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.

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.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

scholarly journals GEKKO Optimization Suite

Processes ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 106 ◽  
Author(s):  
Logan Beal ◽  
Daniel Hill ◽  
R. Martin ◽  
John Hedengren
Keyword(s):  
Optimal Control ◽  
Optimization Problems ◽  
Object Oriented ◽  
Differential Algebraic Equations ◽  
Mixed Integer ◽  
Modeling Languages ◽  
Algebraic Equations ◽  
Simulation And Optimization ◽  
Algebraic Modeling Languages ◽  
Real Time Optimization

This paper introduces GEKKO as an optimization suite for Python. GEKKO specializes in dynamic optimization problems for mixed-integer, nonlinear, and differential algebraic equations (DAE) problems. By blending the approaches of typical algebraic modeling languages (AML) and optimal control packages, GEKKO greatly facilitates the development and application of tools such as nonlinear model predicative control (NMPC), real-time optimization (RTO), moving horizon estimation (MHE), and dynamic simulation. GEKKO is an object-oriented Python library that offers model construction, analysis tools, and visualization of simulation and optimization. In a single package, GEKKO provides model reduction, an object-oriented library for data reconciliation/model predictive control, and integrated problem construction/solution/visualization. This paper introduces the GEKKO Optimization Suite, presents GEKKO’s approach and unique place among AMLs and optimal control packages, and cites several examples of problems that are enabled by the GEKKO library.

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.

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