Covariance Control Theory Research Articles

Pole Location and Input Constrained Robust Fuzzy Control for T-S Fuzzy Models Subject to Passivity and Variance Requirements

In this paper, a robust fuzzy controller for stochastic nonlinear systems subject to multiple performance constraints is discussed. To solve the problem of stochastic behaviors in nonlinear systems, the covariance control theory and passive control theory are applied based on the concept of energy. Additionally, the pole placement method is considered for the better transient behaviors of the system responses. However, it is known that the control inputs and maximum overshoot of the system responses may become bigger at the same time when the settling time or converge rate of the system is required to be faster. Due to this reason, the input constraint is also considered in the fuzzy controller design method to limit the value of the control gain. Moreover, an effective robust control method is applied to deal with the perturbation of the nonlinear systems. Based on the above performance constraints, the sufficient conditions can be obtained to achieve the stability in the sense of mean square and the multi-performance requirements. Finally, simulation results of the nonlinear synchronous generator system are presented to verify the feasibility and efficiency of the proposed control method.

A Covariance Feedback Approach to Covariance Control of Nonlinear Stochastic Systems

In this paper, the covariance control algorithm for nonlinear stochastic systems using covariance feedback is studied. Covariance control of nonlinear systems scenario involves the theory of covariance control based on the idea of the covariance feedback. Therefore, the proposed covariance control algorithm is derived for our case, firstly by applying the covariance control method and linear approximation of nonlinear systems, and then it is achieved by adopting this method for a class of nonlinear stochastic systems by using feedback linearization idea and a covariance feedback controller. The effectiveness of the proposed covariance feedback algorithm is studied using numerous simulations concerning different nonlinear case studies.

Mixed norm H 2/H ∞ and entropy covariance control: a convex optimisation approach

ABSTRACT In this paper, we develop a covariance control problem to address a tradeoff between performance and disturbance attenuation. In particular, we formulate a mixed-norm and entropy covariance control problem that guarantees that the state covariance of an uncertain dynamical system driven by white noise is upper bounded in the sense of the cone of nonnegative definite matrices by a given threshold matrix via state feedback as well as output feedback control. This is accomplished by combining covariance control theory and mixed norm control theory. By using suitable transformations involving dynamic weighting on the complimentary sensitivity system transfer function, the proposed formulation is applicable to robustness problems involving nominal performance subject to a robust stability requirement. The proposed formulation allows for solutions via semidefinite programming. Finally, two illustrative numerical examples are provided to show the efficacy of the proposed approach.

Fuzzy Control with Pole Assignment and Variance Constraints for Continuous-time Perturbed Takagi-Sugeno Fuzzy Models: Application to Ship Steering Systems

The stability analysis and controller design of stochastic systems have become much more important because the stochastic behaviors usually exist in practical nonlinear systems. In this paper, a robust fuzzy controller design approach is proposed with multiple constraints, including state variance constraints, output variance constraints, and pole placement constraints. At first, nonlinear systems are expressed as the Takagi-Sugeno fuzzy model, and the parallel distributed compensation method is applied to design the robust fuzzy controllers. Next, considering the stability analysis and the performance constraints of perturbed Takagi-Sugeno fuzzy models, Lyapunov conditions are developed based on covariance control theory, pole placement theory and robust control theory. By constructing the stability conditions with multiple constraints, the proposed fuzzy control problem can be effectively transferred into the linear matrix inequality problem. It can be solved by the convex optimal programming algorithm. At last, a nonlinear ship steering system is selected to verify the effectiveness and applicability of the proposed robust fuzzy controller design method.

Multi-constrained fuzzy intelligent control for uncertain discrete systems with complex noises: an application to ship steering systems

ABSTRACTIn this paper, a performance constrained fuzzy controller design is investigated for the nonlinear uncertain discrete-time ship steering system with complex noises. The performance constraints considered in this paper include individual state variance constraint and passivity constraint. The nonlinear discrete-time ship steering system considered in this paper is represented by a discrete-time Takagi–Sugeno fuzzy model with perturbations, additional noise and multiplicative noise. According to this complex nonlinear discrete-time stochastic system, a fuzzy controller design approach is investigated based on Lyapunov theory, passivity theory and covariance control theory. Some sufficient conditions are derived to satisfy stability constraint, individual state variance constraint and passivity constraint, simultaneously. These sufficient conditions are constructed as linear matrix inequality forms that can be solved by the convex optimal programming algorithm. Finally, some simulation results are provided to verify the validity and applicability of the proposed fuzzy control approach with multiple performance constraints.

Fuzzy Stabilization for Nonlinear Discrete Ship Steering Stochastic Systems Subject to State Variance and Passivity Constraints

For nonlinear discrete-time stochastic systems, a fuzzy controller design methodology is developed in this paper subject to state variance constraint and passivity constraint. According to fuzzy model based control technique, the nonlinear discrete-time stochastic systems considered in this paper are represented by the discrete-time Takagi-Sugeno fuzzy models with multiplicative noise. Employing Lyapunov stability theory, upper bound covariance control theory, and passivity theory, some sufficient conditions are derived to find parallel distributed compensation based fuzzy controllers. In order to solve these sufficient conditions, an iterative linear matrix inequality algorithm is applied based on the linear matrix inequality technique. Finally, the fuzzy stabilization problem for nonlinear discrete ship steering stochastic systems is investigated in the numerical example to illustrate the feasibility and validity of proposed fuzzy controller design method.

Multi-objective control for stochastic model reference systems based on LMI approach and sliding mode control concept

In this article, a multi-objective control u(t) is designed for stochastic model reference systems to achieve the following three objectives simultaneously: the pole placement constraint, H ∞-norm constraint and individual error state variance constraint. Using the invariance property of sliding mode control, the reference model input and the plant error term will disappear on the sliding mode of the error system. By combining the upper bound covariance control theory, pole placement skill and H ∞-norm control theory, a controller, in which the control feedback gain matrix is synthesised utilising linear matrix inequality (LMI) approach, is derived to achieve the above multiple objectives. Furthermore, a practical example for the problem of ship yaw-motion systems is adopted to illustrate the proposed method.

Variance-Constrained Multiobjective Control and Filtering for Nonlinear Stochastic Systems: A Survey

The multiobjective control and filtering problems for nonlinear stochastic systems with variance constraints are surveyed. First, the concepts of nonlinear stochastic systems are recalled along with the introduction of some recent advances. Then, the covariance control theory, which serves as a practical method for multi-objective control design as well as a foundation for linear system theory, is reviewed comprehensively. The multiple design requirements frequently applied in engineering practice for the use of evaluating system performances are introduced, including robustness, reliability, and dissipativity. Several design techniques suitable for the multi-objective variance-constrained control and filtering problems for nonlinear stochastic systems are discussed. In particular, as a special case for the multi-objective design problems, the mixedH2/H∞control and filtering problems are reviewed in great detail. Subsequently, some latest results on the variance-constrained multi-objective control and filtering problems for the nonlinear stochastic systems are summarized. Finally, conclusions are drawn, and several possible future research directions are pointed out.

State covariance assignment problem

Covariance control theory provides a parameterisation of all controllers which assign a specified state covariance matrix to the closed-loop system according to different requirements on system performance. In this study, an innovative scheme for covariance system description based on the original linear stochastic system is proposed. The main idea of the proposed scheme is based on a special rearrangement of the corresponding covariance matrix Riccati equation to a new linear deterministic state space system. The variances of the states and cross-covariance between each two states of the original system would be the states of the new covariance system. By some mathematical manipulations, the covariance assignment problem is reformulated as a standard disturbance rejection problem. Since the new covariance system is linear and deterministic, all conventional and well-defined control strategies can be applied on it. Results are presented by a covariance feedback control law based on an integral control action applied to the new covariance system. The control law causes a stable closed-loop system and assigns a pre-specified state covariance matrix to the states of the closed-loop system.

Multi-objective controller design for bi-linear stochastic systems via sliding mode control concept

A sliding mode controller u( t) is designed to achieve the following three objectives simultaneously: eigenvalue placement, H∞ norm constraint, and individual state variance constraint for bi-linear stochastic systems. By using the invariance property of sliding mode control, the matched bi-linear term of the system disappears on the sliding mode. With the aid of the upper bound covariance control theory, the controller u( t) is derived in which the control feedback gain matrix G is synthesized for achieving the above multiple objectives.

Dynamics and variance control of hot mill loopers

Poor control of hot strip mill loopers degrades strip width and gauge, and may even lead to mill breakdowns due to instability. In this study, dynamics of the looper–strip system and the control challenges it poses are discussed, and covariance control theory is applied to variance control design for loopers. Since looper disturbances have a deterministic nature, and their accurate modelling is a challenge that is not easily addressed, the problem represents a case in which a variance controller has to be designed without an explicit disturbance model. Control performance is assessed on a full-stand, nonlinear high fidelity finishing mill simulator, and comparisons to a conventional control system are provided.

Synthesis of Covariance Controllers on the Basis of the System Embedding Technology

A new approach to the synthesis of stochastic control systems, which is called the theory of covariance control, is set out in this paper. Its essence involves securing a prescribed steady-state value of the state covariance matrix of a linear system with the aid of feedback. The necessary and sufficient conditions of the attainability of the state covariance matrix are defined and formulas for the entire set of stabilizing covariance controllers are derived.

Fuzzy covariance control for a class of continuous perturbed nonlinear stochastic systems

This paper addresses the analysis and design of a fuzzy controller for a class of continuous perturbed nonlinear stochastic systems. The nonlinear systems are modeled by the Takagi‐Sugeno (T‐S) fuzzy models. The conventional LMI‐based fuzzy control method is inconvenient for directly assigning the common positive definite covariance matrix. Hence, this paper tries to develop a useful methodology to allow designers to assign a common positive definite covariance matrix for the closed‐loop system. Applying the theory of covariance control, a fuzzy controller is developed to achieve the stable conditions for the assigned common positive definite covariance matrix. Finally, a numerical example is given to demonstrate the usefulness of the proposed approach.

Covariance control with observed-state feedback gains for continuous nonlinear systems using T-S fuzzy models

The design problem of state variance constrained control for stochastic systems has received rather extensive attention in recent years. This paper solves the state variance constrained controller design problem by using the covariance control theory with observed-state feedback gains for continuous Takagi-Sugeno (TS) fuzzy models. By incorporating the technique of state estimation into the practical covariance control theory, a variance constrained control methodology is developed for the continuous TS fuzzy models. Finally, a numerical example is shown to demonstrate the efficiency and applicability of the proposed approach.

Multivariable performance-constrained sliding mode control for ship yaw-motion systems with perturbations

A performance-constrained sliding mode controller will be proposed to satisfy the pole assignment, H∞ norm, and individual variance constraints for linear perturbed MIMO systems. Application of this controller involves a simulation of the linearized perturbed ship steering yaw-motion systems which will be provided for demonstration of the effectiveness of the present methodology. The control law developed in this paper is designed in the framework of a combination of the concept of the sliding mode control with the theory of upper bound covariance control. Copyright © 2000 John Wiley & Sons, Ltd.

ROBUST SHAPE CONTROL OF A FLEXIBLE STRUCTURE

This paper deals with a robust shape control of a flexible structure against parameter variations. Based on a covariance control theory with the principle of detailed balance, a new linear state feedback control method is proposed, where the state vector is composed of the position, velocity, and integral variables. The robustness of the controlled system is acquired by approximate orthogonality between the eigenvectors of the system, and is confirmed with a condition number of the model matrix of the closed-loop system numerically. The usefulness of the proposed shape control method is also verified by hardware experiment.

Minimum-energy covariance controllers

Covariance control theory provides a parametrization of all controllers that assign a given state covariance matrix to the closed-loop system. Such controllers are not unique. In this note we obtain the analytic expressions for the covariance controllers that minimize the required control effort. These are solutions to a minimum-effort/guaranteed-performance design problem, in the sense that the required control effort is minimized subject to performance constraints guaranteed by a specified state covariance matrix assignment. In a different interpretation of the results, the solutions to a minimum-effort controller redesign problem are obtained. Both continuous-time and discrete-time results are presented.

Extension of the covariance control principle to nonlinear stochastic systems

An approach for the constrained variance control of nonlinear stochastic systems, which uses the theory of covariance control of linear stochastic systems, is proposed. The nonlinear stochastic systems are linearised by the use of describing functions. Two cases of nonlinear systems are considered: zero-mean nonlinear stochastic systems and nonzero-mean nonlinear stochastic systems. The application of this approach to a position servomechanism is illustrated by a numerical example.

Covariance control with variance constraints for continuous perturbed stochastic systems

This paper proposes an approach which deals with the variance constraints for the perturbed stochastic systems. The purpose of this approach is to develop a novel methodology, which is based on the theory of covariance control, to solve the constrained variance design problem for the linear perturbed stochastic systems. Particular attention is paid to the case in which there are only uncertain perturbations in the state dynamic matrix. Moreover, an example is given to illustrate the power of the technique.

Constrained variance design for bilinear stochastic continuous systems

Recently, a theory for the control of covariance has been introduced for linear continuous systems. The theory was motivated mainly by the fact that performance requirements of many engineering control problems are naturally described in terms of the root-mean-squared (RMS) values of the system states or outputs. However, the primary purpose of this paper is to deal with the theory of covariance control for a general class of bilinear stochastic continuous systems. Moreover, on the basis of the theory of the state covariance assignment, this paper will address a methodology which deals more directly with these RMS values.