Anticipatory iterative learning control for linear motor positioning accuracy by use of hybrid particle swarm optimization

Abstract

The conventional Iterative Learning Control (ILC) process can potentially excite rich frequency contents based on past error history and injects them into the updated learning process. Nevertheless, the learnable error signals should be extracted, and the non-learnable error signals should be separated by adaptive bandwidth filter before the updated command is injected into next repetition. This paper proposes a new particle swarm optimization (PSO) algorithm by combining several hybrid terms adopted from literatures for the new synergy properties of local and global searches based on cognitive and social terms. This algorithm is used to adjust the three proportional– integral– derivative controller gains, the Anticipatory Iterative Learning Control (AILC) learning gain, and the cutoff bandwidth of the Butterworth filter associated with AILC. Developed hybrid PSO algorithm is devised for the updating velocity terms. The new synthesis AILC control law with adaptive learning gain, bandwidth-tuning filter and PID control gains were optimized by PSO and facilitated the improvement of the learning process and positioning accuracy. Numerical simulations were conducted and compared with the literature’s P-type AILC for a linear synchronous motor. The tracking error through the adaptive learning processes was reduced successfully by shaping a new updated input trajectory at every repetition. The experimental results confirm the effectiveness of the new PSO– AILC for ultra-fine positioning in a one-axis linear motor.

Keywords

Anticipatory iterative learning control linear synchronous motor particle swarm optimization ultra-fine positioning

1 Introduction

Proportional– Integral– Derivative control was a well known conventional positioning controller design and used practically in industrial applications. The performance response of a PID controller that using time or frequency domain shaping techniques may be satisfied in static error compensation but is not necessarily the best when system dynamics requires high performance. Since any constraint on settling time, overshoot/undershoot, disturbances and change of dynamic characteristics is system dependent. Iterative learning control (ILC) has been used for over several decades. It is good for some unknown systems that render the system-to-be-controlled as the black box and then operate in a repetitive manner. Its scheme is based on a batch updated control input shaped trajectory and the generation of applied learning signals in current cycle. It can be implemented online or obtained offline using error information from previous repetitions. It intends to iteratively reduce the tracking error over a finite time interval by previous experience into the next input. Hence, noncausal computations are allowed in terms of time based on previous cycles. Published works are referred as learning compensators in front of the law of the ILC. That purpose is to fix the phase-delay problem caused by non-learnable signals of an original ILC for bettering high-frequency response thinking. The first one is the linear phase-lead learning compensator that provides linear increase in phase compensation with frequency, which is excessive at high frequency. Thus, an appropriated and well devised cutoff bandwidth in front of the ILC’s law must be introduced to stop the phase compensation after some high frequencies regime. A linear anticipatory operator developed by Wang was adopted [2]. The AILC features phase-lead characteristics and compensated phase-lag characteristics of a process. However, the learning gain with associated bandwidth filtering could be updated by the next iteration in Huang et al. [14] since the updated command should be shaped before injecting into next repetition. The second one is the use of partial isometry of the Markov parameter matrix system, which is calculated by singular value decomposition in Jang et al. [3]. The computation burden of the learning law is heavy if the matrix size is large. The third is the phase-cancelation learning law that constructs its learning compensator (matrix) by calculating the inverse discrete Fourier transform from the phase characteristics of the plant. The intense computation in real-time implementation is a burden. Computational enhancement was proposed in Lee-Glauser et al to reduce the calculation cost [4].

Essentially the ILC learning has the potential to excite rich frequency contents and try to learn the error signals. Error compensation based on ILC from the time domain to frequency-response thinking has been studied over three decades [2 , 16]. However, the learnable and unlearnable error signals should be separated. New morphing input command should be a better shaped control signal for next repetition. Before the phase margin causes any trouble that is producing a high frequency error condition should be avoided. Therefore, injected learnable error signals through an adaptive bandwidth tuning should be facilitated in devised learning law at every repetition. Thus the tracking error will be effectively reduced as repetitions. The cutoff filter bandwidth associated with the ILC’s updated error history should be manipulated at every repetition. This achieves the improved shaped input of compensated errors at frequency response thinking. Therefore, the adaptive learning gain, three PID gains and suitable cutoff bandwidth filter by the aid of parameter optimization method at every repetition time was not comprehensively discussed and implemented. In this paper with the aid of hybrid particle swarm optimization (PSO) terms is applied for synergistic effects for positioning accuracy of a linear synchronous motor AILC system.

Kennedy and Eberhart first introduced the scheme of particle swarm optimization [7]. It was developed through simulation of a simplified swarm social behavior and was robust in solving continuous nonlinear optimization problems. PSO is one of the population-based optimization algorithms as a swarm intelligence technique. It is fascinated by much attention in recent years due to the following features. First, the algorithm requires only a few lines of coding. Second, its search technique uses the values of the objective function instead of gradient information. Third, it is computationally inexpensive due to its very low memory and CPU speed requirements. Fourth, compared with conventional deterministic methods, it can efficiently solve a problem without requiring linearity, differentiability, or other assumptions. Finally, its solution does not depend on the initial states of the particles, which is advantageous in nonlinear engineering system problems based on optimization approaches.

Updating the velocity term of PSO algorithm was with three terms. Shi and Eberhart proposed an empirical study by linear weighting term on the first velocity term [8]. There are another two different coefficient terms for particle’s updating velocity terms based on the advantages of the contribution and modulation of diverse ability for local and global searches. These second and third term corresponds to the function of cognitive and social meaning components. The cognitive component (the second term) represented the personal thinking of each particle and encouraged the particle to move toward their own best position. The social component (the third velocity term) represented the collaborative effect of the particles in finding the global optimal solution. It contributed the continual effect by pulling the particles toward the global best particle.

Although in 2004, Retnaweera et al. proposed time varying acceleration coefficients instead of constant coefficients in front of the second and third velocity terms [9]. Nevertheless, the particle population may lose diversity and is more likely to confine the search around local minima if committed too early in global search. Therefore, introducing another extra updating velocity term was proposed by He et al. [10]. They formulated the fourth term using the contribution of a random particle in the updating velocity terms. The introduced passive congregation was from a biological thinking. It is an important biological force that preserves swarm integrity and the information can be transferred among individuals in the swarm. The reason for the addition of this fourth term is that the information sharing among conspecifics may not be broadcast by the global best position.

This study adopts aforementioned algorithm concept for PSO and devises a new hybrid velocity terms to tune the gains of the PID controller, AILC, and the Butterworth filter bandwidth at every repetition. The root mean square (RMS) tracking error serves as the minimization criterion by launching PSO mechanism at every AILC repetition. Rejected non-learnable error signals facilitate the decreased error at the next repetition by changed Butterworth filter bandwidth at every AILC repetition. Thus, improving the learning control process using adaptive cutoff bandwidth tuning by PSO is promising from repetition to repetition and will demonstrate the superiority over the fix gain AILC. Developed hybrid PSO algorithm with the structure of A-type ILC controller termed as HPSO-AILC will numerically simulate first and compare with the P-type AILC termed as HPSO-PILC. Reducing the error frequency response starting from lower frequency to higher frequency by consecutive learning repetition based on the PSO algorithm was effective and successful. Finally, an in-lab experiment for a signal axis linear motor system will be verified to reach the good tracking accuracy in time response. Improved precise positioning for a signal linear motor axis in real-time experimentation is demonstrated successfully.

2 Theoretical PSO background with hybrid technique using extra term and acceleration coefficients

2.1 Basic PSO

The PSO algorithm originates the simulation for the behavior of a swarm such as a flock of flying birds as a simplified social system. Each individual is a particle and is treated as a point in a multi-dimensional space. Each particle aiming a target changes its position based on the current velocity, position, and its best previous position. Adjusting the position is based on its own experience and the experiences of its neighbors and among the group. Heuristically, each particle remembers its past experience, and it can effectively discover the search space to recognize the most promising solution state. The conventional PSO algorithm in Kennedy and Eberhart is expressed as follows [7]

$\begin{matrix} V_{id} (t + 1) = w (t) \times V_{id} (t) \\ + c_{1} \times Rand () \times (P_{id} - X_{id} (t)) \\ + c_{2} \times Rand () \times (P_{gd} - X_{id} (t)) \end{matrix}$ (1) $X_{id} (t + 1) = X_{id} (t) + V_{id} (t + 1),$ (2) where V_id (t) is the current velocity of the ith particle; i = 1, …, n, n is the population size. A parameter called inertia weight w (t), shown in front of the first term on the right-hand side, is adopted to balance the global and local search abilities. Rand () represents a uniform random number between zero and one, and subscript d represents the dimension of the particle. P_id represents the best previous position of the ith particle, P_gd represents the best previous position among all particles in the swarm, and X_id represents the current position of the ith particle. Second constant c₁ and third constant c₂ associated with Rand () represent the weight of the stochastic acceleration terms. Both constants stand for the pull of each particle toward the P_id and P_gd positions, respectively. Low values allow particles to roam far from the target regions before being tugged back. Alternatively, high values result in rapid movement toward or past the target regions. w (t) is defined as in Shi et al. [8].

$\begin{matrix} w (t) = w_{min} + \frac{{iter}_{max} - k_{search - {iteration}^{th}}}{{iter}_{max}} \\ (w_{max} (t) - w_{min} (t)), \end{matrix}$ (3) where iter_max represents the maximum iteration, k_{search-iteration^th} is the current search iteration. A small inertia weight facilitates a local search, whereas a large inertia weight facilitates a global search. By decreasing the inertia weight from a relatively large value to a small value, the PSO tends to have a greater global search ability at the beginning of the run and greater local search ability near the end of the run. This research puts w_max (t) to be 1.0 and w_min (t) to be 0.2. Here, the inertia weight value will decrease linearly as repetition goes.

2.1 2.2. Modified hybrid PSO with extra roaming contribution and acceleration coefficient on velocity term

To optimize the bettering parameters for AILC control algorithm, hybrid PSO by adding the fourth term with extra roaming contribution for velocity search and acceleration coefficient on on each velocity term are described as follows.

The first current velocity term (V_id (t)) in Equation (1) by the conventional PSO algorithm is multiplied by the inertia weight as in Equation (3).

By Ko at al. [13], two constants of c₁ and c₂ in Equations (4) and (5) are proposed to be time-varying acceleration coefficients in the second and third velocity terms respectively. The concept of time-changeable acceleration coefficients on the updating velocity terms form was proposed early in 2004 by Retnaweera et al [9]. As for c₁ and c₂, rather than using fixed gain PSO, two terms of Equations (4) and (5) represent the cognitive and social components as follows [13]. $c_{1} = c_{1 \min} + {(\frac{{iter}_{\max} - k}{iter})}^{α} \times (c_{1 \max} - c_{1 \min})$ (4) $c_{2} = c_{2 \max} + {(\frac{{iter}_{max} - k}{iter})}^{β} \times (c_{2 mix} - c_{2 \max})$ (5) where the nonlinear power coefficients of α and β are real number. A note is made here that with constant value by 1 for the each α and β coefficient, this corresponds to the special case in Ratnaweera’s study [9].

Besides, He et al. [10] introduced the fourth term with c₃ × Rand () × (R_id - X_id (t)).

This concerns the contribution of a random particleR_id on the updating velocity term. The reason of adding this passive congregation term to the standard PSO updating velocity term is that the information can be transferred among individuals of the swarm. Therefore, the hybrid velocity updating formula for Equation (1) by using the time-varying acceleration in Equations (4), (5) and (8) are integrated as following:

$\begin{matrix} V_{id} (t + 1) = w (t) \times V_{id} (t) \\ + c_{1} \times Rand () \times (P_{id} - X_{id} (t)) \\ + c_{2} \times Rand () \times (P_{gd} - X_{id} (t)) \\ + c_{3} \times Rand () \times (P_{id} - X_{id} (t)) \end{matrix}$ (6) $X_{id} (t + 1) = X_{id} (t) + V_{id} (t + 1),$ (7) where $c_{1} = c_{1 \min} + {(\frac{{iter}_{max} - k}{iter})}^{α} \times (c_{1 \max} - c_{1 \min})$ $c_{2} = c_{2 \max} + {(\frac{{iter}_{max} - k}{iter})}^{β} \times (c_{2 mix} - c_{2 \max})$ $c_{3} = c_{3 \min} + {(\frac{{iter}_{max} - k}{iter})}^{γ} \times (c_{3 \max} - c_{3 \min})$ (8)

In summary, in this paper, the PSO represented in Equations (6)–(8) is faciliated for the basic PSO [7] for cognitive and social search with introducing a random particle technique by [9]. The new formulated coefficients of the constants c₁, c₂ and c₃ in Equation (8) stand for: (1) the initial hign gain (c_1max) with decreasing gain for the first cognitive component in local searching, (2) the initial low gain (c_2max) with increasing gain for second social component in global searching, and the initial high gain (c_3max) with decreasing gain for the fourth passive congregation term of the velocity updating formula, respectively. In this research, we set the nonlinear power coefficients of α, β and γ are all equal to 1. Since our preliminary numerical simulation for tracking position accuracy of a simple synchronous linear motor system shows that the root means square error and tracking convergence by using Equations (6)–(8) are better than using Equations (1) and (2).

In summary, this study proposes a hybrid but new PSO calculation algorithm from the original form of Equation (1), develops the concept of time-varying acceleration coefficients on the updating velocity terms from Equations (4) and (5) [13], formulates an extra updating velocity term adopted in Equation (8), and traces the bounded constraints based on the experimental experience in Huang and Lee [12]. The synergy of the developed hybrid PSO velocity terms will optimizes the gains and cutoff frequency bandwidth tuning in a better manner because the positioning error is featured with rich frequency contents at every repetition. Adaptive parameters tuning by proposed PSO for AILC is adopted. Improvement in AILC morphing command for an in-lab synchronous linear motor system will be demonstrated with the aid of proposed hybrid PSO.

3 Controller

3.1 Basic PID controller with associated PSO’s gains

The traditional PID controller has the following form in the time domain:

$u (t) = k_{p} e (t) + k_{i} \int_{0}^{t} e (t) dt + k_{d} \frac{de (t)}{dt},$ (9)

where e (t), u(t), k_p, k_i, and k_d represent the feedback error, control variable, proportional gain, integral gain, and derivative gain, respectively. Because each coefficient of the PID controller adds some special effects to the output response of the feedback system, varying well-tuned gains through the PSO will provide dynamics errors compensation in practice.

3.2 A-type ILC with variant butterworth low-pass filter bandwidth tuning by PSO

A state space representation for a linear time-invariant continuous system can be model as: ${\begin{matrix} \dot{x} (t) = Ax (t) + Bu (t) + w (t) \\ y (t) = Cx (t) + v (t) \end{matrix}$ (10) where x represents n-dimensional state space matrix, u is the input command, y is the output, and w and v are the disturbances. The output of the j^th iteration is converted via the Laplace transform as follows.

$\begin{matrix} Y_{j} (s) & = G_{p} (s) U_{j} (s) + C {(sI - A)}^{- 1} x (0) \\ + C {(sI - A)}^{- 1} W (s) + V (s), \end{matrix}$ (11) where G_p (s) = C (sI - A) ^-1B is the system transfer function, x (0) is the initial position of the system. In practice, the initial condition will not be the same when repetitive motion control is implemented. The trajectory error of the jth iteration is E_j (s) = Y_d (s) - Y_j (s), where Y_d (s) is the desired Laplace transform of the output y_d (t) in finite time [0, T].

A simple but effective A-type ILC law [2] in discrete time can be represented as $u_{j} (kT) = u_{j - 1} (kT) + φ e_{j - 1} ((k + Δ) T)$ (12) $e_{j - 1} (kT) = y_{d} (kT) - y_{j - 1} (kT),$ (13) where u_j (kT) represents the input command at time k and iteration j, e_j-1 (kT) is the difference between the actual and desired output at time k and iteration j - 1, Δ is the lead time, and φ is the learning gain. Most often Δ is equal to one with one time step delay in the output y_j-1 ((k + 1) T) at time (k + 1) T with sampling time T.

The Laplace transform of Equation (12) in continuous domain is as follows. $U_{j} (s) = U_{j - 1} (s) + φ e^{Δ s} E_{j - 1} (s)$ (14)

Using Equations (11) and (14) they yields $\begin{matrix} Y_{j} (s) - Y_{j - 1} (s) & = & G_{p} (s) [U_{j} (s) - U_{j - 1} (s)] \\ = & φ e^{Δ s} G_{p} (s) E_{j - 1} (s) \end{matrix}$

On the other hand, by subtracting the desired output of Y_d (s) at every other repetition, we have $Y_{j} (s) - Y_{j - 1} (s) = - [E_{j} (s) - E_{j - 1} (s)]$ Thus $E_{j} (s) = [1 - φ e^{Δ s} G_{p} (s)] E_{j - 1} (s)$

Therefore, the convergence condition of the trajectory error as follows [2]. $| 1 - φ e^{j Δ w} G_{p} (j ω) | < 1$ (15)

The closed loop frequency response system transfer function with existing PID controller can be represented as G_p (jw) = N_p (ω) exp (jθ_p (ω)) with N_p (ω) as amplitude characteristic, while θ_p (ω) is the phase characteristic. To view the learning process in the continuous frequency domain, the convergence condition in Equation (20) will be $| 1 - φ N_{p} (ω) e^{j (θ_{p} (ω) + Δ ω)} | < 1$ (16) i.e.

$\begin{matrix} 1 & - & φ N_{p} (ω) cos (θ_{p} (ω) + Δ ω) \\ - & j φ N_{p} (ω) \sin (θ_{p} (ω) + Δ ω) | < 1 \end{matrix}$ (17)

Then it holds as

$φ N_{p} (ω) < 2 φ \cos (θ_{p} (ω) + Δ ω)$ (18)

Based on the assumption of positive learning gain and the lead time in Equation (18), following inequality made by the theorem of Wang 2005 [2] with uncertainty factor ɛ is defined as

$| θ_{p} (ω) + Δ ω | < 90^{\circ} - ɛ$ (19)

To ensure system robustness, this research sets ɛ to be 10°.

In practice, the simplest form of ILC has unacceptable learning transients [1, 14]. A zero-phase low-pass filter is introduced for the elimination of the bad ILC transients. Without the lead time consideration, with Δ = 0 (called P-type ILC), the z-transform of the learning law in Equation (14) introduced by zero-phase Butterworth low-pass filter Q(z) can be expressed as

$U_{j} (z) = U_{j - 1} (z) + φ Q (z) E_{j - 1} (z)$ (20)

The convergence in the frequency domain [5] is given by $| 1 - φ Q (z) G_{plant} (z) | < 1 \forall ω$ (21)

For A-type ILC (AILC) with a lead time design, the convergence condition for Equation (21) in the frequency domain is expressed as

$| 1 - φ Q (e^{j ω}) e^{j Δ w} G_{p} (j ω) | < 1$ (21)

Here, a continuous sixth-order (n = 6) Butterworth low-pass filter (B (jw)), initially designed to have as flat a frequency response as possible in the passband with cutoff bandwidth frequency fc equal to 30 Hz and sampling rate fs of 1200 Hz, was chosen in this study. The nth order normalized Butterworth filter is expressed by Fig. (22). w_c is equal to f_c multiplied by 2π. $| B_{n} {(jw)}^{2} | = \frac{1}{1 + {(\frac{w}{w_{c}})}^{2 n}}$ (22) $n = 1, 2, 3, \dots$

Most often, the frequency content of the initial error emphasizes a middle range of frequencies, where the range is determined by the feedback control system’s bandwidth coupled with the frequency content of the desired trajectory and the disturbances. From the point of view of the steady-state frequency response, the largest components of the error remaining from the use of the feedback control are at frequencies that can be quickly learned. Therefore, the tracking error can be reduced as the number of repetitions increases. However, while the learning control process is deployed, the positioning errors featured with frequency contents at every repetition should be not be compensated by new non-learnable injection signals. The cutoff bandwidth frequency should have different windows through the PSO tuning process on account of the error profile. Initially, the ILC tries to compensate the feedback error made by the conventional PID controller in the first repetition. Mostly, the feedback control errors preserve the commanded signal error history that features the commanded frequency. These frequencies, most likely at the low-frequency levels, can be learned by the ILC. The ILC tries to compensate the whole or major frequency contents at the beginning of the learning process because the Integrated Absolute Error (IAE) cost function serves as the minimization criterion for the transient response error when the PSO is launched in every repetition. Therefore, based on the initial tracking error profile, the PSO search engine is deployed to reduce the filter bandwidth with low frequency at the initial run and to adaptively change and increase the ILC’s bandwidth frequency as the repetitions progress.

Based on the PSO algorithm, each individual (particle) has five members, and these members are assigned real values. If m individuals are present in a population, then the dimension of the population is m × 5. The matrix representation with a total of 20 particles in a population for three PID control gains, ILC gain and cutoff bandwidth frequency fc of the zero phase Butterworth filter is expressed as follows: $K_{m} = [k_{p}, k_{i}, k_{d}, φ, f_{c}] m = 1, 2 \dots, 20$ (23)

The fitness functions in PSO are established and then serve as the basis for the individual and global optimal particle updates, causing the initial solution to evolve toward the optimal solution. A clear algorithm to show how the HPSO-AILC method works and how the HPSO algorithm is used to find the main parameters of the AILC is represented in appendix.

4 Results and discussion

4.1 Numerical simulation for a single axis linear motor

To validate the new PSO method for ILC control, an in-laboratory one-axis linear synchronous motor (LSM) motion is used for the numerical simulation and experimentation. The transfer function of the moving coil is approximated by a first order lag system with resistance R₂₅ and inductance L. The open loop transfer function G with input voltage V_a is approximated by a third order lag system [12]. $G (s) = \frac{w (s)}{V_{a} (s)} = \frac{K_{f}}{(M + m) s (R_{25} + sL) + K_{f} K_{v}},$ (24) where M is the payload mass on the forcer (moving part of the LSM), m is the linear motor forcer mass, R₂₅ is the armature resistance, L is the inductance, K_f is the thrust constant, K_v is the contrary EMF constant, V_a (s) is the input voltage.

The continuous-time transfer function of the linear motor [12] was expressed as follows. $G (s) = \frac{72.6}{0.003588 s^{2} + 11.316 s + 72.6}$ (25)

Figure 1 shows the satisfied convergence condition in Equation (19) that made by different selection of the lead time starting from 0 second to 0.002. When Δ = 0s (like the P-type ILC), the value of the θ_p (ω) + Δω reaches -80° at 5.88 Hz, and then continue to decline but out of the convergence boundary when ɛ is to be 10°. When Δ is positive, as increasing Δω, the phase of the controlled system plant θ_p (ω) will be compensated. As shown in Fig. 1, when Δ is equal to 0.005, the satisfied convergence condition Equation (19) can reach higher frequency within –80° to 80° boundary.

Fig.1

Different lead time (Δ) selection for system plant.

Followed by the AILC convergence function in Equation (18), the Δ plugged in Equation (18) is chosen as 0.005, then the convergence condition for the range of learning gain φ is plotted in Fig. 2. Figure 2 shows the learning gain φ must be less than 2 for satisfying the convergence condition in Equation (18) before 25 Hz. To verify the above situation. Numerical simulation for the sets of learning gain as 1.9, 2.0 and 2.1 respectively are plotted in Fig. 3. It shows the system response can’t converge once the learning gain is φ > 2. Theoretically, larger learning factors φ should have a higher convergence rate in the early iterations. In Fig. 3, when φ= 2, the convergence rate is less than the φ= 1.9. Nevertheless with φ= 2.1, the learning process behaves apparently unstable phenomenon. This concludes for convergence condition the φ must be less than 2 for this LSM with uncertainty phase to be 10° when AILC is deployed.

Fig.2

Convergence condition for learning gain φ of 0.5,1, and 2.

Fig.3

RMS error with iteration as different learning gain.

Figure 4 shows the block diagram for the HPSO-ILC-PID control structure. This indicates the four hybrid terms for PSO updating velocity was implemented for parameters optimization. Twenty particles are randomly chosen for PSO simulation and set in a five-dimension space. Each particle is characteristic with 5 parameters as: (1) the proportional control gain constant K_P (2) the integral control gain constant K_I (3) the differential control gain constant K_D (4) the learning control gain φ and (5) the bandwidth of the cutoff frequency, Fc. The initial value of the parameters forK_P, K_I and K_D is between 0 and 25, whereas the φ is between 0 and 2, the cutoff bandwidth frequency fc is 30 Hz.

Fig.4

Illustration for the HPSO-ILC-PID control block diagram.

In Computer Numerical Control (CNC) machine industry, a cycloid signal profile by approximating a repetitive motion trajectory of a positioning stage is shown in Fig. 5. The associated jerk (differentiation of the acceleration) of the Fig. 5 is plotted in Fig. 6. What CNC machining concerns is the jerk dynamic response of the positioning table that causes result of the surface roughness. Low jerk effect exhibits the good quality of the CNC machines since well controlled motion trajectory in practical operation produces good surface roughness and geometrical accuracy. Thus cycloid is frequently used as the tracking control command to the AILC. Due to the mechanical inertia effect, the LSM motion stage can’t immediately increase to the commanded speed instantaneously. As indicated by C1-C5 in Fig. 5, the positioning curve was divided into five segments. Characteristic of the first C1 regime is based on slow speed/acceleration and then accelerates in the beginning of C2 regime and decelerates to the highest speed in C3 regime, and similarly in C4 and C5 regime in reverse command.

Fig.5

Position command of a cycloid curve for numerical simulation and experiments.

Fig.6

Illustration for the jerk input shape to controlled system both for numerical simulation and experiment.

The devised cycloid in rising height h and accelerated duration from t to constant height (h) within the duration β is defined as follows: $y = \frac{h}{2 π} [\frac{2 π t}{β} - \sin \frac{2 π t}{β}]$ (26)

In this research, the parameters are with h = 1 and β = 5.

Figure 7 shows numerical simulation for the RMS tracking errors by 200 times AILC iteration followed by every HPSO. The associated parameters are shown in Table 1. The best PID gains, ILC gain and associated bandwidth frequency was picked in each HPSO searching algorithm. Then these five best parameters are fed into the AILC by 200 iterations. As we can see that the RMS tracking errors are diminished as AILC repetition when each HPSO algorithm holds the best parameters. Error convergence was achieved and decreases to zero successfully.

Fig.7

The tracking error of the output response by PSO-AILC.

Table 1

Formula of the developed HPSO algorithm

Proposed Hybrid PSO (HPSO)	Formula
Velocity	V_id (t + 1) = w (t) × V_id (t) + c₁ × Rand () × (P_id - X_id (t)) + c₂ × Rand () × (P_gd - X_id (t))
	+ c₃ × Rand () × (R_id - X_id (t))
Inertia weight	$w = w_{max} - \frac{w_{max} - w_{min}}{N_{max}} \times N_{iter}$
Position	X_id (t + 1) = X_id (t) + V_id (t + 1)
Acceleration factor	$c_{1} (k) {= c}_{1 min} + (\frac{{iter}_{max} - k}{{iter}_{max}}) \times (c_{1 max} - c_{1 min})$
	$c_{2} (k) {= c}_{2 max} + (\frac{{iter}_{max} - k}{{iter}_{max}}) \times (c_{2 min} - c_{2 max})$
	$c_{3} (k) {= c}_{3 min} + (\frac{{iter}_{max} - k}{{iter}_{max}}) \times (c_{3 max} - c_{3 min})$
Conditions	Number of particles:20
	•Dimension of particles:5
	•Velocity extreme value: V_max = 1V_min = -1
	•Position extreme values of three PID parameters:
	•X_{p _max} = 1, X_{p _min} = 0, X_{i _max} = 0.1, X_{i _min} = 0, X_{d _max} = 10, X_{d _min}= 0
	•W_max= 1, W_min= 0.2
	•weighting: w = 1 0.2
	•Learning factor:
	c_{1_max} = 3.15, c_{1_min} = 1.05
	c_{2_max} = 3.15, c_{2_min} = 1.05
	c_{3_max} = 3.15, c_{3_min} = 1.05

As mentioned, the P-type ILC learning has the potential to excite rich frequency contents and try to learn the error signals. Rejected non-learnable error signals by PSO will facilitate the decreased error at the next repetition as demonstrated in Fig. 7. The blue curve (Δ= 0 s) in Fig. 8 is the tracking error made by P-type ILC without phase compensation at each repetition, while the red curve (Δ= 0.005 s) performs the decreasing RMS tracking error as repetition with the design of HPSO-AILC. As a result, using AILC lead time design improves the learning control process. One of the major concerns in machine tool industry is the jerk phenomena. As depicted in Fig. 6, the jerk will affect the machining surface roughness that depends on the controlled system dynamics. Figure 9 shows the RMS tracking error by HPSO-ILC-PID (Δ= 0) and the HPSO-AILC-PID (Δ= 0.005 s). It is observed that the position error mainly occurs around the discontinuous section (t = 0 s and t = 5 s) of the jerk curve. Compared with the P-type ILC, the AILC improves the learning process based on HPSO bandwidth tuning and the lead time contribution. Figure 10 shows the frequency response of the error spectrum when the trajectory error was shown in Fig. 9 by Fast Fourier Transform. The error spectrum around 7 Hz was suppressed significantly. To conclude, the A-type ILC (red line) reduces the overall error amplitude both in Fig. 9 in time domain and Fig. 10 in frequency domain.

Fig.8

The RMS tracking error of the P type ILC (Δ= 0) and different lead time values of AILC.

Fig.9

(a). The RMS tracking error by HPSO-ILC-PID (Δ= 0); (b). The RMS tracking error by HPSO-AILC-PID.

Fig.10

Comparison of the error frequency response by P-type ILC and A-type ILC with HPSO.

Fig.11

The time-frequency plot for the tracking error by P-type ILC-HPSO control.

With adaptive cutoff bandwidth frequency by PSO, Fig. 8 shows the potential of rejecting non-learnable error signals from repetition to repetition. The benefit of using the HPSO bandwidth tuning is demonstrated in Figs. 11 and 12 in time-frequency analysis plot. The color on the right of each graphic represents the amplitude of the corresponding error signal. Compared with the P-type ILC, the A-type ILC has the effect on reducing the amplitude of the error at any time and for each frequency content. From another perspective, the error can be reduced from 10^-5.6 to 10^-7.5 based on the maximum error signals. In comparison with the time-frequency plot in P-type ILC-HPSO and A-type ILC-HPSO, proposal controller exhibits the good learning ability even though the error frequency is above 95 Hz.

Fig.12

The time-frequency plot for the tracking error by A-type ILC-HPSO control.

With the help of HPSO algorithm, the P-type ILC controller has a little bump in the magnitude of the bode plot before the 40 rad/s (about 7 Hz). Fig. 13 shows the bode plot for ILC and AILC based on applying the PSO algorithm. The P-type ILC (blue line) presents a lower amplitude about -25dB at the 100 rad/s, while the HPSO-AILC (red line) improves the frequency response up to 150 rad/s. Through HPSO algorithm the AILC validates good frequency response and enlarge the bandwidth when sweep sine signals were injected into the LSM control system.

Fig.13

Bode diagram of the HPSO-AILC and HPSO-PILC systems.

Figure 14 shows the apparatus setup for the positioning motion of the LSM [12]. The in-lab LSM is driven by a three-phase Y-connected with specification of 1.5kW/60Hz/110V/10.2A. It is forced by a position-controlled voltage via pulse-width-modulation. A closed-loop feedback signals corresponding to the displacement of the stage was measured by a linear encoder sensor (Renishaw-RGH41X50D05A). The PC based model 626 card is interfaced with the motion of stage in real time. Experimental computer programs were carried out using C language and the maximum displacement height in Fig. 5 is 1 cm. Heuristically, velocity updating value in PSO plays important role in real time implementation. For real-time experiments, the V_max and V_min are not easily determined in advance. In real time experiments, each particle will move and fly out of the optimal solution if the V_max is set too large, even though it is just right after the first iteration. An unconstrained V_max value in updating the velocity and position will cause the system to destabilize and diverge. The concept of reducing the range of V_max and V_min is like the gain scheduling design by conventional root locus design process. If the speed of the particles is confined and is relatively conservative by V_max = 0.001 and V_min = -0.001 in this work, the instability problem of PSO application can be resolved. Nevertheless, this solution causes another problem in slow convergence and difficulty in finding the optimal solution, since it is time consuming to reduce the from +1 to +0.001 by a factor of 1000. In this study, the real-time motion of the LSM. The displacement is out of the maximum positioning limit and the onset of unstable oscillation occurs if the values of V_max and V_min are set to out of the range of +1 and –1.

Fig.14

Schematic illustration for the experiment configuration [12].

For real time implementation, as indicated in Table 1 for the pre-defined velocity extreme value, Fig. 15 shows the RMS error of the LSM motion reaches about 40 μm when maximum input commanded height is 1 cm. Though using PSO for ILC experimentation must be better than the conventional P-type ILC controller. The proposed HPSO-AILC exhibits more superiority over the HPSO-PILC in reducing tracking error by using 10 PSO iteration numbers and 5 PSO particles for searching the global best parameters. For real time experimentation, five PSO particles are suitable for 5 parameters optimization in this experiment.

Fig.15

Experimental result for the RMS error at each iteration.

5 Conclusions

This paper has synthesized a new synergy of velocity updating technique for applying PSO to adjust the gains of the PID controller, AILC, and cutoff frequency of a zero-phase Butterworth filter. Convergence condition by using the AILC is designed first for a LSM motion stage with some phase uncertainty. The simulation results demonstrate the proposed HPSO– AILC-PID controller can provide the ability to reduce the tracking error significantly when these gains were adaptively tuned under the hybrid PSO algorithm at every repetition. The error can be reduced monotonically at every repetition. From numerical simulations, developed controller demonstrates error suppression characteristics in time, frequency, and time-frequency responses, successfully. The closed loop feedback control system with HPSO-AILC structure in Bode plot conserves larger bandwidth frequency response. The proposed HPSO– AILC controller enhances good tracking control results compared with the P-type AILC and much better than the conventional P-type ILC. The learning process is illustrated for reducing the RMS tracking errors at every repetition considerably. Real time experimental result verified that the HPSO based AILC controller consolidates high precision accuracy.

Footnotes

Appendix

The proposed HPSO algorithm works as follows:

Initialization

Randomly initialize the positions and velocities of the m particles, setting c1, c2, c3, V_max, V_min, X_max, X_min, w_max, w_min, f_id, f_gd, N_iter, N_max, and f_oc. Note that V_max and V_min represent the maximum and minimum velocity, X_max and X_min represent the maximum and minimum position, w_max and w_min represent the maximum and minimum inertia weight, respectively. The n indicates the whole particle number, n1 is the number of Pbest (the best previous position yielding the best fitness value for the ith particle), n2 is the number of Gbest (the best position discovered by the whole population), f_id is the fitness of the Pbest, f_gd is the fitness of the Gbest, and f_oc is the solution of optimal conditions.

Evaluation

Evaluate the fitness of the particle f_i [n].

Find the Pbest

For i = 1,…,n

If (f_i [i] < f_id) P_id [i] = f_i [i]

n1 = n1+1

End If

End For

Find the Gbest

For i = 1,…,n1

If (P_id [i] < f_gd)

P_gd [i] = P_id [i]

n2 = n2+1

End If

End For

Termination Check

For i = 1,…,n2

If (P_gd [i] < f_oc)

Go to Step 10

End If

End For

Update the Velocity and Position

V_id (t + 1) = w (t) × V_id (t) + c₁ × Rand () × (P_id - X_id (t)) + c₂ × Rand () × (P_gd - X_id (t)) + c₃ × Rand () × (R_id - X_id (t))

X_id (t + 1) = X_id (t) + V_id (t + 1)

maximum velocity

If(V_id[i] > V_max)

V_id[i] =V_max

Else If(V_id[i] < V_min)

V_id[i] =V_min

End If

End For

maximum position

For i = 1,...,n

If(X_id [i] > X_max)

X_id [i]=Rand ()

Else If(X_id [i] < X_min)

X_id [i]=Rand ()

End If

End For

Update the Inertia weight

$w = w_{max} - \frac{w_{max} - w_{min}}{N_{max}} \times N_{iter}$

Termination Check

If (N_iter! = N_max)

Go to Step 2

End If

End HPSO

Acknowledgments

This work was supported by MOST Grant 104-2221-E-018-015 for which the authors are very much grateful.

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