Elsevier

Automatica

Volume 47, Issue 9, September 2011, Pages 1949-1956
Automatica

Brief paper
Equilibrium-independent passivity: A new definition and numerical certification

https://doi.org/10.1016/j.automatica.2011.05.011Get rights and content

Abstract

We extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium-independent passivity, a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point. We provide a necessary input/output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems. An example from network stability analysis is presented which demonstrates the utility of this new definition. We then proceed to numerical certification of equilibrium-independent passivity using sum-of-squares programming. Finally, through numerical examples we show that equilibrium-independent passivity is less restrictive than incremental passivity.

Section snippets

Introduction and motivation

Since Willems’ seminal paper (Willems, 1972), dissipation inequalities have been studied extensively as a means for reducing high-order systems to interconnections of manageable subsystems in order to more easily ascertain the behavior of the full system (Hill and Moylan, 1976, Hill and Moylan, 1977, Moylan and Hill, 1978). Specifically, the particular dissipation inequality associated with passivity has proved useful in analyzing cascade and feedback systems, even in the nonlinear case when

Notation and assumptions

All norms in this paper are Euclidean norms. We will use Σ to refer to a general dynamical system of the form ẋ=f(x,u)y=h(x,u) with x(t)XRn,u(t)URm,y(t)YRm, f locally Lipschitz, and h continuous. When special structure, e.g., SISO (m=1) or scalar state (n=1), is imposed on Σ it will be made clear.

Basic assumptions: We make two assumptions that are basic to the subsequent discussion. First, we assume that there exists a nonempty set UU such that for every uU there exists a unique xX

A new definition

Definition 1

The system Σ is equilibrium-independent passive (EIP) on U if for every uU there exists a once-differentiable and positive definite storage function Su:XR such that Su(x)|x=0 and Ṡu(x,u)xSu(x)f(x,u)(uu)T(yy) for all uU,xX.

It is understood that in this definition x=kx(u),y=h(x,u), and y=ky(u). We will henceforth make the notational convenience of dropping the subscript from the storage function and referring to it simply as S. When they are clear from the context, we

Examples

In this section we present some properties of EIP systems and accompanying illustrative examples.

Certification of EIP by sum-of-squares programming

We now turn our attention to systems for which X=Rn,U=U=Rm, and Y=Rm. We explore the problem of computing a storage function that certifies equilibrium-independent passivity when the existence of such a function is not obvious. Our objective can be summarized as follows: given a dynamical system Σ, find a positive definite function S=S(x) with S(x)=0 such that (uu)T(yy)xSf(x,u)0x,u,x,u such that f(x,u)=0. It is to be understood that y and y are shorthand for y=h(x,u) and y=h(x,

Certification of EIP for systems in normal form

We now show that the search for a storage function certifying EIP is simplified when the system model is in the following normal form ẏ=f(y,z)+g(y)uż=f̃(y,z) with y(t)R and g(y)>0. In this form the state x(t)[y(t)z(t)]TRnis decomposed into the scalar output state y(t) and the internal state z(t)Rn1. (This notation is chosen to preserve the convention that x represents the entire state and y represents the output.)

We only consider polynomial f,g,f̃. Let S(y,z) (positive definite, S(y,z)=

Comparison to incremental passivity

In this section we explore the relationship between equilibrium-independent passivity and incremental passivity, which is defined formally in Stan and Sepulchre (2007) as follows (with some minor notational changes for consistency with the preceding):

Definition 5

A system Σ is incrementally passive if for all Tf>0 and any pair of trajectories xa,xb, with corresponding inputs and outputs ua,ub and ya,yb, there exists a positive definite storage function SΔ with SΔ(0)=0 such that SΔ(Δx(Tf))SΔ(Δx(0))ITf=0Tf

Quadratic equilibrium-independent dissipation inequalities

The concept of equilibrium-independent passivity extends naturally to more general supply rates of the form s(u,y)=[uuyy]TM[uuyy]. Following the strategy of Lemma 1, we can derive a general necessary condition on the steady-state input/output map for a system to be equilibrium-independent dissipative with respect to a quadratic supply rate. Let Σ be equilibrium-independent dissipative with respect to the supply rate s(u,y)=[uuyy]T[QRRTS][uuyy], that is, there exists a positive

Conclusions

We have defined EIP as a system property and have presented conditions on system dynamics which guarantee EIP for certain systems. We have also proposed a computational method with which to certify the EIP of a system, and have demonstrated some useful properties pertaining to the interconnection of EIP systems.

Since the ideas that motivated this study have proved successful in analyzing a certain class of these interconnections which occur regularly in biological systems and communication

George H. Hines graduated with his B.S. in Engineering and Applied Science (Aeronautics) from Caltech in 2008. He is currently a Ph.D. student at UC Berkeley under the supervision of Profs. Andy Packard and Kameshwar Poolla. His research explores the epigenetic influence of environmental cues on the dynamics of gene expression.

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George H. Hines graduated with his B.S. in Engineering and Applied Science (Aeronautics) from Caltech in 2008. He is currently a Ph.D. student at UC Berkeley under the supervision of Profs. Andy Packard and Kameshwar Poolla. His research explores the epigenetic influence of environmental cues on the dynamics of gene expression.

Murat Arcak received the B.S. degree in Electrical and Electronics Engineering from the Bogazici University, Istanbul, in 1996, and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of California, Santa Barbara, in 1997 and 2000, under the direction of Petar Kokotovic. He was an assistant professor (2001–2006) and an associate professor (2006–2008) at the Rensselaer Polytechnic Institute, Troy, NY. He is now an associate professor in the Electrical Engineering and Computer Sciences Department at the University of California, Berkeley. Dr. Arcak’s research is in nonlinear control theory and its applications, with particular interest in networked dynamical systems. He is a member of SIAM, a senior member of IEEE, and an associate editor for the IFAC journal Automatica. He received a CAREER Award from the National Science Foundation in 2003, the Donald P. Eckman Award from the American Automatic Control Council in 2006, and the SIAG/CST Prize from the Society for Industrial and Applied Mathematics in 2007.

Andrew K. Packard joined UC Berkeley Mechanical Engineering in 1990. His technical interests include quantitative nonlinear systems analysis and optimization and data structure issues associated with large-scale collaborative research for predictive modeling of complex physical processes. He is an author of the Robust Control toolbox distributed by Mathworks. The Meyer sound X-10 loudspeaker utilizes novel feedback control circuitry developed by his UCB research group. He is a recipient of the campus Distinguished Teaching Award, the 1995 Eckman Award, the 2005 IEEE Control System Technology Award, and a 2007 IEEE Fellow.

Supported by the National Science Foundation under grant ECCS 0852750, the Air Force Office of Scientific Research under grant FA9550-09-1-0092, and the NASA Langley NRA contract NNH077ZEA001N: “Analytical Validation Tools for Safety Critical Systems.” The technical contract monitor is Dr. Christine Belcastro. The material in this paper was partially presented at the American Control Conference, June 30–July 2, 2010, St. Louis, Missouri, USA. This paper was recommended for publication in revised form by Associate Editor Alessandro Astolfi under the direction of Editor Andrew R. Teel.

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