The Selection of Performance Indices for Optimal Control Problems

1973 ◽  
Vol 95 (1) ◽  
pp. 24-27 ◽  
Author(s):  
R. L. Woods
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Optimal Control Problems ◽  
Performance Indices ◽  
Proper Selection ◽  
Control Effort ◽  
Time Integral ◽  
Quadratic Index ◽  
Selection Of ◽  
A Performance

This paper investigates the proper selection of a performance index to satisfy a word statement of desired effect. A performance index formed as the time integral of Un is solved for various values of n. When viewed in this light, the classical quadratic index (n = 2) gives a seemingly arbitrary solution. From this investigation the true optimal control solution can be ascertained. For the different values of n, solutions for totally different performance can be obtained. Performance indices combining two effects of minimum control effort and minimum response error are also investigated. The differences of a summed combination and a multiplicative combination are studied.

The archived-based genetic programming for optimal design of linear/non-linear controllers

2020 ◽  
Vol 42 (8) ◽  
pp. 1475-1491
Author(s):  
Adel Mohammadi ◽  
Nader Nariman-Zadeh ◽  
Ali Jamali
Keyword(s):  
Optimal Control ◽  
Genetic Programming ◽  
Analytical Solutions ◽  
Performance Index ◽  
Optimal Control Problems ◽  
Linear Time ◽  
Control Problems ◽  
Control Effort ◽  
Linear Time Invariant ◽  
Quadratic Performance

Evaluation of control signal function is one of the critical subjects in the optimal control problems. The optimal control is usually obtained by optimizing a performance index that is a weighted combination of control effort and state trajectories in the quadratic form, typically known as quadratic performance index (QPI). For the simple case of linear time-invariant (LTI) systems, problems are commonly solved using the well-established governing Riccati equation; however, obtaining the analytical solutions for linear time-variant (LTV) and nonlinear systems has always been highly debated in the optimal control problems. In this study, a newly developed type of Genetic Programming called the archived-based genetic programming (AGP) is presented. Using this algorithm, the analytical solutions for any type of optimal control problems can be obtained faster and more efficiently than the ordinary GPs. Subsequently, due to the inefficiency of QPI in capturing the general behavior of signals, a new performance index named the absolute performance index (API) is proposed in this study. Since the developed AGP algorithm could find the analytical solutions irrespective of the conventional mathematical calculations, it can be effectively implemented to solve the introduced API measures. According to the analytical results, it is observed that in a given problem, the solutions of API are more compatible with the design goals compared with QPI. Furthermore, it is shown that some new forms of the control signals such as impulse solutions, which may not be obtained using QPI, can only be estimated using API in defining the optimal control problems.

A Non-Normality Measure of the Condition Number for Monitoring and Control

1997 ◽  
Vol 119 (2) ◽  
pp. 217-222 ◽  
Author(s):  
Kunsoo Huh ◽  
Jeffrey L. Stein
Keyword(s):  
Optimal Control ◽  
Condition Number ◽  
Minimization Problem ◽  
Monitoring And Control ◽  
Proper Selection ◽  
Modal Structure ◽  
L2 Norm ◽  
And Control ◽  
Normality Measure ◽  
Selection Of

Because the behavior of the condition number can have highly steep and multi-modal structure, optimal control and monitoring problems based on the condition number cannot be easily solved. In this paper, a minimization problem is formulated for κ2(P), the condition number of an eigensystem (P) of a matrix in terms of the L2 norm. A new non-normality measure is shown to exist that guarantees small values for the condition number. In addition, this measure can be minimized by proper selection of controller and observer gains. Application to the design of well-conditioned controller and observer-based monitors is illustrated.

Selection of Rotorcraft Models for Application to Optimal Control Problems

2008 ◽  
Vol 31 (5) ◽  
pp. 1386-1399 ◽  
Author(s):  
Chang-Joo Kim ◽  
Sang Kyung Sung ◽  
Soo Hyung Park ◽  
Sung-Nam Jung ◽  
Kwanjung Yee
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Selection Of

scholarly journals Selection of arbitrarily scaled cross-sections in bending stiffness design

2003 ◽  
Vol 217 (2) ◽  
pp. 113-125
Author(s):  
D Pasini ◽  
D J Smith ◽  
S C Burgess
Keyword(s):  
General Solution ◽  
Cross Sections ◽  
Performance Index ◽  
Selection Procedure ◽  
Performance Indices ◽  
Cross Sectional ◽  
Buckling Instability ◽  
Stiffness Design ◽  
Selection Of

Performance indices can be used to model the relative structural efficiency of different cross-sectional shapes. Performance indices have been previously defined mainly for structural cross-sections that are scaled proportionally in size. This paper extends the method of performance indices by allowing scaling of cross-sections in any direction. A novel feature of the method described in this paper is the inclusion of the space envelope as a design parameter. The first part of the paper gives a derivation of the general solution for the performance index. The second part presents a graphical selection procedure and discusses the efficiency limits of cross-sections due to buckling instability. It concludes with a case study to demonstrate the method.

Optimum Control of Gyroscopic Systems

2010 ◽  
Vol 164 ◽  
pp. 121-126
Author(s):  
Rafał Hein ◽  
Cezary Orlikowski
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Degree Of Freedom ◽  
Vibration Level ◽  
Optimum Control ◽  
Mechatronic Systems ◽  
Minimum Value ◽  
Gyroscopic Systems ◽  
Weight Coefficients ◽  
A Performance

The problem of optimal control of transverse rotor vibrations with gyroscopic interactions has been described and solved in the paper. An integral performance index has been defined for such system in order to minimize vibration level of a chosen rotor point. For this reason, an efficient way of establishing weight coefficients of integral performance index for multi-degree-of-freedom system with gyroscopic interactions has been described. Presented method enables to determine such weighing coefficients that selected modal forms would be assumed before dynamics properties and a performance index would took the minimum value. Computation results demonstrate that the proposed method is efficient and may be applied for complex mechatronic systems.

Performance indices for arbitrarily scaled rectangular cross-sections in bending stiffness design

2002 ◽  
Vol 216 (2) ◽  
pp. 101-113
Author(s):  
D Pasini ◽  
S C Burgess ◽  
D J Smith
Keyword(s):  
Cross Sections ◽  
Performance Index ◽  
Graphical Method ◽  
Bending Stiffness ◽  
Performance Indices ◽  
Cross Sectional ◽  
Stiffness Design ◽  
Design Case Study ◽  
Selection Of

Performance indices are presented for the selection of optimal rectangular beams in bending stiffness design. Previous studies have developed performance indices for only three design cases: proportional scaling of width and height, constrained height and constrained width. This paper extends the methodology of the performance index to any arbitrary direction of scaling. The performance index has the form Eq/p, where q is a function only of the scaling vector between two cross-sectional envelopes of different materials. The paper also presents a graphical method for determining the performance of rectangular beams in stiffness design. The perormance index and the graphical method are applied to a design case study.

Comparison on wavelets techniques for solving fractional optimal control problems

2016 ◽  
Vol 24 (6) ◽  
pp. 1185-1201 ◽  
Author(s):  
PK Sahu ◽  
S Saha Ray
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Dynamic Constraints ◽  
Orthonormal Wavelets ◽  
Fractional Optimal Control Problems ◽  
Wavelet Methods

This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.

scholarly journals A numerical method for solving a class of fractional optimal control problems using Boubaker polynomial expansion scheme

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4485-4502 ◽  
Author(s):  
N. Singha ◽  
C. Nahak
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Control Variable ◽  
The State ◽  
Polynomial Expansion ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

We construct a numerical scheme for solving a class of fractional optimal control problems by employing Boubaker polynomials. In the proposed scheme, the state and control variables are approximated by practicingNth-order Boubaker polynomial expansion. With these approximations, the given performance index is transformed to a function of N + 1 unknowns. The objective of the present formulation is to convert a fractional optimal control problem with quadratic performance index into an equivalent quadratic programming problem with linear equality constraints. Thus, the latter problem can be handled efficiently in comparison to the original problem. We solve several examples to exhibit the applicability and working mechanism of the presented numerical scheme. Graphical plots are provided to monitor the nature of the state, control variable and the absolute error function. All the numerical computations and graphical representations have been executed with the help of Mathematica software.

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