Optimal parallel-distributed-compensation controller design for a class of time-varying Takagi–Sugeno fuzzy model–based time-delay systems by using the orthogonal function approach

Abstract

This study proposes a method of designing quadratic optimal fuzzy parallel-distributed-compensation controllers for a class of time-varying Takagi–Sugeno fuzzy model–based time-delay control systems used to solve the finite-horizon optimal control problem. The proposed method fuses the orthogonal function approach and the improved hybrid Taguchi-genetic algorithm. The Taguchi-genetic algorithm only requires algebraic computation to perform the algorithm used to solve time-varying Takagi–Sugeno fuzzy model–based time-delay feedback dynamic equations. The fuzzy parallel-distributed-compensation controller design problem is simplified by using the Taguchi-genetic algorithm to transform the static parameter optimization problem into an algebraic equation. The static optimization problem can then be solved easily by using the improved hybrid Taguchi-genetic algorithm to find the quadratic optimal parallel-distributed-compensation controllers of the time-varying Takagi–Sugeno fuzzy model–based time-delay control systems. The applicability of the proposed integrative method is demonstrated in a real-world design problem.

Keywords

Finite horizon time-varying Takagi–Sugeno fuzzy model time delay orthogonal functions hybrid Taguchi-genetic algorithm quadratic-optimal control

1. Introduction

Takagi and Sugeno (TS) proposed a simple but effective method of using fuzzy model–based expressions to represent nonlinear control systems. There are many successful applications in the design problems of the nonlinear control systems (Babuska,1998; Farinwata et al., 2000; Li, 2006; Mardani et al., 2019; Song et al., 2019; Tanaka and Wang,2001; Tong et al., 2004; Wang et al., 2019; and the references therein). On the other hand, time delay is a common cause of instability and poor performance in a TS fuzzy model, especially in dynamic systems. Therefore, recent studies have investigated methods of improving stability in TS fuzzy model–based time-delay control systems (see, e.g. Cao and Frank (2000, 2001), Er and Lin (2002), Park et al. (2003), Shin et al. (2003), Chen and Liu (2005), Chang and Chang (2000), Ali et al. (2019), Moodi and Kazemy (2019), Wang and Lam (2019), and the references therein). The design of a control system, however, must consider factors other than stability. An effective design usually requires integration of a quadratic optimal controller to achieve the minimum quadratic integral performance criterion (Goodwin et al., 2001). The linear matrix inequality–based (LMI-based) methods have been effectively used to design controllers (Baleghi and Shafiei, 2019; Yang et al., 2018; Zhang et al., 2017). The LMI methods have also been used to design quadratic optimal parallel distributed compensation (PDC) controllers for time-invariant TS fuzzy model–based time-delay control systems by minimizing the upper bound of a quadratic infinite-horizon integral performance index (Chou et al., 2006; Guan and Chen, 2004).

However, the authors of the above studies agree that, for some time-varying nonlinear time-delay control systems, the fuzzy membership function absorption time varying is very difficult to apply in time-invariant exact TS fuzzy model–based time-delay control systems. Therefore, minimizing the quadratic integral performance index in the design of a time-varying TS fuzzy model–based time-delay control system with quadratic optimal PDC controllers is worth studying. However, the LMI method (Chou et al., 2006; Guan and Chen, 2004) cannot be used to design time-varying TS fuzzy model–based time-delay control systems with quadratic-optimal PDC controllers. Although genetic algorithms (GAs) can be used to solve complex static optimization problems (Chou and Cheng, 2019), they are very difficult to use for dynamic optimization when designing quadratic optimal PDC controllers for time-varying TS fuzzy model–based time-delay control systems. To the best of our knowledge, no studies have proposed designs for time-varying TS fuzzy model–based time-delay control systems with quadratic optimal PDC controllers. Even though the authors have previously obtained the preliminary research results (Hsu et al., 2007), the optimal design algorithm is incomplete. Therefore, this issue is still worth to further study in both completeness and depth for the optimal design algorithm. Hence, by proposing an improved optimal design algorithm named the improved hybrid Taguchi-genetic algorithm (IHTGA), this study proposes an integrative method for time-varying TS fuzzy model–based time-delay control systems with quadratic optimal PDC controllers under the constraint of a minimum quadratic integral performance index. Besides, the systematic and step-by-step design procedures are also proposed in this study.

The proposed method integrates the orthogonal function approach (OFA) and IHTGA. OFA simplifies the problem of optimizing a time-varying TS fuzzy model–based time-delay control system by transforming the dynamic optimization problem into a static optimization problem expressed as an algebraic equation. Notably, however, when solving the dynamic optimization problem described above, OFA alone cannot find the fuzzy PDC controllers, and using only GA to find the fuzzy PDC controllers is difficult. Therefore, this study developed an integrative method of solving this design problem. The IHTGA is to improve the HTGA which was presented by Tsai et al. (2004). Tsai et al. (2004) and Ho et al. (2016) previously have demonstrated that the HTGA outperformed the conventional GA. Finally, the feasibility of the proposed integrative method is demonstrated in a real world application. Table 1 describes the notations of this study.

Table 1.

Summary of notations.

Notation	Definition
${\tilde{R}}^{i}$	The i-th implication
$x (t)$	The state vector
$A_{i} (t)$	The i-th consequent time-varying matrices
$B_{i} (t)$	The i-th consequent time-varying matrices
N	The number of fuzzy rules
$u (t - τ)$	The input vector with time delay
$τ$	The known time delay
$z_{i} (t)$	The i-th premise variables
$M_{i j}$	The fuzzy sets to the j-th premise variables in the i-th implication
$ϕ (t)$	The arbitrary known time-function vector
$u (t)$	The fuzzy PDC controller
$F_{i}$	The i-th local feedback gain matrices
J	The quadratic finite-horizon integral performance index
$t_{f}$	The short time interval selected for independent variable t
Q	The positive integer specified by the designer
$\bar{Q}$	The symmetric positive-semidefinite matrix
$\bar{R}$	The symmetric positive-definite matrix
$γ$	The specified positive integer
M	The number of terms required for the orthogonal functions
$T (t)$	The orthogonal basis vector
H	The time-delay operational matrix
$R$	The operational matrix of integration

2. Problem statement

For construction of a sector nonlinearity fuzzy model (Tanaka and Wang, 2001), we can use the following form

\begin{array}{l} {\tilde{R}}^{i} : IF z_{1} is M_{i 1} and \dots and z_{g} is M_{i g}, \\ THEN \dot{x} (t) = A_{i} (t) x (t) + B_{i} (t) u (t - τ) \end{array}

(1)

where the initial state vectors are

x (0)

and

x (t) = ϕ (t)

(for t < 0),

{\tilde{R}}^{i} (i = 1, 2, \dots, N)

is the i-th implication, N is the number of fuzzy rules,

x (t) = {[x_{1} (t), x_{2} (t), \dots, x_{n} (t)]}^{T}

is the n-dimensional state vector,

A_{i} (t)

and

B_{i} (t)

(i = 1, 2, \dots, N)

are the

n \times n

and

n \times p

consequent time-varying matrices, respectively,

u (t - τ) = {[u_{1} (t - τ), u_{2} (t - τ), \dots, u_{p} (t - τ)]}^{T}

is the p-dimensional input vector with time delay,

τ

is the known time delay,

z_{i} (i = 1, 2, \dots, g)

are the premise variables,

M_{i j}

(

i = 1, 2, \dots, N

and

j = 1, 2, \dots, g

) are the fuzzy sets of using sector nonlinearity in fuzzy model construction (Tanaka and Wang, 2001), and

ϕ (t)

is an n-dimensional arbitrary known time-function vector.

The time-varying TS fuzzy model–based time-delay dynamic system inferred from equation (1) is expressed as

\dot{x} (t) = \sum_{i = 1}^{N} h_{i} (z) (A_{i} (t) x (t) + B_{i} (t) u (t - τ))

(2)

in which

z = [z_{1}, z_{2}, \dots, z_{g}]

h_{i} (z) = w_{i} (z) / \sum_{i = 1}^{N} w_{i} (z)

w_{i} (z) = \prod_{i = 1}^{g} M_{i j} (z_{j})

, and

M_{i j} (z_{j})

are the grades of membership of

z_{j}

in fuzzy sets

M_{i j}

The fuzzy PDC controller (Tanaka and Wang, 2001) is given as

u (t) = - \sum_{i = 1}^{N} h_{i} (z) F_{i} x (t)

(3)

where

F_{i} (i = 1, 2, \dots, N)

denote the

p \times n

local feedback gain matrices.

Following the approach given by Tanaka and Wang (2001), the proposed model description can be viewed as parameter-dependent interpolation between linear models; however, the exact classification of the resultant system depends on the nature of the parameters. If each $z_{i}$ is function of the state variables, then the TS fuzzy model describes an autonomous nonlinear system (Tanaka and Wang, 2001). Therefore, following some previous studies (Liu and Hu, 2003; Luo et al., 2008; Sheng et al., 2019; Sakthivel et al., 2020; Tanaka and Wang, 2001; Wang and Lam, 2018), substituting equation (3) into (2) gives

\dot{x} (t) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z) h_{j} (z) (A_{i} (t) x (t) - B_{i} (t) F_{j} x (t - τ))

(4)

Equation (4) is used to design the fuzzy PDC controller in equation (3) by minimizing the following quadratic finite-horizon integral performance index

\begin{array}{l} J = \int_{0}^{q t_{f}} [x^{T} (t) \bar{Q} x (t) + u^{T} (t) \bar{R} u (t)] d t \\ = \sum_{k = 0}^{q - 1} \int_{k t_{f}}^{(k + 1) t_{f}} [x^{T} (t) \bar{Q} x (t) + u^{T} (t) \bar{R} u (t)] d t \end{array}

(5)

where

t_{f}

denotes a short time interval selected for independent variable t, q is a positive integer specified by the designer,

\bar{Q}

is a symmetric positive-semidefinite matrix, and

\bar{R}

is a symmetric positive-definite matrix. Here, the time interval of interest is designated as

t = 0

t = q t_{f}

, where

t = 0

is the initial time and

t = q t_{f}

is the final time of the control period.

3. Quadratic finite-horizon optimal PDC controller design

Now, consider the time interval $k t_{f} \leq t \leq (k + 1) t_{f}$ , where $t_{f} = τ / γ$ is chosen for the independent variable t, and let us define

t = k t_{f} + η

(6)

and

x_{k} = x (k t_{f})

(7)

where

γ

is a specified positive integer with

γ < q

k = 0, 1, 2, \dots, q - 1

, and

0 \leq η \leq t_{f}

The state vector $x (t)$ , time-function vector $ϕ (t - τ)$ , and consequent time-varying matrices $A_{i} (t)$ and $B_{i} (t)$ , within $k t_{f} \leq t \leq (k + 1) t_{f}$ , can be approximated by their respective truncated orthogonal functions

x (t) \approx \sum_{s = 0}^{m - 1} x_{s}^{(k)} T_{s} (t) = {\tilde{x}}^{(k)} T (t)

(8)

ϕ (t - τ) \approx \sum_{s = 0}^{m - 1} ϕ_{s}^{(k)} T_{s} (t) = {\tilde{ϕ}}^{(k)} T (t)

(9)

A_{i} (t) \approx \sum_{s = 0}^{m - 1} A_{i s}^{(k)} T_{s} (t)

(10)

and B_{i} (t) \approx \sum_{s = 0}^{m - 1} B_{i s}^{(k)} T_{s} (t)

(11)

where m is the number of terms required for the orthogonal functions (OF),

T (t) = {[T_{0} (t), T_{1} (t), \dots, T_{m - 1} (t)]}^{T}

denotes the

m \times 1

orthogonal basis vector,

T_{i} (t) (i = 0, 1, 2, \dots, m - 1)

denote the OF,

x_{s}^{(k)} (s = 0, 1, \dots, m - 1)

and

ϕ_{s}^{(k)} (s = 0, 1, \dots, m - 1)

are the

n \times 1

coefficient vectors,

{\tilde{x}}^{(k)} = [x_{0}^{(k)}, x_{1}^{(k)}, \dots, x_{m - 1}^{(k)}]

and

{\tilde{ϕ}}^{(k)} = [ϕ_{0}^{(k)}, ϕ_{1}^{(k)}, \dots, ϕ_{m - 1}^{(k)}]

are the

n \times m

expansion coefficient matrices of

x (t)

and

ϕ (t - τ)

, respectively, and

A_{i s}^{(k)}

(n \times n)

and

B_{i s}^{(k)}

(n \times p)

(s = 0, 1, \dots, m - 1)

are the coefficient matrices of

A_{i} (t)

and

B_{i} (t)

, respectively.

Before inferring the consequent output within $k t_{f} \leq t \leq (k + 1) t_{f}$ , let the value of $h_{i} (z)$ be $h_{i} (z_{k})$ for $t$ at $k t_{f}$ . Then

x (t) - x (k t_{f}) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (\int_{k t_{f}}^{t} A_{i} (t) x (t) d t - \int_{k t_{f}}^{t} B_{i} (t) F_{j} ϕ (t - τ) d t)

(12)

where

h_{i} (z_{k}) = h_{i} (z (k t_{f}))

h_{j} (z_{k}) = h_{j} (z (k t_{f}))

, and

k = 0, 1, 2, \dots, γ - 1

Next, considering $τ \leq k t_{f} \leq t \leq (k + 1) t_{f}$ and integrating equation (4) from $t = k t_{f}$ to $t = t$ within $k t_{f} \leq t \leq (k + 1) t_{f}$ , we can get

x (t) - x (k t_{f}) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (\int_{k t_{f}}^{t} A_{i} (t) x (t) d t - \int_{k t_{f}}^{t} B_{i} (t) F_{j} x (t - τ) d t)

(13)

where

k = γ, γ + 1, \dots, q - 1

Now, the attribute is obtained by using the following orthogonal function (Hsu et al., 2006)

T (t - τ) = H T (t)

(14)

In addition, equations (8)–(11), $A_{i} (t) x (t)$ , $B_{i} (t) F_{j} ϕ (t - τ)$ , and $B_{i} (t) F_{j} x (t - τ)$ can be represented by their respective orthogonal-function representations

A_{i} (t) x (t) \approx \sum_{s = 0}^{m - 1} y_{i s}^{(k)} T_{s} (t) = {\tilde{y}}_{i}^{(k)} T (t)

(15a)

B_{i} (t) F_{j} ϕ (t - τ) \approx \sum_{s = 0}^{m - 1} φ_{i j s}^{(k)} T_{s} (t) = {\tilde{φ}}_{i j}^{(k)} T (t)

(15b)

and B_{i} (t) F_{j} x (t - τ) \approx \sum_{s = 0}^{m - 1} v_{i j s}^{(k)} T_{s} (t) = {\tilde{v}}_{i j}^{(k)} T (t)

(15c)

where

y_{i s}^{(k)}

φ_{i j s}^{(k)}

, and

v_{i j s}^{(k)}

are the

n \times 1

coefficient vectors and

{\tilde{y}}_{i}^{(k)} = [y_{i 0}^{(k)}, y_{i 1}^{(k)}, \dots, y_{i, m - 1}^{(k)}], {\tilde{φ}}_{i j}^{(k)} = [φ_{i j 0}^{(k)}, φ_{i j 1}^{(k)}, \dots, φ_{i j, m - 1}^{(k)}]

, and

{\tilde{v}}_{i j}^{(k)} = [v_{i j 0}^{(k)}, v_{i j 1}^{(k)}, \dots, v_{i j, m - 1}^{(k)}]

are the

n \times m

coefficient matrices, in which

y_{i s}^{(k)} = \sum_{a = 0}^{m - 1} \sum_{b = 0}^{m - 1} α_{a b s} A_{i a}^{(k)} x_{b}^{(k)}

(16a)

φ_{i j s}^{(k)} = \sum_{a = 0}^{m - 1} \sum_{b = 0}^{m - 1} α_{a b s} B_{i a}^{(k)} F_{j} ϕ_{b}^{(k)}

(16b)

v_{i j s}^{(k)} = \sum_{a = 0}^{m - 1} \sum_{b = 0}^{m - 1} \sum_{c = 0}^{m - 1} α_{a c s} h_{b c} B_{i a}^{(k)} F_{j} x_{b}^{(k)}

(16c)

where

h_{b c}

denotes the elements of the time-delay operational matrix H given as described in Hsu et al. (2006).

Application orthogonal functions have the following integrated property (Ho and Chou, 2006; Hsu et al., 2006)

\int_{k t_{f}}^{t} T (t) d t = R T (t)

(17)

and equations (7), (8), and (15), equations (12) and (13) can be rewritten as

\begin{array}{l} {\tilde{x}}^{(k)} - [x_{k}, 0, 0, \dots, 0] = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) {\tilde{y}}_{i}^{(k)} R - \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) {\tilde{φ}}_{i j}^{(k)} R \\ (for k = 0, 1, 2, \dots, γ - 1) \end{array}

(18a)

and

\begin{array}{l} {\tilde{x}}^{(k)} - [x_{k}, 0, 0, \dots, 0] = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) {\tilde{y}}_{i}^{(k)} R - \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) {\tilde{v}}_{i j}^{(k)} R \\ (for k = γ, γ + 1, \dots, q - 1) \end{array}

(18b)

where R is the

m \times m

operational matrix of integration for the orthogonal functions given as described in Ho and Chou (2006) and Hsu et al. (2006).

Equations (16a), (16b), and (16c) can be rewritten as

{\hat{y}}_{i}^{(k)} = {\hat{A}}_{i}^{(k)} {\hat{x}}^{(k)}

(19a)

{\hat{φ}}_{i j}^{(k)} = {\hat{W}}_{i j}^{(k)} {\hat{ϕ}}^{(k)}

(19b)

and

{\hat{v}}_{i j}^{(k)} = {\hat{V}}_{i j}^{(k)} {\hat{x}}^{(k)}

(19c)

where

{\hat{x}}^{(k)} = {[x_{0}^{{(k)}^{T}}, x_{1}^{{(k)}^{T}}, \dots, x_{m - 1}^{{(k)}^{T}}]}^{T}

(20a)

{\hat{ϕ}}^{(k)} = {[ϕ_{0}^{{(k)}^{T}}, ϕ_{1}^{{(k)}^{T}}, \dots, ϕ_{m - 1}^{{(k)}^{T}}]}^{T}

(20b)

{\hat{y}}_{i}^{(k)} = {[y_{i 0}^{{(k)}^{T}}, y_{i 1}^{{(k)}^{T}}, \dots, y_{i, m - 1}^{{(k)}^{T}}]}^{T}

(20c)

{\hat{φ}}_{i j}^{(k)} = {[φ_{i j 0}^{{(k)}^{T}}, φ_{i j 1}^{{(k)}^{T}}, \dots, φ_{i j, m - 1}^{{(k)}^{T}}]}^{T}

(20d)

{\hat{v}}_{i j}^{(k)} = {[v_{i j 0}^{{(k)}^{T}}, v_{i j 1}^{{(k)}^{T}}, \dots, v_{i j, m - 1}^{{(k)}^{T}}]}^{T}

(20e)

{\hat{A}}_{i}^{(k)} = [\begin{matrix} \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a 00} & \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a 10} & \dots & \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a, m - 1,0} \\ \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a 01} & \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a 11} & \dots & \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a, m - 1,1} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a, 0, m - 1} & \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a, 1, m - 1} & \dots & \sum_{a = 0}^{m - 1} A_{i a}^{(k)} α_{a, m - 1, m - 1} \end{matrix}]

(20f)

{\hat{W}}_{i j}^{(k)} = [\begin{matrix} \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a 00} & \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a 10} & \dots & \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a, m - 1,0} \\ \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a 01} & \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a 11} & \dots & \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a, m - 1,1} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a, 0, m - 1} & \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a, 1, m - 1} & \dots & \sum_{a = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a, m - 1, m - 1} \end{matrix}]

(20g)

and

{\hat{V}}_{i j}^{(k)} = [\begin{matrix} \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c 0} h_{0 c} & \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c 0} h_{1 c} & \dots & \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c 0} h_{m - 1, c} \\ \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c 1} h_{0 c} & \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c 1} h_{1 c} & \dots & \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c 1} h_{m - 1, c} \\ ⋮ & ⋮ & \dots & ⋮ \\ \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c, m - 1} h_{0 c} & \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c, m - 1} h_{0 c} & \dots & \sum_{a = 0}^{m - 1} \sum_{c = 0}^{m - 1} B_{i a}^{(k)} F_{j} α_{a c, m - 1} h_{m - 1, c} \end{matrix}]

(20h)

In addition, the Kronecker product and equations (19a), (19b), and (19c) can be used to rewrite equations (18a) and (18b) as

\begin{array}{l} {\hat{x}}^{(k)} - {\hat{x}}_{k} = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{A}}_{i}^{(k)} {\hat{x}}^{(k)}) - \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{W}}_{i j}^{(k)} {\hat{ϕ}}^{(k)}) \\ (for k = 0, 1, 2, \dots, γ - 1) \end{array}

(21a)

and

\begin{array}{l} {\hat{x}}^{(k)} - {\hat{x}}_{k} = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{A}}_{i}^{(k)} {\hat{x}}^{(k)}) - \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{V}}_{i j}^{(k)} {\hat{x}}^{(k)}) \\ (for k = γ, γ + 1, \dots, q - 1) \end{array}

(21b)

where

I_{n}

denotes the

n \times n

identity matrix,

\otimes

denotes the Kronecker product (Barnett, 1979), and

{\hat{x}}_{k} = {[x_{k}^{T}, 0^{T}, 0^{T}, \dots, 0^{T}]}^{T}

(22)

Finally, the explicit form for the solution ${\tilde{x}}^{(k)}$ can be directly derived by equations (21a) and (21b) as

\begin{array}{l} {\hat{x}}^{(k)} = {[I_{m n} - \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{A}}_{i}^{(k)})]}^{- 1} \cdot [{\hat{x}}_{k} - \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{W}}_{i j}^{(k)} {\hat{ϕ}}^{(k)})] \\ (for k = 0, 1, 2, \dots, γ - 1) \end{array}

(23a)

and

\begin{array}{l} {\hat{x}}^{(k)} = {[I_{m n} - (\sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) (R^{T} \otimes I_{n}) ({\hat{A}}_{i}^{(k)}) - (R^{T} \otimes I_{n}) ({\hat{V}}_{i j}^{(k)}))]}^{- 1} \cdot {\hat{x}}_{k} \\ (for k = γ, γ + 1, \dots, q - 1) \end{array}

(23b)

where

I_{m n}

denotes the

m n \times m n

identity matrix. The algebraic formulae in equations (23a) and (23b) can be used to calculate

x_{s}^{(k)} (s = 0, 1, \dots, m - 1)

within any time interval

t_{f} \leq t \leq (k + 1) t_{f}

(k = 0, 1, 2, \dots, q - 1)

. This implies that

{\tilde{x}}^{(k)}

can be obtained by equations (23a) and (23b). The flowchart of the proposed model only involves the algebraic computation as shown in Figure 1.

In addition, substituting equation (3) and the truncated OF representation of $x (t)$ in equation (8) into the quadratic finite-horizon integral performance index in equation (5) gives a quadratic finite-horizon integral performance index J in the following algebraic form

J = \sum_{k = 0}^{q - 1} trace [W {({\tilde{x}}^{(k)})}^{T} (\bar{Q} + \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) F_{i}^{T} \bar{R} F_{j}) ({\tilde{x}}^{(k)})]

(24)

where the constant matrix W is the product integration matrix of two OF basis vectors (Ho and Chou, 2006).

${\tilde{x}}^{(k)} (k = 0, 1, 2, \dots, q - 1)$ can then be calculated with ${F_{1}, F_{2}, \dots, F_{N}}$ in equations (18a) and (18b) by performing only the algebraic computation. The value obtained by equation (24) is actually determined by ${F_{1}, F_{2}, \dots, F_{N}}$ , which means

J = G (f_{111}, f_{112}, \dots, f_{N p n})

(25)

where

f_{i j k}

(

i = 1, 2, \dots, N

j = 1, 2, \dots, p

and

k = 1, 2, \dots, n

) denote the elements of

F_{i}

. Therefore, the fuzzy PDC controller optimal design problem equates to the minimum problem

\min J = G (f_{111}, f_{112}, \dots, f_{N p n})

(26)

subject to

| f_{i j k} | \leq C_{i j k}

, for

i = 1, 2, \dots, N

j = 1, 2, \dots, p

, and

k = 1, 2, \dots, n

, where

C_{i j k}

are positive real numbers given in the practical application, respectively. Therefore, for the quadratic finite-horizon optimal PDC controller design problem, OFA can be used to express the static optimization problem as an algebraic equation according to the time-varying TS fuzzy model–based time-delay control system. This greatly simplifies the problem of optimizing a fuzzy PDC controller. Although any evolutionary optimization approaches can be used in this study and the HTGA has been applied in many different cases, the HTGA still gives better and robust results than those of the existing improved GA reported in the literature (Ho et al., 2016). Genetic algorithm is a well-known optimization method. To reduce the length of the article, for the detailed explanation of the genetic algorithm, the readers may refer to the work by Tsai et al. (2004). Furthermore, in this study, by improving the HTGA, an IHTGA is proposed and used to obtain a better performance index. The IHTGA is used to solve the static optimization problem as shown in equation (26). The IHTGA improves the HTGA by using the generation-based crossover rate

α (\tilde{g}) = α_{min} + (({\tilde{g}}_{max} - \tilde{g}) / {\tilde{g}}_{max}) \times (α_{max} - α_{min})

and the generation-based mutation rate

β (\tilde{g}) = β_{min} + (({\tilde{g}}_{max} - \tilde{g}) / {\tilde{g}}_{max}) \times (β_{max} - β_{min})

, where

α_{max} = 0.9

α_{min} = 0.7

β_{max} = 0.03

β_{min} = 0.01

, as well as

\tilde{g}

and

{\tilde{g}}_{max}

are the current generation and the maximum of generation, respectively. In this study, the maximum generation number is 250. On the other hand, the Taguchi method is a well-known local optimization approach and has been merged into the GA by Tsai et al. (2004) for obtaining both better and robust results. To reduce the length of the article, for the detailed explanation of the hybrid Taguchi-genetic algorithm (HTGA), the readers may refer to the work by Tsai et al. (2004), where the Taguchi method has been explained in detail therein.

4. Illustrative example

Figure 2 shows the illustrative example in which Y is the amplitude, $ω$ is the oscillation frequency, and $\bar{y} (t)$ is the displacement of the pivot point in the vertical direction (Rao, 1995). The pendulum pivot point vibrates as

\bar{y} (t) = Y cos (ω t)

(27)

The nonlinear equation for the motion of the time-varying pendulum system with time-delay torque applied to the system mass can be derived as

\bar{m} l^{2} \ddot{θ} (t) + \bar{m} l (\bar{g} - \ddot{\bar{y}} (t)) sin (θ (t)) = u (t - τ)

(28)

where

\bar{m}

is the system mass of the pendulum system, l is the length of the pendulum system,

\bar{g}

is the acceleration of gravity,

\ddot{\bar{y}} (t)

is the acceleration of the pivot point in the vertical direction,

θ (t)

is the angle of the pendulum system and changes between

- π / 2

and

π / 2

\ddot{θ} (t)

is the angular acceleration of the pendulum system,

u (t - τ)

is the torque with time delay applied to the system mass, and

τ

is the known time delay.

Equation (28) can be rewritten as

\ddot{θ} (t) = - (\frac{\bar{g}}{l} - \frac{ω^{2} Y}{l} cos ω t) sin (θ (t)) + \frac{1}{\bar{m} l^{2}} u (t - τ)

(29)

If the state variables are defined as $x_{1} (t) = θ (t)$ and $x_{2} (t) = {\dot{x}}_{1} (t) = \dot{θ} (t)$ , the state-space equations can be obtained as

{\dot{x}}_{1} (t) = x_{2} (t)

(30a)

and

{\dot{x}}_{2} (t) = - (\frac{\bar{g}}{l} - \frac{ω^{2} Y}{l} cos (ω t)) sin (x_{1} (t)) + \frac{1}{\bar{m} l^{2}} u (t - τ)

(30b)

where

x_{1} (t) = θ (t) \in [- π / 2, π / 2]

By defining $z_{1} = sin (x_{1} (t))$ and $z_{2} = cos (ω t) sin (x_{1} (t))$ , four-rule membership functions can be defined as

M_{11} (z_{1}) = M_{21} (z_{1}) = {\hat{M}}_{1} (z_{1})

(31a)

M_{31} (z_{1}) = M_{41} (z_{1}) = {\hat{M}}_{2} (z_{1})

(31b)

M_{12} (z_{2}) = M_{32} (z_{2}) = {\hat{N}}_{1} (z_{1})

(31c)

M_{22} (z_{2}) = M_{42} (z_{2}) = {\hat{N}}_{2} (z_{2})

(31d)

{\hat{M}}_{1} (z_{1}) = {\begin{matrix} \frac{z_{1} - (2 / π) {sin}^{- 1} z_{1}}{(1 - 2 / π) {sin}^{- 1} z_{1}}, z_{1} \neq 0 \\ 1, otherwise \end{matrix}

(31e)

{\hat{M}}_{2} (z_{1}) = {\begin{matrix} \frac{{sin}^{- 1} z_{1} - z_{1}}{(1 - 2 / π) {sin}^{- 1} z_{1}}, z_{1} \neq 0 \\ 0, otherwise \end{matrix}

(31f)

{\hat{N}}_{1} (z_{2}) = {\begin{matrix} \frac{cos (ω t) (z_{2} - (2 / π) {sin}^{- 1} z_{2})}{(1 - 2 / π) {sin}^{- 1} z_{2}}, z_{2} \neq 0 \\ cos (ω t), otherwise \end{matrix}

(31g)

and

{\hat{N}}_{2} (z_{2}) = {\begin{matrix} \frac{cos (ω t) ({sin}^{- 1} z_{2} - z_{2})}{(1 - 2 / π) {sin}^{- 1} z_{2}}, z_{2} \neq 0 \\ 0, otherwise \end{matrix}

(31h)

In this example, equations (30a) and (30b) give

Model Rule 1 $({\tilde{R}}^{1})$

\begin{array}{l} IF z_{1} is M_{11} (z_{1}), \\ THEN \dot{x} (t) = A_{1} (t) x (t) + B_{1} (t) u (t - τ) \end{array}

(32a)

Model Rule 2 $({\tilde{R}}^{2})$

\begin{array}{l} IF z_{1} is M_{21} (z_{1}), \\ THEN \dot{x} (t) = A_{2} (t) x (t) + B_{2} (t) u (t - τ) \end{array}

(32b)

where

z_{1} = sin (x_{1} (t))

x (t) = {[x_{1} (t), x_{2} (t)]}^{T}

x (0) = {[π / 6, 0]}^{T}

x_{1} (t) \in [- π / 2, π / 2]

A_{1} (t) = [\begin{matrix} 0 & 1 \\ - ((\bar{g} / l) - (ω^{2} Y / l) cos (ω t)) & 0 \end{matrix}]

A_{2} (t) = [\begin{matrix} 0 & 1 \\ (2 / π) ((\bar{g} / l) - (ω^{2} Y / l) cos (ω t)) & 0 \end{matrix}]

B_{1} (t) = B_{2} (t) = [\begin{matrix} 0 \\ 1 / \bar{m} l^{2} \end{matrix}]

τ = 0.5

\bar{m} = 1 kg

l = 1 m

ω = 1 Hz

Y = 0.01 m

\bar{g} = 9.8 m / s^{2}

M_{11} (z_{1}) = {\begin{matrix} \frac{z_{1} - (2 / π) {sin}^{- 1} z_{1}}{(1 - 2 / π) {sin}^{- 1} z_{1}}, z_{1} \neq 0 \\ 1, otherwise \end{matrix}

and

M_{21} (z_{1}) = {\begin{matrix} \frac{{sin}^{- 1} z_{1} - z_{1}}{(1 - 2 / π) {sin}^{- 1} z_{1}}, z_{1} \neq 0 \\ 0, otherwise \end{matrix}

The quadratic finite-horizon integral performance index is

\begin{array}{l} J = \int_{0}^{10} [x^{T} (t) \bar{Q} x (t) + u^{T} (t) \bar{R} u (t)] d t \\ = \sum_{k = 0}^{q - 1} \int_{k t_{f}}^{(k + 1) t_{f}} [x^{T} (t) \bar{Q} x (t) + u^{T} (t) \bar{R} u (t)] d t \end{array}

(33)

in which

q = 1000

t_{f} = 0.01

\bar{Q} = diag {1, 1}

, and

\bar{R} = 1

If $m = 4$ and $| f_{i j k} | \leq 5$ ( $i = 1, 2,$ $j = 1$ and $k = 1, 2$ ), the proposed approach gives $J = 0.4765,$ $F_{1} = [\begin{matrix} - 0.9464 & - 1.2572 \end{matrix}]$ , and $F_{2} = [\begin{matrix} - 0.9533 & - 1.2689 \end{matrix}]$ . The responses for angle $x_{1} (t)$ in Figure 3 and for angular velocity $x_{2} (t)$ in Figure 4 confirm that the control results are satisfactory in this example, in which the orthogonal function is the shifted Chebyshev function. Under the same performance index in equation (33) and executing five independent runs, in Tables 2 and 3, it can be shown that we can obtain a better average performance index of $J = 0.4766$ by using the IHTGA than the average result with $J = 0.4935$ by using the HTGA as well as the proposed IHTGA, which gives a more robust result.

Table 2.

Five different sets of the quadratic-optimal controllers and the performance indices obtained by using the IHTGA in five independent runs.

IHTGA
No.	F ₁	F ₂	J
1	[−0.9483,–1.2573]	[−0.9715, –1.2833]	0.4766
2	[−0.9464, –1.2572]	[−0.9533, –1.2689]	0.4765
3	[−0.9448, –1.2573]	[−0.9687, –1.2849]	0.4766
4	[−0.9468, –1.2573]	[−0.9552, –1.2676]	0.4766
5	[−0.9455, –1.2572]	[−0.9595, –1.2732]	0.4766
Average (standard deviation)			0.4766 (4.4721 × 10⁻⁵)

IHTGA: improved hybrid Taguchi-genetic algorithm.

Table 3.

Five different sets of the quadratic optimal controllers and the performance indices obtained by using the HTGA in five independent runs.

HTGA
No.	F ₁	F ₂	J
1	[−1.0534, –1.2762]	[−1.1370, –1.4834]	0.4921
2	[−1.1247, –1.2962]	[−1.0053, –1.7077]	0.4978
3	[−1.0809, –1.2630]	[−1.2278, –1.4958]	0.4907
4	[−1.0719, –1.2770]	[−1.2458, –1.5402]	0.4929
5	[−1.0453, –1.2879]	[−1.0558, –1.3710]	0.4942
Average (standard deviation)			0.4935 (2.6987 × 10⁻³)

HTGA: hybrid Taguchi-genetic algorithm.

5. Conclusions

When solving the quadratic finite-horizon optimal control problem, OFA can be used to express static parameter optimization problems as algebraic equations based on time-varying TS fuzzy model–based time-delay control systems. Because the proposed integrative method only requires algebraic computation, it greatly simplifies the process of designing quadratic-optimal PDC controllers for time-varying TS fuzzy model–based time-delay control systems. The IHTGA easily solves the static optimization problem in time-varying TS fuzzy model–based time-delay control systems with quadratic optimal PDC controllers. The illustrative example demonstrates the applicability of the proposed integrative method and the proposed IHTGA having a better and more robust result.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Science and Technology, Taiwan under grants MOST 108-2221-E-037-007 and MOST 108-2218-E-992-304, and the “Intelligent Manufacturing Research Center” (iMRC) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan.

ORCID iDs

Ming-Ren Hsu

Wen-Hsien Ho

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Optimal parallel-distributed-compensation controller design for a class of time-varying Takagi–Sugeno fuzzy model–based time-delay systems by using the orthogonal function approach–assisted genetic algorithm

Abstract

Keywords

1. Introduction

2. Problem statement

3. Quadratic finite-horizon optimal PDC controller design

4. Illustrative example

5. Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References