Robust Exact Finite-Time Output Based Control using High-Order ...

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International Journal of Systems Science Vol. 00, No. 00, Month 200x, 1–16

RESEARCH ARTICLE Robust Exact Finite-Time Output Based Control using High-Order Sliding Modes Marco Tulio Anguloa∗ , Leonid Fridmana and Arie Levantb a

Departamento de Control, Facultad de Ingenier´ıa, UNAM, M´exico; b School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel. (Received 00 Month 200x; final version received 00 Month 200x) Linear time-invariant systems with matched perturbations are exactly stabilized in finite time by means of dynamic output-feedback control under the assumptions of a permanent complete vector relative degree and bounded perturbations. The approach makes use of global high-order sliding-mode controllers and differentiators. A criterion of the differentiator convergence is developed for the detection of a proper time of turning on the controller. A gain adaptation strategy is proposed for both controller and differentiator. The performance with noisy discrete sampling is studied.

Keywords: high-order sliding mode; differentiator; finite-time stability;

1.

INTRODUCTION

Hybrid systems with strictly positive dwell time can be effectively controlled, provided the convergence takes place between successive switching times or impulses. Such controllers are to be robust, and preferably exact (i.e. insensitive) w.r.t. disturbances and model uncertainties. The output measurements should be the only available information in real-time. A class of controllers with such properties might be sliding mode (SM) controllers, see for instance Choi (2002), Andrade da Silva et al. (2009) for recent approaches to the output-feedback design problem using linear matrix inequalities. Traditional (first order) SM control can only ensure the insensitivity w.r.t. any bounded disturbances acting in the control channel (matched disturbances, see, for example, Utkin (1992)), and, unfortunately, do not provide for the finitetime exact convergence. On the other hand, recently introduced High-Order SM (HOSM) controllers (see e.g. Levant (2001), Fridman and Levant (2002), Levant (2003, 2005)) also provide for the exact finite-time output stability. Additionally they also allow chattering attenuation, (see Levant (2007)). HOSM controllers were originally designed for single-input single-output (SISO) systems. A family of predesigned controllers is constructed corresponding to the relative degree of the output. They ensure finite-time-stable exact output regulation in spite of the presence of bounded matched disturbances. Each controller makes use of the successive output derivatives w.r.t. to time of the up-to-the-relative-degree-minus-one order. Using the same HOSM methodology, the derivatives can be real-time evaluated by means of a robust exact differentiator (Levant (2003)) converging in finite-time. This way, the HOSM controller is turned on once “enough time” has passed for the finite-time estimation of the derivatives to be achieved. ∗ Corresponding

author. Email: [email protected]

ISSN: 0020-7721 print/ISSN 1464-5319 online c 200x Taylor & Francis

DOI: 10.1080/0020772YYxxxxxxxx http://www.informaworld.com

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The extension of HOSM algorithms to multi-input multi-output (MIMO) systems was done in Bartolini et al. (2000), Edwards et al. (2008) and Defoort et al. (2009). In Bartolini et al. (2000) the whole state is assumed to be known, and a design methodology is introduced to replace the traditional relay control (i.e. first-order sliding modes) with a second order SM controller. Thus, the proposed control is robust and exact, but it does not provide for finite-time state stabilization and the whole state is used in the control, not only the output. An output-based controller is considered in Edwards et al. (2008) and Defoort et al. (2009) in the case when the vector relative degree exists. In Edwards et al. (2008), only asymptotic stability is ensured, the disturbance needs to be smooth, and the system is to be Bounded-Input Bounded-State stable with respect to it. In Defoort et al. (2009) the control gains’ matrix is to be approximately known, and a MIMO tracking problem is solved in finite-time. The needed derivatives of the output are assumed known, and it is mentioned that they can be calculated by means of the robust exact differentiator (Levant (2003)). Hence, the problem of global or semi-global output-feedback finite-time exact stabilization of a disturbed LTI system with matched bounded disturbances is still open. Differentiators with variable gains (Levant (2006)) and non-homogeneous arbitrary-order HOSM controllers (Levant and Michael (2008)) are used to provide for global robust observation and stabilization. A mathematically established criterion is developed to detect the time of the differentiator convergence in the presence of noises and discrete sampling in order to apply the designed output-feedback controller. Some loose restrictions on the initial state are needed to choose the initial values of the differentiator gains.

2.

PROBLEM STATEMENT

Consider a system of the form x˙ = Ax + B[u + w(t)] , y = Cx

(1)

where x ∈ Rn , u ∈ Rm , y ∈ Rm , w ∈ Rm are the state, control input, measured output and disturbance signals, respectively. We assume that only the output y(t) is available for feedback and that the disturbance satisfies the condition kw(t)k ≤ W + , with W + being a known constant. In addition, it is assumed that the system has full vector relative degree, i.e. vector relative degree (Isidori 1995, pp. 220) (to be recalled later on) and the sum of the relative degrees of the output components equals the dimension of the state space. Since the control objective is to stabilize the system at x = 0 using only the measured output y(t), both the controllability and the observability of the system in spite of the present disturbances are necessary. These requirements are satisfied due to the assumption of full vector relative degree. Indeed, it is known that if a vector relative degree exists, the observability in the presence of “unknown inputs” turns out to be equivalent to the condition of full vector relative degree, see for instance Fridman et al. (2007). We will recall this in the form of Lemma 4.2 to be introduced later on. At the same time this condition implies that the system can be transformed into the standard controllability form. Note that any nonlinear system with full vector relative degree can be completely linearized by a static feedback Isidori (1995), taking the output and its derivatives as state variables. Uncertainties matched with the inputs do not interfere with this procedure. Thus, the results of this paper can be readily extended to the case of nonlinear full-relative-degree systems with uniformly bounded matched disturbancies. Nevertheless, the paper deals with the linear presentation of the system, since such models are very common in applications, and some further development in the paper is based on it.

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3

DIFFERENTIATOR-BASED OUTPUT-FEEDBACK CONTROL Convergence criterion for HOSM differentiators

Throughout this paper we assume that all the required derivatives are available for feedback in real-time by means of a real-time exact robust HOSM differentiator (Levant (2003)). Let f (t) ∈ R be a function to be differentiated, then the k-th order differentiator takes the form 1

k

z˙0 = ν¯0 = −λk L k+1 |z0 − f | k+1 sign(z0 − f ) + z1 , 1

z˙1 = ν¯1 = −λk−1 L k |z1 − ν¯0 |

k−1 k

sign(z1 − ν¯0 ) + z2 ,

.. .

(2) 1

1

z˙k−1 = ν¯k−1 = −λ1 L 2 |zk−1 − ν¯k−2 | 2 sign(zk−1 − ν¯k−2 ) + zk , z˙k = −λ0 L sign(zk − ν¯k−1 ), where zi is the estimation of the true signal f (i) (t). The differentiator provides for the finitetime exact estimation under ideal conditions when neither noise nor sampling are present. The only needed information is an a-priory known upper bound L for |f (k+1) |. Then a parametric sequence {λi } > 0, i = 0, 1, . . . , k, is recursively built, which provides for the convergence of the differentiator of each order k. In particular, the parameters λ0 = 1.1, λ1 = 1.5, λ2 = 2, λ3 = 3, λ4 = 5, λ5 = 8 are enough for the construction of differentiators up to the 5-th order. In the presence of input noises or discrete sampling, this differentiator provides for the best possible asymptotic accuracy (Kolmogorov (1962), Levant (2003)). Nevertheless, the exact estimates of the derivatives are only available after a finite-time transient. For control purposes, the rational solution is first to wait until the finite-time exact estimate of the derivatives is ready, and only then to turn on the controller. Until now this procedure was performed waiting “enough time” to ensure the differentiator convergence. Is it possible to check in real-time whether the HOSM differentiator has converged? This question is answered in the following theorem. Theorem 3.1 : Consider the HOSM differentiator (2) of order k, where f (t) is the signal to be differentiated. Assume that the parameters {λi } provide for the finite-time convergence of differentiator (2) for any k in the absence of noises. Let (k+1)

f (t) = f0 (t) + η(t), |f0

(t)| < L, |η(t)| ≤ kη Lξ k+1 ,

(3)

where f0 (t) is an unknown basic signal, η(t) is a Lebesgue-measurable sampling noise, ξ is a positive parameter. Suppose also that f is sampled with the time step τ > 0, and τ ≤ kτ ξ, with kη , kτ being some positive constants. Then for any positive constants γ 0 , γ 1 , ..., γ k and any kf , 0 < kf < γ 0 , there exist kη , kτ , γ t > 0, such that if the inequality |z0 − f (t)| ≤ kf Lξ k+1

(4)

holds during the time interval of the length γ t ξ (or only at the sampling instants within the same interval) then starting from the beginning of this interval the inequalities (i)

|zi − f0 (t)| ≤ γ i Lξ k−i+1 , i = 0, 1, ...k

(5)

hold and are kept forever. Note that in any case the final accuracy of the form (5) is obtained in finite transient time, which is independent of ξ (Levant (2003)). In particular, this shows the applicability of the

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Theorem. Obviously, one can arbitrarily increase γ t and the time-interval length γ t ξ and decrease kf , kη , kτ preserving the statement of the Theorem. In practice, this means that one can simply take a constant length of the observation time interval and a sufficiently small sampling interval, then one has to choose sufficiently small kf , such that keeping (4) during the considered time interval implies establishment of (5). Remark. Exact estimations are obtained in the limit case with ξ = 0 and measurements continuous in time Levant (2003). In that case the convergence time from the detected vicinity (5) is proportional to ξ Levant (2005). Therefore, with properly chosen parameters kη , kτ , γ t > 0, if (4) is kept during the time γ t ξ then the exact differentiation is observed further on (i.e. the convergence time is already included in γ t ξ). (i)

(i+1)

Proof Denote σ i = (zi − f0 )/L, ~σ = (σ 0 , ..., σ k ). Subtracting f0 from the both sides of the equation on zi of (2) and dividing by L, obtain the differential inclusion k

σ˙ 0 = −λk |σ 0 − η(t)/L| k+1 sign(σ 0 − η(t)/L) + σ 1 , σ˙ 1 = −λk−1 |σ 1 − σ˙ 0 |

k−1 k

sign(σ 1 − σ˙ 0 ) + σ 2 ,

.. .

(6) 1 2

σ˙ k−1 = −λ1 |σ k−1 − σ˙ k−2 | sign(σ k−1 − σ˙ k−2 ) + σ k , σ˙ k ∈ −λ0 sign(σ k − σ˙ k−1 ) + [−1, 1]. Excluding derivatives in the right-hand side and using |η(t)| ≤ kη Lξ k+1 , obtain the non-recursive form k ˜ k σ 0 + [−kη ξ k+1 , kη ξ k+1 ] k+1 sign(σ 0 + [−kη ξ k+1 , kη ξ k+1 ]) + σ 1 , σ˙ 0 ∈ −λ k−1 ˜ k−1 σ 0 + [−kη ξ k+1 , kη ξ k+1 ] k+1 sign(σ 0 + [−kη ξ k+1 , kη ξ k+1 ]) + σ 2 , σ˙ 1 ∈ −λ .. .

(7)

1 ˜ 1 σ 0 + [−kη ξ k+1 , kη ξ k+1 ] k+1 sign(σ 0 + [−kη ξ k+1 , kη ξ k+1 ]) + σ k , σ˙ k−1 ∈ −λ ˜ 0 sign(σ 0 + [−kη ξ k+1 , kη ξ k+1 ]) + [−1, 1]. σ˙ k ∈ −λ The right hand side of (7) is minimally enlarged in order to provide for the convexity and upper-semicontinuity of the obtained differential inclusion (Filippov (1960)). The sampling of f corresponds to the time varying delay of the right-hand side not exceeding kt ξ. Denoting (7) by 

~σ ∈ Σ(~σ , ξ), obtain that the system with sampling corresponds to 

~σ ∈ Σ(~σ (t − [0, kt ξ]), ξ).

(8)

The following Lemma is proved after the Theorem proof. Lemma 3.2: For any positive T and γ, there exist such δ > 0 that for any sufficiently small ξ and any solution of (8) it follows from the inequality |σ 0 | ≤ δ being kept during the time period T that also the inequalities |σ i | ≤ γ, i = 1, ..., k, are kept during that time period. Fix some T > 0 and ξ 1 > 0 such that with any ξ < ξ 1 all trajectories of (8) starting within the region |σ i | ≤ γ i , i = 0, ..., k, finish within the same region in time T . It is possible due to the continuous dependence of the solutions on ξ (Filippov (1960)) and the finite-time stability

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of the undisturbed system. Then there exist such δ, 0 < δ < γ 0 , and ξ 0 , ξ 0 < min(δ, ξ 1 ), that with any ξ < ξ 0 all trajectories of (8) keeping the inequality |σ 0 | ≤ δ during the time interval [0, T ], also keep |σ i | ≤ γ = min(γ i ), i = 1, ..., k, during the same interval. Thus, keeping the inequality |(z0 − f (t))/L| ≤ δ 0 = δ − ξ 0 implies that |σ 0 | < δ < γ 0 and also implies that |σ i | ≤ γ i , i = 1, ..., k, are kept during the time interval. The time T was chosen so that if at any moment t the inequalities |σ i | ≤ γ i , i = 0, ..., k, hold, then they hold also at t + T . Since the inequalities are kept during the whole time interval [0, T ], they are kept forever. Show now that with any δ 1 , 0 < δ 1 < δ < γ 0 , and sufficiently small kt it is enough to check |σ 0 | < δ 1 at sampling times only to provide for |σ 0 | < δ 0 during the whole time interval. Indeed, it follows from the discrete-time sampling variant of the above Lemma. In order to prove it, one just needs to replace the division of the time interval [0, T ] considered in the proof with the natural division of [0, T ] into the sampling intervals of the length not exceeding kτ ξ. Due to the homogeneity features (Levant (2005)) under the time-coordinate-parameter transformation Gκ : (t, σ 0 , σ 1 , ..., σ k , ξ) 7−→ (κt, κ k+1 σ 0 , κ k σ 1 , ..., κσ k , κξ). solutions of (8) transfer to solutions of the same inclusion, but with the new disturbance parameter κξ. Transformation Gκ , κ = ξ/ξ 0 transfers solutions of system (8) with the disturbance parameter ξ 0 onto the solutions of system (8) with the disturbance parameter ξ. As a result get that with arbitrary ξ keeping the inequality |(z0 − f (t))/L| ≤ δ 0 (ξ/ξ 0 )k+1 during the time interval [0, (ξ/ξ 0 )T ] implies that also |σ i | ≤ γ i (ξ/ξ 0 )k−i+1 , i = 1, ..., k, are kept. Multiplying by L and choosing appropriate values of kf , γ t , obtain the statement of the Theorem.  Proof of Lemma 3.2. Suppose that the Lemma statement is not true. Take a sequence δ s → 0. Then there exists a sequence ξ s → 0, such that for each s there is a solution ~σ s (t) of (8) with ξ = ξ s , for which |σ 0 | ≤ δ is kept during the time period T , but the inequalities |σ i | ≤ γ, i = 1, ..., k, are not kept. For simplicity suppose that each solution is defined on the segment [0, T ]. Note that, due to the convergence of the differentiator, with small ξ solutions for which |σ 0 | ≤ δ, |σ i | ≤ γ, i = 1, ..., k, are kept always exist (Levant (2005)). Show that k~σ s (t)k remain uniformly bounded. Indeed, divide the segment [0, T ] in 3k equal sub-segments, and fix some δ such that δ s ≤ δ. Taking into account |σ 0 | ≤ δ and applying the mean-value Lagrange theorem to the end points of segment triplets, get 3k−1 points where |σ˙ 0 | is bounded by 3k−1 δ/T . It follows now from (7) that also σ 1 is bounded at the same points. Once more applying the Lagrange theorem get 3k−2 points, where σ 2 is bounded by a constant known in advance, etc. At the last step get 30 = 1 points, where σ k is bounded by a constant known in advance. Now taking this point as an initial one, and integrating the last equation of (7), get that σ k is uniformly bounded over the whole segment [0, T ]. Integrating the last but one equation starting from anyone of founded 3 points, get that σ k−1 is uniformly bounded over the whole segment [0, T ], etc. Thus, k~σ s (t)k remain uniformly bounded, which means that also the right-hand side of (7) is uniformly bounded. Therefore, solutions ~σ s (t) are bounded, and have a joint Lipschitz constant. Due to the Arcela Theorem there exists a uniformly convergent subsequence ~σ sl , l → ∞. The limit function ~σ ∗ has to satisfy the undisturbed system k

˜ k |σ 0 | k+1 sign σ 0 + σ 1 , σ˙ 0 = −λ k−1

˜ k−1 |σ 0 | k+1 sign σ 0 + σ 2 , σ˙ 1 = −λ .. .

(9) 1

˜ 1 |σ 0 | k+1 sign σ 0 + σ k , σ˙ k−1 = −λ ˜ 0 sign σ 0 + [−1, 1], σ˙ k ∈ −λ

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and features the property σ 0 ≡ 0 during the time interval [0, T ]. But it follows from the first equation that σ 1 ≡ 0. Now it follows from the second equation that σ 2 ≡ 0, etc. In such a way get ~σ ∗ ≡ 0. That contradicts the assumption that none of solutions ~σ s (t) keep the inequalities |σ i | ≤ γ, i = 1, ..., k.  3.2

Output-feedback application of the criterion

Consider the output-feedback application of the differentiator. Let yi = ci x, i = 1, . . . , m, be the outputs of system (1). Assume that a differentiator of the order ri − 1 has been used for each output, in order to estimate the required derivatives. Denote by zi,0 the variable z0 of the differentiator applied to the i-th output. Theorem 3.1 provides an easy way to check whether the i-th differentiator has converged by verifying that |z0,i − yi | ≤ kf,i Li τ ri i is kept in some time interval kt,i τ i . It is natural to estimate the constants kf,i and kt,i by simulation. Notice in addition that this criterion is very robust, since the value of ξ in Theorem 3.1 can be enlarged voluntarily without changing the values of the noise magnitude or the sampling step. Also the length of the time interval γ t ξ can be voluntarily enlarged preserving the theorem statement. Thus, the resulting control gets the form  ¯(t) if |z0,i − yi | ≤ kf,i Li τ ri i in time interval [t − ki,t τ i , t] u i = 1, . . . , m. , u(t) =  0 otherwise

(10)

where u ¯(t) is the control calculated using the estimated derivatives.

4.

CONTROL DESIGN

Let us recall the notion of vector relative degree from (Isidori 1995, pp. 220). Definition 4.1: The output y = Cx of system (1) has vector relative degree (r1 , . . . , rm ) if ci Ak B = 0, and

k = 0, 1, . . . , ri − 2, i = 1, 2, . . . , m,   c1 Ar1 −1 B  c2 Ar2 −1 B    rank(Q) = m, Q :=   ∈ Rm×m . ..   . cm Arm −1 B

Using this definition, we have the following well-known property. Lemma 4.2: (See for instance Fridman et al. (2007)) Assume that system (1) has full vector relative degree, i.e. r1 + · · · + rm = n. Then the following relation is valid 

Yr,1  Yr,2  x = M  ..  . Yr,m

    := M Yr , 

 Yr,i

  :=  

yi y˙ i .. . (ri −1)

   , 

(11)

yi

where M is full rank matrix. In fact, matrix M is composed of some rows of the observability matrix of the (A, C) pair. Notice that under the conditions of this last lemma, the control objective of making x = 0 can be reformulated as the problem of designing u(t) providing for y(t) ≡ 0, ∀t ≥ T .

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Now, let us introduce the following controller (ri −1)

vi = −αi Φi (Yr )Hri (yi , y˙ i , . . . , yi

Φi (Yr ) := ki,1 kYr k + ki,2 ,

),

i = 1, . . . , m,

(12)

where vi is the i-th row of v = Qu, {αi , ki,1 ki,2 } are positive gains of the controller (to be tuned), Hri is the ri -th order SM control algorithm with “gain robust parameters” and Φi is the so-called “gain function”, see Levant and Michael (2008) for further details. Theorem 4.3 : Consider system (1), assume that it has a full vector relative degree, i.e. r1 + · · · + rm = n, and that kw(t)k ≤ W + , with W + being a known constant. Then it is finite-time stabilizable to x = 0 by the controller (12), provided that αi is a large enough constant and ki,1 > kci Ari k, ki,2 > kci Ari −1 BkW + . Proof Differentiate each output yi until an input appears and group them together as (r )

y1 1 = c1 Ar1 x + c1 Ar1 −1 B[u + w(t)] (r )

y2 2 = c2 Ar2 x + c2 Ar2 −1 B[u + w(t)] .. . (rm ) ym = cm Arm x + cm Arm −1 B[u + w(t)].

By the assumption of the full vector relative degree, the matrix Q is square and of the full rank. Introduce the input transformation u = Q−1 v, so that (r )

y1 1 = c1 Ar1 x + v1 + c1 Ar1 −1 Bw .. . (rm ) ym = cm Arm x + vm + cm Arm −1 Bw.

Now each output yi satisfies the problem formulation of Levant and Michael (2008) with hi := ci Ari x + ci Ari −1 Bw,

i = 1, . . . , m.

Bounds for this last equation are easy to be obtained as |hi | ≤ kci Ari kkM kkYr k + kci Ari −1 BkW + . Choosing Φi (Yr ) := ki,1 kYr k + ki,2 , and, for instance, selecting ki,1 > kci Ari k, ki,2 > kci Ari −1 BkW + and αi large enough (to compensate for M ), obtain αi Φi > |hi |. Choosing [Levant and Michael (2008)] the controller (ri −1)

vi = −αi Φi (Yr )Hri (yi , y˙ i , . . . , yi

),

obtain that in finite-time the following equality is kept (ri −1)

{yi , y˙ i , . . . , yi

} ≡ 0,

∀t ≥ T,

for i = 1, . . . , m, that is {y1 , . . . , ym } ≡ 0, ∀t ≥ T . Due to the condition r1 + · · · + rm = n and Lemma 4.2, the equality y(t) = 0, t ≥ T implies that x(t) ≡ 0, ∀t ≥ T . 

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4.1

Differentiator gain adaptation

To evaluate controller (12), a gain Li is needed for each differentiator in order to estimate derivatives of every output yi up to the ri − 1 order. By formulas (5), in the presence of noise or discretization, the lower the gain Li the better is the obtained precision. In addition, if the system is assumed to satisfy the conditions of Lemma 4.2, there is a oneto-one correspondence between the state and the output and its derivatives. Therefore, if the initial condition of the state is very large, the differentiators’ gains are to be large as well, but once the trajectories of the system are near the origin, the gains can be significantly reduced. Thus, it is reasonable to make these gains variable in time. It was shown by Levant (2006) that Li can be any continuous function of time, but in order to guarantee robustness properties, it is required in addition that the logarithmic derivative |L˙ i (t)/Li (t)| be uniformly bounded. Proposition 4.4: Consider system (1) and assume that a) it has a full vector relative degree i.e. r1 + · · · + rm = n, + and x+ are known constants, b) kw(t)k ≤ W + and kx0 k ≤ x+ 0 where W 0 c) ku(t)k ≤ ρ1 kxk + ρ2 for some known constants ρ1 and ρ2 , Introduce the following adaptation algorithm for the gain Li (t) of each differentiator (1) Set Li (t) = L0,i for 0 ≤ t ≤ t1 , with t1 the time instant when the convergence of all differentiator is detected and with L0,i a large enough constant. (2) Set Li (t) = li,1 (kYr k + li,2 ) for t > t1 with
li,1 > kci Ari k + kci Ari −1 Bkρ1 kM k,

li,2 >

kci Ari −1 Bk(ρ2 + W + ) , li,1

and Yr as in (11) constructed using the estimations from the same differentiators. Then finite-time estimation of the derivatives of each output yi up to the order ri − 1 is ensured, and in addition, the logarithmic derivative L˙ i (t)/Li (t) is uniformly bounded for each i = 1, . . . , m. Proof Consider the two steps of the algorithm separately (1) We need to show that with a large enough gain L0,i , each differentiator converges. In this time interval the control input is set to u(t) ≡ 0. Since we want to estimate derivatives of (r ) each output up to the ri − 1 the gain L0,i should satisfy |yi i (t)| ≤ L0,i in a time interval [0, tf ] and guarantee the convergence of the differentiator. We can get an upper bound for the solution x(t) in the time interval [0, tf ] as follows. Using the well-known general solution for a linear system, considering u(t) ≡ 0, kw(t)k ≤ W + and that keAt k ≤ ekAkt (evident from the series definition of the exponential) it yields to kx(t)k ≤ keAt x0 k +

Z

t

keA(t−s) Bw(s)kds

0

≤e

kAkt

+

kAkt

Z

t

kx0 k + W kBke

e−kAks ds

0

≤ ekAkt x+ 0 +

 W + kBk  kAkt e −1 kAk (ri )

Let X + = supt∈[0,tf ] kx(t)k with tf < ∞. Now each output satisfy yi ci Ari −1 Bw(t) so (ri )

|yi

(t)| = kci Ari kX + + kci Ari −1 BkW + ,

∀t ∈ [0, tf ]

= ci Ari x +

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Since the gain of each differentiator can be selected large enough to provide any convergence time (see Levant (2003)), the gain L0,i can be selected large enough such that the differentiator converges before tf . For instance, selecting L0,i > qi kci Ari kX + + kci Ari −1 BkW + with qi a large enough constant. (2) Once the differentiators have converged, the signal Yr is available. Since kuk ≤ ρ1 kxk + ρ2 and kw(t)k ≤ W + the selection of Li (t) = li,1 (kYr k + li,2 ) with
li,1 > kci Ari k + kci Ari −1 Bkρ1 kM k,

li,2 >

kci Ari −1 Bk(ρ2 + W + ) , li,1

(r )

ensures that Li (t) is also an upper bound for yi i (t) from t1 and afterward. Now, all that is missing is to check that its logarithmic derivative is uniformly bounded. Since x = M Yr we can rewrite Li (t) as Li (t) = β i,1 (kxk + β i,2 ) for some constant β i,1 , β i,2 > 0. This last expression can be written as 

Li (t) − β i,2 β i,1

2

= xT x

Direct differentiation of this last expression gives  2

Li (t) − β i,2 β i,1

 ˙ Li (t) = xT (A + AT )x + 2xT B(u + w) β i,1 2 ˜ ≤ λmax (A)kxk + 2kxkkBk(ρ1 kxk + ρ2 ) + 2kxkkBkW +  
˜ + 2ρ1 kBk kxk2 + 2kBk ρ2 + W + kxk, ≤ λmax (A)

where A˜ = A + AT . Noticing that obtain



Li (t) β i,1

 − β i,2 = kxk we can divide both sides by kxk to


L˙ i (t) 1 ˜ + 2ρ1 kBk kxk + kBk ρ2 + W + . λmax (A) ≤ β i,1 2 Now computing L˙ i (t)/β i,1 L˙ i (t) = ≤ Li (t) Li (t)/β i,1

1 2



 ˜ + 2ρ1 kBk kxk + kBk (ρ2 + W + ) λmax (A) kxk + β i,2

,

obtain that the logarithmic derivative is uniformly bounded by a suitable constant. 

Remark. Since the trajectories of the system cannot scape to infinity in finite-time under the considered assumptions, at the first stage of the algorithm the constant L0,i can be taken very large, providing for the fast and reliable differentiator convergence. Thus there is no problem of choosing the initial value of L0,i . Any large value will suffice. For practical implementation, in the second part of the algorithm it is enough to consider Li (t) = li,1 (kYr k + li,2 ) with large enough constants li,1 and li,2 . The gain li,2 should be tuned in accordance to the disturbance amplitude, but, if no information is available, it can be selected as li,2 = 1 and only tune li,1 large enough. Also note that the gain of the differentiator decreases as the state approaches zero, which reduces chattering. On the other hand notice that since |Hri (y, y, ˙ . . . , y (ri −1) )| ≤ 1, the

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nonlinear controller v(t) selected as (12) has a norm of the form considered in assumption c) of the proposition. On the other hand, in the presence of measurement noises and discrete sampling, exact differentiation is impossible. However, due to the differentiators’ robustness and the corresponding error asymptotics, the vector Yr is known approximately, and its usage is still possible, due to the robustness of the whole construction.

4.2

System performance in the presence of noise and sampling

Theorem 4.3 guaranties the semi-global finite-time stabilization in the case when the sampling is continuously performed and no measurement noises are present. Now consider the effects of sampling step τ and the measurement noises not exceed li,1 (kYr k + li,2 )ε in absolute value, the following theorem (an immediate consequence of Levant (2005)) holds Theorem 4.5 : Let the initial values of x belong to some compact set. Then with sufficiently small ξ and ε ≤ kε Lξ max(r1 ,...,rm ) , τ ≤ kτ ξ, kε , kτ being some positive constants, the closed loop system stabilizes in finite time to a vicinity of the origin defined by the inequalities (j)

|yi | ≤ µi,j ξ ri −j , j = 0, 1, ..., ri − 1, i = 1, 2, ..., m, where µi,j are some positive constants. A drawback of the proposed controller is its relatively slow convergence with large initial conditions. The restriction is easily removed combining a linear controller with the above nonlinear one. Thus our plan is to wait for the detection of the convergence of the differentiator, then to use a linear controller that guarantees that the trajectories of the system enter a predefined ball centered at the origin, then we switch to the HOSM controller. This way a faster transient can be obtained with smaller gains of the controller. This is certainly useful in practical applications. Let t1 be the moment when the mutual convergence of all differentiators was detected. From that moment on the parameters of the differentiators are variable (see the section above). Now, it is possible to apply a linear controller ¯ r , u = −Kx = −KY

(13)

such that the resulting closed loop system is stable, i.e. A−BK is Hurwitz. The linear controllers can be freely chosen provided that the parameters li,1 , li,2 of the differentiator are taken sufficiently large with respect to this choice. Since the unperturbed system is exponentially stable, practical stability of the origin is achieved in the presence of a uniformly bounded perturbation, that is, for any chosen R > 0 there exists a (large enough) “control gain” such that in time instants t ≥ t2 the inequality kx(t)k ≤ R is true, or equivalently ¯ kYr k ≤ M −1 R = R,

∀t ≥ t2 .

From that time on, the nonlinear controller (12) may be applied. Thus, the following theorem holds. Theorem 4.6 : With sufficiently small ξ and ε ≤ kε ξ max(r1 ,...,rm ) , τ ≤ kτ ξ, kε , kτ being some positive constants, the combined controller, (13) with t1 ≤ t < t2 and (12) afterwards, stabilizes (1) in finite time to a vicinity of the origin defined by the inequalities (j)

|yi | ≤ µi,j ξ ri −j , j = 0, 1, ..., ri − 1, i = 1, 2, ..., m, for t ≥ t2 and where µi,j are some positive constants.

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The only restrictions on ξ are now that the above accuracy defines a region inside the ball kxk ≤ R and linear controllers (13) should provide for the convergence to that ball. Both restrictions can be analytically checked. Note in addition that, since only one switch between the linear controller and the HOSM controller takes place, the previously obtained stability of the system is preserved.

5.

EXAMPLE

Consider 

   01 0 00 00 1 1 1 0 0 0 0        [u + w(t)], 1 0 x˙ =  0 0 −1 0 1  x +     0 1 −1 1 1  0 0 0 1 −1 0 0 01   10000 y= x 00010 The disturbance is chosen for simulation purposes as 

 0.5 + 0.2sq(0.33t) w(t) = , 0.2 + 0.7 sin(4t) where 0.2sq(0.33t) is a symmetric square wave of amplitude 0.2 and period of 0.33 seconds. The relative degree vector is (r1 , r2 ) = (3, 2) and it is clear that it is full and well-defined since Q = I2×2 . Accordingly, Yr = M x, with Yr := [y1 , y˙ 1 , y¨1 , y2 , y˙ 2 ]T and 

 10000 0 1 0 0 0   . 1 1 1 0 0 M =   0 0 0 1 0 01011 ¯ r is designed using the standard LQ methodology with The linear controller u = −Kx = −KY −3 ¯ ¯ = 30I2×2 resulting in weight matrices Q = 10diag{100, 10 , 10−2 , 10−3 , 100} and R 

 6.3448 9.8541 4.4102 1.4123 0.5116 ¯ = K . −1.6018 −0.1048 0.5116 5.3399 7.7657 The neighborhood R of the origin in which the HOSM controller is turned on is characterized by kYr k1 ≤ 18. The required differentiators are of order 2 and 1 for each output, respectively. The selected parameters are presented in Table 1. The controllers are selected as non-homogeneous quasi-continuous ones u ¯1 := −α1 ν 31 Φ1 (Yr )H3 (y1 , y˙ 1 , y¨1 ), 3

y¨1 + 2ν 12 = −α1 ν 31 Φ1 (Yr )

3 2

2

y˙ 1 +ν 1 |y1 | 3 sign y1 h i1 2 |y˙ 1 |+ν 1 |y1 | 3 2

h

|¨ y1 | + 2ν 1 |y˙ 1 | + ν 1 |y1 |

2 3

i1 , 2

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Parameter order Li,0 li,1 li,2 kt kf τ Table 1.

paper-final1

First differentiator (i = 1) 2 200 15 15 100 7 · 10−2 10−2

Second differentiator (i = 2) 1 100 10 10 100 3 10−2

Differentiators parameters

40

20

0

−20

−40

−60

−80

−100

−120

0

5

10

15

Figure 1. Closed loop trajectories x(t) without measurement noise.

with α1 = 8, ν 1 = 1.2, to regulate the convergence rate, see Levant and Michael (2008), and Φ1 (Yr ) = kYr k2 + 1. The second controller is designed as u ¯2 := −α2 ν 22 Φ2 (Yr )H2 (y2 , y˙ 2 ), =

y˙ 2 −α2 ν 22 Φ2 (Yr )

1

+ ν 2 |y2 | 2 sign y2 1

|y˙ 2 | + ν 2 |y2 | 2

,

with α2 = 15, ν 2 = 0.9 and Φ2 (Yr ) := kYr k2 + 1. The overall controller is constructed as in (10). Two simulations were performed, one with bounded measurement noise y(t) = Cx(t) + η(t) satisfying kη(t)k1 ≤ 2 · 10−3 for all t ≥ 0 (simulated using a uniform random number generator) and the other without measurement noise. Both simulations were carried out using using Euler integration with 10−4 time step. Simulations results without measurement noise are presented in Figures 1 for the state x(t), Figure 2 for the control signal u(t) and differentiators gain Li (t) and Figure 3 for the signal of the detection of the convergence of all differentiators and the turn on signal of the linear controller. Figure 1 shows that the closed loop state trajectories reach the origin in finite-time in spite of the (non vanishing) disturbance. In Figures 4, 5 and 6 the corresponding simulations with measurement noise are presented. Comparing the closed loop trajectories under measurement noise of Figure 4 with those of Figure 1 without noise, the system’s trajectories are shown to remain in a vicinity of the original constraint x = 0 in spite of measurement noise, as Theorem 4.6 claims. The effect of the measurement noise in the control signal u(t) and the differentiators’ gains is evident by comparing Figure 5 with the corresponding signals in Figure 2. However, it is worth to remark that in

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13

100 0

0

−100 −200

−200

−400

−300 −400

−600 −800

−500 0

5

10

15

−600

0

5

a)

10

15

10

15

b)

1400

1000

1200

800

1000 800

600

600

400

400 200

200 0

0

5

10

15

0

0

5

c)

d)

Figure 2. Control signals and differentiator’s gains without measurement noise: a)u1 (t), b) u2 (t), c) L1 (t), d) L2 (t).

1 0.8 0.6 0.4 0.2 0 0

5

10

15

0

5

10

15

1 0.8 0.6 0.4 0.2 0

Figure 3. Differentiators convergence detection signal (top) and signal when the linear controller is active (bottom). Simulation results without measurement noise. Value 0 indicates false, value 1 indicates true.

Figure 5 the effect of noise in the linear controller and in the HOSM controller is comparable in spite of the discontinuous nature of the last one. Finally, Figure 6 shows that the detection of the convergence of the differentiator is still possible in spite of noise and, in fact, remains almost the same as in the simulation results without noise.

6.

Conclusions

Linear time-invariant systems with matched perturbations are exactly finite-time stabilized by means of global HOSM controllers and differentiators: • a criterion for the online detection of the convergence time of the differentiators is proposed (section 3.2), • a semi-global output based controller is designed for linear time invariant systems with matched bounded uncertainty ensuring finite-time exact state stabilization (section 4), • an adaptation algorithm for the gains of the differentiators and controllers is suggested (section 4.1).

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M. T. Angulo, L. Fridman and A. Levant 40

20

0

−20

−40

−60

−80

−100

0

5

10

15

Figure 4. Closed loop state trajectories x(t) with measurement noise. 200

100 0

0

−100 −200 −200 −400

−600

−300

0

5

10

15

−400

0

5

a)

10

15

10

15

b)

1200

800

1000 600 800 600

400

400 200 200 0

0

5

10

15

0

0

5

c)

d)

Figure 5. Control signals and differentiator’s gains with measurement noise: a)u1 (t), b) u2 (t), c) L1 (t), d) L2 (t). 1 0.8 0.6 0.4 0.2 0 0

5

10

15

0

5

10

15

1 0.8 0.6 0.4 0.2 0

Figure 6. Differentiators convergence detection signal (top) and signal when the linear controller is active (bottom). Both with noise measurements. Value 0 indicates false, value 1 indicates true.

on the 25, es-2011 [21] January 10:36 International of Systems Science paper-final1 B. Drazenovic, “TheJournal invariance conditions in variable structure systems,” Automatica, vol. 5, pp. 287–295, 1969. identified

n estimated

of an apthe simuf both conding-mode vation and s and senr time consame time plant deteralidated in ered really and sensor

is that the Thus, Berlin,even Gernoises and romechanical eads to the athematical order sliding a,attering vol. 43, pp. in

ring in relay

ation , no. issues 12, pp. chattering le for disconSection IV friction oscilhattering –126, 2000.in e observers,” ol.

and J. Barbot,

main Analysis

ity in Control ondon, U.K.:

control syssecond-order a, vol. 43, pp. nce by second l,orvol. 43, no. uncertain differentiator lti-input chatpp. 413–426, Trans. Autom. n of unknown in, Germany: Structure and d L. Fridman, w motions in 3–130. 003. ce analysis of uators,” IEEE n. 2007. f chattering in Autom. Con-

order sliding rol, 2002, pp.

: Theory and

order sliding matched un23, no. 3, pp.

onholonomic es,” Int. J. Ro8. s Right-Hand c Publishers,

IEEE Trans. 1. s with inertial 2003.

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Alejandra Ferreira was born in Mexico in 1976. 1389 She received the B.Sc. degree from National Audiscrete noisyofsampling is presented • the performance with tonomous University Mexico (UNAM), Mexico (section 4.2). City, Mexico, in 2004, where she is pursuing the [16] A. T. Fuller, “Relay control systems optimizedcontrol. for various performance Ph.D. degree in automatic criteria automation and control,” Proc. First World ConIn remote 2000–2005, she inwas with theIFAC Instrumentation gress, Moscow, Russia, 1960, vol. 1, pp. of 510–519. Department, Institute Astronomy, UNAM. Her [17] K. Furuta and Y. Pan, “Variableinterests structure include control with sliding sector,” professional electronics design, Automatica, vol. 36, pp. 211–228, observation, and2000. identification of linear systems, [18] A. Isidori, Nonlinear Systems, ed. New York: slidingControl mode control, and second its applications. Springer Verlag, 1989. [19] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int. J. Control, vol. 58, pp. 1247–1263, 1993. Francisco Javier Bejarano received the Master [20] A. Levant, “Robust exact differentiation via sliding mode technique,” and Doctor degrees in automatic control from the Automatica, vol. 34, no. 3, pp. 379–384, 1998. CINVESTAV-IPN, Mexico City, Mexico, in “Air2003 [21] A. Levant, A. Pridor, R. Gitizadeh, I. Yaesh, and J. Z. Ben-Asher, and 2006, under the direction of Prof. A. Poznyak craft pitch control via second-order sliding technique,” AIAA J. Guid., Control Dyn., vol.and 23,Dr. no. L. 4, Fridman. pp. 586–594, 2000. stayed one year at the ENSEA, France, two [22] A. Levant, “Higher He order sliding modes, differentiation and and outputyears at UNAM, Mexico with respective posdoctoral feedback control,” Int. J. Control, vol.the 76, M.S. no. 9/10, 924–941, 2003. Marco Tulio Angulo received inpp. 2009 from the Department of Control at UNAM, positions. approach He has published nine sliding papers in internaLevant, “Homogeneity to high-order mode deM´e[23] xico.A.Currently, he is pursuing a Ph.D in the same institution. His research interests include journals. sign,” Automatica,tional vol. 41, no. 5, pp. 823–830, 2005. the[24] existing tradeoffs between robustness and exactness in observation and control, particularly A. Levant, “Quasi-continuous high-order sliding-mode controllers,” in the real-time and the observability IEEE Trans. differentiation Autom. Control, vol.problem, 50, no. 11, pp. 1812–1816, Nov. 2005. properties of disturbed systems. [25] A. Levant, “Construction principles of 2-sliding mode design,” Automatica, vol. 43, no. 4, pp. M. 576–586, 2007. Leonid Fridman (M’98) received the M.S. de[26] A. Levant, “Chattering in Proc. ControlState Conf., Kos, gree inanalysis,” mathematics fromEur. Kuibyshev UniverGreece, Jul. 2007.sity, Samara, Russia, in 1976, the Ph.D. degree in ap[27] A. Levant and L. Alelishvili, “Integralfrom high-order slidingof modes,” IEEE plied mathematics the Institute Control SciTrans. Autom. Control, 52, no.Russia, 7, pp. 1278–1282, ence, vol. Moscow, in 1988, andJul. the2007. Dr.Sci. de[28] A. Levant and L. Fridman, “Accuracy homogeneous sliding modes gree in control scienceoffrom Moscow State University in the presence ofoffast actuators,” and IEEE Trans. Autom. Control, to be Mathematics Electronics, Moscow, Russia, in published. 1998. [29] A. Pisano and E. Usai, “Output-feedback control an the underwater veFrom 1976 to 1999, he was of with Department hicle prototype byofhigher-order modes,” Automatica,and vol.Civil 40, Mathematics,sliding Samara State Architecture no. 9, pp. 1525–1531, 2004. Academy. From 2000 to 2002, he was Engineering Y. B. Shtessel and I. A. Shkolnikov, space vehicle with[30] the Department of Postgraduate Study and“Aeronautical Investigationsand at the Chihuahua Leonid Fridman (M’98) received theJ. M.S. degree inno.mathematics from Kuibyshev State Unicontrol in dynamic sliding manifolds,” Control, vol. 9/10, Institute of Technology, Chihuahua, Mexico. InInt. 2002, he joined the76, Department pp. 1000–1017, 2003. inEngineering of Control, Division of Electrical Engineering Faculty, National mathematics from the Institute of versity, Samara, Russia, 1976, theofPh.D. degree in applied [31] Y. B.University Shtessel, I.ofA.Mexico Shkolnikov, and México. D. J. B. “An asympAutonomous (UNAM), HeBrown, isthe an Editor of three Control Science, Moscow, Russia, in M. 1988, and Dr.Sci. degree in control science from Moscow totic second-order smooth slidingmode modecontrol. control,”He Asian Control, vol. books and five special issues on sliding has J. published over State University of Mathematics and Electronics, Moscow, Russia, in 1998. From 1976 to 1999, 5, no.papers. 4, pp. 498–5043, 2003. 200 technical His research interests include variable structure systems [32] J.-J. E. Slotine and W. Li, Applied Nonlinear Control. London, U.K.: he with the Department of Mathematics, Samara State Architecture and Civil Engineering andwas singular perturbations. Prentice-Hall Inc,he1991. Dr. Fridman an Associate Editor ofthe the International Journal of System SciAcademy. Inis 2002, joined Department of Control, Division of Electrical Engineering of [33] V. I. Utkin, Sliding Modes in of Control and Optimization. , Berlin, Gerence and Conference Editorial Board IEEE Control Systems Society, Member Engineering Faculty, National Autonomous University of M´exico (UNAM), M´exico. He is an Springer Verlag, 1992. of TC onmany: Variable Structure Systems and Sliding mode control of IEEE Control Editor of three books and five special issues on sliding mode control. He has published over [34] V. Utkin and H. Lee, “Chattering analysis,” in Advances in Variable Systems Society. Structure and Sliding Mode Control, Lecture Notes in Control and In200 technical papers. His research interests include variable structure systems and singular performation C. is Edwards, C. Fossas, Editor and L. Fridman, turbations. Dr.Sciences, Fridman an Associate of the Eds. International Journal of System Science Berlin, Germany: Springer Verlag, 2006, vol. 334, pp. 107–123.

and Conference Editorial Board of IEEE Control Systems Society, Member of TC on Variable Structure Systems and Sliding mode control of IEEE Control Systems Society.

Arie Levant (M’08) (formerly L.V. Levantovsky) received the M.S. degree in differential equations from Moscow State University, Moscow, Russia, in 1980, and the Ph.D. degree in control theory from the Institute for System Studies (ISI), USSR Academy of Sciences, Moscow, Russia, in 1987. From 1980 to 1989, he was with ISI, Moscow. From 1990 to 1992, he was with the Mechanical Engineering and Mathematical Departments, Ben-Gurion University, Beer-Sheva, Israel. From 1993 to 2001, he was a Senior Analyst at the Institute for Industrial Mathematics, Beer-Sheva. Since 2001, he has been a Arie Levant (M’08) L.V. Levantovsky) Senior Lecturer at the Applied(formerly Mathematics Department, Tel-Aviv received University, the M.S. degree in differential equaTel-Aviv, Israel. Since January he has beenMoscow, an Associate Professor. His tions from Moscow State2009 University, Russia, in 1980, and the Ph.D. degree in control professional activities have been concentrated in nonlinear control theory, theory from the Institute for System Studies (ISI), USSR Academy of Sciences, Moscow, Russia, stability theory, singularity theory of differentiable mappings, image processing and numerous practical research projects in these and other fields. His current research interests are in high-order sliding-modes and their applications to control and observation, real-time robust exact differentiation and nonlinear robust output-feedback control.

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REFERENCES

in 1987. From 1980 to 1989, he was with ISI, Moscow. From 1990 to 1992, he was with the Mechanical Engineering and Mathematical Departments, Ben-Gurion University, Beer-Sheva, Israel. From 1993 to 2001, he was a Senior Analyst at the Institute for Industrial Mathematics, Beer-Sheva. Since 2001, he was a Senior Lecturer at the Applied Mathematics Department, TelAviv University, Tel-Aviv, Israel. Since January 2009 he is Associate Professor. His professional activities have been concentrated in nonlinear control theory, stability theory, singularity theory of differentiable mappings, image processing and numerous practical research projects in these and other fields. His current research interests are in high-order sliding-modes and their applications to control and observation, real-time robust exact differentiation and nonlinear robust output-feedback control. References

Andrade da Silva, J.M., Edwards, C., and Spurgeon, S.K. (2009), “Sliding-Mode OutputFeedback Control based on LMIs for Plants with Mismatched Uncertainties,” IEEE Transactions on Industrial Electronics, 56(9), 3675–3683. Bartolini, G., Ferrara, A., Usai, E., and Utkin, V. (2000), “On multi-input chattering-free second order sliding mode control,” IEEE Transactions on Automatic Control, 45(9), 1711–1717. Choi, H.H. (2002), “Variable structure output feedback control design for a class of uncertain dynamic systems,” Automatica, 38(2), 335 – 341. Defoort, M., Floquet, T., Kokosy, A., and Perruquetti, W. (2009), “A novel higher order sliding mode control scheme,” Systems and Control Letters, 58(2), 102 – 108. Edwards, C., Floquet, T., and Spurgeon, S. (2008), “Circumventing the Relative Degree Condition in Sliding Mode Design,” Modern Sliding Mode Control Theory New Perspectives and Applications, Lecture Notes in Control and Information Sciences, 375, 137–158. Filippov, A., Differential equations with discontinuous right-hand side, London: Kruwler (1960). Fridman, L., and Levant, A. (2002), “Higher Order Sliding Modes,” in W. Perruquetti, J. P. Barbot, eds. ”Sliding Mode Control in Engineering”, Marcel Dekker, Inc., pp. 53–101. Fridman, L., Levant, A., and Davila, J. (2007), “Observation of Linear Systems with Unknown Inputs via High-Order Sliding-Mode,” International Journal of Systems Science, 38(10), 773–791. Isidori, A., Nonlinear Control Systems, Secaucus, NJ, USA: Springer-Verlag New York, Inc. (1995). Kolmogorov, A.N. (1962), “On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval,” Amer. Math. Soc. Transl, 2(9), 233–242. Levant, A. (2001), “Universal SISO sliding-mode controllers with finite-time convergence,” IEEE Transactions on Automatic Control, 46(9), 1447–1451. Levant, A. (2003), “High-order sliding modes: differentiation and output feedback control,” International Journal of Control, 76(9-10), 1924–041. Levant, A. (2005), “Homogeneity approach to high-order sliding mode design,” Automatica, 41(5), 823–830. Levant, A. (2005), “Quasi-Continuous High-Order Sliding-Mode Controllers,” IEEE Transactions on Automatic Control, 50(11), 1812–1816. Levant, A., and Michael, A. (2008), “Adjustment of high-order sliding-mode controllers,” International Journal of Robust and Nonlinear Control. Levant, A. (2006), “Exact Differentiation of Signals with Unbounded Higher Derivatives,” in Proc. of the 45th IEEE Conference on Decision and Control, San-Diego, California, pp. 5585 – 5590. Levant, A. (2007), “Chattering analysis,” in European Control Conference 2007, Kos, Greece, pp. 2–5. Utkin, V.I., Sliding Modes in Control Optimization, Springer (1992).