Über die Autorin bzw. den Autor

Rafal Goebel is an assistant professor in the Department of Mathematics and Statistics at Loyola University, Chicago. Ricardo G. Sanfelice is an assistant professor in the Department of Aerospace and Mechanical Engineering at the University of Arizona. Andrew R. Teel is a professor in the Electrical and Computer Engineering Department at the University of California, Santa Barbara.

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"This superb book unifies some of the key developments in hybrid dynamical systems from the last decade and, through elegant and clear technical content, introduces the necessary tools for understanding the stability of these systems. It will be a great resource for graduate students and researchers in the field."--Magnus Egerstedt, Georgia Institute of Technology

"With broad applications for science and engineering, this first-rate book develops solid foundations for a comprehensive theory of hybrid dynamical systems. Diverse literature is brought together for the first time, making a huge body of knowledge conveniently accessible."--Dennis S. Bernstein, University of Michigan

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"This superb book unifies some of the key developments in hybrid dynamical systems from the last decade and, through elegant and clear technical content, introduces the necessary tools for understanding the stability of these systems. It will be a great resource for graduate students and researchers in the field."--Magnus Egerstedt, Georgia Institute of Technology

"With broad applications for science and engineering, this first-rate book develops solid foundations for a comprehensive theory of hybrid dynamical systems. Diverse literature is brought together for the first time, making a huge body of knowledge conveniently accessible."--Dennis S. Bernstein, University of Michigan

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Hybrid Dynamical Systems

PRINCETON UNIVERSITY PRESS

Contents

Preface...........................................................................ix1 Introduction....................................................................11.1 The modeling framework........................................................11.2 Examples in science and engineering...........................................21.3 Control system examples.......................................................71.4 Connections to other modeling frameworks......................................151.5 Notes.........................................................................222 The solution concept............................................................252.1 Data of a hybrid system.......................................................252.2 Hybrid time domains and hybrid arcs...........................................262.3 Solutions and their basic properties..........................................292.4 Generators for classes of switching signals...................................352.5 Notes.........................................................................413 Uniform asymptotic stability, an initial treatment..............................433.1 Uniform global pre-asymptotic stability.......................................433.2 Lyapunov functions............................................................503.3 Relaxed Lyapunov conditions...................................................603.4 Stability from containment....................................................643.5 Equivalent characterizations..................................................683.6 Notes.........................................................................714 Perturbations and generalized solutions.........................................734.1 Differential and difference equations.........................................734.2 Systems with state perturbations..............................................764.3 Generalized solutions.........................................................794.4 Measurement noise in feedback control.........................................844.5 Krasovskii solutions are Hermes solutions.....................................884.6 Notes.........................................................................945 Preliminaries from set-valued analysis..........................................975.1 Set convergence...............................................................975.2 Set-valued mappings...........................................................1015.3 Graphical convergence of hybrid arcs..........................................1075.4 Differential inclusions.......................................................1115.5 Notes.........................................................................1156 Well-posed hybrid systems and their properties..................................1176.1 Nominally well-posed hybrid systems...........................................1176.2 Basic assumptions on the data.................................................1206.3 Consequences of nominal well-posedness........................................1256.4 Well-posed hybrid systems.....................................................1326.5 Consequences of well-posedness................................................1346.6 Notes.........................................................................1377 Asymptotic stability, an in-depth treatment.....................................1397.1 Pre-asymptotic stability for nominally well-posed systems.....................1417.2 Robustness concepts...........................................................1487.3 Well-posed systems............................................................1517.4 Robustness corollaries........................................................1537.5 Smooth Lyapunov functions.....................................................1567.6 Proof of robustness implies smooth Lyapunov functions.........................1617.7 Notes.........................................................................1678 Invariance principles...........................................................1698.1 Invariance and ω-limits.................................................1698.2 Invariance principles involving Lyapunov-like functions.......................1708.3 Stability analysis using invariance principles................................1768.4 Meagre-limsup invariance principles...........................................1788.5 Invariance principles for switching systems...................................1818.6 Notes.........................................................................1849 Conical approximation and asymptotic stability..................................1859.1 Homogeneous hybrid systems....................................................1859.2 Homogeneity and perturbations.................................................1899.3 Conical approximation and stability...........................................1929.4 Notes.........................................................................196Appendix: List of Symbols.........................................................199Bibliography......................................................................201Index.............................................................................211

Chapter One

The model of a hybrid system used in this book is informally presented in this section. The focus is on the data structure and on modeling. Several examples are given, including models of hybrid control systems. The model of a hybrid system is then related to other modeling frameworks, such as hybrid automata, impulsive differential equations, and switching systems. A formal presentation of the model, together with a rigorous definition of the solution, is postponed until Chapter 2.

1.1 THE MODELING FRAMEWORK

The model of a hybrid system used in this book can be represented in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)

A reader less familiar with set-valued mappings and differential or difference inclusions may choose to keep in mind a less general representation involving equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

This representation suggests that the state of the hybrid system, represented by x, can change according to a differential inclusion [?? [member of] F(x) or differential equation [??] = f(x) while in the set ITLITL, and it can change according to a difference inclusion x⁺ [member of] G(x) or difference equation x⁺ = g(x) while in the set D. The notation [??] represents the velocity of the state x, while x⁺ represents the value of the state after an instantaneous change.

A rigorous statement of what constitutes a model of a hybrid system and what is a solution to the model is postponed until Chapter 2. This chapter focuses on modeling of various hybrid systems in the form (1.1) or (1.2).

To shorten the terminology, the behavior of a dynamical system that can be described by a differential equation or inclusion is referred to as flow. The behavior of a dynamical system that can be described by a difference equation or inclusion is referred to as jumps. This leads to the following names for the four objects involved in (1.1) or (1.2):

• ITLITL is the flow set.

• F (or f) is the flow map.

• D is the jump set.

• G (or g) is the jump map.

This book discusses hybrid systems in finite-dimensional spaces, that is, the flow set ITLITL and the jump set D are subsets of an n-dimensional Euclidean space Rⁿ. For consistency in the model, it will be required that the function f, respectively g, be defined on at least the set ITLITL, respectively D. In the case of set-valued flow and jump maps, it will be required that F, respectively G, have nonempty values on ITLITL, respectively D.

As the model in (1.2) or (1.1) suggests, the flow set, the flow map, the jump set, and the jump map can be specialized to capture the dynamics of purely continuous-time or discrete-time systems on Rⁿ. The former corresponds to a flow set equal to Rⁿ and an empty jump set, while the latter can be captured with an empty flow set and a jump set defined as Rⁿ.

1.2 EXAMPLES IN SCIENCE AND ENGINEERING

Many mechanical systems experience impacts. Examples range from elaborate systems such as walking robots, through colliding billiard balls or the Newton's cradle, to a seemingly simple bouncing ball. Such systems flow in between impacts. A rough approximation of the impacts suggests considering them as instantaneous, and hence, as leading to jumps in the state of the system. Consequently, systems with impacts can be viewed as hybrid systems.

The first example is the mentioned bouncing ball. This example, and some of the later ones in this chapter, reappear throughout the book as illustrations of various properties and results.

Example 1.1. (Bouncing ball) Consider a point-mass bouncing vertically on a horizontal surface. In between impacts the point-mass flows, experiencing acceleration due to gravity. At impacts, when the point-mass hits the surface, the change in velocity is approximated as being instantaneously reversed and possibly diminished in magnitude due to dissipation of energy.

The state of the point-mass can be described with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where x₁ represents the height above the surface and x₂ represents the vertical velocity. It is natural to say that flow is possible when the point-mass is above the surface, or when it is at the surface and its velocity points up. Hence, the flow set is

C = {x [member of] R² : x₁ > 0 or x₁ = 0, x₂ ≥ 0}.

The choice of a flow map is delicate at one point in ITLITL, that is, at x = 0. First, it is natural to say that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where -γ is the acceleration due to gravity. Second, it is natural to say that f(0) = 0; it has to be accepted, though, that the resulting flow map f is not continuous at 0. Impacts happen when the point-mass is on the surface with negative velocity. Hence, the jump set is

D = {x [member of] R² : x₁ = 0, x₂ < 0}.

The jump map is given, for some λ [member of] (0, 1), by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

An alternative choice for g is the vector -λx since this function agrees with g(x) on the set D. Figure 1.1 illustrates the data of the bouncing ball system.

In the bouncing ball model above, every jump is followed by a period of flow. In other words, consecutive jumps do not happen. Consecutive jumps can happen in other systems with impacts, like in a model of Newton's cradle. Newton's cradle consists of at least three identical steel balls, each of which is suspended on a pendulum. At the stationary state, the balls are aligned along a horizontal line. Lifting a ball from one end of the alignment and releasing it leads to a collision of the lifted ball with the remaining balls. After the collision, the ball that was lifted and released becomes stationary and the ball on the other end of the alignment swings up. One way to model this interaction is to consider a sequence of collisions between pairs of adjacent balls.

A number of biological systems, such as groups of fireflies or crickets, are able to produce synchronized behavior, flashing or chirping, respectively, through a dynamical mechanism that can be viewed as hybrid.

Example 1.2. (Flashing fireflies) The timing of flashes of a firefly is determined by the firefly's internal clock. In between flashes, the internal clock gradually increases. When it reaches a threshold, a flash occurs and the clock is instantly reset to 0. In a group of fireflies, the flash of one firefly affects the internal clock of all other fireflies. That is, when a firefly witnesses a flash from another firefly, its internal clock instantly increases to a value closer to the threshold.

To model the internal clocks of n fireflies, normalize units so that each firefly's internal clock, denoted xi, takes values in the interval [0, 1], i.e., every threshold is 1. The flow set is then

C = [0, 1)ⁿ := {x [member of] Rⁿ : x_i [member of] [0, 1), i = 1, 2, ..., n}.

In between the flashes, every clock state flows toward the threshold according to the differential equation [??]_i = f_i(x_i), where f_i : [0, 1] [right arrow] R_>0, i = 1, 2, ..., n, is continuous. This defines the flow map f.

Jumps occur when one of the internal clocks reaches the threshold. Thus, the jump set is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One method to model the (instantaneous) changes in internal clocks during a flash is through the jump map defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where ε > 0. This indicates that the internal clock x_i of a firefly witnessing a flash increases to (1 + ε)x_i, unless this would result in reaching or exceeding the threshold, in which case the internal clock is reset to 0 together with the internal clock of the flashing firefly. Figure 1.2 illustrates the evolution of the clock variable x for n = 2 and n = 10 when f_i = 1 for each i.

Example 1.3. (Power control with a thyristor) Consider the electric circuit in Figure 1.3(a) for controlling the power delivered to a load. The load consists of a resistor R and an inductance L that is connected to a power source through a thyristor with a gate control port. A simple model describing the operation of the thyristor is as follows. When in conduction mode, which can be triggered through the gate port, the thyristor allows flow of current from anode to cathode, which are the terminals denoted as a+ and c- in Figure 1.3(a), respectively. It will turn off once the current from anode to cathode becomes zero. The load current is denoted by i_L, its voltage by v_L, and the capacitor's voltage by v_o. The sinusoidal input voltage with angular frequency ω is denoted by v_s and is generated by the output v_s = z₁ of the system

z₁ = ωz₂, z₂ = -ωz₁. (1.3)

By defining the state of the system to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the continuous dynamics are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These equations can be derived applying electrical circuit theory for each mode of operation. Note that q = 0 indicates that the discrete state remains constant during flows, and that τ = 1 enforces that τ counts the flow time in between switches. Assuming that when the thyristor is in off mode the load current is zero, two conditions trigger switches of the thyristor mode:

• When the thyristor is off (q = 0, i_L = 0), the firing angle has been reached (τ ≥ α/ω), and the capacitor voltage is positive (v₀ > 0), then switch to on (q = 1).

• When the thyristor is on, the load current is zero and decreasing (i_L = 0, i_L < 0), then switch to off.

These conditions can be captured with the flow and jump sets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the jump map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

At every jump, q is toggled and the timer is restarted to trigger the next jump to on mode at the programmed firing angle. The top plot in Figure 1.3(b) shows the input voltage with ω = 0.1/(2π) rad/sec and the resulting load's current with a firing angle of 20ω rad, while the bottom plot shows the associated logic and timer states.

1.3 CONTROL SYSTEM EXAMPLES

The control of a continuous-time system with state feedback faces both practical and theoretical obstacles: precise information about the state may not be available at all times, even if frequent measurements of the state are available; the behavior of the closed-loop system may be very sensitive to errors in the state measurements; or satisfactory performance of the closed-loop system may not be achievable by using just one state-feedback controller. These issues provide motivation for the use of hybrid control, several simple instances of which are described below.

Example 1.4. (Sample-and-hold control) Given a continuous-time control system and a state-feedback controller, associating with each state of the system the control to be applied there, a sample-and-hold implementation of the feedback is essentially as follows:

• sample: measure the state of the system, and use the feedback controller to obtain the control value based on the measurements;

• hold: apply the computed constant control value for certain amount of time;

and repeat the procedure infinitely many times. The processes of sampling and computing the control can be modeled as an instantaneous event. This leads to a continuous behavior of the closed-loop system in between the sampling times, according to the continuous-time dynamics of the control system and the constant value of the control, and an instantaneous change at every sampling time, when the control value is instantly updated.

A schematic example of a sample-and-hold control system is in Figure 1.4, where a digital device controls an analog plant. The basic operation of the system is as follows. The output of the plant is sampled by an analog-to-digital converter, denoted A/D. The digitized output is processed by the algorithm, and the result is applied to the plant through a digital-to-analog converter, denoted D/A. For a periodic A/D sampler and a zero-order hold (ZOH) type of D/A, the output samples and control input updates occur at a fixed sampling period T.

(Continues...)

Excerpted from Hybrid Dynamical Systemsby Rafal Goebel Ricardo G. Sanfelice Andrew R. Teel Copyright © 2012 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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