Automatica 42 (2006) 755–762www.elsevier.com/locate/automatica

Brief paper

Simultaneous state and input estimation of hybrid systemswith unknown inputs�

Luis Pina, Miguel Ayala Botto∗

Department of Mechanical Engineering, Technical University of Lisbon, Instituto Superior Técnico, GCAR/IDMEC, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Received 14 July 2004; received in revised form 20 July 2005; accepted 20 December 2005Available online 9 February 2006

Abstract

This paper addresses the problem of the simultaneous state and input estimation for hybrid systems when subject to input disturbances. Theproposed algorithm is based on the moving horizon estimation (MHE) method and uses mixed logical dynamical (MLD) systems as equivalentrepresentations of piecewise affine (PWA) systems. So far the MHE method has been successfully applied for the state estimation of linear,hybrid, and nonlinear systems. The proposed extension of the MHE algorithm enables the estimation of unknown inputs, or disturbances, actingon the hybrid system. The new algorithm is shown to improve the convergence characteristics of the MHE method by reducing the delayof convergent estimates, while assuring convergence for every possible sequence of input disturbances. To ensure convergence the system isrequired to be incrementally input observable, which is an extension to the classical incremental observability property.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Hybrid systems; Moving horizon estimation; State estimation; Unknown input estimation; Input observability

1. Introduction

Hybrid systems are defined as systems composed of con-tinuous and discrete variables (Antsaklis, 2000). Many differ-ent modelling approaches for hybrid systems have been devel-oped such as piecewise affine (PWA) systems (Sontag, 1981),mixed logical dynamical (MLD) systems (Bemporad & Morari,1999a), linear complementarity (LC) systems (Heemels, Schu-macher, & Weiland, 2000), and max–min-plus-scaling (MMPS)systems (Schutter & van den Boom, 2001). Despite the dif-ferent characteristics of these models, Heemels, Schutter, andBemporad (2001) proved their equivalence which allows to in-terchange analysis and synthesis tools between them. Specifi-cally, MLD systems are described by interdependent physicallaws (with linear dynamics), logic rules (if–then–else rules) andoperating constraints, and are very pertinent in the real-world

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Gang Taounder the direction of Editor Robert R. Bitmead.

∗ Corresponding author. Tel.: +351 21 841 90 28; fax: +351 21 849 80 97.E-mail addresses: luispina@dem.ist.utl.pt (L. Pina), ayalabotto@ist.utl.pt

(M.A. Botto).

0005-1098/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2005.12.014

engineering environment. In fact many industrial plants exhibitan hybrid behavior because they are composed by continuousdynamics at the lower level and are controlled, at the higherlevel, by logical components such as on/off switches or valves,gears or speed selectors, where evolutions are defined by log-ical rules. This modelling framework presents a compromisebetween applicability and complexity and is most suitable tobe introduced in optimization problems, so it will be used inthe implementation of the proposed algorithm.

The problem of state estimation for hybrid systems has beentackled by many authors using several distinct approaches.The main difference is related to the knowledge of the activemode: some approaches consider only state uncertainty withknown mode, while others assume that both the mode andthe state are unknown. Observer design in the case of knownmode has been addressed in Alessandri and Coletta (2001) andAlessandri and Coletta (2003), where a linear matrix inequality(LMI) based algorithm is used to compute the stabilizing gainsof the Luenberger observers, and in Böker and Lunze (2002)where a set of Kalman filters is designed for each dynamic ofthe hybrid system. In Balluchi, Benvenuti, Benedetto, and San-giovanni-Vincentelli (2001) an algorithm is presented that esti-mates both the mode and the state, but a dwell time is required

756 L. Pina, M.A. Botto / Automatica 42 (2006) 755–762

between mode changes in order to assure a correct mode esti-mation. Other approaches estimate the state before estimatingthe mode, but they are too conservative and can only be appliedto very simple systems (Juloski, Heemeles, Boers, & Verschure,2003; Juloski, Heemels, & Weiland, 2002). To overcome theseproblems, the moving horizon estimation (MHE) method wasproposed and successfully applied to various types of dynamicalsystems, for instance, constrained linear systems (Rao, Rawl-ings, & Lee, 2001), hybrid systems (Bemporad, Mignone, &Morari, 1999; Ferrari-Trecate, Mignone, & Morari, 2002) andnonlinear systems (Rao, Rawlings, & Mayne, 2003). The MHEmethod simultaneously estimates the state and the mode of thesystem and is based on a moving fixed-size estimation windowwhich bounds the size of the optimization problem. Since onlya subset of the information is actually used, stability questionsmay arise.

The optimization problems to be solved in a MHE scheme forhybrid systems are of a mixed-integer nature containing bothdiscrete and continuous variables. Except for special structures,these problems have a NP-complete complexity. In Fletcherand Leyffer (1995) several numerical experiences demonstratethat branch-and-bound techniques provided the best results overother competitive approaches. For hybrid estimation some veri-fication techniques (Bemporad & Morari, 1999b) are also avail-able to reduce the number of possible combinations of the dis-crete variables.

In this paper, the MHE method is extended such that un-known inputs or disturbances acting on the hybrid system canbe also estimated. State estimation for dynamic systems withunknown inputs is an interesting research topic in the fieldof robust and fault-tolerant control, as pointed out in Patton(1997). Unknown inputs can result either from model uncer-tainty or due to the presence of unknown external excitation.This problem is usually referred as the unknown input observerdesign. Different approaches to solve this problem for linearsystems can be found in the literature, such as the geomet-ric method derived in Bhattacharyya (1978), algebraic meth-ods presented in Darouach, Zasadzinski, and Xu (1994), Houand Muller (1992) and the generalized inverse approach de-veloped in Kurek (1983), Miller and R. (1982). Most of theseapproaches are based on a change of coordinates and sepa-ration of the system into two subsystems allowing for Lu-enberger’s observer design. Classical theory on unknown in-put observers design suggest the use of full-order observerswhen both the observer and the system have the same di-mension (Darouach et al., 1994), and reduced-order observerswhen the observer has a lower order than the system (Hou& Muller, 1992, Wang, Davison, & Dorato, 1975). These ob-servers have been extensively applied in the field of fault de-tection and isolation (FDI), for instance in Hou and Muller(1994), Viswanadham and Srichander (1987), although in somesituations only a part of the state is required for the residualgenerator, and so only a subset of the states and inputs needsto be estimated. However, the extension of these methods tohybrid systems suffers from the combinatorial growth of theproblem, the same drawback found in the switching Kalmanfiltering design.

Input estimation can be viewed as an inversion problem forwhich some work have been done for linear systems (Silverman,1969) and for some classes of nonlinear systems (Hirschorn,1979). In Mendel (1977), Kalman filter techniques were suc-cessfully applied to estimate the disturbances of the system.Some work has also been done in the simultaneous state andinput estimation (Corless & Tu, 1998; Ha & Trinh, 2004) forsome classes of nonlinear systems but again all these workscannot be efficiently applied to hybrid systems.

The problem of input estimation (or reconstruction) re-quires the evaluation of the observability of the system inputs.Roughly, input observability means that changes in the in-puts have to be reflected in the system outputs. In Hou andPatton (1998) a rank test on the system matrices is used toensure input observability of a linear system, but for hybridsystems a different input observability test must be used. Thispaper presents an extension to the input reconstruction of thestate observability test previously described in Bemporad, Fer-rari-Trecate, and Morari (2000). All previous developmentsusing MHE methods were primarily concerned with state esti-mation disregarding disturbance estimation, thus being unableto deal, for instance, with constant input disturbances. Theextension of the MHE method proposed in this paper willaccurately estimate the state for every possible sequence ofdisturbances. Moreover, it will be shown that the estimation ofsome of these disturbances will improve the quality of the stateestimates itself by reducing the delay of convergent estimates.This result is obtained without considering any disturbancesforecast, but rather estimating them based only on sufficientinformation available in the output. Besides, if measurementnoise is present, a maximum likelihood estimation will begiven where the uncertainty is originated only from the mea-surement noise and not from the input disturbances, as was thecase for the existing algorithms.

This paper is organized as follows. In Section 2, the problemof state estimation for hybrid systems with unknown inputsis formulated. Then in Section 3, the sufficient conditions forthe convergence of the MHE method with unknown inputs arestated. Finally in Section 4 some conclusions are drawn.

2. State estimation for hybrid systems with unknowninputs

This paper considers hybrid systems modelled in the PWAform as presented in Sontag (1981):

�:x(t + 1) = Aix(t) + Biu(t) + Wiw(t) + fi

y(t) = Cix(t) + Diu(t) + gi + v(t)

iff

[x(t)

u(t)

w(t)

]∈ �i , (1a,b)

where x(t) ∈ X ⊂ Rnx is the state of the system, u(t) ∈ U ⊂Rnu is the known input, w(t) ∈ W ⊂ Rnw is the unknowninput or disturbance, y(t) ∈ Rny is the output, v(t) ∈ V ⊂Rnv is the measurement noise, Ai , Bi , Wi , fi , Ci , Di and gi

are real matrices and vectors with appropriate dimensions thatdefine each linear dynamics of the system. The sets X, U, W

L. Pina, M.A. Botto / Automatica 42 (2006) 755–762 757

and V are assumed to be polytopes and can be represented bya set of linear inequalities, Hjx(t)�hj . Moreover, W and V

contain the origin. The index i represents the current systemmode (i ∈ {1, . . . , s}) and the partitions �i (t) are polytopes(i.e., closed and bounded polyhedra defined by a finite numberof linear inequalities) in the input + state + disturbance space.

Despite their intuitive representation, PWA systems are notsuitable to be integrated into optimization routines thus theirequivalent MLD representation will be adopted throughout. Thetranslation from a PWA to an MLD model is straightforwardas presented in Bemporad and Morari (1999a). Besides, thesoftware package HYSDEL (Torrisi, Bemporad, & Mignone,2000) uses high level language to generate models of hybridsystems, either in PWA or MLD format. The equivalence be-tween PWA, MLD and other representations of hybrid systemsis thoroughly described in Heemels et al. (2001). The MLDrepresentation for hybrid systems has the following structure(Bemporad & Morari, 1999a):

x(t + 1) = Azx(t) + Bzu(t) + Wzw(t) + f �(t), (2a)

y(t) = Czx(t) + Dzu(t) + g�(t) + v(t), (2b)

Exx(t) + Euu(t) + Eww(t) + E��(t) + Ezx zx(t)

+ Ezuzu(t) + Ezwzw(t) + Evv(t) + E0 �0, (2c)

where A� [A1 A2 . . . As], B � [B1 B2 . . . Bs], C �[C1 C2 . . . Cs], D � [D1 D2 . . . Ds], W � [W1W2 . . . Ws], f � [f1 f2 . . . fs], g � [g1 g2 . . . gs],while zx(t) = �(t) ⊗ x(t), zu(t) = �(t) ⊗ u(t) and zw(t) =�(t) ⊗ w(t) are the state, input and disturbance auxiliaryvariables, respectively, with �(t) ∈ {0, 1}s being the logi-cal auxiliary variable that defines the mode of the system.This representation, however, is slightly different from thatadopted in Bemporad and Morari (1999a) as it separates theauxiliary variables, one for each continuous variable, state,known input, unknown input. Due to the well-posednessof (2a)–(2c) (Bemporad & Morari, 1999a), i.e., due to thefact that all auxiliary variables are uniquely defined for all(x(k), u(k), w(k)), the knowledge of the initial state, dis-turbances and control inputs is sufficient for simulating thedynamic behavior of the system. System (2) is only appar-ently linear since the nonlinearity is hidden in inequalities(2c) and introduced in the dynamics through the auxiliaryvariables. These inequalities should be understood componen-twise and represent the partitions �i of (1) and possible safetyand/or performance constraints, usually called the operationalconstraints, acting on the state + input + disturbance space,where the matrices involved are real matrices of appropriatedimensions.

The problem of state estimation in the presence of unknowninputs, at time instant T, can be formulated as the solution ofthe following optimization problem:

minx0,{wk}T −1

k=0

J (0, T , N, x(0|T −1),{w(k|T −1)}T −2−Nk=0 ,x0,{wk}T −1

k=0 )

(3)

subject to the system dynamics (2). For the general case, thecost functional J can be defined as follows:

J (t1, t2, N, x(t1|t2 − 1), {w(k|t2 − 1)}t2−2−Nk=t1

, xt1 , {wk}t2−1k=t1

)

=t2∑

k=t1

‖vk‖2R +

t2−2−N∑k=t1

‖wk − w(k|t2 − 1)‖2Q

+ ‖xt1 − x(t1|t2 − 1)‖2S , (4)

with t1 � t2. If t2 =0, x(t1|t2 −1)= x0 which is the initial guessfor the state and vk=y(k)−Czx(k)−Dzu(k)−g�(k) as given bythe dynamics of system (2b). The first five arguments of the costfunctional correspond to the following parameters: the initialtime instant, t1, the current time instant, t2, the convergencedelay, N, the state estimate at time instant t1 computed at timeinstant t2 − 1, x(t1|t2 − 1), and the estimate of the sequence ofunknown inputs from time instant t1 to t2 − 2 − N , computedat time instant t2 − 1, {w(k|t2 − 1)}t2−2−N

k=t1. The remaining two

arguments are the optimization variables which will be the stateand input sequence estimates at time instant t2. The norms R,Q and S define the type of mixed integer programming (MIP)used to solve optimization (3) and will be defined in Section 3.Obviously the cost functional J (·) of (3) is obtained from (4)with t1 = 0 and t2 = T .

The convergence delay, N, defines the delay of the conver-gent estimates and, as will be shown later, will ensure conver-gence of the estimation process. The main difference betweenthis method and the MHE method proposed in Rao et al. (2001)and Ferrari-Trecate et al. (2002) is related to the introductionof the second summation (replacing the usual

∑t2−1k=t1

‖wk‖2Q)

which allows the estimation of systems unknown inputs. In for-mer MHE schemes, input disturbances were always comparedwith the most likely value, usually zero. In the proposed MHEscheme, input disturbances are compared with the values esti-mated at the previous time instants. Moreover, this differenceis weighted only until time instant t2 − N − 2, since at timeinstant t2 the information from the output can only be used togive an accurate estimate of the disturbances until time instantt2 − N − 1, as will become clear from the input observabilitydefinition. The disturbance at time instant t2 − N − 1 is notcompared with any value, and so it can be freely chosen.

The solution of (3) at time instant T is the unique pair(x(0|T ), {w(k|T )}T −1

k=0 ). Together with the system dynamics(2), this optimal pair generates an estimate for the trajectory ofthe system from 0 to T, {x(k|T )}Tk=0. Notice that the sequence{w(k|T )}T −1

k=T −1−N in the cost functional is not weighted butis still estimated. The estimation problem defined by this op-timization (3) is known as the full information (FI) estimationproblem, since all past information is used, namely the systemoutput from time instant 0 to T.

The FI estimator requires the solution of an optimizationproblem with increasing dimensions as time T grows. In orderto bound the size of the optimization problem, a moving hori-zon approximation strategy will be used. The main idea is toconsider explicitly only a fixed amount of data, while approxi-mately summarizing the discarded data not explicitly accountedfor by the estimator. Consider a fixed time horizon M ∈ N+,

758 L. Pina, M.A. Botto / Automatica 42 (2006) 755–762

and define a new optimization problem based on the FI estima-tor (3) given as follows:

�∗T = min

xT −M,{wk}T −1k=T −M

J (T − M, T, N, x(T − M|T − 1),

{w(k|T − 1)}T −2−Nk=T −M, xT −M, {wk}T −1

k=T −M). (5)

As in the FI estimator, the solution of the MHE (5) at timeinstant T is the unique pair (x(T −M|T ), {w(k|T )}T −1

k=T −M) thatyields the estimated trajectory of the system {x(k|T )}Tk=T −M .Notice that the past information not explicitly accounted for in(5) is introduced through the estimates computed at the previoustime instants, x(T − M|T − 1) and {w(k|T − 1)}T −2−N

k=T −M .

3. Convergence analysis

To analyze the convergence properties of the MHE schemewith unknown inputs (5) one needs first to define what conver-gence means for these type of observers and what propertiesshould the system hold to ensure this convergence.

Assumption 1. The data y(t) are generated by the followingPWA model with no measurement noise:

�:x(t + 1) = Aix(t) + Biu(t) + Wiw(t) + fi

y(t) = Cix(t) + Diu(t) + gi

iff

[x(t)

u(t)

w(t)

]∈ �i . (6a,b)

All variables are the same as in Eq. (1). The equivalent MLDrepresentation of (6) is similar to (2) but without consideringthe output noise v(t).

Remark 1. The convergence of the algorithm to a zero es-timation error will only be ensured if no measurement noiseis present. In the noisy case a maximum likelihood estima-tion based on the norms chosen for the cost functional (4) isobtained. In the algorithm proposed by Ferrari-Trecate et al.(2002) a maximum likelihood estimation was also computedbut the uncertainty would arise from both the input disturbancesand the measurement noise. Notice that for the purposed al-gorithm the uncertainty in the estimation only arises from themeasurement noise. This is a very crucial detail which consid-erably improves the accuracy of the algorithm.

Let x�(t; x0, u, {wj }t−1j=0) and y�(t; x0, u, {wj }t−1

j=0) denotethe state and output solutions of model (6) at time instant tsubject to the initial condition x0, input sequence u and distur-bance sequence {wj }t−1

j=0, respectively. To ensure the feasibilityof the optimization problem (5), the state trajectory must fulfillthe following assumption:

Assumption 2. x�(t, x0, u, {wj }t−1j=0) ∈ X, ∀t ∈ N+, for the

true initial condition x0, input sequence u and disturbance se-quence {wj }t−1

j=0.

This assumption ensures the feasibility of the optimizationproblem (5), since x�(t, x0, u, {wj }t−1

j=0) is a feasible solution.

The definition of convergence for this type of observers readsas follows:

Definition 1. For a given � ∈ {N, N + 1, . . . , M}, the MHEscheme with unknown inputs (5) is �-convergent if,

limT →∞ ‖x(T − �|T ) − x�(T − �, x0, u, {wj }T −�−1

j=0 )‖ = 0. (7)

The definition of �-convergence states that the MHE schemecan track the state trajectory x(T − �) of the system �, atleast asymptotically when T → ∞, independently of the initialguess x0. Notice that � is the convergence delay and that theestimated trajectory from x(T − � + 1|T ) to x(T |T ) is stillestimated, although convergence cannot be guaranteed.

When the MHE scheme is used to estimate the trajectory ofsystem (6), convergence can only be guaranteed if the systemitself enjoys the following properties:

Definition 2 (Bemporad et al., 2000). The system � is incre-mentally observable in T1 steps on X(0) ⊂ X with respect toU ⊂ U if there exist two norms ‖ · ‖S (on Rnx ) and ‖ · ‖R (onRny ) and a positive scalar � such that ∀x1, x2 ∈ X(0), ∀u ∈Uk+1 satisfying Assumption 2 with {w(t)}k−1

t=0 = 0, and

k∑t=0

‖y�(t, x1, u, 0) − y�(t, x2, u, 0)‖R �� · ‖x1 − x2‖S (8)

holds for k = T1 and does not hold for k < T1.

A similar definition of observability was also presented inKeerthi and Gilbert (1988) for nonlinear systems, called theO property. Since the MHE scheme will also estimate the un-known inputs, it is necessary to include them in the previousdefinition:

Definition 3. The system � is incrementally input observablein T2 steps on X(0) ⊂ X with respect to U ⊂ U if it is in-crementally observable in T1 � T2 steps and there exist threenorms ‖ · ‖S (on Rnx ), ‖ · ‖R (on Rny ) and ‖ · ‖Q (on Rnw )

and a positive scalar � such that ∀x1, x2 ∈ X, ∀u ∈ UT1+1,

∀{w1(j)}T1−1j=0 , {w2(j)}T1−1

j=0 ∈ WT1 satisfying Assumption 2,and

T1∑t=0

‖y�(t, x1, u, {w1(i)}t−1i=0) − y�(t, x2, u, {w2(i)}t−1

i=0)‖R

�� ·⎧⎨⎩

T1−1−k∑t=0

‖w1(t) − w2(t)‖Q + ‖x1 − x2‖S

⎫⎬⎭ (9)

holds for k = T2 and does not hold for k < T2.

This property is an extension of the classical incrementalobservability property, now containing the input observability.Notice that the algorithm proposed in Bemporad et al. (2000)to test the incremental observability property can still be used,with appropriate modifications, to test incremental input ob-servability property of the system. The required modifications

L. Pina, M.A. Botto / Automatica 42 (2006) 755–762 759

are basically the introduction of new variables correspondingto the disturbances w1(t) and w2(t) for t = 0, . . . , T1 − 1 − k

and the change of the dynamics equations and constraints ofthe new disturbed MLD formulation (2).

Remark 2. The weighting matrices used in (4) will be theweighting matrices used in the incremental input observabilitytest with scalar � introduced in Q and S by simple multiplica-tion. The value of � can be viewed as a design parameter: asmall value leads to a fast observer with poor filtering capac-ities, while a large value leads to a slower observer but withbetter filtering properties. Notice that � must be smaller than�M , the maximum value of � for which inequality (9) holds.In Bemporad et al. (2000) some practical considerations aremade about both values � and T1.

The observability properties of the system define T1 and T2,which correspond to the time instants required for the outputto provide sufficient information to estimate the full state, andthe unknown input, respectively. The observability definitionsset values of the time delays T1 and T2 required to estimate thestate and the disturbances, so the MHE scheme parameters Mand N must be chosen accordingly:

Assumption 3. The system � is incrementally observable inT1 steps and M � T1 + 1.

Assumption 4. The system � is incrementally input observablein T2 steps and N = T2.

With the former definitions introduced, the main result ofthis paper can now be stated as follows:

Theorem 1. If Assumptions 1–4 hold, the MHE scheme de-fined by the solution of (5) at every time instant T �1 is �-convergent for all N ���M , and the disturbance sequences{w(k|T )}T −N−1

k=T −M converge to the real disturbance sequencewhen T → ∞. The norms and weighting matrices used in (5)are the ones used to establish input incremental observabilityin accordance with Remark 2.

Proof. The proof of Theorem 1 follows after the preliminaryLemmas 1 and 2 concerning the nature of the sequence �∗

T .Lemma 1 states the decreasing nature of the cost functionaland Lemma 2 states its convergence to 0, implying that allterms of the estimation error in the output also converge to0. Then, by the input incremental observability property, theestimation error in the state and unknown inputs must alsoconverge to 0. �

Consider the following optimization problem, similar to �∗T

but with a final state constraint:

�∗T (z) = min

xT −M,{wj }T −1j=T −M

xT =z

J (T − M, T, N, x(T − M|T − 1),

{w(j |T − 1)}T −2−Nj=T −M, xT −M, {wj }T −1

j=T −M). (10)

The following relation holds by the definition of both optimiza-tion problems.

�∗T = min

z�∗

T (z). (11)

Lemma 1. Let Assumptions 1, 3 and 4 hold. Then, ∀T �1 andfor all x(0) ∈ X(0), {w(j)}Tj=0 ∈ WT +1 such that Assumption2 is fulfilled, it holds:

�∗T +1(x�(T + 1, x(T + 1 − M), {w(j)}Tj=T +1−M))

��∗T (x�(T , x(T − M), {w(j)}T −1

j=T −M)). (12)

Proof of Lemma 1. The special case when 1�T < M will beaddressed after the more general case when T �M . Considerthe following optimizers of �∗

T (x�(. . .)):

x(T − M|T ), {w(j |T )}T −1j=T −M ⇒ {v(j |T )}Tj=T −M . (13)

Then, a feasible solution for �T +1(x�(. . .)) can be given by

xT +1−M = x(T + 1 − M|T ) (14a)

�Ai(T −M)

x(T − M|T ) + Bi(T −M)

u(T − M)

+ Wi(T −M)

w(T − M|T ) + fi(T −M)

, (14b)

{wj }T −1j=T +1−M = {w(j |T )}T −1

j=T +1−M, wT = w(T ) (14c)

⇓ ⇓{vj }Tj=T +1−M = {v(j |T )}Tj=T +1−M, vT +1 = 0 (14d)

this feasible solution implies that the minimum value of�∗

T +1(x�(. . .)) will be, by optimality, smaller or at least thesame as the value given by the former feasible solution:

�∗T +1(x�(T + 1, x(T + 1 − M), {w(j)}Tj=T +1−M))

− �∗T (x�(T , x(T − M), {w(j)}T −1

j=T −M))

�T∑

k=T +1−M

‖v(k|T )‖2R + ‖0‖2

R

+T −1−N∑

k=T +1−M

‖w(k|T ) − w(k|T )‖2Q

+ ‖x(T + 1 − M|T ) − x(T + 1 − M|T )‖2S

−T∑

k=T −M

‖v(k|T )‖2R−

T −2−N∑k=T −M

‖w(k|T )−w(k|T −1)‖2Q

− ‖x(T − M|T ) − x(T − M|T − 1)‖2S

= −‖v(T − M|T )‖2R −

T −2−N∑k=T −M

‖w(k|T ) − w(k|T − 1)‖2Q

− ‖x(T − M|T ) − x(T − M|T − 1)‖2S �0. (15)

760 L. Pina, M.A. Botto / Automatica 42 (2006) 755–762

Consider now that 1�T < M . Following the same reasoningone obtains:

�∗T +1(x�(T + 1, x(0), {w(j)}Tj=0))

− �∗T (x�(T , x(0), {w(j)}T −1

j=0 ))

�T∑

k=0

‖v(k|T )‖2R + ‖0‖2

R

+T −1−N∑

k=0

‖w(k|T ) − w(k|T )‖2Q + ‖x(0|T ) − x(0|T )‖2

S

−T∑

k=0

‖v(k|T )‖2R −

T −2−N∑k=0

‖w(k|T ) − w(k|T − 1)‖2Q

− ‖x(0|T ) − x(0|T − 1)‖2S

= −T −2−N∑

k=0

‖w(k|T ) − w(k|T − 1)‖2Q

− ‖x(0|T ) − x(0|T − 1)‖2S �0. �

Lemma 2. Considering the same conditions as in Lemma 1,

limT →∞ �∗

T (x�(T , x(T − M), {w(j)}T −1j=T −M)) = 0.

Proof of Lemma 2. Since �∗T (x�(. . .)) is non-increasing and

lower bounded by zero, it converges. It is now needed to provethat it converges to 0.

Consider the value of �T +1(x�(T + 1, x(T + 1−M),

{w(j)}Tj=T +1−M)) given from the solution obtained at time

instant T1:

�T +1(x�(T + 1, x(T + 1 − M), {w(j)}Tj=T +1−M))

=T∑

k=T +1−M

‖v(k|T )‖2R . (16)

Consider also the solution given by the real states of thesystem, which is obviously feasible:

�T +1(x�(T + 1, x(T + 1 − M), {w(j)}Tj=T +1−M))

=T +1∑

k=T +1−M

‖y(k) − y(k)‖2R

+T −1−N∑

k=T +1−M

‖w(k) − w(k|T )‖2Q

+ ‖x(T + 1 − M) − x(T + 1 − M|T )‖2S . (17)

1 See in Proof of Lemma 1, Eqs. (14) and (15).

By the definition of incremental input observability, using theestimated trajectory at the previous time instant and the realtrajectory, one must have Eqs. (16) � (17):

T∑k=T +1−M

‖y(k|T ) − y(k)‖2R

�T −1−N∑

k=T +1−M

‖w(k|T ) − w(k)‖2Q

+ ‖x(T − M + 1|T ) − x(T − M + 1)‖2S (18)

since the equality only holds when both trajectories are thesame, it is always possible to sensibly reduce the optimumvalue of this cost functional. From Remark 2, the value of thecost functional reduces to a value at least �/�M (which issmaller than 1) times smaller at every time step which impliesits convergence to 0. �

The convergence of �∗T (·) towards 0 implies the convergence

of �∗T to 0 since Eq. (11) holds and �∗

T is non-negative.

Proof of Theorem 1. The following inequalities hold forN ���M:

T∑k=T −M

‖y(k|T ) − y(k)‖2R �

T∑k=T −�

‖y(k|T ) − y(k)‖2R

�T −1∑

k=T −�

‖w(k|T ) − w(k)‖2Q

+ ‖x(T − �|T ) − x(T − �)‖2S , (19)

where the second inequality is derived from the definition ofincremental input observability.

By Lemmas 1 and 2 one obtains:

limT →∞

T∑k=T −M

‖y(k|T ) − y(k)‖2R = 0 (20)

and, by the former inequalities:

limT →∞

T −1−N∑k=T −�

‖w(k|T ) − w(k)‖2Q

+ ‖x(T − �|T ) − x(T − �)‖2S = 0, (21)

which states the �-convergence of the MHE scheme forN ���M . �

Remark 3. In Ferrari-Trecate et al. (2002) some considera-tions were made about the possibility of the state trajectory be-longing to the edge of partitions �i . These considerations arenot considered relevant in this context since they are mainlytechnical, depending on the model used and not on the estima-tion algorithm itself. This paper considers hybrid models withclosed partitions where the state can only belong to one re-gion in accordance with Bemporad and Morari (1999a), whichavoids the afore-mentioned problem.

L. Pina, M.A. Botto / Automatica 42 (2006) 755–762 761

Remark 4. Linear time invariant (LTI) systems can be consid-ered a special case of an hybrid system with only one lineardynamic in the whole state+ input +disturbance space (s =1).This similarity between linear and hybrid systems renders thealgorithm applicable to the estimation of unknown inputs ofLTI systems. However, since in this case the Kalman filter isthe true optimal filter as it uses all the information available (allmeasurements from t = 0 to the actual time) it should producebetter results. Notice that for hybrid systems it is not possibleto use the Kalman filter due to the combinatorial nature of theproblem and so some information must be discarded deteriorat-ing the characteristics of the estimation. Nevertheless, similarsimulation results have been produced with both algorithms forthe case of LTI systems.

This algorithm can also be used for nonlinear systems asthese can be approximated with the desired accuracy by a PWAsystem (Sontag, 1981) and so a MLD system. However, noextensive testing and comparison with other estimation algo-rithms was made for the case of nonlinear systems. In Raoet al. (2003) a moving horizon estimation algorithm is proposedand, the ideas of this paper could be introduced, providing asimultaneous state and unknown input estimator for nonlinearsystems.

4. Conclusions

In this paper, the MHE algorithm is extended to deal withsimultaneous state and input estimation of hybrid systems.This extension improves the convergence characteristics of thestate estimates by reducing the convergence delay and allowingconvergence for every possible sequence of the input distur-bances. The state estimate at the current time instant is muchmore accurate since there are less extrapolation steps from theconvergent estimate to the actual time. A new definition ofobservability for input reconstruction was also given. Futureresearch will focus on broadening this formulation to thecase when both observable and unobservable states and inputdisturbances are present.

Acknowledgments

This research was partially supported by the Fundação paraa Ciência e Tecnologia under POCTI/EME/44336/2002, andGrant SFRH/BD/12208/2003.

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Luis Pina was born in Lisbon in 1979. He grad-uated in Mechanical Engineering at Instituto Su-perior Técnico, Technical University of Lisbon,in 2002. He is currently under a Ph.D. schol-arship from the FCT—Portuguese Science andTechnology Foundation, in the research field ofestimation and control of hybrid systems.

Miguel Ayala Botto was born in Lisbon in1965. He graduated from Instituto SuperiorTécnico, Technical University of Lisbon, in1989, and received his M.Sc. and Ph.D. de-grees in Mechanical Engineering from thesame university in 1992 and 1996, respec-tively. He has held two visiting researchpositions at the Delft University of Tech-nology in The Netherlands. He is currentlyan Associate Professor at the Department ofMechanical Engineering of IST and a seniorresearcher at IDMEC. He is a member of the

IFAC Technical Committee on Discrete Events and Hybrid Systems, headof the Portuguese Association on Automatic Control (the Portuguese IFACNMO), and an Associate Editor of the International Journal of SystemsScience. In 1999 he received the “Heavise Premium” from the Council ofthe IEE—The Institution of Electrical Engineers, UK. His research interestsinclude identification, estimation and control of hybrid systems.