Neural Network Learning and Robust Stabilization of Nonlinear Systems With Dynamic Uncertainties.

The problem of stabilizing a non-linear system whose parameters are not accurately known is discussed. Specifically, it is assumed that the non-linear system σ that needs to be stabilized has a recursive representation of the form x k + 1 =f(x k , u k ), where only a nominal description of the recursion function f is given, and where the actual recursion function may deviate from the nominal one. An explicit procedure for the stabilization of the system σ is derived. The procedure consists of the construction of a pair of dynamic compensators—a precompensator and a feedback compensator—which, when connected in a closed loop around the system σ, yield an internally stable control configuration. This configuration maintains its internal stability as long as the deviation of the recursion function f from nominality is within certain bounds. The compensators consist of recursive systems that can be readily implemented, and which are derived in an explicit form in terms of the nominal recursion function of the system Σ.

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC:15A09, 52A10, 74M05, 93D05.

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.

Input-to-state stability (ISS) has been shown to be a good way to describe the robust stability of nonlinear systems with exogenous disturbances. In this chapter, input-to-state stability of Lur’e singular systems with disturbances is investigated. The notion of input-to-state stability for nonlinear singular systems is defined based on the concept of ISS for standard state-space systems and the characteristics of singular systems. An LMI-based sufficient condition for ISS of Lur’e singular systems is derived by the classical ISS theory and a state feedback controller design method is proposed, such that the closed-loop system is ISS. Furthermore, ISS of Lur’e singular systems whose linear parts of the systems are allowed to be unstable is considered and a stability criterion is proposed.

IQC arguments for analysis of the 3-state Moore-Greitzer compressor system

Robust stabilization via iterative state steering with an application to chained-form systems

The use of avionics networks for flight control system has reduced aircraft weight, system installation, integration and maintenance costs as well as introduced time delays and structured uncertainties. This paper studies the robust stability and stabilisation problem of uncertain networked flight control system with random time delays. Some new sufficient criteria with simple mathematical forms for robust stability and stabilisation of networked flight control system are derived by a free weighting matrix approach. A memoryless state feedback robust controller and the maximum bound of time delay are obtained based on the solution of linear matrix inequalities (LMIs) which are established by the Lyapunov-Krasovskii stability theory. Numerical examples are given to illustrate the effectiveness of design procedure.

Interval Criterion of Robust Stability for Nonlinear System and Its Application to Flight Control

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

Semiglobal Robust Stabilization of Nonlinear Systems without Vanishing Condition

In this note we investigate robustness of optimally controlled systems in presence of nonlinear state-depending perturbations effecting the input additively together with nonlinear change in the control gain. We derive two types of robustness conditions for optimal controller. The first one can be verified a priori and the second — if the optimal control problem is solved. In the next section we propose robust controller design technique without solving optimal control problem. Next we study special systems and give numerical examples.

Unknown input based observer synthesis for an interval type-2 polynomial fuzzy system with time delays and uncertainties

The problem of robust stabilization for nonlinear systems with mismatched uncertainties is further studied by extending the results of Liao (1992) to the case where nonlinear systems either fail to have well-defined relative degree or have unstable zero dynamics. The approach is based on the approximate input/output linearization introduced by Hauser (1992) and an appropriate transformation introduced by Allgower (1997) which transforms the given nonlinear system into a special normal form. The proposed feedback controller guarantees that all the states will remain bounded.

In this chapter, we focus on developing adaptive optimal regulators for a class of continuous-time nonlinear dynamical systems through an improved neural learning mechanism. The main objective lies in that establishing an additional stabilizing term to reinforce the traditional training process of the critic neural network, so that to reduce the requirement with respect to the initial stabilizing control, and therefore, bring in an obvious convenience to the adaptive-critic-based learning control implementation. It is exhibited that by employing the novel updating rule, the adaptive optimal control law can be obtained with an excellent approximation property. The closed-loop system is constructed and its stability issue is handled by considering the improved learning criterion. After that, we apply the adaptation-oriented approximate optimal control strategy to perform robust stabilization when including complex nonlinearity and uncertainty. By considering the dynamical uncertainties, it is proven that the developed near-optimal control law can achieve uniform ultimate boundedness of the closed-loop state vector, thereby guaranteeing a certain extent of robustness for the uncertain nonlinear plant. Simulation for a classical nonlinear system and experiment on an overhead crane are conducted to verify the efficient performance of the present design methods, especially the major role that the stabilizing term performed.

The exact feedback linearization technique with partial state feedback and stable zero dynamics ensures local exponential stability. However, the method requires an accurate model, without uncertainties. The paper proposes an extension of the method for input-affine nonlinear systems with stable zero dynamics and matched uncertainties. Moreover, the weighted closed-loop sensitivity function is imposed using the ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>-gain analysis. The advantage of the proposed extension is that it ensures both robust stability and performance (tailored to reference tracking) for the uncertain nonlinear system affected by the limitations of zero dynamics. The two methodologies are then illustrated on a given academic example.