H ∞ preview tracking control of retarded state‐multiplicative stochastic systems

The problem of infinite‐horizon H _∞ state‐feedback tracking control for linear continuous time‐invariant retarded systems with stochastic parameter uncertainties is investigated. Two tracking patterns are considered depending on the nature of the reference signal; that is, whether it is measured online or previewed in a fixed time‐interval ahead. The stochastic uncertainties appear in the dynamics matrices for both the retarded and the non‐retarded states of the system. The delayed system is transformed via the input–output approach, to an uncertain norm‐bounded system. A new method that efficiently yields a min–max strategy to the solution of each of the aforementioned two cases is suggested where, given a specific reference signal, the controller plays against nature, which chooses the maximizing energy‐bounded disturbance. The theoretical results are demonstrated by two examples that show the impact of the delay length and the preview length on the system performance. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic H/sub /spl infin// tracking with preview for state-multiplicative systems

The problem of finite-horizon H/sub /spl infin// tracking for linear time-varying systems with stochastic parameter uncertainties is investigated. We consider three tracking patterns depending on the nature of the reference signal, i.e., whether it is perfectly known in advance, measured on line or previewed in a fixed time-interval ahead. The stochastic uncertainties appear in both the dynamic and measurement matrices of the system. For each of the above three cases a game theory approach is applied for the state-feedback case where, given a specific reference signal, the controller plays against nature which chooses the initial condition and the energy-bounded disturbance. The problems are solved using an expected value of the standard performance index over the stochastic parameters, where necessary and sufficient conditions are found for the existence of a saddle-point equilibrium. The infinite-horizon time-invariant tracking problem is also solved. The theory developed is demonstrated by a simple tracking example.     

Robust H∞ output‐feedback control of retarded state‐multiplicative stochastic systems

Linear, state‐delayed, continuous‐time systems are considered with both stochastic and norm‐bounded deterministic uncertainties in the state–space model. The problem of robust dynamic H_∞ output‐feedback control is solved, for the stationary case, via the input–output approach where the system is replaced by a nonretarded system with additional deterministic norm‐bounded uncertainties. A delay‐dependent result is obtained which involves the solution of a simple linear matrix inequality. In this problem, a cost function is defined which is the expected value of the standard H_∞ performance cost with respect to the stochastic parameters. A practical example taken from the field of guidance control is given that demonstrates the applicability of the theory. Copyright © 2010 John Wiley & Sons, Ltd.     

Additive‐state‐decomposition‐based tracking control framework for a class of nonminimum phase systems with measurable nonlinearities and unknown disturbances

This paper aims to propose an additive‐state‐decomposition‐based tracking control framework, based on which the output feedback tracking problem is solved for a class of nonminimum phase systems with measurable nonlinearities and unknown disturbances. This framework is to ‘additively’ decompose the output feedback tracking problem into two more tractable problems, namely an output feedback tracking problem for a linear time invariant system and a state feedback stabilization problem for a nonlinear system. Then, one can design a controller for each problem respectively using existing methods, and these two designed controllers are combined together to achieve the original control goal. The main contribution of the paper lies on the introduction of an additive state decomposition scheme and its implementation to mitigate the design difficulty of the output feedback tracking control problem for nonminimum phase nonlinear systems. To demonstrate the effectiveness, an illustrative example is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal Performance in Tracking Stochastic Signal Under Disturbance Rejection

The analysis method of optimal tracking performance is proposed for multiple‐input multiple‐output (MIMO) linear time‐invariant (LTI) systems under disturbance rejection. An H₂ criterion of the error signal between the output of the plant and the reference signal is used as a measure for the tracking performance. Spectral factorization is applied to obtain the optimal solution of the system tracking error. The explicit expressions are derived for this minimal tracking error with respect to random reference signals under disturbance rejection. It is shown that the nonminimum phase zeros, the zero direction, the unstable poles, the pole direction of a given plant, statistical characteristics of the reference input signal, and disturbance signal have a negative effect on a feedback system's ability to reduce the system error with disturbance rejection. The results show that the optimal tracking performance will further be damaged because of disturbance rejection. Some typical examples are given to illustrate the theoretical results.     

Almost fault‐tolerant tracking

A new formulation of the problem of almost fault‐tolerant perfect tracking in presence of disturbances is presented, and necessary and sufficient existence conditions are found, as well as a controller, which consists of a feedback block and two feedforward blocks. The feedback block is aimed for almost disturbance and fault decoupling, and the feedforward blocks are aimed for almost perfect tracking. A two‐step control design procedure such that the feedback block is designed independently of the feedforward blocks is presented. In order to simplify the necessary and sufficient existence conditions, as well as to present a physical interpretation of the existence conditions, the dimension of disturbance and fault vectors is reduced. In order to find a feedback controller, the physical interpretation of the existence conditions is used so that the four‐block‐type problem is transformed to a two‐block‐type problem. Examples are given, which illustrate that the controller of the article is practically realizable.     

Robust H ∞ control of stochastic linear switched systems with dwell time

The theory of H _∞ control of switched systems is extended to stochastic systems with state‐multiplicative noise. Sufficient conditions are obtained for the mean square stability of these systems where dwell time constraint is imposed on the switching. Both nominal and uncertain polytopic systems are considered. A Lyapunov function, in a quadratic form, is assigned to each subsystem that is nonincreasing at the switching instants. During the dwell time, this function varies piecewise linearly in time following the last switch, and it becomes time invariant afterwards. Asymptotic stochastic stability of the set of subsystems is thus ensured by requiring the expected value of the infinitesimal generator of this function to be negative between switchings, resulting in conditions for stability in the form of LMIs. These conditions are extended to the case where the subsystems encounter polytopic‐type parameter uncertainties. The method proposed is applied to the problem of finding an upper bound on the stochastic L₂‐gain of the system. A solution to the robust state‐feedback control problem is then derived, which is based on a modification of the L₂‐gain bound result. Two examples are given that demonstrate the applicability of the proposed theory. Copyright © 2013 John Wiley & Sons, Ltd.     

H∞ prediction and unconstrained H∞ predictive control: multi‐input–multi‐output case

Through the combination of the sequential spectral factorization and the coprime factorization, a k‐step ahead MIMO H_∞ (cumulative minimax) predictor is derived which is stable for the unstable noise model. This predictor and the modified internal model of the reference signal are embedded into the H_∞ optimization framework, yielding a single degree of freedom multi‐input–multi‐output H_∞ predictive controller that provides stochastic disturbance rejection and asymptotic tracking of the reference signals described by the internal model. It is shown that for a plant/disturbance model, that represents a large class of systems, the inclusion of the H_∞ predictor into the H_∞ control algorithm introduces a performance/robustness tuning knob: an increase of the prediction horizon enforces a more conservative control effort and, correspondingly, results in deterioration of the transient and the steady‐state (tracking error variance) performance, but guarantees large robustness margin, while the decrease of the prediction horizon results in a more aggressive control signal and better transient and steady‐state performance, but smaller robustness margin. Copyright © 2001 John Wiley & Sons, Ltd.     

Robust H∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching

This paper is concerned with the problems of robust stochastic stabilization and robust H_∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching. The parameter uncertainties are time‐varying norm‐bounded. For the robust stochastic stabilization problem, the purpose is the design of a state feedback controller which ensures the robust stochastic stability of the closed‐loop system irrespective of all admissible parameter uncertainties; while for the robust H_∞ control problem, in addition to the robust stochastic stability requirement, a prescribed level of disturbance attenuation is required to be achieved. Sufficient conditions for the solvability of these problems are obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, explicit expressions of the desired state feedback controllers are also given. An illustrative example is provided to show the effectiveness of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Output Tracking for One‐Dimensional Schrödinger Equation subject to Boundary Disturbance

In this paper, we are concerned with the output reference signal tracking for a one dimensional Schrödinger equation subject to general harmonic disturbance at one end and the control at the another end. Based on the reference signal, we design a state reference system and the output feedback control to cancel the disturbance and track the output reference signal. We show that the whole closed‐loop system is well‐posed and the performance output is tracking the reference signal. The numerical experiments are carried out to illustrate the approach effective.     

Output regulation for non‐square linear multi‐rate systems

We address the problem of regulating a subset of outputs of a linear time‐invariant plant with multi‐rate measurements so as to achieve asymptotic tracking of an exogenous signal generated by the free motion of a linear time‐invariant system, denoted by exosystem. A solution to this problem is required to yield closed‐loop stability and should be such that output regulation is achieved even in the presence of small plant uncertainties and exogenous disturbances also generated by the exosystem. Contrarily to previous works, we propose a solution to the general case where the plant may have more measured outputs than inputs. We show that this solution allows us to solve simultaneous stabilization and output regulation problems that are not possible to solve through the previous works. Besides incorporating an internal model of the exosystem, the key feature of our proposed controller is that it includes a system that blocks signals generated by the exosystem arriving to the controller from the non‐regulated outputs. Copyright © 2012 John Wiley & Sons, Ltd.     

STABILIZATION AND H∞ CONTROL FOR UNCERTAIN STOCHASTIC TIME‐DELAY SYSTEMS VIA NON‐FRAGILE CONTROLLERS

This paper considers the problems of robust non‐fragile stochastic stabilization and H_∞ control for uncertain time‐delay stochastic systems with time‐varying norm‐bounded parameter uncertainties in both the state and input matrices. Attention is focused on the design of memoryless state feedback controllers which are subject to norm‐bounded uncertainties. For both the cases of additive and multiplicative controller uncertainties, delay‐independent sufficient conditions for the solvability of the above problems are obtained. The desired state feedback controller can be constructed by solving a certain linear matrix inequality.     

On convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems

In this paper, the practical mean-square convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems modelled by the Itô-type stochastic differential equations with vast stochastic uncertainties is developed. We first design an extended state observer (ESO) to estimate both the unmeasured states and the stochastic total disturbance which includes unknown internal system dynamics, external stochastic disturbance without known statistical characteristics, unknown stochastic inverse dynamics, and uncertainty caused by the deviation of control parameter from its nominal value. The stochastic total disturbance is then cancelled (compensated) in the feedback loop. An ESO-based output-feedback control is finally designed analogously as for the system without uncertainties. The practical mean-square reference tracking and practical mean-square stability of the resulting closed-loop system are achieved. The numerical experiments are carried out to illustrate the effectiveness of the proposed approach.     

An optimal control approach to robust tracking of linear systems

In our early work, we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem. If the system is linear, then the optimal control problem becomes a linear quadratic regulator (LQR) problem, which can be solved by solving an algebraic Riccati equation. In this article, we extend the optimal control approach to robust tracking of linear systems. We assume that the control objective is not simply to drive the state to zero but rather to track a non-zero reference signal. We assume that the reference signal to be tracked is a polynomial function of time. We first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic Riccati equation. Because the state feedback is not always available in practice, we also investigated the output feedback. We show that if we place the poles of the observer sufficiently left of the imaginary axis, the robust tracking problem can be solved. As in the case of the state feedback, the observer and feedback can be obtained by solving two algebraic Riccati equations.     

Robust adaptive fault‐tolerant tracking control of multiple time‐delays systems with mismatched parameter uncertainties and actuator failures

This study deals with the problem of robust adaptive fault‐tolerant tracking for uncertain systems with multiple delayed state perturbations, mismatched parameter uncertainties, external disturbances, and actuator faults including loss of effectiveness, outage, and stuck. It is assumed that the upper bounds of the delayed state perturbations, the external disturbances and the unparameterizable time‐varying stuck faults are unknown. Then, by estimating online such unknown bounds and on the basis of the updated values of these unknown bounds from the adaptive mechanism, a class of memoryless state feedback fault‐tolerant controller with switching signal function is constructed for robust tracking of dynamical signals. Furthermore, by making use of the proposed adaptive robust tracking controller, the tracking error can be guaranteed to be asymptotically zero in spite of multiple delayed state perturbations, mismatched parameter uncertainties, external disturbances, and actuator faults. In addition, it is also proved that the solutions with tracking error of resulting adaptive closed‐loop system are uniformly bounded. Finally, a simulation example for B747‐100/200 aircraft system is provided to illustrate the efficiency of the proposed fault‐tolerant design approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust tracking problem for continuous time stochastic uncertain systems

We propose a finite‐horizon robust minimax tracking controller design method for time‐varying continuous time stochastic uncertain systems. The uncertainty in the system is characterized by a set of probability measures under which stochastic noises, driving the system, are defined. A minimax optimal tracking controller is derived from the solution of a risk‐sensitive linear quadratic Gaussian control problem. Also a numerical example is presented to illustrate the characteristics of proposed tracking controller. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Error‐constrained finite‐horizon tracking control with incomplete measurements and bounded noises

This paper is concerned with the finite‐horizon tracking control problem for discrete nonlinear time‐varying systems with state delays, bounded noises and incomplete measurement output. The exogenous bounded noises are unknown and confined to specified ellipsoidal sets. A deterministic measurement output model is proposed to account for the incomplete data transmission phenomenon caused by possible sensor aging or failures. The aim of the addressed tracking control problem is to develop an observer‐based control over a finite‐horizon such that, for the admissible time delays, nonlinearities and bounded noises, both the quadratic tracking error and the estimation error are not more than certain upper bounds that are minimized at every time step. A recursive linear matrix inequality approach is used to solve the problem addressed. The observer and controller parameters are characterized in terms of the solution to a convex optimization problem that can be easily solved by using the semi‐definite programme method. A simulation example is exploited to illustrate the effectiveness of the proposed design procedures. Copyright © 2011 John Wiley & Sons, Ltd.