Dingwen Yu and Jiasheng Zhang

Transcription

1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SIENES Volumer, Numver 3, ages Institute for Scientific omputing an Information RESEARH OF ID ONTROLLER AND FUZZY ID ONTROLLER Dingwen Yu an Jiasheng Zhang Abstract. This chapter covers the some approaches to ID esign an its application. There are Z - N oscillation metho an reaction curve base methos in classical approaches. There are the synthesis of I an ID controllers via pole - assignment techniques. In fuzzy ID auto-tuner conventional gain scheuling involves using extra information from the plant, environment, or user to tune (via scheules ) the gains of a controller. These control esigns provie a set of gains for the controller at each operating conition. We first stuy how the two-link flexible robot operates uner the no control situation. The robot enpoint position shows a significant amount of enpoint oscillation. We can overcome a large overshoot an oscillations near its set-point of open-loop control to use fuzzy control system with a supervisory level (ID controller). The experiment shows that the fuzzy control system with supervisor level (ID controller) gave better results in the case of large, counterrelative, an loae-tip slews. ey Wors. ID controller,fuzzy rules,feeback control system, esign Methos.. Introuction We review a particular control structure that become almost universally use in inustrial control. It is base on a particular fixe-structure controller family, the so-calle ID controller family. Over 9% of the controllers in operation toay are ID controllers. This is because ID controllers are easy to unerstan, easy to explain to others, an easy to implement. Moreover, they are often available at little extra cost since they are often incorporate into the programnabe logic controllers (Ls) that are use to control many inustrial processes. onsier the simple control loop system shown in Figure. Figure. Basic feeback control loop The traitional expression for I an ID controllers can be escribe by their transfer function. Suppose for the sake of iscussion that a ID controller is place in a unity feeback control system where the input to the plant is U (s), the plant output is Y (s), the reference input is R(s), an the error input to the ID controller is Receive April, 5; revise January 5, 6 384

2 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 385 E( s) R( s) Y ( s). The basic form for the ID controller is ( ) U ( t) p E( t) + i E( τ ) τ + t E( t) t where is the proportional gain, is the integral gain, an is the p ervative gain. i roportional action provies a contribution which epens on the instantaneous value of the control error. A proportional controller can control any stable plant, but it provies limite performance an nonzero steay-state errors. This latter limitation is ue to the fact that its frequency response is boune for all frequencies. Integral action, on the other han, gives a controller output that is proportional to the accumulate error, which implies that it is a slow reaction control moe. This characteristic is also evient in its low-pass frequency response. The integral moe plays a funamental role on achieving perfect plant inversion at ω. This forces the steay-state error to zero in the presence of a step reference an isturbance. Derivative action acts on rate of change of the control error. onsequently, it is a fast moe which ultimately isappears in the presence of constant errors. It is sometimes referre to as a preictive moe, because of its epenence on the error tren. The main limitation of the erivative moe, viewe in isolation, is its tenency to yiel large control signals in response to high-frequency control errors, such as errors inuce by set-point changes or measurement noise.. lassical ID control esign Methos One of the traitional ways to esign a ID controller was to empirical tuning rules base on measurements mae on the real plant. The following sections review the best-known of the classical tuning methos.. Ziegler-Nichols (Z-N) Oscillation Metho This proceure is vali only for open-loop stable plants, an it is carrie out by means of the following steps. Set the true plant uner proportional control, with a very small gain. Increase the gain until the loop starts oscillating. Note that linear oscillation is require an that it shoul be etecte at the controller output. Recor the controller critical gain controller output,. an the oscillation perio of Ajust the controller parameters accoring to Table ; there is some controversy regaring the ID parameterization for which the Z-N metho was evelope, but the version escribe here is applicable of best knowlege.

3 386 DINGWEN YU AND JIASHENG ZHANG Table. Ziegler-Nichols tuning, using the oscillation metho.5 i I / ID.6. 3 /.75 Example. onsier a plant with a moel given by () G ( s) ( s + ) 3 Fin the parameters of a ID controller by using the Z-N oscillation metho. Obtain a graph of the response to a unit step input reference an to a unit step input isturbance. We first compute the critical gain an critical frequency ω. These values must satisfy ( 3) G j j + 3 ( ω ) ( ω ) From this equation we obtain 8 an ω 3 c. Hence, the critical perio is If we use the settings in Tab le,an ID controller equation (4) ID ( s) T s ( + + ) T s τ s + r D We obtain the following: ( 5).6* 4.8; Tr.5*.8; T.5*.45 Where ; T Tr. The erivative moe will be attenuate i with time constant τ.. 45 D T. Thus, the final loop-transfer function becomes

4 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 387 (6) G ( s) ( T τ D + τ D ) s + ( + ) + Tr Tr 3 s( τ s + )( s + ) D 5.8s + 9.3s s( s +.)( s + ) With SIMULIN, the loop was simulate with a unit step reference at t an a unit step input isturbance at t. The results are shown in Figure. Figure. Response to step reference an step input isturbance The response shows in Figure exhibits significant overshoot that might be unacceptable in some applications; however, Z-N tuning provies a starting point for finer tuning. Of course, if significant measurement noise is present, then one shoul check whether increasing the erivative action still yiels goo performance. Similar caution applies regaring saturation in the controller output. The heuristic nature of these arguments highlights the limitations of tuning a ID controller in terms of its classical, I, an D tuning parameters.. Reaction curve base methos Many plants, particularly the ones arising in the process inustries can be satisfactorily escribe by the moel in sτ o Oe ( 7) G ( s) where ν o > ν s + o A linearize quantitative version of this moel can be obtaine with an open-loop experiment, by using the following proceure. With the plant in open loop, take the plant manually to a normal operating point. Say that the plant output settles at y ( t) for a constant plant input u ( t). u o At an initial time t, apply a step change to the plant input, from u to u (this shoul be in the range of to % of full scale). o Recor the plant output until it settles to the new operating. Assume that you obtain the curve shown in Figure 3. This curve is known as the process reaction curve. In Figure 3, m.s.t. stans for maximum slope tangent. o y o

5 388 DINGWEN YU AND JIASHENG ZHANG ompute the parameter moel follows: Figure 3. lant step response y y ( 8) ν t t ; τ t t; u u The moel obtaine can be use to erive various tuning methos for ID controllers. One of these methos was propose by Ziegler an Nichols. On their proposal, the esign objective is to achieve a particular amping in the loop response to a step reference. More specifically, the aim is to obtain a ratio of 4: for the first an secon peaks in that response. The suggeste parameters are shown in Table. Table. Ziegler-Nichols tuning by using the reaction curve T T r ν τ I.9ν τ 3τ ID.ν τ τ.5τ The parameter-setting rules propose in Table are applie to the moel (7), where we have again normalize time in elay units. Loop responses for a unit step reference shown in Figure 4 for ifferent values of the ratio τ ν x.

6 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 389 Figure 4. I Z-N tune (reaction curve metho) control loop 3 I an ID synthesis revisite by using pole assignment We place these results in a more moern setting by iscussing the synthesis of I an ID controllers via pole assignment techniques. Throughout this section, we will consier one-.o.f. control loops with I controllers of the form (9) ( s) p + I s an proper ID controllers of the form () ID ( s) p + s i i s + τ s + For future reference, we note the following alternative representation of a ID controller. Lemma. Any controller of the form D () ( s) n s + n s + n s + s () (3) (4) i n n n n n 3 + n ( 5) τ D This proof is ignore.

7 39 DINGWEN YU AND JIASHENG ZHANG Example. A plant has a nominal moel given by (6) G ( s) B A ( s) (s) ( s + ) s + ) Synthesize a ID controller that yiel a close loop ynamics ominate by the factor s + 4 s + 9. Solution. The controller is synthesize by solving the pole assignment equation, with the following a esire close-loop polynomial an quantities ( 7) A cl ( s + 4S + 9)( s + 4) (8) B ( s) ; A s + 3s + Where the factor ( s + 4) has been ae to ensure that the pole-assignment equation has a solution. Note that this factor generates moes that are faster than those originate in factor s + 4s + 9. Solving the pole-assignment equation gives (9) ( s) ( s) L( s) 4s + 59s + 7 s( s + 9) We observe, using Lemma, that (s) is a ID controller with () τ D 5.67; 8;.93;. i An important observation is that the solution to this problem has the structure of ID controller for the given moel G. For a higher-orer, the resulting ( s) G ( s) controller will not, in general, be a ID controller 4 Application of fuzzy ID controllers The two-link flexible robot shown in Figure 5 consists of three main parts () the robot with its sensors, () the computer an the interface to the robot, an (3) the camera with its computer an interface. The robot is mae up of two very flexible links constraine to operate in the horizontal plane. The shouler link is a counterbalance aluminum strip that is riven by a D irect-rive motor with an input voltage v. The elbow link, which is mounte on the shouler link enpoint,is a smaller aluminum strip. The actuator for the elbow link is a geare D motor with an input voltage v. The sensors on the robot are two optical encoers for the motor shaft position Θ an Θ, an two accelerometers mounte on the link enpoints to measure the accelerations a an a.

8 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 39 A line scan camera is use to monitor the enpoint position of the robot for plotting; this ata is not use for feeback. The sampling perio use for all sensors an control upates is 5 millisecons (ms). For comparative purposes, we use the camera ata for robot movements that begin in some position an en in a fully extene position, to approximate equal movements in each joint. When responses are plotte, the final enpoint position is nominally inicate (on the plot) to reflect (approximately) the total movement in egree, of the shouler joint. Figure 5. Two-link flexible robot setup 4. Objectives an open-loop control We will first stuy how the robot operates uner the no control situation; that is, when no external igital control algorithm is applie for vibration compensation. To implement the no control case, we simply apply v ν.365 volts at t v secons an return an ν to zero voltages as soon as the links reach their set-points. Note that for this experiment we monitor the movement of the links but so not use this information as feeback for control. The results of the no control experiment are plotte in Figure 6, where the enpoint position shows a significant amount of enpoint oscillation. As is typical in mechanisms of this sort, inherent moal amping is present. It is well known that the effect of mass loaing slewing flexible beam is to reuce the moal frequencies an this is inee the case for this experiment. Inee, when a 3-gram payloa is attache to the robot enpoint, the first moel frequency of the secon link (enpoint) reuces significantly. This effect causes performance egraation in fixe, linear controller. In Figure 6, as in all plots to follow, enpoint position refers to the position of the elbow link enpoint. Note that the inset shown in Figure 6 epicts the robot slew employe. The two ashe lines escribe the initial position of the links. The arrows show the irection of movement, an

9 39 DINGWEN YU AND JIASHENG ZHANG the soli line shows the final position of the links. Hence, for this open-loop experiment, we wante 9 egrees of movement in each link. In the ieal case the shaft shoul stop moving the instant the voltage signal to the motor amplifier is Figure 6. Enpoint position No control response cut off. But the arm ha been moving at a constant velocity before the signal was cut off, an thus ha a momentum that ragge the shaft past the angle at which it was to stop. This movement epens on the spee at which the arm was moving, which in turn epens on the voltage signal applie. learly, there is a significant nee for vibration amping in enpoint positioning. Quantitatively speaking, in terms of step-type responses (for motions through large angles in each joint), the control objectives are as follows: system settling (elimination of resiual vibrations in enpoint position) within secons of motion initiation, an overshoot minimize to less than 5% eviation from the final esire position. In aition, we wish to achieve certain qualitative aspects such as eliminating jerky movements an having smooth transitions between commane motions. 4. Fuzzy control system with a supervisory level (ID controller) We can overcome a large overshoot an oscillations near its set-point of open-loop control to use fuzzy control system with a supervisory level (ID controller). The simple fuzzy ID controller is shown in Figure 7,which monitors the position error e ( t ) to the shouler motor controller an changes the fuzzy sets an the rules-base of the shouler motor controller. To eliminate the hump, the ID controller is use to vary the spee of the shouler link graually so as to avoi exciting oscillatory moes, which result in excessive enpoint vibrations. The rule-base for the fuzzy ID controller consists of tow single-input e ( t) multiple-output rules:

10 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 393 ) If then use Rule-Base (Table 3) an use expane universes of iscourse. ) If Then use Rule-Base (Table 4) an use compresse universes of iscourse. e ( t) The ID controller expans or compresses the universes of iscourse by simply changing the scaling gains (explaine below). When the universe of iscourse is expane, a coarse control is achieve, an when it is compresse, a fine control is achieves. It is a type of auto - tuning strategy. The form of the premises of the rules of Figure 7. Fuzzy control system with a supervisory level (ID controller) the supervisor guarantees that one (an only one) of the rules will be enable an use at each time step. Since the control objectives can be achieve using only these two rules an since only one rule will be enable at each time step, there is no nee for the use of complex inference strategies in the fuzzy ID controller. Table 3. Rule-Base for coarse control learly, it woul be possible to view the premises of the above rules as linguistic statements then efine membership functions to quantify their meaning, We coul also use membership function to quantify the meaning of the consequents. Then, our supervisor woul be a fuzzy system that woul graually rather than abruptly switch

11 394 DINGWEN YU AND JIASHENG ZHANG between the two conitions. Here, we i not pursue this approach. Instea, we mae sure that the rule-bases were esigne so that a smooth transition woul occur as the supervisor switche between the rules. Table 4. Rule-Base for fine control Rule-base construction The membership function an the rules-base for the elbow link controller are shown in Figure 8. In aition to the rule-base in Table 3, another rule-base was ae for the shouler motor controller. The ID controller therefore switches between these two rule-bases for coarse control an fine control. The membership function for the coarse controller are the fuzzy controller case where the universe of iscourse is [-5, +5] egree for the position error, [-, +] g for the enpoint acceleration, an [-.5, +.5] volts for the out put voltage. The fine controller, with rule-base shown in Table 4, uses the universe of iscourse for the position error is [-5, +5] egrees, an the universe of iscourse is [-.5, +.5] volts. Notice that while going from the coarse to the fine control, the withs of the universes of iscourse for the position error an the output of the shouler link controller have been reuce by a factor of ten, while the with of the universe of iscourse for the enpoint acceleration is reuce by a factor of two. This choice was mae after several experiments where it was foun that when the with of the universe if iscourse for the acceleration was reuce by a large factor, the controller became too sensitive near the set-point. Table 3 an 4 show the rule-base use for the coarse an fine control, respectively. Notice that in row j for the rule-base for fine control there are extra zeros as before to reuce the sensitivity of the controller became to a noisy acceleration signal. The rule-base for coarse control oes not have these zeros as the offset voltage from the accelerometers is of no consequence as long as the controller is operating in this region. Also notice that while the patterns in the boies of the tables shown in Table 4 an 5 are similar, there are ifferences inclue to reflect the best way to control the robot. Notice that the center values in the fine control rule-base change more rapily as we move away from the center of the rule-base as compare to the coarse control rule-base. This causes a bigger change in the output of the controller for smaller changes in the input, resulting in better control over the shaft spee of the motor, preventing it from overshooting its set-point an at the same time causing graual braking of the motor spee. Finally, we note that the fine control rule-base was selecte so that the output is not change too much when the rule-bases are switche promoting a smooth transition between the rule-bases.

12 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 395 Figure 8. Membership functions for the elbow controller Experimental results Experimental results obtaine using this supervisory scheme are shown in Figure 9. The requeste slew is 9 for both links as shown in the inset. The response is relatively Figure 9. Enpoint position for supervisory control fast with very little overshoot. omparing this response to the response obtaine for the open-loop control (see Figure 6), we can see that the hump in the initial portion of the

13 396 DINGWEN YU AND JIASHENG ZHANG graph is almost eliminate. The response of the robot to a counterrelative slew is better than the response when using the open-loop control. From the results obtaine for the supervisory control (ID controller) technique, we see that the results are superior in the cases teste. The supervisor gave better results in the case of large, counterrelative, an loae-tip slews. Overall, we see that rule-base supervision can be use to significant benefit in a practical application. 5 conclusion There are many ID controller esign approaches an its applications. It has been empirically foun that the ID structure often has sufficient flexibility to yiel excellent result in many applications. From a moern perspective, a ID controller is simply a controller (of up-to-secon orer) containing an integrator. Historically, however, ID controllers were tune in terms of, I, an D terms. Various empirical tuning methos can be use to etermine the ID parameters for a given application. They shoul be consiere as a first guess in a search proceure. Except the ID controller esign approaches of the Ziegler-Nichols (Z-N) oscillation metho, the reaction curve base metho, the I an ID synthesis revisite by using pole assignment, the fuzzy tuning of ID controllers an their applications, there are also the esign approaches of the ID synthesis by using the affine parameterization, the aaptive ID control, intelligence ID control, artificial neural network ID control, preication ID control, an so on. Every ID controller esign approach has its characteristic. References [] evin M. assion an Stephen Yurkovich. Fuzzzy control. Tsinghua University ress, [] Graham. Goowin, Stefan F. Graebe an Mario E. Salgao, Tsinghua University ress, [3] Mellquist, M., risafulli, S., an MacLeo, I.M. (997), Derivative filtering issues for inustrial ID controllers. In roceeings of the I.E. Aust. ontrol 97 onference, Syney, pages [4] Voa, A. an Lanau, I.D. (995), A metho for the auto-calibration of ID controllers, Automatica, 3(): [5] Isaksson, A.J. an Graebe, S.F. (999), Moel reuction for ID esign. In proceeings of the th IFA worl congress, Syney, Australia, 5: [6] Rivera, D., Morari, M., an Skogesta, S. (986). Internal moel control. ID controller esign. In. Eng. hem. rocess Des. Dev., 5:5-65. [7] Isaksson, A.J. an Graebe, S.F. (999), Analytical ID parameter expressions for higher orer systems. Automatica. [8].J.Astrom an T. Hagglun (eitors), ID control: Theory, esign, an tuning. Instrument society of America press, Research Triangle ark, N, secon eition, 995. [9] S. hiu, an S. han., Fuzzy controller esign an stability analysis for an aircraft moel. In proc. Of the American ontrol onf., pages 8-86, Boston, June 99. [] J. T. Spooner an.m. assino, Stable irect aaptive control using fuzzy systems an neural networks. In proc. of the IEEE onf. on Decision an ontrol, pages 49-54, New Orleans, LA, December 995. [] H. R. van Nauta Lemke an D.- Z. Wang, Fuzzy ID supervisor, In roc. Of the IEEE onf. on Decision an ontrol, pages 6-68, Fort Lauerale, FL, December 985.

14 RESEARH OF ID ONTROLLERS AND ITS ALIATIONS 397 Dingwen Yu, Northeastern University at Qinhuangao, Qinghuangao, Hebei 664, hina. Jiasheng Zhang, Northeastern University at Qinhuangao Qinhuangao, Hebei 664, hina.