Robust control of a laboratory aircraft model via fast output sampling

A new multimodel approach to robust controller design is illustrated by a practical application: for a laboratory aircraft model, a robust controller is designed for acceptable performance, in normal operating conditions and under propeller failure, simultaneously. Based on a linear model for each operating mode, linear matrix inequality (LMI) formulation of the problem and convex programming are used to search for a state feedback controller that achieves the objective. This state feedback design is then realized simultaneously in both operating modes by a controller that is based on fast output sampling. Robust performance is demonstrated by experimental results.     

Time/fuel optimal control of constrained linear discrete systems

A unified approach to solving three common optimal control problems is presented, for linear systems under general constraints. The problems are: (1) the time optimal control problem; (2) the fuel optimal control problem in fixed time; (3) the time optimal control problem with a fuel constraint. A special purpose linear programming algorithm is used. State variable constraints are efficiently handled by a cutting plane algorithm. An example of a sixth order system with two inputs and two state variable constraints illustrates the method as implemented on a personal computer.     

Algorithms for point-wise state variable constraints in structural optimization

This paper presents algorithms to treat the point-wise state variable constraints in finite dimensional and distributed parameter structural optimization problems. The idea is to impose such constraints at all local maximum points, or over a small region around the maximum points. Therefore, methods for design sensitivity analysis to handle the constraints at some particular point for the distributed parameter problem are presented. The direct differentiation and adjoint variable methods are employed to derive the design sensitivity expressions. For the finite dimensional problem the new idea is easily carried out in optimization process. Simply supported and clamped beams are optimized using the new approach. These are modeled with nonuniform beam elements. Comparisons of finite dimensional and distributed parameter problems are also made.     

Optimal investment policy: An example of a control problem in economic theory

A problem in mathematical economics concerning the optimal investment of resources is solved via the techniques of optimal control theory. Interesting theoretical complications include the simultaneous presence of interdependent control variable inequality constraints, state variable inequality constraints, and singularity conditions. Economic implications of the results are briefly discussed.     

不确定离散切换系统具有极点约束的保性能控制

对一类含有范数有界不确定性的离散切换系统和一个二次型性能指标,研究其具有闭环极点约束的鲁棒状态反馈保性能控制问题．利用二次Lyapunov函数方法和线性矩阵不等式技术,给出了鲁棒保性能控制器存在的一个充分条件,在所构造切换规则下,闭环系统二次D稳定,且满足给定的性能指标．在此基础上,将次优保性能控制器设计问题转化为一组线性矩阵不等式约束下的凸优化问题．数值例子说明了所提方法的有效性．     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

Design sensitivity analysis and optimization of dynamic response

The paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems. A key feature of the paper is the development of procedures to handle point-wise state variable constraints. Difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints. An alternate treatment of such constraints is proposed, developed and evaluated. In this treatment each point-wise state variable constraint is replaced by several constraints that are imposed at all the local max-points for the original constraint function. The differential equations of motion are formulated in the first-order form so as to handle more general problems. The direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point-wise constraints are presented. With the adjoint variable methods, there are two ways of calculating design sensitivity coefficients. The first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation. Since the adjoint variable methods are better for a large class of problems, an efficient computational algorithm with these methods is presented in detail. Optimum results for several problems are obtained and compared with those available in the literature. The new formulation works extremely well as precise optimum designs are obtained.     

Generalized shape optimization using stress constraints under multiple load cases

Optimal shape design using numerical techniques is an increasingly useful engineering tool. Generalized or layout optimal design where the topology of the object is not fixed is one of the emerging applications. These problems are numerically difficult to solve due to the large number of design variables and equality/inequality constraints. Solutions have focused primarily on compliance based minimization under a fixed volume. A more usual engineering approach would be one of minimizing the volume under a stress or deflection constraint. This, however, can lead to problems as stress is a local quantity and volume minimization of multiple load cases under stress constraints may not result in the stiffest design for the remaining material. The approach adopted here is based on a differential rate equation governed by a local operator that defines the state of each element at each time step. This algorithm forms the optimality criteria for the problem. To satisfy the global stress constraints, a feedback derivative is used, analogous to a Lagrange multiplier. The original method for a single load case developed by these authors is extended to deal with multiple load cases. Additionally, a discussion of the global behaviour is included.     

An optimal control approach to dynamic routing in networks

This paper explores the application of optimal control theory to the problem of dynamic routing in networks. The approach derives from a continuous state space model for dynamic routing and an associated linear optimal control problem with linear state and control variable inequality constraints. The conceptual form of an algorithm is presented for finding a feedback solution to the optimal control problem when the inputs are assumed to be constant in time. The algorithm employs a combination of necessary conditions, dynamic programming, and linear programming to construct a set of convex polyhedral cones which cover the admissible state space with optimal controls. An implementable form of the algorithm, along with a simple example, is presented for a special class of single destination networks.     

Discrete optimal control with multiple constraints II The batch crystallisation of sugar

The crystallisation process in the sugar industry is briefly described and the mathematical model describing the operation of a vacuum pan crystalliser is presented. The model consists of two control and five state variables. Associated with the model are two saturation constraints on each of the controls and three state variable inequality constraints. Both the system equations and state constraints are high non-linear.The constraint separation and transformation technique [1] is applied to this problem to find the optimal control policies which will result in a desired crystal size in minimum time when the pan is fed with a feed of either syrup or molasses. The analysis is presented in sufficient detail to show how the constraint separation and transformation technique is applied to a complex non-linear, multi-input, multi-constraint problem. However, the reader is advised to read Ref. [1] before attempting this paper.The solutions to the two problems are presented and comparison is made between the technique used in this paper and the method of penalty functions.     

Constrained Kalman filtering via density function truncation for turbofan engine health estimation

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This article develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter truncates the probability density function (PDF) of the Kalman filter estimate at the known constraints and then computes the constrained filter estimate as the mean of the truncated PDF. The incorporation of state variable constraints increases the computational effort of the filter but also improves its estimation accuracy. The improvement is demonstrated via simulation results obtained from a turbofan engine model. It is also shown that the truncated Kalman filter may provide a more accurate way of incorporating inequality constraints than other constrained filters (e.g. the projection approach to constrained filtering).     

带约束卡尔曼滤波对涡扇发动机状态估计

提出了一种加入线性不等式约束的卡尔曼滤波方法,并用于涡扇发动机的健康状况估计。涡扇发动机数字模型包含10个状态变量、12个量测量、6个控制输入量以及8个健康状况参数。不等式约束不仅保证了状态变量估计在用户自定义的范围内随时间变化平稳缓慢,而且还提高了滤波计算效率,改善了滤波估计精度。同时系统还允许滤波器沿确定的方向修正状态变量估计,以保持状态变量真值恒定。对比传统的无约束卡尔曼滤波,线性化滤波结果显示,该方法对涡扇发动机的健康状况估计尤其行之有效。     

Robust control of linear uncertain systems with regional pole and variance constraints

This paper concerns the design problem of state feedback controllers which guarantee the closed-loop poles within a specified disc and steady-state variances to be less than a set of given upper bounds for linear systems with norm-bounded parameter uncertainties. Using the linear matrix inequality approach, the existence conditions of such controllers are derived. A parametrized representation of the desired controllers is presented in terms of the feasible solutions to a certain linear matrix inequality system. Based on this, a solution to the minimum-effect guaranteed-performance design problem is presented in the sense that the required control effort is minimized subject to performance constraints.     

支持参数和公差设计的多学科协同建模

分析了多学科协同设计的耦合关系，提出支持参数和公差设计的多学科协同建模．首先。获得设计初始阶段各领域专家给出的约束条件和设计变量的大致范围；然后，抽取所有领域中在参数和公差设计阶段的耦合参数和相关约束；最后，构建产品的变动约束网络模型．在多学科协同建模中，考虑了不等式约束条件的转化、微分方程形式约束的转化、参数和公差的综合，以及与三维CAD系统的集成；为进一步实现参数和公差协同求解奠定了基础。     

State estimation of recurrent neural networks with time-varying delay: A novel delay partition approach

The state estimation problem is studied in this paper for a class of recurrent neural networks with time-varying delay. A novel delay partition approach is developed to derive a delay-dependent condition guaranteeing the existence of a desired state estimator for the delayed neural networks. The design of the gain matrix of the state estimator can be achieved by solving a linear matrix inequality, where no slack variable is involved. A numerical example is finally provided to show the advantage of the proposed approach over some existing results.     

基于线性矩阵不等式的不确定关联系统的分散鲁棒镇定

应用线性矩阵不等式（LMI）方法研究不确定性关联大系统的分散便棒镇定问题。系统中不稳定项具有数值界,可不满足匹配条件。基于不确定项的表达形式,给出了其可分散状态反馈镇定的充分条件,即一组LMIs有解。在此基础上,通过求第一凸优化问题,提出了具有较小反馈增益的分散稳定化状态反馈控制律的设计方法。仿真示例说明了该方法的有效性和优越性。     

Fault diagnosis of nonlinear systems using structured augmented state models

This paper presents an internal model approach for modeling and diagnostic functionality design for nonlinear systems operating subject to single- and multiple-faults. We therefore provide the framework of structured augmented state models. Fault characteristics are considered to be generated by dynamical exosystems that are switched via equality constraints to overcome the augmented state observability limiting the number of diagnosable faults. Based on the proposed model, the fault diagnosis problem is specified as an optimal hybrid augmented state estimation problem. Sub-optimal solutions are motivated and exemplified for the fault diagnosis of the well-known three-tank benchmark. As the considered class of fault diagnosis problems is large, the suggested approach is not only of theoretical interest but also of high practical relevance.     

A suboptimal control algorithm for constrained problems using cubic splines

A suboptimal control algorithm for linear-quadratic regulator problems with state variable inequality constraints (SVIC) is developed. The state and control variables are approximated by cubic splines on an uniform mesh. Through collocation at the knots, the dynamic equations and SVIC are reduced to a set of linear algebraic equations and the suboptimal control is constructed from the solution of a quadratic programming problem with sparse matrices. The number of non-zero storage elements required for these matrices varies linearly with the number of mesh points.Computational experience for specific examples is presented and compared with other approaches described in the literature. Good to excellent accuracy is obtained with modest computational requirements. Memory considerations and on-line implementation are discussed. From both the computational and storage aspects, the approach offers an effective alternative for SVIC problems. Extensions of the algorithm to more general control problems are suggested.     

Robust control of a class of under-actuated mechanical systems with model uncertainty

Robust control of under-actuated mechanical systems (UMSs) with model uncertainty is still a challenging problem. For UMSs, the model parametric uncertainties make it difficult to precisely calculate the isolated equilibrium point corresponding to a fixed input. Without an accurate destination state, many set-point control methods cannot eliminate the positioning errors. An improved sliding mode control (ISMC) method is proposed to solve the robust control problem for a class of UMSs with model uncertainty and input disturbance. A balance variable is introduced in the sliding surface design to compensate for the disturbance caused by the inaccurate destination state, and the ISMC method is proposed to make the system state reach the sliding surface in finite time. Linear matrix inequality approach and particle swarm optimisation algorithm are applied to design the sliding mode surface parameters. The simulation results on an UMS are presented to show the effectiveness of the proposed scheme.     

Practical application of a penalty function approach to constrained minimax optimization

A practical, penalty function approach to solving constrained minimax problems is applied here. In essence, this approach reformulates the constrained minimax problem as an unconstrained minimax problem. A recently proposed optimization algorithm called grazor search is used to solve the reformulated unconstrained minimax problem. The proposed approach can handle inequality constraints-parameter constraints in particular. A practical transmission-line filter example with parameter constraints illustrates the results.