An intelligent CMAC-PD torque controller with anti-over-learning scheme for electric load simulator

Abstract

The hybrid architecture of a cerebellar model articulation controller (CMAC) and proportional derivative (PD) can effectively reduce the loading error and restrain the surplus torque of electric load simulators. However, due to the over-learning problem of the CMAC, the practical application of the CMAC-PD for electric load simulators is greatly constrained. This paper analyses the over-learning problem of CMAC and proposes a novel learning scheme including control error and an inhibiting over-learning item. An intelligent CMAC-PD controller with novel anti-over-learning scheme is derived by using the gradient descent algorithm. Both simulation and experimental results demonstrate that the proposed CMAC-PD hybrid controller has good robustness, can effectively eliminate disturbances and restrain the over-learning phenomenon of the CMAC.

Keywords

Cerebellar model articulation controller (CMAC)electric load simulator hybrid controller learning scheme over-learning torque control

Introduction

The electric load simulator (Figure 1) is an electric torque servo system, which can reproduce the aerodynamic hinge moment of control surfaces during flight testing of unmanned aerial vehicles (UAVs) under laboratory conditions. Nowadays, the brushless torque motor has become the mainstream choice in electric load simulators with the advantages of small volume, low inertia, fast response and low maintenance (Yang and Han, 2014).

Figure 1.

Block diagram of electric load simulator (top view) (Yang and Han, 2014).

As a torque control system, the electric load simulator is a typical passive loading system, which has the surplus torque generated on the shaft by the active motion of actuator when the loading instruction is zero and no compensation or control measures are taken. Conventional control methods include the structure invariance principle (Alberto et al., 2007), synchronous compensator (Wei et al., 2008), adaptive control (Pradhan and Subudhi, 2012) and H _∞ control (Hui et al., 2012), etc., which all need a relatively accurate mathematical model. Considering the intrinsic non-linear factors like friction, mechanical clearance and interference of surplus torque in electric load simulators, a new control strategy is required. Due to the advantages of quick learning, a simple structure, a non-existent local optimum and good real-time performance, the cerebella model articulation controller (CMAC) has found successful applications in control problems, model-free function approximation, robot localization and fault diagnosis (Almeida and Simoes, 2005; Bahuguna et al., 2009; Hsu et al., 2009; Pa et al., 2013; Yu et al., 2008). The most commonly used control architecture is a hybrid controller of CMAC and PD proposed by Miller et al. (1990). The hybrid controller can effectively reduce the loading error and restrain the surplus torque of electric load simulators, but the over-learning of the CMAC will occur due to cumulative errors (Chen and Chang, 1996; Jiang and Lin, 2000) when a CMAC is applied to control systems with a continuously variable signal as the input signal.

Current research is mainly focused on the convergence of the CMAC for function approximation (Lin and Chiang, 1997; Teddy et al., 2007), but our simulation showed that the convergence conditions on the learning rate of the CMAC for function approximation does not apply to the control system. Although the existing literature gave various improvements on CMAC (Cao and Tu, 2012; Cheng, 2011; Teddy et al., 2008), most of them focus on theoretical analysis and simulation. These methods still cannot meet the requirements of practical implementation of a CMAC-PD controller for electric load simulators due to the over-learning problem of the CMAC. The adaptive laws of the CMAC are also used to guarantee the stability of control system (Hsu, 2012; Lin and Li, 2013; Lin and Peng, 2005). However, some system dynamics are simplified and ignored for deriving the adaptive laws and only simulation results were given.

This paper proposes an intelligent CMAC-PD hybrid torque controller with a novel anti-over-learning scheme for electric load simulators. The performance of the proposed intelligent controller is verified by both simulation and experiment. The paper is organized as follows. The next section presents the mathematical model of the electric load simulator. Design of the intelligent CMAC-PD hybrid controller with the novel learning scheme is given and simulation results of a conventional CMAC-PD controller and the improved one are provided. Experimental implementation is shown and finally some conclusions are presented.

Mathematical model of electric load simulator

The simulation model of electric load simulator was presented by Han and Yang (2013) and Yang and Han (2014), and is shown in Figure 2.

Figure 2.

Mathematical model of electric load simulator.

The transfer function of electric load simulator is given by

\begin{matrix} \frac{T_{f}}{U_{in}} = \\ \frac{K_{v} K_{c} K_{m} K_{f}}{JL s^{3} + (BL + JR + J K_{c}) s^{2} + (BR + B K_{c} + K_{m} K_{e} + K_{f} L) s + K_{f} R + K_{c} K_{f}} \end{matrix}

(1)

where $U_{in}$ denotes the input voltage and $T_{f}$ denotes the load torque of motor. $θ_{r}$ is the angular velocity disturbance of steering engine. $K_{v}, K_{c}, K_{e}, K_{m}$ and $K_{f}$ are proportional coefficients.

The main parameters are listed in Table 1.

Table 1.

Main parameters of electric load simulator.

Moment of inertia J	0.0010 kg·m²
Armature resistance, R	3.17 Ω
Torsional stiffness of torque sensor, K_f	46.11 N·m/rad
Coefficient of torque, K_m	1.18 N·m/A
Armature inductance, L	0.041 H
Damping coefficient of motor, B	0.35 N·m·s

The transfer function of electric load simulator can be expressed as

\frac{T_{f}}{U_{in}} = \frac{3433}{4.09 \times 10^{- 5} s^{3} + 0.10 s^{2} + 32.04 s + 3835}

(2)

Design of intelligent CMAC-PD hybrid torque controller

Structure of CMAC-PD torque controller

The structure of the CMAC neural network is composed of the input space, associate memory, physical memory and network output (Figure 3). The input state space X is multidimensional. Each input state variable is quantized and mapped in the virtual associative memory space. The entire associative memory space could be regarded as a discrete data set $A$ . With virtual to physical memory mapping, all active memory cells in $A$ are mapped to another discrete data set $w$ , which stands for the physical memory space. Then, the CMAC sums the mapped data up as its output and feeds the error between the actual and the desired outputs ( $u_{n}$ and d) back to the physical memory cells equally, which is so-called weight adjustment.

Figure 3.

The structure of the cerebellar model articulation controller (CMAC) neural network.

The structure of the CMAC-PD torque controller is shown in Figure 4. The CMAC controller and PD controller run in parallel. The CMAC realizes the feed-forward control to guarantee the response speed, tracking effect and reduce the control overshoot; the PD controller is used to improve control performance. With the conduct of the CMAC learning process, the PD output gradually decreases while the CMAC output increases and dominates the control output. The output of the intelligent controller in Figure 4 can be given by

u = u_{p} + u_{n}

(3)

Figure 4.

Structure of the cerebellar model articulation controller–proportional derivative (CMAC-PD) hybrid torque controller.

where $u_{p}$ is the output of PD controller and $u_{n}$ is the output of the CMAC.

Over-learning problem of CMAC

It is commonly considered that the interaction of the CMAC and PD may be the cause for the over-learning of the CMAC. We analyse both the CMAC-PD hybrid controller (Figure 4) and the standalone CMAC controller (Figure 5). The loading instruction follows $r_{in} = 10 \sin (2 π \cdot f \cdot n \cdot T_{s}) N \cdot m$ , where the loading frequency $f$ is 1 Hz, the sample time $T_{s}$ is 0.001 s and $n$ represents the simulation step.

Figure 5.

Standalone cerebellar model articulation controller (CMAC) controller.

The design parameters of controller are listed in Table 2. Two weights, No. 30 and No. 50, were randomly selected from the CMAC-PD controller and the standalone CMAC controller.

Table 2.

Design parameters of controller.

CMAC	Quantization parameter, N	100
	Generalization parameter, C	15
	Learning rate, $η$	0.2
PD	Proportional gain, $K_{p}$	3
	Differential gain, $K_{d}$	0.1

CMAC, cerebellar model articulation controller; PD, proportional derivative.

Figures 6(a) and 7(a) show the variation of the selected weight values respectively. Figures 6(b) and 7(b) depict their control error curves. The absolute values of No. 30 and No. 50 CMAC weights of both the CMAC-PD controller and the standalone CMAC controller rise steadily, which lead to the over-learning of CMAC. They follow the same error trend, despite the fact that CMAC combined with PD can strengthen the system stability and increase the convergence speed to some extent compared with the standalone CMAC, but the over-learning of the CMAC will still be inevitable and lead to system instability with the increasing cumulative errors.

Figure 6.

Control results of the cerebellar model articulation controller–proportional derivative (CMAC-PD) controller: (a) variation of No. 30 and No. 50 CMAC weights; (b) control error curve.

Figure 7.

Control results of standalone cerebellar model articulation controller (CMAC) controller: (a) variation of No. 30 and No. 50 CMAC weights; (b) control error curve.

We can conclude that the interaction of the CMAC and PD is not the cause of over-learning and the major cause depends on the CMAC itself. Only the over-learning of the standalone CMAC will be analysed.

The main direct factors affecting the CMAC control are the learning rate and the weight updating method.

Many researchers use the matrix theory and the general convergence principle of iterations for linear equations to analyse the convergence of the CMAC, and obtain the condition on the learning rate $0 < η < 2$ , guaranteeing the convergence when the input vector of the CMAC is one-dimensional (Chen and Chang, 1996; Cheng, 2011). The error is defined as

e = d - Aw = d - u_{n}

(4)

where $d$ is the input of the CMAC (Figure 5), $A$ is the association matrix, $w$ is the weight memory space and $Aw$ is the output of the CMAC, i.e. $u_{n}$ in Figure 5.

However, when a CMAC is applied to an actual control system (Figure 5), the control error is changed to

e = d - y

(5)

where y is the output of the control plant, not the CMAC (Figure 5). So the convergence condition of $0 < η < 2$ may not be applied to the actual control system. We find the control will still be divergent, even if the learning rate meets the condition $0 < η < 2$ .

We use a Lyapunov function to analyse theoretically the convergence of CMAC with the system error in Equation (5).

A Lyapunov function is commonly defined as

V (k) = \frac{1}{2} e^{2} (k)

(6)

The weights updating formula of conventional CMAC is expressed as

w (k + 1) = w (k) + \frac{η}{C} e (k) A^{T} (k)

(7)

and the error difference can be obtained by

\begin{matrix} Δ e (k) & = e (k + 1) - e (k) \\ = [\frac{\partial e (k)}{\partial w}]^{T} Δ w (k) \\ = [- y_{u} (k) \frac{\partial u_{n} (k)}{\partial w (k)}]^{T} Δ w (k) \\ = [- y_{u} (k) A^{T} (k)]^{T} \frac{η}{C} e (k) A^{T} (k) \\ = - y_{u} (k) η e (k) \end{matrix}

(8)

where $u_{n} (k)$ is the output of CMAC and $y_{u} (k) = \partial y (k) / \partial u_{n} (k)$ represents the sensitivity of the plant with respect to its input.

Then,

\begin{matrix} Δ V (k) & = V (k + 1) - V (k) \\ = Δ e (k) [e (k) + \frac{1}{2} Δ e (k)] \\ = - y_{u} (k) η e (k) [e (k) - \frac{1}{2} y_{u} (k) η e (k)] \\ = - \frac{1}{2} y_{u} (k) η e^{2} (k) [2 - y_{u} (k) η] \end{matrix}

(9)

Theoretically, if $y_{u} (k)$ and $η$ can meet the condition $0 < y_{u} (k) < 2 / η$ , then $Δ V (k) < 0$ and the convergence of the CMAC could be guaranteed, but $y_{u} (k)$ can be positive or negative and $η$ is positive. This implies the commonly used Lyapunov function in Equation (6) is not valid for our system.

A valid Lyapunov candidate function should include the tracking error, the weight estimation error and all other dynamics included in the electric load simulator. As a passive loading control system moved along with the actuator, however, the dynamics in the load simulator is difficult to describe mathematically. As a result, we will improve the weight updating method of the CMAC to resolve the over-learning instead of analysis of the learning rate.

Novel learning scheme of CMAC

We proposed a CMAC-PD torque controller with a fast learning capacity and improved output smoothness for an electric load simulator (Yang and Han, 2014). A novel scalar cost function of the CMAC consisted of an error item and a weight-smoothing item to guarantee a fast learning capacity and improve output smoothness of the CMAC. Although the stability of the CMAC was guaranteed at the same time, the weight updating method of the CMAC updates all weights each sampling time. This means that the improved CMAC is no longer a local neural network. The matrix operation is very time consuming and influences the real-time performance of control.

The learning scheme of the CMAC is derived from the scalar cost function. In order to minimize the system error, limit the output of CMAC and avoid the over-learning a novel scalar cost function is proposed as

E = \frac{1}{2} e^{2} + \frac{1}{2} {(d - u_{n})}^{2}

(10)

where $e$ represents the torque tracking error of electric load simulator in our case, $u_{n}$ is the output of CMAC and $d$ is the loading instruction and the input of CMAC (Figure 4). This scalar cost function implies that the system error $e$ can be minimized and that, at the same time, the difference between the loading instruction $d$ and the output of the CMAC $u_{n}$ can be minimized. Thus, $u_{n}$ is restricted to follow the input instruction.

The second item in Equation (10) can also be interpreted as the difference between input and output energy of the CMAC. If we can keep their balance, then the over-learning problem of CMAC will be restrained. The conventional CMAC, by contrast, has only an error item in the scalar cost function. Its output is not restricted and is easy to diverge due to cumulative errors.

Using the gradient descent algorithm, the novel weights updating formula can be derived as

w (k + 1) = w (k) + \frac{η}{C} (e - u_{n} + d) A^{T}

(11)

This novel updating formula can not only avoid the over-learning of the CMAC, but also improve the learning speed. Compared with the updating method in Yang and Han (2014), this improved CMAC is still a local neural network, which can enhance the real-time ability.

Simulation results and analysis

A comparative simulation of the CMAC-PD hybrid controller is performed using a conventional CMAC learning scheme (Equation 7) and novel scheme (Equation 11).

The design parameters of controller are listed in Table 2. Considering the quantization precision and calculation, N=100 can satisfy our requirements. In addition, a large generalization parameter will make the mapping less accurate while a small generalization parameter will lose the generalization ability (Smith, 1998), C=15 makes the activated cells occupy about 13% of the memory space (N+C=115), so that it is available to act as the generalization parameter. The learning rate of the CMAC is between 0 and 2 (Smith, 1998). The parameters of PD are chosen following the control variables method, first the proportional gain K_p then the differential gain K_d .

The loading instruction follows $r_{in} = 10 \sin (2 π \cdot f \cdot n \cdot T_{s}) N \cdot m$ with a loading frequency of 5 Hz. The sample time $T_{s}$ is 0.001 s and $n$ represents the simulation step.

Tracking performance

Figure 8 shows the tracking error of different controllers. The maximum absolute values of error in each cycle are used, which contains 1/(f·Ts) simulation steps. The maximum absolute value of errors in the first cycle of the CMAC-PD controller using a conventional learning scheme and a novel one are 1.7340 N·m, and 1.1912 N·m, respectively. The error of the conventional CMAC-PD controller can converge to 0.7065 N·m at first, but it diverges quickly. The error of the CMAC-PD controller using the novel learning scheme can converge to 0.6794 N·m. Simulation results show that the novel learning scheme can effectively improve the learning speed and control precision compared with the conventional one.

Figure 8.

Tracking errors of the cerebellar model articulation controller–proportional derivative (CMAC-PD) controller with different learning schemes: (a) using the conventional learning scheme; (b) using the novel learning scheme.

Inhibiting effect of over-learning

To test the inhibiting over-learning effect of the CMAC-PD controller using the novel CMAC learning scheme, a long period simulation test is taken. The results are shown in Figure 9. The novel learning scheme could make the weights fluctuate in a certain range (Figure 9a). Thus, the error will be convergent (Figure 9b). Then, the novel learning scheme can effectively restrain the over-learning phenomenon.

Figure 9.

Inhibiting over-learning effect of the novel learning scheme: (a) variation of the cerebellar model articulation controller (CMAC) weights; (b) control error curve.

In conclusion, simulation results have shown that the CMAC-PD controller using the novel learning scheme can not only improve control precision and learning speed but also effectively inhibit the over-learning phenomenon.

Experimental Implementation

The proposed control solution is conducted at a test bed in the laboratory (Figure 10). The test bed mainly consists of a brushless DC torque motor, a processor board DSP F28335, a torque sensor, a motor driver and a control computer. As the core of control, a DSP F28335 processor is very suitable for motor control due to its good performance: core frequency up to 150 MHz, high-performance static CMOS technology, a high-performance 32-Bit CPU and 12-Bit ADC, etc.

Figure 10.

Block diagram of electric load simulator.

The parameters of PD are $K_{p}$ =2.2 and $K_{d}$ =0.1; the parameters of CMAC are selected as N=100, C=15 and $η$ =0.1. The loading instruction follows $r_{in} = 5 \sin (2 π \cdot f \cdot t) N \cdot m$ with the loading frequencies of 0.5, 1.0, 1.5, 2.0, 2.5 and 3 Hz, respectively.

The experimental results of the CMAC-PD controller using different learning schemes are shown in Figures 11 –13. For the CMAC-PD controller using a conventional learning scheme, the maximum absolute value of error in each cycle converges to 0.57, 0.60, 0.86 and 1.02 N·m at the loading frequency of 0.5, 1.0, 1.5 and 2.0 Hz, and the system diverges at frequencies of 2.5 and 3.0 Hz. For the CMAC-PD using the novel learning scheme, the maximum absolute value of error in each cycle converges to 0.48, 0.54, 0.73, 0.79, 1.05 and 1.07 N·m at a frequency of 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0 Hz, respectively. The tracking accuracy changes with the frequency of the input signals. The experimental results demonstrate that the novel learning scheme can improve learning speed and control precision compared with the conventional one.

Figure 11.

Tracking results of the cerebellar model articulation controller–proportional derivative (CMAC-PD) controller at loading frequency of 1 Hz: (a) using the conventional learning scheme; (b) using the novel learning scheme.

Figure 12.

Tracking results of the cerebellar model articulation controller–proportional derivative (CMAC-PD) controller at loading frequency of 2 Hz: (a) using the conventional learning scheme; (b) using the novel learning scheme.

Figure 13.

With loading instruction $r_{in} = 5 \sin (2 π \cdot 1 \cdot t) N \cdot m$ and no disturbances, the experimental result of the CMAC-PD controller using a conventional learning scheme is shown in Figure 14. After 80 s, the system begins to diverge due to over-learning of the CMAC. With the same loading instruction $r_{in} = 5 \sin (2 π \cdot 1 \cdot t) N \cdot m$ and four times the torque disturbances, the experimental result of the CMAC-PD controller using the novel learning scheme is shown in Figure 15. The proposed control algorithm can effectively inhibit disturbances and over-learning phenomenon, and no divergence occurs during the long period loading.

Figure 14.

Over-learning phenomenon of the cerebellar model articulation controller–proportional derivative (CMAC-PD) controller using the conventional learning scheme.

Figure 15.

Inhibiting over-learning and disturbances test of the cerebellar model articulation controller–proportional derivative (CMAC-PD) controller using the novel learning scheme.

Conclusions

In this paper, we proposed an intelligent CMAC-PD hybrid torque controller with a novel anti-over-learning scheme for electric load simulators. The novel learning scheme is derived from a new scalar cost function, which contains an error item and an inhibiting over-learning item, by using the gradient descent algorithm.

The simulation and experimental results have shown that the CMAC-PD controller using the novel learning scheme can effectively inhibit over-learning of the CMAC, improve learning speed, eliminate disturbances and ensure control accuracy. The proposed control strategy is also suitable for real-time control of other non-linear systems.

Footnotes

Declaration of conflicting interest

The authors declare that there is no conflict of interest.

Funding

The authors would like to express their gratitude for the financial support by National Natural Science Foundation of China (No. 51277008).

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