An exact inversion method for multi-input-single-output decomposable TS fuzzy systems

Abstract

Almost all exact inversion methods provide inverse solutions for only one input variable of fuzzy systems. These methods have certain limitations on the fuzzy system structure such as monotonic rule bases, singleton rule consequents, and invertibility check. These requirements limit the modeling capabilities of the fuzzy systems and also may result in poor application performances. In this study, an exact analytical inversion method is presented for multi-input-single-output decomposable TS fuzzy systems with either singleton or linear consequents. In the proposed method, fuzzy system structures do not need to have monotonic rule bases, singleton rule consequents, or any invertibility conditions. Thus, more flexible fuzzy systems can be used in inverse model based applications. The proposed method provides a simple and systematic way to obtain unique inverse solutions of all input variables simultaneously with respect to any desired system output value. For this purpose, an inversion trajectory approach that guarantees the existence and uniqueness of the inverse solutions is introduced. The inversion trajectory consists of a set of paths defined on the specific edges of universe of discourses of the decomposed fuzzy subsystems. Using this approach, the inverse definition of the overall fuzzy system can easily be derived only by inverting the related decomposed fuzzy subsystems on this inversion trajectory and then combining their inverse definitions. In this way, the inverse definition of the overall fuzzy system is obtained as consisting of analytical solutions of linear and quadratic equations for singleton and linear consequent cases, respectively. Simulation studies are given for the inversion of two and three-input-single-output fuzzy systems, and the exactness and effectiveness of the proposed method are demonstrated.

Keywords

Fuzzy systems decomposability inversion multivariable systems Takagi-Sugeno fuzzy systems

1 Introduction

Inverse model based approaches are widely used in many applications such as control, diagnosis, forecasting, decision making, etc [1 , 40]. In these approaches, the application performance directly depends on the accuracy of the derived system model and also its invertibility. However, it is not an easy task to obtain useful mathematical models with high accuracy. Fuzzy systems provide an easy and powerful tool for the modeling of complex systems due to their universal approximation property [12 , 24]. Thanks to this property, fuzzy systems are effectively used in many model based applications [11 , 27]. Therefore, an inversion of a fuzzy system becomes an important issue in inverse model based applications. In literature, various inversion methods are proposed for fuzzy systems.

The methods proposed for the inversion of fuzzy systems can be classified into two main categories as approximate inversion methods [2 , 34] and exact inversion methods [4 , 33]. The main disadvantage of the approximate inversion methods is that their approximate solutions reduce robustness and stability. Additionally, most of the approximate inversion methods are based on iterative search algorithms such as Levenberg–Marquardt algorithm [9], Ridders algorithm [22], Newton method [2], genetic algorithms [3], and Big Bang-Big Crunch algorithm [34]. These methods need significant computation time due to their iterative nature and the convergence could not be guaranteed in each sampling time. Therefore, these methods could not be easily applicable to real-time applications. The main advantage of these approximate inversion methods is that they mostly do not need any certain limitations on the fuzzy system structure, and there is no need to check any invertibility condition.

Exact inversion methods can easily be applied to real-time systems due to their low computation time compared to iterative approximate inversion methods. Additionally, they provide higher robustness depending on the exactness of the inverse solutions. Moreover, theoretical analyses can be performed for the systems including exact inverse models of fuzzy systems. Some of the exact inversion methods use the direct rule by rule inversion techniques [28 –31], and they need certain limitations on the fuzzy system structure such as strong triangular input partitioning, singleton rule consequents, and monotonic rule base. In order to eliminate the need for the monotonic rule base, various exact inversion methods are proposed based on the decomposability property of fuzzy systems [33 , 39]. However, these methods have still limitations such as strong triangular input partitioning and singleton rule consequents. In [4], an exact inversion method is given for decomposable fuzzy systems with linear consequents. The inversion method which provides inverse solutions for fuzzy systems having more flexible structures is introduced in [5]. There also exist exact and approximate inversion methods proposed for interval type-2 fuzzy systems [6 , 36]. In these inversion methods proposed for decomposable fuzzy systems, multiple solutions are usually obtained, and a selection criterion is required to choose one of the solutions for any application purpose [4 , 32].

The above mentioned inversion methods provide inverse solutions for only one of input variables of fuzzy systems. Therefore, these methods are not suitable for solving inversion problems of multi-input-single-output (MISO) fuzzy systems. In [8], an exact inversion method is proposed for MISO TS fuzzy systems. This method uses the rule by rule inversion approach and provides exact inverse solutions for all input variables of MISO fuzzy systems. However, it is only applicable to decomposable fuzzy systems having monotonic rule bases and singleton consequents. Additionally, the invertibility of the fuzzy system needs to be checked in this method. These requirements limit the modeling capabilities of the fuzzy systems and also may result in poor application performances.

In this study, an exact analytical inversion method is proposed for MISO decomposable TS fuzzy systems with either singleton or linear consequents by using the newly introduced inversion trajectory approach. The proposed method provides unique inverse solutions for all input variables simultaneously with respect to any desired system output value. In the proposed inversion method, an appropriate inversion trajectory is determined with certain criteria on the multi-dimensional universe of discourse of the fuzzy system by evaluating system output values corresponding to vertices of all fuzzy input meshes. This inversion trajectory consists of a set of paths defined on the specific edges of the multi-dimensional universe of discourse. The combination of the related output intervals corresponding to these specific edges gives the overall output interval of the fuzzy system. In this way, the existence and uniqueness of the inverse solutions are guaranteed. Then, the inverse definition of the overall fuzzy system can easily be derived only by inverting the fuzzy subsystems on the inversion trajectory and then combining their inverse definitions. Here, the fuzzy subsystems are defined by using the matrix multiplication representation in order to provide compactness and computational simplicity in the inversion procedure. Simulation studies are done to illustrate the exactness and effectiveness of the proposed method.

The main contributions of this study can be summarized as follows. 1) The proposed method provides inverse solutions for all input variables of a MISO decomposable TS fuzzy system simultaneously. 2) The existence and uniqueness of the inverse solutions are guaranteed by the newly introduced inversion trajectory approach. 3) The exactness of inverse solutions is guaranteed since the solutions are derived in an analytical way. 4) The proposed method has the potential to improve the accuracy and robustness of inverse fuzzy model based applications by allowing the use of more flexible fuzzy systems. 5) The proposed inversion procedure is simple and straightforward. Therefore, an inverse mapping of a fuzzy system is easily derived. 6) The proposed method is very suitable for real time applications since the overall inverse mapping consists only of the parametric solutions of the first or second order equations.

The rest of the paper is organized as follows: In Section 2, preliminaries are given. In Section 3, the proposed inversion method is introduced. In Section 4, simulation studies are given to illustrate the exactness and effectiveness of the proposed method. Finally, conclusions are outlined in Section 5.

2 Preliminaries

2.1 Decomposable TS fuzzy systems

The fuzzy systems having special input partitioning can be decomposed into several fuzzy subsystems [19]. These fuzzy subsystems are the simplest fuzzy systems that have only two fuzzy sets for each input variable, and thus, their rule bases consist of 2ⁿ rules. For any given input vector, the output of the overall decomposable fuzzy system is always generated by only one of its fuzzy subsystems. Therefore, the decomposition property provides a way to evaluate the overall fuzzy system as the collection of its fuzzy subsystems.

Let us consider a TS fuzzy system with n input variables, x_k ∈ X_k, k = 1, . . . , n, and one output variable, y ∈ Y, and assume that the input variables are defined by using the strong triangular partitioning shown in Fig. 1.

Fig. 1

Strong triangular fuzzy partition.

The rule structure of this TS fuzzy system can be defined as [33].

$\begin{matrix} R^{i_{1}, i_{2}, \dots, i_{n}} : IF x_{1} is A_{1}^{i_{1}} and x_{2} is A_{2}^{i_{2}} \dots and x_{n} is A_{n}^{i_{n}} \\ THEN y = f^{i_{1}, i_{2}, \dots, i_{n}} (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}$ (1) where $A_{k}^{i_{k}}$ , k = 1, . . . , n is the triangular fuzzy set representing the input x_k and f^{i₁,i₂,…,i_n} is the rule consequent crisp function. If N_k fuzzy sets are used for the description of x_k, i_k∈ I_k = { 1, 2, …, N_k }, k = 1, . . . , n the entire rule base has $N = \prod_{k = 1, \dots, n} N_{k}$ rules with the corresponding index set I = I₁ × I₂ × … × I_n. For singleton and linear rule consequents, f^{i₁,i₂,…,i_n} can be defined as f^{i₁,i₂,…,i_n} = q^{i₁,i₂,…,i_n} and $f^{i_{1}, i_{2}, \dots, i_{n}} = q_{0}^{i_{1}, i_{2}, \dots, i_{n}} + q_{1}^{i_{1}, i_{2}, \dots, i_{n}} x_{1} + \dots + q_{n}^{i_{1}, i_{2}, \dots, i_{n}} x_{n}$ , respectively. Here, $q_{j}^{i_{1}, i_{2}, \dots, i_{n}}$ (j = 0, …, n), denotes the related coefficients of f^{i₁,i₂,…,i_n}. As it is seen from Fig. 1, the strong triangular input partitioning guarantees that always only two adjacent fuzzy sets $A_{k}^{i_{k}}$ and $A_{k}^{i_{k} + 1}$ are activated for any value of x_k by satisfying that $A_{k}^{i_{k}} (x_{k}) + A_{k}^{i_{k} + 1} (x_{k}) = 1$ .

This TS fuzzy system can be decomposed into fuzzy subsystems TS^{i₁,i₂,…,i_n} described by the following 2ⁿ rules $\begin{matrix} R_{v_{1}, v_{2}, \dots, v_{n}}^{i_{1}, i_{2}, \dots, i_{n}} : \\ IF x_{1} is A_{1}^{i_{1} + v_{1}} and x_{2} is A_{2}^{i_{2} + v_{2}} \dots and x_{n} is A_{n}^{i_{n} + v_{n}} \\ THEN y = f^{i_{1} + v_{1}, i_{2} + v_{2}, \dots, i_{n} + v_{n}} (x_{1}, x_{2}, \dots, x_{n}), \\ v_{k} \in {0, 1}, k = 1, \dots, n \end{matrix}$ (2)

Here, i₁, i₂, …, i_n and v₁, v₂, …, v_n indicate the fuzzy subsystems to which the rule belongs and the activated rules of the related fuzzy subsystem, respectively. This decomposability property ensures that always only one fuzzy subsystem generates the output of the overall decomposable fuzzy system for given any input vector x ∈ X^{i₁,i₂,…,i_n}, that is $TS (x) = {TS}^{i_{1}, i_{2}, \dots, i_{n}} (x)$ (3)

The output of any TS fuzzy subsystem with singleton consequents, ${TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x)$ for x ∈ X^{i₁,i₂,…,i_n}, can be formulated as [19] ${TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x) = g_{n}^{T} (x) {(Ω^{i_{1}, i_{2}, \dots, i_{n}^{T}})}^{- 1} q^{i_{1}, i_{2}, \dots, i_{n}}$ (4)

Here, g (x), Ω, and q = [q₁, q₂, …, q_2ⁿ] ^T are the generator vector, the fundamental matrix, and the vector containing related rule consequent parameters of the fuzzy subsystem ${TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}}$ , respectively. 2ⁿ components of the generator vector g (x) are the elements of the polynomial (1 + x₁) × ⋯ × (1 + x_n) written in the expanded additive form. The fundamental matrix Ω is given as $Ω = {[g (γ_{1}), \dots, g (γ_{2^{n}})]}_{2^{n} \times 2^{n}}$ (5) where, $γ_{j} \in {a_{1}^{i_{1}}, a_{1}^{i_{1} + 1}} \times \dots \times {a_{n}^{i_{n}}, a_{n}^{i_{n} + 1}}$ , j = 1, … 2ⁿ, i_k = 1, …, N_k - 1, k = 1, … n is the element of the cartesian product of boundary values of X^{i₁,i₂,…,i_n}.

The strong triangular partitioning (since $a_{k}^{i_{k} + 1} - a_{k}^{i_{k}} \neq 0$ ) guarantees the existence of the inverse of the fundamental matrix Ω^-1 in the decomposable fuzzy system design. The generator vector g and the inverse of the fundamental matrix Ω^-1 are easily calculated recursively as follows $g_{k + 1} = [\begin{matrix} g_{k} \\ x_{k + 1} g_{k} \end{matrix}]$ (6) $\begin{matrix} Ω_{k + 1}^{- 1} = \frac{1}{L_{k + 1}} [\begin{matrix} \begin{matrix} a_{k + 1}^{i_{k + 1} + 1} & - 1 \end{matrix} \\ \begin{matrix} - a_{k + 1}^{i_{k + 1}} & 1 \end{matrix} \end{matrix}] \otimes Ω_{k}^{- 1}, \\ k = 0, 1, \dots, n - 1 . \end{matrix}$ (7)

Here, the initial values are g₀ = 1 and $Ω_{0}^{- 1} = 1$ . $L_{k} = a_{k}^{i_{k} + 1} - a_{k}^{i_{k}}$ is the length of the $i_{k}^{th}$ interval in the universe of discourse of x_k, and ⊗ represents the Kronecker product. Depending on the recursive nature of g and Ω^-1, TS fuzzy subsystems with large number of inputs can simply and effectively be defined in the matrix multiplications form.

In a similar way, the output of any TS fuzzy subsystem with linear consequents, ${TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}} (x)$ , can be formulated as ${TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}} (x) = g_{n}^{T} (x) {(Ω^{i_{1}, i_{2}, \dots, i_{n}^{T}})}^{- 1} Q^{i_{1}, i_{2}, \dots, i_{n}} [\begin{matrix} 1 \\ x \end{matrix}]$ (8) where, x = [x₁ x₂ … x_n] ^T and Q^{i₁,i₂,…,i_n} is the rule consequent matrix of ${TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}}$ as $Q = [\begin{matrix} q_{1, 0} & q_{1, 1} & \dots & q_{1, n} \\ q_{2, 0} & q_{2, 1} & \dots & q_{2, n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ q_{2^{n}, 0} & q_{2^{n}, 1} & \dots & q_{2^{n}, n} \end{matrix}]$ (9)

Here, q_i,j (i = 1, … 2ⁿ, j = 1, …, n) denotes the coefficient of the rule consequent function related to each input variable.

2.2 Exact inversion of decomposable TS fuzzy subsystems with respect to the single input variable

Let us consider a decomposed TS fuzzy subsystem with singleton consequents ${TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}}$ providing the output interval ${TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x) = [\begin{matrix} α & β \end{matrix}]$ where $α = min ({TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x))$ and $β = max ({TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x))$ . Without loss of generality, it is assumed that the inversion variable is $x_{k} \in [\begin{matrix} a_{k}^{i_{k}} & a_{k}^{i_{k} + 1} \end{matrix}]$ , and the input variables are sorted in the following order: x₁, x₂, …, x_k-1, x_k+1, …, x_n-1, x_n, x_k.

The inverse definition of the decomposed fuzzy subsystem ${TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}}$ with respect to the inversion variable x_k is obtained as follows [4]: $\begin{matrix} x_{k} = \frac{{TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x) - a^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}}{b^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}}, x_{k} \in [\begin{matrix} a_{k}^{i_{k}} & a_{k}^{i_{k} + 1} \end{matrix}], \\ {TS}_{sc}^{i_{1}, i_{2}, \dots, i_{n}} (x) \in [\begin{matrix} α & β \end{matrix}] . \end{matrix}$ (10)

Here $a^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = g_{n - 1}^{T} K_{1}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ and $b^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = g_{n - 1}^{T} K_{2}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ . $K_{1}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ and $K_{2}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ are the partitioned matrices of $K_{sc}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {(Ω^{i_{1}, i_{2}, \dots, i_{n}, i_{k}^{T}})}^{- 1} q^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ as $K_{sc}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {[\begin{matrix} {[K_{1}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}]}_{(2^{n} / 2) \times 1} \\ {[K_{2}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}]}_{(2^{n} / 2) \times 1} \end{matrix}]}_{2^{n} \times 1}$ (11)

Similarly, the inverse definition of the decomposed fuzzy subsystem with linear consequents ${TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}}$ are obtained as follows [4]: $x_{k_{1, 2}} = \frac{- b^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} \pm \sqrt{Δ^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}}}{2 a^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}}$ (12)

where $a^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = g_{n - 1}^{T} K_{22}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ (13) $b^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = g_{n - 1}^{T} (K_{21}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} L_{x} + K_{12}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}})$ (14) $Δ^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {(b^{i_{1}, i_{2}, \dots, i_{n}, i_{k}})}^{2} - 4 a^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} c^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}$ (15) $c^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = g_{n - 1}^{T} K_{11}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} L_{x} - {TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}} (x)$ (16)

$K_{l c}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {(Ω^{i_{1}, i_{2}, \dots, i_{n}, i_{k}^{T}})}^{- 1} Q^{i_{1}, i_{2}, \dots, i_{n}, i_{k}} = {[\frac{{[K_{11}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}]}_{(2^{n} / 2) \times n} | {[K_{12}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}]}_{(2^{n} / 2) \times n}}{{[K_{21}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}]}_{(2^{n} / 2) \times n} | {[K_{22}^{i_{1}, i_{2}, \dots, i_{n}, i_{k}}]}_{(2^{n} / 2) \times n}}]}_{2^{n} \times (n + 1)}$ (17)

$L_{x} = {[1, x_{1}, \dots, x_{k - 1}, x_{k + 1}, \dots, x_{n - 1}, x_{n}]}^{T}$ (18)

Here, x_{k
₁} and x_{k
₂} denote two possible inverse solutions. However, the valid inverse solution is only x_k which satisfies the condition of $x_{k} \in [\begin{matrix} a_{k}^{i_{k}} & a_{k}^{i_{k} + 1} \end{matrix}]$ for ${TS}_{lc}^{i_{1}, i_{2}, \dots, i_{n}} (x) \in [\begin{matrix} α & β \end{matrix}]$ .

3 Exact inversion of MISO decomposable TS fuzzy systems

When a fuzzy subsystem TS^{i₁,i₂,…,i_n} is considered, the universe of discourse of the fuzzy subsystem X^{i₁,i₂,…,i_n} defines a hypercuboid in the multi-dimensional universe of discourse X $\begin{matrix} X^{i_{1}, i_{2}, \dots, i_{n}} = [a_{1}^{i_{1}}, a_{1}^{i_{1} + 1}] \times \dots \times [a_{n}^{i_{n}}, a_{n}^{i_{n} + 1}], \\ i_{k} = 1, \dots, N_{k} - 1, \\ k = 1, \dots, n . \end{matrix}$ (19)

The vertices of this hypercuboid X^{i₁,i₂,…,i_n} can be defined as follows: $\begin{matrix} γ_{j} \in {a_{1}^{i_{1}}, a_{1}^{i_{1} + 1}} \times \dots \times {a_{n}^{i_{n}}, a_{n}^{i_{n} + 1}}, j = 1, \dots 2^{n}, \\ i_{k} = 1, \dots, N_{k} - 1, \\ k = 1, \dots n . \end{matrix}$ (20)

Figure 2 illustrates a hypercuboid X^{i₁,i₂,…,i_n} for n = 3 and its vertices γ_j, j = 1, … 8.

Fig. 2

Hypercuboid universe of discourse X^{i₁,i₂,i₃}.

Let us consider an edge L_j,j+1 of X^{i₁,i₂,…,i_n} and its corresponding vertices γ_j and γ_j+1. It can be assumed that the input variables on L_j,j+1 are $(x_{1}, . . ., x_{k}, . . ., x_{n}) = (a_{1}^{i_{1}}, . . ., [a_{k}^{i_{k}} a_{k}^{i_{k} + 1}], . . ., a_{n}^{i_{n}})$ where $γ_{j} = (a_{1}^{i_{1}}, . . ., a_{k}^{i_{k}}, . . ., a_{n}^{i_{n}})$ and $γ_{j + 1} = (a_{1}^{i_{1}}, . . ., a_{k}^{i_{k} + 1}, . . ., a_{n}^{i_{n}})$ , and y^{i₁,i₂,…,i_n} (γ_j) < y^{i₁,i₂,…,i_n} (γ_j+1). For the input values in $[\begin{matrix} γ_{j} & γ_{j + 1} \end{matrix}]$ , the fuzzy subsystem generates the corresponding output value $y^{i_{1}, i_{2}, \dots, i_{n}} = [\begin{matrix} y^{i_{1}, i_{2}, \dots, i_{n}} (γ_{j}) & y^{i_{1}, i_{2}, \dots, i_{n}} (γ_{j + 1}) \end{matrix}]$ depending on the strong triangular input partitioning. Here, the output values y^{i₁,i₂,…,i_n} (γ_j) and y^{i₁,i₂,…,i_n} (γ_j+1) for adjacent vertices of X^{i₁,i₂,…,i_n} determine the boundary values of the fuzzy subsystem output on the edge L_j,j+1. In other words, on each edge, the output value of the fuzzy subsystem TS^{i₁,i₂,…,i_n} varies between the corresponding output values of two adjacent vertices. Additionally, only one input value, $x_{k} = [a_{k}^{i_{k}} a_{k}^{i_{k} + 1}]$ , varies on each edge of X^{i₁,i₂,…,i_n}, while the other input variables remain constant as $(x_{1}, . . ., x_{k - 1}, x_{k + 1}, . . ., x_{n}) = (a_{1}^{i_{1}}, . . ., a_{k - 1}^{i_{k - 1}}, a_{k + 1}^{i_{k + 1}}, . . ., a_{n}^{i_{n}})$ . Therefore, the inverse solution for any desired fuzzy system output value $y_{d} \in y^{i_{1}, i_{2}, \dots, i_{n}} = [\begin{matrix} y^{i_{1}, i_{2}, \dots, i_{n}} (γ_{j}) & y^{i_{1}, i_{2}, \dots, i_{n}} (γ_{j + 1}) \end{matrix}]$ can easily be calculated by inverting the fuzzy subsystem TS^{i₁,i₂,…,i_n} with respect to the input variable $x_{k} = [a_{k}^{i_{k}} a_{k}^{i_{k} + 1}]$ . In this inversion, only the L_j,j+1 edge of X^{i₁,i₂,…,i_n} is considered. By utilizing this approach, a special trajectory consisting of specific edges of X^{i₁,i₂,…,i_n} can be determined in X so that, the combination of corresponding output intervals of these specific edges on this trajectory covers the output interval of the overall fuzzy system. This trajectory can be named an inversion trajectory, and provides a way to obtain an inverse mapping of a MISO decomposable TS fuzzy system. The inversion trajectory can be defined as follows.

Definition. An inversion trajectory is a set of paths defined on specific edges of fuzzy input meshes X^{i₁,i₂,…,i_n} in the multi-dimensional universe of discourse X, which defines a one-to-one input-output mapping for the overall fuzzy system.

To guarantee the existence and uniqueness of the inverse solutions, the inversion trajectory must be determined with respect to certain criteria by evaluating fuzzy system output values y (γ_j) corresponding to vertices of all X^{i₁,i₂,…,i_n}. The inversion trajectory determination criteria are given in Table 1.

Table 1

Inversion trajectory determination criteria

#	Criterion
1	The inversion trajectory must start from γ_{j ^} and end at γ_{k ^} corresponding to min(y) = y (γ_{j ^}) and max(y) = y (γ_{k ^}), respectively.
2	There must be no discontinuity between the output interval values of the chosen intermediate paths of the inversion trajectory, and the combination of output intervals corresponding to the inversion trajectory must completely cover the output interval value of the whole fuzzy system.
3	In the case that there is an overlap between the output interval values of two adjacent paths, the overlapping output values must be included in only one of these paths.

In addition to the criteria given in Table 1, fewer paths can be chosen in the multi-dimensional universe of discourse and these paths can be chosen from fewer fuzzy subsystems as much as possible. In this way, an inverse model with a simpler structure can be obtained. Therefore, an optimization algorithm can be used for the determination of the inversion trajectory, especially for the higher dimensional fuzzy systems.

The output interval of the overall fuzzy system consists of the combination of output intervals of specific fuzzy subsystems on the predetermined inversion trajectory. Therefore, the inverse mapping of the overall fuzzy system can easily be obtained by combining the inverse mappings of only these specific fuzzy subsystems. The inverse mapping of each related fuzzy subsystem is calculated with respect to the related single inversion variable $x_{k}^{*} = [\begin{matrix} a_{k}^{i_{k}^{*}} & a_{k}^{i_{k}^{*} + 1} \end{matrix}]$ by considering only the corresponding edge on the inversion trajectory. This approach provides significant computational simplicity for the inversion procedure and the overall inversion definition is obtained in a simple structure. Additionally, the inversion trajectory approach guarantees, in advance, the existence and uniqueness of the inverse solutions since it provides a one-to-one input-output mapping for the overall fuzzy system. Thus, there is no need to examine all fuzzy subsystems in order to find an inverse solution for the desired output value.

The proposed inversion procedure for MISO decomposable TS fuzzy systems is given in Table 2. By using the proposed procedure, an inverse mapping of a given fuzzy system can easily be determined, and unique inverse solutions for all input variables are obtained simultaneously for any desired system output value.

Table 2

Proposed inversion procedure for MISO decomposable TS fuzzy systems

Step	The proposed inverse mapping algorithm
1	Calculate the fuzzy system outputs y (γ_j) corresponding to the vertices of all X^{i₁,i₂,…,i_n} where
	$\begin{matrix} γ_{j} \in {a_{1}^{i_{1}}, a_{1}^{i_{1} + 1}} \times \dots \times {a_{n}^{i_{n}}, a_{n}^{i_{n} + 1}}, i_{k} = 1, \dots, N_{k} - 1, \\ k = 1, \dots n . \end{matrix}$
2	Determine vertices γ_{j ^} and γ_{k ^} corresponding to the minimum and maximum output values as y_min = y (γ_{j ^}) and y_max = y (γ_{k ^}), respectively.
3	Determine the inversion trajectory T_inv ={ L₁, L₂, . . . , L_m } starting from γ_{j ^} and ending at γ_{k ^} considering the trajectory determination criteria given in Table 1.
4	For each path L_i (i = 1, …, m), determine the fuzzy subsystem ${TS}^{i_{1}, i_{2}, \dots, i_{n}}$ to be activated and the corresponding variable input $x_{k}^{} \in [\begin{matrix} a_{k}^{i_{k}^{}} & a_{k}^{i_{k}^{*} + 1} \end{matrix}]$ . Set the constant values of the other inputs $x_{i} = a_{i}^{i_{1}}$ or $x_{i} = a_{i}^{i_{1} + 1}$ , i = 1, . . . , n, i ≠ k.
5	For each path L_i (i = 1, …, m), derive the inverse solution definition with respect to the inversion variable $x_{k}^{} \in [\begin{matrix} a_{k}^{i_{k}^{}} & a_{k}^{i_{k}^{} + 1} \end{matrix}]$ as $x_{{kL}_{i}}^{} = f (y_{d})$ by using Eqs. (10) and (12) for fuzzy subsystems with singleton consequents and linear consequents, respectively, and set the local solution $S_{L_{i}} = (x_{1}, \dots, x_{{kL}_{i}}^{*}, \dots, x_{n})$ .
6	By combining all S_{L _i}, set the solution set
	$S = {\begin{matrix} S_{L_{1}} = (x_{1}, \dots, x_{{kL}_{1}}^{}, \dots, x_{n}) for y_{d} = [y (γ_{L_{11}}) y (γ_{L_{12}})], \\ x_{{kL}_{1}}^{} \in [\begin{matrix} a_{k}^{i_{k}^{}} & a_{k}^{i_{k}^{} + 1} \end{matrix}] \\ ⋮ \\ S_{L_{m}} = (x_{1}, \dots, x_{{kL}_{m}}^{}, \dots, x_{n}) for y_{d} = [y (γ_{L_{m 1}}) y (γ_{L_{m 2}})], \\ x_{{kL}_{m}}^{} \in [a_{k}^{i_{k}^{}} a_{k}^{i_{k}^{} + 1}] \end{matrix}$
	where γ_{L _i1} and γ_{L _i2} are two vertices on path L_i.

In order to illustrate the determination of a proper inversion trajectory, it can be assumed that n = 3 and the universe of discourse X^{i₁,i₂,i₃} of a fuzzy subsystem can be given as in Fig. 3. In this figure, the overall fuzzy system output values y (γ_j), j = 1, … 8, corresponding to the vertices of the cuboid X^{i₁,i₂,i₃} are given in red circles. The minimum and maximum output values of the overall fuzzy system can be defined as min(y) = y (γ₁) and max(y) = y (γ₇), and it is assumed that there is an overlap between $[\begin{matrix} y (γ_{1}) & y (γ_{5}) \end{matrix}]$ and $[\begin{matrix} y (γ_{2}) & y (γ_{7}) \end{matrix}]$ as y (γ₁) < y (γ₅) < y (γ₂) < y (γ₇). In this case, the output interval value of this fuzzy subsystem is $y^{i_{1}, i_{2}, i_{3}} = y = [\begin{matrix} y (γ_{1}) & y (γ_{7}) \end{matrix}]$ .

For this case, the inversion trajectory can be determined as follows.

According to Criterion-1, the inversion trajectory must start from γ₁ and end at γ₇.

According to Criterion-2, since $y = [\begin{matrix} y (γ_{1}) & y (γ_{7}) \end{matrix}]$ and y (γ₁) < y (γ₅) < y (γ₂) < y (γ₇), the inversion trajectory is determined as composed of L₁ and L₂ paths, T_inv ={ L₁, L₂ }, as shown in Fig. 3. The fuzzy subsystem output values are $y_{L_{1}} = [\begin{matrix} y (γ_{1}) & y (γ_{2}) \end{matrix}]$ and $y_{L_{2}} = [\begin{matrix} y (γ_{5}) & y (γ_{7}) \end{matrix}]$ on L₁ and L₂ paths, respectively. Thus, on the inversion trajectory T_inv ={ L₁, L₂ }, the output interval value of the fuzzy subsystem is $y_{T_{inv}} = y_{L_{1} \cup L_{2}} = [\begin{matrix} y (γ_{1}) & y (γ_{7}) \end{matrix}]$ without discontinuity.

According to Criterion-3, since there is an overlap between $y_{L_{1}} = [\begin{matrix} y (γ_{1}) & y (γ_{2}) \end{matrix}]$ and $y_{L_{2}} = [\begin{matrix} y (γ_{5}) & y (γ_{7}) \end{matrix}]$ as $y_{overlap} = [\begin{matrix} y (γ_{5}) & y (γ_{2}) \end{matrix}]$ , the output intervals of L₁ and L₂ paths are considered as $y_{L_{1}} = [\begin{matrix} y (γ_{1}) & y (γ_{5}) \end{matrix}]$ and $y_{L_{2}} = (\begin{matrix} y (γ_{5}) & y (γ_{7}) \end{matrix}]$ , respectively.

Fig. 3

Fuzzy subsystem output values at vertices of the cuboid X^{i₁,i₂,i₃}.

Additionally, although the inversion trajectory with higher number of paths can be defined, it is determined as consisting of the minimum number of paths.

After the determination of the inversion trajectory, only L₁ path is valid for the desired output values $y_{d} = [\begin{matrix} y (γ_{1}) & y (γ_{5}) \end{matrix}]$ , and the inverse solution is obtained by inverting the fuzzy subsystem with respect to $x_{1} \in [\begin{matrix} a_{1}^{i_{1}} & a_{1}^{i_{1} + 1} \end{matrix}]$ while x₂ and x₃ are constants as $x_{2} = a_{2}^{i_{2}}$ and $x_{3} = a_{3}^{i_{3}}$ . In a similar manner, only L₂ path is valid for the desired output values $y_{d} = (\begin{matrix} y (γ_{5}) & y (γ_{7}) \end{matrix}]$ , and the inverse solution is obtained by inverting the fuzzy subsystem with respect to $x_{3} \in [\begin{matrix} a_{3}^{i_{3}} & a_{3}^{i_{3} + 1} \end{matrix}]$ while x₁ and x₂ are constants as $x_{1} = a_{1}^{i_{1}}$ and $x_{2} = a_{2}^{i_{2} + 1}$ .

As it is seen from the inversion procedure given in Table 2, the proposed inversion method provides inverse solutions for all input variables of the fuzzy system simultaneously. Additionally, this method does not need binding limitations on the fuzzy system structure such as satisfying invertibility conditions, having singleton rule consequents and monotonic rule bases. Therefore, the proposed method provides a way to use more flexible fuzzy systems in inversion based applications.

4 Simulation studies

In this section, illustrative examples are given in order to show the exactness and effectiveness of the proposed inversion procedure. For this purpose, the inversion validation structure given in Fig. 4 is utilized [4–6 , 35]. In these examples, two-input-single-output and three-input-single-output decomposable TS fuzzy systems are considered, and their inverse mappings are obtained by using the proposed inversion procedure. The inverse mapping of the fuzzy system provides the proper input values which will drive the fuzzy system output y to the desired output y_d. In simulations, sinusoidal functions are used as the desired output.

Fig. 4

Block diagram used for the validation of the proposed inversion method.

4.1 Inversion of a two-input-single-output decomposable TS fuzzy system

To illustrate the exact inversion of a two-input-single-output decomposable TS fuzzy system, the fuzzy system given in Fig. 5 can be considered. The input-output mapping of the fuzzy system is shown in Fig. 6. The corresponding decomposition of the fuzzy system and the output values y (γ_j) (given in red circles), j = 1, … 12, corresponding to the vertices of all rectangular fuzzy meshes X^i₁,i₂ are given in Fig. 7.

Fig. 5

Two-input-single-output TS fuzzy system structure (a) input membership functions (b) the rule table.

Fig. 6

Input-output mapping of the two-input-single-output TS fuzzy system.

Fig. 7

Decomposition of the two-input-single-output TS fuzzy system and the output values with respect to the vertices of all rectangular fuzzy meshes X^i₁,i₂.

Figures 6 7 show that the output interval of the fuzzy system is $y = [\begin{matrix} - 3 & 6 \end{matrix}]$ . Therefore, considering the inversion trajectory determination criteria given in Table 1, the inversion trajectory must start from (x₁, x₂) = (1, - 1) or (x₁, x₂) = (4, 1) where y = -3 and end at (x₁, x₂) = (0, 2) where y = 6. In this example, the inversion trajectory is chosen as composed of L₁ and L₂ paths, T_inv ={ L₁, L₂ }, as shown with blue lines in Fig. 7. On L₁ and L₂ paths, TS^2,1 and TS^2,2 fuzzy subsystems are activated, respectively. On L₁ path, TS^2,1 output interval value is $y_{L_{1}} = [\begin{matrix} - 3 & 5 \end{matrix}]$ , while $x_{1} \in [\begin{matrix} 0 & 1 \end{matrix}]$ and x₂ = -1. Similarly, on L₂ path, TS^2,2 output interval value is $y_{L_{2}} = [\begin{matrix} 4 & 6 \end{matrix}]$ , while $x_{1} \in [\begin{matrix} 0 & 1 \end{matrix}]$ and x₂ = 2. Thus, on the inversion trajectory T_inv ={ L₁, L₂ }, the obtained output interval value becomes $y_{T_{inv}} = y_{L_{1} \cup L_{2}} = [\begin{matrix} - 3 & 6 \end{matrix}]$ which is exactly equal to the overall fuzzy system output interval $y = [\begin{matrix} - 3 & 6 \end{matrix}]$ . Therefore, evaluating the inverse definitions of only TS^2,1 and TS^2,2 fuzzy subsystems is sufficient for the calculation of necessary input values that will drive the fuzzy system output to the desired output values.

In this case, there is an overlap between the output values of $y_{L_{1}} = [\begin{matrix} - 3 & 5 \end{matrix}]$ and $y_{L_{2}} = [\begin{matrix} 4 & 6 \end{matrix}]$ as $y_{overlap} = y_{L_{1} \cap L_{2}} = [\begin{matrix} 4 & 5 \end{matrix}]$ . In order to prevent the existence of multiple solutions for the desired output values y_d = y_overlap, only L₁ path is considered for $y_{d} = [\begin{matrix} - 3 & 5 \end{matrix}]$ and the inverse solution of the fuzzy system is obtained by evaluating the inverse definition of TS^2,1 with respect to the inversion variable x₁ for x₂ = -1. In a similar manner, for $y_{d} = (\begin{matrix} 5 & 6 \end{matrix}]$ , only L₂ path is considered and the inverse solution of the fuzzy system is obtained by evaluating the inverse definition of TS^2,2 with respect to the inversion variable x₁ for x₂ = 2.

When TS^2,1 fuzzy subsystem for L₁ path is considered, since the inversion variable is x₁, the input variables are ordered as x = [x₂, x₁] ^T and L_x vector becomes L_x = [1, x₂] ^T. The related boundary points are $[\begin{matrix} a_{1}^{1} & a_{1}^{2} \end{matrix}] = [\begin{matrix} - 1 & 1 \end{matrix}]$ and $[\begin{matrix} a_{2}^{2} & a_{2}^{3} \end{matrix}] = [\begin{matrix} 0 & 1 \end{matrix}]$ . The consequent parameter matrix Q^2,1 is $Q^{2, 1} = [\begin{matrix} q_{1, 0} & q_{1, 1} & q_{1, 2} \\ q_{2, 0} & q_{2, 1} & q_{2, 2} \\ q_{3, 0} & q_{3, 1} & q_{3, 2} \\ q_{4, 0} & q_{4, 1} & q_{4, 2} \end{matrix}] = [\begin{matrix} 1 & - 4 & 5 \\ 3 & - 2 & - 6 \\ - 2 & 1 & 0 \\ - 2 & 3 & 2 \end{matrix}]$ (21)

For x₂ = -1; L_x vector, the generator vector g (x), and the inverse of the fundamental matrix ${(Ω_{2}^{2, 1^{T}})}^{- 1}$ of TS^2,1 are calculated by using Eqs. (18), (6), and (7) as $L_{u} = {[\begin{matrix} 1 & x_{2} \end{matrix}]}^{T} = {[\begin{matrix} 1 & - 1 \end{matrix}]}^{T}$ (22) $g (x) = g_{2} = {[\begin{matrix} g_{1} & x_{1} g_{1} \end{matrix}]}^{T} = {[\begin{matrix} 1 & - 1 & x_{1} & - x_{1} \end{matrix}]}^{T}$ (23) ${(Ω_{2}^{2, 1^{T}})}^{- 1} = \frac{1}{2} {[\begin{matrix} 1 & - 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & 1 & 1 \end{matrix}]}^{T}$ (24)

K^2,1 matrix is calculated as follows $\begin{array}{l} K^{2, 1} = {(Ω^{2, 1^{T}})}^{- 1} Q^{2, 1} = [\frac{\begin{matrix} 2 & - 3 \\ 1 & 1 \end{matrix} | \begin{matrix} - 0.5 \\ - 5.5 \end{matrix}}{\begin{matrix} - 4 & 5 \\ - 1 & 0 \end{matrix} | \begin{matrix} 1.5 \\ 6.5 \end{matrix}}] \\ = [\frac{K_{11}^{2, 1} | K_{12}^{2, 1}}{K_{21}^{2, 1} | K_{22}^{2, 1}}] \end{array}$ (25)

Then, the inverse mapping definition of TS^2,1 in terms of x₁ is obtained as $x_{1_{1, 2}}^{2, 1} = \frac{- b^{2, 1} \pm \sqrt{Δ^{2, 1}}}{2 a^{2, 1}} = \frac{3 \pm \sqrt{9 + 20 (5 - {TS}^{2, 1})}}{- 10}$ (26)

Since $x_{1} \in [\begin{matrix} 0 & 1 \end{matrix}]$ must be satisfied, the valid inverse solution is $x_{1} = x_{1_{2}}^{2, 1} = \frac{3 - \sqrt{9 + 20 (5 - {TS}^{2, 1})}}{- 10}$ (27)

Thus, considering ${TS}^{2, 1} (x) = y_{d}$ , the inverse solution set for L₁ path is obtained as $\begin{matrix} S_{L_{1}} = (x_{1}, x_{2}) = (\frac{3 - \sqrt{9 + 20 (5 - y_{d})}}{- 10}, - 1) \\ for y_{d} = [\begin{matrix} - 3 & 5 \end{matrix}] \end{matrix}$ (28)

When TS^2,2 fuzzy subsystem for L₂ path is considered, the order of the input variables and L_x vector do not change since the inversion variable is again x₁. By using the same procedure, the inverse solution set for L₂ path is obtained as $S_{L_{2}} = (x_{1}, x_{2}) = (\frac{\sqrt{2 (6 - y_{d})}}{2}, 2) for y_{d} = (\begin{matrix} 5 & 6 \end{matrix}]$ (29)

Finally, considering the inversion trajectory T_inv ={ L₁, L₂ }, the inverse mapping of the overall TS fuzzy system is obtained as $\begin{matrix} S = (x_{1}, x_{2}) \\ = {\begin{matrix} S_{L_{1}} = (\frac{3 - \sqrt{9 + 20 (5 - y_{d})}}{- 10}, - 1) \\ for y_{d} = [\begin{matrix} - 3 & 5 \end{matrix}], x_{1} \in [\begin{matrix} 0 & 1 \end{matrix}] \\ S_{L_{2}} = (\frac{\sqrt{2 (6 - y_{d})}}{2}, 2) \\ for y_{d} = (\begin{matrix} 5 & 6 \end{matrix}], x_{1} \in [\begin{matrix} 0 & 1 \end{matrix}] \end{matrix} \end{matrix}$ (30)

The graphical representation of the inverse mapping of the overall TS fuzzy system is given in Fig. 8.

Fig. 8

Inverse mapping of the two-input-single-output TS fuzzy system.

Let us assume that the desired output of the TS fuzzy system is chosen as y_d (k) = 4.5 sin(2πk/ 65) + 1.5. By using the inverse mapping definition of the fuzzy system obtained in Eq. (30), the calculated x₁ and x₂ inverse solution values with respect to the given y_d are shown in Fig. 9(a). The error between the desired output and the fuzzy system output is shown in Fig. 9(b). Fig. 9(b) clearly shows that the fuzzy system output tracks the desired output y_d with zero error. This validates the exactness of the inversion.

Fig. 9

(a) Calculated x₁ and x₂ input values of the fuzzy system (b) the error between the desired output and the fuzzy system output.

4.2 Inversion of a three-input-single-output decomposable TS fuzzy system

To show the derivation of the inverse mapping of a three-input-single-output decomposable TS fuzzy system, the fuzzy system with input membership functions given in Fig. 10 can be considered. The rule table of the fuzzy system is shown in Table 3. The corresponding decomposition of the fuzzy system and the output values y (γ_j), j = 1, … 27, (shown in red circles) corresponding to the vertices of all cuboid fuzzy meshes X^{i₁,i₂,i₃} are given in Fig. 11.

Fig. 10

Input membership functions of the three-input-single-output TS fuzzy system.

Table 3

The rule table of the three-input-single-output TS fuzzy system

x ₁	x ₂	x ₃	Rule consequents
$A_{1}^{1}$	$A_{2}^{1}$	$A_{3}^{1}$	-3 +0.6x₁ - 0.5x₂ + 0.2x₃
$A_{1}^{1}$	$A_{2}^{1}$	$A_{3}^{2}$	0.5 + x₂ + 0.4x₃
$A_{1}^{1}$	$A_{2}^{1}$	$A_{3}^{3}$	0.25 - 0.5x₁ + 0.15x₂ + 0.2x₃
$A_{1}^{1}$	$A_{2}^{2}$	$A_{3}^{1}$	0.4 + x₁ + 0.5x₂ - 0.6x₃
$A_{1}^{1}$	$A_{2}^{2}$	$A_{3}^{2}$	-0.5 + 0.2x₁ + 0.8x₂
$A_{1}^{1}$	$A_{2}^{2}$	$A_{3}^{3}$	0.6 + 0.5x₁ + 0.2x₂ + 0.1x₃
$A_{1}^{1}$	$A_{2}^{3}$	$A_{3}^{1}$	-0.8 + 0.5x₁ + 0.4x₂ - 0.5x₃
$A_{1}^{1}$	$A_{2}^{3}$	$A_{3}^{2}$	-2.2 + 0.1x₁ + 0.1x₃
$A_{1}^{1}$	$A_{2}^{3}$	$A_{3}^{3}$	-2 + x₁ + 2x₂ + 0.5x₃
$A_{1}^{2}$	$A_{2}^{1}$	$A_{3}^{1}$	0.5x₁ + 0.8x₃
$A_{1}^{2}$	$A_{2}^{1}$	$A_{3}^{2}$	0.4
$A_{1}^{2}$	$A_{2}^{1}$	$A_{3}^{3}$	-0.5 + 0.4x₁ + 0.8x₂ + 0.2x₃
$A_{1}^{2}$	$A_{2}^{2}$	$A_{3}^{1}$	1.5 + 0.5x₁ - 0.2x₂ + 0.8x₃
$A_{1}^{2}$	$A_{2}^{2}$	$A_{3}^{2}$	-4 -0.4x₁ + 0.6x₂ + x₃
$A_{1}^{2}$	$A_{2}^{2}$	$A_{3}^{3}$	1.4 - x₁ - x₂ + 0.5x₃
$A_{1}^{2}$	$A_{2}^{3}$	$A_{3}^{1}$	-0.8 + 0.2x₁ - 0.4x₂ - 0.1x₃
$A_{1}^{2}$	$A_{2}^{3}$	$A_{3}^{2}$	0.6 - 0.5x₁ + 0.2x₂ - 0.8x₃
$A_{1}^{2}$	$A_{2}^{3}$	$A_{3}^{3}$	-1 +0.1x₁ + 0.2x₂ - 0.6x₃
$A_{1}^{3}$	$A_{2}^{1}$	$A_{3}^{1}$	0.5x₁ + 0.8x₂ + 0.1x₃
$A_{1}^{3}$	$A_{2}^{1}$	$A_{3}^{2}$	-3 + x₁ + 2x₂
$A_{1}^{3}$	$A_{2}^{1}$	$A_{3}^{3}$	0.5 + 0.5x₁ + x₂ - x₃
$A_{1}^{3}$	$A_{2}^{2}$	$A_{3}^{1}$	0.5 + x₁ + 0.6x₂ + 0.4x₃
$A_{1}^{3}$	$A_{2}^{2}$	$A_{3}^{2}$	0.8 + 1.5x₁ - 0.5x₂ - 0.6x₃
$A_{1}^{3}$	$A_{2}^{2}$	$A_{3}^{3}$	0.6 + 0.5x₁ + 0.4x₂ - x₃
$A_{1}^{3}$	$A_{2}^{3}$	$A_{3}^{1}$	2 + 0.1x₁ + 0.4x₂ + 0.2x₃
$A_{1}^{3}$	$A_{2}^{3}$	$A_{3}^{2}$	-1 +0.8x₁ + 0.5x₂ + x₃
$A_{1}^{3}$	$A_{2}^{3}$	$A_{3}^{3}$	2 - x₁ + 0.5x₂ + 0.2x₃

Fig. 11

Decomposition of the three-input-single-output TS fuzzy system and the output values with respect to the vertices of all cuboid fuzzy meshes X^{i₁,i₂,i₃}.

Figure 11 shows that the fuzzy system output interval is $y = [\begin{matrix} - 4.5 & 3.8 \end{matrix}]$ . Therefore, the inversion trajectory must start from (x₁, x₂, x₃) = (- 2, - 1, - 4) where y = -4.5 and end at (x₁, x₂, x₃) = (2, 0, 0) where y = 3.8. Here, L₁ and L₂ paths are chosen to construct the inversion trajectory, T_inv ={ L₁, L₂ }, as shown with blue lines in Fig. 11. TS^1,1,1 and TS^2,1,1 (or TS^2,1,2) fuzzy subsystems are activated on L₁ and L₂ paths, respectively.

Over L₁ path, TS^1,1,1 output interval value is $y_{L_{1}} = [\begin{matrix} - 4.5 & 0.8 \end{matrix}]$ , while $x_{2} \in [\begin{matrix} - 1 & 0 \end{matrix}]$ , x₁ = -2 and x₃ = -4. Similarly, TS^2,1,1 output interval value is $y_{L_{2}} = [\begin{matrix} - 3 & 3.8 \end{matrix}]$ on L₂ path, while $x_{2} \in [\begin{matrix} - 1 & 0 \end{matrix}]$ , x₁ = 2 and x₃ = 0. Therefore, the output interval value $y_{T_{inv}} = y_{L_{1} \cup L_{2}} = y = [\begin{matrix} - 4.5 & 3.8 \end{matrix}]$ is obtained on the inversion trajectory T_inv ={ L₁, L₂ }. The inverse mapping of the overall fuzzy system can be calculated by evaluating the inverse definitions of only TS^1,1,1 and TS^2,1,1 fuzzy subsystems given in Fig. 12.

Fig. 12

TS^1,1,1 and TS^2,1,1 fuzzy subsystems and the output values with respect to the vertices of the cuboid fuzzy meshes X^1,1,1 and X^2,1,1.

Since, there is an overlap between the output values of y_{L
₁} and y_{L
₂} as $y_{overlap} = y_{L_{1} \cap L_{2}} = [\begin{matrix} - 3 & 0.8 \end{matrix}]$ , only L₁ path is considered for $y_{d} = [\begin{matrix} - 4.5 & - 3 \end{matrix}]$ and only L₂ path is considered for $y_{d} = (\begin{matrix} - 3 & 3.8 \end{matrix}]$ .

When TS^1,1,1 fuzzy subsystem for L₁ path is considered, since the inversion variable is x₂, the order of the input variables is x = [x₁, x₃, x₂] ^T, and L_x vector is L_x = [1, x₁, x₃] ^T. The related boundary points are $[\begin{matrix} a_{1}^{1} & a_{1}^{2} \end{matrix}] = [\begin{matrix} - 2 & 0 \end{matrix}]$ , $[\begin{matrix} a_{2}^{1} & a_{2}^{2} \end{matrix}] = [\begin{matrix} - 4 & 0 \end{matrix}]$ and $[\begin{matrix} a_{3}^{1} & a_{3}^{2} \end{matrix}] = [\begin{matrix} - 1 & 0 \end{matrix}]$ . The consequent parameter matrix Q^1,1,1 of TS^1,1,1 is $Q^{1, 1, 1} = {[\begin{matrix} - 3 & 0 & 0.5 & 0.4 & 0.4 & 1.5 & - 0.5 & - 4 \\ 0.6 & 0.5 & 0 & 0 & 1 & 0.5 & 0.2 & - 0.4 \\ 0.2 & 0.8 & 0.4 & 0 & - 0.6 & 0.8 & 0 & 1 \\ - 0.5 & 0 & 1 & 0 & 0.5 & - 0.2 & 0.8 & 0.6 \end{matrix}]}^{T}$ (31)

For x₁ = -2 and x₃ = -4; L_x vector, the generator vector g (x), and the inverse of the fundamental matrix ${(Ω_{3}^{1, 1, 1^{T}})}^{- 1}$ of TS^1,1,1 are obtained as follows $L_{x} = {[\begin{matrix} 1 & x_{1} & x_{3} \end{matrix}]}^{T} = {[\begin{matrix} 1 & - 2 & - 4 \end{matrix}]}^{T}$ (32) $\begin{matrix} g (x) = g_{3} = {[\begin{matrix} g_{2} & x_{2} g_{2} \end{matrix}]}^{T} \\ = {[\begin{matrix} 1 & - 2 & - 4 & 8 & x_{2} & - 2 x_{2} & - 4 x_{2} & 8 x_{2} \end{matrix}]}^{T} \end{matrix}$ (33) ${(Ω_{3}^{1, 1, 1^{T}})}^{- 1} = \frac{1}{8} {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & 1 \\ 0 & 0 & 0 & 0 & - 8 & - 4 & - 2 & - 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & - 2 & - 1 & 0 & 0 & - 2 & - 1 \\ 0 & - 4 & 0 & - 1 & 0 & - 4 & 0 & - 1 \\ 8 & 4 & 2 & 1 & 8 & 4 & 2 & 1 \end{matrix}]}^{T}$ (34)

K^1,1,1 matrix is obtained as follows $\begin{matrix} K^{1, 1, 1} = {(Ω^{1, 1, 1^{T}})}^{- 1} Q^{1, 1, 1} \\ = [\frac{\begin{matrix} - 4 & - 0.4 & 1 \\ - 1.75 & - 0.3 & 0.5 \\ - 0.375 & - 0.225 & 0.05 \\ - 0.575 & - 0.0125 & - 0.05 \end{matrix} | \begin{matrix} 0.6 \\ - 0.1 \\ 0.2 \\ 0.0625 \end{matrix}}{\begin{matrix} - 4.4 & - 0.4 & 1 \\ - 1.7 & - 0.3 & 0.7 \\ - 0.475 & - 0.1 & 0.25 \\ - 0.1875 & - 0.025 & - 0.075 \end{matrix} | \begin{matrix} 0.6 \\ 0.4 \\ 0.2 \\ 0.25 \end{matrix}}] \\ = [\frac{K_{11}^{1, 1, 1} | K_{12}^{1, 1, 1}}{K_{21}^{1, 1, 1} | K_{22}^{1, 1, 1}}] \end{matrix}$ (35)

Then, the inverse mapping of TS^1,1,1 in terms of x₂ is calculated as $\begin{matrix} x_{2_{1, 2}}^{1, 1, 1} = \frac{- b^{1, 1, 1} \pm \sqrt{Δ^{1, 1, 1}}}{2 a^{1, 1, 1}} \\ = \frac{- (6.3) \pm \sqrt{{(6.3)}^{2} - 4 (0.8 - {TS}^{1, 1, 1})}}{2} \end{matrix}$ (36)

Considering $x_{2} \in [\begin{matrix} - 1 & 0 \end{matrix}]$ , the inverse solution is obtained as $x_{2}^{1, 1, 1} = x_{2_{1}}^{1, 1, 1} = \frac{- (6.3) + \sqrt{{(6.3)}^{2} - 4 (0.8 - {TS}^{1, 1, 1})}}{2}$ (37)

Therefore, for ${TS}^{1, 1, 1} (x) = y_{d}$ , the inverse solution set for L₁ path is $\begin{matrix} S_{L_{1}} = (x_{1}, x_{2}, x_{3}) \\ = (- 2, \frac{- (6.3) + \sqrt{{(6.3)}^{2} - 4 (0.8 - y_{d})}}{2}, - 4) \\ for y_{d} = [\begin{matrix} - 4.5 & - 3 \end{matrix}] \end{matrix}$ (38)

For TS^2,1,1 fuzzy subsystem on L₂ path, the inversion variable is again x₂. Therefore, the order of the input variables and L_x vector do not change. By using the same procedure, the inverse solution set for L₂ path is calculated as follows $\begin{matrix} S_{L_{2}} = (x_{1}, x_{2}, x_{3}) \\ = (2, \frac{- (4.3) + \sqrt{{(4.3)}^{2} + 10 (3.8 - y_{d})}}{- 5}, 0) \\ for y_{d} = (\begin{matrix} - 3 & 3.8 \end{matrix}] \end{matrix}$ (39)

Finally, by combining the obtained solution sets, the inverse mapping of the overall fuzzy system is obtained as follows $\begin{matrix} S = (x_{1}, x_{2}, x_{3}) \\ = {\begin{matrix} S_{L_{1}} = (- 2, \frac{- (6.3) + \sqrt{{(6.3)}^{2} - 4 (0.8 - y_{d})}}{2}, - 4) \\ for y_{d} = [\begin{matrix} - 4.5 & - 3 \end{matrix}], x_{2} \in [\begin{matrix} - 1 & 0 \end{matrix}] \\ S_{L_{2}} = (2, \frac{- (4.3) + \sqrt{{(4.3)}^{2} + 10 (3.8 - y_{d})}}{- 5}, 0) \\ for y_{d} = (\begin{matrix} - 3 & 3.8 \end{matrix}], x_{2} \in [\begin{matrix} - 1 & 0 \end{matrix}] \end{matrix} \end{matrix}$ (40)

The inverse mapping of the overall TS fuzzy system is shown in Fig. 13 graphically.

Fig. 13

Inverse mapping of the three-input-single-output TS fuzzy system.

Here, let us assume that the desired output is y_d (k) = 4.15 sin(2πk/ 65) - 0.35. In this case, the calculated inverse solution values x₁, x₂, and x₃ with respect to y_d are shown in Fig. 14. The comparison of the fuzzy system output and the desired output is given in Fig. 14(b). Fig. 14(b) indicates that the output of the fuzzy system tracks the desired output exactly.

Fig. 14

(a) Calculated x₁, x₂, and x₃ input values (b) the comparison of the fuzzy system output and the desired output.

5 Conclusion

In this study, an exact analytical inversion method is presented for multi-input-single-output decomposable TS fuzzy systems with either singleton or linear consequents. The proposed method is based on the newly introduced inversion trajectory approach which guarantees the existence and uniqueness of the inverse solutions. The proposed method provides a simple and systematic way to obtain unique inverse solutions for all input variables simultaneously with respect to any desired system output value. The exactness and the effectiveness of the method are shown in illustrative examples.

The only limitation of the proposed method is that the TS fuzzy system to be inverted must provide the decomposability property. Apart from this limitation, the proposed method does not require any limitation on decomposable fuzzy systems such as having singleton consequents and monotonic rule bases, satisfying invertibility conditions, and providing inverse solutions for only one input variable. Therefore, the accuracy and robustness of inverse fuzzy model based applications can be improved by using the proposed method which allows the use of more flexible fuzzy systems. Additionally, the proposed inversion method is very suitable to be used in real-time applications since the obtained inverse definitions and their implementations are simple and straightforward.

In future studies, the proposed method will be extended to multi-input-multi-output decomposable fuzzy systems since these fuzzy systems can always be represented as a collection of multi-input-single-output fuzzy systems.

References

Rotshtein

A.P.

and Rakytyanska

H.B.

, Fuzzy evidence in identification, forecasting and diagnosis, Springer Publishing Company, 2012.

Varkonyi-Koczy

A.R.

, Peceli

, Dobrowiecki

T.P.

and Kovacshazy

, Iterative fuzzy model inversion, in: Proc. IEEE Conf. Fuzzy Syst., IEEE World Congress on Computational Intelligence, Alaska, USA, 1998, vol. 1, pp. 561–566.

Varkonyi-Koczy

A.R.

, Almos

and Kovacshazy

, Genetic algorithms in fuzzy model inversion, in: Proc. Int. Conf. Fuzzy Systems, FUZZ-IEEE ’99, Seoul, Korea, 1999, vol. 3, pp. 1421–1426.

Ulu

, Exact analytical inverse mapping of decomposable TS fuzzy systems with singleton and linear consequents, Applied Soft Computing 23 (2014), 202–214.

Ulu

, Güzelkaya

and Eksin

, Exact analytical inversion of TSK fuzzy systems with singleton and linear consequents, Int J Approx Reason 55(6) (2014), 1357–1368.

Ulu

, Exact analytical inversion of interval type-2 TSK fuzzy logic systems with closed form inference methods, Applied Soft Computing 37 (2015), 60–70.

Kien

C.V

, Anh

H.P.H.

and Son

N.N.

, Inverse–adaptive multilayer T–S fuzzy controller for uncertain nonlinear system optimized by differential evolution algorithm. Soft Comput 24 (2020), 14073–14089.

and Shin

Y.C.

, A Fuzzy inverse model construction method for general monotonic multi-input-single-output (MISO) systems, IEEE Transactions on Fuzzy Systems 16(5) (2008), 1216–1231.

Filev

D.P.

, Inversion of fuzzy models—practical issues, in: Proc. 7th IEEE Internat. Conf. on Fuzzy Systems (FUZZ’IEEE 98), Anchorage, USA, 1998, pp. 1658–1663.

10.

Jarrett

, Hüyük

and Schaar

M.V.D.

, Inverse Decision Modeling: Learning Interpretable Representations of Behavior, in: Proc. The 38th International Conference on Machine Learning, PMLR 139 (2021), 4755–4771.

11.

Singh

D.J.

, Verma

N.K.

, Ghosh

A.K.

and Malagaudanavar

, An application of interval type-2 fuzzy model based control system for generic aircraft, Applied Soft Computing 121 (2022), 108721.

12.

Tikk

, Kóczy

L.T.

and Gedeon

T.D.

, A survey on universal approximation and its limits in Soft Computing techniques, International Journal of Approximate Reasoning 33(2) (2003), 185–202.

13.

Tikk

, Baranyi

and Patton

R.J.

, Approximation properties of TP model forms and its consequences to TPDC design framework, Asian Journal of Control 9(3) (2007), 221–231.

14.

, Jiang

and Shi

, Nonlinear actuator fault estimation observer: an inverse system approach via a T-S fuzzy model, International Journal of Applied Mathematics and Computer Science 22(1) (2012), 183–196.

15.

Yang

, Wang

and Wang

, Fuzzy modeling and prediction of combustion layers^′ temperature for power plant, Journal of Intelligent & Fuzzy Systems 41(6) (2021), 7081–7091.

16.

Ying

, Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno fuzzy systems with linear rule consequent, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans 28(4) (1998), 515–520.

17.

Ying

, Interval Type-2 Takagi–Sugeno fuzzy systems with linear rule consequent are universal approximators, in: Proc. Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS’09, 2009, pp. 1–5.

18.

Abonyi

, Andersen

, Nagy

and Szeifert

, Inverse fuzzy-process-model based adaptive control, Mathematics and Computers in Simulation 51(1–2) (1999), 119–132.

19.

Kluska

, Analytical methods in fuzzy modeling and control, Springer Publishing Company, Rzeszow, 2012.

20.

Castro

J.L.

, Fuzzy logic controllers are universal approximators, IEEE Trans SMC 25 (1995), 629–635.

21.

Youssef

K.H.

, Yousef

H.A.

, Sebakhy

O.A.

and Wahba

M.A.

, Adaptive fuzzy APSO based inverse tracking-controller with an application to DC motors, Expert Systems with Applications 36(2) (2009), 3454–3458.

22.

Siminski

, Ridders algorithm in approximate inversion of fuzzy model with parametrized consequences, Expert Systems with Applications 51 (2016), 276–285.

23.

Chen

K.S.

and Yu

C.M.

, Fuzzy Decision-making Model for Process Quality Improvement of Machine Tool Industry Chain, Journal of Intelligent & Fuzzy Systems 42(3) (2022), 1547–1558.

24.

Wang

L.-X.

and Mendel

J.M.

, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transactions on Neural Networks 3(5) (1992), 807–814.

25.

Fischer

and Isermann

, Robust hybrid control based on inverse fuzzy process models, in: Proc. 5th IEEE Int. Conf. Fuzzy Systems (FUZZ’IEEE 96), New Orleans, USA, 1996, pp. 1210–1216.

26.

Hussein

M.T.

, Gafer

A.S.

and Fadhel

E.Z.

, Robot manipulator inverse kinematics using adaptive neuro-fuzzy inference system, Journal of Engineering Science and Technology 15(3) (2020), 1984–1998.

27.

Asad

M.U.

, et al., Disturbance observer supported fuzzy model based controller with application to bilateral tele-operation systems, Journal of Intelligent & Fuzzy Systems Pre-press (2022), 1–9.

28.

Baranyi

, Bavelaar

, Koczy

L.T.

and Titli

, Inverse rule base of various fuzzy interpolation techniques, in: Proc. 7th IFSA World Congress, Prague, Czech Republic, 1997, pp. 121–126.

29.

Baranyi

, Bavelaar

, Babuska

, Koczy

L.T.

, Titli

and Verbruggen

H.B.

, A method to invert a linguistic fuzzy model, Internat J Systems Sci 29(7) (1998), 711–721.

30.

Babuska

, Fuzzy modeling for control, Kluwer Academic Publishers, Boston, 1998.

31.

Babuska

, Sousa

and Verbruggen

H.B.

, Model based design of fuzzy control systems, in: Proc. third European congress on intelligent techniques and Soft Computing, EUFIT ’95, Aachen, Germany, 1995, pp. 837–841.

32.

Boukezzoula

, Galichet

and Foulloy

, Nonlinear internal model control: application of inverse model based fuzzy control, IEEE Transactions on Fuzzy Systems 11(6) (2003), 814–829.

33.

Galichet

, Boukezzoula

and Foulloy

, Explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents, Fuzzy Sets and Systems 146(3) (2004), 421–436.

34.

Kumbasar

, Eksin

, Guzelkaya

and Yesil

, Adaptive fuzzy model based inverse controller design using BB-BC optimization algorithm, Expert Systems with Applications 38(10) (2011), 12356–12364.

35.

Kumbasar

, Eksin

, Guzelkaya

and Yesil

, Exact inversion of decomposable interval type-2 fuzzy logic systems, International Journal of Approximate Reasoning 54(2) (2013), 253–272.

36.

Kumbasar

, Eksin

, Guzelkaya

and Yesil

, Interval type-2 fuzzy inverse controller design in nonlinear IMC structure, Engineering Applications of Artificial Intelligence 24(6) (2011), 996–1005.

37.

Rauma

and Kurki

, Fuzzy logic applications in diagnosing mechatronic systems, in: Fuzzy Systems Design: Social and Engineering Applications, Springer Publishing Company, 1998, pp. 309–322.

38.

Shiwen

, Dianwei

, Xiaoyu

, Guang

and Jianjun

, Design of Networked Control Systems Using the Fuzzy Inverse Model, in Proc. 37th Chinese Control Conf.,Wuhan, 2018, pp. 6303–6308.

39.

Zeng

X.J.

and Singh

M.G.

, Decomposition property of fuzzy systems and its applications, IEEE Trans Fuzzy Systems 4(2) (1996), 149–165.

40.

Rayouf

, Ghorbel

and Braiek

N.B.

, A new Hammerstein model control strategy: feedback stabilization and stability analysis, Int J Dynam Control 7 (2019), 1453–1461.