Decentralized fuzzy DSC for uncertain large-scale interconnected switched systems in non-strict feedback form with input saturation

Abstract

This paper demonstrates a fuzzy decentralized dynamic surface control (DSC) scheme for switched large-scale interconnected nonlinear systems under arbitrary switching, which contains non-strict feedback form and unknown input saturation uncertainties. An auxiliary design system is established to handled input saturation. Uncertainties of non-strict feedback form are learned by fuzzy logic systems (FLSs) approximators, DSC method is designed to conquer “explosion of complexity” inherented by repeated differential of virtute controller in backstepping approach. Ii is shown that based on common Lyapunov function (CLF) design and analysis scheme, all the closed-loop systems signals are uniformly ultimately bounded (UUB), simulation results are provided to demonstrate the effectiveness of this proposed strategy.

Keywords

DSC scheme large-scale switched nonlinear systems(LSSNs)input saturation non-strict feedback (NSF) form

1 Introduction

The approximation-based adaptive control algorithms of Non-Strict Feedback (NSF) Form nonlinear systems have developed as a significantly challenging problem in practical engineering [1 –12]. NSF contained all the states functions has a more general form, traditional strict-feedback and pure-feedback forms as its special forms, how to design and compensate this performance has attracted a lot of attention [2 –13]. Variable separation technique [2 –4] as a typical scheme to deal with the NSF structure has been researched a lot these years, such as adaptive fuzzy tracking control scheme and multi input multi output (MIMO) nonlinear systems, even stochastic nonlinear NSF systems were analyzed in [2 –4], subsequently, adaptive neural full-state constraints control with unmodeled dynamics and input time-delay systems were considered in [5, 6]. Neural network (NN) reinforcement learning (RL) control has extended to discrete-time systems [7]. fuzzy logic systems (FLSs) approximation-based observer control, asymptotically non-singular convergent tracking error control, and finite-time large-scale nonlinear systems have extended to NSF structure [8 –12]. But these results does not consider the more complex practical switched nonlinear systems.

As a classical hybrid system, switched nonlinear system arise in many practical fields, including autonomous systems, network protocol routings and so on, where they would been modeled with switched topology systems [13 –22]. Subsequently, finite-time fault-tolerant control technique has been developed in [13, 14] addressed switched stochastic nonlinear systems, event-triggered control (ETC) was redescribed for practical time-delay systems with finite-time output feedback stabilization [15] and linear programming [16]. [17] generated the consensus interconnected switched multi-agent systems, output feedback ETC scheme is put forward for switched sensor failures nonlinear systems [18], adaptive NN approximation-based periodic ETC output-feedback for switched nonlinear systems was illustrated in [19], decentralised control for LSSs with backlash like hysteresis and output constraints [20]. Switched-observer-based ETC funnel [21] and dynamic NN technique [22] are considered for switched nonlinear systems respectively. And these results mainly focused on the switched nonlinear systems in the form of strict-feedback or pure-feedback, not consider the more complex NFS form. In [23 –26] for switched nonlinear systems in NFS, several adaptive fuzzy control approaches were proposed.

Nevertheless, these above backstepping-based techniques results computational expansion problem due to the repeated differentiation of virtual controllers. D. Swaroop proposed a DSC technique by introducing a first-order low-pass filter [27], and J. Farrell explored the command filtered backstepping [28] scheme, which would effectively alleviate the computational explosion. Subsequently, dynamic surface control (DSC) scheme was extended to systems with full state constraints and unmodeled dynamics [29] and systems in NFS form [30 –34]. [35] exploited DSC to MIMO output-feedback switched nonlinear systems with unknown gain signs, [36] considered command filtered approach for stochastic time-delayed NFS with unknown input saturation and [37] discussed DSC with switched large-scale systems, However, these results based-on DSC or command filtered approaches did not consider Large-scale Switched Nonlinear systems(LSSNs) or did not consider NFS form with the uncertainty of input saturation, in addition, these results with input saturation uncertainty [38, 39] did not consider LSSNs or DSC scheme. And for the NFS [3 –7] based-on variable separation technique which required unknown functions satisfying the property of monotonically increasing.

Consider the above-mentioned factors, a fuzzy decentralized tracking DSC architectural is investigated for the LSSNs in NFS form with uncertainty input saturation. During the controller and adaptive laws design process, NFS form, uncertain input saturation, general interconnections, and arbitrary switchings are discussed simultaneously. The main contributions can be summarized as follows:

1) This paper analyzes a more general systems which contain interconnected switched systems in NSF form and uncertainty of input saturation. Obviously, the systems investigated in [10 , 37] would be regard as particular cases of this system.

2) This proposed technique does not require the unknown function to be restrictively monotonically increasing [3–7 , 40] and existing bounding functions as their bounds, when dealing with NSF structured systems with non-affine states, which is more general for application in practical engineering.

3) By employing the fuzzy approximation-based DSC technique, the virtual controllers design calculation process is simplified and the computational expansion problem can be successfully resolved compared with [10 , 39].

2 Problem statements and preliminaries

2.1 Preliminaries formulation and system descriptions

Consider a class of LSSNs with input saturation in NSF form, which composed of N interconnected relevant subsystems:

${\begin{matrix} \begin{matrix} {\dot{x}}_{ij} & = g_{ij}^{σ (t)} ({\bar{x}}_{ij}) x_{i (j + 1)} + f_{ij}^{σ (t)} (X) + h_{ij}^{σ (t)} (\bar{y}) \\ + d_{ij}^{σ (t)} (t); j = 1, 2, \dots n_{i} - 1 \\ {\dot{x}}_{{in}_{i}} & = g_{{in}_{i}}^{σ (t)} ({\bar{x}}_{{in}_{i}}) u_{i}^{σ (t)} (v_{i}^{σ (t)}) (t) + f_{{in}_{i}}^{σ (t)} (X) \\ + h_{{in}_{i}}^{σ (t)} (\bar{y}) + d_{{in}_{i}}^{σ (t)} (t) \\ y_{i} & = x_{i 1}, i = 1, 2, \dots, N \end{matrix} \end{matrix}$ (1) where ${\bar{x}}_{ij} = [x_{i 1}, x_{i 2}, \dots, x_{ij}]^{T} \in R^{j}, i = 1, 2, \dots, N,$ j = 1, 2, …, n_i, x_i = [x_i1, x_i2, …, x_{in
_i}] ^T ∈ R^{n
_i}, y_i ∈ R are system state vector and system output of the ith subsystem respectively. $f_{ij}^{σ (t)} (\cdot),$ $g_{ij}^{σ (t)} (\cdot)$ are uncertain smooth nonlinear functions with $f_{ij}^{σ (t)} (0) = 0, g_{ij}^{σ (t)} (0) = 0$ , and $d_{ij}^{σ (t)} (t) (j = 1, 2, \dots, n_{i})$ is external uncertainties. ${\bar{x}}_{i} = [x_{1}^{T}, x_{2}^{T}, \dots, x_{i}^{T}]^{T}$ , $X = [x_{1}^{T}, x_{2}^{T}, \dots, x_{N}^{T}]^{T}$ , disturbances ( $| d_{ij}^{σ (t)} (t) | \leq {\bar{d}}_{ij}^{σ (t)}$ , and ${\bar{d}}_{ij}^{σ (t)}$ is an unknown constant). $h_{ij}^{σ (t)} (\bar{y}), \bar{y} = [y_{1}, y_{2}, \dots, y_{N}]^{T}$ are interconnected functions among the ith subsystems and the other systems with $h_{ij}^{σ (t)} (\bar{y}) (0) = 0$ . The function σ (t) : [0, ∞) → M = {1, 2, …, M} denotes a switched signal, which is a piecewise continuous function, In addition, σ (t) = k denotes that the kth subsystem is active, for conveniences, the corresponding smooth functions $f_{ij}^{σ (t)} (\cdot) = f_{ij (k)} (\cdot),$ $g_{ij}^{σ (t)} = g_{ij (k)} (\cdot),$ $h_{ij}^{σ (t)} = h_{ij (k)} (\bar{y}) (\cdot), d_{ij}^{σ (t)} = d_{ij (k)} (\cdot)$ .

The nonlinearity input saturation $u_{i}^{σ (t)} (v_{i}^{σ (t)})$ would be described as follows [41]. $u_{i}^{σ (t)} (v_{i}^{σ (t)}) = sat (v_{i}^{k}) = {\begin{matrix} sign (v_{i}^{k}) μ_{N}, | v_{i}^{k} | \geq μ_{N} \\ v_{i}^{k}, | v_{i}^{k} | < μ_{N}, σ (t) = k \end{matrix}$ (2)

Here, u_N is a known parameter which denotes the boundary of $u_{i}^{σ (t)} (v_{i}^{σ (t)} (t))$ . From [40] and [41], we known that when $| v_{i}^{σ (t)} (t) | = u_{N}$ , there is a sharp corner. In order to avoid this situation, the following smooth function will approximate the saturation function as below: $c (v_{i}^{σ (t)}) = u_{N} tanh (\frac{v_{i}^{σ (t)}}{u_{N}}) = \frac{u_{N} (e^{\frac{v_{i}^{σ (t)}}{u_{N}}} - e^{\frac{- v_{i}^{σ (t)}}{u_{N}}})}{e^{v_{i}^{σ (t)} / u_{N}} + e^{- v_{i}^{σ (t)} / u_{N}}}$ Then $sat (v_{i}^{σ (t)})$ is defined as $sat (v_{i}^{σ (t)}) = c (v_{i}^{σ (t)}) + ρ (v_{i}^{σ (t)})$ , satisfying $ρ (v_{i}^{σ}) = | sat (v_{i}^{σ} (t)) - c (v_{i}^{σ}) | \leq u_{N} (1 - \tanh (1)) = E_{0}$ , which denotes the upper bound of $ρ (v_{i}^{σ (t)})$ , combined with the mean-value theorem, the function $c (v_{i}^{σ})$ can be expressed as $c (v_{i}^{σ (t)}) = c (v_{0}) + c_{μ} (v_{i}^{σ} - v_{0})$ , where μ is a positive constant with 0 < μ < 1 and c_μ is defined by

$c (v_{i}^{σ (t)}) = \frac{\partial c (v_{i}^{σ (t)})}{\partial v_{i}^{σ (t)}} |_{v_{μ}} = \frac{4}{e^{\frac{v_{i}^{σ (t)}}{u_{N}}} + e^{\frac{- v_{i}^{σ (t)}}{u_{N}}}} |_{v_{μ}}$ where $v_{μ} = μ v_{i}^{σ (t)} + (1 - μ) v_{0}$ , choose v₀ = 0, then the above equation can be rewritten as $c (v_{i}^{σ (t)}) = c_{μ} v$ . Therefore, the nonlinearity input saturation function would be concluded as: $u_{i}^{σ (t)} (v_{i}^{σ (t)} (t)) = c_{μ} v_{i}^{σ (t)} + ρ (v_{i}^{σ (t)})$ (3)

Remark 1. Similar to the related results [13 –17], we assume that the system state were to be continuous solution but not jumped at the switching signal, c (v_i) is non-decreasing and 0 < c₀ ≤ c_μ ≤ c₁, c_μ is bounded, c₀ and c₁ are constants.

Asusmption 1 [10] . Assume that $\forall {\bar{x}}_{ij (k)} \in R, i = 1, 2, \dots, N, j = 1, 2, \dots, n_{i}, k \in {1, 2, \dots, M}$ , without loss of generality, there exist positive parameters g_i0(k), g_id(k), smooth functions $g_{ij (k)} ({\bar{x}}_{i}),$ satisfying $0 < g_{i 0 (k)} \leq | g_{ij (k)} ({\bar{x}}_{ij}) | < \infty$ , $0 < g_{id (k)} \leq | {\dot{g}}_{ij} ({\bar{x}}_{ij}) | < \infty$ .

Asusmption 2 [10]. For the interconnected functions $h_{ij}^{σ (t)} (\bar{y})$ , there exist smooth nonnegative functions $q_{ij}^{σ (t)} (\cdot)$ and constants $η_{ij}^{σ (t)}$ , satisfying $| h_{ij}^{σ (t)} (\bar{y}) | \leq \sum_{l = 1}^{N} η_{ijl}^{σ (t)} q_{ijl}^{σ (t)} (| y_{il} |),$ where i = 1, 2, …, N, j = 1, 2, …, n_i, $η_{ij}^{σ (t)}$ denote the interaction strength, with $q_{ij}^{σ (t)} (0) = 0$ .

Remark 2. Based on Assumption 1, there exist smooth function ${\tilde{q}}_{ij}^{σ (t)} (| y_{i} |)$ with initial condition $q_{ij}^{σ (t)} (0) = 0$ , satisfying $q_{ij}^{σ (t)} (| y_{i} |) = | y_{i} | {\tilde{q}}_{ij}^{σ (t)} (| y_{i} |)$ . The LSSNs in the form NSF structure with input saturation here are more practical and typical models in the industrial process. The condition of the switched nonlinear systems in [23 –25] were the special case of (1) in which N = 1, and in [17 , 37], the form is discussed in large-scale situation with strict feedback ones, and which was the case of the non-strict feedback form. [23 –26] were in NSF structures, but did not adopt DSC technique. [31 , 34] were considered the DSC for NSF systems, but not switched ones.

Fig. 1

Block diagram of the presented DSC decentralized control design.

The control objective is to design adaptive robust DSC decentralized fuzzy controller $u_{i}^{σ (t)} (v_{i}^{σ (t)} (t))$ for the LSSNs in NSF form (1), such that 1) the system output y_i can track the desired trajectory reference signal y_ir; 2) all the signals in the closed-loop system are uniformly ultimately bounded (UUB); where reference signals y_ir are continuous and available, and its kth order derivative $y_{ir}^{(k)}, k = 1, 2, \dots, n_{i}$ are assumed continuous.

2.2 Fuzzy logic systems (FLSs)

FLSs is employed to compensate the unknown smooth functions. The lth fuzzy if-then rule defined as

$R^{l} : If x_{1} (t) is A_{1}^{l} and if x_{2} (t) is A_{2}^{l} and \dots and$ if x_m (t) is $A_{n}^{l} . Then y is B^{l}$ .i = 1, 2, … m, l = 1, 2 ⋯ , p.

Where $A_{i}^{l}$ and B^l are fuzzy membership functions, FLSs is designed based on singleton fuzzification, production inferencing, center-average-weighted defuzzification [1, 2], and FLSs is formulated as $y (x) = \frac{\sum_{l = 1}^{p} {\bar{y}}^{l} (\prod_{i = 1}^{n} μ_{A_{i}^{l}} (x_{i}))}{\sum_{l = 1}^{p} \prod_{i = 1}^{n} μ_{A_{i}^{l}} (x_{i})} = θ^{T} ω (x)$ (4) where ${\bar{y}}_{l}$ satisfies $μ_{B^{l}} ({\bar{y}}_{l}) = 1$ ; $θ^{T} = [{\bar{y}}_{1}, {\bar{y}}_{2}, \dots, {\bar{y}}_{p}]^{T}$ is adjustable parameter vector. ξ (x) = [ξ¹, ξ², …, ξ^p] ^T, where fuzzy basis function ξ^l is defined by

$ξ^{l} (x) = \frac{\prod_{i = 1}^{n} μ_{A_{i}^{l}}}{\sum_{l = 1}^{p} \prod_{i = 1}^{n} μ_{A_{i}^{l} (x_{i})}}$

Then, the FLSs (2) can be rewritten as follows: $y (x) = θ^{T} ξ (x)$

Lemma 1 [1]: Let f (x) be a continuous function defined on a compact set Ω. Then, for any constant ɛ > 0, there exists a FLSs (3) such that $sup_{x \in ξ} | (f (x) - θ^{T} ξ (x)) | \leq ɛ$

Remark 3. Different from the previous NSF structure results [3–7 , 41], the presented technique in this paper will eliminate restrictive conditions |f_ij(k) (x) | ≤ φ_j (||x||) , j = 1, 2, …, n_i of unknown function in the former NSF form, which limited the applicability in practical engineering. This paper will exploited the approximation of FLSs and decentralized DSC output tracking algorithm to these nonlinear LSSNs, witch will avoid the computational expansion problem. And many systems can be transformed through modeling or based on coordinate transformation and state feedback transforming into LSSNs in NSF form, such as: intelligent traffic management systems, motion robots, aircraft and automotive engine control, computing mechanical and electrical systems and chemical systems.

3 Adaptive robust fuzzy DSC control design and performance analysis

3.1 Adaptive fuzzy DSC designing performance

The adaptive fuzzy DSC tracking controller design is based on the following transformations coordinates for i = 1, 2, … N, j = 2, 3, … n_i, k ∈ {1, 2, …, M}: ${\begin{matrix} z_{i 1 (k)} = S_{i 1 (k)} = x_{i 1} - y_{ir} = y_{i} - y_{ir} \\ S_{ij (k)} = x_{ij} - {\bar{α}}_{ij (k)}, \\ P_{ij (k)} = {\bar{α}}_{ij (k)} - α_{i (j - 1) (k)} \end{matrix}$ (5) where z_i1(k) is the tracking error of the ith subsystem and S_i1(k) denotes the first surface error, S_ij(k) denote the surface error, P_ij(k) is the first-order filter error. The concrete design procedure is as follows:

Step1. Consider the first subsystem in plant (1) for ${\dot{x}}_{i 1 (k)} = g_{i 1}^{σ (t)} (x_{i 1}) x_{i 2} + f_{i 1}^{σ (t)} (X) + h_{i 1}^{σ (t)} (\bar{y}) + d_{i 1}^{σ (t)}$ , together with the first error surface and tracking error S_i1(k) = z_i1(k) = x_i1 - y_ir, and along the time derivative of S_i1(k) is $\begin{matrix} {\dot{z}}_{i 1 (k)} = {\dot{S}}_{i 1 (k)} = & g_{i 1}^{σ (t)} (x_{i 1}) x_{i 2} + f_{i 1}^{σ (t)} (X) - {\dot{y}}_{ir} \\ + h_{i 1}^{σ (t)} (\bar{y}) + d_{i 1}^{σ (t)} \end{matrix}$ (6) Here we can view x_i2 as a virtual control input. Since f_i1(k) (X) and g_i1(k) (x_i1) are unknown smooth functions, FLSa $θ_{i 1 (k)}^{* T} ξ_{i 1 (k)}$ would be used to approximate it. Let define $H_{i 1 (k)} (Z_{i 1 (k)}) = g_{i 1}^{- 1} (x_{i 1}) [f_{i 1 (k)} (X) - {\dot{y}}_{ir}]$ where $Z_{i 1 (k)} (x_{i 1}, X, y_{ir (k)}, {\dot{y}}_{ir (k)})$ , then H_i1(k) (Z_i1(k)) will be approximated by FLSs $H_{i 1 (k)} (Z_{i 1 (k)}) = θ_{i 1 (k)}^{* T} ξ_{i 1 (k)} (Z_{i 1 (k)}) + ɛ_{i 1 (k)} (Z_{i 1 (k)})$ (7)

where $θ_{i 1 (k)}^{* T} = arg min sup_{X \in U} {H_{i 1 (k)} (Z_{i 1 (k)}) - θ_{i 1 (k)}^{* T} ξ_{i 1 (k)} (Z_{i 1 (k)})}$ . Substituting (13) into (12) results in $\begin{matrix} {\dot{S}}_{i 1 (k)} = & g_{i 1 (k)} (x_{i 1}) [x_{i 2} + θ_{i 1 (k)}^{* T} ξ_{i 1 (k)} (Z_{i 1 (k)}) \\ + ɛ_{ij (k)} (Z_{i 1 (k)})] + h_{i 1 (k)} (\bar{y}) + d_{i 1 (k)} \end{matrix}$ (8) Define the first virtual controller α_i1(k) and adaptation laws θ_i1(k), δ_i1(k) are considered as follows: $\begin{matrix} α_{i 1 (k)} = & - \frac{{\hat{ɛ}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(1)}} S_{i 1 (k)} - \frac{S_{i 1 (k)}}{g_{i 10}} \sum_{l = 1}^{N} | y_{l} | η_{i 1 l} {\tilde{q}}_{i 1 l} (| y_{i} |) \\ - \frac{{\hat{θ}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(2)}} S_{i 1 (k)} | | ξ_{i 1 (k)} (Z_{i 1 (k)}) | |^{2} - c_{i 1} S_{i 1 (k)} \end{matrix}$ (9) ${\dot{\hat{θ}}}_{i 1 (k)} = {\hat{θ}}_{i 1 (k)} - \frac{Γ_{i 1 (k)}^{(1)}}{4 τ_{i 1 (k)}^{(1)}} S_{i 1 (k)}^{2} | | ξ_{i 1 (k)} (Z_{i 1 (k)}) | |^{2}$ (10) ${\dot{\hat{ɛ}}}_{i 1 (k)} = {\hat{ɛ}}_{i 1 (k)} - \frac{Γ_{i 1 (k)}^{(2)}}{4 τ_{i 1 (k)}^{(2)}} S_{i 1 (k)}^{2}$ (11) where $c_{i 1 (k)} = \frac{1}{2} - \frac{g_{i 1 d}}{g_{i 10}^{2}} - \frac{2 d_{i 1 (k)}^{*}}{g_{i 10}},$ parameters c_i1(k) > 0, $τ_{i 1 (k)}^{(1)} > 0$ , $τ_{i 1 (k)}^{(2)} > 0$ , $Γ_{i 1 (k)}^{(1)} > 0$ , $Γ_{i 1 (k)}^{(2)} > 0$ are positive design constants to be designed. ${\hat{θ}}_{i 1 (k)}, {\hat{ɛ}}_{i 1 (k)}$ are adaptive adjusted parameters to be designed later.

Introduce a new variable ${\bar{α}}_{i 2 (k)}$ and let α_i1(k) pass through a first-order filter ${\bar{α}}_{i 2 (k)}$ with a constant T_i2(k) to obtain ${\bar{α}}_{i 2 (k)}$ . $\begin{matrix} T_{i 2 (k)} {\dot{\bar{α}}}_{i 2 (k)} + {\bar{α}}_{i 2 (k)} = α_{i 1 (k)}, {\bar{α}}_{i 2 (k)} (0) = α_{i 1 (k)} (0) \end{matrix}$ (12) From (5) and (12), the first filter error $P_{i 2 (k)} = {\bar{α}}_{i 2 (k)} - α_{i 1 (k)}$ , it follows that $\begin{matrix} {\dot{P}}_{i 2 (k)} & = {\bar{α}}_{i 2 (k)} - {\dot{α}}_{i 1 (k)} \\ = - \frac{P_{i 2 (k)}}{T_{i 2 (k)}} - B_{i 2 (k)} [{\bar{S}}_{i 2 (k)}, y_{ir}, {\dot{y}}_{ir}, α_{i 1 (k)}, {\dot{α}}_{i 1 (k)}] \end{matrix}$ (13) where $B_{i 2 (k)} [{\bar{S}}_{i 2 (k)}, y_{ir}, {\dot{y}}_{ir}, α_{i 1 (k)}, {\dot{α}}_{i 1 (k)}]$

$= - c_{i 1} {\dot{S}}_{i 1 (k)} - \frac{{\dot{\hat{ɛ}}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(1)}} S_{i 1 (k)} - \frac{{\hat{ɛ}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(1)}} {\dot{S}}_{i 1 (k)}$

$- \frac{1}{g_{i 10}} \sum_{l = 1}^{N} | {\dot{y}}_{l} | η_{i 1 l} {\tilde{q}}_{i 1 l} (| y_{i} |)$

$- \frac{1}{g_{i 10}} \sum_{l = 1}^{N} | y_{l} | η_{i 1 l} {\dot{\tilde{q}}}_{i 1 l} (| y_{i} |)$

$- \frac{{\dot{\hat{θ}}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(2)}} S_{i 1 (k)} | | ξ_{i 1 (k)} (Z_{i 1 (k)}) | |^{2}$

$- \frac{{\hat{θ}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(2)}} {\dot{S}}_{i 1 (k)} | | ξ_{i 1 (k)} (Z_{i 1 (k)}) | |^{2}$

$- \frac{2 {\hat{θ}}_{i 1 (k)}}{4 τ_{i 1 (k)}^{(2)}} S_{i 1 (k)} | | {\dot{ξ}}_{i 1 (k)} (Z_{i 1 (k)}) | | .$

Step j: (2 ≤ j ≤ n_i - 1) Consider the jth part in plant (1) for the ith subsystem, ${\dot{x}}_{ij} = g_{ij}^{σ (t)} (x_{ij}) x_{i (j + 1)} + f_{ij}^{σ (t)} (X) + h_{ij}^{σ (t)} (\bar{y}) + d_{ij}^{σ (t)}$ , together with the jth error surface $S_{ij (k)} = x_{ij} - {\bar{α}}_{ij}$ and along the time derivative of S_ij(k) is $\begin{matrix} {\dot{S}}_{ij} = & g_{ij}^{σ (t)} ({\bar{x}}_{ij}) x_{i (j + 1)} + f_{ij}^{σ (t)} (X) - {\dot{\bar{α}}}_{ij (k)} \\ + h_{ij}^{σ (t)} (\bar{y}) + d_{ij}^{σ (t)} \end{matrix}$ (14) Since f_ij(k) (X) and $g_{ij (k)} ({\bar{x}}_{ij})$ are unknown smooth functions, FLSs $θ_{ij (k)}^{* T} ξ_{ij (k)}$ would be used to approximate it. Let define $H_{ij (k)} (Z_{ij (k)}) = g_{ij (k)}^{- 1} ({\bar{x}}_{ij}) [f_{ij (k)} (X) - {\dot{\bar{α}}}_{ij (k)}]$ where $Z_{ij (k)} ({\bar{x}}_{ij}, X, {\bar{α}}_{ij (k)}, {\dot{\bar{α}}}_{ij (k)})$ , then H_ij(k) (Z_ij(k)) will be approximated by FLSs. $\begin{matrix} H_{ij (k)} (Z_{ij (k)}) & = θ_{ij (k)}^{* T} ξ_{ij (k)} (Z_{ij (k)}) \\ + ɛ_{ij (k)} (Z_{ij (k)}) j = 2, 3, \dots, n_{i} \end{matrix}$ (15) where $θ_{ij (k)}^{* T} = arg min sup_{X \in U} {H_{ij (k)} (Z_{ij (k)}) - θ_{ij (k)}^{* T} ξ_{ij (k)} (Z_{ij (k)})}$ Substituting (15) into (14) results in $\begin{matrix} {\dot{S}}_{ij (k)} = & g_{ij (k)} ({\bar{x}}_{ij}) [x_{i (j + 1) (k)} + θ_{ij (k)}^{* T} ξ_{ij (k)} (Z_{ij (k)}) \\ + ɛ_{ij (k)} (Z_{ij (k)})] + h_{ij (k)} (\bar{y}) + d_{ij (k)} \end{matrix}$ (16) The virtual controller α_ij(k) and adaptation laws θ_i(k), ɛ_ij(k) are considered as follows: $\begin{matrix} α_{ij (k)} & = - c_{ij} S_{ij (k)} - \frac{S_{ij (k)}}{g_{ij 0}} \sum_{l = 1}^{N} | y_{l} | η_{ijl} {\tilde{q}}_{ijl} (| y_{i} |) \\ - \frac{{\hat{θ}}_{ij (k)}}{4 τ_{ij (k)}^{(2)}} S_{ij (k)} | | ξ_{ij (k)} (Z_{ij (k)}) | |^{2} \\ - S_{i (j - 1) (k)} - \frac{{\hat{ɛ}}_{ij (k)}}{4 τ_{ij (k)}^{(1)}} S_{ij (k)} \end{matrix}$ (17) ${\dot{\hat{θ}}}_{ij (k)} = {\hat{θ}}_{ij (k)} - \frac{Γ_{ij (k)}^{(1)}}{4 τ_{ij (k)}^{(1)}} S_{ij (k)}^{2} | | ξ_{ij (k)} (Z_{ij (k)}) | |^{2}$ (18) ${\dot{\hat{ɛ}}}_{ij (k)} = {\hat{ɛ}}_{ij (k)} - \frac{Γ_{ij (k)}^{(2)}}{4 τ_{ij (k)}^{(2)}} S_{ij (k)}^{2}$ (19) where $c_{ij (k)} = \frac{1}{2} - \frac{g_{ijd}}{g_{ij 0}^{2}} - \frac{2 d_{ij (k)}^{*}}{g_{ij 0}},$ parameters c_ij(k) > 0, $τ_{ij (k)}^{(1)} > 0$ , $τ_{ij (k)}^{(2)} > 0$ , $Γ_{ij (k)}^{(1)} > 0$ , $Γ_{ij (k)}^{(2)} > 0$ are positive design constants to be designed. ${\hat{θ}}_{ij (k)}, {\hat{ɛ}}_{ij (k)}$ are adaptive adjusted parameters to be designed later. Introduce a new variable ${\bar{α}}_{ij (k)}$ and let α_i(j-1)(k) pass through a first-order filter ${\bar{α}}_{ij (k)}$ with a constant T_ij(k) to obtain ${\bar{α}}_{ij (k)}$ . $\begin{matrix} T_{ij (k)} {\dot{\bar{α}}}_{ij (k)} + {\bar{α}}_{ij (k)} = α_{i (j - 1) (k)}, \\ {\bar{α}}_{ij (k)} (0) = α_{i (j - 1) (k)} (0) \end{matrix}$ (20)

From (5) and (20), the jth filter error $P_{ij (k)} = {\bar{α}}_{ij (k)} - α_{i (j - 1) (k)}$ , it follows that $\begin{matrix} {\dot{P}}_{ij (k)} & = {\bar{α}}_{ij (k)} - {\dot{α}}_{i (j - 1) (k)} \\ = - \frac{P_{ij (k)}}{T_{ij (k)}} - B_{ij (k)} [{\bar{S}}_{ij (k)}, α_{i (j - 1) (k)}, {\dot{α}}_{i (j - 1) (k)}] \end{matrix}$ (21) where $B_{ij (k)} [{\bar{S}}_{ij (k)}, α_{i (j - 1) (k)}, {\dot{α}}_{i (j - 1) (k)}]$

$= - c_{ij} {\dot{S}}_{ij (k)} - \frac{{\dot{\hat{ɛ}}}_{ij (k)}}{4 τ_{ij (k)}^{(1)}} S_{ij (k)} - \frac{{\hat{ɛ}}_{ij (k)}}{4 τ_{ij (k)}^{(1)}} {\dot{S}}_{ij (k)}$

$- \frac{1}{g_{ij 0}} \sum_{l = 1}^{N} | {\dot{y}}_{l} | η_{ijl} {\tilde{q}}_{ijl} (| y_{i} |)$

$- \frac{1}{g_{ij 0}} \sum_{l = 1}^{N} | y_{l} | η_{ijl} {\dot{\tilde{q}}}_{ijl} (| y_{i} |)$

$- \frac{{\dot{\hat{θ}}}_{ij (k)}}{4 τ_{ij (k)}^{(2)}} S_{ij (k)} | | ξ_{ij (k)} (Z_{ij (k)}) | |^{2}$

$- \frac{{\hat{θ}}_{ij (k)}}{4 τ_{ij (k)}^{(2)}} {\dot{S}}_{ij (k)} | | ξ_{ij (k)} (Z_{ij (k)}) | |^{2}$

$- \frac{2 {\hat{θ}}_{ij (k)}}{4 τ_{ij (k)}^{(2)}} S_{ij (k)} | | {\dot{ξ}}_{ij (k)} (Z_{ij (k)}) | | .$

and $B_{ij (k)} [{\bar{S}}_{ij (k)}, α_{i (j - 1) (k)}, {\dot{α}}_{i (j - 1) (k)}]$ is continuous function.

Step n_i: Similar to the derivations in Step j for the ith subsystems, and using mathematical induction.

Consider the n_ith part in plant (1) of the ith subsystem, and $S_{{in}_{i} (k)} = x_{{in}_{i}} - {\bar{α}}_{{in}_{i}}$ along the time derivative of S_{in_i(k)} is $\begin{matrix} {\dot{S}}_{{in}_{i}} = & g_{{in}_{i}}^{σ (t)} ({\bar{x}}_{{in}_{i}}) [c_{μ} v_{i}^{σ (t)} + ρ (v_{i}^{σ (t)})] \\ + f_{{in}_{i}}^{σ (t)} (X) - {\dot{\bar{α}}}_{{in}_{i}} + h_{{in}_{i}}^{σ (t)} (\bar{y}) + d_{{in}_{i}}^{σ (t)} \end{matrix}$ (22) When σ (t) = k, since f_{in_i(k)} (X) and $g_{{in}_{i} (k)} ({\bar{x}}_{{in}_{i}})$ are unknown smooth functions, fuzzy logic systems $θ_{{in}_{i} (k)}^{* T}$ ξ_{in_i(k)} would be used to approximate it like (15). Substituting (15) into (14) results in $\begin{matrix} {\dot{S}}_{{in}_{i} (k)} = & g_{{in}_{i} (k)} ({\bar{x}}_{{in}_{i}}) [c_{μ} v_{i}^{σ (t)} + ρ (v_{i}^{σ (t)}) \\ + θ_{{in}_{i} (k)}^{* T} ξ_{{in}_{i} (k)} (Z_{{in}_{i} (k)}) + ɛ_{{in}_{i} (k)} (Z_{{in}_{i} (k)})] \\ + h_{{in}_{i} (k)} (\bar{y}) + d_{{in}_{i} (k)} \end{matrix}$ (23) The actual controller $v_{i}^{σ (t)}$ and adaptation laws θ_{in_i(k)}, ɛ_{in_i(k)}, (i = 1, 2, …, N) are considered as follows: $\begin{matrix} v_{i}^{σ (t)} & = - c_{{in}_{i}} S_{{in}_{i} (k)} - \frac{{\hat{ɛ}}_{{in}_{i} (k)}}{4 τ_{{in}_{i} (k)}^{(1)}} S_{{in}_{i} (k)} \\ - \frac{S_{{in}_{i} (k)}}{g_{{in}_{i} 0}} \sum_{l = 1}^{N} | y_{l} | η_{{in}_{i} l} {\tilde{q}}_{{in}_{i} l} (| y_{i} |) \\ - \frac{{\hat{θ}}_{{in}_{i} (k)}}{4 τ_{{in}_{i} (k)}^{(2)}} S_{{in}_{i} (k)} | | ξ_{{in}_{i} (k)} (Z_{{in}_{i} (k)}) | |^{2} \end{matrix}$ (24) ${\dot{\hat{θ}}}_{{in}_{i} (k)} = {\hat{θ}}_{{in}_{i} (k)} - \frac{Γ_{{in}_{i} (k)}^{(1)}}{4 τ_{{in}_{i} (k)}^{(1)}} S_{{in}_{i} (k)}^{2} | | ξ_{{in}_{i} (k)} (Z_{{in}_{i} (k)}) | |^{2}$ (25) ${\dot{\hat{ɛ}}}_{{in}_{i} (k)} = {\hat{ɛ}}_{{in}_{i} (k)} - \frac{Γ_{{in}_{i} (k)}^{(2)}}{4 τ_{{in}_{i} (k)}^{(2)}} S_{{in}_{i} (k)}^{2}$ (26)

Introduce a new variable ${\bar{α}}_{{in}_{i} (k)}$ and let α_{i(n_i-1)(k)} pass through a first-order filter ${\bar{α}}_{{in}_{i} (k)}$ with a constant T_{in_i(k)} to obtain ${\bar{α}}_{{in}_{i} (k)}$ . $\begin{matrix} T_{{in}_{i} (k)} {\dot{\bar{α}}}_{{in}_{i} (k)} + {\bar{α}}_{{in}_{i} (k)} = α_{i (n_{i} - 1) (k)} \\ {\bar{α}}_{{in}_{i} (k)} (0) = α_{i (n_{i} - 1) (k)} (0) \end{matrix}$ (27)

From (5) and (27), the n_ith filter error $P_{{in}_{i} (k)} = {\bar{α}}_{{in}_{i} (k)} - α_{i (n_{i} - 1) (k)}$ , it follows that $\begin{matrix} {\dot{P}}_{{in}_{i} (k)} = {\bar{α}}_{{in}_{i} (k)} - {\dot{α}}_{i (n_{i} - 1) (k)} \\ = - \frac{P_{{in}_{i} (k)}}{T_{{in}_{i} (k)}} \\ - B_{{in}_{i} (k)} [{\bar{S}}_{{in}_{i} (k)}, α_{i (n_{i} - 1) (k)}, {\dot{α}}_{i (n_{i} - 1) (k)}] \end{matrix}$ (28) where $B_{{in}_{i} (k)} [{\bar{S}}_{{in}_{i} (k)}, α_{i (n_{i} - 1) (k)}, {\dot{α}}_{i (n_{i} - 1) (k)}]$ is similar as $B_{ij (k)} [{\bar{S}}_{ij (k)}, α_{i (j - 1) (k)}, {\dot{α}}_{i (j - 1) (k)}]$ for j = n_i.

Remark 4. This presented decentralized DSC-based recursive design scheme with the derivative of the filtered in the virtual control can avoid computational expansion problem [3–7 , 41] inherited by recursive differentiation of α_ij(k). Moreover, this paper considered the more complex LSSNs in NSF form.

The main result of this paper can be summarized by the following theorem.

Theorem 1. Consider the systems (1), under Assumption 1, for the closed-loop systems (17)-(19), consisting of virtual control variables (6) and (9), actual controller (12), filters (7), (10), then, for any initial condition satisfying: $Π_{ij} = {\sum_{l = 1}^{j} S_{il (k)}^{2} (0) + \sum_{l = 2}^{j} P_{il (k)}^{2} (0) < M_{0}}$ (29) where M₀ assumed to be a positive constant, then choose the suitable control parameters c_ij(k), T_ij(k), such that all the signals in the closed-loop system are UUB. Moreover, the ultimate boundedness of the above closed-loop signals can be tuned arbitrarily small by choosing suitable design parameters.

Proof. Consider the following Lyapunov function $V_{i} = \sum_{j = 1}^{n_{i}} (\frac{S_{ij (k)}^{2}}{g_{ij (k)} ({\bar{x}}_{ij})} + \frac{{\tilde{θ}}_{ij (k)}^{2}}{2 Γ_{ij (k)}^{(1)}} + \frac{{\tilde{ɛ}}_{ij (k)}^{2}}{2 Γ_{ij (k)}^{(2)}}) + \sum_{j = 2}^{n_{i}} \frac{P_{ij (k)}^{2}}{2 T_{ij (k)}}$ (30) Where $Γ_{ij (k)}^{(1)} > 0, Γ_{ij (k)}^{(2)} > 0, T_{ij (k)} > 0$ are design parameters, S_ij(k) are surface error, P_ij(k) are filter error, ${\tilde{θ}}_{ij (k)}^{2}, {\tilde{ɛ}}_{ij (k)}^{2}$ are the adaptive estimation error. Since M₀ > 0 is a positive constant, the sets $Π_{ij} = {\sum_{l = 1}^{j} S_{il (k)}^{2} (0) + \sum_{l = 2}^{j} P_{il (k)}^{2} (0) < M_{0}}$ is a compact, therefor, B_ij(k) has its maximum on compact Π_ij denoted by M_ij(k) . Based on surface error $S_{i (j + 1) (k)} = x_{i (j + 1)} - {\bar{α}}_{ij (k)}$ and filter error $P_{ij (k)} = {\bar{α}}_{ij (k)} - α_{ij (k)}$ , the derivative of V_i along the solution.

$\begin{matrix} {\dot{V}}_{i} & = \sum_{j = 1}^{n_{i}} (\frac{S_{ij (k)} {\dot{S}}_{ij (k)}}{g_{ij (k)}} - \frac{S_{ij (k)}^{2} {\dot{g}}_{ij (k)}}{g_{ij (k)}^{2}}) + \sum_{j = 1}^{n_{i}} (\frac{{\tilde{θ}}_{ij (k)} {\dot{\hat{θ}}}_{ij (k)}}{Γ_{ij (k)}^{(1)}} \\ + \frac{{\tilde{ɛ}}_{ij (k)} {\dot{\hat{ɛ}}}_{ij (k)}}{Γ_{ij (k)}^{(2)}}) + \sum_{j = 2}^{n_{i}} \frac{P_{ij (k)} {\dot{P}}_{ij (k)}}{T_{ij (k)}} \\ = S_{i 1 (k)} [θ_{i 1 (k)}^{*} ξ_{i 1 (k)} + ɛ_{i 1 (k)} + S_{i 2 (k)} + α_{i 1 (k)}] \\ + \sum_{j = 2}^{n_{i} - 1} (S_{ij (k)} [θ_{ij (k)}^{*} ξ_{ij (k)} + ɛ_{ij (k)} + S_{i (j + 1) (k)} \\ + P_{i (j + 1) (k)} + α_{ij (k)}] - \sum_{j = 1}^{n_{i}} \frac{S_{ij (k)}^{2} {\dot{g}}_{ij (k)}}{g_{ij (k)}^{2}} \\ + \sum_{j = 1}^{n_{i}} \frac{S_{ij (k)}}{g_{ij (k)}} [h_{ij (k)} (\bar{y}) + d_{ij (k)}]) \\ + \sum_{j = 2}^{n_{i}} (P_{ij (k)} [- \frac{P_{ij (k)}}{T_{ij (k)}} - B_{ij (k)}]) \\ + \sum_{j = 1}^{n_{i}} (\frac{{\tilde{θ}}_{ij (k)} {\dot{\hat{θ}}}_{ij (k)}}{Γ_{ij (k)}^{(1)}} + \frac{{\tilde{ɛ}}_{ij (k)} {\dot{\hat{ɛ}}}_{ij (k)}}{Γ_{ij (k)}^{(2)}}) \\ + S_{{in}_{i} (k)} [c_{μ} v_{i}^{σ (t)} + ρ (v_{i}^{σ (t)}) \\ + θ_{{in}_{i} (k)}^{* T} ξ_{{in}_{i} (k)} (Z_{{in}_{i} (k)}) + ɛ_{{in}_{i} (k)} (Z_{{in}_{i} (k)})] \end{matrix}$ Combined with Youngs’inequalities, we obtain the following equations. $\begin{matrix} \sum_{j = 1}^{n_{i}} S_{ij (k)} θ_{ij (k)}^{* T} ξ_{ij (k)} (Z_{ij (k)}) \\ \leq \sum_{j = 1}^{n_{i}} \frac{S_{ij (k)}^{2} ({\tilde{θ}}_{ij (k)} + {\hat{θ}}_{ij (k)}) | | ξ_{ij (k)} (Z_{ij (k)}) | |^{2}}{4 τ_{ij (k)}^{(1)}} \\ + \sum_{j = 1}^{n_{i}} τ_{ij (k)}^{(1)} \end{matrix}$ (31) $\begin{matrix} \sum_{j = 1}^{n_{i}} S_{ij (k)} ɛ_{ij (k)} (Z_{ij (k)}) \\ \leq \sum_{j = 1}^{n_{i}} \frac{S_{ij (k)}^{2} ({\tilde{ɛ}}_{ij (k)} + {\hat{ɛ}}_{ij (k)})}{4 τ_{ij (k)}^{(1)}} + \sum_{j = 1}^{n_{i}} τ_{ij (k)}^{(2)} \end{matrix}$ (32) where $τ_{ij (k)}^{(1)} > 0, τ_{ij (k)}^{(2)} > 0$ are design positive constants. Together with Assumption 1 and above inequalities, we have $\begin{matrix} \sum_{j = 1}^{n_{i}} \frac{S_{ij (k)} (h_{ij (k)} (\bar{y}) + d_{ij (k)})}{g_{ij (k)}} \\ \leq \sum_{j = 1}^{n_{i}} \frac{S_{ij (k)}^{2}}{g_{ij 0}} (\sum_{l = 1}^{N} | y_{l} | η_{ijl} {\tilde{q}}_{ijl} (| y_{i} |) + d_{ij (k)}^{*}) \end{matrix}$ (33) $\begin{matrix} \sum_{j = 2}^{n_{i}} S_{ij (k)} P_{ij (k)} \leq \sum_{j = 2}^{n_{i}} (\frac{S_{ij (k)}^{2}}{2} + \frac{P_{ij (k)}^{2}}{2}) \end{matrix}$ (34) Together with the virtual controller α_i1(k), α_ij(k), j = 2, 3, …, n_i - 1, adaptive adjust parameters ${\hat{θ}}_{ij (k)},$ ${\hat{ɛ}}_{ij (k)}$ and the above inequalities, we obtain, $\begin{matrix} {\dot{V}}_{i} & \leq \sum_{j = 1}^{n_{i}} (- c_{ij (k)} S_{ij (k)}^{2} + \frac{{\tilde{θ}}_{ij (k)} {\hat{θ}}_{ij (k)}}{Γ_{ij (k)}^{(1)}} + \frac{{\tilde{ɛ}}_{ij (k)} {\hat{ɛ}}_{ij (k)}}{Γ_{ij (k)}^{(2)}}) \\ + \sum_{j = 2}^{n_{i}} (\frac{S_{ij (k)} S_{i (j + 1) (k)}}{g_{ij (k)}} + P_{ij (k)} [- \frac{P_{ij (k)}}{T_{ij (k)}} \\ - B_{ij (k)}] + \frac{P_{ij (k)}^{2}}{2}) + τ_{i 0 (k)}^{(1)} + τ_{i 0 (k)}^{(2)} \\ + S_{{in}_{i} (k)} [c_{μ} v_{i}^{σ (t)} + ρ (v_{i}^{σ (t)})] \end{matrix}$ (35) where $τ_{i 0 (k)}^{(1)} = \sum_{j = 1}^{n_{i}} τ_{ij (k)}^{(1)}, τ_{i 0 (k)}^{(2)} = \sum_{j = 1}^{n_{i}} τ_{ij (k)}^{(2)}$ , and j = 2, 3, …, n_i - 1, S_ij(k)S_i(j+1)(k) will be designed by the next virtual controller and actual controller, when j = n_i, S_{i,n_i+1(k)} = 0 .

By completing the squares, we have $\sum_{j = 1}^{n_{i}} \frac{1}{Γ_{ij (k)}^{(1)}} {\tilde{θ}}_{ij (k)} {\hat{θ}}_{ij (k)} \leq \sum_{j = 1}^{n_{i}} (- \frac{{\tilde{θ}}_{ij (k)}^{2}}{Γ_{ij (k)}^{(1)}} + \frac{θ_{ij (k)}^{2}}{Γ_{ij (k)}^{(1)}})$ $\sum_{j = 1}^{n_{i}} \frac{1}{Γ_{ij (k)}^{(2)}} {\tilde{ɛ}}_{ij (k)} {\hat{ɛ}}_{ij (k)} \leq \sum_{j = 1}^{n_{i}} (- \frac{{\tilde{ɛ}}_{ij (k)}^{2}}{Γ_{ij (k)}^{(2)}} + \frac{ɛ_{ij (k)}^{2}}{Γ_{ij (k)}^{(2)}})$ $\sum_{j = 2}^{n_{i}} P_{ij (k)} B_{ij (k)} \leq \sum_{j = 2}^{n_{i}} (\frac{M_{ij (k)}^{2}}{2} + \frac{P_{ij (k)}^{2}}{2})$ $S_{{in}_{i} (k)} ρ (v_{i}^{σ (t)}) \leq \frac{S_{{in}_{i} (k)}^{2}}{2} + \frac{E_{0}^{2}}{2}$ Therefore, by the above inequalities, together with actual controller $v_{i}^{σ (t)}$ , (35) can be rewritten as $\begin{matrix} {\dot{V}}_{i (k)} & \leq \sum_{j = 1}^{n_{i}} (- \frac{c_{ij (k)} S_{ij (k)}^{2}}{2}) - \sum_{j = 1}^{n_{i}} (\frac{{\tilde{θ}}_{ij (k)}^{2}}{2 Γ_{ij (k)}^{(1)}} - \frac{{\tilde{ɛ}}_{ij (k)}^{2}}{2 Γ_{ij (k)}^{(2)}}) \\ + \sum_{j = 2}^{n_{i}} \frac{ρ_{ij (k)} P_{ij (k)}^{2}}{2} + \sum_{j = 1}^{n_{i}} D_{ij (k)} \end{matrix}$ (36) where $ρ_{ij (k)} = 1 - \frac{1}{T_{ij (k)}}$ , $D_{ij (k)} = τ_{ij (k)}^{(1)} + τ_{ij (k)}^{(2)} + \frac{θ_{ij (k)}^{2}}{Γ_{ij (k)}^{(1)}} + \frac{ɛ_{ij (k)}^{2}}{Γ_{ij (k)}^{(2)}} + \frac{M_{ij (k)}^{2}}{2} + \frac{E_{0}^{2}}{2}$ .

Define c_i(k) = min {c_i1, c_i2, …, c_{in
_i}, ρ_i1, ρ_i2, …, ρ_{in
_i}, i = 1, 2, …, N}, $D_{i (k)} = \sum_{j = 1}^{n_{i}} D_{ij (k)}$ , (36) yields: $\begin{matrix} {\dot{V}}_{i (k)} \leq - c_{i (k)} V_{i} + D_{i (k)} \end{matrix}$ (37) Now, define the Lyapunov function for the whole systems; $\begin{matrix} V = \sum_{i = 1}^{N} V_{i} = & \sum_{i = 1}^{N} \sum_{j = 1}^{n_{i}} (\frac{S_{ij (k)}^{2}}{g_{ij (k)} ({\bar{x}}_{ij})} + \frac{{\tilde{θ}}_{ij (k)}^{2}}{2 Γ_{ij (k)}^{(1)}} + \frac{{\tilde{ɛ}}_{ij (k)}^{2}}{2 Γ_{ij (k)}^{(2)}}) \\ + \sum_{i = 1}^{N} \sum_{j = 2}^{n_{i}} \frac{P_{ij (k)}^{2}}{2 T_{ij (k)}} \end{matrix}$ (38) Based on (37), along the time derivative of Lyapunov equation (38), the following inequality holds: $\begin{matrix} \dot{V} = & \sum_{i = 1}^{N} V_{i} \leq - \sum_{i = 1}^{N} c_{i (k)} V_{i} + \sum_{i = 1}^{N} D_{i (k)} \\ \leq - c_{0} V + d_{0} \end{matrix}$ (39) where $c_{0} = \sum_{i = 1}^{N} c_{i (k)},$ $d_{0} = \sum_{i = 1}^{N} D_{i (k)}$ .

Multiplying both sides of the above Equation (39) by e^{c
₀
t}, and it can be rewritten as $\begin{matrix} \frac{dV (t) e^{c_{0} t}}{dt} \leq d_{0} e^{c_{0} t} \end{matrix}$ (40) Then, integrating the above equation over [0, t], we can obtain $\begin{matrix} 0 \leq V (t) \leq \frac{b_{0}}{a_{0}} + [V_{(0)} - \frac{b_{0}}{a_{0}}] e^{- a_{0} t} \end{matrix}$ (41) If we note that 0 < e^-a₀t < 1 and $\frac{b_{0}}{a_{0}} e^{- a_{0} t} > 0$ , then we have $\begin{matrix} 0 \leq V (t) \leq \frac{b_{0}}{a_{0}} + V_{(0)} \end{matrix}$ (42) Therefore, based on the definition of V in (38), we can conclude that $| S_{ij (k)} | \leq \sqrt{\frac{b_{0}}{a_{0}} + [V (0) - \frac{b_{0}}{a_{0}}] e^{- a_{0} t}}$ . Therefor, it can be shown that all signals surface error S_ij, filter error P_ij(k), adaptive estimation error ${\tilde{θ}}_{ij (k)}, {\tilde{ɛ}}_{ij (k)}, i = 1, 2, \dots, N, j = 1, 2, \dots, n_{i}$ are bounded, since θ_ij(k), ɛ_ij(k) are constants, ${\hat{θ}}_{ij (k)}, {\hat{ɛ}}_{ij (k)}$ are bounded. Virtual controller α_ij(k) are also bounded because of S_ij(k), P_ij(k), θ_ij(k), ɛ_ij(k) are bounded. Hence, we conclude that x_ij are bounded. Furthermore, based on (42), it is easily obtained that $V (t) \leq \frac{b_{0}}{a_{0}}, t \to \infty$ So, the surface error S_ij(k) and filter error P_ij(k) can eventually converge to the compact set of Π_ij. In other words, the UUB of the above closed-loop signals can be guaranteed. This completes the proof. □

Remark 5. Based on the above stability analysis, the main parameters design and controller design procedure contain: first, we select appropriately design parameters for the jth, j = 1, 2, …, n_i - 1 equation of the ith subsystem, such that $c_{ij (k)} > 0, Γ_{ij (k)}^{(1)} > 0, T_{ij (k)} > 0, Γ_{ij (k)}^{(2)} > 0, τ_{ij (k)}^{(1)} > 0, τ_{ij (k)}^{(2)} > 0$ and determine virtual control function α_ij(k), and parameters adaptation law ${\hat{θ}}_{ij (k)}, {\hat{ɛ}}_{ij (k)}$ . then: select appropriately design parameters for the n_ith of the ith subsystem, such that $c_{0} > 0, c_{i} > 0, D_{i (k)}, c_{{in}_{i} (k)} > 0, Γ_{{in}_{i} (k)}^{(1)} > 0, T_{{in}_{i} (k)} > 0, Γ_{{in}_{i} (k)}^{(2)} > 0, τ_{{in}_{i} (k)}^{(1)} > 0, τ_{{in}_{i} (k)}^{(2)} > 0$ and determine actual control function $v_{i}^{σ (t)}$ , and adaptive parameter law ${\hat{θ}}_{{in}_{i} (k)}, {\hat{ɛ}}_{{in}_{i} (k)}$ . Here we note that from $| S_{ij (k)} | \leq \sqrt{\frac{b_{0}}{a_{0}} + [\frac{V (0) a_{0} - b_{0}}{a_{0}}]}$ ×e^-2a₀t, increasing the design parameters $c_{ij (k)}, Γ_{ij (k)}^{(1)},$ $Γ_{ij (k)}^{(2)}$ or decreasing the design parameters $τ_{ij (k)}^{(1)},$ $τ_{ij (k)}^{(2)}$ can make tracking error S_i1 be smaller.

4 Simulation example

In this subsection, based on two inverted pendulums composed of nonlinear spring friction and damper connections [41], a simulation example will illustrate the effectiveness and feasibility of this proposed fuzzy decentralized DSC-filter switched tracking control technique. $\begin{matrix} B_{1} \ddot{q_{1}} = m_{1} gr sin q_{1} - 0.5 Er cos (q_{1} - q) - M_{f 1} + u_{1} \\ B_{2} \ddot{q_{2}} = m_{2} gr sin q_{2} - 0.5 Er cos (q_{2} - q) - M_{f 2} + u_{2} \end{matrix}$ (43) where q₁ and q₂ denote the angular position, B₁ = B₂ = 1kg · m² denote the inertia moments, m₁ = 2kg and m₂ = 2.5kg are the masses, $r = 0.5 m, E = k (p - l) + b \dot{p}$ denote the connection points spring force and damper at the link points [41], and P is the link points distance: $\begin{matrix} p = \sqrt{d^{2} + dr (sin q_{1} - sin q_{2}) + \frac{r^{2}}{2} [1 - cos (q_{2} - q_{1})]} \end{matrix}$ (44) where d = 0.5m, k = 150N/m and $b = 1 N \dot{s} / m .$ The relative position of angular q can be defined as follow: $\begin{matrix} q = {tan}^{- 1} (\frac{\frac{r}{2} (cos q_{2} - cos q_{1})}{d + \frac{r}{2} (sin q_{2} - sin q_{1})}) \end{matrix}$ (45)M_f1 and M_f2 are assumed to a friction model of Lugre defined as: $M_{fi} = s_{0} {\dot{γ}}_{i} + s_{1} {\dot{γ}}_{i} + s_{2} {\dot{q}}_{i}, i = 1, 2$ ${\dot{γ}}_{i} = {\dot{q}}_{i} - \frac{s_{0} | {\dot{q}}_{i} |}{M_{c} + (M_{s} - M_{c}) exp (- | \dot{q} / {\dot{q}}_{2} |^{2})}$ where $s_{0} = 1 N \cdot m, s_{1} = 1 N \cdot ms, s_{2} = 1 N \cdot ms, {\dot{q}}_{s} = 0.1 rd / s, M_{s} = 2 N \cdot m$ and M_c = 1N · m. In this simulation, we regard this inverted pendulum system with uncertainty disturbances, since many mechanical systems are required to be described as their behavior. Define $x_{11} = q_{1}, x_{12} = {\dot{q}}_{1}, x_{21} = q_{2}, x_{22} = {\dot{q}}_{2}$ . Then, the mechanical system (1) can be transformed into the following structure:

${\begin{matrix} {\dot{x}}_{i 1} & = f_{i 1}^{σ (t)} (x) + g_{i 1} (x_{i 1}) x_{i 2} + h_{i 1}^{σ (t)} (\bar{y}) + d_{i 1}^{σ (t)} \\ {\dot{x}}_{i 2} & = f_{i 2}^{σ (t)} (x) + g_{i 2} ({\bar{x}}_{i 2}) t) u_{i}^{σ (t)} (v_{i}^{σ (t)}) (t) \\ y_{i} & = x_{i 1} . \end{matrix}$ (46) Where switching signal function σ : [0, + ∞) → M = 1, 2 is assumed to be a piecewise continuous function. $f_{11} (x) = 0, h_{11} = 0.1 x_{21}, d_{11} (t) = 1 + t^{2}, g_{11} (x_{11}) = 3 x_{11}^{1 / 3}, f_{12} (x) = g_{12} [m_{1} gr x_{11} - 0.5 {Erx}_{11} - q - T_{f 1}^{2} 3 x_{1}, g_{12} = \frac{1}{J_{1}},$ $h_{12} (y) = x_{11} x_{21}^{3}, d_{12} (t) = 1 + t^{2} + t^{3} .$ f₂₁ (x) =0, f₂₂ (x) = g₂₂ [m₂grx₂₁ - 0.5Frx₂₁ - q - T_f2] , $h_{21} (y) = x_{1}^{2} x_{2},$ $g_{22} = \frac{1}{J_{2}}$ $h_{22} (y) = x_{11}^{2} x_{21}^{1 / 3}, d_{21} (t) = 1 + t^{2} + t^{3}, d_{22} (t) = t^{3} .$ The input saturation $u_{i}^{σ (t)} (v_{i}^{σ (t)})$ satisfies: $u_{i}^{σ (t)} (v_{i}^{k}) = sat (v_{i}^{k)}) = {\begin{matrix} sign (v_{i}^{σ (t)}) μ_{N}, | v_{i}^{k} | \geq μ_{N} \\ v_{i}^{σ (t)}, | v_{i}^{k} | < μ_{N}, σ (t) = k \end{matrix}$ with μ_N = 2.

The objective of this simulation for this second-order system is to apply the proposed decentralized fuzzy DSC-based tracking approach to design controllers and adaptive laws:

1) All the signals in the closed-loop system are UUB,

2) The output y₁ = x₁₁, y₂ = x₂₁ would track the reference signal y_1r = 0.5 sin(t) +0.5 sin(0.5t) and y_2r = sin(t) + cos(0.5t) very well.

Fuzzy sets are defined over interval [-5, 5] the partitioning points are chosen as 5, 3, 1, 0, - 1, - 3, - 5, and the fuzzy membership functions are chosen as $μ_{F_{ij}^{1}} = exp [\frac{(- 0.5 (x_{ij} + 5)^{2})}{7}], μ_{F_{ij}^{2}} = exp [\frac{(- (x_{ij} + 3)^{2})}{7}],$ $μ_{F_{ij}^{3}} = exp [(- (x_{ij} + 1)^{2}) / 7], μ_{F_{ij}^{4}} = exp [(- (x_{ij} + 0)^{2}) / 7],$ $μ_{F_{ij}^{5}} = exp [(- (x_{ij} - 1)^{2}) / 7], μ_{F_{ij}^{6}} = exp [(- (x_{ij} - 3)^{2}) / 7],$ $μ_{F_{ij}^{7}} = exp [(- (x_{ij} - 5)^{2}) / 7], i = 1, 2, j = 1, 2 .$

Select the appropriately design parameters such that c₁₁ = c₁₂ = c₂₁ = c₂₂ = 0.5, $Γ_{11}^{(1)} = 0.1, T_{11} = 0.2,$ $Γ_{11}^{(2)} > 0, τ_{11}^{(1)} = 0.1, τ_{11}^{(2)} = 0.2,$ $Γ_{21}^{(1)} = 0.1, T_{21} = 0.2, Γ_{21}^{(2)} > 0, τ_{21}^{(1)} = 0.1, τ_{21}^{(2)} = 0.2,$ $Γ_{12}^{(1)} = 0.5, Γ_{12}^{(2)} = 0.4, T_{12} = 0.8, τ_{12}^{(1)} = 0.1, τ_{12}^{(2)} = 0.1$ and $Γ_{22}^{(1)} = 0.5, Γ_{22}^{(2)} = 0.4, T_{22} = 0.8, τ_{22}^{(1)} = 0.1, τ_{22}^{(2)} = 0.1$ . c₀ = 0.1, c₁ = 0.2, c₂ = 0.1, D₁ = 0.5, D₂ = 0.3 . E₀ = 0.9, g₁₀ = g₂₀ = 0.2, g_1d = g_2d = 0.3 . Determine virtual control function α_i1, i = 1, 2, k = 1, 2, and parameters adaptation laws ${\hat{θ}}_{i 1 (k)}, {\hat{ɛ}}_{i 1 (k)}, i = 1, 2, k = 1, 2$ .

Choose the design initial values as x₁₁ (0) = x₁₂ (0) = x₂₁ (0) = x₂₂ (0) =0.01, the initial adaptive laws conditions are as ${\dot{\hat{θ}}}_{11} (0) = {\dot{\hat{θ}}}_{12} (0) = 1.2$ , ${\dot{\hat{θ}}}_{21} (0) = {\dot{\hat{θ}}}_{22} (0) = 0.9$ , ${\dot{\hat{ɛ}}}_{11} (0) = {\dot{\hat{ɛ}}}_{12} (0) = 1.5$ , ${\dot{\hat{ɛ}}}_{21} (0) = {\dot{\hat{ɛ}}}_{22} (0) = 1.5$ .

According to the design procedure of decentralized fuzzy DSC-based scheme in section 3, it is illustrated that the prescribed fuzzy approximation-based DSC architecture can guarantee the boundedness of the whole LSSNs in NSF form with input saturation.

The simulation results are demonstrated in Figs. 2–7. Figures 2 and 3 illustrate the output trajectory y₁, y₂ can track the trajectory of reference signal y_1d, y_2d and the tracking trajectory of tracking error. Figures 4 and 5 demonstrate the responses of adaptive estimation of the adjusted parameters ${\hat{θ}}_{11}, {\hat{θ}}_{12}, {\hat{θ}}_{21}, {\hat{θ}}_{22}$ , ${\hat{ɛ}}_{11}, {\hat{ɛ}}_{12}$ , ${\hat{ɛ}}_{21}, {\hat{ɛ}}_{22}$ , from which, we could see that they are bounded. Figure 6 exhibits the trajectory of plant input u₁ (v₁ (t)),u₂ (v₂ (t)). The designed controller can be easily applied to cope with input saturation from Fig. 6. The trajectory of switching signal σ (t) was expressed in the Fig. 7. According to these simulation figures, it can be resulted that all the signals are UUB, and the simulation results investigated the presented approximation-based DSC decentralized design technique is effective.

Fig. 2

Trajectory of reference signal y_1r, output y₁.

Fig. 3

Trajectory of reference signal y_2r, output y₂.

Fig. 4

Trajectory of the estimation of ${\hat{θ}}_{11}, {\hat{θ}}_{12}, {\hat{θ}}_{21}, {\hat{θ}}_{22}$ .

Fig. 5

Trajectory of the estimation of ${\hat{ɛ}}_{11}, {\hat{ɛ}}_{12}$ , ${\hat{ɛ}}_{21}, {\hat{ɛ}}_{22}$ .

Fig. 6

Trajectory of control input u₁ (v₁ (t)),u₂ (v₂ (t)).

Fig. 7

Switching signal σ (t).

5 Conclusion

The decentralized FLSs-approximation DSC-based tracking approach for the LSSNs in NSF structure with input saturation under arbitrary switching has been investigated in this paper. The unknown uncertainties of the NSF structure were approximated by FLSs, auxiliary design system was adopted to handle input saturation. The UUB of all states can be guaranteed based on constructing CLF and Lyapunov theorem analysis, system output would track the desired signal very well. This proposed technique can eliminate computational expansion problem inherented by recursive differentiated virtute controller and can compensate uncertainties, input saturation and deal with the NSF forms of included non-affine structure states, which is more general for application in practical engineering. Furthermore, simulation results demonstrated the feasibility of the presented scheme. In future, we will combine the deep learning techniques [42] and DSC algorithm to the nonlinear switching systems or design switching signals and dwell time or consider the infinite-time condition and constraint problem in future.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (Nos. 12102236, 11971065, 62025303), in part by Natural Science Foundation of Shanxi Province (Nos. 20210302124258, 202203021211334).

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