Adaptive Zero Phase Error Tracking Controller with Precision Tracking Performance

Copyright © IFAC System Identification, Kitakyushu, Fukuoka, Japan, 1997

ADAYnVE ZERO PHASE ERROR TRACKING CONTROLLER WITH PRECISION TRACKING PERFORMANCE

Manabu Yamada, Zaier Riadh and Naoki Mizuno

Department ofMechanical Engineering, Nagoya Institute of Technology, Showa, Nagoya 466, JAPAN e-mail: [email protected], [email protected], [email protected]

Abstract: This paper deals with a discrete-time tracking control where the desired output to be tracked is partially known a priori. The problem of designing prefilter which provides the overall system with the following frequency characteristics is considered: 1) The phase is zero for all frequencies, 2) The gain at given particular frequencies is set to unity, 3) The maximum error over a given frequency range between the gain and unity is less than an arbitrary given positive number. The prefilter satisfying this problem is given in an explicit form, and is implemented in adaptive control scheme. Keywords: discrete-time systems, non-minimum phase systems, frequency responses, adaptive control, dead-beat control, tracking characteristics.

1. INTRODUCTION

(1993) proposed the optimal ZPETC minimizing the integral of the squared error between the gain and unity over a given frequency range. However, the design procedure requires to solve an optimization problem with a troublesome inequality constraint. Torfs, et al. (1992) proposed a ZPETC based on an expansion of the inverse system in power series. However, if the power series diverges, the gain becomes worse than that of Tomizuka (1987). To overcome this problem Funahashi, et af. (1995) proposed a ZPETC so that the convergence of the power series is always guaranteed. Moreover Yamada et a/. (1997) presented a simple design method of obtaining the ZPETC such that the maximum error between the gain and unity is less than an arbitrary given positive number.

In this paper, a tracking control problem for a discrete time non-minimum phase system is considered. When the desired trajectory is entirely known in advance, the perfect tracking can be achieved (Jayasuriya and Tornizuka, 1993). This paper deals with the tracking control problem in which the desired trajectory is not entirely known in advance but a finite steps of future desired output is assumed known. This problem is often fonnulated as a design problem of preview feedforward controllers in the framework of frequency domain. Along this line, the most attractive feedforward controller is the Zero Phase Error Tracking Controller, abbr., ZPETC, which has been first proposed by Tomizuka (1987). The ZPETC can provide the overall system from the desired output to the controlled one with frequency characteristics such that the phase is zero for all frequencies and the gain is unity at only zero frequency. However, there has been no discussion nor description of the gain characteristics except at zero frequency. Therefore, the resulting control system may has undesirable gain characteristics. In order to improve the gain characteristics, many new types of ZPETC's have been proposed until now. Funahashi and Yamada

On the other hand, it has been widely accepted that the parameter uncertainty or changes in the plant may cause significant deterioration of the tracking performance. The ZPETC's proposed by Tornizuka (1987), Funahashi and Yamada (1993) and Funahashi et al. (1995) were made adaptive by Tsao and Tornizuka (1987), Yamada et al. (1995) and Yamada et af. (1996), respectively.

837

coprime and A(O) = 1 and B(O) 7;. o. Decompose

The aim of this paper is to propose an adaptive ZPETC to simultaneously ensure deadbeat tracking performance for a priori known desired output and achieve a specified tracking one for unknown desired one. As shown by Tsao (1994), this problem is effective when the class of the desired output to be tracked is partially known a priori or when the desired output is involved while some fundamental frequencies are given a priori. This problem is formulated as the problem of designing adaptive prefilter which provides the resulting overall system with the following attractive frequency characteristics: 1) The phase is zero for all frequencies, 2) The gain is set to unity at given particular frequencies, 3) The maximum gap over a given frequency range between the gain and unity is less than an arbitrary given positive value. The point to be stressed is that this problem generalizes the previous ones (Funahashi, et aI., 1995; Yamada et aI. , 1996, 1997) in terms of the class of the desired output to ensure deadbeat tracking performance. The contribution of this paper is to present the prefilter satisfying this problem in an explicit form. Moreover the proposed design method is straightforward and is implemented in adaptive control scheme. By computer simulations it is confirmed that more excellent both tracking performance and frequency characteristics than the previous ones can be obtained.

(2) where B+(z-l)eR[z-l] contains all asymptotically stable zeros and B - (Z-I) eR[z-l] contains the other zeros, e.g., unstable zeros.

In this paper, a design of a prefilter satisfying the following problem is considered. Main Problem: Find a stable feedforward controller Cr(z) that satisfies the following specifications:

(S 1) The phase shift of the frequency characteristics of the overall transfer function G(z) is equal to zero for all frequencies, i. e., G( e jllJT ) ? 0 for V (i) ER, where T denotes the sampling period. (S2) The gain of the frequency characteristics of the overall transfer function G(z) is equal to unity at zero frequency and other given particular frequencies, (i);, i = l···m , i.e., (4) (S3) For a given frequency range Q E [0, wo], the maximum gap over a given frequency range between the gain and unity is less than an arbitrary given positive number ho > 0 , i,e.,

The following notations are used throughout this paper. N and R denote the set of natural numbers and real numbers, respectively. R[Z·I] and R(z) denote the set of real polynomials in Z-l and real rational functions in z , respectively. Rp(z) is the set

(5) Remark: The specifications (SI) and (S2) ensure dead-beat tracking performance for any sinusoidal desired outputs with given frequencies w,' s as

of proper elements in R(z)

2. PROBLEM FORMULATION

explained later. The specifications (SI) and (S3) mean that the greatest limiting value of the controlled error for a sinusoidal desired output with unit amplitude, consisting of frequency in 0 is less than a given number ho.

Figure 1 depicts the overall structure of the singleinput single-output discrete time tracking control system studied in this paper. yAk), y(k) and

G/z- 1 ) eRp(z) are the desired output, the controlled

The ideas and procedures to solve the above problem are as follows. First, the existence condition and design method of prefilters C1(z·1) satisfying the specification (S2) are shown by using a Diophantine equation. Secondly, a stable prefilter C2(z) satisfYing the specification (SI) subject to specification (S2) is shown. Finally, a stable C3(z) satisfYing the specification (S3) subject to both specifications (SI) and (S2) is added by an expansion of the inverse of the system, see (Funahashi, et aI. , 1995; Yamada, et aI., 1997).

output and a given discrete time plant, respectively. G(z) denotes the overall transfer function from yAk) toy(k) . The feedback controller C,,(Z-I) eR/z)

and

(3)

the

feedforward

one

C,(z) ER(z) are to be designed for regulation and

tracking, respectively. It is assumed that the feedback controller Cy (Z-I) has already been designed so that the closed loop transfer function Gclo.ed(Z-I)

is

asymptotically stable, and it is given by -LB( -I)

Gclared(Z-I) z

~

A(z- )

eRp(z),

3. ZPETC WITH DEABEAT TRACKING PERFORMANCE

(1)

First, all asymptotically stable zeros and poles are

838

Theorem 1: For any x E:3 and NE N, the frequency transfer function G(z) of eq.(l8) satisfies both specifications (SI) and (S2).

e(k) = y .(k)-

j6T (k) = [- y(k+Lp -I) . .. - y(k+Lp -N p )' u(k) ... u(k-Mp)j

(k)-

Proof The proof is straightforward, and it is omitted. Q.E.D.

y.

-

(k) _

r/J(k)

A(k) = tr F(k) . tr F(O)

Secondly, as shown in Fig.2, the control input to the discrete time plant is determined by

Remark 2: The proposed controller of eq.(13)

L + (degB-(z) + degCl (z») (N + 1)

y(k)

max(lI;(k)I~I)

- max(llr/J(q~l)

Remark 1: The merit of the FIR filter of eq.(17) is that there always exists a finite number N satisfying eq.(5) for an arbitrary given ho > o. The algorithm for obtaining the minimum odd number N and x E:3 is given by Yamada et af. (1997).

requires

er (k -1)(k - Lp)

preview

steps of the desired output. where Q(Z-I) and R(z-I) are stabilizing controllers

Therefore the proposed design algorithm for obtaining the proposed controller satisfYing the specifications (S1), (S2) and (S3) can be summarized as follows:

and Cr(z,k) is given by eq.(13) replacing B-(Z-I) by Bp(z-\k) .

Step 1: Calculate Co(z) from eq.(6). Step 2: Find the minimum degree solution C1(z) by solving the Diophantine equation of eq.(8). Step 3: Find the FIR filter C2 (z) from eq.(lI). Step 4: Find the FIR filter C3(z) by following the same algorithm given by Yamada et af. (1997). As a result, from eq.(13), the proposed controller is obtained.

6. NUMERICAL EXAMPLE To illustrate the effectiveness of the main result of this paper consider the following nominal plant of a two axis positioning table (Sugiyama, 1984).

G (Z-I)= z-2(0.01869+0.06260z- l ) p

5. ADAPTIVE CONTROL SCHEME

closed loop transfer function is obtained as follows: G

(_'

)

_

z- Bp (z -, ,k) A

(

N

whereAAz-i,k) =1 +

) '

COl' CO2 and COo are set to 0,18.2 and 157 [radlsec], respectively. Figure 5 shows the gain plots of the overall frequency transfer functions, G( ej.,r ), with the proposed ZPETC's C,(z) of N = 0, 1 and 3. This figure demonstrates that the maximum gap between the gain and unity over n = [0, coo] becomes strictly

M

Ia (k)z-i and BAz-I,~=ibik)z-i.

i=1

p

i=O

Define

smaller as N increases. Figure 6 shows the gain plots of the overall frequency transfer functions with the previous ZPETC's presented by Tomizuka (1987), Yamada et af. (1995) for /=2 and Yamada et af. (1996) for N=5. Comparing with Fig.5, it is shown that the proposed ZPETC presents more excellent frequency characteristics than the previous ones. Note that the ZPETC presented by Yamada et al. (1996) for N=5 requires the same preview steps of the desired output as the one proposed in this paper for N= 1. Concerning the controller proposed by Yamada et af. (1995), due to the troublesome inequality constraint, the controller for / > 2 can not be implemented in adaptive control scheme. Figure 7 shows the

The measured signals are u(k) andy(k). The parameter vector 8( k) is estimated from the measured signals by the following normalized least squares adaptation algorithm, see (Tsao and Tomizuka, 1987). e(k);:: e(k-l)

+

z-2(Om869+0.062roz-') 1- 207z-1 + L25z-2 -O.065z-3 -O.l2Iz" +O.OI97z- s

Figure 3 shows the desired output to be tracked and Fig. 4 shows its frequency spectrum, which has two big peaks at co = 0 and 18.2 [rad / sec] . Hence

(19)

L,

Ap z ~ , k

z _


estimated as follows. A

'

where the sampling period is T = 0.01 sec. By using feedback controllers Q( z -I) and R( z -I), the stable

In this section, an adaptive control system based on the proposed ZPETC is constructed. Figure 2 shows the overall structure of the proposed adaptive control system. Let the discrete time plant 6 p CZ -1 , k) to be

G p z ,k -

1-1.7994z-1 +O.7994z- 2

F(k-l)(k-L p)e(k) 1+ cI>T (k - Lp )F(k -1)(k - Lp)

1 { FCk-l)
where F(O) = F(O/ > 0 and

840