Jakob Löber - Optimal Trajectory Tracking of Nonlinear Dynamical Systems

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Springer International Publishing AG 2017

Jakob Lber Optimal Trajectory Tracking of Nonlinear Dynamical Systems Springer Theses Recognizing Outstanding Ph.D. Research 10.1007/978-3-319-46574-6_1

1. Introduction

Jakob Lber 1

(1)

Institute for Theoretical Physics, EW 7-1, Technical University of Berlin, Berlin, Germany

Jakob Lber

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Science often begins with the discovery of physical phenomena. The second step is to describe, understand, and predict them, often in terms of mathematical theories. The final step is to take advantage of the discovered phenomena. This last step is the topic of control theory.

To familiarize the reader with the broad topic of control theory, and to classify the different methods of and objectives for control, we discuss a simple example. Imagine you want to balance a broom standing upright on the palm of your hand. How would you proceed? You monitor the angle of the broom with respect to the floor. In case the broom leans to one side, which it will do over and over again, you move your hand to counteract the brooms movement. Your eyes act as a detector for the brooms orientation, while your hand acts as a control with the objective of enforcing an upright broom position as good as possible.

In a more technical language, the upright broom is called an inverted pendulum and is a classic problem in control theory (Chen ). A pendulum is a weight suspended from a pivot with a fixed length. It may perform motion along the entire circle, including the upright position. Due to friction, an uncontrolled pendulum oscillates with decreasing amplitude until it ends up in the stable equilibrium position directly below the pivot. To control the pendulum, we apply a force whose strength and direction over time can be prescribed to at least some extent.

In mathematical terms, the task of balancing the broom in an upright position corresponds to the stabilization of the pendulums unstable equilibrium position directly above the pivot. The disturbances due to e.g. thermal motion of the surrounding molecules, which drive the pendulum out of the unstable equilibrium position, are essentially unpredictable, at least within the simple pendulum model. For a successful stabilization, we must monitor the state of the pendulum, i.e., its position or velocity or both, and apply the control force accordingly. Such a control is called a closed-loop or feedback control, which generally necessitates a continuous or at least repeated monitoring of the system state (Sontag ).

In contrast to closed-loop control, open-loop control does not require state monitoring. These methods are applicable if the system dynamics, including all parameter values, are known sufficiently well and the impact of perturbations can be neglected. Thus the time dependence of the control force can be computed in advance to achieve a prescribed control target. For the pendulum, the target could be a given position over time of the pendulum bob. An example of open loop control is a kick of defined strength applied during each oscillation period to counteract friction and sustain the oscillation with a given amplitude.

Note that open-loop control methods are usually unsuitable to stabilize a stationary point because of the unpredictable nature of the disturbances driving the system out of the unstable state. A notable exception are methods like periodic forcing (Schll and Schuster ). The Kapitza pendulum is a pendulum with a vertically vibrating pivot point. For a certain range of frequencies and amplitudes of the forcing, the upper stationary point is stabilized. However, this method of control is not exact in the sense that the bob does not strictly reach the equilibrium state but oscillates around the stationary point with an amplitude and frequency depending on the details of the forcing. In contrast to the methods of mathematical control theory, where the control objective is usually formulated in a mathematically rigorous manner, the actual solution stabilized by periodic forcing is not prescribed and more difficult to characterize.

Besides a distinction of open and closed-loop control, control methods can be classified in many additional ways. The stabilizing feedback control acts only if the pendulum leaves the unstable stationary point, and vanishes otherwise. This is an example of a non-invasive control, whose amplitude approaches zero upon reaching the control target. On the other hand, invasive controls are nonzero at all times. Another classification is optimal and non-optimal control. Optimal controls aim at finding the best control in the sense of minimizing a given performance criterion. An example in the context of the pendulum is to prescribe the position and velocity of the bob during a time interval. In general, it is impossible for the bob to follow an arbitrary combination of position and velocity simply because the velocity is always given as the time derivative of the position. Nevertheless, an optimal control can be found which brings the controlled position and velocity as closely as possible to the desired position and velocity. This control task is called optimal trajectory tracking.

Other classifications of control systems arise with respect to their mathematical structure. We distinguish between dynamical systems, which have only time as the independent variable, and spatio-temporal systems, which additionally depend on spatial coordinates. Furthermore, control systems can be linear and nonlinear. In contrast to uncontrolled systems, which can only be linear or nonlinear in the system state, controlled systems can also be classified with respect to the linearity or nonlinearity in the control. Controlled dynamical systems which are linear in control but allowed to be nonlinear in state are known as affine systems.

1.1 Affine Control Systems

The subject of this thesis are controlled dynamical systems of the form

(1.1)

(1.2)

Here, t is the time, is called the state vector with n components and denotes the transposed of vector . The dot

(1.3)

denotes the time derivative of . The vector with components is the vector of control or input signals . The nonlinearity

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