PD control of inverted pendulum based on adaptive fuzzy compensation

Abstract

As a typical, nonlinear system with instability, high order, strong coupling and multiple variations, the inverted pendulum model is the focus and research object of many experts and scholars in the control field. We propose the PD control method of the system based on the adaptive fuzzy compensation in order to weaken the impact of uncertainty in regards to external disturbances, as well as improve the precise control of the inverted pendulum system when the parameters are unknown. It uses the fuzzy method to conduct fuzzy approximation on the nonlinear system of the inverted pendulum by modeling the nonlinear inverted pendulum system to weaken the impact of uncertainty and achieve complete compensation for the nonlinear system. Then, it establishes the adaptive PD fuzzy controller, forms the adaptive control law, applies the Lyapunov function to verify the stability and robustness of the system and finally achieves the intelligent, optimal control of the system. Simulation results show that the control method can achieve the optimal control of the tracking error and parameter error, has a better anti-interference ability and assures system stability.

Keywords

Inverted pendulum adaptive fuzzy control compensation nonlinear system

1 Introduction

The complex, nonlinear inverted pendulum system can effectively reflect many typical control problems. Therefore, it is widely used to test whether a new control method is qualified with a strong processing ability of nonlinear and instability problems. Currently, the inverted pendulum control method includes state feedback, a linear quadratic regulator (LQR) of the modern control theory, root locus, a proportional integral derivative (PID) of the classic control theory, as well as the fuzzy control, neural networks, support vector machine and other intelligent methods. Furthermore, the PID method is the most widely used in the inverted pendulum control system because the theory is mature and easy to implement. Nevertheless, PID control has poor generalization abilities and a lower robustness. Therefore, the fuzzy control is increasingly combined for use in such control systems. Traditional fuzzy control has a deficiency; a state feedback controller was designed in the inverted pendulum, and then the dimension was reduced. In order to eliminate the uncertainty parameters, perturbations and load disturbance of the control system affected by the emergence of environmental parameters, the error could be online identification and dynamic compensation. It could realize fast robust adaptive control. The variable universe adaptive fuzzy control [11, 17] and a RBF network based compensation Fuzzy adaptive control algorithm was proposed [7] to realize fuzzy compensation. It could distinguish different disturbance compensation term to approach, may well restrain friction, disturbance and load changes of the nonlinear factors. An adaptive fuzzy PID controller was designed [20] to make non-linear decoupling for the non-linear, strong coupling and multivariable-plane inverted pendulum system of the mathematical model; the system has good stability and robustness. A complex control scheme to calculate the nominal torque of the controller, as well as the additional compensation for the adaptive fuzzy controller was designed [6] to ensure the asymptotic stability of the unknown system parameters for the closed-loop control system. The furnace desulfurization dynamics process has the characteristics of a large delay, large inertia, time-varying and non-linear. Therefore, we proposed a furnace desulfurization control system based on the fuzzy PID adaptive compensation of the incremental method [9]. An adaptive control method based on fuzzy compensation [10] was proposed to solve the nonlinear characteristics of the electrohydraulic servo system. By applying the variable structure sliding-mode control theory and fuzzy control to design an adaptive fuzzy controller [8], it can weaken the influences of uncertainties in external disturbance and fix the problem of the precise control of unknown parameters in the inverted pendulum system. An adaptive fuzzy PID self-tuning parameter controller [4] was designed for the second order inverted pendulum to achieve a good balance. A PD algorithm based on the adaptive fuzzy compensation was proposed [18, 19] to use in the online approach friction model and the model identification, the results realize a closed-loop system boundlessness of tracking errors.

A cascade adaptive fuzzy sliding-mode control, including inner and outer control loops, [3, 15] was proposed to investigate the stabilizing and tracking control of a nonlinear, two-axis, inverted-pendulum servomechanism. The adaptive network based an interval type-2 fuzzy logic controller design for a single, flexible link carrying a pendulum [13] to determine the motion of the flexible link and determine uncertain oscillation of the link before experimental works. The paper [16] presented the implementation of the control of an inverted pendulum system by using the Adaptive Neural-Fuzzy. It is based on the expert experiences that optimize the parameters of the fuzzy controller by training and learning sample data. A novel, online-motion planning method for a double-pendulum overhead crane system was proposed [14]. Simulation results indicate that the proposed control method achieves good performance and strong robustness.

The inverted pendulum system is widely used in many fields. These include aerospace, mechanical equipment, deep sea technology, compressors and turbo machinery. It has been well studied for its nonlinear characteristic, and it can be used as an uncertain linear system and a discrete system for research and analysis [1, 2]. When the system receives external disturbance and uncertain factors, a robust fuzzy sliding-mode control [5] will be presented to solve these problems.

The fuzzy control method is poor, and when the system parameters change or an unknown condition occurs, the control effect significantly deteriorates. Combining with the advantages of the adaptive control, we propose an adaptive control based on fuzzy compensation. Based on traditional PD control, this paper conducts fuzzy design on system variables, sets up different membership functions, forms an adaptive fuzzy control system and conducts optimization and control on the nonlinear inverted pendulum.

2 Problem description

The second-order nonlinear system: $\ddot{x} = f (x, \dot{x}) + g (x, \dot{x}) u$ (1)

Where, f represents the unknown, nonlinear function; on behalf of modeling the uncertain part and the interference; g represents the linear function; u ∈ Rⁿ is the input, y ∈ Rⁿ is the output. ${\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = f (x_{1}, x_{2}) + g (x_{1}, x_{2}) u \\ y = x_{1} \end{matrix}$ (2)

The position command is x_d, assuming e = x_d - y = x_d - x₁, in which $E = (e \dot{e})^{T}$ .

Choose K = [k_p k_d] ^T,and make all the roots of polynomials s² + k_ds + k_p = 0 on the left-half part of the complex plane.

Take the control law: $u^{*} = \frac{1}{g (x)} [- f (x) + {\ddot{x}}_{d} + K^{T} E]$ (3)

According to the formula (3) and the equation (1), we can get the closed-loop system equation: $\ddot{e} + k_{d} \dot{e} + k_{p} e = 0$ (4)

By choosing K, when t→ ∝, e (t) →0, $\dot{e} (t) \to 0$ . That is, the system output y and its derivative progressively converge to the desired output x_d and its derivatives.

If the non-linear function f (x) is known, we can eliminate its nonlinear property by selecting the control input u, and then design a controller based on the linear control theory.

3 Adaptive fuzzy controller design and analysis

If f (x) is unknown, the control law in equation (3) is difficult to achieve. We can use the fuzzy system $\hat{f} (x)$ to replace f (x) and realize the adaptive compensation.

3.1 Basic fuzzy system

The unknown function can be the fuzzy approximation by using the universal approximation theorem that was proposed by Wang Lixin [12]. The fuzzy approximation can form the fuzzy system $\hat{f} (x | θ_{f})$ by the following two-step structure, for example $\hat{f} (x | θ_{f})$ approach f (x).

Step 1. The variable x_i (i = 1, 2) that defines p_i fuzzy sets $A_{i}^{l_{i}} (l_{i} = 1, 2, \dots, p_{i})$ .

Step 2. The following fuzzy rules $Π_{i = 1}^{2} p_{i}$ to construct a fuzzy system $\hat{f} (x | θ_{f})$ .

$\begin{matrix} R^{(j)} : IF x_{1} is A_{1}^{l_{1}} AND x_{2} is A_{2}^{l_{2}} \\ THEN f is E^{l_{1} l_{2}} \end{matrix}$ (5)

Where, l₁ = 1, 2, i = 1, 2.

The product inference engine, singleton fuzzifier and center average defuzzifier are used to calculate the fuzzy system outputs. $\hat{f} (x | θ_{f}) = \frac{\sum_{l_{1} = 1}^{p_{1}} \sum_{l_{2} = 1}^{p_{2}} {\bar{y}}_{f}^{l_{1} l_{2}} (Π_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i}))}{\sum_{l_{1} = 1}^{p_{1}} \sum_{l_{2} = 1}^{p_{2}} (Π_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i}))}$ (6)

Where, $μ_{A_{i}^{l_{i}}}$ is the membership function of x_i.

The fuzzy inference process adopts the following two steps: (1) using product inference rules to implement the inference premise; using a singleton fuzzifier to evaluate ${\bar{y}}_{f}^{l_{1} l_{2}}$ ; (2) using the product inference engine rules to implement the inference of the rules’ premise and the rules’ conclusion, as well as parallel computing of all fuzzy rules; we used the average defuzzifier to get the fuzzy system output $\hat{f} (x | θ_{f})$ .

Suppose ${\bar{y}}_{f}^{l_{1} l_{2}}$ is a free parameter and put it in the set $θ_{f} \in R^{Π_{i = 1}^{p_{i}}}$ . Introduce the column vector ɛ (x), the formula (6) will become: $\hat{f} (x | θ_{f}) = θ_{f}^{T} ɛ (x)$ (7)

Where, ɛ (x) is the $Π_{i = 1}^{2} p_{i}$ dimensions column vector, the element l₁, l₂ is $ɛ_{l_{1} l_{2}} (x) = \frac{Π_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i})}{\sum_{l_{1} = 1}^{p_{1}} \sum_{l_{2} = 1}^{p_{2}} (Π_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i}))}$ (8)

3.2 Design of adaptive fuzzy controller

Using the fuzzy system approach, the control law equation (3) will become: $u = \frac{1}{g (x)} [- \hat{f} (x | θ_{f}) + {\ddot{x}}_{d} + K^{T} E]$ (9) $\hat{f} (x | θ_{f}) = θ_{f}^{T} ɛ (x)$ (10)

Where, ɛ (x) is the fuzzy vector, the parameter θ_f changes with the adaptive law.

The adaptive control law can be designed as follows: ${\hat{θ}}_{f} = - {rE}^{T} Pb ɛ (x)$ (11)

4 Stability analysis

Next, we refer to the indirect, adaptive fuzzy control method, which was mentioned by Wang Lixin [20], and analyze the closed-loop system stability.

Combing the equation (9) and the equation (1), we can obtain the dynamic equation of the fuzzy closed-loop system: $\ddot{e} = K^{T} E + [\hat{f} (x | θ_{f}) - f (x)]$ (12)

Let $Λ = [\begin{matrix} 0 & 1 \\ k_{p} & k_{d} \end{matrix}], b = [\begin{matrix} 0 \\ 1 \end{matrix}]$ (13)

The dynamic Equation (12) can be written in vector form: $\dot{E} = Λ E + b [\hat{f} (x | θ_{f}) - f (x)]$ (14)

Set the optimal parameter as: $θ_{f}^{*} = arg min_{θ_{f} \in Ω_{f}} [sup | \hat{f} (x | θ_{f}) - f (x) |]$ (15)

Where,Ω_f is the collection of θ_f, θ_f ∈ Ω_f.

Then the definition of the minimum approximation error is: $ω = \hat{f} (x | θ_{f}^{*}) - f (x)$ (16)

The formula (14) can be expressed as: $\dot{E} = Λ E + b {[\hat{f} (x | θ_{f}) - \hat{f} (x | θ_{f}^{*})] + ω}$ (17)

According to the formula (10) and the formula (17), we can obtain the closed-loop dynamic equation: $\dot{E} = Λ E + b [{(θ_{f} - θ_{f}^{*})}^{T} ɛ (x) + ω]$ (18)

This equation clearly describes the relationship between the tracking error E and control parameter θ_f. Adaptive law establishes a regulatory mechanism; it can make the tracking error E and the parameters error $θ_{f} - θ_{f}^{*}$ minimize.

Suppose the Lyapunov function: $V = \frac{1}{2} E^{T} PE + \frac{1}{2 γ} {(θ_{f} - θ_{f}^{*})}^{T} (θ_{f} - θ_{f}^{*})$ (19)

Where, γ is the positive constant, P is the positive definite matrix and satisfies the Lyapunov equation: $Λ^{T} P + P Λ = - Q$ (20)

Where, Q is an arbitrary 2 × 2 positive-definite matrix, which is given by the formula (13).

Suppose: $V_{1} = \frac{1}{2} E^{T} PE, V_{2} = \frac{1}{2 γ} {(θ_{f} - θ_{f}^{*})}^{T} (θ_{f} - θ_{f}^{*})$

To facilitate the equation, we may suppose $M = b [{(θ_{f} - θ_{f}^{*})}^{T} ɛ (x) + ω] .$

Formula (18) is changed to: $\dot{E} = Λ E + M$ (21)

So we can get the results:

$\begin{matrix} {\dot{V}}_{1} & = & \frac{1}{2} {\dot{E}}^{T} PE + \frac{1}{2} E^{T} P \dot{E} \\ = & \frac{1}{2} (E^{T} Λ^{T} + M^{T}) PE + \frac{1}{2} E^{T} P (Λ E + M) \end{matrix}$ (22)

Combining the formulas (21), we can get the formula:

$\begin{matrix} {\dot{V}}_{1} & = & \frac{1}{2} E^{T} (Λ^{T} P + P Λ) E + \frac{1}{2} M^{T} PE + \frac{1}{2} E^{T} PM \\ {\dot{V}}_{1} & = & \frac{1}{2} {\dot{E}}^{T} PE + \frac{1}{2} E^{T} P \dot{E} \\ = & \frac{1}{2} (E^{T} Λ^{T} + M^{T}) PE + \frac{1}{2} E^{T} P (Λ E + M) \\ = & - \frac{1}{2} E^{T} QE + E^{T} PM \end{matrix}$ (23)

Substitute M into formula (23), and consider the formula: $E^{T} Pb {(θ_{f} - θ_{f}^{*})}^{T} ɛ (x) = {(θ_{f} - θ_{f}^{*})}^{T} E^{T} Pb ɛ (x),$

we can get: $\begin{matrix} {\dot{V}}_{1} & = & - \frac{1}{2} E^{T} QE \\ + E^{T} P {b [{(θ_{f} - θ_{f}^{*})}^{T} ɛ (x) + ω]} \\ = & - \frac{1}{2} E^{T} QE + {(θ_{f} - θ_{f}^{*})}^{T} E^{T} Pb ɛ (x) + E^{T} Pb ω \\ {\dot{V}}_{2} & = & \frac{1}{γ} {(θ_{f} - θ_{f}^{*})}^{T} {\dot{θ}}_{f} \end{matrix}$

Then obtain the derivatives V: $\begin{matrix} \dot{V} & = & {\dot{V}}_{1} + {\dot{V}}_{2} = - \frac{1}{2} E^{T} QE \\ + {(θ_{f} - θ_{f}^{*})}^{T} E^{T} Pb ɛ (x) \\ + E^{T} Pb ω + \frac{1}{γ} {(θ_{f} - θ_{f}^{*})}^{T} {\dot{θ}}_{f} \\ = & - \frac{1}{2} E^{T} QE + \frac{1}{γ} {(θ_{f} - θ_{f}^{*})}^{T} \\ [γ E^{T} Pb ɛ (x) + {\dot{θ}}_{f}] + E^{T} Pb ω \end{matrix}$

If we combine the adaptive law (11) and ${\hat{θ}}_{f} = - γ E^{T} Pb ɛ (x)$ , we can have: $\begin{matrix} \dot{V} & = & {\dot{V}}_{1} + {\dot{V}}_{2} = - \frac{1}{2} E^{T} QE + {(θ_{f} - θ_{f}^{*})}^{T} E^{T} Pb ɛ (x) \\ + E^{T} Pb ω + \frac{1}{γ} {(θ_{f} - θ_{f}^{*})}^{T} {\dot{θ}}_{f} \\ = & - \frac{1}{2} E^{T} QE + \frac{1}{γ} {(θ_{f} - θ_{f}^{*})}^{T} [γ E^{T} Pb ɛ (x) + {\dot{θ}}_{f}] \\ + E^{T} Pb ω - \frac{1}{2} E^{T} QE \leq 0 \end{matrix}$ is known. By selecting the very small, minimum approximation error ω of the fuzzy systems, we can guarantee $\dot{V} \leq 0$ .

5 Simulation example

Supposing the controlled object is a single-stage inverted pendulum, then the dynamic equation is expressed as: ${\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = f (x) + g (x) u \end{matrix}$

Where, $\begin{matrix} f (x) & = & \frac{g sin x_{1} - {mlx}_{2}^{2} cos x_{1} sin x_{1} / (m_{c} + m)}{l (4 / 3 - m {cos}^{2} x_{1} / (m_{c} + m))}, \\ g (x) & = & \frac{cos x_{1} / (m_{c} + m)}{l (4 / 3 - m {cos}^{2} x_{1} / (m_{c} + m))}, \end{matrix}$ x₁ and x₂ are the swinging angle and the swing speed, g = 9.8 m/s², m_c s the mass of the cart, m_c = 1 kg, m is the mass of the pendulum, m = 0.1 kg, l is the half length of the pendulum, l = 0.5 m, u is the control input.

The position instruction is x_d (t) =0.1 sin(πt). The variables x₁ and x₂ are defined as five fuzzy sets. If p₁ = p₂ = 5, then $Π_{i = 1}^{2} p_{i} = p_{1} p_{2} = 25$ .

The following five membership functions are: $\begin{matrix} μ_{NM} (x_{i}) & = & exp [- ((x_{i} + π / 6) / (π / 24))^{2}], \\ μ_{NS} (x_{i}) & = & exp [- ((x_{i} + π / 12) / (π / 24))^{2}], \\ μ_{Z} (x_{i}) & = & exp [- (x_{i} / (π / 24))^{2}], \\ μ_{PS} (x_{i}) & = & exp [- ((x_{i} - π / 12) / (π / 24))^{2}], \\ μ_{PM} (x_{i}) & = & exp [- ((x_{i} - π / 6) / (π / 24))^{2}] . \end{matrix}$

Then there will be 25 fuzzy rules for the approximating fuzzy system.

Based on above the fuzzy controller designed and the adaptive law, the Simulink fuzzy system model is established in Fig. 1.

The membership function can be designed by the program, and its graph shown in Fig. 2.

The initial state of the pendulum is [π/60, 0] and the initial value of θ_f is 0.1. Then, we used the control law (9) and the adaptive law (11) to simulate the nonlinear inverted pendulum. Let $Q = [\begin{matrix} 500 & 0 \\ 0 & 500 \end{matrix}]$ , k_d = 20, k_p = 10, the adaptive parameter γ = 100. Then we can get the simulation results, which are shown in Figs. 3–5.

Seen from Fig. 3, the beginning speed and position tracking have the slightest errors. Then, the tracking trajectory and the desired trajectory almost completely overlap. These results show that the tracking effect is positive. From Fig. 4, when applying the external interference in the control system and using the adaptive fuzzy compensation control, the impacts will be well reduced and realize a good tracking. The compensation result based on the adaptive fuzzy output tracking control is shown in Fig. 5. When the system has fuzzy compensation, the practical output curve and the estimation output curve will give a better fit, and the tracking results is good. The above simulation results show that the system can get an effective tracking performance, and can also obtain good stability and robustness by using the adaptive control based on fuzzy compensation.

6 Conclusion

We study the control problem of the inverted pendulum system to apply the adaptive fuzzy compensation technique into the nonlinear response model of the inverted pendulum and propose a robust tracking controller based on the adaptive fuzzy compensation. Applying the Lyapunov stability theory can prove the feedback of the adaptive fuzzy control algorithm of this state; it can also ensure that the closed-loop system is uniformly and ultimately bounded under the condition of reasonable control parameters. Therefore, we can achieve the good tracking control of the nonlinear inverted pendulum system. The controller takes into account the effects of the modeling errors and external interference by using the fuzzy system to approximate the unknown non-linearity in the system. It combines with the swing angle and swing speed of the inverted pendulum system and introduces the adaptive fuzzy compensation. It can make the output signal meet the requirements of the actual system. The simulation results show that the controller is insensitive to parameter perturbation, has strong anti-interference performance, good robust stability and control precision and is obviously superior than the traditional PD control.

References

Paula

A.S.D.

and Savi

M.A.

, Chaos and transient chaos in an experimental nonlinear pendulum, Journal of Sound & Vibration294(3) (2006), 585–595.

Paula

A.S.D.

and Savi

M.A.

, controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method, Chaos Solitons & Fractals42(5) (2009), 2981–2988.

Tao

C.W.

, Taur

, Chang

J.H.

, et al., Adaptive fuzzy switched swing-up and sliding control for the double-pendulum-and-cart system, IEEE Transactions on Systems Man & Cybernetics Part B Cybernetics A Publication of the IEEE Systems Man & Cybernetics Society40(1) (2010), 241–252.

Song

G.J.

, Research on Double Inverted Pendulum Control Using Adaptive Fuzzy PID, Journal of Huaqiao University (Natural Science)37(1) (2016), 74–78.

Yau

H.T.

, Chaos synchronization of two uncertain chaotic nonlinear gyros using fuzzy sliding mode control, Mechanical Systems & Signal Processing22(2) (2008), 408–418.

Liang

and Chen

, Fuzzy logic adaptive compensation control for space manipulator to track desired trajectory in joint space, Journal of System Simulation23(3) (2011), 577–582.

Lei

J.L.

and Dou

M.F.

, Adaptive Fuzzy Control for BLDCM in Near Space Based on RBF Neural Network Compensation, Journal of Northwestern Polytechnical University32(3) (2014), 394–399.

Zhang

J.L.

and Zhang

, Research on Adaptive Fuzzy Sliding Mode Control for Uncertain Inverted Pendulum System, Computer Simulation30(10) (2013), 341–344.

Bai

J.Y.

, Zhang

Z.H.

and Zhang

P.H.

, Application of fuzzy adaptive compensation control in furnace desulfurization system, Process Automation Instrumentation37(3) (2016), 65–68.

10.

J.W.

and Zhao

K.D.

, The position control of electrohydraulic servosystem in simulator via an adaptive fuzzy controller, Machine Tools & Hydraulics4(2) (2003), 85–87.

11.

Dong

L.H.

, Research on the robust adaptive fuzzy control of manipulators based on fuzzy compensation, Computer Engineering and Science34(1) (2012), 169–173.

12.

Wang

L.X.

, Fuzzy system and fuzzy control tutorial, Tsinghua University press, Beijing, (2003), 234–238.

13.

Tinkir

, Onen

, Kalyoncu

, et al., Adaptive network based interval type-2 fuzzy logic controller design for a single flexible link carrying a pendulum, The International Conference on Computer and Automation Engineering, IEEE, (2010), 112–116.

14.

Zhang

, Ma

, Chai

, et al., A novel online motion planning method for double-pendulum overhead cranes, Nonlinear Dynamics85(2) (2016), 1079–1090.

15.

Wai

R.J.

, Kuo

M.A.

and Lee

J.D.

, Design of cascade adaptive fuzzy sliding-mode control for nonlinear two-axis inverted-pendulum servomechanism, IEEE Transactions on Fuzzy Systems16(5) (2008), 1232–1244.

16.

Yang

X.H.

, Liu

H.S.

, Liu

G.P.

, et al., Control Experiment of the Inverted Pendulum Using Adaptive Neural-Fuzzy Controller, Proceedings of the 2010 International Conference on Electrical and Control Engineering, IEEE Computer Society, (2010), 629–632.

17.

Hong

X.Y.

and Liu

C.Y

, Theory-region changed adaptive fuzzy control of triple inverted pendulum based on LQR, Control Engineering of China13(5) (2006), 445–448.

18.

Wang

Y.F.

and Chai

T.Y.

, Adaptive Fuzzy Control Method for Dynamic Friction Compensation, Proceedings of the CSEE25(2) (2005), 139–143.

19.

Wang

Y.F.

and Chai

T.Y.

, Compensating modeling and control of robot joint friction based on adaptive fuzzy systems, Chinese Journal of Scientific Instrument27(2) (2006), 186–190.

20.

Wang

Z.T.

, Tao

, Wang

J.J.

and He

, Modeling and simulation of the planar inverted pendulum based on fuzzy adaptive PID controller, Journal of Hangzhou Dianzi University30(4) (2010), 86–91.