تجزیه و تحلیل از ورودی به حالت ثبات برای سیستم های غیر خطی زمان گسسته از طریق برنامه ریزی پویا

The input-to-state stability (ISS) property for systems with disturbances has received considerable attention over the past decade or so, with many applications and characterizations reported in the literature. The main purpose of this paper is to present analysis results for ISS that utilize dynamic programming techniques to characterize minimal ISS gains and transient bounds. These characterizations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimization problems in nonlinear dissipative systems theory (including L2L2-gain analysis and nonlinear H∞H∞ theory). As such, the results presented address an obvious gap in the literature.

Among the many stability properties for systems with disturbances, the input-to-state stability (ISS) property proposed by Sontag (1989) deserves special attention. Indeed, ISS is fully compatible with Lyapunov stability theory (Sontag & Wang, 1995) while its other equivalent characterizations relate it to robust stability, dissipativity and input–output stability theory (Sontag, 2000; Sontag & Wang, 1996). The ISS property has found its main application in the ISS small gain theorem that was first proved by Jiang, Teel, and Praly (1994). Several different versions of the ISS small gain theorem that use different (equivalent) characterizations of the ISS property and their various applications to nonlinear controller design can be found in Jiang and Mareels (1997); Jiang, Mareels, and Wang (1996); Teel (1996) and references defined therein. The ISS property and the ISS small gain theorems naturally lead to the concept of nonlinear disturbance gain functions or simply “nonlinear gains”. In this context, obtaining sharp estimates for the nonlinear gains is an important issue. Indeed, the better the nonlinear gain estimate that we can obtain, the larger the class of systems to which the ISS small gain results can be applied. Currently, the main tool for estimating the nonlinear gains are the so-called ISS Lyapunov functions that typically produce rather conservative estimates (over bounds) for the ISS nonlinear gains. It is the main purpose of this paper to present several results that provide a constructive framework based on dynamic programming for obtaining minimum ISS nonlinear gains . These results are related to optimization based methods in nonlinear dissipative systems theory, such as L2L2-gain analysis and nonlinear H∞H∞ theory (see, Helton & James, 1999 and references defined therein), as well as recently developed optimization based L∞L∞ methods (see, Fialho & Georgiou, 1999; Huang & James, 2003 and references defined therein). Needless to say, the optimization approach that we take in this paper can inflict a heavy (and sometimes infeasible) computational burden on the user. This is a reflection of the intrinsic complexity of the problem that we are trying to solve. We present results only for discrete-time nonlinear systems since many calculations and technical details are in this way simplified. The paper is organized as follows. In Section 2 we present several equivalent definitions of the ISS property and state a result from the literature that motivates our definitions and results. A fundamental dynamic programming equation that we need to state our main results is given in Section 3. Sections 4, 5 and 6 contain results on minimum nonlinear gains for different equivalent definitions of the ISS property. Two related ISS properties are analysed in Section 7 using the techniques of Sections 5 and 6. Several illustrative examples are presented in Section 8 and the paper is closed with conclusions in Section 9.

We have presented results for verifying different characterizations of ISS via dynamic programming. Formulas for minimum nonlinear gains and bounds on transients for different characterizations are presented. A discussion on how these results can be used to analyse input-to-output stability and incremental input-to-state stability is also given. We illustrated our approach by three examples. The aim of this paper is to present a constructive formulation for finding minimal ISS gains and transient bounds. The results of this paper provide a framework for generating numerical algorithms for calculating ISS gains and transient bounds. The detailed development and analysis of numerical methods is an important topic for future investigation and is outside the scope of the present paper. Our example, however, indicates the potential benefits of this numerical approach and motivates careful investigation of numerical issues.