scholarly journals Quasi-Optimum Control of a Second Order Vibratory System with Bounded Control

1968 ◽  
Vol 34 (259) ◽  
pp. 436-444
Author(s):  
Yoshikazu SAWARAGI ◽  
Toshiro ONO
Keyword(s):  
Second Order ◽  
Optimum Control ◽  
Bounded Control ◽  
Vibratory System

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

Решения гамильтоновой системы с двумерным управлением в окрестности особой экстремали второго порядка

2021 ◽  
Vol 76 (5(461)) ◽  
pp. 201-202
Author(s):  
Мария Игоревна Ронжина ◽  
Mariya Igorevna Ronzhina ◽  
Лариса Анатольевна Манита ◽  
Larisa Anatol'evna Manita ◽  
Лев Вячеславович Локуциевский ◽  
...  
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Infinite Number ◽  
Finite Time ◽  
Second Order ◽  
Two Dimensional ◽  
Optimal Solutions ◽  
Countable Number ◽  
Bounded Control ◽  
Singular Extremal

We consider a Hamiltonian system that is affine in two-dimensional bounded control that takes values in an ellipse. In the neighborhood of a singular extremal of the second order, we find two families of optimal solutions: chattering trajectories that attain the singular point in a finite time with a countable number of control switchings, and logarithmic-like spirals that reach the singular point in a finite time and undergo an infinite number of rotations.

scholarly journals Maximum principle and its extension for bounded control problems with boundary conditions

2004 ◽  
Vol 2004 (35) ◽  
pp. 1855-1879 ◽  
Author(s):  
Olga Vasilieva
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Second Order ◽  
Necessary Condition ◽  
Control Problems ◽  
Bounded Control ◽  
Singular Controls ◽  
Bounded Control Problems ◽  
The Maximum Principle ◽  
First Order

This note is focused on a bounded control problem with boundary conditions. The control domain need not be convex. First-order necessary condition for optimality is obtained in the customary form of the maximum principle, and second-order necessary condition for optimality of singular controls is derived on the basis of second-order increment formula using the method of increments along with linearization approach.

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.

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.

Non-Linear Random Vibrations Using Second-Order Adjoint and Projected Differentiation Methods

2021 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Random Variables ◽  
Random Excitation ◽  
Second Order ◽  
Output Process ◽  
Random Vibrations ◽  
Order Accuracy ◽  
Vibratory System ◽  
Non Linear

Abstract This paper proposes a new computationally efficient methodology for random vibrations of nonlinear vibratory systems using a time-dependent second-order adjoint variable (AV2) method, and a second-order projected differentiation (PD2) method. The proposed approach is called AV2-PD2. The vibratory system can be excited by stationary Gaussian or non-Gaussian random processes. A Karhunen-Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. A second-order adjoint approach is used to obtain the required first and second-order output derivatives accurately by solving as many sets of equations of motion (EOMs) as the number of KL random variables. These derivatives are used to compute the marginal CDF of the output process with second-order accuracy. Then, a second-order projected differentiation method calculates the autocorrelation function of each output process with second-order accuracy, at an additional cost of solving as many sets of EOM as the number of outputs of interest, independently of the time horizon (simulation time). The total number of solutions of the EOM scales linearly with the number of input KL random variables and the number of output processes. The efficiency and accuracy of the proposed approach is demonstrated using a non-linear Duffing oscillator problem under a quadratic random excitation.

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