set-theoretic approaches in analysis, estimation and

Set-Theoretic Approaches in

Analysis, Estimation and Control

of Nonlinear Systems

Benoit Chachuat ∗ Boris Houska ∗∗∗ Radoslav Paulen ∗∗

Nikola Peric ∗ Jai Rajyaguru ∗ Mario E. Villanueva ∗

∗ Centre for Process Systems Engineering, Department of ChemicalEngineering, Imperial College London, South Kensington

Campus,London SW7 2AZ, UK.∗∗ Process Dynamics and Operations Group, Department of

Biochemical and Chemical Engineering, Technische UniversitatDortmund, Emil-Figge-Str. 70, 44221 Dortmund, Germany.

∗∗∗ School of Information Science and Technology, ShanghaiTechUniversity, 319 Yueyang Road, Shanghai 200031, China.

Abstract: This paper gives an overview of recent developments in set-theoretic methods fornonlinear systems, with a particular focus on the activities in our own research group. Centralto these approaches is the ability to compute tight enclosures of the range of multivariatesystems, e.g. using ellipsoidal calculus or higher-order inclusion techniques based on multivariatepolynomials, as well as the ability to propagate these enclosures to enclose the trajectories ofparametric or uncertain differential equations. We illustrate these developments with a range ofapplications, including the reachability analysis of nonlinear dynamic systems; the determinationof all equilibrium points and bifurcations in a given state-space domain; and the solution ofset-membership parameter estimation problems. We close the paper with a discussion abouton-going research in tube-based methods for robust model predictive control.

Keywords: affine-set parameterization; differential equations; reachable set; bifurcationanalysis; set-membership estimation

1. INTRODUCTION

Many engineering design and control problems can beformulated, analyzed or solved in a set-theoretic frame-work. By set-theoretic we refer here to any method whichexploits properties of suitably chosen sets or constructedsets in the state space (Blanchini and Miani, 2008). Indesigning a control system for instance, the constraints,uncertainties and design specifications all together aredescribed naturally in terms of sets; and in measuringthe effect of a disturbance on a system’s response or inbounding the error of an estimation algorithm likewise,sets play a central role. A number of key set-theoreticconcepts have been proposed in the early 1970s, but theirsystematic applications were not possible until enoughcomputational capability became widely available.

Today, many such methods and tools are available forthe estimation and control of linear systems – Witnessfor instance the popularity of Matlab’s Multi-ParametricToolbox (Herceg et al., 2013). To name but a few features,they support the construction and a variety of opera-tions on convex sets, the synthesis and implementationof explicit model predictive control (MPC) for linear timeinvariant or piecewise affine systems, the construction ofmaximal invariant sets or Lyapunov functions for piecewiseaffine systems, etc (see, e.g., Kurzhanski and Valyi, 1997;Blanchini and Miani, 2008).

The focus in this paper is on set-theoretic methods fornonlinear systems which can offer computational guaran-tees or certificates, with a strong emphasis on recent de-velopments in our own research group. As such, the papermay not be considered a review or survey paper, althougheffort is made to point the reader to relevant, alternativeapproaches or related work throughout. Despite enormousprogress in recent years, set-theoretic methods and toolsfor nonlinear systems are not as developed as their linearcounterparts, and they still constitute a widely open fieldof research when it comes to computational efficiency;see, e.g., Streif et al. (2013) for a recent survey (withapplications to biochemical networks).

A key enabler for set-theoretic methods is the ability toenclose the range of nonlinear multivariate systems, andthe class of factorable functions—namely, those functionswhich can be represented by means of a finite compu-tational graph—has attracted much attention. Since theinvention of interval analysis by Moore more than 50 yearsago, many computational techniques have been developedto construct tight enclosures for the range of factorablefunctions. The focus in the first part of the paper (Sect. 2)is on so-called affine set-parameterizations, which encom-pass both convex and nonconvex set parameterizations.Among the alternatives to enclose the range of functionsor sets defined by equalities and/or inequalities, mentionshould be made of the sum-of-squares (SOS) approaches

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which provide nested sequences outer-approximations ashierarchies of linear matrix inequality (LMI) relaxations(Lasserre, 2009); see also Parrilo (2003).

Questions of reachability and invariance have been studiedextensively in the dynamics and control literature due totheir wide range of applications. Direct characterizationof reachability concepts is one of the topics addressed byviability theory (Aubin, 1991), and the development ofcomputational tools supported by these theories is an on-going effort. In Sect. 3, we describe a number of directapproaches that rely on the discrete- or continuous-timepropagation of affine set-parameterizations. Similar to thefactorable case above, alternative approaches recently de-veloped involve constructing outer-approximations of thereachable set of (possible controlled and constrained) dy-namic systems as hierarchies of linear matrix inequality(LMI) relaxations (Henrion and Korda, 2014). Moreover,so-called indirect approaches have also been developed,which formulate reachability questions as optimal con-trol (or game theory) problems and determine reachable-set outer-approximations by Hamilton-Jacobi projections(Mitchell and Tomlin, 2003; Lygeros, 2004). Related toreachability analysis, the problems of computing the regionof attraction of a target set or the maximum invariant setare also essential and long-standing challenges in dynamicsystem and control theory (Henrion and Korda, 2014;Korda et al., 2014).

A great variety of algorithms, including complete searchmethods for problems in global optimization, constraintsatisfaction or robust estimation, hinge on the ability toenclose the range of functions or the reachable set ofdynamic systems. Our focus in the second part of thepaper (Sect. 4) is more specifically on two selected prob-lems: (i) the determination of all equilibrium points andbifurcations of a nonlinear dynamic system (Monnigmannand Marquardt, 2002; Waldherr and Allgower, 2011);and (ii) the solution of set-membership parameter esti-mation problems in dynamic systems (Walter and Piet-Lahanier, 1990). Finally, we conclude the paper with adiscussion about the application of set-theoretic methodsin tube-based methods for robust model predictive control(Sect. 5).

1.1 Case Study Problem Definition

The modeling of bioprocesses often gives rise to challengingdynamic systems, whereby differential equations describ-ing species mass balances in the system are coupled withalgebraic equations describing charge balance or other fastphenomena that are assumed to be at equilibrium (orquasi steady-state). Throughout this paper, we considera two-reaction model of anaerobic digestion inspired from(Bernard et al., 2001)—All the parameter values are re-ported in Table A.1 (Appendix) for the sake of repro-ducibility.

The model involves an acetogenesis step, where organiccompounds (S1) are converted into VFA (S2), followed bya methanogenic step; these steps are associated with thebacterial populations X1 and X2, respectively:

• Acetogenesis: k1 S1µ1(·)X1−→ X1 + k2 S2 + k4 CO2

• Methanogenesis: k3 S2µ2(·)X2−→ X2 + k5 CO2 + k6 CH4

The biological kinetics for the reactions are:

µ1(S1) = µ1S1

S1 +KS1, µ2(S2) = µ2

S2

S2 +KS2 +S22

KI2

.

Under the assumptions that the liquid phase is perfectlymixed and that only a fraction α of the biomass is notattached onto a support inside the digester, the speciesbalance equations for the state variables S1, X1, S2, X2,inorganic carbon (C) and alkalinity Z are given by:

S1 = D(Sin1 − S1)− k1µ1(S1)X1 (1)

X1 = (µ1(S1)− αD)X1 (2)

S2 = D(Sin2 − S2) + k2µ1(S1)X1 − k3µ2(S2)X2 (3)

X2 = (µ2(S2)− αD)X2 (4)

Z = D(Z in − Z) (5)

C = D(C in − C)− qCO2 + k4µ1(S1)X1 (6)

+ k5µ2(S2)X2 ,

with D, the dilution rate; and Sin1 , Sin

2 , C in and Z in,the inlet concentrations. Note that these balances neglectgaseous emissions other than CO2 and methane. Underthe assumption that the pH range is between 6-8 (normaloperation), a charge-balance equation gives:

CO2aq = C + S2 − Z .

Finally, assuming that the partial pressures of CO2 (PCO2)and methane (PCH4 ) quickly reach equilibrium and thatthe gas phase behaves ideally, we have:

Ptot − PCO2

qCH4

=PCO2

qCO2

, (7)

where the liquid-gas transfer rate of CO2 and methane aregiven by:

qCH4 = α11µ2(S2)X2

qCO2 = kLa(

CO2aq −KHPCO2

)

with KH, Henry’s constant for CO2; and kLa, the liquid-gas transfer coefficient.

Notice that (7) leads to a quadratic equation in PCO2 ,and therefore the model (1)-(7) comes in the form of a(semi-explicit index-1) DAE system. Nonetheless, (7) hasa unique nonnegative root in the form:

PCO2 =φCO2 −

√

φ2CO2

− 4KHPtCO2aq

2KH

with: φCO2 := CO2aq +KHPt +k6kLa

µ2(S2)X2,

which can be used to formulate an equivalent ODE system.

Overall, this anaerobic digestion model is challenging as itfeatures complex dynamics due to pH self-regulation andliquid-gas transfer. Moreover, the processes span multipletime scales, with fast dynamics acting on a time-scale ofminutes/hours, and slow dynamics acting on a time-scaleof days.

1.2 Notation

The set of compact subsets of Rn is denoted by Kn, andthe subset of compact convex subsets of Kn, by Kn

C. The

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diameter diam (Z) of a set Z ∈ Kn is defined as

diam (Z) := maxz,z′∈Z

‖z − z′‖ ,

for any given norm on Rn, and the support functionV [Z] : Rn → R of Z as

∀c ∈ Rn , V [Z](c) :=max

z

cTz | z ∈ Z

. (8)

The Minkowski sum W ⊕ Z, the Minkowski differenceW ⊖Z and the Haussdorf distance dH(W,Z) between twocompact sets W,Z ∈ Kn are given by

W ⊕ Z := w + z | w ∈ W, z ∈ Z ,

W ⊖ Z := w − z | w ∈ W, z ∈ Z ,

dH(W,Z) := max

maxw∈W

minz∈Z

‖w − z‖ ,maxz∈Z

minw∈W

‖w − z‖

.

In particular, if W ⊆ Z we have

dH(W,Z) = maxz∈Z

minw∈W

‖w − z‖ .

The set of n-dimensional interval vectors is denoted byIR

n. The midpoint and radius of an interval vector P :=[

pL, pU]

∈ IRn are defined as mid(P ) := 1

2 (pU + pL) and

rad(P ) := 12 (p

U − pL), respectively. The n-by-n matrixdiag rad (P ) ∈ Rn×n is a diagonal matrix whose elementsare the components of rad(P ).

The set of n-dimensional positive semi-definite symmetricmatrices is denoted by Sn+. An ellipsoid with shape matrixS ∈ Sn+ and centered at the origin is denoted by

E(S) :=

S12 v | v ∈ R

n, vTv ≤ 1

.

2. THE BUILDING BLOCKS

Central to set-theoretic approaches described in thispaper is the selection of parameterizations that candescribe/approximate compact subset in Rn. To keepour considerations general, we consider affine set-parameterizations (Houska et al., 2014), a particular classof computer-representable sets in the form (Em,Dn,m)such that:

∀Q ∈ Dn,m, ImEm(Q) := Q [ b 1 ]T | b ∈ Em ,

with Em ⊆ Rm, m ≥ 1, the so-called basis set; andDn,m ⊆ Rn×(m+1), n ≥ 1, associated domain set.

Usual convex sets such as intervals, ellipsoids or polytopescan be characterized using affine set-parameterizationswith convex basis sets—see top row of Fig. 1. In particular,the parameterization (Eball

m ,Rn×(m+1)) with Eballm := ξ ∈

Rm | ‖ξ‖2 ≤ 1 describes ellipsoids in Rn.

Nonconvex sets too can be represented in terms ofaffine set-parameterizations. For instance, the affine set-

parameterization (Epol(q)m ,Rn×α(q)

m ) with nonconvex basis

set Epol(q)m := P

(q)m (ξ) | ξ ∈ [−1, 1]m, where P

(q)m (ξ) ∈

Rα(q)m is the vector containing the first α

(q)m monomials in

ξ in lexicographic order,

P (q)m (ξ) := 1, ξ1, . . . , ξm, ξ21 , ξ1ξ2, . . . , ξ

2m, ξ31 , . . . , ξ

qm ,

describes the image set of qth-order polynomials in m vari-ables. Naturally, alternative polynomial parameterizationsare possible with different polynomial bases, such as the

Chebyshev polynomials of the first kind defined by therecurrence relation

T0(ξ) = 1, T1(ξ) = ξ

Tk+1(ξ) = 2ξTk(ξ)− Tk−1(ξ), k ≥ 1 ,

which leads to:

P (q)m (ξ) := 1, T1(ξ1), . . . , T1(ξm), T2(ξ1), T1(ξ1)T1(ξ2), . . . ,

T2(ξm), T3(ξ1), . . . , Tq(ξm) .

More involved representations of nonconvex sets can beconstructed by combining convex and nonconvex basis setsin turn. Polynomial image sets combined with intervals orellipsoids, for instance, are illustrated in the bottom rowof Fig. 1. These are commonly referred to as Taylor orChebyshev models in the literature—see Sect. 2.2.

An important property of affine set-parameterizationsis invariance under affine transformation, that is, theproperty of a parameterized set’s image under any affinetransformation to be exactly representable on the samebasis set. Among the foregoing examples, the classesof ellipsoids, polytopes and polynomial image sets areall invariant under affine transformation, and so is anyfinite combination of these parameterizations. In contrast,the class of interval boxes is not invariant under affinetransformation given that the rotation of an interval boxmay yield another interval box whose edges are no longeraligned with the original axes—this is one of the mainsources of the wrapping effect in interval analysis. Itshould also be clear that any affine set-parameterizationobtained from the combination with interval boxes willfail to be invariant under affine transformation, includingTaylor/Chebyshev models with interval remainder terms.

2.1 Affine Set-Parameterization Extensions

Factorable functions cover an extremely inclusive class offunctions which can be represented finitely on a computerby means of a code list or a computational graph involvingatom operations. These are typically unary and binaryoperations within a library of atom operators, which canbe based for example on the C-code library math.h.Natural interval extensions and their variants (Mooreet al., 2009) were among the first techniques developed forbounding the range of factorable functions. The conceptof interval extension in interval analysis extends readily toaffine set-parameterizations.

Given a function g : Rn → Rp and two affine set-parameterizations (Em,Dn,m) and (Em,Dp,m), we call thefunction gEm : Dn,m → Dp,m an Em-extension of g if

∀Q ∈ Dn,m, ImEm

(

gEm(Q))

⊇ g(ImEm(Q)) ,

with g(ImEm(Q)) := g(z) | z ∈ ImEm

(Q) . A keyproperty of an affine set-parameterization extension is howmuch overestimation it carries with respect to the imageset of the original function (Fig. 2). In particular, theextension gEm is said to have Hausdorff convergence orderβ ≥ 1, if

∀Q ∈ Dn,m, dH(

ImEm

(

gEm(Q))

, g(ImEm(Q))

)

∈ O(diam(ImEm(Q))β) . (9)

In general, extensions that have Hausdorff convergenceorder two (or higher) may not exist when the underlyingaffine set-parameterizations is not invariant under affinetransformation.



Interval Ellipsoid Zonotope

Domain Dintervaln := (diag(w), c) | w ∈ Rm

+ , c ∈ Rm Rn×(m+1) Rn×(m+1)

Basis Eboxm := ξ ∈ Rm | ‖ξ‖∞ ≤ 1 Eball

m := ξ ∈ Rm | ‖ξ‖2 ≤ 1 Eboxm

Polynomial Polynomial⊕ Interval Polynomial⊕Ellipsoid

Domain Rn×α(q)m Rn×α

(q)m × Dinterval

n Rn×(α(q)m +n+1)

Basis Epol(q)m := Ml,q(ξ) | ξ ∈ [−1, 1]m E

pol(q)m × Ebox

m Epol(q)m × Eball

m

Fig. 1. Illustration of affine set parameterizations with convex (top row) and nonconvex (bottom row) basis sets.

g1

g2

g(ImEm(Q))

ImEm

(

gEm(Q))

dH(

ImEm

(

gEm(Q))

, g(ImEm(Q))

)

Fig. 2. Illustration of an extension and the correspondingoverestimation in terms of the Hausdorff distance.

2.2 Practical Construction of High-Order Inclusions

The construction of extensions can be automated for fac-torable functions using a variety of arithmetics, whichcan be conveniently implemented in computer programs.Unlike interval arithmetic, Taylor and Chebyshev modelarithmetics can be used to construct extension functionsthat enjoy higher-order Hausdorff convergence (see Fig. 3).The idea is to propagate the polynomial part (expressedeither in monomial or Chebyshev basis) by symbolic cal-culations wherever possible, and processing the intervalremainder term as well as the higher-order terms accordingto the rules of interval arithmetic.

Given a (q + 1)-times continuously-differentiable functionf : Rn → R on the set P ∈ IR

n, a qth-order Taylor modelof f on P at a point p ∈ P is the pair (Pq

f,P ,Rqf,P ) of a

qth-order multivariate polynomial Pqf,P : Rn → R with an

interval remainder Rqf,P ∈ IR satisfying 1

1 Multi-index notation: A multi-index γ is a vector in Nn, n > 0.The order of γ is |γ| :=

∑n

i=1γi. Given a point p ∈ Rn, pγ is

a shorthand notation for the expression∏n

i=1pγii , and Tγ(p) for

∏n

i=1Tγi (pi). Moreover, ∂γf is a shorthand notation for the partial

derivative ∂|γ|f

∂pγ11

···∂pγnn

.

f

Pqf,P

Pqf,P + rq,Lf,P

Pqf,P + rq,Uf,P

Pqf,P (p) ⊕Rq

f,P

ppL pU

Fig. 3. Illustration of Taylor/Chebyshev models.

∀p ∈ P , f(p)− Pqf,P (p) ∈ Rq

f,P and

Pqf,P (p) =

∑

γ∈Nn,|γ|≤q

∂γf(p)

γ!(p− p)γ .

Likewise, (Pqf,P ,R

qf,P ) is called a qth-order Chebyshev

model of f on P if

∀p ∈ P , f(p)− Pqf,P (p) ∈ Rq

f,P and

Pqf,P (p) =

∑

γ∈Nn,|γ|≤q

aγ(P )Tγ(ps) ,

with aγ(P ) ∈ O(diam (P )|γ|) and psi :=

p−mid(Pi)rad(Pi)

.

Rules for binary sum, binary product and univariate com-position between Taylor models or Chebyshev models havebeen described and analyzed, e.g., in Makino and Berz(2003); Bompadre et al. (2013); Dzetkulic (2014); Ra-jyaguru et al. (2014). As well as enabling the computationof Taylor and Chebyshev models for factorable functions,these rules guarantee high-order convergence of the re-mainder term to zero with the diameter of the parameterhost set P as

Rqf,P ∈ O(diam (P )

q+1) . (10)

Naturally, the same approach applies to vector-valuedfunctions f : Rn → Rp by treating each function compo-



nent separately, also retaining the high-order convergenceproperty (10).

In connection to the affine set-parameterization formalismintroduced in Sect. 2.1, Taylor and Chebyshev model arith-metics support the constructions of extensions for a vector-

valued function f : Rn → Rp in the form fE

pol(q)m ×E

boxm :

Rn×α(q)m × Dinterval

n → Rp×α(q)m × Dinterval

p . In the sense of(9) nonetheless, such extensions may only have Hausdorffconvergence order β = 1 regardless of the polynomialorder q, since the underlying affine set-parameterizationis not invariant under affine transformation. Note thatthis result is not in contradiction with (10), which onlyguarantees (q+1)th-order convergence for extensions from

(Epol(q)m ,Rn×α(q)

m ) into (Epol(q)m × Eball

m ,Rp×α(q)m × Dinterval

p ).

A procedure that makes use of Taylor/Chebyshev modelarithmetic for the construction of extensions that arequadratically convergent was proposed by Houska et al.(2013); see also Houska et al. (2014). This construc-tion is based on Taylor/Chebyshev models with ellip-soidal remainder terms as the pair (Pq

f,P , Sqf,P ), with Sq

f,P

the shape matrix of the ellipsoidal remainder E(Sqf,P ),

thus yielding extensions in the form fEpol(q)m ×E

ballm :

Rn×(α(q)m +n+1) → Rp×(α(q)

m +p+1).

Finally, because the image set of a multivariate polynomialof order 2 or higher is nonconvex in general, applicationsof Taylor/Chebyshev models often call for constant/affinebounds or convex/polyhedral enclosures of such sets.

• Tight interval bounds can be obtained using LMImethods (Lasserre, 2009). Other ways of deriving rig-orous interval bounds involve exact bounding of thepolynomial’s first- and second-order terms (Makinoand Berz, 2003; Lin and Stadtherr, 2007b) or ex-pressing the polynomial in Bernstein bases (Lin andRokne, 1995).

• Affine bounds can be obtained likewise by retainingthe first-order term, while bounding all of the otherterms using one of the foregoing approaches.

• Polyhedral enclosures can be obtained on appli-cation of the reformulation-linearization technique(RLT) by Sherali and Fraticelli (2002); Sherali et al.(2012). Other approaches to convex/polyhedral en-closures include the decomposition/relaxation/outer-approximation technique (Smith and Pantelides,1999; Tawarmalani and Sahinidis, 2004) as wellas McCormick’s relaxation technique (McCormick,1976; Mitsos et al., 2009).

Both the Taylor and Chebyshev model arithmetics,along with the aforementioned bounding and relax-ation approaches to enclose the range of these estima-tors, are implemented in MC++, our in-house librarythat is freely available from https://bitbucket.org/omega-icl/mcpp. The following example illustrates someof these features in connection to the anaerobic digestionmodel in Sect. 1.1.

Case Study 1. We consider the nonlinear relationship be-tween the concentration S2 and the partial pressure PCO2

in the anaerobic digestion model in Sect. 1.1. After sub-stitution of the various subexpressions, we obtain:

f(PCO2 , S2) = KHP2CO2

+ Pt(C + S2 − Z)

+

C + S2 − Z +KHPt +k6kLa

µ2S2X2

S2 +KS2 +S22

KI2

PCO2

The parameter values are those given in Table A.1 andthe other states are taken as C = 30 [mmol L−1], Z =50 [mmol L−1] and X2 = 4 [g(cell) L−1]. Moreover, all thecomputations are carried out with MC++.

The set of points (PCO2 , S2) ∈ Y := [0.1, 1]×[0, 15] satisfy-ing f(PCO2 , S2) = 0 is represented in solid black line on theplots of Fig. 4. We investigate several parameterizations ofthis (nonconvex) set in the form of Chebyshev models oforders q = 2, . . . , 4. The blue, green and purple lines onthe left plot of Fig. 4 enclose the set of points (PCO2 , S2)such that:

Pqf,Y (PCO2 , S2) + r = 0 , for some r ∈ Rq

f,Y ,

with (Pqf,Y ,R

qf,Y ) a qth-order Chebyshev model of f on

Y . These parameterizations provide tighter and tighterapproximations as the order q increases. For comparison,convex and concave relaxations obtained from the Mc-Cormick relaxation of f on Y are shown in dashed lineson the figure. It is clear that, for this example, Chebyshevmodels of order 3 or higher provide tighter enclosuresthan McCormick relaxations, mainly due to their abilityto capture nonconvexity.

The middle and right plots of Fig. 4 show polyhedral enclo-sures f(PCO2 , S2) = 0, as constructed by constructing con-vex/concave bounds of the multivariate polynomial usingMcCormick’s approach (middle plot) or by extracting thefirst-order term of the Chebyshev models and boundingthe remaining terms (right plot). These simple boundsappear to be better than, or at least comparable to, thosecomputed directly fromMcCormick relaxations. Moreover,the Chebyshev-derived bounds are found to progressivelyoutperform the latter as the parameter range is reduced(results not shown). Note that tighter polyhedral enclo-sures could also be obtained by accounting for the depen-dencies in higher-order terms of the Chebyshev models.⋄

3. REACHABILITY ANALYSIS OF NONLINEARDYNAMIC SYSTEMS

The problem addressed in this section is the computa-tion of time-varying enclosures X(t, P ) ⊇ X(t, P ), whereX(t, P ) := x(t, p) | p ∈ P stands for the reachable set ofparametric dynamic systems in the form

∀t ∈ [0, T ] , x(t, p) = f(x(t, p), p) (11)

with x(0, p) = x0(p) .

In this formulation, the state x : [0, T ] × P → Rnx isregarded as a function of the uncertain parameter vectorp ∈ P ⊆ Rnp along the time horizon [0, T ]. A majorcomplication in computing the enclosures X(t, P ) is thatthese functions do not have a factorable representationin general, and therefore the bounding/relaxation tech-niques introduced in Sect. 2 may not be applied directly.Nonetheless, algorithms for bounding the solution set ofsuch parametric dynamic systems can take advantage ofthe fact that the right-hand side function f and the initialvalue function x0 are usually factorable.



0

2

4

6

8

10

12

14

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S2

PCO2

0

2

4

6

8

10

12

14

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1S2

PCO2

0

2

4

6

8

10

12

14

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

solution set

McCormick relax.

2nd-ord. Cheb. mod.3rd-ord. Cheb. mod.4th-ord. Cheb. mod.

S2

PCO2

Fig. 4. Result of various parameterizations in enclosing the nonlinear relationship between S2 and PCO2 in the two-stepanaerobic digestion model. Left plot: full Chebyshev model; Middle plot: Chebyshev-based convex/concave bounds;Right plot: Chebyshev-based affine bounds.

Existing methods for set-valued ODE integration can bebroadly classified as discrete-time and continuous-time.Discrete-time methods proceed by first discretizing theintegration horizon into finite steps, and then propagatinga reachable set enclosure through each step. Many suchintegrators go back to the original work by Moore (1979),who presented a simple test for checking the existence anduniqueness of ODE solutions over a finite time step usinginterval analysis. This test was later incorporated into analgorithm that implements a two-phase approach (Lohneret al., 1992; Nedialkov et al., 1999): (i) determine a step-size and an a priori enclosure of the ODE solutions overthe current step; then, (ii) propagate a tightened enclosureuntil the end of that step. Both phases typically rely on ahigh-order Taylor expansion of the ODE solutions in time,and the enclosures are propagated by a variety of affineset-parameterization extensions (Berz and Makino, 2006;Lin and Stadtherr, 2007b; Neher et al., 2007). Recently,Houska et al. (2013) have proposed a reversed, two-phasealgorithm that starts by constructing a predictor of thereachable set and then determines a step-size for whichthis predictor yields a valid enclosure.

In contrast, continuous-time methods involve constructingan auxiliary system of ODEs whose solution is guaranteedto enclose the reachable set of the original ODEs. Thesemethods are inspired from the theory of differential in-equalities (Walter, 1970; Scott et al., 2012), viability the-ory (Aubin, 1991), or other set-theoretic methods such asellipsoidal calculus (Kurzhanski and Valyi, 1997; Houskaet al., 2012). Recently, Villanueva et al. (2014) have devel-oped a unifying framework based on a generalized differen-tial inequality for continuous-time propagation of convexand non-convex enclosures of the reachable set of uncer-tain ODEs. Other recent developments of continuous-timemethods are concerned with enclosing the reachable setof implicit differential equations (Scott and Barton, 2013;Rajyaguru et al., 2015).

The following subsections further detail both approachesand their properties, with a focus on the authors’ owncontributions and using the set-theoretic concepts andtools in Sect. 2. Specifically, given a parameterization Qp

of the parameter set P on the affine set (Em,Dnp,m),we describe techniques for constructing a matrix valuedfunction Qx : [0, T ] → Dnx,m such that

∀t ∈ [0, T ], ImEm(Qx(t)) ⊇ X(t, P ) . (12)

An application is presented for the two-step anaerobicdigestion model at the end of the section.

3.1 Discrete-Time Set Propagation

Many discrete-time methods consider a Taylor expansionin time of the ODE solutions. Assuming that x(·, p) is thesolution of (11) up to time t ∈ [0, T ) for a given parameterp, and provided that this solution can be extended untilt+h with h ∈ (0, T−t], the application of Taylor’s theoremfor an s-th order expansion gives

x(t+ h, p) =s

∑

i=0

hiφi(x(t, p), p) + hs+1φs+1(x(τ, p), p)

for some τ ∈ [t, t+h]; and with φ0, φ1, . . . , φs+1 the Taylorcoefficient functions of the solution, defined recursively as

φ0(x, p) := x and φi(x, p) :=1

i

∂φi−1

∂x(x, p) f(x, p).

In reversing the two phases of the traditional discrete-time approach, the algorithm by Houska et al. (2013)removes the need for an a priori enclosure of the solu-tion and also provides a natural mechanism for step-sizeselection. The propagation starts from a parameterizationQx(0) := xEm

0 (Qp), with xEm

0 an extension of the initial-value function x0, so that ImEm

(Qx(0)) ⊇ X(0, P ). Then,the following two steps are applied repeatedly:

(1) Given a parameterization Qx(t) at some t ∈ [0, T )such that ImEm

(Qx(t)) ⊇ X(t, P ), compute a predic-tor Qx(t + h) of the solution for all h ∈ (0, T − t]as:

Qx(t+ h) :=

s⊎

i=0

hiφEm

i (Qx(t), Qp) ⊎ hTOLQunit

for a pre-specified tolerance TOL > 0 and Qunit ∈Dnx,m; and where φEm

i are extensions of the Taylorcoefficient functions φi for each i = 0, . . . , s and ⊎stands for the extension of the addition operator.

(2) Determine a step-size h such that the predictorQx(t+h) is guaranteed to yield a valid enclosure of thereachable set, ImEm

(Qx(t+ h)) ⊇ X(t + h, P ), forall h ∈ [0, h]; that is,

∀τ ∈ [t, t+ h], ImEm

(

(τ − t)s φEm

s+1(Qx(τ), Qp))

⊆ TOL ImEm(Qunit) ,

with φEm

s+1 an extension of the Taylor coefficient func-tion φs+1.



A practical way of finding a valid step-size h is given inHouska et al. (2013, 2014).

Especially appealing within this approach is the inherentflexibility of the algorithm, which can be used with anyaffine set-parameterization, including the propagation ofconvex sets (e.g., interval boxes, ellipsoids) and noncon-vex sets (e.g., polynomial image sets derived from Tay-lor/Chebyshev models).

3.2 Continuous-Time Set Propagation

This approach involves constructing auxiliary ODEs thatdescribe the coefficient Qx of a parameterization of thestate variables; that is,

∀t ∈ [0, T ] , Qx(t) = F (Qx(t), Qp) , (13)

with initial parameterization Qx(0) := xEm

0 (Qp). Clearly,a possible choice for the right-hand-side function F in(13) is the extension fEm of the original right-hand sidef in (11). However, it is useful in practice to accountfor certain facet constraints that mitigate the growth ofthe reachable set enclosure; this includes the method ofstandard differential inequalities as well as ellipsoidal setpropagation techniques. Recently, Villanueva et al. (2014)have formulated a generalized differential inequality (GDI)that contains the usual facet constraint methods as specialcases. This GDI describes sufficient conditions on thesupport function V [X(t, P )] of a convex enclosure X(t, P )of the reachable set as follows:

a.e. t ∈ [0, T ] , ∀c ∈ Rnx ,

V [X(t, P )](c) ≥ maxξ,ρ

cTf(ξ, ρ)

∣

∣

∣

∣

∣

∣

ξ ∈ X(t, P )cTξ = V [X(t, P )](c)ρ ∈ P

with V [X(0, P )](c) ≥ maxρ

cTx0(ρ)∣

∣ ρ ∈ P

.

Interestingly, the GDI also supports the constructionof nonconvex enclosures, e.g., in the form of Tayloror Chebyshev models with convex remainder bounds(Pq

x,P (t, ·),Rqx,P (t)) for each t ∈ [0, T ], so that:

X(t) := Pqx,P (t, p) | p ∈ P ⊕Rq

x,P (t) ⊇ X(t, P ) . (14)

The polynomial part Pqx can be directly obtained via

the solution of the auxiliary ODEs (13) in the desired

polynomial image basis Epol(q)m and with F := fE

pol(q)m .

In the case of Taylor models, for instance, these ODEscorrespond to the sensitivities of (11) up to order q.

Besides the polynomial part, ways of propagating convexremainder enclosures in the form of interval bounds orellipsoids are also described in Villanueva et al. (2014).An interval remainder Rq

x(t) :=[

rLx (t), rUx (t)

]

can bepropagated by integrating the following 2 × nx system ofauxiliary ODEs, for all i = 1, . . . , nx:

rLi (t) = minξ,ρ

fi(Pqx(t, ρ) + ξ, ρ)

− Pqxi(t, ρ)

∣

∣

∣

∣

∣

∣

∣

ξi = rLxi(t)

ξ ∈ [rLx (t), rUx (t)]

ρ ∈ P

rUi (t) = maxξ,ρ

fi(Pqx(t, ρ) + ξ, ρ)

− Pqxi(t, ρ)

∣

∣

∣

∣

∣

∣

∣

ξi = rUxi(t)

ξ ∈ [rLx (t), rUx (t)]

ρ ∈ P

.

Likewise, an ellipsoidal enclosure E(Sqx(t)) can be prop-

agated by integrating the following nx × nx system ofauxiliary ODEs:

Sqx(t) =A(t)Sq

x(t) + Sqx(t)A(t)

T +

nx∑

i=1

κi(t)Sqx(t)

+ diag(κ(t))−1 diag rad(Ωqf [S

qx(t)])

2 ,

with A(t) =(

∂f∂x (P

qx(t, p), p)

)

. Here, the nonlinearity

bounder Ωqf [S] ∈ IR

nx must satisfy

f(Pqx(t, ρ) + r, ρ)− Pq

x(t, ρ)−∂f

∂x(Pq

x(t, ρ), ρ) r ∈ Ωqf [S],

for all (r, ρ) ∈ E(S) × P , and its construction can beautomated using interval analysis for instance. Moreover,the scaling function κ : [0, T ] → R

nx

++ can be chosen insuch a way as to minimize tr(Sq

x(t)).

Continuous-time methods based on differential inequalitiesare appealing in that the auxiliary ODEs can be solvedusing off-the-shelf ODE solvers. However, because of theneed for applying a numerical discretization and since theright-hand side function defining the auxiliary ODE istypically non-differentiable, it is hard to give any guaranteeon the discretization error. This non-differentiability canalso impair the step-size control mechanism of the numer-ical integration algorithm, and in principle it should beaddressed in the framework of hybrid discrete-continuoussystems (see, e.g., Singer, 2004).

3.3 Properties

A common feature of discrete- and continuous-time setpropagation methods is their ability to propagate setsdescribed by a variety of affine-set parameterizations,including both convex and nonconvex sets. Despite thefact that these methods rely on rather different ideas,they share several important properties regarding theirconvergence order and stability.

We have already stressed the importance of high-orderinclusions in Sect. 2. As far as the convergence of reach-able set enclosures is concerned, the computed enclosuresImEm

(Qx(t)) can, under certain conditions and at eachtime t ∈ [0, T ], inherit the convergence order of functionextensions in the chosen affine set-parameterization Em. Inthe particular case of qth-order Taylor/Chebyshev mod-els with convex remainder bounds, (Pq

x,P (t, ·),Rqx,P (t)),

computer implementations can be devised such that (Vil-lanueva et al., 2014):

• for continuous-time set propagation,

∀t ∈ [0, T ], Rqx,P (t) ∈ O(diam (P )

q+1) ;

• for discrete-time set propagation,

∀t ∈ [0, T ], Rqx,P (t) ∈ O(diam (P )

q+1) +O(TOL) .

Notwithstanding these high-order convergence properties,both continuous and discretized set-valued integrationmethods are subject to the wrapping effect, which typicallyresults in the diameter of the reachable set enclosurediverging to infinity, even on finite time horizons—theso-called ‘bound explosion’ phenomenon. For stable ODEsystems in particular, a rather natural requirement wouldappear to be that the computed enclosures are themselves



stable, at least for small enough initial value or uncertainparameter sets.

Recently, Houska et al. (2014) have derived sufficient con-ditions for the discrete-time method outlined in Sect. 3.1to be locally asymptotically stable, in the sense that thecomputed enclosures are guaranteed to remain stable oninfinite time horizons when applied to a dynamic system inthe neighborhood of a locally asymptotically stable peri-odic orbit (or equilibrium point). The key requirement hereis quadratic Hausdorff convergence of function extensionsin the chosen affine set-parameterization, since the enclo-sures may not contract fast enough to neutralize the wrap-ping effect otherwise. Note that this result also appliesto continuous-time set propagation. This stability analysissheds light on the fundamental reason why those currentlyavailable set-valued ODE integrators relying on intervalarithmetics in one way or another fail to be locally asymp-totically stable, regardless of the size of the uncertaintyset; and even state-of-the-art integrators based on Taylormodels with interval remainders, such as VSPODE (Linand Stadtherr, 2007b) or COSY Infinity (Makino andBerz, 2005), may not stabilize the reachable set enclosuresof asymptotically stable dynamic systems, despite the factthat they implement advanced heuristics for rotating thebasis of the interval remainder.

One way to promote asymptotic stability involves propa-gating Taylor/Chebyshev models with ellipsoidal remain-der bounds, for which extensions with quadratic Haus-dorff convergence can be constructed (see Sect. 2.2). Bothdiscrete- and continuous-time set-valued ODE integratorsimplementing this approach based on MC++ are madefreely available at https://bitbucket.org/omega-icl/eqbnd. The following example illustrates these stabilityconsiderations.

0.6

0.7

0.8

0.9

1

1.1

1.2

1

2

3

4

5

0 5 10 15 20

t [day]

q = 1 q = 2 q = 3 q = 4

S2[m

molL

−1]

X2[g(cell)L−1]

Fig. 5. Projections ontoX2 (top plot) and S2 (bottom plot)of the reachable set X(t, P ) (shaded area) and of theenclosures X(t, P ) computed with Taylor models oforder q = 1, . . . , 4 with interval remainders (dashedlines) and ellipsoidal remainders (solid lines) in thetwo-step anaerobic digestion model.

Case Study 2. We consider the nonlinear ODE model (1)-(6) of anaerobic digestion, with uncertain initial conditionsas X1(0) ∈ 0.5± 4% [g(cell) L−1], X2(0)± 4% [g(cell) L−1]

and C(0) ∈ 40 ± 4% [mmol L−1]. The rest of the initialconditions are given by S1(0) = 1 [g(COD)L−1], S2(0) =5 [mmol L−1] and Z(0) = 50 [mmol L−1], in addition tothe parameter values in Table A.1.

Projections onto the variables X2 and S2 of the actualreachable set and of the bounds derived from Taylormodels of order q = 1, . . . , 4 with either interval orellipsoidal remainders are shown in Fig. 5. We note that itonly takes a few seconds to generate the bounds in all ofthe cases for this problem; see Villanueva et al. (2014) fora detailed numerical comparison.

For Taylor models with interval remainders, increasing theorder q delays the blow up time significantly, up to aboutt ≈ 4 [day] with 4th-order Taylor models. In comparison,bounds derived from the propagation of Taylor modelswith ellipsoidal remainders are found to be superior forq ≥ 2 here. In particular, Taylor models with ellipsoidalremainders of order q ≥ 4 are found to stabilize thereachable set enclosure for this level of uncertainty; that is,the bounds converge to the actual steady-state values ast → ∞. Stabilizing the enclosures with lower-order Taylormodels would require reducing the level of uncertainty. ⋄

4. APPLICATIONS OF SET-THEORETICAPPROACHES FOR COMPLETE SEARCH

A great variety of algorithms, including complete searchmethods for problems in global optimization, constraintsatisfaction or robust estimation, hinge on the ability toconstruct tight enclosures for the range of (nonlinear andnon-necessarily factorable) functions, such as the methodsdescribed in Sect. 2 and Sect. 3. Our focus in this sectionis on set-inversion techniques (Moore, 1992; Jaulin andWalter, 1993), which enable approximation of sets definedin implicit form, such as

Pe := p ∈ P0 | g(p) ∈ Γ , (15)

using subpavings (sets of non-overlapping boxes). Obvi-ously, these techniques bear many similarities with branch-and-bound search in global optimization (Neumaier, 2004;Tawarmalani and Sahinidis, 2004). They are similar inflavor to set-oriented numerical methods, as developed,e.g., by Dellnitz et al. (2001).

Given a compact set P0 ⊂ Rnp , a set Γ ⊂ Rng , anda continuous function g : Rnp → Rng , set-inversionalgorithms compute partitions Pin and Pbnd such that

⋃

P∈Pin

P ⊆ Pe ⊆⋃

P∈Pin∪Pbnd

P , (16)

with Pbnd sufficiently small. A prototypical algorithm isgiven below.



Algorithm 1. Set-inversion algorithm.

Input: Termination tolerances ǫbox ≥ 0 and ǫbnd ≥ 0Initialization: Set partitions Pbnd = P0 and Pin = ∅;Set iteration counter k = 0

Main Loop:(1) Select a parameter box P in the partition Pbnd and

remove it from Pbnd

(2) Compute an enclosure g(P ) ⊇ g(p) | p ∈ P(3) Exclusion Tests:

(a) If g(P ) ⊂ Γ, insert P into Pin

(b) Else if g(P ) ∩ Γ = ∅, fathom P(c) Else bisect P and insert subsets back into Pbnd

(4) If width(P ) ≤ ǫbox for all P ∈ Pbnd or Vbnd :=∑

P∈Pbndvolume(P ) ≤ ǫbnd, stop

(5) Increment counter k+=1; Return to step 1Output: Partitions Pin and Pbnd; Iteration count k

Clearly, Step 2 of Algorithm 1 calls for a procedure capableof computing an enclosure of the image set g(p) | p ∈P for a given parameter subpartition P . Should thefunction g be factorable, such enclosures can be obtainedby considering an extension gEm on a suitable affine set-parameterization basis Em, as explained in Sect. 2. If g isdefined implicitly via the solution of differential equations,the set-propagation techniques described in Sect. 3 canbe used instead. Both cases are considered subsequently,with applications to bifurcation analysis (Sect. 4.1) andset-membership parameter estimation (Sect. 4.2).

The use of Taylor/Chebyshev models to construct enclo-sures g is appealing in that it can capture the parametricdependencies in the actual solution set Pe (besides enjoy-ing higher-order convergence properties). Given a param-eter box P := [pL, pU] and a qth-order Taylor/Chebyshevmodel enclosure g(P ) := Pq

g,P (p) | p ∈ P ⊕ Rqg,P , the

lower and upper parameter bounds pLj and pUj for eachj = 1, . . . , np can be tightened by solving optimizationproblems of the form Paulen et al. (2015):

pL/Uj = min /maxp pj s.t. Pq

g,P (p) ∈ Γ⊖Rqg,P . (17)

This way, a reduced box P is obtained after solving2× np optimization problems—one problem for the lowerbound and one for the upper bound of each parameter.As written, the bound-reduction problems (17) are ingeneral nonconvex for q ≥ 2. Instead of attempting tosolve these problems directly, one can construct polyhedralrelaxations in the form of linear programs (LPs), similarto the approach used for bound contraction in branch-and-bound search (see, e.g., Neumaier, 2004; Tawarmalaniand Sahinidis, 2004). In a related approach, Lin andStadtherr (2007a) and Kletting et al. (2011) use constraintpropagation in the domain reduction procedure.

In effect, the domain-reduction procedure can be per-formed as an extra step in Algorithm 1, between Steps 2and 3. In the case that the reduction of a parameter box Pis larger than a given threshold, it can be repeated multipletimes. It is important to bear in mind that repeating thereduction several times involves recomputing the enclo-sures g(P ) of the model outputs on the reduced box Pthough. This defines a clear trade-off between the extracomputational burden and the reduction in the size of

the partition Pbnd, which is of course problem-dependent.Paulen et al. (2015) have also presented a simple approachto avoid recomputing the enclosures g(P ) as soon as theremainder term in the Taylor/Chebyshev model is withina given threshold.

4.1 Bifurcation Analysis

Locating the equilibrium and bifurcation points of a non-linear dynamic system occurring within a given state-space domain is an important problem in control. Theset-inversion algorithm described above provides a meansfor addressing this problem rigorously. Not only is this apowerful alternative to the classical continuation methods(Allgower and Georg, 1990), but it also enables bifurcationanalysis with respect to multiple parameters simultane-ously; see, e.g., Hasenauer et al. (2009); Waldherr andAllgower (2011); Smith et al. (2014) for recent relatedstudies.

The equilibrium manifold of a dynamic system given inthe form of parametric ODEs (11) is defined as the pairof points (p, x) ∈ Rnp+nx such that f(x, p) = 0. Theproblem of approximating this manifold as close as possiblecan be therefore cast as the set-inversion problem (15) byconsidering the extended parameter-state space Rnx+np as

Pe ×Xe := (p, x) ∈ P0 ×X0 | f(x, p) = 0 . (18)

Once a subpaving of Pe ×Xe has been constructed usingAlgorithm 1, so that (16) is satisfied with Pbnd sufficientlysmall, one can then infer the stability of the equilibriumpoints contained in each subpartition of Pbnd and isolatethose subpartitions which may contain a bifurcation point.

Recall that a dynamic system is stable if and only if thedeterminant of its nx-by-nx Hurwitz matrix,

H(x, p) :=

c1(x, p) c3(x, p) c5(x, p) · · · 0c0(x, p) c2(x, p) c4(x, p) · · · 0

0 c1(x, p) c3(x, p) · · · 00 c0(x, p) c2(x, p) · · · 0...

......

. . ....

0 0 0 · · · cnx(x, p)

,

and all its leading principal minors are positive, whereci(x, p) are the coefficients of the characteristic polynomial

of the Jacobian matrix Jf := ∂f∂x ,

det(Jf (x, p)− λI) =: λnx +

nx∑

i=1

ci(x, p)λnx−i .

In practice, symbolic expressions for the coefficients cican be obtained using the Faddeev-Leverrier algorithm(Helmberg et al., 1993). Moreover, the Hurwitz matrixcan be reduced to an upper triangular form, say U , usinga modified Neville elimination algorithm in O(n2

x) opera-tions (Gasca and Pena, 1992). The following stability testscan be performed sequentially on a subpartition P ×X byevaluating the elements ci and Uii in a given arithmetic,such as interval arithmetic or Taylor/Chebyshev modelarithmetic (see Sect. 2):

(1) If none of the coefficients ci or diagonal elements Uii

are nonnegative, then all the equilibrium points in theconsidered subpartition are stable;

(2) If at least one element ci or Uii is negative, then all theequilibrium points in the subpartition are unstable;



(3) Otherwise (that is, if any one of the elements ci orUii have zero in range), the subpartition may containa bifurcation point.

The types of bifurcation points that can be detected thisway include:

• Steady-state bifurcation points, whereby the Jaco-bian matrix Jf has a zero eigenvalue; their occurrencecan be tested by checking if any element ci or Uii

vanishes.• Hopf bifurcation points, which occur in the presenceof a limit cycle when Jf has one conjugate pair ofeigenvalues with zero real part and the other eigen-values all have negative real parts; their occurrencecan be tested by checking that

cnx> 0, ∆nx−1 = 0, ∆nx−2 > 0, . . . ,∆1 > 0,

with ∆i the ith principal minor of H (El Kahoui andWeber, 2000).

Note that enclosures of the bifurcation points can also bedirectly computed on appending the foregoing bifurcationcharacterization constraints to the equilibrium constraintsin (18). This removes the need for computing the fullequilibrium manifold.

Case Study 3. We consider the problem of determiningthe equilibrium points of the two-step anaerobic digestionmodel (1)-(7), along with characterizing their stability.The focus is on varying the dilution rate D as the bi-furcation parameter in the range P0 := [10−3, 1.5], andthe state-space domain for [X1 X2 S1 S2 Z C PCO2 ] isX0 := [0, 0.6]×[0, 0.8]×[0, 5.]×[0, 80]×[0, 80]×[0, 80]×[0, 1].As previously, the model parameter values are those listedin Table A.1.

The results produced by the set-inversion algorithm areshown in Fig. 6, with an indication of the stable andunstable parts of the equilibrium manifold. In order tocompute the enclosures in Step 2 of Algorithm 1, we useaffine relaxations derived from 2nd-order Chebyshev mod-els of the equilibrium constraints, similar to the approachdescribed in Case Study 1 above. Moreover, we performdomain reduction at each node via the solution of auxiliaryLPs in order to refine the equilibrium set approximation.The set-inversion algorithm is interrupted after 40,000iterations, with a corresponding runtime of a few minutesonly.

Note that only steady-state bifurcations occur in thisproblem, which correspond to a change in stability of theequilibrium manifold. One such bifurcation occurs herearound D = 1 [day], where both a stable branch and anunstable branch merge into a single stable branch. Anotherbifurcation is seen to occur around D = 1.07 [day] with achange in stability along a single branch. ⋄

4.2 Set-Membership Parameter Estimation

Among the available techniques to account for uncertaintyin parameter estimation, set-membership estimation, alsoknown as guaranteed parameter estimation, aims to deter-mine all the parameter values of a model that are consis-tent with a set of measurements under given uncertaintyscenarios. Initially developed for algebraic models in the

early 1990s (Moore, 1992; Jaulin and Walter, 1993), thesetechniques were later extended to dynamic models usingODE bounding techniques (e.g., Jaulin, 2002; Raissi et al.,2004). Our focus in this subsection shall be on the latterand we consider the case where the uncertainty enters theestimation problem in the form of bounded measurementerrors.

Given a nonlinear dynamic system in the form of paramet-ric ODEs (11), consider a set of output functions

y(t, p) = g(x(t, p), p) ∈ Rny . (19)

Now, for a set of output measurements ym(ti) at Ntime points t1, . . . , ti, . . . , tN ≤ T , classical parameterestimation seeks to determine one particular instance p⋆

of the parameter values for which the (possibly weighted)normed difference between these measurements and thecorresponding model outputs y is minimal. We note bythe way that such optimization problems are typicallynonconvex and call for global optimization, for which onecan use branch-and-bound search in combination withthe ODE bounding techniques described earlier in Sect. 3(Esposito and Floudas, 2000; Singer and Barton, 2006).

In contrast, guaranteed (bounded-error) parameter estima-tion accounts for the fact that the actual process outputs,yp, are only known within some bounded measurementerror e ∈ E := [eL, eU ], so that

yp(ti) ∈ ym(ti) + [eL, eU ] =: Yp(ti) .

The main objective then is to estimate the set Pe of allpossible parameter values p such that y(ti, p) ∈ Yp(ti) forevery i = 1, . . . , N . Clearly, this problem can be cast asthe following set inversion:

Pe :=

p ∈ P0

∣

∣

∣

∣

∀i = 1, . . . , N,ym(ti) + eL ≤ y(ti, p) ≤ ym(ti) + eU

.

(20)

Depicted in red on the top plot of Fig. 7 is the set ofall output trajectories satisfying y(ti, p) ∈ Yp(ti) withi = 1, . . . , N , and on the bottom plot the correspondingset Pe projected onto the (p1, p2) space.

A practical limitation for the guaranteed parameter esti-mation problem as given in (20) is the need for consistentmeasurement data and bounds throughout the entire timeseries; otherwise, there may not be any model responsematching the output measurements within the specified er-ror bounds, and the parameter set Pe is empty. In most ap-plications based on real data, this calls for data preprocess-ing, for instance using data reconciliation techniques, inorder to get rid of the outliers. For instance, these outlierscould be due to over-optimistic noise bounds or to sensorfailures at given time instants. To handle this situation, itis possible to ‘protect’ the estimator against a prespecifiednumber of outliers, by allowing for a number of outputvariables to be outside of their prior feasible intervals(see, e.g., Jaulin et al., 2001; Kieffer and Walter, 2005).Another situation whereby the parameter set Pe may beempty is in the presence of significant model mismatch,which calls for the development of further robustificationstrategies or alternative guaranteed parameter estimationparadigms (Csaji et al., 2012; Kieffer and Walter, 2014).

In a related work, Rumschinski et al. (2010) apply a set-inversion approach in order to discriminate between com-



0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2 0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2 0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2 0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2 0.25

0.3

0.35

0.4

0.45

0 0.2 0.4 0.6 0.8 1 1.2

DDD

DDD

X1

X2

S1

S2

CPCO

2

Fig. 6. Set of stable (blue) and unstable (red) equilibrium points of the two-step anaerobic digestion model with respectto the dilution rate D.

yi(t)

t

Y (·, P3)

Y (·, P2)

Y (·, P1)

p1

p2P2 ∈ Pin

P1 /∈ Pin ∪ Pbnd

Pe

P3 ∈ Pbnd

P0

Fig. 7. Illustration of guaranteed parameter estimationconcepts in the space of output trajectories (top plot)and in the parameter space (bottom plot).

peting model hypotheses and to provide guaranteed outerestimates on the model parameters that are consistentwith the experimental measurements. They consider non-linear dynamic systems with polynomial/rational right-hand sides, and construct SDP relaxations in order to

Table 1. Dilution rate and inlet concentrationprofiles in guaranteed estimation problem.

Input Day 1 Day 2 Day 3 Day 4

D [/day] 0.25 1.00 1.00 0.25Sin1 [g(COD)/L] 2.38 2.38 4.76 2.38Sin2 [mmol/L] 80.0 80.0 160.0 80.0

Z in [mmol/L] 50.0 50.0 100.0 50.0Cin [mmol/L] 5.0 5.0 10.0 5.0

carry out the exclusion tests after discretizing the differ-ential equations. In a follow up work, Streif et al. (2012)have developed the Matlab toolbox ADMIT automatingthe analysis.

Case Study 4. We consider the problem of computingguaranteed parameter sets for the nonlinear ODE model(1)-(6). Three outputs are considered to carry out the es-timation, namely S1, S2, and C, with measurements every4 hours over a 4-day period. Pseudo-experimental data aregenerated by simulating a nominal model with parametervalues from Table A.1 and using the dilution rate andinfluent concentrations in Table 1. Moreover, the effect ofmeasurement noise is simulated by rounding the nominaloutputs up or down to the nearest values by retaining,respectively, 2, 1 and 1 significant digits; then, measure-ment error ranges of, respectively, ±0.01, ±0.1 and ±0.1are added to these values. Regarding the parameters, thefocus is on estimating the kinetic parameters describingbiomass growth, namely µ1 ∈ [1.15, 1.25], KS1 ∈ [6.7, 7.3],µ2 ∈ [0.735, 0.75], KS2 ∈ [9.2, 9.5], and KI2 ∈ [235, 265].

The results produced by the set-inversion algorithm areshown in Fig. 8. The shape of the guaranteed parameterset is indeed characteristic of the large correlations be-tween the parameters µ1 and KS1 , which indicates thatµ1(S1) ≈

µ1

KS1X1 in this case. Likewise, a large correlation

is observed between µ2, KS2 and KI2 . We also note that



the nominal parameter values (red cross on Fig. 8) lieinside the approximation of the set Pe.

In order to compute the reachable set enclosures we use4th-order Taylor models with ellipsoidal remainders, inagreement with the results of the reachability analysisconducted in Case Study 2. Moreover, we perform domainreduction at each node via the solution of auxiliary LPsconstructed from the polyhedral relaxations of the Taylormodels of the predicted outputs at each measurementtime, and we apply up to 10 passes as long as the rangereduction for any parameter exceeds 20%. Finally, we donot recompute the reachable sets on children nodes whenall the Taylor model remainders at their parent nodes haveconverged to within 10−4. With these settings, the set-inversion algorithm reaches a tolerance of ǫbnd = 5 · 10−5

after 3,130 iterations and a runtime of about 35 minutes.See Paulen et al. (2015) for further numerical comparisons,including the solution of a related guaranteed parameterestimation problem in 7 parameters. ⋄

6.6 6.8 7 7.2 7.4

1.16

1.18

1.2

1.22

1.24

0.735 0.74 0.745 0.75 0.7556.6

6.8

7

7.2

7.4

9.25 9.3 9.35 9.4 9.450.735

0.74

0.745

0.75

240 245 250 255 2609.25

9.3

9.35

9.4

9.45

µ1

KS1

KS1

µ2

µ2

KS2

KS2 KI2

Fig. 8. Projections onto the subspaces (KS1 ,µ1), (µ2,KS1),(KS2 ,µ2) and (KI2 ,KS2) of the guaranteed parametersets for the two-step anaerobic digestion model.

5. SUMMARY AND FUTURE DIRECTIONS

This paper has presented an overview of some of the set-theoretic methods developed in our research group forthe analysis of uncertain/parametric nonlinear dynamicsystems. These methods are heavily dependent on ourability to compute tight bounds on the range of factorablefunctions and on the reachable set of dynamic systems.In particular, we have presented a formalism based on socalled affine-set arithmetics and set propagation, and wehave applied this formalism so as to cast problems in bifur-cation analysis and set-membership parameter estimationas set-inversion problems. This way, these problems can beaddressed both rigorously and efficiently using completesearch methods.

Besides complete search applications, it is important toreemphasize that set-theoretic methods also hold manypromises in other areas of mathematics and engineering,such as fault diagnosis (Scott et al., 2014) or modelpredictive control (MPC). To reinforce this latter point,

we provide in this following subsection a formulationfor robust MPC, which makes use (and extends) thegeneralized differential inequality presented in Sect. 3.

5.1 Tube-Based Methods for Robust MPC

Model predictive controllers are a class of feed-back con-trollers, which proceed by solving an optimal control prob-lem to predict the future behavior of a dynamic systemon a finite time-horizon. The computed optimal input isthen applied to the real process until the next measure-ment arrives and the process is then repeated, followinga receding horizon approach. In this procedure, the futurebehavior of the system is optimized without accounting forexternal disturbances or model-plant mismatch, althoughthese uncertainties are the only reason why feedback isneeded at all.

For applications with hard constraints, MPC implementa-tions can lead to constraint violations, and it is for thoseapplications that robust MPC can be used to correct theoptimistic predictions of MPC. Of the possible approachesfor implementing robust MPC (see, e.g., Bertsekas, 2007;Dadhe and Engell, 2008; Goulart and Kerrigan, 2006), ourfocus here is on tube-based MPC (Langson et al., 2004),whereby the predicted trajectories used in traditionalMPC are replaced by robust forward invariant tubes; thatis, tubes enclosing all the response trajectories for a chosencontrol law regardless of the particular realization of theuncertainty.

Consider a controlled dynamic system with time-varyinguncertainties in the form

∀t ∈ [0, T ], x(t) = f(x(t), u(t), w(t)),

where u(t) ∈ U ⊆ Knu and w(t) ∈ W ⊆ Knw denotethe control and disturbances, respectively. By letting Ydenote the set of all robust forward invariant tubes, wecan formulate the tube-based MPC problem as:

minY ∈Y

∫ t+T

t

ℓ(Y (τ))dτ s.t.

Y (τ) ⊆ Fx,

Y (t) = xt,

where the feasibility constraints Fx := x ∈ Rnx |g(x) ≤ 0are enforced for all τ ∈ [t, t + T ]; the set-valued functionℓ : Π(Rnx) → R is the objective function in the MPCcontroller; and xt denotes the current state measurementat time t.

This formulation shows deep connections between forwardinvariant tubes and the reachable-set enclosure X(·, P ),e.g., computed via the generalized differential inequalitypresented in Sect. 3. In particular, we envision extensionsof the work by Villanueva et al. (2014) towards thefollowing min-max differential inequality:

a.e. t ∈ [0, T ] , ∀c ∈ Rnx ,

V [X(t)](c) ≥ minν

maxξ,ω

cTf(ξ, ν, ω)

∣

∣

∣

∣

∣

∣

∣

ξ ∈ X(t)cTξ = V [X(t)](c)ν ∈ Uω ∈ W

,

which defines a sufficient condition for the set-valuedfunction X : [0, T ] → Π(Rnx) to be a robust forwardinvariant tube.

Our on-going investigations aim at developing efficientand tractable numerical procedures for constructing such



robust forward invariance tubes, as well as tractable algo-rithms for the on-line solution of the corresponding tube-based MPC problems.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge financial support fromMarie Curie Career Integration Grant PCIG09-GA-2011-293953, the Engineering and Physical Sciences ResearchCouncil (EPSRC) under Grant EP/J006572/1, and fromthe Centre of Process Systems Engineering (CPSE) ofImperial College London. MEV thanks CONACYT fordoctoral scholarship. NP and JR thank EPSRC and theDepartment of Chemical Engineering at Imperial CollegeLondon for doctoral training awards. RP acknowledges thesupport of European Commission under grant agreement291458 (MOBOCON).

Appendix A. CASE STUDY MODEL PARAMETERS

The parameter values used in the two-step anaerobicdigestion model in Sect. 1.1 can be found in Table A.1.

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Table A.1. Parameter values in two-step anaerobic digestion model Bernard et al. (2001).

Parameter Value Parameter Value

µ1 1.2 day−1 k1 42.14 g(COD) g(cell)−1

KS17.1 g(COD) L−1 k2 116.5 mmol g(cell)−1

µ2 0.74 day−1 k3 268.0 mmol g(cell)−1

KS29.28 mmol L−1 k4 50.6 mmol g(cell)−1

KI2 256 mmol L−1 k5 343.6 mmol g(cell)−1

kLa 19.8 day−1 k6 453.0 mmol g(cell)−1

KH 16 mmol L−1 atm−1 Sin1 5 g(COD)L−1

Pt 1 atm Sin2 80 mmol L−1

α 0.5 − Zin 50 mmol L−1

D 0.4 day−1 Cin 0 mmol L−1

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