Compensation of plant variations in optimal control systems

Control of a linear plant, with bounded control input, may be implemented by constructing a control law generator which produces the optimal control as a function of the state variables. If the plant parameters differ from their nominal values, then maintaining optimal control by changing the control law generator is inconvenient since the control law is usually nonlinear. It is shown that in certain cases optimal control can be maintained without changing the control law generator. This is accomplished by using a linear transformation of state variables as the input to the control law generator. The variations of the plant are compensated for by changing the linear transformation. The conditions under which this is possible are established in this paper. The advantage of this system is that a change in a linear function is easier to implement than a change in a nonlinear function. It is shown how this system can be incorporated into an adaptive system which compensates for plant variations.     

Adaptive control design under structured model information limitation: A cost-biased maximum-likelihood approach

Networked control strategies based on limited information about the plant model usually result in worse closed-loop performance than optimal centralized control with full plant model information. Recently, this fact has been established by utilizing the concept of competitive ratio, which is defined as the worst-case ratio of the cost of a control design with limited model information to the cost of the optimal control design with full model information. We show that an adaptive controller, inspired by a controller proposed by Campi and Kumar, with limited plant model information, asymptotically achieves the closed-loop performance of the optimal centralized controller with full model information for almost any plant. Therefore, there exists, at least, one adaptive control design strategy with limited plant model information that can achieve a competitive ratio equal to one. The plant model considered in the paper belongs to a compact set of stochastic linear time-invariant systems and the closed-loop performance measure is the ergodic mean of a quadratic function of the state and control input.     

Relatively optimal control and its linear implementation

Motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality, roughly optimality for a specific (nominal) plant initial condition. We consider a generic discrete-time finite-horizon constrained optimal control problem for linear systems, and we seek for a state feedback (possibly dynamic) controller. As a fundamental requirement, we do not admit preactions or controller-state initialization based on the plant initial state and we assume our controller to be time-invariant. In particular, we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory. A relatively optimal control is a stabilizing controller such that, if initialized at its zero state, produces the optimal (constrained) trajectory for the nominal initial condition of the plant. We show that one of such controllers is linear, dead-beat, and its order is equal to the length of the horizon minus the plant order, thus, of complexity which is known a priori. Some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed. We show that, by means of the proposed approach, we can face several problems such as optimal point-to-point operations, optimal impulse response and optimal tracking.     

LQG control of networked control systems with access constraints and delays

We explore the LQG control of a networked control system (NCS) in which a linear plant is controlled remotely over a network or other shared communication medium. The medium provides a limited number of simultaneous connections, so that only a subset of the plant's sensors and actuators may communicate with the controller at any one time, subject to known transmission delays. Instead of insisting on jointly optimal control and medium access policies, we reduce the infinity of possible access sequences down to those which preserve the stabilisability and detectability of the underlying plant, and are periodic. Our choice of communication and NCS model effect a kind of ‘decoupling’ of the LQG problem, in the sense that the medium access policy can be selected independently of the controller. This guarantees the existence of a stabilising LQG controller which is optimal for the communication policy of choice, and which is then combined with a delay compensator. We include simulations that illustrate our approach.     

A separation theorem for stochastic control problems with point-process observations

The exact solution is derived for a stochastic optimal control problem involving a linear stochastic plant, quadratic costs, and nonlinear, nongaussian observations. The observations are in the form of a point process in which each point has both a temporal and a spatial coordinate. The state of the stochastic plant influences the intensity of the observed time-space point process. The solution to this dual control problem can be realized with a separated estimator-controller in which the estimator is nonlinear, mean-square optimal, and finite dimensional, and the controller is the certainty equivalent linear controller. Motivation for the stochastic optimal control problem studied here is given in terms of position sensing and tracking for quantum-limited optical communication problems.     

Optimal digital feedback control systems counting computation time of control laws

Since microprocessors are easily obtained, multivariable control theory can be applied to many practical control problems, for example control of robots. However, when the time constant of the plant is short and the dynamic order of the plant is high, the time delay due to the computation time of the control law cannot be neglected. In this paper, the author proposes design methods for linear optimal regulators and linear optimal servosystems in which the delay arising from the computation time of the processors is counted properly. From the theoretical point of view, the results are interesting since all the control laws derived in this paper are obtained using only conventional results of optimal regulator theory.     

The design of optimal compensators for linear constant systems with inaccessible states

This paper considers the problem of designing an optimal linear time-invariant dynamic compensator for the regulation of annth-order linear time-invariant plant. The usual quadratic cost on the state and control is averaged over initial plant state values. The globally optimal compensator gains and dynamic order are determined by showing that this problem is mathematically identical to a steady-state stochastic control problem whose optimal solution is known.     

Optimal dynamic quantizers for discrete-valued input control

This paper discusses an optimal design problem of dynamic quantizers for a class of discrete-valued input systems, i.e., linear time-invariant systems actuated by discrete-valued input signals. The quantizers considered here are in the form of a linear difference equation, for which we find a quantizer such that the system composed of a given linear plant and the quantizer is an optimal approximation of the given linear plant in the sense of the input-output relation. First, we derive a closed form expression for the performance of a class of dynamic quantizers. Next, based on the performance analysis, an optimal dynamic quantizer and its performance are provided. This result further shows that even for such discrete-valued input systems, a controller can be easily designed by the existing tools for the linear system design such as robust control theory. Finally, the relation among the optimal dynamic quantizer and two other quantizers, i.e., the receding horizon quantizer and the ΔΣ modulator, is discussed.     

A method of realizing quasi-time optimal control by means of an approximate switching surface†

The optimal switching surface, required for the time optimal control of a linear plant with real poles, is approximated by a polynomial expansion of the state variables. The expansion is obtained by least error squared fitting. Experimental results, obtained by using the approximate switching surface to control a third-order plant, are presented.     

Synthesis of optimal control using neural network with mixed structure

This paper proposes firstly to use a neural network with a mixed structure for learning the system dynamics of a nonlinear plant, which consists of multilayer and recurrent structure. Since a neural network with a mixed structure can learn time series, it can learn the dynamics of a plant without knowing the plant order. Secondly, a novel method of synthesizing the optimal control is developed using the proposed neural network. Procedures are as follows: (1) Let a neural network with a mixed structure learn the unknown dynamics of a nonlinear plant with arbitrary order, (2) after the learning is completed, the network is expanded into an equivalent feedforward multilayer network, (3) it is shown that the gradient of a criterion functional to be optimized can be easily obtained from this multilayer network, and then (4) the optimal control is generated by applying any of the existing non-linear programming algorithm based on this gradient information. The proposed method is successfully applied to the optimal control synthesis problem of a nonlinear coupled vibratory plant with a linear quadratic criterion functional.     

Constrained predictive control for multivariable systems with application to power systems

Generalized predictive control (GPC) and dynamic performance predictive control (DPC) algorithms are introduced for industrial applications. Constraints on plant input rate, plant absolute input and plant absolute output can be implemented and are demonstrated on an application of these algorithms. A standard quadratic programming algorithm performs the calculation of the optimal control. A MATLAB/Simulink toolbox environment has been developed where controllers can be designed, linear and non-linear plant models can be embedded, discrete- and continuous-time loop parts can be mixed and simulation results can be managed and evaluated by graphical and statistical tools. This package utilises a graphical user interface. Finally, a case study design example is presented where a linear gas turbine model for power generation is examined with constrained GPC and DPC, and the advantages and drawbacks of the approach are the discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

线性采样系统二次型最优控制器的MATLAB实现

当被控对象具有较大滞后特性时通常运用采样控制方式进行控制，而线性二次型最优控制又以其综合性、灵活性和实用性受到重视和发展。MATLAB软件的Simulink模块可以快速方便地对控制系统进行建模和仿真。文中通过实例给出了完整的线性采样系统二次型最优控制器的设计过程，简述了连续系统离散化以及线性二次型最优控制器的基本原理、设计方法，并提出了一个Simulink环境下的仿真模型，通过得到的闭环系统的阶跃响应分析了参数变化对最优控制系统的影响。整个设计表明了运用Smulink对采样系统进行设计和分析是完全可行的。     

A nonlinear regulator design in the presence of systemuncertainties using multilayered neural network

The authors present a novel nonlinear regulator design method that integrates linear optimal control techniques and nonlinear neural network learning methods. Multilayered neural networks are used to add nonlinear effects to the linear optimal regulator (LOR). The regulator can compensate for nonlinear system uncertainties that are not considered in the LOR design and can tolerate a wider range of uncertainties than the LOR alone. The salient feature of the regulator is that the control performance is much improved by using a priori knowledge of the plant dynamics as the system equation and the corresponding LOR. Computer simulations are performed to show the applicability and the limitations of the regulator.     

Optimal anti-windup synthesis for repetitive controllers

Repetitive controllers use internal models that provide very high gain at a selected fundamental frequency and its harmonics, additionally, some of the internal models may result unstable, as in the high order repetitive control approach. These characteristics make the repetitive control system susceptible to exhibit wind-up when actuator saturation occurs. This paper proposes an anti-windup scheme for repetitive control based on the model recovery anti-windup strategy. The proposed scheme provides low order, low computational burden and also isolation of the controller from the saturation effects. The anti-windup compensator is constructed from the plant model and provides an additional linear feedback path aimed at enhancing system performance. This feedback path is designed to obtain a deadbeat behaviour, which makes the system recovery faster. Finally, internal stability and deadbeat features are designed in a compact procedure based on linear matrix inequalities and an optimal linear quadratic design. Experimental validation of the proposed anti-windup compensator is provided using a mechatronic plant.     

Optimal control of saturable linear plants for minimum integral-square-error and similar performance criteria†

The paper deals with the problem of minimizing integral–square–error and similar performance indices for linear single–variable plants with saturating control input. It is shown that the optimal control can be obtained by treating it as the limit as υ→∞ of the Sequence of controls which minimize the performance index on the class of all bang–bang controls containing not more than υ switches. These optimal υ switch controls are shown to satisfy a modified maximum principle.The method enables the computation of optimal controls from any given initial plant state. The particular problems of minimizing integral–modulus–error for a two–integrator plant and integral–square–error for a three–integrator plant are worked out in detail so as to illu3trata the techniques involved.     

An Optimization Approach to Petri Net Monitor Design

This note addresses the problem of enforcing generalized mutual exclusion constraints on a Petri net plant. First, we replace the classical partition of the event set into controllable and uncontrollable events from supervisory control theory, by associating a control and observation cost to each event. This leads naturally to formulate the supervisory control problem as an optimal control problem. Monitor places which enforce the constraint are devised as a solution of an integer linear programming problem whose objective function is expressed in terms of the introduced costs. Second, we consider timed models for which the monitor choice may lead to performance optimization. If the plant net belongs to the class of mono-T-semiflow nets, we present an integer linear fractional programming approach to synthesize the optimal monitor so as to minimize the cycle time lower bound of the closed loop net. For strongly connected marked graphs the cycle time of the closed-loop net can be minimized     

非定常参考模型的模型参考自适应控制系统（MRACS）设计

针对定常参考模型ＭＲＡＣＳ无法实现最优性能指标控制问题,提出了采用非定常参考模型的ＭＲＡＣＳ设计方法,使被控对象跟随最优性能指标下线性时变或非线性参考模型的设计问题得以实现。     

A deterministic theory of estimation and control

A feedback control system can be structured for linear nonstationary process and measurement systems comprising a deterministic filter whose output is the independent variable of a linear control law. Subject to uniform controllability and observability, the filter and control gains can be specified to provide arbitrary and separable stability properties. If the filter gain is selected to produce a stabilizing effect on the state estimate, and the control gain is selected to produce a stabilizing effect on the process, the filter and control gains are shown to satisfy matrix Riccati differential equations. This suggests the use of stochastic optimal control theory when there is no quantitative measure of optimality, but it is desirable to assure the qualitative property that feedback be stabilizing. A concise derivation of the Kalman-Bucy filter is included in an appendix to illustrate the facility of approaching optimal estimation problems with the methods of stability theory.     

Linear filtering with adaptive adjustment of the disturbance covariation matrices in the plant and measurement noise

For filtering a nonstationary linear plant under the unknown intensities of input signals such as plant disturbances and measurement noise, a new algorithm was presented. It is based on selecting the vectors of values of these signals compatible with the observed plant output and minimizing the error variances of the last predicted measurement. The measurement prediction is determined from the Kalman filter where the input signals are assumed to be white noise and the covariance matrix coincides with the empirical covariance matrix of the selected vectors. Numerical modeling demonstrated that the so-calculated filter coefficients are close to the optimal ones constructed from the true covariance matrices of plant disturbances and measurement noise. The approximate Newton method for minimization of the prediction error variance was shown to agree with the solution of the auxiliary optimal control problem, which allows to make one or some few iterations to find the point of minimum.     

Optimal control of a servo derived from nonquadratic performance criteria

Optimal control theory is applied to a serve problem described by linear plant dynamics and a nonquadratic performance index. The performance index consists of the sum of positive semidefinite homogeneous multinomial forms of the error between plant states and a given input. It is shown that by adding conveniently chosen higher multinomial forms to the performance index, the optimal control law can be obtained in closed-form as a nonlinear function of the error between the controllable states and the input vector.