Difference feedback can stabilize uncertain steady states

The paper is concerned with the stabilization of uncertain steady states by the state difference feedback. The feedback method has a peculiar feature that it uses only the difference between the present state x(t) and the past state x(t-T), considering exact information on the steady state is unavailable. Hitherto a condition is known under which such stabilization can not be realized. The article conversely shows that the state difference feedback can stabilize only if the exclusion condition is not true. Furthermore a dynamic output difference feedback is shown to be able to stabilize under quite a mild condition that the steady state is not associated with zero eigenvalues. The ability of the method is illustrated by using a cart-pendulum system which moves along a one dimensional varying slope     

State feedback for non-linear control systems

A theory of static state feedback for non-linear discrete-time systems is developed. The theory applies to non-linear systems possessing a recursive representation of the form x _k+1 =?(x _k , u_k ), where ? is a continuous function, and it deals with the construction of continuous state feedback functions that internally stabilize a given system. The theory yields an explicit method for the computation of stabilizing feedback functions, and several examples of the computation of such functions are provided.     

Internal model control with feedback compensation for uncertain non-linear systems

Control of non-linear systems by model-based strategies is often affected by model uncertainties. For these systems, internal model control with feedback compensation (IMC-FC), which consists of a non-linear model control and an error feedback loop, to achieve disturbance attenuation and offset-free performance, is proposed in this study. The matching conditions for the uncertainties of non-linear systems are not necessary, and adjustable parameters can easily be tuned to satisfy the particular specification. The underlying theoretical approach for the feedback compensation is the Lyapunov stability theory. The effectiveness of the proposed approach is illustrated by a simulation example on the composition control of a continuous stirred tank reactor. The theoretical analysis and simulation results show that the proposed IMCFC can reduce the effect of modelling uncertainties such that the performance of the control system is greatly improved.     

Locally optimal feedback strategy for non-linear systems with uncertain parameters

This paper broadens the classical optimal control approach to the optimal regulation of a non-measurable, but parametrized uncertainty. A locally optimal feedback strategy (LOFS) is derived by minimizing a measure of the performance index sensitivity with respect to the uncertainty parameter vector. In the state feedback case the LOFS can be characterized by an extension of the well-known Matrix-Riccati equation. Conditions for stability and finiteness of the resulting linear feedback law are derived for the stationary LQ case with uncertain parameters. Different numerical algorithms also covering the output feedback case are presented. Examples demonstrate the superior performance of the proposed feedback for problems with a significant parametric uncertainty.     

On the output feedback stability for non-linear uncertain control systems

In this paper, we will study the stability problem by output feedback of a class of uncertain control systems, whose nominal part is linear and whose uncertain part is norm-bounded by a known function. We propose an output controller that forces the state to the origin, in the global exponential stability sense.     

Adaptive feedforward and feedback control of non-linear time-varying uncertain systems

An adaptive regulation scheme is proposed for a class of non-linear time-varying systems with parametric uncertainties. The proposed approach is based upon a combination of the adaptive backstepping design method and a feedforward control scheme to design a non-linear adaptive feedforward and feedback controller, such that robust output tracking can be achieved even in the presence of structured uncertainties, as well as time-varying, measurable disturbances. Although the systematic design procedure does not a priori satisfy the feedback linearizable system with triangular structures, however, the constructed condition must be satisfied to ensure that the control scheme has a stable inversion. Under the feasibility condition, the states of the resulting closed-loop system would be guaranteed boundedness and converge to a bounded set. Finally, the proposed methodology is illustrated by a chemical reactor example.     

Procedure for simultaneously stabilizing a collection of non-linear systems using linear state feedback control

The problem of simultaneously stabilizing a finite collection of non-linear systems is considered. A new and simple approach to designing a single state (linear) feedback control which simultaneously stabilizes this collection of non-linear systems is presented. The concepts of the function norm and the Bellman-Gronwall inequality are used to investigate the conditions for simultaneous stabilization. A design procedure for single state feedback control, is also presented. Several examples are given to demonstrate the validity of our results.     

State feedback control of networked systems with uncertain plant

This article, considers the problem of state feedback control of networked systems with an uncertain plant. The signals for feedback periodically switch between the plant state and the state of a model of the plant according to whether the plant state is available from the communication network or not. The model is used to generate control signals when the plant state is not available from the network. A sufficient condition for the robust exponential stability of the closed-loop system is derived in terms of the network dwell time and the system parameters. Examples are also worked out to demonstrate numerical procedures for designing state feedback controller of the system based on the obtained results. Simulations show the feasibility and efficiency of the proposed methods.     

Application of the quantitative synthesis of feedback systems with uncertain non-linear plants

An improvement of the quantitative feedback theory for uncertain non-linear single-input-single-output (SISO) systems is obtained, by using a new procedure to convert the uncertain non-linear plant into an “equivalent” linear time-invariant plant set. The method is applied to the control of uncertain Van der Pol plant and Duffing plant. The results, compared with those obtained by following the classical approach, show the superiority of the new technique in reducing the “cost of feedback” in bandwidths of the compensation, for the same extent of plant uncertainty.     

不确定广义系统鲁棒镇定控制器的设计

讨论了线性定常广义系统的稳定性, 给出了这种系统H_∞ 范数形式的稳定性判据, 在此基础上, 利用线性矩阵不等式 (LMI)方法讨论了含有不确定参数的线性广义控制系统的鲁棒镇定问题, 并相应地给出了鲁棒镇定控制器的设计.     

A procedure for simultaneously stabilizing a collection of single input linear systems using non-linear state feedback control

This paper considers the problem of simultaneously stabilizing a finite collection of single input linear systems. This stabilization is achieved using a single non-linear state feedback controller. The stability of the resulting closed loop system is established using a collection of quadratic Lyapunov functions. The main result of this paper is a sufficient condition for simultaneous stabilizability. Furthermore, when this sufficient condition is satisfied, the paper gives a formula for constructing the stabilizing feedback control law. The paper includes an example in which the stabilization procedure is applied to the stabilization of an F4E fighter aircraft.     

Parallelotopic and practical observers for non-linear uncertain systems

For a class of dynamical systems, with uncertain non-linear terms considered as unknown inputs, we give suffcient conditions for observability. We show also that there does not exist any exact observer independent of the unknown inputs. Under the additional assumption that the uncertainty is bounded, we build practical observers whose error converges exponentially towards an arbitrarily small neighbourhood of the origin. Under the hypothesis that bounds are available for the uncertain terms, we build parallelotopic observers providing time-varying bounds for the state variables, even when the system is not observable for unknown inputs. These results are illustrated with a biological model of a structured population.     

Quantized feedback control for linear uncertain systems

This paper studies robust control problems under the setting of quantized feedback. We consider both the static and dynamic logarithmic quantizers. In the static quantization case, the quantizer has an infinite number of levels, and the design problem is to find the minimal quantization density required to achieve a given control objective. In the dynamic quantization case, the problem is to minimize the number of quantization levels to achieve a given control objective. We present a number of results for different controller‐quantizer configurations. These results are developed using the so‐called sector bound approach for quantized feedback control, which was initiated by the authors previously for systems without uncertainties. Copyright © 2009 John Wiley & Sons, Ltd.     

State feedback control for uncertain switched systems with interval time‐varying delay

Stability criteria and hybrid controllers' design problems for a class of uncertain switched systems with interval time‐varying delay are considered in this paper. Based on the average dwell time method, by choosing a new appropriate Lyapunov‐Krasovskii functional which fully utilizes the information of both the lower and upper bounds of the interval time‐varying delay, new delay‐range‐dependent stability criteria and stabilization conditions are first derived in terms of linear matrix inequalities. Moreover, in order to obtain much less conservative results, a tighter bounding for some terms is estimated and no redundant matrix variable is introduced. Finally, two numerical examples are given to demonstrate the applicability and the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Designing stabilizing control of uncertain systems by quasiconvexoptimization

The design of stabilizing linear output feedback control of uncertain systems is a computationally demanding optimization problem as local minima may exist which are distinct from the global minimum. However, in some special cases, quasiconvexity can be proven through a simple redefinition of variables and/or a reformulation of the problem in the dual form. Such special cases include state feedback control for both discrete and continuous time systems     

Optimal feedback control of non-linear systems

A method is proposed for computing optimal nonlinear feedback control laws. It is shown that the feedback loop satisfies a system of quasi-linear partial differential equations. This system is of first order when the dimensions of the state and the control vector are the same. Hereby, it degenerates into algebraic equations when the performance index does not depend on the control. These important results offer new ways for determining optimal feedback laws. Connections with the Volterra series and the Hamilton-Jacobi-Bellman equation are treated. Some examples are given to illustrate the advantages of this method.     

State feedback control of real-time discrete event systems with infinite states

In this paper, we study a state feedback supervisory control of timed discrete event systems (TDESs) with infinite number of states modelled as timed automata. To this end, we represent a timed automaton with infinite number of untimed states (called locations) by a finite set of conditional assignment statements. Predicates and predicate transformers are employed to finitely represent the behaviour and specification of a TDES with infinite number of locations. In addition, the notion of clock regions in timed automata is used to identify the reachable states of a TDES with an infinite time space. For a real-time specification described as a predicate, we present the controllability condition for the existence of a state feedback supervisor that restricts the behaviour of the controlled TDES within the specification.     

Further results on stabilizing controllers for discrete non-linear stochastic systems

More relationships that exist among stabilizability conditions of controllers designed for a class of discrete non-linear stochastic systems are described. The conditions are in terms of both mean square and almost sure asymptotic stability of the controlled systems and provide converse results to those presented earlier     

Optimal feedback control of non-linear systems

A fundamental method to solve functional optimization control problems is achieved through Hamilton-Jacobi theory. This method leads to feedback laws satisfying the HJB equation in a sense. On the basis of this approach, a design technique, applicable to a large class of optimization problems, is described. The technique is developed for systems of vth order. An example demonstrates the effectiveness of the determined controller to stabilize the inverted pendulum.