Continuous Terminal Sliding-Mode Controller

Continuous Terminal Sliding-Mode Controller S. Kamal a , J. A. Moreno b , A. Chalanga a , B. Bandyopadhyay a , L. Fridman c a b

Systems and Control Eng., Indian Institute of Technology, Bombay, India

Instituto de Ingenier´ıa, Universidad Nacional Aut´ onoma de M´exico (UNAM), Coyoac´ an, D.F., 04510, Mexico

c Institute for Automation and Control, Graz University of Technology, Graz, Austria on leave of Department of Control Engineering and Robotics, Facultad de Ingenier´ıa, Universidad Nacional Aut´ onoma de M´exico (UNAM), Coyoac´ an D.F., 04510, Mexico

Abstract For uncertain systems with relative degree two, a continuous homogeneous sliding-mode control algorithm is proposed. This algorithm ensures finite-time convergence to the third-order sliding set, using only information about the output and its first derivative. We prove the convergence of the suggested algorithm via a homogeneous, continuously differentiable and strict Lyapunov function. Key words: Third-order sliding-mode; Finite time stability; Lyapunov stability.

1

Introduction

Sliding-mode control (SMC) [31,9] is one of the most efficient control techniques for controlling plants under heavy uncertainty conditions. The goal of such controllers is to (theoretically exactly) compensate a matched uncertainty by keeping some properly chosen (sliding) variables at zero. To reach this goal, theoretically infinite switching frequency is required [31]. Such control is not desirable from the implementation point of view due to the oscillations caused by the highfrequency switching, which results in dangerous system vibrations (chattering) (see e.g. [31,9,6]). In the last two decades, a number of methods have been proposed to diminish the chattering effect (see [12] and references cited therein). One of the most popular is the ? The authors want to acknowledge the joint scientific program Mexico-India: Conacyt, Project 193564, and DST project DST/INT/MEX/RPO-02/2008. This work was also supported in part by DGAPA-UNAM, projects PAPIIT IN113216 and IN113614, Conacyt projects 241171, 132125 and 261737, and Fondo de Cooperaci´ on II-FI, Project IISGBAS-100-2015. Email addresses: [email protected] (S. Kamal), [email protected] (J. A. Moreno), [email protected] (A. Chalanga), [email protected] (B. Bandyopadhyay), [email protected] (L. Fridman).

Preprint submitted to Automatica

higher-order sliding modes (HOSM) approach [18,3,20]. HOSM algorithms, for systems with relative degree r, ensure finite-time convergence to zero of the output σ and its first (r − 1) derivatives. The asymptotic accuracy of the r-th HOSM was analyzed in [18], where it was shown that the best possible asymptotic accu
racy with e.g. sampling interval τ is σ (j) = O τ r−j , j = 0, 1, · · · , r − 1. Homogeneous HOSM controllers of order r provide for this accuracy [22], so that they have this optimal precision of the output tracking with respect to the sampling step, measurement noises, and fast actuators’ dynamics [18,20,21,24]. The main drawback is that the control signal is discontinuous, which produces chattering.

The super-twisting algorithm (STA) plays a special role among the sliding-mode controllers. Differently from the other HOSM algorithms, it was designed for systems with relative degree one with respect to the output σ, and has the following advantages: (i) it compensates matched Lipschitz uncertainties/perturbations, while first-order sliding-mode can compensate discontinuous and uniformly bounded uncertainties/perturbations; (ii) it provides finite-time convergence to σ = σ˙ = 0 simultaneously; (iii) it requires only the information of the output σ; (iv) it generates a continuous control signal and, consequently, reduces chattering effects; (v) it has sliding accuracy of order one with respect to σ˙ and two with respect to σ.

17 January 2016

erties of the CTSM algorithm we use a continuouslydifferentiable, homogeneous and strict Lyapunov function.

The first proof of the convergence of the STA was based on the idea of majorant curves [18,19]. Later, proofs based on Lyapunov functions were found [26,30,29,27]. One can apply the STA to alleviate the chattering in systems with relative degree two (see for example [12] and references therein). It is natural in this case to select the sliding variable as s = σ˙ + cσ, c > 0 [7]. With this choice of s, the STA ensures the uncertainties/perturbations compensation, finite-time convergence to the set s(t) = s(t) ˙ = 0, but the states σ, and σ˙ converge only asymptotically to the origin.

Notation and Definitions In this paper the following notation is used: dzcp = |z|p sign(z) where z ∈ R and p ∈ R. Therefore dzc2 = |z|2 sign(z) 6= z 2 . Note that, if p is an odd number, the two expressions bzep = z p are equivalent. In particular, dzc0 = sign(z), dzc0 |z|p = dzcp , dzcp dzcq = |z|p+q . Homogeneous functions and systems have appealing properties and they play an important role in HOSM and in this paper. Classical homogeneity corresponds to the scaling property of a scalar function f (kx) = k δ f (x), for all k > 0 and some δ ∈ R. From [2] we recall some definitions of (weighted) homogeneous functions and vector fields and, from [22,28], the corresponding concepts for vector-set fields. A function f : Rn → R (or a multivalued function F : Rn ⇒ R, F (x) ⊂ R) is called homogeneous of degree δ with the dilation dk : (x1 , x2 , · · · , xn ) 7→ (k r1 x1 , k r2 x2 , · · · , k rn xn ), where r1 , · · · , rn are some positive numbers (called weights) if, for any k > 0, the identity f (dk x) = k δ f (x) holds (respectively, F (dk x) = k δ F (x)).

Arbitrary order sliding modes approach [3,20,21] can also be used to attenuate the chattering in the control of systems with relative degree two with respect to σ. In order to adjust the chattering and, at the same time generate a continuous signal, one rises artificially the relative degree to three by obtaining σ ¨ , and uses thirdorder nested or quasi-continuous SMC on the new control variable u. ˙ This ensures the finite-time convergence to the origin of σ and σ˙ and compensates Lipschitz perturbations. The disadvantage of this approach is that the values of σ ¨ are needed in order to realize third order SMC [20,21]. For systems with known control gain the knowledge of σ ¨ means the knowledge of the perturbations or uncertainties. In this case the perturbations/uncertainties could be compensated directly.

A vector field f : Rn → Rn (or a vector-set field F : Rn ⇒ Rn , F (x) ⊂ Rn ) is homogeneous of degree δ if for any k > 0 the identity f (dk x) = k δ dk f (x) holds (respectively, F (dk x) = k δ dk F (x)). If a vector field has homogeneity degree δ 6= 0, then it can be always scaled to ±1 by an appropriate proportional change of the weights r1 , · · · , rn .

Therefore, the problem of designing a controller which is continuous and, at the same time, ensures finite-time convergence to the third-order sliding-mode, using only information about σ, σ, ˙ is an important task. The first results solving this problem were obtained in [4,10], where a combination of the super-twisting and twisting or continuous algorithms were reported, ensuring finite-time convergence of σ, σ˙ to the origin.

2

Problem Statement and Main Result

The goal of this paper is to propose a homogeneous control algorithm for uncertain second-order plants, having the following advantages:

We consider a perturbed second-order plant described as

• it compensates Lipschitz uncertainties/perturbations; • it provides finite-time convergence to third-order sliding-mode set, and therefore provides sliding accuracy of order three with respect to σ; • it generates a continuous control signal; • it requires only the knowledge of the output σ and σ; ˙

x˙ 2 = u + µ (t) ,

x˙ 1 = x2

(1)

where x1 , x2 ∈ R are the states, u ∈ R is the control and µ (t) is the perturbation, which is a Lipschitz continuous time signal with Lipschitz constant ∆, i.e. |µ˙ (t)| ≤ ∆ almost everywhere. The problem is to design a (time) continuous control law such that the output σ = x1 and its derivative σ˙ = x2 converge in finite time and remain in zero σ = σ˙ ≡ 0 despite of the perturbation µ (t). Moreover, after a finite time the control should compensate for the perturbation, i.e. u(t) ≡ −µ(t), so that also σ ¨ ≡ 0, establishing a third order sliding mode.

This algorithm can be considered as a combination of super-twisting and terminal sliding-mode. This result has been announced in [13, Section 5.2] as Continuous Nonsingular Terminal Sliding Mode Algorithm, but without providing a convergence proof, which is the main result of the present paper.

The problem is solved by the dynamic feedback control law 1 2 u = −k1 L 3 dφL (x1 , x2 )c 3 + z (2) 0 z˙ = −k2 L dφL (x1 , x2 )c ,

In this light we denote the proposed controller as Continuous Terminal Sliding Mode (CTSM) Controller. To prove the finite-time convergence and robustness prop-

2

where

Step 1: The role of L

α

3 2

, (3) 1 dx2 c L2 is a continuously differentiable function of the state. In turn, the parameters ki and L are positive gains to be designed. When L = 1 we denote φ1 (x1 , x2 ) simply as φ (x1 , x2 ). By defining x3 , z +µ, the closed loop (1)-(2) yields the (discontinuous) 3rd-order differential equation φL (x1 , x2 ) = x1 +

By the change of variables x →

1

2

system (4) becomes

x˙ 1 = x2 1

x˙ 2 = −k1 dφ (x1 , x2 )c 3 + x3 µ˙ (t) 0 x˙ 3 = −k2 dφ (x1 , x2 )c + . L

x˙ 1 = x2 x˙ 2 = −k1 L 3 dφL (x1 , x2 )c 3 + x3

1 Lx

(5)

Since stability for both systems (4) and (5) is equivalent, we will consider the latter one in the rest of the proof.

(4)

0

x˙ 3 = −k2 L dφL (x1 , x2 )c + µ˙ (t) . Step 2: Lyapunov function candidate Since the righthand side of (4) is discontinuous, its solutions will be understood in the sense of Filippov [11]. Notice that the Filippov differential inclusion corresponding to the discontinuous and uncertain system (4) is homogeneous of degree (scaled to) δ = −1 and has weights r = [3, 2, 1].

We propose V (x) below as a Lyapunov function candidate for system (5) 5 5 2 1 5 V (x) = β |x1 | 3 + x1 x2 + α |x2 | 2 − 3 x2 x33 + γ |x3 | , 5 k1 (6) which is homogeneous (of degree δV = 5) and continuously differentiable. We will show that, by selecting β > 0 and γ > 0 sufficiently large, V (x) is positive definite and there are two class K∞ functions µ1,2 (kxk) such that µ1 (kxk) ≤ V (x) ≤ µ2 (kxk).

In the main result of the paper (Theorem 1 below) we will use the following definition [22]: The origin x = 0 of a differential inclusion x˙ ∈ F (x) (a differential equation x˙ = f (x)) is called globally uniformly finite-time stable if it is Lyapunov stable and, for any R > 0, there exists T > 0 such that any trajectory with initial condition kx0 k < R reaches 0 at time T and x(t) ≡ 0 for all t ≥ T .

For this, recall Young’s inequality [14]: for any real values p > 1 and q > 1 such that p1 + 1q = 1, and any positive p q real numbers a, b, c, the inequality ab ≤ cp ap + c−q bq holds. From (6) it follows that

Theorem 1 Consider the third-order system (4) with a uniformly bounded signal |µ˙ (t) | ≤ ∆. Then, for every ∆ ≥ 0 and α > 0, there exist (positive) values of the gains (k1 , k2 , L) such that the state x converges to zero globally, uniformly and in finite-time, despite any bounded perturbation |µ˙ (t) | ≤ ∆. In Section 3 we prove Theorem 1 by means of a continuously differentiable and homogeneous Lyapunov function. The proof also provides a method to find values of the gains that satisfy the conditions of Theorem 1. This issue will be discussed in Section 4.

5 5 2 −5 3 5 |x1 | |x2 | ≤ c13 |x1 | 3 + c1 2 |x2 | 2 5 5 5 2 −5 3 5 3 5 |x2 | |x3 | ≤ c2 2 |x2 | 2 + c23 |x3 | , 5 5

An immediate consequence of Theorem 1 is that the states x1 , x2 converge in finite-time to the origin; the control signal u (t) in (2) is continuous; and −z(t) converges in finite-time to the Lipschitz perturbation µ(t).

3

3

5 2α |x2 | |x3 | 5 |x2 | 2 − + γ |x3 | 5 k13 (7) and applying Young’s inequality we obtain 5

V ≥ β |x1 | 3 − |x1 | |x2 | +

with c1 , c2 > 0. Replacing them in (7) we obtain     5 5 3 5 2 −5 −5 1 V ≥ β − c13 |x1 | 3 + α − c1 2 − c2 2 3 |x2 | 2 5 5 k1   3 1 35 5 + γ− c |x3 | . 5 k13 2

Proof of Theorem 1.

We prove the theorem by means of a continuouslydifferentiable, homogeneous Lyapunov function. We recall that, due to the homogeneity of the system, the stability property is global. For simplicity we split the proof in the following steps:

Notice that V is positive definite if all coefficients in the previous inequality are positive. This can be achieved by

3

selecting e.g. c1 =

4 α


25

, c2 =



4 αk13

 25

2

older Furthermore, since the function d·c 3 is globally H¨ 1 continuous with H¨older constant 2 3 , for all (φ, x2 ) holds

, and

  23 4 , α  2 3 4 3 5 k1 γ > . 5 α 3 β> 5

l k2 3 2 φ − α dx2 c 2 3 + α 23 x2 ≤ 2 31 |φ| 3 .

(8)

These relations allow us to write    1 3 V˙ ≤ −k1 φ − dξ3 c dφc 3 − ξ3 + W (φ, x2 , ξ3 ) ,(12)

(9)

The fact that V is bounded above is proved analogously.

where W is the continuous and homogeneous function Step 3: Time derivative of the Lyapunov function   2 2 5 ∆ 5 1 2 4 3 γ κ |ξ3 | W , 2 3 β |φ| |x2 | − βα 3 − 1 |x2 | + 5¯ 3 3 λ   2 2 5 2 3 +3κ |ξ3 | γ¯ dξ3 c − dφc − x2 3

The derivative of V along the trajectories of (5) is given by the following (multivalued) function, where the arguments are omitted for the sake of readability

! 2 ∆ 2 2    3 γ κ |φ| 3 |ξ3 | . +3κ |ξ3 | |x2 | − 5¯ 1 2 x dx c 5β 3 3 3 3 λ − + x V˙ = −k1 φ − dφc dx c x + 2 1 2 k13 k1 3    In what follows we distinguish two cases for φ. 1 µ˙ 2 2 0 −k2 |x3 | 5γ dx3 c − 3 3 x2 dφc + k1 Lk2  5 l k 23   3 1 n o 3 3 3 β φ − α dx2 c 2 V˙ = −k1 φ − dξ3 c dφc 3 − ξ3 + The case φ = dξ3 c . On the surface M = φ = dξ3 c 3      1 3 ˙ 5 µ˙ 2 2 0 + x2 ) x2 − 3κ |ξ3 | γ¯ dξ3 c − x2 dφc + . (10) the first term in (12) vanishes and VM ≤ WM φ, x2 , dφc , 3 λ which is given by Here we defined

h

k2 x3 ,κ, , γ¯ , γk15 , λ , Lk2 , ξ3 , k1 k1

2 3

i

"

WM = − |φ| , |x2 | P where 

and we used the equality x1 = φ − α dx2 c , obtained from (3) by taking L = 1. Note that, for any value of k1 , there is a one to one relationship between (k2 , γ, L) and (κ, γ¯ , λ). This implies that it is possible to fix the values of (κ, γ¯ , λ) independently of k1 . We want to show that for any given (positive) values of κ, α, ∆ there are positive and sufficiently large values of β, γ¯ , k1 , L such that V˙ is negative definite for any perturbation |µ˙ (t)| ≤ ∆.

P =

5κ¯ γ 1− − 21



5 13 32 β

# ,

|x2 |

(11)

3 2

2

|φ| 3

∆ λ




+ 3κ 1 +

F ∆ λ




,



2 5 3 3 βα

  . −1

W , and therefore V˙ , is negative on the set M if λ>∆ 3 β> 2 5α 3 

Next, we find a continuous and homogeneous function that is an upper bound for V˙ . For this we use the following equalities

γ¯ >

l k 23 k 23 3 3 5 l 5 β φ − α dx2 c 2 + x2 = β φ − α dx2 c 2 + 3 3   5 2 2 α 3 x2 − βα 3 − 1 x2 , 3

(13) (14)
 2

5 31 32 β

+ 3κ 1 + ∆ λ .
5 2 ∆ 3 − 1 20κ 1 − λ βα 3

3

(15)

The case φ 6= dξ3 c . Now we show that by selecting k1 large, it is possible to render V˙ negative definite also outside the manifold M. For this we write    1 3 V˙ MC ≤ − (k1 − ψ (φ, x2 , ξ3 )) φ − dξ3 c dφc 3 − ξ3 ,

 2 2 5 5  5 2 2 γ¯ dξ3 c − x2 = γ¯ dξ3 c − dφc 3 − x2 + γ¯ dφc 3 . 3 3 3

4

In conclusion, by classical Lyapunov theorems for Differential Inclusions [2, Theorem 4.1][8, Proposition 14.1] it follows that x = 0 is a (strongly, i.e. for every trajectory) Uniformly Globally Asymptotically Stable equilibrium point for (4). Since the system is homogeneous of negative degree it follows that (4) is globally uniformly finite-time stable at 0 [22, Theorem 1]. Alternatively, this latter conclusion can be drawn directly from the inequality (18) and the Homogeneity Principle obtained in [28]. From (18) we can also obtain an estimation of the convergence time.

where the function ψ (φ, x2 , ξ3 ) is defined as ψ (φ, x2 , ξ3 ) , 

W (φ, x2 , ξ3 )  . 1 3 φ − dξ3 c dφc 3 − ξ3

(16)

We notice that ψ (φ, x2 , ξ3 ) is a homogeneous function of degree δψ
= 0, i.e. ∀k > 0, ψ (φ, x2 , ξ3 ) = ψ k 3 φ, k 2 x2 , kξ3 . This implies that all the values ψ (φ, x2 , ξ3 )ncan take are taken on the homogeneous unit o 2

2

sphere S = (φ, x2 , ξ3 ) ∈ R3 | |φ| 3 + |x2 | + |ξ3 | = 1 , which is compact. We restrict the domain of ψ to S. Since the numerator W of ψ is continuous, and the denominator is continuous and positive everywhere, except on set M ∩ S, ψ is continuous everywhere, except on M ∩ S. Moreover, since under conditions (13)-(15) W is negative on M it follows that ψ is upper semicontinuous. Therefore, it is bounded from above on S [25, Theorem 3.7, Chap. 3, Section 3, p. 76]. Consequently, ψ is bounded form above in R3 and, for sufficiently large values of k1 , i.e. k1 > max {ψ (φ, x2 , ξ3 )} , S

4

Procedure 1 in Section 3 describes a method to calculate values of the gains (k1 , k2 , α, L) of the controller (2) and parameters of the Lyapunov function, such that the conditions of the Theorem 1 are satisfied. The most difficult step being the maximization of the upper semicontinuous function ψ in (17), which can be easily done numerically.

(17)

By using this procedure we are able to calculate different values of the gains (k1 , k2 , α, L), with L = 1, that fulfill the conditions of the Theorem. In Table 1 we list 4 sets of gains obtained in this form for a perturbation size of ∆ = 1. For an arbitrary perturbation of size ∆ we just need to scale the parameter L = ∆ and take the gains 2 1 as (k1 L 3 , k2 L, α/L 2 ).

V˙ is negative definite. Note that despite ξ3 = x3 /k1 , the max ψ in (17) does not depend on k1 , since ψ is homogeneous of degree 0. Alternatively, the negative definiteness of V˙ can be derived from the following classical result on homogeneous functions [15, Theorem 4.4, Sec. 5] (see also [1]): Lemma 2 Let η : Rn → R and γ : Rn → R+ be two continuous homogeneous functions of degree m > 0. Suppose that γ(x) ≥ 0. If η(x) > 0 for all x 6= 0 such that γ(x) = 0, then there exists a real number λ∗ such that, for all λ ≥ λ∗ η(x) + λγ(x) > 0 n for all x ∈ R \ {0}.

Since V˙ is bounded above by a continuous, negative definite and homogeneous function of degree δV˙ = 4 (see the rhs of (12)) and V is a continuous, positive definite and homogeneous function of degree δV = 5, it follows from [5, Lemma 4.2] that there exists η > 0 such that .

1

2

3

4

k1

4.4

4.5

7.5

16

k2

2.5

2

2

7

α

20

28.7

7.7

1

Note that the homogeneous Lyapunov function V is C 1 and its derivative V˙ along the trajectories of the homogeneous Differential Inclusion x˙ ∈ F (x) corresponding to (4) is upper bounded by a negative definite, homogeneous, and continuous function (see the righthand of (12)). This implies that V˙ remains negative definite outside of a (sufficiently small) neighborhood of the origin x = 0 for small perturbations of F (x), causing small changes in its graph in the Hausdorff metrics [22]. Therefore, using Lyapunov-like results (see [17, Theorem 4.18]), one concludes that the trajectories of the perturbed system will be globally uniformly ultimately bounded. As a consequence, using the Lyapunov Function, we obtain (qualitatively) the same results corresponding to Corollary 1 and Theorem 2 in [22]. For example, if we consider noisy measurements of xi of magnitude τ ri (with r1 = 3, r2 = 2 and r3 = 1 and small

Procedure 1 Select some positive values of κ, α, ∆. Choose β so that (8) and (14) hold; λ that satisfies (13); and γ¯ that satisfies (9) and (15). Finally, choose k1 such that (17) holds. By using (11) we get a set of values for k1 , k2 , α and L for which the convergence properties of (4) apply.

4 5

Set

∆ 1 1 1 1 Table 1 Sets of gain values obtained with Procedure 1 for L = 1.

We propose the following procedure to select gains fulfilling the conditions of Theorem 1:

V˙ ≤ −ηV

Discussion on the Proposed Controller

(18)

5

5

τ > 0) then, for the solutions of the perturbed system (4) the inequalities |xi | < νi τ ri will be established in finite time with some positive constants νi independent of τ and of the initial conditions. This corresponds to the best possible asymptotic accuracy of third order HOSM [18], [22].

Example

We illustrate our results by means of some simulations. The time evolution of the states of system (4) are presented in Figure 1, where the perturbation is µ(t) = 2+2 cos(t)+0.2 sin(10t) and gains are selected as (see Set 4 in Table 1) k1 = 16, k2 = 7, α = 1, with L = 15 > ∆ = 4. There we can see that all the states converge to zero, despite the perturbation. Figure 2 shows the precision of each of the states when the simulation step of the Euler algorithm is set to τ = 10−4 (Fig. 2a) or τ = 10−5 (Fig. 2b); whence we can calculate the (precision) coefficients: ν3 = 800, ν2 = 800 and ν1 = 3000. They evidence the precision corresponding to a third-order sliding mode.

On system (4), a third-order sliding-mode [22] is attained in finite time, since σ = x1 , σ˙ = x2 and σ ¨ = x˙ 2 = 1 2 3 3 −k1 L dφL (x1 , x2 )c + x3 are continuous functions of the state variables. After a finite time, σ = σ˙ = σ ¨ = 0, and σ (3) is discontinuous due to the discontinuity of x˙ 3 . This implies that with controller (2) the precision, given by |x1 | < ν1 τ 3 , |x2 | < ν2 τ 2 , is higher (for small τ ) than with any second-order sliding-mode algorithm (as e.g. the twisting controller) with precision |x1 | < ν¯1 τ 2 , |x2 | < ν¯2 τ .

Figure 3 depicts the phase portrait of the plant states x1 and x2 , along with the switching curve φ = 0. It is noteworthy that the trajectory reaches the switching surface and then slides along it, until the origin is reached in finite time. This is also clear from behavior of φ(t) in Figure 1. Such a behavior is similar to the one of the classical second-order sliding-mode known as Terminal (or Prescribed) Controller [18,23]. Figure 4 shows the continuous control signal provided by controller (2). The same figure shows the behavior of the perturbation µ(t) and the (finite-time) convergence of the controller state z (see (2)) to −µ(t). Thus, the controller is able to completely compensate the perturbation µ since it is able to estimate it. This is also a feature of the Super-Twisting Algorithm (STA).

The class of perturbations that are supported by the proposed controller is quite different from the class supported by a second-order sliding-mode algorithm, as for example the twisting controller applied on the secondorder plant (1). For the latter, µ(t) is just supposed to be a uniformly bounded Lebesgue measurable function, so it does not need to be continuous. The control variable u in this case is bounded but discontinuous. In contrast, for controller (2) µ is supposed to be a Lipschitz continuous function of time, but it is not required to be bounded. The resulting control variable u is therefore a continuous function of time and it does need to be bounded. These two sets are different. For example, a twisting controller can compensate a function switching arbitrarily between +1 and −1 but it cannot compensate a ramp function, while the proposed controller is unable to compensate the switching signal but is able to compensate the ramp.

Figure 5 presents again the phase portrait of plant’s states x1 and x2 of the same controller (2), but with different gains k1 = 6, k2 = 6, α = 1/6, which have been obtained in [16] by means of an alternative Lyapunov function. The trajectories in Figure 5 have a rather undamped behavior compared to the ones in Figure 3. For the former situation the switching surface φ = 0 does not seem to be a sliding surface, a behavior typical of the SOSM known as Twisting Controller. However, as shown in [23], this is also a possible behavior of the Terminal (or Prescribed) Controller. Thus, the proposed CTSM Controller presents both types of behaviors of the Terminal Controller, i.e. a Sliding (Fig. 3) and a Twistinglike (Fig. 5) behavior. We note that this twofold dynamic behavior can neither be determined from the Lyapunov function nor it is relevant for the presented Lyapunov analysis.

The main idea of the proposed controller (2) is to add a discontinuous integral term that is able to estimate the perturbation µ(t) and to fully compensate it. Without the discontinuous integral term, controller (2) becomes continuous and it is able to stabilize the origin (x1 , x2 ) = (0, 0), but it will not be able to (fully) compensate the perturbation.

An alternative strategy that provides a continuous control signal and third-order sliding-mode precision, is to add an integrator at the input of the plant, and then apply a third-order sliding-mode algorithm [22]. However, this alternative requires to feedback all three resulting states, while the proposed algorithm (2) only requires the feedback of the two states x1 , x2 . Of course, if only x1 is available for measurement the use of a robust differentiator [20] will provide an exact estimate of the velocity x2 in finite-time.

6

Conclusions

In this paper the Continuous Terminal Sliding Mode Controller, for the uncertain systems with relative degree two, is proposed as a combination of the STA and terminal sliding-mode control. This algorithm ensures finite-time convergence to the third-order sliding set with respect to the output, using only information

6

Phase portrait of x and x

Time behavior of the states and φ

1

2

x1

2

x vs. x

0

1

x

1.5

x

−1

φ

−2

3

1

2

φ=0

2

0.5

x2

x, φ

−3 0

−5

1

−0.5

x 10

−4

φ

−5

−1 −1.5

0

−6

−1 0

−2 0

1

−7

5

2

3

4

−8

5

0

1

2

t

Fig. 1. Time evolution of the sates x1 , x2 , x3 and of the switching variable φ under a non vanishing perturbation µ(t). −5

1

5

0.08

0.8

4

0.06

0.6

3

0.04

0.4

2

0.02

0.2

1

3

4

5

Fig. 3. Phase portrait of plant’s states x1 and x2 , and locus of the switching curve φ = 0, showing a Sliding-Like behavior of the CTSM controller.

−9

x 10

0.1

x1

x 10

Control Input and perturbation

0 −0.2

−1

−0.04

−0.4

−2

−0.06

−0.6

−3

−0.08

−0.8

−4

−0.1 3

4 t

−1 3

5

4 t

5

4 3

0

−0.02

u/5 µ −z

5

u, µ, z

0

x1

x2

x3

6

2 1 0

−5 3

4 t

5

−1 −2 0

(a)

1

2

3

4

5

t −12

0.008

0.8

4

0.006

0.6

3

0.004

0.4

2

0.002

0.2

1

0

5

x1

1

x2

x3

−7

x 10

0.01

0 −0.2

−1

−0.004

−0.4

−2

−0.006

−0.6

−3

−0.008

−0.8

−4

4 t

5

−1 3

4 t

5

Fig. 4. Time behavior of the continuous control signal u, and (finite-time) estimation of the perturbation µ(t) by the controller state −z(t).

to the third-order sliding-mode, and its precision.

0

−0.002

−0.01 3

x 10

−5 3

References [1] Andrieu, V.; Praly, L. and Astolfi, A. (2008). Homogeneous approximation, recursive observer design and output feedback. SIAM J. Control Optim. 47(4): 1814–1850. 4 t

[2] Baccioti, A. and Rosier, L. (2005). Liapunov functions and stability in control theory. 2nd ed. New York, Springer-Verlag.

5

(b)

[3] Bartolini, G.; Ferrara, A. and Usai, E. (1998). Chattering avoidance by second-order sliding-mode control. IEEE Transactions on Automatic Control, vol. 43, no. 2, pp. 241– 246.

Fig. 2. Precision of the state variables x1 , x2 , x3 corresponding to a 3-order Sliding Mode for (4).

[4] Basin, M.V. and Rodriguez-Ramirez, P. A. (2014). Supertwisting Algorithm for Systems of Dimension More Than One, IEEE Transaction on Industrial Electronics, Vol. 61, no. 11, pp. 6472-6480.

about the outputs and its derivative. A strict, homogeneous, continuously-differentiable Lyapunov function is proposed to prove the convergence of the suggested algorithm. The simulations presented in the paper illustrate the finite-time convergence of the system’s states

[5] Bhat, S.P. and Bernstein, D.S. (2005). Geometric homogeneity with applications to finite-time stability. Math. Control Signals Systems 17, pp. 101–127.

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Phase portrait of x and x 1

[18] Levant, A. (1993). Sliding order and sliding accuracy in sliding-mode control, International Journal of Control, vol. 58, no. 6, pp. 1247–1263.

2

x vs. x

2

1

2

φ=0

[19] Levant, A. (1998). Robust exact differentiation via sliding modes techniques. Automatica, Vol. 34, No. 3, pp. 379–384.

1 0

[20] Levant, A. (2003). Higher-order sliding-modes, differentiation and output-feedback control, Int. J. Control, vol. L. 76, no. 9/10, pp. 924–941.

x

2

−1 −2

[21] Levant, A. (2005). Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Autom. Control, vol. 50, no. 11, pp. 1812–1816.

−3 −4

[22] Levant, A. (2005). Homogeneity approach to high-order sliding mode design. Automatica, No. 41, pp. 823–830.

−5 −6 −1

0

1

2

x1

3

4

[23] Levant, A. (2007). Construction principles of 2-sliding mode design. Automatica., vol. 43, no. 4 pp. 576–586.

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[24] Levant, A. and Fridman, L. (2010). Accuracy of homogeneous Sliding Modes in the Presence of Fast Actuators, IEEE Transactions on Automatic Control, vol. 55, no. 3, pp. 810814.

Fig. 5. Phase portrait of plant’s states x1 and x2 , and locus of the switching curve φ = 0, showing a Twisting-Like behavior of the CTSM controller.

[25] McShane, E. J. and Botts, T. A. (2005). Real Analysis. N.Y., Dover Publications Inc.

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[7] Chalanga, A., Kamal, S., Fridman, L., Bandyopadhyay, B., Moreno, J.A. (2014). How to implement Super-Twisting Controller based on sliding mode observer? In: Proc. of 13th IEEE Workshop on Variable Structure Systems, Nantes. doi:10.1109/VSS.2014.6881145

[27] Moreno, J. and Osorio, M. (2012). Strict Lyapunov functions for the Super-Twisting algorithm, IEEE Transactions on Automatic Control, vol. 57, no. 4, pp. 1035–1040.

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[9] Edwards, C. and Spurgeon, S. (1998). Sliding Mode Control. London: Taylor and Francis. [10] Edwards, C. and Shtessel, Y.B. (2014). Continuous higher order sliding mode control based on adaptive disturbance compensation. In 13th International Workshop on Variable Structure Systems (VSS). pp. 1–5.

[29] Orlov, Y., Aoustin, Y. and Chevallereau, C. (2011). Finite time stabilization of a perturbed double integrator- Part I: Continuous sliding mode-based output feedback synthesis. IEEE Tran. on Automatic Control. 56(3): 614-618.

[11] Filippov, A.F. (1988). Differential equations with discontinuous right-hand side. Dordrecht, The Netherlands: Kluwer.

[30] Polyakov, A. and Poznyak, A. (2009). Reaching time estimation for super-twisting second-order sliding-mode controller via Lyapunov function designing. IEEE Transactions on Automatic Control, vol. 54, no. 8, pp. 1951–1955.

[12] Fridman, L. (2011). Sliding mode enforcement after 1990: main results and some open problems. In: Sliding Modes after the First Decade of the 21st Century. LNCIS vol. 412. L. Fridman, J. Moreno and R. Iriarte (eds.). Berlin: Springer. pp. 3-57.

[31] Utkin, V.; Guldner, J.; Shi, J. (2009). Sliding Mode Control in Electro-Mechanical Systems. 2nd ed. Boca Raton: CRC Press.

[13] Fridman, L.; Moreno, J.; Bandyopadhyay, B.; Kamal, S. and Chalanga, A. (2015). Continuous Nested Algorithms: The Fifth Generation of Sliding Mode Controllers. In: Recent Advances in Sliding Modes: From control to intelligent mechatronics. Series: Studies in Systems, Decision ¨ and Control, vol 24, X. Yu and M. Onder Efe (eds.), Springer International Publisher, Switzerland, pp. 5-35. DOI: 10.1007/978-3-319-18290-2. eBook ISBN: 978-3-319-18290-2 http://www.springer.com/gp/book/9783319182896 [14] Hardy, G. H.; Littlewood, J. E. and P´ olya, G. (1951). Inequalities. London: Cambridge Unversity Press. [15] Hestenes, M. R. (1966). Calculus of variations and optimal control theory. New York: John Wiley & Sons. [16] Kamal, S.; Chalanga, A.; Moreno, J.A.; Fridman, L. and Bandyopadhyay, B. (2014). Higher-order super-twisting algorithm, Proc. of 13th IEEE Workshop on Variable Structure Systems, Nantes. DOI: 10.1109/VSS.2014.6881129. [17] Khalil, H.K. (2002). Nonlinear Systems. Third ed. Upsaddle River, New Jersey: Prentice–Hall.

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