Finite-Time Stability: An Input-Output Approach
Description
Systematically presents the input-output finite-time stability (IO-FTS) analysis of dynamical systems, covering issues of analysis, design and robustness
The interest in finite-time control has continuously grown in the last fifteen years. This book systematically presents the input-output finite-time stability (IO-FTS) analysis of dynamical systems, with specific reference to linear time-varying systems and hybrid systems. It discusses analysis, design and robustness issues, and includes applications to real world engineering problems.
While classical FTS has an important theoretical significance, IO-FTS is a more practical concept, which is more suitable for real engineering applications, the goal of the research on this topic in the coming years.
Key features:
- Includes applications to real world engineering problems.
- Input-output finite-time stability (IO-FTS) is a practical concept, useful to study the behavior of a dynamical system within a finite interval of time.
- Computationally tractable conditions are provided that render the technique applicable to time-invariant as well as time varying and impulsive (i.e. switching) systems.
- The LMIs formulation allows mixing the IO-FTS approach with existing control techniques (e. g. H8 control, optimal control, pole placement, etc.).
This book is essential reading for university researchers as well as post-graduate engineers practicing in the field of robust process control in research centers and industries. Topics dealt with in the book could also be taught at the level of advanced control courses for graduate students in the department of electrical and computer engineering, mechanical engineering, aeronautics and astronautics, and applied mathematics.
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Persons
FRANCESCO AMATO is Professor of Bioengineering, Dean of the School of Computer and Biomedical Engineering and the Coordinator of the Doctorate School in Biomedical and Computer Engineering at the University of Catanzaro, Italy. The scientific activity of Francesco Amato has developed in the fields of systems and control theory; robust control, finite-time stability and control, control of nonlinear quadratic systems with applications to the contexts of aircraft control, computational biology and bioengineering. He has published around 250 papers in international journals and conference proceedings and two monographs with Springer Verlag entitled "Robust Control of Linear Systems subject to Uncertain Time-Varying Parameters" and "Finite-Time Stability and Control".
GIANMARIA DE TOMMASI is Associate Professor with the Department of Electrical Engineering and Information Technology, University of Naples Federico II, Italy. Since 2002, he has been a Visiting Researcher with the Joint European Torus (JET) Tokamak, Oxfordshire, U.K., where he has participated in various projects connected to the JET plasma current and shape control system. He has authored more than 100 journal and conference papers, and is a co-author of the monograph "Finite-Time Stability and Control" (Springer). His current research interests include control of nuclear fusion devices, fault detection for discrete event systems, identification of discrete event systems modeled with Petri nets, and stability on finite-time horizon of hybrid systems. Dr. De Tommasi is a member of the IEEE Control System Society Conference Editorial Board, and has been Guest Editor of the Fusion Engineering and Design special issue titled "Design and Implementation of Real-Time Systems for Magnetic Confined Fusion Devices".
ALFREDO PIRONTI is a Full Professor of System and Control Theory in the Department of Electrical and Information Technology Engineering, University of Naples Federico II, Italy. He spent several periods as visiting researcher at the Max Planck Institute for Plasma Physics in Garching (Germany), the Center for Control Engineering and Computation (University of California at Santa Barbara), the ITER Joint Work Site of Naka (Japan), and the EFDA-JET site of Culham (UK). His research interests include application of feedback control to nuclear fusion problems, robust control of uncertain systems, and differential games theory. In 2005 he was guest editor for the IEEE Control Systems Magazine journal, where he contributed to two special issues focused on the control of plasmas in tokamak machines. He is the author of more than 200 papers published in international journals, books, and conference proceedings.
Content
Preface xi
List of Acronyms xiii
1. Introduction 1
1.1 Finite-Time Stability (FTS) 1
1.2 Input-Output Finite-Time Stability 6
1.3 FTS and Finite-Time Convergence 10
1.4 Background 10
1.4.1 Vectors and signals 10
1.4.2 Impulsive dynamical linear systems 12
1.5 Book Organization 13
2. Linear Time-Varying Systems: IO-FTS Analysis 15
2.1 Problem Statement 15
2.2 IO-FTS for W2 Exogenous Inputs 16
2.2.1 Preliminaries 16
2.2.2 Necessary and sücient conditions for IO-FTS for W2 exogenous inputs 22
2.2.3 Computational issues 25
2.3 A Sücient Condition for IO-FTS for W8 Inputs 26
2.4 Summary 29
3. Linear Time-Varying Systems: Design of IO Finite-Time Stabilizing Controllers 33
3.1 IO Finite-Time Stabilization via State Feedback 34
3.2 IO-Finite-Time Stabilization via Output Feedback 36
3.3 Summary 42
4. IO-FTS with Nonzero Initial Conditions 45
4.1 Preliminaries 45
4.2 Interpretation of the Norm of the Operator LSNZ 48
4.3 Sücient Conditions for IO-FTS-NZIC 52
4.4 Design of IO Finite-Time Stabilizing Controllers NZIC 55
4.4.1 State feedback 56
4.4.2 Output feedback 57
4.5 Summary 58
5. IO-FTS with Constrained Control Inputs 61
5.1 Structured IO-FTS and Problem Statement 61
5.2 Structured IO-FTS Analysis 63
5.3 State Feedback Design 65
5.4 Design of an Active Suspension Control System Using Structured IO-FTS 67
5.5 Summary 70
6. Robustness Issues and the Mixed H8/FTS Control Problem 71
6.1 Preliminaries 72
6.1.1 System setting 72
6.1.2 IO-FTS with an H8 bound 73
6.2 Robust and Quadratic IO-FTS with an H8 Bound 77
6.2.1 Main result 78
6.2.2 A numerical example 80
6.3 State Feedback Design 82
6.3.1 Numerical example: Cont'd 85
6.4 Case study: Quadratic IO-FTS with an H8 Bound of the Inverted Pendulum 86
6.5 Summary 88
7. Impulsive Dynamical Linear Systems: IO-FTS Analysis 89
7.1 Background 90
7.1.1 Preliminary results for the W2 case 90
7.2 Main Results: Necessary and Sücient Conditions for IO-FTS in Presence of W2 Signals 91
7.3 Example and Computational Issues 96
7.4 Main Result: A Sücient Condition for IO-FTS in Presence of W8 Signals 98
7.4.1 An illustrative example 99
7.5 Summary 100
8. Impulsive Dynamical Linear Systems: IO Finite-Time Stabilization via Dynamical Controllers 103
8.1 Problem Statement 103
8.2 IO Finite-Time Stabilization of IDLSs: W2 Signals 104
8.2.1 A numerical example 107
8.3 IO Finite-Time Stabilization of IDLSs: W8 Signals 108
8.3.1 Illustrative example: Cont'd 110
8.4 Summary 111
9. Impulsive Dynamical Linear Systems with Uncertain Resetting Times 113
9.1 Arbitrary Switching 113
9.2 Uncertain Switching 114
9.3 Numerical Example 116
9.3.1 Known resetting times 117
9.3.2 Arbitrary switching 118
9.3.3 Uncertain switching 118
9.4 Summary 119
10. Hybrid Architecture for Deployment of Finite-Time Control Systems 121
10.1 Controller Architecture 121
10.2 Examples 123
10.2.1 Hybrid active suspension control 123
10.2.2 Lateral collision avoidance system 124
10.3 Summary 129
A. Fundamentals on Linear Time-Varying Systems 131
B. Schur Complements 137
C. Computation of Feasible Solutions to Optimizations Problems
Involving DLMIs 139D. Solving Optimization Problems Involving DLMIs using MATLAB® 145
E. Examples of Applications of IO-FTS Control Design to Real-World Systems 151
References 159
Index 167
1
Introduction
This first chapter has the twofold objective of introducing the framework of input-output finite-time stability (IO-FTS), together with the notation that will be used throughout the book, and providing some useful background on the analysis of the behavior of dynamical systems.
In order to introduce the topics dealt with in this monograph, we first recall the concept of state FTS, and then we will extend it to the input-output case, both with zero and nonzero initial conditions. The former extension correspond to the concept of IO-FTS, while the latter represents a generalization of the finite-time boundedness (FTB) concept, namely IO-FTS with nonzero initial conditions (IO-FTS-NZIC).
Roughly speaking, FTS involves the behavior of the system state for an autonomous dynamical system with nonzero initial conditions, while IO-FTS looks at the input-output behavior of the system, with zero initial conditions. IO-FTS-NZIC mixes the two concepts, considering the input-output finite-time control problem with a nonzero initial condition. The common points to these definitions is that they are defined over a finite-time interval and that quantitative bounds are given for the admissible signals during this interval.
1.1 Finite-Time Stability (FTS)
The concept of finite-time stability (FTS) dates back to the fifties, when it was introduced in the Russian literature ([1-3]); later, during the sixties, this concept appeared for the first time in Western journals [4-6].
Given the dynamical system
1.1where , we can give the following formal definition, which restates the original definition in a way consistent with the notation adopted in this monograph; in the following we consider the finite-time interval , with .
Definition 1.1 (FTS, [2, 4, 8])
Given the time interval , a set , and a family of sets , system (1.1) is said to be finite-time stable with respect to (wrt) if
1.2where, with a slight abuse of notation, denotes the solution of (1.1) starting from at time .?
Note that, in general, the set , called outer (or trajectory) set, possibly depends on time; obviously must contain theinner (or initial) set , for well-posedness of Definition 1.1.
An issue that is important to clarify is why the property expressed by (1.2) is called FTS.
In order to answer this question, we recall the classical definition of Lyapunov stability (LS, [32, Ch. 4]; see also Appendix A.3). Let be an equilibrium point for system (1.1), i.e., for all . The equilibrium point is said to be stable in the sense of Lyapunov if for each , there exists a positive scalar , possibly depending on and , such that , implies
and this holds for all .
The key points in the above definition are: an equilibrium point is stable if, once an arbitrary value for has been fixed, which defines a ball centered in , then it must be possible to build an inner ball (of radius ) such that, whenever the initial condition is inside such ball, the trajectory of the system starting from does not exit the outer ball (of radius ). Moreover this property holds for an infinite time horizon, that is, for all between and infinity.
Note that LS is a qualitative concept, that is, both the inner and the outer ball are not quantified; therefore, LS can be regarded as a structural property: a given equilibrium point is either stable or it is not.
Now let us come back to Definition 1.1; even in this case there is an inner set , usually centered at an equilibrium point of system (1.1), and an outer set . FTS requires that, whenever the trajectory of (1.1) starts inside the inner set, it does not exit the outer set. From this point of view, Definition 1.1) mimics the one of LS, and this justifies the use of the term stability. However, differently from LS, this is only required over a finite interval of time, which should be possibly short with respect to steady state; i.e., FTS can be used to shape the behavior of the system during the transients.
Another important point is that FTS is a quantitative concept, since the inner and the outer set are specified once and for all. Therefore the same system can be finite-time stable for some choice of , , and , and non-finite-time stable for a different choice of these parameters.
It is worth noting that, in principle, FTS does not necessarily requires that the inner set contains any equilibrium point for system (1.2); however, this particular case will not be dealt with in this book, where we shall consider ellipsoidal sets centered at the origin of the state space.
A direct consequence of the discussion above is that FTS and LS are independent concepts; referring to a linear system, to simplify the terminology (see Appendix A.3), a system can be finite-time stable, despite not being stable in the sense of Lyapunov, and vice versa. While LS deals with the behavior of a system within a sufficiently long (in principle infinite) time interval, FTS is a more practical concept, useful to study the behavior of the system within a finite (possibly short) interval, and therefore it finds application whenever it is desired that the state variables do not exceed a given threshold (for example to avoid saturation or the excitation of nonlinear dynamics) during the transients.
In the following, we shall focus on linear time-varying (LTV) systems
1.3with piecewise continuous; note that the assumption on piecewise continuity of guarantees existence and uniqueness of the solution of system (1.3) starting from , at time (see Appendix A). Moreover, if we consider ellipsoidal state sets, i.e.,
Definition 1.1 can be rewritten as follows.
Definition 1.2 (FTS for LTV Systems [12, 33, 34])
Given the time interval , a positive definite matrix , and a continuous, positive definite matrix-valued function defined over , such that , system (1.3) is said to be finite-time stable wrt if
1.4?
As said above, the assumption that in Definition 1.2 is needed to guarantee that the initial closed ellipsoid is a proper subset of the open ellipsoid , hence guaranteeing the well-posedness of the definition itself.
A graphical explanation of the FTS concept is reported in Figure 1.1 for a second-order system with a constant matrix-valued function . In particular, if a system is FTS, then all the trajectories starting within the ellipse defined by should be like the one depicted in green in Figure 1.1. Conversely, two trajectories that are not FTS are reported in red.
Figure 1.1 Given a time interval , and the two ellipsoidal domains delimited by and by the constant matrix , a second-order system is finite-time stable if all the trajectories over the considered time interval are like the one reported in light gray. Furthermore, in dark gray are reported two examples of trajectories that are not finite-time stable.
In the following, we consider a numerical example.
Example 1.1 (Lyapunov stability and finite-time stability for LTI systems)
This introductory example shows the difference between the two concepts of LS and FTS for a second-order linear time-invariant (LTI) system. To this aim, let us consider the following autonomous LTI system
1.5System (1.5) is clearly Lyapunov stable, being negative the maximum real part of the eigenvalues of the matrix .
While LS is a structural property of an LTI system, FTS it is not. Indeed, given the time interval if we specify the two weighting matrices in Definition 1.2 as follows
it turns out that system (1.5) is not FTS wrt , as it is clearly shown in Figure 1.2, since there is at least one state trajectory that starts within the initial domain defined by and that goes outside the ellipsoidal domain specified by during the time interval .
On the other hand, if we consider a different time interval for the FTS analysis, e.g., by letting , systems (1.5) turns out to be FTS wrt . Indeed, in the last case, it can be shown that all the state trajectories of (1.5) that start within the initial domain defined by remain within the target ellipsoidal domain defined by (one possible way to check FTS is to solve the feasibility problem reported in [35], Theorem 2.1-(v)]).
Figure 1.2 Free response of the LS stable LTI system (1.5) when the initial state is set equal to . Although the considered LTI system is Lyapunov stable, the same system can be either FTS or not, depending on the FTS parameters.
Let us now consider the following Lyapunov unstable system
1.6Also in this case FTS is not related to LS, as it is shown in Figure 1.3, where the initial ellipsoidal domain is given by
the finite-time interval is taken equal to , and two different time-invariant target domains, defined by
are chosen. It is straightforward to check...
