Smallest Ellipsoid Containing p-Sum of Ellipsoidswith Application to Reachability Analysis

Abhishek Halder

Abstract—We study the problem of ellipsoidal bounding ofconvex set-valued data, where the convex set is obtained bythe p-sum of finitely many ellipsoids, for any real p ≥ 1.The notion of p-sum appears in the Brunn-Minkowski-Fireytheory in convex analysis, and generalizes several well-known set-valued operations such as the Minkowski sum of the summandconvex sets (here, ellipsoids). We derive an outer ellipsoidalparameterization for the p-sum of a given set of ellipsoids, andcompute the tightest such parameterization for two optimalitycriteria: minimum trace and minimum volume. For such optimalparameterizations, several known results in the system-controlliterature are recovered as special cases of our general formula.For the minimum volume criterion, our analysis leads to afixed point recursion over a scalar that parameterizes the shapematrix of the outer ellipsoid. This recursion is proved to becontractive, and found to converge fast in practice. We applythese results to compute the forward reach sets for a linearcontrol system subject to different convex set-valued uncertaintymodels for the initial condition and control, generated by varyingp ∈ [1,∞]. Our numerical results show that the proposed fixedpoint algorithm offers more than two orders of magnitude speed-up in computational time for p = 1, compared to the existingsemidefinite programming approach without significant effect onthe numerical accuracy. For p > 1, the reach set computationresults reported here are novel. Our results are expected to beuseful in real-time safety critical applications such as decisionmaking for collision avoidance of autonomous vehicles, wherethe computational time-scale for reach set calculation needs tobe much smaller than the vehicular dynamics time-scale.

Keywords: Ellipsoidal calculus, Firey p-sum, outer approxi-mation, optimal ellipsoid, reach sets.

I. INTRODUCTION

Computing an ellipsoid that contains given set-valued data,is central to many applications such as guaranteeing collisionavoidance in robotics [1]–[3], robust estimation [4]–[7], sys-tem identification [8]–[10], and control [11], [12]. To reduceconservatism, one requires such an ellipsoid to be “smallest”according to some optimality criterion, among all ellipsoidscontaining the data. Typical examples of optimality criteria are“minimum volume” and “minimum sum of the squared semi-axes”. A common situation arising in practice is the following:the set-valued data itself is described as set operations (e.g.union, intersection, or Minkowski sum) on other ellipsoids. Inthis paper, we consider computing the smallest ellipsoid thatcontains the so-called p-sum of finitely many ellipsoids, wherep ∈ [1,∞].

As a set operation, the p-sum of convex sets returns a newconvex set, which loosely speaking, is a combination of theinput convex sets. The notion of p-sum was introduced by

Abhishek Halder is with the Department of Applied Mathematics, Univer-sity of California, Santa Cruz, CA 95064, USA, ahalder@ucsc.edu

Firey [13] to generalize the Minkowski sum† of convex bodies,and was studied in detail by Lutwak [14], [15], who termed theresulting development as Brunn-Minkowski-Firey theory (alsoknown as Lp Brunn-Minkowski theory, see e.g., [16, Ch. 9.1]).In this paper, we derive an outer ellipsoidal parameterizationthat is guaranteed to contain the p-sum of given ellipsoids, andthen compute the smallest such outer ellipsoid.

Since the p-sum subsumes well-known set operations likethe Minkowski sum as special case, we recover known resultsin the systems-control literature [17]–[20] about the minimumtrace and minimum volume outer ellipsoids of such sets, byspecializing our optimal parameterization of the p-sum ofellipsoids. Furthermore, based on our analytical results, wepropose a fixed point recursion to compute the minimumvolume outer ellipsoid of the p-sum for any real p ∈ [1,∞],that has fast convergence. The proposed algorithm not onlyenables computation for the novel convex uncertainty models(for p > 1), it also entails orders of magnitude faster runtimecompared to the existing semidefinite programming approachfor the p = 1 case. Thus, the contribution of this paperis twofold: (i) generalizing several existing results in theliterature on outer ellipsoidal parameterization (Section III) ofa convex set obtained as set operations on given ellipsoids, andanalyzing its optimality (Section IV); (ii) deriving numericalalgorithms (Section V) to compute the minimum volume outerellipsoid containing the p-sum of ellipsoids.

A salient feature of the proposed minimum volume (con-strained to a parametric family that we construct) outer el-lipsoid algorithm is that it does not require extra parameter-ized ellipsoids, unlike many other algorithms available in theliterature [20], [30]. Put differently, the proposed algorithmsdirectly process the data of the problem to return the outerapproximation without any extra construction.

To illustrate the numerical algorithms derived in this paper,we compute (Section VI) the smallest outer ellipsoidal ap-proximations for the (forward) reach sets of a discrete-timelinear control system subject to set-valued uncertainties inits initial conditions and control. When the initial conditionand control sets are ellipsoidal, they model weighted normbounded uncertainties, and at each time, we are led to computethe smallest outer ellipsoid for the Minkowski sum (p = 1case). It is found that by specializing the proposed algorithmsfor p = 1, we can lower the computational runtime by morethan two orders of magnitude compared to the current state-of-the-art, which is to reformulate and solve the same viasemidefinite programming. For p > 1, the initial conditionsand controls belong to p-sums of ellipsoidal sets, which

†The Minkowski sum of two compact convex sets X ,Y ⊂ Rd is the setX +1 Y := {x + y | x ∈ X ,y ∈ Y} ⊂ Rd.

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are convex but not ellipsoidal, in general. In this case too,the proposed algorithms enable computing the smallest outerellipsoids for the reach sets.

II. PRELIMINARIES

1) Convex Geometry: The support function hK(·) of acompact convex set K ⊂ Rd, is

hK(y) := sup{〈x,y〉 | x ∈ K, y ∈ Rd

}, (1)

where 〈·, ·〉 denotes the standard Euclidean inner product. Thefunction hK : K 7→ R, and can be viewed geometricallyas the (signed) perpendicular distance from the origin to thesupporting hyperplane of K, which has the outer normal vectory. Thus, the support function returns negative value if andonly if the normal vector y points into the open halfspacecontaining the origin. The support function hK(·) can also beseen as (see e.g., [21, Theorem 13.2]) the Legendre-Fenchelconjugate of the indicator function of the set K, and thusuniquely determines the set K.

The following properties of the support function are well-known:(i) convexity: hK(y) is a convex function in y ∈ Rd,(ii) positive homogeneity: hK(αy) = αhK(y) for α > 0,(iii) sub-additivity: hK(y+z) ≤ hK(y)+hK(z) for y, z ∈ Rd,(iv) inclusion: given compact convex sets K1 and K2, theinclusion K1 ⊆ K2 holds if and only if hK1

(y) ≤ hK2(y)

for all y ∈ Rd,(v) affine transformation: hAK+b(y) = hK(A>y) + 〈b,y〉,for A ∈ Rd×d, b ∈ Rd,y ∈ Rd.

Definition 1. (p-Sum of convex sets) Given compact convexsets K1,K2 ⊂ Rd, their p-sum [13] is a new compact convexset K ⊂ Rd defined via its support function

hK(y) =(hpK1

(y) + hpK2(y)) 1

p , 1 ≤ p ≤ ∞, (2)

and we write K := K1 +p K2.

Special cases of the p-sum are encountered frequently inpractice. For example, when p = 1, the set K = K1 +1 K2 isthe Minkowski sum of K1 and K2, and

hK1+1K2(·) = hK1

(·) + hK2(·).

When p = ∞, the set K = K1 +∞ K2 is the convex hull ofthe union of K1 and K2, and

hK1+∞K2(·) = max{hK1(·), hK2(·)

}.

For 1 ≤ p < q ≤ ∞, we have the inclusion [13, p. 20] (SeeFig. 1):

K1 ∪ K2 ⊆ K1 +∞ K2 ⊆ . . . ⊆ K1 +q K2 ⊆ K1 +p K2

⊆ . . . ⊆ K1 +1 K2, (3)

which follows (Appendix A) from the support function in-equality

hK1+qK2(·) ≤ hK1+pK2

(·), 1 ≤ p < q ≤ ∞.

Fig. 1: The p-sum of two input ellipses E1 and E2 (filledgray) are shown for p = 1, 2,∞. For varying p ∈ [1,∞],the p-sum E1 +p E2 defines a nested sequence of convex sets– the outermost being the Minkowski sum (p = 1, dashedboundary), and the innermost being the convex hull of theunion (p =∞, solid boundary) of E1 and E2. In this case, thep-sum is an ellipse only for p = 2 (dash-dotted boundary).

From Definition 1, it is easy to see that the p-sum is commu-tative and associative, that is,

K1 +p K2 = K2 +p K1, (4a)(K1 +p K2) +p K3 = K1 +p (K2 +p K3) , (4b)

where the compact sets K1,K2,K3 are convex. Furthermore,linear transformation is distributive over p-sum, i.e.,

A (K1 +p K2) = AK1 +p AK2, A ∈ Rd×d, (5)

which is immediate from the aforesaid property (v) andequation (2). We mention here that it is often convenient toexpress hK(·) as function of the unit vector y/ ‖ y ‖2 in Rd(see Fig. 2).

Remark 1. At first glance, it might seem odd that theMinkowski sum is defined pointwise as X+1Y := {x+y | x ∈X ,y ∈ Y}, which remains well-defined for X ,Y compact (notnecessarily convex), but the p-sum in Definition 1 is given viasupport functions and requires the summand sets to be convex.Indeed, a pointwise definition for the p-sum was proposed in[22] for compact summand sets X and Y , given by X+pY :={(1 − µ)1/p

′x + µ1/p′y | x ∈ X ,y ∈ Y, 0 ≤ µ ≤ 1},

where p′ denotes the Holder conjugate of p, i.e., 1p + 1

p′ = 1.This pointwise definition was shown to reduce to Definition 1provided the compact summands X ,Y are also convex.

2) Ellipsoids: Let Sd+ be the cone of d × d symmetricpositive definite matrices. For an ellipsoid with center q ∈ Rdand shape matrix Q ∈ Sd+, denoted by

E (q,Q) := {x ∈ Rd : (x− q)>Q−1 (x− q) ≤ 1},

Fig. 2: The support function hE1+pE2(·) for the p-sum of twoinput ellipses E1 and E2 are shown for different values ofp ≥ 1, where Ei :=

{(x, y) ∈ R2 | x2

a2i+ y2

b2i≤ 1

}, i = 1, 2,

with a1 = 4, b1 = 7, a2 = 1, b2 = 14. The correspondingp-sum sets for p = 1, 2,∞ are as in Fig. 1. In this case,

hE1+pE2(θ) =(∑

i=1,2(a2i cos2 θ + b2i sin2 θ)p/2)1/p

for p ∈[1,∞), and hE1+∞E2(θ) = maxi=1,2(a2i cos2 θ+b2i sin2 θ)1/2,where θ ∈ [0, 2π).

(1) reduces to

hE(q,Q)(y) = 〈q,y〉 +√〈Qy,y〉. (6)

Furthermore, the volume of the ellipsoid E (q,Q) is given by

vol (E (q,Q)) =vol(Bd1)

√det (Q−1)

=π

d2

Γ(d2 + 1

)√

det (Q), (7)

where Bd1 denotes the d-dimensional unit ball, and Γ(·) denotesthe Gamma function, i.e., Γ(s) :=

∫∞0ηs−1 exp (−η) dη,

provided the real part of s > 0.An alternative ellipsoidal parameterization can be ob-

tained via a matrix-vector-scalar triple (A, b, c) encoding thequadratic form, i.e., E(A, b, c) := {x ∈ Rd : x>Ax+2x>b+c ≤ 0}. The following relations among (A, b, c) and (q,Q)parameterizations for ellipsoid will be useful in the later partof this paper:

A = Q−1, b = −Q−1q, c = q>Q−1q − 1, (8a)

Q = A−1, q = −Qb. (8b)

III. PARAMETERIZED OUTER ELLIPSOID

Given two ellipsoids E(q1,Q1), E(q2,Q2) ⊂ Rd, considertheir p-sum

E(q1,Q1) +p E(q2,Q2), 1 ≤ p ≤ ∞, (9)

which is convex but not an ellipsoid in general (see Fig. 1and 2). We want to determine an ellipsoid E(q,Q) ⊂ Rd, asfunction of the input ellipsoids, that is guaranteed to contain

the p-sum (9). For this to happen, we must have (from (2) andproperty (iv) in Section II.1)

hpE(q,Q)(y) ≥ hpE(q1,Q1)(y) + hpE(q2,Q2)

(y),

(6)⇒(y>q +

√y>Qy

)p≥∑

i=1,2

(y>qi +

√y>Qiy

)p, (10)

for all y ∈ Rd. In the rest of this paper, we make the followingassumption.

Assumption 1. The center vectors for the summand ellipsoidsin a p-sum are assumed to be zero, i.e., q1 = q2 = 0.

Under Assumption (1), and p = 2, it follows from (6) and(10) that (9) is an ellipsoid E (0,Q1 + Q2).

For p 6= 2, the convex set (9) is again not an ellipsoid ingeneral, but we can parameterize an outer ellipsoid E (0,Q) ⊇E(0,Q1) +p E(0,Q2) as follows.

For α, γ > 1, let

Q := αQ1 + γQ2, and g2i := y>Qiy ≥ 0, i = 1, 2. (11)

For the time being, we think of α, γ > 1 as free parameters.We will see that it is possible to re-parameterize Q in termsof a single parameter β > 0, by expressing both α and γ asappropriate functions of β, while guaranteeing the inclusionof the p-sum in E (0,Q).

With the standing assumptions q1 = q2 = 0, we can re-write (10) as

(αg21 + γg22

) p2 ≥ gp1 + gp2 ,

⇒(αg21 + γg22

)p ≥ g2p1 + g2p2 + 2gp1gp2 . (12)

To proceed further, we need the following Lemma.

Lemma 1. A convex function f(·) with f(0) = 0 is super-additive on [0,∞), i.e., f(x+y) ≥ f(x)+f(y) for all x, y ≥0.

Proof. For 0 ≤ λ ≤ 1, by convexity

f(λx) = f(λx+ (1− λ)0) ≤ λf(x) + (1− λ)f(0) = λf(x).

Therefore, we get

f(x) = f

(x

x+ y(x+ y)

)≤ x

x+ yf(x+ y),

and

f(y) = f

(y

x+ y(x+ y)

)≤ y

x+ yf(x+ y).

Adding the last two inequalities, f(x)+f(y) ≤ f(x+y).

In Theorem 1 below, we use Lemma 1 to derive an explicitparameterization of the shape matrix Q(β) ∈ Sd+, β > 0, thatguarantees an outer ellipsoidal containment of the p-sum

E(0,Q1) +p E(0,Q2).

Theorem 1. Given a scalar β > 0, and a pair of matricesQ1,Q2 ∈ Sd+, let

Q(β) =

(1 +

1

β

)1p

Q1 + (1 + β)1p Q2, p ∈ [1,∞) \ {2}. (13)

Then E (0,Q(β)) ⊇ E(0,Q1) +p E(0,Q2).

Proof. Since f(x) := xp is convex on [0,∞) for 1 ≤ p <∞,Lemma 1 yields

(αg21 + γg22

)p ≥ αpg2p1 + γpg2p2 ,

or equivalently,(αg21 + γg22

)p= αpg2p1 + γpg2p2 + ξ, where ξ ≥ 0. (14)

Combining (12) and (14), we obtain

(αp − 1) (gp1)2

+ (γp − 1) (gp2)2 − 2gp1g

p2 + ξ ≥ 0. (15)

Since α > 1, we have αp > 1 for p ≥ 1. Therefore,multiplying both sides of (15) by (αp − 1) > 0, and thenadding and subtracting (gp2)2, we get

((αp − 1) gp1 − gp2)2

+ ((αp − 1) (γp − 1)− 1) (gp2)2

+ (αp − 1) ξ ≥ 0. (16)

A sufficient condition to satisfy the inequality (16) is to chooseα, γ > 1 such that

(αp − 1) (γp − 1) ≥ 1.

Letting αp − 1 := β−1 and γp − 1 = β, β > 0, and using(11), we arrive at (13).

Remark 2. The parameterization (13) generalizes the outerellipsoidal parameterization containing the Minkowski sum(p = 1 case), well-known in the systems-control literature (seee.g., [17, p. 104], [18], [19]). For this special case p = 1, adiscussion about equivalent parameterizations can be foundin [23, Section II.A].

IV. OPTIMAL PARAMETERIZATION

To reduce conservatism, it is desired that the parameterizedouter ellipsoid E (0,Q(β)) in Theorem 1 containing the p-sum, be as tight as possible. One way to promote “tightness”is by minimizing the sum of the squared semi-axes lengthsof E (0,Q(β)), which amounts to minimizing trace (Q(β))over β > 0. Another possible way to promote “tightness” isby minimizing the volume of E (0,Q(β)), which, thanks to(7), amounts to minimizing log det (Q(β)). We next analyzethese optimality criteria. A discussion on different optimalitycriteria promoting different notions of “ellipsoidal tightness”can be found in [30, p. 226].

A. Minimum Trace Outer Ellipsoid

We consider the optimization problem

minimizeβ>0

trace (Q(β)) , (17)

where Q(β) is given by (13), and let

β∗tr := arg minβ>0

trace (Q(β)) .

Setting∂

∂βtrace (Q(β)) = 0, and using the linearity of

trace operator, straightforward calculation yields

β∗tr =

(trace (Q1)

trace (Q2)

) p1+p

, p ∈ [1,∞) \ {2}, (18)

and

∂2

∂β2trace (Q(β))

∣∣∣β=β∗tr

=1

p

(1

p+ 1

)(β∗tr)

− 1p−2 (β∗tr + 1)

1p−1 trace (Q1) > 0.

The formula (18) generalizes the previously known formulafor p = 1 case (minimum trace ellipsoid containing theMinkowski sum of two given ellipsoids) reported in [19,Appendix A.2] and in [17, Lemma 2.5.2(a)].

B. Minimum Volume Outer Ellipsoid

Now we consider the optimization problem

minimizeβ>0

log det (Q(β)) , (19)

where Q(β) is given by (13), and let

β∗vol := arg minβ>0

log det (Q(β)) . (20)

To simplify the first order condition of optimality∂∂β log det (Q(β)) = 0, we notice that the matrix R :=

Q−11 Q2 is diagonalizable [23, Section III.A, Lemma 1], anddenote its spectral decomposition as R := SΛS−1. Further,let the eigenvalues of R be {λi}di=1, which are all positive (seethe discussion following Proposition 1 in [23]). Then directcalculation gives

∂

∂βlog det (Q(β)) = trace

((Q(β))−1

∂

∂βQ(β)

)

= − 1

pβ(1 + β)trace

((I + β

1pR)−1 (

I − β3− 1pR))

= − 1

pβ(1 + β)trace

((I + β

1p Λ)−1 (

I − β3− 1p Λ))

, (21)

wherein the last step follows from substituting I = SS−1,R = SΛS−1, and using the invariance of trace of matrixproduct under cyclic permutation.

Therefore, from (21), the first order optimality condition∂∂β log det (Q(β)) = 0 is equivalent to the following nonlinearalgebraic equation:

d∑

i=1

1− β3− 1pλi

1 + β1pλi

= 0, p ∈ [1,∞) \ {2}, (22)

to be solved for β > 0, with known parameters λi > 0,i = 1, . . . , d. If (22) admits unique positive root (which seemsnon-obvious, and will be proved next), then it would indeedcorrespond to the argmin in (20) since

∂2

∂β2log det (Q(β))

∣∣∣∣β>0

=1

pβ(1 + β)

d∑

i=1

(2− 1p )β2λ2i +

[(3− 1

p )β2− 1p + 1

pβ1p−1]λi

(1 + βλi)2

> 0, for p ∈ [1,∞) \ {2}. (23)

In the following Theorem, we establish the uniqueness of thepositive root.

Theorem 2. Given λi > 0, i = 1, . . . , d, equation (22) invariable β admits unique positive root.

Proof. We start by rewriting (22) as

d∑

i=1

(β3− 1

p − 1

λi

) d∏

j=1

j 6=i

(β

1p +

1

λj

)= 0. (24)

Now let er ≡ er(

1λ1, . . . , 1

λd

)for r = 1, . . . , d, denote the rth

elementary symmetric polynomial [24, Ch. 2.22] in variables1λ1, . . . , 1

λd, that is,

er ≡ er(

1

λ1, . . . ,

1

λd

):=

∑

1≤i1<i2...<ir≤d

1

λi1 . . . λir.

For example,

e1 =∑

1≤i≤d

λ−1i , e2 =∑

1≤i<j≤d

(λiλj)−1, ed =

∏

1≤i≤d

λi

−1

,

and e0 = 1 by convention. For all r = 1, . . . , d, we haveer > 0 since λ1, . . . , λd are all positive. Letting ζ := β

1p , we

write (24) in the expanded form

ζ3p−1{ d∑

r=1

(d− r + 1) er−1ζd−r}−

d∑

r=1

rerζd−r = 0. (25)

We notice that (25) is a polynomial in ζ, in which thecoefficients undergo exactly one change in sign. Therefore, byDescartes’ rule of sign, the equation (25) (equivalently (24) or(22)) admits unique positive root.

Next, we give an algorithm to compute the unique positiveroot β∗vol of the equation (22), and show how the same can beused for reachability analysis for linear control systems.

V. ALGORITHMS AND APPLICATIONS

A. Computing the Minimum Volume Outer Ellipsoid

Thanks to the parameterization (13), computing the min-imum volume outer ellipsoid (MVOE) for the p-sumE(0,Q1)+pE(0,Q2), is reduced to computing β∗vol introducedin the previous Section. Motivated by the observation that thefirst order optimality criterion (22) can be rearranged as

β3− 1p

d∑

i=1

λi

1 + β1pλi

=1

1 + β1pλi

, (26)

we consider the fixed point recursion

βn+1 = g(βn) :=

d∑

i=1

1

1 + β1pn λi

d∑

i=1

λi

1 + β1pn λi

p3p−1

, (27)

where n = 0, 1, 2, . . ., and p ∈ [1,∞) \ {2}. Furthermore,g : R+ 7→ R+, i.e., the map g is cone preserving. In thefollowing Theorem, we show that the fixed point recursion(27) converges to a unique positive root (Fig. 3), and is in

Fig. 3: The unique positive fixed point β∗vol for the mapβ 7→ g(β) given by (27), is shown for p = 1, 1.5, 2, 2.5, 3, 10,all with d = 3, and with parameters {λ1, λ2, λ3} = {5, 0.6, 3}.For any given p ≥ 1, the fixed point β∗vol is the point of inter-section between the dashed straight line and the correspondingsolid curve g(β).

fact contractive in the Hilbert metric [25, Ch. 2]. Thus, therecursion (27) is indeed an efficient numerical algorithm tocompute β∗vol. The recursion (27) and the contraction proofbelow subsume our previous result [23, Section IV.B] for thep = 1 case (computing MVOE for the Minkowski sum).

Theorem 3. Starting from any initial guess β0 ∈ R+, therecursion (27) with fixed p ∈ [1,∞) \ {2}, converges to aunique fixed point β∗vol ∈ R+, i.e., lim

n→∞gn(β0) = β∗vol.

Proof. For λi, x > 0, consider the positive functions fi :=1/(1 + λix

1/p), where i = 1, . . . , d, p ∈ [1,∞)\{2}, and let

φ(x) := xp

3p−1 , and ψ(x) :=

∑i fi∑i λifi

.

Clearly, φ(x) and ψ(x) are both concave and increasing in R+,and therefore [26, p. 84] so is g(βn) = φ(ψ(βn)) as a functionof βn, n = 0, 1, 2, . . .. Consequently (see e.g., the first step inthe proof of Theorem 2.1.11 in [27]) the map g is contractivein Hilbert metric on the cone R+. By Banach contractionmapping theorem, g admits unique fixed point β∗vol ∈ R+,and lim

n→∞gn(β0) = β∗vol.

The rate-of-convergence for (27) is fast in practice, see Fig.4. Next, we show how the p-sum computation may arise inthe reachability analysis for linear systems.

B. Application to Reachability Analysis for Linear Systems

Consider a discrete-time linear time-invariant (LTI) system

x(t+ 1) = Fx(t) + Gu(t), x ∈ Rnx , u ∈ Rnu , (28)

Fig. 4: Starting from the same initial guess β0 = 0.647584,the iterates for (27) are shown to converge (within an errortolerance of 10−5) in few steps for p = 1, 1.5, 10. The aboveresults are for the same parameters as in Fig. 3.

with set-valued uncertainties in initial condition x(0) ∈ X0 ⊂Rnx , and control u(t) ∈ U(t) ⊂ Rnu . For t = 0, 1, 2, . . ., wewould like to compute a tight outer ellipsoidal approximationof the reach set X (t) 3 x(t), when the sets X0 and U(t) aremodeled as p1 and p2-sum of ellipsoids, respectively, i.e.,

X0 = E (x0,Q01)+p1 . . .+p1E (x0,Q0m) , (29a)U(t)= E (uc(t),U1(t))+p2 . . .+p2E (uc(t),Un(t)) , (29b)

where p1, p2 ≥ 1. Here, x0 and uc(t) are the nominal initialcondition and control, respectively. Further, Q0i ∈ Snx

+ fori = 1, . . . ,m, and for all t = 0, 1, 2, . . ., the matrices Uj(t) ∈Snu+ for j = 1, . . . , n. The case m = n = 1 corresponds

to the situation when both the initial condition and controlhave ellipsoidal uncertainties, and has appeared extensivelyin systems-control literature [4], [5], [17]–[19], [28]–[33]. Byvarying p1, p2 ∈ [1,∞] in (29), one obtains a large class ofconvex set-valued uncertainty descriptions.

In the absence of control (u(t) ≡ 0), the set X (t) re-mains a p1-sum of ellipsoids whenever X0 is a p1-sum ofellipsoids as in (29a). This follows from the support functionhX (t)(y) = hΦ(t,0)X0

(y) = hX0

((Φ(t, 0))>y

), y ∈ Rnx ,

where Φ(t, 0) = F t is the state transition matrix of (28).When the control u(t) is not identically zero, then the

solution x(t) = F tx(0) +∑t−1k=0 F

t−k−1Gu(k) correspondsto the support function

hX (t)(y) = hX0

((F>)ty

)+

t−1∑

k=0

hU(k)(G>(F>)t−k−1y

). (30)

From (5) and (30), it is evident that X (t) is the 1-sum(Minkowski sum) of (t + 1) convex sets, one of which isp1-sum of ellipsoids, and each of the remaining t sets are p2-sum of ellipsoids, where p1, p2 ≥ 1. Clearly, this remains trueeven when the system matrices F ,G are time-varying. We

mention here that the general idea of using support functionsfor reachability analysis has appeared before in the literature[34], [35].

In the following Section, we apply the formula (18) and thefixed point algorithm (27) to compute the minimum trace andminimum volume outer ellipsoidal approximations of the reachsets of (28), constrained in the parametric family (13). Withslight abuse of nomenclature, we will hereafter refer theseellipsoids as MTOE and MVOE, respectively. These shouldbe understood as the optimal within the parametric family(13). The prototype algorithms for computing the same for theconvex set E (0,Q1) +p E (0,Q2), Q1,Q2 ∈ Sd+, are givenbelow. These serve as the building blocks for computing theparametric MTOE and MVOE for the reach set of (28) subjectto (29), which we illustrate next. For the results reported inthis paper which use Algorithm 2, we have set tol = 10−5,and MaxIter = 100.

Algorithm 1 Algorithm to compute the parametric MTOE forthe p-sum E (0,Q1) +p E (0,Q2), p ∈ [1,∞) \ {2}.

1: procedure MTOE(Q1,Q2, p) . Q1,Q2 ∈ Sd+2: β∗tr ← (trace (Q1) /trace (Q2))

p1+p

3: Q← (1 + 1/β∗tr)1p Q1 + (1 + β∗tr)

1p Q2

4: return E (0,Q) . The parametric MTOE5: end procedure

Algorithm 2 Algorithm to compute the parametric MVOE forthe p-sum E (0,Q1) +p E (0,Q2), p ∈ [1,∞) \ {2}.

1: procedure MVOE(Q1,Q2, p, tol, MaxIter) .Q1,Q2 ∈ Sd+, tol is numerical tolerance, MaxIter ismaximum number of iterations

2: {λi}di=1 ← spectrum(Q−11 Q2

)

3: β ← random positive number . Initialize parameter4: ε← 1 . Initialize error5: j ← 1 . Initialize iteration index6: while ((ε > tol) & (j < MaxIter)) do7: βnew ← g(β) . g(·) as in (27), needs {λi}di=1, p8: ε← |βnew − β|9: β ← βnew

10: end while11: β∗vol ← β

12: Q← (1 + 1/β∗vol)1p Q1 + (1 + β∗vol)

1p Q2

13: return E (0,Q) . The parametric MVOE14: end procedure

VI. NUMERICAL SIMULATIONS

We consider the planar discrete-time linear system(x1(t+ 1)x2(t+ 1)

)=

(1 h0 1

)(x1(t)x2(t)

)+

(h h2/20 h

)(u1(t)u2(t)

), (31)

which can be seen as the sampled version of the continuous-time system x1 = x2 + u1, x2 = u2, with sampling periodh > 0. To illustrate the proposed algorithms, we will computethe reach sets of (31) for various convex uncertainty models ofthe form (29) for X0 and U(t). For this numerical example, we

Fig. 5: For dynamics (31), the MTOEs and MVOEs of the reach sets X (t) given by (32) with parameters (35), are shown fort = 1, . . . , 10, along with the (t+ 1) summand ellipses in the Minkowski sum (32) for each t. The MTOEs admit analyticalsolution (18). We compute the MVOEs via three different methods: SDP computation (33)-(34), a root bracketing techniqueto solve (24) proposed in [23, Section IV.A], and recursion (27).

set the nominal initial condition x0 ≡ 0, and nominal controluc(t) ≡ 0. For other examples, see Appendix B and C.

A. The case m = n = 1

For m = n = 1 in (29), both X0 and U(t) are ellipsoidal,and hence the reach set X (t) for (31), is the Minkowski sumof (t+ 1) ellipsoids. Consequently, we are led to compute theMTOE and MVOE of the Minkowski sum‡

X (t) = F tE (0,Q0) +1

t−1∑

k=0

F t−k−1GE (0,U(t)) . (32)

Notice that the MTOE admits analytical solution (18), appliedpairwise to the (t + 1) summand ellipsoids in (32), withp = 1. For this case, the current state-of-the-art for MVOEcomputation is to reformulate the same as a semidefinite pro-gramming (SDP) problem via the S-procedure (see e.g., [36,Ch. 3.7.4]). Specifically, given (t+ 1) ellipsoids E(qi,Qi) orequivalently E(Ai, bi, ci) in Rnx , i = 1, . . . , t+1, to compute

‡The symbol “Σ” in (32) stands for the 1-sum.

the MVOE containing their Minkowski sum E(q1,Q1) +1

. . .+1 E(qt+1,Qt+1), one solves the SDP problem:

minimizeA0,b0,τ1,...,τt+1

log detA−10 (33)

subject to

A0 � 0, (34a)τi ≥ 0, i = 1, . . . , t+ 1, (34b)E>0 A0E0 E>0 b0 0b>0 E0 −1 b>0

0 b0 −A0

−

t+1∑

i=1

τi

Ai bi 0

b>i ci 00 0 0

� 0,

(34c)

where Ei is the binary matrix of size nx×(t+1)nx that selectsthe i-th vector, i = 1, . . . , t+ 1, from the vertical stacking of(t+ 1) vectors, each of size nx × 1; and

E0 :=

t+1∑

i=1

Ei, Ai := E>i AiEi, bi := E>i bi, i = 1, . . . , t+1.

The argmin pair (A∗0, b∗0) associated with the SDP (33)-(34),

results the optimal ellipsoid

ESDPMVOE := E (qSDP,QSDP) ,

where, using (8b), QSDP := (A∗0)−1, and qSDP := −QSDPb∗0.

Our intent is to compare ESDPMVOE with Eproposed

MVOE , where

EproposedMVOE := E (q1 + . . .+ qt+1,Q(β∗vol)) ,

and the parametric form of Q(β∗vol) is given by (13) withp = 1, applied pairwise to the given set of shape matrices{Q1, . . . ,Qt+1}. The numerical value of β∗vol is computedfrom the fixed point recursion (27) with p = 1, solved pairwisefrom the set {Q1, . . . ,Qt+1}. In other words, due to the asso-ciate property of p-sum, the ellipsoid Eproposed

MVOE is obtained byapplying Algorithm 2 pairwise to the set {Q1, . . . ,Qt+1}, andthen translating the resulting ellipsoid by vector q1+. . .+qt+1.

We will see that the MVOE algorithms proposed hereinhelp in reducing computational time, compared to the SDP ap-proach, without sacrificing accuracy. For comparing numericalperformance, we implemented both the SDP (via cvx [37])and our proposed algorithms in MATLAB 2016b, on 2.6 GHzIntel Core i5 processor with 8 GB memory.

For the dynamics (31), we set h = 0.3, and

Q0 = I2, U(t) =(1 + cos2(t)

)diag([10, 0.1]), (35)

and for each t = 1, 2, . . ., compute the MTOE and MVOEof the reach set (32). For MVOE computation, we use threedifferent methods: using the SDP (33)-(34), using a root-bracketing algorithm proposed in [23, Section IV.A] to solve(24), and by using the fixed point recursion (27) proposedherein. In Fig. 5, the corresponding MTOEs and MVOEs, aswell as the summand ellipses in the Minkowski sum (32) areshown for t = 1, . . . , 10. In this paper, we do not emphasizethe root bracketing method for MVOE computation given in[23, Section IV.A] since that is a custom method for nx = 2,while the SDP (33)-(34) and the fixed point recursion (27) arevalid in any dimensions.

To assess the quality of the outer ellipsoidal approximationsshown in Fig. 5, we compare the volumes (in our two-dimensional example, areas) of the MVOEs computed via thethree different methods in Table I. The columns in Table Icorrespond to different time steps while the rows correspondto the three different methods mentioned above. From Table I,notice that the MVOE volumes are not monotone in time forany given method (see e.g., the columns for t = 7 and t = 8)since the shape matrices U(t) in (35) are periodic. We noticethat the volumes listed in the second and third row in TableI are in close agreement, while they are slightly conservativecompared to the same in the first row, which are computed bysolving the SDP (33)-(34). Furthermore, the relative numericalerror seems to grow (albeit slowly) with t (with increasingnumber of summand ellipsoids).

By looking at the computational accuracy comparisons fromTable I, it may seem that the SDP approach is superior tothe algorithms proposed herein. However, the corresponding

Fig. 6: Comparison of the computational times for MVOEcalculations in Fig. 5 and Table I. The MVOE computationaltimes for the proposed fixed point recursion method ( ,third row in Table I) are orders of magnitude faster than thesame for the SDP computation ( , first row in Table I).A root bracketing technique to solve (24) proposed in [23,Section IV.A] requires more computational time ( , secondrow in Table I) than iterating the recursion (27), but is fasterthan solving the SDP (33)-(34).

computational runtimes plotted in Fig. 6 reveal that solving thefixed point recursion (27) entails orders of magnitude speed-upcompared to solving the SDP. Given that the growing interestsin reach set computation among practitioners are stemmingfrom real-time safety critical applications (e.g., decision mak-ing for collision avoidance in a traffic of autonomous andsemi-autonomous cars or drones), the issue of computationalruntime becomes significant. Such applications indeed requirecomputing the reach set over a short physical time horizon(typically a moving horizon of few seconds length); the MVOEcomputational time-scale, then, needs to be much smaller thanthe dynamics time-scale. The results in Fig. 6 show thatthe proposed algorithms can be useful in such context asthey offer significant computational speed-up without muchconservatism.

B. The Case m,n > 1

We now consider the case m,n > 1 in (29) with m 6= n,and compute the MVOEs and MTOEs for the reach set X (t)of (31) with h = 0.3, as before. Specifically, in (29), wefix m = 2, n = 3, p1 = 2.5, and p2 = 1.5. In words,the set of uncertain initial conditions X0 is modeled as 2.5-sum of two ellipsoids, i.e., X0 = E (0,Q01) +2.5 E (0,Q02);the control uncertainty set U(t) is modeled as 1.5-sum ofthree (time-varying) ellipsoids, i.e., U(t) = E (0,U1(t)) +1.5

E (0,U1(t)) +1.5 E (0,U3(t)). The shape matrices for the setof uncertain initial conditions are randomly generated positive

hhhhhhhhhhhvol(MVOE)Physical time step

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = 9 t = 10

vol(ESDP

MVOE

)using (33)-(34) 8.6837 14.5461 27.9035 31.9097 35.0421 61.0650 65.3182 59.1310 100.8786 111.2311

vol(Eproposed

MVOE

)using [23, Section IV.A] 8.6837 14.6765 28.7263 33.2574 36.8740 65.1379 70.1631 63.8502 109.2246 120.8542

vol(Eproposed

MVOE

)using (27) 8.6837 14.6765 28.7263 33.2574 36.8740 65.1379 70.1632 63.8502 109.2246 120.8542

TABLE I: Comparison of the volumes of the MVOEs for the reach set X (t) in (32) at t = 1, 2, . . . , 10, corresponding to thedynamics (31) with parameters given by (35), computed via three methods: by solving the SDP (33)-(34) (first row), by usinga pairwise root bracketing technique reported in [23, Section IV.A] (second row), and by iterating the fixed point recursion(27) with tolerance 10−5 (third row). The corresponding MVOEs are depicted in Fig. 5. Overall, the MVOE volumes in thesecond and third row are in close agreement, while they are slightly conservative than the SDP results in the first row.

definite matrices:

Q01 =

(2.2259 0.19920.1992 2.4357

),Q02 =

(2.3111 0.67680.6768 2.1848

). (36)

The shape matrices for the set of uncertain controls are chosenas

Uj(t) =(1 + cos2 (jt)

)diag([10, 0.1]), j = 1, 2, 3. (37)

Then, the reach set X (t) for (31) at any time t > 0, equals

F t

{E (0,Q10) +2.5 E (0,Q20)

}+1

t−1∑k=0

F t−k−1G

{E (0,U1(t))

+1.5 E (0,U2(t)) +1.5 E (0,U3(t))

}, (38)

which, due to (5), is the 1-sum§ of (t+1) convex sets, one ofthem being the 2.5-sum of two ellipsoids, and the remainingt of them each being the 1.5-sum of three ellipsoids.

Unlike the case in Section VI-A where the SDP approachis known in the literature for computing an MVOE of the1-sum, and served as a baseline algorithm to compare theperformance of our proposed algorithms, to the best of ourknowledge, no such algorithm is known for the general p-sumcase. Our proposed algorithms are generic enough to enablethe MVOE/MTOE computation in this case. Specifically, ateach time t, we first compute the MVOEs (resp. MTOEs) foreach of the (t + 1) summand convex sets in (38) by solving(27) (resp. (18)) pairwise, and then compute the MVOE (resp.MTOE) of the 1-sum of the resulting MVOEs (resp. resultingMTOEs) using the same. In Fig. 7, we show the MVOEs andMTOEs thus computed, for the reach set (38) at t = 1, . . . , 10.

The volumes (in our two dimensional numerical example,areas) of the MVOEs shown in Fig. 7, are listed in Table II.The corresponding computational times are shown in Fig. 8.Notice that the MVOE computational times reported in Fig.8 are about two orders of magnitude slower than the samereported in Fig. 6. This is expected since the results in Fig.8 correspond to computing MVOEs of the 1-sums while the

§The summation symbol “Σ” in (38) stands for the 1-sum.

same in Fig. 8 correspond to computing MVOEs of the mixedp-sums (in our example, 1-sums, 1.5 sums and 2.5 sums)at any given time t > 0, and is indeed consistent with theobservation made in Fig. 4 that the rate-of-convergence ofrecursion (27) decreases with increasing p ≥ 1. Nevertheless,the computational times shown in Fig. 8 are still smaller thanthe computational times for the SDP approach in the 1-sumcase shown in Fig. 6.

C. Quality of Approximation

It is natural to investigate the quality of approximations forthe MTOEs and MVOEs reported herein with respect to theactual reach sets, in terms of the “shapes” of the true andapproximating sets. For example, it would be undesirable ifthe MTOE/MVOE computation promotes “skinny ellipsoids”which are too elongated along some directions and too com-pressed along others, when the same may not hold for theactual reach sets. This motivates us to compute the (two-sided)Hausdorff distance δH(t) (which is a metric) at each time stept, between the true reach set X (t) and its approximating outerellipsoid E(t), given by

δH(t) := max

{sup

x(t)∈X (t)

infx(t)∈E(t)

‖ x(t)− x(t) ‖2 ,

supx(t)∈E(t)

infx(t)∈X (t)

‖ x(t)− x(t) ‖2}. (39)

Remark 3. We clarify here that the MVOE for any compactconvex set (in our case, the reach set) is guaranteed tobe unique, and is referred to as the Lowner-John ellipsoid[38], [39]. Exact computation of the Lowner-John ellipsoid,however, leads to semi-infinite programming [26, Ch. 8.4.1],and for most convex sets such as the Minkowski sum ofellipsoids, has no known exact SDP representation. Thus, theellipsoid ESDP

MVOE described in Section VI.A, and computed bysolving (33)-(34), is an SDP relaxation of the true MVOE; seee.g., [40, Ch. 3.7.4.1]. In the following, we will compare theHausdorff distance between X (t) and its approximating outerellipsoid E(t), where E(t) is either the MTOE EMTOE(t), orone of the MVOEs: ESDP

MVOE(t) and EproposedMVOE (t).

Fig. 7: For dynamics (31), the MTOEs and MVOEs of the reach sets X (t) given by (38) with parameters (36) and (37), areshown for t = 1, . . . , 10, along with the (t+ 1) summand MVOEs in the 1-sum (38) for each t. As before, the MTOEs admitanalytical solution (18), applied pairwise. We compute the MVOEs via the pairwise recursion (27).

hhhhhhhhhhhvol(MVOE)Physical time step

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8 t = 9 t = 10

vol(Eproposed

MVOE

)using (27) 57.7493 99.3984 182.9045 206.0490 266.6789 383.9408 387.4037 461.7879 610.9069 666.9160

TABLE II: Volumes of the MVOEs for the reach set X (t) in (38) at t = 1, 2, . . . , 10, corresponding to the dynamics (31) withparameters given by (36) and (37), computed via fixed point recursion (27) with tolerance 10−5. The corresponding MVOEsare shown in Fig. 7.

For the setup considered in (28) and (29), the reach set X (t)is guaranteed to be convex, which allows us to transcribe (39)in terms of the support functions:

δH(t) = sups∈Snx−1

∣∣hE(t)(s) − hX (t)(s)∣∣, (40)

where Snx−1 denotes the Euclidean unit sphere embedded inRnx . Thanks to property (iv) in Section II.1, the absolute valuein (40) can be dropped. Furthermore, suppose that both X (t)and E(t) are centered at origin, as in Section VI; in particular,E(t) ≡ E

(0, Q(t)

). Using (2), (6), (29) and (30), we can

then rewrite (40) as

δH(t)

= sups∈Snx−1

(s>Q(t)s

)12 −

{(m∑

i=1

(s>F tQ0i

(F>)ts)p1

2

)1p1

+

t−1∑

k=0

n∑

j=1

(s>F t−k−1GUj(t)G

> (F>)t−k−1

s)p2

2

1p2}.

(41)

In words, (41) is the two-sided Hausdorff distance betweenthe reach set of (28) and its outer ellipsoidal approximation attime t, provided X0 is the p1-sum of m centered ellipsoids,

Fig. 8: Computational times for the MVOE calculations in Fig.7 and Table II using recursion (27).

and U(t) is the p2-sum of n time-varying centered ellipsoids,where p1, p2 ≥ 1.

To illustrate the use of (41), let us consider m = n = 1 asin Section VI.A. In this case, (41) can be expressed succinctlyas

δH(t) = sups∈Snx−1

∥∥Q 12 (t)s

∥∥2−

t∑

k=0

∥∥M12

k (t)s∥∥2, (42)

where Mk := F t−k−1GU(t)G>(F>)t−k−1 ∈ Snx

+ for k =

0, 1, . . . , t− 1, and Mt := F tQ0

(F>)t ∈ Snx

+ . From (42), asimple upper bound for δH(t) follows.

Proposition 1. (Upper bound for δH(t) when m = n = 1)

δH(t) ≤∥∥∥∥Q

12 (t)−

t∑

k=0

M12

k (t)

∥∥∥∥2

. (43)

Proof. For s ∈ Snx−1 , by the (repeated use of) reversetriangle inequality, we have

∥∥Q 12 (t)s

∥∥2−

t∑

k=0

∥∥M12

k (t)s∥∥2≤∥∥∥∥∥

(Q

12 (t)−

t∑

k=0

M12

k (t)

)s

∥∥∥∥∥2

≤∥∥∥∥Q

12 (t)−

t∑

k=0

M12

k (t)

∥∥∥∥2

,

where the last step follows from the sub-multiplicative prop-erty of the matrix 2-norm. Since this holds for any s ∈ Snx−1,the same holds for the optimal s in (42). Hence the result.

We can use the bound derived in Proposition 1 as aconservative numerical estimate for δH(t). In Fig. 9, we usethe data from Section VI-A to plot the upper bound (43)for the Hausdorff distance between the reach set of (28) andits MVOE approximations. The plot shows that the MVOEscomputed using the proposed algorithms result in a lowerupper bound than the MVOE computed using the standard

1 2 3 4 5 6 7 8 9 10

Physical time step t

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Upper

bou

nd

for

Hau

sdor

↵m

etri

c� H

(t)

Fig. 9: The upper bound (43) for the Hausdorff distanceδH(t) between the reach set of (28) subject to ellipsoidaluncertainties (the case m = n = 1), and its various MVOEapproximations. The data for this plot are same as in SectionVI-A, and correspond to Fig. 5, 6 and Table I. The MVOEfrom the SDP computation ( ) results in a larger upperbound than the MVOEs computed using the fixed point recur-sion ( ) proposed herein, and the root-bracketing algorithm( ) proposed in [23].

SDP approach. This indicates that the proposed MVOEsapproximate the reach set quite well compared to the SDPapproach, even though their volumes are slightly larger thanthe SDP MVOEs, as noted in Table I.

D. Computational Complexity

The runtime for computing the approximate MVOE for thep-sum of a given number of summand ellipsoids using theproposed fixed point recursion (27) scales as O

(d3)

wheneach of the (non-degenerate) summand ellipsoids are in Rd.This is because the fixed point recursion needs the generalizedeigenvalues {λi}di=1 of the matrix pencil formed by a pair ofshape matrices associated with the summand ellipsoid pair – atask that takes O

(d3)

worst-case runtime, and dominates theremaining O(d) computation. In particular, the computationat time t in Section VI-A requires finding the approximateMVOE of the Minkowski sum of t+ 1 summand ellipsoids inRnx , resulting in O

(tn3x)

runtime. Likewise, the computationat time t in Section VI-B requires finding the approximateMVOE of the Minkowski sum of t+1 summand convex sets inRnx , where one of the summand is the p1-sum of m ellipsoids,and each of the remaining t convex sets are the p2-sum of nellipsoids, thus resulting in O

((m+ (t− 1)n)n3x

)runtime.

This can be interpreted as follows. Since m and n encodethe complexity of the set-valued initial condition and controluncertainties, the worst-case runtime scales linearly with thecomplexity of the set-valued uncertainty description, and cubicin state dimension.

VII. CONCLUSIONS

Computing a tight ellipsoidal outer approximation of convexset-valued data is necessary in many systems-control problemssuch as reachability analysis, especially when the convex set isdescribed as set operations on other ellipsoids. Depending onthe set operation, isolated results and algorithms are known inthe literature to compute the minimum trace and minimumvolume outer ellipsoidal approximations. In this paper, weunify such results by considering the p-sum of ellipsoids,which for different p ∈ [1,∞], generate different convex setsfrom the summand ellipsoids. Our analytical results lead toefficient numerical algorithms, which are illustrated by thereach set computation for a discrete-time linear control system.A specific direction of future study is to extend the reach setcomputation for hybrid systems, by computing the intersectionof the guard sets with p-sum of ellipsoids. Comparing theproposed algorithms with existing reachability computationtools on benchmark problems will be pursued in our follow-upwork. We hope that the models and methodologies presentedhere, will help to expand the ellipsoidal calculus tools [17],[18], [20], [32], [33], [42] for systems-control applications, andmotivate further studies to leverage the reported computationalbenefits in real-time applications.

ACKNOWLEDGEMENT

The author is grateful to Suvrit Sra for suggesting [41] thefixed point iteration (27) for p = 1. This research was partiallysupported by a 2018 Faculty Research Grant awarded by theCommittee on Research from the University of California,Santa Cruz, and by a 2018 Seed Fund Award from CITRISand the Banatao Institute at the University of California. Theauthor is indebted to Lia Gianfortone for discussions and helpwith preliminary numerical investigation.

APPENDIX

A. Set Inclusion for the p-Sum

In the following, we formally state and prove the setinclusion relation for the p-sum.

Proposition 2. Given compact convex sets K1,K2, we haveK1 +q K2 ⊆ K1 +p K2 for 1 ≤ p < q <∞.

Proof. We proceed in two steps: first, we show thathK1+qK2

≤ hK1+pK2, 1 ≤ p < q < ∞ holds; second, we

show that for any two compact convex sets A,B, the supportfunction inequality hA ≤ hB is equivalent to A ⊆ B.

From Definition 1, showing the first step amounts to proving((hK1(x))

q+ (hK2(x))

q)1/q ≤ ((hK1(x))

p+ (hK2(x))

p)1/p

for 1 ≤ p < q < ∞, which clearly holds from monotonicity.For the second step, we recall that the Legendre-Fenchelconjugate of a function f(x), denoted as f∗(y), is given by

f∗(y) := supx〈y,x〉 − f(x).

As a consequence of the above definition, f ≥ g ⇒ f∗ ≤ g∗.Furthermore, recall the well-known fact that for a proper,closed, convex function f , its bi-conjugate f∗∗ = f . Now,notice that for any two compact convex sets A,B, the support

Figure 4.3: Overlaid analytical (outlined, no fill) and approximated (filled, no outline)

solutions for reach sets of the double integrator (4.1).

grator at di�erent times t 2 [0, 10]. As designed, solutions generated by the ellipsoidal

algorithm always outwardly bound the actual reach set.

For a quantitative measure of the performance of the ellipsoidal algorithm, we

compare the areas of the analytical reach sets,

area(X [t]) = 2/3 µ2t3, (4.4)

calculated in Appendix A, with the areas of the ellipsoidal over-approximations, cal-

culated with Eqn. (2.2) as

area(E(q, Q)) = ⇡p

det (Q). (4.5)

On the left of Figure 4.4 is a plot of the ratios between the areas of the actual

and simulated reach sets for several reachable tube approximations, each computed

27

Figure 4.1: Left: The analytical reach set of the double integrator system at t = 1

with control |u| 2 and initial state (x01, x

02) = (0, 0). Right: A discretization of the

analytical reachable tube of the double integrator with control |u| .3 and initial

state (x01, x

02) = (0, 0). The tube describes the reachable states over 60s, evaluated at

single-second intervals.

Fig. 4.1(Left)). The parametric equation for the upper

x+1 (t) = x0

1 + x02t� µ(t2/2� �2) x+

2 (t) = x02 � 2µ� � µt (4.2)

and the same for the lower boundary is given by

x�1 (t) = x0

1 + x02t + µ(t2/2� �2), x�

2 (t) = x02 + 2µ� + µt, (4.3)

with the parameter � constrained to the interval [�t, 0] [13, p. 111]. A reach set

defined by these curves is shown on the left in Figure 4.1. The union of these reach

sets over an interval of time is the reachable set eX for the system. A wireframe plot

for the three-dimensional set eX is shown on the right in Figure 4.1.

Slight reformulation of the system is necessary to approximate the reachable tube

of the system with the proposed ellipsoidal algorithm. The scalar interval |u| µ

25

Fig. 10: Left: The reach set for the double integrator x = u at t = 1 with initial condition

(x10, x20) ⌘ (0, 0), and control bound: |u| 2. Right: 33 different snapshots of the reach sets

(black solid outlines) for the double integrator with |u| 2, t 2 [0, 10], superimposed with their

respective approximate MVOEs (filled, no outline) at those times. For the MVOE computation,

the initial set-valued ellipsoidal uncertainty is taken as E (0, 10�4I2), where 0 denotes the 2⇥ 1

zero vector, and I2 is the 2⇥2 identity matrix. This small ellipsoidal initial condition uncertainty

serves as a proxy for the fixed initial condition (0, 0) used in plotting the true reach sets (solid

outlines), and allows using the proposed fixed point algorithm.

The same for the lower boundary (x�1 (t), x�

2 (t)) is

x�1 (t) = x10 + x20t� µ

✓t2

2� �2

◆, (45a)

x�2 (t) = x20 � 2µ� � µt. (45b)

These analytical expressions not only help in visually confirming the performance of the

proposed algorithm, but also allow us to quantify the ratio vol(EMVOE(t))/vol(X (t)), where

the numerator equals ⇡p

det(Q(t)), the matrix Q(t) being the 2⇥2 shape matrix obtained from

our MVOE algorithm at time t, and the denominator has closed-form expression 2µ2t3/3, where

|u| µ. We found the said volume ratio to be � 85% for all 33 snapshots in Fig. 10.

In general, the true reach set is not available, and one resorts to indirect methods (see Section

VI.C) to assess the quality of the outer-approximation.

REFERENCES

[1] Y-K. Choi, J-W. Chang, W. Wang, M-S. Kim, and G. Elber, “Continuous Collision Detection for Ellipsoids”, IEEE

Transactions on Visualization and Computer Graphics, Vol. 15, No. 2, pp. 311–325, 2009.

Fig. 10: Left: The reach set for the double integrator x = uat t = 1 with initial condition (x10, x20) ≡ (0, 0), and controlbound: |u| ≤ 2. Right: 33 different snapshots of the reach sets(black solid outlines) for the double integrator with |u| ≤ 2,t ∈ [0, 10], superimposed with their respective approximateMVOEs (filled, no outline) at those times. For the MVOEcomputation, the initial set-valued ellipsoidal uncertainty istaken as E

(0, 10−4I2

), where 0 denotes the 2×1 zero vector,

and I2 is the 2 × 2 identity matrix. This small ellipsoidalinitial condition uncertainty serves as a proxy for the fixedinitial condition (0, 0) used in plotting the true reach sets (solidoutlines), and allows using the proposed fixed point algorithm.

functions hA, hB are the conjugates of the respective setindicator functions 1A,1B, i.e., hA = 1∗A, and hB = 1∗B.Here,

1A(x) :=

{0 if x ∈ A,+∞ otherwise.

The set indicator function 1B(x) is defined likewise. Thus,

A ⊆ B ⇒ 1A ≥ 1B ⇒ 1∗A ≤ 1∗B ⇒ hA ≤ hB.Conversely,

hA ≤ hB ⇒ h∗A ≥ h∗B ⇒ 1A ≥ 1B ⇒ A ⊆ B.This proves the second step, i.e., A ⊆ B ⇔ hA ≤ hB. Weconclude the proof by settingA ≡ K1+qK2 and B ≡ K1+pK2

for 1 ≤ p < q <∞, and then combining the two steps.

B. Approximate MVOE for the Double IntegratorWe performed the approximate MVOE computation for the

reach set of the double integrator x = u with bounded control|u| ≤ µ, µ > 0, using the proposed Algorithm 2. Fig. 10shows that the resulting approximate MVOEs obtained via theproposed algorithm provide tight outer-approximations of thetrue reach sets. This example is special in the sense that thetrue reach set starting from (x10, x20) ≡ (0, 0) with the controlbound |u| ≤ µ can be analytically computed (see e.g., [20, p.111]) as follows. Let the parameter σ ∈ [−t, 0]. the parametricequation for the upper boundary

(x+1 (t), x+2 (t)

)is

x+1 (t) = x10 + x20t+ µ

(t2

2− σ2

), (44a)

x+2 (t) = x20 + 2µσ + µt. (44b)

The same for the lower boundary(x−1 (t), x−2 (t)

)is

x−1 (t) = x10 + x20t− µ(t2

2− σ2

), (45a)

x−2 (t) = x20 − 2µσ − µt. (45b)

Fig. 11: The computational times for the MVOE calculationsreported in Appendix C using the proposed Algorithm 2.

These analytical expressions not only help in visually con-firming the performance of the proposed algorithm, but also al-low us to quantify the ratio vol (EMVOE(t)) /vol (X (t)), wherethe numerator equals π

√det (Q(t)), the matrix Q(t) being

the 2 × 2 shape matrix obtained from our MVOE algorithmat time t, and the denominator has closed-form expression2µ2t3/3, where |u| ≤ µ. We found the said volume ratio tobe ≥ 85% for all 33 snapshots in Fig. 10.

In general, the true reach set is not available, and one resortsto indirect methods (see Section VI-C) to assess the qualityof the outer-approximation.

C. Large Scale Example of Approximate MVOE Computation

To illustrate the scalability of the proposed algorithm, wenow provide a large scale numerical example for MVOE com-putation. Specifically, we consider a continuous time lineartime invariant (LTI) model of the first assembly stage of theInternational Space Station (ISS), also known as the Russianservice module 1R or ISS-I model [43]–[45]. The modelconsists of 270 states and 3 controls; the model data are avail-able at http://verivital.com/hyst/benchmark-large-scale/. Fromthis model data, we generate the corresponding discrete-timeLTI system matrices F (of size 270 × 270) and G (of size270 × 3) with sampling time h = 0.05 seconds. As inSection VI.A, we consider m = n = 1, for which the reachstate X (t) ⊂ R270 is of the form (32) for the ellipsoidaluncertainties in initial condition modeled via the shape matrixQ0, and the (time-varying) ellipsoidal actuation uncertaintiesmodeled via the shape matrices U(t). We set Q0 = I270, andU(t) = (1 + cos2(t))diag (rand(3, 1)), that is, time-varyingpositive scaling of random positive diagonal matrices. We usedAlgorithm 2 to compute the approximate MVOEs of X (t)for each t = 1, 2, . . . , 100. As in Section VI.A, this for anygiven t, amounts to computing the MVOE of the 1-sum oft+ 1 summand ellipsoids. The resulting computational timesare shown in Fig. 11.

REFERENCES

[1] Y-K. Choi, J-W. Chang, W. Wang, M-S. Kim, and G. Elber, “ContinuousCollision Detection for Ellipsoids”, IEEE Transactions on Visualizationand Computer Graphics, Vol. 15, No. 2, pp. 311–325, 2009.

[2] A. Best, S. Narang, D. Manocha, “Real-time Reciprocal CollisionAvoidance with Elliptical Agents”, 2016 IEEE International Conferenceon Robotics and Automation (ICRA), pp. 298–305, 2016.

[3] Y. Yan, Q. Ma, G.S. Chirikjian, “Path Planning Based on Closed-formCharacterization of Collision-free Configuration-spaces for EllipsoidalBodies, Obstacles, and Environments”, Proceedings of the 1st Interna-tional Workshop on Robot Learning and Planning in Conjunction with2016 Robotics: Science and Systems, pp. 13–19, 2016.

[4] F.C. Schweppe, “Recursive State Estimation: Unknown but BoundedErrors and System Inputs”, IEEE Transactions on Automatic Control,Vol. 13, No. 1, pp. 22–28, 1968.

[5] D. Bertsekas, and I. Rhodes, “Recursive State Estimation for A Set-Membership Description of Uncertainty”, IEEE Transactions on Auto-matic Control, Vol. 16, No. 2, pp. 117–128, 1971.

[6] F.M. Schlaepfer, and F.C. Schweppe, “Continuous-time State EstimationUnder Disturbances Bounded by Convex Sets”, IEEE Transactions onAutomatic Control, Vol. 17, No. 2, pp. 197–205, 1972.

[7] Y.N. Reshetnyak, “Summation of Ellipsoids in the Guaranteed Estima-tion Problem”, Journal of Applied Mathematics and Mechanics, Vol. 53,No. 2, pp. 193–197, 1989.

[8] E. Fogel, “System Identification via Membership Set Constraints withEnergy Constrained Noise”, IEEE Transactions on Automatic Control,Vol. 24, No. 5, pp. 752–758, 1979.

[9] G. Belforte, B. Bona, and V. Cerone, “Parameter Estimation Algorithmsfor A Set-membership Description of Uncertainty”, Automatica, Vol. 26,No. 5, pp. 887–898, 1990.

[10] R.L. Kosut, M.K. Lau, S.P. Boyd, “Set-membership Identification ofSystems with Parametric and Nonparametric Uncertainty”, IEEE Trans-actions on Automatic Control, Vol. 37, No. 7, pp. 929–941, 1992.

[11] A.B Kurzhanski, and P. Varaiya, “Optimization of Output FeedbackControl under Set-membership Uncertainty”, Journal of OptimizationTheory and Applications, Vol. 151, No. 1, pp. 11–32, 2011.

[12] D. Angeli, A. Casavola, G. Franze, and E. Mosca, “An EllipsoidalOff-line MPC Scheme for Uncertain Polytopic Discrete-time Systems”,Automatica, Vol. 44, No. 12, pp. 3113–3119, 2008.

[13] WM.J. Firey, “p-Means of Convex Bodies”. Mathematica Scandinavica,Vol. 10, pp. 17–24, 1962.

[14] E. Lutwak, “The Brunn-Minkowski-Firey Theory I: Mixed Volumes andthe Minkowski Problem”. Journal of Differential Geometry, Vol. 38, no.1, pp. 131–150, 1993.

[15] E. Lutwak, “The Brunn-Minkowski-Firey Theory II: Affine and Geo-minimal Surface Areas”. Advances in Mathematics, Vol. 118, no. 2, pp.244–294, 1996.

[16] R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, Encyclope-dia of Mathematics and Its Applications, No. 151, Cambridge UniversityPress, 2014.

[17] A.B. Kurzhanski, and I. Valyi, Ellipsoidal Calculus for Estimation andControl, Systems & Control: Foundations and Applications, BirkhauserBoston and International Institute for Applied Systems Analysis, 1997.

[18] F.C. Schweppe, Uncertain Dynamic Systems. Prentice Hall, EnglewoodCliffs, New Jersey, 1973.

[19] D.G. Maksarov, and J.P. Norton, “State Bounding with Ellipsoidal SetDescription of the Uncertainty”, International Journal of Control, Vol.65, No. 5, pp. 847–866, 1996.

[20] A.B. Kurzhanski, and P. Varaiya, Dynamics and Control of TrajectoryTubes. Theory and Computation. Systems & Control: Foundations &Applications. Springer, 2014.

[21] R.T. Rockafellar, Convex Analysis. Princeton Landmarks in Mathemat-ics, Princeton University Press, 1970.

[22] E. Lutwak, D. Yang, and G. Zhang, “The Brunn-Minkowski-FireyInequality for Nonconvex Sets”, Advances in Applied Mathematics, Vol.48, No. 2, pp. 407–413, 2012.

[23] A. Halder, “On the Parameterized Computation of Minimum VolumeOuter Ellipsoid of Minkowski Sum of Ellipsoids”, 2018 IEEE Confer-ence on Decision and Control, available online:https://arxiv.org/pdf/1803.08157.pdf, 2018.

[24] G.H. Hardy, J.E. Littlewood, and G. Polya, Inequalities. 2nd ed. Cam-bridge, United Kingdom: Cambridge University Press, 1988.

[25] B. Lemmens, and R. Nussbaum, Nonlinear Perron-Frobenius Theory.Cambridge Tracts in Mathematics, Vol. 189, Cambridge UniversityPress, 2012.

[26] S. Boyd, and L. Vandenberghe, Convex Optimization, Cambridge Uni-versity Press, 2004.

[27] U. Krause, Positive Dynamical Systems in Discrete Time: Theory,Models, and Applications. Studies in Mathematics, Vol. 62, Walter deGruyter GmbH & Co KG, 2015.

[28] F.L. Chernousko, “Optimal Guaranteed Estimates of Indeterminacieswith the Aid of Ellipsoids. I”, Engineering Cybernetics, Vol. 18, No.3, pp. 1–9, 1980.

[29] C. Durieu, E. Walter, and B. Polyak, “Multi-Input Multi-Output Ellip-soidal State Bounding”, Journal of Optimization Theory and Applica-tions, Vol. 111, No. 2, pp. 273–303, 2001.

[30] F.L. Chernousko, and A.I. Ovseevich, “Properties of the Optimal Ellip-soids Approximating the Reachable Sets of Uncertain Systems”, Journalof Optimization Theory and Applications, Vol. 120, No. 2, pp. 223–246,2004.

[31] A.A. Kurzhanskiy, and P. Varaiya, “Ellipsoidal Toolbox (ET)”, 45thIEEE Conference on Decision and Control, pp. 1498–1503, 2006.

[32] A.A. Kurzhanskiy, and P. Varaiya, “Ellipsoidal Techniques for Reacha-bility Analysis of Discrete-time Linear Systems”, IEEE Transactions onAutomatic Control, Vol. 52, No. 1, pp. 26–38, 2007.

[33] P.V. Gagarinov, “Computation of Alternated Reachability Tubes forLinear Control Systems Under Uncertainty”, Moscow University Com-putational Mathematics and Cybernetics, Vol. 36, No. 4, pp. 169–177,2012.

[34] H.S. Witsenhausen, “Sets of Possible States of Linear Systems GivenPerturbed Observations”, IEEE Transactions on Automatic Control, Vol.13, No. 5, pp. 556–558, 1968.

[35] A. Girard, and C. Le Guernic, “Efficient Reachability Analysis forLinear Systems using Support Functions”, IFAC Proceedings Volumes,Vol. 41, No. 2, pp. 8966–8971, 2008.

[36] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory, SIAM Studies in Applied

Mathematics, Vol. 15, 1994.[37] M. Grant, and S. Boyd. CVX: Matlab Software for Disciplined Convex

Programming, version 2.0 beta. http://cvxr.com/cvx, September2013.

[38] F. John, “Extremum Problems with Inequalities as Subsidiary Condi-tions”. In Studies and Essays presented to R. Courant on his 60thBirthday, pp. 187–204, Interscience Publishers, 1948.

[39] H. Busemann, “The Foundations of Minkowskian Geometry”. Commen-tarii Mathematici Helvetici, Vol. 24, No. 1, pp. 156–187, 1950.

[40] A. Ben-Tal, and A. Nemirovski, Lectures on Modern Convex Optimiza-tion: Analysis, Algorithms, and Engineering Applications. MPS-SIAMSeries on Optimization, Society for Industrial and Applied Mathematics,2001.

[41] S. Sra (https://mathoverflow.net/users/8430/suvrit),Positive root of a polynomial, URL (version: 2018-03-23): https://mathoverflow.net/q/260067

[42] L. Ros, A. Sabater, and F. Thomas, “An Ellipsoidal Calculus Basedon Propagation and Fusion”, IEEE Transactions on Systems, Man, andCybernetics, Part B (Cybernetics), Vol. 32, No. 4, pp. 430–442, 2002.

[43] S. Gugercin, A.C. Antoulas, and N. Bedrossian, “Approximation of theInternational Space Station 1R and 12A Flex Models”. Proceedings ofthe 40th IEEE Conference on Decision and Control (CDC), Vol. 2, pp.1515–1516, 2001.

[44] Y. Chahlaoui, and P. Van Dooren, “Benchmark Examples for ModelReduction of Linear Time-Invariant Dynamical Systems”. in P. Benner,D.C. Sorensen, V. Mehrmann (eds), Dimension Reduction of Large-ScaleSystems, Lecture Notes in Computational Science and Engineering,Springer, Berlin, Heidelberg, Vol. 45, pp. 379–392, 2005.

[45] H-D. Tran, L.V. Nguyen, and T.T. Johnson, “Large-scale Linear Systemsfrom Order-Reduction (Benchmark Proposal)”. Proceedings of the 3rdApplied Verification for Continuous and Hybrid Systems Workshop(ARCH), Vienna, Austria, pp. 60–67, 2016.