Robust and adaptive control of a beam deflector

610 zyxw zyx zyxwvutsrqpon IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 13, NO 7 , JULY 1988 Robust and Adaptive Control of a Beam Deflector PER-OLOF GUTMAN, MEMBER, IEEE, HANOCH LEVIN, LINDA NEUMANN, TUVIA SPRECHER, Abstract-A DC-motor driven beam deflector is to be controlled such that the beam moves with constant velocity for 12 ms out of a 20 ms period before swinging back to the original position. One control system must serve any deflector without retuning. The problem is difficult because the dynamics vary from one deflector to another. It is found that a closed-loop system with poles in a Bessel filter configuration will satisfy the specifications. A straightforward poleplacement design is insufficiently robust. Redesigning it yields greater, but still insufficient, robustness against the known extent of plant parameter variation. The problem is solved by adding an adaptive control loop. Laboratory tests show performance to specification for a number of different deflector cases. This study raises some questions concerning robust and adaptive control. AND ELI VENEZIA U MOTOR Fig. 1. A schematic diagram of the beam deflector set-up. see Fig. 1. The control system includes a reference signal generator, the feedback control circuits, and a power amplifier. I. INTRODUCTION The beam deflector is part of an optical recording system that HE last two decades have seen a vigorous research in adaptive presents the data as television pictures. The reference signal is periodical (50 Hz). During 16 ms the control, and lately various adaptive control methods have found their way into practical use [I]. At the same time, classical signal is linear with constant velocity; the remaining 4 ms allows control design methods have retained their popularity, and they for a sinusoidal swingback to the original position. The reference have been improved to yield robust control systems; see, e.g., [2]. signal is displayed in Fig. 3. There are no significant noise (By a robust controller we mean a fixed parameter controller that sources in the system. The purpose of the control system is to make the prism move gives satisfactory control for a set of plants.) Robust control design methods have also been suggested based on state-space linearly, i.e., with constant velocity, over a prescribed angle for at least 12 ms during each period. The control system must be such designs and optimal control [3]. Naturally, a debate has started such questions as when a robust that deflector changes or exchanges may occur without the need of design is more advantageous than adaptive control, what the retuning the parameters. The problem is difficult because the deflectors exhibit different limitations of the methods are, and how they can be combined [4]. This paper should be seen as a case study of the application of dynamics: the DC-gain, the dominant spring resonance, and highrobust and adaptive control. First, the closed-loop specification frequency mechanical resonances vary. It could be argued that the tolerances were derived, and the extent of plant uncertainty was control problem would have been easier if the spring had been established. Then a straightforward pole-placement design [8] was excluded, since most of the dynamics variations stem from it. modified with the Horowitz robust design method [2], with the Then, however, the DC-motor would have had to be larger, to aim of making the control system stand up to all plant cases. This give the power needed for the rapid sinusoidal swingback. The plant transfer functions were measured with spectrum and was found impossible, and an adaptive control loop [5] had to be frequency analysis. A typical transfer function is found in Fig. 2. added. The solution presented here is similar to designs proposed The primary spring resonance resides at 136 Hz, and secondary during the late 1950’s [17], one of which was actually flight tested resonances appear at 2, 3.7, and 6.4 kHz. By measuring a large with disastrous results. Then, adaptive control seems to have been number of plants it was found that the DC-gain varies in the range designed with enthusiastic heuristics. Today, 25 years later, a case [6, 201, the main resonance (which is due to the spring) has a like ours is based on enthusiasm, but also on solid analysis and damping factor { E [0.032, 0.0631 and a natural frequency w E [819, 10101 rad/s, and that the secondary resonances each have a theory. The paper is organized as follows. Section I1 contains the spread of a couple of hundred rad/s. The exact reasons for the problem definition, establishes the plant transfer function and the three secondary resonances beyond IO00 Hz were not investiextent of plant uncertainty, and discusses the desired closed-loop gated. They are probably due to vibration modes in the mechanispecifications. Section 111 covers the pole-placement design, and cal components. An uncertain plant model was defined in the the robust redesign. Section IV describes the adaptive gain robust design package HORPAC [9]; see Table I. No nonlinear analysis was performed, since the plant was control. Section V contains a few measurements and the final discussion. The Appendix contains a brief summary of the adequately described by the above set of linear transfer functions. A simplified linear model, taking only into account the gain Horowitz design method in the single input-single output case. variation and the main resonance, is 11. SYSTEM AND PERFORMANCE REQUIREMENT k The beam deflector to be controlled consists of a DC-motor P(s)= 2r s2 working against a spring, a prism, and an angular position sensor; T zyxwvut zyxwvutsrqp zyxwvu zyx zyxwvutsrqponm zyxwvut zyxwvut 1 Manuscript received August 4, 1986; revised October 5, 1987 and March ri - - w The reference signal is displayed in Fig. 3. Harmonic analysis reveals components at n - 5 0 Hz, n = I , 2, . * with approxi- 0018-9286/88/0700-0610$01.OO r r- S + T \k E [ 6 , 201, {=0.04,w=854. 24, 1987. Paper recommended by Associate Editor, B. Friedland. The authors are with El-Op Electro-Optics Industries, Ltd., Rehovot, Israel. IEEE Log Number 8821356. +-w O 1988 IEEE zyxwv I -- I 1 zyxwvutsrqponm zyxwvutsrqponmlk zyxwvutsrqpo zyxwvutsrq zyxwvu 611 GUTMAN et al. : ROBUST AND ADAPTIVE CONTROL OF A BEAM DEFLECTOR dB PNASE I. . 1 1 I II 7 40 20 GAIN 0 - 20 I I zyxwvutsr zyxwvuts zyxwvut id00 100 10 I [Nx] 101 TABLE I THE UNCERTAIN MODEL DEFINITION IN HORPAC. FWZQUENCIES ARE IN R A D I S . # STANDS FOR NUMBEROF CASES CONSIDERED FOR THE PARAMETER IN QUESTION. POLE # 2: SHOULD BE INTERPRETED AS A CERTAIN POLE AT -20 OOO W / S . CPOLE # 3: SHOULD BE INTERPRETED AS A COMPLEX POLE PAIR, WHERE THE RELATIVE DAMPING HAS THE NOMINAL VALUE = 0.045, THE MINIMUM = 0.032, AND THE W M U M = 0.063 WITH ”HREE CASES CONSIDERED, AND WHERE THE NATURAL FREQUENCY HAS THE NOMINAL = 850 RAD/S, THE hfEJMUM = 819 W / S , AND THE MAXIMUM = 1010 RAD/S WITH THREE CASES CONSIDERED Begin: Plant 1 Gain 1: Nom 18.5 Integrators Delay Nom Uin # Max 6.0 20. 4 # Min Max (I ........................... Nom Min Max x Czero Damp Nom Min Max X zero Freq Nom Min x Max _ _ _ _ _ - - - _ _ _ _ _ _ - - - _ _ _ - - - - - - Nom Min Max -2.00E+04 -2.00E+04 -2.00E+04 x Cpole Damp Nom Min Max 3: 4.50E-02 3.20E-02 6.3OE-02 4: 0.25 0.25 0.25 5: 2.00E-02 2.00E-02 2.00E-02 6: 2.00E-02 2.00E-02 2.00E-02 # Pole 2: End: Plant 1 1 3 1 1 1 # Freq Non Min Max 8.503+02 8.19E+02 1.01E+03 1.263+04 1.23E+04 1.35E+04 4.00E+04 3.60E+04 4.40E+04 2.32E+04 2.10E+04 2.55E+04 3 3 3 3 zyxwvut mately exponentially decreasing power for frequencies above 100 characterized by their linear phase lag [7, p. 2281. The pole Hz. location for a second-order Bessel filter with a bandwidth of 200 The question is: what type of closed-loop network will retain Hz is found in Fig. 4 and is (1088 628j). The linear phase lag the “linearity” of the signal described above, and formally property was found to be important to preserve output linearity. defined here. Hence, if the dominant poles of the closed-loop system can be Definition I : The plant output signal is said to be linear if its made to form a suitable Bessel filter configuration for all position versus time graph is a straight line for at least 12 ms per uncertain plant cases, then the control problem is solved. 20 ms period. A deviation from the straight line corresponding to 0.2 percent of the peak-peak amplitude of the signal is allowed. III. POLE-PLACEMENT DESIGN AND ROBUSTREDESIGN Note that the problem is not a tracking problem, i.e., the In view of the results of Section II, a pole-placement design was position versus time graphs of the reference and the output need not overlap. A time shift is allowed, and the amplitude of the tried [8]. A state-space representation of (1) is, e.g., output signal may be different from the amplitude of the input signal. It is important, however, that the linearity and amplitude of the output signal is preserved for all uncertain plant cases. By simulations with the simulation language SMNON [6], it was found that Bessel filters of order 2 or more with a bandwidth 2 200 Hz will make the output signal linear. Bessel filters are * zyxwvutsrqp zyxw zyxwvuts zyxwvuts zyxw zyx 612 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 33, 1988 linearity of y . If k / g is large enough, the closed-loop system is unstable. In a similar way it is calculated that the effect of the uncertainty in w and on the relative damping of the closed-loop poles is smaller than the effect of the DC-gain uncertainty. w E [819, 10101 causes {, E [0.784, 0.9681, while { E [0.032, 0.0631 will cause {, E [0.839, 0.8811. In spite of the lack of robustness of the controller (3), it might be of interest to implement it. Since xI is not available for measurement, it has to be estimated. A Kalman filter cannot be used since k is unknown, but differentiating y gives an estimate of x1:PI = y l w 2 . A high-frequency pole s = - M has to be inserted to avoid pure differentiation. The Laplace transform of (3) is then 10 [v 1 zyxwvutsrqponmlkjih c 5. NO. 7, JULY 0. - 5. -10 40 6a 50 70 [SEC] 80 Fig. 3. The reference signal ( l ) , and the output (2) from a fourth-order Bessel filter with bandwidth = 205 Hz. The output signal is linear for 15 ms. c, where urefin (3) is chosen such that the PD-feedback acts on the error Yref With g = 18.5 and M = 3oo00, (9) becomes U(s)= with w = 854, { = 0.04, and k E [6, 201. y is the angular position, and U the plant control input. Then x2 is the scaled position x2 = y / 0 2 and xI the scaled velocity x1 = y / w 2 . Now find the feedback control U = - 1,Xl - 12x2 + Uref 0.87, W E = 1257. ( Y(S)- YIef(S)). (10) If (10) is viewed as a one-loop controller, then it is rewritten as (3) such that the closed-loop system has the poles of a second-order Bessel filter with a bandwidth = 200 Hz { S ( S 2 + ~{B’BWBS+ Wi=o}, zyxwvu 1.5 10-4s 1+s / 3 m (4) Inserting (3) into (2) and equating the closed-loop characteristic polynomial with the characteristic polynomial of the Bessel filter (4) gives the following equation for II and 12: The Horowitz robust control design method [2], mechanized in the interactive program package HORPAC [9] was used for the robust redesign. A summary of the Horowitz design method is found in the Appendix. Another reference containing a readable review of the method is [14]. The pole-placement design (1 l), applied on the uncertain plant defined in Table I was checked against Horowitz bounds for the compensated nominal open loop, derived from closed-loop specifications (A.2) based on admissible Bessel filters. The nominal plant is given in Table I; notice especially the nominal gain. In Fig. 5 the compensated nominal open loop is displayed in a Nichols chart. The heavily marked “ellipsoidical” highfrequency Horowitz bound is defined by the second equation in (A.2) for w S- U,. Its gain extent reflects the plant gain uncertainty. An admissible nominal open loop must stay outside this bound. Clearly, the pole-placement design violates the highfrequency bound in such a way that robustness against DC-gain variations is not present (cf. Appendix). Several attempts to find a robust design were made, but they led to high-order compensators with unrealizable lead networks, and an unrealistically large bandwidth. We settled for a more modest redesign zyxwvutsr S ~ + ( ~ { U + ~ I I ) S + ( SW2 +~ 2+{ ~~ ~~ ~~S)+ w i . ( 5 ) However, k is unknown. To solve (S), the plant gain has to be assumed to b e g E [6,20], i.e., in the range of the true plant gain. Then If g # k , the closed-loop poles will not attain the prescribed Bessel filter configuration, but rather U(s)= 1 +sz/w; 1 +s2/w: 1 +s/w1+s2/w: 1 +s/w2+s2/w; 2.063(1 +s/2778) (1 +s/1oooO)(1 +s/37071) (7) Equation (7) is the characteristic equation of the closed-loop system (2), (3), (6). Clearly, g E [0.3, 3.331 (8) for all possible combinations of g and k. The root locus of (7), (8) when k / g is varied is found in Fig. 4. We conclude from the root locus, that even for moderate differences between g and k , the deviation from a Bessel pole pattern is considerable, and spoils the 0.34(1 +s/455) 1 + s/28369 . Y(s)+0.245 . ~ 1 + 0.245 4.3 1 +s/3707 with wI = 2 kHz and w2 = 3.7 kHz. The lower part of Fig. 8 presents the block diagram of the control system. The first two factors in (12) represent notch filters at the nominal locations of the first two secondary resonances. The terms inside the square brackets approximately yield PD-action and should be compared to (IO). The remaining factors outside the brackets give the desired loop shaping in the crossover frequency 1 1 zyxwvutsrqponmlkj zyxwvutsrqponmlkjihg zyxwvutsrqpon zyx GUTMAN et al.: ROBUST AND ADAPTIVE CONTROL OF A BEAM DEFLECTOR 33 v -9000 2 - -7000 I -5000 _ . 3000 613 - -- Fig. 4. Root locus for ( 7 ) and (8). The Bessel poles assumed for k / g = 1 are marked by 0. The real endpoints are assumed for k / g = 3.33, and the complex endpoints for k / g = 0.3. zyxwvutsrqponmlkji zyxwvutsrqponmlkji d0 so 30 IO so 40 40 30 30 zyxwvutsrqponmlkjihgfedcb zyxwvutsrqponmlkjihgfe 40 20 50 0 -10 -20 -30 - 360’ 20 20 IO IO 0 0 -10 -I 0 -20 - 20 -30 -270. -180. -90- 00 Fig. 5 . Nichols chart of the open loop G(s).P,,,,(s) with G(s) from (11) the nominal from Table I. The marked “ellipsoidical” curve is and Pn,,m(s) the high-frequency Horowitz bound. Dashed lines are low-frequency Horowitz bounds. Frequency markings on the open loop are in rad/s. range, while the factor preceding Yrefrepresents the prefilter. The Nichols chart of the nominal open loop is found in Fig. 6. The sensitivity to DC-gain variations remains. The robustness in other respects is, however, somewhat improved: the range of secondary frequencies ( > 2 kHz) is outside the high-frequency bound which means that the secondary resonances are well attenuated even when they vary. The open-loop phase for the interval [1000, 30001 rad/s is approximately constant = - 115” which gives the same phase margin and crossover characteristics even when the DC-gain is varying. In fact, each attempt to redesign was accompanied with - 360’ ~ -270. -180- -90- 30 0. Fig. 6. Nichols chart of the open loop G(s).P.,,(s) where G(s) is the feedback part of (12), and P,,,(s) the nominal from Table 1. The marked “ellipsoidical” curve is the high-frequency Horowitz bound. Dashed lines are low-frequency Horowitz bounds. Frequency markings on the open loop are in radis. laboratory experiments, and the control (12) is considerably less sensitive to parameter variations, and yields a better linearity than (10). A Bode diagram of the closed loop is found in Fig. 7. It could be argued that one or two integrators should have been includzd in the feedback compensator. Besides the fact that true tracking was not required (see Section U),integrators were found to cause overshoots that destroyed the linearity. In summary, the robust redesign gave a closed-loop system, still sensitive to plant gain variations. Manual tuning of the loop gain (at point A in Fig. 8) had to be done to compensate for 614 zyxwv zyxw zyxwvutsrqpon IEEE TWNSACTIONS ON AUTOMATIC CONTROL, VOL. 33, NO. 7,JULY 1988 zyxwvutsr zyxwvutsrqp Fig. 7. Bode diagram of a well-tuned robustly redesigned closed-loop transfer function from reference yEfto output y ( t ) . AGC R E F = I8V I ADAPTIVE GAIN CONTROL PREFILTER OFF S E T x 0 . 2 5 V DETECTOR 1. zyxwvu zyxw I Fig. 8. Block diagram of the final system. The two uppermost blocks represent the adaptive gain control included in Section IV. Notice points A and B. varying plant DC-gain, thus placing the dominant closed-loop poles in an approximate Bessel filter configuration. The loop gain influenced the output signal linearity, and also, in a monotonous way, the output signal peak-peak value. Every well-tuned system had the same loop gain, sufficient linearity, and the same required peak-peak value. IV. ADAWIVE GAINCONTROL For a well-tuned system, z( t ) = zo. Linearizingfaround z ( t ) zo, and regarding it as a sampled data system, gives the simplified model = 1 z ( i ) =P- . (u(i-d)), i = 1, 2, ... (14) with l/p = the unknown gain and d the unknown delay. Precisely for such and related systems there exists a satisfactory adaptive control theory [5]. In [5] the following controller is proposed: zyxwvutsrqp zyxwvutsrqp zy The observation in Section III that the robustly redesigned system could be manually tuned, with the output peak-peak monotonously depending on the loop gain, led to the following view of the closed-loop control system (with the (manual) gain factor as input, and the peak-peak as output): z(O=f(4t-6)) (13) where z ( t ) = the output signal peak-peak, u ( t ) = the input factor, multiplying the loop gain, 6 = an unknown time delay, and f a monotonous, positive, and unknown function. In the lower portion of Fig. 8 (i.e., excluding the two uppermost blocks in the figure), u ( t ) is seen at point A , and z ( t ) at point B. -1 I1 - - where zc(i) is the reference for the output peak-peak, and Po is an initial estimate of 0. In [5] it is proved that (14), (15) gives a globally, asymptotically stable closed-loop system iff 0 < pO/P < 2. With z,(i) = zo = 18 [Volt] for all i, (15) gives U( 1 i ) = u ( i - d ) - Po[z( i ) - z,]. (16) zyxwvutsrqponmlkj zyxwvutsrqponm zyxwvutsrqpon zyxwvutsr GUTMAN et ai.: ROBUST AND ADAPTIVE 615 CONTROL OF A BEAM DEFLECTOR VOLTS 1-10 o s O O zyxwvutsrqpon zyxwvutsrqponmlk zyxwvutsrqp os 15 I 25 2 SEC 3 Fig. 9. Position output y ( t ) (point B in Fig. 8), and output from the adaptive gain control (point A in Fig. 8) for the final control system with adaptive gain control, at start-up. -10 zyxwvutsrqponml zyxwvutsrqpo , + U 0 01 -1- I 002 a03 004 005 0 06 Q07 SEC -., 008 Fig. 10. Position output y ( t ) at steady state for the final control system, with and without adaptive gain control. The output from the system with AGC is linear for 14.2 ms per period. This is simply an integrating controller. Equation (16) was implemented and the block diagram of the final system is found in Fig. 8. d was set to 1 [sampling period], i.e., 20 ms, and Po was tuned manually, once and for all, to a suitable value. V. MEASUREMENTS AND DISCUSSION In this section a few measurements of the final system are presented. Figs. 9 and 10 show the performance of the complete control system from start-up to steady state. Fig. 9 gives the output y ( t ) and the output from the adaptive gain control u(i). The adaptive gain control is turned on at about 0.8 s. The dip in u ( i ) reflects the fact that at start-up, u ( i ) is forced to low values in order to avoid destabilizingly high gains. Thus, u ( i ) converges to its steady-state value from below. Fig. 10 is a blowup of a steadystate portion of Fig. 9, and we see that 14.2 ms linearity in the output is achieved. For comparison, the output of an untuned system without adaptive gain control is included in Fig. IO: notice that neither linearity nor peak-peak meet the specifications. Figs. 11 and 12 show the result of an artificial gain change in the plant. The plant was extended with a potentiometer in series with which the “plant” gain could be changed. With this “plant” initially in steady-state, the “plant” gain was changed in two steps (within 1 s) to one third of its original value. In Fig. 11 the control input u ( t ) and position output y ( t ) of such a system without adaptive gain control are shown. Clearly, the signal amplitudes 616 zyx zyxwvutsrqpo zyxwvutsrqponmlkjihgf zyxwvutsrqponmlkjihg zyxwvuts IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 33, NO. 7, JULY 1988 Y 4 VOLTS 6 U 2 -. 2 E zyxwvutsrqponmlkjih y zyxwvutsrqpo -I I 2 0 8 .6 .4 SEC I Fig. 1 1 . Position output y ( f )and control input u ( t ) for the final control system without AGC when the plant gain is changing in two steps to 1/3 of its original value. 0 2 I 3 4 5 6 SEC 7 Fig. 12. Position output y ( t ) and AGC output for the final control system with AGC, when the plant gain is changing in two steps to 113 of its original value. decrease immediately at the instants of the “plant” gain changes, and linearity suffers. In Fig. 12, the same experiment is repeated with the automatic gain control: we see how the output amplitude converges to the required reference within 6 s, and how the output from the automatic gain control compensates for the “plant” gain change. Experiments on a number of deflectors showed that the control system specifications were fully satisfied. In this paper the design process has been presented as a smooth and logical flow from one stage to another. As in all real design projects it was not so; many attempts in different directions were made, and many dead ends encountered. From the experiments a few things stand out. 1) Designs where robustness and sensitivity were taken into -T I[--- . account were consistently better than other designs. This observation could, e.g., be compared to [ 111 where it is stated that pure pole-placement is very sensitive to plant parameter variations. 2) It was not possible to design a fixed parameter linear controller that met the robustness specifications. Such a controller would have been impossible to build with analog components, would have had an unrealistically high bandwidth, and would have presupposed a less uncertain, but unavailable, model of the highfrequency behavior. 3) The fixed parameter linear controller was supplemented with an adaptive gain control. The AGC worked considerably better together with robustly redesigned fixed parameter controllers than with others. These observations raise questions concerning the practical z zyxwvutsrqponmlkjih zyxwvutsrqponmlk zyxwvutsrq GUTMAN et al.: ROBUST AND ADAPTIVE CONTROL OF A BEAM DEFLECTOR zyxwvutsrqp zyxwvutsrqponmlkj - PREFILTER REFERENCE FEEDBACK COMPENSATOR F is) 617 G 1s) OUTPUT zyxwvuts I. Fig. 13. The canonical two degree-of-freedom structure for SISO systems where both the plant output and the reference are measured. should roll off as fast as possible in order to reduce the influence of high-frequency measurement noise. I B( w ) - A ( w ) I defines the tolerance; for w L U, the tolerance is said to be free. From the disturbance rejection specified in (A.2), follow the gain and phase margins, the sensitivity function, and the error transfer function. According to Bode's relations [ 151, the disturbance rejection must be greater than one for some frequencies. Therefore, x > 1 is chosen in (A.2). Nothing prevents the user from specifying phase bounds for the closed loop, or requiring a frequency-dependent disturbance rejection x ( w ) . However, it must be made sure that the specifications satisfy Bode's relations. APPENDIX 2) Plant Uncertainty: Determine the set of transfer functions THEHOROWITZ DESIGNMETHODIN THE BASICSINGLE-INPUT that defines the plant. For each frequency s = j w the transfer SINGLEOUTPUT(SISO) CASE functions give rise to a set of complex numbers. This set is called It is assumed that the uncertain plant P ( s ) to be controlled is the template for w ; see Fig. 14. The template is most convendescribed by a set of linear, time-invariant transfer functions with iently displayed in a Nichols chart (see, e.g., [ 16, p. 3351) since it no zeros in the right half of the complex plane. When parametriz- includes loci for the closed-loop transfer function G ( s ) P ( s ) /1( + ing this set, it is assumed that the parameters belong to bounded G(s)P(s)).The inverted Nichols chart gives loci for 1/(1 + intervals. It is also assumed that all the transfer functions have the G(s)P(s)),and the x-locus (A.2) is easily included in the regular same sign of the high-frequency gain, since this is a necessary Nichols chart. The interior of the x-locus is called the disturbance rejection set and is defined as { s i 1/(1 + G ( s ) P ( s ) )2 x}; see condition for simultaneous stabilizability; see, e.g., [4] or [ 151. The method in the basic case is described in [2]. It is easily Fig. 14. For design purposes, it suffices in general to compute extended to cascaded SISO systems, to nonminimum phase templates for 5-10 frequencies. 3) Horowitz Bounds: Study Fig. 14. The drawn template systems [lo], and to discrete-time systems. Certain classes of time-varying systems [lo], nonlinear systems, MIMO-systems, represents the uncompensated open loop for w = 4 rad/s. Reading off the Nichols chart we see that the closed-loop gain will vary 10 and others can also be dealt with. Assuming that both the output and the reference signals are - (-37))dB = 37 dB and that the template is outside the available for measurement, a two degree-of-freedom system [ 151 forbidden disturbance rejection set. The closed-loop gain variais postulated; see Fig. 13. The feedback compensator G(s) is used tion should be compared to the tolerance for this frequency; see to reduce the sensitivity to disturbances and to variations in the Step 1 above. Now consider the compensated open loop plant P(s),while the prefilter F(s)is used to shape the closed- G(s)P(s).Its template for w will be identical in shape, size, and loop transmission from reference to output. Let the closed loop be orientation to the template of P ( j w ) in the Nichols chart (since G(s) is precisely known) but slided vertically and horizontally according to the gain and phase of G( j w ) . The closed-loop gain variation can be altered at will by changing the open loop G ( j w ) P (j w ) . For instance, a sufficiently high open-loop gain will decrease the closed-loop gain variation to an arbitrary small Notice that if the control system is linear, the canonical two value. Clearly, certain open-loop templates G( jw)P( j w ) satisfy degree-of-freedom system structure in Fig. 13 is equivalent to all the tolerance and disturbance specification (A.2), others do not. The complex number for the nominal open loop other two degree-of-freedom structures, e.g., with a compensator in the feedback path, or a feedforward block directly from the G( j w ) P n o m ( j w ) , corresponding to one plant transfer function reference to the plant input. This is easily seen by using block case, can be taken as the representative for the open-loop template. Then, for each frequency, the complex plane will be diagram algebra. The design procedure proceeds in five steps. I) Closed-Loop Specijications: The closed-loop specifica- divided into two sets; one set where the nominal open loop resides when the tolerance and disturbance rejection specifications are tions are given in the following form: satisfied, and one set when they are not. The boundary between these two sets is called the Horowitz bound for U . Hence, the Horowitz bound denotes, for each frequency, the boundary for the allowed compensated nominal open loop. In Figs. 5 and 6 Horowitz bounds are drawn. The Horowitz bounds for frequencies w 2 w, are closed curves Upper and lower bounds for the gain of the closed-loop transfer function are specified in (A.2) for frequencies w < U,. For higher around the (0 dB, - 180") point. They are called high-frequency frequencies the open-loop and closed-loop transfer functions bounds, and reflect the fact that for these frequencies, only the limitations of robust control, adaptive versus robust control, the robustness of adaptive controllers, the adaptive modification of robust controllers, and the combination of robust and adaptive control. These questions are the subject of vigorous research; see, e.g., [4], [12], and [131. From this case study, where we successfully designed a combination of robust and adaptive control that completely satisfies the performance specifications, we venture to hypothesize that a successful adaptive controller must be based on, and smoothly converge to, a robust design. zyxwvutsrqpo zyxwvutsrqp 618 zyxw zyx zyxwvutsrqpon IEEE TRANSACTIONS ON AUTOMATIC CONTROL, - 360. VOL. 3i3, NO. 7, JULY I1988 zyxwv zyxwvut zyxwvutsrqponm zyxwvutsrqp zyxwvutsrq -270’ 0 -30. Fig. 14. The template for the plant P ( s ) = k/(l + Ts)’,k E [ I , 41, T E [0.5, 21, for w = 4 rad/s (s = j w ) , in the Nichols chart (dB versus degrees). The nominal plant case Pno,,,(j.4), marked with 0, corresponds to k = 1, T = 2. The disturbance rejection locus I1/(1 G(s)P(s))l= 6 dB is heavily marked. + disturbance rejections specification (A.2) is active. Hence, if the compensated nominal open loop is outside the high-frequency bound, the template will not intersect the disturbance rejection set; cf. Figs. 14, 5, and 6. 4) Feedback Compensator: In Step 3 the design problem for the set of transfer functions P ( s ) was transformed into a design problem for the nominal plant transfer function Pno,(s) by the computation of the constraining Horowitz bounds for the compensated nominal open loop. So, e.g., by classical methods, find a stable, minimum-phase compensation network G ( s ) such that G(s)P,,,(s) satisfies the Horowitz bounds. Independently, you must ascertain that the generalized Nyquist stability criterion [161 is satisfied for each plant case. The attempt to design G ( s )will fail if the closed-loop tolerance specifications and the extent of plant uncertainty are incompatible. Conditions for the design method to work are found in [2], [4], and [lo]. In particular, the plant phase uncertainty must not be more than (360 - 2 ~ ) for ” frequencies around the crossover frequency (the phase margin cp and the crossover frequency are implicitly given by the closed-loop specifications and the plant uncertainty). For instance, a couple of varying resonances and antiresonances will violate the phase uncertainty condition. The Horowitz design method gives a clear picture of the tradeoff between closed-loop specifications and plant uncertainty. If you fail to design G(s), you have either to loosen the specifications, or achieve greater knowledge about the appropriate plant parameters. In the present case study, it was impossible to find a G(s)that simultaneously satisfied the closed-loop specifications, and preserved the necessary gain margin. Since the specifications could not be changed, we solved the gain problem by the adaptive gain controller. A particular case is when the plant uncertainty is less than the required tolerance. If, in addition, the plant is stable and disturbances are negligible, an open-loop control could be considered. Then go directly to Step 5 . 5) Prefilter: After the design of the feedback compensator G ( s ) , the closed loop G,(s) = G ( s ) P ( s ) / ( l + G(s)P(s)) satisfies the tolerance specifications. However, the gain I G,(s)l might not coincide with the specifications (A.2). Therefore, a prefilter might be needed. By classical filter design methods, find a stable, minimum-phase prefilter F(s)that satisfies (A.2). This can always be done, since only the gain of T(s)is specified; see [15] and [4]. Notice that the precisely known F(s) does not contribute to the closed-loop variations. This concludes the design procedure. An interactive, graphic computer program such as HORPAC [9] highly facilitates the design process. The closed-loop system should be simulated to check that the responses are acceptable. Design examples are found in, e.g., [2] and [4]. zyxwvutsrqp zyxwvutsrqponm -TT’ - ACKNOWLEDGMENT The authors would like to mention the excellent conditions at El-Op Electro-Optics Industries that are conducive to a rare combination of practical control design work and research. The first author is grateful to the Cederbaum family at whose home the original manuscript was prepared. Thanks go to A. Berg and G. Abraham of the Technion who expertly typed the first version of the manuscript, and to E. Tal of El-Op who beautifully edited the figures. Finally, the authors appreciate the valuable suggestions made by the reviewers. REFERENCES K. J. Astrom, “Theory and applications of adaptive control-A survey,” Automatica, vol. 19, pp. 471-487, 1983. I. M. Horowitz and M. Sidi, “Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances,” Int. J. Contr., vol. 16, no. 2, pp. 287-309, 1972. J. C. Doyle and G . Stein, “Multivariable feedback design: Concept for a classical/modern synthesis.” IEEE Trans. Automat. Contr., vol. AC-26, pp. 4-16, 1981 zyxwvutsrqponmlkjihg zyxwvutsrqponmlk zyxwvutsrqpo zyxwvutsrqponm zyxwvutsrqpo GUTMAN et al.. ROBUST AND ADAPTIVE CONTROL OF A BEAM DEFLECTOR 141 K. J. Astrom, L. Neumann, and P. 0. Gutman, “A comparison between robust and adaptive control of uncertain systems,’’ in Proc. 2nd IFAC Workshop on Adaptive Syst. Contr. and Signal Processing, Lund, Sweden, July 1-3, 1986, pp. 37-42. 151 C. Mannerfelt, “Robust design with simplified models,” Ph.D. dissertation, Dep. Automat. Contr., Lund Inst. Technol., Lund, Sweden, CODEN: LUTFD2/(TFRT-l021)/1-153/, 1981. H. Elmqvist, “Simnon: User’s manual,” Dep. Automat. Contr., Lund Inst. Technol., Lund, Sweden, Rep. 7502, 1975. L. R. Rabiner and B. Gold., Theory and Applications of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. P. 0. Gutman and L. Neumann, “HORPAC-An interactive program package for robust control systems design,” in Proc. 2nd IEEE Contr. Syst. Society Symp. Comput. Aided Control Syst. Design, Santa Barbara, CA, Mar. 13-15, 1985. I. M. Horowitz, “A synthesis theory for linear time varying feedback systems with plant uncertainty,” IEEE Trans. Automat. Contr., vol. AC-20, pp. 454-464, 1975. A. J . Laub, A. Linnemann, and M. Wette, “Algorithms and software for pole assignment by state feedback,” in Proc. 2nd IEEE Contr. Syst. Society Symp. Cornput. Aided Contr. Syst. Design, Santa Barbara, CA, Mar. 13-15, 1985. 0. Yaniv, P. 0. Gutman, and L. Neumann, “An algorithm for adaptation of a robust controller to reduced plant uncertainty,” in Proc. 2nd IFAC Workshop on Adaptive Syst. Contr. and Signal Processing, Lund, Sweden, July 1-3, 1986, pp. 37-42. G. Kreisselmeyer and B. D. 0. Anderson, “Robust model reference adaptive control,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 127-133, 1986. M. Sidi, “Feedback synthesis with plant ignorance, nonminimumphase, and time-domain tolerances,” Automatica, vol. 12, pp. 265- 619 Hanoch Levin was born in Riga, Latvia, on September 15, 1956 He graduated in 1982 from the Tel Aviv University Technical College, Tel Aviv, Israel, as a Practical Engineer, majoring in electronics. Since 1984 he has been with the Control Systems Group, El-Op Electro-Optics Industries, Rehovot, Israel, specializing in the electronic design of servo systems His main field of interest is multiaxes stabilized systems zyxwvutsr zyxwvutsrqp zyxwvutsrqponm 271 1976. _ . _ ,.~ 1151 I. M. Horowitz, Synthesis of Feedback Systems. New York: Academic, 1963. [I61 J . J . D’Azzo and C. H . Houpis, Linear Control System Analysis and Desian. Conventional and Modern, 2nd edition. New York: McGiaw-Hill, 1981. [I71 E. Mishkin and L. Braun, Eds., Adaptive Control Systems. New York: McGraw-Hill, 1961. ~. Linda Neumann was born in Rochester, NY, in 1948 She received the B.A. degree in mathematics from Mt. Holyoke College, South Hadley, MA, in 1969, and the Ph D degree in noncommutative ring theory from the Weizmann Institute of Science, Rehovot, Israel, in 1977 From 1978 to 1984 she worked with Prof Isaac Horowitz at the Weizmann Institute, doing research in robust control design. Since 1984 she has been doing control design and analysis at the Control Systems Group, El-Op Electro-Optics Industries, Rehovot, Israel She was appointed Head of the Group in 1986. Her research interests include computer-aided design of robust control systems and adaptive robust control. Per-Olof Gutman (S’79-M’82) was born in Hoganas, Sweden, on May 21, 1949. He received the Civ.-Ing degree in engineering physics in 1973, the Ph.D degree in automatic control in 1982, and the title of docent in automatic control in 1988, all from the Lund Institute of Technology, Lund, Sweden He also studied at the University of California, Los Angeles, as a Fulbright grant recipient, and received the M.S.E degree in 1977. From 1973 to 1975 he taught mathematics in Tanzania. From 1983 to 1984 he held a post-doctoral position in the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel. Since 1984 he has been with the Control Systems Group, El-Op Electro-Optics Industries, Rehovot, Israel, where he is currently a Senior Expert. His research interests include target traclung, robust and adaptive control, control of nonlinear systems, and computer-aided design. Tuvia Sprecher was born in Holland in 1946. He received the B.Sc. degree in 1968 and the M.Sc. degree in 1971, both in electrical engineering, from Delft University of Technology, Delft, Holland, specializing in control systems As a Senior Lecturer, he taught electronics and control at the Jerusalem College of Technology, Israel, from 1973 to 1982. He was with EL-DE, Jerusalem, from 1978 to 1982, and Tadiran Electronics Systems, Holon, Israel, from 1982 to 1983. Since 1983 he has been with El-Op Electro-Optics Industries, Rehovot, Israel, where he set up, and until 1986, headed, an active Control Systems Group, while developing line-of-sight stabilization technology. He is currently engaged in project management, and is interested in applying classical control techniques to the management of large projects. Eli Venezia was born in Tel Aviv, Israel, on November 15, 1961. Before joining the Control Systems Group, El-Op Electro-Optics Industries, Rehovot, Israel, in 1984, he was employed at an electronics laboratory of the Israeli Air Force. His function at El-Op is the control system designer’s, specializing in servo design and systems engineering. He is currently persuing the B.Sc. degree in electronics at the Tel Aviv University Technological Institute, Holon, Israel.