The e ect of actuator saturation on the performance of PD-controlled servo systems

Transcription

1 Mechatronics 9 (1999) 497±511 The e ect of actuator saturation on the performance of PD-controlled servo systems Michael Goldfarb*, Taweedej Sirithanapipat Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA Received 26 June 1998; accepted 30 December 1998 Abstract Perhaps the most often utilized means of closed-loop control of a servo system is proportional-derivative (PD) control. Linear analysis methods suggest the best tracking performance is achieved at maximum possible proportional and derivative gains. Maximum gains, however, drive the actuators into saturation, which renders the system nonlinear and the linear analysis invalid. This paper investigates the e ect of actuator saturation on servo system tracking performance by formulating a frequency-based tracking performance measure roughly equivalent to the linear system 3 db bandwidth. The proposed measure utilizes a series of band-limited pseudo-random tracking inputs to characterize the `bandwidth' of the (nonlinear) saturating system. Numerical simulations based on this measure show that, for a servo system that exhibits actuator saturation, the best tracking performance is not achieved at maximum gain. Instead, performance improves up to a given gain, then begins to recede as the gain is increased further. The simulations also show that avoiding actuator saturation to ensure linear behavior signi cantly sacri ces tracking performance. The measure of tracking performance is compared with the 3 db bandwidth utilized in linear analysis techniques, and the two are shown to be well correlated. # 1999 Elsevier Science Ltd. All rights reserved. * Corresponding author. Fax: address: goldfarb@vuse.vanderbilt.edu (M. Goldfarb) /99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 498 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± Introduction The servo control problem is generally characterized by a rotational system with a position output related to a torque input through the dynamics of inertia, friction, and sometimes gravity. The objective of a servo control system is typically to obtain accurate position tracking while maintaining system stability. Several control techniques exist to achieve this objective. Perhaps the most often utilized approach is proportional-derivative (PD) control, which utilizes a control signal consisting of a weighted sum of the position and velocity errors. If not used in a direct fashion, the PD-control structure often forms the basis for other commonly-used servo control approaches. For example, a common and closelyrelated approach is computed torque control, which utilizes a model of the control plant to cancel (typically inertial and gravitational) nonlinearities, then essentially utilizes a PD-type control on the residual linear system. Additionally, sliding mode control can also be considered a variant of PD-control, especially for cases in which the control gains are large relative to the parametric uncertainty of the model. The fundamental structure of these and many other servo controllers is in essence a PD control structure. A signi cant asset of a PD-controlled servo system is its stable behavior. Demonstrating the stability of the PD-control of a servo system with linear Fig. 1. Block diagram representing (a) PD control of a servo system and (b) PD control of servo system with actuator saturation.

3 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± mechanics is trivial. More signi cantly, several researchers, including Slotine and Li [1] and Yigit [2], have demonstrated via Lyapunov methods that the PDcontrol of a servo system with nonlinear inertial and gravitational mechanics (e.g., a robot manipulator) also exhibits asymptotic stability. 2. Linear analysis and PD control Probably the two most signi cant issues in servo control system design are the closed-loop system stability and the tracking performance. For a linear (or linearized) system, the analysis and design of a PD-control system can be guided by frequency-based techniques. For example, PD control of the linear system with inertia and damping, as represented by the block diagram of Fig. 1(a), would yield a closed-loop characteristic equation of: Js 2 B K d s K p ˆ 0 1 where J is the rotational inertia, B the damping ratio, s the LaPlace variable, K p the proportional control gain, and K d the derivative control gain. This system is stable for any K p >0, and as both K d and K p approach positive in nity, the poles of the system converge to negative in nity, resulting in an instantaneous response and thus perfect tracking. Some prior publications in the engineering literature o er recommendations for the selection of PD control gains that are consistent with the aforementioned linear analysis. Shahruz et al. [3] utilize the theory of singularly perturbed systems to assert that, for PD control of a generalized multi-rigid-link robot manipulator: ``large gain matrices K p and K d make the robot manipulator perform better.'' Xu et al. [4] also recommend that PD gains should be selected ``as high as possible up to the point that sensor noise causes instability.'' The notion of optimal performance at in nitely large gains is based on linear system behavior. The use of large gains, however, inevitably introduces saturation, which in turn violates the assumption of linearity. The use of large gains (and in particular in nite gains) is therefore in direct con ict with the assumption of linear system behavior. Speci cally, as control gains become large, some component of the control system (typically the motor servo-ampli er) is forced into saturation, thus introducing a distinct nonlinearity (as illustrated in Fig. 1(b)) that in turn invalidates the aforementioned linear analysis. This problem can be circumvented by limiting the proportional and derivative gains to a level that avoids actuator saturation. This approach, however, sacri ces actuator e ort (and thus tracking performance) to gain linear behavior. Such strategy is in direct con ict with the notion of optimal tracking performance, which intuitively should take full advantage of system resources, and thus deliberately push the actuators into saturation.

4 500 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± Nonlinear analysis of saturation in control Several researchers have analyzed the stability of control systems in the presence of actuator saturation. Kim and Bien [5] present su cient conditions for robust stability in the presence of actuator saturation. Glattfelder and Schaufelberger [6], Hui and Chan [7], among others, have investigated the stability e ects resulting from integrator wind-up in the presence of saturating actuators. Sourlas et al. [8] determined that actuator saturation improves disturbance rejection, which is an intuitively reasonable result. Several researchers have proposed speci c control methodologies that provide enhanced stability or tracking performance in the presence of actuator saturation. Frankena and Sivan [9], Ryan [10], and Bernstein [11] derived control laws within an optimal control framework that included a bounded control constraint to account for actuator saturation. Derivation of the optimal control laws, however, required the assumption that the open-loop system is asymptotically stable, which is quite limiting, since servo systems are typically either marginally stable or unstable in position. Paden and Tomizuka [12] derived a control law for fast point to point position control in the presence of actuator saturation, which is concerned with step inputs, but not with tracking per se. Kabuli and Kosut [13] derived a control law for trajectory tracking in the presence of actuator saturation that incorporates a bang-bang type control e ort. Goto et al. [14] incorporated model inversion to pre-compute a control command based on a known desired trajectory to improve tracking in the presence of actuator saturation. Shewchun and Feron [15] proposed `high performance bounded control', a control approach that varies the feedback gains in order to purposefully avoid actuator saturation. As demonstrated by the simulations presented in this paper, such an approach may be considerably sub-optimal for purposes of tracking performance. None of the previously mentioned works has focused on the e ects of actuator saturation on the tracking performance of PD-controlled servo systems. Rather than propose special control methodologies to accommodate actuator saturation, this paper treats the more common scenario of incurring actuator saturation during the PD-control of a servo system. Speci cally, the paper derives a generalized measure of tracking performance, and utilizing this measure, incorporates numerical simulations to demonstrate the relationship between loop gain and tracking performance in PD-controlled servo systems with actuator saturation. 4. A measure of tracking performance Linear control system design often incorporates frequency-based measures, such as pole locations or frequency response, to assess tracking performance. A frequency-based measure typically used in the linear case is the 3 db bandwidth, which is the frequency at which the system output is generally agreed to no longer adequately track the reference input. Control gains for optimal tracking

5 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± Fig. 2. The band-limited psuedo-random input for frequency-based characterization of tracking performance is generated by passing white noise through a second-order critically-damped low pass lter with cuto frequency of o c.

6 502 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497±511 performance are typically selected to maximize the bandwidth while maintaining unity input/output gain below this frequency. A linear system can be characterized completely in the frequency domain because the behavior of such a system can be described solely as a function of the input frequency. That is, the output of a linear system will be of the same frequency as the input, subject to a frequency-dependent gain and phase. A linear system can therefore be characterized by these two components. The performance of a nonlinear system, however, cannot be similarly characterized, since the output generally depends upon the input amplitude and frequency, in addition to the system location in the state space. Characterization of a nonlinear system thus requires input that is su ciently amplitude and frequency rich to assess the tracking performance throughout the state space. The authors have derived a generalized measure of tracking performance for nonlinear systems to evaluate the e ect of actuator saturation on the performance of a PD-controlled servo system. The tracking performance characterization is based loosely on the construction of a linear system Bode plot. Rather than utilize a sinusoidal input with a single amplitude and frequency, however, the characterization technique utilizes a band-limited pseudo-random input that is devised by passing white noise through a second-order critically-damped low-pass lter, given by: H s ˆ 1 s o c 2 2 The resulting signal contains primarily frequencies up to the cuto frequency o c, which are approximately uniformly distributed, and signi cantly less frequency content above the cuto, as illustrated in Fig. 2. For purposes of performance characterization, the `frequency' of this signal is identi ed by that of the lter cuto, o c. The amplitude of the output is also normalized for the appropriate bounds of the coordinate space (or workspace, as the case may be). Fig. 3 shows examples of these band-limited psuedo-random signals for cuto frequencies of 0.1, 1.0, and 10.0 Hz, and for signal amplitudes normalized to a workspace of Rather than using a combination of magnitude and phase, the tracking performance at a given `frequency' is characterized by the root-mean-square (RMS) tracking error, which is normalized to the RMS of the input signal and referred to as the normalized RMS tracking error. More speci cally, the RMS error is given by: e rms ˆ 1 T T 0 q y y d 2 dt where y is the output, y d is the input command, and T is the tracking duration; the RMS of the input signal is given by: 3

7 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± Fig. 3. Band-limited psuedo-random input signals for cuto frequency of 0.1, 1.0, and 10.0 Hz. The signal amplitudes were normalized for a desired workspace of The minimum value of the 10 Hz signal is 458. The data shown do not appear to reach this value because they were thinned for graphical presentation.

8 504 u rms ˆ 1 T T M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497±511 q y d y o 2 dt 0 where y o is any DC component of the input signal; and nally, the normalized RMS tracking error given by: 4 e rms ˆ erms u rms 5 Perfect tracking performance is indicated by a normalized RMS tracking error of zero, while no response whatsoever to the input command would result in a normalized RMS tracking error of one. In a manner similar to a linear Bode plot, this measure is taken across the frequency spectrum (by varying the lter cuto, o c ) to construct a measure of tracking performance across the `frequency' spectrum. A typical loop shape generated by this performance measure is illustrated in Fig. 4. As shapes are generally di cult to compare, one can choose a value of the normalized RMS error that represents the extent of good tracking, and associate a bandwidth with that value. Selection of the value of error that corresponds to the bandwidth is somewhat arbitrary (as is the choice of 3 db), but also conceptually unimportant. For the purposes of this work, the `bandwidth' of the system is characterized by a normalized RMS tracking error of one half, Fig. 4. An example of a loop shape obtained by the normalized RMS error method, along with an indication of the `bandwidth' determined by a normalized RMS error of 0.5.

9 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± which is depicted in the normalized RMS error loop shape plot of Fig. 4 and corresponds to a time-based performance such as that shown in Fig. 5. Having established a tracking performance measure, the tracking performance of a servo system can be mapped as the control system gains are varied. 5. The e ects of actuator saturation The previously described tracking performance measure is utilized on a simulated single degree-of-freedom servo system in order to demonstrate the e ect of actuator saturation on a PD controlled servo system. The simulated system, which is assumed to move in the horizontal plane without signi cant bearing friction, is described (without saturation) by the closed-loop characteristic equation: Js 2 K d s K p ˆ 0 6 Also, to simplify the illustration, the authors reduce the two degree-of-freedom gain selection problem (selection of K p and K d ) to a single degree-of-freedom problem by constraining the (linear) closed-loop characteristic equation to a damping ratio of unity. The derivative gain is therefore given by: p K d ˆ 2 JK p 7 Fig. 5. Time-based tracking of a signal that represents a performance corresponding to a normalized RMS error of 0.5.

10 506 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497±511 As a result, the gain selection can be discussed in terms of a single loop gain, K p. To enable comparison, the closed-loop servo system was characterized for both the purely linear case (no actuator saturation) and the case with saturation. The model parameters are the rotational inertia, given by J=0.06 kg m 2, and the actuator saturation level, which was set at T max =7.5 Nm, both selected to model an existing servo system. The system was simulated at nine gain sets, as shown in Table 1. Tracking performance at each gain set was measured at twelve input frequencies (as determined by o c ) between 0.1 and 100 Hz. The trajectory inputs were all normalized to a maximum amplitude of positive or negative 458. Tracking inputs for 0.1, 1.0, and 10.0 Hz are shown in Fig. 3. The dependence of the normalized RMS error on the duration of the input was investigated by comparing the measured error for several di erent inputs lasting in duration from 1±10 s. The results showed that, for the system simulated, the normalized RMS error for the lowest tested frequency (0.1 Hz) converged after approximately 5 s. Each normalized RMS error was therefore taken for a 10 s input duration, to ensure convergence. The tracking performance of the linear (no saturation) and nonlinear (with actuator saturation) closed-loop systems, as given by the previously described frequency-based normalized RMS error method, is shown in Figs. 6 and 7, respectively. These gures are roughly analogous to a set of linear system Bode plots (each plot representing a di erent set of gains), but characterize tracking performance across the frequency spectrum with normalized RMS error rather than as a combination of magnitude and phase. As expected with the linear system, the overall tracking performance improves (smaller normalized RMS error) with increasing gain. Unlike the linear system, however, the overall performance of the saturating (nonlinear) system improves with increasing gain up to a point, at which point the system hits a `performance wall', beyond which the performance degrades with increased loop gain. The distinction between linear and nonlinear (saturating) behavior is additionally illustrated by comparing the 0.5 normalized RMS error `bandwidth' Table 1 PD gain sets corresponding to a closed-loop damping ratio of unity. The gain set numbers given correspond to the labels in Figs. 6 and 7 Gain set K p K d

11 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± Fig. 6. Normalized RMS error plot characterizing tracking performance across the frequency spectrum for the linear system of Fig. 1(a) at several di erent gain sets. Each plot is labeled with the corresponding gain set number, as given in Table 1. The 0.5 normalized RMS error `bandwidth' is the frequency at which each curve crosses the 0.5 RMS error level. Note that, as expected for the linear system, the tracking performance improves with increased loop gain. for each case, as shown in Fig. 8. This plot shows clearly that the linear and nonlinear systems behave similarly at low gains (K p R1), but quite di erently at higher gains, where the actuator saturation becomes signi cant. The plot also shows the optimal gain, as given by the 0.5 normalized RMS error measure, is at K p 110, which corresponds to a `bandwidth' of approximately 2.2 Hz. Note that if the PD control gains were chosen to avoid saturation in the interest of linearity (K p 1 1), the closed-loop bandwidth would be less than half of the maximum possible. This is not surprising, as optimal performance should maximally utilize the system resources, and thus regularly push the actuators into saturation. Fig. 9 illustrates the time-based di erence in performance between avoiding actuator saturation (K p 11) and utilizing actuator saturation (K p 110). The simulations therefore show two signi cant features of a PD-controlled servo system with actuator saturation. First, avoiding actuator saturation to ensure linear behavior signi cantly sacri ces system performance. Second, the best tracking performance is not necessarily achieved at the maximum obtainable gain. Rather, performance improves up to a given gain, then begins to recede as the gain is increased further. Though the PD control structure is quite simple, the results have conceptual implications for feedback linearization and sliding mode

12 508 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497±511 Fig. 7. Normalized RMS error plot characterizing tracking performance across the frequency spectrum for the nonlinear (saturating) system of Fig. 1(b) at several di erent PD gain sets. Each plot is labeled with the corresponding gain set number, as given in Table 1. Note that the performance improves up to a given gain (gain set 5), at which point the system hits a `performance wall', then begins to recede as the gain is increased further. approaches as well, since for a servo system, all three control approaches are of a similar structure. 6. Correlation of proposed measure with linear bandwidth As a matter of interest, the 0.5 normalized RMS error bandwidth found for the linear system (shown in Fig. 8) can be compared to the 3 db bandwidth. The 3 db bandwidth of the closed-loop linear servo system of Eq. (6) can be found from the closed-loop transfer function, given by: H s j sˆjo ˆ 1 J K p K d s s 2 K d J s K p J 8 and solving for kh jo k ˆ 3 db ˆ 0:708, which simpli es to the following equation:

13 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497± Fig. 8. Summary of tracking performance vs loop gain for linear and nonlinear (saturating) systems. The optimal gain for the system with actuator saturation is K p 1 10, which corresponds to a `bandwidth' of approximately 2.2 Hz. Note that if the PD control gains were chosen to avoid saturation in the interest of linearity (K p 11), the closed-loop bandwidth would be less than half of the maximum possible. K o 4 2 d J 2 2K p o 2 K 2 p J J 2 ˆ 0 9 where o is the 3 db bandwidth in rad/s. Table 2 shows the closed-loop system bandwidths for several gain sets as characterized by both the 3 db and 0.5 Table 2 Comparison of 3 db bandwidth and 0.5 normalized RMS error bandwidth for a linear system K p K d 3 db BW (Hz) 0.5 norm RMA error BW (Hz)

14 510 M. Goldfarb, T. Sirithanapipat / Mechatronics 9 (1999) 497±511 Fig. 9. Time-based tracking history for input signal with f c =2 Hz for (a) K p 11 (avoiding saturation to maintain linear behavior) and for (b) K p 110 (utilizing actuator saturation). Note also that the tracking in (b) yields a normalized RMS error of 0.5, and thus corresponds closely to the `bandwidth' of the system, as de ned by the devised characterization. normalized RMS error performance measures. As shown in the table, both measures are fairly well correlated. 7. Conclusion A generalized measure has been proposed for tracking performance characterization and utilized to demonstrate the e ect of actuator saturation on PD-controlled servo systems. Numerical simulations of a simple servo system with actuator saturation give rise to two signi cant observations. First, and not surprisingly, avoiding actuator saturation signi cantly impairs tracking performance. Second, and more interestingly, large gains can degrade the tracking performance of a PD-controlled servo system. References [1] Slotine JJ, Li W. Applied nonlinear control. Prentice Hall, [2] Yigit A. On the stability of PD control for a two-link rigid- exible manipulator. ASME Journal of Dynamic Systems, Measurement, and Control 1994;116:208±15. [3] Shahruz SM, Langari G, Tomizuka M. Design of robust PD-type control laws for robotic manipulators with parametric uncertainties. In: Proceedings of the American Control Conference, p. 2967±8. [4] Xu Y, Hollerbach JM, Ma D. A nonlinear PD controller for force and contact transient control. IEEE Control Systems Magazine, p. 15±21. [5] Kim JH, Bien Z. Robust stability of uncertain linear systems with saturating actuators. IEEE Transactions on Automatic Control 1994;39:202±7.

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