interval observer design based on nonlinear hybridization

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2002; 00:1–6 Prepared using acsauth.cls [Version: 2002/11/11 v1.00]

Interval observer design based on nonlinear hybridization andpractical stability analysis

N. Meslem1 and N. Ramdani2,∗

1 INRA Jouy-en-Josas, France, e-mail: [email protected] CERTES Universite Paris-Est Creteil, France, e-mail: [email protected]

SUMMARY

In this paper we design an interval observer for nonlinear uncertain continuous-time dynamical systems, in theunknown-but-bounded error framework. We show how to generate two coupled hybrid observers, which yield thebest conservative upper and lower component-wise bounds for the state, provided bounds are available for initialconditions and uncertainties. The interval observer depends on a tuning gain, which is chosen in order to satisfythe practical stability of a given hybrid dynamical system. An example is studied to emphasize the performance ofour method.

Copyright c© 2002 John Wiley & Sons, Ltd.

KEY WORDS: Bounded-error state estimation; Interval observers; Nonlinear continuous-time systems; Hybridsystems; Practical stability

1. INTRODUCTION

State estimation with uncertain nonlinear continuous-time dynamical systems is an issue of importancein many fields. Classical point state estimation algorithms such as the extended Luenberger observer,the extended Kalman filter [1, 2], the high gain observer [3] and the sliding mode observer [4, 5] aremore or less robust with respect to disturbances and measurement noises, however they often fail togive satisfactory estimations in the presence of uncertainties in model parameters [6].

When all uncertain quantities, measurement noise, modeling error, model and input uncertainties,remain within a bounded set with known bounds, the estimation problem no longer has an uniquesolution and must be tackled with set-membership estimation techniques. Clearly, if the solution setcan be accurately estimated on-line, it would undoubtedly help developing new robust control anddiagnosis methods.

In the literature there are two approaches for set-membership state estimation (SME). They aim atproviding sets containing all the state vectors consistent with the model structure, uncertainties in initial

∗Correspondence to: Nacim Ramdani, IUT Creteil-Vitry, 122 rue Paul Armangot, 94400 Vitry-sur-Seine, France. E-mail:[email protected]

Received 31 October 2009Copyright c© 2002 John Wiley & Sons, Ltd. Revised 20 May 2010

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state and parameters and also with the experimental data and their feasible domains. These approachesare founded on reachability computation for uncertain nonlinear systems using novel arithmetics ongeometrical representations such as ellipsoids [7, 8], parallelotopes [9], zonotopes [10], boxes orinterval vectors [11]. In this paper, we will consider the latter representation.

The first type of SME approach addresses the case of state estimation from discrete-timemeasurement and was introduced in [12] and improved by [13, 14, 15, 16, 17, 18]. It relies on twosteps: (i) a prediction step, which consists in computing an over-approximation for the reachable setof the uncertain nonlinear system by using set integration methods for uncertain differential equations[19, 20, 21, 22]; (ii) a correction step, which uses consistency techniques to prune at each measurementtime instant inconsistent state vectors from the conservative estimation of the reached state set [11]. Inthe sequel of the paper, this approach will not be considered.

The second type of SME approach deals with the case of state estimation from continuous-timemeasurement. It was proposed for the first time in [23, 24] to address a particular class of uncertainsystems, namely bioreactors. The main concept of this approach consists in designing an intervalobserver which yields the best conservative upper and lower component-wise bounds for the stateprovided known bounds for initial conditions and uncertainties are available. Thereafter, severaltechniques were developed to improve the performance of this interval observer such as using lineartransformation, running in parallel bundle of interval observers and realizing regular re-initialization[25, 26, 27, 28].

The purpose of this paper is to introduce a generic method for designing interval observers for abroad class of uncertain nonlinear systems. Our method is based on (i) an enhanced version of thehybrid bounding method for computing over-approximation of the reachable set of uncertain nonlinearsystems [22]; and on (ii) a suitable choice of the observation gain matrix to guarantee the convergenceof the observation error, namely keeping the estimated state trajectories within a given tight boundedbox. To prove the latter point, we will use the notion of practical stability for hybrid and switchingsystems developed in [29]. Note that, our design method does not need the cooperative property of thelinear part of the proposed interval observer nor the boundedness of its nonlinear part, contrarywiseto the methods available in the literature. However, we assume that the nonlinear part is Lipschitzcontinuous with respect to the state vector and also with respect to the uncertain parameters vector.

The paper is divided into four sections. In Section 2, we briefly recall the bounding method foruncertain nonlinear systems, then we state in Section 3 our main result regarding the design of intervalobservers. An illustrative example is given in Section 4 to show the potential of the proposed hybridinterval observer. Appendix II recalls Muller’s theorem. Finally, in Appendix III, we prove the mainresult of the paper using the practical stability concept, and we show how to compute a suitableobservation gain matrix.

2. BOUNDING METHOD FOR UNCERTAIN NONLINEAR SYSTEMSThe main techniques used by our interval observer design approach is the hybrid bounding methodfor computing an over-approximation of the state flow generated by uncertain nonlinear dynamicalsystems [22, 30]. It is recalled in this section.

Copyright c© 2002 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2002; 00:1–6Prepared using acsauth.cls

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2.1. Bounding using Muller’s theorem

Consider the nonlinear uncertain continuous-time dynamical system

x = f(x,p, t), x0 ∈ [x0,x0]⊂ D⊆ Rn, p ∈ [p,p] = P⊂ Rnp and t ≥ 0. (1)

The vector field f(., ., .) is continuous over the domain D×P×R+. The initial state vector and theuncertain parameters vector are defined as interval vectors [11] with dimensions n and np respectively.

The core idea of Muller’s theorem [31, 32], which is given in Appendix II for completeness,consists in building a lower and an upper coupled time-continuous dynamical systems which involve nouncertainty and enclose in a guaranteed way all the state trajectories generated by the original uncertainsystem (1). In the sequel, we will write the coupled deterministic bounding systems as follows

x = f(x,x,p,p, t)x = f(x,x,p,p, t)

x(t0) = x0,x(t0) = x0

(2)

where x(t) denotes the minimal solution and x(t) the maximal solution of (1).The question which arises now is how to obtain the bounding systems (2) from the uncertain system

(1). In fact, when all the components of f(., ., .) are monotonic with respect to each uncertain parameterpk and each state variable x j, it is quite easy to obtain these bounding systems as proposed in [32].

Rule 1 (Monotonicity test) [32]: For instance, in order to build the upper system, that is the onewhich yields the maximal solution x(t), one must replace in the formal expression of fi(., ., .), i ∈{1, . . . ,n},

1. xi by xi,2. x j by x j if ∂ fi(.,.,.)

∂x j≥ 0 or by x j if ∂ fi(.,.,.)

∂x j< 0, j ∈ {1, . . . ,n} and j 6= i

3. pk by pk if ∂ fi(.,.,.)∂ pk

≥ 0 or by pk if ∂ fi(.,.,.)∂ pk

< 0, k ∈ {1, . . . ,np}.

The components of the lower system, that is the one which yields the lower solution x(t) are derived byreversing monotonicity conditions.

A detailed presentation of the use of this rule can be found in [33, 22]. Nevertheless, to illustrate itsuse, let us consider the uncertain linear system

x =

[−1 0.8−4 2

]x+[

p0

]u(t) (3)

where u(t) is an unit step (u(t) = 1, t ≥ t0). The initial state vector x(t0) ∈ [x(t0),x(t0)]⊂ D= R2 andthe uncertain parameter p ∈ [p, p] ⊂ R. For this system, it is clear that all the partial derivatives withrespect to the state variables and the uncertain parameter have constant sign. Then, we can use Rule 1to obtain the following coupled systems that yield bounding systems for (3)

[xx

]=

−1 0.8 0 00 2 −4 00 0 −1 0.8−4 0 0 2

[ xx

]+

p0p0

u(t). (4)


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Remark 1. When the systems under study are order-preserving monotone dynamical systems [34], wecan build the lower and the upper systems separately [30]. Thus, the bounding systems are decoupledand can be written as follows {

x = f(x,p,p, t), x(t0) = x0x = f(x,p,p, t), x(t0) = x0.

(5)

In next section, we show how to use Rule 1 for bounding systems for which the sign of some partialderivatives change within the reachable set.

2.2. Bounding using a nonlinear hybridization

In practice, one does not expect the signs of the partial derivatives of nonlinear systems to remainconstant over the reachable set. Therefore, it is of interest to partition the state space D of the uncertainsystem – in fact it is not necessary to partition whole state space, but an over-approximation of thereachable set only – into cells that only intersect on their boundaries, and within which the partialderivatives have constant signs. Then, one can use Rule 1 over each cell to build suitable boundingsystems. Finally, as time spans [t0, t f ], the collection of these bounding systems can be used togenerate an over-approximation of the set reachable by the original uncertain system. In other words,this nonlinear hybridization method for reachability computation introduced in [22] uses a switchingsystem H composed of a collection of time-continuous bounding dynamical systems

H = {(M1,D1),(M2,D2), . . . ,(Mq,Dq), . . .}, q ∈ {1, . . . , l} (6)

where each subsystem Mq is obtained by using Rule 1 on each domain Dq. They are described by thefollowing state representation

Mq ≡{

x(t) = fq(x,x,p,p, t)x(t) = fq(x,x,p,p, t)

}for q ∈ L = {1, . . . , l}. (7)

The switching sequence σ ={(t0,q0), (t1,q1), . . . ,(ts,qs), . . .

}(where qs ∈ L and t0 ≤ t1 ≤ ·· · ≤ ts ≤

. . . ) of the switching system H that specifies that the local bounding system Mqs is active during[ts, ts+1) is generated by the sign change of the partial derivatives

Cp(i,k)(x,p, t) = ∂ fi(x,p, t)/∂ pk, i ∈ {1, . . . ,n}, k ∈ {1, . . . ,np}Cx(i, j)(x,p, t) = ∂ fi(x,p, t)/∂x j, i, j ∈ {1, . . . ,n}, i 6= j. (8)

These partial derivatives act as guard conditions that drive the discrete transitions from one boundingsystem to another one. For each discrete transition corresponds a reset function given by

(x(ts),x(ts)) = Jqs−1,qs

(x(t−s ),x(t−s )

)= (x(t−s ),x(t−s )). (9)

We can now define the state-dependent switching law γ over time interval τ = [t0, t f ] as follows

Definition 1 (State-dependent switching law γ) Given a time interval τ , a switching law γ over τ isa mapping: Rn×Rnp → ∑τ which specifies a nonZeno switching sequence σ ∈ ∑τ for any boxes ofinitial state [x0] and uncertain parameters [p]. Here ∑τ , {switching sequence σ over τ}.

To summarize, for a given initial state box [x0] and interval vector of uncertain parameters [p], theexecution of the hybrid deterministic automaton (6) characterizes a conservative over-approximationof the reachable set of the nonlinear continuous-time dynamical system (1).


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2.3. Practical bounding methods

When one uses our hybrid bounding approach with uncertain dynamical systems, one often meetssituations where for some time instants t, the sign of some partial derivatives (8) are not constant overthe over-approximation [x(t),x(t)] of the reachable set, hence Rule 1 cannot be used. In the sequel, weintroduce propositions that can be used to overcome this issue.

Let us consider the components fi(.,z j, .) for which the range of (∂ fi/∂ z j)(.,z j, .) = Cz(i, j)(.,z j, .)over [z j,z j] contains zero, where i ∈ {1, . . . ,n} and z j denotes either a state variable x j j 6= i or anuncertain parameter pk.

Property 1 (P1) ([18] and the references therein) Any Lipschitz function fi can be written as the sumof one decreasing function d(.,z j, .) and one increasing function c(.,z j, .), i.e.:

∀z j ∈ [z j,z j] = [z j], fi(.,z j, .) = d(.,z j, .)+ c(.,z j, .) (10)

Hence,∀z j ∈ [z j,z j], di(.,z j, .)+ ci(.,z j, .)≤ fi(.,z j, .)≤ di(.,z j, .)+ ci(.,z j, .). (11)

In fact, there are several ways of decomposing function fi. Some decompositions may introduce largeoverapproximation when writing inequalities (11), hence very loose bounds for the computed reachableset. Therefore, we must look for decompositions that do not introduce the latter over-approximations,decompositions which are obtained using what we define as a pragmatic way. In other words, to obtainthe inequalities (11), one would, for instance, analyze the monotonicity of some subexpressions of fiinstead of using Lipschitz constants. Furthermore, it is also interesting to investigate the possibilityto decompose, in a pragmatic way, function fi as the product of one increasing function and onedecreasing one, which we summarize in the following hypothesis (H1).

Hypothesis 1 (H1) Function fi(.,z j, .) can be written in a pragmatic way as the product of onedecreasing function di(.,z j, .) and one increasing function ci(.,z j, .), viz.

∀z j ∈ [z j,z j], fi(.,z j, .) = di(.,z j, .)× ci(.,z j, .). (12)

Proposition 2 (P2) If (H1) is true, then according to the sign of di(.,z j, .) and ci(.,z j, .) on the boundedinterval [z j], it is always possible to frame fi(.,z j, .) by two elements of the following set,

B = {di(.,z j, .)× ci(.,z j, .); di(.,z j, .)× ci(.,z j, .); di(.,z j, .)× ci(.,z j, .);

di(.,z j, .)× ci(.,z j, .); di(.,z′j, .)× ci(.,z′′j , .); min( fi(.,z j, .), fi(.,z j, .))} (13)

where z′j, z′′j belong to [z j,z j] if ci(.,z j, .) and di(.,z j, .) do not have constant sign on [z j,z j], and satisfyrespectively ci(.,z′j, .) = 0, di(.,z′′j , .) = 0.

Proof. Consider the case where the sign of di(.,z j, .) and ci(.,z j, .) is not constant on the boundedinterval [zi]. Then, since ci(.,z j, .) is increasing and di(.,z j, .) is decreasing it is clear that the minimalvalue of fi(., ., .) is negative and given by

fi= min{ci(.,z j, .)×di(.,z j, .), ci(.,z j, .)×di(.,z j, .)} = min{ fi(.,z j, .), fi(.,z j, .)} (14)

Now, to obtain the maximal value of fi(., ., .) which is obviously positive, let us characterize the domain[z j]′ ∈ [z j] over which fi(., ., .) is positive. To do so, we must compute z′j and z′′j for which ci(.,z′j, .) = 0


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and di(.,z′′j , .) = 0. Then we obtain [z j]′ = [z′j,z

′′j ] or [z j]

′ = [z′′j ,z′j]. Thus, since ci(.,z j, .) and di(.,z j, .)

have the same sign on [z j]′, by direct computation we can upper bound fi(.,z j, .) by

fi(.,z j, .)≤ di(.,z′j, .)× ci(.,z′′j , .) = f i. (15)

Hence (14) and (15) belong to B. To complete the proof of the proposition we can use a similarreasoning to show that the set B contains all the possible lower and upper bounds of fi(.,z j, .) whateverthe signs of di(.,z j, .) and ci(.,z j, .). �

Next example will illustrate the use of property P1 and proposition P2 within our hybrid boundingapproach.

Example 1. Consider the nonlinear continuous-time dynamical system{x1 = x2x2 = −2x2

1 exp(x2−1) (16)

where initial state vector is uncertain (x1(t0),x2(t0)) ∈ [x1(t0)]× [x2(t0)] ⊂ R2. To characterize ina guaranteed way the reachable set, we will build the hybrid automaton according to the sign ofthe partial derivatives. For all x in R2, we have Cx(1,2)(x) = ∂ x1/∂x2 = 1, Cx(2,1)(x) = ∂ x2/∂x1 =−4x1 exp(x2− 1), and sign(Cx(2,1)(x)) = −sign(x1). Then, given a time instant t, we can build thefollowing locally bounding systems, according to the sign of Cx(2,1)(x) on [x1](t):

• If for all x1 ∈ [x1](t), x1 < 0 then sign(Cx(2,1)(x)) is positive. By using the monotonicity test given inRule 1 we obtain the first bounding system M1,

M1 =

x1 = x2x2 = −2x2

1 exp(x2−1)x1 = x2x2 = −2x2

1 exp(x2−1)

(17)

• If for all x1 ∈ [x1](t), x1 > 0 then sign(Cx(2,1)(x)) is negative and thanks to Rule 1 we obtain the secondbounding system M2,

M2 =

x1 = x2x2 = −2x2

1 exp(x2−1)x1 = x2x2 = −2x2

1 exp(x2−1)

(18)

• Now, if 0 ∈ [x1](t) the sign of Cx(2,1)(x) changes on ([x1](t), [x2](t)) and Rule 1 cannot be used.We can use Proposition P2 to frame x2. First, we write x2 = d2(x1)× c2(x1,x2), where c2(x1,x2) =x1 exp(x2 − 1) is an increasing function of x1 and d2(x1) = −2x1 is a decreasing function of x1.Second, since x2 is negative, the signs of c2(., .) and d2(.) are not constant over ([x1](t), [x2](t)), andc2(0,x2) = d2(0) = 0, we can use the following double inequality to frame x2 on [x1](t) as suggestedin Proposition P2: ∀(x1,x2) ∈ [x1](t)× [x2](t), −2min{x2

1,x21}exp(x2−1)≤ x2 ≤ 0. Finally, thanks to

Rule 1 and Proposition P2, we derive the third bounding system M3 as follows:

M3 =

x1 = x2x2 = 0x1 = x2x2 = −2min{x2

1,x21}exp(x2−1)

(19)


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Hence, for any initial state vector box [x](t0), an over-approximation of the reachable set of (16)is obtained by running the hybrid system H = {M1,M2,M3} governed by the automaton depicted onFigure 1. Each bounding system Mq, q ∈ {1,2,3}, is activated on specific state region delimited by theguard condition Cx(2,1)(x) as illustrated on Figure 2. On Figure 3, a zoom around the guard conditionis depicted to illustrate the fact that bounding system M3 is necessary to cross the guard set.

Finally, note that when one uses our hybrid bounding method, the cardinality of the collection of allpossible bounding dynamical systems (as defined by (6)) depends on the number of partial derivativesthat can change sign onto the reachable set. When the sign of a partial derivative over the over-approximation [x(t),x(t)] of the reachable set is either positive or negative, Rule 1 can be used, butwhen the partial derivative changes sign over [x(t),x(t)], property P1 or proposition P2 must be used.There are three cases, hence the cardinality of H

l = card(H ) = 3d ,

where d is the number of these partial derivatives.

3. HYBRID INTERVAL OBSERVER

Consider the uncertain continuous-time dynamical system described by the following nonlinear stateequations

x = A(t)x+Φ(x,p, t)y = C(t)xp ∈ [p]x(t0) ∈ [x0]

(20)

where A(t) and C(t) are matrices with dimension n×n and m×n respectively, and where n, m and npare respectively, the dimensions of the state vector x ∈ Rn, the output vector y ∈ Rm and the uncertainparameter vector p ∈ [p] ⊂ Rnp . Φ(., ., .) is a n-dimensional vector of Lipschitz continuous functions(w.r.t both parameter and state variables) defined on Rn×Rnp ×R+.

Remark 2. In practice, formulation (20) is not restrictive, since we can rewrite without loss ofgenerality a given nonlinear equation x = f(x,p, t) as

f(x,p, t) = A(t)x+(f(x,p, t)−A(t)x

)= A(t)x+Φ(x,p, t). (21)

For instance, matrix A(t) can be taken as the gradient ∇f(x1,p1, t) where x1(t) is the trajectory solutionof x = f(x,p, t) when x1(t0) = Mid([x0]) and p1 = Mid([p]). We denote by Mid(.) the midpoint of aninterval vector [11]. Moreover, this formulation encloses situations where the state equations dependof some uncertain inputs which can be viewed as uncertain parameters.

We assume that measurements yM(t) are subject to additive errors, that are unknown but bounded withknown bounds. Thus the feasible domains for measurements are given by the following boxes

Y(t) = [yM(t)−b(t),yM(t)+b(t)] (22)

where b(.) are vectors of nonnegative entries b j(.), j ∈ {1, . . . ,m}, that correspond to a priori knownmeasurement errors bounds.

Here we give some matrix definitions useful to state our main result.


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Definition 2. For an arbitrary real n× n matrix A we let A be the n× n Metzler matrix (i.e. with alloff-diagonal elements non-negative), defined by

ai j =

{ai j, if i = j|ai j|, if i 6= j. (23)

Definition 3. Let A be an arbitrary square real matrix of order n. We denote by A its associatedsquare matrix of order 2n, obtained when applying the two first points of Rule 1 to the dynamicalsystem x = Ax, x0 ∈ [x0], viz.

(x, x)T = A (x,x)T . (24)

To be clear, reconsider system (3) of subsection 2.1. Then, when we apply the two first items of Rule 1with the square matrix

A =

[−1 0.8−4 2

],

we obtain

A =

−1 0.8 0 00 2 −4 00 0 −1 0.8−4 0 0 2

.Definition 4. Let K be an n×m arbitrary real matrix. Denote by K its associated real matrix withdimension 2n× 2m obtained when we use the third item of Rule 1 to bound the algebraic systemz =−Ky when y ∈ [y], viz.

(z,z)T = K (y,y)T . (25)

Now, let (26) be a Luenberger-like observer for system (20) when we assume that its parameters areknown perfectly and its measurements are exact,{

ˆx = A(t)x+Φ(x,p, t)+K(t)(y(t)−y(t))y = C(t)x (26)

where K(.) is the observation gain matrix with dimension n×m. Hence, it is clear that for any matrixK(.), any x0 ∈ [x0] and any p ∈ [p], the solution of system (20) is also a solution of the Luenberger-like observer (26). By using our nonlinear hybridization bounding method we can derive the hybridobserver (27) that generates a guaranteed enclosure of all the possible state trajectories of the uncertainnonlinear system (20).[

xx

]= [Aq(t)+Cq(t)]

[xx

]+

[Φq(x,x,p,p, t)Φq(x,x,p,p, t)

]+Kq(t)

[yM(t)yM(t)

], (27)

where q is taken in the set L = {1, . . . , l}, which contains the indexes of the continuous modesconstituting the hybrid system (27). Matrices Aq(.) and Cq(.) are of dimension 2n × 2n, builtrespectively from matrices A(.) and K(.)C(.) according to definition 3. Also, matrices Kq(.) are ofdimension 2n×2m, built from matrix −K(.) according to definition 4.

We can now state our main result.


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Theorem 1. For all q ∈ L, define Eq(t) = A(t)+Kq(t)C(t), and λ0(t) = max{λiq(t),0} a piecewisecontinuous positive function, where λiq(t) are the eigenvalues of matrices Eq(t) associated to matricesEq(t) according to definition 2.

If λ0(t) satisfies ∫ t f

t0λ0(z)dz <

12, (28)

then the hybrid system (27) is an interval observer for the uncertain system (20) with the followingproperties satisfied ∀ x(t0) ∈ [x(t0),x(t0)], ∀ p ∈ [p,p], ∀ t ≥ t0,

Positivity:(x(t)−x(t)≥ 0

)∧(x(t)−x(t)≥ 0

)(29)

Convergence: limt→+∞

‖x(t)−x(t)‖= w(‖p−p‖,‖b‖) (30)

where w(., .) is a scalar increasing function w.r.t. to both the width of the parameters uncertainty box,and measurements errors bounds. Here ‖.‖ denotes the maximal norm of vectors.

Proof. see Appendix III. �In summary, we have given a condition that ensures that the hybrid interval observer (27) yields

on-line, an upper x(t) and a lower x(t) state trajectories that enclose in guaranteed way all thepossible state vectors generated by the uncertain system (20) and consistent with the feasible domainsfor experimental data. Note that our interval observer design method needs not the matrices A(.)+Kq(.)C(.) to be cooperative nor the nonlinear function Φ(., ., .) to be bounded. These relaxations are ofimportance and should allow the design of interval observers for a broader class of uncertain nonlinearsystems, including those considered in [23, 24, 26, 27, 35].

4. EXAMPLE

As an illustrative example, we consider a three-states biochemical reactor given in [36], where the statevariables are the biomass x1, the substrate x2 and the product x3. The process can be modeled by thefollowing ordinary differential equations (31) and the output equations (32) :

x1x2x3

=

−D 0 00 −D 0β 0 −D

x1x2x3

+ µ(x2,x3)x1

Dx2 f − µ(x2,x3)x1Y

αµ(x2,x3)x1

(31)

[y1y2

]=

[0 1 00 0 1

] x1x2x3

(32)

where the specific growth rate µ(., .) is a function of both the substrate and the product concentrations

µ(x2,x3) =µm(1− (x3/x3m)

)x2

ks + x2. (33)

We assume that the maximum growth rate µm, the saturation parameter ks and the input substratex2 f are unknown but belong to the following intervals [0.35,0.45] h−1, [1.4,1.6] g/L and [17,23] g/Lrespectively. We denote (p1, p2, p3)

T = (µm,ks,x2 f )T . The initial state variables are also unknown


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but within the interval vector [20,100]× [10,100]× [100,200] (g/L)3. The vector of measurements(y1m(t),y2m(t)

)T is corrupted by noise and we assume that the actual output vector(y1(t),y2(t)

)T

belongs to a bounded feasible domain,(y1(t),y2(t)

)T ∈ [y1m(t)− b1(t),y1m(t) + b1(t)]× [y2m(t)−b2(t),y2m(t)+b2(t)], where b1(t)= 0.05 ·y1m(t) and b2(t)= 0.05 ·y2m(t) are the maximal measurementerrors. Other parameters and inputs are assumed to be known exactly with the following values Y = 0.4,β = 0.2 h−1, D = 0.4 h−1, α = 2.2, and x3m = 50 g/L.

4.1. Designing the hybrid interval observer

4.1.1. Tuning the observation gain matrix. Note that, for this example, all the entries of matrices

A =

−D 0 00 −D 0β 0 −D

and C =

[0 1 00 0 1

], (34)

are taken constants. Let us define E = A+KC. Then, let us look for conditions on the entries of thegain matrix K such that matrix E, associated to E according to definition 2, satisfies condition (28)needed to apply Theorem 1. To do so, we must compute the eigenvalues of E as functions of matrix Kentries. We have

det|λ I− E|= det

∣∣∣∣∣∣λ +D −|k1| −|k4|0 λ +D− k2 −|k5|−β −|k3| λ +D− k6

∣∣∣∣∣∣= (λ +D)

((λ +D− k2)(λ +D− k6)−|k3||k5|

)−β

[|k1||k5|+(λ +D− k2)|k4|

)(35)

Then, if we set k4 = k5 = 0 we obtain the following eigenvalues for E

λ1 =−D, λ2 = k2−D, λ3 = k6−D. (36)

It is obviously sufficient to take k2 < 0 and k6 < 0 so that (28) is satisfied. Indeed, for k2 < 0 and k6 < 0,we have λ0 = max{λ1,λ2,λ3,0}= 0, and hence (28) is always true, i.e.

∫ t ft0 λ0dz = 0 < 1

2 .

Remark 3. Note that we need not to assume matrix A+KC to be cooperative to build the intervalobserver (42). Hence, k1 and k3 can take any value in R.

4.1.2. Building the bounding systems. Obviously, we can deal with the linear part and the nonlinearpart of the bioreactor separately to build the bounding systems. We start by bounding the linear part ofbioreactor (31). To do so, we use Rule 1 and we obtain

[A +C ] =

−D k1 0 0 0 00 −D+ k2 0 0 0 0β 0 −D+ k6 0 k3 00 0 0 −D k1 00 0 0 0 −D+ k2 00 k3 0 β 0 −D+ k6

(37)

Now, to build bounding functions for the nonlinear part of the bioreactor (31)

Φ(x,p) =

µ(x2,x3)x1

Dx2 f − µ(x2,x3)x1Y

αµ(x2,x3)x1

(38)


11

we must analyze the signs of its partial derivatives with respect to the state variables and uncertainparameters on the state space. Note that, for this example, the state variables xi, i ∈ {1,2,3}, arealways positive because for xi = 0, we have xi ≥ 0, namely the faces xi = 0 are repulsive. Forexample, we have for x2 = 0, x2 = Dx2 f > 0. By direct computation we can show that the signof the partial derivatives (∂φ1/∂x2)(x,p)), (∂φ1/∂ p1)(x,p)), (∂φ2/∂ p2)(x,p)), (∂φ3/∂x1)(x,p)),(∂φ3/∂x2)(x,p)), and (∂φ3/∂ p1)(x,p)) is equal to the sign of (1− x3

x3m). Similarly, the sign of the

partial derivatives (∂φ1/∂ p1)(x,p)), (∂φ2/∂x1)(x,p)), (∂φ3/∂ p2)(x,p)), and (∂φ2/∂ p1)(x,p)) isequal to the sign of ( x3

x3m− 1). Finally, the above partial derivatives signs depend on the sign of the

unique expression 1− x3x3m

, for x3 ∈ [x3](t). All the other partial derivatives have constant sign. Finally,because we have dealt with the linear and nonlinear parts separately, and because we have detectedthat partial derivatives signs are driven by a single function, we can reduce the cardinal number ofthe collection of continuous bounding dynamical systems of hybrid system (6) to 31 = 3 instead of39 = 19683. Hence the nonlinear part of the bioreactor (31) has three bounding functions according tothe sign of 1− x3(t)

x3m, for a given time instant t in [t0, t f ]:

• If (∀x3 ∈ [x3](t), x3 ≤ x3m ) by using only Rule 1, we obtain as first bounding functions

Φ1(x,x,p,p) =

µm(1−

x3x3m

)x2

ks+x2x1

Dx2 f − 1Y

µm(1−x3

x3m)x2

ks+x2x1

αµm(1−

x3x3m

)x2

ks+x2x1

Φ1(x,x,p,p) =

µm(1−

x3x3m

)x2

ks+x2x1

Dx2 f − 1Y

µm(1−x3

x3m)x2

ks+x2x1

αµm(1−

x3x3m

)x2

ks+x2x1

(39)

• If (∀x3 ∈ [x3](t), x3 > x3m) by using only Rule 1, we obtain as second bounding functions

Φ2(x,x,p,p) =

µm(1−

x3x3m

)x2

ks+x2x1

Dx2 f − 1Y

µm(1−x3

x3m)x2

ks+x2x1

αµm(1−

x3x3m

)x2

ks+x2x1

Φ2(x,x,p,p) =

µm(1−

x3x3m

)x2

ks+x2x1

Dx2 f − 1Y

µm(1−x3

x3m)x2

ks+x2x1

αµm(1−

x3x3m

)x2

ks+x2x1

(40)

• If (x3m ∈ [x3](t)) then 1 − x3x3m

has not constant sign on [x3](t), therefore Rule 1 is notapplicable. We must use property P1, i.e. the following decomposition of the specificgrowth rate µ(x2,x3) = µ1(x2,ks,µm)+ µ2(x2,x3,ks,µm), where µ1(x2,ks,µm) = µm

x2ks+x2

andµ2(x2,x3,ks,µm) =−µm

x3x2x3m(ks+x2)

, and we obtain the third bounding function


12 AA

Φ3(x,x,p,p) =

µmx2ks+x2

x1−µmx3x2

x3m(ks+x2)x1

Dx2 f − 1Y (

µmx2

ks+x2x1−

µmx3x2x3m(ks+x2)

x1)

α(µmx2ks+x2

x1−µmx3x2

x3m(ks+x2)x1)

Φ3(x,x,p,p) =

µmx2

ks+x2x1−

µmx3x2x3m(ks+x2)

x1

Dx2 f − 1Y (

µmx2ks+x2

x1−µmx3x2

x3m(ks+x2)x1)

α(µmx2

ks+x2x1−

µmx3x2x3m(ks+x2)

x1)

(41)

Finally, with the observation gain matrix

K =

k1 0k2 0k3 k6

,we obtain the following hybrid interval observer for bioreactor (31)

[xx

]=

−D k1 0 0 0 00 −D+ k2 0 0 0 0β 0 −D+ k6 0 k3 00 0 0 −D k1 00 0 0 0 −D+ k2 00 k3 0 β 0 −D+ k6

[

xx

]

+

[Φq(x,x,p,p)Φq(x,x,p,p)

]+

0 0 −k1 0−k2 0 0 0−k3 −k6 0 0−k1 0 0 0

0 0 −k2 00 0 −k3 −k6

y1m(t)+b1(t)y2m(t)+b2(t)y1m(t)−b1(t)y2m(t)−b2(t)

, (42)

where q∈ L = {1, 2, 3} and k2, k6 must be negative. The switching sequence σ is generated accordingto the sign of 1− x3(t)

x3m, when x3(t) is taken in [x3](t).

4.2. Simulation.

Simulated measurements are obtained by running system (31) under the following conditions: initialstate variables are taken as x1(0) = 65 g/L, x2(0) = 50 g/L, and x3(0) = 150 g/L, and parameters aretaken as µm = 0.4 h−1, ks = 1.5 g/L and x2 f = 20 g/L. For the hybrid interval observer we have chosenthe following values for the entries of the observation gain matrix: k1 = 0, k4 = 0, k5 = 0, k2 = −15,k3 =−10, and k6 =−20.

The blue bold continuous curves on Figures 4, 5 and III.2.2 show a guaranteed enclosure of allthe possible state trajectories of (31) consistant with all the uncertainties and the feasible domainsfor measurements. The red continuous curves denote the supposed actual state vector of (31). Figure7 shows the autonomous switching signal which denotes the actual switching sequence between thethree bounding subsystems.


13

5. CONCLUSION

We have introduced a new method for designing interval observers for uncertain nonlinear dynamicalsystems based on an improved version of our hybrid method for computing an over-approximation ofreachable sets. We have also proved the positivity of the observation error and its practical stabilitythanks to a suitable tuning of the observation gain matrix. Another contribution of the work consistsin replacing the cooperativity, stability and boundedness assumptions by an unique practical stabilitycondition. In future works, we plan to use this type of interval observer for model-based fault detection.

APPENDIX

II. An adapted version of Muller’s theorem

The theorem below presents an adapted version of Muller’s theorem as given in [32]. It is given herefor completeness.

Theorem 2 (Muller’s theorem [31, 32, 37]) If xi(t) and xi(t) satisfy the following inequalities for alli ∈ {1, . . . ,n}

• the left Dini derivatives D−xi(t) and D−xi(t) and the right Dini derivatives D+xi(t) and D+xi(t) ofxi(t) and xi(t) are such that

D±xi(t)≤ minD(t)

fi(x,p, t) (43)

D±xi(t)≥ maxD(t)

fi(x,p, t) (44)

where D(t) is the subset of D(t) defined by

Di :

xi = xi(t)

x j(t)≤ x j ≤ x j(t), j 6= i

p≤ p≤ p(45)

and where D(t) is the subset of D(t) defined by

Di :

xi = xi(t)

x j(t)≤ x j ≤ x j(t), j 6= i

p≤ p≤ p(46)

Then for all x0 ∈ [x0,x0], p ∈ [p,p], the solution x(t) of system (1) exists and remains bounded by

x(t)≤ x(t)≤ x(t), ∀ t0 ≥ t (47)

In addition, if for all p ∈ [p0,p0], function f(x,p, t) is Lipschitz with respect to x over D, then thissolution is unique for any given p.

III. PROOF OF THE MAIN RESULTS

In this Appendix, we prove the results stated in Section 3, namely the positivity and the convergenceof the proposed hybrid interval observer.


14 AA

III.1. Proof of positivity

Before proving the positivity of the observation error, let us state the following propositions.

Proposition 3 (P3) Let A be an arbitrary square real matrix of order n and

• A is its associated square matrix of order 2n obtained according to definition 3,• ˜A is the associated square matrix to A according to definition 2.

Then we have for all x ∈ [x,x][A−A

][xx

]+

[−A 00 A

][xx

]= ˜A

[x−xx−x

], (48)

where A and A are submatrices of A with dimension n×2n such that A =

[AA

].

Proof. For any i ∈ {1, . . . ,n}, we have from the left side of (48) the following expression

Aiixi + ∑j∈S

Ai jx j +n

∑j=1

Ain+ jx j−Aiixi−∑j∈S

Ai jx j (49)

where S= {1, . . . , i−1, i+1, . . . ,n}, Aii = Aii, and ∀ j ∈ S, Ai j ≥ 0 and Ai(n+ j) ≤ 0. We define by S1the subset of S such that for all j ∈ S1

Ai j 6= 0 and Ai j = Ai j > 0.

In the same way, we define by R1 the subset of {S−S1} such that for all j ∈ R1

Ai(n+ j) 6= 0 and Ai(n+ j) = Ai j < 0.

Then (49) becomes

Aii(xi− xi)+ ∑j∈S1

Ai j(x j− x j)+ ∑j∈R1

−Ai(n+ j)(x j− x j).

Moreover, since for all j ∈ S, Ai j ≥ 0 and Ai(n+ j) ≤ 0, we obtain

Aii(xi− xi)+ ∑j∈S1

|Ai j|(x j− x j)+ ∑j∈R1

|Ai(n+ j)|(x j− x j).

which is exactly the ith line of the right side of (48). Similarity we can proove this equality for anyi ∈ {n+1, . . . ,2n}. This completes the proof. �

Definition 5. For an arbitrary real n×m matrix A, let B = |A| be the n×m matrix defined by

bi j = |ai j|, ∀i ∈ {1, . . . ,n}, ∀ j ∈ {1, . . . ,m}. (50)

Proposition 4 (P4) Let −K be an n×m arbitrary real matrix, and K its associated real matrix withdimension 2n×2m obtained according to definition 4. Then the following equality holds[

K−K

][yy

]+

[K 00 −K

][yy

]= |K |

[y−yy−y

](51)

where K and K are submatrices of K with dimension n×2m.


15

Proof. We can prove this proposition with the same reasoning as the one in the proof of Proposition P3except that we set S= {1, . . . ,n}. �

Now, let us define by e(t) = x(t)−x(t) and e(t) = x(t)−x(t) respectively the upper and the lowerobservation errors, where x(t) and x(t) are the solutions of the hybrid coupled observer (27) and x(t) isthe actual solution of (20). Then, we will show that for all t ≥ t0, both errors are positive for any initialstate vector x0 ∈ [x0,x0], and any vector of parameters p ∈ [p].Thanks to Proposition P3 and Proposition P4, and by simple computation we can describe the dynamicsof the observation errors by[

ee

]= Eq(t)

[ee

]+

[Φq(x,x,p,p, t)−Φq(x,p, t)Φq(x,p, t)−Φq(x,x,p,p, t)

]+ |Kq|

[b(t)b(t)

], (52)

where q ∈ L = {1, . . . , l} andEq(t) = Aq(t)+Cq(t). (53)

Then, the dynamics of the observation errors has the following properties: ∀x0 ∈ [x0], ∀p ∈ [p], ∀q ∈ Land ∀t ≥ t0

• the matrices Eq(t) are Metzler,• all the entries of matrices |Kq| are positive,• in line with Rule 1, Property P1 and Proposition P2, one has

Φq(x,x,p,p, t)−Φq(x,p, t)≥ 0 and Φq(x,p, t)−Φq(x,x,p,p, t)≥ 0.

As a consequence, the theory of monotone cooperative systems [34] allows us to state that:

• if e(t0) ≥ 0 and e(t0) ≥ 0, then ∀x0 ∈ [x0], ∀p ∈ [p], and ∀t ≥ t0 the observation errors e(t) ande(t) remain positive.

As a consequence, we can summarize the proof of positivity in the following proposition.

Proposition 5 (P5) The hybrid interval observer (27) satisfies by construction property (29), for anyKq.

III.2. Proof of convergence

The aim of interval observers is to yield an enclosure of the state trajectories with a convergent size,namely property (30). Then, it is of interest to use the notion of practical stability to obtain conditionson the gain matrices Kq which guarantee this convergence.

III.2.1. Practical stability for hybrid systems. Here, we will use theorems stated in [29] which givesufficient conditions for ε-practical stability of hybrid systems, that is conditions which keep statetrajectories within given bounds. We recall these theorems and we show how to use them to computesuitable observation gain matrices Kq which ensure the practical stability of the observation error.

Definition 6 (ε-practical stability over τ) Assume that a time interval τ and switching law γ over τ

are given. Given an ε > 0, the hybrid system H is said to be ε-practically stable over τ under γ if thereexists a δ > 0 such that ‖y(t)‖ ≤ ε, ∀t ∈ τ whenever ‖y(t0)‖ ≤ δ . Here y(t) denotes the continuousstate trajectory of H .


16 AA

Definition 7 (ε-practical Lyapunov-like function) Given a time interval τ and switching law γ overτ , a continuously differentiable real-valued function V (y(t), t) satisfying V (0, t) = 0, ∀t ∈ τ is a ε-practical Lyapunov-like function over τ under γ if there exists a Lebesgue integrable function ϕ(y(t), t),positive constants µ1 and µ2, such that for any trajectory y(t) generated by γ that starts from y(t0) andits corresponding switching sequence σ = ((t0,q0), (t1,q1), . . . ,(ts,qs), . . .)∈∑τ , qs ∈ L, the followingholds

1. V (y(t), t)≤ ϕ(y(t), t), a.e. t ∈ τ;2. V

(Jqs,qs+1(y(t

−s )), t−s )≤ µ1V (y(t−s ), t−s ) at any switching instant ts;

3.∫ t

t0 µN(z,t)1 ϕ(y(z),z)dz < inf‖x‖≥ε V (y(t), t)−µ2µ

N(t0,t)1 , ∀t ∈ τ .

N(ta, tb) denotes the number of switching during the time interval τ , and reset functions Jqs,qs+1 aredefined by (9).

Note that V (., .) is not a Lyapunov function in the usual sense, since no definiteness condition isimposed on it or its derivatives.

Theorem 3 ([29]) Given a time interval τ and switching law γ over τ , an hybrid system H is ε-practically stable over τ under γ , if there exists an ε-practical Lyapunov-like function V (y(t), t) overτ under γ .

In the next sections, we show how to apply the above results to the dynamics the observation error.

III.2.2. Practical stability of the observation error. First we can easily prove that the dynamics of thetotal observation error e(t) = e(t)+ e(t) = x(t)−x(t) is given by the hybrid system below

e = Eq(t)e+(Φq(x,x,p,p, t)−Φq(x,x,p,p, t)

)+ |Kq|b(t) (54)

where q ∈ L = {1, . . . , l} and Eq(t) = A(t)+KqC(t). We can rewrite this hybrid system as follows

e = Eq(t)e+hq(t), q ∈ L = {1, . . . , l}. (55)

where hq(t) = Φq(z1,r1, t)−Φq(z2,r2, t)+ |Kq|b(t), and zT1 = (x,x), zT

2 = (x,x), rT1 = (p,p), rT

2 =(p,p).

Since Φ(., ., .) is Lipschitz continuous w.r.t state and parameter vectors, we denote by Lx(t) theLipschitz constant of Φ w.r.t x(t) and by Lp(t) its Lipschitz constant w.r.t p, and we set hx(t) =Lx(t)‖z1− z2‖+‖Kq‖‖b(t)‖. Let λ0(t) = max{λiq(t),0} where λiq(t) are the eigenvalues of Eq(t).

We can now address the proof of convergence.

Proposition 6 (P6) The hybrid system (55) is ε-practically stable over τ under γ if

∀t ∈ τ,∫ t

t0λ0(z)dz <

12

(56)

which can be satisfied if there exists some gain matrices Kq such that the matrices Eq(t) are stable. Inaddition, the lower bound on ε such that system (55) is ε-practically stable over τ under γ is given by

∀t ∈ τ, ε = w(‖p−p‖,‖b(t)‖)>(∫ t

t0 hx(z)dz+‖p−p‖∫ t

t0 Lp(z)dz)( 1

2 −∫ t

t0 λ0(z)dz) . (57)


17

Proof. Let us consider V (e, t) = 12 eT e. Form (55) we have

V (e, t) = eT Eq(t)e+ eT hq(t)≤ λ0(t)‖e‖2 +‖e‖

(Lx(t)‖z1− z2‖+‖Kq‖b(t)+Lp(t)‖p−p‖

)≤ λ0(t)‖e‖2 +‖e‖

(hx(t)+Lp(t)‖p−p‖

) (58)

Now, we can define piecewise continuous functions α(t) , λ0(t) and β (t) , hx(t) + Lp(t)‖p− p‖.For any e(t) satisfying ‖e(t)‖ ≤ ε , (58) leads to V (e, t) ≤ ε2α(t) + εβ (t). Let us choose ϕ(e, t) =ε2α(t) + εβ (t) and let µ1 = 1 because V (., .) has the same value after and prior to the switchingbetween two bounding subsystems. According to Theorem 3 and definition 7, system (55) is ε-practically stable over τ under γ if the following inequality holds

∃µ2 > 0, ∀t ∈ τ,∫ t

t0

(ε

2α(z)+ εβ (z)

)dz <

12

ε2−µ2 (59)

which, in accordance with the definitions of α(t) and β (t), is equivalent to ∃µ2 > 0, ∀t ∈ τ,(− 1

2+∫ t

t0λ0(z)dz

)ε

2 +(∫ t

t0hx(z)dz+‖p−p‖

∫ t

t0Lp(z)dz

)ε <−µ2. (60)

Let us choose µ2� 1, then we only need to have

∀t ∈ τ,(− 1

2+∫ t

t0λ0(z)dz

)ε +(∫ t

t0hx(z)dz+‖p−p‖

∫ t

t0Lp(z)dz

)< 0. (61)

Then, since ε must be positive, the last inequality (61) is equivalent to the requirement (57) which istrue if and only if (56) is satisfied. This completes the proof. �

Finally, Propositions P5 and P6 prove the main result, i.e. Theorem 1.

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19

Figure 2. The state space partition with respect to the guard condition x1 = 0. The blue and green state boxesrepresent an over-approximation of the reachable set of (16) with [15,20]× [2,4] as initial condition and T = [0,20]as simulation period. For all state boxes [x](t) where x1(t) < 0, the activated bounding system is M1, and for allstate boxes [x](t) where x1(t) > 0, the activated bounding system is M2. Finally for the state boxes [x](t) which

contain 0, the activated bounding system is M3.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−7

−6.5

−6

−5.5

−5

−4.5

x 2(t)

Figure 3. A zoom around the state region where the reached set of (16) crosses the guard condition. The bluediscontinuous rectangular represents the state box when the reached set computed by the bounding system M2intersects for the first time the guard condition, this event will activate the bounding system M3. On the other hand,the black continuous rectangular represents the state box when the reached set computed by the bounding systemM3 leaves completely the guard condition. This event deactivates the use of M3 and simultaneously activates thebounding system M1. Consequently, the green boxes show the sliding of all the state vectors before and after the

intersection with the set defined by the guard condition.


20 AA

0 2 4 6 8 10 12 14 16 18 20

0

10

20

30

40

50

60

70

80

90

100

110

t(h)

Bio

mas

s x 1(t

)

Figure 4. Lower and upper estimates of the biomass state variable.


21

0 2 4 6 8 10 12 14 16 18 2010

20

30

40

50

60

70

80

90

100

t(h)

Sub

stra

te x

2(t)

Figure 5. Lower and upper estimates of the substrate state variable.


22 AA

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160

180

200

t(h)

Pro

duct

x3(t

)

Figure 6. Lower and upper estimates of the product state variable.


23

Figure 7. Switching sequence.


interval observer design based on nonlinear hybridization

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