Digital Redesign of Continuous Time Controller by Multirate Sampling and High Order Holds Yuping Gu and Masayoshi Tomizuka Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720-1740
[email protected] [email protected] Josep Tornero System Engineering and Control, Universidad Politecnica de Valencia, Valencia Spain, P.O.Box 22012 E-46071
[email protected]
Abstract This paper presents the digital redesign of continuous time controller when the measurement sampling period is relatively large and standard discretization methods are most likely not able to provide a suitable digital controller from either stability or performance point of view. An analysis tool is provided to improve the performance of digital control systems by the following two measures: 1) update the controller output N times faster than the measurement sampling frequency, and 2) utilize past sampled outputs to predict the evolution of the output error between two consecutive sampling instances. The analysis tool is developed by regarding the overall system as a multirate sampled data system. An illustrative example is given to show the utilization of the developed analysis tool.
keyword: multi-rate control, digital control
1 Introduction Digital redesign is to obtain a digital controller by discretising a predesigned analogue controller [1] [2] [3]. One of the important advantages of this approach is the wealth of continuous time design methods. Moreover, the sampling period can be selected after the analogue control system is designed and, thus, the continuous time closed-loop bandwidth is known. The performance of this method is signi cantly a ected by the selected discretisation method and the selected sampling interval. Standard methods such as bilinear 1
transformation often require a high sampling rate to retain performance and closed-loop stability. In certain cases, however, the sampling rate is constrained by either the computational speed of the microprocessor for digital control or by the measurement scheme, and it has to be selected low. For example, the sampling rate may be constrained because of intensive signal processing of row measured data. Feedback control based on vision feedback is such an example [4]. Computer Hard Disk Drives (HDDs) represent another example. In sector servo of HDDs, a circular disk is divided into equally sized angular pieces (called sectors) and position information is written on the disk surface such that the position error signal (PES) can be measured once in each sector [5]. The primary purpose of HDDs is to store data and not to store position information. Thus, it is an interesting question how we may reduce the number of sectors, i.e. the measurement frequency, while maintaining the track following error at an adequate level for ensuring no recording errors. When the sampling time is relatively large, standard digital design methods most likely fail and even may not make sense. In this paper, we investigate the digital design based on multi-rate control, which update the controller output N times faster than the measurement sampling frequency, and high order holds, which utilize past sampled outputs to predict the evolution of the output error between two consecutive sampling instances. The controlled plant and the output feedback controller is rst expressed in the continuous time domain. Then, Zero Order Hold (ZOH ) equivalent of the controller and plant are obtained for representation of the feedback system in the discrete time domain. The use of ZOH for the plant is justi ed based on the standard assumption that the controller output is given to the plant via D/A converter. For multi-rate control, it will be assumed that N digital controllers obtained as redesigned continuous time controllers are successively applied over N time steps, which equally divide one measurement sampling period. In the previous paper [6] , we have shown a multi-rate controller whose input is held constant during one output measurement sampling period. This corresponds to multi-rate control with zero order hold. When the overall system is simulated, however, the control input under this scheme is often oscillatory between output measurement instances. Thus, if one objective of multi-rate control is to smoothen the control input, it is not necessarily the right approach. In this paper, we examine multi-rate control with higher order holds. 2
The remainder of this paper is organized as follows. The problem is stated in Section 2 along with all necessary equations for design and analysis. In Section 3, an illustrative example is given to examine the open and closed loop characteristics of the multi-rate controller with higher order holds. Concluding remarks are given in Section 4.
2 Multi-rate Sampling with High Order Holds Assume that the plant to be controlled in continuous time domain is described by
x_ p (t) = Ap;t xp (t) + Bp;t u(t) y(t) = Cp;t xp (t) + Dp;t u(t)
(1)
where u(t) is the m-dimensional control input vector, y(t) is the p-dimensional output and xp (t) is the
np -dimensional state space vector of the plant. The continuous controller has been designed in order to t speci cations as
x_ c (t) = Ac;t xc (t) + Bc;t e(t) u(t) = Cc;t xc (t) + Dc;te(t)
(2)
where xc (t) is the nc-dimensional state space vector of the controller, e(t) = yd (t) ? y(t), and yd (t) is the desired trajectory. The multi-rate system, where the plant input is updated at T and the plant output is sampled at NT , is shown in Fig. 1. In order to make the multi-rate nature of the overall system explicit, we yd(t)
+
NT
-
y(t)
Multi-rate Controller
Gp(s)
ZOHT NT
Figure 1: Continuous time plant under multi-rate control use signal representations with two time indexes: i.e. x(k; i), u(k; i), y(k; i), and so on, where the rst time index k is for the instances of slow sampling rate and the second time index i (0; 1; 2; :::; N ? 1) is for the 3
instances of fast sampling rate within one slow sampling period. Note that the i-th instance within the k-th slow sampling period is at t = (kN + i)T . The discretized plant is represented by,
xp (k + 1; 0) = ANp xp (k; 0) +
NX ?1
ApN ?j?1 Bp u(k; j )
j=0 y(k; 0) = Cp xp (k; 0) + Dp u(k; 0)
(3)
where
Ap = e A T ; p;t
Bp =
Z T 0
eA Bp;t d p;t
and the transfer function from u(k; i) to y(k; 0) is Gp;i (z N ), which is described by
Gp;i (z N ) = Cp (z N I ? ANp )?1 ANp ?i?1 Bp + Dp i
(4)
i = 1 for i = 0 and i = 0 for i = 1; : : : ; N ? 1 The controller given by Eq. (2) is discretized in the following way. The error signal e(k; 0) = e(kNT ) =
yd(kNT ) ? y(kNT ) is the natural choice as the input to the controller at the sampling instance of the plant output (except for the delayed rst order hold discussed below). While there is no new information from the plant between two consecutive sampling instances, we may specify the input of the controller as we wish. While an obvious choice is to use e(kNT ) for i = 1:::; N ? 1, any other choices will be possible. Thus, we denote the input at (kN + i)T by e(k; i). Then, supposing that the continuous time controller (2) is preceded by a zero order hold with duration T , the discretized controller is represented by
xc (k + 1; 0) = ANc xc (k; 0) +
NX ?1 j=0
u(k; i) = Cc Aic xc (k; 0) + Cc +Dc e(k; i)
ANc ?1?j Bc e(k; j ) i?1 X j=0
Aic?1?j Bce(k; j ) (5)
where
Ac = e A T ; c;t
Bc = 4
Z T 0
eA Bc;t d c;t
By taking Z transformation of Eq. (5), we can get
Ui (z N ) = Cc Aic (z N I ? ANc )?1 +Cc
i?1 X j=0
NX ?1 j=0
ANc ?1?j Bc Ej (z N )
Aic?1?j Bc Ej (z N ) + Dc Ei (z N )
(6)
i = 0; :::; N ? 1 where
Ui (z N ) = Ei (z N ) =
1 X k=0
1 X k=0
u(k; i)z ?kN e(k; i)z ?kN
Noting that
Aic (z N I ? ANc )?1 = (z N I ? ANc )?1 Aic we combine the rst and second term of Eq. (6) and obtain
Ui (z N ) = Cc (z N I ? ANc )?1 f +(z N I ? ANc ) +DcEi (z N )
i?1 X j=0
+z N
j=0
ANc +i?1?j Bc Ej (z N )
Aic?1?j Bc Ej (z N )g
= Cc (z N I ? ANc )?1 f i?1 X
NX ?1
NX ?1 j=i
ANc +i?1?j Bc Ej (z N )
Aic?1?j Bc Ej (z N )g
j=0 +DcEi (z N )
(7)
By changing index, Eq. (7) can be expressed as
Ui (z N ) =
NX ?1 N N ? 1 Cc (z I ? Ac ) f Ajc Bc EN +i?1?j (z N ) j=i iX ?1 +z N Ajc Bc Ei?1?j (z N )g j=0 +DcEi (z N )
i = 0; :::; N ? 1 5
(8)
Note that Eq. (8) is a general expression for any e(k; i), i = 0; :::; N ? 1. It is natural to make e(k; i) a linear function of fe(j; 0); j kg. In this case, we can write
Ei (z N ) = G i (z N )E0 (z N ) i = 0; :::; N ? 1
(9)
where G i (z N ) is the hold which maps e(k; 0) to e(k; i). By plugging Eq. (9) into Eq. (8), the transfer function from e(k; 0) to u(k; i), which is EU0((zz )) , can be expressed as i
N
N
NX ?1
Gc;i (z N ) = Cc (z N I ? ANc )?1 f +z N
i?1 X j=0
j=i
Ajc Bc G N +i?1?j (z N )
Ajc Bc G i?1?j (z N )g
+DcG i (z N )
(10)
i = 0; :::N ? 1 By writing
G i;j (z N ) = G N +i?1?j (z N )
ji
= z N G i?1?j (z N )
jinfinity continuous
magnitude(dB)
100
50
0
−50 −2 10
−1
10
0
10
−100
phase(deg)
−150
−200
−250 −2
10
−1
10 Frequency(rad/sec)
0
10
Figure 2: Open loop frequency response for continuous and ZOH multi-rate system additional phase delay of the plant induced by sampling and the phase margin of the open loop system with the discrete time controller obtained by the prewarped bilinear transformation is about only 10 degree.
In the limit of N ! 1, the multirate controller with FOH approximate the original continuous system remarkably well up to about 1 rad/sec. On the other hand, the multirate controller with ZOH cannot provide the phase lead comparable to that of the continuous time controller in the limit of N ! 1, and the phase margin is only about 14.4 degree.
The multirate controller with N = 2 stabilizes the closed loop system (both the ZOH and FOH case). The di erence between the ZOH case and the FOH case is not appreciable.
When N is increased to 4, the di erence between the ZOH case and the FOH case is clear, and the FOH multirate controller provides a better approximation of the continuous time compensator with 11
N=1 N=2 N=4 N−>infinity continuous
magnitude(dB)
100
50
0
−50 −2 10
−1
0
10
10
−100
phase(deg)
−150
−200
−250 −2
−1
10
0
10 Frequency(rad/sec)
10
Figure 3: Open loop frequency response for continuous and FOH multi-rate system a larger phase margin than the ZOH multirate controller. 0.5
0.4
0.3
0.2
control input
0.1
0
−0.1
−0.2
−0.3
foh zoh
−0.4
−0.5
0
5
10
15 20 time(*0.75sec)
25
30
35
Figure 4: Open loop control input for ZOH and FOH multi-rate system N = 2 with sinusoidal input Figure 4 shows the output of the multirate controller (N = 2) with ZOH or FOH for a sinusoidal input with a frequency of 0.3 rad/sec. Notice that the output of the FOH multirate controller has a signi cantly reduced level of high frequency oscillation than the ZOH multrate controller. 12
3.2
Closed Loop Behavior
10
8
6
output
4
2
0
−2
−4
single rate at low freq. FOH multi−rate (N=2) single−rate at fast freq.
−6
−8
0
5
10
15
20 25 time(sec)
30
35
40
45
Figure 5: Closed loop response of single and multi-rate controller(step input) 15
10
control input
5
0
−5
single rate at low freq. FOH multi−rate (N=2) single−rate at fast freq. −10
0
5
10
15
20 25 time(sec)
30
35
40
45
Figure 6: Control input of closed loop single and multi-rate controller(step input) Figure 5 shows the closed loop response to a step desired output signal. Three cases are shown in the gure: the single-rate discrete time feedback system (sampling period=1.5 sec), the single-rate discrete time feedback system (sampling period=0.75 sec) and the FOH multi-rate feedback system (T = 0:75 sec, N = 2). Fig. 6 shows the corresponding control input to the plant. Notice that despite of the large measurement sampling period of 1.5 sec, the response under FOH multi-rate control closely approximate that of single-rate control with the shorter period of 0.75 sec.
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4 Conclusion In this paper, we studied the digital redesign of continuous time controller when the measurement sampling period is relatively large. In order to recover the performance of the continuous time controller lost by a large sampling period, we updated the controller output at a frequency N times faster than the measurement frequency, and at the same time estimated the evolution of the error signal between two consecutive sampling instants by utilizing present and past error measurements. A general equation for analyzing such multirate control systems has been presented. An illustrative example showed that multirate control and high order holds are both useful for recovering the performance of the original continuous time compensator.
References [1] N. Hori, T. Mori, and P. N. Nikiforuk, \A new perspective for discrete-time models of a continuous-time system," IEEE Transactions on automatic control, vol. 37, no. 7, pp. 1013{1017, July 1992. [2] A. H. D. Markazi and N. Hori, \Discretisation of continuous-time control systems with guaranteed stability," IEE Proc. on Control Theory and Applications, vol. 142, no. 4, pp. 323{328, 1995. [3] R. A. Comeau and N. Hori, \State space forms for higher order discrete-time models," Systems and Control Letters, vol. 34, pp. 24{31, 1998.
[4] M. Nemani, T.-C. Tsao, and S. Hutchinson, \Multirate analysis and design of visual feedback servo cont rol systems," ASME Journal of Dynamic Systems, Measurement, and Control, vol. 116, no. 1, pp. 45{55, 1994. [5] W.-W. Chiang, \Multi-rate state-space digital controller for sector servo systems," Proceedings of the 29th IEEE Conference on Decision and Control, pp. 1902{1907, Honolulu, HI, December 1990.
[6] J. Tornero, Y. Gu, and M. Tomizuka, \Analysis of multi-rate discrete equivalent of continuous controller," The 1999 American Control Conference, San Diego, CA, July 1999.
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