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Robust stabilization of uncertain 2-D discrete-time delayed systems using sliding mode control Imran Ghous a, Zhaoxia Duan b,∗, Jahanzeb Akhtar a, Muhammad Jawad a a Department
of Electrical and Computer Engineering, COMSATS University Islamabad (Lahore Campus), Lahore 54000, Pakistan b College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, People’s Republic of China Received 20 August 2018; received in revised form 15 January 2019; accepted 26 February 2019 Available online xxx
Abstract This paper aims to solve the problem of sliding mode control for an uncertain two-dimensional (2-D) systems with states having time-varying delays. The uncertainties in the system dynamics are constituted of mismatched uncertain parameters and the unknown nonlinear bounded function. The proposed problem utilizes the model transformation approach. By segregating the proper Lyapunov– Krasovskii functional in concert with the improved version of Wirtinger-based summation inequality, sufficient solvability conditions for the existence of linear switching surfaces have been put forward, which ensure the asymptotical stability of the reduced-order equivalent sliding mode dynamics. Then, we solve the controller synthesis problem by extending the recently proposed reaching law to 2-D systems, whose proportional part is appropriately scaled by the factor that does not depend on some constant terms but rather on current switching surface’s value, which in turn ensures the faster convergence and better robustness against uncertainties. Finally, the proposed results have been validated through an implementation to a suitable physical system. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Time-delay frequently appears in diverse engineering systems, like telecommunication systems, chemical engineering, metal rolling systems, and network control systems, etc. [1,2]. ∗
Corresponding author. E-mail address:
[email protected] (Z. Duan).
https://doi.org/10.1016/j.jfranklin.2019.02.041 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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Similarly, practical systems are nonlinear and contain the uncertainties (matched and unmatched) and disturbances because of the erroneous modeling, measurement faults and external factors [3,4]. The presence of nonlinearities, time-delays, uncertainties, and disturbances may cause degradation in system’s performance and may even lead to instability [5–10]. Moreover, during the recent years, a surge in the popularity of sliding mode methodology can be observed evidently as it is considered advantageous due to the fact that it provides robustness against uncertainties, non-linearities, and disturbances. Other advantages include finite-time convergence and reduced-order compensated dynamics [11]. Sliding mode control (SMC) have applications in many research areas such as electric vehicles, spacecraft, aircraft, underwater vehicles, robotic manipulators, power systems and automotive engines [12]. In general, the SMC design procedure follows two basic approaches. The first approach initially states the control law and then proves that it guarantees the sliding motion’s stability for the system under consideration. The second approach utilizes the reaching law scheme, which initially specifies the required sliding variable evolution and then drives the control law which guarantees that the system under investigation follows the specified portrait of a particular variable [13]. The Lyapunov approach is utilized frequently for the stability analysis, in the SMC design [14]. For one-dimensional (1-D) systems, with uncertainties and delays and also without delays, the SMC design problem for the systems with continuous and discrete dynamics have been vastly researched in the literature (Please see [15–19] and references therein). Different from 1-D systems where the state variables are the only function of time, there exists a type of systems, known as multidimensional systems where the states of the system are the function of more than one variable (time, space and so on.). 2-D systems are a special type of multidimensional systems where the state depends on variables (two independent ones) and have applications in several engineering fields such as sorption process, thermal processes, multidimensional filtering and image processing [20,21]. Stability analysis and controller synthesis problems for such systems with and without delays, and uncertainties have been notably inspected in the references [22–26]. Other interesting relevant studies can also found in [27–29]. As far as, the SMC design problem for 2-D systems is concerned, relatively fewer results can be found in the literature. The first effort in this regard has appeared in [30], where initially the authors have extended the Gao’s reaching law [31] for 2-D discrete systems without delays. By using the concept adopted in [32], they have substituted the positive constant ε by s2 (k), in order to ensure convergence for system trajectories. Then, the problem was also solved by the Choi’s method [33]. Later, the similar problem for 2-D Roesser model without delays was also solved [34]. Similarly, the reference [35], solved the SMC design problem for first Fornasini and Marchesini delay free model by utilizing the 1-D vectorial form approach. It can be noticed from the afore-mentioned discussion that the practical factors like the state delays, mismatched uncertainties and the unknown nonlinear function (matched and bounded) have not been taken into consideration while investigating the SMC design problem for 2-D systems. That would obviously make the investigation of SMC design problem more challenging. Therefore, the research gap highlighted above motivates us to study the robust SMC design problem for 2-D discrete delayed systems, which to the best of our knowledge has not been researched before. The following points summarize the contribution of this work. Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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• SMC design for 2-D discrete systems with delays, mismatched uncertainties and matched unknown nonlinear function by linear matrix inequality (LMI) approach: This work utilizes the model transformation approach [30], which transforms the original system into the commonly known ‘regular’ form by adding a non-singular transformation matrix. For 2-D systems of discrete nature having delays, mismatched uncertainties and matched unknown nonlinear function, LMI based conditions have been derived, by selecting a proper Lyapunov–Krasovskii functional (LKF) together with improved version of Wirtinger-based summation inequality (WSI), such that the existence of switching surfaces can be ensured with the asymptotical stability of the reduced-order sliding mode dynamics. • Extension of reaching law: The dynamics of the switching function are generally specified by the reaching law, which is actually differential or difference equation. In reaching phase, the dynamics quality of SMC can be controlled by appropriate selection of its parameters. Authors in [30], extended the concept of Gao’s reaching law to 2-D discrete system based on the results given in [32]. Motivated by the results in [36], this work extends the proposed reaching law to the 2-D systems, where the proportional part is substituted by a factor that is switching function depended (current value) instead of some constant term. The results are finally validated with a suitable example. The following structure will be followed in the remaining paper. Section 2 describes the system dynamics and formulates the problem. The design of robust SMC for 2-D delayed systems is discussed in Section 3 whereas Section 4 presents a suitable example. Section 5, concludes the contribution of this paper. Notations: Standard notations are being utilized throughout this manuscript. AT ∈ Rb × a means the transpose of a matrix A ∈ Ra × b ; B > 0, I and 0 represent the positive definite symmetric matrix, the identity matrix and the zero matrix with suitable dimensions, respectively; . means the Euclidean norm vector; | . |: → [0, ∞) represents the absolute value, with being the set of real numbers; diag{.} symbolizes the block diagonal matrix; the asterisk (∗ ) illustrates the term induced by the symmetry. If the reader finds that the dimensions of matrices have not been specified explicitly, it should be assumed that they are compatible and consistent for the computation of algebraic operations. 2. Preliminaries and problem statement Let us refer to the following 2-D uncertain system with delays expressed in the state-space form by the Roesser model: x + (h, v ) = M¯ x (h, v ) + M¯ d x (dh (ι), dv (κ ) ) + C [u (ι, κ ) + N N (x (h, v ), ι, κ )],
(1)
with
M¯ M¯ 2 h ( ι + 1, κ ) h (ι, κ ) C1 , x (h, v ) = , M¯ = ¯ 1 , , x + (h, v ) = C = v (ι, κ + 1 ) v (ι, κ ) C2 M3 M¯ 4 M¯ d1 M¯ d2 h (ι − dh (ι), κ ) ¯ and x (dh (ι), dv (κ ) ) = . Md = ¯ v (ι, κ − dv (κ ) ) Md3 M¯ d4 where, h (ι, κ ) ∈ Rnh and v (ι, κ ) ∈ Rnv describe the horizontal and vertical states, respectively; x (h, v ) ∈ Rn shows the whole state with n = nh + nv ; u(ι, κ) ∈ Rq symbolizes the control input; N (x (h, v ), ι, κ ) ∈ Rl is the nonlinear function; C and N are some appropriately dimensioned Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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known matrices; M¯ and M¯ d are the uncertain matrices of suitable dimensions and have the following form: M¯ = M + M, M¯ d = Md + Md , (2) 1 where M = D (ι, κ )Ga , Md = D (ι, κ )Gb , = , Ga = Ga1 Ga2 , Gb = 2 Gb1 Gb2 and M, Md , , Ga and Gb are real matrices with suitable dimensions; D(ι, κ) is unknown norm-bounded function and satisfies DT (ι, κ)D(ι, κ) < I; dh (ι) and dv (κ ) are the time-varying delays in the corresponding states and satisfy: hL ≤ dh (ι) ≤ hU , vL ≤ dv (κ ) ≤ vU The boundary conditions of the system (1) are as follows: ⎧ h (ι, κ ) = ρικ , ∀0 ≤ κ ≤ m1 , ⎪ ⎪ ⎪ ⎪ ι = −hU , −hU + 1, −hU + 2, . . . , 0, ⎪ ⎪ ⎪ ⎪ ⎪ v ⎪ ⎪ (ι, κ ) = σικ , ∀0 ≤ ι ≤ m2 , ⎪ ⎪ ⎪ ⎨ κ = −vU , −vU + 1, −vU + 2, . . . , 0, ρ00 = σ00 , ⎪ ⎪ ⎪h (ι, κ ) = 0, ∀κ ≥ m1 , ⎪ ⎪ ⎪ ⎪ ⎪ ι = −hU , −hU + 1, −hU + 2, . . . , 0, ⎪ ⎪ ⎪ ⎪ v (ι, κ ) = 0, ∀ι ≥ m2 , ⎪ ⎪ ⎩ κ = −vU , −vU + 1, −vU + 2, . . . , 0,
(3)
(4)
where m1 < ∞ and m2 < ∞ are some constants that are positive; ρ ικ and σ ικ are some given vectors. In this paper, the aim is to investigate the SMC design problem using the model transformation method. To this end, let us rewrite the system model (1) as follows: M¯ 1 M¯ 2 h (ι, κ ) M¯ d1 M¯ d2 h (ι − dh (ι), κ ) h ( ι + 1, κ ) = ¯ + ¯ v (ι, κ + 1 ) M3 M¯ 4 v (ι, κ ) Md3 M¯ d4 v (ι, κ − dv (κ ) ) C1 0nh ×q v C1 h u (ι, κ ) + u (ι, κ ) + N N (h (ι, κ ), ι, κ ) + 0nv ×q C2 0(nv ×q )×(q×l ) 0 + (nh ×q )×(q×l ) N N (v (ι, κ ), ι, κ ). (5) C2 This paper assumes that: Assumption 1. The pairs (M1 , C1 ) and (M4 , C2 ) in system (5) are stabilizable. Assumption 2. Uncertainties M, Md and the nonlinear function N (x (h, v ), ι, κ ) are unknown but bounded. Assumption 3. Matrices C1 and C2 have full column rank and satisfy nh > q and nv > q, respectively. In light of Assumptions 1 and 3, it can be stated that there will exist matrices B1 and B2 , which are non-singular and satisfy:
0(nh −q )×q 0(nv −q )×q B1C1 = , B2C2 = , C˜1 C˜2 Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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where, C˜1 ∈ Rq×q and C˜2 ∈ Rq×q are non-singular matrices. We choose transformation matrices as follows for convenience purpose: T2T U2T B1 = , B = , 2 T1T U1T with T1 ∈ Rnh ×q , T2 ∈ Rnh ×(nh −q ) , U1 ∈ Rnv ×q and U2 ∈ Rnv ×(nv −q ) are the two sub-blocks of a unitary matrix obtained as the result of singular value decomposition of C1 and C2 , respectively, which is: ϒ VT, C1 = T1 T2 0nh ×(nh −q ) 1
VT, C2 = U1 U2 0nv ×(nv −q ) 2 where ϒ ∈ Rq × q , ∈ Rq × q are positive definite diagonal matrices and V1 ∈ Rq × q , V2 ∈ Rq × q are the unitary matrices. The system (5) can be exhibited into the ‘regular’ form by introducing the state transformation ℵ(ι, κ ) = diag{B1 , B2 }x (h, v ) as follows: h M˜ 1 M˜ 2 ℵh (ι, κ ) M˜ d1 M˜ d2 ℵh (ι − dh (ι), κ ) ℵ ( ι + 1, κ ) = + ℵv (ι, κ + 1 ) M˜ 3 M˜ 4 ℵv (ι, κ ) M˜ d3 M˜ d4 ℵv (ι, κ − dv (κ ) ) BC 0 B1C1 N N ℵh (ι, κ ), ι, κ + 1 1 uh (ι, κ ) + nh ×q uv (ι, κ ) + 0nv ×q B2C2 0(nv ×q )×(q×l )
0(nh ×q )×(q×l ) N N (ℵv (ι, κ ), ι, κ ), (6) + B2C2 with M˜ 1 = B1 M¯ 1 B1−1 , M˜ 2 = B1 M¯ 2 B2−1 , M˜ 3 = B2 M¯ 3 B1−1 , M˜ 4 = B2 M¯ 4 B2−1 , M˜ d1 = B1 M¯ d1 B1−1 , M˜ d2 = B1 M¯ d2 B2−1 , M˜ d3 = B2 M¯ d3 B1−1 and M˜ d4 = B2 M¯ d4 B2−1 . Considering the M¯ and M¯ d defined in (2), we further denote that: M 1 = B1 M1 B−1 , M 2 =
1
B1 M2 B2−1 , M 3 = B2 M3 B1−1 , M 4 = B2 M4 B2−1 , M d1 = B1 Md1 B1−1 , M d2 = B1 Md2 B2−1 , M d3 =
B2 Md3 B1−1 , M d4 = B2 Md4 B2−1 , M 1 = B1 M1 B1−1 , M 2 = B1 M2 B2−1 , M 3 = B2 M3 B1−1 ,
M 4 = B2 M4 B2−1 , M d1 = B1 Md1 B1−1 , M d2 = B1 Md2 B2−1 , M d3 = B2 Md3 B1−1 ,
M d4 = B2 Md4 B2−1 . These notations will be used in the derivation of results in Theorem 1 and Theorem 2. It follows from Eq. (6) that ⎡ h ⎤ ⎡ ⎤⎡ h ⎤ ⎡ ⎤ M˜ 1a M˜ 1b M˜ 2a M˜ 2b M˜ d1a M˜ d1b M˜ d2a M˜ d2b ℵ1 ( ι + 1, κ ) ℵ1 (ι, κ ) ⎢ℵh (ι + 1, κ )⎥ ⎢M¯ ⎥⎢ h ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ 1c M˜ 1d M˜ 2c M˜ 2d ⎥⎢ℵ2 (ι, κ )⎥ ⎢M˜ d1c M˜ d1d M˜ d2c M˜ d2d ⎥ ⎢ v ⎥=⎢ ⎥⎢ v ⎥+⎢ ⎥ ⎣ℵ1 (ι, κ + 1 )⎦ ⎣M˜ 3a M˜ 3b M˜ 4a M˜ 4b ⎦⎣ℵ1 (ι, κ )⎦ ⎣M˜ d3a M˜ d3b M˜ d4a M˜ d4b ⎦ ℵv2 (ι, κ + 1 ) ℵv2 (ι, κ ) M˜ 3c M˜ 3d M˜ 4c M˜ 4d M˜ d3c M˜ d3d M˜ d4c M˜ d4d ⎡ h ⎤ ⎡ ⎤ ⎡ ⎤ 0(nh −q )×q 0(nh −q )×q ℵ1 (ι − dh (ι), κ ) ⎢ ℵh (ι − d (ι), κ ) ⎥ ⎢ C˜ ⎥ ⎢ 0 ⎥ h ⎢ ⎥ ⎢ ⎥ h ⎢ q×q ⎥ v 1 × ⎢ v2 ⎥+⎢ ⎥u (ι, κ ) + ⎢ ⎥u (ι, κ ) ⎣ℵ1 (ι, κ − dv (κ ) )⎦ ⎣0(nv −q )×q ⎦ ⎣0(nv −q )×q ⎦ ℵv2 (ι, κ − dv (κ ) ) 0q×q C˜2 Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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⎡
⎤ ⎡ ⎤ 0(nh −q )×q 0(nh −q )×q ⎢ C˜ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 0q×q ⎥ 1 +⎢ ⎥ × N N ℵh (ι, κ ), ι, κ + ⎢ ⎥N N (ℵv (ι, κ ), ι, κ ). ⎣0(nv −q )×q ⎦ ⎣0(nv −q )×q ⎦ 0q×q C˜2 (7) where ℵh1 (ι, κ ) ∈ Rnh −q , ℵh2 (ι, κ ) ∈ Rq , ℵv1 (ι, κ ) ∈ Rnv −q , ℵv2 (ι, κ ) ∈ Rq , M˜ 1a = T2T M¯ 1 T2 , M˜ 1b = T2T M¯ 1 T1 , M˜ 2a = T2T M¯ 2U2 , M˜ 2b = T2T M¯ 2U1 , M˜ 3a = U2T M¯ 3 T2 , M˜ 3b = U2T M¯ 3 T1 , M˜ 4a = U2T M¯ 4U2 , M˜ 4b = U2T M¯ 4U1 , M˜ d1a = T2T M¯ d1 T2 , M˜ d1b = T2T M¯ d1 T1 , M˜ d2a = T2T M¯ d2U2 , M˜ d2b = T2T M¯ d2U1 , M˜ d3a = U2T M¯ d3 T2 , M˜ d3b = U2T M¯ d3 T1 , M˜ d4a = U2T M¯ d4U2 , M˜ d4b = U2T M¯ d4U1 , C˜1 = ϒV1T , C˜2 = V2T . In what follows, some essential definitions and lemmas are presented that will be used in the establishment of the results later. Definition 1 [20]. An overall closed-loop system of the form (6), a decomposition of which is given by system (7), is said to be asymptotically stable under any boundary conditions of the form (4), if lim ℵ(℘) = 0,
℘→∞
T where ℵ(℘) = {ℵ(ι, κ ) : ι + κ = ℘}, ℵ(ι, κ ) = ℵhT (ι, κ ) ℵvT (ι, κ ) with the Euclidean norm of the state defined by ℵ(℘) = supℵ∈ℵ(℘) ℵ, and the following control law ensures the reaching law condition of sliding mode for the specified switching functions: ⎧ h+ ui (ι, κ ), when sh (ι, κ ) > 0, ⎪ ⎪ ⎪ ⎨uh− (ι, κ ), when sh (ι, κ ) < 0, i ui (ι, κ ) = i = 1, 2, . . . , q. ⎪ uiv+ (ι, κ ), when sv (ι, κ ) > 0, ⎪ ⎪ ⎩ v− ui (ι, κ ), when sv (ι, κ ) < 0, Definition 2. The quasi-sliding mode of the 2-D system (1) is said to be in ε 1 and ε 2 vicinities of sh (ι, κ ) = 0 and sv (ι, κ ) = 0, sliding surfaces, respectively, such that h s (ι, κ ) ≤ ε1 , |sv (ι, κ )| ≤ ε2 , where the symbols ε1 and ε2 represent the widths of quasi-sliding mode band in the horizontal and vertical directions, respectively. Lemma 1 [37]. For a matrix R > 0, positive integers τ 2 > τ 1 and any sequence of discrete-time variable x: Z[τ 1 , τ 2 ], the below mentioned inequality holds:
T R τb 0 1 1 1 δ T ( j )Rδ( j ) ≥ a +1 R 0 3 ττbb −τ τb − τa 2 2 −τa −1 j=τa
where 1 = x (τb ) − x (τa ), and 1 = x (τb ) + x (τa ) − 2
τ b
x( j ) j=τa τb −τa +1 .
Lemma a matrix Q > 0 and any matrix X, the term (α) = α−dL −1 2 T [38]. For α−d (α)−1 T δ j Q δ( j + ( ) ) j=α−d (α) j=α−dU δ ( j )Qδ( j ), containing the time-varying state delay d(α) (satisfying dL ≤ d(α) ≤ dU ) can be estimated as follows:
T dU −d (α) F 0 1 T Q˜ X J1 J1 1 dLU + ς (α) (α) ≥ ς (α) d (α)−dL J2 J ∗ Q˜ 0 F dLU 2 2 d LU
Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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where
−1 −1 T T ˜ ˜ ˜ ˜ ˜ Q = diag{Q, 3Q}, F1 = Q − X Q X , F2 = Q − X Q X, J1 =
f2 − f3 , f2 + f3 − 2 f6 T ς (α) = x T (α), x T (α − dL ), x T (α − d (α) ), x T (α − dU ), v1T (α), v2T (α), v3T (α) , v1 (α) =
α 1 1 x ( j ), v2 (α) = hL + 1 j=α−d d (α) − hL + 1 L
v3 (α) =
α−h L
x ( j ),
j=α−d (α)
α−d (α) 1 f3 − f4 , dLU = dU − dL and x ( j ), J2 = f3 + f4 − 2 f7 hU − d (α) + 1 j=α−h U
fk = 0n×(k−1)n , In×n , 0n×(7−k )n , k = 1, 2, . . . , 7. Lemma 3 [39]. For some matrices ω1 = ω1T , ω2 and ω3 having appropriate dimensions, the inequality stated below holds: ω1 + ω2 (s )ω3 + ω3T T (s )ω2T < 0, for all (s), satisfying T (s)(s) ≤ I, subject to the existence of some scalar λ > 0 such that ω1 + λ−1 ω2 ω2T + λω3T ω3 < 0. 3. Robust SMC design This section describes the SMC design problem for uncertain 2-D discrete state delayed systems in two steps. At first, switching surface is designed using the LMI technique such that the system under investigation achieves desirable properties (stability and tracking) when restricted to the switching surface. Then, a robust SMC law has been derived which globally altogether forces the system state trajectories to already defined switching surface and maintains their presence there afterward. 3.1. Designing the switching surface We consider the switching function along the horizontal direction and vertical direction as follows:
ℵh1 (ι, κ ) h h s (ι, κ ) = E¯1 ℵ (ι, κ ) = E1 I , (8) ℵh2 (ι, κ ) s (ι, κ ) = E¯2 ℵv (ι, κ ) = E2 v
I
ℵv1 (ι, κ ) ℵv2 (ι, κ )
,
(9)
where, E¯1 ∈ Rq×nh , E¯2 ∈ Rq×nv , E1 ∈ Rq×(nh −q ) and E2 ∈ Rq×(nv −q ) are the real valued matrices to be determined. When the trajectories of the system are driven onto the switching surface then sh (ι, κ ) = 0 and sv (ι, κ ) = 0. Therefore, the following reduced-order Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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system is easy to obtain from system (7) upon the substitution of ℵh2 (ι, κ ) = −E1 ℵh1 (ι, κ ) and ℵv2 (ι, κ ) = −E2 ℵv1 (ι, κ ) in the first and third equation.
ℵh1 (ι + 1, κ ) M˜ 1a − M˜ 1b E1 M˜ 2a − M˜ 2b E2 ℵh1 (ι, κ ) = v ℵv1 (ι, κ + 1 ) M˜ 3a − M˜ 3b E1 M˜ 4a − M˜ 4b E2 ℵ1 (ι, κ ) M˜ d1a − M˜ d1bE1 M˜ d2a − M˜ d2b E2 ℵh1 (ι − dh (ι), κ ) . + ˜ Md3a − M˜ d3bE1 M˜ d4a − M˜ d4b E2 ℵv1 (ι, κ − dv (κ ) ) That is h h M˜ 1a M˜ 2a M˜ 1b M˜ 2b 0 E1 ℵ1 ( ι + 1, κ ) ℵ1 (ι, κ ) = − ˜ × 0 E2 ℵv1 (ι, κ + 1 ) ℵv1 (ι, κ ) M˜ 3a M˜ 4a M3b M˜ 4b h M˜ d1a M˜ d2a M˜ d1b M˜ d2b E1 0 ℵ1 (ι − dh (ι), κ ) − + ℵv1 (ι, κ − dv (κ ) ) M˜ d3a M˜ d4a M˜ d3b M˜ d4b 0 E2 ℵh ι, κ ℵh ι − d ι), κ ) ) h( 1( 1( ˜ ˜ + , = M˜ a − M˜ b E M − M E da db ℵv1 (ι, κ ) ℵv1 (ι, κ − dv (κ ) ) where
M˜ 1a M˜ 2a M˜ 1b M˜ 2b M˜ ˜ ˜ ˜ , Mb = ˜ , Mda = ˜ d1a Ma = ˜ ˜ ˜ M3a M4a M3b M4b Md3a ˜ ˜ M Md2b 0 E and E = 1 . M˜ db = ˜ d1b 0 E2 Md3b M˜ d4b
(10)
M˜ d2a , M˜ d4a
Remark 1. The switching functions stated in Eq. (8) and Eq. (9) actually represent a doubleplane function along the corresponding directions. In the following, we shall derive some conditions which will ensure the asymptotical stability of sh (ι, κ ) = 0 and sv (ι, κ ) = 0. 3.2. Analyzing the sliding mode dynamics This subsection is dedicated to the stability analysis of the reduced-order system (10). The purpose of Theorem 1 stated below is to present some sufficient conditions in the form of LMIs that will ensure the asymptotical stability of the system (10) and as a result, the parameters of the switching functions described in Eqs. (8) and (9) can be obtained. Theorem 1. Given the positive integers hL , vL , hU , vU and matrices H1 > 0, H2 > 0, H3 > 0; system (10) with time-varying delay defined in Eq. (3) is robustly h v asymptotically h sta℘ , S , Sv , ble if there exist positive definite symmetric matrices ℘ = diag , ℘ S = diag h v h v h v h v Z1 = diag Z1 , Z1 , Z2 = diag Z2 , Z2 , R = diag R , R , Q = diag Q , Q and any matrices E = diag{E1 , E2 }, X = diag X h , X v having suitable dimensions and a scalar λ > 0 such the below stated inequalities are satisfied:
11a 12 < 0, (11) ∗
22
11b ∗
12 < 0,
22
(12)
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with ⎡
⎤ ⎡ ⎤ ¯ − 5b (dL ) J1T X fs1T ¯ − 5b (dU ) J1T X fs1T
11a = ⎣ X T J1 −Q˜ 0 ⎦, 11b = ⎣ X T J1 −Q˜ 0 ⎦, ¯ ¯ fs1 0 fs1 0
℘
℘ ⎡ ¯ ⎡ ⎤ 0 λdL ϒaT
R dL fs2T dLU fs2T 0 fs3T ⎢ ¯ 0 λdLU ϒaT
Q 0 0 0 ⎦, 22 = ⎢
12 = ⎣ 0 T T ⎣ λdL ϒa λdLU ϒa −λI 0 0 λϒaT 0 0 0 0 T T T ¯
= − f1 ℘f1 + 2 + 3 + 4a − 4b , 2 = f1 S f1 − f3 S f3 ,
⎤ 0 0 ⎥ ⎥, 0 ⎦ −λI
3 = f1T (Z1 + Z2 ) f1 − f2T (Z1 + Z2 ) f2 − f4T (Z1 + Z2 ) f4 , ¯ ℘ = η1T ℘η1 − 2η1 ,
¯ R = η2T Rη2 − 2η2 ,
4a = dLU f1T S f1 ,
¯ Q = η3T Qη3 − 2η3 , fs3 = Ga (ϒa − ϒb E ) f1 + Gb (ϒa − ϒb E ) f3 ,
T J1 2Q˜ X J1 , fs1 = M a − M b E f1 + M da − M db E f3 , 5b (dL ) = J2 ∗ Q˜ J2 0q×q f2 − f3 f3 − f4 T1 , J2 = , ϒb = q×q , J1 = f2 + f3 − 2 f6 f3 + f4 − 2 f7 0 U1 T Q˜ X J1 J , d3 = d iag d3h Inh , d3v Inv , Q˜ = d iag{Q, 3Q}, 5b (dU ) = 1 J2 ∗ 2Q˜ J2 fs2 = M a − M b E − I f1 + M da − M db E f3 , fk = 0nh ×(k−1)nh , 0nv ×(k−1)nv , Inh ×nh , Inv ×nv , 0nh ×(7−k )nh , 0nv ×(7−k )nv , k = 1, 2, . . . , 7, T 0 f1 − f2 R f1 − f2 ,
4b = f1 + f2 − 2 f5 0 3d3 R f 1 + f 2 − 2 f 5 0(nh −q )×(nv −q ) TT , dL = diag hL Inh , vL Inv , ϒaT = (nv −q )2×(nh −q ) 0 U2T
M 1a M 2a dU = diag hU Inh , vU Inv , dLU = diag hLU Inh , vLU Inv , M a = , M 3a M 4a
M 1b M 2b M d1a M d2a M d1b M d2b Mb = , M da = and M db = M 3b M 4b M d3a M d4a M d3b M d4b Proof. Let us choose the following LKF for the system (10): V ℵh , ℵv = V h ℵh (ι, κ ) + V v (ℵv (ι, κ ) ),
(13)
with 3 3 V h ℵh (ι, κ ) = Vmh (ℵ(ι, κ ) ), V v (ℵv (ι, κ ) ) = Vmv (ℵ(ι, κ ) ) and m=1
V1h (ℵ(ι, κ ) ) = ℵhT (ι, κ )℘h ℵh (ι, κ ) +
m=1 ι−1
ℵhT (α, κ )S h ℵh (α, κ ),
α=ι−dh (ι)
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V2h (ℵ(ι, κ ) ) =
ι−1
ℵhT (α, κ )Z1h ℵh (α, κ ) +
α=ι−hL
V3h (ℵ(ι, κ ) ) =
ι−1
ℵhT (α, κ )Z2h ℵh (α, κ ),
α=ι−hU
−hL
ι−1
ℵhT (α, κ )S h ℵh (α, κ ) + hL
g=−hU +1 α=ι+g
+ hLU
ι
0
δ hT (α, κ )Rh δ h (α, κ )
g=−hL +1 α=ι+g
−hL
ι
δ hT (α, κ )Qh δ h (α, κ ),
g=−hU +1 α=ι+g
V1v (ℵ(ι, κ ) ) = ℵvT (ι, κ )℘v ℵv (ι, κ ) +
κ−1
ℵvT (ι, β )S v ℵv (ι, β ),
β=κ−dv (κ )
V2v (ℵ(ι, κ ) ) =
κ−1
ℵvT (ι, β )Z1v ℵv (ι, β ) +
β=κ−vL
V3v (ℵ(ι, κ ) ) =
−vL
κ−1
ℵvT (ι, β )Z2v ℵv (ι, β ),
β=κ−vU κ−1
g=−vU +1 β=κ+g
+ vLU
−vL
0
ℵvT (ι, β )S v ℵv (ι, β ) + vL
κ
δ vT (ι, β )Rv δ v (ι, β )
g=−vL +1 β=κ+g κ
δ vT (ι, β )Qv δ v (ι, β ),
g=−vU +1 β=κ+g
where ℘h > 0, Sh > 0, Z1h > 0, Z2h > 0, Rh > 0, Qh > 0, ℘v > 0, S v > 0, Z1v > 0, Z2v > 0, Rv > 0, Qv > 0, hLU = hU − hL , vLU = vU − vL , δ h (α, κ ) = ℵh (ι + 1, κ ) − ℵh (ι, κ ) and δ v (ι, β ) = ℵv (ι, κ + 1 ) − ℵv (ι, κ ). The finite difference along the horizontal and vertical directions can be evaluated as follows: V ℵh , ℵv = V h ℵh (ι + 1, κ ) − V h ℵh (ι, κ ) + (V v (ℵv (ι, κ + 1 ) ) − V v (ℵv (ι, κ ) ) ). (14) Computation of V ℵh , ℵv , which is defined in Eq. (14) yields: V ℵh , ℵv = ξ T (ι, κ )[ 1 + 2 + 3 + 4a + 5a ]ξ (ι, κ ) − hL
ι
δ hT (α, κ )Rh δ h (α, κ ) − hLU
α=ι−hL +1
− vL
κ β=κ−vL +1
ι−h L
δ hT (α, κ )Qh δ h (α, κ )
α=ι−hU +1
δ vT (ι, β )Rv δ v (ι, β ) − vLU
κ−v L
δ vT (ι, β )Qv δ v (ι, β ),
(15)
β=κ−vU +1
where
T T
1 = f1T M˜ a − M˜ b E ℘ M˜ a − M˜ b E f1 + f3T M˜ da − M˜ db E ℘ M˜ da − M˜ db E f3 T + 2 f1T M˜ a − M˜ b E ℘ M˜ da − M˜ db E f3 − f1T ℘f1 , fs = M˜ a − M˜ b E − I f1 2 + M˜ da − M˜ db E f3 , and 5a = fsT dL2 R + dLU Q fs . Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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Let us denote ℵh (ι, κ ) = ℵh , ℵh (ι − hL , κ ) = ℵhL , ℵhT (ι − dh (ι), κ ) = ℵhd (ι) , ℵh (ι − hU , κ ) = ℵUh , ℵv (ι, κ ) = ℵv , ℵv (ι, κ − vL ) = ℵvL , ℵv (ι, κ − dv (κ ) ) = ℵvd (κ ) , ℵv (ι, κ − vU ) = ℵUv , hT hT hT hT hT T ℵh (α, κ ) = ℵhα , ℵv (ι, β ) = ℵvβ , ξ h (ι, κ ) = ℵhT , ℵhT , L , ℵd (ι) , ℵU , η1 , η2 , η3 hT vT T hT vT T hT vT T hT T ℵ = ℵ ,ℵ , ℵL = ℵL , ℵL , ℵU = ℵU , ℵU , ℵd = ℵd (ι) , ℵvT d (κ ) , ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ι−h ι κ v h h L ℵ ℵ ℵ β α ⎦ α ⎦, η1v = ⎣ ⎦, η1h = ⎣ , η2h = ⎣ h + 1 d ι) − h + 1 v + 1 ( L h L L β=κ−vL α=ι−hL α=ι−dh (ι) ⎡ ⎤ ⎡ ⎤ ι−d ι) ( κ−v h L ℵvβ ℵhα ⎦, η3h = ⎣ ⎦, η2v = ⎣ d κ − v + 1 h − d ι) + 1 ) ( ( v L U h β=κ−dv (κ ) α=ι−hU ⎡ ⎤ κ−d κ ( ) v ℵvβ ⎦, η1 = η1hT , η1vT T , η2 = η2hT , η2vT T , η3 = η3hT , η3vT T , η3v = ⎣ v − d κ + 1 ) ( v β=κ−vU U T vT vT vT vT vT T ξ v (ι, κ ) = ℵvT , ℵvT and ξ (ι, κ ) = ℵT , ℵTL , ℵTd , ℵUT , η1T , η2T , η3T . L , ℵd (κ ) , ℵU , η1 , η2 , η3 Utilization of Lemma 1, to evaluate the Rh and Rv dependent summation terms in Eq. (15), one obtains: ι h h hL δ hT (α, κ )Rh δ h (α, κ ) ≥ ξ hT (ι, κ ) 4b ξ (ι, κ ), (16) α=ι−hL +1 κ
vL
v v δ vT (ι, β )Rv δ v (ι, β ) ≥ ξ vT (ι, κ ) 4b ξ (ι, κ ),
(17)
β=κ−vL +1
In the similar fashion, by the utilization of Lemma 2 to evaluate the Qh and Qv terms in Eq. (15), we have: ι−h L
hLU
δ hT (α, κ )Qh δ h (α, κ ) ≥ ξ hT (ι, κ ) 5hb (dh (ι) )ξ h (ι, κ ),
(18)
δ vT (ι, β )Qv δ v (ι, β ) ≥ ξ vT (ι, κ ) 5vb (dv (κ ) )ξ v (ι, κ ),
(19)
α=ι−hU +1 κ−v L
vLU
β=κ−vU +1
with h T h h h h hL + 1 Q˜ Xh 0 J1 d1 F1 J1 h + = h , h h h , d3 = h ˜ J2 0 d F J ∗ Q hL − 1 2 2 2 v T v v v v Q˜ Xv 0 f2h − f3h J1 d1 F1 J1 h v , 5b (dv (κ ) ) = v + , J1 = h J2 0 d2v F2v J2v f2 + f3h − 2 f6h ∗ Q˜ v
5hb (dh (ι) )
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f3h − f4h f2v − f3v f3v − f4v v v J = J = , , , 1 2 f2v + f3v − 2 f6v f3v + f4v − 2 f7v f3h + f4h − 2 f7h vU − dv (κ ) vL + 1 d1v = , Q˜ h = diag Qh , 3Qh , Q˜ v = diag{Qv , 3Qv }, d3v = , vLU vL − 1 −1 hU − dh (ι) dh (ι) − hL d1h = , d2h = , F1h = Q˜ h − X h Q˜ h X hT , hLU hLU −1 F2h = Q˜ h − X hT Q˜ h X h , fkh = 0nh ×(k−1)nh , Inh ×nh , 0nh ×(7−k )nh , fkv = 0nv ×(k−1)nv , Inv ×nv , 0nv ×(7−k )nv , k = 1, 2, . . . , 7, T v dv ( κ ) − vL 0 f1v − f2v R f1v − f2v v , d2v =
4b = v , f1 + f2v − 2 f5v 0 3d3v Rv f1v + f2v − 2 f5v vLU −1 −1 F1v = Q˜ v − X v Q˜ v X vT , F2v = Q˜ v − X vT Q˜ v Xv T h 0 fh − fh R f1h − f2h h . and 4b = h 1 h 2 h f1 + f2 − 2 f5 0 3d3h Rh f1h + f2h − 2 f5h J2h =
Thus, by utilization of inequalities established in Eqs. (16)–(19), the inequality (15) becomes: := ξ T (ι, κ ) ξ (ι, κ ),
(20)
with
= 1 + 2 + 3 + 4a + 5a − 4b − 5b (d ), d1 = d iag d1h Inh , d1v Inv , −1 d2 = d iag d2h Inh , d2v Inv , F1 = Q˜ − X Q˜ X T , F2 = Q − X T (Q )−1 X, and
T Q˜ J
5b (d ) = 1 J2 ∗
X dF + 1 1 0 Q˜
0 d2 F2
J1 . J2
By the Schur complement lemma and using the inequalities stated below H1T ℘ H1 ≥ 2H1 − ℘−1 , H2T RH2 ≥ 2H2 − R−1 , H3T QH3 ≥ 2H3 − Q−1 , for any matrices H1 > 0, H2 > 0 and H3 > 0, the relation in Eq. (20) is equivalent to: ¯ + 1 D (ι, κ )2 + T2 DT (ι, κ )T1 < 0, (21)
T T where 1 = 0 0 T ϒa dL T ϒa dLU T ϒa and 2 = fs3 0 0 0 0 . It is implied by Lemma 3 that Eq. (21) can be guaranteed if there exists a scalar λ > 0, such that: ¯ + λ1 T1 + λ−1 T2 2 < 0.
(22)
Thus, it follows from Schur compliment that Eqs. (11)–(12) are equivalent to Eq. (22). By Eq. (14), we may state that: h h V ℵ (ι + 1, κ ) − V h ℵh (ι, κ ) + (V v (ℵv (ι, κ + 1 ) ) − V v (ℵv (ι, κ ) ) ) < 0, Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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which is V h ℵh (ι + 1, κ ) + V v (ℵv (ι, κ + 1 ) ) < V h ℵh (ι, κ ) + V v (ℵv (ι, κ ) ).
13
(23)
By consideration of boundary conditions defined in Eq. (4), the following can be obtained as the result of the summation of terms to the left-hand side of Eq. (23) from Z to 0 with reference to ι and from 0 to Z with reference to κ, where Z is a non-negative integer: V h ℵh (Z + 1, 0 ) + V v (ℵv (Z, 1 ) ) + V h ℵh (Z, 1 ) + V v (ℵv (Z − 1, 2 ) ) + ... + V h ℵh (2, Z − 1 ) + V v (ℵv (1, Z ) ) + V h ℵh (1, Z ) + V v (ℵv (0, Z + 1 ) ) = V h ℵh (Z + 1, 0 ) + V h ℵh (Z, 1 ) + · · · + V h ℵh (2, Z − 1 ) + V h ℵh (1, Z ) + V v (ℵv (Z, 1 ) ) + V v (ℵv (Z − 1, 2 ) ) + · · · + V v (ℵv (1, Z ) ) + V v (ℵv (0, Z + 1 ) ) h h V ℵ (ι, κ ) + V v (ℵv (ι, κ ) ) . = (24) ι+κ=Z+1
Similarly, the summation of terms to the right hand side of Eq. (23) from Z to 0 with reference to ι and from 0 to Z with reference to κ results: V h ℵh (Z, 0 ) + V v (ℵv (Z, 0 ) ) + V h ℵh (Z − 1, 1 ) + V v (ℵv (Z − 1, 1 ) ) + ... + V h ℵh (1, Z − 1 ) + V v (ℵv (1, Z − 1 ) ) + V h ℵh (0, Z ) + V v (ℵv (0, Z ) ) = V h ℵh (Z, 0 ) + V h ℵh (1, Z − 1 ) + · · · + V h ℵh (Z − 1, 1 ) + V h ℵh (0, Z ) + V v (ℵv (Z, 0 ) ) + V v (ℵv (Z − 1, 1 ) ) + · · · + V v (ℵv (1, Z − 1 ) ) + V v (ℵv (0, Z ) ) h h V ℵ (ι, κ ) + V v (ℵv (ι, κ ) ) . = (25) ι+κ=Z
Thus, Eq. (23) together with Eq. (24) and Eq. (25) implies that: h h h h V ℵ (ι, κ ) + V v (ℵv (ι, κ ) ) < V ℵ (ι, κ ) + V v (ℵv (ι, κ ) ) , ι+κ=Z +1
ι+κ=Z
which simply implies that the finite difference along the horizontal and vertical directions goes on decreasing along the state trajectories of the system under consideration. Thus, by the (Definition 1), it can also be deduced that the system (10) is asymptotically stable. This completes the discussion of our proof. Remark 2. It is worth mentioning that the results derived in the Theorem 1 utilize the improved version of WSI, which is well known in the literature to overcome the conservativeness of the Jensen’s inequality [37,38]. Moreover, the consideration of practical factors such as delays, uncertainties, and nonlinearities make our established results more general and different from other similar works (see [30,34,35]). 3.3. Extension of reaching law and designing the controller After appropriately designing the sliding surface by utilizing the results established in Theorem 1, the next step is to design an SMC which is presented in this subsection. Recently, it was highlighted by the authors in [36] that the consideration of the reaching law as selected in [31] becomes impractical for the systems having some constraints imposed on the rate of change of the input and the output signals because of the direct dependence of the rate of change of switching function s(ι, κ) upon the current value of switching function itself. Thus, Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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motivated by the above discussion we aim to extend the reaching law [36], for 2-D systems, that is: h h s ( ι + 1, κ ) s (ι, κ ) 0 1 − m1 sh (ι, κ ) τ1 sgn sh (ι, κ ) = − . (26) s v ( ι + 1, κ ) sv (ι, κ ) 0 1 − m2 [sv (ι, κ )] τ2 sgn (sv (ι, κ ) ) having τ 1 > 0, τ 2 > 0 and functions mr : → [0, 1), r = 1, 2, are defined below: sh0 sv0 m1 sh (ι, κ ) = h , m sv (ι, κ ) = v , s (ι, κ ) + sh0 2 |s (ι, κ )| + sv0 where sh0 and sv0 are positive constants. Theorem 2. Assume that the inequalities (11)–(12) have h v h av set of feasible solutions h v ℘ , S , Z1 , Z1 , ℘ = diag , ℘ S = diag , S Z = diag Z2 = diag Z2h , Z2v , R= 1 h v h v d iag R , R , Q = d iag Q , Q and any matrices E = diag{E1 , E2 }, X = diag X h , X v , with the switching functions along the horizontal and vertical directions defined in Eqs. (8) and (9), respectively. Then, the horizontal and vertical trajectories of the states associated to the closed-loop system defined in Eq. (7) are pushed to the sliding surfaces sh (ι, κ ) = 0 and sv (ι, κ ) = 0, respectively with the below stated control laws: uh (ι, κ ) = − C˜1−1 E¯1 M 1 ℵh (ι, κ ) + E¯1 M 2 ℵv (ι, κ ) − 1 − m1 sh (ι, κ ) sh (ι, κ ) v h v h + τ1 sgn sh (ι, κ ) + 1hα + 2α + d1 α + d2α + nα h v +πd2 + πnh sgn sh (ι, κ ) . (27) + π1h + π2v + πd1 uv (ι, κ ) = − C˜2−1 E¯2 M 3 ℵh (ι, κ ) + E¯2 M 4 ℵv (ι, κ ) − 1 − m2 sv (ι, κ ) sv (ι, κ ) v h v v + τ2 sgn (sv (ι, κ ) ) + 3hα + 4α + d3 α + d4α + nα h v +πd4 + πnv sgn (sv (ι, κ ) ) . + π3h + π4v + πd3
(28)
where L1h + U1h 1h − L1h 2v + U2v v , π1h = U , 2α = L , 2 2 2 2v − L2v d1h + Ud1h d1h − Ld1h d2v + Ud2v h h v π2v = U , d1 = L , πd1 = U , d2 = L , 2 2 2 2 d2v − Ld2v nh + Unh nh − Lnh 3h + U3h v πd2 = U , nh = L , πnh = U , 3hα = L , 2 2 2 2 3h − L3h 4v + U4v 4v − L4v d3h + Ud3h v h π3h = U , 4α = L , π4v = U , d3 = L , 2 2 2 2 d3h − Ld3h d4v + Ud4v d4v − Ld4v vh + Uvh h v v πd3 = U , d4 = L , πd4 = U , nv = L , 2 2 2 2 vh − Lvh πnv = U , E¯1 = E1 I , E¯2 = E2 I . 2 There exist some known parameters L1h , U1h , L2v , U2v , Ld1h , Ud1h , Ld2v , Ud2v , nh nh L , U which represent the bounds of E¯1 M 1 ℵh (ι, κ ), E¯1 M 2 ℵv (ι, κ ), E¯1 M˜ d1 ℵh (ι − dh (ι),κ ), E¯1 M˜d2 ℵv (ι, κ − dv(κ) ) and C˜1N N ℵh (ι, κ ), ι, κ , respectively. Similarly, L3h , U3h , L4v , U4v , Ld3h , Ud3h , Ld4v , Ud4v , Lnv , Unv , are the bounds of 1hα =
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h v ¯ ˜ E¯2 M 3 ℵh (ι, κ ), E¯2 M 4 ℵv (ι, κ ), E¯2 M˜ d3 ℵ h(ι − dh (ι),κ ), Eh2Md4 ℵ (ι, hκ − dv (κ ) ) and v ˜ s sgn s2 · · · sgn sqh , sgn C2 N N (ℵ (ι, κ ), ι, κ ), respectively; sgn s (ι, κ ) = 1 sgn s2v · · · sgn sqv , sh = s1h s2h · · · sqh , sgn (sv (ι, κ ) ) = sgn s1v sv = s1v s2v · · · sqv .
Proof. Considering the inequality of reaching condition for the SMC as proposed in [31], we may write from Eq. (26) (considering the horizontal direction only): sh (ι, κ ) = sh (ι + 1, κ ) − sh (ι, κ ) ≤ −m1 sh (ι, κ ) sh (ι, κ ) − τ1 sgn sh (ι, κ ) , if sh (ι, κ ) > 0, sh (ι, κ ) = sh (ι + 1, κ ) − sh (ι, κ ) ≥ −m1 sh (ι, κ ) sh (ι, κ ) − τ1 sgn sh (ι, κ ) , if sh (ι, κ ) < 0, Similarly, for vertical direction, it follows from Eq. (26) that: sv (ι, κ ) = sv (ι, κ + 1 ) − sv (ι, κ ) ≤ −m2 sv (ι, κ ) sv (ι, κ ) − τ2 sgn (sv (ι, κ ) ), if sv (ι, κ ) > 0, sv (ι, κ ) = sv (ι, κ + 1 ) − sv (ι, κ ) ≥ −m2 sv (ι, κ ) sv (ι, κ ) − τ2 sgn (sv (ι, κ ) ), if sv (ι, κ ) < 0, It follows from Eq. (8) that sh (ι, κ ) = E¯1 ℵh (ι + 1, κ ) − E¯1 ℵh (ι, κ ).
(29)
Similarly, from Eq. (9) we may write: sv (ι, κ ) = E¯2 ℵv (ι, κ + 1 ) − E¯2 ℵv (ι, κ ).
(30)
By Eqs. (6)–(9), the following relations can be easily derived from Eqs. (29) and (30): sh (ι, κ ) = E¯1 M˜ 1 ℵh (ι, κ ) + E¯1 M˜ 2 ℵv (ι, κ ) + E¯1 M˜ d1 ℵh (ι − dh (ι), κ ) + E¯1 M˜ d2 ℵv (ι, κ − dv (κ ) ) + C˜1 uh (ι, κ ) + N N ℵh (ι, κ ), ι, κ − E¯1 ℵh (ι, κ ), (31) sv (ι, κ ) = E¯2 M˜ 3 ℵh (ι, κ ) + E¯2 M˜ 4 ℵv (ι, κ ) + E¯2 M˜ d3 ℵh (ι − dh (ι), κ ) + E¯2 M˜ d4 ℵv (ι, κ − dv (κ ) ) + C˜2 (uv (ι, κ ) + N N (ℵv (ι, κ ), ι, κ ) ) − E¯2 ℵv (ι, κ ). (32) As stated in Assumption 2, that the parameters M1 , M2 , Md1 , Md2 , M3 , M4 , Md3 , Md4 , N(ℵh (ι, κ), ι, κ) and N (ℵv (ι, κ ), ι, κ ) are bounded, and by utilization of improved version of Razumikhin theorem [40] for any solutions ℵh (ι − dh (ι), κ ) and ℵv (ι, κ − dv (κ ) ) of Eqs. (9)–(10), respectively; there exists constants o1 > 1 and o2 > 1, such that below stated inequalities are satisfied: ! h ! ! ! !ℵ (ι − dh (ι), κ )! < o1 !ℵh (ι, κ )!, ℵv (ι, κ − dv (κ ) ) < o2 ℵv (ι, κ ), for all ι and κ with time-varying delay satisfying (3). Without any loss in gener ality, it can be assumed that 1h (ι, κ ) = E¯1 M 1 ℵh (ι, κ ), 2v (ι, κ ) = E¯1 M 2 ℵv (ι, κ ), h v d1 d2 nh (ι, κ ) = (ι, κ ) = E¯1 M˜ d1 ℵh (ι − dh (ι), κ ), (ι, κ ) = E¯1 M˜ d2 ℵv (ι, κ − dv (κ ) ), h C˜1 N N ℵ (ι, κ ), ι, κ , 3h (ι, κ ) = E¯2 M 3 ℵh (ι, κ ), 4v (ι, κ ) = E¯2 M 4 ℵv (ι, κ ), h h v v ¯ ˜ ¯ ˜ d3 (ι, κ ) = E2 Md3 ℵ (ι − dh (ι), κ ), d4 (ι, κ ) = E2 Md4 ℵ (ι, κ − dv (κ ) ), nv (ι, κ ) = Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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C˜2 N N (ℵv (ι, κ ), ι, κ ) and they will also have upper and lower bounds. Let these bounds are: h v L1h ≤ 1h (ι, κ ) ≤ U1h , L2v ≤ 2v (ι, κ ) ≤ U2v , Ld1h ≤ d1 (ι, κ ) ≤ Ud1h , Ld2v ≤ d2 (ι, κ ) ≤ d2v nh h nh 3h h 3h 4v v 4v d3h h U , L ≤ n (ι, κ ) ≤ U , L ≤ 3 (ι, κ ) ≤ U , L ≤ 4 (ι, κ ) ≤ U , L ≤ d3 (ι, κ ) ≤ v Ud3h , Ld4v ≤ d4 (ι, κ ) ≤ Ud4v and Lvh ≤ nv (ι, κ ) ≤ Uvh . where the inequality h h h T 1h 1h T · · · q, 2,L · · · q,L L1h = 11,L ≤ 1h (ι, κ ) = 1h,1 2, 1 1 h 1h 1h T 2,U · · · q,U ≤ U1h = 11,U , 1h 1h implies that that i,L ≤ i,h 1 (ι, κ ) ≤ i,U , i = 1, 2, . . . , q; similar notations will be adopted for v h v h h v 2 (ι, κ ), d1 (ι, κ ), d2 (ι, κ ), n (ι, κ ), 3h (ι, κ ), 4v (ι, κ ), d3 (ι, κ ), d4 (ι, κ ) and nv (ι, κ ). Thus, in view of Eq. (27), the equality (31) can be written as: h v sh (ι, κ ) = 1h (ι, κ ) + 2v (ι, κ ) + d1 (ι, κ ) + d2 (ι, κ ) + nh (ι, κ ) h h h v h v h − m1 s (ι, κ ) s (ι, κ ) − τ1 sgn s (ι, κ ) − 1hα + 2α + d1 α + d2α + nα h v +πd2 + πnh sgn sh (ι, κ ) . (33) − π1h + π2v + πd1
We have
1h (ι, κ ) ≤ 1hα + π1h sgn sh (ι, κ ) , sh (ι, κ ) > 0; 1h (ι, κ ) ≥ 1hα + π1h sgn sh (ι, κ ) , sh (ι, κ ) < 0; v + π2v sgn sh (ι, κ ) , sh (ι, κ ) > 0; 2v (ι, κ ) ≤ 2α v + π2v sgn sh (ι, κ ) , sh (ι, κ ) < 0; 2v (ι, κ ) ≥ 2α h h h h h d1 (ι, κ ) ≤ d1 α + πd1 sgn s (ι, κ ) , s (ι, κ ) > 0; h h h h h d1 (ι, κ ) ≥ d1 α + πd1 sgn s (ι, κ ) , s (ι, κ ) < 0; v v v + πd2 sgn sh (ι, κ ) , sh (ι, κ ) > 0; d2 (ι, κ ) ≤ d2α v v v + πd2 sgn sh (ι, κ ) , sh (ι, κ ) < 0; d2 (ι, κ ) ≥ d2α h + πnh sgn sh (ι, κ ) , sh (ι, κ ) > 0; nh (ι, κ ) ≤ nα h + πnh sgn sh (ι, κ ) , sh (ι, κ ) < 0. nh (ι, κ ) ≥ nα
Similarly, keeping in view Eq. (28), the Eq. (32) can be written as: h v sv (ι, κ ) = 3h (ι, κ ) + 4v (ι, κ ) + d3 (ι, κ ) + d4 (ι, κ ) + nv (ι, κ ) v v v h v v − m2 s (ι, κ ) s (ι, κ ) − τ2 sgn (sv (ι, κ ) ) − 3hα + 4α + d3 α + d4α + nα h v +πd4 + πnv sgn (sv (ι, κ ) ). (34) − π3h + π4v + πd3
We have 3h (ι, κ ) ≤ 3hα + π3h sgn (sv (ι, κ ) ), sv (ι, κ ) > 0; 3h (ι, κ ) ≥ 3hα + π3h sgn (sv (ι, κ ) ), sv (ι, κ ) < 0; v 4v (ι, κ ) ≤ 4α + π4v sgn (sv (ι, κ ) ), sv (ι, κ ) > 0; v 4v (ι, κ ) ≥ 4α + π4v sgn (sv (ι, κ ) ), sv (ι, κ ) < 0; h h h v v d3 (ι, κ ) ≤ d3 α + πd3 sgn (s (ι, κ ) ), s (ι, κ ) > 0;
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h h h v v d3 (ι, κ ) ≥ d3 α + πd3 sgn (s (ι, κ ) ), s (ι, κ ) < 0; v v v d4 + πd4 sgn (sv (ι, κ ) ), sv (ι, κ ) > 0; (ι, κ ) ≤ d4α v v v d4 (ι, κ ) ≥ d4α + πd4 sgn (sv (ι, κ ) ), sv (ι, κ ) < 0; v nv (ι, κ ) ≤ nα + πnv sgn (sv (ι, κ ) ), sv (ι, κ ) > 0; v nv (ι, κ ) ≥ nα + πnv sgn (sv (ι, κ ) ), sv (ι, κ ) < 0.
In what follows, for simplicity, we move forward with our discussion by considering the horizontal state only, the same analogy can be followed for the vertical state. It is quite straightforward to notice from Eq. (33) that sh (ι, κ) and sh (ι, κ) have opposite signs regardh v less of the values of uncertainties 1h (ι, κ ), 2v (ι, κ ), d1 (ι, κ ), d2 (ι, κ ), and the disturbance h n (ι, κ ). Moreover, the reaching condition defined in Eq. (26) is also satisfied by the closedloop system (7). Furthermore, to satisfy the crossing and re-crossing, the following condition must be met for the quasi-sliding mode: sgn sh (ι + 2, κ ) = −sgn sh (ι + 1, κ ) = sgn sh (ι, κ ) . It follows from Eq. (33) that: sh (ι + 2, κ ) = 1 − m1 sh (ι + 1, κ ) sh (ι + 1, κ ) − τ1 sgn sh (ι + 1, κ ) h v + 1h (ι + 1, κ ) + 2v (ι + 1, κ ) + d1 (ι + 1, κ ) + d2 (ι + 1, κ ) + nh (ι + 1, κ ) h v h v h − 1α + 2α + d1 α + d2α + nα h h v +πd2 + πnh sgn sh (ι + 1, κ ) , − π1 + π2v + πd1
which results in the following after some computations: sh (ι + 2, κ ) = sgn sh (ι, κ ) 1 − m1 sh (ι + 1, κ ) × 1 − m1 sh (ι, κ ) sh (ι, κ ) h v + τ1 m1 sh (ι + 1, κ ) +m1 sh (ι + 1, κ ) π1h + π2v + πd1 + πd2 + πnh h v + 1 − m1 sh (ι + 1, κ ) 1h (ι, κ ) + 2v (ι, κ ) + d1 (ι + 1, κ ) + d2 ( ι + 1, κ ) h h v h v h + n (ι + 1, κ ) − 1α + 2α + d1α + d2α + nα h + 1h (ι + 1, κ ) + 2v (ι + 1, κ ) + d1 ( ι + 1, κ ) v h h v h v h + d2 (ι + 1, κ )+n (ι + 1, κ ) − 1α + 2α + d1 α + d2α + nα . h v h Due to the fact that the parameters 1h (ι, κ ), 2v (ι, κ ), d1 (ι, κ ), d2 (ι, κ ) and nh (ι, κ ) h h h v v v h bounded, v the valuesv of 1 v(ι, κ ) − 1αh ≤ π1 , h 2(ι, κ ) h− 2α ≤ π2 , d1 (ι, κ ) − d1α ≤ h n (ι, κ )− nα ≤ πn . Thus, πd1 , d2 (ι, κ ) − d2α ≤ πd2 , and to meet the condition of quasi-sliding mode, which is sgn sh (ι + 2, κ ) = sgn sh (ι, κ ) for arbitrarily small magnitude of sh (ι, κ), the following should be satisfied: " # h τ 1 1 h v −1 . π1 + π2v + πd1 + πd2 + πnh < (35) 2 1 − m1 s h ( ι + 1, κ ) The right side of Eq. (35) tends to zero as m1 sh (ι + 1, κ ) approaches zero. Moreover, the inequality (35) must mode band. When the system is in the be satisfied only in the sliding sliding mode sgn sh (ι + 1, κ ) = −sgn sh (ι, κ ) , sh (ι, κ) = 0, the bound of the quasi-sliding mode band (QSMB) (stated in (Definition 2)) can be obtained from Eq. (33) as follows [41]: h v ε1 < τ1 + 2 π1h + π2v + πd1 + πd2 + πnh ,
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that means m1 s h ( ι + 1, κ ) >
sh0 h . v h + πv + πh + s τ1 + 2 π1 + π2 + πd1 h0 n d2
By Eqs. (35)–(36), we get: s 2 h0 h v h v 2 π1h + π2v + πd1 + πd2 + πnh < τ1 − π1h − π2v − πd1 − πd2 − πnh . 2 As τ 1 > 0, so for Eq. (37) to be satisfied the below stated conditions must hold: h v sh0 > 2 π1h + π2v + πd1 + πd2 + πnh . 2 h v 4 π1h + π2v + πd1 + πd2 + πnh τ1 > . h − 2π v − 2π h sh0 − 2π1h − 2π2v − 2πd1 n d2
(36)
(37)
(38)
(39)
Thus, by using Eqs. (38)–(39), the parameters sh0 and τ 1 of the controller derived in Eq. (27) can be specified to ensure the quasi-sliding mode’s existence. In the similar fashion, conditions for finding sv0 and τ 2 can be found. This completes our proof. that for improved convergence and minimal chattering Remark 3. It is important to highlight the τ 1 should be small and m1 sh (ι + 1, κ ) should be large such that (35) is satisfied. Remark 4. In Eq. (35), assume that m1 sh (ι + 1, κ ) = m1 (constant instead of switching m1 function dependent) then the value of τ 1 should be selected accordingly such that 2(τ11−m > 1) h v h v h π1 + π2 + πd1 + πd2 + πn is satisfied. 4. Simulation results In practical situations, the Darboux equation stated below, is well know to represent the dynamics of many processes such as water stream heating, sorption process, gas absorption, and air drying [20,21]: ∂ 2 s (ς , τ ) ∂s (ς , τ ) ∂s (ς , τ ) = m1 + m2 + m0 s ( ς , τ ) ∂ς∂τ ∂τ ∂ς + md s (ς , τ − d (τ ) ) + c[u (ς , τ ) + N n (s (ς , τ ), ς , τ )],
(40)
where s(ς , τ ) is the function that is not known and defined at ς (space) that belongs to [0, ς f ] and τ (time) that belongs to [0, ∞); m1 , m2 , m0 , md , c and N are some real coefficients; d(τ ) means the time delay. The input function is represented by u(ς , τ ); n(s(ς , τ ), ς , τ ) being some nonlinear function representing the disturbance and unmodelled dynamics in the system model and we define: ∂s (ς , τ ) − m 2 s ( ς , τ ). ∂τ Eq. (41) helps us to transform the system (40) as follows:
∂r (ς,τ ) m1 m1 m2 + m0 r ( ς , τ ) m ∂ς + d s (ς , τ − d (τ ) ) ∂s (ς,τ ) = 1 m2 s (ς , τ ) 0 ∂τ c + [u (ς , τ ) +N n (s (ς , τ ), ς , τ )]. 0 r (ς , τ ) =
(41)
(42)
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As discussed in [20], by taking h (ι, κ ) = r (ι, κ ) = r (ις , κτ ) and v (ι, κ ) = s (ι, κ ) = s (ις , κτ ) the following model of the form of system (1) is easy to obtain from Eq. (42):
h ( ι + 1, κ ) 1 + m1 ς (m1 m2 + m0 )ς h (ι, κ ) = v (ι, κ + 1 ) τ 1 + m2 τ v (ι, κ ) md ς cς v (ι, κ − d (κ ) ) + + [u (ι, κ ) + N n (s (ι, κ ), ι, κ )]. 0 0
(43)
Assume m1 = −1, m2 = −2, m0 = −1, τ = 0.6, c = 1.42, N = 1 and thus system (1) with the following dynamics can be easily obtained from Eq. (43): ⎡
M
Gb
M
Md
⎤ ⎡ ⎤ ⎡ 0.3 0.3 0.7 0.7 0 0 ⎢0.3 0.3 ⎥ ⎢1⎥ ⎢0 0. 7 0. 7 ⎥ ⎢ ⎥ ⎢ =⎢ ⎣0.6 0.6 −0.2 −0.2⎦, C = ⎣0⎦, Md = ⎣0 0.6 0.6 −0.2 −0.2 1 0 ⎡ ⎤ 0.1 0 ⎢0.3 0.1⎥ 0 0.2 ⎥, Ga = 0.35 0.28 , =⎢ ⎣0 0.1⎦ 0.2 0.3 0.4 0.4 0.2 0.1 0 0.35 0.29 0.1 , = 0.3 0.1 0.11 0.3 ⎡ ⎤ 0.01 sin θ 0.02 cos θ 0.01 sin θ 0.03 sin θ ⎢0.03 cos θ 0.04 sin θ 0.05 cos θ 0.04 cos θ ⎥ ⎥ =⎢ ⎣0.04 cos θ 0.03 cos θ 0.03 cos θ 0.01 sin θ ⎦, 0.02 sin θ 0.05 cos θ 0.02 sin θ 0.04 sin θ ⎡ ⎤ 0.01 sin θ 0.03 sin θ 0.01 sin θ 0.02 cos θ ⎢0.05 cos θ 0.04 cos θ 0.03 cos θ 0.04 sin θ ⎥ ⎥ =⎢ ⎣0.03 cos θ 0.01 sin θ 0.04 cos θ 0.03 cos θ ⎦, 0.02 sin θ 0.04 sin θ 0.02 sin θ 0.05 cos θ
0 0 0 0
0.14 0.14 0 0
⎤ 0.14 0.14⎥ ⎥, 0 ⎦ 0
N (h (ι, κ ), ι, κ ) = 0.4 sin (h1 (ι, κ ) ) and N (v (ι, κ ), ι, κ ) = 0.5 sin (v1 (ι, κ ) ), with θ = (0.01π (ι + κ ) ). It can be observed from above specified dynamics the system has so called ‘regular’ form. Therefore, it is not necessary to take into account the model transformation, that is ℵh1 (ι, κ ) = h1 (ι, κ ), ℵh2 (ι, κ ) = h2 (ι, κ ), ℵv1 (ι, κ ) = v1 (ι, κ ) and ℵv2 (ι, κ ) = v2 (ι, κ ). Take the parameters hL = 2, vL = 3, hU = 3, vU = 4, H1 = diag{0.26, 0.26}, H2 = diag{90, 90} and H3 = diag{99, 99}. Then, by solving of inequalities (11)–(12) stated in Theorem 1, we may obtain λ= 0.3665 and the following switching function parameters: E1 = 0.8947 and E2 = 1.0553. Using Eqs. (8)–(9), we can obtain following: Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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Table 1 Maximum upper bound of delay (hU = vU ) for fixed values of hL = vL = 2 and the parameters specified in Example 1, along with the feasibility of inequalities (11)–(22). Upper bound Feasibility
5
10
15
20
25
30
31.8
32 ×
sh (ι, κ ) = 0.8947h1 (ι, κ ) + h2 (ι, κ ), sv (ι, κ ) = 1.0553v1 (ι, κ ) + v2 (ι, κ ). On the other hand, let ! ! ! ! L1h = −!E¯1 1 !Ga1 h (ι, κ ), U1h = !E¯1 1 !Ga1 h (ι, κ ), ! ! ! ! L2v = −!E¯1 1 !Ga2 v (ι, κ ), U2v = !E¯1 1 !Ga2 v (ι, κ ), ! ! ! ! ! ! ! ! Ld1h = −!E¯1 M˜ d1 h (ι, κ )!, Ud1h = !E¯1 M˜ d1 h (ι, κ )!, ! ! ! ! ! ! ! ! Ld2v = −!E¯1 M˜ d2 v (ι, κ )!, Ud2v = !E¯1 M˜ d2 v (ι, κ )!, ! ! ! ! ! ! ! ! Lnh = −!C˜1 N N (h (ι, κ ), ι, κ )!, Unh = !C˜1 N N (h (ι, κ ), ι, κ )!, ! ! ! ! L3h = −!E¯2 2 !Ga1 h (ι, κ ), U3h = !E¯2 2 !Ga1 h (ι, κ ), ! ! ! ! L4v = −!E¯2 2 !Ga2 v (ι, κ ), U4v = !E¯2 2 !Ga2 v (ι, κ ), ! ! ! ! ! ! ! ! Ld3h = −!E¯2 M˜ d3 h (ι, κ )!, Ud3h = !E¯2 M˜ d3 h (ι, κ )!, ! ! ! ! ! ! ! ! Ld4v = −!E¯2 M˜ d4 v (ι, κ )!, Ud4v = !E¯2 M˜ d4 v (ι, κ )!, ! ! ! ! ! ! ! ! Lnv = −!C˜2 N N (v (ι, κ ), ι, κ )!, Unv = !C˜2 N N (v (ι, κ ), ι, κ )! then by utilizing the results given in Theorem 2, the desired controllers specified in Eqs. (27) and (28), can be obtained. Moreover, keeping in mind (38) and (39), we choose τ1 = 5, τ2 = 6, sh0 = 15, sv0 = 20 and the following boundary conditions: $ $ 2 0 ≤ κ ≤ 10 3 0 ≤ κ ≤ 12 h1 (0, κ ) = , h2 (0, κ ) = , 0 κ > 10 0 κ > 12 $ $ 4 0 ≤ ι ≤ 11 8.5 0 ≤ ι ≤ 15 v1 (ι, 0 ) = , v2 (ι, 0 ) = , 0 ι > 11 0 ι > 15 The trajectories of the resultant the closed-loop system are depicted in Figs. 1–4 while the control laws (27) and (28) are plotted in Figs. 5, 6. The convergence of the horizontal and vertical states can be evidently observed in Figs. 1–4, which in turn verifies the importance of established results. Moreover, Table 1 shows the maximum upper bound of delay (hU = vU ) for fixed values of hL = vL = 2 and same parameters as specified above, along with the feasibility of inequality (11)–(22).
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Fig. 1. The evolution of horizontal state h1 (ι, κ).
Fig. 2. The evolution of horizontal state h2 (ι, κ). Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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Fig. 3. The evolution of vertical state v1 (ι, κ ).
Fig. 4. The evolution of vertical state v2 (ι, κ ).
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Fig. 5. The control law uh (ι, κ).
Fig. 6. The control law uv (ι, κ ). Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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5. Conclusions The paper has solved the SMC design problem for 2-D delayed systems with uncertainties and nonlinearities. The model transformation approach has been utilized together with WSI instead of conventional Jensen’s inequality (usually degenerates conservative results) in establishing the LMI based solvability conditions for the existence of the switching function, which ensures the asymptotical stability of the reduced-order system. Then, we extended the recently proposed reaching law concept to 2-D systems that ensure the faster convergence better robustness against uncertainties. The results have been finally validated with suitable example. An interesting future work could be to extend the current results for the case when system is required to be exponentially stable [29]. Acknowledgment This work was supported by National Natural Science Foundation of China under Grant No. 61703137 and the Fundamentals Research Funds for the Central Universities under Grant No. 2019B14814. References [1] M.S. Koo, H.L. Choi, J.T. Lim, Stabilisation of feedback linearisable uncertain nonlinear systems with time delay using scaled sliding surface, IET Control Theory Appl. 2 (11) (2008) 974–979. [2] J.P. Richard, Comments on time-delay systems: an overview of some recent advances and open problems, Automatica 39 (10) (2003) 1667–1694. [3] P. Zhang, J. Hu, H. Zhang, D. Chen, Robust H∞ control for delayed systems with randomly varying nonlinearities under uncertain occurrence probability via sliding mode method, Syst. Sci. Control Eng. 6 (1) (2018) 160–170. [4] M. Yan, Y. Shi, Robust discrete-time sliding mode control for uncertain systems with time-varying state delay, IET Control Theory Appl. 2 (8) (2008) 662–674. [5] Y. Wang, H. Shen, H.R. Karimi, D. Duan, Dissipativity-based fuzzy integral sliding mode control of continuous-time t-s fuzzy systems, IEEE Trans. Fuzzy Syst. 26 (3) (2018) 1164–1176. [6] Y. Wang, Y. Gao, H.R. Karimi, H. Shen, Z. Fang, Sliding mode control of fuzzy singularly perturbed systems with application to electric circuit, IEEE Trans. Syst. Man Cybern. 48 (10) (2018) 1667–1675. [7] H. Hou, Q. Zhang, Novel sliding mode control for multi-input multi-output discrete-time system with disturbance, Int. J. Robust Nonlinear Control 28 (8) (2018) 3033–3055. [8] B.L. Zhang, Q.L. Han, X.M. Zhang, X. Yu, Sliding mode control with mixed current and delayed states for offshore steel jacket platforms, IEEE Trans. Control Syst. Technol. 22 (5) (2014) 1769–1783. [9] X.M. Zhang, Q.L. Han, Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems, Automatica 57 (2015) 199–202. [10] S. Xiao, L. Xu, H.B. Zeng, K.L. Teo, Improved stability criteria for discrete-time delay systems via novel summation inequalities, Int. J. Control Autom. Syst. 16 (4) (2018) 1592–1602. [11] B.L. Zhang, Q.L. Han, X.M. Zhang, Recent advances in vibration control of offshore platforms, Nonlinear Dyn. 89 (2) (2017) 755–771. [12] H.H. Choi, Variable structure control of dynamical systems with mismatched norm-bounded uncertainties: an LMI approach, Int. J. Control 74 (13) (2001) 1324–1334. [13] A. Bartoszewicz, P. Lesniewski, New reaching law for quasi-sliding mode control of discrete-time systems, in: IEEE 52nd Annual Conference on Decision and Control (CDC), Florence, Italy, 2013, pp. 2881–2887. [14] J. Fei, C. Lu, Adaptive sliding mode control of dynamic systems using double loop recurrent neural network structure, IEEE Trans. Neural Netw. Learn. Syst. 29 (4) (2018) 1275–1286. [15] H. Li, J. Wang, H. Du, H.R. Karimi, Adaptive sliding mode control for Takagi–Sugeno fuzzy systems and its applications, IEEE Trans. Fuzzy Syst. 26 (2) (2018) 531–542. Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041
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[16] B. Jiang, H.R. Karimi, Y. Kao, C. Gao, Takagi–Sugeno modelbased sliding mode observer design for finite-time synthesis of semi-Markovian jump systems, IEEE Trans. Syst. Man Cybern. Syst. (2018), doi:10.1109/TSMC. 2018.2846656. [17] W. Perruquetti, J.P. Barbot, Sliding Mode Control in Engineering, Marcel Dekker, New York, 2002. [18] D. Efimov, A. Polyakov, L. Fridman, W. Perruquetti, J.P. Richard, Delayed sliding mode control, Automatica 64 (2016) 37–43. [19] K. Abidi, J.X. Xu, Y. Xinghuo, On the discrete-time integral sliding-mode control, IEEE Trans. Autom. Control 52 (4) (2007) 709–715. [20] T. Kaczorek, Two-Dimensional Linear Systems, Springer-Verlag, Berlin, Germany, 1985. [21] W. Lu, A. Antoniou, Two-dimensional Digital Filters, Marcel Dekker, New York, 1992. [22] Z. Fei, S. Shi, C. Zhao, L. Wu, Asynchronous control for 2-d switched systems with mode-dependent average dwell time, Automatica 79 (2017) 198–206. [23] Z. Duan, Z. Xiang, Finite frequency H∞ control of 2-d continuous systems in Roesser model, Multidim. Syst. Signal Process. 28 (4) (2017) 1481–1497. [24] Z. Duan, I. Ghous, B. Wang, J. Shen, Necessary and sufficient stability criterion and stabilization for positive 2-d continuous-time systems with multiple delays, Asian J. Control 21 (4) (2019) 1–12. [25] I. Ghous, Z. Xiang, Robust state feedback H∞ control for uncertain 2-d continuous state delayed systems in the Roesser model, Multidimens. Syst. Signal Process. 27 (2) (2016) 297–319. [26] I. Ghous, Z. Xiang, H.R. Karimi, H∞ control of 2-d continuous Markovian jump delayed systems with partially unknown transition probabilities, Inform. Sci. 382 (2017) 274–291. [27] D. Zhang, P. Shi, L. Yu, Containment control of linear multiagent systems with aperiodic sampling and measurement size reduction, IEEE Trans. Neural Netw. Learn. Syst. 29 (10) (2018) 5020–5029. [28] D. Zhang, L. Liu, G. Feng, Consensus of heterogeneous linear multiagent systems subject to aperiodic sampleddata and dos attack, IEEE Trans. Cybern. (2018) 1–11, doi:10.1109/TCYB.2018.2806387. [29] C.J. Fallaha, M. Saad, H.Y. Kanaan, K. Al-Haddad, Sliding-mode robot control with exponential reaching law, IEEE Trans. Ind. Electron. 58 (2) (2011) 600–610. [30] L. Wu, H. Gao, Sliding mode control of two-dimensional systems in Roesser model, IET Control Theory Appl. 2 (4) (2008) 352–364. [31] W. Gao, H. Wang, A. Homaifa, Discrete-time variable structure control systems, IEEE Trans. Ind. Electron. 42 (2) (1995) 117–122. [32] Y. Zheng, Y.W. Jing, G.H. Yang, Design of approximation law for discrete-time variable structure control systems, in: 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 2006, pp. 4969–4973. 13–15 December [33] H.H. Choi, A new method for variable structure control system design: a linear matrix inequality approach, Automatica 33 (11) (1997) 2089–2092. [34] H. Adloo, P. Karimaghaee, A.S. Sarvestani, An extension of sliding mode control design for the 2-d systems in Roesser model, in: 48th IEEE Conference on Decision and Control (CDC) held Jointly with 28th Chinese Control Conference (CCC), Shanghai, China, 15–18, 2009, pp. 7753–7758. [35] A.R. Argha, L. Li, S.W. Su, A new approach to applying discrete sliding mode control to 2-d systems, in: 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013, pp. 10–13. 3584–3589 [36] A. Bartoszewicz, P. Lesniewski, New switching and nonswitching type reaching laws for SMC of discrete time systems, IEEE Trans. Control Syst. Technol. 24 (2) (2016) 670–677. [37] A. Seuret, F. Gouaisbaut, E. Fridman, Stability of discrete-time systems with time-varying delays via a novel summation inequality, IEEE Trans. Autom. Control 60 (10) (2015) 2740–2745. [38] C.K. Zhang, Y. He, L. Jiang, M. Wu, An improved summation inequality to discrete-time systems with time– varying delay, Automatica 74 (2016) 10–15. [39] L. Xie, Output feedback H∞ control of systems with parameter uncertainty, Int. J. Control 63 (4) (1996) 741–750. [40] B. Xu, Y. Liu, An improved razumikhin-type theorem and its applications, IEEE Trans. Autom. Control 39 (4) (1994) 839–841. [41] A. Bartoszewicz, Remarks on discrete-time variable structure control systems, IEEE Trans. Ind. Electron. 43 (1) (1996) 235–238.
Please cite this article as: I. Ghous, Z. Duan and J. Akhtar et al., Robust stabilization of uncertain 2-D discretetime delayed systems using sliding mode control, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.02.041