Control of nonlinear systems by alternative design of fuzzy logic controllers subject to Lyapunov stability theory

Abstract

In this article, a set of fuzzy logic controller is alternatively designed for control of nonlinear systems in Lyapunov sense. Through referencing the statuses of error states and error dynamics, the corresponding fuzzy if–then rules can be constructed based on Lyapunov Stability Theory following the proposed standard design procedure, where simple quadratic form is decided for candidate Lyapunov function. Different from previous research achievements, the resulting controllers are simple constant forces, and for control of two nonlinear systems with different structures, the design procedure is still the same without any further information. Furthermore, appropriate and adjustable control forces with less maximum magnitude are stably supplied through exciting the weighting through the constructed fuzzy if–then rules, which leads the proposed fuzzy logic controllers compensate control system more flexible and in real time. Two examples, control of two identical and different nonlinear systems, are illustrated for simulation examples. In comparison to the previous work, the simulation results reveal the effectiveness, flexibility, as well as convenient design of the proposed method.

Keywords

Intelligent control fuzzy logic theory Lyapunov stability theory

Introduction

The concept of fuzzy logic is proposed by Prof. LA Zadeh in 1965,¹ which has received much attention as a powerful tool for interdisciplinary applications, especially in the field of fuzzy logic control (FLC)system.^2–9 FLC has been designed and further applied to intelligent mechanical systems for many years, such as Wang et al.¹⁰ applied the multiple-input and multiple-output (MIMO) fuzzy control technology to self-assembling of swarm modular robots, Chang and Tong¹¹ designed a set of adaptive fuzzy control approach for permanent magnet synchronous motors, Li et al.¹² proposed the FLC-based output feedback control problem for interval type-2 fuzzy systems of a mass–spring–damping system, and Hao and Kan¹³ proposed self-tuning fuzzy proportional–integral–derivative control in hydraulic crane control system.

Many great researchers devoted themselves to create novel approaches to improve the performance of FLC system and ensure their stability, for instances, Lee¹⁴ developed a set of systematic method to design proportional–integral (PI)-types FLC system in 1993, which focuses on linear system analysis; for nonlinear systems, Ying¹⁵ presented a practical design method to study the stability of FLC system in 1994; and Choi et al.¹⁶ developed a single control input to ensure the stability of FLC system in 2000.

Yau and Shieh¹⁷ proposed a robust and effective fuzzy logic controller to force two identical chaotic systems with two and three orders to be synchronized in 2008, which constructs the fuzzy if–then rules subject to a common Lyapunov function such that the error dynamics satisfies stability in the Lyapunov sense. In Yau and Shieh,¹⁷ the resulting controllers are in nonlinear form, and for control of nonlinear systems with different structures, expert experience is required. Based on this kernel designing spirit, Li and Ge¹⁸ and Li¹⁹ proposed efficient design strategies of fuzzy logic controllers to synchronize different chaotic systems with adjustable constant forces in 2011.

Inspired via Prof. Yau and Shieh, in this article, we aim to propose an alternative design of fuzzy logic controllers subject to Lyapunov stability theory with hieratical structure, which also constructs the fuzzy if–then rules according to the control Lyapunov function with quadratic form. The error derivative is set as the antecedent part, and the corresponding control input is considered to be the consequent part in the development of the fuzzy if–then rules. Via detecting the real-time sign of error states, the boundary values of the error derivatives are applied to be the control forces to further lead the error dynamics to achieve asymptotically stable. The proposed fuzzy logic controllers are simple constants, and the design procedure can be applied to different kinds of control systems. More importantly, more concise control forces are developed in the fuzzy if–then rules to enforce systems tracing the correct trajectory effectively.

The rest of this article is organized as follows: In section “Materials and Methods,” the control scheme as well as control flowchart are introduced. In section “Results and discussion,” simulation results and comparison are given, and in section “Conclusion,” a brief description of conclusion is provided.

Materials and methods

Chaos control scheme

Consider the following nonlinear system with n order

\overset{\cdot}{x} = f (t, x) + u

(1)

where $x = [x_{1}, x_{2}, \dots, x_{n}]^{T} \in R^{n}$ is the system state vector, $f : R_{+} \times R^{n} \to R^{n}$ represents the vector functions covering with those nonlinear and linear terms, and $u = [u_{1}, u_{2}, \dots, u_{n}]^{T} \in R^{n}$ is the fuzzy logic controller needed to be designed.

The goal system which can be either chaotic or regular is

\overset{\cdot}{y} = g (t, y)

(2)

where $y = [y_{1}, y_{2}, \dots, y_{n}]^{T} \in R^{n}$ is the system state vector and $g : R_{+} \times R^{n} \to R^{n}$ refers to the vector functions covering with those nonlinear and linear terms.

In order to enforce the control system states x to trace the trajectory of the goal system states y with real-time control inputs, we define the error states as $e = y - x = [e_{1}, e_{2}, \dots, e_{n}]^{T}$ . The chaos control is accomplished in the following sense

lim_{t \to \infty} e = lim_{t \to \infty} (y - x) = 0

(3)

From equation (3) we have the following error dynamics

\overset{\cdot}{e} = \overset{\cdot}{y} - \overset{\cdot}{x} = g (t, y) - f (t, x) - u

(4)

According to Lyapunov second stability theorem, we have the following control Lyapunov function with quadratic form to further derive the fuzzy logic controllers

\begin{matrix} V = f (e_{1}, \dots, e_{m}, \dots, e_{n}) = \frac{1}{2} (e_{1}^{2} + \dots + e_{m}^{2} + \dots + e_{n}^{2}) > 0 \\ = V_{1} + \dots + V_{m} + \dots + V_{n} \end{matrix}

(5)

where e_m = y_m —x_m represents any error state in the designed Lyapunov function in equation (5) and V_m ₌ (e_m)²/2 refers to each subspace of the Lyapunov function. The derivative of the Lyapunov function can be described as follows

\overset{\cdot}{V} = e_{1} {\overset{\cdot}{e}}_{1} + \dots + e_{m} {\overset{\cdot}{e}}_{m} + \dots e_{n} {\overset{\cdot}{e}}_{n}

(6)

If the controllers included in ${\overset{\cdot}{e}}_{1} \dots {\overset{\cdot}{e}}_{m} \dots {\overset{\cdot}{e}}_{n}$ can be suitably designed to achieve the target $\overset{\cdot}{V} = {\overset{\cdot}{V}}_{1} \dots + {\overset{\cdot}{V}}_{m} \dots + {\overset{\cdot}{V}}_{n} < 0$ , then the error dynamic system is asymptotically stable, that is, the control task can be achieved. The design procedure in detail is presented in the following section.

Design procedure of fuzzy logic controllers

FLC system considers a number of system boundary situations and designs the corresponding control strategy following the domain knowledge and design experience from experts, which generates the associated fuzzy if–then rules and membership functions. The fuzzy rules base consists of collection of fuzzy if–then rules expressed as the form if a is A and then b is B, where “if a is A” is the antecedent part and “b is B” is the consequent part; a and b denote linguistic variables and A and B represent linguistic values which are characterized by membership functions.

We use the error derivatives $\overset{\cdot}{e} (t) = {[{\overset{\cdot}{e}}_{1}, {\overset{\cdot}{e}}_{2}, \dots, {\overset{\cdot}{e}}_{m}, \dots, {\overset{\cdot}{e}}_{n}]}^{T}$ as the antecedent part, and the corresponding control input u is considered to be the consequent part of the FLC system in this article. In particular, u is designed as a constant column vector given below to stabilize the error dynamics in equation (4)

u = {[u_{1}, u_{2}, \dots, u_{m}, \dots, u_{n}]}^{T}

(7)

Assuming the upper bound and lower bound of ${\overset{\cdot}{e}}_{m}$ as Z_m and –Z_m, respectively, the Fuzzy logic controllers can be designed step by step as follows:

If $e_{m}$ is detected as positive $(e_{m} > 0)$ , we have to design a controller for ${\overset{\cdot}{e}}_{m} < 0$ , and then, ${\overset{\cdot}{V}}_{m} = e_{m} {\overset{\cdot}{e}}_{m} < 0$ can be achieved. Therefore, we have the following ith if–then fuzzy rules as

Rule 1 : if {\overset{\cdot}{e}}_{m} is M_{m 1}, then u_{m 1} = P_{m}

(8)

Rule 2 : if {\overset{\cdot}{e}}_{m} is M_{m 2}, then u_{m 2} = 0

(9)

Rule 3 : if {\overset{\cdot}{e}}_{m} is M_{m 3}, then u_{m 3} = e_{m}

(10)

If $e_{m}$ is detected as negative $(e_{m} < 0)$ , we have to design a controller for ${\overset{\cdot}{e}}_{m} > 0$ , and then, ${\overset{\cdot}{V}}_{m} = e_{m} {\overset{\cdot}{e}}_{m} < 0$ can be achieved. Therefore, we have the following ith if–then fuzzy rules as

Rule 1 : if {\overset{\cdot}{e}}_{m} is M_{m 1}, then u_{m 1} = - P_{m}

(11)

Rule 2 : if {\overset{\cdot}{e}}_{m} is M_{m 2}, then u_{m 2} = 0

(12)

Rule 3 : if {\overset{\cdot}{e}}_{m} is M_{m 3}, then u_{m 3} = e_{m}

(13)

If $e_{m}$ approaches to zero, then the synchronization is nearly achieved. Therefore, we have the following ith if–then fuzzy rules as

Rule 1 : if {\overset{\cdot}{e}}_{m} is M_{m 1}, then u_{m 1} = e_{m}

(14)

Rule 2 : if {\overset{\cdot}{e}}_{m} is M_{m 2}, then u_{m 2} = e_{m}

(15)

Rule 3 : if {\overset{\cdot}{e}}_{m} is M_{m 3}, then u_{m 3} = e_{m}

(16)

where $M_{m 1} = M_{m 2} = | {\overset{\cdot}{e}}_{m} | / Z_{m}$ and $M_{m 3} = (Z_{m} - | {\overset{\cdot}{e}}_{m} |) / Z_{m}$ , $M_{m 1}$ , $M_{m 2}$ , and $M_{m 3}$ refer to the membership functions of positive (P), negative (N), and zero (Z) separately, which are presented in Figure 1. P_m is the control force with constant form, which is suggested to set a little bit larger than the boundary values to guarantee ${\overset{\cdot}{e}}_{m} = - sgn (e_{m})$ , that is, P_m ≧ Z_m.

Figure 1.

Design of membership functions for each situationof ${\overset{\cdot}{e}}_{m}$ .

For each case, $u_{mi}$ , i = 1–3 is the ith output of ${\overset{\cdot}{e}}_{m}$ which is a constant controller. The centriod defuzzifier evaluates the output of all rules as follows

u_{m} = \frac{\sum_{i = 1}^{3} M_{mi} \times u_{mi}}{\sum_{i = 1}^{3} M_{mi}}

(17)

The fuzzy rule base is listed in Table 1, in which the input variables in the antecedent part of the rules are ${\overset{\cdot}{e}}_{m}$ and the output variable in the consequent part is $u_{mi}$ .

Table 1.

Rule table of FLC.

Rule	Antecedent part	Consequent part
Rule	${\overset{\cdot}{e}}_{m}$	$u_{mi}$
1	Negative (N)	$u_{m 1}$
2	Positive (P)	$u_{m 2}$
3	Zero (Z)	$u_{m 3}$

FLC: fuzzy logic controller.

It is worth to mention that for equations (9) and (12), the control forces are designed to be zero, it is because the subspace of derivatives of Lyapunov function has satisfied the condition of negative definite ${\overset{\cdot}{V}}_{m} = e_{m} {\overset{\cdot}{e}}_{m} < 0$ , redundant control forces may lead the error sates jumping outside the trajectory approaching to the original point. After designing appropriate fuzzy logic controllers and blending the control force through the defuzzification process in equation (17), a negative definite of derivatives of Lyapunov function $\overset{\cdot}{V}$ can be obtained and the asymptotically stability of Lyapunov theorem can be achieved.

Results and discussion

In this section, two simulation examples are illustrated to investigate the performance of the proposed control scheme. In Example 1, two identical nonlinear systems with different initial conditions as well as different parameters are proposed, in which the design process of fuzzy logic controllers is revealed and explained step by step. Furthermore, in Example 2, control of two different nonlinear systems with distinct structures is studied, which indicates that we can use the same process to design the fuzzy logic controllers achieving various kinds of control goals.

Example 1

Two identical systems with different settings

Consider the following classical Lorentz system in equation (18) as the goal system

{\begin{matrix} {\overset{\cdot}{y}}_{1} (t) = a (y_{2} (t) - y_{1} (t)) \\ {\overset{\cdot}{y}}_{2} (t) = c y_{1} (t) - y_{1} (t) y_{3} (t) - y_{2} (t) \\ {\overset{\cdot}{y}}_{3} (t) = y_{1} (t) y_{2} (t) - b y_{3} (t) \end{matrix}

(18)

where y₁, y₂, and y₃ are state variables and a, b, and c are system parameters. When the initial conditions and parameters are set to be (y₁₀, y₂₀, and y₃₀) = (0.1, 0.2, and 0.3) and a = 10, b = 8/3, and c = 28, chaotic behaviors appear, which are shown in Figure 2.

Figure 2.

Phase portrait of the classical Lorentz system.

Also, the classical Lorentz system with FLC input is given below in equation (19)

{\begin{matrix} {\overset{\cdot}{x}}_{1} (t) = a_{1} (x_{2} (t) - x_{1} (t)) + u_{1} \\ {\overset{\cdot}{x}}_{2} (t) = c_{1} x_{1} (t) - x_{1} (t) x_{3} (t) - x_{2} (t) + u_{2} \\ {\overset{\cdot}{x}}_{3} (t) = x_{1} (t) x_{2} (t) - b_{1} x_{3} (t) + u_{3} \end{matrix}

(19)

where a₁, b₁, and c₁ are system parameters, which are set to be different from those of goal system to show the effectiveness and feasibility of the proposed control scheme, a₁ = 8, b₁ = 8/3, and c₁ = 10. The initial conditions are designed as (x₁₀, x₂₀, and x₃₀) = (10, −10, and 10), and u₁, u₂, and u₃ are fuzzy logic controllers.

Following the control scheme mentioned in section “Materials and Methods,” the error dynamic system can be obtained as follows

{\begin{matrix} {\overset{\cdot}{e}}_{1} = a (y_{2} - y_{1}) - (a_{1} (x_{2} - x_{1}) + u_{1}) \\ {\overset{\cdot}{e}}_{2} = c y_{1} - y_{1} y_{3} - y_{2} - (c_{1} x_{1} - x_{1} x_{3} - x_{2} + u_{2}) \\ {\overset{\cdot}{e}}_{3} = y_{1} y_{2} - b y_{3} - (x_{1} x_{2} - b_{1} x_{3} + u_{3}) \end{matrix}

(20)

Equation (20) can be further rearranged into the following simple form

{\begin{matrix} {\overset{\cdot}{e}}_{1} = ({\overset{\cdot}{e}}_{1})_{o} - u_{1} \\ {\overset{\cdot}{e}}_{2} = ({\overset{\cdot}{e}}_{2})_{o} - u_{2} \\ {\overset{\cdot}{e}}_{3} = ({\overset{\cdot}{e}}_{3})_{o} - u_{3} \end{matrix}

(21)

where $({\overset{\cdot}{e}}_{1})_{o}$ , $({\overset{\cdot}{e}}_{2})_{o}$ , and $({\overset{\cdot}{e}}_{3})_{o}$ refer to the original error dynamics without any control input in each iteration. This simple mathematics expression brings us a clearer view to design the effective fuzzy logic controllers. Choosing Lyapunov function with quadratic form, we have the following derivative of the Lyapunov function

\begin{matrix} \overset{\cdot}{V} = e_{1} {\overset{\cdot}{e}}_{1} + e_{2} {\overset{\cdot}{e}}_{2} + e_{3} {\overset{\cdot}{e}}_{3} \\ = e_{1} (({\overset{\cdot}{e}}_{1})_{o} - u_{1}) + e_{2} (({\overset{\cdot}{e}}_{2})_{o} - u_{2}) + e_{3} (({\overset{\cdot}{e}}_{3})_{o} - u_{3}) \end{matrix}

(22)

According to the control scheme, it is necessary to figure out the information of upper bounds and lower bounds of the original error dynamics in order to design the appropriate membership function to adjust the control forces more flexible, and the related information is investigated in Figure 3.

Figure 3.

The ranges of the original error dynamics.

In Figure 3, the upper bounds and lower bounds of the original error dynamics are given, which can be described as

{\begin{matrix} - Z_{1} \leq | {({\overset{\cdot}{e}}_{1})}_{o} | \leq Z_{1} \\ - Z_{2} \leq | {({\overset{\cdot}{e}}_{2})}_{o} | \leq Z_{2} \\ - Z_{3} \leq | {({\overset{\cdot}{e}}_{3})}_{o} | \leq Z_{3} \end{matrix}

(23)

where Z₁ = 200, Z₂ = 400, and Z₃ = 400, which is considered to be the boundary values of control force adjusted by the fuzzy logic weighting (membership functions).

In order to design fuzzy logic controllers, the derivative of Lyapunov function in equation (22) is divided into three subspaces

\begin{matrix} \overset{\cdot}{V} = e_{1} (({\overset{\cdot}{e}}_{1})_{o} - u_{1}) + e_{2} (({\overset{\cdot}{e}}_{2})_{o} - u_{2}) + e_{3} (({\overset{\cdot}{e}}_{3})_{o} - u_{3}) \\ = {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{V}}_{2} + {\overset{\cdot}{V}}_{3} \end{matrix}

(24)

where

{\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} (({\overset{\cdot}{e}}_{1})_{o} - u_{1}) \\ {\overset{\cdot}{V}}_{2} = e_{2} (({\overset{\cdot}{e}}_{2})_{o} - u_{2}) \\ {\overset{\cdot}{V}}_{3} = e_{3} (({\overset{\cdot}{e}}_{3})_{o} - u_{3}) \end{matrix}

(25)

According to the design procedure of fuzzy logic controllers described in section “Materials and methods,” for ${\overset{\cdot}{V}}_{1}$ , we have the FLC force as P₁ = Z₁ = 200; for ${\overset{\cdot}{V}}_{2}$ , we have the FLC force as P₂ = Z₂ = 400; and for ${\overset{\cdot}{V}}_{3}$ , the FLC force is set to be P₃ = Z₃ = 400, and as a consequence, each subspace of the derivative of Lyapunov function is negative

{\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} (- δ_{1}) \\ {\overset{\cdot}{V}}_{2} = e_{2} (- δ_{2}) \\ {\overset{\cdot}{V}}_{3} = e_{3} (- δ_{3}) \end{matrix}

(26)

where $δ_{1}$ , $δ_{2}$ , and $δ_{3}$ are positive, and then, we have the following derivative of Lyapunov function with negative definite

\overset{\cdot}{V} = - e_{1} δ_{1} - e_{2} δ_{2} - e_{3} δ_{3}

(27)

The simulation results are shown in Figure 4, where the fuzzy control inputs are operated after 30 s. Obviously, the fuzzy if–then rules created in our control scheme can lead the error dynamics to be asymptotically stable.

Figure 4.

The time history of error states with fuzzy logic controllers.

Comparing to the previous study by Li,¹⁹ the proposed method in this article has less redundant control forces, which effectively reduces the chance that unnecessary control force leads the sate trajectories (approaching to the original point) to jump outside the region of Lyapunov-sense stability; in addition, the simulation results give a strong evidence that the control performance of the proposed method is as good as the results of Li,¹⁹ and the comparison is shown in Figure 5.

Figure 5.

The time history of error states with fuzzy logic controllers in Li¹⁹ and in this article.

Example 2

Two different nonlinear dynamic systems

Chen and Lee²⁰ presented a new chaotic system in 2004, which is now called Chen–Lee system. The system is described by the following nonlinear differential equations and denoted as the goal system

{\begin{matrix} {\overset{\cdot}{z}}_{1} (t) = - z_{2} (t) z_{3} (t) + a_{2} z_{1} (t) \\ {\overset{\cdot}{z}}_{2} (t) = z_{1} (t) z_{3} (t) + b_{2} z_{2} (t) \\ {\overset{\cdot}{z}}_{3} (t) = \frac{z_{1} (t) z_{2} (t)}{3} + c_{2} z_{3} (t) \end{matrix}

(28)

where z₁, z₂, and z₃ are state variables, and a₂, b₂, and c₂ are system parameters. When the initial conditions and parameters are set to be (z₁₀, z₂₀, and z₃₀) = (0.5, 0.7, and 1.5) and (a₂, b₂, and c₂) = (5, −10, and −3.8), system equation (28) is a chaotic attractor as shown in Figure 6. It is clear that the Chen–Lee system is a regular chaotic system.

Figure 6.

Phase portrait of the Chen–Lee system.

Also, the classical Lorentz system with the corresponding FLC input in equation (19) is considered to be the control system. Following the control scheme mentioned in section “Materials and methods,” the error dynamic system can be described as follows

{\begin{matrix} {\overset{\cdot}{e}}_{1} = ({\overset{\cdot}{e}}_{1})_{o} - u_{1} \\ {\overset{\cdot}{e}}_{2} = ({\overset{\cdot}{e}}_{2})_{o} - u_{2} \\ {\overset{\cdot}{e}}_{3} = ({\overset{\cdot}{e}}_{3})_{o} - u_{3} \end{matrix}

(29)

where $({\overset{\cdot}{e}}_{1})_{o}$ , $({\overset{\cdot}{e}}_{2})_{o}$ , and $({\overset{\cdot}{e}}_{3})_{o}$ refer to the original error dynamics without any control input in each iteration

{\begin{matrix} ({\overset{\cdot}{e}}_{1})_{o} = - z_{2} (t) z_{3} (t) + a_{2} z_{1} (t) - (a_{1} (x_{2} - x_{1})) \\ ({\overset{\cdot}{e}}_{2})_{o} = z_{1} (t) z_{3} (t) + b_{2} z_{2} (t) - (c_{1} x_{1} - x_{1} x_{3} - x_{2}) \\ ({\overset{\cdot}{e}}_{3})_{o} = z_{1} (t) z_{2} (t) / 3 + c_{2} z_{3} (t) - (x_{1} x_{2} - b_{1} x_{3}) \end{matrix}

(30)

Choosing Lyapunov function with quadratic form, we have the following Lyapunov function derivatives

\begin{matrix} \overset{\cdot}{V} = e_{1} {\overset{\cdot}{e}}_{1} + e_{2} {\overset{\cdot}{e}}_{2} + e_{3} {\overset{\cdot}{e}}_{3} \\ = e_{1} (({\overset{\cdot}{e}}_{1})_{o} - u_{1}) + e_{2} (({\overset{\cdot}{e}}_{2})_{o} - u_{2}) + e_{3} (({\overset{\cdot}{e}}_{3})_{o} - u_{3}) \\ = {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{V}}_{2} + {\overset{\cdot}{V}}_{3} \end{matrix}

(31)

where ${\overset{\cdot}{V}}_{1}, {\overset{\cdot}{V}}_{2}, {\overset{\cdot}{V}}_{3}$ are subspaces of Lyapunov function derivative

{\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} (({\overset{\cdot}{e}}_{1})_{o} - u_{1}) \\ {\overset{\cdot}{V}}_{2} = e_{2} (({\overset{\cdot}{e}}_{2})_{o} - u_{2}) \\ {\overset{\cdot}{V}}_{3} = e_{3} (({\overset{\cdot}{e}}_{3})_{o} - u_{3}) \end{matrix}

(32)

In order to design the appropriate membership function to adjust the control forces more flexible, the information of upper bounds and lower bounds of $({\overset{\cdot}{e}}_{1})_{o}$ , $({\overset{\cdot}{e}}_{2})_{o}$ , and $({\overset{\cdot}{e}}_{3})_{o}$ are provided in Figure 7.

Figure 7.

The ranges of the original error dynamics.

The information of the upper bounds and lower bounds of the error derivatives are given in Figure 7; however, the ranges of the error derivatives are large in this case, so that the control forces P_m should be set to be very strong, which is not a benefit to control systems, especially for those mechanical or electrical systems. As a consequence, in this case, we consider to capture the boundary information of the error derivatives after 50 s (operation time), which brings us the steady error states of the two chaotic systems. The simulation results are given in Figure 8, and it is clear to show that the trajectories of the error derivatives are limited to smaller ranges.

Figure 8.

The modified ranges of the error derivatives.

In Figure 8, the upper bounds and lower bounds of the original error dynamics are given, which can also be described as

{\begin{matrix} - Z_{1} \leq | {({\overset{\cdot}{e}}_{1})}_{o} | \leq Z_{1} \\ - Z_{2} \leq | {({\overset{\cdot}{e}}_{2})}_{o} | \leq Z_{2} \\ - Z_{3} \leq | {({\overset{\cdot}{e}}_{3})}_{o} | \leq Z_{3} \end{matrix}

(33)

where Z₁ = 500, Z₂ = 500, and Z₃ = 500, which is considered to be the boundary values of control force adjusted by the fuzzy logic weighting (membership functions).

According to the design procedure of fuzzy logic controllers described in section “Materials and methods,” for ${\overset{\cdot}{V}}_{1}$ , we have the FLC force as P₁ = Z₁ = 500; for ${\overset{\cdot}{V}}_{2}$ , we have the FLC force as P₂ = Z₂ = 500; and for ${\overset{\cdot}{V}}_{3}$ , the FLC force is set to be P₃ = Z₃ = 500, and as a consequence, each subspace of the derivative of Lyapunov function is all negative

{\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} (- δ_{1}) \\ {\overset{\cdot}{V}}_{2} = e_{2} (- δ_{2}) \\ {\overset{\cdot}{V}}_{3} = e_{3} (- δ_{3}) \end{matrix}

(34)

where $δ_{1}$ , $δ_{2}$ , and $δ_{3}$ are positive, and we have the following derivative of Lyapunov function with negative definite

\overset{\cdot}{V} = - e_{1} δ_{1} - e_{2} δ_{2} - e_{3} δ_{3}

(35)

The simulation results are shown in Figure 9, where the fuzzy control inputs are operated after 50 s (which is more reasonable to control the dynamic systems under steady states). Obviously, the fuzzy if–then rules created in our control scheme can lead the error dynamics to be asymptotically stable.

Figure 9.

The time history of error states with fuzzy logic controllers.

In this example, control of different nonlinear systems through the proposed control scheme is discussed, which try to demonstrate that the control scheme is feasible to different kinds of control goals. Following the same design process, the control targets can be achieved without considering those system properties, different structure of states equation, and so on. Moreover, those redundant control forces are further removed, and through the delay-controlling process, effective as well as economic fuzzy control forces can be obtained to achieve the control goal.

Conclusion

A Lyapunov-sense fuzzy logic controller with adjustable constant force is discussed in this article. Through the analysis of real-time status of error states, an error derivative–based FLC system is constructed. The error derivative is designed to be the antecedent part, and the corresponding control input is considered as the consequent part, and the boundary values of the error derivative are used to be the feedback control force in each fuzzy if–then rules; with the blending of each control force via the defuzzification process, the expressing strength of the proposed fuzzy controller can be adjusted to be a more appropriate control force leading the error dynamics to achieve asymptotical stability. The proposed fuzzy logic controller is a simple, convenient, and more flexible design, and the redundant forces in the constructed fuzzy rules are reduced. Simulation results demonstrate the effectiveness and feasibility of the proposed method.

Footnotes

Acknowledgements

The authors would like to thank anonymous reviewers for their valuable suggestions and comments to improve this article.

Handling Editor: Stephen D Prior

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 study was funded in part by the Ministry of Science and Technology (MOST 106-2221-E-027-028, 105-2218-E-027-005, 104-2622-E-027-003-CC2, and 105-2221-E-027-068 and 106-222-1-E-027-009).

ORCID iD

Shih-Yu Li

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