is given to demonstrate the validity of our results.

This paper investigates the problem of designing a linear memoryless state feedback control to stabilize a class of linear uncertain systems with state delays. Each uncertain parameter and each delay under consideration may take arbitrarily large values. In such a situation, the locations of uncertain entries in the system matrices play an important role. It has been shown that it is a necessary and sufficient condition for the stabilization of time-varying or time-invariant uncertain systems without delays to have a particular geometric configuration called an ASC or a GASC, respectively. However, those results are inapplicable to systems that contain delays in the state variables. The objective of this paper is to show that if time-varying uncertain systems with time-varying delays or time-invariant uncertain systems with time-invariant delays have an ASC or a GASC, respectively, then the systems are stabilizable no matter how large the bounds of delays and uncertain parameters may be. However, we restrict our attentions to 3-dimensional systems for simplicity. The results shown here imply that the stabilizability conditions are not deteriorated by the existence of time delays.

Stabilization irrespective of bounds of uncertain variations for linear uncertain systems with delays

Time-delay in dynamical systems is often a source of instability and poor performance which presents in many applications. This paper deals with the robust control problem for class of uncertain linear neutral systems with multiple state and state derivatives delays. The parametric uncertainties are time varying and unknown but norm bounded. In this paper by introducing a new Lyapunov functional, the stability condition is extended to structured uncertain neutral systems. so new ( Descriptor ) model transformation and a corresponding Lyapunov functional are introduced for stability analysis of systems with discrete and distributed multiple delay.Sufficient conditions are given in terms of linear matrix inequalities ( LMI ) and refer to neutral systems with discrete and distributed delays. Based on the stability condition, designing delay dependent / independent state feedback control is formulated. Solving the LMI problems, a robust memoryless state feedback control law is designed for all admissible uncertainties. The results depend on the size and varying rate of the delays.In this paper the presented model transformation and Lyapunov function can be applied further to H∞ control of linear uncertain systems with multiple state delays. Two examples are provided to show the effectiveness of the proposed strategy .

This paper investigates the problem of designing a linear memoryless state feedback control to stabilize a class of linear uncertain systems with state delays. Each uncertain parameter and each delay under consideration might vary with time in an arbitrarily large range. In such a situation, the locations of uncertain elements in the system matrices play an important role. Wei introduced the concept of antisymmetric stepwise configuration (ASC) and proved that it is a necessary and sufficient condition for linear uncertain systems to be quadratically stabilizable using linear state feedback control to have this configuration. However, his method is inapplicable to systems that contain delays in the state variables. On the other hand, Amemiya developed conditions for the stabilization of linear uncertain systems with state delays using linear memoryless state feedback control. This paper presents development of the conditions of this problem that have been obtained to date. Fundamentally, it is proved that having an ASC is also a sufficient condition for the stabilization of linear uncertain delay systems. For systems satisfying the stabilizability conditions, a simple control design procedure is also provided and illustrated by an example.

In this paper, a design problem of the robust finite time functional observers in the uncertain linear systems is investigated. The aim is to design a robust functional observer which can estimate the state linear functions of the uncertain linear systems almost in a predefined finite time. Based on the parametric solutions for a class of Sylvester matrix equation, this paper presents the parametric expressions of all the gain matrices for the finite time functional observers in the linear systems without uncertainties, in which the free parameters can offer all the design degrees of freedom needed in the robust control system design. By using the offered design degrees of freedom, the robust indexes can be parameterised and the design problem of the robust finite time functional observers for the uncertain linear systems can be changed into a minimisation problem with some constraints. By solving the changed minimisation problem, a corresponding algorithm to design the robust finite time functional observers in the uncertain linear systems is proposed. Finally, a numerical example and its simulation results show the simplicity and effectiveness of the proposed design method of robust finite time functional observers in the uncertain linear systems.

This paper considers a problem of controllability for a class of linear uncertain systems. The uncertain systems under consideration contain norm bounded time-varying uncertainty. The paper introduces a new notion of controllability referred to as feedback controllability. Roughly speaking, an uncertain system is feedback controllable if there exists a time varying linear state feedback control such that with any initial condition, the closed loop system state converges to zero in a finite time. The main result of the paper shows that if the uncertain system satisfies a certain matching condition then the system will be feedback controllable. This matching condition is also known to be a sufficient condition for the stabilizability of the uncertain system.

An algorithm for an adaptive memoryless feedback control law, that is easy to implement in practice, is presented, which is designed to estimate bounded uncertainty in a class of linear uncertain time-delay systems. This estimated uncertainty is then used to cancel the effect of the uncertainty in the system so that it is then relatively straightforward to design `standard' controllers in order to modify the behaviour of the uncertainty-free system. Here, a class of linear uncertain time-delay systems of the retarded type, in which the uncertainty is an additive perturbation of a known (nominal) linear model, is studied. The time-delays are assumed to be time-varying and unknown, but bounded. Thus, the main advantage of using this adaptive memoryless feedback control law is that the uncertainty can be cancelled from the system and, if further design objectives are to be realized (for example, with respect to a tracking problem), the controls can be designed on the information from the nominal model, i.e. the model without uncertainty, only, and not on the uncertain model.

A robust guaranteed cost consensus problem of discrete-time high-order uncertain linear multi-agent systems with directed graphs is studied in this paper. Firstly, the robust guaranteed cost consensus problem of high-order uncertain linear multi-agent systems is transformed to a robust guaranteed control problem of an uncertain system. Secondly, a sufficient LMI condition insuring the guaranteed cost consensus of discrete-time high-order uncertain linear multi-agent systems is derived, and convergence results are given as consensus function sequences of discrete-time high-order uncertain linear multi-agent systems. At last, a numerical example is provided to demonstrate the correctness and effectiveness of the theoretical results.