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Application of iterative feedback tuning (IFT) to speed and position control of a servo drive
ARTICLE IN PRESS Control Engineering Practice 17 (2009) 834–840
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Application of iterative feedback tuning (IFT) to speed and position control of a servo drive S. Kissling a, Ph. Blanc a, P. Myszkorowski b, I. Vaclavik a, a b
University of Applied Sciences of Business and Engineering, Route de Cheseaux 1a, 1401 Yverdon-les-Bains, Switzerland ¨ ller (Suisse) SA, 1401 Yverdon-les-Bains, Switzerland Baumu
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2008 Accepted 12 February 2009 Available online 13 March 2009
Iterative feedback tuning (IFT) is applied to tune the cascade speed and position controller of a permanent magnet (PM) servo drive. Several variants of experimentally obtained criteria for the test bed are analyzed. The results of the IFT controller, which was easily implemented on a programmable logic controller (PLC), are compared with the results of three tuning schemes commonly used in the industry. Experimental tests on the PM motor with different types of load show that IFT performance in transient conditions is as good as or better than other methods. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Iterative feedback tuning Speed and position motor controller PM servo drive
1. Introduction Permanent magnet (PM) brushless motors and their control drives are used in a wide range of industrial applications, including packaging, web handling, printing, and metal cutting. Tuning the speed and position control parameters of these highly dynamic servo drives is a time-consuming task. A variety of approaches to tuning have been reported in the literature. Specifically, manual tuning, Ziegler–Nichols (ZN) tuning (Shinskey, 1994), and robust control tuning (Grear, Cafuta, Kumin, & Jezernik 1999) are widely used in industrial applications. System parameters, such as load inertia and friction, change with operating conditions; for this reason, it is highly desirable to have an automatic tuning method in place (Thirusakthimurugan & Dananjayan, 2006). A genetic algorithm has been applied for tuning a speed controller (de Soussa, Caux, Fadel, & Lima, 2007). In Pritschow and Bretschneider (1999), it is suggested that both position and speed controllers be tuned simultaneously; however, the time required for minimizing the performance criterion is long (15 min). Lin, Shieh, Shyu, and Huang (2004) analyzed speed and position cascade controller tuning for linear induction motors, but an accurate knowledge of the controlled system was prerequisite for the method to work. Another possible approach to automatic tuning of speed and position motor control parameters is iterative feedback tuning
Corresponding author. Tel.: +41 24 5576 192; fax: +41 24 5576 320.
E-mail address: [email protected] (I. Vaclavik). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.02.005
(IFT), developed by Hjalmarsson, Gunarsson, and Gevers (1994) and Hjalmarsson, Gevers, Gunarsson, and Lequin (1998). Hildebrand, Lecchini, Soilari, and Gevers (2005) analyzed the IFT asymptotic accuracy. Veres and Hjalmarsson (2002) considered the robustness of the IFT controller tuning method. Lequin, Gevers, Mossberg, Bosmans, and Triest (2003) compared IFT with classical tuning rules. Several applications of the IFT algorithm with experimental results have been reported, including control of profile cutting machines (Graham, Young, & Xie, 2007), control of photoresistant film thickness (Tay, Young, Ho, Deng, & Lok, 2006), and control of two-mass spring systems with friction (Hamamoto, Fukuda, Sugie, & Triest, 2003; Nilkhamhang and Sano 2005). Lee, Tan, Lim, and Dou (2000) and Wu and Ding (2007) presented the application of IFT for linear motor control parameters. Interested readers are referred to other papers compiled in a special section devoted to IFT that appeared in the November 2003 issue of Control Engineering Practice. In the present paper, the IFT algorithm is adapted to the tuning of a typical cascade structure, consisting of speed and position controllers of a PM servo drive. The remainder of the paper is organized as follows: The servo drive controller and the experimental platform are presented in Section 2. IFT is introduced in Section 3. Programmable logic controller (PLC) implementation of the basic idea for the servo drive controller is presented in Section 4. Practical experiences regarding the test bed and a comparison of the performances resulting from three types of controller tuning are reported in Section 5. Finally, some concluding remarks are presented in Section 6.
ARTICLE IN PRESS S. Kissling et al. / Control Engineering Practice 17 (2009) 834–840
2. System description
discrete norms defined by
The controlled system used was a two-mass system in which the first mass represents the motor and the second mass represents the load. The shaft with rigid coupling is considered inertia free. Fig. 1 shows a schematic diagram of this platform, which enables modification of the motor and load inertia, Jm and Jl, respectively. The drive of each motor is able to supply sufficient accelerating current and continuous current (IRMS) for the application’s dutycycle requirements. Moreover, no application-dependent motion constraints, such as position-related constraints, were taken into account during the experiments. The angular motor position was measured by an incremental rotary encoder with 32 768 counts/rev or, equivalently, a resolution of 0.110 arcsec. The motor parameters are presented in Table 1. The experimental platform (Fig. 2) was manufactured by Baumu¨ller SA. The experiments were performed at different inertia ratios of Jl/Jm ¼ a, where a ¼ 0, 5, 11. At the inertia ratio of Jl/Jm ¼ 11, the values B ¼ 0 and 1.4 103 Nm s/rad of viscous friction were examined.
jjxðq¯ Þjj1 ¼
2.1. Closed-loop servo drive model The commonly used control strategy for PM servo drives is cascade control, which consists of an inner current, a speed loop and an outer position loop. The cascaded position, speed, and current controllers are designed to ensure separation of the position, speed, and current loops in a frequency domain with a minimum of interactions. Only the speed and position controllers are adjusted with IFT. In the block diagram illustrated in Fig. 3, the mechanical system is represented as a one-mass system with total inertia Jl+Jm, due to the rigid coupling between the motor and the load. The speed controller is of type PI; the position controller is a simple gain Kp.
jjxðq¯ Þjj2 ¼
N 1 X jxðq¯ ; t k Þj N k¼1
835
and
N 1 X ðxðq¯ ; t k ÞÞ2 N k¼1
!1=2 .
Finally, L, L1 , and * denote the Laplace transform, the inverse Laplace transform, and the convolution product, respectively. 3.2. Principle As in Hjalmarsson et al. (1998), a closed-loop system with a one-degree-of-freedom controller is considered, as shown in Fig. 4. In this loop, Gc ðq¯ ; sÞ and Ga(s) are the transfer functions of the controller and the controlled system, and eðq¯ Þ ¼ w yðq¯ Þ, w, yðq¯ Þ, and uðq¯ Þ are the error, the reference, the output (motor speed or position), and the command signal, respectively. A classical method for determining the optimal control parameters is to introduce a criterion of the following type: JðpÞ ¯Þ ¼ l ðq
1 ðjjeðq¯ Þjjpp þ l jjuðq¯ Þjjpp Þ; p
p ¼ 1 or 2,
(1)
where l is a positive constant that expresses the relative importance of the command signal. In its initial formulation, by Hjalmarsson et al. (1994), IFT consists of introducing the criterion J ðpÞ ¯ Þ with p ¼ 2 and using l ðq the gradient method to compute a sequence of iterates q¯ ð1Þ ; q¯ ð2Þ ; . . . starting with the initial guess q¯ ð0Þ such that the criterion Jð2Þ ¯ ðiÞ Þ l ðq ð q Þ, expected to be global. The converges to a minimum of J ð2Þ l ¯
3. Principle of IFT 3.1. Notations xðq¯ Þ is used to denote a temporal signal depending on q¯ ¼ ½r1 ; r2 ; . . . ; rm T , where ri are the parameters of controller PI or P. For a given time, T40, and a positive integer, N, set Dt ¼ T/N, tk ¼ k Dt for k ¼ 0,1,y,N and introduce jjxðq¯ Þjjp , p ¼ 1, 2, two
Fig. 1. Mechanics of the test machine with motor, load, and coupling.
Fig. 2. Photo of the experimental platform used for the implementation of IFT.
Table 1 Motor parameters. Motor type
Nominal speed (rpm)
Nominal torque Tn (Nm)
Torque constant (Nm/A)
Motor inertia Jm (kg cm2)
DSD028M44U60-3
6000
0.9
0.38
0.2
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Fig. 3. Block diagram of the feedback system with PM motor.
and @Yðq¯ Þ @ðln Gc ðq¯ ÞÞ ð1Þ ¼ Y ðq¯ Þ, @q¯ @q¯
(5)
where Eðq¯ Þ ¼ W Yðq¯ Þ and Y ð1Þ ðq¯ Þ ¼ Fig. 4. Closed-loop system.
Gc ðq¯ ÞGa Eðq¯ Þ. 1 þ Gc ðq¯ ÞGa
(6)
A similar computation shows that the Laplace transform Uðq¯ Þ ¼ Lðuðq¯ ÞÞ satisfies computation of q¯ ðiþ1Þ uses the relation
@Uðq¯ Þ @ðln Gc ðq¯ ÞÞ ð1Þ ¼ U ðq¯ Þ, @q¯ @q¯
ð2Þ
q¯ ðiþ1Þ ¼ q¯ ðiÞ gi
@Jl ðq¯ ðiÞ Þ, @q¯
(2)
where gi is an appropriately chosen positive number. It is worthwhile to note that the IFT algorithm does not require knowledge of the transfer function Ga(s), which is generally unknown. As a consequence, the criterion J cannot be explicitly described as a function of the parameters, regardless of their number. The IFT algorithm provides a very elegant way of estimating the quantity @J ð2Þ ¯ ðq¯ ðiÞ Þ by performing two experiments on the l =@q closed-loop system. Details of this algorithm are presented below for the sake of completeness. Since @yðq¯ ; t k Þ eðq¯ ; t k Þ @q¯ k¼1 ! N X @uðq¯ ; t k Þ þl , uðq¯ ; t k Þ @q¯ k¼1
@J ð2Þ l ðqÞ ¼ 1 ¯ @q¯ N
N X
(3)
the functions @yðq¯ Þ=@q¯ and @uðq¯ Þ=@q¯ must be computed. Dropping the symbol s for ease of notation, the Laplace transforms Yðq¯ Þ ¼ Lðyðq¯ ÞÞ and W ¼ LðwÞ are related as follows: Yðq¯ Þ ¼
Gc ðq¯ ÞGa W. 1 þ Gc ðq¯ ÞGa
(4)
And, therefore, @Yðq¯ Þ @Gc ðq¯ Þ=@q¯ Gc ðq¯ ÞGa ¼ W. Gc ðq¯ Þ ð1 þ Gc ðq¯ ÞGa Þ2 @q¯ By applying the identity z=ð1 þ zÞ2 ¼ z=ð1 þ zÞð1 z=ð1 þ zÞÞ to the previous relation with z ¼ Gc ðq¯ ÞGa , together with Eq. (4), the following expressions are obtained: @Yðq¯ Þ @ðln Gc ðq¯ ÞÞ Gc ðq¯ ÞGa ¼ Eðq¯ Þ 1 þ Gc ðq¯ ÞGa @q¯ @q¯
(7)
where U ð1Þ ðq¯ Þ ¼
Gc ðq¯ Þ Eðq¯ Þ. 1 þ Gc ðq¯ ÞGa
(8)
Comparing Eqs. (4), (6), and (8) makes clear that yð1Þ ðq¯ Þ ¼ L1 ðY ð1Þ ðq¯ ÞÞ and uð1Þ ðq¯ Þ ¼ L1 ðU ð1Þ ðq¯ ÞÞ are the output and the command signal, respectively, for the input eðq¯ Þ. This observation suggests the following procedure to achieve the (i+1)th iteration of the gradient method: 1. Perform an experiment with the reference input signal w and with the parameters q¯ ðiÞ , and record the output signal yðq¯ ðiÞ Þ and the command signal uðq¯ ðiÞ Þ. 2. Perform an experiment with the reference input eðq¯ ðiÞ Þ ¼ w yðq¯ ðiÞ Þ, and record the output signal yð1Þ ðq¯ ðiÞ Þ and the command signal uð1Þ ðq¯ ðiÞ Þ. 3. Use Eqs. (5) and (7) to compute ! @yðq¯ ðiÞ Þ ¯ ðiÞ ÞÞ 1 @ðln Gc ðq ¼L nyð1Þ ðq¯ ðiÞ Þ, @q¯ @q¯ ! @uðq¯ ðiÞ Þ @ðln Gc ðq¯ ðiÞ ÞÞ ¼ L1 nuð1Þ ðq¯ ðiÞ Þ. @q¯ @q¯ 4. Use Eq. (3) to compute @J ð2Þ ¯ ðq¯ ðiÞ Þ, and use Eq. (2) to compute l =@q q¯ ðiþ1Þ . Criterion (1) with p ¼ 1 is analyzed in Kammer, De Bruyne, and Bitmead (1999). A variant with p ¼ 2, in which weighted discrete norms appear, is given in Lequin et al. (2003). 3.3. Adaptation of the experimental platform In order to obtain the fundamental closed-loop control representation (Fig. 4) used for the development of the IFT
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(ISEC), integrated square error (ISE), and integrated absolute error (IAE) (see Kammer et al., 1999), respectively, were adopted for tuning the parameters of the speed controller. The seemingly high value of l ¼ 2000 was due to the units chosen for the error (rad/s) and the command signal (Nm). For a better understanding of the criterion behavior T JðpÞ l ð½K v ; T v Þ, considered as a function of Kv and Tv, three graphs are shown in Fig. 6. These were obtained by computing the criteria for (Kv, Tv) belonging to an admissible grid of parameters. 4.2. Methodology for adjusting the speed controller parameters The method for computing the optimal parameters Kv and Tv is a consequence of the following remarks, which are based on observation of the graphs of the criteria as shown in Fig. 6a–c: Fig. 5. (a) Motor speed controller loop and (b) motor position controller loop.
algorithm, the actual control structure (Fig. 3) must first be reduced to the system shown in Fig. 5a for the speed controller and the system shown in Fig. 5b for the position controller. Due to the cascade structure, first the speed controller is tuned, and then the position controller is tuned. For completeness, note that the derivatives that appear in the aforementioned gradient method are given by @ðln Gcv ðK v ; T v ÞÞ 1 ¼ , @K v Kv @ðln Gcv ðK v ; T v ÞÞ 1 , ¼ @T v T v ð1 þ s T v Þ
T 1. For all fixed Tv, the criterion Jð2Þ 2000 ð½K v ; T v Þ, considered as a function of Kv, has a global minimum. T 2. For all fixed Kv, the criterion J ð2Þ 0 ð½K v ; T v Þ, considered as a function of Tv, has a global minimum. T 3. Remark 2 also applies to the criterion J ð1Þ 0 ð½K v ; T v Þ, and the associated global minimum is more precisely localized.
These observations justify the following iterative method for adjusting the parameters of the speed controller described in Fig. 5a: ð0Þ T 1. Choose an initial guess ½K ð0Þ v ; Tv . 2. Compute successively using a gradient method:
@ðln Gcp ðK p ÞÞ 1 ¼ . @K p Kp
ð2Þ ð0Þ T K ð1Þ v ¼ arg min J 2000 ð½K v ; T v Þ;
4. Implementation
ð1Þ ð1Þ T T ð1Þ v ¼ arg min J 0 ð½K v ; T v Þ;
Kv
Tv
In this section, an implementation of the IFT algorithm for PM motor speed control parameters is proposed. The methodology for position control parameter tuning is similar.
ð2Þ ð1Þ T K ð2Þ v ¼ arg min J 2000 ð½K v ; T v Þ; Kv
4.1. Experimental evaluation of the design criteria ð1Þ ð2Þ T T ð2Þ v ¼ arg min J 0 ð½K v ; T v Þ.
A key step in the design of IFT control is finding a suitable T design criterion. As such, the criterion JðpÞ l ð½K v ; T v Þ, defined by (1), was experimentally evaluated for different values of the two parameters p and l. The choice of these parameters is discussed below. With respect to the norm defined in Section 3.1, the values T ¼ 500 ms and N ¼ 500 were used in the sequel. The starting point for each experiment was defined as follows: (Motor speed, Motor torque) ¼ (0, 0). The speed reference signal oc in Fig. 5a was chosen as a ramp with the slope 666 rad/s2; i.e., the speed 200 rad/s is reached in 300 ms and is held for T ¼ 500 ms. The experimental range of variation of the controller parameters Kv and Tv was based on the closed-loop stability requirement. Ideally, a value of l which would lead to a global minimum of the criterion with respect to Kv and Tv with acceptable gain and phase margins is requested. However, the experimental measures do not show the existence of this ideal l. For this reason, different criteria were used to adjust the speed controller parameters Kv and Tv. T T J ð2Þ and Finally, the criteria J ð2Þ 2000 ð½K v ; T v Þ, 0 ð½K v ; T v Þ, ð1Þ T J 0 ð½K v ; T v Þ, referred to as integral of square error and command
Tv
The experiments presented in Section 5.2 show that these four iterations give very good results. It seems reasonable that the command signal uðq¯ Þ be used solely for adjustment of Kv. In fact, in a PI controller, the proportional part acts on the system dynamic (which has to be limited), whereas the integral part is only used to reduce the static error. For the position controller described in Fig. 5b, similarly to the speed controller, the ISEC criterion Jð2Þ 4 ð½K v ; T v T Þ was used 510 because the controller is a simple gain.
5. Comparison of IFT to other methods In this section, the tracking performances of IFT for different motor loads are compared with the following three methods widely used in industrial applications: ZN, redefined ZN (RZN), and trial-and-error (TE). All methods compared were evaluated via an algorithmic method. Performance improvement via manual tuning may be possible. The control algorithms were implemented on PLC connected to the drive with an internal bus with a sampling time of 1 ms.
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Fig. 6. (a) The ISEC criterion vs. speed controller parameters, (b) the ISE criterion vs. speed controller parameters, and (c) the IAE criterion vs. speed controller parameters.
5.1. PI parameters obtained for the four tuning methods The speed and position control parameters for the different tuning methods and motor loads are given in Table 2.
Table 2 Motor controller parameters for different tuning methods and load types. Tuning method
Without load
ZN RZN TE IFT
Jl ¼5 Jm
ZN
10
RZN TE IFT
6.0 10 19
ZN
5.2. Performances In order to compare the four different controller parameter sets, the root mean square (RMS) of speed and position error signals were experimentally evaluated. The RMS was evaluated for smooth reference signals without jerk. The disturbance attenuation for a load step with a magnitude of 20% of the nominal torque was evaluated through the drop impact defined in Fig. 7. 5.2.1. Speed controller The experimental closed-loop motor speed responses obtained by the four PI tuning methods are shown in Fig. 8. These responses together with performance evaluations in Table 3 give that IFT produces the best speed reference tracking and disturbance and T ð1Þ require 13 and attenuation. The computations of K ð1Þ v v ð2Þ require only 11 iterations, respectively, whereas K v and T ð2Þ v 6 and 1 iterations, respectively. For the auto-tuning procedure
Kv (Nm s/rad) 103
Load
Jl ¼ 11 Jm
Viscous friction
Tv (ms)
Kp (1/s)
21.3 25.1 200 38.5
29.3 7.9 23 30.0
45.9
42.9
54.0 100 16.7
16.7 20 40.3
15
51.9
27.3
RZN TE IFT
8.2 15 35
61.0 60.0 18.5
20.7 25.0 60.4
ZN RZN TE IFT
15 8.2 15 28
51.9 61.0 60.0 21.6
27.3 20.7 25.0 49.7
5.8 3.3 1.0 9.0
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1.2 1
φm [rad]
0.8 0.6 0.4 Reference input ZN RZN ESS IFT
0.2 0 -0.2 0
100 200 300 400 500 600 700 800 900 1000 Time [ms]
Fig. 9. Position responses achieved by the four P controllers of Table 2. Disturbance applied at t ¼ 600 ms.
Fig. 7. Load disturbance specification.
250
Table 4 Performance evaluation average of the different loads for the position controller.
200 Reference input
m [rad/s]
150
ZN RZN TE
100
Method used
ERMS (rad)
DI (rad s)
ZN RZN TE IFT
0.184 0.308 0.230 0.139
0.019 0.0889 0.0484 0.00691
IFT
50
0
-50 0
100 200 300 400 500 600 700 800 900 1000 Time [ms]
Fig. 8. Speed responses achieved by the four PI controllers of Table 2 for load disturbance applied at t ¼ 600 ms.
performance evaluations are presented in Fig. 9 and Table 4. Computation of Kp required 11 iterations, and the position controller auto-tuning took less than 20 s. The RMS of the position tracking error with IFT was 1.3 times lower than that of the ZN method. The disturbance attenuation evaluated by drop impact with IFT was 2.9 times lower than that of the ZN method. The response of the IFT-tuned position controller (Fig. 9) showed a slightly oscillatory character. This confirms that the L2-based criterion is not necessarily sensitive to minor oscillatory responses and indicates that the outcome of an automatic procedure may still be subject to manual improvement.
Table 3 Performance evaluation average of the different loads for the speed controller. Method used
ERMS (rad/s)
DI (rad)
5.3. Comments and observations
ZN RZN TE IFT
6.22 12.2 14.9 2.16
0.540 1.29 7.85 0.228
To insure convergence of the gradient method for searching for a minimum of a given criterion, the coefficient gi, which appears in Eq. (2), must be controlled. In fact, from one system to another, or more simply from one parameter to another, this coefficient may vary widely during the iterations. The following methodology for searching for a minimum is proposed. First, the speed controller parameters are initialized with a low gain value and a high time constant of integration; these values are also recommended by manufacturers of drive controller systems. This insures the stability of the system and gives an indication of how the parameters must be modified in the first iteration: the gain increases, and the time constant of integration decreases. Details of the algorithm are provided for the computation of
of the speed controller, in which several starting points are not necessarily close to optimum values, the optimum controller setting was achieved in less than 40 s. The RMS of the speed tracking error achieved with IFT was 2.9 times lower than that of the ZN method. The disturbance attenuation evaluated by drop impact for IFT was 2.4 times better than that of the ZN method.
5.2.2. Position controller The experimental closed-loop motor position responses obtained by the four P controller tuning methods and the
ð2Þ ð0Þ T K ð1Þ v ¼ arg min J 2000 ð½K v ; T v Þ Kv
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which appears in Section 4.2. ð0Þ ð0Þ T 3 i 1. For i ¼ 1,2,y compute J ð2Þ 2000 ð½ð2Þ K v ; T v Þ if the criterion decreases. 2. Continue on the computation using the gradient method. ð2Þ The computation of T ð1Þ v is similar, and the computations of K v and T ð2Þ only use the gradient method. v For practical applications, in the case of multiple drives, this approach works if the actuators are weakly coupled. Otherwise, for example in printing machines with a strong coupling between different axes, some modifications of the auto-tuning algorithm may be necessary. As Fig. 6 suggests, the criteria used in this paper are unimodal. For such cases, the gradient-based model-free IFT control strategy is well suited and seems to converge well to the global minimum.
6. Conclusion This paper describes the application of a control strategy that combines IFT with a cascade position and speed controller on a PM servo drive. The imposed cascade structure of controllers does not allow for the direct application of robust controller methods (e.g., HN). Therefore, no explicit comparison between IFT and HN was performed, despite the promising results obtained by Lee, Lin, and Lin (2005). The experimental results show that stability and performance are maintained with changes in the motor load. The closed-loop transient responses to a step change were better than those of standard position and speed controllers. The model-independent IFT controller tuning strategy can be easily incorporated in existing control loops. References Graham, A. E., Young, A. J., & Xie, S. Q. (2007). Rapid tuning of controllers by IFT for profile cuttings machines. Mechatronics, 17, 121–128. Grear, B., Cafuta, P., Kumin, L., & Jezernik, K. (1999). Robust servo-drive control for dynamic load perturbations. In Proceedings of the IEEE international symposium on industrial electronics (Vol. 3, pp. 1041–1044).
Hamamoto, K., Fukuda, T., Sugie, T., & Triest, L. (2003). Iterative feedback tuning of controllers for a two-mass-spring system with friction. Control Engineering Practice, 11, 1061–1068. Hildebrand, R., Lecchini, A., Soilari, G., & Gevers, M. (2005). Asymptotic accuracy of iterative feedback tuning. IEEE Transactions on Automatic Control, 50, 1182–1185. Hjalmarsson, H., Gevers, M., Gunarsson, S., & Lequin, O. (1998). Iterative feedback tuning: Theory and application. IEEE Control Systems Magazine, 18, 26–41. Hjalmarsson, H., Gunarsson, S., & Gevers, M. (1994). A convergent iterative restricted complexity control design scheme. In Proceedings of the 33rd IEEE conference on decision and control (pp. 1735–1740), Orlando, FL. Kammer, L. C., De Bruyne, F., & Bitmead, R. R. (1999). Iterative feedback tuning via minimization of the absolute error. In 38th IEEE conference on decision and control (pp. 4619–4624). Lee, T. S., Lin, C. H., & Lin, F. J. (2005). An adaptive H1 controller design for permanent magnet synchronous motor drives. Control Engineering Practice, 13, 425–439. Lee, T. H., Tan, K., Lim, S. Y., & Dou, H. F. (2000). Iterative learning control of permanent magnet linear motor with relay automatic tuning. Mechatronics, 10, 169–190. Lequin, O., Gevers, M., Mossberg, M., Bosmans, E., & Triest, L. (2003). Iterative feedback tuning of PID parameters: Comparison with classical tuning rules. Control Engineering Practice, 11, 1023–1033. Lin, F. L., Shieh, H. J., Shyu, K. K., & Huang, P. K. (2004). On-line gain-tuning IP controller using real-coded genetic algorithm. Electrical Power Systems Research, 72, 157–169. Nilkhamhang, I., & Sano, A. (2005). Iterative tuning algorithm for feedforward and feedback control of two-mass motor system with physical parameter identification. In Proceedings of the 2005 IEEE conference on control applications (pp. 1624–1629), Toronto, Canada. Pritschow, G., & Bretschneider, J. (1999). A self-tuning controller for digitally controlled electromechanical servo drives in machine tools. Annals of the CIRP, 14, 307–312. Shinskey, F. G. (1994). Feedback controllers for the process industries. New York: McGraw-Hill. de Soussa, M. A. T. F., Caux, S., Fadel, M., & Lima, A. M. N., 2007. Computer aided thought process for robust PMSM control. In Proceedings of the international conference on computer as a tool, EUROCON 2007 (pp. 1740–1745), September 9–12, Warsaw. Tay, A., Young, A. J., Ho, W. K., Deng, J., & Lok, B. K. (2006). Control of photoresist film thickness: Iterative feedback tuning approach. Computers and Chemical Engineering, 30, 572–579. Thirusakthimurugan, P., & Dananjayan, P. (2006). A robust auto tuning speed control of permanent magnet brushless dc motor. In Proceedings of the international conference on information and automation, ICIA 2006 (pp. 270–273). Veres, S., & Hjalmarsson, H. (2002). Tuning for robustness and performance using iterative feedback tuning. In Proceeding of the 41st IEEE conference on decision and control (pp. 4682–4687), Las Vegas. Wu, J., & Ding, H. (2007). Iterative learning variable structure controller for highspeed and high-precision point-to-point motion. Robotics and Computer Integrated Manufacturing, doi:10.1016/j.rcim2007.02.021.
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Application of iterative feedback tuning (IFT) to speed and position control of a servo drive S. Kissling a, Ph. Blanc a, P. Myszkorowski b, I. Vaclavik a, a b
University of Applied Sciences of Business and Engineering, Route de Cheseaux 1a, 1401 Yverdon-les-Bains, Switzerland ¨ ller (Suisse) SA, 1401 Yverdon-les-Bains, Switzerland Baumu
a r t i c l e in f o
a b s t r a c t
Article history: Received 7 April 2008 Accepted 12 February 2009 Available online 13 March 2009
Iterative feedback tuning (IFT) is applied to tune the cascade speed and position controller of a permanent magnet (PM) servo drive. Several variants of experimentally obtained criteria for the test bed are analyzed. The results of the IFT controller, which was easily implemented on a programmable logic controller (PLC), are compared with the results of three tuning schemes commonly used in the industry. Experimental tests on the PM motor with different types of load show that IFT performance in transient conditions is as good as or better than other methods. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Iterative feedback tuning Speed and position motor controller PM servo drive
1. Introduction Permanent magnet (PM) brushless motors and their control drives are used in a wide range of industrial applications, including packaging, web handling, printing, and metal cutting. Tuning the speed and position control parameters of these highly dynamic servo drives is a time-consuming task. A variety of approaches to tuning have been reported in the literature. Specifically, manual tuning, Ziegler–Nichols (ZN) tuning (Shinskey, 1994), and robust control tuning (Grear, Cafuta, Kumin, & Jezernik 1999) are widely used in industrial applications. System parameters, such as load inertia and friction, change with operating conditions; for this reason, it is highly desirable to have an automatic tuning method in place (Thirusakthimurugan & Dananjayan, 2006). A genetic algorithm has been applied for tuning a speed controller (de Soussa, Caux, Fadel, & Lima, 2007). In Pritschow and Bretschneider (1999), it is suggested that both position and speed controllers be tuned simultaneously; however, the time required for minimizing the performance criterion is long (15 min). Lin, Shieh, Shyu, and Huang (2004) analyzed speed and position cascade controller tuning for linear induction motors, but an accurate knowledge of the controlled system was prerequisite for the method to work. Another possible approach to automatic tuning of speed and position motor control parameters is iterative feedback tuning
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E-mail address: [email protected] (I. Vaclavik). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.02.005
(IFT), developed by Hjalmarsson, Gunarsson, and Gevers (1994) and Hjalmarsson, Gevers, Gunarsson, and Lequin (1998). Hildebrand, Lecchini, Soilari, and Gevers (2005) analyzed the IFT asymptotic accuracy. Veres and Hjalmarsson (2002) considered the robustness of the IFT controller tuning method. Lequin, Gevers, Mossberg, Bosmans, and Triest (2003) compared IFT with classical tuning rules. Several applications of the IFT algorithm with experimental results have been reported, including control of profile cutting machines (Graham, Young, & Xie, 2007), control of photoresistant film thickness (Tay, Young, Ho, Deng, & Lok, 2006), and control of two-mass spring systems with friction (Hamamoto, Fukuda, Sugie, & Triest, 2003; Nilkhamhang and Sano 2005). Lee, Tan, Lim, and Dou (2000) and Wu and Ding (2007) presented the application of IFT for linear motor control parameters. Interested readers are referred to other papers compiled in a special section devoted to IFT that appeared in the November 2003 issue of Control Engineering Practice. In the present paper, the IFT algorithm is adapted to the tuning of a typical cascade structure, consisting of speed and position controllers of a PM servo drive. The remainder of the paper is organized as follows: The servo drive controller and the experimental platform are presented in Section 2. IFT is introduced in Section 3. Programmable logic controller (PLC) implementation of the basic idea for the servo drive controller is presented in Section 4. Practical experiences regarding the test bed and a comparison of the performances resulting from three types of controller tuning are reported in Section 5. Finally, some concluding remarks are presented in Section 6.
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2. System description
discrete norms defined by
The controlled system used was a two-mass system in which the first mass represents the motor and the second mass represents the load. The shaft with rigid coupling is considered inertia free. Fig. 1 shows a schematic diagram of this platform, which enables modification of the motor and load inertia, Jm and Jl, respectively. The drive of each motor is able to supply sufficient accelerating current and continuous current (IRMS) for the application’s dutycycle requirements. Moreover, no application-dependent motion constraints, such as position-related constraints, were taken into account during the experiments. The angular motor position was measured by an incremental rotary encoder with 32 768 counts/rev or, equivalently, a resolution of 0.110 arcsec. The motor parameters are presented in Table 1. The experimental platform (Fig. 2) was manufactured by Baumu¨ller SA. The experiments were performed at different inertia ratios of Jl/Jm ¼ a, where a ¼ 0, 5, 11. At the inertia ratio of Jl/Jm ¼ 11, the values B ¼ 0 and 1.4 103 Nm s/rad of viscous friction were examined.
jjxðq¯ Þjj1 ¼
2.1. Closed-loop servo drive model The commonly used control strategy for PM servo drives is cascade control, which consists of an inner current, a speed loop and an outer position loop. The cascaded position, speed, and current controllers are designed to ensure separation of the position, speed, and current loops in a frequency domain with a minimum of interactions. Only the speed and position controllers are adjusted with IFT. In the block diagram illustrated in Fig. 3, the mechanical system is represented as a one-mass system with total inertia Jl+Jm, due to the rigid coupling between the motor and the load. The speed controller is of type PI; the position controller is a simple gain Kp.
jjxðq¯ Þjj2 ¼
N 1 X jxðq¯ ; t k Þj N k¼1
835
and
N 1 X ðxðq¯ ; t k ÞÞ2 N k¼1
!1=2 .
Finally, L, L1 , and * denote the Laplace transform, the inverse Laplace transform, and the convolution product, respectively. 3.2. Principle As in Hjalmarsson et al. (1998), a closed-loop system with a one-degree-of-freedom controller is considered, as shown in Fig. 4. In this loop, Gc ðq¯ ; sÞ and Ga(s) are the transfer functions of the controller and the controlled system, and eðq¯ Þ ¼ w yðq¯ Þ, w, yðq¯ Þ, and uðq¯ Þ are the error, the reference, the output (motor speed or position), and the command signal, respectively. A classical method for determining the optimal control parameters is to introduce a criterion of the following type: JðpÞ ¯Þ ¼ l ðq
1 ðjjeðq¯ Þjjpp þ l jjuðq¯ Þjjpp Þ; p
p ¼ 1 or 2,
(1)
where l is a positive constant that expresses the relative importance of the command signal. In its initial formulation, by Hjalmarsson et al. (1994), IFT consists of introducing the criterion J ðpÞ ¯ Þ with p ¼ 2 and using l ðq the gradient method to compute a sequence of iterates q¯ ð1Þ ; q¯ ð2Þ ; . . . starting with the initial guess q¯ ð0Þ such that the criterion Jð2Þ ¯ ðiÞ Þ l ðq ð q Þ, expected to be global. The converges to a minimum of J ð2Þ l ¯
3. Principle of IFT 3.1. Notations xðq¯ Þ is used to denote a temporal signal depending on q¯ ¼ ½r1 ; r2 ; . . . ; rm T , where ri are the parameters of controller PI or P. For a given time, T40, and a positive integer, N, set Dt ¼ T/N, tk ¼ k Dt for k ¼ 0,1,y,N and introduce jjxðq¯ Þjjp , p ¼ 1, 2, two
Fig. 1. Mechanics of the test machine with motor, load, and coupling.
Fig. 2. Photo of the experimental platform used for the implementation of IFT.
Table 1 Motor parameters. Motor type
Nominal speed (rpm)
Nominal torque Tn (Nm)
Torque constant (Nm/A)
Motor inertia Jm (kg cm2)
DSD028M44U60-3
6000
0.9
0.38
0.2
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Fig. 3. Block diagram of the feedback system with PM motor.
and @Yðq¯ Þ @ðln Gc ðq¯ ÞÞ ð1Þ ¼ Y ðq¯ Þ, @q¯ @q¯
(5)
where Eðq¯ Þ ¼ W Yðq¯ Þ and Y ð1Þ ðq¯ Þ ¼ Fig. 4. Closed-loop system.
Gc ðq¯ ÞGa Eðq¯ Þ. 1 þ Gc ðq¯ ÞGa
(6)
A similar computation shows that the Laplace transform Uðq¯ Þ ¼ Lðuðq¯ ÞÞ satisfies computation of q¯ ðiþ1Þ uses the relation
@Uðq¯ Þ @ðln Gc ðq¯ ÞÞ ð1Þ ¼ U ðq¯ Þ, @q¯ @q¯
ð2Þ
q¯ ðiþ1Þ ¼ q¯ ðiÞ gi
@Jl ðq¯ ðiÞ Þ, @q¯
(2)
where gi is an appropriately chosen positive number. It is worthwhile to note that the IFT algorithm does not require knowledge of the transfer function Ga(s), which is generally unknown. As a consequence, the criterion J cannot be explicitly described as a function of the parameters, regardless of their number. The IFT algorithm provides a very elegant way of estimating the quantity @J ð2Þ ¯ ðq¯ ðiÞ Þ by performing two experiments on the l =@q closed-loop system. Details of this algorithm are presented below for the sake of completeness. Since @yðq¯ ; t k Þ eðq¯ ; t k Þ @q¯ k¼1 ! N X @uðq¯ ; t k Þ þl , uðq¯ ; t k Þ @q¯ k¼1
@J ð2Þ l ðqÞ ¼ 1 ¯ @q¯ N
N X
(3)
the functions @yðq¯ Þ=@q¯ and @uðq¯ Þ=@q¯ must be computed. Dropping the symbol s for ease of notation, the Laplace transforms Yðq¯ Þ ¼ Lðyðq¯ ÞÞ and W ¼ LðwÞ are related as follows: Yðq¯ Þ ¼
Gc ðq¯ ÞGa W. 1 þ Gc ðq¯ ÞGa
(4)
And, therefore, @Yðq¯ Þ @Gc ðq¯ Þ=@q¯ Gc ðq¯ ÞGa ¼ W. Gc ðq¯ Þ ð1 þ Gc ðq¯ ÞGa Þ2 @q¯ By applying the identity z=ð1 þ zÞ2 ¼ z=ð1 þ zÞð1 z=ð1 þ zÞÞ to the previous relation with z ¼ Gc ðq¯ ÞGa , together with Eq. (4), the following expressions are obtained: @Yðq¯ Þ @ðln Gc ðq¯ ÞÞ Gc ðq¯ ÞGa ¼ Eðq¯ Þ 1 þ Gc ðq¯ ÞGa @q¯ @q¯
(7)
where U ð1Þ ðq¯ Þ ¼
Gc ðq¯ Þ Eðq¯ Þ. 1 þ Gc ðq¯ ÞGa
(8)
Comparing Eqs. (4), (6), and (8) makes clear that yð1Þ ðq¯ Þ ¼ L1 ðY ð1Þ ðq¯ ÞÞ and uð1Þ ðq¯ Þ ¼ L1 ðU ð1Þ ðq¯ ÞÞ are the output and the command signal, respectively, for the input eðq¯ Þ. This observation suggests the following procedure to achieve the (i+1)th iteration of the gradient method: 1. Perform an experiment with the reference input signal w and with the parameters q¯ ðiÞ , and record the output signal yðq¯ ðiÞ Þ and the command signal uðq¯ ðiÞ Þ. 2. Perform an experiment with the reference input eðq¯ ðiÞ Þ ¼ w yðq¯ ðiÞ Þ, and record the output signal yð1Þ ðq¯ ðiÞ Þ and the command signal uð1Þ ðq¯ ðiÞ Þ. 3. Use Eqs. (5) and (7) to compute ! @yðq¯ ðiÞ Þ ¯ ðiÞ ÞÞ 1 @ðln Gc ðq ¼L nyð1Þ ðq¯ ðiÞ Þ, @q¯ @q¯ ! @uðq¯ ðiÞ Þ @ðln Gc ðq¯ ðiÞ ÞÞ ¼ L1 nuð1Þ ðq¯ ðiÞ Þ. @q¯ @q¯ 4. Use Eq. (3) to compute @J ð2Þ ¯ ðq¯ ðiÞ Þ, and use Eq. (2) to compute l =@q q¯ ðiþ1Þ . Criterion (1) with p ¼ 1 is analyzed in Kammer, De Bruyne, and Bitmead (1999). A variant with p ¼ 2, in which weighted discrete norms appear, is given in Lequin et al. (2003). 3.3. Adaptation of the experimental platform In order to obtain the fundamental closed-loop control representation (Fig. 4) used for the development of the IFT
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(ISEC), integrated square error (ISE), and integrated absolute error (IAE) (see Kammer et al., 1999), respectively, were adopted for tuning the parameters of the speed controller. The seemingly high value of l ¼ 2000 was due to the units chosen for the error (rad/s) and the command signal (Nm). For a better understanding of the criterion behavior T JðpÞ l ð½K v ; T v Þ, considered as a function of Kv and Tv, three graphs are shown in Fig. 6. These were obtained by computing the criteria for (Kv, Tv) belonging to an admissible grid of parameters. 4.2. Methodology for adjusting the speed controller parameters The method for computing the optimal parameters Kv and Tv is a consequence of the following remarks, which are based on observation of the graphs of the criteria as shown in Fig. 6a–c: Fig. 5. (a) Motor speed controller loop and (b) motor position controller loop.
algorithm, the actual control structure (Fig. 3) must first be reduced to the system shown in Fig. 5a for the speed controller and the system shown in Fig. 5b for the position controller. Due to the cascade structure, first the speed controller is tuned, and then the position controller is tuned. For completeness, note that the derivatives that appear in the aforementioned gradient method are given by @ðln Gcv ðK v ; T v ÞÞ 1 ¼ , @K v Kv @ðln Gcv ðK v ; T v ÞÞ 1 , ¼ @T v T v ð1 þ s T v Þ
T 1. For all fixed Tv, the criterion Jð2Þ 2000 ð½K v ; T v Þ, considered as a function of Kv, has a global minimum. T 2. For all fixed Kv, the criterion J ð2Þ 0 ð½K v ; T v Þ, considered as a function of Tv, has a global minimum. T 3. Remark 2 also applies to the criterion J ð1Þ 0 ð½K v ; T v Þ, and the associated global minimum is more precisely localized.
These observations justify the following iterative method for adjusting the parameters of the speed controller described in Fig. 5a: ð0Þ T 1. Choose an initial guess ½K ð0Þ v ; Tv . 2. Compute successively using a gradient method:
@ðln Gcp ðK p ÞÞ 1 ¼ . @K p Kp
ð2Þ ð0Þ T K ð1Þ v ¼ arg min J 2000 ð½K v ; T v Þ;
4. Implementation
ð1Þ ð1Þ T T ð1Þ v ¼ arg min J 0 ð½K v ; T v Þ;
Kv
Tv
In this section, an implementation of the IFT algorithm for PM motor speed control parameters is proposed. The methodology for position control parameter tuning is similar.
ð2Þ ð1Þ T K ð2Þ v ¼ arg min J 2000 ð½K v ; T v Þ; Kv
4.1. Experimental evaluation of the design criteria ð1Þ ð2Þ T T ð2Þ v ¼ arg min J 0 ð½K v ; T v Þ.
A key step in the design of IFT control is finding a suitable T design criterion. As such, the criterion JðpÞ l ð½K v ; T v Þ, defined by (1), was experimentally evaluated for different values of the two parameters p and l. The choice of these parameters is discussed below. With respect to the norm defined in Section 3.1, the values T ¼ 500 ms and N ¼ 500 were used in the sequel. The starting point for each experiment was defined as follows: (Motor speed, Motor torque) ¼ (0, 0). The speed reference signal oc in Fig. 5a was chosen as a ramp with the slope 666 rad/s2; i.e., the speed 200 rad/s is reached in 300 ms and is held for T ¼ 500 ms. The experimental range of variation of the controller parameters Kv and Tv was based on the closed-loop stability requirement. Ideally, a value of l which would lead to a global minimum of the criterion with respect to Kv and Tv with acceptable gain and phase margins is requested. However, the experimental measures do not show the existence of this ideal l. For this reason, different criteria were used to adjust the speed controller parameters Kv and Tv. T T J ð2Þ and Finally, the criteria J ð2Þ 2000 ð½K v ; T v Þ, 0 ð½K v ; T v Þ, ð1Þ T J 0 ð½K v ; T v Þ, referred to as integral of square error and command
Tv
The experiments presented in Section 5.2 show that these four iterations give very good results. It seems reasonable that the command signal uðq¯ Þ be used solely for adjustment of Kv. In fact, in a PI controller, the proportional part acts on the system dynamic (which has to be limited), whereas the integral part is only used to reduce the static error. For the position controller described in Fig. 5b, similarly to the speed controller, the ISEC criterion Jð2Þ 4 ð½K v ; T v T Þ was used 510 because the controller is a simple gain.
5. Comparison of IFT to other methods In this section, the tracking performances of IFT for different motor loads are compared with the following three methods widely used in industrial applications: ZN, redefined ZN (RZN), and trial-and-error (TE). All methods compared were evaluated via an algorithmic method. Performance improvement via manual tuning may be possible. The control algorithms were implemented on PLC connected to the drive with an internal bus with a sampling time of 1 ms.
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Fig. 6. (a) The ISEC criterion vs. speed controller parameters, (b) the ISE criterion vs. speed controller parameters, and (c) the IAE criterion vs. speed controller parameters.
5.1. PI parameters obtained for the four tuning methods The speed and position control parameters for the different tuning methods and motor loads are given in Table 2.
Table 2 Motor controller parameters for different tuning methods and load types. Tuning method
Without load
ZN RZN TE IFT
Jl ¼5 Jm
ZN
10
RZN TE IFT
6.0 10 19
ZN
5.2. Performances In order to compare the four different controller parameter sets, the root mean square (RMS) of speed and position error signals were experimentally evaluated. The RMS was evaluated for smooth reference signals without jerk. The disturbance attenuation for a load step with a magnitude of 20% of the nominal torque was evaluated through the drop impact defined in Fig. 7. 5.2.1. Speed controller The experimental closed-loop motor speed responses obtained by the four PI tuning methods are shown in Fig. 8. These responses together with performance evaluations in Table 3 give that IFT produces the best speed reference tracking and disturbance and T ð1Þ require 13 and attenuation. The computations of K ð1Þ v v ð2Þ require only 11 iterations, respectively, whereas K v and T ð2Þ v 6 and 1 iterations, respectively. For the auto-tuning procedure
Kv (Nm s/rad) 103
Load
Jl ¼ 11 Jm
Viscous friction
Tv (ms)
Kp (1/s)
21.3 25.1 200 38.5
29.3 7.9 23 30.0
45.9
42.9
54.0 100 16.7
16.7 20 40.3
15
51.9
27.3
RZN TE IFT
8.2 15 35
61.0 60.0 18.5
20.7 25.0 60.4
ZN RZN TE IFT
15 8.2 15 28
51.9 61.0 60.0 21.6
27.3 20.7 25.0 49.7
5.8 3.3 1.0 9.0
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1.2 1
φm [rad]
0.8 0.6 0.4 Reference input ZN RZN ESS IFT
0.2 0 -0.2 0
100 200 300 400 500 600 700 800 900 1000 Time [ms]
Fig. 9. Position responses achieved by the four P controllers of Table 2. Disturbance applied at t ¼ 600 ms.
Fig. 7. Load disturbance specification.
250
Table 4 Performance evaluation average of the different loads for the position controller.
200 Reference input
m [rad/s]
150
ZN RZN TE
100
Method used
ERMS (rad)
DI (rad s)
ZN RZN TE IFT
0.184 0.308 0.230 0.139
0.019 0.0889 0.0484 0.00691
IFT
50
0
-50 0
100 200 300 400 500 600 700 800 900 1000 Time [ms]
Fig. 8. Speed responses achieved by the four PI controllers of Table 2 for load disturbance applied at t ¼ 600 ms.
performance evaluations are presented in Fig. 9 and Table 4. Computation of Kp required 11 iterations, and the position controller auto-tuning took less than 20 s. The RMS of the position tracking error with IFT was 1.3 times lower than that of the ZN method. The disturbance attenuation evaluated by drop impact with IFT was 2.9 times lower than that of the ZN method. The response of the IFT-tuned position controller (Fig. 9) showed a slightly oscillatory character. This confirms that the L2-based criterion is not necessarily sensitive to minor oscillatory responses and indicates that the outcome of an automatic procedure may still be subject to manual improvement.
Table 3 Performance evaluation average of the different loads for the speed controller. Method used
ERMS (rad/s)
DI (rad)
5.3. Comments and observations
ZN RZN TE IFT
6.22 12.2 14.9 2.16
0.540 1.29 7.85 0.228
To insure convergence of the gradient method for searching for a minimum of a given criterion, the coefficient gi, which appears in Eq. (2), must be controlled. In fact, from one system to another, or more simply from one parameter to another, this coefficient may vary widely during the iterations. The following methodology for searching for a minimum is proposed. First, the speed controller parameters are initialized with a low gain value and a high time constant of integration; these values are also recommended by manufacturers of drive controller systems. This insures the stability of the system and gives an indication of how the parameters must be modified in the first iteration: the gain increases, and the time constant of integration decreases. Details of the algorithm are provided for the computation of
of the speed controller, in which several starting points are not necessarily close to optimum values, the optimum controller setting was achieved in less than 40 s. The RMS of the speed tracking error achieved with IFT was 2.9 times lower than that of the ZN method. The disturbance attenuation evaluated by drop impact for IFT was 2.4 times better than that of the ZN method.
5.2.2. Position controller The experimental closed-loop motor position responses obtained by the four P controller tuning methods and the
ð2Þ ð0Þ T K ð1Þ v ¼ arg min J 2000 ð½K v ; T v Þ Kv
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which appears in Section 4.2. ð0Þ ð0Þ T 3 i 1. For i ¼ 1,2,y compute J ð2Þ 2000 ð½ð2Þ K v ; T v Þ if the criterion decreases. 2. Continue on the computation using the gradient method. ð2Þ The computation of T ð1Þ v is similar, and the computations of K v and T ð2Þ only use the gradient method. v For practical applications, in the case of multiple drives, this approach works if the actuators are weakly coupled. Otherwise, for example in printing machines with a strong coupling between different axes, some modifications of the auto-tuning algorithm may be necessary. As Fig. 6 suggests, the criteria used in this paper are unimodal. For such cases, the gradient-based model-free IFT control strategy is well suited and seems to converge well to the global minimum.
6. Conclusion This paper describes the application of a control strategy that combines IFT with a cascade position and speed controller on a PM servo drive. The imposed cascade structure of controllers does not allow for the direct application of robust controller methods (e.g., HN). Therefore, no explicit comparison between IFT and HN was performed, despite the promising results obtained by Lee, Lin, and Lin (2005). The experimental results show that stability and performance are maintained with changes in the motor load. The closed-loop transient responses to a step change were better than those of standard position and speed controllers. The model-independent IFT controller tuning strategy can be easily incorporated in existing control loops. References Graham, A. E., Young, A. J., & Xie, S. Q. (2007). Rapid tuning of controllers by IFT for profile cuttings machines. Mechatronics, 17, 121–128. Grear, B., Cafuta, P., Kumin, L., & Jezernik, K. (1999). Robust servo-drive control for dynamic load perturbations. In Proceedings of the IEEE international symposium on industrial electronics (Vol. 3, pp. 1041–1044).
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