A Technique of Quasi-Optimum Control

1966 ◽  
Vol 88 (2) ◽  
pp. 437-443 ◽  
Author(s):  
B. Friedland
Keyword(s):  
Nonlinear System ◽  
Analytic Solution ◽  
Control Variable ◽  
Control Law ◽  
System Of Equations ◽  
Point Boundary ◽  
Optimum Control ◽  
Bounded Control ◽  
Linear System Of Equations ◽  
The Matrix

To find the optimum control law u = u(x) for the process x˙ = f(x, u), the Hamiltonian H = p′f is formed. The optimum control law can be expressed as u = u* = σ(p, x), where u* maximizes H. The transformation from the state x to the “costate” p entails the analytic solution of the nonlinear system: x˙ = f(x, σ(p, x)); p˙=−fx′p with boundary conditions at two points. Since such a solution generally can not be found, we seek a quasi-optimum control law of the form u = σ(P + Mξ, x), where x = X + ξ with ‖ξ‖ small, and X, P are the solutions of a simplified problem, obtained by setting ξ = 0 in the above two-point boundary-value problem. We assume that P(X) is known. It is shown that the matrix M satisfies a Riccati equation, −M˙ = MHXP + HPXM + MHPPM + HXX, and can be computed by solving a linear system of equations. A simple example illustrates the application of the technique to a problem with a bounded control variable.

New Results in Quasi-Optimum Control

2006 ◽  
Vol 129 (1) ◽  
pp. 96-99
Author(s):  
Bernard Friedland
Keyword(s):  
Boundary Value Problem ◽  
Control Law ◽  
Point Boundary ◽  
Optimum Control ◽  
Bounded Control ◽  
Actual Problem ◽  
State Variable ◽  
State Variable Constraints ◽  
Linear Correction ◽  
New Applications

A technique of quasi-optimum control, developed by the author in 1966, has as its goal to permit one to use the apparatus of optimum control theory without having to solve the two-point boundary value problem for the actual problem. This is achieved by assuming the actual problem is “near” a simplified problem the solution of which was known. In this case, the control law adds a linear correction to the costate of the simplified problem. The linear correction is obtained as the solution of a matrix Riccati equation. After a review of the theory, several new applications of the technique are provided. These include mildly nonlinear processes, processes with bounded-control, and processes with state-variable constraints.

Some Notes on the Grassmann Manifolds and Nonlinear System

2013 ◽  
Vol 419 ◽  
pp. 611-615
Author(s):  
Chun Na Zeng ◽  
Peng Tao
Keyword(s):  
Differential Geometry ◽  
Control System ◽  
Nonlinear System ◽  
Control Theory ◽  
Boundary Problem ◽  
Periodic Solutions ◽  
Grassmann Manifold ◽  
Quotient Space ◽  
Point Boundary ◽  
The Matrix

The differential geometry as a new tool has been introduced to research the control system, especially the nonlinear system. In this paper, by considering how to construct a manifold from a quotient space, we investigate the structure of Grassmann manifold concretely. This is beneficial to study the problem of finding periodic solutions of the matrix Riccati equations of control theory and the two point boundary problem.

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.

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].

scholarly journals New robust bounded control for uncertain nonlinear system using mixed backstepping and lyapunov redesign

2019 ◽  
Vol 9 (2) ◽  
pp. 1090
Author(s):  
Muhammad Nizam Kamarudin ◽  
Sahazati Md. Rozali ◽  
T. Sutikno ◽  
Abdul Rashid Husain
Keyword(s):  
Nonlinear System ◽  
Average Power ◽  
Variable Structure ◽  
Linear Matrix ◽  
Control Law ◽  
Uncertain Nonlinear System ◽  
Bounded Control ◽  
Control Laws ◽  
Structure Control ◽  
Lyapunov Redesign

<p>This paper presents a new robust bounded control law to stabilize uncertain nonlinear system with time varying disturbance. The design idea comes from the advantages of backstepping with Lyapunov redesign, which avoid the needs of fast switching of discontinuous control law offered by its counterpart - a variable structure control. We reduce the conservatism in the design process where the control law can be flexibly chosen from Lyapunov function, hence avoiding the use of convex optimization via linear matrix inequality (LMI) in which the feasibility is rather hard to be obtained. For this work, we design two type control algorithms namely normal control and bounded control. As such, our contribution is the introduction of a new bounded control law that can avoid excessive control energy, high magnitude chattering in control signal and small oscillation in stabilized states. Computation of total energy for both control laws confirmed that the bounded control law can stabilize with less enegry consumption. We also use Euler's approximation to compute average power for both control laws. The robustness of the proposed controller is achieved via saturation-like function in Lyapunov redesign, and hence guaranting asymptotic stability of the closed-loop system.</p>

A Full-System Approach of the Elastohydrodynamic Line/Point Contact Problem

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
W. Habchi ◽  
D. Eyheramendy ◽  
P. Vergne ◽  
G. Morales-Espejel
Keyword(s):  
Finite Element ◽  
Nonlinear System ◽  
System Approach ◽  
Jacobian Matrix ◽  
Point Contact ◽  
System Of Equations ◽  
Nonlinear System Of Equations ◽  
Full System ◽  
Element Discretization ◽  
Fully Coupled

The solution of the elastohydrodynamic lubrication (EHL) problem involves the simultaneous resolution of the hydrodynamic (Reynolds equation) and elastic problems (elastic deformation of the contacting surfaces). Up to now, most of the numerical works dealing with the modeling of the isothermal EHL problem were based on a weak coupling resolution of the Reynolds and elasticity equations (semi-system approach). The latter were solved separately using iterative schemes and a finite difference discretization. Very few authors attempted to solve the problem in a fully coupled way, thus solving both equations simultaneously (full-system approach). These attempts suffered from a major drawback which is the almost full Jacobian matrix of the nonlinear system of equations. This work presents a new approach for solving the fully coupled isothermal elastohydrodynamic problem using a finite element discretization of the corresponding equations. The use of the finite element method allows the use of variable unstructured meshing and different types of elements within the same model which leads to a reduced size of the problem. The nonlinear system of equations is solved using a Newton procedure which provides faster convergence rates. Suitable stabilization techniques are used to extend the solution to the case of highly loaded contacts. The complexity is the same as for classical algorithms, but an improved convergence rate, a reduced size of the problem and a sparse Jacobian matrix are obtained. Thus, the computational effort, time and memory usage are considerably reduced.

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