scholarly journals On the optimal control of the n-fold integrator

2017 ◽  
Vol 21 (10) ◽  
pp. 114-133
Author(s):  
Yu.N. Gorelov
Keyword(s):  
Optimal Control ◽  
Boundary Conditions ◽  
Impulsive Control ◽  
Singular Solutions ◽  
Quadratic Functional ◽  
Arbitrary Boundary ◽  
The Maximum Principle ◽  
Problem Of Moments ◽  
Time Optimal ◽  
General Relations

 The optimal control problem n-fold integrator with arbitrary boundary conditions and functionals of type norms in spaces of Lq[t0; tf ], q = 1; 2;1 is considered. First, it is the problem of minimizing the total controling impulse, which boils down to L1- problem of moments; secondly, the problem of minimizing the maximum values of the control parameter (represented as L1-problem of moments), and, finally, it is the problem of minimizing ”generalized work control” (as L2-problem of moments). Solving problems is obtained by using the method of moments in the form of the maximum principle by N.N. Krasovsky. It is shown that optimal control in the first problem is approximated by a -impulsive control. Conditions for the existence of regular and singular solutions to this problem depending on the boundary conditions are also specified. The general solution of the second problem, which is the conditions for existence of regular and singular solutions and not equivalence with the mutual problem of time-optimal control is obtained. Examples of solution for the considered control tasks are given. In case of a quadratic functional general relations required for constructing a program optimal control were obtained.

scholarly journals Jerk Limited Time Optimal Control of Flexible Structures

2001 ◽  
Author(s):  
Marco Muenchhof ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Boundary Conditions ◽  
Control Algorithm ◽  
Flexible Structures ◽  
Numerical Results ◽  
Time Optimal Control ◽  
Limited Time ◽  
Benchmark Problem ◽  
Time Optimal ◽  
Geometric Boundary

Abstract This paper deals with the design of jerk-limited time-optimal control sequences for rest-to-rest maneuvers of flexible structures. The resulting jerk profiles will be either bang-bang or bang-off-bang. To ensure quiescent states at the end of the maneuver, a pole cancellation technique will be used. Further constraints account for the geometric boundary conditions. This paper will also investigate the development of the control profile upon variation of the maximum allowable amount of jerk. The last section presents numerical results. The proposed control algorithm is implemented for the Floating Oscillator benchmark problem.

Strong Lagrange duality and the maximum principle for nonlinear discrete time optimal control problems

2019 ◽  
Vol 25 ◽  
pp. 20
Author(s):  
Eero V. Tamminen
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Discrete Time ◽  
Optimal Control Problems ◽  
Discrete Maximum Principle ◽  
Control Problems ◽  
Convexity Condition ◽  
Lagrange Duality ◽  
The Maximum Principle ◽  
Time Optimal

We examine discrete-time optimal control problems with general, possibly non-linear or non-smooth dynamic equations, and state-control inequality and equality constraints. A new generalized convexity condition for the dynamics and constraints is defined, and it is proved that this property, together with a constraint qualification constitute sufficient conditions for the strong Lagrange duality result and saddle-point optimality conditions for the problem. The discrete maximum principle of Pontryagin is obtained in a straightforward manner from the strong Lagrange duality theorem, first in a new form in which the Lagrangian is minimized both with respect to the state and to the control variables. Assuming differentiability, the maximum principle is obtained in the usual form. It is shown that dynamic systems satisfying a global controllability condition with convex costs, have the required convexity property. This controllability condition is a natural extension of the customary directional convexity condition applied in the derivation of the discrete maximum principle for local optima in the literature.

Time Optimal Control for a Class of Common Random Disturbances

1970 ◽  
Vol 92 (2) ◽  
pp. 197-203
Author(s):  
R. Oldenburger ◽  
N. P. Smith
Keyword(s):  
Optimal Control ◽  
Control Function ◽  
Statistical Properties ◽  
Time Optimal Control ◽  
System Variable ◽  
Approximation Accuracy ◽  
The Maximum Principle ◽  
Time Optimal ◽  
Second Order Systems ◽  
Almost All

This paper concerns the time optimal control of a system variable where the controlling input is bounded, as is usually the case, and the system is subject to arbitrary disturbances. An arbitrary disturbance is made up of uncontrollable portions followed by controllable sections. In industrial practice controllers are sized, as for example as to power, to fit the system so that the disturbances encountered are primarily made up of uncontrollable sections followed by controllable portions of sufficient duration for the controller to bring the system to equilibrium. The control designer wishes to have optimal control for any disturbance made up of such an uncontrollable portion followed by a sufficiently long controllable section. Here this problem is solved with the aid of the maximum principle for the class of second order systems which describe almost all governor-engine applications to first approximation accuracy. Previous attempts to solve this problem involved assuming statistical properties of the disturbance thus severely restricting the class of applications. Here only those statistical properties required to implement optimal control are determined. A single control function is derived which suffices to yield optimal trajectories.

scholarly journals Jerk Limited Time Optimal Control of Flexible Structures

2003 ◽  
Vol 125 (1) ◽  
pp. 139-142 ◽  
Author(s):  
Marco Muenchhof ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Rigid Body ◽  
Control Problem ◽  
Flexible Structures ◽  
Time Optimal Control ◽  
Limited Time ◽  
Time Optimal

This paper addresses the problem of designing jerk limited time-optimal control profiles for rest-to-rest maneuvers of flexible structures. The variation of the structure of the jerk profile as a function of the permissible jerk is studied. An optimal control problem is formulated which includes constraints to cancel the poles corresponding to the rigid body and flexible modes of the system and to satisfy the boundary conditions of the rest-to-rest maneuver. The proposed technique is illustrated on the benchmark Floating Oscillator problem where the jerk profile is parameterized as a bang-off-bang or bang-bang profile.

scholarly journals Boundary conditions optimal control

1989 ◽  
Vol 30 (3) ◽  
pp. 343-349 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Control Problem ◽  
Fixed Time ◽  
Rigorous Approach ◽  
Time Problem ◽  
Time Optimal

AbstractA simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.

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