Optimal Regulator Problem Research Articles

Optimal Regulation and Passivity Results for Axisymmetric Rigid Bodies Using Two Controls

Wepresent a partial solution to theproblem of optimal feedback reorientation of the symmetry axisofan axially symmetric rigid body. The performance index is quadratic in the state and the control variable, and the optimal reorientation maneuver requires the use of only two control torques. Because of the passivity characteristics and the cascade structure of the system we e rst state two optimal regulation problems for the dynamics and the kinematics subsystems, separately. In this case one is able to e nd explicit solutions to the associated Hamilton ‐ Jacobi equations. For thecompletesystem, wepresent solutions fortwo partial cases. Thee rst case is when there is no penalty on the control input. In this case, one can asymptotically recover the cost for the kinematics by making the dynamics sufe ciently fast. The second case investigates restrictions imposed by optimality considerations on the aforementioned control law to avoid high gain.

Supervised learning control of a generic missile

This paper makes use of artificial neural networks to perform the control function in the lateral autopilot of a generic missile. The neural autopilot controller is formulated as a three layer, non-linear, discrete time-series neural network whose parameters are adjusted under a supervised learning environment via the Levenberg-Marquardt (LM) optimisation algorithm. As opposed to using heuristic reasoning for selecting the network input and output parameters, a novel technique utilising a priori knowledge of the baseline controller is used. The current fin position command is modelled as a function of the past value of commanded fin position, current and past values of lateral acceleration command, achieved fin position, measured lateral acceleration and body rate. The necessary training and recall data are obtained from time-histories of missile dynamic response when the autopilot control function is being performed by a baseline controller obtained by solving a discrete optimal Linear Quadratic Regulator (LQR) problem. Lateral acceleration time histories for two flight speeds for a range of incidences are used to illustrate performances. The neural controller is validated using a test lateral acceleration command sequence which differs significantly from that used for generating the network training data. The effectiveness of the proposed neural controller design can be judged from an examination of these digital simulation response results. The dynamic performance of the neural controller is also compared with that of the baseline LQR controller.

Self-tuning Optimal Regulation of Respiratory Motor Output by Hebbian Covariance Learning

The respiratory motor system is a specialized musculoskeletal system that is controlled by a small assembly of neuronal clusters in the brainstem. Its prime function is to maintain CO 2, O 2, and pH homeostasis in arterial circulation through the motor act of breathing. A longstanding dilemma is that during muscular exercise homeostatic regulation occurs automatically without any apparent feedback or feedforward signals, whereas the homeostasis is readily abolished by exogenous chemical challenge. Recently, it has been proposed that these seemingly incongruous behaviors of the respiratory controller may be a manifestation of self-tuning adaptive control. This hypothesis is supported in part by recent discoveries of various memory systems in the brainstem controller including short- and long-term potentiation and depression of synaptic transmission as well as the dramatic abolishment of homeostatic regulation in mutant mice with targeted genetic disruption of the NMDA receptors. In this paper, we propose a model of self-tuning homeostatic regulation based upon a synaptic adaptation rule—Hebbian covariance learning rule—that has been suggested to underlie many forms of learning and memory in the higher brain. We show that such an adaptation rule may be a useful neuronal substrate for reinforcement learning in optimization tasks. The model demonstrates how spontaneous oscillations and/or random-like fluctuations of neural activity may be exploited by the controller to adaptively regulate and optimize motor output. Such a self-tuning neural control paradigm, which operates without the need for any internal model of the external environment, is generally applicable to a class of steady-state optimal regulation problems with infinite time horizon. An interesting implication of the present results is that brain intelligent control can occur at a subconscious level without the need for voluntary intervention from the higher brain. Copyright © 1996 Elsevier Science Ltd.

Regulation of multinational firms with two active governments: A common agency approach

When a government agency imposes cost-based taxes/ regulations on a multinational with private cost information, it may initiate countervailing regulations by another of the governments with which the multinational interacts. We analyze the problem of optimal regulation of a multinational under incomplete cost information (via trade taxes) by multiple governments as a problem of common agency with adverse selection. By focusing on the game played by the competing governments we characterize the equilibrium trade taxes and show that the non-cooperative behavior of the governments not only reduces aggregate national welfare but also reduces firm profits.

最適レギュレータの極限形式と拘束条件付力学系の制御問題

This paper is concerned with an application of optimal regulator to control for mechanical systems. The properties of limiting form is discussed in detail, which is derived by letting the weighting matrix of state vector tend to infinity in standard optimal regulator problem. In this study, the limiting form is derived under the general initial conditions. Then it is shown that the limiting form includes delta functions and its derivatives. Further, it is pointed out that the limiting form leads to the solution for the output zeroing problem. Control inputs are bounded in some initial conditions, although the feedback gain increases with the weighting matrix. This clarifies the reason why the linear regulator can be applied to the control for certain mechanical systems with a constraint. A simulation is carried out to evaluate the effectiveness of this control method.

A gradient flow approach to computing a non-linear discrete time quadratic optimal feedback gain matrix

In this paper we propose an approach to solving infinite planning horizon quadratic optimal regulator problems with linear static state feedback for discrete time systems. The approach is based on solving a sequence of approximate problems constructed by combining a finite horizon problem with an infinite horizon linear problem. A gradient-flow algorithm is derived to solve the approximate problems. As part of this, a new algorithm is derived for computing the gradient of the cost functional, based on a system of difference equations to be solved completely forward in time. Two numerical examples are presented.

Optimal and minimum-energy optimal tracking of discrete linear time-varying systems

Based on the theory of generalized inverses and Bellman's dynamic programming approach, two forms of solutions to the optimal and minimum-energy optimal tracking problems are presented for discrete linear time-varying systems. A further and detailed study on the optimal and minimum-energy optimal terminal control problem is given, where simple necessary and sufficient conditions for output controllability and reachability are obtained. As natural consequences, all the above solutions are reduced to give solutions to the optimal and minimum-energy optimal regulation problems. In each case, simple expressions for the minimum tracking error and minimum control energy are derived.

Stabilization of Passenger Car-Caravan Combination Using Four Wheel Steering Control

SUMMARY This study aims to stabilize the trailer at high speed. The behavior of passenger cars with four wheel steering system, vehicles with rear wheel steering and of trailer and passenger car are similar. This is regarded as an optimal regulator problem with linear equation of motion, and a state variables feedback control system is adopted. The problem of stability at high speed on a straight course can be solved. Therefore, the passenger car-trailer system can be stabilized. Furthermore, this study indicates the way forward to stabilize a passenger car-trailer system.

Linear Quadratic Optimal Control of a Wave Equation with Boundary Damping and Pointwise Control Input

The linear quadratic optimal regulator problem on a wave equation in a bounded subset of Rn for n = 2, 3 with boundary damping and pointwise control input is formulated as a linear quadratic control problem with unbounded input/output operators using a variational approach. A dual control system is defined which allows us to investigate the regularity of the solution of the generalized wave equation under pointwise control. Finally, the linear quadratic optimal control problem is solved in the infinite time interval and, the existence and the regularity of the solution of the Riccati operator equation are obtained.

Gradient-flow approach for computing a nonlinear-quadratic optimal-output feedback gain matrix

In this paper, an approach is proposed for solving a nonlinear-quadratic optimal regulator problem with linear static state feedback and infinite planning horizon. For such a problem, approximate problems are introduced and considered, which are obtained by combining a finite-horizon problem with an infinite-horizon linear problem in a certain way. A gradient-flow based algorithm is derived for these approximate problems. It is shown that an optimal solution to the original problem can be found as the limit of a sequence of solutions to the approximate problems. Several important properties are obtained. For illustration, two numerical examples are presented.

Guaranteed cost LQG control of uncertain linear systems

The paper presents a Riccati equation approach to the output feedback quadratic stabilisation of an uncertain system. The uncertain systems under consideration depend on a norm-bounded time-varying matrix of uncertain parameters. A feature of the approach taken is that it involves the minimisation of a certain bound on an LQG cost function. The results were obtained by combining results on an optimal guaranteed cost linear quadratic regulator problem for uncertain systems and an optimal guaranteed cost state estimation problem for uncertain systems.

High Precision Control for Electrical Machine with Periodical Disturbances based on Optimal Repetitive Control.

It is well known that the speed control system, even though which is constructed with an excellent control method, has an unexpected steady speed error when it is suffered by a periodical load disturbance. As the method of eliminating the speed fluctuation caused by such kind of disturbance, a repetitive control is the most effective. But the repetitive control method is only adoptive for a constant speed control system.In this paper, a novel control system designing method which is effective to both the constant speed control system and variable speed driving system is shown. The fact that a periodical disturbance depends on time function and also rotational angle is utilized to design the control system. From the result, the numerical modeling of the controlled object is derived in the rotational angle function domain. For the obtained numerical model, an extended state variable equation based on the original system is presented by using the control technique named “Error System Technique”. And then the optimal feedback gains are decided by solving the optimal regulator problem for the extended state equation. Therefore, a stability of the system is ensured.The proposed control method is applied to DC motor variable speed control system and the experimental setup is constructed. Finally, the effectiveness of the proposed method is confirmed by the result of simulations and experiments.

On the polynomial equations for the continuous-time LQ optimization problem

This paper discusses the role and the relationships among three coupled diophantine polynomial equations involved in the solution of the optimal continuous-time LQ regulation problem. The emphasis is on establishing conditions under which the LQ regulator can be obtained via the solution of a single uncoupled diophantine equation. This analysis is motivated by the fact that, although recent papers have clarified the matter for the discrete-time domain, some subtle differences exist in the continuous-time domain. As a consequence, the extension of the discrete-time results to the continuous-time context requires a deeper analysis, which is the main aim of this paper.

The linear-quadratic optimal regulator for continuous-time descriptor systems: a dynamic programming approach

In this paper we consider the linear-quadratic optimal regulator problem for continuous-time descriptor system by the dynamic programming approach. A generalized Riccati equation is derived by making use of the Hamilton-Jacobi equation for the descriptor system. Sufficient conditions for the existence of a solution of the generalized Riccati differential (algebraic) equation are studied. It is shown that the existence of the solution of the generalized Riccati differential (algebraic) equation depends on the solutions of a standard Riccati algebraic equation and a standard Riccati differential (algebraic) equation. As in the regular state-space system, the solution of the generalized Riccati equation is used in the construction of the optimal feedback gains.

Vibration Control of Driver-Vehicle Coupling System Using Optimal Regulator.

The purpose of this investigation is to develop a method of vibration analysis of a driver-vehicle coupling system installed active suspensions, so as to improve the riding comfort and the driving quality. In the present paper, we attempted to develop a method which can simulate vibrational responses of a coupling system, which comprises a vehicle and a driver, controlled by active suspensions. Feedback gains were defined by solving the matrix Riccati equation as the optimal regulator problem. Joint parameters, such as stiffness and damping constants, of the driver model were identified by making use of some experimental results. Dynamic analysis of using the gains were performed on vibration of the system moving on an irregular road. We proved the effect of the dimension of the road, the position of the driver and the control gain on the riding comfort and the driving quality. It was also revealed that the improvement of both the riding comfort and the driving quality is a conflicting requirement.

Simulation of Vibration Control of a Flexible Rotor Using Optimal Regulator.

The purpose of this study is to develop an analytical method which can simulate vibration phenomena of a flexible rotor controlled by active suspensions. Feedback gains were defined by solving the matrix Riccati equation as the optimal regulator problem. We also used a finite-element method (FEM) in the formation of a rotating shaft in order to assume an arbitrary shape and higher vibration modes. We show the calculated results of vibration modes of a model rotor, whirling eigen-frequencies vs. spin speeds of the rotor, time history responses in the case that the rotor was controlled by the active suspensions, and frequency responses. The capability for development of a control device of a flexible rotor system was demonstrated.

Optimal Regulator for Descriptor Systems

An optimal regulator problem is considered for descriptor systems. The performance index is quadratic, in which the weight matrix for the descriptor variable is chosen so that only the dynamic part of the system is observed. The necessary and sufficient condition for the existence of an optimal stabilizing feedback is presented in terms of positive semidefinite solutions of a generalized Riccati equation. The optimal feedback gain is computed without transforming the descriptor equation into a state equation.

Existence of a Low-Order Integral Manifold and Non-linear Regulator Problems

A set of sufficient conditions which confine the solution of weak non-linear differential equations to some low order stable integral manifold are derived by applying the fixed point theorem. These conditions can be weakened substantially if the initial values of the differential equations are confined to some bounded region. These results are then applied to weak non-linear optimal regulator problems. Sufficient conditions are derived for the existence of a non-linear feedback control law which meets the neccesary conditions for optimality. An algorithm for calculating the feedback control law is proposed. Simulations using the algorithm exhibited favorable results.

On the nonlinear optimal regulator problem

The linear quadratic regulator is one of the most widely used tools for control systems design. Many real world systems, however, are inherently nonlinear and can only be optimally regulated using a nonlinear controller. This is, in general, much more difficult to achieve than the linear quadratic case. In this paper we show that a nonlinear feedback controller can be synthesized by appropriate training of a feedforward neural network to satisfy the corresponding nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equation in a restricted domain of the state space. Asymptotic stability of the closed-loop system is shown to be a natural consequence of the HJB equation.

Sensor/actuator placement for flexible space structures

A new approach for the placement of sensors and actuators in the active control of flexible space structures is developed. The approach converts the discrete nature of sensor and actuator positioning problems to a nonlinear programming optimization through approximation of the control forces and output measurements by spatially continuous functions. The locations of the sensors and actuators are optimized in order to move the transmission zeros of the system further to the left of the imaginary axis. This criterion for sensor/actuator placement can be useful for optimal regulation and tracking problems, as well as for low authority controller designs. Two performance metrics are considered for the optimization and are applied to the sensor/actuator positioning of a large-order flexible space structure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>