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Abstract

A complementary computational approach is proposed for the time-varying Takagi–Sugeno fuzzy model system (TVTSFMS). The proposed approach integrates orthogonal-functional approach (OFA), hybrid Taguchi genetic algorithm (HTGA), and a stabilizability condition (SC) for use in designing stable and quadratic-optimal parallel-distributed-compensation (SQOPDC) controllers for optimal control problems. First, the SC was set according to linear matrix inequalities (LMIs). Next, OFA was used to derive an algorithm that only required algebraic computation to solve the TVTSFMS. Finally, The HTGA could be used to search the SQOPDC controller for the TVTSFMS. The SQOPDC controller obtained by the proposed complementary computational approach was evaluated in a case study of a vibratory pendulum design; the successful design verified the usability of the proposed hybrid intelligent computing method.

Keywords

Takagi-Sugeno fuzzy model orthogonal-functional approach hybrid Taguchi-genetic algorithm time-varying parallel-distributed-compensation controller

Introduction

The plant in an actual control system is mostly nonlinear. Therefore, scholars have developed many different approaches to designing controllers in nonlinear control systems in order to solve controller design problems in actual control systems.¹ In the Takagi–Sugeno fuzzy model, each fuzzy rule for the local linear control system is represented by a linear dynamic equation.^2–7 The overall nonlinear control system is then formalized by combining the fuzzy rules. Therefore, linear control theory is generally used in controller design. For nonlinear time-varying systems, it is difficult to design the controllers. Using Takagi–Sugeno fuzzy model (TSFM), the overall nonlinear time-varying system can be formalized by combining the fuzzy rules such that the linear control theory can be used in the controller design facilitating the controller easy implementation in practical situation. However, an important research issue is the design of stable and quadratic-optimal parallel-distributed-compensation (SQOPDC) controllers for a time-varying Takagi–Sugeno fuzzy model system (TVTSFMS) with a minimized performance index. Though the PI/PD/PID controller can be easily implemented in practical situation, it is very difficult to design the PI/PD/PID controller for the time-varying Takagi–Sugeno fuzzy model system (TVTSFMS) with a minimized performance index. In general, linear matrix inequalities (LMIs) are applied to design the SQOPDC controller of time-invariant Takagi–Sugeno fuzzy model systems. However, the LMIs are not directly applicable for solving the problem of designing an SQOPDC controller for a TVTSFMS.^8–14 Therefore, the objective of this study was to develop a complementary computational approach in which orthogonal-functional approach (OFA) is used to convert the SQOPDC controller design problem of the TVTSFMS into an algebraic computation problem.¹⁵ To simplify the controller design problem, the proposed approach also integrates hybrid Taguchi genetic algorithm (HTGA)^16–19 and LMIs in the design process to ensure that the TVTSFMS can be stabilized in a closed loop. Here it should be noticed that, for the TVTSFMS, the OFA or the LMI technique cannot be applied alone to find the SQOPDC controllers, and it is also very difficult to apply the genetic algorithms alone to find the SQOPDC controllers. So, in this paper, we complementarily fuse the OFA, the HTGA and the LMI technique to solve the design problem to be studied. The proposed integrative method fusing the OFA, the HTGA and the LMI technique belongs to the hard-computing-assisted soft-computing category, where the OFA and the LMI technique belong to the hard computing constituents and the HTGA is one of the soft computing constituents. That is, the main contribution of this paper is to integrate the OFA, the HTGA, and an LMI-based stabilizability condition (SC) for use in designing SQOPDC controllers for optimal control problems. A system block for the closed-loop system is shown in Figure 1.

Figure 1.

Stable and quadratic-optimal parallel-distributed-compensation controller of the time-varying Takagi–Sugeno fuzzy model system.

Problem statement

A TVTSFMS can be expressed as follows:

${\tilde{R}}^{i}$ : IF $z_{1} (t)$ is $M_{i 1}$ and … and $z_{g} (t)$ is $M_{i g}$

THEN \dot{x} (t) = A_{i} (t) x (t) + B_{i} (t) u (t),

(1)

with the initial state vector

x (0),

where

{\tilde{R}}^{i}

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

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

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

denotes the n-dimensional state vector,

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

denotes the p-dimensional input vector,

z_{i} (t)

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

are the premise variables,

A_{i} (t)

and

B_{i} (t)

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

are, respectively, the

n \times n

and

n \times p

consequent time-varying matrices, and

M_{i j}

(i = 1, 2, \dots, N a n d j = 1, 2, \dots, g)

are the fuzzy sets.

The TVTSFMS in equation (1) has the following two characteristics: (i) time-varying elements $a_{i j k} (t)$ in $A_{i} (t)$ belonging to $[{\underline{a}}_{i j k}, {\bar{a}}_{i j k}],$ where ${\underline{a}}_{i j k}$ and ${\bar{a}}_{i j k}$ are given constants and (ii) time-varying elements $b_{i j k} (t)$ in $B_{i} (t)$ belonging to $[{\underline{b}}_{i j k}, {\bar{b}}_{i j k}],$ where ${\underline{b}}_{i j k}$ ${\bar{b}}_{i j k}$ are given constants.

Equation (1) can be rewritten as

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

(2)

in which

z (t) = {[z_{1} (t), z_{2} (t), \dots, x_{g} (t)]}^{T}

denotes the

g

-dimensional premise vector,

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

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

, and

M_{i j} (z_{j} (t))

are the grades of membership of

z_{j} (t)

in the fussy sets

M_{i j}

(i = 1, 2, \dots, N a n d j = 1, 2, \dots, g)

. It can be seen that, for all t,

h_{i} (z (t)) \geq 0

and

\sum_{i = 1}^{N} h_{i} (z (t)) = 1 .

Here, we consider

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

is the SQOPDC controller²⁰ and where

F_{i}

is the local feedback gain matrices, which are expressed as

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

(3)

In equation (3), the stabilizability problem is whether $F_{i}$ of the SQOPDC controller meets the pre-specified SC. Therefore, the following sections propose an SC based on LMI to formalize the instances of equation (3) that can be stabilized.

Theorem

To stabilize SQOPDC controller $F_{i}$ of the TVTSFMS in equation (3), there exists a symmetric positive definite matrix P that enables the controller to satisfy the following LMI

U_{i j l}^{T} P + P U_{i j l} < 0,

(4)

where

U_{i j l} = \sum_{α = 0}^{n} \sum_{β = 0}^{n} ε_{i α β} (t) E_{i α β}^{j} - \sum_{α = 0}^{n} \sum_{β = 0}^{p} η_{i α β} (t) L_{i j α β} | \begin{array}{l} ε_{i α β} (t) = {\underline{ε}}_{i α β} or {\bar{ε}}_{i α β} \\ η_{i α β} (t) = {\underline{η}}_{i α β} or {\bar{η}}_{i α β} \end{array}

(5)

are constant matrices in which

E_{i α β}^{j} = E_{i α β}

denotes an

n \times n

constant matrix with 1 in the αβ-th entry and 0 elsewhere and in which

{\underline{η}}_{i α β} = ({\underline{b}}_{i α β} - {\bar{b}}_{i α β}) / 2,

{\bar{η}}_{i α β} = ({\bar{b}}_{i α β} - {\underline{b}}_{i α β}) / 2,

L_{i j α β} = V_{i α β} F_{j},

and

V_{i α β}

denote an

n \times p

constant matrix with 1 in the αβ-th entry and 0 elsewhere.

Proof: See Appendix.

Therefore, the question considered in this study is how to specify a value for $F_{i}$ of the SQOPDC controller such that (i) equation (3) meets the requirements of the LMIs in equation (4) and (ii) Control performance under the initial state conditions can be optimized by minimizing the following performance index

J = \sum_{k = 0}^{q - 1} \int_{k t_{f}}^{(k + 1) t_{f}} [x^{T} (t) Q x (t) + x^{T} (t) Q x (t)] d t,

(6)

where

t_{f}

is the shortest time interval.

Therefore, the SQOPDC controller design process is specifying $F_{i}$ and then executing the following two steps to optimize stability and control:

Step 1: Verify that the SC of the LMI in equation (4) is the constraint condition.

Step 2: For the TVTSFMS, minimize J in equation (6).

That is, the design problem of the SQOPDC controller for the TVTSFMS is a constrained-optimization problem. In the next section, we will complementarily fuse the OFA, the HTGA and the presented LMI-based SC to solve the SQOPDC controller design problem of the TVTSFMS in equation (1), where the performance index J in equation (6) subject to the constraint of SC in equation (4) is considered to be directly minimized, where if there is no solution for equation (4), a penalty is given in the HTGA.

SQOPDC controller design

First, we define

t = k t_{f} + η,

(7)

and

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

(8)

Next, an orthogonal function (OF) can be used to obtain

x (t) = \sum_{s = 0}^{m - 1} x_{s}^{(k)} P_{s} (t) = {\tilde{x}}^{(k)} P (t),

(9)

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

(10)

and

B_{i} (t) = \sum_{s = 0}^{m - 1} B_{i s}^{(k)} P_{s} (t),

(11)

where

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

represents the basis vector of OF,

P_{i} (t)

represents OF, and

x_{s}^{(k)}

A_{i s}^{(k)}

and

B_{i s}^{(k)}

represent the coefficient matrices.

By substituting $u (t) = - \sum_{i = 1}^{N} h_{i} (z (t)) F_{i} x (t)$ and $x (t) = {\tilde{x}}^{(k)} P (t)$ into equation (6), performance index J can be rewritten as

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

(12)

where

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

and where constant matrix

W = \int_{k t_{f}}^{(k + 1) t_{f}} P (t) P^{T} (t) d t

.¹³

From the orthogonal condition and the properties of OF, the product of any two OFs can be acquired and can be expressed by the following formula¹⁵

P_{a} (t) P_{b} (t) = \sum_{s = 0}^{m - 1} ξ_{a b s} P_{s} (t) .

(13)

The consequent output only requires inference in an extremely short time interval. Therefore, integrating equation (2) yields

x (t) - x (k t_{f}) = \sum_{i = 1}^{N} h_{i} (z_{k}) [\int_{k t_{f}}^{t} A_{i} (t) x (t) d t + \int_{k t_{f}}^{t} B_{i} (t) u (t) d t] .

(14)

The coefficient matrix of equation (14) can then be expressed as

{\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)} H + \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z_{k}) h_{j} (z_{k}) {\tilde{v}}_{i j}^{(k)} H .

(15)

Given a set of local feedback gain matrices ${F_{1}, F_{2}, \dots, F_{N}}$ , ${\tilde{x}}^{(k)}$ can then be acquired by the algebraic calculation.

The above algebraic calculation clearly indicated that specifying a ${F_{1}, F_{2}, \dots, F_{N}}$ enables the calculation of ${\tilde{x}}^{(k)}$ , which in turn facilitates the use of equation (12) to calculate the performance index. Therefore, this control problem can be converted into the following a static-constrained optimization problem

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

(16)

If (i) constraints $| f_{i j k} | \leq C_{i j k},$ are met and if (ii) the LMI of equation (4) is established (where $C_{i j k}$ is a given positive real number based on actual engineering considerations), the OFA and LMI can be used to obtain a constrained algebraic equation that represents the stability of the TVTSFMS and the SQOPDC control design problem. This representation substantially simplifies the SQOPDC control problem. The HTGA can then be used to determine the optimal solution for equation (16).^16–19

Remark 1

From the results mentioned above, we can see that, by using the OFA, we can transform the SQOPDC control problem for the TVTSFMS into a static optimization problem represented by the algebraic equations. Then, by incorporating the LMI-based SC for this TVTSFMS, the static optimization problem becomes a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based SC. This means that the main characteristic and contribution of this technique, bringing the OFA and the LMI-based SC together, is that it reduces the mixed H₂/LMI PDC controllers design problem to that of solving a static constrained-optimization problem of algebraic form; thus greatly simplifying the mixed H₂/LMI PDC controllers design problem and facilitating the work of applying genetic algorithms to solve the SQOPDC controllers design problem. The proposed integrative method, that complementarily fuses the OFA, the HTGA and the LMI-based SC, belongs to the hard-computing-assisted soft-computing category.

Remark 2

For the problem of designing the SQOPDC controllers of the TVTSFMS under the criterion of minimizing a performance index. But, the LMI-based approaches^8–14 cannot be applied to find the SQOPDC controllers of the TVTSFMS under the criterion of directly minimizing a performance index. In addition, though the genetic algorithms can solve the complex static-optimization problems that are not easy to analyze mathematically, it is also very difficult to only use the genetic algorithms to solve the dynamic-optimization problem of designing the SQOPDC controllers of the TVTSFMS. Hence, summing up the above statements and reasons, we can see that it is worth while to present a complementarily integrative approach fusing the HTGA, the LMI-based SC to find the SQOPDC controllers of the TVTSFMS under the criterion of minimizing a performance index.

Illustrative examples

Consider a vibratory pendulum system (Figure 2),²¹ which can be represented by a two-rule TVTSFMS:

Figure 2.

Vibratory pendulum system.

${\tilde{R}}^{1}$ : IF $z_{1} (t)$ is $M_{11} (z_{1} (t)),$

THEN \dot{x} (t) = A_{1} (t) x (t) + B_{1} (t) u (t)

(17)

${\tilde{R}}^{2}$ : IF $z_{1} (t)$ is $M_{21} (z_{1} (t)),$

THEN \dot{x} (t) = A_{2} (t) x (t) + B_{2} (t) u (t)

(18)

where

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

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

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

\bar{\bar{m}} = 1 kg,

\bar{\bar{l}} = 1 m,

ω = 1 Hz,

Y = 0.01 m,

and

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

In equation (17), the proposed complementary computational approach is used to design the SQOPDC controller, to obtain a symmetric positive definite matrix P in equation (4), and to minimize J in equation (16). The OF type considered in this case was the Legendre function.

Table 1 lists the average, median, minimum, maximum, and standard deviation of the performance indices obtained in 100 computations performed using the proposed HTGA. For comparison, the table also displays the results for 100 computations performed using a traditional genetic algorithm (TGA).²²

Table 1.

Results of performance comparison of hybrid Taguchi genetic algorithm and traditional genetic algorithm.

Mean performance index		Median performance index		Standard deviation in performance index		Minimal performance index		Maximal performance index
HTGA	TGA	HTGA	TGA	HTGA	TGA	HTGA	TGA	HTGA	TGA
2.8213	3.1634	2.8210	2.9947	0.0011	0.3952	2.8201	2.8385	2.8240	4.4668

HTGA: hybrid Taguchi genetic algorithm; TGA: traditional genetic algorithm

Table 1 and Figure 3 show the results of the performance comparisons of the HTGA and TGA. Compared to the TGA, the HTGA had (i) superior average and median performance indices, (ii) smaller SDs in performance indices, and (iii) superior convergence. Accordingly, HTGA yielded and more stable solutions. Therefore, compared with TGA, the HTGA is more effective for routine use in designing an SQOPDC controller.

Figure 3.

Convergence performance comparison of hybrid Taguchi genetic algorithm and traditional genetic algorithm.

Figure 4 shows the $x_{1} (t),$ $x_{2} (t)$ and $u (t)$ responses of the vibratory pendulum system obtained after implementation of the SQOPDC controller designed using the HTGA $(F_{1} = [\begin{matrix} 0.2661 & 1.0889 \end{matrix}]$ and $F_{2} = [\begin{matrix} 0.2528 & 1.1583 \end{matrix}]$ ). Its symmetric positive definite matrix was $P = [\begin{matrix} 15.7691 & 0.5082 \\ 0.5082 & 1.8590 \end{matrix}] .$ Figure 5 also illustrates the $x_{1} (t)$ and $x_{2} (t)$ responses of the vibratory pendulum system without the SQOPDC controller.

Figure 4.

The $x_{1} (t)$ (rad) (solid line) and $x_{2} (t)$ (rad/s) (dashed line) obtained by the stable and quadratic-optimal parallel-distributed-compensation controller in response to control input $u (t)$ (N-m) (dashed-dotted line).

Figure 5.

The $x_{1} (t)$ (rad) (solid line) and $x_{2} (t)$ (rad/s) (dashed line) obtained without the stable and quadratic-optimal parallel-distributed-compensation controller.

Figures 4 and 5 and the above results confirm that, by integrating OFA, HTGA, and LMI, the proposed complementary computational approach effectively stabilized the TVTSFMS, minimized the performance index, and achieved a satisfactory control effect.

Conclusions

This study developed an algebraic algorithm for using OFA to solve a TVTSFMS controller design problem. An algebraic algorithm was integrated in HTGA to design an SQOPDC controller for a TVTSFMS constrained by LMIs to minimize the performance index. The OFA was also used to convert the performance index to algebraic form. Application of the proposed approach in the design of a vibratory pendulum system verified the effectiveness of the approach for designing the SQOPDC controller for the TVTSFMS.

Footnotes

Acknowledgements

The authors of this study would like to express their gratitude to Professor Jyh-Horng Chou (Department of Electrical Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, Taiwan) for his constructive suggestions and guidance in this research. The authors also thank to 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, R.O.C.

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 in part by the Ministry of Science and Technology, Taiwan, R.O.C., under Grant Number MOST 109-2221-E-037-005.

ORCID iD

Wen-Hsien Ho

Appendix

Proof of Theorem

Since (i) time-varying elements $a_{i j k} (t)$ in $A_{i} (t)$ belong to $[{\underline{a}}_{i j k}, {\bar{a}}_{i j k}],$ and since (ii) time-varying elements $b_{i j k} (t)$ in $B_{i} (t)$ belong to $[{\underline{b}}_{i j k}, {\bar{b}}_{i j k}],$ the TVTSFMS in equation (3) can be rewritten as (19)

\dot{x} (t) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z (t)) h_{j} (z (t)) (A_{i} + \sum_{α = 1}^{n} \sum_{β = 1}^{n} ε_{i α β} (t) E_{i α β} - (B_{i} + \sum_{α = 1}^{n} \sum_{β = 1}^{p} η_{i α β} (t) V_{i α β}) F_{j}) x (t)

where constant matrices

A_{i}

and

B_{i}

contain elements

a_{i j k} = ({\underline{a}}_{i j k} + {\bar{a}}_{i j k}) / 2

and

b_{i j s} = ({\underline{b}}_{i j s} + {\bar{b}}_{i j s}) / 2

, respectively.

The TVTSFMS in equation (19) can then be rewritten as (20)

\dot{x} (t) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z (t)) h_{j} (z (t)) (\sum_{α = 0}^{n} \sum_{β = 0}^{n} ε_{i α β} (t) E_{i α β}^{j} - \sum_{α = 0}^{n} \sum_{β = 0}^{p} η_{i α β} (t) L_{i j α β}) x (t),

where

E_{i 00}^{j} = L_{i j 00} = {\tilde{A}}_{i j},

and

{\tilde{A}}_{i j} = A_{i} - B_{i} F_{j};

E_{i α β}^{j} = E_{i α β}

and

L_{i j α τ} = V_{i α τ} F_{j} .

The $U_{i j l}$ are constant matrices defined as (21)

U_{i j l} = \sum_{α = 0}^{n} \sum_{β = 0}^{n} ε_{i α β} (t) E_{i α β}^{j} - \sum_{α = 0}^{n} \sum_{β = 0}^{p} η_{i α β} (t) L_{i j α β} | \begin{array}{l} ε_{i α β} (t) = {\underline{ε}}_{i α β} or {\bar{ε}}_{i α β} . \\ η_{i α β} (t) = {\underline{η}}_{i α β} or {\bar{η}}_{i α β} . \end{array}

(22)

\sum_{α = 0}^{n} \sum_{β = 0}^{n} ε_{i α β} (t) E_{i α β}^{j} - \sum_{α = 0}^{n} \sum_{β = 0}^{p} η_{i α β} (t) L_{i j α β} = \sum_{l = 1}^{c} θ_{i j l} (t) U_{i j l},

where

θ_{i j l} (t) \geq 0

\sum_{l = 1}^{c} θ_{i j l} (t) = 1,

and

c = 2^{n (n + p)} .

Equation (19) then becomes (23)

\dot{x} (t) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} h_{i} (z (t)) h_{j} (z (t)) (\sum_{l = 1}^{c} θ_{i j l} (t) U_{i j l}) x (t) .

Where

V (x (t)) = x^{T} (t) P x (t)

is a candidate quadratic Lyapunov function for the system in equation (23) (24)

\dot{V} (x (t)) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} \sum_{l = 1}^{c} h_{i} (z (t)) h_{j} (z (t)) θ_{i j l} (t) x^{T} (t) (U_{i j l}^{T} P + P U_{i j l}) x (t) .

For the specified feedback gain matrices $F_{i}$ , if there exists a symmetric positive definite matrix P

$U_{i j l}^{T} P + P U_{i j l} < 0$ , then $\dot{V} (x (t)) < 0$ , and $\forall x (t) \neq 0$ . (25)

Therefore, we can conclude that, to design an SQOPDC controller for the TVTSFMS (equation (3)), a series of $F_{i}$ in a symmetric positive definitive matrix P must be specified that simultaneously satisfies the LMI of equation (4). This completes the verification.

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