Direct Optimal Control of Nonlinear Systems Via Hamilton’s Law of Varying Action

1995 ◽  
Vol 117 (3) ◽  
pp. 262-269 ◽  
Author(s):  
E. Adigu¨zel ◽  
H. O¨z
Keyword(s):  
Optimal Control ◽  
Equations Of Motion ◽  
A Priori ◽  
Integral Form ◽  
Performance Measure ◽  
Algebraic Equations ◽  
Nonlinear Feedback ◽  
State Equations ◽  
Generalized Coordinates ◽  
Direct Optimal Control

Based on direct application of Hamilton’s Law of Varying Action in conjunction with an assumed-time-modes approach for both the generalized coordinates and input functions, a direct optimal control methodology is developed for the control of nonlinear, time varying, spatially discrete mechanical systems. Expansion coefficients of admissible time modes for the dependent variables of the dynamic system and those for the inputs constitute the states and controls, respectively. This representation permits explicit a priori integration in time of the energy expressions in Hamilton’s law and leads to the algebraic equations of motion for the system which replace the conventional differential state equations; therefore the customary extremum principles of calculus of variations involving differential form constraints are also bypassed. Similarly, the standard integral form of the quadratic regulator performance measure employed in the formulation of the optimality problem is transformed into an algebraic performance measure via assumed-time-modes expansion of the generalized coordinates and the control inputs. The proposed methodology results in an algebraic optimality problem from which a closed-form explicit solution for the nonlinear feedback control law is obtained directly. Simulations of two nonlinear nonconservative systems are, included.

The Optimal Control Approach to Dynamical Inverse Problems

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Wolfgang Steiner ◽  
Stefan Reichl
Keyword(s):  
Optimal Control ◽  
Inverse Problems ◽  
Practical Importance ◽  
Differential Algebraic Equations ◽  
Performance Measure ◽  
State Variables ◽  
Algebraic Equations ◽  
State Equations ◽  
Solution Approach ◽  
Control Approach

This paper considers solution strategies for “dynamical inverse problems,” where the main goal is to determine the excitation of a dynamical system, such that some output variables, which are derived from the system’s state variables, coincide with desired time functions. The paper demonstrates how such problems can be restated as optimal control problems and presents a numerical solution approach based on the method of steepest descent. First, a performance measure is introduced, which characterizes the deviation of the output variables from the desired values, and which is minimized by the solution of the inverse problem. Second, we show, how the gradient of this error functional can be computed efficiently by applying the theory of optimal control, in particular by following an idea of Kelley and Bryson. As the major contribution of this paper we present a modification of this method which allows the application to the case where the state equations are given by a set of differential algebraic equations. This situation has great practical importance since multibody systems are mostly described in this way. For comparison, we also discuss an approach which bases an a direct transcription of the optimal control problem. Moreover, other methods to solve dynamical inverse problems are summarized.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

scholarly journals Time-optimal control of infinite order hyperbolic systems with time delays

2009 ◽  
Vol 19 (4) ◽  
pp. 597-608 ◽  
Author(s):  
Adam Kowalewski
Keyword(s):  
Optimal Control ◽  
Time Delays ◽  
Infinite Order ◽  
Hyperbolic Systems ◽  
Integral Form ◽  
Adjoint System ◽  
Time Optimal Control ◽  
Optimal Controls ◽  
State Equations ◽  
Time Optimal

Time-optimal control of infinite order hyperbolic systems with time delaysIn this paper, the time-optimal control problem for infinite order hyperbolic systems in which time delays appear in the integral form both in state equations and in boundary conditions is considered. Optimal controls are characterized in terms of an adjoint system and shown to be unique and bang-bang. These results extend to certain cases of nonlinear control problems. The particular properties of optimal control are discussed.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sa¨nger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Secondly, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Natural Coordinates in the Optimal Control of Multibody Systems

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
Peter Betsch ◽  
Ralf Siebert ◽  
Nicolas Sänger
Keyword(s):  
Optimal Control ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Open Loop ◽  
Algebraic Equations ◽  
Natural Coordinates ◽  
Joint Forces ◽  
New Energy

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Dwelling on the Connection Between SO(3) and Rotation Matrices in Rigid Multibody Dynamics – Part 2: Comparison Against Formulations Using Euler Parameters or Euler Angles

2021 ◽  
Author(s):  
Jay Taves ◽  
Alexandra Kissel ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Group Structure ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Implicit Time ◽  
Euler Angles ◽  
Algebraic Equations ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Orientation Matrix

Abstract We compare three solution approaches that use the index 3 set of differential algebraic equations (DAEs) to solve the constrained multibody dynamics problem through straight discretization via an implicit time integrator. The first approach is described in a companion paper and dwells on the connection between the orientation matrix and the SO(3) group. Its salient point is that the orientation matrix A is a problem unknown, directly computed without resorting to the use of other position-level generalized coordinates such as Euler angles or Euler parameters. The second approach employs Euler angles as part of the position-level generalized coordinates, and uses them to subsequently evaluate the orientation matrix A. The third approach replaces the Euler angles with Euler parameters (quaternions). The numerical integration method of choice in this contribution is first order implicit Euler. We report a similar number of iterations for convergence for all solution implementations (called rA, rε, and rp); we also observed an approximately twofold speedup of rA over rp and rε. The tests were carried out in conjunction with three models: simple pendulum, slider crank, and four-link mechanism. These simulation results were obtained using two Python simulation engines that were developed independently as part of this formulation comparison undertaking. The codes are available in a GitHub public repository and were developed to provide two different perspectives on the formulation performance issue. The improvements in simulation speed are traced back to a simpler form of the equations of motion and more concise Jacobians that enter the numerical solution. It remains to investigate whether these speed gains carry to higher order integration formulas, where the underlying Lie-group structure of SO(3) brings additional complexity in the rA solution.

Application of Singular Value Decomposition for Analysis of Mechanical System Dynamics

1985 ◽  
Vol 107 (1) ◽  
pp. 82-87 ◽  
Author(s):  
N. K. Mani ◽  
E. J. Haug ◽  
K. E. Atkinson
Keyword(s):  
Singular Value Decomposition ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Singular Value ◽  
Algebraic Equations ◽  
Generalized Coordinates ◽  
Singular Value Decomposition Method ◽  
System Information ◽  
Value Decomposition ◽  
Algebraic Constraint

A singular value decomposition method for efficient solution of mixed differential-algebraic equations of motion of mechanical systems is developed. Differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates that are related through nonlinear algebraic constraint equations. Singular value decomposition of the constraint Jacobian matrix is used to define a new set of generalized coordinates that are partitioned into optimal independent and dependent sets. Integration of only independent generalized coordinates generates all system information. A numerical example is presented to demonstrate effectiveness of the method.

scholarly journals A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Marwan Abukhaled ◽  
Ibrahim Sadek
Keyword(s):  
Optimal Control ◽  
Operational Matrix ◽  
Performance Measure ◽  
Point Of View ◽  
Computational Method ◽  
Algebraic Equations ◽  
State Control ◽  
Lumped Parameter ◽  
Linear Algebraic Equations ◽  
Quadratic Cost Function

A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

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