A time-varying general type-II fuzzy sliding mode controller for a class of nonlinear power systems

Abstract

In this paper, in order to control a class of nonlinear uncertain power systems, a new simple indirect adaptive general type-II fuzzy sliding mode controller (IDAGT2FSMC) is proposed. For handling dynamic uncertainties, the proposed controller utilizes the advantages of general type-2 fuzzy logic systems (GT2FLS) to approximate unknown nonlinear actions and noisy data. Implementing general type-2 fuzzy systems is computationally costly; therefore, to decrease computational burden, the proposed method uses a recently introduced α-plane representation so that GT2FLS can be seen as a composition of several interval type-2 fuzzy logic systems (IT2FLS) with a corresponding level of α for each. The globally asymptotic stability of the closed-loop system is mathematically proved. To evaluate the superiority of the proposed controller, performance of the proposed method is compared with those of Indirect Adaptive type-1 Fuzzy Sliding Mode (IDAFSM) controller, Indirect Adaptive Interval Type-II Fuzzy Sliding Mode (IDAT2FSM) controller, conventional Sliding Mode controller (SMC) and PID controller results which are all among the most recent methods applied to the issue in question. Finally, the proposed method is applied to an uncertainly chaotic power system as a case study. Simulation indicates the effectiveness of the proposed controller while facing of dynamic uncertainties and external disturbances.

Keywords

Mathematical modeling chaotic power systems general type-II fuzzy adaptive control sliding mode control nonlinear robust control

1 Introduction

In the modern industry, most of the systems and plants are intensively nonlinear, which motivates researches to expand and develop nonlinear control systems [1 –5]. Indeed, this trend has attracted lots of attention in both industrial and academic communities. As a result, a large number of researchers pay much attention to the Sliding Mode Control theorem as a popular branch of nonlinear controllers. Sliding-mode control demonstrates several advantages including: simple structure, robustness against uncertainties of parameters and the exogenous disturbance; moreover, its stability proof and analysis can be simply done by Lyapunov stability theorem. Additionally, many researchers have tried to integrate fuzzy logic and sliding mode technique to take advantage of both approaches and achieve better control performance of the control systems. This integration approach is known as fuzzy sliding-mode control [6, 7].

Sliding mode control is a robust controller that so far has been used to control the systems facing uncertainty. However, sliding mode controllers face a problem which is known as chattering. This phenomenon occurs as because sliding mode controllers are switching based controller [8]. Researchers have presented the adaptive fuzzy sliding mode controllers, to overcome chattering problems. But each adaptive fuzzy sliding mode controllers demonstrate different behaviors due to their specific the type of sliding mode algorithm implemented, the type of fuzzy system, the approximating fuzzy system factor and the adaptive law. Generally, the adaptive fuzzy sliding mode controllers are divided in two categories: direct adaptive fuzzy sliding mode controller and indirect adaptive fuzzy sliding mode controller. In direct adaptive fuzzy sliding mode control, control input coefficients are tuned using online adaptive law to reduce the tracking error [9] and in indirect adaptive fuzzy sliding mode control, the system parameters could be tuned according to approximating system dynamics and also by adaptive law to reduce the tracking error [10].

Lately, a large number of researches have been devoted to general type-2 fuzzy sets and systems because they are able to deal with uncertainties and disturbances of systems [11 –14]. Type-2 fuzzy set was first proposed by Zadeh as a development of type-1 fuzzy sets [13]. Since then, type-2 fuzzy logic systems, especially IT2FLSs, due to their computational simplicity have been successfully applied to different engineering applications. Such experiments has indicated that IT2FLSs outperform type-1 fuzzy logic systems (T1FLS) while facing different uncertainties such as dynamic uncertainties, rule uncertainties, external disturbances and noises, See [14]. The rules which are created in a fuzzy logic system using available data and information may be uncertain. Unlike interval type-2 fuzzy sets (IT2FS) and type-1 fuzzy sets (T1FS), General type-2 fuzzy sets can deal with rule uncertainties properly. As general type-2 fuzzy sets and systems are computationally complex, just IT2FLSs have been mainly used in literatures so far. But, Liu [15] proposed a useful way to compute centroid and type reduction of GT2FLS using a newly introduced plane representation theorem.

Adaptive control is a technique known to be able to handle parameter uncertainties. In this paradigm, to adapt with system uncertainties, an adaptation law is introduced which adjusts the parameters of the controller against system uncertainties and disturbances. It should be noted that, unfortunately, adaptive control generally guarantees parameter convergence only if parameter changes are slow enough [16]. To address this problem, Shahnazi and Akbarzadeh in 2008 [17] introduced a PI Adaptive fuzzy controller which could efficiently reject fast and large disturbances. Lin et al. in 2009 [18] introduced a direct adaptive interval type-2 fuzzy logic controller in a way that H^∞ tracking performance could be satisfied for a general class of multi-input multi-output nonlinear systems. The results showed that the adaptive interval type-2 fuzzy logic control (AIT2FLC) can handle unpredicted uncertainties very well, while adaptive type-1 fuzzy logic control (AT1FLC) requires more time and control effort to handle them. Pan et al. in 2011 [19] introduced a fire-rule-based direct adaptive interval type-2 fuzzy controller which uses an H^∞ term to reduce the fuzzy estimation error.

In this paper, to achieve the above targets in the proposed control, a simple and new indirect adaptive general type-II fuzzy sliding mode controller is introduced whose stability is proved under Lyapunov stability criteria. The proposed method can be easily applied to nonlinear systems of order n. Furthermore, the globally asymptotic stability of the closed-loop system is mathematically proved. The proposed controller uses the advantage of GT2FLS, sliding mode technique and adaptive controller to cope with uncertainties and unknown disturbances. In addition, to achieve high performance, an H^∞ compensator is added to weaken noises, external disturbances and fuzzy approximation errors. To verify the effectiveness of the proposed method, simulation results on a chaotic nonlinear power system is presented. Simulation results indicate the superiority of the proposed controller over Indirect Adaptive type-1 Fuzzy Sliding Mode (IDAFSM) controller, Indirect Adaptive Interval Type-II Fuzzy Sliding Mode (IDAT2FSM) controller, conventional Sliding Mode controller (SMC) and PID controller.

2 General type-2 fuzzy systems

A GT2FS in a universal set X can be defined as $\tilde{A} = \int_{x \in X} μ_{\tilde{A}} (x) / x$ (1) $μ_{\tilde{A}} (x) = \int_{u \in J_{x}} (f_{x} (u)) / u, J_{x} \in [0, 1]$ (2) where $μ_{\tilde{A}} (x)$ is called a secondary membership function (MF) and f_x (u) is called secondary grade; J_x is the domain of the secondary MF which is called primary membership and u is a fuzzy set in [0, 1]. Figure 1 illustrates a GT2FS where the upper and lower MFs are triangular and its secondary MF is also triangular. When f_x (u) =1, a IT2FS is obtained that demonstrate a uniform uncertainty in the primary membership function and is simply described by its lower ${\underline{μ}}_{\tilde{A}} (x)$ and upper ${\bar{μ}}_{\tilde{A}} (x)$ MFs. Many researchers use interval type-2 fuzzy sets instead of general type-2 fuzzy sets because of their calculation simplicity and also type reduction [4 , 14].

Lately, Liu [15] presented a new method for GT2FSs which is theoretically and computationally effective. Because this method resembles the α-cut for type-1 fuzzy sets, it is denoted as α-plane for type-2 fuzzy sets. ${\tilde{A}}_{α}$ is the denotation of An α-plane representation for a GT2FS $\tilde{A}$ . It is the union of all primary MFs whose secondary grades are greater than or equal to the special value α: ${\tilde{A}}_{α} = \int_{x \in X} μ_{{\tilde{A}}_{α}} (x) / x$ (3) $μ_{{\tilde{A}}_{α}} (x) = \int_{u \in J_{x}} (f_{x} (u) \geq α) / u, J_{x} \in [0, 1]$ (4)

Then a GT2FS $\tilde{A}$ based on α-plane representation theorem can be demonstrated in the following form: $\tilde{A} = \underset{α \in [0, 1]}{U} α / {\tilde{A}}_{α}$ (5)

It is a beneficial representation because $α / {\tilde{A}}_{α}$ can be seen as an IT2FS with the secondary grade of level α. As a result, several IT2FSs may be made from the decomposition of a general type-2 fuzzy set with a corresponding level of α for each, where α = {0, 1/K, …, (K - 1)/K, 1}. In simpler terms, a general type-2 fuzzy logic system can be seen as a huge collection of IT2FLSs with one IT2FLS for each value of α. However, Liu [20] showed that using only 5 to 10 α-plane can result in the required accuracy for centroid calculation. Figure 2 illustrates the new designing for a general type-2 fuzzy system based on α-plane representation.

In general, a GT2FLS is made of a fuzzifier; fuzzy rule-based; fuzzy inference engine; type reducer and defuzzifier. Fuzzifier maps real values into fuzzy sets. Singleton fuzzifier whose output is a single point of a unity membership grade is used in this paper because it is simple. Fuzzy rule base includes fuzzy IF-THEN rules. In the following The jth rule in the GT2FLS is shown: $\begin{matrix} R^{j} : If x_{1} is {\tilde{F}}_{1}^{j} and x_{2} is {\tilde{F}}_{2}^{j} and \dots x_{n} is {\tilde{F}}_{n}^{j} then y is {\tilde{G}}^{j} \\ j = 1, 2 \dots M \end{matrix}$ (6)

Where x_i (i = 1, 2, …, n) and y are the input and output of the GT2FLS, ${\tilde{F}}_{i}^{j}$ and ${\tilde{G}}^{j}$ are general type-2 antecedent and the consequent sets. A mapping from input GT2FSs to output GT2FSs is given by the inference engine that merges rules. Because α-plane representation for fuzzy set is used, the firing set for each related IT2FS is shown as following: $F_{α}^{j} (X) = [{\underline{f}}_{α}^{j} (X), {\bar{f}}_{α}^{j} (X)]$ (7) ${\underline{f}}_{α}^{j} (X) = {\underline{μ}}_{{\tilde{F}}_{1_{α}}^{j}} * {\underline{μ}}_{{\tilde{F}}_{2_{α}}^{j}} * \dots * {\underline{μ}}_{{\tilde{F}}_{n_{α}}^{j}}$ (8) ${\bar{f}}_{α}^{j} (X) = {\bar{μ}}_{{\tilde{F}}_{1_{α}}^{j}} * {\bar{μ}}_{{\tilde{F}}_{2_{α}}^{j}} * \dots * {\bar{μ}}_{{\tilde{F}}_{n_{α}}^{j}}$ (9)

Here ${\underline{f}}_{α}^{j} (X)$ and ${\bar{f}}_{α}^{j} (X)$ are the lower and upper MFs of the jth rule with level of α, and * indicates product t-norm. A type reducer changes the output of the inference engine which is a type-2 fuzzy set into a type-1 fuzzy set before defuzzification. Five kinds of reducers are presented in [21] which are based on calculating the centroid of an IT2FS. Due to uniformly secondary grade of IT2FLS, the output of the type reduction in IT2FLS can be defined only with its left-end point y_r and right-end point y_r.

KM iterative algorithms, introduced two algorithms for calculating these two end points in [22], is presented by Mendel and Karnik. In comparison to the other type reduction methods, center of sets (COS) is used more widely because of its computation simplicity by the KM iterative algorithm [23]. If singleton fuzzifier is used, product inference engine and COS type reducer, left and right end points for each part of GT2FLS based on α- representation theorem can be shown as follows: $y_{l_{α}} = \frac{\sum_{j = 1}^{L} {\bar{f}}_{α}^{j} θ_{l_{α}}^{j} + \sum_{j = L + 1}^{M} {\underline{f}}_{α}^{j} θ_{l_{α}}^{j}}{\sum_{j = 1}^{L} {\bar{f}}_{α}^{j} + \sum_{j = L + 1}^{M} {\underline{f}}_{α}^{j}} = θ_{l_{α}}^{T} ξ l_{α}$ (10) where in this formula $θ_{l_{α}}^{j}$ is the left-end point of jth consequent set with level of α, $θ_{l_{α}} = [θ_{l_{α}}^{1}, \dots, θ_{l_{α}}^{M}]^{T}$ , $ξ_{l_{α}}^{j} = [\frac{{\underline{f}}_{α}^{j}}{D_{l_{α}}}, \frac{{\bar{f}}_{α}^{j}}{D_{l_{α}}}]$ , $D_{l_{α}} = \sum_{j = 1}^{L} {\bar{f}}_{α}^{j} + \sum_{j = L + 1}^{M} {\underline{f}}_{α}^{j}$ and $ξ_{l_{α}} = {[ξ_{l_{α}}^{1}, \dots, ξ_{l_{α}}^{M}]}^{T}$ . In addition, $y_{r_{α}} = \frac{\sum_{j = 1}^{R} {\underline{f}}_{α}^{j} θ_{r_{α}}^{j} + \sum_{j = R + 1}^{M} {\bar{f}}_{α}^{j} θ_{r_{α}}^{j}}{\sum_{j = 1}^{R} {\underline{f}}_{α}^{j} + \sum_{j = R + 1}^{M} {\bar{f}}_{α}^{j}} = θ_{r_{α}}^{T} ξ_{r_{α}}$ (11)

Where $θ_{r_{α}}^{j}$ is the right end point of jth consequent set with level of α, $θ_{r_{α}} = [θ_{r_{α}}^{1}, \dots, θ_{r_{α}}^{M}]^{T}$ , $ξ_{r_{α}}^{j} = [\frac{{\underline{f}}_{α}^{j}}{D_{r_{α}}}, \frac{{\bar{f}}_{α}^{j}}{D_{r_{α}}}]$ $D_{r_{α}} = \sum_{j = 1}^{R} {\underline{f}}_{α}^{j} + \sum_{j = R + 1}^{M} {\bar{f}}_{α}^{j}$ and $ξ_{r_{α}} = {[ξ_{r_{α}}^{1}, \dots, ξ_{r_{α}}^{M}]}^{T}$ . In the meanwhile, performing KM iterative algorithm can specify R and L for each individual IT2FLS of level α. From the combination of all of these obtained intervals into a type-1 fuzzy set like Fig. 3, a crisp output can be obtained using centroid defuzzification as:

$\begin{matrix} y & = \frac{\sum_{α} α (y_{l_{α}} + y_{r_{α}})}{2 \sum_{α} α} = \frac{\sum_{α} α (y_{l_{α}} + y_{r_{α}})}{K + 1}, \\ α & = {0, 1 / K, \dots, (K - 1) / K, 1} \end{matrix}$ (12)

Where K+1 shows the number of the α-planes or in other words it determines the number of individual IT2FLSs.

3 Sliding mode control

The general dynamic of SISO nonlinear system can be described as [8]: $\begin{matrix} x^{(n)} = f (\underline{X}, t) + f (\underline{X}, t) u + d (\underline{X}, t) \\ y = x \end{matrix}$ (13) where $f (\underline{X}, t)$ and $g (\underline{X}, t)$ are not exactly known, but bounded nonlinear functions; d (t) is an external disturbance and the bound, D, is unknown; y and u are output and input of the nonlinear system, respectively; and $\underline{X} = [x, \dot{x}, \dots, x^{(n - 1)}] = [x_{1}, x_{2}, \dots, x_{n}]$ is the nth order state vector of the system which is considered to be accessible for measurement [14 , 25]. The following assumptions are necessary, in order to plan the intelligent control-input:

Assumption 1. An unknown constant D exists which shows a limit for external disturbance, i.e. $| d (t) | \leq D$ (14)

Assumption 2. The system where described in Equation (13) is controllable, i.e. $g (\underline{X}, t) \neq 0$ . Without loss of generality, the $g (\underline{X}, t)$ is bounded by $0 < \underline{g} < g (\underline{X}, t) < \bar{g}$ where $\underline{g}$ and $\bar{g}$ are positive constants.

The goal of controller is to calculate a feedback control $u (\underline{X})$ whereas the state vector $\underline{X}$ of the nonlinear system, is shown in Equation (13), follows the preferred state vector $\underset{-_{d}}{X} = [x_{d}, {\dot{x}}_{d}, \dots, x_{d}^{(n - 1)}]$ in the existence of uncertainties and disturbances; so, in the following, the tracking error is: $\underline{E} = \underline{X} - \underset{-_{d}}{X} = [e, \dot{e}, \dots, e^{(n - 1)}]^{T}$ (15)

Now, a sliding surface can be defined in the state error space as follows: $S (E) = C_{-}^{T} \underline{E} = e^{(n - 1)} + c_{n - 1} e^{(n - 2)} + \dots + c_{1} e$ (16)

Where the coefficients $\underline{C} = [c_{n}, \dots, c_{1}]$ are taken whereas the polynomial h (λ) = λ^(n-1) + c_n-1λ^(n-2) + … + c₁(λ shows a Laplace operator) is Hurwitz [24]. A sufficient condition that the control system is stable is given in [8] as: $\dot{S} \leq - η sgn (s)$ (17) where η is a constant positive parameter. Equation (18) shows the time derivative of both sides of Equation (16), $\begin{matrix} \dot{S} = \sum_{i = 1}^{n - 1} c_{i} e^{i} + x^{(n)} - x_{d}^{(n)} = \sum_{i = 1}^{n - 1} c_{i} e^{i} + \\ f (\underline{X}, t) + (g (\underline{X}, t) u + d (\underline{X}, t) - x_{d}^{(n)} \end{matrix}$ (18)

By substituting Equations (18) into (17), sliding condition can be presented as:

$\begin{matrix} \sum_{i = 1}^{n - 1} c_{i} e^{i} + f (\underline{X}, t) + (g (\underline{X}, t) u + d (\underline{X}, t) \\ - x_{d}^{(n)} \leq - η sgn (s)) \end{matrix}$ (19)

Now, the control problem is to find the intelligent control input u^* whereas the sliding condition in Equation (19) is satisfied. If $d (\underline{X}, t), g (\underline{X}, t)$ and $f (\underline{X}, t)$ are known, the intelligent sliding mode control law can be defined as below: $\begin{matrix} u^{*} = \frac{1}{g (\underline{X}, t)} [- \sum_{i = 1}^{n - 1} c_{i} e^{i} - f (\underline{X}, t) - d (\underline{X}, t) \\ + x_{d}^{(n)} - η_{Δ} sgn (s)] \end{matrix}$ (20) where η_Δ > η > 0.

4 Indirect adaptive general typ-II fuzzy control design

When $d (\underline{X}, t), g (\underline{X}, t)$ and $f (\underline{X}, t)$ are known, by substitute $f (\underline{X}, t)$ , $g (\underline{X}, t)$ by the GT2FLS $\hat{f} (\underline{X} | θ_{f})$ , $\hat{g} (\underline{X} | θ_{g})$ , the resulting control input can be written as follow: $\begin{matrix} u^{*} = \frac{1}{\hat{g} (\underline{X} | θ_{g})} [- \sum_{i = 1}^{n - 1} c_{i} e^{i} - \hat{f} (\underline{X} | θ_{f}) + x_{d}^{(n)} \\ - (η_{Δ} + D) sgn (s)] \end{matrix}$ (21) where $\hat{f} (\underline{X} | θ_{f}) = \sum_{α} α ({\hat{f}}_{l_{α}} + {\hat{f}}_{r_{α}}) / K + 1 = θ_{f}^{T} ξ_{f}$ and $\hat{g} (\underline{X} | θ_{g}) = \sum_{α} α ({\hat{g}}_{l_{α}} + {\hat{g}}_{r_{α}}) / K + 1 = θ_{g}^{T} ξ_{g}$ .

Theorem 1. By choosing the fuzzy based adaptive laws as below, and the system that described in (13) with the control input u in (21), the closed loop system outputs are regularly bounded and the tracking error will diverge to zero asymptotically. ${\dot{θ}}_{fl} = γ_{1} s ξ_{fl}$ (22) ${\dot{θ}}_{fr} = γ_{2} s ξ_{fr}$ (23) ${\dot{θ}}_{gl} = γ_{3} s ξ_{gl} u$ (24) ${\dot{θ}}_{gr} = γ_{4} s ξ_{gr} u$ (25)

Proof. By introducing the optimal parameters of fuzzy systems as below: $θ_{f}^{*} = {arg}_{θ f φ f} min [x R^{n} sup | \hat{f} (\underline{X} | θ_{f}) - f (\underline{X}, t)]$ (26) $θ_{g}^{*} = {arg}_{θ g φ g} min [x R^{n} sup | \hat{g} (\underline{X} | θ_{g}) - g (\underline{X}, t)]$ (27)

Where φ_f and φ_g are demarcated as φ_f = { θ_f∈Rⁿ||θ_f| ≤ M_f } and φ_g ={ θ_g ∈ Rⁿ||θ_g| ≤ M_g }, where M_f and M_g are positive constants. The minimum estimate error is written as: $ω = [f (\underline{X}, t) - \hat{f} (\underline{X} | θ_{f}^{*})] + [g (\underline{X}, t) - \hat{g} (\underline{X} | θ_{g}^{*})] u$ (28)

Then by replacing Equations (21 and 28) into (18), the derivative of sliding surface can be shown as: $\begin{matrix} \dot{S} = \sum_{i = 1}^{n - 1} c_{i} e^{i} + f (\underline{X}, t) + g (\underline{X}, t) u + d (\underline{X}, t) - x_{d}^{(n)} \\ = f (\underline{X}, t) - \hat{f} (\underline{X} | θ_{f}) \\ + (g (\underline{X}, t) - \hat{g} (\underline{X} | θ_{g})) u \\ + d (\underline{X}, t) - (η_{Δ} + D) sgn (s) \\ = \hat{f} (\underline{X} | θ_{f}^{*}) - \hat{f} (\underline{X} | θ_{f}) + (\hat{g} (\underline{X} | θ_{g}^{*}) \\ - \hat{g} (\underline{X} | θ_{g})) u + d (\underline{X}, t) - (η_{Δ} + D) sgn (s) + ω \\ = (θ_{f^{T}}^{*} ξ_{f} - θ_{f^{T}} ξ_{f}) + (θ_{g^{T}}^{*} ξ_{g} - θ_{g}^{T} ξ_{g}) u + d (\underline{X}, t) \\ - (η_{Δ} + D) sgn (s) + ω \end{matrix}$ (29) In addition, $\begin{matrix} = φ_{f}^{T} ξ_{f} + φ_{g}^{T} ξ_{g} u + d (\underline{X}, t) - (η_{Δ} + D) sgn (s) + ω \\ = \frac{\sum_{α \in [0, 1]} α (φ_{{fl}_{a}}^{T} ξ_{{fl}_{α}} + φ_{{fr}_{a}}^{T} ξ_{{fr}_{α}})}{K + 1} \\ + \frac{\sum_{α \in [0, 1]} α (φ_{{gl}_{a}}^{T} ξ_{{gl}_{α}} + φ_{{gr}_{a}}^{T} ξ_{{gr}_{α}})}{K + 1} + d (\underline{X}, t) \\ - (η_{Δ} + D) sgn (s) + ω \end{matrix}$ (30)

Where $φ_{f} = θ_{f}^{*} - θ_{f}$ and $φ_{g} = θ_{g}^{*} - θ_{g}$ . Then, the function of the Lyapunov can be written as:

$\begin{matrix} V = \frac{1}{2} S^{2} + \frac{1}{4 γ_{1}} φ_{fl}^{T} φ_{fl} + \frac{1}{4 γ_{2}} φ_{fr}^{T} φ_{fr} \\ + \frac{1}{4 γ_{3}} φ_{gl}^{T} φ_{gl} + \frac{1}{4 γ_{4}} φ_{gr}^{T} φ_{gr} \end{matrix}$ (31)

Where γ₁ and γ₂ are positive factor. The time derivation of Equation (31) is: $\begin{matrix} \dot{V} = s \dot{s} + \frac{1}{2 γ_{1}} φ_{fl}^{T} {\dot{φ}}_{fl} + \frac{1}{2 γ_{2}} φ_{fr}^{T} {\dot{φ}}_{fr} + \frac{1}{2 γ_{3}} φ_{gl}^{T} {\dot{φ}}_{gl} \\ + \frac{1}{2 γ_{4}} φ_{gr}^{T} {\dot{φ}}_{gr} \\ = s (φ_{f}^{T} ξ_{f} + φ_{g}^{T} ξ_{g} u + d (\underline{X}, t) - (η_{Δ} + D) sgn (s) + ω) \\ + \frac{1}{2 γ_{1}} φ_{fl}^{T} {\dot{φ}}_{fl} + \frac{1}{2 γ_{2}} φ_{fr}^{T} {\dot{φ}}_{fr} + \frac{1}{2 γ_{3}} φ_{gl}^{T} {\dot{φ}}_{gl} + \frac{1}{2 γ_{4}} φ_{gr}^{T} {\dot{φ}}_{gr} \\ = \frac{1}{2 γ_{1}} φ_{f}^{T} ({\dot{φ}}_{fl} + γ_{1} s ξ_{fl}) + \frac{1}{2 γ_{2}} φ_{fr}^{T} ({\dot{φ}}_{fr} + γ_{2} s ξ_{fr}) \\ + \frac{1}{2 γ_{3}} φ_{gl}^{T} ({\dot{φ}}_{gl} + γ_{3} s ξ_{gl} u) + \frac{1}{2 γ_{4}} φ_{gr}^{T} ({\dot{φ}}_{gr} + γ_{4} s ξ_{gr} u) \\ + sd (\underline{X}, t) - s (η_{Δ} + D) sgn (s) + s ω \end{matrix}$ (32)

By replacing ${\dot{φ}}_{fl} = - {\dot{θ}}_{fl}$ , ${\dot{φ}}_{fr} = - {\dot{θ}}_{fr}$ , ${\dot{φ}}_{gl} = - {\dot{θ}}_{gl}$ and ${\dot{φ}}_{gr} = - {\dot{θ}}_{gr}$ into Equations (22–25) and then into (32), then we can obtain: $\begin{matrix} \dot{V} = sd (\underline{X}, t) - s (η_{Δ} + D) sgn (s) + s ω \\ = sd (\underline{X}, t) - | s | (η_{Δ} + D) sgn (s) + s ω \\ \leq - | s | η_{Δ} + s ω \end{matrix}$ (33)

By using of the approximation theorem [26 –31], it can be expected that the term sω should be near to zero in Fuzzy Logic System, $\dot{V} \leq - | s | η_{Δ} \leq 0$ (34) when $\underline{C}$ is Hurwitz and by using corollary of Barbalat’s lemma [10], we can conclude that $lim_{t \to \infty} | e (t) | = 0$ , so $lim_{t \to \infty} | s (e) | = 0$ . Finally, The proof is satisfied for proposed method. The overall scheme of the proposed method is shown in Fig. 4.

5 Simulation and results

I. Non-linear dynamic of Power Chaos system:

Recently, CHAOTIC OSCILLATION DAMPING OF POWER SYSTEM (CODOPS) has become one of the most interesting topics of dynamic systems [32 –35]. An interconnected power system will be introduced in this section; then, its dynamic model will be extracted. Next, the proposed controller will be applied to the chaotic power system containing model uncertainties and external disturbances. Figure 5 shows a simple interconnected power system.

In the Fig. 4, G₁ and G₂ are the equivalent generators of systems 1 and 2, respectively; 3 and 4 are the main transformers of systems 1 and 2. Then, 5, 6 and 7 are considered as the load, circuit breakers and tie line, respectively. This system exhibits some important aspects of the behavior of multi-machine systems [32 –34].

Different forms of chaotic systems with linear or nonlinear damping specifications have been represented in the literature [2, 4]. In the state space form, the CODOPS equation (called swing equations) can be described as [32 –35]: $\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - \frac{1}{H} P_{s} sin (x_{1}) - \frac{D}{H} x_{2} + \frac{1}{H} P_{m} \\ + \frac{1}{H} P_{e} cos (β t) + u^{*} (t) + d (t) \end{matrix}$ (35) where in Equation (35), x₁ is the relative angle between voltages of generators 1 and 2, P_s is the maximum electric power transmitting from generator 1 to 2, H and D are inertia and damping coefficient of generator, respectively. P_m is the machinery power of generator 1, u^* (t) is the input signal and d (t) is an external disturbance with unknown but bounded amplitude. This chaotic power system exhibits complex dynamics for values of constant values of H = 0.04, β = 10, D = 0.15, P_s = 3 and P_m = 2 and P_e > |34.715D - 3.1483P_m|≅1.0894 [11]. Figure 6 shows irregular motion of this system without controller for P_e = 2 and initial states of (x₁ (0) , x₂ (0)) = (1, - 0.5).

Matlab software is used to simulate the proposed controller. To solve the system of deferential, Fourth-order Runge-Kutta method is utilized with sampling time of 0.01 Sec. Now, in order to demonstrate the efficiency of the IDAGT2FSMC, the proposed method is applied to the second order nonlinear chaotic Power system [32 –35] as a case study and the results are compared with indirect adaptive type-I fuzzy sliding mode controller (IDAT1FSMC) in [14], indirect adaptive interval type-II fuzzy sliding mode controller (IDAIT2FSMC) in [24] traditional Sliding Mode controller (SMC) and PID controller.

Simulations results indicate that the system output y mimics the desired output y_d = 5 sin(0.65πt) –3 cos(0.4πt) and a square wave disturbance of amplitude 5 and period 14s, separately. whereas the initial state is (x₁ (0) , x₁ (0)) = (1, - 0.5). Let P_e as 1.5 < P_e < 2 that sup(P_e) =2. Furthermore, in order to create a more difficult challenge for the proposed control, the external disturbance d is added to the output of the system, which is a Gaussian noise with zero average and variance of 0.009. Figure 7 shows the membership functions for α = 0 and α = 1. Accordingly, to construct the controller, 9 rules are required [30]. The initial consequent parameters ${\dot{θ}}_{fl} (0), {\dot{θ}}_{fr} (0), {\dot{θ}}_{gl} (0)$ and ${\dot{θ}}_{gr} (0)$ are chosen zero.

Figure 8 demonstrates the trajectory of the output x₁. This figure indicates that the proposed IDAGT2FSMC controller not only demonstrates better tracking than the IDAT1FSMC control and IDAIT2FSMC control method introduced in [14, 24], but also it is robust against the system uncertainties and external disturbance. Figure 9 demonstrates the error signal. As shown in Fig. 9, the error effort signal in IDAGT2FSMC controller is smother than IDAT1FSMC and IDAIT2FSMC control methods.

6 Conclusion

In this article, by applying a combination of indirect adaptive methods with general type-II fuzzy sliding mode control, a novel robust and simple control strategy is proposed for a class of nonlinear power systems dealing with external disturbances. Mathematical analysis guarantees the stability of the closed loop system in sense of Lyapunov and using this method outcome H^∞ tracking performance. The control design methodology is tested on a chaotic power system, which represents a large power system. To achieve optimal performance based on Lyapunov analysis, the free parameter of the controller can be tuned online by adaptive laws. The new technique was very simple and free of complex computations. The simulation results show that the proposed indirect adaptive general type-2 fuzzy sliding mode controller achieves a desirable control performance while facing various uncertainties. It is more robust against sudden and fast external disturbances, when compared with the IDAT1FSMC, IDAIT2FSMC, SMC and PID controllers.

References

Yin

, Zhong

and Chen

, Design of sliding mode controller for a class of fractional-order chaotic systems, Commun Nonlinear Sci Numer Simul17(1) (2012), 356–366.

Niknam

and Khooban

M.H.

, Fuzzy sliding mode control scheme for a class of non-linear uncertain chaotic systems, IET Sci Meas Technol7(5) (2013), 249–255. doi: 10.1049/iet-smt.2013.0039

Khooban

M.H.

, Design an intelligent proportional-derivative PD feedback linearization control for nonholonomic-wheeled mobile robot, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology26(4) (2014), 1833–1843.

Sha Hsadeghi

, Khooban

M.H.

and Niknam

, A robust and simple optimal type II fuzzy sliding mode control strategy for a class of nonlinear chaotic systems, Journal of Intelligent and Fuzzy Systems: Applications in Engineering and Technology27(4) (2014), 1849–1859.

Soltanpour

M.R.

, Khooban

M.H.

and Soltani

, Robust fuzzy sliding mode control for tracking the robot manipulator in joint space and in presence of uncertainties, Robotica32(03) (2014), 433–446.

Soltanpour

M.R.

, Khooban

M.H.

and Khalghani

M.R.

, An optimal and intelligent control strategy for a class of nonlinear systems: Adaptive fuzzy sliding mode, Journal of Vibration and Control (2014), 1077546314526920.

Alfi

, Akbarzadeh Kalat

and Khooban

M.H.

, Adaptive fuzzy sliding mode control for synchronization of uncertain non-identical chaotic systems using bacterial foraging optimization, Journal of Intelligent and Fuzzy Systems26(5) (2014), 2567–2576.

Jean-Jacques

Slotine E.

and Li

Weiping

, Applied nonlinear control, Vol. 60. Englewood Cliffs, NJ: Prentice-Hall, 1991.

Lin

Chih-Min

and Hsu

Chun-Fei

, Adaptive fuzzy sliding-mode control for induction servomotor systems, Energy Conversion, IEEE Transactions on19(2) (2004), 362–368.

10.

Wang Ahmad

, Rad

and Chan

P.T.

, Indirect adaptive fuzzy sliding mode control: Part I: Fuzzy switching, Fuzzy Sets and Systems122(1) (2001), 21–30.

11.

Chiclana

and Zhou

S.-M.

, Type-reduction of general type-2 fuzzy sets: The type-1 OWA approach, International Journal of Intelligent Systems28(5) (2013), 505–522.

12.

Ghaemi

Mostafa

, Seyyed Kamal Hosseini-Sani and Mohammad Hassan Khooban, Direct adaptive general type-2 fuzzy control for a class of uncertain non-linear systems, IET Science, Measurement & Technology8(6) (2014), 518–527.

13.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning–I, Information Sciences8(3) (1975), 199–249.

14.

Khooban

Mohammad Hassan

and Niknam

Taher

, A new and robust control strategy for a class of nonlinear power systems: Adaptive general type-II fuzzy, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering229(6) (2015), 517–528.

15.

Liu

, An efficient centroid type-reduction strategy for general type-2 fuzzy logic system, IEEE Comput Intell Soc, Walter J Karplus Summer Res Grant Rep, 2006.

16.

Astrom

K.J.

and Wittenmark

, Adaptive control: Second Edition, Addison-Wesley Pub Co, Newyork, 1994, pp. 19–41.

17.

Akbarzadeh-T

M.R.

and Shahnazi

, Direct adaptive fuzzy PI sliding mode control of systems with unknown but bounded disturbances, Iranian Journal of Fuzzy Systems3(2) (2006), 33–51.

18.

Lin

Tsung-Chih

, Liu

Han-Leih

, and Kuo

Ming-Jen

, Direct adaptive interval type-2 fuzzy control of multivariable nonlinear systems, Engineering Applications of Artificial Intelligence22(3) (2009), 420–430.

19.

Pan

, Er

M.J.

, Huang

, et al., Fire-rule-based direct adaptive type-2 fuzzy H∞ tracking control, Eng Appl Artif Intell24(7) (2011), 1174–1185.

20.

Liu

, An efficient centroid type-reduction strategy for general type-2 fuzzy logic system, Inf Sci178(9) (2008), 2224–2236.

21.

Karnik

N.N.

and Mendel

J.M.

, Type-2 fuzzy logic systems: Type-reduction, IEEE Int Conf on Systems, Man, and Cybernetics, 19982042 (1998), 2046–2051.

22.

Karnik

N.N.

and Mendel

J.M.

, Centroid of a type-2 fuzzy set, Inf Sci132(1– 4) (2001), 195–220.

23.

Karnik

N.N.

, Mendel

J.M.

and Qilian

, Type-2 fuzzy logic systems, IEEE Trans Fuzzy Syst7(6) (1999), 643–658.

24.

Ghaemi

Mostafa

and Mohammad Reza Akbarzadeh

Totonchi

, Indirect Adaptive Interval Type-2 Fuzzy PI Sliding Mode Control for a Class of Uncertain Nonlinear Systems, Iranian Journal of Fuzzy Systems11(5) (2014), 1–21.

25.

Ghaemi

Mostafa

, Mohammad Reza Akbarzadeh

Totonchi

and Mohsen

JalaeianFarimani

, Optimization of membership functions for an indirect adaptive interval type-2 fuzzy sliding mode control using particle swarm, Hydrology Research11(5) (2014), 1–21.

26.

Wang

L.-X.

, Stable adaptive fuzzy control of nonlinear systems, Fuzzy Systems, IEEE Transactions on1(2) (1993), 146–155.

27.

Soltanpour

Mohammad Reza

Khooban

Mohammad Hassan

and Niknam

Taher

, A robust and new simple control strategy for a class of nonlinear power systems: Induction and servomotors, Journal of Vibration and Control (2014), 1077546314543729.

28.

Soltanpour

Mohammad Reza

, Otadolajam

Pooria

and Khooban

Mohammad Hassan

, Robust control strategy for electrically driven robot manipulators: Adaptive fuzzy sliding mode, IET Science, Measurement & Technology9(3) (2014), 322–334.

29.

Veysi

Mohammad

, Soltanpour

Mohammad Reza

and Khooban

Mohammad Hassan

, A novel self-adaptive modified bat fuzzy sliding mode control of robot manipulator in presence of uncertainties in task space, Robotica, pp. 1–20.

30.

Khooban

Mohammad Hassan

and Soltanpour

Mohammad Reza

, Swarm optimization tuned fuzzy sliding mode control design for a class of nonlinear systems in presence of uncertainties, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology24(2) (2013), 383–394.

31.

Soltanpour

Mohammad Reza

, Zolfaghari

Behrouz

, Soltani

Mahmoodreza

and Khooban

Mohammad Hassan

, Fuzzy sliding mode control design for a class of nonlinear systems with structured and unstructured uncertainties, International Journal of Innovative Computing, Information and Control9(7) (2013), 2713–2726.

32.

Chen

Xingwu

, Zhang

Weinian

and Zhang

Weidong

, Chaotic and subharmonic oscillations of a nonlinear power system, Circuits and Systems II: Express Briefs, IEEE Transactions on52(12) (2005), 811–815.

33.

Noroozi

Navid

, Khaki

Behnam

and Seifi

Alireza

, Chaotic oscillations damping in power system by finite time control theory, International Review of Electrical Enginerring-IREE, 3(6) (2008), 1032–1038.

34.

Luo

Xiao Shu

, Passivity-based adaptive control of chaotic oscillations in power system, Chaos, Solitons & Fractals31(3) (2007), 665–671.

35.

Khooban

Mohammad Hassan

and Niknam

Taher

, A new intelligent online fuzzy tuning approach for multi-area load frequency control: Self Adaptive Modified Bat Algorithm, International Journal of Electrical Power & Energy Systems71 (2015), 254–261.