HIGHER ORDER SLIDING MODES AND ARBITRARY-ORDER EXACT ROBUST DIFFERENTIATION

HIGHER ORDER SLIDING MODES AND ARBITRARY-ORDER EXACT ROBUST DIFFERENTIATION A. Levant Institute for Industrial Mathematics, 4/24 Yehuda Ha-Nachtom St., Beer-Sheva 843, Israel Fax: +972-7-232 and E-mail: levant@bgumail.bgu.ac.il Keywords: Variable Structure Control, Trajectory Tracking in Non-linear Systems, Stabilization of Non-linear Systems, Robust Filtering, Non-linear Observers. Abstract An arbitrary-order finite-time-convergent exact robust differentiator is constructed based on higher-order sliding mode (HOSM) technique. Being used in a feedback together with previously proposed HOSM controllers, it produces a universal controller, formulated in input-output terms only, which causes the output of any uncertain smooth SISO minimum-phase dynamic system with known relative degree to vanish in finite time. That allows exact tracking of arbitrary real-time smooth signals. The control can be made arbitrarily smooth, providing for arbitrarily-high trackingaccuracy order with respect to the sampling step. Only one parameter is to be tuned. Introduction Control under heavy uncertainty conditions remains one of the main research fields of the modern control theory. One of the most simple and effective ways to withstand the uncertainty is based on the sliding-mode technique [3]. Sliding modes keep equality of some output variable to zero. With being the deviation of some real-time given signal from the output, the standard sliding mode provides actually for full output control in the case when the relative degree is (i.e. the control appears explicitly already in the first total derivative of ). In order to apply that approach with higher relative degrees some auxiliary output is to be found with relative degree. Another restriction is that standard sliding modes may cause possibly dangerous system vibrations (the so-called chattering effect). These restrictions are removed by higher-order sliding modes (HOSM) [4, 6, ]. Arbitrary-order sliding mode controllers [8, 0] provide for full output control of any uncertain smooth SISO minimumphase dynamic system with known relative degree r, provided r- exact successive output derivatives,,, (r-) being estimated in real time. The auxiliary-constraint construction is avoided, the convergence time is finite and may be made arbitrarily small, while only one scalar parameter needs to be adjusted [8, 0]. The control can be made an arbitrarily smooth function of time, totally removing the chattering effect and providing for ultimate accuracy in realization. Hence, the real-time robust exact differentiation becomes the main problem of output-feedback HOSM control design. Even the most modern differentiators [3, 2, 4] do not provide for exact differentiation with finite-time convergence or require some knowledge on the noise and would actually eliminate the above-listed advantages of the HOSM controllers [0]. The derivatives may be calculated by successive implementation of a robust exact first-order differentiator [7] with finite-time convergence. That differentiator is based on 2-sliding mode and is proved to feature the best possible asymptotics in the presence of infinitesimal Lebesgue-measurable measurement noises if the second time derivative of the unknown base signal is bounded. The accuracy of the mentioned differentiator is proportional to ε /2, where ε is the maximal measurement-noise magnitude. Therefore, having been n times successively implemented, that differentiator will provide for nth-order differentiation accuracy of the order of ε (2-n ). Hence, the differentiation accuracy deteriorates rapidly. On the other hand, it is proved [7] that when the Lipschitz constant of the nth derivative of the unknown clear-of-noise signal is bounded by a given constant, the best possible differentiation accuracy of the ith derivative is proportional to ε (n+-i)/(n+), i = 0,,, n. Therefore, a special differentiator is to be designed for each differentiation order. Such second-order differentiator was proposed in [8, 9]. Having been used in combination with the above-mentioned r-sliding controller, an (r-)th-order asymptotically optimal differentiator would provide for sup to be proportional to the maximal measurement error [8]. An arbitrary order asymptotically-optimal robust exact differentiator with finite-time convergence is presented for the first time in this paper. Computer simulation and a model example of the differentiator usage in a closed loop are demonstrated. In spite of the good features of the proposed differentiator the author does not claim to solve immediately a very complicated practical problem of the input-based output

control under heavy uncertainty conditions. Nevertheless, he believes that the proposed theoretical solution indicates a way promising future practical success. At the same time the proposed differentiator is possibly the best solution for the single problem of real-time numerical differentiation. 2 Arbitrary-order differentiator Let input signal f(t) be a function defined on [0, ) consisting of a bounded Lebesgue-measurable noise with unknown features and an unknown base signal f 0 (t) with the nth derivative having a known Lipschitz constant C > 0. The problem is to find real-time robust estimations of f 0 (t), f (n) 0 (t),, f 0 (t) being exact in the absence of measurement noise. Two similar recursive schemes of the differentiator are proposed here. Let an (n-)th-order differentiator D n- (f( ), C) i produce outputs D n-, i = 0,,, n -, being estimations of f 0, f 0, f (n-) (n-) 0,, f 0 for any input f(t) with f 0 having Lipschitz constant C > 0. Then the nth-order differentiator i has the outputs z i = D n, i = 0,,, n, defined as follows: z 0 = v, v = -λ 0 z 0 - f(t) n/( n + ) sign(z 0 - f(t)) + z, z = D n- 0 (v( ), C),, zn = D n- n- (v( ), C). Here D 0 (f( ), C) is a simple nonlinear filter () D 0 : z = -λ sign(z - f(t)), λ > C. (2) Thus, the first-order differentiator coincides here with the differentiator published in [7]: z 0 = v, v = -λ 0 z 0 - f(t) /2 sign(z 0 - f(t)) + z, (3) z = -λ sign(z - v) = -λ sign(z 0 - f(t)). Another recursive scheme is based on the differentiator (3) as the basic one. Let D ~ n (f( ), C) be such a new (n-)thorder differentiator, n, D (f( ), C) coinciding with the ~ differentiator D (f( ), C) given by (3). Then the new scheme is defined as z 0 = v, v = -λ 0 z 0 - f(t) n/( n + ) sign(z 0 - f(t)) + w 0 + z, w 0 = -α 0 z 0 - f(t) (n - )/( n +) sign(z 0 - f(t)); ~ 0 n z = D (v ( ), C),, z n = D (v( ), C). n ~ n The resulting 2nd-order differentiator was presented at ECC 99, CDC 00 [8, 9]: z 0 = v 0, v 0 = -λ 0 z 0 - f(t) 2/3 sign(z 0 - f(t)) + w 0 + z, w 0 = -α 0 z 0 - f(t) /3 sign(z 0 - f(t)); (4) z = v, v = -λ z - v 0 /2 sign(z - v ) + w, w = -α sign(z - v 0 ); z 2 = w. Similarly, a 2nd-order differentiator from each of these sequences may be used as a base for a new recursive scheme. An infinite number of differentiator schemes may be constructed in this way. The only requirement is that the resulting systems be homogeneous in a sense described further. While the author has checked only the above two schemes (), (2) and (3), (4), the hypothesis is that all such schemes produce working differentiators, provided suitable parameter choice. Differentiator () takes on the form z 0 = v 0,v 0 = -λ 0 z 0 - f(t) n/( n + ) sign(z 0 - f(t)) + z, z = v,v = -λ z - v 0 (n-)/ n sign(z 0 - v 0 ) + z 2, z i = v i, v i = -λ i z i - v i- (n - i)/ ( n - i + ) sign(z i - v i- ) + z i+, (5) z n- = v n-, v n- = -λ n- z n- - v n-2 / 2 sign(z n- - v n-2 ) + z n, z n = -λ n sign(z n - v n- ), Theorem. The parameters being properly chosen, the following equalities are true in the absence of input noises after a finite time of a transient process for both differentiator schemes: z 0 = f 0 (t); z i = v i- = f 0 (t), i =,, n. Moreover, the corresponding solutions of the dynamic systems are Lyapunov stable, i.e. finite-time stable [2]. The Theorem means that the equalities z i = f 0 (t) are kept in 2-sliding mode, i = 0,, n-. Theorem 2. Let the input noise satisfy the inequality f(t) - f 0 (t) ε. Then the following inequalities are established in finite time for some positive constants µ i, ν i depending exclusively on the parameters of differentiators () or (4): z i - f 0 (t) µi ε (n - i +)/(n + ), i = 0,, n; v i - f 0 (i+) (t) νi ε (n - i )/(n + ), i = 0,, n-. Consider the discrete-sampling case, when z 0 (t j ) - f(t j ) is substituted for z 0 - f(t) with t j t < t j+, t j+ - t j = τ > 0. Theorem 3. Let τ > 0 be the constant input sampling interval in the absence of noises. Then the following inequalities are established in finite time for some positive constants µ i, ν i depending exclusively on the parameters of differentiators () or (4): z i - f 0 (t) µi τ n - i +, i = 0,, n; v i - f 0 (i+) (t) νi τ n - i, i = 0,, n -. In particular, the nth derivative error is proportional to τ. The latter Theorem means that there are a number of real sliding modes of different orders [6]. Nevertheless, nothing can be said on the derivatives of v i and of z i, because they are not continuously differentiable functions.

Homogeneity of the differentiators. The differentiators are invariant with respect to the transformation (t, f, z i, v i, w i ) ( ηt, η n+ f, η n-i+ z i, η n-i v i, η n-i w i ). The parameters α i, λ i are to be chosen recursively in such a way that α, λ,, α n, λ n provide for the convergence of the (n-)th order differentiator with the same Lipschitz constant C, and α 0, λ 0 be sufficiently large (α 0 is chosen first). The best way is to choose them by computer simulation. Substituting f(t)/c for f(t) and taking new coordinates z i = C z i, v i = C v i, w i = Cw i achieve the following proposition. Proposition. Let parameters α 0i, λ 0i, i = 0,,., n, of differentiators (), (2) or (3), (4) provide for exact n-th order differentiation with C =. Then the parameters α i = α 0i C 2/(n-i+), λ i = λ 0i C /(n-i+) are valid for any C > 0 and provide for the accuracy z i - f 0 (t) µi C i/(n + ) ε for some µ i. (n - i +)/(n + ) Remark. It is easy to check that differentiator (5) may be rewritten in the non-recursive form z 0 = -κ 0 z 0 - f(t) n/( n + ) sign(z 0 - f(t)) + z, z i = -κ i z 0 - f(t) (n - i)/( n + ) sign(z 0 - f(t)) + z i+, i =,, n- z n = -κ n sign(z 0 - f(t)) for some positive κ i calculated on the basis of λ 0,, λ n. 3 Proofs Consider for simplicity differentiator (5). The proof for differentiator (4) is very similar. Introduce functions 0 = z 0 - f 0 (t), = z - f (n) 0 (t),, n = z n - f 0 (t), ξ = f(t) - f0 (t). Then any solution of (5) satisfies the following differential inclusion understood in the Filippov sense [5]: 0 = -λ 0 0 + ξ(t) n/( n + ) sign( 0 + ξ(t)) +, (6) = -λ - 0 (n-)/ n sign( - 0 ) + 2, n n n = -λ n- n- - 2 / 2 sign( n- - 2 ) + n, -λ n sign( n - ) + [-C, C], n n where ξ(t) [-ε, ε] is a Lebesgue-measurable noise function. It is important to mark that (6), (7) do not remember anything on the unknown input basic signal f 0 (t). System (6) is homogeneous in sense [2] with ε = 0, its trajectories are invariant with respect to the transformation (7) G η : (t, i, ξ, ε) ( ηt, η n-i+ i, η n+ ξ, η n+ ε). (8) The Theorems are proved by induction. Define the main features of differential inclusion (6), (7) which hold with a proper choice of the parameters λ i. Proposition 2. Let ξ δ (t) satisfy the condition that the integral ξ δ (t) dt over a time interval δ is less than some fixed K > 0. Then for any 0 < S i < S i, i = 0,, n, each trajectory of (6), (7) starting from the region i S i does not leave the region i S i during this time interval if δ is sufficiently small. Proposition 3. For each set of numbers S i > 0, i = 0,, n, there exist such numbers Σ i > S i, k i >0 and T > 0, ε M 0 that for any ξ(t) [-ε, ε], ε ε M, any trajectory of (6), (7) starting from the region i S i enters within the time T, and without leaving the region i Σ i, the region i k i ε n-i+ and stays there forever. It is easy to see that these Propositions are true with n = 0, λ 0 > C. Prove these Propositions by induction. Let them be true for the differentiator of the order n -. Prove them for the nth order differentiator () with sufficiently large λ 0. Proof of Proposition 2. Choose some S Mi, S i < S i < S Mi, i = 0,, n. Then 0 λ 0 ξ(t) + 0 n/( n + ) + λ 0 ξ(t) n/( n + ) + λ 0 S M0 n/( n + ) + SM. Thus, according to the H lder inequality 0 dt λ 0 δ /( n + ) ( ξ(t) dt) n/( n + ) n/( n + ) + δ(λ 0 S M0 + SM ). The remark that 0 serves as the input disturbance for the (n-)th order system (7) and satisfies the conditions of Proposition 2 finishes the proof. Lemma. If for some S i < S i, i = 0,, n, and T > 0 any trajectory of (6), (7) starting from the region i S i enters within the time T the region i S i and stays there forever, then the system (6), (7) is finite-time stable. The Lemma is a simple consequence of the invariance of (6), (7) with respect to transformation (8). The convergence time is estimated as a sum of a geometric series. Proof of Proposition 3. Consider first the case ε M = 0. Choose some larger region i S i, S i < S i, i = 0,, n, and let Σ i > S i, i = 0,, n -, be some upper bounds chosen with respect to Proposition 3 for (7) and that region. It is easy to check that for any q > with sufficiently large λ 0 the trajectory enters the region 0 (qσ / λ 0 ) ( n + )/n in arbitrarily small time. During that time 0 does not change its sign. Therefore, 0 disturbance - 0 dt S 0 for sufficiently large λ 0. Thus, the entering the subsystem (7) satisfies

Proposition 2 and the inequalities i S i are kept. From that moment on 0 3 Σ is kept with a properly chosen q. Differentiating (6) with ξ = 0 achieve 0 = -λ 0 0 -/( n + ) 0 +, where according to (7) the inequality 4λ Σ + Σ 2 holds. Thus, with sufficiently large λ 0 within arbitrarily small - time the inequality λ 0 (4λ Σ + Σ 2 ) (qσ / λ 0 ) /n is established. Its right-hand side may be made arbitrarily small with large λ 0. Thus, due to Proposition 3 for (n-)-order system (7) and to the Lemma, the statement of Proposition 3 is proved with ε M = 0. Let now ε M > 0. As follows from the continuous dependence of the solutions of a differential inclusion on the right-hand side [5] with sufficiently small ε M all trajectories concentrate within a finite time in a small vicinity of the origin. The asymptotic features of that vicinity with ε M changing follow now from the homogeneity of the system. Theorems and 2 are simple consequences of Proposition 3 and the homogeneity of the system. To prove Theorem 3 it is sufficient to consider 0 = -λ 0 0 (t j ) n/( n + ) sign( 0 (t j )) +, (9) instead of (6) with t j t < t j+ = t j + τ. The resulting hybrid system (9), (7) is invariant with respect to the transformation (t, τ, i ) ( ηt, ητ, η n-i+ i ). (0) On the other hand, it may be considered as system (6), (7) with arbitrarily small ε for any fixed region of the initial values when τ is sufficiently small. Applying successively Proposition 3 and the homogeneity reasoning with transformation (0) achieve Theorem 3. Remark. It is easy to see that the above proof may be transformed in order to obtain a constructive upper estimation of the convergence time. Also the accuracy may be estimated by means of the suggested inductive approach. 4. Numeric differentiation results Following are equations of the 5th-order differentiator with simulation-tested coefficients for C =. As mentioned, it contains also differentiators of the lower orders: z 0 = v 0, v 0 = -50 z 0 - f(t) 5/6 sign(z 0 - f(t)) + z, () z = v, v = -30 z - v 0 4/5 sign(z - v 0 ) + z 2, (2) z 2 = v 2, v 2 = -6 z 2 - v 3/4 sign(z 2 - v ) + z 3, (3) z 3 = v 4, v 4 = -8 z 3 - v 2 2/3 sign(z 4 - v 3 ) + z 4, (4) z 4 = v 4, v 4 = -4 z 4 - v 3 /2 sign(z 4 - v 4 ) + z 5, (5) z 5 = -2 sign(z 5 - v 4 ); (6) These parameters can be easily changed, for the differentiator is not very sensitive to their values. The tradeoff is as follows: the larger the parameters, the faster the convergence and the higher sensitivity to input noises and the sampling step. Differentiator () - (6) and its 3rd-order sub-differentiator (3) - (6) were used for simulation. Initial values of the differentiator state were taken zero with exception for the initial estimation of f, which is taken equal to the initial measured value of f. The base input signal f 0 (t ) = sin 0.5t + cos t. (7) was taken for the differentiator testing. The higher derivatives of f 0 (t ) do not exceed.25 in absolute value. Third order differentiator. The measurement step τ = 5 0-4 was taken, noises are absent. The attained accuracies are 9.40 0 -, 2.97 0-7, 2.40 0-4, 0.040 for the signal tracking, the first, second and third derivatives respectively. The derivative tracking deviations changed to 9.44 0-5, 2.97 0-0, 2.42 0-6 and 0.0040 respectively after τ was reduced to 5 0-5. That corresponds to Theorem 3. Fifth order differentiator.. The attained accuracies are.46 0-3, 7.6 0-0, 9.86 0-7, 3.76 0-4, 0.0306 and 0.449 for tracking the signal, the first, second, third, fourth and fifth derivatives respectively with τ = 5 0-4. The derivative tracking deviations changed to 4.44 0-6, 6.34 0-2, 2.24 0-8, 2.44 0-5, 0.00599 and 0.265 respectively after τ was reduced to 5 0-5. The latter results are demonstrated in Fig.. Reduction of τ no longer leads to noticeable improvement of the fifth-derivative tracking. Following is the explanation. The demonstrated asymptotics is not trivially obtained by computer simulation. For example, the computer realization accuracy of the harmonic functions is not sufficient here. In other words, in computer G sin t cos t, and even the long GW double precision does not help here. For that reason the input was modeled as a solution of the corresponding differential equation and most calculations were carried out with the long double precision. Nevertheless, there is such a value of the sampling interval τ = τ that any further reduction of τ no longer leads to any considerable accuracy improvement. In particular, τ is about 0-4 with the differentiation order 4 and τ 0-3 with the differentiation order 5, when the Lipschitz constant C of the 4th and 5th derivative respectively is C =. While preparing this paper the author wanted to demonstrate 0th order differentiation, but found that differentiation of the order exceeding 5 is unlikely to be performed with the standard software. Further calculations are to be carried out with precision higher than the standard long double precision (28 bits per number).

K m u (r) r K M, L a L for some Km, K M, L > 0. The controller has the form u = - α sign(φ(β,, β r-,,,, (r-) )). Here β,, β r- may be chosen in advance, α is adjusted mainly by computer simulation, not using the unknown exact values of K m, K M, L. The identity (r) = u u (r) r + L a implies (r) L + αk M, which allows implementation of the proposed (r-)th-order differentiator [8]. A list of such controllers is presented in [8, 0]. Fig. : 5th-order differentiation Sensitivity to noises. The main problem of the differentiation is certainly its well-known sensitivity to noises. As we have seen, even small computer calculation errors appear to be a considerable noise in the calculation of the fifth derivative. Recall that, when the nth derivative has the Lipschitz constant and the noise magnitude is ε, the best possible accuracy of the ith order differentiation, i n, is kε (n-i+)/(n+) [7], where k > is a constant independent on the differentiation realization. That is a minimax (worst case) evaluation. Since differentiator () - (6) assumes this Lipschitz input condition, it satisfies this accuracy restriction as well (see also Theorems 2, 3). In particular, with the noise magnitude ε = 0-6 the 5th derivative error exceeds ε /6 = 0.. For comparison, if successive first order differentiation were used, the respective error would be at least ε (2-5 ) = 0.649 and some additional conditions on the input signal would be required. Taking 0% as a border, achieve that the direct successive differentiation does not give reliable results starting with the order 3, while the proposed differentiator may be used up to the order 5. The lack of place does not allow to list here detailed simulation results with noisy inputs. With the noise magnitude 0.0 and the noise frequency about 0000 the 5th order differentiator produces estimation errors 0.000200, 0.036, 0.84, 0.649,.05 and 0.740 for signal (7) and its 5 derivatives respectively. The author found that the second differentiation scheme (4) provides for slightly better accuracies. 5. Output-feedback HOSM control Universal HOSM controllers [8, 0] are used in order to make vanish in finite time the output of any uncertain minimum-phase system x = a(t,x) + b(t,x)u with an output = (t, x). Here x R n, a, b, are unknown smooth functions, u R, n is also uncertain. The relative degree r of the system is assumed to be constant and known. It is supposed that 0 < Fig. 2: Kinematic car model Consider a simple kinematic model of car control [] y = v cos ϕ, y 2 = v sin ϕ, ϕ = v/l tan θ, (8) θ = ξ, ξ = u, (9) where x and y are Cartesian coordinates of the rear-axle middle point, ϕ is the orientation angle, v is the longitudinal velocity, l is the length between the two axles and θ is the steering angle (Fig. 2). The system is artificially extended by (9) in order to provide for smoother performance, u is the resulting control. The task is to steer the car from a given initial position to the trajectory given by the equation y 2 = g(y ), while y and y 2 are assumed to be measured in real time. Define s = y 2 - g(y ), Let v = const = 0 m/s, l = 5 m, g(y ) = 0 sin 0.05y + 5, y = y 2 = ϕ = θ = ξ = 0 at t = 0. Suppose that x, y, ϕ, θ are available, and apply the universal 4-sliding controller [8, 0] with α = 40 u = - 40 sign{ D 3 3 (s( ))+3 (D3 2 (s( ))) 6 + (D3 (s( ))) 4 + s 3 ) /2 sign[ D 3 2 (s( ))+(D3 (s( )) 4 + s 3 ) /6 sign(d3 (s( )) +0.5 s 3/4 sign s )]}, where D 3 (s( )), D3 2 (s( )), D3 3 (s( )) are the outputs of the third order differentiator (6) with parameters λ 0 = 5, λ = 20, λ 2 = 40, λ 3 = 88. During the first half-second control was not applied. The trajectory of the car is shown in Fig. 3 and the corresponding steering angle θ in Fig. 4. The tracking accuracy.63 0-4 was attained with τ = 0.00.

Thus, it may be replaced by sign (r-2), where (r-2) is the first difference between successive discrete measurements. In such a way the second order differentiator appears to be sufficient in many cases. References 6. Conclusions Fig. 3: Trajectory of the car Fig. 4: Steering angle Arbitrary order real-time exact differentiation with the optimal input-output noise asymptotics is demonstrated. Together with the arbitrary order sliding controllers it provides for full SISO control based on the input measurements only, when the only information on the controlled process is actually its relative degree. The proposed differentiator as well as the resulting closed-loop controller are robust with respect to measurement noises. At the same time this robustness drastically decreases with the growing differentiation order. The reason is not an unsuccessful differentiator structure, but the very nature of higher order differentiation. It is easy to see that most practically important problems of output control are covered by the cases when relative degree r equals 2, 3 and 4. Indeed, according to the Newton law, the relative degree of a spatial variable with respect to a force, being understood as a control, is r = 2. Taking into account some dynamic actuator, achieve relative degree 3. If the actuator input is needed to be a continuous Lipschitz function, the relative degree has to be artificially increased to 4. As was mentioned above, the rth order sliding controller requires availability of,,, (r-). Thus, in practice the need for more than 3 successive output derivatives is very rare. In fact, often only the sign of (r-) is actually required. [] G. Bartolini, A. Ferrara, and E. Usai, Chattering avoidance by second-order sliding mode control, IEEE Trans. Automat. Control, vol. 43, no. 2, pp.24-246, (998). [2] G. Bartolini, A. Pisano and E. Usai, First and second derivative estimation by sliding mode technique, Journal of Signal Processing, vol. 4, no. 2, pp. 67-76, March 2000, Research Institute of Signal Processing, Tokyo, (2000). [3] A. Dabroom, H.K. Khalil, Numerical Differentiation Using High-Gain Observers, Proc. of the 37th Conf. on Decision and Control (CDC'97), San Diego, California, December '97, pp. 4790-4795, (997). [4] S.V. Emelyanov, S.K. Korovin, and L.V Levantovsky, Higher order sliding regimes in the binary control systems, Soviet Physics, Doklady, vol. 3, no. 4, pp. 29-293, (986). [5] A.F. Filippov, Differential Equations with Discontinuous Right-Hand Side, Kluwer, Dordrecht, the Netherlands, (988). [6] A. Levant (L.V. Levantovsky), Sliding order and sliding accuracy in sliding mode control, International Journal of Control, vol. 58, no.6, pp.247-263, (993). [7] A. Levant, Robust exact differentiation via sliding mode technique, Automatica, vol. 34, no.3, pp. 379-384, (998). [8] A. Levant, Controlling output variables via higher order sliding modes, Proc. of the European Control Conference (ECC 99), September 999, Karlsruhe, Germany, (999). [9] A. Levant, Higher order sliding: differentiation and black-box control, Proc. of the 39th IEEE Conf. on Decision and Control (CDC2000), December 2000, Sydney, Australia, pp. 703-708, (2000). [0] A. Levant, Universal SISO sliding-mode controllers with finite-time convergence, to appear in IEEE Transactions on Automatic Control, (200). [] R. Murray and S.Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Automat. Control, vol. 38, no. 5, pp.700-76, (993). [2] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Control Lett., vol. 9, no. 4, pp. 467 473, (992). [3] V.I. Utkin, Sliding Modes in Optimization and Control Problems, Springer Verlag, New York, (992). [4] X. Yu and J.X. Xu, Nonlinear derivative estimator, Electronic Letters, vol. 32, no. 6, (996).