Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems

Automatica 49 (2013) 2767–2773

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Brief paper

Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems✩ Marco Tulio Angulo a,1 , Leonid Fridman b , Jaime A. Moreno c a

Facultad de Ingeniería, UAQ, Mexico

b

Departamento de Control y Robótica, UNAM, Mexico

c

Eléctrica y Computación, Instituto de Ingeniería, UNAM, Mexico

article

info

Article history: Received 20 April 2012 Received in revised form 19 March 2013 Accepted 3 May 2013 Available online 14 June 2013 Keywords: Sliding-mode control

abstract A novel methodology for designing multivariable High-Order Sliding-Mode (HOSM) controllers for disturbed feedback linearizable nonlinear systems is introduced. It provides for the finite-time stabilization of the origin of the state-space by using output feedback. Only the additional assumptions of algebraic strong observability and smooth enough matched disturbances are required. The control problem is solved in two consecutive steps: firstly, designing an observer based on the measured output and, secondly, designing of a full-state controller computed from a new virtual output with vector relative degree. The introduced notion of algebraic strong observability allows recovering the state of the system using the measured output and its derivatives. By estimating the required derivatives through the HOSM differentiator, a finite-time convergent observer is constructed. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction The design of robust controllers is an important topic in automatic control theory. Uncertainty is also manifested by only measuring a subset of the state variables of the system, namely, its ‘‘measured output’’. In such scenario, Sliding-Mode (SM) outputfeedback based controllers have shown to be very successful (Oliveira, Peixoto, Costa, & Hsu, 2010; Oliveira, Peixoto, & Hsu, 2010, 2013; Peixoto, Oliveira, Hsu, Lizarralde, & Costa, 2011). Moreover, modern systems frequently mix continuous and discrete event dynamics for which the stabilization problems are much more intricate. However, hybrid systems with strictly positive dwell-time can be effectively controlled if the controller accomplishes the control objective before the next switching or impulse time. Therefore, robust output-feedback controllers providing finite-time state stabilization become relevant.

✩ We gratefully acknowledge the financial support from projects PAPIIT 17211 and IN111012, CONACyT 56819, 132125, 51244 and CVU 229959, Fondo de Colaboración del II-FI, UNAM, IISGBAS-165-2011. The material in this paper was partially presented at the 11th International Workshop on Variable Structure Systems (VSS’10), June 26–28, 2010, Mexico City, Mexico. This paper was recommended for publication in revised form by Associate Editor Nathan Van De Wouw under the direction of Editor Andrew R. Teel. E-mail addresses: [email protected] (M.T. Angulo), [email protected] (L. Fridman), [email protected] (J.A. Moreno). 1 Tel.: +52 442 186 22 55; fax: +52 55 5623 3600x8052.

0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.05.013

High Order SMs (HOSMs) are useful in this context providing for the finite-time exact2 output stabilization using only output feedback (Levant, 2003). In addition, in the presence of noise and sampling, HOSMs offer better accuracy than first-order SM. However, they were originally designed only for single-input single-output systems, see, e.g., Dinuzzo and Ferrara (2009) and Efimov, Zolghadri, and Raïssi (2011). In Defoort, Floquet, Kokosy, and Perruquetti (2009), HOSM controllers are extended to Multi-Input Multi-Output (MIMO) nonlinear systems under the assumption of vector relative degree with respect to the measured output. This last assumption also requires that the system has the same number of inputs as outputs. Recently, in Angulo, Fridman, and Levant (2012), the authors introduced a methodology for the design of HOSM controllers for MIMO disturbed linear systems under necessary conditions: a known (affine) bound on the disturbance, controllability and strong observability (Hautus, 1983). The proposed methodology allows constructing an output-based HOSM controller guaranteeing the finite-time exact convergence of the whole state of the system. This brief extends the methodology introduced in Angulo et al. (2012) in the context of linear systems to a class of MIMO nonlinear systems. As in the linear case, the measured output does not necessarily has vector relative degree. In this form, HOSMs can

2 Exactness is more than robustness: the effect of (matched) disturbances is completely eliminated after a finite-time transient.

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be applied to a larger class of systems when compared to the strategy of Defoort et al. (2009). Our approach requires introducing a suitable concept of observability despite disturbances. The notion of ‘‘algebraic strong observability’’ corresponds to the possibility of reconstructing the state as a function only of the measured output and a finite number of its derivatives. By using the HOSM differentiator (Levant, 2003) to estimate the required derivatives, a finite-time convergent observer is obtained. By combining the proposed observer with HOSMs controllers and dynamic feedback linearization, the problem of exact finitetime state stabilization of nonlinear systems is solved. The proofs of all Theorems are collected in the Appendix. Main contributions. (1) An algorithm to construct an unknown input observer for nonlinear systems is presented based on the notion of algebraic strong observability and the use of HOSM differentiators; (2) by using the proposed observer, a novel HOSMbased output feedback control strategy for the finite-time state stabilization of (dynamic) feedback linearizable nonlinear systems with bounded matched disturbances is presented.

generality, we assume that q(0) = 0. Note that, in general, the flat output z (t ) may not coincide with the measured output y(t ). The degree S of required smoothness for the disturbance depends on the ‘‘order’’ of the dynamic compensator used to linearize the system, as shown in Section 4. In the particular case when the system can be linearized by a static compensator, the disturbance needs to be only uniformly bounded. The approach we follow to solve the problem involves two steps. First, based on A1, we present an algorithm to compute the function F appearing in Eq. (2). This allows constructing a UIO for the system once the required derivatives are estimated. With Assumptions A2 and A3, it becomes possible to estimate such derivatives using the HOSM differentiator (Levant, 2003). In the second step, by using A4, it is shown that the finite-time state stabilization problem is equivalent to the finite-time stabilization of the flat output. This way, by using the estimated state of the previous step to evaluate the flat output, a multivariable HOSM controller that semi-globally stabilizes the flat output to zero in finite-time is presented.

2. Problem statement

3. Construction of the Unknown Input Observer

Consider x˙ = f (x) + g (x)[u + w],

y = h(x);

x(0) = x0 ,

(1)

where x(t ) ∈ Rn , u(t ) ∈ Rm , w(t ) ∈ Rm and y(t ) ∈ Rp are the state, control input, disturbance and measured output, respectively. The vector fields f , g and the function h are assumed to be smooth. The control objective is to design the control input u(t ), depending only on the output y(t ), to provide finite-time state stability at zero despite the presence of the disturbance w(t ). In order to present the assumptions we have made to solve the control problem, let us first introduce the following concept: Definition 1. System (1) is said to be algebraically strongly observable with respect to the output y(t ) ∈ Rp , if there exists a function F and integers ki , i = 1, . . . , p, such that (k )

(kp )

x = F (y1 , . . . , y1 1 , . . . , yp , . . . , yp

).

(2)

Strong observability was introduced by Hautus (1983) for linear systems to solve the Unknown Input Observer (UIO) design problem. It allows recovering the state of the system through the knowledge of the output and its derivatives only, irrespective of the input. The notion of algebraic strong observability is a natural extension to the case of nonlinear systems. In Section 3, this property will be shown to be instrumental for the design of a UIO for nonlinear systems. The following assumptions about the system will be made throughout this paper: (A1) system (1) is algebraically strongly observable with respect to the measured output y; (A2) for a given degree of smoothness S ∈ Z+ , there exists a constant W + such that ∥w (k) (t )∥ ≤ W + , ∀t ≥ 0 and all k = 0, . . . , S; (A3) for all initial conditions x0 ∈ Rn and disturbances w(·) satisfying A2, the solution3 to system (1) with u(·) ≡ 0 exists for all t ≥ 0 and remains upper-bounded by a possibly unknown number; (A4) there exists a function q : Rn → Rm such that z = q(x) is a flat output for system (1).

Once the function F in (2) is known, the state can be determined when the derivatives of the output are estimated. By means of the HOSM differentiator, these derivatives can be estimated exactly and in finite time. In addition, the HOSM differentiator provides an estimation that is robust to measurement noise and has the best order of precision (Levant, 2003). The following Subsection presents an algorithm to compute the function F for a class of nonlinear systems. Section 3.2, analyzes the use of the HOSM differentiator to estimate the required derivatives. 3.1. An algorithm to construct an Unknown Input Observer Let us consider the following additional assumption: (A5) The Lie derivatives Lg Lkf h(x) are constant for k = 1, . . . , n − 1. This assumption means that in the time derivatives of the measured output, the matrix multiplying the input is constant. This allows writing a constant orthogonal to such matrix. Assumption A5 is only introduced to simplify the presentation of the algorithm: in the general case when A5 is not satisfied, the orthogonal can be computed in the same way but now depending on the state. Let b⊥ denote a left annihilator for matrix b (i.e. b⊥ b = 0). The presented algorithm computes the function F introduced in Eq. (2), enabling the construction of a UIO for system (1). Our algorithm is the ‘‘dual’’ of the standard Structure Algorithm (see Conte, Moog, & Perdon, 2007, p. 76), since it looks for directions orthogonal to the input. Nevertheless, it is closely related to the algorithm presented in Molinari (1976) for linear systems. Step 0. Set yˆ 0 := y and M0 (x) := h(x). ˙ 0 (x) = y˙ˆ 0 = Lf M0 (x) + Lg M0 (x)w := Step 1. Compute M

˙ ˆ 1 (x) and set a1 (x) + b1 (u + w), introduce y˙ˆ 1 := b⊥ 1 M0 (x) = a     yˆ 0 h( x ) = . aˆ 1 (x) y˙ˆ 1 Step k + 1. Compute   ˙ k (x) = yˆ˙ 0 · · · yˆ (k+1) T := ak+1 (x) + bk+1 (u + w), M k

M1 (x) :=

(k+1) ˙ ˆ k+1 (x), and set introduce yˆ k+1 := b⊥ k+1 Mk (x) = a



yˆ 0





h(x)



A flat system is feedback linearizable and, moreover, its state and input can be written as a function of the flat output and its derivatives (Sira Ramírez & Agrawal, 2004). Without loss of

Mk+1 (x) := 

3 Solutions to differential equations (and their associated inclusions) are understood in Filippov’s sense (Filippov, 1960).

By applying the algorithm, one is able to compute a sequence (k) of equations of the form [ˆy0 , y˙ˆ 1 , . . . , yˆ k ]T = Mk (x) for k ≥ 1. The algorithm converges in the following sense.

 ..   ..  .  =  . . (k+1) aˆ k+1 (x) yˆ k+1

M.T. Angulo et al. / Automatica 49 (2013) 2767–2773

Theorem 1. Suppose that the vector fields f , g and the function h are meromorphic, rank h(x) = p and the system is algebraically strongly observable. Under these conditions rank d Mn−p (x) = n for almost all x ∈ Rn and the function F := Mn−−1p as in Eq. (2) exists locally. Recall that a meromorphic function is the quotient of two analytic functions. Computing the inverse function of Mk−1 is, in general, a difficult problem. For this reason, it is useful to perform all possible simplifications at each step of the algorithm, replacing functions of the state by functions of the output. We have illustrated this idea in the example below by introducing the additional variables Nk , k ≥ 1. Example 3.1. Consider the following system not having vector relative degree with respect to the measured output: x˙ 1 = x1 x2 + w1 , x˙ 2 = x33 + w1 ,

x˙ 3 = x4 (1 + x1 ) + w1 , x˙ 4 = sin(x1 x2 ) + w2 ,

y1 = x1 , y2 = x2 .

Applying the algorithm yields: Step 0: Set yˆ 0 = y and N0 (x) =

  x1 x2

simplify it and set M0 (x) = N0 (x).  ˙ 0 (x) = x13x2 + 1 Step 1: Compute M 1 x

= yˆ 0 =



yˆ 0,1 yˆ 0,2



.Trivially



0 0

w = a1 (x)+b1 w. Thus   ˙ 0 (x) = x33 − x1 x2 and 1 , introduce y˙ˆ 1 = −1 1 M  T set N1 (x) := yˆ 0 y˙ˆ 1 .Proceed to simplify it as   yˆ 0  T 1   M1 (x) := ˙yˆ + yˆ 0,1 yˆ 0,2 3 = x1 x2 x3 . 1 3

 b 1 = −1



⊥

Step 2: Compute

 ˙ 1 ( x) =  M



x1 x2 1 + 1 x33 1 x4 (1 + x1 )



0 0 w := a2 (x) + b2 w. 0



−1 0 1 and introduce y¨ˆ 2 = −1 Then b⊥ 2 = = x4 (1 + x1 ) − x1 x2 , 





Set N2 (x) = yˆ 0 , y˙ˆ 1 + yˆ 0,1 yˆ 0,2





 13

0

1

ki

k +1 ζ˙0 = ν0 = −λki Li i |ζ0 − yi | ki +1 sign(ζ0 − yi ) + ζ1 , .. . ζ˙ki −1 = νki −1 1  1 = −λ1 Li2 ζki −1 − νk−2  2 sign(ζki −1 − νki −2 ) + ζki , ζ˙ki = −λ0 Li sign(ζki − νki −1 ),

(3)

(j)

where ζj is the estimation of the true derivative yi (t ). This differentiator provides for the finite-time exact differentiation under ideal conditions of exact measurement in continuous time. The only information needed is an a-priory known upper bound Li (k +1) for |yi i |. Then a parametric sequence {λj } > 0, j = 0, 1, . . . , ki , is recursively built, which provides for the convergence of the differentiators for each order ki . In particular, the parameters λ0 = 1.1, λ1 = 1.5, λ2 = 2, λ3 = 3, λ4 = 5, λ5 = 8 are enough till the 5-th differentiation order. With discrete sampling the differential equations are replaced by their Euler approximations. In addition, the differentiator (3) provides for the best possible order of accuracy in the presence of measurement noises (Levant, 2003). Proposition 1. Suppose that the system (1) satisfies A1, A2 and A3. Then, for any large enough constant Li > 0, the HOSM differentia(j) tor (3) guarantees that ζj (t ) = yi (t ) after a finite-time transient. Moreover, in the presence of measurement noise uniformly bounded by δ , the accuracies

  ki +1−j j |ζj − y(i j) | = O L ki +1 δ ki +1 ,

j = 0, . . . , ki ,

are obtained after a finite-time transient. Using one HOSM differentiator per output, it is possible to obtain all the required derivatives of the measured output.

˙ 1 (x) 1 M 

T

, y¨ˆ 2

2769

and simplify it as

  yˆ 0   1   x1  ˙ 3  yˆ 1 + yˆ 0,1 yˆ 0,2  x2    M2 (x) :=   = x3  .  y¨ˆ 2 + yˆ 0,1 yˆ 0,2  x4

1 + yˆ 0,1 Since rank d M2 (x) = 4 the algorithm stops and the system is algebraically strongly observable. In addition, we have obtained an expression of all states as a function of the measured output and its derivatives.  This last example also shows that the condition of vector relative degree (with respect to the measured output) considered in Defoort et al. (2009) is not necessary for the system to be algebraic strongly observable. 3.2. Estimating the derivatives: a finite-time convergent UIO Once Eq. (2) has been determined using the procedure described in the previous Subsection, the next step to construct the UIO is to evaluate the required derivatives of the measured output. This Subsection briefly describes the use of the HOSM differentiator to estimate them. Let yi (t ) be a component of the measured output to be differentiated ki -times. The ki -th order HOSM differentiator (Levant, 2003) takes the form

Remark 1. The reliable derivatives’ estimations are available after a finite-time transient. Thus, a controller is to be applied only after the transient ends, avoiding unnecessarily increasing the disturbances in the system. This dwell-time strategy improves the stability and performance properties in the output-feedback control of strongly nonlinear systems (Oliveira, Peixoto, Costa et al., 2010; Oliveira et al., 2013). Detecting the convergence of the HOSM differentiators is done by checking how close to zero the differences |ζ0 (t ) − yi (t )| are, for i = 1, . . . , p, as shown in Angulo et al. (2012). This last criterion can be applied in the presence of measurement noise and discretization. It should also be noted that precluding finite-time escape of the trajectories (Assumption A3) is fundamental for the existence of the constants Li , i = 1, . . . , p, required for the HOSM differentiators. When this last assumption is not satisfied, it is still possible to use the HOSM differentiator but requires additional modifications (Oliveira et al., 2013). 4. Control design Once the state of the system has been recovered as shown in the last section, the control problem can be solved by designing u(t ) to keep the identity z (t ) ≡ 0 after a finite-time transient. Proposition 2. For any input w(·) and initial state x0 , the condition z (t ) ≡ 0, t ≥ 0 implies that x(t ) ≡ 0, t ≥ 0. Since the flat output z (t ) ∈ Rm has vector relative degree, each one of its components can be independently controlled by one distinct input.

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Let us recall that an undisturbed system (1) is said to be flat (or linearizable by dynamic state feedback) if there exists a feedback

ξ˙ = M (x, ξ ) + N (x, ξ )v, u = α(x, ξ ) + β(x, ξ )v,

ξ (t ) ∈ Rm¯ ;

(4)

where v(t ) ∈ Rm is the ‘‘new input’’, and a diffeomorphism η = Ξ (x, ξ ) such that the closed loop system reads as η˙ = Aη + Bv, where the pair (A, B) is composed by Brunovsky blocks with only one input per block (Sira Ramírez & Agrawal, 2004). Equivalently, selecting each component of the flat output zi , i = 1, . . . , m, as the first variable of each Brunovsky block, the undisturbed system looks like a chain of integrators (ni )

zi

= vi ,

i = 1 , . . . , m,

(5)

where ni , i = 1, . . . , m, is a list of integers. Note that unlike (Defoort et al., 2009), the relative degree of the measured output plays no role in the control design. The only requirement is the invertibility of the augmented system with respect to the new input v . When the disturbance w is present, it perturbs system (5). The disturbance first appears in the derivative of the flat output (r ) corresponding to its relative degree ri , i.e., zi i = ai (x) + bi (x)(u + w), i = 1, . . . , m. From then on, it needs to be differentiated (ni − ri )-times to linearize the system (ni )

zi

= vi +

Si 

γℓ (x, ξ )w(ℓ) ,

(6)

ℓ=0

where Si = ni − ri and γℓ (x, ξ ), ℓ = 0, . . . , Si , are known functions that can be computed from the parameters of the system. Therefore, the problem is now to design each input vi to provide exact finite-time stability of the output zi . For doing this, we shall assume that the required S = max{S1 , . . . , Sm } derivatives of the disturbance are all uniformly bounded (Assumption A2). Theorem 2. Consider system (1) under Assumptions A2, A3 and A4. Then with any αi > 0 large enough and ϵi > 0, the controller (4) with v = (v1 , . . . , vm ) selected as (ni −1)

vi = −αi Φi (x, ξ )Hni (zi , . . . , zi

),

i = 1, . . . , m,

(7)

where zi is the first variable of the i-th Brunovsky block and Hni is an ni -th order HOSM control algorithm with gain function Φi selected as

Φi (x, ξ ) = ϵi +

S 

γℓ (x, ξ )W + ,

i = 1 , . . . , m,

ℓ=0

provides for the finite-time stabilization of the origin. Moreover, if Assumption A1 is also met and the function F in (2) has been determined then finite-time stability is preserved when all the variables in (7) are replaced by the ones estimated using Proposition 1. In expression (7), Hni denotes an ni -th order HOSM controller. It can be chosen from a predefined list of controllers using solely its order, see Levant and Michael (2009) and Section 5 for an example with n1 = 1, n2 = 2. When the function F appearing in (2) and the feedback linearization of (4) are globally valid, then our approach can provide for semi-global finite-time stability of the origin. This fact comes from noticing that given a ball of initial conditions for the system from which we want to stabilize it, it is always possible to design the gains Li , i = 1, . . . , p, large enough ensuring that the HOSM differentiators converge if the trajectories do not scape a larger ball, see Proposition 1. The proposed methodology can still be applied only when Assumption A3 is satisfied in the sense that the system’s trajectories exist for all t ≥ 0. In such a case, notice that for every

R > 0 and T > 0, there exists r (R, T ) > 0 such that ∥x(t )∥ ≤ R for all t ∈ [0, T ] if ∥x0 ∥ ≤ r for the uncontrolled system. According to Remark 1, the controller is turned on once the observer has converged, so it provides stability of the origin. Therefore, it always exists r0 such that ∥x(t )∥ ≤ R for all t ≥ 0 if ∥x0 ∥ ≤ r0 when the state feedback control is applied. Hence if the observer is designed to be convergent when the trajectories are such that ∥x(t )∥ ≤ R and also such that it converges in a time T0 < T , when the trajectories are still in the set ∥x0 ∥ ≤ r0 , then the controller will assure that they satisfy ∥x(t )∥ ≤ R for all t ≥ 0. Hence, finitetime stability of the origin is obtained provided that the initial conditions of the system are small enough, i.e. ∥x0 ∥ ≤ r (r0 , T0 ). These conditions for the observer can be always satisfied if the gain Li , i = 1, . . . , p, of each differentiator is sufficiently large (Levant, 2003). This fact is illustrated in the example of Section 5. A constructive procedure to obtain the dynamic compensator of Eq. (4) is provided by the Inversion Algorithm (Conte et al., 2007), see also the example in Section 5. Remark 2. For a flat system the input can also be written as a function of the flat output and its derivatives. Together with the strong observability with respect to the measured output, this allows estimating the disturbance using the measured output. The only requirement is to take one more time-derivative. Due to the use of the HOSM differentiator, this adds restrictions in the disturbance and the trajectories of the system: it must be slow enough. Then it becomes possible to compensate the disturbance without applying a discontinuous HOSM controller. This last strategy has been recently explored in Ferreira, Bejarano, and Fridman (2011). 5. Simulation example Consider the following system x 2 + u1 + w 1 , x33 + x3 + u1 + w1 , sin(x1 x2 ) + x3 cos(x3 ) + u2 + w2 ,

x˙ 1 = x˙ 2 = x˙ 3 =

y1 = y2 =

x1 , x2 ,

with w1 = w2 = 0.5 sin(t ) + 0.5 square(0.2t ) the sum of a sinusoid and a square signal of amplitude 0.5 and 0.2 s of period. The system above satisfies A3 only in the sense that their trajectories are bounded for each time instant. This happens since the x33 does not appear in the equation for x3 . The method of Defoort et al. (2009) is not applicable since y does not have vector relative degree. Using the algorithm of Section 3.1 yields



x1

x2





x3 = y1

y2

 φ1 (˙y2 − y˙ 1 − y2 ) ,

where

2/3  √ −231/3 + 21/3 9θ + 12 + 81θ 2 , φ1 (θ ) =  1/3 √ 62/3 9θ + 12 + 81θ 2 and the required derivative was obtained using a first-order HOSM differentiator with gain L = 250. We shall compare the control design for two distinct choices of flat output. For the first choice, we select the measured output as the flat output. In this case, it turns out that a dynamic compensator is required and that the first disturbance needs to be smooth. Since the selected disturbance does not satisfy this last requirement, it prevents applying the proposed methodology using the measured output as flat output. For the second selection we compute a new flat output that linearizes the system using a static compensator. This second choice outperforms the first one: both disturbances only need to be uniformly bounded, the controller is simpler and static, and it requires less derivatives. The simulation results for this second choice are also presented.

M.T. Angulo et al. / Automatica 49 (2013) 2767–2773

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Fig. 1. Simulation result with (bottom) and without (top) measurement noises. (a) System trajectories x(t ). (b) Control input. (c) Top: estimation error for state x3 (t ); bottom: 1 indicates that the differentiator (observer) has converged.

For the first selection we use the measured output as flat output. Computing the second derivative of each output and using the dynamic compensator ξ˙ = −ξ − x33 + x3 + v1 with u1 = ξ ,

u2 = (3x23 + 1)−1 v2 − γ (x) − v1 + ξ + x33 − x3 ,





and γ (x) = (3x23 + 1)[sin(x1 x2 ) + x3 cos(x3 )] yields y¨ 1 = v1 + w1 + w ˙ 1,

y¨ 2 = v2 + (3x23 + 1)w2 + w ˙ 1.

Therefore, we require that w1 , w2 and w ˙ 1 are all uniformly bounded. Since w1 (t ) contains square signals, w ˙ 1 is not uniformly bounded as a function of time and prevents applying the proposed methodology. For the second selection, we apply the procedure of Conte et al. (2007, p. 122) to find a flat output that linearizes the system using a static compensator. If possible, the only restriction would be that the disturbance is uniformly bounded. For finding the desired flat output, compute the subspaces H1 = span{d x}, H2 = span{d x1 − d x2 } and H3 = {0}. This fact confirms that the system can be linearized using a static compensator. Then set ω2 = d x1 − d x2 and compute ω ˙ 2 = d (δ x1 ) − d (δ x2 ) = d x2 − (3x23 + 1)d x3 . Then one may select ω1 = d x2 . This yields H1 = span{d x} = span{ω1 , ω2 , ω ˙ 2 }, and therefore span{d z1 , d z2 } = span{ω1 , ω2 } is a set of flat outputs. That is z1 = x2 and z2 = x1 − x2 . Notice that z can be obtained using y through a nonsingular transformation, so the system is also strongly observable with respect to the output z. Computing the input–output dynamics for these two outputs one obtains z˙1 = x33 + x3 + u1 + w1 , z¨2 = x33 + x3 − γ (x) + u1 + w1 − (3x23 + 1)(u2 + w2 ), u1 = −

− x3 + v 1 ,

u2 = −(3x23 + 1)−1 (−x23 − x3 − γ (x) − u1 + v2 ), yields z˙1 = v1 + w1 , z¨2 = v2 − (3x23 + 1)w2 . Now design the HOSM controller by selecting v1 = −α1 H1 (z1 ) = −α1 sign(z1 ) and v2 = −α2 Φ2 (x3 )H2 (z2 , z˙2 ) where H2 (z2 , z˙2 ) =

6. Conclusions A methodology for the design of output feedback HOSM controllers for (dynamic) feedback linearizable disturbed nonlinear systems was presented. It provides finite-time exact state stabilization using only output feedback, provided that the system is algebraically strongly observable and the disturbances are smooth enough. The introduced notion of ‘‘strong’’ observability allows recovering the state by using only the output and its derivatives. When the derivatives are estimated using the HOSM differentiator, an Unknown Input Observer that converges in finite-time is obtained. Acknowledgment

and setting x33

is a Quasi-Continuous algorithm (Levant & Michael, 2009) with α1 = 1, α2 = 4.5, Φ2 (x3 ) := 3x23 + 0.5. The controller is switched on only when the convergence of the differentiator (i.e., the UIO) is detected according to Angulo et al. (2012). All simulations were carried out using Euler’s method with 10−3 [s] of time step. Two simulations studies were carried out, see Fig. 1. The first one presents the simulation results in the absence of noise. It shows the finite-time exact convergence of the state trajectories to zero. Secondly, measurement noises were added to both measured outputs. The noises were generated using Simulink’s ‘‘Uniform Random Number’’ generator with maximum amplitudes of 0.005 and 0.01 for y1 and y2 , respectively. The simulation results confirms that the robustness of HOSM controllers and differentiators to noises guarantees keeping a bounded state error despite the presence of noise.

1 2

z˙2 + |z2 | sign(z2 )

|˙z2 | + |z2 |

1 2

,

The authors would like to express their gratitude to Prof. Claude H. Moog (IRCCyN, France) for his assistance in the construction of the algorithm of Section 3.1. Appendix. Proofs of theorems and propositions Proof of Theorem 1. Under the assumption that the vector fields are meromorphic, it is possible to use differential–algebraic

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tools to derive structural properties of the system (Conte et al., 2007). This approach introduces finite-dimensional spaces of 1forms over the field of meromorphic functions K of x: Y k := spanK {d y, . . . , d y(k) }, X = spanK {d x}, W = spanK {d w (j) , j ≥ 0}. For our purposes, this means that the statement rank Mk (x) = n holds except (possibly) in a set of zero Lebesgue measure. Under this framework, Definition 1 of algebraic strong observability is equivalent to the condition X ∩ Y k = X with k = max{k1 , . . . , kp }. By construction span{d Mi (x)} = X ∩ Y i . Notice that equality is guaranteed by construction. This way vectors Mi (x) define a chain of subspaces (X ∩ Y 0 ) ⊂ · · · ⊂ (X ∩ Y i ) ⊂ (X ∩ Y i+1 ) ⊂ · · · that is non-decreasing and bounded by X . Therefore, the sequence of subspaces has a unique limit. Since algebraic strong observability implies that X ∩ Y k = X for some finite k, then the upper bound of the sequence is attained and must coincide with the limit of such sequence. Now, since span{d Mk (x)} = X ∩ Y k = X this means that rank d Mk (x) = n. Moreover, the condition X ∩ Y i−1 = X ∩ Y i would imply that X ∩ Y i+1 = X ∩ Y i and that X ∩ Y i−1 is another limit, then the dimension of each subspace should increase by at least one at each step of the algorithm. Then, if we begin with a dimension p, at most n − p steps of the algorithm are required to reach the limit X ∩ Y n−p = X . Finally, since rank d Mn−p (x) = n then the Inverse Function Theorem guarantees that there exists, at least locally, an inverse for Mn−p as claimed.  Proof of Proposition 1. Due to Assumptions A2 and A3, x˙ (t ) is uniformly bounded as a function of time. By Assumption A1, the function F in formula (2) exists. From these two facts and taking one time derivative of F and equating with x˙ (t ) obtain that ( k +1 ) yi i (t ), i = 1, . . . , p, are all uniformly bounded. Therefore, there exist large enough constants Li > 0, i = 1, . . . , p, such that each HOSM differentiator converges. This fact and the accuracy under measurement noise were proven in Levant (2003).  Proof of Proposition 2. By definition, the system is also algebraically strongly observable with respect to the flat output z (t ). Now, on one hand, since q(0) = 0, the condition x(t ) ≡ 0 implies that z (t ) ≡ 0. Any other state trajectory satisfying z (t ) ≡ 0 contradicts the strong observability of the system.  Proof of Theorem 2. Using the diffeomorphism η = Ξ (x, ξ ) and the dynamic compensator (4) yields a set of m disturbed chain of integrators (6). Moreover, the output z has vector relative degree with respect to v . This way the HOSM control design for v can be made exactly as shown in Angulo et al. (2012) to ensure that z (t ) ≡ 0 after a finite-time Tc . From Proposition 2, one concludes that x(t ) ≡ 0, t ≥ Tc . This completes the proof of the first part of the claim. For the second part, we assume that the system is algebraically strongly observable (A1). Then Proposition 1 ensures the finitetime exact estimation of the derivatives and hence, that controller (7) constructed using the estimated variables is applied only when it coincides with the controller constructed using the true variables. After the controller is applied, the correct estimation of the derivatives using the HOSM differentiator is preserved if their gains Li were large enough, as Proposition 1 ensures.  References Angulo, M. T., Fridman, L., & Levant, A. (2012). Output-feedback finite-time stabilization of disturbed lti systems. Automatica, 48(4), 606–611. Conte, G., Moog, C., & Perdon, A. (2007). Algebraic methods for nonlinear control systems (2nd ed.). Springer. Defoort, M., Floquet, T., Kokosy, A., & Perruquetti, W. (2009). A novel higher order sliding mode control scheme. Systems & Control Letters, 58(2), 102–108. Dinuzzo, F., & Ferrara, A. (2009). Higher order sliding mode controllers with optimal reaching. IEEE Transactions on Automatic Control, 54(9), 2126–2136.

Efimov, D. V., Zolghadri, A., & Raïssi, T. (2011). Actuator fault detection and compensation under feedback control. Automatica, 47(8), 1699–1705. Ferreira, A., Bejarano, F., & Fridman, L. (2011). Robust control with exact uncertainties compensation: with or without chattering? IEEE Transactions on Control Systems Technology, 19(5), 969–975. Filippov, A. (1960). Differential equations with discontinuous right-hand side. London: Kluwer. Hautus, M. (1983). Strong detectability and observers. Linear Algebra and its Applications, 50, 353–468. Levant, A. (2003). High-order sliding modes: differentiation and output feedback control. International Journal of Control, 76(9–10), 924–941. Levant, A., & Michael, A. (2009). Adjustment of high-order sliding-mode controllers. International Journal of Robust and Nonlinear Control, 19(15), 1657–1672. Molinari, B. P. (1976). A strong controllability and observability in linear multivariable control. IEEE Transactions on Automatic Control, 91(5), 761–764. Oliveira, T. R., Peixoto, A. J., Costa, R. R., & Hsu, L. (2010). Dwell-time and disturbance monitoring for peaking avoidance and performance improvement in highgain observer based sliding mode control. Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms, 17(6), 839–874. Oliveira, T., Peixoto, A., & Hsu, L. (2010). Sliding mode control of uncertain multivariable nonlinear systems with unknown control direction via switching and monitoring function. IEEE Transactions on Automatic Control, 55(4), 1028–1034. Oliveira, T. R., Peixoto, A. J., & Hsu, L. (2013). Peaking free output-feedback exact tracking of uncertain nonlinear systems via dwell-time and norm observers. International Journal of Robust and Nonlinear Control, 23(5), 483–513. Peixoto, A. J., Oliveira, T. R., Hsu, L., Lizarralde, F., & Costa, R. R. (2011). Global tracking sliding mode control for a class of nonlinear systems via variable gain observer. International Journal of Robust and Nonlinear Control, 21(2), 177–196. Sira Ramírez, H., & Agrawal, S. (2004). ser. control engineering, Differentially flat systems. Marcel Dekker.

Marco Tulio Angulo received his Ph.D. (Summa cum Laude) degree in Electrical Engineering (Automatic Control) from the National Autonomous University of México (UNAM), México, in 2012. Currently, he is a Professor in the Autonomous University of Querétaro (UAQ). He enjoys investigating the tradeoffs between robustness and fragility of various problems in automatic control, optimization of uncertain systems and thermodynamics of networked systems.

Leonid Fridman received the M.S. degree in Mathematics from Kuibyshev (Samara) State University, Samara, Russia, in 1976, the Ph.D. degree in Applied Mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and the Dr. Sci. degree in Control Science from Moscow State University of Mathematics and Electronics, Moscow, Russia, in 1998. From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil Engineering University. From 2000 to 2002, he was with the Department of Postgraduate Study and Investigations at the Chihuahua Institute of Technology, Chihuahua, Mexico. In 2002, he joined the Department of Control Engineering and Robotics, Division of Electrical Engineering, in the Engineering Faculty at the National Autonomous University of Mexico (UNAM), México. His main research interests are variable structure systems. He is an Author and Editor of five books, and a Guest Editor of twelve special issues on sliding mode control. He is an Author of more than 300 technical papers. Prof. Fridman is an Associate Editor of The Journal of the Franklin Institute, and Nonlinear Analysis: Hybrid Systems. He is on the Conference Editorial Board of the IEEE Control Systems Society, a Member of the IEEE Control Systems Society TC on Variable Structure Systems and Sliding Mode Control and a Member of the IFAC TC on Discrete Events and Hybrid Systems. He was a winner of the Scopus prize for the best cited Mexican Scientists in Mathematics and Engineering in 2010. He has worked as an Invited Professor in 17 universities and research centers in Australia, France, Germany, Italy, Israel and Spain.

Jaime A. Moreno was born in Colombia and he received his Ph.D. degree (Summa cum Laude) in Electrical Engineering (Automatic Control) from the Helmut-Schmidt University in Hamburg, Germany in 1995. The Diploma Degree in Electrical Engineering (Automatic Control) from the Universität zu Karlsruhe, Karlsruhe, Germany in 1990, and the Licentiate Degree (with honors) in Electronic Engineering from the Universidad Pontificia Bolivariana, Medellin, Colombia in 1987. He is Full Professor of Automatic Control and the Head of the Electrical and Computing Department at the Institute of Engineering

M.T. Angulo et al. / Automatica 49 (2013) 2767–2773 from the National University of Mexico (UNAM), in Mexico City. From 2008 until 2011 he was the Chair of the IFAC Technical Committee ‘‘Bioprocesses and Biosystems’’. He is the Author and Editor of 8 books, 4 book chapters, 1 patent, and Author and Co-Author of more than 250 papers in refereed journals and conference

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proceedings, and they have received more than 450 citations. His current research interests include robust and non-linear control with application to biochemical processes (wastewater treatment processes), the design of nonlinear observers and higher order sliding mode control.