Evolutionary Algorithms for Minimax Problems in Robust Design

Many robust design problems can be described by minimax optimization problems. Classical techniques for solving these problems have typically been limited to a discrete form of the problem. More recently, evolutionary algorithms, particularly coevolutionary optimization techniques, have been applied to minimax problems. A new method of solving minimax optimization problems using evolutionary algorithms is proposed. The performance of this algorithm is shown to compare favorably with the existing methods on test problems. The performance of the algorithm is demonstrated on a robust pole placement problem and a ship engineering plant design problem.     

A gradient flow approach to on-line robust pole assignment for synthesizing output feedback control systems

Pole assignment is a basic design method for synthesis of feedback control systems. In this paper, a gradient flow approach is presented for robust pole assignment in synthesizing output feedback control systems. The proposed approach is shown to be capable of synthesizing linear output feedback control systems via on-line robust pole assignment. Convergence of the gradient flow can be guaranteed. Moreover, with appropriate design parameters the gradient flow converges exponentially to an optimal solution to the robust pole assignment problem and the closed-loop control system based on the gradient flow is globally exponentially stable. These desired properties make it possible to apply the proposed approach to slowly time-varying linear control systems. Simulation results are shown to demonstrate the effectiveness and advantages of the proposed approach.     

NEW TUNING METHOD FOR PID CONTROL OF A PLANT WITH UNDER‐DAMPED RESPONSE

In this paper, a tuning formula is derived for PID control of plants with under‐damped step responses. The under‐damped systems are modeled by second‐order plus dead time transfer functions. To derive the tuning rule, the dominant pole assignment method was applied to design the PID controllers for a variety of plant models. Then, the correlation between the controller parameters and the parameters of the models was found and the tuning formula derived. Several simulation examples are given to demonstrate the effectiveness and robustness of this formula.     

多变量系统固定状态反馈下的容错极点配置

本文研究了多变量系统对执行器故障的容错极点配置问题。基于n-线性特征系数系统理论及参数空间设计方法,提出了一种容错极点配置的方法。借助这一方法,对于给定的被控对象,可以设计出一个固定的状态反馈控制律,在执行器的各种故障模式下都将系统的闭环极点设置在预定的区域内。     

An Automatic Building Approach To Special Takagi‐Sugeno Fuzzy Network For Unknown Plant Modeling And Stable Control

In previous studies, several stable controller design methods for plants represented by a special Takagi‐Sugeno fuzzy network (STSFN) have been proposed. In these studies, the STSFN is, however, derived directly from the mathematical function of the controlled plant. For an unknown plant, there is a problem if STSFN cannot model the plant successfully. In order to address this problem, we have derived a learning algorithm for the construction of STSFN from input‐output training data. Based upon the constructed STSFN, existing stable controller design methods can then be applied to an unknown plant. To verify this, stable fuzzy controller design by parallel distributed compensation (PDC) method is adopted. In PDC method, the precondition parts of the designed fuzzy controllers share the same fuzzy rule numbers and fuzzy sets as the STSFN. To reduce the controller rule number, the precondition part of the constructed STSFN is partitioned in a flexible way. Also, similarity measure together with merging operation between each neighboring fuzzy set are performed in each input dimension to eliminate the redundant fuzzy sets. The consequent parts in STSFN are designed by correlation measure to select only the significant input terms to participate in each rule's consequence and reduce the network parameters. Simulation results in the cart‐pole balancing system have shown that with the proposed STSFN building approach, we are able to model the controlled plant with high accuracy and, in addition, can design a stable fuzzy controller with small parameter number.     

鲁棒极点配置的优化算法

本文研究了一类线性多变量系统的鲁棒极点配置问题，提出了一种基于目标函数优化的鲁棒极点配置算法，解决问题的基本思想是，利用极点配置的参数化表示结果，以系统输出的Ｌ∞范数作为极点配置寻优的准则函数，在这个准则下，通过解优化问题的途径确定极点配置问题中的自由参数，从而达到鲁棒极点配置的目的。     

具有域极点配置的混合H2 /H∞ 滤波

解决了具有域极点配置的连续时不变系统的混合H₂ /H_∞ 滤波问题. 通过采用线性矩阵不等式 (LMI)方法描述域稳定性限制、H₂ 和H_∞ 优化, 以建立求解这个问题的总框架. 这个问题的可解性的充分必要条件由一组LMI给出. 最后用一个数字例子来说明所给出的设计方法.     

Fuzzy neural networks for tuning PID controller for plants withunderdamped responses

In this paper, the fuzzy neural network (FNN) for tuning proportional-integral-derivative (PID) controller for plants with underdamped step responses is proposed. The underdamped systems are modeled by second-order-plus-dead-time transfer functions. For deriving the FNN, the dominant pole assignment method is applied to design the PID controllers for a batch of test plant models that represent the plants with underdamped responses. Then, a fuzzy neural modeling method is utilized to identify the relationship between the parameters that characterize the plant dynamics and the controller parameters. We then utilize the obtained FNN to tune the PID controller for plants with underdamped responses. Several simulation examples are given to demonstrate the effectiveness and robustness of the FNN obtained     

多变量系统鲁棒性复数极点配置的一种新方法*

本文对线性定常多变量系统的鲁棒性极点配置问题,给出了一种算法。这个算法对指定闭环极点中含共轭复极点的情形,用起来十分方便,它把一个带约束优化问题变成了一个无约束优化问题求解,本文末还给出了一个数值例子。     

An improved sum‐of‐squares based approach to fuzzy tracking control design of nonlinear systems

In this paper, using a more general Lyapunov function, less conservative sum‐of‐squares (SOS) stability conditions for polynomial‐fuzzy‐model‐based tracking control systems are derived. In tracking control problems the objective is to drive the system states of a nonlinear plant to follow the system states of a given reference model. A state feedback polynomial fuzzy controller is employed to achieve this goal. The tracking control design is formulated as an SOS optimization problem. Here, unlike previous SOS‐based tracking control approaches, a full‐state‐dependent Lyapunov matrix is used, which reduces the conservatism of the stability criteria. Furthermore, the SOS conditions are derived to guarantee the system stability subject to a given H_∞ performance. The proposed method is applied to the pitch‐axis autopilot design problem of a high‐agile tail‐controlled pursuit and another numerical example to demonstrate the effectiveness and benefits of the proposed method.     

A generic approach to the design of decentralized linear output- feedback controllers

A sufficient condition for failure-tolerant performance stabilization in a desirable performance region under decentralized linear output-feedback is established. To exploit the flexibility in decentralized control beyond multivariable pole assignment, and to address the subsystem design objectives along with those of the overall system, a generic problem on decentralized linear output-feedback is then defined. The problem is reformulated in terms of a constrained nonlinear optimization problem. The proposed methodology results in the optimal reconciliation of failure-tolerant robust performance of the overall system, and (maximal) robustness, disturbance rejection, noninteractive performance, reliability and low actuator gains in the isolated subsystems in the face of unstructured perturbations in the controller and plant parameters. The effectiveness of the proposed approach is demonstrated by a numerical example.     

Characteristic polynomial assignment for plants with semialgebraic uncertainty: A robust diophantine equation approach

In this paper, we address the problem of robust characteristic polynomial assignment for LTI systems whose parameters are assumed to belong to a semialgebraic uncertainty region. The objective is to design a dynamic fixed‐order controller in order to constrain the coefficients of the closed‐loop characteristic polynomial within prescribed intervals. First, necessary conditions on the plant parameters for the existence of a robust controller are reviewed, and it is shown that such conditions are satisfied if and only if a suitable Sylvester matrix is nonsingular for all possible values of the uncertain plant parameters. The problem of checking such a robust nonsingularity condition is formulated in terms of a nonconvex optimization problem. Then, the set of all feasible robust controllers is sought through the solution to a suitable robust diophantine equation. Convex relaxation techniques based on sum‐of‐square decomposition of positive polynomials are used to efficiently solve the formulated optimization problems by means of semidefinite programming. The presented approach provides a generalization of the results previously proposed in the literature on the problem of assigning the characteristic polynomial in the presence of plant parametric uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.     

Dimensional analysis approach to dominant three-pole placement in delayed PID control loops

PID control loops with time delay are characterized by infinite number of poles but the pole assignment technique for adjusting the controller parameters can be applied to placing three poles only. The dominance of these poles is therefore an essential condition for this application. A novel approach to this problem involves applying dimensional analysis theory to obtain a generalized model of the control loop and then to perform a parameter tuning for its dimensionless representation. A one-row dimensional matrix results from the assumption of the usual dimensionless interpretation of both control error and actuating signals of the controller. Dimensionless similarity numbers of the so-called swingability and laggardness are introduced to specify the plant dynamics in the controller synthesis. A trio of numbers is assigned to become the dominant zeros of the characteristic quasi-polynomial of the control loop and the corresponding PID parameter adjustment is derived in the form of uniform formulae. The tuning of the proper damping and the real pole position ratios is provided by means of an IAE optimization technique. A dominance degree notion is introduced and an argument increment criterion is proposed to check the dominance of any of the pole placement cases. The quality of the disturbance rejection response is taken as the general criterion in the design of the time delay plant control.     

Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment

In this paper, the robust stability of a fractional‐order time‐delay system is analyzed in the frequency domain based on finite spectrum assignment (FSA). The FSA algorithm is essentially an extension of the traditional pole assignment method, which can change the undesirable system characteristic equation into a desirable one. Therefore, the presented analysis scheme can also be used as an alternative time‐delay compensation method. However, it is superior to other time‐delay compensation schemes because it can be applied to open‐loop poorly damped or unstable systems. The FSA algorithm is extended to a fractional‐order version for time‐delay systems at first. Then, the robustness of the proposed algorithm for a fractional‐order delay system is analyzed, and the stability conditions are given. Finally, a simulation example is presented to show the superior robustness and delay compensation performance of the proposed algorithm. Moreover, the robust stability conditions and the time‐delay compensation scheme presented can be applied on both integer‐order and fractional‐order systems.     

Feedback controls for pole subset assignment

A pole assignment problem for multi-input linear systems is discussed. In conventional pole assignment problems, an input is concerned with all poles simultaneously or it assigns only a specified subset of poles. Here we propose a design method which makes each input assign an arbitrary subset of poles. Thus when we replace a pole by using this method, it is enough to reconstruct only the corresponding input.     

Central tendency adaptive pole assignment

The concept of central-tendency adaptive control for achieving minimum variance objectives, proposed by the authors in a companion paper (1989), is applied to adaptive pole assignment. At any iteration, given plant parameter estimate and their uncertainty, a controller is designed which is most likely to achieve the pole assignment objectives. Simulations show a factor-of-100 improvement in transient response over certainty equivalent adaptive pole assignment schemes, for one example     

具有域极点配置的混合H2/H∞滤波

解决了具有域极点配置的连续时不变系统的混合H2/H∞滤波问题.通过采用线性矩阵不等式(LMI)方法描述域稳定性限制、H2和H∞优化,以建立求解这个问题的总框架.这个问题的可解性的充分必要条件由一组LMI给出.最后用一个数字例子来说明所给出的设计方法.     

Least squares pole assignment by memoryless output feedback

In this paper, a pole assignment problem for a linear time-invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed. The cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum. This approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters. The relation between the modified cost functions and the original pole assignment problem is revealed. The proposed approach is compared with other existing approaches and illustrated by numerical results.     

Optimal design of linear control systems by an interactive optimization method

When designing linear control systems, one of the most difficult problems is that the designer almost has no theoretical basis for the determination of proper parameters in order to obtain a system with desired specifications. Poles and directions of eigenvectors in the pole assignment method or weighting matrices of the quadratic criterion function in the optimal regulator method are such parameters. The designer has to determine them by trial-and-error using computer simulation. The purpose of this paper is to propose an approach to helping determine proper parameters in linear control system design by the state space methods. In the case where the desired specifications are not given explicitly, the approach applies an interactive optimization method called the Interactive Simplex method to search the most suitable parameters directly in the parameter space. But, if the specifications are given explicitly, the design problem can be formulated as a multiobjective optimization problem. In this case, weights which indicate relative importance of different specifications are introduced and the Interactive Simplex method is applied in the weight space to indirectly find the most appropriate parameters. The approach is implemented as part of a CAD system. The designer has only to make pairwise comparisons of response curves which are shown on a graphics display terminal in order to obtain the most preferred control system. Two illustrative examples are demonstrated to indicate the efficiency of the approach.     

Dominant pole and eigenstructure assignment for positive systems with state feedback

In this paper, the dominant pole assignment problem, the dominant eigenstructure assignment problem and the robust dominant pole assignment problem for linear time-invariant positive systems with state feedback are considered. The dominant pole assignment problem is formulated as a linear programming problem, and the dominant eigenstructure problem is formulated as a quasiconvex optimisation problem with linear constraints. The robust dominant pole assignment problem is formulated as a non-convex optimisation problem with non-linear constraints which is solved using particle swarm optimisation (PSO) with an efficient scheme which employs the dominant eigenstructure assignment technique to accelerate the convergence of the PSO procedure. Each of the three problems can be further constrained by requiring that the controller has a pre-specified structure, or the gain matrix have both elementwise upper and lower bounds. These constraints can be incorporated into the proposed scheme without increasing the complexity of the algorithms. Both the continuous-time case and the discrete-time case are treated in the paper.