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Enhanced Servo-Control Performance of Dual-Mass Systems Timothy M. O’Sullivan, Christopher M. Bingham, Member, IEEE, and Nigel Schofield
Abstract—This paper provides systematic analysis and controller design methods for high-performance two-mass servo drives with an appraisal of the effects of supplementary filtering elements associated with practical systems. Implementation issues and the resulting performance achievable from proportional– integral, proportional–integral–derivative (PID), and resonance ratio control (RRC) controllers with regard to both closed-loop robustness and control of the process variable (load velocity), in response to a step reference speed or load-side disturbance, are presented. It is shown that the high-frequency gain of the controllers is a critical design variable for determining the resulting robustness of the closed-loop system when subject to unmodeled resonant modes, high-frequency noise from the derivative of quantized sensor signals, and process perturbations, and is strongly influenced by the location and number of filters present in the various feedback loops and, importantly, the ratio of their time constants. A complete design methodology is also presented to assign the time constants of the various loop filters, and their location, using a single user-definable variable, thereby reducing the time-consuming trial-and-error approach commonly employed using conventional tuning procedures. The technique employs both time- and frequency-domain design tools to address the conflicting requirements of robustness and control performance (overshoot, bandwidth, etc.). It is also shown that, since the PID and RRC controllers are closely related, they are theoretically able to impart identical closed-loop input–output dynamics. However, by virtue of the different feedback mechanisms employed, RRC is shown to provide superior closed-loop robustness. This paper demonstrates, and practically validates, the proposed techniques by showing significant performance enhancements from a commercial off-the-shelf servo-drive test platform. Index Terms—Acceleration control, motion control, resonance, robustness, surface acoustic wave devices, torque control, velocity control, vibration control.
N OMENCLATURE Jd Kmd R te td
Load inertia (in kilogram square meters). Interconnecting shaft stiffness (in newton meters per radian). Inertia ratio Jd /Jm . Drive system electromagnetic torque demand (in newton meters). Load-side torque (in newton meters).
Manuscript received August 17, 2004; revised September 11, 2006. Abstract published on the Internet January 27, 2007. This work was supported in part by the U.K. Engineering and Physical Sciences Research Council (EPSRC) and in part by Sensor Technology Ltd., Banbury, U.K. T. M. O’Sullivan and C. M. Bingham are with the Electrical Machines and Drives Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S10 2TN, U.K. (e-mail:
[email protected]). N. Schofield is with the Department of Electrical Engineering and Electronics, University of Manchester, Manchester M60 1QD, U.K. Digital Object Identifier 10.1109/TIE.2007.893048
tmd ωn ωa ωm ωd ωr ωx ωo ω1 , ω2 ζ1 , ζ2 Kp Ki Kd Ks A B τ1 , τ 2 , τ 3 ωr n Gp (s) P (s) G(s) H(s) T (s) Tden (s) U (s) L(s)
Torsional shaft torque (in newton meters). Mechanical resonant frequency (in radians per second). Mechanical antiresonant frequency (in radians per second). Motor angular velocity (in radians per second). Load angular velocity (in radians per second). Reference angular velocity (in radians per second). Load-side closed-loop tracking bandwidth (in radians per second). Cutoff frequency associated with real pole location (in radians per second). Undamped natural frequencies associated with the first and second pair of complex conjugate pole assignments (in radians per second). Damping ratios of the first and second pairs of complex conjugate pole assignments. Proportional gain. Integral gain. Derivative gain. Proportional shaft torque gain. Frequency ratio (ω1 /ωa ). Squared relative damping ratio (ζ2 /ζ1 )2 . Time constants of first-order filters. Filters 1, 2, and 3, respectively (in seconds). Output from controller prefilter. Speed sensor noise input. Transfer function (TF) of controller prefilter. TF of the plant or two-inertia model. TF of the controller equivalent forward path. TF of controller equivalent feedback path. Complementary sensitivity TF or closed-loop TF from speed sensor noise to motor speed. Denominator of complementary sensitivity TF. Closed-loop TF from speed sensor noise to electromagnetic torque demand. Optimal TF based on “integral of time multiplied by absolute error” (ITAE) performance criterion for a step input. I. I NTRODUCTION
Improvements in the power densities of electromagnetic machines and power conversion electronics have allowed electromechanical systems to encroach into application fields that have traditionally been the reserve of pure mechanical and hydraulic systems. In these cases, torque/force production can be considered essentially instantaneous, with the system dynamics
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being dominated by the (often complex) mechanical drive train. Consequently, for complex multicomponent mechanical drive trains incorporating many nonstiff interconnecting shafts and elastic couplings, impulsive transient torque output from the servo drive may excite mechanical torsional resonances that often degrade product performance, induce controller instability, or, in the worst case, cause complete mechanical failure. The dominant fundamental resonant mode for many such systems is typically < 300 Hz, which often overlaps with the closedloop bandwidth imposed by the servo-drive control algorithm. In these cases, the servo-drive system can be modeled using a two-inertia approximation [1]–[8], [10], [12]–[14], [16], [17]. Classically, closed-loop performance specifications are focused on the controller objectives, i.e., the behavior of the process output or load velocity in response to a reference demand or a load-side torque disturbance [2], [10], [13], [16]. However, such dynamics only provide partial information about system behavior. To appreciate the levels of performance achievable in a practical environment, where often destabilizing unmodeled dynamics and sensor noise are present, additional performance issues must be considered in order that a twoinertia model can be confidently employed. Over the past decade, much attention has been focused on the design of extended controllers for systems whose dynamics are approximated by two-inertia models [1]–[8], [10], [12]–[14], [16], [17], the most reported being based on state variable feedback with combined integral action, where unmeasured states are obtained from an observer, and the controller is designed using eigenstructure assignment [3], [10], [12], [16], [17]. More recently, however, due to practical problems associated with “tuning” state-feedback controllers as part of a commercial off-the-shelf product, emphasis has returned to the use of three-term controllers and methodologies to provide maximum benefit by appropriate gain selection. Such schemes generally consist of a proportional (P) and integral (I) structure augmented with an additional feedback signal of either of the following: i) motor acceleration: to form a classical proportional– integral–derivative (PID) controller [2]; ii) estimated shaft torque: using motor acceleration, thereby essentially constituting a PID controller but is commonly referred to as resonance ratio control (RRC) [4]–[6]; iii) derivative of estimated shaft torque [7]; where speed and acceleration feedback are obtained from the derivative and double derivative of sensed motor-shaft position, respectively, generally via the provision of an incremental encoder employing a quadrature position counter. Since perfect derivative action is impractical, controllers i) and ii) employ a first-order filter, which consists of an “idealized” double differentiator followed by a low-pass filter. It is commonly understood that there then exists a tradeoff between reduced filter bandwidth and the deterioration in the derivative action and, hence, controller performance. Finally, even when using relatively low-bandwidth filters, type iii) structures are generally impractical for high-bandwidth systems since they require a triple derivative of the motor position signal. In each case, controller gains are usually chosen assuming ideal derivative
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007
Fig. 1. Mechanical schematic of a two-inertia system. (a) Two-inertia representation of a servo-drive system. (b) Control block diagram of a two-inertia mechanical system.
action, with derivative filtering being subsequently applied for practical realization, with various methodologies employed for choosing and varying the time constant of the filter. It is generally assumed that “controller-generated” noise increases approximately in inverse proportion to the time constant of the filter or proportional to the double-derivative gain [2], [4], [6]. In reality, however, this is an oversimplification since practical controllers often contain several additional low-pass filters within their feedback paths, for instance, the speed filter (which reduces the impact of the first derivative of quantized position data). Nevertheless, a common method for adjusting filter time constants is on a trial-and-error basis, i.e., arbitrarily maximizing the filter time constants until controller-induced performance deteriorates and/or noise levels are suppressed to an acceptable level. Here then, enhanced three-term closed-loop control schemes are investigated. It is shown that modified PID-based structures (which are realized via the provision of a signal proportional to the estimated shaft torque) provide significant practical performance enhancements over classical PID variants by reducing the closed-loop sensitivity to filtering and by the strategic positioning and assignment of additional loop filters. Moreover, it is also shown that significant performance and robustness enhancements can be realized by appropriate design methodologies beyond that achievable from conventional controller structures without significantly trading-off controller complexity, i.e., with careful design, high performance can be achieved with the minimum of alterations to proprietary control architectures. II. T WO -I NERTIA M ECHANICAL S YSTEM Fig. 1(a) shows a schematic of a two-inertia servo-drive system consisting of two lumped inertias Jm and Jd , representing the motor and load, respectively, coupled via a shaft of finite stiffness Kmd , that is subject to torsional torque tmd and excited by a combination of electromagnetic torque te and load-torque perturbations td . The motor angular velocity is denoted ωm , and the load velocity is denoted ωd . Since damping losses are
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Fig. 2.
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Generic closed-loop servo-drive system controller structure.
usually considered to be relatively low, they are neglected without significantly affecting the accuracy of the forgoing analysis [1]–[10], [12]–[14], [16], [17]. Fig. 1(b) shows a dynamic block diagram representation of the system from which TFs describing the relationships between the electromagnetic torque produced by the servo machine and the machine rotor angular velocity (1) and the load angular velocity (2) are obtained as s2 + ωa2 ωm (s) = te (s) Jm s3 + Jm ωn2 s
(1)
ωd (s) ωa2 = te (s) Jm s3 + Jm ωn2 s
(2)
where the antiresonant frequency ωa and the resonant frequency ωn are defined as ωa =
Kmd
1 Jd
√ ωn = ωa R + 1
(3) (4)
and the load-to-motor inertia ratio is R=
Jd . Jm
(5)
III. R OBUST P ERFORMANCE S PECIFICATIONS AND C ONTROL D ESIGN Fig. 2 shows a generic closed-loop control structure. Lowercase variables represent dynamic input/output signals, where td (s) is a torque disturbance applied at the load-side inertia, n(s) is the sensor noise, and ωr (s) is the reference velocity. System outputs are the motor-side velocity ωm (s) and the load´ r (s), the output of Gp (s), side velocity ωd (s). Also shown is ω and te (s), which represents the control input to the plant (electromagnetic torque demand). Uppercase variables represent TFs, where G(s) is the equivalent forward-path controller, P (s) is the plant (the power amplifier and two-inertia model, where the former is assumed ideal), H(s) is the feedback controller, and Gp (s) is the prefilter. Since the disturbance torque is applied to the load-side inertia to represent disturbances acting on the closed loop in a more conventional manner, additional transfer blocks W (s) and 1/W (s) are included. These components only influence the disturbance input and effectively cancel within the loop.
Classically, performance specifications have focused on the behavior of the process output or load velocity in response to a reference demand (tracking performance) described by ωd (s)/ωr (s) or a load-side torque disturbance (regulation performance) described by ωd (s)/td (s) [2], [10], [13], [16]. In addition to control performance, the closed-loop system can also be evaluated in terms of robustness or robust performance, i.e., the degree to which the control performance is insensitive to extraneous inputs such as noise from the feedback sensor(s), high-frequency unmodeled dynamics, and process perturbations. Robust performance can be characterized using the complementary sensitivity function T (s) as T (s) =
GP H(s) ωm (s) = n(s) 1 + GP H(s)
(6)
which relates the closed-loop motor-side speed to the noise input. When plant parameter variations and modeling errors are expressed as a multiplicative uncertainty on the plant input ∆P (s), the system will remain stable if |T (s)∆P (s)| < 1 [18]. It follows then that system robustness improves for smaller T (s). More generally, large variations can be tolerated at frequencies where T (s) is small. Also, the dynamics relating control output to noise input U (s) is given by U (s) =
GH(s) te (s) = n(s) 1 + GP H(s)
(7)
and is used to characterize the noise-attenuating properties of the controller. Since measurement noise, unmodeled plant dynamics, and process perturbations are typically of high frequency, i.e., above the crossover frequency of GP H(s) for this type of system, the high-frequency behavior of U (s) is also an important measure of robustness and is essentially determined by the high-frequency behavior of the TF GH(s) [see (7), where for high frequencies GP H(s) 1], which represents the open-loop TF from noise input to control output. It therefore follows that the amount of sensor noise gives bound to the highfrequency gain of the controller and, therefore, of controller performance, since, ultimately, a tradeoff exists between the attenuating high-frequency noise and the controller-induced performance. The presented structures are termed two-degree-of-freedom controllers since the prefilter block Gp (s) acts only on the reference input. This is advantageous since the loop controllers G(s) and H(s) can be optimized to accommodate disturbances, high-frequency noise, and process uncertainties, while the
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TABLE I MECHANICAL PARAMETERS
prefilter shapes the input–output frequency response and tracking performance. A pole-placement scheme to assign the closed-loop poles of the characteristic equation Tden (s) [the denominator polynomial of T (s)] is now proposed. The characteristic equation is fifth order and has two complex and one real root. The proposed assignment is such that all poles have identical real parts. Classically, therefore, these poles have identical decay coefficients, which optimize the settling time and transient response for a given set of pole locations [2], [15]. Regulation performance is measured from time-domain properties of overshoot, undershoot, and the ITAE [9], [15] in response to a step disturbance input applied to the load-side inertia. The ITAE criteria is chosen in preference to other performance indices since it provides the best selectivity by minimizing overshoot and settling time for a given undershoot (or rise time for tracking responses) [9]. Robustness is evaluated using the highfrequency gain of the controller as a metric with comparative results being considered from the frequency response U (s).
loop system (represented by the noise input n(s) from the speed sensor) and amplified by the derivative components of both the PI and the PID/RRC velocity control loops. The level of noise ultimately determines the magnitude of induced variations in the control signal and the process variable (the load speed). In addition to the controller gain terms shown in Fig. 3, there normally exists a network of first-order low-pass filters arranged in a general structure representative of practical servodrive system realization. Filter 1 represents the derivative filter and is always present in practice. Filters 2 and 3—an additional derivative filter and a motor speed filter, respectively, are generally employed in an attempt to further reduce the effects of sensor noise and unmodeled dynamics. Usually, controller gains are selected by assuming that the additional loop filters are not present or that their inclusion does not unduly affect the controller-induced performance. A common criterion for adjusting the filter time constants is to maximize τ1 , τ2 , and τ3 until controller-induced performance deteriorates and/or noise levels are suppressed to an acceptable level. However, it will be shown that the location of the additional loop filters and the ratio of their respective cutoff frequencies ultimately dominate the resulting noise suppression characteristics of the closed-loop system. B. Tracking Performance Criteria L(s) represents the Laplace transform of the desired timedomain step-tracking response ωd (s)/ωr (s) and is given from the optimal ITAE criterion for a step input applied, in this case, to the controller reference input—ωx represents the required −3-dB tracking bandwidth [9]—as
IV. P RACTICAL C ONTROL S TRUCTURES A. Proportional–Integral (PI) and PID/RRC Controllers A prototype two-mass test facility comprising of 2 × 2.2 kW brushless permanent magnet servo machines has been commissioned to demonstrate the presented performance considerations and tradeoffs. The mechanical parameters of the test rig are given in Table I for specific values of R, which are realized by inserting inertial disks to the motor-side servo machine. Fig. 3(a) shows the structure of a classical PI control scheme with an additional derivative signal proportional to motor acceleration being included to form a PID structure, where Kd is the derivative feedback gain. An alternative structure is given in Fig. 3(b) and is redrawn in Fig. 3(c), which takes the PI elements of Fig. 3(a) and augments them with a feedback signal proportional to the estimated shaft torque tmd (tmd = te − (dωm /dt)Jm ), where the associated feedback gain is Ks . The use of a feedback signal proportional to torsional torque in this manner is commonly referred to as RRC [4], [5]. It can be seen from Fig. 3(a) and (c) that both PID and RRC are essentially equivalent in form, with the motor speed and motor acceleration signals employed by the PID/RRC controllers being obtained from the derivative and double derivative of sensed motorshaft position, respectively, generally via the provision of an incremental encoder. Due to the discrete nature of the feedback signal, in addition to unmodeled dynamics and process perturbations, quantization noise is naturally injected into the closed-
L(s) =
s5
+ 2.8ωx
s4
+
ωx5 . + 5.5ωx3 s2 + 3.4ωx4 s + ωx5 (8)
5.0ωx2 s3
To achieved the desired response, the prefilter is designed as Gp (s)
ωd (s) ωd (s) = = L(s) ωr (s) ωr (s)
giving
Gp (s) = L(s) ·
ωr (s) . ωd (s) (9)
C. Robust Performance Criteria The magnitude of the high-frequency gain of the TF GH(s) is representative of the robust performance of the controller (see Section III). By rearrangement of the structures in Fig. 3 to be consistent with that in Fig. 2, for the PI controller in Fig. 4(a) (where Kd = 0 and τ1 = τ2 = 0), GH(s)PI is given by GH(s)PI =
s(Kp /Ki ) + 1 Ki · . sτ3 + 1 s
(10)
It can be seen that lims→∞ |GH(s)PI | = 0, and the magnitude is attenuated at the rate of −20 dB/dec. However, when Filter 3 is removed, i.e., when τ3 = 0, lims→∞ |GH(s)PI | = Kp . Since the presence of Filter 3 significantly contributes to highfrequency performance, it should be included as part of the
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Fig. 3. Control block diagrams of extended PI controllers for the two-inertia mechanical model. (a) PID controller. (b) RRC controller. (c) RRC controller equivalent structure.
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It is notable that (13)–(15) also represent lims→∞ |GH(s)RRC | by replacing Kd with (−Jm Ks ). Comparing (13) with (14) and (15), the following important points are identified. 1) From (13), the level of noise amplification with a PID or RRC controller is not approximately proportional to the derivative gain (Kd or Ks ), as commonly reported [2], [6]. 2) From (13) and (14) and noting that Kd is negative and Ks is positive, the inclusion of a speed filter (with no additional derivative filter) in either the PID or the RRC controller results in significant high-frequency noise amplification. 3) From (13) and (15), an additional derivative filter (with no speed filter) may not enhance noise attenuation when compared with a single derivative filter. Furthermore, from (11) and (12), both a speed filter and an additional derivative filter are required to ensure lims→∞ |GH(s)| = 0 with an attenuation rate of −20 dB/dec. However, since Kd is negative and Ks is positive, the s3 term on the numerator of (11) and (12) can be cancelled (increasing the attenuation rate to −40 dB/dec) by appropriate selection of filter time constants, i.e.,
Fig. 4. PI controller performance.
controller structure. Similarly, the high-frequency behavior of the control elements of the PID (11) and RRC (12) (GH(s)) structures (shown at the bottom of the page) is strongly influenced by the location and number of filters present in the feedback loop and, importantly, the ratio of their time constants. Since Filter 1 is always present in practice, consider the PID controller when lims→∞ |GH(s)PID | for the three following possible scenarios, as Filters 2 and 3 are selectively included. Case 1) τ2 = τ3 = 0 (derivative filter only) Kd + K p τ 1 . lim |GH(s)PID | = abs s→∞ τ1 Case 2) τ2 = 0 (case 1 + speed filter) Kd . lim |GH(s)PID | = abs s→∞ τ1
lim |GH(s)PID | = abs(Kp ).
PID rop =
Kp τ 1 Kd
RRC rop =
Kp τ 1 . (16) −Jm Ks
The dynamics imparted by Filter 1 are, therefore, considered part of the PID and RRC controller structures. Filters 2 and 3 are then added afterward to increase robustness. This has the advantage of eliminating the normal trail and error approach to “tuning,” since, with the additional filters in place, only τ2 is adjustable (until the control performance is adversely effected) because τ1 is ultimately dependent on the desired closed-loop pole locations and τ3 on (16).
(13) D. Regulation and Robust Performance Tradeoff
(14)
With reference to Fig. 4(a), the closed-loop characteristic PI equation Tden (s) for the PI controller with Filter 3 included (where Kd = 0) is given by PI Tden (s) = Jm τ3 s5 + Jm s4 + Jm ωa2 τ3 (1 + R) + Kp s3
Case 3) τ3 = 0 (case 1 + additional derivative filter) s→∞
τ3 = rop τ2
(15)
+ Jm ωa2 (1 + R) + Ki s2 + Kp ωa2 s + Ki ωa2 . (17)
GH(s)PID =
s3 (Kd τ3 + Kp τ1 τ2 ) + s2 (Kd + Kp (τ1 + τ2 ) + Ki τ1 τ2 ) + s (Kp + Ki (τ1 + τ2 )) + Ki s4 τ1 τ2 τ3 + s3 (τ1 τ2 + τ3 (τ1 + τ2 )) + s2 (τ1 + τ2 + τ3 ) + s
(11)
GH(s)RRC =
s3 (−Jm Ks τ3 + Kp τ1 τ2 ) + s2 (−Jm Ks + Kp (τ1 + τ2 ) + Ki τ1 τ2 ) + s (Kp + Ki (τ1 + τ2 )) + Ki s4 τ1 τ2 τ3 + s3 (τ1 τ2 + τ3 (τ1 + τ2 )) + s2 (τ1 + τ2 + τ3 (1 + Ks )) + s
(12)
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Similarly, PID Tden (s) = Jm τ1 s5 + (Jm + Kp τ1 )s4
+ (Jm ωa2 τ1 (1 + R) + Kp + Ki τ1 )s3 + Kp ωa2 τ1 + Ki )s2 + (Jm ωa2 (1 + R) + (Kp ωa2 + Ki ωa2 τ1 )s + Ki ωa2
(18)
= Jd /Jm , and where Jm = Jm + Kd and R RRC Tden (s) = Jm τ1 s5 + (Jm + Kp τ1 )s4
+ (Jm ωa2 τ1 (1 + R) + Kp + Ki τ1 )s3 + Kp ωa2 τ1 + Ki )s2 + (Jm ωa2 (1 + R) + (Kp ωa2 + Ki ωa2 τ1 )s + Ki ωa2
(19)
= R(1 + Ks ), give T PID (s) and T RRC (s) for the PID where R den den and RRC controllers, respectively, where, in both cases, only Filter 1 is included, i.e., τ2 = τ3 = 0. The generic characteristic equation of a fifth-order polynomial with two pairs of complex and one real root is
s2 + 2ς1 ω1 + ω12
s2 + 2ς2 ω2 + ω22 (s + ω0 ) = 0.
(20)
Expanding (20) and constraining to force the real parts of the roots to be identical, i.e., −ω1 ς1 = −ω2 ς2 = ωo , gives s5 + 5ς1 ω1 s4 + 8ς12 ω12 + ω12 + ω22 s3 + 3ς1 ω1 ω22 + ω12 3ς1 + 4ς13 s2 + ω12 ω22 1 + 2ς12 + 2ς12 ω14 s + ω13 ω22 ς12 = 0.
(21)
Equating the coefficients of (17) and (21), and defining B = (ς2 /ς1 )2 , where, without loss of generality, B ≤ 1, and A = ω1 /ωa , where ωa is the antiresonant frequency, and ω1 is the frequency of the first complex root, the following relationships are obtained: 0.4(1 + B) + 7.2ς12 B 2ς12 (1 + B) + 0.8 A2 3(1 + B) + 4ς12 B − A4 − 5B R= 5B
A2 =
Fig. 5. PI controller performance. (a) Regulation step response. (b) Closedloop frequency gain characteristics of U (s) from sensor noise n to electromagnetic torque te .
(22)
(23)
which satisfy the pole location constraints. Equations (22) and (23) can be solved simultaneously for A and ς1 by first selecting suitable values for B and R (which will be considered in due course).
Fig. 4 shows the resulting performance of the PI controller as B is varied from 0.1 to 1 for variations of inertia ratio R from 0.25 to 0.75. The controller high-frequency noise attenuation performance [sensor noise to controller output, U (s) → GH(s)PI ], which is obtained by evaluating the magnitude of (10) at high frequency, specifically at ten times the mechanical resonant frequency in this case (for the range of B considered), is given in Fig. 4(a). Moreover, Fig. 4(b) and (c) shows, respectively, the ITAE and overshoot and undershoot corresponding to time-domain regulation performance measurements. It can be seen that while undershoot remains relatively constant over the range of B, the overshoot and ITAE increase with decreasing R, i.e., for the PI controller, the achievable level of timedomain performance is limited for low inertia ratios (R ≤ 0.5). Furthermore, for values of B < 0.3, noise attenuation improves at the cost of a significant deterioration in timedomain performance. Moreover, it can be seen that, generally, choosing 0.3 ≤ B ≤ 0.6 provides the best time-domain regulation performance for a given inertia ratio. For ease of practical implementation, it is therefore recommended that B = 0.5. For example, with B = 0.5, Fig. 5(a) shows the resulting motor and load responses for a step load-torque disturbance for cases of R = 0.25 and 0.5, while Fig. 5(b) shows the corresponding closed-loop plots of |U (jω)|. It can be seen that time-domain performance deteriorates for R < 0.5, while noise attenuation remains relatively unaffected at high frequency. Therefore, to improve the regulation performance when low inertia ratios are present, PID or RRC controllers should be considered in preference to the PI controller. By equating the coefficients of
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Fig. 6. PID and RRC controller performance. (a)–(d) R = 0.25. (e)–(h) R = 0.1.
(18) for the PID controller, or (19) for the RRC controller, with (21), A can be constrained as A4 2ς12 (1 + B) + 1 −A2 (B(8ς1 + 1) + 1) + B(1 + R) = 0
(24)
where A is the maximum real root. Equation (24) therefore establishes the same constraints for both PID and RRC controllers. In each case, however, both B and ς1 can now be independently selected. Since (24) applies to both PID and RRC controllers, both can impart identical closed-loop dynamics. The resulting high-frequency behavior described in (13)–(15) (replacing Kd with −Jm Ks for the RRC controller) is also identical for the two structures (for the same closed-loop pole locations). Fig. 6(a) to (d) therefore shows the PID/RRC controller performance as B is varied for a number of damping ratios ς1 (note that ς2 is constrained via the assignment of B) for R = 0.25. This is repeated in Fig. 6(e) to (f) for a lower value of R = 0.1. Of particular note from Fig. 6(a) and (e) is that the high-frequency noise attenuation characteristics tend to a constant value, as dictated by (13). Moreover, it can be seen that choosing low values of B ≤ 0.5 provides improved regulation performance (reduced overshoot and ITAE) at the expense of poorer noise attenuation. By contrast, choosing values of B > 0.5 results in poorer regulation performance but improved noise attenuation. It is therefore recommended as a design compromise that B = 0.8 for the PID and RRC controllers and that ς1 is chosen to provide a satisfactory trade-
off between the regulation performance and the attenuation of high-frequency noise. In general, therefore, the PID and RRC controllers can impart the desired time-domain responses for the whole range of R, but high-frequency noise has greater impact as R reduces, or reduced ITAE (and/or overshoot) is required for a given R. This can be seen graphically from the example results in Fig. 9, where lower values of ς1 provide greater ITAE and overshoot while improved high-frequency noise characteristics (U (s)) are obtained—the results are taken for B = 0.8 and R = 0.25 in each case.
E. Robust Performance Considering the effects of adding the speed and additional derivative filters (Filters 2 and 3) to the PID or RRC controllers, as illustrated in Fig. 3, and from the discussion in Section IV-C, τ2 should be chosen to be as high as possible without unduly compromising overshoot and settling time, and τ3 adjusted according to the ratio rop given by (16). The ultimate value of τ2 is influenced by the ratio of the two time constants of Filters 1 and 2 (τ1 /τ2 ), i.e., if the ratio is large, Filter 1 dominates the time response—which is fixed to assign the required pole locations (that have identical real parts) for both PID and RRC controllers. Thus, in general, larger values of τ1 will enable a greater range of τ2 to be employed, which allows further robustness enhancement without sacrificing regulation performance. However, importantly, it is now shown that the RRC control structure is less sensitive to the inclusion of Filter 2 compared to the PID counterpart, since the RRC
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Fig. 7. Constraint relationships between τ1 and B for the PID and RRC controllers.
Fig. 9. PID/RRC controller performances with additional filters. (a) Regulation step response. (b) Closed-loop frequency gain characteristics of U (s) from sensor noise n to electromagnetic torque te .
τ1RRC , gives τ1PID =
BR (3A2 (1 + B) + 4A2 ς12 B − A4 − 5B) ς1 Aωa
(25)
3 2 τ1RRC A5 ς13 ωa3 − τ1RRC A4 ωa2 2ς12 + 2ς12 B + 1 + τ1RRC Aς13 Bωa + ς12 B = 0
Fig. 8. PID/RRC controller performances. (a) Regulation step response. (b) Closed-loop frequency gain characteristics of U (s) from sensor noise n to electromagnetic torque te .
structure requires a significantly higher value of τ1 to provide the same closed-loop pole locations. Equating (18) and (19) for the PID and RRC controllers with (21) to obtain the respective values of τ1 for each, viz. τ1PID and
(26)
where τ1RRC = maximum real root of (26), B = 0.8, and ς1 are selected to impart the desired level of time-domain performances, as previously illustrated in Fig. 6, and A is obtained by first solving the constraining pole relations for the PID and RRC controllers given in (24). Fig. 7 shows how the value of τ1 is influenced by the selection of B for cases ς1 = 0.3 and 0.5 when R = 0.1 and 0.25. Specifically, Fig. 7(a) shows the relationship for the RRC controller, and Fig. 7(b) for the PID controller, with the results normalized by 1/(τ1 ωa ). It can be seen that the PID controller requires an acceleration signal of increased bandwidth [since 1/(τ1PID ωa ) is higher] when increased damping is required, i.e., when ς1 is increased from 0.3 to 0.5, and/or the inertia ratio R is reduced from 0.25 to 0.1. More generally, a comparison of Fig. 7(a) and (b) shows that the RRC controller requires a shaft torque signal and therefore an acceleration signal (since acceleration is used by the RRC controller to estimate shaft
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TABLE II CONTROLLER GAINS
torque, see Fig. 3) of significantly less bandwidth than the PID controller for the same pole locations. Of particular note, in general, is that the required bandwidth of the estimated torque signal is significantly less than the antiresonant frequency of the mechanical system. To demonstrate the deterioration in performance as τ2 is increased, the time and frequency response (U (s)) results given in Fig. 8 when ς1 = 0.5 are repeated in Fig. 9 for both PID and RRC controllers with the additional two filters, when τ2 = 1/(5Aωa ) and τ3 is tuned according to (16) (where in Fig. 8, B = 0.8, R = 0.25, and τ2 = τ3 = 0). It can be seen that the time-domain performance imparted by the PID controller is adversely influenced by the presence of additional filters while the RRC structure is relatively insensitive. In summary, therefore, while the RRC and PID controllers can, theoretically, impart identical closed-loop dynamics, the RRC structure is relatively insensitive to the phase lag imparted by the additional filters required in practical systems and consequently imparts superior high-frequency robustness without sacrificing the control objectives—the RRC structure should therefore be adopted in preference to PID. For completeness, the controller gains for the PI, PID, and RRC controllers are given in Table II.
V. E XPERIMENTAL V ALIDATION The proposed control techniques, simulation results, and observations are now validated on an experimental test facility [Fig. 10(a)] whose mechanical parameters are given in Table I. The control algorithms, sensor inputs, and control outputs are realized via a DSP-based dSPACE system, as shown in Fig. 10(b). The inline surface acoustic wave torque transducer interconnecting the motor and load is employed for instrument purposes only. The presented results are in response to a step disturbance load torque of 3.5 N · m, where the speed controller refer-
Fig. 10. Experimental facility, components, and control system. (a) Mechanical system overview. (b) Functions of components.
ence is chosen to be constant at 10 rad/s, thereby considering regulation performance of the control schemes. Fig. 11 shows the responses resulting from the use of the PI controller shown
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Fig. 11. Experimental step responses of the PI controller.
in Fig. 3(a) when Kd = 0 and B = 0.5, thereby depicting the situation simulated in Fig. 5. It can be seen that as the inertia ratio reduces (i.e., the motor inertia is increased by the addition of inertial discs) from R = 0.5 to 0.25, the PI controller imparts an underdamped response while exhibiting reasonable noise attenuation characteristics—compare Figs. 4 and 5 at high frequency. To improve damping for small-inertia ratios, PID or RRC controllers are preferred. It has been shown (Section IV) that a tradeoff exists between the PID and RRC controllerinduced damping characteristics and the attenuation of highfrequency noise (see Figs. 6–9) and, consequently, a reduction in controller-induced robustness. Furthermore, stability robustness is shown to be improved with the addition of derivative and speed filters (Filters 2 and 3), and further improved if the ratio of their time constants (τ3 /τ2 ) is chosen to ensure a −40-dB/dec attenuation characteristic (16). However, it was also shown that the time-domain characteristics imparted by the RRC controller are more robust than those of the PID, since it is less sensitive to the inclusion of additional speed and derivative filters—see Figs. 7 and 9. To demonstrate the robustness of the RRC compared to the PID structure, and the tradeoff of time-domain performance and high-frequency noise attenuation, Fig. 12 shows experimental measurements from both control structures for the case when R = 0.25, B = 0.8, τ2 = 1/(5Aωa ), and τ3 is chosen to satisfy (16)—the measurements can be compared with the simulated results in Fig. 9(a) for the case when ς1 = 0.5. From Fig. 12(a), it can be seen that the PID-induced dynamics are oscillatory and exhibit poor noise attenuation characteristics. By contrast, measurements using the RRC structure [Fig. 12(b)] show the expected damping characteristics (for cases of ς1 = 0.5, 0.4, and 0.3), compare with Fig. 8 (where the additional filters where excluded, τ2 = τ3 = 0), verifying the relative insensitivity of the RRC controller to the additional filters. It is also evident from the corresponding measurements of te that as damping reduces, the level of controller noise also reduces, demonstrating the tradeoff between control performance and attenuation of controller noise. For completeness, Fig. 12(c) shows the effect of implementing the RRC controller (when ς1 = 0.5) with a ratio of time constants that do not satisfy (16), i.e., τ3 = Krop τ2 , where K is a scalar. The results show that either an increase or a decrease in
Fig. 12. Experimental step responses of the PID and RRC controllers where R = 0.25 and τ2 = 1/(5Aωa ). (a) PID controller, τ3 = rop τ2 . (b) RRC controller, τ3 = rop τ2 . (c) RRC controller, τ3 = Krop τ2 .
the ratio of time constants (i.e., an increase and a decrease in the cutoff frequency of the speed filter) increases the susceptibility of the system to noise. Fundamentally, this is due to the highfrequency rolloff reducing from −40 dB/dec when K = 1 to −20 dB/dec when K = 1. The results presented in Fig. 13 demonstrate the tracking and regulation performance attributed to RRC. Specifically, measurements are taken in response to a 10-rad/s step reference demand with a step disturbance load torque of 3.5 N·m applied after 0.8 s. The combined tracking and regulation responses demonstrate the flexibility afforded by the twodegree-of-freedom controller (see Section IV-B) to impart independent levels of tracking and regulation performance—as in Fig. 12, R = 0.25, B = 0.8, τ2 = 1/(5Aωa ), and τ3 is
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ACKNOWLEDGMENT The authors would like to thank the U.K. Engineering and Physical Science Research Council (EPSRC) and Sensor Technology Ltd., Banbury, U.K., for the provision of a CASE studentship. R EFERENCES
Fig. 13. Experimental step responses of the RRC controller where R = 0.25, τ2 = 1/(5Aωa ), and τ3 = rop τ2 .
adjusted according to (16). It can also be seen from Fig. 13(a) and (b), which depicts the conditions when a damping ratio of ς1 is designed to be ς1 = 0.3 and 0.5, respectively, that the resulting tracking performance (where the −3-dB tracking bandwidth ωx = ωa ) remains relatively invariant. Fig. 13(c) demonstrates the condition when a 40% increase in tracking bandwidth is assigned for ς1 = 0.5 (i.e., ωx = 1.4 ωa ). It can be seen that while increasing the tracking bandwidth reduces the rise times, as expected, good load-side tracking dynamics remain. However, while maintaining optimum load-side performance, the motor-side tracking dynamics deteriorate as a result of countering the effects of shaft compliance. VI. C ONCLUSION This paper has considered controller design methodologies for enhanced performance of two-mass servo-drive systems, including an appraisal of the effects of supplementary filtering elements associated with practical systems. PI, PID, and RRC controllers have been investigated with regard to both the closed-loop robustness and the control of the process variable (load velocity) in response to a step reference speed or load-side disturbance. In all cases, it is shown that the high-frequency gain of the respective controllers should be considered as a crucial design variable for imposing a sufficient degree of robustness to the closed-loop system when subject to unmodeled resonant modes, high-frequency noise from the derivative of quantized sensor signals, and process perturbations. Moreover, induced performance has been shown to be strongly influenced by both the location and the number of filters present in the various feedback loops and, importantly, the ratio of their time constants. Furthermore, a design methodology is presented to assign the time constants of the various loop filters, and their location, using a single user-definable variable, thereby reducing the time-consuming trial-and-error approach commonly employed using conventional tuning procedures. Although the PID and RRC controllers are able to impart identical closedloop input–output dynamics, RRC is shown to provide superior closed-loop robustness by virtue of the different feedback mechanism employed.
[1] S. N. Vukosavic and M. R. Stojic, “Suppression of torsional oscillations in a high performance speed servo drive,” IEEE Trans. Ind. Electron., vol. 45, no. 1, pp. 108–117, Feb. 1998. [2] G. Zhang and J. Furusho, “Speed control of a two-inertia system by PI/PID control,” IEEE Trans. Ind. Electron., vol. 47, no. 3, pp. 603–609, Jun. 2000. [3] J. K. Ji and S. K. Sul, “Kalman filter and LQ based speed controller for torsional vibration suppression in a 2-mass motor drive system,” IEEE Trans. Ind. Electron., vol. 42, no. 6, pp. 564–571, Dec. 1995. [4] Y. Hori, H. Sawada, and Y. Chun, “Slow resonance ratio control for vibration suppression and disturbance rejection in torsional system,” IEEE Trans. Ind. Electron., vol. 46, no. 1, pp. 162–168, Feb. 1999. [5] K. Yuki, T. Murakami, and K. Ohnishi, “Vibration control of 2-mass resonant system by resonance ratio control,” in Proc. IEEE IECON, 1993, vol. 3, pp. 2009–2014. [6] S. Mortimoto, A. Hamamoto, and Y. Takeda, “Vibration control of twomass system with low inertia ratio considering practical use,” Electr. Eng. Jpn., vol. 125, no. 2, pp. 1–9, 1998. [7] K. Sugiura and Y. Hori, “Vibration suppression in 2- and 3-mass system based on the feedback of imperfect derivative of the estimated torsional torque,” IEEE Trans. Ind. Appl., vol. 43, no. 1, pp. 56–64, Jan./Feb. 1996. [8] T. M. O’Sullivan, N. Schofield, and C. M. Bingham, “Simulation and experimental validation of induction machines dynamics driving multi-inertia loads,” J. Appl. Electromagn. Mech., vol. 19, no. 1–4, pp. 231–236, 2004. [9] R. C. Dorf, Modern Control Systems. Reading, MA: Addison-Wesley, 2000. [10] Y. Hori, H. Iseki, and K. Sugiura, “Basic consideration of vibration suppression and disturbance rejection control of multi-inertia system using SFLAC (state feedback and load acceleration control),” IEEE Trans. Ind. Appl., vol. 30, no. 4, pp. 889–896, Jul./Aug. 1994. [11] R. C. Kavanagh and J. M. D. Murphy, “The effects of quantization noise and sensor nonideality on digital differentiator-based rate measurement,” IEEE Trans. Instrum. Meas., vol. 47, no. 6, pp. 1457–1463, Dec. 1998. [12] J. Keun, J. Dong, C. Lee, and S. K. Sul, “LQG based speed controller for torsional vibration suppression in 2-mass motor drive system,” in Proc. IEEE IECON, 1993, pp. 1157–1162. [13] G. Ellis and R. D. Lorenz, “Resonant load control methods for industrial servo drives,” in Proc. IEEE IAS Annu. Meeting, Rome, Italy, Oct. 2000, pp. 1438–1445. CD proceedings. [14] J. M. Pacas, A. John, and T. Eutebach, “Automatic identification and damping torsional vibration in high-dynamic drives,” in Proc. IEEE ISIE, 2000, vol. 1, pp. 201–206. [15] G. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Englewood Cliffs, NJ: Prentice-Hall, 2002. [16] S. H. Song, J. K. Ji, S. K. Sul, and M. H. Park, “Torsional vibration suppression control in 2-mass system by state feedback speed controller,” in Proc. IEEE CCA, 1993, pp. 129–134. [17] T. Orlowska-Kowalska and K. Szabat, “Sensitivity analysis of state variable estimators for two-mass drive system,” Acta Electrotech. Inform., vol. 4, no. 1, 2004, CD-ROM. [18] K. Dutton, S. Thompson, and W. Barraclough, The Art of Control Engineering. Reading, MA: Addison-Wesley, 1997.
Timothy M. O’Sullivan received the M.Eng. degree in 2000 from the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, U.K., where he is currently working toward the Ph.D. degree. His Ph.D. thesis is on the use of surface acoustic wave (SAW) torque transducers for improving the dynamic response of industrial servo-drive systems.
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Christopher M. Bingham (M’00) received the B.Eng. degree in electronic systems and control engineering from Sheffield City Polytechnic, Sheffield, U.K., in 1989, the M.Sc. (Eng.) degree in control systems engineering from the University of Sheffield, Sheffield, in 1990, and the Ph.D. degree from Cranfield University, Bedfordshire, U.K., in 1994, for research on control systems to accommodate nonlinear dynamic effects in aerospace flight-surface actuators. He was a Post-Doctoral Researcher at Cranfield University until subsequently taking up a research position at the University of Sheffield. Since 1998, he has been a Lecturer in the Department of Electronic and Electrical Engineering, University of Sheffield. His current research interests include traction control/antilock braking systems for electric vehicles, electromechanical actuation of flight control surfaces, control of active magnetic bearings for high-speed machines, sensorless control of brushless machines, and analysis and design of resonant converter systems.
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Nigel Schofield received the B.Eng. degree in electrical power engineering and the Ph.D. degree on the field weakening of brushless permanent-magnet traction machines from the University of Sheffield, Sheffield, U.K., in 1990 and 1997, respectively. From 1993 to 1995, he was a Senior Experimental Officer in the Department of Electronic and Electrical Engineering (EEE) before taking up the post of Design Engineer in industry. From 1997 to 2000, he was a Post-Doctoral Researcher in the Electrical Machines and Drives Research Group, Department of EEE, University of Sheffield, and from 2000 to 2004, a Lecturer in the same department. On July 1, 2004, he was appointed to a Mechatronics Lectureship at the School of Electrical and Electronic Engineering, University of Manchester, Manchester, U.K. His research interests include electromagnetic power trains for all- and hybrid-electric vehicles, the vehicular application of hydrogen fuel cell systems, aerospace machines and actuators, and industrial applications of electromagnetic devices.
