Analysis of Random Reference Tracking in Systems ... - IEEE Xplore
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 11, NOVEMBER 2005
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Analysis of Random Reference Tracking in Systems With Saturating Actuators Yongsoon Eun, Pierre T. Kabamba, and Semyon M. Meerkov Abstract—This note presents a method for analysis of random reference tracking in feedback systems with saturating actuators. The development is motivated by the frequency domain approach to linear systems, where the bandwidth and resonance peak of the sensitivity function are used to predict the quality of step reference tracking. Similarly, based on the so-called saturating random sensitivity function, we introduce tracking quality indicators and show that they can be used to determine both the quality of random reference tracking and the nature of track loss under actuator saturation. The shortcomings of the method are also discussed. Index Terms—Reference tracking, saturating actuators, sensitivity function.
Fig. 1. Feedback system with saturating actuator and reference signal. (a) Feedback system with saturating actuator. (b) Reference signal r (t).
I. INTRODUCTION
this approach has been extended to tracking random inputs by introducing the notion of random sensitivity (RS) function [1]. In the current work, we extend this approach to systems with saturating actuators. This is accomplished by introducing and analyzing the so-called saturating random sensitivity (SRS) function. We provide a method for calculating the SRS using a quasi-linearization technique known as stochastic linearization [2]. Stochastic linearization is an approximation technique similar to the method of describing function [3], which proved to be useful in countless applications as a tool for analysis of limit cycles in systems with nonlinear actuators. Similarly, in this note we utilize stochastic linearization for analysis of reference tracking under actuator saturation.
A. Motivation As it is well known, the quality of reference tracking in linear systems is determined by the loop transfer function. In systems with saturating actuators, this is not the case. Indeed consider, for example, the single-input–single-output (SISO) feedback system and the reference signal shown in Fig. 1, where the latter is a realization of a colored noise process with power spectral density SR (!) = (6)=(1 + (!=0:5)6 ). The quality of tracking for several C (s) and P (s), satisfying C (s)P (s) = (75=s(s + 10)), is illustrated in Fig. 2. (In Fig. 2(a), the reference and the output signals practically coincide.) Clearly, the nature of tracking errors in each of the three cases is qualitatively different, which supports the previous assertion. The track loss in systems with saturating actuators may occur due to a number of different reasons. These include those that occur in linear systems plus those due to actuator saturation. To illustrate these reasons, consider again the system and the reference signal of Fig. 1 and select C (s) and P (s), which result in different patterns of track loss but with the same standard deviation of the tracking error e . The results are shown in Fig. 3 (for e = 0:67). As one can see, track loss in Fig. 3(a)–(c) is due to static unresponsiveness, dynamic lagging, and oscillatory behavior, respectively. These reasons take place in the purely linear case as well (see [1]) and are included here for the sake of completeness. Track loss in Figs. 3(d)–(g) is due to saturation, namely, amplitude truncation without controller wind-up, amplitude truncation with the controller wind-up, nonlinear lagging, and nonlinear oscillations, respectively. The goals of this note are to analyze what determines the quality of tracking in systems with saturating actuators and quantify under which conditions one or another type of track loss takes place. B. Approach In the case of linear systems, the quality of step input tracking is often characterized in the frequency domain by the sensitivity (S ) function, specifically, by its d.c. gain, bandwidth, and resonance peak. Recently, Manuscript received October 16, 2004; revised May 21, 2005. Recommended by Associate Editor S. Tarbouriech. Y. Eun is with the Joseph C. Wilson Center for Research and Technology, Xerox Corporation, Webster, NY 14580 USA. P. T. Kabamba and S. M. Meerkov are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.858668
C. Related Literature and Note Outline Systems with saturating actuators have been studied for a long time (see recent monographs [4]–[6]). However, just a few publications have been devoted to reference tracking. These include [7] where tracking domains have been investigated, [4] where asymptotic output tracking has been studied, [8] where random reference tracking by a servo with a PD-controller has been analyzed, and [9] where the notion of system type has been extended to feedback control with saturating actuators. However, no general methods for analysis of quality of random reference tracking in systems with saturating actuators exist. This note is intended to contribute to this end. As far as the application of stochastic linearization to systems with saturating is concerned, it should be pointed out that it has been used in [10] for the problem of disturbance rejection. While in linear systems this problem is equivalent to the problem of reference tracking, in systems with saturating actuators this is not the case. Therefore, the original contribution of this note is in providing a quantitative method for analysis of quality of random reference tracking in systems with saturating actuators and for determining reasons for track loss. To accomplish this, Section II introduces the SRS and its characteristics: dc gain, bandwidth, resonance frequency, and resonance peak. In Section III, we use these characteristics to define dimensionless tracking quality indicators and diagnostic flow charts. Finally, in Section IV the conclusions are given. Due to space limitations the proofs are not included here and can be found in [11]. II. SATURATING RANDOM SENSITIVITY FUNCTION A. Random Reference Signals Similar to [1], the class of random reference signals, considered in this work, is defined as the scaled steady state output of the third
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Fig. 2. Random reference tracking in systems with saturating actuators and identical loop transfer functions. (a) (b) C (s) = 27 and P (s) = (25=9)=(s(s + 10)). (c) C (s) = (5=s) and P (s) = (15)=(s + 10).
C (s)
= 5 and
P (s)
= (15=s(s + 10)).
Fig. 3. Illustration of reasons for track loss in systems with saturating actuators. (a) C (s) = 4=(s + 10) and P (s) = 6:2=(s + 5). (b) C (s) = 0:4 and P (s) = 0:8125(s + 0:1)=s(s + 0:02) . (c) C (s) = 0:0112(s + 30:533)=s and P (s) = 4=s. (d) C (s) = 100 and P (s) = 11=(s + 3)(s + 7) . (e) C (s) = 2=s and P (s) = 8=(s + 10). (f) C (s) = 25 and P (s) = 2=s(s + 10). (g) C (s) = 4:126(s + 1:102) and P (s) = 0:355=s .
order Butterworth filter driven by a standard zero mean white Gaussian process. The transfer function of this filter is given by
F (s; ) =
3
3
s3 + 2 s2 + 2 2 s + 3
(2.1)
where the dc gain is selected so that, for all 3-dB bandwidths , the standard deviation of the output is 1. Thus, the reference signals considered in this work are given by
r(t) = r r(t; )
(2.2)
where r(t; ) is the output of (2.1) and r is the “amplitude” or, more precisely, the standard deviation of r(t). Clearly, higher order Butterworth filters can be considered instead of (2.1). However, as it turns out, the results remain quite similar to those obtained using (2.1) (see also [1]) and, thus, for the sake of simplicity,
we consider band-limited reference signals (2.2).
r(t) defined by (2.1) and
B. System Model Consider the system shown in Fig. 4(a) with reference signal (2.1), (2.2) and sat (u) defined by
;
sat (u) = u;
0 ;
if < u if 0 u if u < 0
:
(2.3)
Due to the nonlinearity, exact analysis of this system requires solving the Fokker–Plank equation [12], which is possible only in a few special cases. Therefore, a simplification is necessary. As it was pointed out previously, we use for this purpose the method of stochastic linearization [2]. According to this method, the saturation function is replaced by a linear function, the slope of which depends on the standard deviation of the signal at the input of the saturation.
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Fig. 4. Original system and its stochastically linearized counterpart. (a) Original system. (b) Stochastically linearized system.
N = erf
p
F (s; )C (s) 1+NP (s)C (s)
2 r
:
(2.9)
2
As one can see, physically SRS( ; r ) represents the ratio of the standard deviations of the error signal e^(t) and reference signal r(t), i.e., SRS( ; r ) = ^e =r . In some cases, (2.9) does not have a solution for N . As it is pointed out in [2], this is an indication that the system of Fig. 4(a) with input (2.2) does not have a stationary regime. Asymptotic properties of SRS( ; r ) are as follows. Theorem 2.1.: Assume that the closed-loop system of Fig. 4(b) is asymptotically stable for all N 2 (0; 1], and both P (s)C (s) and F (s; )C (s) are strictly proper. Then i) for any > 0,
F (s; ) lim !0 SRS( ; r ) = 1 + P (s)C (s) 2 ; for any r > 0
ii) Fig. 5. Saturating random sensitivity for systems of Fig. 2.
Using stochastic linearization, the nonlinear system of Fig. 4(a) can be replaced by the quasilinear system shown in Fig. 4(b), where the equivalent gain N (u^ ) is given by [10]
N (u^ ) = erf
erf() = p2
p
2u^ exp(0 2 ) d :
(2.4)
0
(2.5)
The system of Fig. 4(b) is quasi-linear since N depends on the standard ^. deviation of u The results reported in this note are obtained using the simplified model of Fig. 4(b). However, the system of Fig. 4(a) is also used—to verify that the results derived are applicable to the original nonlinear system as well. C. Definition and Properties of the Saturating Random Sensitivity Function If N in Fig. 4(b) were a constant gain equal to 1, the sensitivity and the random sensitivity functions of the closed-loop system would be given by [1]
1 1 + P (s)C (s) RS( ) = 1 +FP(s(s; ) ) : C (s ) 2 S (s) =
(2.6) (2.7)
These functions are extended to the case of the quasi-linear system of Fig. 4(b) by defining the saturating random sensitivity (SRS) as follows:
F (s; ) SRS( ; r ) = 1 + NP (s )C ( s )
2
(2.8)
(2.10)
!1
lim SRS( ; r ) = 1;
(2.11)
!0
(2.12)
lim SRS( ; r ) = 1 + NP1(0)C (0)
where N satisfies
N = erf
p
2 r
C (0) 1+NP (0)C (0)
:
(2.13)
Clearly, statement (2.10) implies that for small reference signals,
SRS( ; r ) practically coincides with RS( ). Statement (2.11) indicates that for large the functions SRS( ; r ); RS( ) and S (s) are practically identical, and no tracking takes place. Finally, since N 1, statement (2.12) shows that for low frequencies SRS( ; r ) is typically larger than RS( ) and, thus, the presence of saturation impedes tracking. Figs. 5 and 6 illustrate the SRS functions for all systems of Figs. 2 and 3, respectively. As it will be shown in Section 3, these functions define the nature of tracking and track loss in the corresponding systems. D. Shape Characteristics Although a complete description of SRS( ; r ) requires a two-dimensional surface, a compact (but incomplete) description can be given in terms of characteristics, similar to those used to describe the S (s) and RS( ) functions. Namely, introduce i) saturating random d.c. gain: SRdc = lim 0; 0 SRS( ; r ); ii) saturating random bandwidth: p SR BW (r ) = minf jSRS( ; r ) = 1= 2g; iii) saturating random resonance frequency: SR r (r ) = arg max >0 SRS( ; r ); iv) saturating random resonance peak: SRMr (r ) = sup >0 SRS( ; r ). For the SRS functions of Figs. 6(a) and 6(d), SRdc are 0.67 and 0.019, respectively, while for all others it is 0. Clearly, one might expect
! !
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Fig. 6. Saturating random sensitivity for systems of Fig. 3.
TABLE I SIZE OF TRACKABLE DOMAINS FOR SYSTEMS OF FIGS. 2 AND 3
where r0 is the size of the step and C0 and P0 are d.c. gains of the controller and plant, respectively. Trackable domains for all systems of Figs. 2 and 3 are given in Table I. Clearly, systems of Figs. 2(c), 3(a), (d), and (e) have finite trackable domains and, therefore, their bandwidth must drop to 0 for r sufficiently large, no matter how small
is. The SRS( ; r ) function and its characteristics are used in Section III to quantify the nature of random reference tracking and track loss in systems with saturating actuators. E. Accuracy of SRS and Its Shape Characteristics Fig. 7. Saturating random bandwidth for systems of Figs. 2 and 3. (a) For systems of Fig. 2. (b) For systems of Figs. 3(a)–(d). (c) For systems of Figs. 3(e)–(g).
that tracking of even small and slowly changing signals in the system of Fig. 6(a) is poor, and the track loss is due to static unresponsiveness. The SR BW for all systems of Figs. 2 and 3 are shown in Fig. 7. In all cases SR BW is monotonically decreasing in r , but systems of Figs. 2(c), 3(a), 3(d), and 3(e) result in SR BW with almost infinite roll-off rate. This phenomenon can be explained using the notion of trackable domain (TD) introduced in [9]. Indeed, it has been shown in [9] that the set of step inputs that can be tracked by a system with a saturating actuator and its size can be quantified, respectively, as
TD = r0 2 : jr0 j < C1 + P0 0 1 jTDj = +P C0
0
(2.14)
The saturating random sensitivity function provides an estimate, ^e , of the steady state tracking error, e , in the system of Fig. 4(a) as follows:
^e = r SRS( ; r ):
(2.15)
Unfortunately, the question of accuracy of this estimate has no general answer. In a few special cases, where e can be found exactly by solving the corresponding Fokker–Planck equation, this accuracy has been shown to be quite high (well within 10%) [2], [10]. In other cases, the accuracy of this estimate has been evaluated numerically [10]. It has been observed that the accuracy is high for small and large r (regardless of ) and for large (regardless of r ). For some intermediate values of and r , the approximation could be poor. However, since the four shape characteristics, i.e., SRdc ; SR BW (r ); SR r (r ); SRMr (r ), are defined in the domains where the accuracy is typically high, we use them for tracking quality analysis in systems with saturating actuators.
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Fig. 8. Diagnostic flow charts for analysis of tracking quality in systems with saturating actuators. (a) Diagnostics based on I . (b) Diagnostics based on I . (c) Diagnostics based on I and I .
III. TRACKING QUALITY INDICATORS AND DIAGNOSTIC FLOW CHARTS Tracking quality indicators are functionals intended to qualitatively characterize the SRS( ; r ) along with TD vis-a-vis the class of reference signals to be tracked and, thereby, predict the nature of random reference tracking and track loss in systems with saturating actuators. In this regard they are analogous to the gain and phase margins of linear systems, intended to predict stability robustness, or to the resonance peak and the bandwidth, used to predict the nature of the step response. The tracking quality indicators are introduced as follows:
I0 =
r
jTDj
I1 = SRdc I2 =
(3.1) (3.2)
SR BW (r )
I3 = min SR r (r ) ; SRMr (r ) 0 1
:
Although indicators I1 0 I3 are proper extensions of the corresponding tracking quality indicators for linear systems [1], they may be large due to either the linear or the nonlinear part of the system. The two cases can be discriminated by the value of the equivalent gain, N , defined by (2.9). Specifically if N is close to 1, the phenomenon is caused by the linear part of the system, otherwise, it is due to saturation. For example, if I2 is large and both I3 and N are small, the resulting behavior exhibits nonlinear lagging. Similarly, if I3 is large and N is small, the resulting behavior exhibits nonlinear oscillations. Based on the above discussion, the nature of tracking quality and reasons for track loss can be diagnosed using the flow charts shown in Fig. 8. Each of them includes a qualitative term “large.” Based on our experience, an indicator can be viewed as large if
(3.3)
I0 > 0:4 I1 > 0:1 I2 > 0:4 I3 > 0:2:
(3.4)
Consider, for example, the system of Fig. 4(a) with C (s) = 5=s; P (s) = 15=(s + 10), and r(t) = 1:5 r(t; 10). The tracking
Clearly, I0 quantifies the “size” of the reference signal vis-a-vis the trackable domain; large I0 implies that amplitude truncation must take place. Indicator I1 quantifies the level of static responsiveness; large I1 implies that responsiveness, even to small and slow signals, is poor. Indicator I2 quantifies the bandwidth of the reference signal in units of the closed-loop bandwidth; large I2 implies that dynamic lagging must take place. Finally, I3 characterizes oscillatory properties of the response; large I3 implies that oscillations must be present.
(3.5)
quality indicators for this system and reference signal are
I0 = 1 I1 = 0 I2 = 4:705 I3 = 0:078
(3.6)
while N = 0:47. Thus, using the flowchart of Fig. 8(a) we determine that tracking is poor due to amplitude truncation with wind-up. Using the flowchart of Fig. 8(b), we conclude that there is no loss of tracking due to unresponsiveness. Using the flowchart of Fig. 8(c), we expect
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Fig. 9.
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Tracking r (t) = 1:5 r (t; 10) in system of Fig. 4 (a) with C (s) = 5=s; P (s) = 15=(s + 10) and = 1. TABLE II DIAGNOSED QUALITY OF TRACKING IN SYSTEMS OF FIGS. 2 AND 3
nonlinear lagging. These conclusions are supported by the trace of y (t) [obtained by simulating the system of Fig. 4(a)] shown in Fig. 9. Table II presents the tracking quality indicators and the conclusions as to the nature of tracking and track loss for all systems considered in Section I. Clearly, these conclusions are supported by the traces of Figs. 2 and 3. Remark: The diagnostics approach, described previously, leads to qualitatively correct results in the majority of cases analyzed. However, this is not always the case. Typically, this approach fails when C (s) and P (s) are such that the usual sensitivity function, S (s), does not predict the step response well. An example of this type, where neither the linear nor the saturating cases are well characterized by their sensitivity functions, can be found in [1]. IV. CONCLUSION This note provides a simple method for analysis of random reference tracking in systems with saturating actuators. The method mimics the classical frequency domain approach to step reference tracking in linear systems. Indeed, it is based on the indicators, which are similar to bandwidth and resonance peak, used in the linear case, and which allow one to predict the quality of random reference tracking and nature of track loss in systems with saturating actuators. The limitations of the method are as follows. 1) As it is stated in Theorem 2.1, the stochastically linearized system of in Fig. 4(b) is assumed to be asymptotically stable for all N 2 (0; 1]. This implies that the plants, for which the method is applicable, have no poles in the open right half plane. 2) As it is indicated in the Remark of Section III, the method may lead to erroneous conclusions if the saturating random sensitivity function has a “contrived” behavior (see [1] for an example). In these cases, however, the classical approach, based on the sensitivity function, fails as well. In spite of these shortcomings, the method developed in this note offers control system designers a quick and easy way to predict system performance without resorting to lengthy and expensive numerical sim-
ulations. In addition, it illuminates reasons for track loss, which might be useful for developing improvement measures. REFERENCES [1] Y. Eun, P. T. Kabamba, and S. M. Meerkov, “Tracking of random references: Random sensitivity function and tracking quality indicators,” IEEE Trans. Autom. Control, vol. 48, no. 9, pp. 1666–1671, Sep. 2003. [2] J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization. New York: Wiley, 1990. [3] H. K. Khalil, Nonlinear Systems Analysis. Upper Saddle River, NJ: Prentice-Hall, 2002. [4] A. Saberi, A. A. Stoorvogel, and P. Sannuti, Control of Linear Systems With Regulation and Input Constraints. London, U.K.: SpringerVerlag, 2000. [5] T. Hu and Z. Lin, Control Systems with Actuator Saturation: Analysis and Design. Boston, MA: Birkhäuser, 2001. [6] V. Kapila, Ed., Actuator Saturation Control. New York: Marcel Dekker, 2002. [7] V. A. Yakubovich, S. Nakaura, and K. Furuta, “Tracking domains for unstable plants with saturating-like actuators,” in Proc. Control Decision Conf., 1999, pp. 2750–2755. [8] M. Goldfarb and T. Sirithanapipat, “The effect of actuator saturation on the performance of PD-controlled servo systems,” Mechatron., vol. 9, pp. 497–511, 1999. [9] Y. Eun, P. T. Kabamba, and S. M. Meerkov, “System types in feedback control with saturating actuators,” IEEE Trans. Autom. Control, vol. 49, no. 2, pp. 287–291, Feb. 2004. [10] C. Gökçek, P. T. Kabamba, and S. M. Meerkov, “An LQR/LQG theory for systems with saturating actuators,” IEEE Trans. Autom. Control, vol. 46, no. 10, pp. 1529–1542, Oct. 2001. [11] Y. Eun, P. T. Kabamba, and S. M. Meerkov, “Analysis of random reference tracking in systems with saturating actuators,” Univ. Michigan, Ann Arbor, MI, Control Group Rep. CGR 04-12, 2004. [12] H. Risken, Fokker-Plank Equation: Methods of Solution and Applications. Berlin, Germay: Springer-Verlag, 1989.
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Analysis of Random Reference Tracking in Systems With Saturating Actuators Yongsoon Eun, Pierre T. Kabamba, and Semyon M. Meerkov Abstract—This note presents a method for analysis of random reference tracking in feedback systems with saturating actuators. The development is motivated by the frequency domain approach to linear systems, where the bandwidth and resonance peak of the sensitivity function are used to predict the quality of step reference tracking. Similarly, based on the so-called saturating random sensitivity function, we introduce tracking quality indicators and show that they can be used to determine both the quality of random reference tracking and the nature of track loss under actuator saturation. The shortcomings of the method are also discussed. Index Terms—Reference tracking, saturating actuators, sensitivity function.
Fig. 1. Feedback system with saturating actuator and reference signal. (a) Feedback system with saturating actuator. (b) Reference signal r (t).
I. INTRODUCTION
this approach has been extended to tracking random inputs by introducing the notion of random sensitivity (RS) function [1]. In the current work, we extend this approach to systems with saturating actuators. This is accomplished by introducing and analyzing the so-called saturating random sensitivity (SRS) function. We provide a method for calculating the SRS using a quasi-linearization technique known as stochastic linearization [2]. Stochastic linearization is an approximation technique similar to the method of describing function [3], which proved to be useful in countless applications as a tool for analysis of limit cycles in systems with nonlinear actuators. Similarly, in this note we utilize stochastic linearization for analysis of reference tracking under actuator saturation.
A. Motivation As it is well known, the quality of reference tracking in linear systems is determined by the loop transfer function. In systems with saturating actuators, this is not the case. Indeed consider, for example, the single-input–single-output (SISO) feedback system and the reference signal shown in Fig. 1, where the latter is a realization of a colored noise process with power spectral density SR (!) = (6)=(1 + (!=0:5)6 ). The quality of tracking for several C (s) and P (s), satisfying C (s)P (s) = (75=s(s + 10)), is illustrated in Fig. 2. (In Fig. 2(a), the reference and the output signals practically coincide.) Clearly, the nature of tracking errors in each of the three cases is qualitatively different, which supports the previous assertion. The track loss in systems with saturating actuators may occur due to a number of different reasons. These include those that occur in linear systems plus those due to actuator saturation. To illustrate these reasons, consider again the system and the reference signal of Fig. 1 and select C (s) and P (s), which result in different patterns of track loss but with the same standard deviation of the tracking error e . The results are shown in Fig. 3 (for e = 0:67). As one can see, track loss in Fig. 3(a)–(c) is due to static unresponsiveness, dynamic lagging, and oscillatory behavior, respectively. These reasons take place in the purely linear case as well (see [1]) and are included here for the sake of completeness. Track loss in Figs. 3(d)–(g) is due to saturation, namely, amplitude truncation without controller wind-up, amplitude truncation with the controller wind-up, nonlinear lagging, and nonlinear oscillations, respectively. The goals of this note are to analyze what determines the quality of tracking in systems with saturating actuators and quantify under which conditions one or another type of track loss takes place. B. Approach In the case of linear systems, the quality of step input tracking is often characterized in the frequency domain by the sensitivity (S ) function, specifically, by its d.c. gain, bandwidth, and resonance peak. Recently, Manuscript received October 16, 2004; revised May 21, 2005. Recommended by Associate Editor S. Tarbouriech. Y. Eun is with the Joseph C. Wilson Center for Research and Technology, Xerox Corporation, Webster, NY 14580 USA. P. T. Kabamba and S. M. Meerkov are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.858668
C. Related Literature and Note Outline Systems with saturating actuators have been studied for a long time (see recent monographs [4]–[6]). However, just a few publications have been devoted to reference tracking. These include [7] where tracking domains have been investigated, [4] where asymptotic output tracking has been studied, [8] where random reference tracking by a servo with a PD-controller has been analyzed, and [9] where the notion of system type has been extended to feedback control with saturating actuators. However, no general methods for analysis of quality of random reference tracking in systems with saturating actuators exist. This note is intended to contribute to this end. As far as the application of stochastic linearization to systems with saturating is concerned, it should be pointed out that it has been used in [10] for the problem of disturbance rejection. While in linear systems this problem is equivalent to the problem of reference tracking, in systems with saturating actuators this is not the case. Therefore, the original contribution of this note is in providing a quantitative method for analysis of quality of random reference tracking in systems with saturating actuators and for determining reasons for track loss. To accomplish this, Section II introduces the SRS and its characteristics: dc gain, bandwidth, resonance frequency, and resonance peak. In Section III, we use these characteristics to define dimensionless tracking quality indicators and diagnostic flow charts. Finally, in Section IV the conclusions are given. Due to space limitations the proofs are not included here and can be found in [11]. II. SATURATING RANDOM SENSITIVITY FUNCTION A. Random Reference Signals Similar to [1], the class of random reference signals, considered in this work, is defined as the scaled steady state output of the third
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Fig. 2. Random reference tracking in systems with saturating actuators and identical loop transfer functions. (a) (b) C (s) = 27 and P (s) = (25=9)=(s(s + 10)). (c) C (s) = (5=s) and P (s) = (15)=(s + 10).
C (s)
= 5 and
P (s)
= (15=s(s + 10)).
Fig. 3. Illustration of reasons for track loss in systems with saturating actuators. (a) C (s) = 4=(s + 10) and P (s) = 6:2=(s + 5). (b) C (s) = 0:4 and P (s) = 0:8125(s + 0:1)=s(s + 0:02) . (c) C (s) = 0:0112(s + 30:533)=s and P (s) = 4=s. (d) C (s) = 100 and P (s) = 11=(s + 3)(s + 7) . (e) C (s) = 2=s and P (s) = 8=(s + 10). (f) C (s) = 25 and P (s) = 2=s(s + 10). (g) C (s) = 4:126(s + 1:102) and P (s) = 0:355=s .
order Butterworth filter driven by a standard zero mean white Gaussian process. The transfer function of this filter is given by
F (s; ) =
3
3
s3 + 2 s2 + 2 2 s + 3
(2.1)
where the dc gain is selected so that, for all 3-dB bandwidths , the standard deviation of the output is 1. Thus, the reference signals considered in this work are given by
r(t) = r r(t; )
(2.2)
where r(t; ) is the output of (2.1) and r is the “amplitude” or, more precisely, the standard deviation of r(t). Clearly, higher order Butterworth filters can be considered instead of (2.1). However, as it turns out, the results remain quite similar to those obtained using (2.1) (see also [1]) and, thus, for the sake of simplicity,
we consider band-limited reference signals (2.2).
r(t) defined by (2.1) and
B. System Model Consider the system shown in Fig. 4(a) with reference signal (2.1), (2.2) and sat (u) defined by
;
sat (u) = u;
0 ;
if < u if 0 u if u < 0
:
(2.3)
Due to the nonlinearity, exact analysis of this system requires solving the Fokker–Plank equation [12], which is possible only in a few special cases. Therefore, a simplification is necessary. As it was pointed out previously, we use for this purpose the method of stochastic linearization [2]. According to this method, the saturation function is replaced by a linear function, the slope of which depends on the standard deviation of the signal at the input of the saturation.
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Fig. 4. Original system and its stochastically linearized counterpart. (a) Original system. (b) Stochastically linearized system.
N = erf
p
F (s; )C (s) 1+NP (s)C (s)
2 r
:
(2.9)
2
As one can see, physically SRS( ; r ) represents the ratio of the standard deviations of the error signal e^(t) and reference signal r(t), i.e., SRS( ; r ) = ^e =r . In some cases, (2.9) does not have a solution for N . As it is pointed out in [2], this is an indication that the system of Fig. 4(a) with input (2.2) does not have a stationary regime. Asymptotic properties of SRS( ; r ) are as follows. Theorem 2.1.: Assume that the closed-loop system of Fig. 4(b) is asymptotically stable for all N 2 (0; 1], and both P (s)C (s) and F (s; )C (s) are strictly proper. Then i) for any > 0,
F (s; ) lim !0 SRS( ; r ) = 1 + P (s)C (s) 2 ; for any r > 0
ii) Fig. 5. Saturating random sensitivity for systems of Fig. 2.
Using stochastic linearization, the nonlinear system of Fig. 4(a) can be replaced by the quasilinear system shown in Fig. 4(b), where the equivalent gain N (u^ ) is given by [10]
N (u^ ) = erf
erf() = p2
p
2u^ exp(0 2 ) d :
(2.4)
0
(2.5)
The system of Fig. 4(b) is quasi-linear since N depends on the standard ^. deviation of u The results reported in this note are obtained using the simplified model of Fig. 4(b). However, the system of Fig. 4(a) is also used—to verify that the results derived are applicable to the original nonlinear system as well. C. Definition and Properties of the Saturating Random Sensitivity Function If N in Fig. 4(b) were a constant gain equal to 1, the sensitivity and the random sensitivity functions of the closed-loop system would be given by [1]
1 1 + P (s)C (s) RS( ) = 1 +FP(s(s; ) ) : C (s ) 2 S (s) =
(2.6) (2.7)
These functions are extended to the case of the quasi-linear system of Fig. 4(b) by defining the saturating random sensitivity (SRS) as follows:
F (s; ) SRS( ; r ) = 1 + NP (s )C ( s )
2
(2.8)
(2.10)
!1
lim SRS( ; r ) = 1;
(2.11)
!0
(2.12)
lim SRS( ; r ) = 1 + NP1(0)C (0)
where N satisfies
N = erf
p
2 r
C (0) 1+NP (0)C (0)
:
(2.13)
Clearly, statement (2.10) implies that for small reference signals,
SRS( ; r ) practically coincides with RS( ). Statement (2.11) indicates that for large the functions SRS( ; r ); RS( ) and S (s) are practically identical, and no tracking takes place. Finally, since N 1, statement (2.12) shows that for low frequencies SRS( ; r ) is typically larger than RS( ) and, thus, the presence of saturation impedes tracking. Figs. 5 and 6 illustrate the SRS functions for all systems of Figs. 2 and 3, respectively. As it will be shown in Section 3, these functions define the nature of tracking and track loss in the corresponding systems. D. Shape Characteristics Although a complete description of SRS( ; r ) requires a two-dimensional surface, a compact (but incomplete) description can be given in terms of characteristics, similar to those used to describe the S (s) and RS( ) functions. Namely, introduce i) saturating random d.c. gain: SRdc = lim 0; 0 SRS( ; r ); ii) saturating random bandwidth: p SR BW (r ) = minf jSRS( ; r ) = 1= 2g; iii) saturating random resonance frequency: SR r (r ) = arg max >0 SRS( ; r ); iv) saturating random resonance peak: SRMr (r ) = sup >0 SRS( ; r ). For the SRS functions of Figs. 6(a) and 6(d), SRdc are 0.67 and 0.019, respectively, while for all others it is 0. Clearly, one might expect
! !
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Fig. 6. Saturating random sensitivity for systems of Fig. 3.
TABLE I SIZE OF TRACKABLE DOMAINS FOR SYSTEMS OF FIGS. 2 AND 3
where r0 is the size of the step and C0 and P0 are d.c. gains of the controller and plant, respectively. Trackable domains for all systems of Figs. 2 and 3 are given in Table I. Clearly, systems of Figs. 2(c), 3(a), (d), and (e) have finite trackable domains and, therefore, their bandwidth must drop to 0 for r sufficiently large, no matter how small
is. The SRS( ; r ) function and its characteristics are used in Section III to quantify the nature of random reference tracking and track loss in systems with saturating actuators. E. Accuracy of SRS and Its Shape Characteristics Fig. 7. Saturating random bandwidth for systems of Figs. 2 and 3. (a) For systems of Fig. 2. (b) For systems of Figs. 3(a)–(d). (c) For systems of Figs. 3(e)–(g).
that tracking of even small and slowly changing signals in the system of Fig. 6(a) is poor, and the track loss is due to static unresponsiveness. The SR BW for all systems of Figs. 2 and 3 are shown in Fig. 7. In all cases SR BW is monotonically decreasing in r , but systems of Figs. 2(c), 3(a), 3(d), and 3(e) result in SR BW with almost infinite roll-off rate. This phenomenon can be explained using the notion of trackable domain (TD) introduced in [9]. Indeed, it has been shown in [9] that the set of step inputs that can be tracked by a system with a saturating actuator and its size can be quantified, respectively, as
TD = r0 2 : jr0 j < C1 + P0 0 1 jTDj = +P C0
0
(2.14)
The saturating random sensitivity function provides an estimate, ^e , of the steady state tracking error, e , in the system of Fig. 4(a) as follows:
^e = r SRS( ; r ):
(2.15)
Unfortunately, the question of accuracy of this estimate has no general answer. In a few special cases, where e can be found exactly by solving the corresponding Fokker–Planck equation, this accuracy has been shown to be quite high (well within 10%) [2], [10]. In other cases, the accuracy of this estimate has been evaluated numerically [10]. It has been observed that the accuracy is high for small and large r (regardless of ) and for large (regardless of r ). For some intermediate values of and r , the approximation could be poor. However, since the four shape characteristics, i.e., SRdc ; SR BW (r ); SR r (r ); SRMr (r ), are defined in the domains where the accuracy is typically high, we use them for tracking quality analysis in systems with saturating actuators.
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Fig. 8. Diagnostic flow charts for analysis of tracking quality in systems with saturating actuators. (a) Diagnostics based on I . (b) Diagnostics based on I . (c) Diagnostics based on I and I .
III. TRACKING QUALITY INDICATORS AND DIAGNOSTIC FLOW CHARTS Tracking quality indicators are functionals intended to qualitatively characterize the SRS( ; r ) along with TD vis-a-vis the class of reference signals to be tracked and, thereby, predict the nature of random reference tracking and track loss in systems with saturating actuators. In this regard they are analogous to the gain and phase margins of linear systems, intended to predict stability robustness, or to the resonance peak and the bandwidth, used to predict the nature of the step response. The tracking quality indicators are introduced as follows:
I0 =
r
jTDj
I1 = SRdc I2 =
(3.1) (3.2)
SR BW (r )
I3 = min SR r (r ) ; SRMr (r ) 0 1
:
Although indicators I1 0 I3 are proper extensions of the corresponding tracking quality indicators for linear systems [1], they may be large due to either the linear or the nonlinear part of the system. The two cases can be discriminated by the value of the equivalent gain, N , defined by (2.9). Specifically if N is close to 1, the phenomenon is caused by the linear part of the system, otherwise, it is due to saturation. For example, if I2 is large and both I3 and N are small, the resulting behavior exhibits nonlinear lagging. Similarly, if I3 is large and N is small, the resulting behavior exhibits nonlinear oscillations. Based on the above discussion, the nature of tracking quality and reasons for track loss can be diagnosed using the flow charts shown in Fig. 8. Each of them includes a qualitative term “large.” Based on our experience, an indicator can be viewed as large if
(3.3)
I0 > 0:4 I1 > 0:1 I2 > 0:4 I3 > 0:2:
(3.4)
Consider, for example, the system of Fig. 4(a) with C (s) = 5=s; P (s) = 15=(s + 10), and r(t) = 1:5 r(t; 10). The tracking
Clearly, I0 quantifies the “size” of the reference signal vis-a-vis the trackable domain; large I0 implies that amplitude truncation must take place. Indicator I1 quantifies the level of static responsiveness; large I1 implies that responsiveness, even to small and slow signals, is poor. Indicator I2 quantifies the bandwidth of the reference signal in units of the closed-loop bandwidth; large I2 implies that dynamic lagging must take place. Finally, I3 characterizes oscillatory properties of the response; large I3 implies that oscillations must be present.
(3.5)
quality indicators for this system and reference signal are
I0 = 1 I1 = 0 I2 = 4:705 I3 = 0:078
(3.6)
while N = 0:47. Thus, using the flowchart of Fig. 8(a) we determine that tracking is poor due to amplitude truncation with wind-up. Using the flowchart of Fig. 8(b), we conclude that there is no loss of tracking due to unresponsiveness. Using the flowchart of Fig. 8(c), we expect
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Tracking r (t) = 1:5 r (t; 10) in system of Fig. 4 (a) with C (s) = 5=s; P (s) = 15=(s + 10) and = 1. TABLE II DIAGNOSED QUALITY OF TRACKING IN SYSTEMS OF FIGS. 2 AND 3
nonlinear lagging. These conclusions are supported by the trace of y (t) [obtained by simulating the system of Fig. 4(a)] shown in Fig. 9. Table II presents the tracking quality indicators and the conclusions as to the nature of tracking and track loss for all systems considered in Section I. Clearly, these conclusions are supported by the traces of Figs. 2 and 3. Remark: The diagnostics approach, described previously, leads to qualitatively correct results in the majority of cases analyzed. However, this is not always the case. Typically, this approach fails when C (s) and P (s) are such that the usual sensitivity function, S (s), does not predict the step response well. An example of this type, where neither the linear nor the saturating cases are well characterized by their sensitivity functions, can be found in [1]. IV. CONCLUSION This note provides a simple method for analysis of random reference tracking in systems with saturating actuators. The method mimics the classical frequency domain approach to step reference tracking in linear systems. Indeed, it is based on the indicators, which are similar to bandwidth and resonance peak, used in the linear case, and which allow one to predict the quality of random reference tracking and nature of track loss in systems with saturating actuators. The limitations of the method are as follows. 1) As it is stated in Theorem 2.1, the stochastically linearized system of in Fig. 4(b) is assumed to be asymptotically stable for all N 2 (0; 1]. This implies that the plants, for which the method is applicable, have no poles in the open right half plane. 2) As it is indicated in the Remark of Section III, the method may lead to erroneous conclusions if the saturating random sensitivity function has a “contrived” behavior (see [1] for an example). In these cases, however, the classical approach, based on the sensitivity function, fails as well. In spite of these shortcomings, the method developed in this note offers control system designers a quick and easy way to predict system performance without resorting to lengthy and expensive numerical sim-
ulations. In addition, it illuminates reasons for track loss, which might be useful for developing improvement measures. REFERENCES [1] Y. Eun, P. T. Kabamba, and S. M. Meerkov, “Tracking of random references: Random sensitivity function and tracking quality indicators,” IEEE Trans. Autom. Control, vol. 48, no. 9, pp. 1666–1671, Sep. 2003. [2] J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization. New York: Wiley, 1990. [3] H. K. Khalil, Nonlinear Systems Analysis. Upper Saddle River, NJ: Prentice-Hall, 2002. [4] A. Saberi, A. A. Stoorvogel, and P. Sannuti, Control of Linear Systems With Regulation and Input Constraints. London, U.K.: SpringerVerlag, 2000. [5] T. Hu and Z. Lin, Control Systems with Actuator Saturation: Analysis and Design. Boston, MA: Birkhäuser, 2001. [6] V. Kapila, Ed., Actuator Saturation Control. New York: Marcel Dekker, 2002. [7] V. A. Yakubovich, S. Nakaura, and K. Furuta, “Tracking domains for unstable plants with saturating-like actuators,” in Proc. Control Decision Conf., 1999, pp. 2750–2755. [8] M. Goldfarb and T. Sirithanapipat, “The effect of actuator saturation on the performance of PD-controlled servo systems,” Mechatron., vol. 9, pp. 497–511, 1999. [9] Y. Eun, P. T. Kabamba, and S. M. Meerkov, “System types in feedback control with saturating actuators,” IEEE Trans. Autom. Control, vol. 49, no. 2, pp. 287–291, Feb. 2004. [10] C. Gökçek, P. T. Kabamba, and S. M. Meerkov, “An LQR/LQG theory for systems with saturating actuators,” IEEE Trans. Autom. Control, vol. 46, no. 10, pp. 1529–1542, Oct. 2001. [11] Y. Eun, P. T. Kabamba, and S. M. Meerkov, “Analysis of random reference tracking in systems with saturating actuators,” Univ. Michigan, Ann Arbor, MI, Control Group Rep. CGR 04-12, 2004. [12] H. Risken, Fokker-Plank Equation: Methods of Solution and Applications. Berlin, Germay: Springer-Verlag, 1989.
