Input Estimation for Output Tracking in NonlinearSystems

Koshy GeorgeDepartment of Telecommunication Engineering, P.E.S. Institute of Technology,

100 Feet Ring Road, BSK III Stage, Bangalore 560 085, IndiaEmail: kgeorge@pes.edu

Abstract— A novel technique to achieve output tracking viainput estimation applicable to nonminimum phase nonlinearsystems is presented. The earlier approaches to stable inver-sion is directly applicable only to nonminimum phase systemswith hyperbolic internal dynamics. Systems with nonhyperbolicdynamics required modification of the internal dynamics. Themethod presented in this paper overcomes this drawback.

Keywords—nonlinear systems, tracking, extended Kalmanfilter

I. INTRODUCTION

A fundamental problem is that of precision output tracking,and a number of papers in the literature are devoted to thisproblem. A well recognized difficulty is that of achievingoutput tracking with systems that are nonminimum phase [1].The conditions for asymptotic tracking of a given trajectoryare outlined in [2] which were later generalised to nonlinearsystems in [3]. These approaches asymptotically track anymember in a given family of signals generated by an ex-osystem. The stable inversion approach via a dichotomic splitof the system equations of a nonminimum phase plant wasintroduced in [4] to avoid the use of exosystems, and, in thecase of nonminimum phase systems, mitigate the poor transientperformance by using pre-actuation.

The method of dichotomies [4] is directly applicable onlyto systems with hyperbolic internal dynamics, i.e., systemswith no zeros on the imaginary axis. Moreover, the period ofpre-actuation required to overcome the effect of nonminimumphase zeros is related to the proximity of such zeros to theimaginary axis [5]: the closer these are to the imaginaryaxis, the longer the pre-actuation. It is immediately evidentthat if the input is computed via this method, systems withnonhyperbolic internal dynamics require infinite pre-actuation.For linear systems, a trade-off between precision tracking andthe required period of pre-actuation was introduced in [5] bymodifying the internal dynamics. This trade-off was extendedin [6] to nonlinear systems.

The Stable Dynamic Model Inversion (SDI) technique fordiscrete-time linear systems was first proposed in [7] to over-come the drawbacks of existing techniques. This techniqueplaced no restriction on the location of zeros and effectivelyrequired no pre-actuation for systems with nonminimum phasezeros. Besides, the method is as well applicable to systemswith nonhyperbolic internal dynamics. Moreover, SDI is ableto account for system/measurement noise when the desired

trajectories are generated by data acquisition runs with thephysical system at hand [8]. The SDI technique minimizes amulti-objective criterion with a trade-off between the effect ofexogenous disturbances on the desired trajectory, and that ofthe controllable (unknown) input to generate a desired trajec-tory. The SDI technique for LTI systems appears to fail onlywhen the given system has zeros at unity, and a numericallyefficient way to deal with this problem was introduced in[9] by dislocating the zeros in the neighbourhood of z = 1.The nonminimum phase zeros amongst such dislocated zerosintroduce very little pre-actuation, and no modification ofinternal dynamics is required.

In this paper, we extend the above SDI technique forestimating the input required by a nonlinear discrete-timesystem to track desired output trajectories. (This extensionwas indicated in [9].) The technique essentially amounts tobuilding an Extended Kalman Filter (EKF) for an augmentedstate space model, and hence estimating the inputs necessaryfor desired trajectories. We then apply this method to threeexamples: (1) an induction motor example considered in [10]to demonstrate the efficacy even under worst case initialconditions; (2) a two carts and an inverted pendulum exampleintroduced in [6] and show that no modification is necessaryfor systems with nonhyperbolic internal dynamics; and, (3) acar-like robot with nonholonomic constraints [11].

Evidently, the approach suggested here is based on theassumption of perfect knowledge of the system. In a morerealistic scenario of parametric variations and unmodelleddynamics, this approach appears rather ill-posed. Therefore, inpractical applications, the estimated input using our approachbecomes a feed-forward signal used in conjunction with a moreconventional feedback control law [4], [12]. For instance, in[12] the advantages of a sliding mode control law and the inputestimated using our approach have been combined for robusttracking in linear systems.

II. INPUT ESTIMATION FOR NONLINEAR SYSTEMS

Suppose that a discrete time nonlinear system is representedby the following state space description:

xk+1 = f(xk, uk)yk = h(xk, uk) (1)

where xk ∈ IRn, uk ∈ IRm, and yk ∈ IRp. We assume thatf(xk, uk) and h(xk, uk) are differentiable with respect to both

1–4244–0342–1/06/$20.00 c© 2006 IEEE ICARCV 2006

xk and uk. Given measurements y, our objective is to obtainan estimate u of the input u by obtaining a suitable ‘inversesystem’ Σinv as illustrated in Fig. 1, where Σ is representedby (1). We state our problem more precisely:

�� �

� ��

u

w v

Σ Σinv uyy

Fig. 1. Input Estimation.

Problem 1: Given measurements yk obtain an estimate uk

of the input such that the output yk of system (1) with thisinput uk satisfies the following:

limk→∞

‖yk − yk‖ = 0 (2)

In general, there are two approaches for the design ofobservers for nonlinear systems. The first one is based on anonlinear state transformation using Lie algebra. In principle,the original system is transformed into a canonical form andstate observers are designed using linear techniques in the newcoordinates. Due to the conservative nature of the underlyingassumptions, this procedure is applicable only to a restrictedclass of nonlinear systems. However, under these conditions,global and exponential convergence is guaranteed; see [13],[14], and the references cited therein. The second approachis based on a linearised model and is without any canonicaltransformation. Despite the convergence of this method beingonly local, this method is widely used in practice and generallygives good results under less restrictive conditions than the firstapproach [15], [16]. The Extended Kalman Filter is perhapsone of the most popular techniques for the estimation of thestate of a nonlinear system. A number of papers have dealt withconvergence issues in the past; more recently, it was shown in[17], [10] that the ‘covariance’ matrices Rk and Qk play acentral role to improve convergence.

A. The Extended Kalman Filter

The Extended Kalman Filter (EKF) consists of applying theclassical Kalman filter equations to the first order approxima-tion of the nonlinear model about the last estimate. However,since the EKF equations are only approximate ones, thecorresponding propagation equations are reasonably accurateonly if the estimate belongs to a neighbourhood of the actualstate. We now summarize the EKF [18] for the ‘noisy’ systemassociated with (1):

xk+1 = f(xk, uk) + wk

yk = h(xk, uk) + vk (3)

where the covariances of the ‘noise’ sequences wk and vk aredesign parameters.

Suppose that, at the time of measurement k, we have ana priori estimate of the state, xk|k−1, and the covariance

of the error in the a priori estimate, Pk|k−1. We obtain themeasurement update as follows:

Kk = Pk|k−1H′k

(HkPk|k−1H

′k +Rk

)−1

xk|k = xk|k−1 +Kk

{yk − h

(xk|k−1, uk

)}Pk|k = (I −KkHk)Pk|k−1

where

Hk∆=∂h(xk, uk)

∂xk

∣∣∣∣xk|k−1,uk

With this information, we obtain the time update as follows:

xk+1|k = f(xk|k, uk)Pk+1|k = FkPk|kF ′

k +Qk

where

Fk∆=∂f(xk, uk)

∂xk

∣∣∣∣xk|k,uk

For linear stochastic systems, optimal filtering in the maximallikelihood sense is obtained when Qk and Rk are respectivelythe covariance matrices of the system and measurement noises.We note that for nonlinear noisy systems, optimality has notbeen proved; however, in general, we consider Qk and Rk

as covariance matrices. The choice of Qk and Rk play animportant role in improving the convergence of the extendedKalman filter when used as an observer for the system (1) [17],[10]. Let X ′ denote the transpose of a matrix X . We have thefollowing result:

Theorem 2 ([17], [10]): Let αk and βk be diagonal matri-ces chosen such that the following inequalities are satisfied forsome ζ ∈ (0, 1):

αkR−1k αk − αkR

−1k −R−1

k αk +R−1k HkPk|kH ′

kR−1k ≤ 0

F ′kβk

(FkPk|kF ′

k +Qk

)βkFk − (1 − ζ)P−1

k|k ≤ 0

Further let Fk be a bounded nonsingular matrix, and let thereexist real numbers κ1 and κ2 such that for all k ≥M − 1 wehave

κ1I ≤ O′e(k−M+1, k)R(k−M+1, k)Oe(k−M+1, k) ≤ κ2I

(4)where

Oe(k −M + 1, k) =

Hk−M+1

Hk−M+2Fk−M+1

...Hk−1Fk−2Fk−3 · · ·Fk−M+1

HkFk−1Fk−2 · · ·Fk−M+1

R(k −M + 1, k) = Diag(R−1

k−M+1 · · · R−1k

)Then, the EKF for system (1) ensures that

limk→∞

(xk − xk|k) = 0 (5)

The range of values for the diagonal elements of the matricesαk and βk are discussed in [17], [10]. However, since theactual values that help satisfy the inequalities (4) are unknown,an a priori choice of Rk is to set it much larger thanHkPk|k−1H

′k. Since high values of Rk lead to small values of

the gain Kk, and hence a slow convergence rate, a trade-offbetween stability of the EKF and the rate of convergence wasproposed in [17]: Choose positive constants ζ1 and ζ2 suchthat

Rk = ζ1HkPk|k−1H′k + ζ2I (6)

For Qk = 0, it was proved in [17] that the diagonal elementsof βk must lie in [−1, 1]; for nonzero Qk, the following choicewas proposed in [10]:

Qk = ζ3e′kekI + ζ4I (7)

Here ek = yk − h(xk|k−1, uk

); ζ3 is chosen sufficiently large

and positive and ζ4 is a small enough positive scalar. Weremark that (4) represents an uniform observability conditionthat guarantees the boundedness of Pk|k [19].

B. Input Estimation

The key idea for system inversion in the sense of estimatingthe necessary input for desired output trajectories is to augmentthe state space model of the given system by a model for theinput history. In this paper, we assume that the input belong tothe class of all functions which are solutions of autonomous,possibly nonlinear, difference equations of the form

φk+1 = s(φk)uk = g(φk) (8)

with φk ∈ IRmφ and the initial condition φ0 unknown. Weassume that both s(φk) and g(φk) are differentiable withrespect to φk. (We note that such a generator of possible inputfunctions is similar to the exosystem considered for nonlinearcontinuous time systems [20].) We then build an EKF for this

augmented state space model. Denoting xa,k =(xk

φk

), we

have the following augmented state space model from eqns. (1)and (8):

xa,k+1 = fa (xa,k)yk = ha(xa,k) (9)

where fa(xa,k) =(f (xk, g(φk))

s(φk)

), and ha(xa,k) =

h (xk, g(φk)). Let

Ha,k =∂ha(xa,k)∂xa,k

∣∣∣∣xa,k|k−1

∆=(Hx,k Hφ,k

)

Fa,k =∂fa(xa,k)∂xa,k

∣∣∣∣xa,k|k

∆=(Fx,k Fφ,k

0 Sk

)

denote the linearisation of the augmented state space model (9)about the estimates. We remark that Fφ,k is bounded if bothFu,k = ∂f(xk,uk)

∂ukand Gk = ∂g(φk)

∂φkare bounded. The

following result is then immediate:Lemma 3: For the given system (1) let Fx,k and Fu,k

be bounded with Fx,k nonsingular. Then, for the augmentedsystem (9), Fa,k is bounded and nonsingular if Gk and Sk arebounded with Sk nonsingular.

In order to obtain an EKF that converges, we require thefollowing standard [17], [10] assumptions:

1) Fx,k and Fu,k are bounded with Fx,k nonsingular. Fur-ther, Gk and Sk are bounded with Sk nonsingular.

2) Let αk and βk be diagonal matrices chosensuch that the following inequalities are satisfiedfor some ζ ∈ (0, 1): αkR

−1k αk − αkR

−1k −

R−1k αk + R−1

k Ha,kPa,k|kH ′a,kR

−1k ≤ 0 and

F ′a,kβk

(Fa,kPa,k|kF ′

a,k +Qa,k

)βkFa,k − (1 −

ζ)P−1a,k|k ≤ 0.

3) The augmented system (9) is uniformly observable; i.e.,there exist real numbers κ1 and κ2 such that for all k ≥M − 1 we have κ1I ≤ O′

a,e(k−M + 1, k)R(k −M +1, k)Oa,e(k −M + 1, k) ≤ κ2I where

Oa,e(k −M + 1, k) =

Ha,k−M+1

Ha,k−M+2Fa,k−M+1

...Ha,kFa,k−1 · · ·Fa,k−M+1

and R(k −M + 1, k) is as defined earlier.4) h(xk, uk) is locally Lipschitz continuous; i.e., there

exists finite constant κh such that ‖h(xk, uk) −h(zk, vk)‖ ≤ κh (‖xk − zk‖ + ‖uk − vk‖), ∀ k ≥ 0,∀ (xk, uk), (zk, vk) ∈ Brh

where Brhis a ball of radius

rh in IRn+m.5) g(φk) is locally Lipschitz continuous; i.e., there ex-

ists finite constant κg such that ‖g(φk) − g(ψk)‖ ≤κg (‖φk − ψk‖), ∀ k ≥ 0, ∀ φk, ψk ∈ Brg

where Brgis

a ball of radius rg in IRmφ .The following result then follows rather straightforwardly:

Theorem 4: Given measurements yk and an EKF for theaugmented system (9) that satisfy assumptions 1–5, we have,

limk→∞

‖yk − yk‖ = 0

where yk = h(xk|k, g(φk|k)

).

Proof: By Theorem 1, assumptions 1–3 guarantee the stabilityand convergence of the EKF for the augmented system (9).Thus,

limk→∞

(xa,k − xa,k|k) = 0

Moreover, since g(φk) is locally Lipschitz, we have

‖uk − uk‖ = ‖g(φk) − g(φk|k)‖≤ κg

(‖φk − φk|k‖

)

Taking limits on both sides yields

limk→∞

‖uk − uk‖ = 0

Finally,

‖yk − yk‖ = ‖h(xk, uk) − h(xk|k, uk)‖≤ κh

(‖xk − xk|k‖ + ‖uk − uk‖)

follows from the fact that h(xk, uk) is locally Lipschitz. Takinglimits gives us the required result.We remark that the range of values for the diagonal elementsof αk and βk can be obtained similar to [17], [10]. However,

as suggested in [10], it is sufficient to choose Rk as in (6),and Qk as in (7).

C. A Heuristic Approach

When the representation of the input as in (8) is unknownwe now discuss a second, more ad hoc manner of estimatinginputs for desired output trajectories. Here we represent theinput by a simple dynamical model and rely on the robustnessof the Kalman filter to compensate for the modelling errors.One example of a simple (stochastic) representation of theinput is:

uk+1 = uk + ηk (10)

where ηk is a ‘noise’ sequence with ‘covariance’ Qη �= 0.Thus, the augmented system (9) is approximated by,(

xk+1

uk+1

)=

(f(xk, uk)

uk

)+

(0ηk

)

yk ≈ h (xk, uk)

Such a choice for the input model history yielded a particularsolution to SDI of linear systems in [7], [8]. Moreover, whenthe choice of input is as given in (10) it as well accords withthe experience gained through the system inversion procedureapplied to several examples of nonlinear systems. Three ex-amples illustrating key features of the outlined procedure arereported in the next section.

III. APPLICATION EXAMPLES

A. Induction Motor

We now use our procedure with the input uk modelled asin (10) and Rk and Qk in the EKF respectively as in (6) and(7) for the estimation of the necessary inputs for a two phaseinduction motor. The fifth order nonlinear model provided in[10] has been the object of a large number of applications,especially in control system designs. (For a discussion of themodel we refer to [21].) For convenience, we repeat here thestate space model in stator fixed reference frame [10]:

x1,k+1 = x1k + h

(−γx1k +

K

Trx3k +Kpx5kx4k

+1σLs

u1k

)

x2,k+1 = x2k + h

(−γx2k +

K

Trx4k −Kpx3kx5k

+1σLs

u2k

)

x3,k+1 = x3,k + h

(M

Trx1,k − 1

Trx3,k − px4,kx5,k

)

x4,k+1 = x4,k + h

(M

Trx2,k − 1

Trx4,k − px3,kx5,k

)

x5,k+1 = x5,k + h

(pM

JLr(x3,kx2,k − x1,kx4,k) − TL

J

)

y1,k = x1,k

y2,k = x2,k

In this model x1 and x2 represent the stator currents, x3 andx4 the rotor fluxes, and x5 the rotor speed vector. The inputsare the stator voltages control vector, and TL is the load torque.The rotor time constant Tr = Lr

Rr, parameters σ = 1− M2

LsLr,

K = MσLsLr

, and γ = Rs

σLs+ RrM2

σLsL2r

; here, Rr and Rs arethe rotor and stator resistances, Lr and Ls are the rotor andstator inductances. Moreover, p is the number of pole pairsand J is the rotor moment of inertia. We choose the samenumerical values of these parameters as in [21]. The followinginputs were considered in [10]: u1,k = 350 cos(0.03k) andu2,k = 300 sin(0.03k). These inputs and the outputs of themodel with these inputs are recorded for reference; we use theoutputs as desired trajectories to estimate the inputs using ourinput estimation procedure.

The inputs are estimated for two different sets of initialconditions: In Case 1 we assume that the initial conditions arechosen properly, and the input estimation error u = u − uis shown in Fig. 4(a). In Case 2 we assume that the EKF isinitialized rather poorly (same as that considered in [10]), andthe input estimation error is shown in Fig. 4(b). In both cases,we observe that the errors converge to zero rather quickly, withthe convergence being naturally faster in Case 1.

B. Two carts and Inverted Pendulum

It is well-known that the inverted pendulum on a carthas hyperbolic nonminimum phase internal dynamics [22]. Inorder to introduce nonhyperbolicity, an extra cart was addedin [6] as shown in Fig. 2. Here, both carts have the samemass M , and m is the mass of the block on the pendulum.The pendulum has length l, and the spring constant is K. Letx1 and x2 represent the position of the two carts, and θ theangle the pendulum makes with the vertical. The differentialequations that describe the motion was provided in [6] whichwe repeat here for convenience:

(M +m)x1(t) +ml cos θ(t)θ(t)+K(x1(t) − x2(t)) −mlθ2(t) sin θ(t) = u(t)

ml cos θ(t)x1(t) +mlθ2(t) −mgl sin θ(t) = 0Mx2(t) −K(x1(t) − x2(t)) = 0 (11)

The input to the system is the applied force u; the position

u � M

������

�m

φ

����������������������������

K

M

Fig. 2. A two carts and an inverted pendulum example.

of the first cart x1 is the output of the system, and the chosendesired output trajectory in [6] is as shown in Fig. 5(a). It was

shown in [6] that the linearised internal dynamics has two poleson the imaginary axis (corresponding to the dynamics of thesecond cart) leading to nonhyperbolic internal behavior. (Theother two poles correspond to the dynamics of the invertedpendulum.) We note that these two poles on the imaginaryaxis preclude the direct use of stable inversion technique bythe method of dichotomies developed in [4]. In [6], the desiredtrajectory is modified by an additional input; in turn, thisadditional input (computed by a pole-placement algorithm)removes the nonhyperbolicity of the internal dynamics. In ourscheme, we first obtain a discretised model, and estimate theinput after augmenting this discretised state space model withthe input model considered earlier. We note that we require nomodification of the internal dynamics.

We can rewrite the differential equations that govern thedynamics in the following standard state space model:

xc(t) = fc(xc(t), u(t))y(t) = h(xc(t))

where xc =(x1 x1 x2 x2 θ θ

)′is the state vector.

The following discrete time equivalent state space model isobtained by using the forward rectangular rule:

xk+1 = xk + ∆fc(xk, uk)yk = h(xk)

where ∆ is the sampling interval, xk = xc(k∆), uk = u(k∆),and yk = y(k∆).

An EKF is built for the augmented state space model,and the input necessary for the desired output trajectory isestimated: We choose wk in (3) to be zero, and Qη , thecovariance corresponding to ηk to be unity. Finally, Rk ischosen as per (6). The result of applying the estimated inputto the given plant model is as shown in Fig. 5. (Even thoughour input-estimation technique yields a discrete-time signal,we assume that ∆ is sufficiently small so that the signalwith a zero-order hold approximates well the correspondingcontinuous-time signal.) We observe from Fig. 5(a) that thedesired and the actual trajectories of the position of the firstcart match rather well: albeit not quite evident from theresponse figure, we note that the tracking error is within 1% ofthe maximum value of the desired output, and is comparableto that obtained in [6]. The position of the second cart is asshown in Fig. 5(b), and can be observed to be similar to thatobtained in [6]. We note that the inverse solution by the methodof dichotomies [4] is iterative in nature, and its applicationto the example considered here necessitated modification ofthe internal dynamics [6]. On the contrary, it is evident thatour scheme is noniterative and required no modification of theinternal dynamics.

C. Car-like Robot with Nonholonomic Constraints

Nonholonomic systems are characterised by constraintequations involving the time-derivatives of the system con-figuration variables. These equations are nonintegrable, andtypically arise when the number of control inputs are less

����������

��

��������������

����

�

��

��

��

��

��

���φ

����������

�

x

y

Fig. 3. A car-like robot with nonholonomic constraints.

than the number of configuration space variables. Given apath in the configuration space, we consider here the problemof determining the input required for a car-like robot withnonholonomic constraints to track the path. The model andthe path that we consider here are from [11]:

x = cos θu1, y = sin θu1,

φ = u2, θ =1l

tanφu1. (12)

The robot, shown in Fig. 3, is modelled with the front and rearpairs of wheels acting as single wheels at the mid-point of theaxles. The wheels are allowed to roll and spin, but not slip.The configuration (x, y, θ, φ) is parametrised by the locationof the rear wheels, the angle of the car-body with respect tothe horizontal, and the steering angle with respect to the carbody. The inputs to the system are the forward velocity of therear wheel u1, and the velocity of the steering wheel, u2. Thetrajectory considered here moves the car from (−5, 1, 0.05, 1)to (0, 0.5, 0, 0) [11].

We first discretise the model in a manner similar to thatconsidered for the earlier example, and then apply our inputestimation technique. The trajectory is shown in Fig. 6. Thethree figures that depict the states φ, θ and y versus x showa good correspondence with those shown in [11]. The othertwo figures show the input estimation errors. Different inputsare required in [11] for each of the segments A, B and C ofthe trajectory; thus, each input is the concatenation of threesinusoids. Consequently, an error is observed in our approachat the instants when the sinusoids change. Nonetheless, theefficacy of the approach is evident as the estimation procedurerecovers from these changes.

IV. CONCLUSIONS

A conceptually and algorithmically simple input estimationtechnique for nonlinear systems was introduced. This approachcombined an existing stable dynamic inversion technique forlinear systems and the Extended Kalman Filter. The efficacy ofthe method is observed from its application to three differentexamples. In particular, this method is easily applicable for

nonminimum phase nonlinear systems with nonhyperbolic in-ternal dynamics without any modification of internal dynamicswhich was necessary with an earlier method.

REFERENCES

[1] L. Qiu and E. J. Davison, “Performance limitations of non-minimumphase systems in the servomechanism problem,” Automatica, vol. 29,no. 2, pp. 337–349, 1993.

[2] B. A. Francis and W. M. Wonham, “The internal model principle ofcontrol theory,” Automatica, vol. 12, pp. 457–465, 1976.

[3] A. Isidori and C. I. Byrnes, “Output regulation of nonlinear systems,”IEEE Transactions on Automatic Control, vol. 35, no. 2, pp. 131–140,February 1990.

[4] S. Devasia, D. Chen, and B. Paden, “Nonlinear inversion-based outputtracking,” IEEE Transactions on Automatic Control, vol. 41, no. 7, pp.930–942, July 1996.

[5] S. Devasia, “Output tracking with nonhyperbolic and near nonhyperbolicinternal dynamics: Helicopter hover control,” Journal of Guidance,Control, and Dynamics, vol. 20, no. 3, pp. 573–580, May-June 1997.

[6] ——, “Approximated stable inversion for nonlinear systems with nonhy-perbolic internal dynamics,” IEEE Transactions on Automatic Control,vol. 44, no. 7, pp. 1419–1425, July 1999.

[7] K. George, M. Verhaegen, and J. M. A. Scherpen, “Stable inversion ofMIMO linear discrete time non-minimum phase systems,” in Proceed-ings of the 7th Mediterranean Conference on Control and Automation,Haifa, Israel, June 1999, pp. 267–281.

[8] ——, “A systematic and numerically efficient procedure for stabledynamic model inversion of LTI systems,” in The 38th IEEE Conferenceon Decision and Control, Phoenix, Arizona, U.S.A., December 1999.

[9] ——, “Time waveform replication,” Control Laboratory, Faculty ofInformation Technology and Systems, Delft University of Technology,Delft, The Netherlands, Tech. Rep. Deliverable 10a: A Workpackage 2Report for SCOOP, October 1999.

[10] M. Boutayeb and D. Aubry, “A strong tracking extended Kalmanobserver for nonlinear discrete-time systems,” IEEE Transactions onAutomatic Control, vol. 44, no. 8, pp. 1550–1556, August 1999.

[11] R. M. Murray and S. S. Sastry, “Nonholonomic motion planning:Steering using sinusoids,” IEEE Transactions on Automatic Control,vol. 38, no. 5, pp. 700–716, May 1993.

[12] G. Monsees, K. George, J. M. A. Scherpen, and M. Verhaegen, “Afeedforward-feedback interpretation of a sliding mode control law,”in Proceedings of the 7th Mediterranean Conference on Control andAutomation, Haifa, Israel, June 1999, pp. 2384–2398.

[13] F. Deza, E. Busvelle, J. P. Gauthier, and D. Rakotopara, “High gainestimation for nonlinear systems,” Systems and Control Letters, vol. 18,no. 4, pp. 295–299, 1992.

[14] J. P. Gauthier, H. Hammouri, and S. Othman, “A simple observer fornonlinear systems applications to bioreactors,” IEEE Transactions onAutomatic Control, vol. 37, no. 6, pp. 875–880, June 1992.

[15] A. H. Jaswinski, Stochastic Processes and Filtering Theory. New York,U.S.A.: Academic Press, 1970.

[16] R. K. Mehra, “A comparison of several nonlinear filters for reentryvehicle tracking,” IEEE Transactions on Automatic Control, vol. 16, pp.307–319, 1971.

[17] M. Boutayeb, H. Rafaralahy, and M. Darouach, “Convergence analysisof the extended Kalman filter used as an observer for nonlinear determin-istic discrete-time systems,” IEEE Transactions on Automatic Control,vol. 42, no. 4, pp. 581–586, April 1997.

[18] B. D. O. Anderson and J. B. Moore, Optimal Filtering. New Jersey,U.S.A.: Prentice-Hall, Inc., 1979.

[19] J. J. Deyst and C. F. Price, “Conditions for asymptotic stability ofthe discrete minimum-variance linear estimator,” IEEE Transactions onAutomatic Control, vol. 13, no. 6, pp. 702–705, December 1968.

[20] A. Isidori, Nonlinear Control Systems, 3rd ed. London, U.K.: Springer-Verlag, 1995.

[21] R. Marino, S. Peresada, and P. Valigi, “Adaptive input-output linearizingcontrol of induction motors,” IEEE Transactions on Automatic Control,vol. 38, no. 2, pp. 208–221, February 1993.

[22] R. Gurumoorthy and R. Sanders, “Controlling non-minimum phasenonlinear systems — The inverted pendulum on a cart example,” inProceedings of the American Control Conference, vol. 1, San Francisco,California, U.S.A., June 1993, pp. 680–685.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

100

200

300

400

Time (secs)

(a) Case 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

100

200

300

400

Time (secs)

(b) Case 2

Fig. 4. Input estimation errors ‖u‖ = ‖u − u‖: (a) Case 1: with a goodknowledge of initial state. (b) Case 2: with a poor knowledge of initial state.

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

100

Time (secs)

Ou

tpu

t x

1 (

m)

DesiredActual

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

100

Time (secs)

x2 (

m)

Fig. 5. Trajectory tracking for the two carts and an inverted pendulumexample.

−6 −4 −2 0 2 4 6−0.4

−0.2

0

0.2

0.4

0.6

x

φ

−6 −4 −2 0 2 4 60

0.5

1

1.5

2

AB

C

x

y

−6 −4 −2 0 2 4 6

−0.2

−0.1

0

0.1

0.2

0.3

x

θ

0 1 2 3−5

0

5

Err

or

in u

1

0 1 2 3−4

−2

0

Time (secs)

Err

or

in u

2

Fig. 6. Trajectory tracking for nonholonomic robot.