On the Design of Digital Tracking Controllers

Transcription

1 Masayoshi Tomizuka Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA On the Design of Digital Tracking Controllers This paper describes my work on the design of digital tracking controllers over the past two decades. In tracking control, the control object must be moved along a time varying desired output and the transient path error must be minimized. Tracking control is discussed for the following two typical situations: one is the situation where the desired output is known in advance and the other is where the desired output itself is not known but has a certain known property, e.g., it is periodic with a known period. Control algorithms for dealing with these situations will be reviewed and applications to mechanical systems are discussed. 1 Introduction In servo control, two fundamental problems are the pointto-point control problem and the tracking (path following) control problem. The point-to-point control problem is concerned with moving the control object from one point to another. In this problem, the controller is required to provide a small final positioning error and superior regulation. A regulation function is required to keep the object at a desired position in the presence of disturbances. In tracking control, the control objet must be moved along the desired trajectory. Tracking control is extremely important in many mechanical systems: e.g., automated arc welding using a robot arm and high speed machining of complex shaped workpieces. The design of controllers for mechanical systems must consider (1) stringent performance specifications in terms of tracking error and speed, (2) nonlinear characteristics such as Coulomb friction, stiction and actuator saturation, (3) time varying characteristics, and (4) the use of microprocessors in implementation. The control algorithms presented in this paper directly address the first and fourth points. The second and third points become important in applying the algorithms to actual systems. This paper describes my work on the design of digital tracking controllers over the past twenty years. In view of the fourth point mentioned above, I will describe discrete time control algorithms only. My involvement in tracking control goes back to my Ph.D. dissertation, "On Finite Preview Problems and Its Application to Man Machine Systems" (1973). When the desired output is known in advance, i.e., "previewable," the controller does not have to be constrained to work only on the error signal. Prior information can be utilized in a variety of ways to recover the inherent limitation in tracking, i.e., dynamic lag of the control object.one of my doctoral dissertation committee members, T. B. Sheridan, had three models of preview control (1966), but a good model from a viewpoint of optimal control was missing. In the early 1970s, the optimal tracking problem in the linear quadratic optimal control context was well understood (e.g., Athans and Falb, 1966; An- Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Revised manuscript received by the Dynamic Systems and Control Division January 6, derson and Moore, 1971). However, its solution required the knowledge of future information related to the desired output signal over the entire problem duration. In the finite preview problem (Tomizuka, 1973; Tomizuka and Whitney, 1975; Tomizuka et al., 1984), the basic assumptions were (1) future information is available only in a local sense, i.e., over a time interval, [k<i<k + N p ] (k + N p <N), where k is the present time, N p is the preview length and N is the problem duration, and (2) the optimality must be judged based on a performance index defined over the problem duration. More recently, I introduced the idea of zero phase error tracking control (ZPETC) as a generalization of feedforward control based on the idea of inverse systems (1987). The ZPETC can be applied to systems whose inverses are unstable. Other researchers have utilized future values of a desired output for attaining an improved tracking performance, e.g., independent tracking and regulation approach (Landau and Lozano, 1981) and an adaptive LQ controller (Samson, 1982). In some applications, such as computer disk file systems, the tracking error, that is, the difference between the desired output and actual output, may be the only signal available for the controller (Chew and Tomizuka, 1989) and the desired output itself is not known in advance. However, it may be known that the desired output possesses a certain known property, e.g., it is periodic with a known period. Such properties can be utilized to design controllers which may adsorb the tracking error in an asymptotic sense. The repetitive controller (Tomizuka, Tsao and Chew, 1989) is such a controller. As far as I known, repetitive control was originated by Japanese researchers (Inoue et al., 1981; Omata et al., 1985). My graduate students and I have devoted a significant amount of effort to better understanding and establishing methodologies for repetitive control over the past decade because periodic desired outputs and disturbances are quite common in mechanical systems. In the following section, the design of tracking controllers for known (i.e., previewable) desired outputs will be discussed. The design of controllers which are solely based on the tracking error will be discussed in Section 3. Conclusions and further remarks including future research directions will be given in Section /Vol. 115, JUNE 1993 Transactions of the ASME Copyright 1993 by ASME

2 y d m Feedforward Controller r(k) + i 4, Feedback Controller GdoiatfU" 1 Controlled Plant Fig. 1 Two-degree-of-freedom control system y(k) I ; as a natural extension of the linear quadratic (LQ) optimal control and can be found in a number of standard text books. The continuous time optimal tracking problem can be found in Athans and Falb (1966) and the discrete time version can be found in Anderson and Moore (1971). Since the optimal tracking problem is the basis of other optimal tracking controllers, it is summarized below. 2 Tracking Control for Known Desired Output When the desired output is known in advance, its future values should be utilized to enhance the tracking performance. The overall control system normally takes a two-degree-offreedom structure in Fig. 1. The feedback controller must be designed to provide an adequate level of regulation as well as robustness. The feedforward controller is used as a prefilter for the desired output signal to compensate for the dynamic lag of the closed-loop systems and to achieve a high tracking performance. The most intuitive prefilter for the purpose is the mathematical inverse of the closed-loop system for perfect tracking control (PTC), explained below. Let the discrete time transfer function for the closed-loop system be expressed as J closed <*-'): z~ d B c {z-') where z~ l denotes a one sampling delay, z~ d represents a d- step delay normally caused by the delay in the plant and computation, B c (z~ l )=b c0 + b cl z~ i +- +b cm z~ m, b c0 7i0 A c (z~ l ) = l-a ci z~ 1 a cn z~" Our goal is to let the output y( k) follow the desired output )>d{k). To achieve this objective, the reference input for the closed-loop, r(k), is determined by processingy d (k) by z d A c (z- 1 ) Gff(Z ) = [G d osed(2 )]" ' B e (z~ l (2) ) so that the overall transfer function from y d (k) to y{k) is unity. Equation (2) implies that the closed-loop poles and zeros should be canceled by the zeros and poles of the feedforward controller. Notice that this feedforward controller is in unrealizable form, and it must be implemented as r(k)=~^ry d (k + d) (3) B c (z ) Equation (3) clearly indicates the necessity of previewing the desired output: i.e., the desired output must be known in advance. This requirement is not unrealistic in many servo problems such as control of machining centers. Gjy(z~') in Eq. (2) is unstable and is unusable if any closedloop zero appears on or outside the unit circle in the z-plane; in digital control, it is well known that the unstable discrete time zero(s) appear when fast sampling is applied to a continuous time plant with a relative degree greater than or equal to two (Astrom and Wittenmark, 1984). For example, a pure double integrator plant G p (s) = 1 Is 2 generates a zero at 1. The ZPETC described later in this section was developed to deal with unstable or uncancellable zeros of the closed-loop system. The prefilter can be designed based on the linear optimal control theory instead of pole/zero cancellation. While the PTC and ZPETC approaches attempt perfect or near perfect tracking of the desired output, the optimal preview approach minimizes combinations of the tracking error and control effort. Chronologically, the tracking controller design based on optimal control precedes the development of the design based on the mathematical inverse, though the latter appears to be more intuitive and straightforward. 2.1 Tracking Controller Design Based on Optimal Control. The optimal tracking control problem was formulated (1) Optimal Tracking Problem. In the discrete time optimal tracking problem, the controlled plant is described by x(k+l) = Ax{k)+Bu(k) (4) y(*) = Cx(*) (5) where x(k),u{k) andy(fc) are the w-dimensional state vector, w-dimensional input vector and /--dimensional output vector, respectively, and A, B, and C have appropriate dimensions. Given a desired output sequence \y d {k) \0<k<N], the optimal control u opt (k) must minimize the quadratic performance index N-\ J=e T (N)Sz(N)+Yj Se r (0Qe(()+u 7 '(()Ru(/)! (6) ;=o where e(k)=y d (k)-y(k), N is the specified problem duration, S and Q are positive semidefinite, and R is positive definite. The solution to this optimal control problem is given by u P t (k)=-[b T H(k+l)B + R]- l B T [H(k+l)Ax(k) where H(fc) is the solution of the Riccatti equation, H(k)=A T {H(k+ 1)-H(A: + \)B[B T H(k + 1)B and g(k) is given by + g(* + J)l (7) + Rr'B 7 H(A:+l))A + C 7 QC (8) H(N)=S (9) g(a:)=(a-b[b r H(fc+l)B + Rr 1 B 7 H( +l)a} 7 g(fc+l) -C T Qy d (k) (10) g(n)=-c 7 'Sv d (JV) (11) Equations (8) and (10) can be solved by off-line backward recursion. Notice that the performance index (6) trades off the tracking error against the input magnitude. It is straightforward to incorporate other quantities such as incremental changes in other state variables and/or control inputs in the performance index Optimal Preview Problem. Equation (7) implies that u pt (k) depends on {y d (i) \k<i<n], which is not practical especially when N is large. It is often the case that the desired output is known in advance only in a local sense: i.e., [y d (i)\k<i<k + N p, k + N p <N] is known, where N p is the preview (or look ahead) steps. For the situation where the knowledge on the desired output is local, Hayase and Ichikawa (1969) suggested formulating the optimal tracking problem over the local time interval and applying its solution in a successive manner. However, their solution was only suboptimal when viewed over the entire problem duration, N. The major motivation for the finite preview control theory (Tomizuka, 1973) was to find the tracking control law, which depends only on local future values of the desired output, and yet is optimal in a certain sense over the entire problem duration. The optimal preview control approaches are reviewed below. The performance index (6) depends on \y d (i) I0</<7V). Therefore, if u opt ( ) is to be based on the minimization of (6) when only {y d (i)\k<i<k + N p } {N P <N) is previewable at time k, we need to supply somehow the missing information, {y d (i) \N P + l</<a r ). The missing information can be supplied either from a probabilistic point of view (Tomizuka, 1973) Journal of Dynamic Systems, Measurement, and Control JUNE 1993, Vol. 115/413

3 y d (k) A Deterministic Future (Previewable) Fig. 2 Probabilistic Future k (now) k + N p Time Knowledge of future desired output or a deterministic point of view (Tomizuka and Rosenthal, 1979; Tomizuka et al., 1980). To formulate the preview problem as a stochastic control problem, the desired output y d (i) is modeled as the output of a command signal generator (linear shaping filter) x d (k+l) = A d x d (k)+b d w d (k) (12) y d (k)=c d x d {k) (13) where a ^-dimensional vector x d (k) is the state vector of the shaping filter, an s-dimensional vector w d {k) is zero-mean, white and Gaussian, and A d, B d and C d have appropriate dimensions. This shaping filter is running N p time steps ahead of the controlled plant allowing the controller to have an N p - step preview of the desired output. Namely, at time k, the entire future of the desired output consists of two parts (Fig. 2): one is the deterministic future, the other the probabilistic future. The optimal preview control can be determined by minimizing the following performance index. N-l J=E e T (N)Se(N) + J] e r (()Qe(()+n T (i)ru(i)i (14) where the expectation is taken over the random quantity, w d (k). The problem is now well formulated. A more general formulation and its solution, which include the effects of measurement delay and noise, are in Tomizuka and Whitney (1975). The stationary solution is practically important and it minimizes, /' = lim J/N N-a> where /is given by Eq. (14). It is written as u opt (k)=-g x x(k) + 2 G y Al)y d (k + l)+g xd x d (k + N p ) /=i (15) (16) The first term in Eq. (16) is the feedback control term and the feedback gain is given by G* = [B r H s B + R] 'B r H s A (17) where H s is the steady-state solution of the Riccati Eq. (8). The second and third terms in Eq. (16) are the preview control terms and provide feedforward control. G yd (l)'s satisfy the relation (Tomizuka and Whitney, 1975), G, d (/) = [B r H J B + R]- 1 B r H, d (/+l) (18) H yd V+l)=[A-BG x ] T H yd (l) (19) where A c = A - B G x is the closed-loop system matrix for the controlled plant under the feedback control. Since the closedloop system is asymptotically stable under normally satisfied conditions, Eq. (19) implies that the future desired output y d (k+l) has diminishing importance as / is increased. The third term in Eq. (16) may require the use of an estimator since x rf is not necessarily accessible. However, the magnitude of G xd 414/Vol. 115, JUNE 1993 is small for a large N p. Note that the output of the preview controller is a moving average of a future desired output sequence and that the number of preview steps required to achieve almost all of the possible benefits of preview depends on the eigenvalues of A c or Q and R matrices in the performance index (14). The properties of the preview control terms as explained above imply that any reasonable assumption for [yd{i)\i>k + N p ] is acceptable when N p is sufficiently large, and has motivated' deterministic preview control approaches (Tomizuka and Rosenthal, 1979; Tomizuka et al., 1980). In particular, it has been found to be practical to assume that the desired output does not change beyond the preview time interval: i.e., y d (j+ l) = y,/(/) for />k + N p where k is the present time. 2.2 Tracking Controller Design Based on Pole/Zero/Phase Cancellation. In the tracking controller design based on optimal control, the tracking error was traded off against the input magnitude. However, in mechanical systems control, the error is of major concern especially when the desired output is precisely known in advance. In such cases, the idea of inverse systems discussed at the beginning of this section is more appealing than the optimal preview controller. As pointed out already, such controllers must deal with unstable or uncancellable zeros of the controlled plant. In order to deal with such zeros, we factorize B c (z~ l ) into two parts and write B c (z- l )=B;(z- l )B c + (z- 1 ) (20) where B? (z~ [ ) contains uncancellable zeros, which include unstable zeros, and B? (z _1 ) contains cancelable zeros. The ZPET feedforward controller (Tomizuka, 1987) is dz->) z d A c (z- { )B; (z) 'fl+(z-')fl-(l) 2 (21) where B? (z) is obtained by replacing every z~' in 5<T (z - ') by z. G ZPET (z"') is unrealizable, and this feedforward control law is implemented in the following form:... A c (z- l )Bc*jz- 1 ),..,, rw= B^Z-^B-W 1 y {k+d+s) (22) «where B c ~*(z~ 1 ) = z~ s Bc (z) and 5 is the order of B^, that is, the number of uncancellable zeros. From Eqs. (1) and (22), for zero initial conditions, the plant output is y(k) = [B- (z~')b7 {z)/b- (l) 2 ]y d (k) (23) The following relation can be easily verified by direct substitution of z = e J " T, 0 < u < T/ T'where Tis the sampling period. Im[B-(e JuT )B^(e JuT )/B-(l) 2 }=0, 0<o><ir/T (24) Equation (24) implies that, when y(k) is sinusoidal, there is no phase shift between^(k) and y(k). The ZPETC cancels the poles and cancellable zeros of the closed-loop system and compensates for phase shifts induced by uncancellable zeros. Because of this property, G ZPET (z~') is called the zero phase error tracking controller (ZPETC). The ZPETC requires (d + s)-step preview of the desired output while the number of preview steps to attain almost all of the benefit of preview in the optimal preview control depended on Q and R matrices in'eq. (14). Notice that B^{\) 2 is used as a scaling factor in the ZPET in Eq. (22). The reason for this scaling factor is that in many mechanical systems, uncancellable zeros appear in the left half side of z-plane and the the frequency responses of building blocks of zero phase error transfer functions, (z-1) (z~ l - f)> satisfy the properties summarized in the following theorem. Theorem 1. For any zero in the left-half side of z-plane, Transactions of the ASME

4 I(e- JoiT r_)(e^'-f_)ki(l-f-) z l for 0so)<7r/r (25) In particular, if f _ is real, I (e-"" 7 " f _) (e~ iwl - f _) I is wonotonically decreasing. For any zero in the right-half side of z-plane, f +, l(e^r-f + )(e^r~r + )l<l(-l-r + ) 2 r for 0<o J <7r/7 (26) /n particular, if f + is rea/, o-/" 7- - f + )(e- y " r -r + )l jsmon- otonically increasing. This theorem assures that the static gain from j> rf to j> under ZPETC remains unity. As mentioned above, in mechanical systems, uncancellable zeros appear often in the left-half side of z-plane. This will make the overall transfer function from yd to y possess low pass filter characteristics. If the bandwidth of this equivalent filter is narrow relative to the frequency contents of y d (k), it is possible to take advantage of a real zero in the right-half side of z-plane (latter half of Theorem 1) to widen the bandwidth. In this case, the ZPETC in Eq. (21) is modified to ru" 1 )- z d A c (z~ l )B; (z) (z~ b)(z-b) B+( Z - l )B;(l) 2 (l-b) 2 0<b<l 1 (27) The additional component, (z~ l -b) (z-b)/(\ -b) 2, was called the error canceling filter (E-filter) by Haach and Tomizuka (1991), where a guideline for selection of b is provided. When the ZPETC is used, the tracking error, e{k) =y<i(k)-y(k), is e(k)-- l- B;( Z - 1 )B;(Z) B7(iy yak) = G c (z-')y d (k) (28) Equation (28) allows the designer to predict a tracking error given a desired output sequence. For G e (z ')ineq. (28) (Tomizuka, 1992), dz G e(z -1 )lz=i = 1 Be or This implies that G e (z G e (l) = -l dbc (z~ dz~ + B, (*-') B; (Z) (zf dz ') can be expressed as G e (z- l ) = (z-l) 2 [e s^2z? e l z = i (29) = 0 (30) + e 0 + e_!z l + eaz 2 +< + e,_ 2 z~1 (31) Noting that the z-transforms of step and ramp signals are, respectively, Y d (z) =cz/(z- 1) and Y d (z) =c x z/(z- l) 2, Eq. (31) implies that if the desired output is a step or ramp signal, the tracking error under ZPET control converges to zero in a finite number of time steps. An extension of the ZPETC has been reported by Funahashi and Yamada (1992). The idea of ZPETC can be generalized to multi-input multi-output systems (Tsao, 1988). As expected, the tracking controller based on pole/zero/phase cancellations is sensitive to modeling uncertainties. The adaptive ZPETC has been considered by Tsao and Tomizuka (1987). 2.3 Applications to Mechanical Systems. Applications of optimal preview control to mechanical systems include weld seam tracking problems (Tomizuka et al., 1980; Tomizuka et al., 1984) and robotic systems (Tomizuka and Janczak, 1985; Pak and Tuner, 1986). The continuous time version of the optimal preview controller has been successfully applied to vehicle lateral guidance in highway automation (Peng and Tomizuka, 1991; Peng et al., 1992). In these problems, the optimal preview controller has been shown to make the controlling input smooth while maintaining the tracking error in an acceptable range. It should be noted that in these applications the basic theory had to be modified to handle physical nonlinearities and specific control objectives. Typical modifications included the use of gain scheduling (Tomizuka and Janczak, 1985; Peng et al., 1992) and the use of a frequency shaped performance index (Peng and Tomizuka, 1991). The effectiveness of ZPETC for mechanical systems has been demonstrated on mechanical systems such as a vertical machining center (Suzuki and Tomizuka, 1991; Tung and Tomizuka, 1992), direct drive robot manipulators (Horowitz et al., 1987) and a high speed positioning table (Endo et al., 1992). In these applications, a superior tracking performance was obtained by first compensating nonlinear characteristics and various uncertainties of the mechanical systems and then applying the ZPETC. For the machining center problem, nonlinear friction forces had to be percompensated so that the dynamics of the closed-loop system could be approximated by a linear transfer function. Furthermore, Tung and Tomizuka (1992) have shown that a ZPET controller can be effectively designed based on a low-order closed-loop transfer function model, which captures the low frequency dynamics. In an extreme case that a first-order model is sufficient for covering the frequency range of interest, the feedforward controller design is simplified because the location of zeros is no longer an issue. ZPETC acts as an excellent feedforward controller for direct drive robot arms, the nonlinear dynamics of which are compensated by adaptive control (Horowitz et al., 1987). 3 Tracking Control for a Class of Unmeasurable Desired Output Tracking controllers in the previous section require that the desired output is known in advance and is given to the controller independent from the actual plant output. When this is not the case, the controlling input must be based on the tracking error. Let the controlled plant be described by A(z^)y{k)=z~ d B( z - l )u(k)+d(k) (32) where d(k) is the disturbance, B(z- l )=b 0 + b i z ' + b m z-'", b 0 *0 -a z~" the sys In terms of the tracking error, e(k) =y(k) -y d (k), tem is represented as where A(z- l ) = l-a iz ~ l A(z' l )e(k)=z~ d B(z' l )u(k) + w(k) (33) w(k)=d{k)-a{z- l )y d (k) (34) In this section, we consider a class of w(k), i.e., a class of desired outputs and disturbances, characterized by W(z~ 1 )- BAz~ x ) A d (z- 1 ) (35) The characteristic roots of A d {z~ ) are normally on the unit circle. For example, Eq. (35) implies t\\sx y d (k) is a step signal for A d {z~ i )=l-z~\ a ramp signal for A d (z~ x ) = {\ -Z~ l f, or a periodic signal for^4 rf (z~') = 1 -z~ N where Ndenotes the period. Without the loss of generality, it is assumed that Eq. (32) is asymptotically stable. This assumption implies that an appropriate stabilizing controller has been applied when the plant is unstable. It is also assumed that B(z~') and A d (z~ x ) are coprime. Under these assumptions, a variety of approaches are possible for the design of controllers which assure lime(a:)=0 (36) Journal of Dynamic Systems, Measurement, and Control JUNE 1993, Vol. 115/415

5 >Q->- ^fm e[h) e(k) ^Out u(k) S<! P = xb ck~' Fig. 3 Feedback control system with internal model One approach is to utilize the internal model principle (Francis and Wonham, 1975) and tc construct the feedback control system as shown in Fig. 3. A d (z~ ] ) in the controller is the internal model. Notice that for the closed-loop system in Fig. 3, e(k) is S(z^)A e{k) = d (z~') S( z - l )A d (z~ l )A(z- l )+z~ d R{z- l )B{ Z - x ) w(k) (37) where the initial condition has been ignored. Recall that the characteristic roots of A d (z~ [ ) are normally on the unit circle. Therefore, in view of the closed-loop transfer function in Eq. (37), the closed-loop system cannot be made asymptotically stable unless A d (z~ [ ) and B(z~ l ) are coprime; notice that common characteristic roots of A d {z~ l ) and B (z~') become closed-loop poles. Under the coprimeness assumption, the closed-loop system can be stabilized; for example, an arbitrary pole placement is possible based on the general property of the Diophantine equation (Astrom and Wittenmark, 1984), S{z~ l )A d (z- l )A(z- 1 )+z~ d R(z- l )B(z' 1 )=D(z- i ) (38) Furthermore, noting that the cancellation of A d (z~ 1 ) takes place in Eq. (37) for w(k) characterized by Eq. (35), the asymptotic regulation or tracking (36) is achieved. These results have been further refined to the discrete time repetitive controller in Tomizuka et al. (1989). 3.1 Discrete Time Repetitive Controller. The repetitive controller is for periodic desired outputs and disturbances with a known period: namely, A d (z~^) r = 1 -z~ N where the period, N, is known. Notice that the internal model for this case is a periodic signal generator. In the development of the repetitive controller, the following points required special attentions: 1. Amount of Real Time Computation. In repetitive control, N can be large. While A d (z~ [ )= l-z~ N in the denominator of the controller can be efficiently implemented, the amount of real time computation becomes too excessive if the order of R{z~ l ) is high, which is the case if the closed-loop poles are arbitrarily assigned and the Diaphantine Eq. (38) is solved for R(z~ l ) and S(z _1 ). In Tomizuka et al. (1989), it has been suggested to set the controller in the following form. Gr/iz- 1 )- R(z~ l ) ~{\- Z - N )S(z^) k rz - N+d+s A(z~ l )B-*( Z - x )_ (1-Z- /V )J5 + (Z-')(3 2 ' /3 = max -(e- jw5r ) (39) where B + U~'), B (z~') and B * (z~') are similarly defined as for the ZPETC. The amount of real time computation to implement (39) depends on the plant complexity, but not on N. The control gain, k r must be between 0 and 2 for asymptotic stability of the repetitive control system. 2. Stability Robustness. The internal model, l-z~ N, possesses N characteristic roots all on the unit circle, which is the stability boundary for the discrete time system. This makes the stability of the overall system highly sensitive to unmodeled dynamics of the controlled plant. The robustness problem has been resolved by modifying the internal model so that the Fig. 4 Implementation diagram for discrete time repetitive controller with modified internal model characteristic roots, especially high frequency characteristic roots, are shifted to the inside of the unit circle. Strictly speaking, the modified internal model is no longer a periodic signal generator. In particular, Tsao and Tomizuka (1988) have shown that the following internal model offers flexibility in trading off the tracking performance against robustness. where A d (z ) = l-g(z,z )z~ (40) q(z, Z [ )=a m z m + a,-. i z m + +a x z + a 0 + a[z - l +---+a m z- m a,->0 2a, + 2a, _ a! + a 0 =l (41) Notice that g(z, z~ l ) is a low pass filter possessing the zero phase shift characteristics. Since the implementation diagram for the repetitive controller with the modified internal model has not been given elsewhere, it is shown in Fig Fluctuation of Period. In practice, the period of w (k) may fluctuate. While this is an interesting topic for further research, this problem has been bypassed in implementation by adjusting the sampling time so that N remains fixed. For example, in disk file systems, the desired trajectory for the read/write head and disturbance are periodic due to the eccentricity, i.e., the center of rotation of the disk and the center of a selected servo track do not agree (Chew and Tomizuka, 1990). In this problem, A? remains fixed by synchronizing the sampling of error with the angle of the spindle axis. 3.2 Applications of Discrete Time Repetitive Control. The discrete time repetitive control algorithm has been successfully applied to computer disk file systems (Chew and Tomizuka, 1990), noncircular machining (Tsao and Tomizuka, 1988) and robot manipulators (Tsai et al., 1988). Notice that the repetitive controller has a learning ability, so that the system performance is improved each time the controller experiences a repetitive situation. This property has been utilized by Tung et al, (1991) to learn an appropriate control signal for overcoming low velocity friction, for which it is hard to make a precise model. 4 Conclusions and Further Remarks In this paper, the design of digital tracking controllers has been presented for the following two cases: (1) the desired output is known in advance, and (2) the desired output is not directly measured but is known to possess a certain property. For the first case, the design can be based on either the linear optimal control theory (optimal preview controller) or pole/ zero/phase cancellation (ZPETC). For the second case, the internal model based controller can achieve asymptotic regulation and tracking of the desired output. An interesting special case of the internal model based controller is the discrete time repetitive controller. Summary of these control algorithms are given below. The optimal preview controller minimizes the performance index which includes the tracking error term and the input term. The weights of these terms determine the number of 416 /Vol. 115, JUNE 1993 Transactions of the ASME

6 preview steps for attaining almost all of the possible benefit of preview. «ZPETC is a good approximation for the inverse of the closedloop system when the system possesses unstable zeros. The number of preview steps required for implementation of ZPETC depends on the number of delay steps in the system, d, and the number of uncancellable zeros, 5. The optimal preview controller and ZPETC are based on the linear control theory. Therefore, in applications, they must be combined with various linearization techniques. 8 The discrete time repetitive control is effective when the desired output and disturbance are periodic and their period is known. For stability robustness, the internal model needs to be modified so that their characteristic roots are inside the stability region. 8 The optimal preview controller, ZPETC and discrete time repetitive controller have been successfully tested on a variety of mechanical systems. Some further remarks are In tracking control discussed in this paper, the desired output has been given as a time sequence. Such a sequence may be obtained by decomposing a desired spatial trajectory to servo axes. In this case, the desired output sequence for each axis depends on the tracking speed along the trajectory. The tracking problem is more complicated when the tracking speed is adjustable. Huang and Tomizuka (1990) applied the fuzzy set theory to deal with such situations. This type of tracking is called self-paced tracking (Sheridan, 1966). It is an interesting area for further research. The optimal preview control and generalized predictive control (GPC) (Clark et al., 1987) have some common aspects. The optimal preview control emphasizes the improvement of tracking performance by utilizing the future values of the desired output, and the GPC is motivated by attaining robustness by incorporating the predicted future system behavior. In fact, it is possible to formulate the two problems in a unified mathematical framework (Egami et al., 1992; Konno et al., 1992). ZPETC is sensitive to modeling errors. The frequency spectrum of the desired output is normally band limited, and it is important to develop a mathematical model of the closedloop plant in the relevant frequency range. It should be noted that the tracking performance under ZPETC depends on the mathematical model as well as on identification algorithms for model development (Tung and Tomizuka, 1992). The discrete time repetitive control algorithm based on the internal model principle is one of the possible approaches for dealing with periodic signals. For other approaches, see Kempf et al. (1992). As the title of my dissertation (Tomizuka, 1993) suggests, I was a member of the Man Machine Systems Laboratory of MIT and worked on preview control from a human machine systems point of view. After joining the University of California at Berkeley, I have found that "tracking" is one of the most important features for mechanical systems such as a robot manipulator and a machining center. In tracking arbitrary shaped desired outputs, the system remains continuously transient and never reaches the steady-state, which is a very unique aspect in mechanical systems control and provides a challenge to mechanical control engineers. 5 Acknowledgments I would like to acknowledge the contributions of my numerous graduate students over the past twenty years to the research summarized in this paper. References Anderson, B. D, O., and Moore, J. B., 1971, Linear Optimal Control, Prentice Hall, Englewood Cliffs, NJ. Astrom, K., and Wittenmark, B., 1984, Computer Controlled Systems, Prentice Hall, Englewood, Cliffs, NJ. Athans, M., and Falb, P., 1964, Optimal Control, McGraw Hill, New York. Chew, K. K., and Tomizuka, M., 1990, "Digital Control of Repetitive Errors in Disk Drive Systems," IEEE Control Magazine, Vol. 10, No. 1, Jan., pp Clark, D. W., Mohatadi, C, and Tuffs, P. S., 1987, "Generalized Predictive Control Parts I and II," Automatica, Vol. 23-2, pp Egami, T., Tsuchiya, T., Aida, K., and Kitamori, T., 1992, "Preview Servo System Design and Relation with GPC System," Transactions of the Society of Instrument and Control Engineers, Vol. 28, No. 8, pp (in Japanese). Francis, B. A., and Wonham, W. M., 1975, "The Internal Model Principle for Linear Multivariable Regulators," Applied Mathematics and Optimization, Vol. 2, pp Funahashi, Y., and Yamada, M., 1992, "Generalization of Zero Phase Error Tracking Controller," Transactions of the Society of Instrument and Control Engineers, Vol. 28, No. 1, Jan., pp (in Japanese). Haack, B., and Tomizuka, M., 1991, "The Effect of Adding Zeros to Feedforward Controllers," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 113, Mar., pp Hayase, M., and Ichikawa, K., 1969, "Optimal Servosystem Utilizing Future Values of Desired Function," Trans, of the Society of Instrument and Control Engineers, Vol. 5, No. 1, Mar., pp (in Japanese). Horowitz, R., Tsai, M. C, Anwar, G., and Tomizuka, M., 1987, "Model Reference Adaptive Control of a Two Axis Direct Drive Robot Manipulator Arm," Proceedings of the 1987 IEEE International Conference on Robotics and Automation, pp Huang, L.-J., and Tomizuka, M., 1990, "A Self-Paced Fuzzy Tracking Controller for Two-Dimensional Motion Control," IEEE Trans, on Systems, Man and Cybernetics, Vol. 20, No. 5, Sept., pp Inoue, T., Iwai, andnakano, M., 1981, "High Precision Control of Playback Servo Systems," Transactions of Institute of Electrical Engineers, Japan, Series C, Vol. 101, No. 4, pp (in Japanese). Kempf, C, Messner, W., Tomizuka, M., and Horowitz, R., 1992, "A Comparison of Four Discrete Time Repetitive Control Algorithms," Proceedings of the 1992 American Control Conference, June, pp Konno, Y., Hashimoto, H., and Tomizuka, M., 1992, "On Prediction and Preview in Control Systems," Proceedings of the 21st SICE Symposium on Control Theory, SICE, May, pp (in Japanese). Omata, T., Nakano, M., and Inoue, T., 1984, "Application of Repetitive Control Method to Multivariable Systems," Transactions of the Society of Instrument and Control Engineers, Vol. 20, No. 9, pp (in Japanese). Pak, H. A., and Tuner, P. J., 1986, "Optimal Tracking Controller Design for Invariant Dynamics Direct-Drive Arms," ASME JOURNAL OF DYNAMIC SYS TEMS, MEASUREMENT, AND CONTROL, Vol. 108, Mar., pp Peng, H., and Tomizuka, M., 1991, "Preview Control for Vehicle Lateral Guidance in Highway Automation," Proceedings of the 1991 American Control Conference, pp Peng, H., et al., 1992, "A Theoretical and Experimental Study on Vehicle Lateral Control," Proceedings of the 1992 American Control Conference, pp Samson, C, 1982, "An Adaptive LQ Controller for Non-Minimum-Phase Systems," International lournal of Control, Vol. 35, No. 1, pp Sheridan, T. B., 1966, "Three Models of Preview Control," IEEE Transactions on Human Factor in Electronics, Vol. HFE-7, No. 2, June, pp Suzuki, A., and Tomizuka, M., 1991, "Design and Implementation of Digital Servo Controller for High Speed Machine Tools," Proceedings of the 1991 American Control Conference, June, pp Tomizuka, M., 1973, "The Optimal Finite Preview Problem and its Application to Mari-Machine Systems," Ph.D. dissertation, Mass. Inst, of Tech., Cambridge, MA, Sept. Tomizuka, M., and Whitney, D. E., 1975, "The Discrete Optimal Finite Preview Control Problem (Why and How is Future Information Important?)," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 97, No. 4, Dec., pp Tomizuka, M., and Rosenthal, D. E., 1979, "On the Optimal Digital State Vector Feedback Controller with Integral and Preview Actions," ASME JOUR NAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 101, No. 2, June, pp Tomizuka, M., Dornfeld, D., and Purcell, M., 1980, "Application of Microcomputers to Automated Weld Quality Control," ASME JOURNAL OF DY NAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 102, No. 2, Mar., pp Tomizuka, M., Dornfeld, D., Bian, X.-Q., and Cai. H.-C, 1984, "Experimental Evaluation of the Preview Servo Scheme for a Two-Axis Positioning System," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 106, No. 1, Mar., pp Tomizuka, M., and Janczak, D., 1985, "Linear Quadratic Design of Decoupled Preview Controllers for Robotic Arms," Internationa/Journal of Robotics Research, Vol. 4, No. 1, pp Tomizuka, M., 1987, "Zero Phase Error Tracking Algorithm for Digital Control," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. 109, Mar., pp Tomizuka, M., Tsao, T.-C, and Chew, K.-K., 1989, "Discrete-Time Domain Journal of Dynamic Systems, Measurement, and Control JUNE 1993, Vol. 115/417

7 Analysis and Synthesis of Repetitive Controllers," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL, Vol. Ill, Sept., pp Tomizuka, M., 1992, "On Compensation of Phase Shift in Discrete-Time Linear Controller Design," Proceedings of the Symposium on Fundamentals of Discrete-Time Systems, June. Tsai, M.-C, Anwar, G., and Tomizuka, M., 1988, "Discrete Time Repetitive Control for Robot Manipulators," Proceedings of the 1988 IEEE Int. Conference on Robotics and Automation, April, pp Tsao, T.-C, and Tomizuka, M., 1987, "Adaptive Zero Phase Error Tracking Algorithm for Digital Control," ASME JOURNAL OF DYNAMIC SYSTEMS, MEAS UREMENT, AND CONTROL, Vol. 109, Dec, pp Tsao, T.-C, 1988, "Digital Tracking Control and Its Application to Noncircular Machining," Ph.D. dissertation, Univ. of California at Berkeley, Tsao, T.-C, and Tomizuka, M., 1988, "Adaptive and Repetitive Digital Control Algorithms for Noncircular Machining," Proceedings of the 1988 American Automatic Control Conference, June, pp Tung, E., Anwar, G., and Tomizuka, M., 1991, "Low Velocity Friction Compensation and Feedforward Solution Based on Repetitive Control," Proceedings of the 1991 American Control Conference, June, pp Tung, E., and Tomizuka, M., 1992, "Application of Frequency-Weighted Least Squares System Identification to Feedforward Tracking Controller Design," Proceedings of the 1992 Japan /USA Symposium on Flexible Automation, ASME, Vol. 1, pp (also to appear in ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL). 418 /Vol. 115, JUNE 1993 Transactions of the ASME