On a Problem of Letov in Optimal Control

1965 ◽  
Vol 87 (1) ◽  
pp. 81-89 ◽  
Author(s):  
C. D. Johnson ◽  
W. M. Wonham
Keyword(s):  
Optimal Control ◽  
Present Note ◽  
Control Variable ◽  
Control Law ◽  
Initial Value ◽  
Bounded Control ◽  
Quadratic Performance Index ◽  
Linear Plant ◽  
Quadratic Performance ◽  
Optimal Regulator

In a series of papers [1, 2], A. M. Letov discussed an optimal regulator problem for a linear plant with bounded control variable and quadratic performance index. This problem was also discussed by Chang [3]. Krasovskii and Letov observed later [4] that the solution proposed in [1, 2, and 3] may be correct only for special choices of the initial value of the state vector. In the present note, further aspects of the solution in the general case are described and three examples are given. The possible existence of a regime of unsaturated-nonlinear optimal control is demonstrated. The presence of this regime in the optimal control law was apparently overlooked in [1–4].

Optimal Bounded Control of Linear Sampled-Data Systems With Quadratic Loss

1965 ◽  
Vol 87 (1) ◽  
pp. 135-141 ◽  
Author(s):  
G. W. Deley ◽  
G. F. Franklin
Keyword(s):  
Optimal Control ◽  
Piecewise Linear ◽  
Control Variable ◽  
Piecewise Linear Function ◽  
Control Law ◽  
Quadratic Loss ◽  
Data Systems ◽  
Sampled Data ◽  
Bounded Control ◽  
Unbounded Control

A method is presented for the computation of optimal control for linear sampled-data systems when the control variable is a bounded scalar. It is shown that for this problem the optimal control is a piecewise linear function of the state and may be computed by piecewise iteration of suitable recurrence relations. The optimal control is presented in terms of the control coefficients (matrices) and the regions to which they apply. No solution other than computer storage is suggested for the synthesis of these controls. In the second section of the paper, it is shown that the method applies with trivial modification to the random-input, random-observation noise case. The optimal control law has the same form as the deterministic case with the conditional expectation used in the control law in place of the stale itself. A simple deterministic example computed on an IBM 1620 is presented. As might be expected, the computer capacity required for the problem is intermediate between the unbounded control case, where the control is linear, and more general problems.

scholarly journals Suboptimal Regulation of a Class of Bilinear Interconnected Systems with Finite-Time Sliding Planning Horizons

2008 ◽  
Vol 2008 ◽  
pp. 1-26 ◽  
Author(s):  
M. de la Sen ◽  
Aitor J. Garrido ◽  
J. C. Soto ◽  
O. Barambones ◽  
I. Garrido
Keyword(s):  
Control Law ◽  
State Variables ◽  
Linear Quadratic ◽  
Equivalent Representation ◽  
Quadratic Performance Index ◽  
Matrix Sequence ◽  
Control Scheme ◽  
Quadratic Performance ◽  
System Equations ◽  
Riccati Matrix

This paper focuses on the suboptimization of a class of multivariable discrete-time bilinear systems consisting of interconnected bilinear subsystems with respect to a linear quadratic optimal regulation criterion which involves the use of state weighting terms only. Conditions which ensure the controllability of the overall system are given as a previous requirement for optimization. Three transformations of variables are made on the system equations in order to implement the scheme on an equivalent linear system. This leads to an equivalent representation of the used quadratic performance index that involves the appearance of quadratic weighting terms related to both transformed input and state variables. In this way, a Riccati-matrix sequence, allowing the synthesis of a standard feedback control law, is obtained. Finally, the proposed control scheme is tested on realistic examples.

Pointwise-Optimal Control of Robotic Manipulators

1988 ◽  
Vol 110 (2) ◽  
pp. 210-213 ◽  
Author(s):  
S. Tadikonda ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Adverse Effects ◽  
Sampling Period ◽  
Approximate Expression ◽  
Robotic Manipulators ◽  
Joint Angles ◽  
Time Step ◽  
Quadratic Performance Index ◽  
Linear Regulator ◽  
Quadratic Performance

A method is presented for the pointwise-optimal control of robotic manipulators along a desired trajectory. An approximate expression for the manipulator response is used to minimize a quadratic performance index with a linear regulator and tracking criterion, during each sampling period. The delay associated with implementation of the control action is analyzed, and its adverse effects are eliminated by estimation of the joint angles and torques one time step ahead.

Optimization of Weighting Parameters for an Active Suspension System of a Vehicle

2008 ◽  
Author(s):  
Ahmad A. Fayed ◽  
Mohamed B. Trabia ◽  
Mohamed M. ElMadany
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Ride Comfort ◽  
Active Suspension ◽  
Suspension System ◽  
Performance Criteria ◽  
Penalty Function Method ◽  
Quadratic Performance Index ◽  
Quadratic Performance ◽  
Optimization Loop

Optimal control schemes are usually employed to minimize different performance criteria of active suspension system of a vehicle such as, ride comfort and road safety. These factors are usually combined into a single quantity using proper weighting parameters that depend on the designer’s preferences. Generally, the selection of these weighting parameters is based on trial and error, which can be a time-consuming and computationally-intensive process. This paper proposes the use of an approach based on nested optimization loops to automate the selection process of these weighting parameters. The objective of the inner optimization loop is to minimize of the quadratic performance index associated with the original active suspension problem while the objective of the outer optimization loop is to minimize driver’s acceleration, for ride comfort, while maintaining both tire deflection and suspension deflection within acceptable limits. The design variables are the weighting parameters associated with the quadratic performance index used in the optimal control of active suspension. A modified form of Hooke-Jeeves algorithm is used to handle this problem while the penalty function method is used to handle the constraints. Simulation results show that this approach can improve the design process for active suspension of vehicles.

Perturbation Analysis of Optimal Integral Controls

1984 ◽  
Vol 106 (1) ◽  
pp. 114-116 ◽  
Author(s):  
G. L. Slater
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Guidance System ◽  
Primary System ◽  
Eigenvalues And Eigenvectors ◽  
Closed Loop System ◽  
Outer Loop ◽  
Integral Control ◽  
Quadratic Performance Index ◽  
Quadratic Performance

The application of linear optimal control to the design of systems with integral control action on specified outputs is considered. Using integral terms in a quadratic performance index, an asymptotic analysis is used to determine the effect of variable quadratic weights on the eigenvalues and eigenvectors of the closed loop system. It is shown that for small integral terms the placement of integrator poles and gain calculation can be effectively decoupled from placement of the primary system eigenvalues. This technique is applied to the design of integral controls for a STOL aircraft outer loop guidance system.

When Is a Linear Control System Optimal?

1964 ◽  
Vol 86 (1) ◽  
pp. 51-60 ◽  
Author(s):  
R. E. Kalman
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Control Variable ◽  
Point Of View ◽  
Complete Analysis ◽  
Control Law ◽  
Quadratic Loss ◽  
Mathematical Form ◽  
State Variables

The purpose of this paper is to formulate, study, and (in certain cases) resolve the Inverse Problem of Optimal Control Theory, which is the following: Given a control law, find all performance indices for which this control law is optimal. Under the assumptions of (a) linear constant plant, (b) linear constant control law, (c) measurable state variables, (d) quadratic loss functions with constant coefficients, (e) single control variable, we give a complete analysis of this problem and obtain various explicit conditions for the optimality of a given control law. An interesting feature of the analysis is the central role of frequency-domain concepts, which have been ignored in optimal control theory until very recently. The discussion is presented in rigorous mathematical form. The central conclusion is the following (Theorem 6): A stable control law is optimal if and only if the absolute value of the corresponding return difference is at least equal to one at all frequencies. This provides a beautifully simple connecting link between modern control theory and the classical point of view which regards feedback as a means of reducing component variations.

Using Constrained Bilinear Quadratic Regulator for the Optimal Semi-Active Control Problem

2017 ◽  
Vol 139 (11) ◽  
Author(s):  
I. Halperin ◽  
G. Agranovich ◽  
Y. Ribakov
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Control Design ◽  
Control Signal ◽  
Structural Vibration ◽  
Feedback Form ◽  
Quadratic Performance Index ◽  
Active Feedback ◽  
Quadratic Performance ◽  
Bilinear Quadratic Regulator

Semi-active systems provide an attractive solution for the structural vibration problem. A useful approach, aimed to simplify the control design, is to divide the control system into two parts: an actuator and a controller. The actuator generates a force that tracks a command which is generated by the controller. Such approach reduces the complexity of the control law design as it allows for complex properties of the actuator to be considered separately. In this study, the semi-active control design problem is treated in the framework of optimal control theory by using bilinear representation, a quadratic performance index, and a constraint on the sign of the control signal. The optimal control signal is derived in a feedback form by using Krotov's method. To this end, a novel sequence of Krotov functions which suits the multi-input constrained bilinear-quadratic regulator problem is formulated by means of quadratic form and differential Lyapunov equations. An algorithm is proposed for the optimal control computation. A proof outline for the algorithm convergence is provided. The effectiveness of the suggested method is demonstrated by numerical example. The proposed method is recommended for optimal semi-active feedback design of vibrating plants with multiple semi-active actuators.

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