Disturbance-observer based sliding mode control for fuzzy singularly perturbed systems

Abstract

This paper presents a disturbance-observer based sliding mode control (SMC) for fuzzy singularly perturbed systems (SPSs) with uncertainties and disturbances. Firstly, we designed a linear sliding surface. The sliding surface parameter matrix is determined by solving linear matrix inequalities (LMIs). The stability of the sliding mode is proved by a Lyapunov function. Secondly, a disturbance observer is designed to estimate the disturbance, and the obtained disturbance estimate is incorporated in the design of SMC. The reachability condition under the fuzzy SMC law is shown to be satisfied. Finally, simulation results show the feasibility and effectiveness of the proposed control method.

Keywords

Singularly perturbed systems (SPSs)sliding mode control (SMC)linear matrix inequality (LMI)T-S fuzzy model

1 Introduction

In many industrial systems such as robots, large nuclear reactors and power systems, contain small parasitic parameters which may lead to high dimensionality and ill-conditioned numerical issues in dynamic system modeling, system analysis and control method design process [1 –3]. In order to avoid the above mentioned problems, such systems are typically modeled as SPSs with a singular perturbation parameter ɛ. In recent years, some scholars have reported excellent research results on SPSs [2 –5]. The common methods are the quasi-steady state method [6, 7] and block diagonalization method [8, 9], which decompose SPSs into slow subsystem and fast subsystem, then design controller for each of the two subsystems and finally construct a composite control law for the overall system by combining the two controllers. These methods of decomposing into subsystems can avoid high dimensional and ill-conditioned numerical problems, but non-standard SPSs which cannot be decomposed into subsystems is not applicable [10]. Thus, based on full-order system model, an alternative approach called descriptor system was proposed [11].

In the robust control field, sliding mode control (SMC) is a common and effective method, especially for systems with uncertainties. SMC can make the system move along a specified trajectory under certain characteristics [12]. Many scholars have studied SMC of linear SPSs [13 –18]. In [16], for the fast subsystem, a feedback controller was designed and the integral SMC method was adopted for the slow subsystem. In [17], SMC method was studied for SPSs with matched disturbances. In [18], SMC was proposed for a three– time– scale system. These studies are focused on SMC of linear SPSs. However, there are relatively few studies on nonlinear SPSs [19, 20]. Lin [19] discussed an adaptive SMC based on neural network composite method for SPSs. In [20], the design of an exponentially stabilized controller based on observer for nonlinear SPSs was studied.

The Takagi-Sugeno (T-S) fuzzy modeling method is effective in representing nonlinear systems [21]. The nonlinear system can be approximated by a series of linear subsystems. The virtues of T– S fuzzy model are as follows: 1) T-S models can generate relatively complex nonlinear functions. This can effectively reduce fuzzy rules when dealing with multivariable systems. 2) The T-S fuzzy system can utilize many sophisticated linear model control theories to analyze and synthesize complex nonlinear models. In the past few decades, T-S fuzzy modeling method has been widely studied in the field of nonlinear SPSs. In [22], a robust H_∞ control based on fuzzy Lyapunov function and non-PDC control were proposed for fuzzy SPSs. In [23], for fuzzy SPSs, a H_∞∼ control approach based on LMI was proposed. In [24], for a class of fuzzy SPSs, a state feedback control method was proposed to achieve multi-objective control. Asemani and Majd [25] studied a H_∞ controller based on multiple Lyapunov functions for fuzzy SPSs. In recent years, many scholars have studied SMC of T-S fuzzy systems and some excellent results have been published. In [26], for the T-S fuzzy systems, under the condition of unmatched uncertainties and external disturbances, a composite adaptive SMC method was proposed. In [27], a speed control strategy for the T-S fuzzy model of SPMSM was designed. Hong et al. [28] investigated an adaptive integral SMC for general T-S fuzzy systems, the fuzzy integral sliding surface was designed. However, there are few people who are concerned about the SMC method for the T-S fuzzy SPSs. Wang et al. [29] developed an integral SMC strategy for T-S fuzzy SPSs. However, it is still an open problem for designing a disturbance-observer based SMC for T-S fuzzy SPSs.

This paper will develop a new SMC strategy for fuzzy SPSs with uncertainties and disturbances. Firstly, a linear sliding surface is constructed and the parameter matrix of the sliding mode controller is determined by solving of a set of LMIs. The stability of the sliding mode is proved by a Lyapunov function. After that, designing an ɛ-dependent fuzzy disturbance observer to obtain the estimate of the matched disturbance, and the estimate value is included in the design process of SMC. It is further shown that the reaching condition under the effect of the fuzzy SMC can be satisfied. Finally, this paper provides two examples to verify the validity of the proposed control method.

Notions. A^T is the transposed matrix of the matrix A, A^-¹ is the inverse matrix of matrix A, r = { 1, 2, …, r } , I_m is the identity matrix with the dimension m × m .

2 Problem statement and preliminaries

Consider the T-S fuzzy SPSs described as follows

Fuzzy Rule i:

IF θ₁ (t) is M_i1, θ₂ (t) is M_i2, . . . , and θ_p (t) is M_ip,

THEN $E_{ɛ} \dot{x} (t) = (A_{i} + Δ A_{i}) x (t) + B (u (t) + f (t))$ (1) where ɛ > 0 is a small singular perturbation parameter, $E_{ɛ} = diag {\begin{matrix} I, & ɛ I \end{matrix}}$ , θ_i (t) are premise variables, M_ij (i = 1, 2, … r, j = 1, 2, … p) are fuzzy sets. x (t) ∈ Rⁿ is the state variable, u (t) ∈ R^m is the control input, A_i ∈ R^n×n and B ∈ R^n×m are known system matrices, rank(B)=m. ΔA_i and f (t) ∈ R^m represent the uncertainties and disturbances, respectively.

The overall T-S fuzzy SPSs is written as follows: $E_{ɛ} \dot{x} (t) = \sum_{i = 1}^{r} h_{i} (θ) [(A_{i} + Δ A_{i}) x (t) + B (u (t) + f (t))]$ (2) where $h_{i} (θ (t)) = \prod_{l = 1}^{p} M_{il} (θ_{l} (t)) / \sum_{i = 1}^{r} \prod_{l = 1}^{p} M_{il}$ (θ_l (t)), $\sum_{i = 1}^{r} h_{i} (θ) = 1$ .

Assumption 1. The matched disturbance f (t) satisfies ∥f (t) ∥ ⩽ μ and $∥ \dot{f} (t) ∥ ⩽ χ$ , μ and χ are two positive constants.

Assumption 2. The unmatched uncertainties ΔA_i (t) are norm-bounded, i.e., $Δ A_{i} = S_{1 i} Δ F_{i} S_{2 i}$ where S_1i and S_2i are known constant matrices, and ΔF_i satisfying $Δ F_{i}^{T} Δ F_{i} ⩽ I$ .

Remark 1. Uncertainties and disturbances widely exist in practical control systems and Assumption 1 and 2 are common in the literature [29]. In addition, the fuzzy SPS (1) with common input matrix B represents a typical class of SPSs that can be used to model motor systems, circuit systems etc. [30].

The following lemma is given, which will be used for subsequent process.

Lemma 1. [26] Let S_1i, S_2i and ΔF_i be real appropriate dimensioned matrices, and $Δ F_{i}^{T} Δ F_{i} ⩽ I$ . Then, for any scalar δ_i > 0, we have $S_{1 i} Δ F_{i} S_{2 i} + S_{2 i}^{T} Δ F_{i} S_{1 i}^{T} ⩽ δ_{i}^{- 1} S_{1 i} S_{1 i}^{T} + δ_{i} S_{2 i}^{T} S_{2 i}$

3 Main results

This section will design a linear sliding surface. Because the f (t) is unknown, we will also design a disturbance observer to obtain the estimate of f (t) and incorporate it into the design process of the controller. And at last, stability of the system is proven.

3.1 Disturbance observer and sliding motion analysis

Motivated by [31] a fuzzy disturbance observer for system (2) is designed and the estimate value of the disturbance is obtained. $\begin{matrix} \dot{d} (t) = Z \hat{f} (t) + \sum_{i = 1}^{r} h_{i} (θ) Z B^{+} ((A_{i} + Δ A_{i}) x (t) + Bu (t)) \\ \hat{f} (t) = d (t) - Z B^{+} E_{ɛ} x (t) \end{matrix}$ (3) where B⁺ = (B^TB) ^-1B^T, $\hat{f} (t)$ represents the estimate value of f (t), d (t) is the internal variable vector and Z is a Hurwitz matrix, $\tilde{f} = f - \hat{f}$ denotes the disturbance estimation error.

Theorem 1. For the disturbance observer (3), the disturbance estimation error satisfies $∥ \tilde{f} ∥ ⩽ β$ , where β is a positive scalar.

Proof. Based on Equation (3), we have $\dot{d} (t) = Z B^{+} E_{ɛ} \dot{x} (t) - Z \tilde{f} (t)$ (4) $\dot{\hat{f}} (t) = \dot{d} (t) - Z B^{+} E_{ɛ} \dot{x} (t)$ (5)

Substituting Equation (4) into Equation (5) yields $\dot{\hat{f}} = - Z \tilde{f}$ (6)

Thus, we have $\dot{\tilde{f}} = \dot{f} + Z \tilde{f}$ (7) $\tilde{f} (t) = e^{Z t} f (0) + \int_{0}^{t} e^{Z (t - s)} \dot{f} (s) ds$ (8) $\begin{matrix} ∥ \tilde{f} (t) ∥ ⩽ ∥ e^{Z t} f (0) ∥ + ∥ \int_{0}^{t} e^{Z (t - s)} \dot{f} (s) ds ∥ \\ ⩽ ∥ e^{Z t} ∥ ∥ f (0) ∥ + \int_{0}^{t} ∥ e^{Z (t - s)} ∥ ∥ \dot{f} (s) ∥ ds \\ ⩽ c μ e^{\frac{λ_{max} (Z)}{2} t} + c χ \int_{0}^{t} e^{\frac{λ_{max} (Z)}{2} (t - s)} ds \\ = c μ e^{\frac{λ_{max} (Z)}{2} t} + c χ \frac{2}{λ_{max} (Z)} (e^{\frac{λ_{max} (Z)}{2}} - 1) \\ ⩽ c μ - c χ \frac{2}{λ_{max} (Z)} \end{matrix}$ (9)

Define $β = c μ - c χ \frac{2}{λ_{max} (Z)}$ . It’s easy to see that $∥ \tilde{f} (t) ∥ ⩽ β$ .

Remark 2. The disturbance observer (3) considers not only the singular perturbation characteristics but also the T-S fuzzy system structure, which is completely different from the design of the disturbance observer for the normal systems [32]. Such a configuration aims to avoid the ill-conditioned numerical issues in the control design.

Remark 3. Compared with [29], the obtained disturbance estimate rather than the bound of the disturbance is used in the SMC. As a result, the proposed disturbance observer-based SMC law in this paper can lead to shorter response times, less overshoots, which are demonstrated by simulation results.

We design a linear sliding surface for the system (2) as $σ = {SE}_{ɛ} x = [\begin{matrix} S_{1} \\ S_{2} \end{matrix}] E_{ɛ} x = 0$ (10) where S is a parameter matrix of the linear sliding surface.

To design the parameter matrix S, define a transformation matrix T satisfying Equation (11) $T = [\begin{matrix} F \\ S \end{matrix}] = [\begin{matrix} (Γ^{T} E_{ɛ} X_{ɛ} Γ)^{- 1} Γ^{T} \\ (B^{T} X_{ɛ} B)^{- 1} B^{T} X_{ɛ} \end{matrix}]$ (11) where Γ ∈ R^n×q satisfies Γ^TB = 0 and $X_{ɛ} = [\begin{matrix} X_{1} & ɛ X_{2}^{T} \\ X_{2} & X_{3} \end{matrix}]$ .

Thus, we have $TB = [\begin{matrix} 0 \\ I_{m} \end{matrix}]$ (12)

Under the transformation $[\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] = {TE}_{ɛ} x (t)$ (13)

system (2) becomes $\begin{matrix} [\begin{matrix} {\dot{x}}_{1} (t) \\ {\dot{x}}_{2} (t) \end{matrix}] = {TE}_{ɛ} \dot{x} (t) \\ = \sum_{i = 1}^{r} h_{i} (θ) [T (A_{i} + Δ A_{i}) x (t) + TB (u + f (t))] \\ = \sum_{i = 1}^{r} h_{i} (θ) [T (A_{i} + Δ A_{i}) ({TE}_{ɛ})^{- 1} [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] \\ + [\begin{matrix} 0 \\ I_{m} \end{matrix}] (u + f (t))] \\ = [\begin{matrix} \sum_{i = 1}^{r} h_{i} (θ) A_{11 i} & \sum_{i = 1}^{r} h_{i} (θ) A_{12 i} \\ \sum_{i = 1}^{r} h_{i} (θ) A_{21 i} & \sum_{i = 1}^{r} h_{i} (θ) A_{22 i} \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] \\ + [\begin{matrix} 0 \\ I_{m} \end{matrix}] (u + f (t)) \end{matrix}$ (14) where A_11i = (Γ^TE_ɛX_ɛΓ) ^-1Γ^T (A_i + ΔA_i) X_ɛΓ, x₂ = σ $[\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] = {TE}_{ɛ} x (t)$ and x₂ = σ.

When the system states go into sliding mode, by setting $\dot{σ} = σ = 0$ , system (14) degenerates to: ${\dot{x}}_{1} (t) = \sum_{i = 1}^{r} h_{i} (θ) (Γ^{T} E_{ɛ} X_{ɛ} Γ)^{- 1} Γ^{T} (A_{i} + Δ A_{i}) X_{ɛ} Γ x_{1} (t)$ (15)

Theorem 2. If there exist a matrix Γ with Γ^TB = 0 and a lower triangular matrix $X = [\begin{matrix} X_{1} & 0 \\ X_{2} & X_{3} \end{matrix}]$ (16) with 0 < X₁ ∈ R^n₁×n₁ and 0 < X₃ ∈ R^n₂×n₂, satisfying the following LMI $[\begin{matrix} Γ^{T} A_{i} X Γ + Γ^{T} X^{T} A_{i}^{T} Γ + δ_{i} Γ^{T} S_{1 i} S_{1 i}^{T} Γ & Γ^{T} X^{T} S_{2 i}^{T} \\ S_{2 i} X Γ & - δ_{i} I \end{matrix}] < 0$ (17) then there exists an ɛ * >0, for any ɛ ∈ (0, ɛ *], the system (15) is asymptotically stable.

Proof. Due to X₁ > 0 and X₃ > 0, there exists a scalar ɛ₁ > 0 such that $X_{1} - ɛ X_{2}^{T} X_{3}^{- 1} X_{2} > 0$ (18) for all ɛ ∈ (0, ɛ₁]. Using Schur complement, inequality (18) yields $E_{ɛ} X_{ɛ} = [\begin{matrix} X_{1} & ɛ X_{2}^{T} \\ ɛ X_{2} & ɛ X_{3} \end{matrix}] > 0, \forall ɛ \in (0, ɛ_{1}]$ (19) where $X_{ɛ} = [\begin{matrix} X_{1} & ɛ X_{2}^{T} \\ X_{2} & X_{3} \end{matrix}] = X + ɛ X_{0}$ , and $X_{0} = [\begin{matrix} 0 & X_{2}^{T} \\ 0 & 0 \end{matrix}]$ .

Define a Lyapunov function $V = x_{1}^{T} (t) {Px}_{1} (t)$ with P = Γ^TE_ɛX_ɛΓ being positive definite.

Then, $\begin{matrix} \dot{V} = 2 x_{1}^{T} P {\dot{x}}_{1} \\ = 2 \sum_{i = 1}^{r} h_{i} (θ) x_{1}^{T} (t) P (Γ^{T} E_{ɛ} X_{ɛ} Γ)^{- 1} Γ^{T} (A_{i} + Δ A_{i}) X_{ɛ} Γ x_{1} (t) \end{matrix}$ (20)

Using Lemma 1, it is easy to see that Equation (20) is rewritten as $\begin{matrix} \dot{V} = 2 \sum_{i = 1}^{r} h_{i} (θ) x_{1}^{T} (t) Γ^{T} (A_{i} + Δ A_{i}) X_{ɛ} Γ x_{1} (t) \\ = \sum_{i = 1}^{r} h_{i} (θ) x_{1}^{T} (t) [Γ^{T} (A_{i} + Δ A_{i}) X_{ɛ} Γ \\ + Γ^{T} X_{ɛ}^{T} (A_{i} + Δ A_{i})^{T} Γ] x_{1} (t) \\ ⩽ \sum_{i = 1}^{r} h_{i} (θ) x_{1}^{T} [Q + δ_{i} Γ^{T} X_{ɛ}^{T} S_{2 i}^{T} S_{2 i} X_{ɛ} Γ + δ_{i}^{- 1} Γ^{T} S_{1 i} S_{1 i}^{T} Γ] x_{1} \end{matrix}$ (21) where $Q = Γ^{T} A_{i} X_{ɛ} Γ + Γ^{T} X_{ɛ}^{T} A_{i}^{T} Γ$ .

We define $Φ = [\begin{matrix} Γ^{T} A_{i} X Γ + Γ^{T} X^{T} A_{i}^{T} Γ + δ_{i} Γ^{T} S_{1 i} S_{1 i}^{T} Γ & Γ^{T} X^{T} S_{2 i}^{T} \\ S_{2 i} X Γ & - δ_{i} I \end{matrix}]$ (22) $Φ_{0} = [\begin{matrix} Γ^{T} A_{i} X_{0} Γ + Γ^{T} X_{0}^{T} A_{i}^{T} Γ + δ_{i} Γ^{T} S_{1 i} S_{1 i}^{T} Γ & Γ^{T} X_{0}^{T} S_{2 i}^{T} \\ S_{2 i} X_{0} Γ & 0 \end{matrix}]$ (23)

If the inequality (17) is satisfied, then there is a small constant ɛ₂ > 0 such that Φ + ɛΦ₀ < 0 for any given ɛ ∈ (0, ɛ₂], which shows that $\dot{V}$ < 0 for all ɛ ∈ (0, ɛ₂] and x₁ ≠ 0. Let ɛ * = min {ɛ₁, ɛ₂}. Then we have E_ɛ X_ɛ > 0 and $\dot{V}$ < 0 for any ɛ ∈ (0, ɛ *] and x₁ ≠ 0. Thus, sliding-mode dynamics (15) asymptotically stable can be guaranteed. The proof is completed.

The upper bound estimate of ɛ* is also an important problem for SPSs. By the method of [33], the theorem 3 is given.

Theorem 3. If there exists a scalar η > 0, a matrix W > 0, and the matrix X defined in (16), such that the following LMIs satisfied $W < η X_{1}$ (24)

$[\begin{matrix} W & X_{2}^{T} \\ X_{2} & X_{3} \end{matrix}] > 0$ (25) $Φ < 0$ (26) $Φ_{0} < - η Φ$ (27) where Φ and Φ₀ are defined in (22) and (23), for all ɛ ∈ (0, ɛ *], the system (15) is asymptotically stable and ɛ * = η^-1.

Remark 4. Stability bound is an important performance index for SPSs. A trial and error method was given in [29] by solving a set of LMIs depending on a predefined upper bound $\bar{ɛ}$ for the singular perturbation parameter ɛ. Theorem 2 provides an ɛ-independent sufficient condition for the existence of stability bound, by which the stability bound estimation problem is reduced to solving a generalized eigenvalue problem. The obtained method is more computationally efficient and exquisite than the method in [29].

3.2 T-S fuzzy SMC law design

The work in the previous section has two parts. One part is to design a disturbance observer to obtain the estimate value of f (t), the other part is to design a linear sliding surface, and system asymptotically stable can be guaranteed. The next section is focused on designing control law and ensuring all the system states can reach the sliding mode surface.

When the system states are in the sliding surface, we have $\begin{matrix} \dot{σ} (x) = {SE}_{ɛ} \dot{x} (t) \\ = \sum_{i = 1}^{r} h_{i} (θ) S [(A_{i} + Δ A_{i}) x + B (u (t) + f (t))] \\ \sim = 0 \end{matrix}$ (28)

We use the sliding mode equivalent control method, the solution to Equation (28) yields $u_{eq} = - \sum_{i = 1}^{r} h_{i} (θ) S (A_{i} + Δ A_{i}^{T} - W_{i}) x - \hat{f}$ (29) where $W_{i} = δ_{i}^{- 1} S_{1 i} S_{1 i}^{T} + δ_{i} S_{2 i}^{T} S_{2 i}$ .

We choose the switching law as $u_{sw} = - (λ + β) \frac{σ}{∥ σ ∥}$ (30)

where λ > 0 is a constant.

Therefore, the SMC law can be organized as $u (t) = - \sum_{i = 1}^{r} h_{i} (θ) S (A_{i} + Δ A_{i}^{T} - W_{i}) x - \hat{f} - (λ + β) \frac{σ}{∥ σ ∥}$ (31)

Theorem 4. For the system (2), when we design the SMC law as shown in the formula (31), it can satisfy the reachability condition of SMC.

Proof. We defined a Lyapunov function $V (t) = \frac{1}{2} σ^{T} σ$ (32)

Calculating the time-derivative of V (t) along (28) yields $\begin{matrix} \dot{V} = σ^{T} \dot{σ} = σ^{T} \sum_{i = 1}^{r} h_{i} (θ) S [(A_{i} + Δ A_{i}) x + B (u + f)] \\ = σ^{T} \sum_{i = 1}^{r} h_{i} (θ) [{SA}_{i} x + S Δ A_{i} x - {SA}_{i} x - S Δ A_{i}^{T} x \\ - {SW}_{i} x - \hat{f} - (λ + β) \frac{σ}{∥ σ ∥} + f] \\ = \sum_{i = 1}^{r} h_{i} (θ) [σ^{T} \tilde{f} - (λ + β) ∥ σ ∥] \\ ⩽ - \sum_{i = 1}^{r} h_{i} (θ) λ ∥ σ ∥ \\ < 0, \forall σ \neq 0 \end{matrix}$ (33) which implies that the reachability condition of SMC is satisfied. The proof is completed.

4 Examples

This section will verify the validity of the proposed method by two examples. The first example considers an electric circuit nonlinear system [29], and the second example considers a permanent magnet synchronous motor (PMSM).

Example 1. Consider the nonlinear circuit shown in Fig. 1, we can see that the feasibility and effectiveness of the proposed method through the simulation results over that in [29].

Fig.1

Electric circuit.

According to the knowledge of circuit principle, circuit system state equation can be written ${\begin{matrix} L {\dot{I}}_{L} (t) = - I_{L} (t) R - V_{c} + U \\ C {\dot{V}}_{c} (t) = I_{L} - \frac{1}{5} (V_{c}^{3} - V_{c}) + α U \end{matrix}$ (34) where ɛ = C, α = 0.5, R = 2Ω and L = 0.1H.

Define x₁ (t) = L · I_L (t), x₂ (t) = V_c (t), and select the voltage U as the control input. The membership functions are chosen as $h_{1} (x_{2} (t)) = 1 - \frac{x_{2}^{2} (t)}{9}, h_{2} (x_{2} (t)) = \frac{x_{2}^{2} (t)}{9}$

In this example, uncertainties are assumed to satisfy $Δ A_{1} = Δ A_{2} = [\begin{matrix} 3 cos (2 t) & 0 \\ 0 & 2 sin (2 t) \end{matrix}]$ , the disturbance is f (x (t) , t) =2 sin(2t) x₁ (t) +2 sin(2t) x₂ (t) and $x (0) = {[\begin{matrix} 0 & - 3 \end{matrix}]}^{T}$ is initial condition.

Considering circuit system uncertainties, the system (34) can be modeled under x₂ (t) ∈ [3, - 3].

Fuzzy Rule 1: IF x₂(t) is about h₁(x₂(t))

THEN

$E_{ɛ} \dot{x} (t) = (A_{1} + Δ A_{1}) x (t) + B_{1} (u (t) + f (x (t), t))$

Fuzzy Rule 2: IF x₂(t) is about h₂(x₂(t)),

THEN $E_{ɛ} \dot{x} (t) = (A_{2} + Δ A_{2}) x (t) + B_{2} (u (t) + f (x (t), t))$ (35) where $A_{1} = [\begin{matrix} - 20 & - 1 \\ 10 & 0.2 \end{matrix}], A_{2} = [\begin{matrix} - 20 & - 1 \\ 10 & - 1.6 \end{matrix}], B = [\begin{matrix} 1 \\ 0.5 \end{matrix}]$

In this example, according to the Lemma 1, we choose the corresponding parameter matrices as

$\begin{matrix} S_{11} = S_{12} = [\begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}] S_{21} = S_{22} = [\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}] \\ Δ F_{1} = Δ F_{2} = [\begin{matrix} cos (2 t) & 0 \\ 0 & sin (2 t) \end{matrix}], δ_{1} = δ_{2} = 0 . 045 \end{matrix}$ (36)

Let $Γ = {[\begin{matrix} - 1 & 2 \end{matrix}]}^{T}$ . Then, according to the theorems 2 and 3, the parameter matrices are obtained as $X = [\begin{matrix} 0.0091 & - 0.0022 \\ - 0.0217 & 0.0552 \end{matrix}], S = [\begin{matrix} - 0.1596 & 2.3192 \end{matrix}]$ $Z = - 1200, λ + β = 35, η = 0 . 0086, ɛ * = 116 . 28$

The simulation results of the nonlinear circuit under the control law (31) are shown in Figs. 2– 5. The state curves are given in Figs. 2 and 3. It can be seen that the T-S model-based SMC method proposed in the paper makes the closed-loop system asymptotically stable. Moreover, the proposed disturbance observer-based SMC law can lead to shorter response times, less overshoots than the integral SMC (ISMC) presented in [29] because the estimate rather than the bound of the disturbance is used in this paper. The control signal u (t) and the switching surface σ (t) response curves are shown in the Figs. 4 and 5, respectively. It can be clearly seen that the sliding motion can be achieved in a limited time.

Fig.2

Responses of x₁ (t) with ɛ = 0.1.

Fig.3

Responses of x₂ (t) with ɛ = 0.1.

Fig.4

Responses of u (t).

Fig.5

Responses of σ (t).

Example 2. To prove the effectiveness and accuracy of the proposed approach, a PMSM is considered. In this case, the small parameter ɛ is known. The mathematical model of PMSM under dq-axis can be represented by $\dot{w} = k_{1} i_{q} - k_{2} w - k_{3} T_{l}$ $ɛ \dot{iq} = - iq - k_{4} w + k_{5} Vq - \frac{L}{R} w \underset{d}{i} + \frac{L}{R} f_{1} (t)$ $ɛ \dot{id} = - id + k_{5} Vd + \frac{L}{R} wiq + \frac{L}{R} f_{2} (t)$ (37) where $k_{1} = \frac{3}{2} \frac{1}{J} np φ f, k_{2} = \frac{F}{J}, k_{3} = \frac{1}{J}$ $k_{4} = \frac{φ f}{R}, \sim k_{5} = \frac{1}{R}, ɛ = \frac{L}{R}$

The specific parameter values of PMSM are listed in Table 1.

Table 1

Motor parameters

PMSM	n _p	R(Ω)	L(mH)	F(N·m·s)	φ _f (Wb)	J(kg·m²)
Value	4	2.875	15	0.005	0.175	0.029

Rule i (i = 1, 2, . . . r): If w is W_i, Then $E_{ɛ} \dot{x} = \sum_{i = 1}^{r} h_{i} (w) [(A_{i} + Δ A_{i}) x + B (u + f (t))]$ (38) where $A_{i} = [\begin{matrix} - 0.172414 & 36.2069 & 0 \\ 0 & 0 & - \frac{1}{191.67} Wi \\ 0 & \frac{1}{191.67} Wi & - 1 \end{matrix}]$ $\begin{matrix} E_{ɛ} = [\begin{matrix} 1 & 0 & 0 \\ 0 & \frac{1}{191.67} & 0 \\ 0 & 0 & \frac{1}{191.67} \end{matrix}], f (t) = [\begin{matrix} f_{1} (t) \\ f_{2} (t) \end{matrix}] \\ B = [\begin{matrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{matrix}], u = [\begin{matrix} uq \\ ud \end{matrix}], x = [\begin{matrix} we \\ iqe \\ id \end{matrix}] \end{matrix}$

We can approximate the dynamics model of PMSM with the following linear subsystem

Fuzzy rule 1: IF w is about W₁ (W₁ = -500) , THEN

$E_{ɛ} \dot{x} = (A_{1} + Δ A_{1}) x (t) + B (u + f (t))$

Fuzzy rule 2: IF w is about W₂ (W₂ = 500) , THEN

$E_{ɛ} \dot{x} = (A_{2} + Δ A_{2}) x (t) + B (u + f (t))$ (39)

where $\begin{matrix} A_{1} = [\begin{matrix} - 0.172414 & 36.2069 & 0 \\ 0 & 0 & 2.609 \\ 0 & - 2.609 & - 1 \end{matrix}] \\ A_{2} = [\begin{matrix} - 0.172414 & 36.2069 & 0 \\ 0 & 0 & - 2.609 \\ 0 & 2.609 & - 1 \end{matrix}] \\ B^{T} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{matrix}$

In this example, according to Lemma 1, the corresponding parameter matrix is selected as follows: $\begin{matrix} S_{1 i} = [\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], S_{2 i} = [\begin{matrix} 5 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \\ Δ F_{1} = Δ F_{2} = [\begin{matrix} sin (2 t) & 0 & 0 \\ 0 & cos (2 t) & 0 \\ 0 & 0 & cos (2 t) \end{matrix}] \end{matrix}$ (40) and the membership functions are $h_{1} = \frac{e^{- μ (w - w_{1})^{2}}}{e^{- μ (w - w_{1})^{2}} + e^{- μ (w - w_{2})^{2}}}$ $h_{2} = \frac{e^{- μ (w - w_{2})^{2}}}{e^{- μ (w - w_{1})^{2}} + e^{- μ (w - w_{2})^{2}}}$

The exogenous disturbances are f₁ (t) =5000 sin(20t) and f₂ (t) =3000 sin(20t). Let $Γ = {[\begin{matrix} 1 & 0 & 0 \end{matrix}]}^{T}$ and μ = 0.1/500² .

Then, solves the LMIs in theorem 2 and 3 yields the following solutions $\begin{matrix} X = [\begin{matrix} 0.0516 & 0.2382 & 0 \\ 0.2382 & 0.1233 & 0 \\ 0 & 0 & 0.1846 \end{matrix}], S = [\begin{matrix} 1.932 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], \\ Z = [\begin{matrix} - 1000 & 0 \\ 0 & - 800 \end{matrix}], λ + β = 8500, \\ η * = 3 . 3971, ɛ * = 0.2944 \end{matrix}$

The simulation results under the control law are shown in Figs. 6– 9. The given speed is w = 60 rad/s, and the actual angular speed of the PMSM can quickly follow the given value under disturbances. Figure 7 shows the three-phase stator current. Figures 8 and 9 show the actual disturbance signal and the observed disturbance value. It can be seen from the figure that the observer designed in this paper can estimate the disturbance with high accuracy.

Fig.6

Angular speed of motor.

Fig.7

Stator three-phase current.

Fig.8

The estimate of f₁ (t).

Fig.9

The estimate of f₂ (t).

5 Conclusion

In this paper, the SPSs based on T-S fuzzy model with uncertainties and disturbances have been studied. The proposed SMC strategy based on disturbance observer not only effectively makes the closed-loop system asymptotically stable, but also reduces the influence of disturbance on the system. The simulation examples also show that the proposed method is meaningful for nonlinear SPSs affected by disturbances and uncertainties.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61873272 and 61603392, and Fundamental Research Funds for the Central Universities under Grant 2017QNB16.

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