Design of robust controllers for uncertain dynamical systems

The general design approach for robust controllers of uncertain dynamical systems is considered. The design approach, utilizing only the bound of uncertainty, can be used to construct various classes of controllers which render the systems practically stable. No statistical information of the uncertainty is required, and the system performance is described in a deterministic manner     

Near optimal interval observers bundle for uncertain bioreactors

In this paper we design an interval observer for the estimation of unmeasured variables of uncertain bioreactors. The observer is based on a bounded error observer, as proposed in [Lemesle, V., & Gouzé, J.-L. (2005). Hybrid bounded error observers for uncertain bioreactor models. Bioprocess and Biosystems Engineering, 27, 311-318], that makes use of a loose approximation of the bacterial kinetics. We first show how to generate guaranteed upper and lower bounds on the state, provided that known intervals for the initial condition and the uncertainties are available. These so-called framers depend on a tuning gain. They can be run in parallel and the envelope provides the best estimate. An optimality criterion is introduced leading to the definition of an optimal observer. We show that this criterion provides directly a gain set containing the best framers. The method is applied to the estimation of the total biomass of an industrial wastewater treatment plant, demonstrating its efficiency.     

Stabilizing controllers using observers for uncertain systems with delays

In this paper, we study the problem of designing stabilizing controllers using observers for a class of uncertain systems with state and input delays. The uncertainties are assumed to be norm bounded. By employing a Lyapunov-Krasovskii functional approach, it is proven that the closed-loop observer-based controlled system is asymptotically stable. The observer and state feedback controllers are determined by solving two linear matrix inequalities which are independent of the delay factors. A simulation example is given to illustrate the theoretical developments.     

Robust observers for neutral jumping systems with uncertain information

In this paper, the problem of designing observer for a class of uncertain neutral systems. The uncertainties are parametric and norm-bounded. Both robust observation and robust H_∞ observation methods are developed by using linear state-delayed observers. In case of robust observation, sufficient conditions are established for asymptotic stability of the system, which is independent of time delay. The results are then extended to robust H_∞ observation which renders the augmented system asymptotically stable independent of delay with a guaranteed performance measure. Furthermore, a memoryless state-estimate feedback is designed to stabilize the closed-loop neutral system. In all cases, the gain matrices are determined by linear matrix inequality approach. Two numerical examples are presented to illustrate the validity of the theoretical results.     

Fault detection for discrete-time LPV systems using interval observers

This paper is concerned with the fault detection (FD) problem for discrete-time linear parameter-varying systems subject to bounded disturbances. A parameter-dependent FD interval observer is designed based on parameter-dependent Lyapunov and slack matrices. The design method is presented by translating the parameter-dependent linear matrix inequalities (LMIs) into finite ones. In contrast to the existing results based on parameter-independent and diagonal Lyapunov matrices, the derived disturbance attenuation, fault sensitivity and nonnegative conditions lead to less conservative LMI characterisations. Furthermore, without the need to design the residual evaluation functions and thresholds, the residual intervals generated by the interval observers are used directly for FD decision. Finally, simulation results are presented for showing the effectiveness and superiority of the proposed method.     

Estimation of uncertain models of activated sludge processes with interval observers

This paper deals with the designs of observers (software sensors) for uncertain models of wastewater treatment. We assume that the model (4 state variables) is such that the growth rate is unknown and other parameters (e.g. the organic concentration in the influent) are uncertain, i.e. we only know their upper and lower bounds. Given the measurements of the dissolved oxygen concentration, we build interval observers, giving dynamic bounds containing the variables to estimate. The width of these bounds are related to the width of the uncertainty bounds of the parameters. In some cases, we show that we can estimate exactly the unknown variables despite the uncertainties on the model.     

Construction of interval observers for continuous-time systems with discrete measurements

We consider continuous-time systems with input, output and additive disturbances in the particular case where the measurements are only available at discrete instants and have disturbances. To solve a state estimation problem, we construct continuous–discrete interval observers that are asymptotically stable in the absence of disturbances. These interval observers are composed of two copies of the studied system and of a framer, accompanied with appropriate outputs which give, componentwise, upper and lower bounds for the solutions of the studied system.     

Nonlinear observers for a class of continuous-time uncertain state delayed systems

This paper is concerned with the problem of observer design for a class of time-delay nonlinear systems with parameter uncertainties. The purpose of this problem is to design the gain-scheduled state observers such that, for the addressed nonlinearities as well as all admissible parameter uncertainties in state and output equations, the observation process remains globally exponentially stable, independently of the time delay. The nonlinearities are assumed to satisfy the global L ipschitz conditions, and the parameter uncertainties are allowed to be time varying, unstructured and norm bounded. An effective matrix inequality methodology is developed to solve the proposed problem. W e derive the conditions for the existence of the desired robust nonlinear observers, and then characterize the analytical expression of these observers. Two numerical examples demonstrate the validity and applicability of the present approach.     

Asymptotic stability of controlling uncertain dynamical systems

Asymptotic stabilization of general uncertain dynamical systems is investigated. A new class of continuous feedback controls is proposed to guarantee asymptotic stability for any uncertain systems whose nominal system is uniformly asymptotically stable. The analysis is based on a new theoretical result on asymptotical stability. The required information about uncertain dynamics in the system is merely that the uncertainties are bounded in euclidean norm by a known function of the system state.     

Parameter and state estimation for uncertain linear systems by multiple observers

A parameter and state estimation problem is considered for uncertain linear time-invariant systems. Under a certain condition, it is shown that the state of the system can be asymptotically described by a linear combination of state estimates generated by suitable multiple observers, where the weights of the linear combination are the parameters to be estimated. Using this property, a computation method is proposed for simultaneous estimation of the parameters and the state from the output data of the plant and multiple observers.     

State observation of nonlinear uncertain dynamical systems

This note proposes new types of observers for nonlinear dynamical systems subjected to bounded nonlinearities or uncertainties. The design of these observers utilizes techniques related to variable structure systems theory. A measure for the rate at which the estimates converge to the actual states is derived.     

Optimal-observer design for linear dynamical systems with uncertain parameters

This paper presents a design technique for the synthesis of robust observers for linear dynamical systems with uncertain parameters. The perturbations under consideration are modelled as unknown but bounded disturbances and an ellipsoidal set-theoretic approach is used to formulate the optimal-observer design problem. The optimal criterion introduced here is the minimization of the ‘size’ of the bounding ellipsoid of the estimation error. A necessary and sufficient condition for this optimal design problem is presented. The results are stated in terms of a reduced-order observer with constant gain matrix, which is then determined by solving a matrix Riccati-type equation. Furthermore, a gradient-search algorithm is presented to find the optimal solution when the free parameter that enters in the construction of the bounding ellipsoids of the estimation error is considered as a design parameter. The effectiveness of the proposed approach is illustrated through a numerical example.     

Ellipsoidal bounds for uncertain linear equations and dynamical systems

In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the -procedure. This formulation leads to convex optimization problems that can be essentially solved in O(n³)—n being the size of unknown vector—by means of suitable interior point barrier methods, as well as to closed form results in some particular cases. We further show that the uncertain linear equations paradigm can be directly applied to various state-bounding problems for dynamical systems subject to set-valued noise and model uncertainty.     

Design of observers for non-linear time-varying systems

Novel stable observers are proposed for discrete, time-varying systems having time-varying, sector-bounded non-linearities. First, the form of the non-linearity is assumed to be known for all time and a stable non-linear filter is presented. Then assuming that the form of the non-linearity is not known but only a sector bound is given, we propose a linear filter and an upper bound on the sector constant to obtain a stable observer. The convergence conditions for the proposed observers are given directly in terms of the design parameters and are, therefore, very easy to check.     

Adaptive hybrid intelligent control for uncertain nonlinear dynamical systems

A new hybrid direct/indirect adaptive fuzzy neural network (FNN) controller with a state observer and supervisory controller for a class of uncertain nonlinear dynamic systems is developed in this paper. The hybrid adaptive FNN controller, the free parameters of which can be tuned on-line by an observer-based output feedback control law and adaptive law, is a combination of direct and indirect adaptive FNN controllers. A weighting factor, which can be adjusted by the tradeoff between plant knowledge and control knowledge, is adopted to sum together the control efforts from indirect adaptive FNN controller and direct adaptive FNN controller. Furthermore, a supervisory controller is appended into the FNN controller to force the state to be within the constraint set. Therefore, if the FNN controller cannot maintain the stability, the supervisory controller starts working to guarantee stability. On the other hand, if the FNN controller works well, the supervisory controller will be deactivated. The overall adaptive scheme guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. Two nonlinear systems, namely, inverted pendulum system and Chua's (1989) chaotic circuit, are fully illustrated to track sinusoidal signals. The resulting hybrid direct/indirect FNN control systems show better performances, i.e., tracking error and control effort can be made smaller and it is more flexible during the design process.     

Design of a class of new nonlinear disturbance observers based on tracking differentiators for uncertain dynamic systems

The main contribution of this paper is to present a general design method of new nonlinear disturbance observer (NDO) based on tracking differentiator (TD) for uncertain dynamic system. The stability and convergence of the proposed NDO can be guaranteed by TD. This new NDO can be used to estimate many types of uncertain disturbances, and can overcome the disadvantages of existing NDOs that need the priori information concerning the upper and lower bounds of the disturbance and its ith derivative’s Lipschitz upper bound. It can be also applied in uncertain dynamic system for various purposes such as disturbance estimate and compensation, solving the problem of control input constraint, and reducing even eliminating chattering of control input. Simulation results are presented to show the effectiveness of the developed NDO.     

Deterministic control for a new class of uncertain dynamical systems

We study the problem of stabilization of nonlinear multivariable uncertain systems. We show that for a nonlinear plant with nonlinear input, a class of stabilizing continuous-type controllers can he constructed.