scholarly journals On the optimal control of affine nonlinear systems

2005 ◽  
Vol 2005 (4) ◽  
pp. 465-475 ◽  
Author(s):  
M. Popescu ◽  
A. Dumitrache
Keyword(s):  
Optimal Control ◽  
Nonlinear Systems ◽  
Lie Algebra ◽  
Closed Form Solution ◽  
Bilinear System ◽  
Open Loop ◽  
Form Solution ◽  
Loop Control ◽  
Nilpotent Operators ◽  
Affine Nonlinear Systems

The minimization control problem of quadratic functionals for the class of affine nonlinear systems with the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control corresponding to the first-, second-, and third-order nilpotent operators is determined. In this paper, we have considered the minimum fuel problem for the multi-input nilpotent control and for a scalar input bilinear system for such systems. For the multi-input system, usually an analytic closed-form solution for the optimal controlui∗(t)is not possible and it is necessary to use numerical integration for the set ofmnonlinear coupled second-order differential equations. The optimal control of bilinear systems is obtained by considering the Lie algebra generated by the system matrices. It should be noted that we have obtained an open-loop control depending on the initial value of the statex0.

Modeling and Control of Surface Quality in Chemical Mechanical Planarization (CMP)

2017 ◽  
Author(s):  
Pavan Poosarla ◽  
Hamid Emadi ◽  
Abhijit Chandra ◽  
Sourabh Bhattacharya
Keyword(s):  
State Space ◽  
Control Strategies ◽  
Chemical Mechanical Planarization ◽  
State Space Model ◽  
Closed Form Solution ◽  
Surface Profile ◽  
Open Loop ◽  
Form Solution ◽  
Machining Process ◽  
Loop Control

Obtaining uniform surface finish across large length scales is extremely important in Chemical Mechanical Planarization (CMP). Existing control strategies use results from model simulations to propose open-loop control strategies to reduce the step height on surfaces being polished. In the present work, we propose a strategy to control the surface profile of substrate during CMP process. The evolution of the surface profile is predicted using the state space model of the polishing process. The resulting state space equation is solved and a closed form solution of the surface profile is obtained as a function of time. Based on the solution, we provide a fundamental limitation for the machining process in terms of the extent of planarization that can be achieved for a given material budget.

Adjoint-Based Optimization Procedure for Active Vibration Control of Nonlinear Mechanical Systems

2017 ◽  
Vol 139 (8) ◽  
Author(s):  
Carmine M. Pappalardo ◽  
Domenico Guida
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Control Problem ◽  
Numerical Solution ◽  
Optimal Control Theory ◽  
Numerical Procedure ◽  
Nonlinear Optimal Control ◽  
Open Loop ◽  
Adjoint Equations ◽  
Loop Control

In this paper, a new computational algorithm for the numerical solution of the adjoint equations for the nonlinear optimal control problem is introduced. To this end, the main features of the optimal control theory are briefly reviewed and effectively employed to derive the adjoint equations for the active control of a mechanical system forced by external excitations. A general nonlinear formulation of the cost functional is assumed, and a feedforward (open-loop) control scheme is considered in the analytical structure of the control architecture. By doing so, the adjoint equations resulting from the optimal control theory enter into the formulation of a nonlinear differential-algebraic two-point boundary value problem, which mathematically describes the solution of the motion control problem under consideration. For the numerical solution of the problem at hand, an adjoint-based control optimization computational procedure is developed in this work to effectively and efficiently compute a nonlinear optimal control policy. A numerical example is provided in the paper to show the principal analytical aspects of the adjoint method. In particular, the feasibility and the effectiveness of the proposed adjoint-based numerical procedure are demonstrated for the reduction of the mechanical vibrations of a nonlinear two degrees-of-freedom dynamical system.

scholarly journals Nonlinear Optimal Control Law of Autonomous Unmanned Surface Vessels

2020 ◽  
Vol 10 (5) ◽  
pp. 1686
Author(s):  
Yung-Yue Chen ◽  
Chun-Yen Lee ◽  
Shao-Han Tseng ◽  
Wei-Min Hu
Keyword(s):  
Optimal Control ◽  
Closed Form ◽  
Closed Form Solution ◽  
Nonlinear Optimal Control ◽  
Form Solution ◽  
Control Law ◽  
Tracking Problem ◽  
Optimal Tracking ◽  
Surface Vessels ◽  
Marine Surface

For energy conservation, nonlinear-optimal-control-law design for marine surface vessels has become a crucial ocean technology for the current ship industry. A well-controlled marine surface vessel with optimal properties must possess accurate tracking capability for accomplishing sailing missions. To achieve this design target, a closed-form nonlinear optimal control law for the trajectory- and waypoint-tracking problem of autonomous marine surface vessels (AUSVs) is presented in this investigation. The proposed approach, based on the optimal control concept, can be effectively applied to generate control commands on marine surface vessels operating in sailing scenarios where ocean environmental disturbances are random and unpredictable. In general, it is difficult to directly obtain a closed-form solution from this optimal tracking problem. Fortunately, by having the adequate choice of state-variable transformation, the nonlinear optimal tracking problem of autonomous marine surface vessels can be converted into a solvable nonlinear time-varying differential equation. The solved closed-form solution can also be acquired with an easy-to-implement control structure for energy-saving purposes.

Extended Finite Time Inverse Optimal Control of Nonlinear Systems

2011 ◽  
Vol 403-408 ◽  
pp. 1499-1502
Author(s):  
Xin Jun Ren ◽  
Yan Jun Shen
Keyword(s):  
Optimal Control ◽  
Nonlinear Systems ◽  
Finite Time ◽  
Lyapunov Functions ◽  
Control Law ◽  
Inverse Optimal Control ◽  
Closed Loop System ◽  
Affine Nonlinear Systems ◽  
Definition Of ◽  
The Cost

In this paper, we use the definition of control Lyapunov functions to study finite time inverse optimal control for affine nonlinear systems. Based on control Lyapunov functions, a finite time universal control formula is presented, which can ensure the closed-loop system is finite time stable. From this, less conservative conditions for the finite time inverse optimal control are derived. We design a finite time inverse optimal control law, which minimizes the cost functional. A numerical example verifies the validity of the proposed method.

scholarly journals Control of Three Degrees-of-Freedom Wave Energy Converters Using Pseudo-Spectral Methods

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Ossama Abdelkhalik ◽  
Shangyan Zou ◽  
Rush Robinett ◽  
Giorgio Bacelli ◽  
David Wilson ◽  
...  
Keyword(s):  
Optimal Control ◽  
Wave Energy ◽  
Parametric Excitation ◽  
Degrees Of Freedom ◽  
Closed Form Solution ◽  
Form Solution ◽  
Programming Approach ◽  
Wave Energy Converters ◽  
Pseudo Spectral ◽  
Three Degrees Of Freedom

Abstract This paper presents a solution to the optimal control problem of a three degrees-of-freedom (3DOF) wave energy converter (WEC). The three modes are the heave, pitch, and surge. The dynamic model is characterized by a coupling between the pitch and surge modes, while the heave is decoupled. The heave, however, excites the pitch motion through nonlinear parametric excitation in the pitch mode. This paper uses Fourier series (FS) as basis functions to approximate the states and the control. A simplified model is first used where the parametric excitation term is neglected and a closed-form solution for the optimal control is developed. For the parametrically excited case, a sequential quadratic programming approach is implemented to solve for the optimal control numerically. Numerical results show that the harvested energy from three modes is greater than three times the harvested energy from the heave mode alone. Moreover, the harvested energy using a control that accounts for the parametric excitation is significantly higher than the energy harvested when neglecting this nonlinear parametric excitation term.

scholarly journals Closed-form solution of discrete-time optimal control and its convergence

2018 ◽  
Vol 12 (3) ◽  
pp. 413-418 ◽  
Author(s):  
Hehong Zhang ◽  
Yunde Xie ◽  
Gaoxi Xiao ◽  
Chao Zhai
Keyword(s):  
Optimal Control ◽  
Closed Form ◽  
Discrete Time ◽  
Closed Form Solution ◽  
Form Solution ◽  
Time Optimal Control ◽  
Time Optimal

A CLOSED-FORM SOLUTION OF THE EVOLUTION OPERATOR IN TWO-LEVEL SYSTEMS

1994 ◽  
Vol 08 (08n09) ◽  
pp. 505-508 ◽  
Author(s):  
XIAN-GENG ZHAO
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Evolution Operator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Two Level Systems ◽  
Special Case

It is demonstrated by using the technique of Lie algebra SU(2) that the problem of two-level systems described by arbitrary time-dependent Hamiltonians can be solved exactly. A closed-form solution of the evolution operator is presented, from which the results for any special case can be deduced.

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