Gain scheduled control of IPMC actuators with ‘model-free’ iterative feedback tuning

Sensors and Actuators A 164 (2010) 137–147

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Gain scheduled control of IPMC actuators with ‘model-free’ iterative feedback tuning A.J. McDaid ∗ , K.C. Aw, S.Q. Xie, E. Haemmerle Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, 20 Symonds Street, Auckland, New Zealand

a r t i c l e

i n f o

Article history: Received 19 July 2010 Received in revised form 26 August 2010 Accepted 30 September 2010 Available online 8 October 2010 Keywords: Ionic Polymer-Metal Composites (IPMC) Iterative feedback tuning (IFT) Gain schedule (GS) Control Adaptive Actuator

a b s t r a c t IPMCs are actuators with significant potential in robotics and biomedical applications. In order to make full use of their potential, their complex actuation behaviour must be effectively controlled. Previous efforts to control these actuators relied heavily on the development of an accurate model of the IPMC. After 15 years of modelling research, not one such accurate model exists. This paper presents a nonlinear controller which is adaptively tuned using a model free approach called iterative feedback tuning (IFT). Automatic tuning of the controller, without the any knowledge of the system is achieved. By eliminating the need for a plant model, the developed control system represents major progress towards the implementation of IPMCs into real world applications. The controller is tuned to achieve high position accuracy on both the micro and macro scale, where most previous controllers limit their operating range to remain in a small linear region. The proposed controller continuously adapts its parameters to handle the IPMC nonlinearity. The IFT algorithm has successfully tuned the control parameters with a performance increase of up to 65% from an initial arbitrary un-tuned controller. Experimental results show the newly proposed controller achieves much higher performance when compared with a conventional linear controller. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Ionic Polymer Metal Composites (IPMCs) are a type of flexible transducer which can be directly used as both sensors and actuators through their inherent coupling of the electrical and mechanical domains [1]. As a result of increased research and commercial interest, IPMC technology is rapidly evolving. IPMCs exhibit significant potential in a wide range of applications, including robotics [2,3], biomimetics [2,4], medical devices [4,5] as well as replacements for traditional actuators, such as pneumatics and servo motors. IPMC transducers have intrinsic properties making them desirable when compared with conventional actuators, including low power consumption, very small mass, flexibility and biocompatibility as well as the ability to fabricate them in a range of different geometries for specific applications [6]. However, there are still a number of performance concerns which need to be overcome before they can be implemented in real world applications [7]. In order to harness the wide-ranging advantages of IPMC actuators and to aid their successful implementation into real systems, the actuation response of an IPMC must be effectively controlled. However, this is not a trivial task due to their complex behaviour.

∗ Corresponding author. Tel.: +64 9 373 7599x87555; fax: +64 9 373 7479. E-mail address: [email protected] (A.J. McDaid). 0924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2010.09.023

Currently most IPMC controllers are tuned using an approximate model; the system performance then assessed through simulation before implementing the controller in the real system. There are a few major issues with this method. Firstly, the development of a suitable IPMC model which will accurately represent the real system is extremely complex and time consuming. Although there has been considerable research into the material behaviour and actuation mechanisms of IPMCs over the last 15 years, still no complete and widely accepted model has been developed to predict the mechanical output and even the underlying mechanisms for actuation are not fully understood. This is partly due to the very complex nonlinear, time-varying and environmentally sensitive nature of the composite material [8–10]. Secondly, once a controller has been developed in simulation, it then must be further tuned on the real system to account for variability between simulation and the actual response. No model exists which takes into account the full time-varying behaviour of an IPMC which occurs as a result of dehydration and redistribution of ions in the polymer. Solely implementing a time-invariant model based control system is insufficient when actuating for a period of time. The controller must be adaptively tuned online to obtain successful system performance. Traditionally this tuning would require an experienced operator with knowledge of the system relying mainly on intuition. The developed controller would not then be directly transferable to other IPMC samples, so each individual sample must be manually tuned.

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Iterative feedback tuning (IFT) is an automatic tuning method which seeks to optimize the controller parameters in the system. The IFT system presented in this paper enables the IPMC to be adaptively tuned without the need of any model or knowledge of the system. One simply sets the IPMC then runs the tuning algorithm to obtain an optimally tuned system. This controller tuning method eliminates the need for an accurate model in order to accurately control the IPMC actuator. This is a completely new way of thinking for smart materials as up until now research emphasis has been on the modeling of materials in order to be able to control their behavior. This new model-free approach to controller design presents a major step forward for IPMC technology, towards wide acceptance as a viable alternative to traditional actuators. This control system is particularly intended for robotic and biomimetic applications and therefore is tuned for step changes set point and operation at low frequencies for a large range of displacement outputs on both the micro (<1 mm) and macro (>1 mm) scale. In order to achieve this, a gain scheduled (GS) nonlinear control architecture is proposed. This allows the control parameters to vary depending on the states of the system. This controller therefore adapts itself in order to tackle the nonlinearities and time-variance to achieve accurate positioning over a large displacement range. The proposed controller provides a valuable tool for the development of systems with integrated IPMC actuators.

2. Gain scheduled nonlinear controller There have been many attempts by previous researchers to control the bending actuation of an IPMC transducer. Linear time invariant (LTI) models have been used to develop LTI controllers. These have been implemented in a real system with limited success [11,12]. The major problem, which was quickly realized, is that when you linearise the highly nonlinear and time-variant IPMC plant, the performance of the developed controller will dramatically decrease when the system operates further away from the equilibrium point, i.e. at high displacements. Therefore in order to accurately control the complex system over a large range of conditions more advanced and robust controllers are needed, for example those developed by Ahn et al. [13], Brufau-Penella et al. [14] and Lavu et al. [15]. GS is a nonlinear control scheme commonly used in research and industrial applications such as flight control, vehicle control and automotive engine control. In traditional GS control, the nonlinear plant is linearised at a number of finite operating or equilibrium points. Linear controllers are then developed at each operating point, resulting in a set of linear controllers which exhibit good performance at each point. The control parameters for each of the linear controller are then interpolated between their operating points, based on an appropriate design schedule. The resulting GS controller has the same architecture as the linear controllers but with parameters that are continuously varying with respect to the chosen schedule, resulting in a nonlinear controller. Two research papers [16,17] present a history and in depth details about GS controllers. The two major guidelines for the choice of the scheduling variables for the controller are: (i) the scheduling variable should encapsulate the nonlinearities of the plant, and (ii) the scheduling variables should vary slowly compared with the plant itself [18,19]. A number of different techniques have been implemented in literature including; scheduling parameters on the reference trajectory and on the plant output [20]; scheduling the gain, poles and zeros of the controller transfer function [21] and; linear interpolation of the elements in a state space representation of a system [22]. This approach to designing a nonlinear controller has the major advantage of utilizing simple linear control techniques, which are

r(t) -

e(t)

GC(ρ)

u(t)

GIPMC

y(t)

ρ fGS(r) Fig. 1. Block diagram of proposed GS controller.

extremely well understood and hence a large number of approaches exist. The main disadvantage is that no real performance or stability and robustness guaranties exist at present, except where the scheduled parameters are varying slowly [19,23]. Most research into the control and modeling of IPMCs has been limited to small ranges of deflection in an attempt to reduce the nonlinearity inherent in the IPMC and improve the accuracy and repeatability of experiments. However if IPMCs are to be implemented and functional in real applications they must have excellent performance over a large operating region. It has been shown by Kothera [24] that it is advantageous to use different controller parameters if actuating to different target displacements. Therefore it was decided to implement a GS controller in order to accurately control the position of the IPMC over a large range of displacements, for both micro and macro movements. The motivation for using this type of controller is that well understood linear control techniques could be utilized and also that the IPMC has been shown to be nonlinear based on the level of actuation [25,26], lending itself well to a scheduled gain controller. The scheduling variable should then be based on the level of actuation (either voltage input or displacement output) to encapsulate the nonlinearities of the plant. Using the voltage input is not a good choice because when implemented, the controller output and hence IPMC input can change very rapidly to try and track a reference. The IPMC dynamics are relatively slow in comparison to these fast input changes therefore the only stability and robustness condition for the system, that the scheduled variable changes slowly, will not be guaranteed. It therefore seems appropriate to schedule the gains on the IPMC displacement, however there are issues surrounding this: (i) the output may contain a significant amount of noise, especially when actuating the IPMC at micro targets, therefore the controller gains will vary rapidly and cause large jerky and even unstable oscillatory motions and (ii) there will be implementation issues as the system would need to be tuned at a number of different outputs, but the actual output changes dynamically. The proposed solution to this is to schedule for the controller gains based on a function of the reference trajectory. The motive for this is that in position tracking applications, the output signal follows the reference trajectory and hence typically the output is just a low pass filtered version of the reference [27]. So actually the parameters will vary with the IPMC displacement, despite actually being scheduled on the reference signal. Also because the controller is being designed for changes in set point the assumption of slowly varying control parameters will hold true as the parameters are scheduled on reference trajectory and the reference trajectory in a stair-step function are piece-wise constant. The designed GS controller is shown schematically in Fig. 1 with the associated variables. Gc () represents the controller transfer function, GIPMC is the plant. fGS (r) is a function that schedules the controller parameters, , based on the reference input, r, and outputs them to the controller. A Proportion-Integral-Derivative (PID) control algorithm has been selected for the linear controllers. This was chosen for its simplicity to implement, ease of use with the IFT algorithms and because it has shown to provide good results for position control of IPMCs over a limited range [11] and as such should provide the

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139

based on a least squares fit as shown below in Eq. (1).

r(t) -

e(t)

GC(ρ)

u(t)

GIPMC

y(t)

1  (˜yt ())2 2N N

J() =

(1)

t=1

Fig. 2. Control system used to tune the IPMC controller using IFT.

basis for a stable and robust GS controller with good performance. The steps to realize the GS controller are first, develop linear controllers for different reference inputs across the desired operating range using IFT then interpolate the gains (Kp , Ki and Kd ) as function of reference position, finally implement and test the controller.

where  is a vector of the controller parameters, N is the total number of time steps for a given experiment and y˜ t () = yt () − r is the system error at discrete time step t. The criteria function is used to find a local minimum of the controller, in the presented case, find the minimum tracking error over the entire time frame of the experiment. In order to locate the minimum of the criteria its differential must be equal to zero as in Eq. (2). ∂J() =0 ∂

(2)

3. Iterative feedback tuning

In order to find the solution to this, the gradient of the criteria is found by Eq. (3):

Currently most IPMC controllers are tuned in simulation using an approximate plant model. The performance is then assessed, also in simulation, before implementing the controller on the real system. One major issue with this method is the development of a suitable IPMC model which is complex and time consuming as the IPMC is extremely non-linear, time-variant and environmentally sensitive. In addition the controller must be further fine-tuned on the real system to account for variability between the model and the real plant and the controller is also sample specific so it cannot be used on a different IPMC sample. Consequently it is highly desirable to develop an automatic tuning method. The development of an IFT routine will allow the IPMC controller to be automatically tuned without the need for any model or knowledge of the system. IFT was originally proposed in 1994 by Hjalmarsson et al. [28] which is an iterative optimization approach to designing controllers through the objective of minimizing a controller design criterion of an unknown plant. This relatively new tuning method tests the response of the actual system to determine new updated and improved control parameters. As the updated parameters are based on experiments on the actual system, the approach is model free. IFT has been implemented demonstrating good results in both laboratory and industrial applications such as control of profile cutting machines [29], speed and position control of servo drive [30], temperature regulation in a distillation column [31] and control of photo resistant film thickness [32]. The major advantages over other tuning methods include: it is automatic and therefore no need for experienced operator; model free so no knowledge of the system required; it can be implemented as an online tuning method; it can be used to tune many types of controllers; and different design criterion can be used to optimize the operating performance depending on the application requirements. A full description of the IFT algorithm can be found in [31] and a concise explanation of the key details for implementation is presented in the following section.

∂J() 1 = N ∂

3.1. Procedure and algorithms for implementation IFT is a time domain approach whose objective is to minimize a cost function or design criterion based on the controller performance in order to obtain an optimally tuned system. The simple control system shown in Fig. 2 will be used to tune the IPMC system. There are a number of different design criteria which have been proposed in the literature based on tracking error and control effort. Depending on the application requirements, the system performance can be modified by placing a weighting filter on the design criteria when tuning the controller to ensure more emphasis is placed on either the transient or steady state. The design criteria that will be used for the IPMC controller is a quadratic function

N



t=1

∂yt () y˜ t () ∂

 (3)

then by using the iterative algorithm in Eq. (4), the solution for  can be found to obtain the minimum error for the system. This is essentially a gradient search algorithm: i+1 = i − yRi−1

∂J(i ) ∂

(4)

where Ri is an appropriate positive definite matrix which determines the search direction for the optimization, i is the iteration number and  is a positive real scalar which controls the step size. Using the identity matrix for Ri gives a negative gradient direction. It is commonly accepted in literature [27,29,31] that using the Gauss–Newton approximation of the Hessian of J() for Ri gives improved results, this becomes more important when the sample size is small. The Hessian is given below in Eq. (5): 1 Ri = N N

t=1





∂yt (i ) ∂yt (i ) ∂ ∂

T 

(5)

For the given control system used for tuning the IPMC system, Fig. 2, it can be shown that: y(i ) =

GC (i )GIPMC r 1 + GC (i )GIPMC

And therefore ∂y(i ) 1 ∂GC (i ) = GC (i ) ∂ ∂

(6)

G ( G C i IPMC )

1 + GC (i )GIPMC



(r − y(i ))

(7)

Eq. (7) can be physically interpreted as the amount the system output changes with respect to the controller parameters, hence by calculating this one can theoretically update the control parameters to change the system output as desired. In order to find the solution to these two independent experiments are carried out on the system. The first experiment is conducted under normal operating conditions, with an external deterministic reference signal, r, applied at the input, and the output y is recorded. Now the two signals are compared and the error is calculated from (r − y(i )), which is then injected as the reference signal for the second experiment. The output from the plant for this 2nd experiment gives the term in the square brackets in Eq. (7). The two terms can be found by differentiating the controller itself and hence ∂y(i )/∂ can be established. Using this result the Hessian can be calculated from Eq. (5) and also ∂J()/∂ can be found leading to the new updated controller parameters, i+1 , which will give an improved controller for the system. This procedure is then repeated for the desired number of iterations, or until the desired system performance is achieved.

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Fig. 4. Schematic diagram of the laser setup and target on IPMC cantilever.

Fig. 3. Test rig used for the IPMC experiments.

In [28] it is briefly discussed that provided the initial control system is stable, if the step size is small enough and the data set is relatively large then the search will always be in the negative direction, ensuring convergence to the local minimum of the design criteria. 4. Implementation 4.1. Experimental setup The experiments were undertaken using a custom test rig as shown in Fig. 3. The rig supports 2 copper clamps which act as electrodes to pass the voltage to the IPMC. A Nafion® 110 based IPMC was used, with Pt electrodes. The IPMC was 35 mm long, 10 mm wide with a thickness of 200 ␮m. The clamped length was 5 mm. This relatively long length of IPMC was chosen because most research has been carried out with shorter lengths of IPMC as actuation response is more linear with shorter IPMCs [33,34]. This research is attempting to tackle the nonlinearity so therefore a long length was used. Also shorter IPMCs cannot achieve large displacements so a long length will be needed to ensure that both micro and macro displacements can be investigated. With the specific IPMC used for this research, up to a 3 mm displacement will be input as the target reference. The IPMC and clamps are placed in a container of de-ionized water in order to avoid rapid dehydration and potential damage to the IPMC. Control electronics and a National Instruments DAQ card are used to interface between the Matlab environment running the Simulink model on the PC and the IPMC actuator. A Banner LG10A65PU laser sensor with a 10 ␮m resolution was used to measure the displacement of the IPMC. It was set up as per Fig. 4 to measure the linear displacement at a distance of 25 mm from the base, which corresponds to 5 mm back from the tip. The laser is placed this far back from the tip to ensure that at high IPMC displacements/curvature the laser target will not leave the tip of the IPMC. There is an appreciable level of noise in the system, with a maximum ±30 ␮m mainly due to the laser sensor resolution and the quality of the reflected beam back to the laser sensor. A contribution to the noise may also be due to the control electronics and DAQ card. At displacements less than 200 ␮m there is a low signal-to-noise (SNR) and this will prevent accurate tuning at low displacements because of the lack of information obtained for the IFT algorithm.

IFT still tunes the controller, but the results are inconsistent and are highly dependent on the noise contribution. Despite the inability to reliably tune at these displacements the controller can still accurately track to these targets but the output signal includes a noise error of ±30 ␮m. 4.2. IPMC control system In order to develop the GS controller, a set of linear PID controllers must be developed. This is achieved using the IFT algorithm to tune the Kp , Ki and Kd values at a number of different target displacements across the range of operation of the IPMC. Fig. 5 depicts the control system that was developed and implemented in Simulink for tuning the control parameters. The switch in the system is used to control what signal is injected as the reference input. To tune for a specific displacement, a step input of that displacement is set as the reference trajectory, r1 for the first experiment. For the second tuning or gradient experiment the switch is turned and the system has the reference input, r2 = (r − y(i )), which is the error of the first experiment. After these two experiments, the control parameters are updated and then the next iteration is run. The formulas for the above blocks are given below in Eqs. (8)–(11). T is the sample time for the controller which has been chosen as 0.1 s for this system as the IPMC has slow dynamics and therefore 10 Hz is fast enough to accurately capture the changes in output. Lp is the low pass filter constant, set to 0.85, this is used to suppress any unwanted high frequency inputs to the IPMC caused by noise or other disturbances. The saturation block was placed in the loop and set to ±3 V to ensure that no excessive voltage was input to the IPMC preventing any damage. PID(z) =

(Kp + Ki T + Kd /T )z 2 − (Kp + 2Kd /T )z + Kd /T

LPF(z) =

z2 − z 1 − Lp z − Lp

(8) (9)

2z T (z + e−2 )

(10)

GC (z) = PID(z) · LPF(z) · ZOH(z)

(11)

ZOH(z) =

The terms 1/Gc(i) ∂Gc(i) /∂ for the IPMC control system, which are needed to solve Eq. (7), are given below in Eqs. (12)–(14). 1 ∂GC (z) z2 − z = 2 GC (z) ∂Kp (Kp + Ki T + Kd /T )z − (Kp + 2Kd /T )z + Kd /T

(12)

1 ∂GC (z) Tz 2 = GC (z) ∂Ki (Kp + Ki T + Kd /T )z 2 − (Kp + 2Kd /T )z + Kd /T

(13)

A.J. McDaid et al. / Sensors and Actuators A 164 (2010) 137–147

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Fig. 5. Schematic diagram of the IPMC control system used to tune the set of linear controllers.

1 ∂GC (z) 1 z 2 − 2z + 1 (14) = T (Kp + Ki T + Kd /T )z 2 − (Kp + 2Kd /T )z + Kd /T GC (z) ∂Kd 4.3. IFT settings Due to the desired applications in robotics and biomimetics, the control system was designed to be accurate for changes in set point, in terms of both the transient and steady state response. In order to tune for this, the reference trajectory was a stair-step function to the desired set point to be tuned. The experiments will be 60 s long consisting of, positive the desired displacement for the first 15 s, then step back to zero displacement until 30 s, then negative the desired displacement until 45 s and finally to zero displacement until 60 s. This reference will ensure that the IPMC has been tuned in both directions, as it has been shown that due to imperfect fabrication techniques the IPMC can have different performance in different directions, and will also tune for 4 transient periods as well as steady state behavior. The initial controller parameters, Kp , Ki and Kd , were chosen arbitrarily after a few tests to ensure a stable system, but one which exhibited poor performance. Kp was set to ensure that at the first time step, the error for the largest desired displacement would not saturate the control output. Integral gain was chosen conservatively so as not to introduce too much oscillation but still ensuring zero steady state error. The derivative gain in a PID controller contributes based on the change in error, and therefore will amplify any high frequency noise that may be present in the laser sensor or control electronics. It was desired to control the IPMC to micron displacements, where the noise starts to become an appreciable part of the feedback signal, so a high Kd value is likely to introduce large high frequency oscillation and possibly make the system unstable. Also it has been shown by Liu in 2010 [35] that PI controllers can exhibit good response in controlling IPMCs. For these reasons the authors were confident to start with a PI controller by setting the derivative gain to 0 and let the tuning algorithm decide how much derivative action to include. The chosen initial values were, Kp = 1000; Ki = 500; Kd = 0. In order to ensure convergence to the local minimum of the design criteria the data set has been chosen large, 600 samples (60 s at 10 Hz), and the step size for the control parameters must be chosen relatively small. The step size must be small enough to ensure the controller does not ‘jump too far’ and result in an unstable sys-

tem, but be large enough so that there is a rapid convergence to the minimum design criteria, otherwise too many iterations will be needed, making the algorithm impractical. The step sizes chosen for the IPMC system were Kp = 1; Ki = 1; and Kd = 0.5. As a rule for the IPMC system the step size was chosen so that control parameters would update by no more than 100% of the previous value. From the experiments undertaken it has been shown that this step size will ensure that the system will remain stable, but also achieve a rapid convergence within 5 iterations. The value for step size of the derivative gain was chosen as half of that for the proportional and integral gain because for a large increase in derivative term it is possible the system may iterate to an unstable system at low deflections in the presence of large noise input. 5. Results of IFT With the setup completed the IFT algorithm was run on the IPMC using the initial controller values and step sizes. The IFT algorithm was run to tune the IPMC controller for displacements in the micro and macro range. Previous controller has not been able to achieve this due to the nonlinear nature of the IPMC and the varying dynamics of the IPMC at varying displacements in these two ranges. The controller was tuned for reference signals ranging from 100 ␮m to 3 mm. The time response for the 3 mm target displacement is shown in Fig. 6. First is the initial output using the arbitrary PI controller (Fig. 6(a)), then 5 consecutive updated controllers are developed and tested on the system (Fig. 6(b)–(f)). It is clear to see that the initial controller had large oscillation in the first quarter, large overshoot starting at 30 s and also some overshoot and oscillation when finally returning to zero displacement. Even after just one iteration of the control parameters there was an obvious improvement in the response, with the oscillations in the first 15 s reduced significantly as well as the overshoot at 30 s. After each iteration, the controller parameters were updated and it can be seen from the time response that the IPMC output drastically improved. After 5 iterations the performance of the controller for a 3 mm displacement had significantly improved. The improvement can be quantified as a 56% of the design criteria J(). This same procedure was undertaken for 100 ␮m, 200 ␮m, 300 ␮m, 500 ␮m, 1 mm, 1.5 mm, 2 mm, 2.5 mm and 3 mm displacements. It was found that at 100 ␮m and 200 ␮m despite being able to track accurately to the target displacements, the noise level

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Initial

6

(b) Displacement (mm)

Displacement (mm)

(a)

4 2 0 -2 -4 -6

0

10

20

30

40

50

4 2 0 -2 -4 -6

60

Iteration 1

6

0

10

20

Time (s) Iteration 2

6

(d) Displacement (mm)

Displacement (mm)

(c)

4 2 0 -2 -4 -6

0

10

20

30

40

50

Displacement (mm)

Displacement (mm)

(f)

2 0 -2 -4 0

10

20

30

60

40

50

60

40

50

60

2 0 -2 -4 0

10

20

30

Time (s)

4

-6

50

4

-6

60

Iteration 4

6

40

Iteration 3

6

Time (s)

(e)

30

Time (s)

40

50

60

Iteration 5

6 4 2 0 -2 -4 -6

0

10

20

Time (s)

30

Time (s)

Fig. 6. Time response over 5 iterations of the controller for 3 mm step displacement.

from the sensor and electronics was too large to accurately tune for these displacements. The tuned parameters were very inconsistent because of the low signal-to-noise ratio (SNR). This was confirmed by the fact that the IFT algorithm kept the proportional gain at these displacements at zero or very low to ensure that noise was not amplified. The controller gains and design criteria are plotted in Fig. 7 for a small, 300 ␮m, medium, 1.5 mm and large, 3 mm displacement after each iteration to show how they are being dynamically updated through this tuning algorithm. It can be seen that the design criteria was converging to an optimal solution. The full set of results for the final tuned values for each displacement is given in Table 1 below, along with the percentage improvement of the controller design criterion. It is very clear to see the major success of the IFT algorithm in tuning the system from an arbitrary controller without the need of any plant model.

The results for the controller parameters are plotted in Fig. 8 to display how the gains change after each iteration and at each target displacment. If a smaller or larger step size was chosen these plots would be either smoother or more bumpy, respectively. The effect of target displacement on the tuning of the control papameters can clearly be seen as the initial controller for all reference displacements are the same, yet the controller tunes to very different states for the different reference inputs.

6. Development of gain schedule The schedule for the controller gains has been chosen based on a function of the reference trajectory. Its architecture is shown in Fig. 1 and the task now is to find the function fGS (r) for the controller.

Table 1 Summary of the results for IFT at different target displacements. Target displacement (mm)

Final Kp

Final Ki

Final Kd

Initial J (×10−9 )

Final J (×10−9 )

Improvement (%)

0.3 0.5 1 1.5 2 2.5 3

3092.56 2636.35 2019.93 1480.53 1488.18 1577.95 980.449

2813.93 2791.43 1904.23 1903.93 1585.71 1128.05 1040.13

273.26 483.04 436.34 476.95 539.94 708.01 354.75

3.2411 11.383 29.289 58.403 91.605 296.87 288.56

1.9618 3.9523 12.650 26.322 52.129 113.38 127.93

39.47 65.28 56.80 54.93 43.09 61.81 55.66

A.J. McDaid et al. / Sensors and Actuators A 164 (2010) 137–147

3000

3.0E-09

2500

2.5E-09

Design Criteria (J)

3.5E-09

Contoller Parameters

(a) 3500

2000

Kp

1500

Ki 1000

Kd

2.0E-09 1.5E-09 1.0E-09 5.0E-10

500 0 Inial

143

1

2

3

4

0.0E+00 Inial

5

1

2

3

4

5

4

5

4

5

Iteraon

Iteraon

(b) 2000

7.0E-08

1800

6.0E-08

1400 1200

Kp

1000

Ki

800

Design Criteria (J)

Contoller Parameters

1600

Kd

600

5.0E-08 4.0E-08 3.0E-08 2.0E-08

400 1.0E-08

200 0 Inial

1

2

3

4

0.0E+00 Inial

5

1

2

Iteraon 3.5E-07

1000

3.0E-07

Kp

800

Design Criteria (J)

Contoller Parameters

(c) 1200

Ki 600

Kd

400 200 0 Inial

3

Iteraon

2.5E-07 2.0E-07 1.5E-07 1.0E-07 5.0E-08

1

2

3

4

5

0.0E+00 Inial

1

2

Iteraon

3

Iteraon

Fig. 7. Controller parameters and design criteria after each iteration for (a) 300 ␮m, (b) 1.5 mm and (c) 3 mm reference input.

The IPMC PID controller has been tuned using IFT for a range of different target values, resulting in a set of tuned linear controllers for different reference trajectories. Now in order to turn the finite linear controllers into a continuous controller, the control parameters must be interpolated in order to schedule the gain continuously over the operating range. The tuned controller gains Kp , Ki and Kd are plotted in Fig. 9 below as a function of target displacement. There is a clear trend in the relationship with the control parameters and target displacements. After some analysis it was found that a logarithmic fit would give the best correlation between these parameters. These trends for the control parameters are plotted on the graph below and the relationships are presented below in Eqs. (15)–(17). Kp = −839.7 ln(r) − 3757.6

(15)

Ki = −808.0 ln(r) − 3560.2

(16)

Kd = 84.73 ln(r) + 1038.1

(17)

The controller was also tuned for references of 100 ␮m and 200 ␮m but because of the low SNR the parameters were very inconsistent. The tuned values for Kp and Ki at 100 ␮m and 200 ␮m were in the region of those obtained for 300 ␮m and the values for Kd were slightly lower than that for 300 ␮m at these small targets. For this reason it was decided to restrict the controller parameters at their 300 ␮m values i.e. if the target displacement was less than 300 ␮m then use the values scheduled for 300 ␮m. This is also necessary to prevent Kp and Ki from increasing extremely high because their schedule approaches the zero reference asymptotically.

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Fig. 8. (a) Kp , (b) Ki and (c) Kd , plotted for each tuning iteration, as a function of target displacement.

An interesting observation is that if the IPMC was linear, then the controller gains would be constant across all target displacements. This itself validates that the IFT algorithm has successfully tackled the nonlinear characteristics of the IPMC. It can also be observed that the IFT algorithm realizes that the SNR is low at small displacements and consequently reduces KP accordingly as not to amplify the noise at these levels. Now the schedule to change the gains has been developed, fGS (r), the nonlinear GS controller is complete and ready for implementation.

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7. Results for gain scheduled controller The developed nonlinear GS controller was tested for a number of different reference trajectories and its performance compared to a conventional PID controller. It was decided to use the tuned controller parameters for 1.5 mm displacement for the PID controller, as this is in the middle of the range of IPMC operation. The design criteria for IFT, Eq. (1), was used as a quantitative measure of the performance of the controllers and qualitative measures such as overshoot and settling time are also analyzed. It was clear that after tuning the GS controller for different step inputs, as per Table 1, that the GS controller would outperform the PID controller for all step target displacements except for at 1.5 mm, where the two controllers will be the same. Consequently no comparisons for the step inputs are presented. In order to test the performance of the GS controller versus the conventional PID controller for changes in set point, which is what the controller is designed for, a random stair-step sequence of varying amplitude was input. Both the micro and macro targets were used. The results of this test are shown in Fig. 10. By examining the plot it can be seen that the GS controller had a much smaller overshoot at all of the set point changes, except for at micro targets (30 s and 90 s). This is due to the fact that the proportional gain is high when the target displacement is low as seen in Fig. 9. This suggests that the cut-off region in the schedule, which was set at 300 ␮m, may be too low. Using a higher cut-off (say 500 ␮m) will restrict the Kp and Ki values at micro displacements and the GS controller may not over shoot as much. The settling time after a set point change is better for the GS controller in all cases, even at micro targets, where there is more overshoot. Comparing the overall error for the controllers using the design criteria the GS controller is 17% better.

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4

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J(GS) = 1.02e−7 In order to demonstrate the versatility of the GS controller to other input signals, sinusoid reference trajectories were tested versus the conventional PID controller. This will result in dynamically varying control parameters as the reference signal is continuously changing. A number of experiments were undertaken to assess the performance under different conditions inside the desired operating range of the IPMC. This will also test the robustness of the GS controller with respect to the GS guideline that the scheduled parameters must vary slowly. Fig. 11(a) shows the 33 × 10−2 Hz signal with a micro amplitude of 500 ␮m. It can be seen that both controllers follow the reference very well, despite a relatively high level of noise. By inspection the performance of both controllers are comparable, but using the design criteria it can be seen that the GS controller does perform better, J(PID) = 1.32e−9 and J(GS) = 1.15e−9 .

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Fig. 11(b) shows the 33 × 10−3 Hz signal with a large amplitude of 3 mm. Both controllers track the reference extremely well and again by inspection the performance of both controllers are comparable. Using the design criteria, J(PID) = 6.55e−9 and J(GS) = 5.35e−9 so again the GS controller does perform better. Fig. 12(b) shows the performance with reference amplitude of 3 mm at 0.1 Hz. Similar to the 500 ␮m reference it is clear to see that the standard PID controller exhibits overshoot. This high level of overshoot again results in an output lag. By inspection the GS controller performs a great deal better and this can be confirmed by the design criteria, J(PID) = 13.6e−8 and J(GS) = 6.96e−8 . It has been shown that the designed controllers can accurately track a dynamic reference with a time period of 30 s, so a faster signal was tested to push the guideline of slowly varying parameters. It was decided that a time period of 10 s should be tested as this is nearing the limit of the speed of the IPMC. Fig. 12(a) shows the 0.1 Hz performance at a 500 ␮m amplitude for the two controllers. It can be seen that the standard PID controller has a considerable level of overshoot and then consequently lags the reference signal. It is clear that the GS controller is performing better and this is confirmed by the design criteria, J(PID) = 16.3e−9 and J(GS) = 2.35e−9 .

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The IFT algorithm has the major advantage of being model-free which is extremely useful for IPMCs and other systems where a model of the system is not available and developing one may be too complex. IFT is also an automatic tuning process and therefore removes the need of a skillful operator. With regard to IPMCs this is a major step forward, as modeling has been a large research focus for 15 years and there is still no widely accepted model for the actuation response. This will have a major impact in this field and will aid in the implementation of IPMCs into real systems, where up until now have been restricted due to limitations in modeling and control. This algorithm is transferable to any different material combination and geometry of IPMC. The main disadvantage is the

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number or experiments that need to be performed on the actual system which may be undesirable in some applications. If some prior knowledge or primitive model of the system is available to set the initial controller parameters, the number of iterations can be dramatically reduced. IFT is based on the optimization of a specific control design criteria. It has been shown that the for a time invariant system if the IFT step size and starting controller are chosen appropriately the system will converge to this optimum state. The IPMC is a time-varying system and as such the plant dynamics will change between and even within an iterative experiment. As such it is not possible to confirm that the controller will converge to an optimal state, as the entire system will constantly be changing. To overcome this tuning must be either done online, throughout the normal operation, or must be undertaken at timely intervals, i.e. when the IPMC dynamics vary so the controller performance no longer meets the design specifications, the system should be tuned again. The IPMC system was tuned for 5 iterations only. This was done to restrict the number and time of experiments to try and keep the system as time invariant as possible, as the IPMC performance does fluctuate with time. If the system had been tuned over more iterations, a more optimal controller may have been achieved, although Fig. 7(a)–(c) demonstrate that the design criterion converges very quickly after only 5 iterations. It should also be noted that the gradient search algorithm used for this implementation of IFT will converge to a local minimum. If the initial controller was at another state far away from the one used, then the controller may have converged to a different minimum resulting in an alternate optimally tuned controller. The final GS controller exhibits overshoot to set point changes as a trade-off with settling time. This characteristic of the system can easily be altered by placing a weighting filter, wt in the design criteria, Eq. (18), when tuning the controller to ensure more emphasis on either the transient or steady state performance. To place more influence on the steady state, wt can be set to zero in the transient or initial time period after a set point change then to one after that. A number of variations of wt can be implemented to penalize different performance characteristics, such as overshoot, etc. 1  wt (˜yt ()2 ) 2N n

J() =

(18)

t=1

The hysteresis behaviour is not directly addressed by this controller, as some model or knowledge of the system would be necessary to account for this [36], which is specifically what the IFT algorithm is avoiding. Despite this the controller may well instinctively compensate for some part of the hysteresis when it is automatically tuned due to its nonlinear nature. Despite the fact no complete model exists for IPMCs, some model-based control methods have shown reasonable performance using only an approximate model [13,14,37]. These methods can operate well over a short period of time, but if the system is to run continuously the IPMCs dynamics vary far from their initial state and a model following controller can become unacceptable. Over a long period of time the system dynamics drift far from the developed model making the effort spent on modeling the system redundant. The system dynamics can change so much that they will even shift outside the acceptable range of a robust control design. The IFT algorithm overcomes these issues as it will adaptively tune towards an optimal state whatever the system dynamics change to. 9. Conclusions A GS controller has been developed using a PID architecture with the controller parameters, Kp , Ki and Kd scheduled based on the input reference trajectory. This was done to improve performance

over a large range of displacements, both micro and macro, because the nonlinearity of the IPMC is based on actuation level. An automatic, model-free tuning method called IFT was implemented to tune the controller parameters at different target displacements. The IFT algorithm iterates the parameters to optimize the system, based on a design criterion using a least squares fit of the IPMC output. A schedule for the control parameters with respect to the target displacement has been developed by interpolating between the optimally tuned parameters. The performance of the GS controller was tested against a standard PID controller and has shown improved performance at large and small deflections and at different frequencies. Overall a model-free approach has been developed, which provides an adaptive approach for IPMC research as this eliminates the need for models in control. This will help aid in the implementation of IPMCs into real world applications. Acknowledgement The authors would like to thank Prof. Kwang Kim of the Mechanical Engineering Department, University of Nevada, Reno, USA for providing the IPMC transducers used in this research. References [1] M. Shahinpoor, K.J. Kim, Ionic polymer–metal composites: I. Fundamentals, Smart Materials and Structures 10 (4) (2001) 819–833. [2] K.J. Kim, S. Tadokoro, Electroactive Polymers for Robotic Applications: Artificial Muscles and Sensors, Springer, London, 2007. [3] K. Yun A Novel Three-Finger IPMC Gripper for Microscale Applications, Ph.D. Thesis Texas A&M University, 2006. [4] M. Shahinpoor, Electromechanics of ionoelastic beams as electrically controllable artificial muscles, in: Smart Structures and Materials: Electroactive Polymer Actuators and Devices, SPIE, Newport Beach CA, 1999. [5] H.H. Lin, B.K. Fang, M.S. Ju, C.C.K. Lin, Control of ionic polymer-metal composites for active catheter systems via linear parameter-varying approach, Journal of Intelligent Material Systems and Structures 20 (3) (2009) 273–282. [6] Y. Bar-Cohen, Electroactive Polymer (EAP) Actuators as Artificial Muscles. Reality, potential, and challenges, American Institute of Aeronautics and Astronautics. Paper #2001-1492, 2001. [7] S. Manley, A.J. McDaid, K.C. Aw, E. Haemmerle, S. Xie, Experimental Performance and Feasibility of a Miniature Single-Degree-Of-Freedom Rotary Joint with Integrated IPMC Actuator, Electroactive Polymers and Devices, SPIE San Diego, USA (2009). [8] C. Bonomo, L. Fortuna, P. Giannone, S. Graziani, S. Strazzeri, A nonlinear model for ionic polymer metal composites as actuator, Smart Materials and Structures 16 (1) (2007) 1–12. [9] G.D. Bufalo, L. Placidi, M. Porfiri, A mixture theory framework for modelling the mechanical actuation of ionic polymer metal composites, Smart Materials and Structures 17 (2008). [10] K. Yagasaki, H. Tamagawa, Experimental estimate of viscoelastic properties for ionic polymer-metal composites, Physical Review 70 (5) (2004). [11] K. Yun, W.J. Kim, Microscale position control of an electroactive polymer using an anti-windup scheme, Smart Materials and Structures 15 (4) (2006) 924–930. [12] R.C. Richardson, M.C. Levesley, M.D. Brown, J.A. Hawkes, K. Watterson, P.G. Walker, Control of ionic polymer metal composites, IEEE/ASME Transactions on Mechatronics 8 (2) (2003) 245–253. [13] K.K. Ahn, D.Q. Truong, D.N.C. Nam, J.I. Yoon, S. Yokota, Position control of ionic polymer metal composite actuator using quantitative feedback theory, Sensors and Actuators A: Physical 159 (2010) 204–212. [14] J. Brufau-Penella, K. Tsiakmakis, T. Laopoulos, M. Puig-Vidal, Model reference adaptive control for an ionic polymer metal composite in underwater applications, Smart Materials and Structures 17 (4) (2008). [15] B.C. Lavu, M.P. Schoen, A. Mahajan, Adaptive intelligent control of ionic polymer–metal composites, Smart Materials and Structures 14 (4) (2005) 466–474. [16] D.J. Leith, W.E. Leithead, Survey of gain-scheduling analysis and design, International Journal of Control 18 (11) (2000) 1001–1025. [17] W.J. Rugh, J.S. Shamma, Research on gain scheduling, Automatica 36 (10) (2000) 1401–1425. [18] W.J. Rugh, Analytical framework for gain scheduling, IEEE Control System Magazine 11 (1) (1991) 79–84. [19] J.S. Shamma, M. Athans, Gain scheduling: potential hazards and possible remedies, IEEE Control System Magazine 12 (3) (1992) 101–107. [20] J.S. Shamma, M. Athans, Analysis of gain scheduled control for nonlinear plants, IEEE Transactions on Automatic Control 35 (8) (1990) 898–907.

A.J. McDaid et al. / Sensors and Actuators A 164 (2010) 137–147 [21] R.A. Nichols, R.T. Reichert, W.J. Rugh, Gain scheduling for H∞ controllers: a flight control example, IEEE Transactions Control Systems Technology 1 (2) (1993) 69–79. [22] R.A. Hyde, K. Glover, The application of scheduled H∞ controllers to a VSTOL aircraft, IEEE Transactions on Automatic Control 38 (7) (1993) 1021–1039. [23] J.S. Shamma, M. Athans, Guaranteed properties of gain scheduled control of linear parameter-varying plants, Automatica 27 (3) (1991) 559–564. [24] C. S. Kothera Characterization, Modeling, and Control of the Nonlinear Actuation Response of Ionic Polymer Transducers, Ph.D. Thesis, Virginia Polytechnic Institute and State University, 2005. [25] A.J. McDaid, K.C. Aw, E. Haemmerle, S. Xie, A nonlinear scalable model for designing ionic polymer–metal composite actuator systems, Proceedings of SPIE – The International Society for Optical Engineering (2009) 74930P. [26] A.J. McDaid, K.C. Aw, E. Haemmerle, S. Xie, A conclusive scalable model for the complete actuation response for IPMC transducers, Smart Materials and Structures 19 (7) (2010). [27] H. Hjalmarsson, Iterative feedback tuning – an overview, International Journal of Adaptive Control and Signal Processing 16 (5) (2002) 373–395. [28] H. Hjalmarsson, S. Gunnarsson, M. Gevers, A convergent iterative restricted complexity control design scheme, in: Proceedings of the 33rd IEEE Conference on Decision and Control vol. 2, 1994, pp. 1735–1740. [29] A.E. Graham, A.J. Young, S. Xie, Rapid tuning of controllers by IFT for profile cutting machines, Mechatronics 17 (2007) 121–128. [30] S. Kissling, P. Blanc, P. Myszkorowski, I. Vaclavik, Application of iterative feedback tuning (IFT) to speed and position control of a servo drive, Control Engineering Practice 17 (7) (2009) 834–840. [31] H. Hjalmarsson, M. Gevers, S. Gunnarsson, O. Lequin, Iterative feedback tuningtheory and applications, IEEE Control Systems 26 (1998). [32] A. Tay, K.H. Weng, D. Jiewen, K.L. Boon, Control of photoresist film thickness: Iterative feedback tuning approach, Computers & Chemical Engineering 30 (3) (2006) 572–579. [33] M. Anton, A. Aabloo, A. Punning, M. Kruusmaa, A mechanical model of a nonuniform ionomeric polymer metal composite actuator, Smart Materials and Structures 17 (2) (2008). [34] A. Hunt, A. Punning, M. Anton, A. Aabloo, M. Kruusmaa, A multilink manipulator with IPMC joints, SPIE-Electroactive Polymers and Devices (EAPAD) (2008). [35] D. Liu, Design and Control of an IPMC Actuated Single Degree of Freedom Rotary Joint, Master’s Thesis, The University of Auckland, New Zealand, 2010. [36] Z. Chen, L. Hao, D. Xue, X. Xu, Y. Liu, Modeling and control with hysteresis and creep of IPMC actuator, in: Proceeding of Control and Decision Conference, July 2008, 2008, pp. 865–870.

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[37] S. Kang, J. Shin, S.J. Kim, H.J. Kim, Y.H. Kim, Robust control of ionic polymer–metal composites, Smart Materials and Structures 16 (6) (2007) 2457–2463.

Biographies Andrew McDaid received his Bachelor of Engineering Degree, with 1st class honours in mechatronics, from The University of Auckland, New Zealand in 2007. He is currently a University of Auckland Doctoral Scholar studying towards a PhD in Mechanical Engineering. His main research area is smart/functional materials for sensors and actuators and their implementation into real world applications. His other research interests include intelligent mechatronics systems and devices, especially at the micro/nano scale as well as bio-mechatronics and bio-medical robotics. Kean C. Aw is a senior lecturer in the Department of Mechanical Engineering at University of Auckland. He gained his M.Sc. in Advanced Manufacturing Systems from Brunel University and earned his Ph.D. in Applied Physics. He has over a decade of industrial experiences at Intel, Altera and Navman before joining the university. He teaches under-graduate courses and supervises post-graduate research in Mechatronics Engineering. His current research interests are bio-mechatronics, micro-systems and smart materials. Shane Xie received his Ph.D. from Huazhong University of Science and Technology (China), and University of Canterbury (New Zealand). He is an associate professor at University of Auckland, New Zealand. His research interests include intelligent mechatronics systems, vision techniques and applications, smart sensors and actuators, Bio-mechatronics and Biomedical robotics. He has more than 15 years of teaching and research experience in mechatronics and robotics. He is the editor of two international journals, and is an editorial board member and scientific advisory member for many international journals and conferences. He has published more than 150 papers in refereed international journals and conferences. Enrico Haemmerle is an associate professor and head of the Mechatronics Engineering group at the University of Auckland, New Zealand and director of Paric Limited, an indoor position location technology company. He lectures in Technology Management and Mechatronics Engineering subjects, is an experienced product designer and has research interests in medical devices, zinc oxide and piezoelectric micro and nano systems as well as computer vision systems.