Dynamic Output Feedback Stabilization of Non-uniformly Observable Dissipative Systems

This technical note deals with asymptotic stabilization of general nonlinear discrete-time systems by dynamic output feedback control. First, a sufficient condition for the dynamic output feedback stabilization with the closed-loop domain of attraction (DOA) estimation is given. For a given Lyapunov function of the plant state and the controller state, the negative-definite domain (NDD) in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space is proposed, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z_+,u,z$</tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y$</tex-math></inline-formula> denote the next controller state, the current control input, the current controller state and the current plant output, respectively. All points <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+;u;z;y)$</tex-math></inline-formula> in the NDD can make the time difference of the Lyapunov function negative-definite for the controller state <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> and all plant states in a set of plant states, which is determined by the controller state <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> and the plant output <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y$</tex-math></inline-formula>. The NDD in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space could be viewed as an unstructured dynamic output feedback stabilization controller set. And any invariant subset of the projection of the NDD in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space onto the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x)$</tex-math></inline-formula>-space is an estimate of the DOA for the closed-loop, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> denotes the current plant state. Second, based on the NDD, the controller design problem is converted to a set approximation problem in Euclidean space. Using numerical set approximation methods, dealing with strong nonlinearities is no longer the bottleneck. An estimation method for the unstructured dynamic output feedback stabilization controller set is proposed based on interval analysis, which is a kind of guaranteed numerical method for approximating sets.

The aim of the paper is the output feedback stabilization of a class of single-input single-output (SISO) nonlinear systems. We deal with a class of systems, so-called triangular form systems, that may have unobservable and unstabilizable approximate linearization about the origin. This kind of systems, referred to as singular triangular forms, may neither be stabilized by a smooth state feedback nor smoothly detectable. However, under certain simple conditions, it is known to be stabilizable by a static state feedback. A solution to continuous dynamic output feedback stabilization is proposed. This is based on the construction of the continuous observer and nonlinear analogue of the well known separation principle. The resulting observer may be computed and tuned in a straightforward constructive way. Numerical simulation of selected examples are performed to underline the applicability of the presented approach and to illustrate good convergence of both observer and controller system.

New results on dynamic output state feedback stabilization of some class of time-varying nonlinear Caputo derivative systems

This correspondence paper addresses the problem of output feedback stabilization of control systems in networked environments with quality-of-service (QoS) constraints. The problem is investigated in discrete-time state space using Lyapunov's stability theory and the linear inequality matrix technique. A new discrete-time modeling approach is developed to describe a networked control system (NCS) with parameter uncertainties and nonideal network QoS. It integrates a network-induced delay, packet dropout, and other network behaviors into a unified framework. With this modeling, an improved stability condition, which is dependent on the lower and upper bounds of the equivalent network-induced delay, is established for the NCS with norm-bounded parameter uncertainties. It is further extended for the output feedback stabilization of the NCS with nonideal QoS. Numerical examples are given to demonstrate the main results of the theoretical development.

Further results on feedback stabilization control design of Boolean control networks

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.

This paper suggests sufficient conditions for asymptotically stable dynamical output feedback controller design based on the circle criterion. It is shown that a dynamic output feedback stabilization problem with impending problems of finite escape time, previously attacked by observer-based design, can be successfully solved using circle criterion design. Stability of the closed-loop system is global and robust to parameter uncertainty.

This paper is concerned with stability analysis and output feedback control design of fuzzy bilinear systems with time-varying delay. First, we consider the stability of the closed-loop systems by assuming a special form of a fuzzy controller. A delay-dependent stability condition, which is less conservative than delay-independent one, is given in terms of linear matrix inequality (LMI). Based on such a stability condition, we propose a state feedback stabilizing control design for fuzzy bilinear systems with time-varying delays. Next, we consider a fuzzy observer design for the same class of systems, and propose an observer design based on another delay-dependent condition. Then, we show that state feedback stabilizing controller and observer make an output feedback stabilizing controller. Finally, we give a numerical example to illustrate our design procedures and to show the effectiveness of our approach.

Using the Lyapunov function method, this study investigates both state and output feedback stabilisation control design problems for fractional order non‐linear systems in the lower triangular form, and presents a number of new results. First, some new properties for Caputo fractional derivative are presented. Second, by introducing appropriate transformations of coordinates, the feedback stabiliser design problem is converted into the determination of finding some parameters, which can be obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, both state and output feedback stabilisers are explicitly designed to make the closed‐loop system asymptotically stable. The study of an illustrative example shows that the obtained results are effective in designing feedback stabilisers for fractional order non‐linear systems in the lower triangular form.

The paper mainly deals with the problem of finite-time stabilization of linear time-varying systems. A dynamic output feedback controller is designed, which is able to stabilize the linear time-varying systems in finite time. By virtue of extended piecewise constant method, novel criteria for the existence of a dynamic output feedback controller is established in terms of linear matrix inequalities. Compared with the existing method, the proposed method is more efficient from a computational point of view. A simulation is given to illustrate the effectiveness of the obtained result.

This paper investigates the dynamic output feedback control for a class of nonlinear time-delay system with the triangular structure-type uncertain nonlinearity satisfying the linear growth condition. A dynamic observer-based output feedback robust stabilizing controller is presented on the basis of Lyapunov-Razumikhin function design strategy. A numerical example is given to illustrate the effectiveness of this method.

This paper addresses an output feedback stabilization problem of a class of nonlinear systems with nonlinearities satisfying the non‐decreasing propertyvialimited‐capacity communication channels. Using ideas from absolute stability theory, we propose an algorithm that consists of an observer and a coder/decoder‐controller pair. Sufficient conditions for the stabilization problem are provided and presented in terms of linear matrix inequalities. Copyright © 2006 John Wiley &amp; Sons, Ltd.

LMI-based Dynamic Output Feedback Stabilization of Uncertain Systems

The output feedback stabilization of polytopic-type uncertain discrete systems with interval-like time-varying state and input delays is studied. Based on a new bounding inequality technique, combining a parameter-dependent Lyapunov functional, a stability criterion is firstly presented in terms of a set of simple convex feasibility tests. Then, the output feedback stabilization conditions are formulated in the form of non-convex matrix inequalities, of which a feasible solution can be obtained by solving an LMI-based minimization problem. The newly proposed inequality lies in the partitioning idea of the varying interval and shows its more tightness over some existing bounding techniques. No free weighting matrix is involved. Two illustrative examples are finally given to verify the advantage and effectiveness of the proposed method.

This paper considers the problem of global stabilization by output feedback for a class of uncertain nonholonomic systems. By combining the output feedback domination design and nonsmooth change of coordinates together, we explicitly construct a simple output feedback controller, which can universally stabilize a whole family of uncertain nonholonomic systems. The effectiveness of the proposed approach is illustrated by solving the output feedback stabilization for wheeled mobile robots.