Elsevier

Automatica

Volume 45, Issue 7, July 2009, Pages 1601-1610
Automatica

Box invariance in biologically-inspired dynamical systems

https://doi.org/10.1016/j.automatica.2009.02.028Get rights and content

Abstract

A dynamical system is box invariant if there exists a box-shaped positively invariant region. We show that box invariance can be checked in cubic time for linear and affine systems, and that it remains decidable for classes of nonlinear systems of interest (with polynomial structure). We present results on the robustness of box invariance for linear systems using spectral properties of Metzler matrices. We also present sufficient conditions for establishing box invariance of switched and hybrid systems. In general, we argue that box invariance is a characteristic of many biologically-inspired dynamical models.

Introduction

An invariant set is a subset of the state space of a dynamical system with the property that, if the system state is in this set at some time, then it will stay in the set indefinitely in the future (Blanchini, 1999). An invariant set is extremely useful from the perspective of formal analysis and verification (Clarke, Grumberg, & Peled, 2000). The task in formal verification is to show that none of the trajectories of a given dynamical system violate a given property, such as a liveness or safety property, or in the opposite instance to find “witnesses” that do not abide by such properties. Safety specifications form an important class of properties, which encode the condition that a system can never reach a given subset of “unsafe” or “bad” states. Direct verification of safety properties is difficult because computing the set of reachable states is often infeasible. However, an invariant set can be used to verify a safety property by showing that it encloses all reachable states, but none of the unsafe states. From a dual perspective, invariants can be used to look at reachability properties, where the objective is to verify if any trajectory of the system, starting from a region of the state space, will reach a target set (which is again a subset of the state space). The concept of invariance can also be related to certain notions of stability (Podelski & Wagner, 2006). This motivates the need to develop effective and constructive approaches to discover invariant sets for dynamical systems—and especially invariant sets with simple shapes.

Positively invariant sets can be obtained by exploiting the property that their boundaries may correspond to level surfaces of a proper Lyapunov-like function. This approach has been the source of several results on the existence of positively invariant sets (Blanchini, 1999, Kiendl et al., 1992). However, this is quite restrictive in general, since systems that are not stable can still have useful invariant sets.

In this paper, we focus on positively invariant sets that are in the form of a box, that is, a hyper-rectangular region specified by giving (upper and lower) bounds for each state variable. The concept of box invariance is related to a number of studies in the literature (Blanchini, 1999) (see Section 2.1). For instance, Kiendl et al. (1992) look at the use of vector norms to study stability. The notions that are developed in the present study are related to that of component-wise stability (Pastravanu and Voicu, 2003, Voicu, 1984), as well as to the concepts of practical stability and Lagrange stability (Passino, Burgess, & Michel, 1995).

The study of several systems, especially models drawn from the domain of systems biology, has suggested that they frequently admit box-shaped, positively invariant sets. This seems natural in retrospect since state variables often correspond to physical quantities that are naturally constrained and tend to either degrade, or remain conserved. In this paper, we are interested in the practical aspects of the notion of box invariance. In particular, we focus on how complex it is to check for box invariance of a dynamical model, as well as to construct a particular box, whenever possible. More precisely, we show that it is computationally feasible to check if a dynamical system is invariant with respect to a box set, and to explicitly find out box invariant sets for a large class of dynamical systems (in particular, biological ones). Because of the discussed connections with other notions in systems theory, it is then argued that box invariance is an ideal concept for building analysis and verification tools to investigate such systems.

Outline. We formally define the notion of box invariance in Section 2. Next, we present necessary and/or sufficient characterizations of this notion for linear (Section 3), affine (Section 3.3), and classes of nonlinear systems (Section 4) that are especially meaningful for models of biological systems. Box invariance of linear systems is strongly related to the theory of Metzler matrices, as explained in Section 3.1. Using this connection, we perform robustness analysis of box invariant systems in Section 3.2. In Section 5, we extend the study to the more general case of switched and hybrid systems. All throughout, we will present computational complexity results and illustrate the introduced concepts using examples from systems biology.

Section snippets

The concept of box invariance

We consider general and uncontrolled dynamical systems of the form ẋ=f(x),xRn. We assume the basic boundedness and Lipschitz properties that ensure the existence of a unique solution of the vector field, given any possible initial condition. A rectangular box around a point x0 is specified using two diagonally opposite points l and u, where l<x0<u (interpreted component-wise) and is defined as Box(l,u){xlxu}. Such a box has 2n faces consisting of n lower and n upper faces. The jth lower

Box invariant linear and affine systems

Given a linear system and a box around its equilibrium point, the problem of checking whether the system is box invariant with respect to the given box can be solved by verifying the related condition only at the 2n vertices of the box (rather than on all the points of the surface of the box). The set of vertices, Vert(l,u), of the box Box(l,u) is defined as Vert(l,u)={xxi=lixi=ui,i}.

Proposition 1

A linear systemẋ=Ax,xRn, is box invariant if there exist two pointsl(R)nandu(R+)nsuch that for each point

Box invariant nonlinear systems

In this Section, we extend the study of box invariance to nonlinear systems. While for the linear and the affine cases box invariance can be characterized with necessary and sufficient conditions, in the more general nonlinear case we will present only sufficient conditions. Again, our focus will be on the computational aspects.

Polynomial systems. Dynamical models in biology, especially those drawn from biochemical relations, commonly take the form of polynomial systems, ẋ=p(x), where p(x) is

Box invariance for hybrid systems

We extend the notion of box invariance to models, known as hybrid or switched, which are compositions of different dynamical systems. As in the case of the preceding nonlinear studies, we will only derive sufficient conditions.

Definition 4

Hybrid System, (Lygeros, Johansson, Simic, Zhang, & Sastry, 2003)

A hybrid system is a tuple =(Q,E,D,G,R,F), where

  • Q={1,,m} is a finite set of discrete states

  • EQ×Q is a set of edges, where e=(e(1),e(2))E

  • D={Di}iQ is a set of domains, where DiRn

  • G={Ge}eE is a set of guards, where GeDe(1)

  • R={Re}eE is a set of identity reset maps

  • F={fi

Conclusions

With a focus on computational aspects related to the characterization of box invariance, this paper has obtained necessary and sufficient conditions for linear system, and sufficient conditions for classes of nonlinear (in particular, monotone multi-affine) and hybrid systems. We observed that the Metzler structure helps in obtaining efficient computational procedures for analyzing dynamical systems.

Since robustness is a central issue for biological systems, we also presented results on

Alessandro Abate received the Laurea degree in Electrical Engineering from the University of Padova in 2002, and the M.S. and Ph.D. degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley, in 2004 and 2007 respectively. He is currently a Postdoctoral Researcher at the Department of Aeronautics and Astronautics at Stanford University.

His research interests are in the analysis, control, and verification of probabilistic and hybrid systems, and their

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    Alessandro Abate received the Laurea degree in Electrical Engineering from the University of Padova in 2002, and the M.S. and Ph.D. degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley, in 2004 and 2007 respectively. He is currently a Postdoctoral Researcher at the Department of Aeronautics and Astronautics at Stanford University.

    His research interests are in the analysis, control, and verification of probabilistic and hybrid systems, and their application in systems biology.

    Ashish Tiwari received his B.Tech and Ph.D. degrees in Computer Science from the Indian Institute of Technology, Kanpur and the State University of New York at Stony Brook in 1995 and 2000, respectively. He is currently a member of the formal methods group in the Computer Science Laboratory at SRI International. His research interests are in automated deduction, decision procedures, program analysis, and formal technologies for analysis and verification of hybrid system models of embedded software, controlsystems, and biological systems.

    Dr. Tiwari co-chaired the International Workshop on Hybrid Systems in 2006 and the workshop on Automated Deduction: Decidability, Complexity and Tractability in 2007. He has served on the program committee of the major conferences on automated deduction, verification, logic, and hybrid systems.

    Shankar Sastry received a B.Tech. from the Indian Institute of Technology, Bombay, 1977, an M.S. in EECS, M.A. in Mathematics and Ph.D. in EECS from UC Berkeley, 1979, 1980, and 1981 respectively. S. Shankar Sastry is currently dean of the College of Engineering. He was formerly the Director of CITRIS (Center for Information Technology Research in the Interest of Society) and the Banatao Institute @ CITRIS Berkeley. He served as chair of the EECS department from January, 2001 through June 2004. In 2000, he served as Director of the Information Technology Office at DARPA. During 1996–1999, he was the Director of the Electronics Research Laboratory at Berkeley, an organized research unit on the Berkeley campus conducting research in computer sciences and all aspects of electrical engineering. He is the NEC Distinguished Professor of Electrical Engineering and Computer Sciences and holds faculty appointments in the Departments of Bioengineering, EECS and Mechanical Engineering. Prior to joining the EECS faculty in 1983 he was a professor at MIT.

    Research supported by the grants CCR-0225610 and DAAD19-03-1-0373, and in part by NSF CNS-0720721, and NASA NNX08AB95A. The material in this paper was partially presented at the 2nd ADHS in 2006, and at the 46th CDC in 2007. This paper was recommended for publication in revised form under the direction of Editor Andrew R. Teel.

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