Chapter Eleven. Control Vector Lyapunov Functions for Large-Scale Impulsive Systems

2011 ◽  
pp. 271-288
Keyword(s):  
Lyapunov Functions ◽  
Large Scale ◽  
Impulsive Systems ◽  
Control Vector ◽  
Vector Lyapunov Functions

Control Vector Lyapunov Functions for Large-Scale Impulsive Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Lyapunov Theory ◽  
System Stability ◽  
Dynamical System Theory ◽  
Control Vector ◽  
Vector Lyapunov Functions

This chapter extends the notion of control vector Lyapunov functions to impulsive dynamical systems. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In particular, the use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Using control vector Lyapunov functions, the chapter develops a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.

Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Nonlinear Dynamical System ◽  
Control Vector ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions

This chapter introduces the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, showing that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. These control vector Lyapunov functions are used to develop a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. The chapter also describes the connections between the notion of vector dissipativity and optimality of the proposed decentralized feedback control law. The proposed control framework is then used to construct decentralized controllers for large-scale nonlinear dynamical systems with robustness guarantees against full modeling uncertainty.

Finite-Time Stabilization of Large-Scale Systems via Control Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Finite Time ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Control Vector ◽  
Time Stability ◽  
Nonlinear Dynamical ◽  
Thermoacoustic Instabilities ◽  
Finite Time Stability ◽  
Vector Lyapunov Functions

This chapter develops a general framework for finite-time stability analysis based on control vector Lyapunov functions. Specifically, it develops a vector comparison system whose solution is finite-time stable and relates this finite-time stability property to the stability properties of a nonlinear dynamical system using a vector comparison principle. The results are specialized to the case of a scalar Lyapunov function to obtain universal finite-time stabilizers for nonlinear systems that are affine in the control. Finally, the utility of the proposed framework is demonstrated using two numerical examples: the first involves a large-scale dynamical system with control signals for each decentralized control channel as a function of time; the second example considers control of thermoacoustic instabilities in combustion processes.

Large-Scale Impulsive Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Supply Rate ◽  
Nonlinear Dynamical ◽  
Dissipation Inequality ◽  
Vector Lyapunov Functions ◽  
Definition Of ◽  
Stability Results

This chapter develops vector dissipativity notions for large-scale nonlinear impulsive dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear impulsive dynamical systems in terms of a hybrid vector dissipation inequality involving a vector hybrid supply rate, a vector storage function, and an essentially nonnegative, semistable dissipation matrix. The chapter also defines generalized notions of a vector available storage and a vector required supply and shows that they are element-by-element ordered, nonnegative, and finite. Extended Kalman-Yakubovich-Popov conditions, in terms of the local impulsive subsystem dynamics and the interconnection constraints, are developed for characterizing vector dissipativeness via vector storage functions for large-scale impulsive dynamical systems. Finally, using the concepts of vector dissipativity and vector storage functions as candidate vector Lyapunov functions, the chapter presents feedback interconnection stability results of large-scale impulsive nonlinear dynamical systems.

Introduction

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Interconnected Systems ◽  
Large Scale Systems ◽  
Vector Lyapunov Functions ◽  
Finite Time Stabilization ◽  
And Control ◽  
Interconnected Dynamical Systems

This book develops a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, with an emphasis on vector Lyapunov function methods and vector dissipativity theory. It examines large-scale continuous-time interconnected dynamical systems and describes thermodynamic modeling of large-scale interconnected systems, along with the use of vector Lyapunov functions to control large-scale dynamical systems. It also discusses finite-time stabilization of large-scale systems via control vector Lyapunov functions, coordination control for multiagent interconnected systems, large-scale impulsive dynamical systems, finite-time stabilization of large-scale impulsive dynamical systems, and hybrid decentralized maximum entropy control for large-scale systems. This chapter provides a brief introduction to large-scale interconnected dynamical systems as well as an overview of the book's structure.

scholarly journals Vector dissipativity theory for large-scale impulsive dynamical systems

2004 ◽  
Vol 2004 (3) ◽  
pp. 225-262 ◽  
Author(s):  
Wassim M. Haddad ◽  
VijaySekhar Chellaboina ◽  
Qing Hui ◽  
Sergey Nersesov
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Impulsive System ◽  
Vector System ◽  
Solution Properties ◽  
Impulsive Systems ◽  
Large Scale Systems ◽  
Vector Lyapunov Functions ◽  
The Individual ◽  
Stability Results

Modern complex large-scale impulsive systems involve multiple modes of operation placing stringent demands on controller analysis of increasing complexity. In analyzing these large-scale systems, it is often desirable to treat the overall impulsive system as a collection of interconnected impulsive subsystems. Solution properties of the large-scale impulsive system are then deduced from the solution properties of the individual impulsive subsystems and the nature of the impulsive system interconnections. In this paper, we develop vector dissipativity theory for large-scale impulsive dynamical systems. Specifically, using vector storage functions and vector hybrid supply rates, dissipativity properties of the composite large-scale impulsive systems are shown to be determined from the dissipativity properties of the impulsive subsystems and their interconnections. Furthermore, extended Kalman-Yakubovich-Popov conditions, in terms of the impulsive subsystem dynamics and interconnection constraints, characterizing vector dissipativeness via vector system storage functions, are derived. Finally, these results are used to develop feedback interconnection stability results for large-scale impulsive dynamical systems using vector Lyapunov functions.

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