Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems

Orthogonal function approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA) are used to solve quadratic finite-horizon optimal controller design problems in both a fuzzy parallel distributed compensation (PDC) controller and a non-PDC controller (linear state feedback controller) for Takagi–Sugeno (TS) fuzzy-model-based control systems for dynamic ship positioning systems (TS-DSPS). Based on the OFA, an algorithm requiring only algebraic computation is used to solve dynamic equations for TS-fuzzy-model-based feedback and is then integrated with HTGA to design quadratic finite-horizon optimal controllers for TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index, which is also converted to algebraic form by the OFA. Integration of OFA and HTGA in the proposed approach enables use of simple algebraic computation and is well adapted to the computer implementation. Therefore, it facilitates design tasks of quadratic finite-horizon optimal controllers for the TS-DSPS. The applicability of the proposed approach is demonstrated in the example of a moored tanker designed using quadratic finite-horizon optimal controllers.     

Stable and Quadratic Optimal Control for TS Fuzzy-Model-Based Time-Delay Control Systems

For the finite-horizon optimal control problem of the Takagi-Sugeno (TS) fuzzy-model-based time-delay control systems, by integrating the delay-dependent stabilizability condition, the shifted-Chebyshev-series approach (SCSA), and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is presented to design the stable and quadratic optimal parallel distributed compensation (PDC) controllers. In this paper, the delay-dependent stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the SCSA, an algebraic algorithm only involving the algebraic computation is derived in this paper for solving the TS fuzzy-model-based time-delay feedback dynamic equations. In addition, by using the SCSA, the stable and quadratic optimal PDC control problem for the TS fuzzy-model-based time-delay control systems is replaced by a static parameter optimization problem represented by the algebraic equations with constraint of the LMI-based stabilizability condition, thus greatly simplifying the stable and optimal PDC control design problem. The computational complexity for both differential and integral in the stable and optimal PDC control design of the original dynamic systems may therefore be reduced considerably. Then, for the static constrained optimization problem, the HTGA is employed to find the stable and quadratic optimal PDC controllers of the TS fuzzy-model-based time-delay control systems. A design example of the stable and quadratic optimal PDC controllers for the continuous-stirred-tank-reactor system is given to demonstrate the applicability of the proposed integrative approach.     

Robust Quadratic-Optimal Control of TS-Fuzzy-Model-Based Dynamic Systems With Both Elemental Parametric Uncertainties and Norm-Bounded Approximation Error

This paper considers the design problem of the robust quadratic-optimal parallel-distributed-compensation (PDC) controllers for Takagi–Sugeno (TS) fuzzy-model-based control systems with both elemental parametric uncertainties and norm-bounded approximation error. By complementarily fusing the robust stabilizability condition, the orthogonal functions approach (OFA), and the hybrid Taguchi genetic algorithm (HTGA), an integrative method is presented in this paper to design the robust quadratic-optimal PDC controllers such that 1) the uncertain TS-fuzzy-model-based control systems can be robustly stabilized, and 2) a quadratic integral performance index for the nominal TS-fuzzy-model-based control systems can be minimized. In this paper, the robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). By using the OFA and the LMI-based robust stabilizability condition, the robust quadratic-optimal PDC control problem for the uncertain TS-fuzzy-model-based dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition, thus greatly simplifying the robust optimal PDC control design problem. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust quadratic-optimal PDC controllers of the uncertain TS-fuzzy-model-based control systems. Two design examples of the robust quadratic-optimal PDC controllers for an uncertain inverted pendulum system and an uncertain nonlinear mass–spring–damper mechanical system are given to demonstrate the applicability of the proposed integrative approach.      

Numerical solutions of time-varying TS-fuzzy-model-based time-delay dynamic equations via orthogonal functions

In this paper, the approach of orthogonal functions is presented to solve the time-varying Takagi–Sugeno (TS) fuzzy-model-based time-delay dynamic equations (time-varying TSFMTDE). The new method simplifies the procedure of solving the time-varying TSFMTDE into the successive solution of a system of recursive formulae only involving matrix algebra. Based on the presented recursive formulae, an algorithm only including straightforward algebraic computation is also proposed in this paper. The new proposed approach is non-iterative, non-differential, non-integral, straightforward, and well-adapted to computer implementation. Hence, the computational complexity is considerably reduced. The first illustrated example shows that the proposed method, based on the orthogonal functions, can obtain satisfactory results. The second illustrated example, for the pendulum time-delay system with the vibration in the vertical direction on the pivot point having a fuzzy parallel-distributed-compensation controller, is given to demonstrate the application of the proposed approach.     

Shifted-Chebyshev series solutions of Takagi–Sugeno fuzzy-model-based dynamic equations

By the use of the elegant operational properties of the shifted-Chebyshev series, a direct computational algorithm for solving the Takagi–Sugeno (TS) fuzzy-model-based dynamic equations is developed in this paper. The basic idea is that the state variables are expressed in terms of the shifted-Chebyshev series. The new method simplifies the procedure of solving the TS-fuzzy-model-based dynamic equations into the successive solution of a system of recursive formulae taking only two terms of expansion coefficients. Based on the presented recursive formulae, an algorithm only involving the straightforward algebraic computation is also proposed in this paper. The computational complexity can, therefore, be reduced remarkably. The illustrated example shows that the proposed method based on the shifted-Chebyshev series can obtain the satisfactory results.     

Solutions of time-varying TS-fuzzy-model-based dynamic equations using a shifted Chebyshev series approach

In this paper, the approach of a shifted Chebyshev series is proposed to solve the time-varying Takagi–Sugeno (TS) fuzzy-model-based dynamic equations. The new method simplifies the procedure of solving the time-varying TS-fuzzy-model-based dynamic equations into the successive solution of a system of recursive formulae only involving the matrix algebra. Based on the presented recursive formulae, an algorithm involving only straightforward algebraic computation is also proposed in this paper. The new proposed approach is non-iterative, non-differential, non-integral, straightforward, and well adapted to the computer implementation. The computational complexity can therefore be reduced remarkably. The first illustrated numerical example shows that the proposed method based on the shifted Chebyshev series can yield better and more satisfactory results than the Runge–Kutta method. The second illustrated example for the pendulum system with the vibration in the vertical direction on the pivot point is given to demonstrate the application of the proposed approach.     

Synthesis of minimax optimal controllers for uncertain time-delay systems with structured uncertainty

This paper is concerned with the design of robust state feedback controllers for a class of uncertain time-delay systems. The uncertainty is assumed to satisfy a certain integral quadratic constraint. The controller proposed is a minimax optimal controller in the sense that it minimizes the maximum value of a corresponding linear quadratic cost function over all admissible uncertainties. The controller leads to an absolutely stable closed loop uncertain system and is constructed by solving a finite dimensional parameter-dependent algebraic Riccati equation.     

Design of robust-optimal output feedback controllers for linear uncertain systems using LMI-based approach and genetic algorithm

This paper considers the robust-optimal design problems of output feedback controllers for linear systems with both time-varying elemental (structured) and norm-bounded (unstructured) parameter uncertainties. Two new sufficient conditions are proposed in terms of linear-matrix-inequalities (LMIs) for ensuring that the linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties are asymptotically stable, where the mixed quadratically-coupled parameter uncertainties are directly considered in the problem formulation. A numerical example is given to show that the presented sufficient conditions are less conservative than existing ones reported recently. Then, by integrating the hybrid Taguchi-genetic algorithm (HTGA) and the proposed LMI-based sufficient conditions, a new integrative approach is presented to find the output feedback controllers of the linear systems with both time-varying elemental and norm-bounded parameter uncertainties such that the control objective of minimizing a quadratic integral performance criterion subject to the stability robustness constraint is achieved. A design example of the robust-optimal output feedback controller for the AFTI/F-16 aircraft control system with the time-varying elemental parameter uncertainties is given to demonstrate the applicability of the proposed new integrative approach.     

A non-linear programming approach to the computer-aided design of regulators using a linear-quadratic formulation†

A design technique is proposed for linear regulators in which a feedback controller of fixed structure is chosen to minimize an integral quadratic objective function subject to the satisfaction of integral quadratic constraint functions. Application of a non-linear programming algorithm to this mathematically tractable formulation results in an efficient and useful computer-aided design tool. Particular attention is paid to computational efficiency and various recommendations are made. Two design examples illustrate the flexibility of the approach and highlight the special insight afforded to the designer.     

Discrete-time optimal fuzzy controller design: global conceptapproach

Proposes a systematic and theoretically sound way to design a global optimal discrete-time fuzzy controller to control and stabilize a nonlinear discrete-time fuzzy system with finite or infinite horizon (time). A linear-like global system representation of a discrete-time fuzzy system is first proposed by viewing such a system in a global concept and unifying the individual matrices into synthetic matrices. Then, based on this kind of system representation, a discrete-time optimal fuzzy control law which can achieve a global minimum effect is developed theoretically. A nonlinear two-point boundary-value-problem (TPBVP) is derived as a necessary and sufficient condition for the nonlinear quadratic optimal control problem. To simplify the computation, a multi-stage decomposition of the optimization scheme is proposed, and then a segmental recursive Riccati-like equation is derived. Moreover, in the case of time-invariant fuzzy systems, we show that the optimal controller can be obtained by just solving discrete-time algebraic Riccati-like equations. Based on this, several fascinating characteristics of the resultant closed-loop fuzzy system can easily be elicited. The stability of the closed-loop fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal closed-loop fuzzy system can not only be guaranteed to be exponentially stable, but also stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant closed-loop fuzzy system possesses an infinite gain margin, i.e. its stability is guaranteed no matter how large the feedback gain becomes. An example is given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proven stability properties     

Double-shifted Chebyshev series for convolution integral and integral equations

Double-shifted Chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral, Volterra integral equation, and Fredholm integral equation. This method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form. In addition, the computational complexity can be reduced remarkably. Three examples are illustrated. It is seen that the proposed approach is straightforward and convenient, and converges faster in finding approximations than other existing orthogonal function methods.     

应用于离散模糊系统的最优模糊追踪器设计

针对二阶离散式模糊追踪系统,提出用于构造一种最优模糊追踪器的设计方式, 并用于追踪一类时变目标或任意模型目标.为了能让所提出的设计方法完整并达到总体最小 化,提出了一种近似线性化的离散模糊动态系统的表示法.另外,为了简化计算量,一个多 层分解方式被用于本文所提出的最优设计程序中,模拟结果显示,本文提出的最优模糊追踪 设计方式确能达到预期效果.     

H∞ control of nonlinear continuous-time systems based on dynamical fuzzy models

This paper presents a solution of the H_∞ control problem for a class of continuous-time nonlinear systems. The method is based on a fuzzy dynamical model of the nonlinear system. A suitable piecewise differentiate quadratic (PDQ) Lyapunov function is used to establish asymptotic stability of the closed-loop system. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of suitable algebraic Riccati equations. An example is given to illustrate the application of the method     

Reply to “Further Comment on “Optimal Fuzzy Controller Design: Local Concept Approach””

We proposed two approaches to solve a quadratic optimal problem of Takagi-Sugeno (T-S) fuzzy-model-based nonlinear systems. These two approaches are totally different in both concept and derivation even they deal with the same issue and the adopted notations look similar. Readers are suggested to clear local-concept approach away from their brains and reexamine the global-concept approach. Then, you will find out that in global-concept approach, via the proposed synthetical matrices, a quadratic optimal fuzzy problem is transformed into a general nonlinear quadratic optimal problem; a numerical approach (dynamic decomposition algorithm) is further introduced to speed up numerical solution and to keep the global optimality at the same time. The stability analysis, the derivation of controller, and the minimum energy are derived fully on the basis of nonlinear systems. Their formulation or representation are all in form of entire fuzzy system instead of fuzzy subsystems. Therefore, the mentioned issue is not the case for global approach. Those are only related to local-concept approach. In this article, we clarify the pointed issue and focus on reinforcing our theorem.     

Optimal control for a polynomial system with a quadratic criterion over infinite horizon

This paper proposes an optimal control algorithm for a polynomial system with a quadratic criterion over infinite horizon. The designed regulator gives a closed-form solution to the infinite horizon optimal control problem for a polynomial system with a quadratic criterion. The obtained solution consists of a feedback control law obtained by solving a Riccati algebraic equation dependent on the state. Numerical simulations in the example show advantages of the developed algorithm.     

H infinity Control of nonlinear discrete-time systems based on dynamical fuzzy models

This paper presents a solution to H infinity control problem for a class of discrete-time nonlinear systems. This class of nonlinear systems can be represented by a discretetime dynamical fuzzy model. A suitable quadratic L yapunov function is used to establish asymptotic stability with an l₂-norm bound gamma of the closed-loop system. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of suitable algebraic Riccati equations. An example is given to illustrate the application of the method.     

Numerical computation of sign-indefinite linear quadratic differential games for weakly coupled large-scale systems

In this paper, N-player linear quadratic differential games that are sign-indefinite for infinite horizon weakly coupled large-scale systems are discussed. After establishing the asymptotic structure and local uniqueness of the solution for cross-coupled sign-indefinite algebraic Riccati equations (CSARE), a new algorithm for solving CSARE is provided. It is shown that the proposed algorithm attains linear convergence. Moreover, in order to reduce the computational workspace, the recursive algorithm is combined. Finally, a high-order approximation strategy based on the proposed iterative solutions is described. As a result, it was recently proved that the numerical strategy achieves a high-order approximation of the equilibrium value. As another important feature, when the small parameters are unknown, a parameter-independent strategy is developed.     

Nonlinear robust state feedback control system design for nonlinear uncertain systems

This paper presents a systematic approach to the design of a nonlinear robust dynamic state feedback controller for nonlinear uncertain systems using copies of the plant nonlinearities. The technique is based on the use of integral quadratic constraints and minimax linear quadratic regulator control, and uses a structured uncertainty representation. The approach combines a linear state feedback guaranteed cost controller and copies of the plant nonlinearities to form a robust nonlinear controller with a novel control architecture. A nonlinear state feedback controller is designed for a synchronous machine using the proposed method. The design provides improved stability and transient response in the presence of uncertainty and nonlinearity in the system and also provides a guaranteed bound on the cost function. An automatic voltage regulator to track reference terminal voltage is also provided by a state feedback equivalent robust nonlinear proportional integral controller. Copyright © 2016 John Wiley & Sons, Ltd.     

倒立摆系统的高精度控制器设计与实现

对线性二次调节器LQR算法中Q和R矩阵的参数与控制系统反馈矩阵Ⅸ之间的关系进行了实验研究,并将所获得的最佳反馈矩阵作为所设计的神经网络控制器的权值初始值。该神经网络控制器是带有局部递归神经网络并具有PID结构的控制器,因而设计简单,尤其适合用于多变量非线性时变系统。通过对一级和二级直线倒立摆系统的具体控制器的设计以及实验,将LQR控制器与神经网络控制器分别在无干扰和有干扰情况下的控制效果进行了对比分析,设计并实现了具有控制精度以及鲁棒性比最优线性二次调节器更高的一级和二级直线倒立摆系统。     

Solution of variational problems via general orthogonal polynomials

The general orthogonal polynomials approximation is employed to solve variational problems. The operational matrix of integration is applied to reduce an integral equation to an algebraic equation with expansion coefficients. A simple and straightforward algorithm is then developed to calculate the expansion coefficients of the general orthogonal polynomials. The proposed method is general and various classical orthogonal polynomial approximations of the same problem can be obtained as a special case of the derived results.