Ellipsoidal techniques for reachabilityanalysis∗

A.B.KurzhanskiMoscow State University

P.VaraiyaUniversity of California at Berkeley

December 16, 1998

Abstract

The paper describes the behavior of tight external ellipsoidal approximations of the reachsets and tubes for linear control systems with time-varying coefficients and hard bounds on thecontrols and shows how such approximations may be expressed through ordinary differentialequations with coefficients given in explicit analytical form.

Introduction

Recent activities to promote advanced automation of real-time processes motivate new interest inthe problem of reachability for controlled systems. Reachability is also related to the problemof verification of hybrid systems [3]. Effective and implementable solutions to these problemsmust incorporate adequate procedures for calculating reach sets and reach tubes for continuous-time systems [10], [11]. Another demand for the solution to similar problems comes from intervalanalysis in scientific computation [9].

Among methods for reachability analysis are those based on ellipsoidal techniques [13], [2], [7].Ellipsoidal methods make possible the exact representations of the reach sets and tubes for linearsystems through parametrized families of ellipsoids [7]. However, an important open question is toeffectively single out the tightest ellipsoidal approximations to these reach tubes and thus to opennew routes for deriving adequate numerical error estimates and new methods for nonlinear systems[14].

∗Research supported by National Science Foundation Grant ECS 9725148. We thank Oleg Botchkarev for the figuresand animation.

1

In this paper we deal with ellipsoidal approximation of reach tubes for control systems with lineardynamics and hard bounds on the control. We study the following question: given a reach tube and asmooth curve that runs along its surface, does there exist an ellipsoidal-valued tube that satisfies thefollowing two properties: on one hand, the ellipsoidal tube contains the reach tube, being thereforeits external approximation, while on the other, it touches the reach tube precisely along the givencurve?

The answer to this question is positive. However the properties of the respective ellipsoidal tubesdepend strongly on the given curve. The “good” case is when the given curve may itself be realizedas a trajectory of the original control system. 1 The required ellipsoidal tube is then generated byan ellipsoid-valued map which satisfies the semigroup property and thus generates a generalizeddynamic system. Moreover, the approximating tube is tight in the sense that there exists no otherellipsoidal tube that could be squeezed in between the approximation and the reach tube. Lastly, inthe good case, the parameters of the ellipsoidal approximation are described by a simple ordinarydifferential equation.

Even when givenany smooth curveon the surface of the reach tube, which is not itself a systemtrajectory, there again exists an ellipsoidal tube that touches the reach sets along this curve. Butnow the ellipsoid-valued map may not satisfy the semigroup property and its evolution in time isnot described by equations as simple as in the “good” case. A simplification of the computationalprocedure in this general case to the level of the “good” case results in non-tight approximations.

1 The reachability problem

Consider a controlled system described by the linear differential equation with time-varying coeffi-cients,

x = A(t)x + B(t)u, t0 ≤ t ≤ t1, (1)

wherex ∈ IRn is the state andu ∈ IRm is the control.The matricesA(t), B(t) are continuous andthe system iscompletely controllable(see [8], [4]), equivalently, the quadratic form

W (τ, σ) =∫ τ

σ(l,X(τ, s)B(s)B′(s)X ′(τ, s)l)ds, l ∈ IRn,

is positive definite for anyτ > σ.

HereX(t, s) is the transition matrix for the homogeneous system (1),

∂X(t, s)/∂t = A(t)X(t, s), X(s, s) = I,

1 This happens when the given curve (a system trajectory) develops along the points of support for hyperplanesgenerated by vectors that are realized as the motions of the linear system adjoint to the homogeneous part of the controlsystem under investigation.

2

whereI is the identity matrix. For a time-invariant systemA(t) = A = const, X(t, s) = exp(A(t−s)).

The controlu = u(t) is any measurable function restricted by hard bounds

u(t) ∈ P(t), (2)

for almost allt. The set-valued boundP(t) is a nondegenerate ellipsoid continuous int, P(t) =E(q(t), Q(t)), where

E(q(t), Q(t)) = {u : (u − q(t), Q−1(t)(u − q(t)) ≤ 1}, (3)

with q(t) ∈ IRm (the center of the ellipsoid) and positive definite matrix functionQ(t) ∈ IRm×m

(the matrix of the ellipsoid) continuous int. Thesupport functionof the ellipsoid is

ρ(l|E(q(t), Q(t))) = max{(l, x)|x ∈ E(q(t), Q(t)}

= (l, q(t)) + (l, Q(t)l)1/2.

and the continuity ofQ(t) means that its support functionρ(l|Q(t)) is continuous int uniformly inl with (l, l) ≤ 1.

Definition 1.1 Given position{t0, x0}, the reach set(or “attainability domain”) X (τ, t0, x0) attime τ > t0 from this position is the set

X [τ ] = X (τ, t0, x0) = {x[τ ]}

of all statesx[τ ] = x(τ, t0, x0) reachable at timeτ by system (1), withx(t0) = x0, through allpossible controlsu that satisfy the constraint (2).

The set-valued functionτ 7→ X [τ ] = X (τ, t0, x0) is thereach tube.

Definition 1.2 The reach setX (τ, t0,X 0) (at timeτ , from setX 0 = X (t0)) is the union

X (τ, t0,X0) = ∪{X (τ, t0, x0)|x0 ∈ X 0}.

The set-valued functionτ 7→ X [τ ] = X (τ, t0,X0) is the reach tube from setX 0.

The following properties may be checked directly

Lemma 1.1 The set-valued mapX (t, t0,X 0) satisfies the semigroup property

X (t, t0,X 0) = X (t, τ,X (τ, t0,X 0)). (4)

3

In the sequel it is assumed thatX 0 = E(x0,X0) is an ellipsoid. Its support function is

ρ(l|X 0) = (l, x0) + (l,X0l)1/2.

It is worth noting that the setX [τ ] may also be treated as the cutX [τ ] = X (τ, t0, E(x0,X0)) of thesolution tubeX (·) = {X [t] : t ≥ t0} to the differential inclusion

x ∈ A(t)x + E(q(t), B(t)Q(t)B′(t)), t ≥ t0, x0 ∈ E(x0,X0). (5)

The solution of this differential inclusion may be defined directly as in Section 4 below.

The reach setX (t, t0, E(x0,X0)) may also be presented through a set-valued (“Aumann”) integral[1].

Lemma 1.2 The following relation is true:

X (t, t0, E(x0,X0)) = x?(t) + X(t, t0)E(0,X0) +∫ t

t0X(t, s)E(0, B(s)Q(s)B′(s))ds, (6)

where

x?(t) = X(t, t0)x0 +∫ t

t0X(t, s)B(s)q(s)ds. (7)

A standard calculation using convex analysis yields the following result (see, [5], [7]).

Lemma 1.3 The support function

ρ(l|X (t, t0, E(x0,X0)) = (l, x?(t)) + (l,X(t, t0)X0X ′(t, t0)l)1/2 + (8)

+∫ t

t0(l,X(t, s)B(s)Q(s)B′(s)X ′(t, s)l)1/2ds.

This last representation leads to the next result.

Lemma 1.4 The reach setX [t] = X (t, t0, E(x0,X0)) is a convex compact set inIRn, continuousin t.

Points on the boundary of the reach setX [t] have an important characterization. Consider a pointx∗ on the boundary∂X [τ ] of the reach setX [τ ] = X (τ, t0, E(x0,X0)).2 Then there exists a relatedsupport vectorl∗ such that

(l∗, x∗) = ρ(l∗|X [τ ]). (9)

The controlu = u∗(t) and the initial statex(t0) = x∗0 ∈ E(x0,X0) which transfer system (1) fromstatex(t0) = x∗0 to x(τ) = x∗ are given by the following fact ( see [8], [5]).

2 The boundary∂X [τ ] of setX [τ ] may be here defined as the set∂X [τ ] = X [τ ] \ intX [τ ]. Under the controllabilityassumption of the above setX [τ ] has a non-void interiorintX [τ ] 6= ∅ for τ > t0.

4

Theorem 1.1 Suppose the statex(τ) = x∗ is given andx∗ ∈ ∂X [τ ]. Then the controlu = u∗(t)and the initial statex(t0) = x∗0 which yield a unique trajectoryx∗[t] that reaches pointx∗ = x∗[τ ]from pointx∗0 = x(t0) satisfy the following “maximum principle” for almost allt ∈ [t0, τ ]:

(l∗,X(τ, t)B(t)u∗(t)) = max{(l∗,X(τ, t)B(t)u)|u ∈ E(q(t), Q(t))} = (10)

= (l∗,X(τ, t)B(t)q(t)) + (l∗,X(τ, t)B(t)Q(t)B′(t)X ′(τ, t)l∗)1/2

and the extremal relation

(l∗,X(τ, t0)x∗0) = max{(l∗, x)|x ∈ X(τ, t0)E(x0,X0)} = (11)

= (l∗,X(τ, t0)x0) + (l∗,X(τ, t0)X0X(τ, t0)l∗)1/2.

wherel∗ is the support vector for setX [t] at pointx∗ that satisfies relation (8).

The exact calculation of reach sets, especially in large dimensions, is cumbersome. Among theapproximation methods for these problems are those that rely on ellipsoidal techniques, as given in[7].

Remark 1.1.Due to the controllability assumption we will further assume, without loss of generality,thatB(t) = I. To return to the caseB(t) 6= I it suffices in the sequel to substitute everywhereQ(t)by B(t)Q(t)B′(t). However, in this case, for computational purposes it may be useful to start theapproximation process at timet = t0 + δ, δ > 0, to ensureW (t0 + δ, t0) > 0.

2 Ellipsoidal approximation of reach sets

Although the initial setE(x0,X0)) and the control setE(q(t), Q(t)) are ellipsoids, the reach setX [t] = X (t, t0, E(x0,X0)) will not generally be an ellipsoid.3 As indicated in [7], the reachabilitysetX [t] may be approximated both externally and internally by ellipsoidsE− andE+, with E− ⊆X [t] ⊆ E+. An external approximationE+ is tight, if for any ellipsoidE , X [t] ⊆ E ⊆ E+ impliesE = E+. This paper is concerned with tight external approximations.

It is important to emphasize thatall the considerations of this paper are restricted to the classE = {E+} of ellipsoidsdescribed within the following proposition.

Theorem 2.1 The classE = {E+} consists of ellipsoids that approximate the reach setX [t] exter-nally and are of the formE+[t] = E(x?,X+[t]), wherex?(t) satisfies the equation

x? = A(t)x? + q(t), x?(t0) = x0, t ≥ t0, (12)

which follows from (7) withB(t) = I and

X+[t] = X+(t|p(·)) = (13)

3The sum of two ellipsoids,E1 + E2, is generally not an ellipsoid.

5

=(∫ t

t0p(s)ds + p0(t)

)(∫ t

t0p−1(s)X(t, s)Q(s)X ′(t, s)ds + p−1

0 (t)X(t, t0)X0X ′(t, t0))

,

where functionp(s) > 0, s ∈ [t0, t], is continuous andp0(t) > 0. The following inclusions aretrue:

X [t] ⊆ E(x?(t),X+(t|p(·)), ∀p(s) > 0, t0 ≤ s ≤ t, (14)

Moreover,X [t] = ∩{E(x?(t),X+(t|p(·))|p(s) > 0, s ∈ [t0, t]}. (15)

Relations (14),(15) are true forp(s) ranging over all the continuous positive functions. These resultsfollow from [7, Sections 2.1, 2.7]. What also holds is the following characterization of the tightexternal ellipsoids.

Theorem 2.2 For a given timeτ the tight external ellipsoidsE+[τ ] = E(x?(τ),X+[τ ]), withE+[τ ] ∈ E, are those for which the functionsp andp0 are selected as

p(s) = (l,X(τ, s)Q(s)X ′(τ, s)l)1/2, t0 ≤ s ≤ τ, (16)

p0(τ) = (l,X(τ, t0)X0X(τ, t0)l)1/2, (17)

with vectorl given. Each ellipsoidE+[τ ] touchesX [τ ] at points

{x∗ : (l, x∗) = ρ(l|X [τ ])},so that

ρ(l|X [τ ]) = (l, x?(τ)) + (l,X+[τ ]l)1/2 = ρ(l|E(x?(τ),X+[τ ])).

Note that the result above requires the evaluation of the integrals in (13) for each timeτ and vectorl. If the computation burden for each evaluation of (13) isC, and we estimate the reach tube via(14) forT values of timeτ andL values ofl, the total computational burden would beC × T × L.A solution of the following problem would reduce this burden toC × L.

Problem 2.1.Given a unit-vector functionl∗(t), (l∗, l∗) = 1, continuously differentiable int, findan external ellipsoidE∗

+[t] ⊇ X [t] that would ensurefor all t ≥ t0, the equality

ρ(l∗(t)|X [t]) = ρ(l∗(t)|E+[t]) = (l∗(t), x∗(t)), (18)

so that the supporting hyperplane forX [t] generated byl∗(t), namely, the plane(x−x∗(t), l∗(t)) =0 that touchesX [t] at pointx∗(t), would also be a supporting hyperplane forE∗

+[t] and touch it atthe same point.

In order to solve this problem, we first make use of Theorem 2.1. However, the functionp used forthe parametrization in (13), (16), will now be a function of two variables,s, t, since the requirementis that relation (18) should now hold for allt ≥ t0.

6

Theorem 2.3 With l(t) = l∗(t) given, the solution to Problem 2.1 is an ellipsoidE+[t] =E(x∗(t),X∗

+[t]), whereX∗

+[t] = (19)(∫ t

t0p∗t (s)ds + p∗0(t)

)(∫ t

t0(p∗t (s))

−1X(t, s)Q(s)X ′(t, s)ds + p∗−10 (t)X(t, t0)X0X ′(t, t0)

),

andp∗t (s) = (l∗(t),X(t, s)Q(s)X ′(t, s)l∗(t))1/2, (20)

p∗0(t) = (l∗(t0),X(t, t0)X0X ′(t, t0)l∗(t0))1/2.

The proof is obtained by substituting (20) into (19) and further comparing the result with (8),(18).

Relations (19), (20) need to be solved “afresh” for eacht. It may be more convenient for computa-tional purposes to have them given in the form of recurrence relations generated through differentialequations.

In all the ellipsoidal approximations considered in this paper the center of the approximating ellip-soid is always the same, being given byx?(t) of (12). The discussion will therefore actually concernonly the relations for the matricesX+[t],X∗[t] of these ellipsoids.

3 Recurrence relations

We start with a particular case.

Assumption 3.1 The functionl∗(t) is of the the following forml∗(t) = X(t0, t)l , with l ∈ IRn

given. For the time-invariant casel∗(t) = e−A′(t−t0)l. 4

Thenp∗t (s), p∗0(t),X∗+[t] of (20), (19) transform into

p∗t (s) = (l,X(t0, s)Q(s)X ′(t0, s)l)1/2 = p∗(s); p∗0(t) = (l,X0l)1/2 = p∗0, (21)

and

X∗+[t] = X(t, t0)X+(t)X ′(t, t0), X+[t] =

(∫ t

t0p∗(s)ds + p∗0

)Ψ(t), (22)

whereΨ(t) = (23)

=∫ t

t0(l,X(t0, s)Q(s)X ′(t0, s)l)−1/2X(t0, s)Q(s)X ′(t0, s)ds + (l,X0l)−1/2X0.

4 Under this Assumption the vectorl∗(t) may be expressed as the solution to equation

l∗ = −A′(t)l∗, l∗(t0) = l

which is the adjoint to the homogeneous part of equation (1).

7

In this particular casep∗t (s) does not depend ont (p∗t′(s) = p∗t”(s) for t′ 6= t”) and the lower indext may be dropped.

Direct differentiation ofX+[t] yields

X+[t] = π∗(t)X+[t] + π∗−1(t)X(t0, t)Q(t)X ′(t0, t), X+[t0] = X0, (24)

where

π∗(t) = p∗(t)(∫ t

t0p∗(s)ds + p∗0

)−1

.

Calculating

(l,X+[t]l) =(∫ t

t0p∗(s)ds + p∗0

)(l,Ψ(t)l) = (

∫ t

t0p∗(s)ds + p∗0)

2,

one may observe that

π∗(t) = (l,X(t0, t)Q(t)X ′(t0, t)l)1/2(l,X+[t]l)−1/2. (25)

In order to pass to the matrix functionX∗+[t] we note that

X∗+[t] = A(t)X(t, t0)X+[t]X ′(t, t0) + X(t, t0)X+[t]X ′(t, t0)A(t) + X(t, t0)X+[t]X ′(t, t0).

After a substitution from (24) this gives

X∗+ = A(t)X∗

+ + X∗+A′(t) + π∗(t)X∗

+ + π∗−1(t)Q(t), X∗(t0) = X0. (26)

We summarize these results.

Theorem 3.1 Under Assumption 3.1 the solution to Problem 2.1 is given by the ellipsoidE∗+[t] =

E(x?(t),X∗+[t]), wherex∗(t) satisfies equation (12) andX∗

+[t] is a solution to equations (26), (25).

Since the setX∗+[t] depends on vectorl, we denoteX∗

+[t] = X∗+[t]l.

Theorem 3.2 For anyt ≥ t0 the reach setX [t] may be described as

X [t] = ∩{E(x?,X∗+[t]l)}| l : (l, l) = 1}. (27)

This is a direct consequence of Theorems 1.1, 2.1. Direct differentiation ofπ∗(t) in (25) also yieldsthe next result.

8

Corollary 3.1 With Q(t) differentiable, functionπ∗(t) of Theorem 3.1 satisfies the differentialequation

π∗ = f(t)π∗ − π∗2, π∗(t0) = 1, (28)

where

f(t) =(l,X(t0, t)(−A(t)Q(t) − Q(t)A′(t) + Q(t))X ′(t0, t)l)

2(l,X(t0, t)Q(t)X ′(t0, t)l)1/2,

Corollary 3.2 With A = const.,Q = const., and l an eigenvector ofA′, (A′l = λl) with realeigenvalueλ, functionf(t) is given by

f(t) = −λ(l, eA(t0−t)Q(t)eA′(t0−t)l)1/2.

Thus, if l∗(t) satisfies Assumption 3.1, the complexity of computing a tight, external ellipsoidalapproximation to the reach set for allt, is the same as computing the solution to the differentialequation (26).

For the general case, differentiating relation (19) forX∗+[t], we have

X+[t] =

=(

p∗t (t)+∫ t

t0(∂p∗t (s)/∂t)ds+p∗0

)(∫ t

t0p∗−1

t (s)X(t, s)Q(s)X ′(t, s)ds+p∗−10 (t)X(t, t0)X0X ′(t, t0)

)

+(∫ t

t0p∗t (s)ds + p∗0(t))

((p∗t (t))

−1Q(t) +∫ t

t0(∂((p∗t (s))

−1X(t, s)Q(s)X ′(t, s))/∂t)ds+

+d(X(t, t0)p∗0(t)X′(t, t0))/dt

),

which, using the notationX∗

+[t] = (29)

=(∫ t

t0p∗t (s)ds + p∗0(t)

)(∫ t

t0(p∗t (s))

−1X(t, s)Q(s)X ′(t, s)ds + p∗−10 (t)X(t, t0)X0X ′(t, t0)

),

π∗(t) = p∗t (t)(∫ t

t0p∗t (s)ds + p∗0(t)

),

givesX∗

+ = (30)

= A(t)X∗+ + X∗

+A′(t) + π∗(t)X∗+ + π∗−1(t)Q(t) + X(t, t0)φ(t, l(t), Q(·))X ′(t, t0),

X∗+(t0) = X0,

whereφ(t, l(t), Q(·)) = (31)

=(∫ t

t0(∂p∗t (s)/∂t)ds + p∗0(t)

)(∫ t

t0(p∗t (s))

−1P (t0, s)ds + (p∗0(t))−1P0

)−

9

−(∫ t

t0p∗t (s)ds + p∗0(t)

)( ∫ t

t0(∂(p∗t (s))

−1/∂t)P (t0, s)ds + d(p∗0(t))−1/dtP0

).

andP (t0, s) = X(t0, s)Q(s)X ′(t0, s), P0 = X0.

Under Assumption 3.1 (l∗(t) = X ′(t0, t)l) the terms∂p∗t (s)/∂t = 0, p∗0(t) = 0, so thatφ(t, l(t), Q(·)) = 0.

Theorem 3.3 The solutionE∗[t] = E(x?(t),X∗[t]) to Problem 2.1 is given by the functionx?(t)of (12) and the matrix functionX∗

+[t] that satisfies equation (30) withπ∗(t) defined as in (29) andp∗t (s) as in (20). Under Assumption 3.1 the termφ ≡ 0 in (30).

Throughout the previous discussion we have observed that under Assumption 3.1 the tight externalellipsoidal approximationE(x?,X∗

+(t)) is governed by the simple ordinary differential equation(26). Moreover, in this case the pointsx∗(t) of support for the hyperplanes generated by vectorl(t) run alonga system trajectoryof (1) which is generated by a control that satisfies the maximumprinciple (10) and relation (11) of Theorem 1.1.

In connection with this fact the following question arises:

Problem 3.1. Given a curvel(t) such that the supporting hyperplane generated by vectorsl(t)touches the reach setX [t] at the point of supportxl(t), What should be the curvel(t), so thatxl(t)is a system trajectory for (1) ?

Let us investigate this question. Suppose a curvel(t) is such that

xl(t) = X(t, t0)x0t +

∫ t

t0X(t, τ)ut(τ)dτ (32)

is a vector generated by the pair{x0t , ut(·)}. In order thatxl(t) be a support vector to the supporting

hyperplane generated byl(t) it is necessary and sufficient that this pair satisfy at each timet themaximum principle (10) and the condition (11) under constraints

x0t ∈ E(0,X0), ut(τ) ∈ E(0, Q(τ)),

which means

ut(τ) =Q(τ)X ′(t, τ)l(t)

(l(t),X(t, τ)Q(τ)X ′(t, τ)l(t))1/2, t0 ≤ τ ≤ t, (33)

and

x0t =

X0X ′(t, t0)l(t)(l(t),X(t, t0)X0X ′(t, t0)l(t))1/2

. (34)

Note that sincexl(t) varies int, the variablesx0t , ut(τ) should in general be taken as being depen-

dent ont.

10

Now the question is: what should be the functionl(t) so thatxl(t) of (32) - (34) is a trajectory ofsystem (1)?

Differentiating (32), we come to

xl(t) = A(t)(

X(t, t0)x0τ +

∫ t

t0X(t, τ)ut(τ)dτ

)+ ut(t) + Y (t), (35)

where

Y (t) = X(t, t0)∂x0t /∂t +

∫ t

t0X(t, τ)(∂ut(τ)/∂t)dτ. (36)

In order thatxl(t) be a system trajectory for (1) it is necessary and sufficient thatY (t) ≡ 0. Herethe control at timet is ut(t) and the initial position isx0

t0 .

Let us look for the class of functionsl(t) that yieldY (t) = Yl(t) ≡ 0. 5

SupposeYl(t) ≡ 0. Then for anyt we have

Yl(t) = (37)

= X(t, t0)∂(X0X ′(t, t0)l(t)(p0(t))−1)/∂t +∫ t

t0X(t, τ)∂(Q(τ)X ′(t, τ)l(t)(pt(τ))−1)/∂tdτ,

where, as in (20),pt(s, l(t)) = (l(t),X(t, s)Q(s)X ′(t, s)l(t))1/2, (38)

p0(t, l(t)) = (l(t),X(t, t0)X0X ′(t, t0)l(t))1/2.

Continuing the calculation ofYl(t), after some transformations we come to

Yl(t) = Φ0(t, t0, l(t))(p0(t, l(t)))−3 +∫ t

t0Φ(t, τ, l(t))(pt(τ, l(t)))−3dτ, (39)

whereΦ0(t, t0, l) = P0(t, t0)(A(t)l + l))(l,X(t, t0)X0X ′(t, t0)l)−

−P0(t, t0)l((l′A(t) + l′), P0(t, t0))l),

andΦ(t, τ, l) = P (t, τ)(A(t)′l + l)(l, P (t, τ))l)−

P (t, τ)(A(t)′l + l)(l, P (t, τ l)−−P (t, τ)l)((l′A(t) + l)P (t, τ)l).

Here, in accordance with previous notation,

P0(t, t0) = X(t, t0)X0X ′(t, t0), P (t, τ) = X(t, τ)Q(τ)X ′(t, τ)5 Since we are about to vary the problem parameters depending on choice ofl(·), we denoteY (t) = Yl(t) and

pt(τ ) = pt(τ, l), p0(t) = p0(t, l).

11

Let us now suppose thatl(t) satisfies the equation

l + A(t)l = k(t)l(t), (40)

wherek(t) is a continuous scalar function. Then, through direct substitution, we observe thatYl(t) = 0.

On the other hand, supposeYl(t) ≡ 0. Let us prove that thenl(t) satisfies equation (40). Bycontradiction, suppose a functionl0(t) yieldsYl0(t) ≡ 0, but does not satisfy (40). Then for anytwe should have

(l′0(t)A(t) + l′0(t))Yl0(t) = 0,

so that the expression(l′0(t)A(t) + l′0(t))×

×(Φ0(t, t0, l0(t))(p0(t, l0(t)))−3 +

∫ t

t0Φ(t, τ, l0(t))(pt(τ, l0(t)))−3dτ

)=

=(

(l′0(t)A(t) + l′0(t))P (t, t0)(A′(t)l0(t) + l0(t)))(l0(t), P0(t, t0)l0(t))−

−(l′0(t)P0(t, t0)(A′(t)l0(t) + l0(t))2)

p0(t, l0(t))−3+

+∫ t

t0

(l′0(t)A(t) + l′0)P (t, τ)(A(t)l0(t) + l0(t))(l0(t), P (t, τ)l0(t))−

−((l′0(t)A(t) + l′0(t))P (t, τ)l)2)

(pt(τ, l0(t)))−3dτ,

should be equal to zero. But due to the H¨older inequality (applied in finite-dimensional version forthe first term and in infinite-dimensional version for the integral term) this term is equal to zero ifand only if l0(t) satisfies relation (40):(A′(t)l + l(t)) is collinear withl(t).6 This contradicts ourassumption and proves the following result.

Theorem 3.4 Given a curvel(t), the functionxl(t) of (32)-(34), formed of support vectors to thehyperplanes generated byl(t), is a system trajectory for (1) if and only ifl(t) is a solution to thedifferential equation (40).

Through similar calculations the following assertion may be proved

Theorem 3.5 In order thatX(t, t0)φ(t, l(t), Q(·))X ′(t, t0) ≡ 0 it is necessary and sufficient thatl(t) satisfies equation (40).

6Here we also take into account that(l′(t)A(t) + l(t))Yl(t) is bounded withp0(t) > 0 andpt(τ ) > 0 almosteverywhere due to the strong controllability assumption.

12

We have thus come to the condition thatl(t) should satisfy (40). This equation may be interpretedas follows. Consider the transformation

ln(t) = exp(−∫ t

t0k(s)ds)l(t). (41)

Then

ln(t) = exp(−∫ t

t0k(s)ds)l(t) − k(t) exp(−

∫ t

t0k(s)ds)l(t) =

= (−A′(t) + k(t)I) exp(−∫ t

t0k(s)ds)l(t) − k(t) exp(−

∫ t

t0k(s)ds)l(t) = −A′(t)ln(t)

which means thatln(t) is again a solution to the adjoint equationl = −A(t)l, but in a new scale.However, the nature of transformation (41) is such thatln(t) = γ(t)l(t), γ > 0.

Lemma 3.1 The solutionslk(t) to equation (40) generate the same support hyperplanes to thereach setX [t] and the same points of supportxlk(t), whatever be the functionsk(t) (provided allthese solutions have the same initial conditionlk(t0) = l0).

In particular, withk(t) ≡ 0, we come to the functionsl(t) of Assumption 3.1 and withAl = kl (lis an eigenvector of constant matrixA) - to equationl = 0 and to conditionl(t) = l = const.

The functionl(t) of Assumption 3.1 is such that in general(l(t), l(t)) 6= 1. In order to generate afunction lu(t) with vectorslu(t) of unit length, take the substitutionlu(t) = l(t)((l(t), l(t))−1/2 ,wherel + A′(t)l = 0. Direct calculations indicate the following:

Lemma 3.2 Suppose the functionlu(t) satisfies equation (40) with

k(t) = (l(t), A(t)l(t))/(l(t), l(t)),

wherel(t) satisfies Assumption 3.1. Then(lu(t), lu(t)) ≡ 1.

Remark 3.1. Lemma 3.1 indicates that the necessary and sufficient conditions for the solution ofProblem 3.1 do not lead us beyond the class of functions given by Assumption 3.1.

Relations (30), (12) above describe the evolution of the basic parametersx?(t),X∗+[t] of the ellip-

soidsE∗[t]. It is also instructive to consider an evolution equation with set-valued solutions for theellipsoidE∗[t] itself. We do this next.

13

4 The evolution of approximating ellipsoids

It is known that the evolution of the reach setX [t],X [t0] = E0 = E(x0,X0) may be described by anevolution equation for the “integral funnel” of the differential inclusion (1.5), so that the set-valuedfunctionX [t] satisfies for almost allt the relation

limε→0

ε−1h(X [t + ε], (I + εA(t))X [t] + εE(q(t), Q(t))) = 0, (42)

with initial conditionX[t0] = E0 ([6]).

Hereh(Q,M) stands for theHausdorff distancebetween setsQ,M, which is defined as

h(Q,M) = max{h+(Q,M), h−(Q,M)}, h+(Q,M) = h−(M,Q),

and theHausdorff semidistanceis defined as

h−(M,Q) = maxx

minz

{(x − z, x − z)1/2|x ∈ Q, z ∈ M},Equation (42) has a unique solution.

The idea of constructing external ellipsoidal approximations forX [t] is also reflected in the solutionsto the following evolution equation withellipsoid-valued solutions

limε→0

ε−1h−(E [t + ε], (I + εA(t))E [t] + εE(q(t), Q(t))) = 0 (43)

with initial conditionE [t0] = E(x0,X0) = E0.

Definition 4.1 A set-valued functionE+[t] is said to be the solution to equation (43) if

(i) E+[t] satisfies (43),

(ii) E+[t] is ellipsoid-valued.

Definition 4.2 A solution to equation (43) is said to beminimal if together with conditions (i), (ii)of Definition 4.1 it satisfies condition

(iii) E+[t] is a minimal solution to (43) with respect to inclusion.

Condition(iii) means that there is no other ellipsoidE [t] 6= E+[t] that satisfies both equation (43)and the inclusionsE+[t] ⊇ E [t] ⊇ X [t]. This also means that among the minimal solutionsE+[t] to(43) there are the non-dominated (tight) ellipsoidal tubesE+[t] that containX [t]. For a given initialsetE0 = E [t0] the solutions to (43), as well as its minimal solutions may not be unique.7

Denote an ellipsoid-valued tubeE[t] = E(x?(t),X[t]), that starts at{t0, E0}, E0 = E(x0,X0)}and is generated by given functionsx?(t),X[t] ( matrix X[t] > 0 ) as E[t] = E(t|t0, E0) =E(x?(t),X[t]).

The evolution equation (43) defines a generalized dynamic system in the following sense.7Note that equation (43) is written in terms of Hausdorffsemidistanceh− rather than Hausdorff distanceh as in (42).

14

Lemma 4.1 Each solutionE(t|t0, E0) = E[t] to equation (43) defines a map withthe semigroupproperty

E(t|t0, E0) = E(t|E(τ, |t0, E0)), t0 ≤ τ ≤ t. (44)

This follows from Definition 4.1.

Equation (43) implies that a solutionE[t] to this equation should satisfy the inclusion

E[t + ε] + o(ε)S ⊇ (I + εA(t))E[t] + εE(q(t), Q(t)),

whereε−1o(ε) → 0 with ε → 0 andS is the unit ball in IRn. (HereS = {x : (x, x) ≤ 1}).

Given a continuously differentiable functionl(t), t ≥ t0, and assuming thatE[t] = E(x?(t),X[t])is defined up to timet, let us selectE[t + ε] as an external ellipsoidal approximation of the sum

Z(t + ε) = (I + εA(t))E(x?(t),X[t]) + εE(q(t), Q(t)),

requiring that it touch this sumZ(t + ε) (which, by the way, is not an ellipsoid) according to therelation

ρ(l(t + ε)|E[t + ε]) = ρ(l(t + ε)|Z(t + ε)), (45)

that is at those pointsz(t + ε) whereZ(t + ε) is touched by its supporting hyperplane generated byvectorl(t + ε). Namely,

{z(t, ε) : (l(t + ε), z(t, ε)) = (l(t + ε), x?(t + ε)) + (l(t + ε),X[t + ε]l(t + ε))1/2 =

= (l(t + ε), x?(t + ε)) + (l(t + ε), (I + εA(t))X[t](I + εA(t))′(l(t + ε))1/2+

+ε(l(t + ε), Q(t)l(t + ε))1/2 + o1(ε)}In the latter case, according to [7, sections 2.1, 2.7],E[t + ε] is among the tight external approxi-mations forZ(t + ε) (relative to terms of order higher thanε) 8.

The requirement above is ensured ifE[t + ε] externally approximates the sum

Z(t + ε) = X(t + ε, t)E(x?(t),X[t]) +∫ t+ε

tX(t + ε, s)E(q(s), Q(s))ds

with the same criterion of tightness (45) as in the previous case. Select

X[t + ε] = (1 + p−1(t + ε))X(t + ε, t)X[t]X ′(t + ε, t) + (46)

ε2(1 + p(t + ε))X(t + ε, t)Q(t)X(t + ε, t),

where

p(t + ε) =(l(t + ε),X(t + ε, t)X[t]X ′(t + ε)l(t + ε)1/2)ε(l(t + ε),X(t + ε, t)Q(t)X ′(t + ε)l(t + ε))1/2

. (47)

8Here and in the sequel the terms of typeoi(ε) are assumed to be such thatoi(ε)ε−1 → 0 with ε → 0.

15

Indeed, withp(t + ε) chosen as in (4.6), direct substitution ofp(t + ε) into (4.5) gives

(l(t + ε),X[t + ε]l(t + ε)) = (l(t + ε),X(t + ε, t)X[t]X ′(t + ε, t)l(t + ε))1/2 + (48)

+ε(l(t + ε),X(t + ε, t)Q(t)X ′(t + ε, t)l(t + ε))1/2)2.

Estimating the Hausdorff distance

R(t, ε) = h

(∫ t+ε

tX(t + ε, s)E(0, Q(s))ds, ε2X(t + ε, t)Q(t)X ′(t + ε, t)

),

by direct calculation, we have the next result.

Lemma 4.2 The following estimate holds,

R(t, ε) ≤ Kε2, K > 0. (49)

In view of (47), Lemma 4.2, and the fact thatX(t + ε, t) = X[t] + εA(t) + o2(ε), we arrive at thedesired result below.

Lemma 4.3 Withp(t, ε) selected as in (47), the relations (46)-(49) reflect the equalities

ρ(l(t + ε)|E(0,X[t + ε])) = (50)

= ρ(l(t + ε)|Z[t + ε]) + o3(ε) = ρ(l(t + ε)|Z(t + ε)) + o4(ε).

Lemma 4.3 implies

X[t + ε] = (1 + p−1(t + ε))(I + εA(t))X[t](I + εA(t))′ + (51)

+ε2(1 + p(t + ε))(I + εA(t))′Q(t)(I + εA(t)) + o5(ε).

Substitutingp(t+ε) of (47) into (51), denotingεp(t+ε) = π(t+ε), and expandingX[t+ε],X(t+ε, s), π(t + ε) in ε, one arrives at

X[t + ε] = X[t] + εA(t)X[t] + εX[t]A(t) + π(t)X[t] + π−1(t)Q(t) + o6(ε). (52)

This further gives, by rewriting the last expression, dividing it byε, and passing to the limit withε → 0,

X = A(t)X + XA′(t) + π(t)X + π−1(t)Q(t), (53)

where

π(t) =(l(t), Q(t)l(t))1/2

(l(t),X[t]l(t))1/2. (54)

Starting the construction ofE[t] with E[t0] = E0, we have thus constructed the ellipsoidE[t] asa solution to (43) with its evolution selected by the requirement (45), (48), (50) which resulted inequations (53),(54),X[t0] = X0.

Equation (53) obviously coincides with (30) ifφ = 0. However, as indicated in Sections 2, 3, underAssumption 3.1 the ellipsoidE[t] constructed due to equation (30),φ = 0 (which is also (13) or(26)) touches the reach setX [t] at points generated by support vectorl∗(t) = X(t0, t)l (for anypreassignedl ∈ IRn). It therefore satisfiesfor all t ≥ t0 the requirement (iii) of Definition 4.2.These observations lead to the next result.

16

Theorem 4.1 Under Assumption 3.1 the ellipsoidE[t] = E(x?(t),X[t]) constructed followingequations (12), (53), (54) withX[t0] = X0 is a solution to Problem 2.1, (withl(t) = X(t0, t)l), sothatE[t] = E∗

+[t]. It is also aminimal solution to equation (43).

From Lemma 4.1 it follows that the mappingE[t] = E(t|t0,X0) constructed as in Theorem 4.1,with X[t] = X∗

+[t], satisfies the semigroup property (44). This may be also checked by directcalculation, using (21) - (23).

In the general case the ellipsoidal tubeE∗[t] = E(x?,X∗+[t]) generated by functionX∗

+[t] of (52)is such that it touchesX [t] according to the requirement of Problem 2.1. However the respectivemappingE∗

+[t] = E∗[t] = E∗(t|t0, E0) may not satisfy the semigroup property(44) and thereforemay not satisfy equation(43). This may be shown by direct calculations.

Theorem 4.2 summarizes the discussion of Sections2 − 4.

Theorem 4.2 The solution of Problem2.1 allows the following conclusions:

(i) the ellipsoid E∗[t] = E∗(x?[t],X∗+[t]) constructed from equations (19), (20) forl∗(t) =

l(t),X∗+[t0] = X0, is a solution of Problem 2.1 andX∗

+[t] satisfies equation (30).

(ii) The ellipsoidE[t] = E(x?(t),X[t]) constructed from equations (26), (25) ( or (53), (54)) isa solution to the evolution equation (43) and satisfies the semigroup property (44). It is an upperbound for the solutionE∗

+[t] = E∗(x?(t),X∗+[t]) of Problem 2.1, so that

ρ(l∗(t)|E(x?(t),X[t])) ≥ ρ(l∗(t)|E(x?(t),X∗+[t])), (55)

with equalityX[t] = X∗[t] reached for allt ≥ t0 under Assumption 3.1.

In the latter case the functionE∗+[t] is a minimal solution to the evolution equation (43) and the

mappingE∗[t] = E(x?[t],X∗+[t]) = E∗(t|t0, E0) satisfies the semigroup property (44).

(iii)In order that E(x?[t],X∗+[t] would be described by equations (26), (25) it is necessary and

sufficient thatl(t) satisfies equation (40) with somek(t).

The solutions to problem 2.1 given in the last theorem aretight in the sense that there exists no otherellipsoid that could be squeezed in betweenE(x?(t),X∗

+[t]) andX [t].

Theorem 4.2 describes the evolution of approximating ellipsoidal sets considered asset-valued func-tionsand their relation with the approximated reach set.

17

5 The reach tube

An application of Theorem 1.1 yields the following conclusion.

Theorem 5.1 Suppose Assumption 3.1 is fulfilled. Then the pointsx∗(t) of support for vectorl∗(t),namely, those for which the equalities

(l∗(t), x∗(t)) = ρ(l∗(t)|X [t]) = ρ(l∗(t)|E(x?(t),X∗+[t])) (56)

are true for all t ≥ t0, are reached from initial statex(t0) = x∗0 using controlu = u∗(t) thatsatisfy the maximum principle (10) and condition (11) which now have the form (l∗(t) = X(t0, t)l)

(l,X(s, t0)B(s)u∗(s)) = max{(l,X(s, t0)B(s)u)|u ∈ E(q(t), Q(t))}, (57)

and(l, x∗0) = max{(l, x)|x ∈ E(x0,X0)}, (58)

for any t ≥ t0. For almost alls ≥ t0 the controlu∗(s), s ∈ [t0, t] may be taken to be the same,whatever be the valuet.

From the maximum principle above one may directly calculate the optimal controlu∗(s) which is

u∗(s) =Q(s)B′(s)X ′(s, t0)l

(l,X(s, t0)B(s)Q(s)B′(s)X ′(s, t0)l)1/2, t0 ≤ s ≤ t, (59)

and the optimal trajectory

x∗(t) = x?(t) +∫ t

t0

X(s, t0)B(s)Q(s)B′(s)X ′(s, t0)l(l,X(s, t0)B(s)Q(s)B′(s)X ′(s, t0)l)1/2

ds, t ≥ t0, (60)

where

x∗(t0) =X0l

(l,X0l)1/2+ x0. (61)

These relations are not very convenient for calculation, being expressed in a non-recursive form.

However, one should note that due to (56), thetrajectoryx∗(t) of the Theorem 5.1 also satisfies thefollowing “maximum relation”:

(l∗(t), x∗(t)) = max{(l∗(t), x)|x ∈ E(x?,X∗+[t])}, (62)

which is attained atx∗(t) = x?(t) + X∗

+[t]l∗(t)(l∗,X∗+[t]l∗(t))−1/2, (63)

wherel∗(t) = X ′(t0, t)l and

X[t0] = X0, x∗(t0) = x0 + X0l(l,X0l)−1/2. (64)

This results in Theorem 5.2.

18

Theorem 5.2 The trajectoryx∗(t) that runs along the boundary∂X [t] of the reach setX [t] andtouchesX [t] at the points of support for the vectorl∗(t) = X ′(t, t0)l is given by equalities (63),(64),whereX∗

+[t] = X[t] is the solution to equation (53), (54).

Due to Lemma 3.1 , the last theorem may be complemented by

Corollary 5.1 Given vectorl0 = l∗(t0), the related ellipsoidE(x?,X∗+[t]) and vectorx∗(t) ∈

∂X [t] ,generated by functionl∗(t) of equation (40), do not depend on the choice of functionk(t) inthis equation.

Denotingx∗(t) = x[t, l], we thus come to a two-parameter surfacex[t, l] that defines the boundary∂X of the reachability tubeX = ∪{X[t], t ≥ t0}. With t = t′ fixed andl ∈ S varying, the vectorx[t′, l] runs along the boundary∂X [t′]. On the other hand, withl = l′ fixed and witht varying,the vectorx[t, l′] moves along one of the trajectoriesx∗(t) that touches the reachability setX [t]according to (1.1) with controlu∗(t) of (57), (59). Then

∪{x[t, l]|l ∈ S} = ∂X [t], ∪{x[t, l]|l ∈ S, t ≥ t0} = ∂X

The relation (63) is now given in a recursive form and in the calculation of the curvesx∗[t] and thesurfacex[t, l] one avoids the computation of the controlsu∗(t).

6 Example

For an illustration of the results consider the following system

x1 = x2, x2 = u, (65)

x1(0) = x01, x2(0) = x0

2; |u| ≤ µ, µ > 0.

We have,

x1(t) = x01 + x0

2t +∫ t

0(t − τ)u(τ)dτ,

x2(t) = x02 +

∫ t

0u(τ)dτ.

The support functionρ(l|X [t]) = max{(l, x(t))||u| ≤ µ}

may be calculated directly and is given by the formula

ρ(l|X [t]) = l1x01 + x0

2t +∫ t

0|l1(t − τ) + l2|dτ.

19

The boundary of the reach setX [t] may be calculated from the formula ([7])

minl{ρ(l|X [t]) − l1x1 − l2x2|(l, l) = 1} = 0. (66)

Then the direct calculation of the minimum in the last problem gives a parametric representation forthe boundary∂X [t], by introducing parameterσ = l02/l

01, wherel01, l

02 are the minimizers in (66).

This gives two curvesx1(t) = x0

1 + x02t + µ(t2/2 − σ2), (67)

x2(t) = x02 + 2µσ + µt,

andx1(t) = x0

1 + x02t − µ(t2/2 − σ2), (68)

x2(t) = x02 − 2µσ − µt,

for σ ≤ 0.9

These curves (parametrized byσ) form the upper and lower boundaries ofX [t] with parametertfixed (see Figure 1). Witht increasing this allows to draw the reachability tubeX [t] as a set-valuedfunction of timet, see Figure 2.

The formulal(t) = e−A′tl of Assumption 3.1 here transforms into(l1(t) = l1; l2(t) = l2 − tl1).This allows us to write down the relations for the points of support to the hyperplanes generated byvectorl(t). A substitution into (67), (68) gives

x1l(t) = x01 + x0

2t + µ(t2/2 − (tl1 − l2/l1)2), (69)

x2l(t) = x02 + 2µ(l2 − tl1)/l1 + µt,

andx1(t) = x0

1 + x02t − µ(t2/2 − (tl1 − l2/l1)2), (70)

x2(t) = x02 − 2µ(l2 − tl1)/l1 − µt,

With vector l ∈ IR2 fixed, this gives a parametric family of curvesxl(t) that cover the surface ofthe reach tubeX [·] and are the points of support for the hyperplanes generated by vectorsl ∈ IR2

through formulal(t) = (l1,−tl1 + l2). These curves are shown in thick lines in Figure 2.

Let us now choose two pairs of vectorsl ∈ IR2, for example

l∗ = (−10,−4), l∗ = (10, 4); l∗∗ = (−4, 0), l∗∗ = (4, 0)

.9Note that for the productl1l2 > 0 we haveσ > 0. For such vectors the point of supportxl[t] will be at either of the

vertices of the setX [t].

20

Each of these pairs generates a tight ellipsoidE∗[t] andE∗∗[t] due to equations (26) (or (53)), whichnow have the formE∗[t] = E(0,X∗(t)) where the elementsx∗

i,j of X∗ satisfy the equations

x∗11 = x∗

12 + x∗21 + π(t)x∗

11, x∗12 = x∗

22 + πx∗12,

x∗21 = x∗

22 + π(t)x∗21, x∗

22 = π(t)x∗22 + (π(t))−1,

with X∗(0) = 0 andπ(t) = f1(t)/f2(t),

withf1(t) = µ|l2 − tl1|; f2(t) = (x∗

11l21 + 2x∗

12l1(l2 − tl1) + x∗22(l2 − tl1)2)1/2.

Ellipsoid E∗∗[t] is expressed similarly.

Here E∗[t] touches the reach setX [t] at pointsx∗l (t) generated by vectorl∗ through formulae

(68),(70) andE∗∗[t] touches it at pointsx∗∗l (t) generated by vectorl∗∗ through formulae (67), (69),

(see crossection in Figure 3 fort = 0.5).

For a given timet the intersectionE∗[t] ∩ E∗∗[t] ⊃ X [t]

gives an approximation ofX [t] better than the four supporting hyperplanes that could be placed atthe same points of support as for the ellipsoids. At the same time, by taking more ellipsoids for othervalues ofl we achieve, as a consequence of Theorem 2.1, a more accurate approximation of setX [t]( see Figure 4 for seven ellipsoids) with exact representation achievable if number of appropriatelyselected ellipsoids tends to infinity. An onlineanimationof the reach set is available.10

10http://www.path.berkeley.edu/ varaiya/a2.gif

21

Figure 1. Reach Set

Figure 2. Reach Tube

22

Figure 3. Two Tight Ellipsoids

Figure 4. Several Tight Ellipsoids

23

7 Conclusion

This paper specifies and studies the behavior of the tight external ellipsoidal approximations ofreach sets and reach tubes for linear time-variant control systems. It presents analytical results thatachieve a substantial reduction of the computation burden for calculating these sets as compared todirect methods and gives effective techniques for calculating the reach tubes in a compact recursiveform. The analytical relations developed in this paper open routes to the investigation of preciseerror estimates in ellipsoidal approximations for problems of evolution, estimation and control aswell as to the development of new computational tools for classes of systems more complicated thanthose treated in this paper.

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[12] ROCKAFELLAR, R. T.,Convex Analysis,Princeton University Press, 1970.

[13] SCHWEPPE F.C.Uncertain Dynamic Systems, Prentice Hall, Englewood Cliffs, 1973.

[14] VARAIYA P. Reach Set Computation Using Optimal Control, inProc.of KIT Workshop onVerification of Hybrid Systems, Verimag, Grenoble, 1998.

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