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Ellipsoidal Techniques for Reachability Analysis

of Discrete-Time Linear Systems∗

Alex Kurzhanskiy and Pravin Varaiya Department of Electrical Engineering and Computer Sciences

University of California, Berkeley, CA

June 30, 2005

Abstract

This paper describes the computation of reach sets for discrete-time linear control systems with time-varying coefficients and ellipsoidal bounds on the controls and initial conditions. The algorithms construct external and internal ellipsoidal approximations so that they touch the reach set boundary from outside and from inside. Recurrence relations that describe the time evolution of these approximations are provided. The paper also deals with discrete-time linear systems with singular state transition matrix.

1 Introduction

We extend the use of ellipsoidal methods for continuous-time linear systems developed in [1] to calculate the reach set for discrete-time linear systems. We also deal with systems with a singular state transition matrix and show how to approximate the reach set with any given accuracy using regularization and provide the means for picking the appropriate regulariza- tion parameters that guarantee this accuracy over a specified finite time interval. We make no controllability assumptions regarding the system. Thus, the suggested regularization technique also extends results of [1] to systems that are not completely controllable.

In case of singular state transition matrix, or if the system is not controllable at every time step, the exact representation of the reach set on every time step by external ellipsoidal approximation is not possible. But the exact representation is possible for the regularized reach set that overapproximates the actual one, and whose boundary can be made arbitrarily close to the boundary of the actual one over any given finite time interval. Using internal ellipsoidal approximations, on the other hand, it is always possible to exactly represent

∗Research supported by NSF Grant CCR-00225610.

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the actual reach set. However, the concept of ‘good curves’ that was introduced in [1] and without which the computation process becomes heavy and complicated, can only be applied for systems with nonsingular state transition matrix. Therefore, in order to make use of good curves in the singular case, we perform the internal approximation of the regularized reach set.

2 Related work

Research on dynamical and hybrid systems has produced methods for verification and con- troller synthesis. A common step in most methods is the calculation of reach sets. For example, [2] uses bounded model checking for verification of discrete-time linear hybrid systems. For that, the hybrid system is converted into a discrete transition system. The conversion requires the computation of reach sets of discrete-time linear systems.

Exact computation of reach sets is not generally possible, and approximation techniques are needed. The choice of the reach set representation determines the efficiency of such techniques. The more complex the representation is, the more costly is the storage of the sets, the more elaborate is the computation, but the better is the reach set approximation. Choosing the reach set representation is a compromise between these factors.

The level set method [3, 4] deals with general nonlinear controlled systems and gives exact representation of their reach sets, but the computation is complex, requiring solving the HJB equation and finding the set of states that belong to sub-zero level set of the value function. The method [4] is impractical for systems of dimension higher than three.

Reachability analysis of controlled linear systems with convex bounds on the control and initial conditions reduces to effective manipulation with convex sets, performing set-valued operations such as unions, intersections, geometric sums and differences. Two basic objects are used as convex approximations: various kinds of polytopes, e.g. general polytopes, zonotopes, parallelotopes, rectangular polytopes; and ellipsoids.

Reachability analysis and verification for discrete-time linear systems through general poly- topes are implemented in the Multi Parametric Toolbox (MPT) for Matlab [5], [6]. The reach set at every time step is computed as the geometric sum of two polytopes. The pro- cedure consists in finding the vertices of the resulting polytope and calculating its convex hull. The convex hull algorithm employed by MPT is based on the Double Description method [7] and implemented in the CDD/CDD+ package [8]. Its complexity is V n, where V is the number of vertices and n is the state space dimension. Hence the use of MPT is practical for low dimensional systems. But even in low dimensional systems the number of vertices in the the reach set polytope can grow very large with the number of time steps. For example, consider the system,

xk+1 = Axk + uk,

2

with A =

[ cos 1 − sin 1 sin 1 cos 1

] , uk ∈ {u ∈ R2 | ‖u‖∞ ≤ 1}, and x0 ∈ {x ∈ R2 | ‖x‖∞ ≤ 1}.

Starting with a rectangular initial set, the number of vertices of the reach set polytope is 4k + 4 at the kth step.

In d/dt [9], the reach set is approximated by unions of rectangular polytopes [10]. The

x1(t0) x2(t0)

x2(t1)x1(t1)

x1(t2) x2(t2)

x1(t0) x2(t0)

x2(t1)x1(t1) R (t1)

R (t2)

R [t0,t 1]

R [t1,t 2]

(a)

(c)

(e)

(d)

(f)

(b)

Figure 1: Reach set approximation by union of rectangles.

algorithm works as follows. First, given the set of initial conditions defined as a polytope, the evolution in time of the polytope’s extreme points is computed (figure 1(a)). R(t1) in figure 1(a) is the reach set of the system at time t1, and R[t0, t1] is the set of all points that can be reached during [t0, t1]. Second, the algorithm computes the convex hull of vertices of both, the initial polytope and R(t1) (figure 1(b)). The resulting polytope is then bloated to include all the reachable states in [t0, t1] (figure 1(c)). Finally, this overapproximating polytope is in its turn overapproximated by the union of rectangles (figure 1(d)). The same procedure is repeated for the next time interval [t1, t2], and the union of both rectangular approximations is taken (figure 1(e,f)), and so on. Rectangular polytopes are easy to represent and the number of facets grows linearly with dimension, but a large number of rectangles must be used to assure the approximation is not overly conservative. Besides, the important part of this method is again the convex hull calculation whose implementation relies on the same CDD/CDD+ library. This limits the dimension of the system and time interval for which it is practical to calculate the reach set.

Polytopes can give arbitrarily close approximations to convex sets, but the number of ver- tices can grow prohibitively large and, as shown in [11], the computation of a polytope by its convex hull becomes intractable for large number of vertices in high dimensions.

The method of zonotopes for external approximation of reach sets is presented in [12]. A

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zonotope is a special class of polytopes (see [13]) of the form,

Z = {x ∈ Rn | x = c + p∑

i=1

αigi, − 1 ≤ αi ≤ 1},

wherein c and g1, ..., gp are vectors in Rn. Thus, a zonotope Z is defined by its center c and generator vectors g1, ..., gp. The value p/n is called the order of the zonotope. This compact representation and the fact that they are closed under linear transformation and geometric sum, make zonotopes attractive for reach set computation. The difficulty is that with every integration step the order of the approximating zonotope increases by p/n. The difficulty can be averted by limiting the number of generator vectors, and overapproximating zonotopes whose number of generator vectors exceeds this limit by lower order zonotopes. The zonotope method of [12] provides neither a notion of tightness of approximation nor an estimate of the error resulting from the zonotope order reduction. Further, the method does not indicate any way of control synthesis, i.e. finding a sequence of controls that drives the system to the desired terminal set or point.

CheckMate [14] is a Matlab toolbox that can evaluate specifications for trajectories starting from the set of initial (continuous) states corresponding to the parameter values at the vertices of the parameter set. This provides preliminary insight into whether the specifica- tions will be true for all parameter values. The method of oriented rectangluar polytopes for external approximation of reach sets is introduced in [15]. The basic idea is to con- struct an oriented rectangular hull of the reach set for every time step, whose orientation is determined by the singular value decomposition of the sample covariance matrix for the states reachable from the vertices of the initial polytope. The limitation of CheckMate and the method of oriented rectangles is that only autonomous (i.e. uncontrolled) systems are allowed, and only an external approximation of the reach set is provided.

Requiem [16] is a Mathematica notebook which, given a linear system, the set of initial conditions and control bounds, symbolically computes the reach set exactly, using the ex- perimental quantifier elimination package. Quantifier elimination is the removal of all quan- tifiers (the universal quantifier ∀ and the existential quantifier ∃) from a quantified system. Each quantified formula is substituted with quantifier-free expression with operations +, ×, = and

It is proved in [17] that