WCMAC-based control system design for nonlinear systems using PSO

Abstract

This paper presents a wavelet cerebellar model articulation controller (WCMAC) based adaptive control design for nonlinear systems using a particle swarm optimization (PSO). The WCMAC is the main tracking controller and a robust compensation controller is used to compensate for the residual error. In the WCMAC, the adaptive laws of controller parameters are derived using the gradient descent method. However, the initial values of learning rates of these adaptive laws are very important and they affect much to the performance of control systems. In this paper, the particle swarm optimization algorithm is applied to find the optimal learning rates of the parameter adaptation laws. To show the effectiveness of the proposed approach, numerical simulations of magnetic levitation system and inverted pendulum are provided to confirm the applications of the proposed PSO-WCMAC-based control system. The superiority of the proposed control scheme is also evaluated by quantitative comparison with other control schemes.

Keywords

Wavelet cerebellar model articulation controller particle swarm optimization magnetic levitation system inverted pendulum

1 Introduction

A cerebellar model articulation controller (CMAC) was firstly introduced by Albus in 1975 [1]. It is a type of neural network based on a model of the mammalian cerebellum (associative memory). CMAC has many advantages than other neural network controllers such as fast learning property, simple computation, and good generalization capability, so it is frequently used in real-time control systems [2]. In recent years, many papers have proposed the modifications of CMAC such as self-organizing CMAC, function-link CMAC and type-2 fuzzy CMAC to enhance the performance [3 –7]. The conventional neural networks (NNs) often use a Gaussian function as the adaptive function; however, some studies have proven that the learning capability of wavelet NNs is more efficient than the conventional Gaussian function NNs [8, 9]. In this study, the considered wavelet CMAC combines the advantages of CMAC and decomposition property of wavelet function to achieve better learning performance so it can be more suitable for control applications [2].

The PSO algorithm is an optimization technique first introduced in 1995 by J. Kennedy and R. Eberhart, it mimics the social behavior of bird flocking or fish schooling [10]. Compared to the typical genetic algorithm (GA), PSO has advantages as easy to implement and few parameters to adjust [11]. In PSO, each single solution is called as a particle. In the beginning, the particles will be initialized with a group of random solutions. In each iteration, the best solution can be shared among other particles and every particle will move to follow the best solution [10]. Recently, PSO algorithm is widely used with some controllers such as PID, SMC, LQR, fuzzy, and neural network to solve nonlinear problems in control and economic dispatch [12 –23]. In 2011, Bingül and Karahan proposed a PSO fuzzy logic controller for 2 DOF robot which used PSO algorithm to tune the antecedent and consequent parameters of fuzzy rules [13]. In 2014, Lu et al. presented a Chaotic PSO for optimal design of PID controller [17]. Following that, in 2015 Wang and Liu introduced PSO-based fuzzy controller for optimal charge pattern of li-ion batteries [18]. Also in 2015, Kapoor and Ohri provided the PSO algorithm for optimal parameters of PID and SMC controller [19]. The advance of these method is simple in calculation and easier for implementation. However, most papers in literature often use offline PSO to search optimal parameters for the controllers, so it led to large dimension of particles and will take long time to satisfy the fitness function. In many adaptive control systems, the learning rates of adaptive laws are very important and they are highly influential to the system performance. Most papers in literature often use a trial-and-error method to determine the learning rates of adaptive laws. However, selection the optimized learning rate is difficult and it often takes long time to train the network. Therefore, in this study, offline and online PSO method is applied to find the optimal learning rates for the mean, the variance of the wavelet Gaussian function in the input membership and also for the weight in memory space layer.

In the proposed control scheme, a PSO-WCMAC is the main tracking controller used to mimic an ideal controller and a robust compensation controller is designed to recover the residual error between the ideal controller and the main controller. Firstly, the PSO run in offline mode to find the best initial value of learning rates of WCMAC. Then, those obtained best values are used for the initials of the particles in online PSO. The developed controller can be applied to control nonlinear systems even without exact information of system dynamics.

Recently, many researchers focus on the magnetic levitation system (MLS) because of its advantages such as no mechanical contact, friction, and noise. MLS has many applications in precise positioning such as maglev train, magnetic bearing, wind tunnel, and conveyor system, etc. Beside MLS, an inverted pendulum is a classical problem in nonlinear control system, which attracted the attention of many researchers. In 2011, Lin et al. published an adaptive PID design for MLS [16]. After that, in 2015, Zhang et al. published robust tracking control for MLS [24]. In 2014, Tao and Su provided the moment adaptive fuzzy control for an inverted pendulum [25]. Mohammad and Ahmad [26] proposed a type-2 fuzzy PID controller for an inverted pendulum system [26]. However, all the methods of these papers are difficult to determine the initial parameter and the control performance can be further improved. In this paper, the developed PSO-WCMAC control system is applied for these two systems.

The motivation of the proposed paper is to apply offline and online PSO algorithm to find the optimal learning rates for adaptive laws of WCMAC. All the adaptive laws are designed based on the gradient descent method and the stability of the control system is proved by the Lyapunov function. The effectiveness of the proposed controller is verified by numerical simulations of magnetic levitation system and inverted pendulum. This paper is organized as follows. The problem formulation of nonlinear control system is presented in Section 2. The design of PSO-WCMAC control system is presented in Section 3. The simulation results are provided in Section 4. Finally, the conclusion is given in Section 5.

2 Problem formulation

Consider an nth order nonlinear system denoted as $x^{(n)} (t) = f (x (t)) + g (x (t)) u (t) + d (t)$ (1) where $u (t) \in R$ and $x (t) \in R$ are the control input and system output, respectively, $d (t) \in R$ is an external disturbance, $f (x (t)) \in R$ and $g (x (t)) \in R$ represent the unknown continuous functions and assumed to be bounded.

When uncertainty is under consideration, (1) can be reformulated as

$\begin{matrix} x^{(n)} (t) & = & f_{0} (x (t)) + Δ f (x (t)) \\ + [g_{0} (x (t) + Δ f (x (t)))] u (t) + d (t) \\ = & f_{0} (x (t)) + g_{0} (x (t)) u (t) + β (x (t)) \end{matrix}$ (2) where f₀ ( x (t)) and g₀ ( x (t)) are the nominal parts of f ( x (t)) and g ( x (t)), Δf ( x (t)) and Δg ( x (t)) are the unknown uncertainties of f ( x (t)) and g ( x (t)), respectively; without loss of generality g₀ ( x (t)) is assumed greater than zero. β ( x (t)) is referred to as the lumped uncertainty and is defined as $β (x (t)) = Δ f (x (t)) + Δ g (x (t)) u (t) + d (t)$ (3)

The goal of the controller is to produce a control signal u (t), which can force the system output x (t) to track the reference signal $x_{d} (t) \in R$ . If the lumped uncertainty β ( x (t)) and the nominal parts f₀ ( x (t)) and g₀ ( x (t)) are known exactly, then an ideal controller can be designed as follows:

$\begin{matrix} u^{*} (t) & = & g_{0}^{- 1} (x (t)) \\ [x_{d}^{(n)} (t) - f_{0} (x (t)) - β (x (t)) + K^{T} e (t)] \end{matrix}$ (4) where $K = {[k_{n}, \dots, k_{2}, k_{1}]}^{T} \in R^{n}$ is the feedback gain vector, which contains real numbers, and $e (t) ≜ {[e (t), \dot{e} (t), \dots, e^{(n - 1)} (t)]}^{T} \in R^{n}$ is the system tracking error vector, where e (t) is defined as $e (t) ≜ x_{d} (t) - x (t) \in R$ (5)

Substituting (4) into (2), the error dynamics can be obtained as $e^{(n)} + K^{T} e = 0$ (6)

From (6), it is obvious that if K is selected to correspond to the coefficients of Hurwitz polynomial, then the tracking error can go to zero $\lim_{t \to \infty} e (t) = 0$ , and then the system can follow the reference signal x_d (t) well. However, the lumped uncertainty β ( x (t)) cannot be precisely known in practical applications. Therefore, the ideal controller is unobtainable. Thus, a PSO-WCMAC control system is proposed in the following section to achieve the desired control performance.

3 PSO-WCMAC

The structure of PSO-WCMAC control system consists of a WCMAC where its learning rates can be updated using PSO and a robust compensation controller. The structure of WCMAC feedback control system is shown in Fig. 1.

Fig.1

Block diagram of PSO-WCMAC control system.

3.1 The WCMAC Model

The structure of WCMAC is shown in Fig. 2 which consists of an input space, an association memory space, a receptive-field space, a weight memory space and an output space. Based on [2, 8], a signal propagation and a basic function in each space are described as follows.

Fig.2

The architecture of the WCMAC controller.

Input space I : For a given $I = [I_{1}, I_{2}, \dots I_{i}, \dots I_{n_{i}}]^{T} \in R^{n_{i}}$ , where I_i represents the ith input and n_i is number of input state variable. Each input state variable I_i can be quantized into discrete regions (called elements or neurons) according to a given control space and it is normalized in an interval value.

Association memory space A: In this space, several elements can be accumulated as a block. Each block performs a receptive-field basis function. The Gaussian membership function (MF) is used as a mother wavelet in the receptive-field, which can be represented as

$\begin{matrix} f_{ijk} (F_{ijk}) \\ = - F_{ijk} \exp (- F_{ijk}^{2} / 2) for i = 1, 2, \dots, n_{i} \\ j = 1, 2, \dots, n_{j} and k = 1, 2, \dots, n_{k} \end{matrix}$ (7) where n_k is number of block in a layer, n_j is number of layer. $F_{ijk} = \frac{s_{i} - m_{ijk}}{v_{ijk}}$ where m_ijk is the mean, v_ijk is the variance of block kth in the jth layer corresponding to the ith input variable.

Receptive-field space φ: The multidimensional receptive-field function is defined as a vector

$\begin{matrix} φ & = & {[φ_{11}, \dots, φ_{1 n_{k}}, \dots, φ_{n_{j} 1}, \dots, φ_{n_{j} n_{k}}]}^{T} \\ \in R^{n_{j} n_{k}} \end{matrix}$ (8) where:

$\begin{matrix} φ_{jk} & = & \prod_{i = 1}^{n_{i}} f_{ijk} (F_{ijk}) \\ = & - \prod_{i = 1}^{n_{i}} (\frac{s_{i} - m_{ijk}}{v_{ijk}}) \exp (\frac{- {(\frac{s_{i} - m_{ijk}}{v_{ijk}})}^{2}}{2}) \end{matrix}$ (9)

Weight memory space W : Each location of φ is corresponding to a particular adjustable value in the weight memory space and can be described as $\begin{matrix} w & = & {[w_{11}, \dots, w_{1 n_{k}}, \dots, w_{n_{j} 1}, \dots, w_{n_{j} n_{k}}]}^{T} \\ \in R^{n_{j} n_{k}} \end{matrix}$ (10) where w_jk is the connecting weight value of the output associated with the jth layer and kth block.

Output space o: The output of a WCMAC is the algebraic sum of the activated weights in the weight receptive-field and is expressed as $u_{WCMAC} = o = w^{T} φ = \sum_{j = 1}^{n_{j}} \sum_{k = 1}^{n_{k}} w_{jk} φ_{jk}$ (11) Initial values of the parameters m_ijk, v_ijk and w_jk of WCMAC can be chosen randomly and it can be updated based on the derived.

3.2 Adaptive law for PSO-WCMAC control system

Assumed that there exists an ideal $u_{WCMAC}^{*}$ to approximate the ideal controller u^* (t) in (4), then $u^{*} (t) = u_{WCMAC}^{*} (w^{*}, m^{*}, v^{*}, t) + ɛ (t)$ (12) where ɛ (t) denotes the approximation error, w^*, m^*, v^* are the optimal parameters of w, m, v, respectively. However, the ideal controller $u_{WCMAC}^{*}$ cannot be obtained. Thus, an estimation WCMAC controller ${\hat{u}}_{WCMAC}$ can be designed to online estimate $u_{WCMAC}^{*}$ , and then the overall controller can be designed as

$\begin{matrix} \hat{u} (t) & = & {\hat{u}}_{WCMAC} (\hat{w}, \hat{m}, \hat{v}, t) + {\hat{u}}_{R} (t) \\ \equiv & {\hat{w}}^{T} \hat{φ} + {\hat{u}}_{R} (t) \end{matrix}$ (13) where ${\hat{u}}_{R}$ is a robust compensation controller and $\hat{w}, \hat{m}, \hat{v}$ are the estimation value of w^*, m^*, v^*, respectively.

A sliding surface s (t) is defined as $s (t) ≜ e^{(n - 1)} + k_{1} e^{(n - 2)} + \dots + k_{n} \int_{0}^{t} e (τ) d τ$ (14)

Taking the derivative of (14) and using (2) and (5), yield

$\begin{matrix} \dot{s} (t) & = & e^{(n)} + K^{T} e \\ = & - f_{0} (x (t)) - g_{0} (x (t)) ({\hat{u}}_{WCMAC} (t) + {\hat{u}}_{R} (t)) \\ + x_{d}^{(n)} - β (x (t)) + K^{T} e \end{matrix}$ (15)

By choosing a Lyapunov cost function $V_{1} (s (t)) = \frac{1}{2} s^{2} (t)$ , then ${\dot{V}}_{1} (s (t)) = s (t) \dot{s} (t)$ . Using (15), yields

$\begin{matrix} {\dot{V}}_{1} (s (t)) & = & s (t) [- f_{0} - g_{0} (x (t)) ({\hat{u}}_{WCMAC} (t) + {\hat{u}}_{R} (t)) \\ + x_{d}^{(n)} - β (x (t)) + K^{T} e] \end{matrix}$ (16)

The aim of PSO-WCMAC is to find the optimal value of $\hat{w}, \hat{m}, \hat{v},$ so that the derivative of cost function ${\dot{V}}_{1} (s (t))$ is minimized to achieve fast convergence of s (t). Using the gradient descent method, $\hat{w}, \hat{m} and \hat{v}$ can be updated by the following equations $\begin{matrix} {\dot{\hat{w}}}_{jk} & = & - {\hat{η}}_{w} \frac{\partial s^{T} \dot{s}}{\partial {\hat{w}}_{jk}} = - {\hat{η}}_{w} \frac{\partial s (t) \dot{s} (t)}{\partial {\hat{u}}_{WCMAC}} \frac{\partial {\hat{u}}_{WCMAC}}{\partial {\hat{w}}_{jk}} \\ = & {\hat{η}}_{w} s (t) g_{0} {\hat{φ}}_{jk} \end{matrix}$ (17) $\begin{matrix} {\dot{\hat{m}}}_{ijk} & = & - {\hat{η}}_{m} \frac{\partial s^{T} \dot{s}}{\partial {\hat{m}}_{ijk}} \\ = & - {\hat{η}}_{m} \frac{\partial s (t) \dot{s} (t)}{\partial {\hat{u}}_{WCMAC}} \frac{\partial {\hat{u}}_{WCMAC}}{\partial {\hat{φ}}_{jk}} \frac{\partial {\hat{φ}}_{jk}}{\partial f_{ijk}} \frac{\partial f_{ijk}}{\partial {\hat{m}}_{ijk}} \\ = & - {\hat{η}}_{m} s (t) g_{0} {\hat{w}}_{jk} {\hat{φ}}_{jk} \frac{1 - F_{ijk}^{2}}{(s_{i} - {\hat{m}}_{ijk})} \end{matrix}$ (18) $\begin{matrix} {\dot{\hat{v}}}_{ijk} & = & - {\hat{η}}_{v} \frac{\partial s^{T} \dot{s}}{\partial {\hat{v}}_{ijk}} \\ = & - {\hat{η}}_{ν} \frac{\partial s (t) \dot{s} (t)}{\partial {\hat{u}}_{WCMAC}} \frac{\partial {\hat{u}}_{WCMAC}}{\partial {\hat{φ}}_{jk}} \frac{\partial {\hat{φ}}_{jk}}{\partial f_{ijk}} \frac{\partial f_{ijk}}{\partial {\hat{v}}_{ijk}} \\ = & - {\hat{η}}_{v} s (t) g_{0} {\hat{w}}_{jk} {\hat{φ}}_{jk} \frac{(1 - F_{ijk}^{2})}{{\hat{v}}_{ijk}} \end{matrix}$ (19) where ${\hat{η}}_{w}, {\hat{η}}_{m}, {\hat{η}}_{v}$ are the learning-rates with positive number and they can be obtained by PSO algorithm; and ${\hat{φ}}_{jk}$ is the estimation of φ_jk.

3.3 Particle swarm optimization (PSO) algorithm

3.3.3.1 1) Offline PSO

The PSO algorithm is shown in Fig. 3. At beginning, the initial values are set randomly for learning rate of the weight, the mean, and the variance ${\hat{η}}_{w}, {\hat{η}}_{m}, {\hat{η}}_{v}$ . The first set values will be applied to WCMAC controller and calculate the fitness function, then the loop will run continuously with the second set values and go on till $n_{p}^{th}$ set values. After that, based on the best position of the particle $(p_{{Best}_{q}^{l}})$ and the best position of the swarm $(g_{{Best}_{q}^{l}})$ each set can adjust their positions. Finally, the main loop will run again with the new $n_{p}^{th}$ set value until reaches the times setup. The formula for updating laws are $p_{q}^{l} (n + 1) = p_{q}^{l} (n) + v_{q}^{l} (n + 1)$ (20) and

$\begin{matrix} v_{q}^{l} (n + 1) & = & v_{q}^{l} (n) + c_{1} * rand (.) \\ * [p_{{Best}_{q}^{l}} - p_{q}^{l} (n)] \\ + c_{2} * rand (.) * [g_{{Best}_{q}^{l}} - p_{q}^{l} (n)] \end{matrix}$ (21) where $v_{q}^{l} (n)$ and $p_{q}^{l} (n)$ denote the current velocity and current position, respectively, rand (.) is a random variable between 0∼1, c₁ and c₂ denote the acceleration factors based on local information and global information, respectively, $p_{{Best}_{q}^{l}}$ is the local best index (best position of the particle), and $g_{{Best}_{q}^{l}}$ is the global best index (best position of the swarm), q = 1, 2, …, n_p (n_p is the population size) and l = 1, 2, …, n_d (n_d is the dimension of each particle).

Fig.3

Flowchart of PSO.

To evaluate the performance of learning rates in PSO algorithm, the fitness function is chosen as $fitness = \exp (\frac{{∥ e (t) ∥}^{2}}{T_{p}^{2}})$ (22) where T_p is a precise of fitness function

3.3.3.2 2) Online PSO

After run offline PSO, the optimal values for the learning rates of the weight, the mean, and the variance are used as the initial values for online PSO. In controller design, the main difference between online and offline PSO is the running time and initial particles. In offline PSO, each set values of learning rates will run for a long time (20 seconds) to calculate the fitness function based on the tracking error. But in online PSO, the running time is short (0.1 seconds) to response to the online real-time feedback control.

3.4 Robust compensation control

The robust compensation controller is designed to cope with the approximation error between the WCMAC and the ideal controller. Assumed that the approximation error can be bounded by 0 ≤ ɛ (t) ≤ E, where E is a positive and assumed to be a constant during the observation. However, this bounds is difficult to obtain, especially in practical systems. So that, the bound estimation is used to obtain the bound of the approximation error which can be defined as $\tilde{E} (t) = E - \hat{E} (t)$ (23) where $\hat{E} (t)$ is the estimated value of uncertainty bound E. The robust controller is designed to compensate the effect of the approximation error and is chosen as $u_{R} (t) = \hat{E} (t) sgn (s (t))$ (24) and the adaptive law is chosen as $\dot{\hat{E}} (t) = η_{D} | s (t) |$ (25)

Using some straightforward manipulation, the error equation can be obtained $\dot{s} = e^{(n)} + K_{-}^{T} \underline{e} = ɛ (t) - u_{R} (t)$ (26)

Define a Lyapunov function as $V_{2} (s (t), \tilde{E} (t)) = \frac{1}{2} s^{2} (t) + \frac{{\tilde{E}}^{2} (t)}{2 η_{D}}$ (27)

Take derivative of Equation (27), then

$\begin{matrix} {\dot{V}}_{2} (s (t), \tilde{E} (t)) \\ = s (t) \dot{s} (t) + \frac{\tilde{E} (t) \dot{\tilde{E}} (t)}{η_{D}} \\ = s (t) (ɛ (t) - u_{R} (t)) + \frac{\tilde{E} (t) \dot{\tilde{E}} (t)}{η_{D}} \\ = s (t) ɛ (t) - \hat{E} (t) | s (t) | + \frac{\tilde{E} (t) \dot{\tilde{E}} (t)}{η_{D}} \end{matrix}$ (28)

Because E is a constant, so $\dot{\tilde{E}} (t) = - \dot{\hat{E}} (t)$ . By applying (25) into (28), obtains

$\begin{matrix} {\dot{V}}_{2} (s (t), \tilde{E} (t)) \\ = s (t) ɛ (t) - \hat{E} (t) | s (t) | - (E - \hat{E} (t)) | s (t) | \\ = s (t) ɛ (t) - E | s (t) | \leq | s (t) | | ɛ (t) | - E | s (t) | \\ = - | s (t) | (E - | ɛ (t) |) \leq 0 \end{matrix}$ (29)

Since ${\dot{V}}_{2} (s (t), \tilde{E} (t))$ is negative semidefinite, that is, $V_{2} (s (t), \tilde{E} (t)) \leq V (s (0), \tilde{E} (0))$ , it implies that s (t) and $\tilde{E} (t)$ are bounded. Let function $Ω \equiv (E - ɛ (t)) s \leq (E - | ɛ (t) |) | s | \leq - {\dot{V}}_{2} (s (t), \tilde{E} (t))$ , and integrate Ω (t) with respect to time, then it is easy to obtain that $\int_{0}^{t} Ω (τ) d τ \leq V_{2} (s (0), \tilde{E} (0)) - V_{2} (s (t), \tilde{E} (t))$ (30)

Because $V_{2} (s (0), \tilde{E} (0))$ is bounded, and $V_{2} (s (t), \tilde{E} (t))$ is non increasing and bounded, so from (30) we can have $lim_{t \to \infty} \int_{0}^{t} Ω (τ) d τ < \infty .$ (31)

Also, $\dot{Ω} (τ)$ is bounded, so by Barbalat’s lemma [27], $\lim_{t \to \infty} Ω = 0$ . That is, s (t) →0 as t→ ∞. As the result, the PSO-WCMAC with robust compensation control system can achieve robust tracking performance. It does not require to know exactly the nominal functions f₀ ( x (t)) and g₀ ( x (t)), because the lumped uncertainty β ( x (t)) can be covered for the mis-modeling between the practical system and these functions. Therefore, the proposed control system can be applied to the nonlinear systems even though without known precise dynamic functions.

4 Simulation results

In order to demonstrate the effectiveness of the proposed approach, the PSO-WCMAC system is applied to control a magnetic levitation system and an inverted pendulum. In the controller design, the parameters of WCMAC and robust controller can be tuned online by adaptive laws and the learning rates can be adjusted by the PSO algorithm.

4.1 Magnetic levitation system

The diagram of the magnetic levitation system is shown in Fig. 4. Using the Newton’s second law, the behavior of the metallic ball is given as [16] $Ma = F (x, I) - {Mg}_{m}$ (32) where:

$\begin{matrix} F (x, I) \\ = \frac{μ_{0} N^{4} I^{2} S}{8} ((x + h) \ln | \frac{R_{2} + \sqrt{R_{2}^{2} + {(x + h)}^{2}}}{R_{1} + \sqrt{R_{1}^{2} + {(x + h)}^{2}}} | \\ + xln | \frac{R_{1} + \sqrt{R_{1}^{2} + x^{2}}}{R_{2} + \sqrt{R_{2}^{2} + x^{2}}} |) \end{matrix}$ where M (Kg) is the mass of metallic ball, a (m/sec²) is the acceleration, g_m (m/sec²) denotes the acceleration due to gravity, x (m) is the distance of the ball from electromagnet, I (A) is the current, F (x, I) is the magnetic control force, μ₀ (Wb/Am) is the permeability of free space, h (m) is the length of coil, N is the number of turns per meter, S (m²) is the material surface crossed by the magnetic flux, R₁ is minimum coil radius, R₂ is maximum coil radius. The specifications of MLS are N = 2850, R₁ = 0.012, R₂ = 0.038, S = 5.515, μ₀ = 4π · 10^-7, M = 0.0216, g_m = 9.8 and w = 0.065.

Fig.4

The construction of a magnetic ball levitatic system [16].

Rewrite (31), then $\ddot{x} = g (x) u (t) + d$ (33) where

$\begin{matrix} g (x) \\ = \frac{μ_{0} N^{4} S}{8 M} ((x + h) \ln | \frac{R_{2} + \sqrt{R_{2}^{2} + {(x + h)}^{2}}}{R_{1} + \sqrt{R_{1}^{2} + {(x + h)}^{2}}} | \\ + xln | \frac{R_{1} + \sqrt{R_{1}^{2} + x^{2}}}{R_{2} + \sqrt{R_{2}^{2} + x^{2}}} |) \end{matrix}$ and u (t) = I², d = - g_m.

For the offline PSO, the parameter of WCMAC will be initialized randomly and it can be updated. The parameters of PSO are chosen as random values initially for the population size (n_p = 20, n_d = 3), and the acceleration factor c₁ = c₂ = 0.07. In online PSO-WCMAC, the initial parameters are set from the best value found from the offline PSO algorithm.

The simulation results of the proposed control system are given in Figs. 5–8. From these figures, it is clear that the system can quickly achieve the steady state tracking performance with very small error and the learning rates can be adjusted via online PSO. The initial position of the metallic sphere is set to –1 mm. In the first simulation, the MLS are controlled by online PSO-WCMAC and normal WCMAC to follow a sine command signal x_d (t) =0.01 * sin(2πft) with f = 0.15Hz as shown in Fig. 5. The online adjustments of the learning rates are shown in Fig. 6. For another case simulation, the same initial condition is used, and the command signal is a trapezoid signal as shown in Fig. 7. The learning-rates are shown in Fig. 8. Besides PSO, the genetic algorithm (GA) is another often used optimization algorithm. The comparison of root mean square error (RMSE) for WCMAC, GA-WCMAC and PSO-WCMAC is shown in Table 1. From these comparisons, it can be found the PSO-WCMAC can achieve smaller tracking error than the WCMAC and the GA-WCMAC. From these simulations, it is shown the proposed control system (PSO-WCMAC) can achieve satisfactory control performance with small tracking error for the MLS.

Fig.5

Simulation result of MLS with sine signal reference.

Fig.6

The learning-rates adjustment via online PSO by sine signal reference.

Fig.7

Simulation result of MLS with trapezoid signal reference.

Fig.8

The learning-rates adjustment via online PSO by trapezoid signal reference.

Table 1

Comparison results in RMSE of MLS

	Sine command	Trapezoid command
WCMAC	0.7595	0.3285
GA-WCMAC	0.0534	0.0417
PSO-WCMAC	0.0278	0.0194

4.2 The Inverted Pendulum System

The inverted pendulum system as used in [25 , 29] is considered and it can be described by the dynamic equations

$\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = f + bu + d \\ f = \frac{{gsinx}_{1} - ({mlx}_{2}^{2} {sinx}_{1} {cosx}_{1}) / (m_{c} + m_{p})}{l [4 / 3 - m_{p} \cos^{2} x_{1} / (m_{c} + m_{p})]} \\ b = \frac{{cosx}_{1} / (m_{c} + m_{p})}{l [4 / 3 - m_{p} \cos^{2} x_{1} / (m_{c} + m_{p})]} \end{matrix}$ (34) where x₁ (rad), x₂ (m/s) and ${\dot{x}}_{2} (m / s^{2})$ are the angle, the velocity, and the acceleration of the pendulum, respectively, g (m/s²) is the gravity, m_p and m_c are the mass of the pole and the mass of the cart (kg), respectively, l (m) is the half length of the pole, d is the external disturbance, and u (V) is the control signal. Firstly, we consider the inverted pendulum without uncertainty and the model parameter are given as in Table 2 as [25].

Table 2

Model parameters of Inverted pendulum

Parameter	Value
m_c (kg)	1.0
m_p (kg)	0.1
l (m)	0.5

The goal of control is to make the inverted pendulum following the reference trajectory $x_{d} (t) = \frac{π}{10} [sin (t) + 0.3 sin (3 t)] (rad)$ . The external disturbance d is (-5N ≤ d ≤ 5N). The simulation result for the initial angle x₁ = 0.2 (rad) is shown in Fig. 9. The adjustment of learning rates are shown in Fig. 10. Figure 11 shows the result for the initial angle x₁ = 0.458 (rad), and the learning rates are shown in Fig. 12.

Fig.9

Simulation result of inverted pendulum with the online PSO-WCMAC for the initial angle = 0.2 (rad).

Fig.10

The learning-rates adjustment via online PSO of inverted pendulum for the initial angle = 0.2 (rad).

Fig.11

Simulation result of inverted pendulum with the online PSO-WCMAC for the initial angle = 0.458 (rad).

Fig.12

The learning-rates adjustment via online PSO of inverted pendulum for the initial angle = 0.458 (rad).

From the simulation results, it can be observed that the system output can follow the reference trajectory well even though the disturbance exists and the learning rates can quickly converge to the optimal value. The root mean square error is calculated after 1 second. They are RMSE = 0.0001 for initial angle = 0.2 (rad) and RMSE = 0.0008 for initial angle = 0.458 (rad). In the same conditions, Table 3 shows the comparison result of RMSE (after 1 second) between the WCMAC, GA-WCMAC, PSO-WCMAC and the results in [25]. From these comparisons, it can be seen that the proposed PSO-WCMAC has achieved smaller tracking error than the other control methods.

Table 3

Comparison results of tracking error in RMSE

	Initial angle 0.2 (rad)	Initial angle 0.458 (rad)
Tradition adaptive fuzzy robust control [25]	1.6618 × 10^-2	1.8393 × 10^-2
Moment adaptive fuzzy with robust supervisory control [25]	1.0534 × 10^-3	1.047 × 10^-3
Moment adaptive fuzzy control with CMAC-based residue compensation [25]	9.747 × 10^-4	9.7341 × 10^-4
WCMAC	2.317 × 10^-3	3.631 × 10^-3
GA-WCMAC	1.687 × 10^-4	9.258 × 10^-4
PSO-WCMAC	1.209 × 10^-4	8.878 × 10^-4

In Figs. 5, 7, 9 and 11, it can be seen u_R provides large contribution for the total control effort in the initial phase since there exists large tracking error in this phase. After that, u_WCMAC plays the main controller to provide desired control effort to track the desired trajectory.

In our research, the offline PSO is trained as 500 epochs with 20 sets value of learning rates. The computing time of the online WCMAC algorithm is 0.000035 second for each epoch and the computing time of PSO-WCMAC algorithm is 0.000042 second. The simulations were done on Windows 7 64-bit and the processor is Core i5-4460 3.2 GHz, RAM 8 GB.

5 Conclusion

In this study, an adaptive PSO-WCMAC was designed and combined with robust compensation controller to make the system convergence and ensure good control performance. It is suitable for a large class of unknown nonlinear systems. The major contributions of this study are the development of a PSO-WCMAC with the adaptive law for updating parameters, and the learning rates can be online optimized to best value based on PSO algorithm. Various simulation results of the magnetic ball and the inverted pendulum have

demonstrated the effectiveness of the proposed approach.

Footnotes

Acknowledgments

This work was supported partially by the National Science Council of the Republic of China under Grant NSC 98-2221-E-155-058-MY3.

References

Albus

J.S.

, Theoretical and experimental aspects of a cerebellar model, Ph. D. dissertation, University of Maryland, 1972.

Lin

C.M.

, Lin

M.H.

and Yeh

R.G.

, Synchronization of unified chaotic system via adaptive wavelet cerebellar model articulation controller, Neural Computing & Application 23 (2013), 965–973.

Lin

C.M.

, Yang

M.S.

, Chao

, Hu

X.M.

and Zhang

, Adaptive filter design using type-2 fuzzy cerebellar model articulation controller, IEEE Transactions on Neural Network Learning Systems 27(10) (2016), 2084–2094.

Yeh

M.F.

and Chang

K.C.

, A self organizing CMAC network with gray credit assignment, IEEE Trans Systems, Man and Cybernetics, Part B, Cybernrtics 36(3) (2006), 623–635.

Lin

C.M.

, Liu

Y.L.

and Li

H.Y.

, SoPC-based function-link cerebellar model articulation control system design for magnetic ball levitation systems, IEEE Trans Industrial Electronics 61(8) (2014), 4265–4273.

Chung

C.C.

, Chen

T.S.

, Lin

L.H.

, Lin

Y.C.

and Lin

C.M.

, Bankruptcy prediction using cerebellar model neural networks, International Journal of Fuzzy Systems 18(2) (2016), 160–167.

Lee

C.H.

, Chang

F.Y.

and Lin

C.M.

, An efficient interval type-2 fuzzy CMAC for chaos time-series prediction and synchronization, IEEE Trans on Cybernetics 44(3) (2014), 329–341.

Hsu

C.F.

, Lin

C.M.

and Lee

T.T.

, Wavelet adaptive backstepping control for a class of nonlinear systems, IEEE Trans Neural Networks 17(5) (2006), 1175–1183.

Mai

T.L.

and Wang

, Adaptive force/motion control system based on recurrent fuzzy wavelet CMAC neural networks for condenser cleaning crawler-type mobile manipulator robot, IEEE Trans Control Systems Technology 22(5) (2015), 1973–1982.

10.

Kennedy

and Eberhart

R.C.

, Particle swarm optimization, IEEE Conference on Neural Network 4 (1995), 1942–1948.

11.

Jones

K.O.

, Comparison of genetic algorithm and particle swarm optimization, International Conference on Computer Systems and Technologies 3 (2005), A1.1–A.1.6.

12.

Baykasoglu

and Gocken

, Solving fully fuzzy mathematical programming model of EOQ problem with a direct approach based on fuzzy ranking and PSO, Journal of Intelligent & Fuzzy Systems 22(5) (2011), 237–251.

13.

Bingül

and Karahan

, A fuzzy logic controller tuned with PSO for 2 DOF robot trajectory control, Expert Systems with Applications 38(1) (2011), 1017–1031.

14.

Mojarrad

H.D.

, Rastegar

and Gharehpetian

G.B.

, Probabilistic interactive fuzzy satisfying generation and transmission expansion planning using fuzzy adaptive chaotic binary PSO algorithm, Journal of Intelligent & Fuzzy Systems 30(3) (2016), 1629–1641.

15.

Premkumar

and Manikandan

B.V.

, GA-PSO optimized online ANFIS based speed controller for brushless DC motor, Journal of Intelligent & Fuzzy Systems 28(6) (2015), 2839–2850.

16.

Lin

C.M.

, Lin

M.H.

and Chen

C.W.

, SoPC based adaptive PID control system design for magnetic levitation system, IEEE Systems Journal 5(2) (2011), 278–287.

17.

Y.Z.

, Yan

and Levy

, Chaotic particle swarm optimization for optimal design of PID controllers in industrial systems, Journal of Intelligent & Fuzzy Systems 30(5) (2016), 3007–3016.

18.

Wang

S.C.

and Liu

Y.H.

, A PSO-based fuzzy-controller searching for the optimal charge pattern of li-ion batteries, IEEE Trans. Industrial Electronics 62(5) (2015), 2983–2993.

19.

Kapoor

and Ohri

, Improved PSO tuned classical controllers (PID and SMC) for robotic manipulator, IJ Modern Education and Computer Science 1 (2015), 47–54.

20.

Erdem

and Altinoz

O.T.

, Tuning of output scaling factor in PI-like fuzzy controllers for power converters using PSO, Journal of Intelligent & Fuzzy Systems 30(5) (2016), 2677–2688.

21.

Ardjani

and Sadouni

, Optimization of SVM multiclass by particle swarm (PSO-SVM), IJ Modern Education and Computer Science 2 (2010), 32–38.

22.

J.Z.

, Wu

Z.X.

, Wang

and Tan

, CPG network optimization for a biomimetic robotic fish via PSO, IEEE Trans Neural Networks and Learning Systems 27(9) (2016), 1962–1968.

23.

Calvini

, Carpita

, Formentini

and Marchesoni

, PSO-based self-commissioning of electrical motor drives, IEEE Trans Industrial Electronics 62(2) (2015), 768–776.

24.

Zhang

, Xian

and Ma

, Continuous robust tracking control for magnetic levitation system with unidirectional input constraint, IEEE Trans Industrial Electronics 62 (2015), 5971–5980.

25.

Tao

and Su

S.F.

, Moment adaptive fuzzy control and residue compensation, IEEE Trans Fuzzy Systems 22(4) (2014), 803–816.

26.

Mohammad

E.B.

and Ahmad

M.E.N.

, Interval type-2 fuzzy PID controller for uncertain nonlinear inverted pendulum system, ISA Transactions 53 (2014), 732–743.

27.

Slotine

J.J.E.

and Li

, Applied nonlinear control, Prentice-Hall, Englewood Cliffs, New Jersey, 1991.

28.

Wang

C.H.

, Liu

H.L.

and Lin

T.C.

, Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems, IEEE Trans Fuzzy Systems 10(1) (2002), 39–49.

29.

Yoo

B.K.

and Ham

W.C.

, Adaptive fuzzy sliding mode control of nonlinear system, IEEE Trans Fuzzy Systems 6(2) (1998), 315–321.