H∞ -Sliding mode control of one-sided Lipschitz nonlinear systems subject to input nonlinearities and polytopic uncertainties

ISA Transactions 90 (2019) 19–29

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

H∞ -Sliding mode control of one-sided Lipschitz nonlinear systems subject to input nonlinearities and polytopic uncertainties ∗

Wajdi Saad a , , Anis Sellami a , Germain Garcia b a b

LISIER, National High School of Engineers of Tunis ENSIT, University of Tunis, 5 Av. Taha Hussein, Tunis, Tunisia Research Laboratory LAAS-CNRS, University of Toulouse, Av. Colonel Roche F-31400, Toulouse, France

highlights • H∞ SMC for uncertain one-sided systems with nonlinear inputs is firstly addressed. • Robustness against unmatched polytopic parameter uncertainties and disturbances. • Less-conservative LMI conditions.

article

info

Article history: Received 4 July 2018 Received in revised form 16 December 2018 Accepted 24 December 2018 Available online 25 January 2019 Keywords: H∞ Nonlinear input One-sided Lipschitz Poly-quadratic stability Sliding mode control

a b s t r a c t This paper is concerned with the control problem for different types of one-sided Lipschitz (OSL) nonlinear systems subject to input nonlinearities, unmatched polytopic uncertainties and external perturbations. The design framework gathers the high robustness qualities of the sliding mode control (SMC) and the H∞ technique. The aims is to consider the bounded L2 disturbance attenuation measure in the analysis of sliding mode dynamics, thus to ameliorate the performance of the SMC system. An integral switching surface function is chosen and related synthesis conditions for the sliding mode dynamics are formulated in terms of linear matrix inequalities (LMIs). Then, an appropriate control law is synthesized such that reachability of the switching manifold is ensured. At last, simulation examples are provided to prove the feasibility of the proposed approach. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction One of powerful approaches to robust controller design is indisputably the famous technique namely, sliding mode control [1,2]. SMC has attractive features such as insensitivity to the matching uncertainties and disturbances, good transient performance and reasonable computational simplicity. In recent years, systems with the well-known Lipschitz condition has received growing research consideration. Numerous SMC methods have been proposed for different types of systems with Lipschitzian nonlinearities, such as networked systems [3], singular systems [4], stochastic systems [5], neutral systems [6,7], Markovian jump systems [8,9], TS fuzzy systems [10,11] and so on. Moreover, the SMC approach has been successfully applied to many practical systems including Lipschitz nonlinearities, such as offshore steel jacket platforms [12], single-link flexible joint robot [3], terminal guidance [13], multimachine power system [14], diesel engine [15], oil catalytic cracking [5], etc. Nevertheless, although many achievements have been obtained in these works, there are still some unstudied problems ∗ Corresponding author. E-mail addresses: [email protected] (W. Saad), [email protected] (A. Sellami), [email protected] (G. Garcia). https://doi.org/10.1016/j.isatra.2018.12.040 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

and conservativeness issues to be worthy of attention in the SMC research field. Generally speaking, it is well recognized that design methods based on the traditional Lipschitz conditions usually applicable for systems with small Lipschitz constants [16,17]. The provided synthesis conditions are generally unfeasible when these constants become large. In fact, the Lipschitz constant is region-based and often dramatically increases as the operating region is enlarged. To relax this problem, the so-called one-sided Lipschitz condition has been first introduced by Hu [18] instead of the Lipschitz counterpart. The OSL constant is always less than or equal to the corresponding Lipschitz constant of a nonlinear function, which makes it much more suitable for describing the influence of nonlinear aspects. The concept has been further extended in [19] where a new condition, namely the quadratically inner-bounded condition, has been proposed to deal with the observer design problem. Following this line, considerable attention has been paid, in the last decade, to the study of OSL nonlinear systems and valuable publications concerning control design have been produced for such systems [20–27]. To name a few, in [20], a stabilization method of OSL systems is proposed. In [22,23], the finite time H∞ control is discussed. In [26], the observer based H∞ control problem for OSL systems is investigated. However, it should be pointed out that

20

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

the polytopic-type parameter uncertainty has not been considered in all available papers concerning OSL systems, even parameter uncertainty has not been taken into account for lot of studies [20, 21,24,25,27]. As the polytopic representation is convex, it can be considered the best ways to preserve the structure information of uncertainty matrices when its model depends affinely on uncertain parameters [28,29]. Moreover, the case when nonlinear input is implicated has not be discussed by all OSL studies. Indeed, in many engineering systems, due to physical limitations, there do exist nonlinearities in the control input such as saturation, quantization, backlash, dead-zones, and so on, naturally originate from actuators in practical realizations [30,31]. The existence of nonlinear input often is a source of degradation, instability and poor performance of controlled systems. Hence, the effects of input nonlinearities must be taken into account when analyzing and implementing any controller scheme. Motivated by the above analysis, this paper aims to discuss the design problem of the combined H∞ -SMC control for varied OSL nonlinear systems with nonlinear inputs, unmatched polytopic uncertainties and disturbances. To the best of author’s knowledge, this issue has never been investigated yet and research on this topic remains open, which makes us proceed to the present work. The major contributions of the current study with respect to the related literature lie in the following points: (i) As far as authors know, the problem of H∞ -SMC for uncertain one-sided Lipschitz nonlinear systems with nonlinear inputs is unavailable in the literature and is addressed for the first time. (ii) Robustness against uncertainties is considered by taking the polytopic parameter uncertainties and the external disturbance to be together unmatched (also the nonlinearity term). (iii) Based on poly-quadratic Lyapunov functions, the present work furnishes less-conservative and sufficient conditions in terms of LMIs to guarantee both asymptotic stability and H∞ disturbance attenuation performance for the sliding mode dynamics. (iv) A simple sliding mode controller is designed to assure the occurrence of the sliding mode in finite time despite of mismatched uncertainties and nonlinearities. The rest of this paper is organized in the following way: in Section 2, the system description and some preliminaries are outlined. The main results are given in Section 3. Section 4 shows the simulation results and Section 5 concludes the paper. Notations. The notations used throughout the paper are fairly standard. ∥·∥ represents the √ Euclidean norm of a vector or induced

∫∞

∥.∥2 dt stands for the L2 norm. Rn 0 denotes the real n-dimensional space. Rm×n is the real (m × n) matrix space. L2 [0, ∞) denotes the space of square-integrable vector functions over [0, ∞). I and 0 signify the identity matrix and the zero matrix with appropriate dimensions. (·)T and (·)−1 indicate transpose and matrix inverse. ‘‘∗’’ denotes the symmetric elements of a symmetric matrix. ⟨x, y⟩ = yT x is the inner product matrix norm and ∥.∥2 =

in the space Rn . Matrices, if their dimensions are not explicitly specified, are assumed to be compatible for algebraic operations. 2. Preliminaries and problem statement

Assumption 1. The matrix B is assumed to be of full column rank, .i.e., rank(B) = m. Assumption 2. The matrix Aη describes polytopic-type uncertainties, i.e., belongs to a polytope-type set Ω with h known vertices. This set is given by

{ Ω = ⟨Aη ⟩ =

h ∑ i=1

Ai ;

h ∑

} ηi = 1, ni ≥ 0

Assumption 3. ϕ (u) is a continuous function containing sector nonlinearity which satisfy ϕ (0) = 0 and

α1 uT u ≤ uT ϕ (u) ≤ α2 uT u,

x˙ (t) = Aη x(t) + Bϕ (u) + φ (x) + E w (t)

(1)

y(t) = Cx(t) + Dw (t).

(2)

where x ∈ Rn is the state, u ∈ Rm is the control input, y ∈ Ry is the output, ϕ (u) is a continuous vector-valued nonlinear function, φ (x) represents the nonlinear dynamics associated with the state vector x(t) with φ (0) = 0 and w (t) describes a set of exogenous disturbance which belongs to L2 [0, ∞) and assumed to be bounded by a non-negative scalar w0 ; that is ∥w (t)∥ ≤ w0 . The following assumptions are made.

(4)

where α1 and α2 are known nonzero positive constants and called the gain reduction tolerances. Now, we recall some preliminary concepts on the nonlinearity. The following definitions about Lipschitz, P one-sided Lipschitz, one-sided Lipschitz, and quadratic inner-boundedness properties for the nonlinear function φ (x) are introduced to further our study. Definition 1 (Lipschitz Condition [32]). The nonlinear function φ (x) is said to be locally Lipschitz in a region ℜ including the origin with respect to x, if there exists a positive constant ρ0 > 0 satisfying

∥φ (x1 ) − φ (x2 )∥ ≤ ρ0 ∥x1 − x2 ∥

(5)

for any x1 , x2 ∈ ℜ. The smallest ρ0 > 0 satisfying (5) is called the Lipschitz constant. The region ℜ is the operational region or the region of interest. If condition (5) is valid everywhere in Rn , then the function is said to be globally Lipschitz. Definition 2 (P-one-sided Lipschitz Condition [33]). The nonlinear function φ (x) is said to be P-one-sided Lipschitz in the region ℜ with respect to x if there exists P > 0 such that

⟨f (x1 ) − f (x2 ), x1 − x2 ⟩ ≤ ρp ∥x1 − x2 ∥2

(6)

holds for any x1 , x2 ∈ ℜ, where f (x) = P φ (x). ρp ∈ R is the P-onesided Lipschitz constant. Definition 3 (One-sided Lipschitz Condition [32]). The nonlinear function φ (x) is said to be one-sided Lipschitz in the region ℜ with respect to x, if there exists a constant ρ1 ∈ R satisfying

⟨φ (x1 ) − φ (x2 ), x1 − x2 ⟩ ≤ ρ1 ∥x1 − x2 ∥2

(7)

for any x1 , x2 ∈ ℜ, where ρ1 is called the one-sided Lipschitz constant. Definition 4 (Quadratic inner-bounded property [32]). The nonlinear function φ (x) is called quadratically inner-bounded in the ˜ with respect to x, if there exists constants ρ2 , ρ3 ∈ R such region ℜ that

(φ (x1 ) − φ (x2 ))T (φ (x1 ) − φ (x2 )) ≤ ρ2 ∥x1 − x2 ∥2 +ρ3 ⟨x1 − x2 , φ (x1 ) − φ (x2 )⟩

Consider the following nonlinear system

(3)

i=1

(8)

˜. holds for any x1 , x2 ∈ ℜ Generally speaking, the OSL property describes a broad family of nonlinear plants and possess inherent advantages with consideration of conservativeness. By including much useful information of the nonlinear part, it extends the well-known Lipschitz property (5) to a more general family of nonlinear systems. Any Lipschitz function is also OSL (e.g. with ρ1 > 0 or ρp > 0 ). However, the converse is not true. The OSL constant can be positive, zero or even negative whereas the Lipschitz constant must be positive. Usually,

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

21

According to the sliding mode theory, when the system operates in the sliding mode it follows that s˙(t) = 0. Thus, the so-called equivalent control can be extracted as

ϕ (u)eq = −Kx(t) − (GB)−1 G[φ (x) + E w (t)]

(12)

¯ = [I − B(GB)−1 G] and substituting (12) into the system Taking G (1) yield the sliding motion as below Fig. 1. Lipschitz, one-sided Lipschitz and quadratically inner-bounded function sets.

the OSL constants can be found to be much smaller than the Lipschitz constant (see the first simulation example in Section 4). Moreover, many forms of nonlinearities satisfy the OSL continuity but not the Lipschitz continuity (see Example 3 in Section 4). Fig. 1 illustrates the relation between the condition sets. For comprehensive study and more details about the various structures, we refer readers to [32,33] and references therein. Before moving on, we also give the following definition about the H∞ disturbance attenuation performance, which plays a key role in deriving the main results in the next sections. Definition 5. The closed-loop system (1)–(2) with control u(t) = −Kx(t) is said to be robust asymptotically stable with H∞ disturbance attenuation performance for all admissible uncertainties (3), if 1. It is asymptotically stable for w (t) = 0. 2. Under the assumption of zero-initial condition x(t0 ) = 0, the following constraint is satisfied

∥y(t)∥2 ≤ γ ∥w(t)∥2 ∀t ≥ t0 ≥ 0

(9)

¯ φ (x) + E¯ w(t), x˙ (t) = A¯ η x(t) + G

(13)

y(t) = Cx(t) + Dw (t).

(14)

¯ where A¯ η = Aη − BK and E¯ = GE. Thus, the motion behavior of the system (1)–(2) in the sliding surface will be analyzed through the resulting sliding-mode dynamics (13)–(14). Suppose that the system is in the sliding mode s(t) = 0 for t ≥ tr , where tr ≥ 0 is the reaching instant. In the sequel, we mainly address to determine the control gain K which ensures the desired performances. 3.1.2. H∞ performance Under different hypothesis, sufficient conditions of robust asymptotic stability with H∞ performance measure for the sliding mode dynamics (13)–(14) are proposed. Hypothesis 1. The nonlinear function φ (x) is OSL in ℜ and ˜. quadratic inner-bounded in ℜ Theorem 1. Consider the integral sliding surface (11). Under Hypothesis 1, for given scalar τ , if positive constants ε1 , ε2 and γ , symmetric positive definite matrices Qi ∈ Rn×n , and any appropriately matrices T and Y can be found to satisfy the following LMIs, for i = 1, . . . , h:

for all nonzero w (t) ∈ L2 [t0 , ∞), where γ > 0 is the L2 gain attenuation level. The following lemma is necessary for the sake of convenience. Lemma 1 (Schur Complement). Let X = X T , Z = Z T and S be matrices of appropriate dimensions, then Z < 0, X − SZ −1 S T < 0 is equivalent to

[

X ST

S Z

]

<0

(10)

3. Main results In this section, under different hypothesis, the main results of this study are presented. The purpose is to contribute to the development of SMC and L2 gain performance for one-sided Lipschitz systems with unmatched uncertainties and nonlinear inputs. The solvability condition for the desired sliding surface is established. To achieve the reaching condition, a robust control law is designed. Additionally, other related cases are discussed. 3.1. SMC-H∞ design 3.1.1. Sliding mode dynamics The design of the sliding surface determines the performance of the SMC system. The surface is designed to enable the system to possess the sliding mode and consider the required behavior. An integral sliding surface is considered here with the following switching functional

⎡

Xi11

⎢ ∗ ⎢ ∗ ⎣ ∗ ∗

Xi = ⎢

Xi12

τT + τTT ∗ ∗ ∗

X 13 −τ T G¯ −2ε2 I

∗ ∗

¯ −T GE ¯ −τ T GE 0

−γ 2 I ∗

CT 0 0 DT −I

⎤ ⎥ ⎥ ⎥<0 ⎦

(15)

with Xi11 = −T Ai − ATi T T + Y + Y T + 2(ε1 ρ1 + ε2 ρ2 )I Xi12 = Qi + T − τ ATi T T + τ Y T ¯ − (ε1 − ε2 ρ3 )I X 13 = −T G then the sliding mode dynamic (13)–(14) is robust asymptotically stable with disturbance attenuation level γ . Moreover, the design matrix K in (11) is given by K = B+ T −1 Y where B+ = (BT B)−1 BT is the Moore–Penrose inverse of matrix B. Proof. From (13), it follows that 2 xT T1 + x˙ T T2

[

][

¯ φ (x) − E¯ w = 0 x˙ − A¯ η x − G

]

(16)

where T1 and T2 are any appropriately matrices. Eq. (16) can rewritten as

υ T X˜ η υ = 0

(17)

with

⎡

X˜ η11

⎢ ∗ ∗ ∗

X˜ η = ⎢ ⎣

X˜ η12 T2 + T2T

−T1 G¯ −T2 G¯

∗ ∗

∗

0

⎤ ⎡ ⎤ −T1 E¯ x ⎥ −T2 E¯ ⎥ , υ = ⎢ x˙ ⎥ ⎣ φ ⎦ ⎦ 0 w 0

t

∫

G(Aη − BK )x(ν )dν

s(t) = Gx(t) −

(11)

0

where the matrix G is to be chosen such that square GB is nonsingular (det(GB) ̸ = 0) and the matrix K will be determined in the course of the design.

X˜ η11 = −T1 A¯ η − A¯ Tη T1T

X˜ η12 = T1 − A¯ Tη T2T

Notice that φ (0) = 0. For any positive constant ε1 , the onesided Lipschitz property given in Definition 3 yields the following

22

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

2ε1 ρ1 I 0 ⎢ υ ⎣ −ε1 I

⎡

T

0

0 0 0 0

⎤

−ε1 I

0 0 ⎥ υ≥0 0 ⎦ 0

0 0 0

(18)

Similarly, from the quadratically inner-bounded property in Definition 4, for any positive constant ε2 , we can obtain 2ε2 ρ2 I 0 T ⎢ υ ⎣ ε2 ρ 3 I 0

⎡

ε2 ρ 3 I

0 0 0 0

0 0 ⎥ υ≥0 0 ⎦ 0

(19)

Now, to establish the H∞ analyzing performance, let us choose the Lyapunov function candidate Vx (t) = xT (t)Qη x(t) where Qη = ∑ h i=1 ηi Qi > 0 and consider the constraint J = V˙ x (t) + yT (t)y(t) − γ 2 w T (t)w (t)

(20)

Then, along the trajectories of (13), we have

(21)

J ≤ υ T Xˆ η υ

(22)

Θi12 τT + τTT ∗ ∗ ∗

Θ 13 −τ T G¯

¯ −T GE ¯ −τ T GE

0

0

∗ ∗

−γ 2 I ∗

CT 0 0 DT −I

⎤ ⎥ ⎥ ⎥≤0 ⎦

(25)

with

Θi11 = −T Ai − ATi T T + Y + Y T + 2ερp I Θi12 = Qi + T − τ ATi T T + τ Y T Θ 13 = −T G¯ − ε P

Proof. In light of P-OSL condition (6), for ε > 0, we have 2ερp I 0 T ⎢ υ ⎣ −ε P 0

0 0 0 0

−ε P 0 0 0

⎤

0 0 ⎥ υ≥0 0 ⎦ 0

(26)

Combining (17), (21) and (26) produces

where

⎢ ∗ Xˆ η = ⎢ ⎣ ∗ ∗

⎢ ⎢ Θi = ⎢ ⎣

Θi11 ∗ ∗ ∗ ∗

⎡

Thus, adding the left-hand sides of (17)–(19) to the right-hand side of (21) results

Xˆ η11

⎡

then the sliding mode dynamic (13)–(14) is robust asymptotically stable with disturbance attenuation level γ and the control gain K in (11) is given by K = B+ T −1 Y .

J = xT Qη x˙ + x˙ T Qη x + (Cx + Dw)T (Cx + Dw) ⎡ T ⎤ C C Qη 0 CT D 0 0 0 ⎢ ∗ ⎥ −γ 2 wT w = υ T ⎣ ⎦υ ∗ ∗ 0 0 2 T ∗ ∗ ∗ −γ + D D

⎡

Theorem 2. Consider the integral sliding surface (11). Under Hypothesis 2, for given scalar τ , if positive constants ε and γ , symmetric positive definite matrices Qi ∈ Rn×n , and any appropriately matrices T and Y can be found to satisfy the following LMIs, for i = 1, . . . , h:

⎤

0 −2ε2 I 0

The nonlinear function φ (x) satisfy the P-OSL

Hypothesis 2. property (6).

inequality

Xˆ η12 T2 + T2T

Xˆ 13 −T2 G¯ −2ε2 I

∗ ∗

∗

ˆηυ J ≤ υT Θ

⎤

−T1 E¯ + C T D ⎥ −T2 E¯ ⎥ ⎦ 0 −γ 2 + DT D

where

⎡

ˆ η11 Θ ⎢ ∗ ˆη = ⎢ Θ ⎣ ∗ ∗

Xˆ η11 = −T1 A¯ η − A¯ Tη T1T + 2(ε1 ρ1 + ε2 ρ2 )I + C T C Xˆ η12 = Qη + T1 − A¯ Tη T2T

¯ − (ε1 − ε2 ρ3 )I Xˆ 13 = −T1 G

∑h

Set T1 = T , T2 = τ T and Y = TBK . By convexity (Xη = i=1 ηi Xi ∑h ˆ and Xˆ η = i=1 ηi Xi ), it can be shown that if the condition (15) is satisfied, Xˆ η < 0 is held by using Lemma 1. Therefore, for υ ̸ = 0 J ≤0

(23)

Integrating (23) from tr to ∞, ∞

∫

yT (t)y(t) − γ 2 w T (t)w (t) dt ≤ 0 , ∀t ≥ tr

(

Vx (∞) − Vx (tr ) +

)

tr

(24) Assuming the zero initial condition x(t0 = tr ) = 0 (Vx (tr ) = 0) and taking into account the positive definitiveness of the func∫∞ tional (limt →∞ Vx (t) ≥ 0), it follows that t yT (t)y(t) dt ≤

∫∞

(27)

r

γ 2 wT (t)w(t)dt; that is ∥y(t)∥2 ≤ γ ∥w(t)∥2 for all t ≥ tr . Additionally, if w (t) = 0 then (23) implies V˙ x (t) + yT (t)y(t) ≤ 0, that is V˙ x (t) ≤ 0, which means that the sliding motion (starts at the tr

moment tr ) is asymptotically stable. □ It should be noted that nonlinear plants in [19,21,34] are only a subclass of OSL nonlinear systems, which are also quadratically inner bounded. Indeed, checking the quadratically inner bounded property is not an easy task, as we shall see in the simulation study. Consequently, for design problems, it will significantly reduce the complexity and conservatism if we remove this constraint. As argued in [33], an alternative solution is using the condition (6) instead of conditions (7) and (8).

ˆ η12 Θ T2 + T2T ∗ ∗

ˆ 13 Θ −T2 G¯ 0

∗

⎤ −T1 E¯ + C T D ⎥ −T2 E¯ ⎥ ⎦ 0 −γ 2 + DT D

ˆ η11 = −T1 A¯ η − A¯ Tη T1T + 2ερp I + C T C Θ ˆ η12 = Qη + T1 − A¯ Tη T2T Θ ˆ 13 = −T1 G¯ − εP Θ

∑h

Set T1 = T , T2 = τ T and Y = TBK . By convexity (Θη = i=1 ηi Θi ∑ ˆ i ), it can be shown that if the condition (25) is ˆ η = hi=1 ηi Θ and Θ ˆ η < 0 is held by using Lemma 1. Thus, by analogy with satisfied, Θ the discussion above (proof of Theorem 1), the asymptotic stability and the H∞ performance for the closed-loop system in the sliding mode are attained. □ Remark 1. Theorems 1 and 2 provide an upper bound on the L2 gain for bounded disturbances. The scalar γ can be included as an optimization variable to obtain a lower bound of the guaranteed H∞ constraint. The minimal bound named γ ∗ can be determined by solving the following convex optimization problem min γ s.t . (15) Qi > 0, ε1 > 0, ε2 > 0, Y , T . Or min γ s.t . (25) Qi > 0, ε > 0, Y , T . 3.1.3. SMC law synthesis In this section, the attention is focused on controller synthesis, by which the reaching condition is satisfied. This condition ensures

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

that the trajectories of the OSL system (1)–(2) will reach the sliding surface (11) in finite time tr and will stay on it thereafter, even when nonlinearities and unmatched uncertainties are implicated. Therefore, the following result is proposed. Theorem 3. Consider the sliding surface function (11) and the gain matrix K obtained in Theorem 1 (or in Theorem 2). For given positive scalars β and µ, the state trajectories of the one-sided Lipschitz nonlinear system (1)–(2) are driven to the pre-designed sliding surface in finite time tr and a sliding motion can be maintained on it thereafter by the control u(t) = −α1−1 σ (t)

s(t)

(28)

∥s(t)∥ + µ

with

    ) ( σ (t) = ∥K ∥ ∥x(t)∥ + (GB)−1 G ∥φ (x)∥ + (GB)−1 GE  w0 + β Proof. From (11), we have s˙(t) = GBϕ (u) + GBKx(t) + Gφ (x) + GE w (t)

(29)

Consider the Lyapunov function Vs (t) =

1 2

sT (t)(GB)−1 s(t)

Remark 2. Matrix G and constants β and µ represent the design parameters in this study. G is chosen so that the square GB is nonsingular (As B has full column rank, the term inverse (GB)−1 always exists). Following the method in [13], the matrix G can be designed by selecting G = BT X with X > 0. It is clear that the convergence speed of the system states is determined by β . The larger the value of β , the faster the convergence of the system trajectories is. From a practical point of view, the choice of β is related to the physical limits of the control and the dynamics of the system. Its value can be properly adjusted in order to keep the reaching time tr as short as possible and a compromise has to be made between the response speed and the control input. To prevent the control signal from chattering, we substitute the s(t) discontinuous term sign(s(t)) by [5,11]. But, in practice, ∥s(t)∥ + µ this approximation leads to obtain a pseudo-sliding mode (the state trajectories no longer slide on the sliding surface, and instead they evolve in the vicinity of the sliding surface). As µ tends to be zero, the performance of the approximated control law can be made arbitrarily close to that of the original control law. It can be used to trade off the requirement of maintaining veritable robustness performance with that of ensuring a smooth control action.

(30) 3.2. Further discussions

By design it is assumed that the square matrix GB is nonsingular. The time derivative of Vs (t), along the trajectory of (11), is given by V˙ s (t) = sT (t)(GB)−1 s˙(t) = sT (t)(GB)−1 [GBϕ (u) + GBKx(t)+ −1 Gφ (x) + GE w (t)] = sT (t)ϕ (u) + sT (t)Kx(t) + sT (t)(GB) G ( T −1 T ∥ ∥ ∥ ∥ ∥ ∥ ×φ(x) + s (t)(GB) GE w (t) ≤ s (t) ϕ (u) + s(t) K x(t)    ) + (GB)−1 GE  ∥φ (x)∥ + (GB)−1 GE  w0 (31) Using (4) and (28) yields uT ϕ (u) = −α1−1 σ (t)

sT (t)

∥s(t)∥ + µ

In this part, further results about control of OSL nonlinear systems are presented. The sliding surface design conditions offered by Theorems 1 and 2 can be easily extended to show an extra LMI criterion of the H∞ control by state feedback for the system (1)–(2) when ϕ (u) = u(t) = −Ks x(t). For all t ≥ 0, consider the nonlinear system

{

x˙ (t) = (Aη − BKs )x(t) + φ (x) + E w (t) y(t) = Cx(t) + Dw (t)

(32)

sT (t)ϕ (u) ≤ −σ (t) ∥s(t)∥ − µσ (t) ≤ −σ (t) ∥s(t)∥

(33)

Theorem 4. Consider the closed-loop system (37). Under Hypothesis 1, for given scalar τ , if positive constants ε1 , ε2 and γs , symmetric positive definite matrices Qi ∈ Rn×n , and any appropriately matrices T and Y can be found to satisfy the following LMIs, for i = 1, . . . , h:

Applying (28) and (33) into (31), one can obtain

⎡

V˙ s (t) ≤ −β ∥s(t)∥ < 0 ; s(t) ̸ = 0

(34)

The reaching condition is assured. By Rayleigh’s inequality, (34) is equivalent to V˙ s (t) ≤ −β ∥s(t)∥ = −β¯

√

Vs (t)

(35)

1 −1 where β¯ = β 2λ− ) > 0. max ((GB) Integrating both sides of (35) from 0 to t > 0, we have

√

Vs (0) ≤ −

β¯ 2

t,

(36)

In fact, suppose that the system states cannot reach the sliding mode s(t) = 0 within finite time, then from

√

√

Vs (0) −

β¯

t, 2 Vs (t) becomes negative with t sufficiently large. This contradicts √ with Vs (t) nonnegative. In this way, considering tr as the time required to attain s(t) = 0. As Vs (t = tr ) = 0, the condition √ λmax ((GB)−1 ) ∥Gx(0)∥ holds. So, the proposed tr ≤ 2β¯ −1 Vs (0) =

√

Vs (t) ≤

⎢ ⎢ Ξi = ⎢ ⎣

Ξi11 ∗ ∗ ∗ ∗

Ξi12 τT + τTT ∗ ∗ ∗

Ξ 13 −τ T −2ε2 I ∗ ∗

−TE −τ TE 0

−γ 2 I ∗

CT 0 0 DT −I

⎤ ⎥ ⎥ ⎥<0 ⎦

(38)

with

Ξi11 = −T Ai − ATi T T + Y + Y T + 2(ε1 ρ1 + ε2 ρ2 )I Ξi12 = Qi + T − τ ATi T T + τ Y T Ξ 13 = −T − (ε1 − ε2 ρ3 )I

√

Vs (t) −

(37)

Now, we are in position to state the following results.

ϕ (u) ≥ α1−1 σ 2 (t)

then

√

23

β

sliding mode control (28) brings the system trajectories onto the predesigned switching manifold in finite time tr and maintains the sliding mode thereafter. □

then the one-sided Lipschitz system (37) is robust asymptotically stabilizable with disturbance attenuation γs by the H∞ control u = −Ks x(t) with Ks = (BT B)−1 BT P −1 Y . Theorem 5. Consider the closed-loop system (37). Under Hypothesis 2, for given scalar τ , if positive constants ε and γs , symmetric positive definite matrices Qi ∈ Rn×n , and any appropriately matrices T and Y can be found to satisfy the following LMIs, for i = 1, . . . , h:

⎡ ⎢ ⎢ Λi = ⎢ ⎣

Λ11 i ∗ ∗ ∗ ∗

Λ12 i τT + τTT ∗ ∗ ∗

Λ13 −τ T

−TE −τ TE

0

0

∗ ∗

−γ 2 I ∗

CT 0 0 DT −I

⎤ ⎥ ⎥ ⎥≤0 ⎦

(39)

24

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29 Table 1 Comparison between Two methods.

with T T T Λ11 i = −T Ai − Ai T + Y + Y + 2ερp I T T T Λ12 = Q + T − τ A T + τ Y i i i Λ13 = −T − ε P

then, for Ks = (BT B)−1 BT P −1 Y , the one-sided Lipschitz system (37) is robust asymptotically stabilizable with disturbance attenuation level γs by the H∞ control law u = −Ks x(t). Remark 3. If the polytopic parameter uncertainties does not affect the system (Aη = A), the LMI synthesis conditions in Theorems 1, 2, 4 and 5 can be solved by considering the single Lyapunov matrix Qη = Q > 0. From a complexity point of view, it is important to note that the provided LMIs are solved by using a semi-definite programming (SDP) solver (‘MOSEK’ [35]) which run under the MATLAB YALMIP environment [36]. In fact, SDP algorithms are polynomial in problem size and that is why SDP is accepted as a good tool. As can be seen, Theorems 1, 2, 4 and 5 include more variables than the existing methods after inserting two auxiliary matrices (T1 and T2 ). Indeed, by adding these variables, more iterations are required before returning solutions, then the algorithm will take more time, but not significant because of the nature of the SDP optimization (use of primal–dual interior-point algorithms). That is the computational complexity is lightly affected; the reader is referred to [37] for a comprehensive discussion about interiorpoint methods as they are out of the scopes of the current study. Nevertheless, it worth pointing out that introducing T1 and T2 removes the product between the Lyapunov matrices and the system dynamic matrices which enables us to employ a multiple Lyapunov function. This feature brings further flexibility and reduces the conservatism imposed, usually, by employing single Lyapunov function in the analysis and synthesis problems of systems with polytopic uncertainties. 4. Illustrative examples In this section, three physical examples are provided to demonstrate the effectiveness of the proposed methods. 4.1. Example 1: Motion of a moving object in Cartesian co-ordinates The motion of a moving object in Cartesian co-ordinates can be described by the system (1)–(2) with

[ Aη =

[ C =

−θ 1

]T

0 1

−1 θ

]

,B =

[

, D = 0, w(t) =

0 1

]

[

−x1 (x21 + x22 ) −x2 (x21 + x22 )

√ ρ3 ⎝ r = min − , 4

4

ρ2 +

[12, Proposition 4]

0,0.25,−1 0,1.5,−2.25 0,−3,−4 0,−10,−16 0,−99,−100

1.3195 1.3365 1.3371 1.2670 1.2334

1.4223 1.4704 No feasible solution No feasible solution No feasible solution

ε1 = 0.3124, ε2 ]= 0.4033[ −0.0009 1.5906 Q1 = , Q2 = 2.5711 0.0000 [ ] [ −0.6591 −0.0049 0.1384 T = ,Y = 0.0196 − 0.9164 − 0.9690 [ ] K = 1.0528 5.6861 , γ = 1.3381.

[

1.9043 −0.0009

0.0000 1.5906

]

0.6044 −5.2296

]

[

0

1

]

∫ x(t) −

t

[

−0.0528

−2.9361

]

x(ν )dν

0

]

√

Theorem 1

0.75 1.6875 3 12 75

then the quadratically inner-bounded property of φ is verified in ˜ . As the system is globally OSL, that is ℜ = R2 , ℜ ∩ ℜ ˜ = ℜ ˜. ℜ ˜ can be made arbitrarily large by choosing Note that the region ℜ appropriate values for ρ2 and ρ3 . To demonstrate the advantages of the proposed method, we present a comparison with the method of Zhang [12], in which the SMC-H∞ problem of Lipschitz nonlinear systems with unmatched uncertainties has been investigated. For comparison purpose, the polytopic uncertainty is not]considered; [ 1 0 θ = 2.75 (see Remark 3). Set X = > 0 and 0.5 0.5 τ = 0.42. Under the sliding mode, the feasible results for the H∞ performance bound γ among the above methods are summarized in Table 1. It can be seen that good values for the disturbance attenuation bound γ are obtained by our approach while no feasible solutions are provided by the approach in [12] when r = 1, r = 2 and r = 5. Due to the fact that ρ0 = 3r 2 , the Lipschitz constant rapidly increases with the increase of r. As mentioned before, the methods based on the traditional Lipschitz condition strongly depends on the Lipschitz constant and more the latter is larger, the more difficult it is to find a feasible solution. Compared to Lipschitz condition, OSL constants are found significantly smaller, which makes them much more appropriate for estimating the influence of non-linear terms. More in comparison, the methods are examined in the case when r = 0.75 and noises [ measurement ] [ are also ] 0.1 0.1 and G = 0 1 . implicated (D ̸ = 0). Let D = For τ = 0.69, by solving the LMI (15), the following feasible values are obtained

s(t) =

This model represents a typical example for one-sided discussion studies (see [38,39] and references therein). As shown in [19,32], the nonlinear function φ (x) is globally one-sided Lipschitz with {ρ1 = 20. Also, the } system is locally Lipschitz on any set ℜ = x ∈ R : ∥x∥ ≤ r and the corresponding Lipschitz constant is { } ˜ = x ∈ R2 : ∥x∥ ≤ r , conseρ0 = 3r 2 . Now consider the set ℜ quently, if

⎛

ρ1 , ρ2 , ρ3

]

The nonlinear function is

φ (x) =

0.5 0.75 1 2 5

γ

ρ0

while, the feasible value of the performance index γ is found to be 1.3465 by Proposition 4 in [12]. The sliding surface function (11) is computed as

, E = I2 ,

0.1 sin(100t) 0.1 cos(100t)

[

ρi , i ∈ {0, 1, 2, 3}

r

ρ32 4

⎞ ⎠;

ρ3 < 0, ρ2 +

ρ32 4

>0

To establish the comparison, the system without nonlinear input is simulated ϕ (u) = u(t); that is α1 = α2 = 1 (as the method in [12] cannot deal with systems including nonlinear inputs). Thence, for β = 0.4 and w0 = 0.1, the SMC law (28) is designed as u(t) = − (5.7827 ∥x∥ + ∥φ∥ + 0.5)

s(t)

∥s(t)∥ + 0.01 [ ]T −0.5 0.5 , the simulation For initial condition x(0) =

results are illustrated in Figs. 2–6. The curves in Fig. 2 represent the behaviors of ∥x(t)∥ for the open-loop system and the controlled systems. Since the eigenvalues of the matrix A are located at −2.8284 and 2.8284, the uncontrolled system is unstable while controlled systems are stable after a short time. It can be seen

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

25

Fig. 5. SMC law u(t).

Fig. 2. Evolution of ∥x(t)∥ for uncontrolled and controlled systems.

Fig. 6. Disturbance attenuation from w (t) to y(t).

Fig. 3. Output y(t).

that x(t) with our control method has a better convergence qualities than that with the method in [12]. Fig. 3 depicts the time response of the system output y(t). It can be observed that the design scheme in this paper produces fast response and minimum overshoot in comparison with [12]. Plots of the sliding variable s(t) and SMC law u(t) are displayed in Figs. 4 and 5 respectively. As can be seen, both closed-loop systems move along the switching surface s(t) = 0. The sliding mode is attained in a finite moments approximated to tr ≈ 0.22 s (under our approach) and tr ≈ 0.53 s (under the approach in [12]). The proposed controller provides best performances. The next simulation compares the disturbance attenuation abilities of the two methods. For each closed-loop system, the level of disturbance attenuation in the sliding mode is measured, at any time T , by the following L2 gain performance index

√∫ g(t) = √∫ Fig. 4. Sliding function s(t).

T tr

T tr

yT (t)y(t)dt

wT (t)w(t)dt

,

∀t ∈ (0, T ]

(40)

where y(t) denotes the output resulting from all w (t) ∈ L2 [0, T ]. The function g(t) represents the ratio between the truncated norms

26

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

Fig. 7. Schematic of the single-link flexible joint robotic system. Table 2 System parameters. Parameter (Units)

Value

Motor inertia Jm , (Kg m2 ) Link inertia Jl , (Kg m2 ) Pointer mass m, (Kg) Link length h, (m) Torsional spring constant k, (N m rad−1 ) Viscous friction coefficient Bm , (N m V−1 ) Amplifier gain kι , (N m V−1 )

3.7 × 10−3 9.23 × 10−3 2.1 × 10−2 1.5 × 10−1 1.8 × 10−1 4.6 × 10−3 8.0 × 10−2

Fig. 8. State vector x(t).

⎡

0 ⎢ −54 A2 = ⎣ 0 21.6

⎡

∥y(t)∥2,T and ∥w(t)∥2,T . It is used to show the effect on the output y(t) from the disturbance input the assumption of [ w(t) ]under T 0 0 zero initial condition x(0) = . Time responses of g(t) for both methods are demonstrated in Fig. 6. It turns out that the squared root of the ratio between the output energy and the disturbance input energy is always bounded for both closed-loop systems as g(t), under the two methods, is always less than the feasible values of γ . But, Fig. 6 clearly shows that our method has better behavior and disturbance attenuation performance than the existing one [12]. It can be concluded that the proposed SMC design scheme provides less-conservative results and has better advantages in comparison with the SMC method in [12]. 4.2. Example 2: Single-link flexible joint robotic system The proposed approach is tested on the model of a singlelink flexible joint robotic system [26,40] as shown in Fig. 7. The dynamics of this system without uncertainties can be described as follows

θ˙m (t) = ωm (t), ω˙ m (t) =

k Jm

(θl (t) − θm (t)) −

Bm Jm

ωm (t) +

kι Jm

u(t),

θ˙l (t) = ωl (t), k

mgh

Jl

Jl

ω˙ l (t) = − (θl (t) − θm (t)) −

sin(θl (t))

where θm (t) and θl (t) are respectively the angles of rotations of the motor and the link. ωm (t) and ωl (t) are their angular velocities. Jm represents the inertia of the actuator (DC motor) and Jl stands for the inertia of the link. The parameters’ definition and their values are given in Table 2. For 1.6 N m rad−1 = k1 ≤ k ≤ k2 = 2 N m rad−1 , the robotic system can be written in the form of (1)– (2) with

⎡

0 ⎢ −43.2 A1 = ⎣ 0 17.3

1 −1.25 0 0

0 43.2 0 −17.3

⎤

0 0 ⎥ , 1 ⎦ 0

⎤

1 −1.25 0 0

0 54 0 −21.6

⎡

⎤

0 0 ⎥ , 1 ⎦ 0

⎤

0 1 [ 1 ⎢ 21.6 ⎥ ⎢ 1.5 ⎥ B=⎣ ,E = ⎣ ,C = 0 ⎦ 0 ⎦ 0 0 0.8

0 1

0 0

0 0

]

, D = 0,

[ ]T φ (x) = 0 0 0 −3.3 sin(x3 ) , w(t) = 0.1 cos(100t), ϕ (u) = (0.4 + 0.2 cos(u))u φ (x) satisfies conditions (7) and (8) with constants ρ1 = 3.3, ρ2 = 10.89 and ρ3 = 0 [40]. The matrix X is selected as

⎡

1 ⎢ 3.2407 X =⎣ 1 0

−10 7.4074 1 0

0 −5.0926 1 0

⎤

0 4.0741 ⎥ ⎦>0 0 1

Set τ = 0.0046. By resolving (15), the sliding mode gain K and the performance bound γ are obtained as follows K =

[

−1.6482

1.1422

1.6527

−1.1768

]

, γ = 1.5167

It should be mentioned that the problem is infeasible (as ρ0 is large) when the approach in [12] is applied. For the designed sliding mode controller in (28), we choose β = 0.1, w0 = 0.1 and µ = 0.02. The simulations are illustrated in Figs. 8–10 with initial condition x(0) = [ −0.3 0 −0.3 0 ]T . The plot of the state vector x(t) is shown in Fig. 8. In Figs. 9 and 10, the plots represent the integral switching surface function s(t) and the SMC input u(t), respectively. As observed, the sliding mode is attained in finite-time approximated to tr ≈ 3.2 s, and after that, the system states have desirable behaviors as designed. The SMC system performs satisfactorily and has good performance characteristics. It is obvious that the proposed scheme works well and provides nice robustness qualities in the presence of unmatched uncertainties and nonlinearities. 4.3. Example 3: Mechanical revolving model Fig. 11 shows the schematic of a physical system [38,41]. J represents inertia of the revolving cylinder; R, the radius of the cylinder; H , the viscous friction coefficient of the revolving part; θ , the angular rotation; F , the force added by a motor whose power

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

27

Table 3 Comparison between different methods. Method

Applicability

Theorem 2 Lipschitz SMC methods in [3–15] OSL control methods in [22,25,26,42]

Applicable Inapplicable Inapplicable

Fig. 9. Switching function s(t).

Fig. 12. State vector x(t).

where ω is the angular velocity and FRω = a, a is a constant. The system can be described by (1)–(2) with appropriate parameters:

[

0 0

1

]

,B = ζ [ ] θ x = , u = ∆F ω

A =

[

0 1

]

,c =

[

ϕ (u) = (0.2 + 0.1cos(u))u , w (t) = Fig. 10. Control input u(t).

Fig. 11. A mechanical revolving model.

1 0

]T

[ , φ (x) =

0 4.25

] ,

x2

0.1

+ 0.1sin(100t) 1 + e4t The uncertain parameter ζ lies in the range of [1, 1.4]. φ (x) is not the usual Lipschitz nonlinearity (5) and does not satisfy the quadratically inner-bounded property (8) in the region ℜ = {x1 ∈ R, x2 > 0}. However, as proved in [41], φ (x) obeys the P-one-sided Lipschitz property (6) with the constant ρp = 0 for any positivedefinite matrix P. In Table 3, the benefit and the superiority of our design approach compared with available literature are clear. As the Lipschitz property (5) is not satisfied, all works dealing with the classical Lipschitz description, including SMC methods, cannot be applied to systems containing such nonlinearities. In contrast to OSL control methods in [22,25,26,42], which need the fulfillment of the inner-bounded property (8), the design condition in Theorem 2 provides feasible and suitable results. [ ] 2 0.5 Taking τ = 0.2 × 10−6 , P = and G = BT and 0.5 2 resolving (25) in Theorem 2, the following values are obtained

is a constant value, while ∆F is input force from the time-varying power. A state space description of this system is given by

⎧ ⎨θ˙ = ω

H

a

J

Jω

⎩ω˙ = − ω +

+

∆FR J

K =

[

129.6962

26.9878

]

, γ = 1.5

Selecting β = 1, w0 = 0.2 and µ = 0.05, the simulations are presented in Figs. 12–14. Fig. 12 shows the plot of the state vector x(t). In Fig. 13, the curves denote the integral switching surface function s(t) and the SMC input u(t), respectively. The sliding mode is attained in finite-time approximated to tr ≈ 0.015 s, and

28

W. Saad, A. Sellami and G. Garcia / ISA Transactions 90 (2019) 19–29

synthesis conditions has been derived both for the sliding surface existence problem and H∞ state-feedback control design. A tractable LMI framework that benefits from the decoupling between Lyapunov and system matrices has been offered. Comparative studies and simulation results have shown the performance and robustness capabilities of the suggested approach. Future recommendation: This study has also opened doors for further research related to the SMC and its applications, such as the synchronization of OSL chaotic systems using the sliding mode method and the fault tolerant SMC design for systems with OSL nonlinearities.

Declarations of interest None. References Fig. 13. Switching function s(t) and Control input u(t).

Fig. 14. Output y(t) and L2 -gain disturbance attenuation g(t).

after that, the system states have desirable behavior as designed. Fig. 14 illustrates the output y(t) and the disturbance attenuation measure g(t). Although the output is affected by the noise w (t), the performance and robustness capabilities of the proposed method is guaranteed. From Fig. 14, it is observed that the L2 gain from w(t) to y(t) defined by the function g(t) in (40) is about 0.44 at T = 2 s, which is less than γ = 1.5. That is, the squared root of the ratio between the output energy and the disturbance input energy is always bounded. The SMC system performs satisfactorily and has good disturbance attenuation performance in despite of uncertainties and noises, which further demonstrates the validity of the results. 5. Conclusion This paper has investigated the problem of SMC with L2 -gain disturbance attenuation constrains for various kinds of one-sided Lipschitz nonlinear systems. Unlike most studies we further consider the involvement of polytopic-type parameter uncertainties and input nonlinearities. Thus, the range of applicability of the proposed scheme becomes large. By the employment of vertexdependent Lyapunov functions, sufficient and less conservative

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