36,809 research outputs found
Optimal trajectory tracking
This thesis investigates optimal trajectory tracking of nonlinear dynamical
systems with affine controls. The control task is to enforce the system state
to follow a prescribed desired trajectory as closely as possible. The concept
of so-called exactly realizable trajectories is proposed. For exactly
realizable desired trajectories exists a control signal which enforces the
state to exactly follow the desired trajectory. For a given affine control
system, these trajectories are characterized by the so-called constraint
equation. This approach does not only yield an explicit expression for the
control signal in terms of the desired trajectory, but also identifies a
particularly simple class of nonlinear control systems. Based on that insight,
the regularization parameter is used as the small parameter for a perturbation
expansion. This results in a reinterpretation of affine optimal control
problems with small regularization term as singularly perturbed differential
equations. The small parameter originates from the formulation of the control
problem and does not involve simplifying assumptions about the system dynamics.
Combining this approach with the linearizing assumption, approximate and partly
linear equations for the optimal trajectory tracking of arbitrary desired
trajectories are derived. For vanishing regularization parameter, the state
trajectory becomes discontinuous and the control signal diverges. On the other
hand, the analytical treatment becomes exact and the solutions are exclusively
governed by linear differential equations. Thus, the possibility of linear
structures underlying nonlinear optimal control is revealed. This fact enables
the derivation of exact analytical solutions to an entire class of nonlinear
trajectory tracking problems with affine controls. This class comprises
mechanical control systems in one spatial dimension and the FitzHugh-Nagumo
model.Comment: 240 pages, 36 figures, PhD thesi
Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems With Additive Noise
In this paper, we study the problem of how to optimally steer the state
covariance of a general continuous-time linear stochastic system over a finite
time interval subject to additive noise. Optimality here means reaching a
target state covariance with minimal control energy. The additive noise may
include a combination of white Gaussian noise and abrupt "jump noise" that is
discontinuous in time. We first establish the controllability of the state
covariance for linear time-varying stochastic systems. We then turn to the
derivation of the optimal control, which entails solving two dynamically
coupled matrix ordinary differential equations (ODEs) with split boundary
conditions. We show the existence and uniqueness of the solution to these
coupled matrix ODEs, and thus those of the optimal control.Comment: 8 pages, 2 figure
Stabilizability and optimal control of switched differential algebraic equations
In this thesis control of dynamical systems with switches is considered. Examples of such systems are electronic circuits and mechanical systems. The switches are induced by abrupt structural changes due to component failure or physical switches. In the case of constraints on the dynamics, the state of the system can only take certain values and not only differential equations are involved in modeling the system, but also algebraic equations. An important question in control problems is often how well a certain controller performs. Some controllers require little energy, but induce undesired behavior of the system, whereas others perform well in terms of the systems behavior but require a lot of energy. It turns out that in general an optimal controller does not exist. However, necessary and sufficient conditions for the existence of optimal controller given a quadratic cost functional are presented in this thesis. Besides quantitative properties also some qualitative properties are investigated. The systems considered exhibit discontinuous behavior and Dirac impulses, whereas especially Dirac impulses are practically undesirable. Dirac impulses occur in practice in the form of hydraulic shocks in fluid networks or sparks in electronic circuits. The possibility to avoid Dirac impulses is also studied and necessary and sufficient conditions are given
Theory of differential inclusions and its application in mechanics
The following chapter deals with systems of differential equations with
discontinuous right-hand sides. The key question is how to define the solutions
of such systems. The most adequate approach is to treat discontinuous systems
as systems with multivalued right-hand sides (differential inclusions). In this
work three well-known definitions of solution of discontinuous system are
considered. We will demonstrate the difference between these definitions and
their application to different mechanical problems. Mathematical models of
drilling systems with discontinuous friction torque characteristics are
considered. Here, opposite to classical Coulomb symmetric friction law, the
friction torque characteristic is asymmetrical. Problem of sudden load change
is studied. Analytical methods of investigation of systems with such
asymmetrical friction based on the use of Lyapunov functions are demonstrated.
The Watt governor and Chua system are considered to show different aspects of
computer modeling of discontinuous systems
Distributions with dynamic test functions and multiplication by discontinuous functions
As follows from the Schwartz Impossibility Theorem, multiplication of two
distributions is in general impossible. Nevertheless, often one needs to
multiply a distribution by a discontinuous function, not by an arbitrary
distribution. In the present paper we construct a space of distributions where
the general operation of multiplication by a discontinuous function is defined,
continuous, commutative, associative and for which the Leibniz product rule
holds. In the new space of distributions, the classical delta-function
extends to a family of delta-functions ,
dependent on the \textit{shape} . We show that the various known
definitions of the product of the Heaviside function and the delta-function in
the classical space of distributions become particular cases of
the multiplication in the new space of distributions, and provide the
applications of the new space of distributions to the ordinary differential
equations which arise in optimal control theory. Also, we compare our approach
of the Schwartz distribution theory with the approach of the Colombeau
generalized functions algebra, where the general operation of multiplication of
two distributions is defined
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