In this paper, the authors deal with one method for the control of sampled-data control systems. In contrast with ordinary sampled-data control systems, the polarity of the control signal is reversed at several instants in every sampling period. By deciding the instants properly, the finite settling time response can be obtained. Especially, for a step input, it is shown that the system error can be reduced to zero in one sampling period, irrespective of the order of the controlled element. Furthermore, the compensator is simpler than that for ordinary sampled-data control systems, because it consists only of a sampler, a hold element, and a reverse element which reverses the polarity of the control signal. Even for a controlled element with a symmetric saturation characteristic, the settling time has a finite value. Moreover, it is shown that the reverse element can be used along with an ordinary compensator containing delay elements, and that the settling time can be made shorter. The response for a random input is also analyzed for a typical sampled-data control system with a reverse element

HAYASHI, Shigenori

HOSHINO, Satoshi

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Sampled-data Control SystemDesign Using Reverse ElementHAYASHI, Shigenori; HOSHINO, SatoshiHAYASHI, Shigenori ...[et al]. Sampled-data Control System Design Using ReverseElement. Memoirs of the Faculty of Engineering, Kyoto University 1961, 23(4): 342-3511961-10-30http://hdl.handle.net/2433/280506342 Sampled-data Control System Design Using Reverse Element By Shigenori HAYASHI* and Satoshi HOSHINO** (Received July 31, 1961) In this paper, the authors deal with one method for the control of sampled-data control systems. In contrast with ordinary sampled-data control systems, the polarity of the control signal is reversed at several instants in every sampling period. By deciding the instants properly, the finite settling time response can be obtained. Especially, for a step input, it is shown that the system error can be reduced to zero in one sampling period, irrespective of the order of the cont-rolled element. Furthermore, the compensator is simpler than that for ordinary sampled-data control systems, because it consists only of a sampler, a hold element, and a reverse element which reverses the polarity of the control signal. Even for a controlled element with a symmetric saturation characteristic, the settling time has a finite value. Moreover, it is shown that the reverse element can be used along with an ordinary compensator containing delay elements, and that the settling time can be made shorter. The response for a random input is also analyzed for a typical sampled-data control system with a reverse element. I. Introduction Sampled-data control systems with a compensator containing delay elements have been treated by many authors, and the mathematical techniques and the theory developed for them are now well known. In this paper, the authors deal with sampled-data control systems of a different kind. In contrast with ordinary sampled-data systems, the polarity of the control signal is reversed at several instants in every sampling period. The instants and the number of reversals are determined by the sampling period and the characteristic of the controlled element. In the following discussion, a com-pensator which reverses the polarity of the control signal is called a reverse element. II. System Design for a Second Order Controlled Element (1) Design Procedure for a Step Input In this section, a design procedure is discussed for the control system shown in Fig. 1. The polarity of the control signal is reversed when t = Ti (measured * Department of Electrical Engineering, II. ** Department of Electronics. Sampled-data Control System Design Using Reverse Element 343 from the beginning of every sampling period) as shown in Fig. 2. ;.; -~ K 1----, "' Reverse Arrplifier Controlled element "IS ~ 01-------r---Fig. 1. Block diagram of a sampled-data control system with a second order controlled element. <S r, -K Fig. 2. Reversal of the control signal in a sampling period. The instant T1 is chosen so that the system error becomes identically zero after one sampling period. The system output c(t) for a step input is (1) where H(t) is a unit step function, and g(t) is the indicial response of the cont-rolled element, which is written as g(t) = t-T(l-e-tlT) According to the design condition mentioned above, we obtain c'(To) = 0 C (T0) = 1 } (2) ( 3) where To is the sampling period. Substitution of Eq. (1) into Eq. (3) yields 1,0,---~--~------ir---, ---I ~ le; 0 1 2 3 4 5 To/,- -(a) 10 5 0 1 ' ' ' .... ~.o ',..fl'J.'Vt -----t 'j 4 Tyr (b) Fig. 3. Conditions for finite setting time response. ----5 344 where Shigenori HAYASHI and Satoshi HOSHINO ram = 0.5(1 + e-a) } _J._ = 1-2m KTo m I-Ti/To a= To/T ( 4) ( 5) ( 6) The relations given by Eq. (4) are plotted in Figs. 3(a) and 3(b) with solid lines. Figure 4 gives the step response of the system output c(t) for a= 1. The polarity of the control signal is reversed when Ti/T0 =0.62, and KTo=4.2. 1 0.51-------------------t--~ 0 <ii .~ 4.2 "' r;, I ~ 0 0.5 0.5 ! 0.62 1.0 1.0 Fig. 4. Step response of the control system in Fig. 1. (a=l.0) (2) Design Procedure for a Ramp Input The discussion in the preceding section indicates that the simple control system shown in Fig. 1 has a finite settling time for a step input. But the system with a finite settling time for a ramp input should have a compensator containing delay elements. Consider the control system shown in Fig. 5, and let the time darivative of the system input be l/T0 • The design conditions corresponding to Eq. (3) are given by Sampled-data Control System Design Using Reverse Element r(t) + Amplifier Reverse Controlled element Fig. 5. Block diagram of a sampled-data control system which is designed for a ramp input. c'(To) = 1/To C (To)= 2 Substitution of Eq. (1) into Eq. (7) yields } e-am = os(1+e-a+l-2m) l · 2+1/a 1 _ 1-2m KTo - 2+1/a 345 ( 7) (8) The relations given by Eq. (8) are plotted in Figs. 3(a) and 3(b) with dotted lines. Figure 6 gives the step response of the system output c(t) for a=l. The polarity of the control signal is reversed when T1/T0 =0.73 and KT0 =6.6. Fig. 6. Velocity response of the control system shown in Fig. 5. (a=l.0). III. System Design for a Third Order Controlled Element Suppose that the transfer function of the controlled element G(s) is given as 346 Shigenori HA y ASHI and Satoshi HOSHINO <ii I( -~ 1 G(s) = s(l+ Ts)(l+ T's) ( 9) V) <> 0 -::: " T, ~ (3 To -I( Fig ... 7. Reversal of the control signal in a sampling period. In this case, the polarity of the control signal is reversed twice in every sampling period as shown in Fig. 7. Similar to the system design discussed in the previous sections for a second order controlled element, the following conditions can be obtained. e-amz-e-am, = 0.5(1-e-0 ) e-bmz-e-bm, = 0.5(1-e-b) } (10) (11) where a= TolT b TolT' m,=1-T,ITo m2 l-T2IT0 (12) (13) (14) (15) The instants T, and T2 are found from Eq. (10) by solving for m, and m2, and using Eqs. (14) and (15). Figure 8 indicates m, and m2 as a function of Toi T and T0I T'. This figure is composed of two regions separated by a straight ..._ ~11-.. ~ '-~r--2.0 3.0 .IP..(or To)_ T T' Fig. 8. Conditions for finite settling time response. m1 ..... ~ <., ,; ~ <S 05 0 05 1.0 •/ t/ To Fig. 9. Step response of a control system with a controlled element of third degree given by Eq. (9). (a=2.0, b=l.0). Sampled-data Control System Design Using Reverse Element 347 line passing through the origin. The value of m1 is shown in the right-hand region, and that of m2 is shown in the upper region. Figure 9 gives the step response of the system output c(t) for a=2.0 and b=l.0. For a ramp input, a similar method can also be applied to design a control system with a finite settling time. Now, it will be shown that the reverse element can be used along with an ordinary compensator containing delay elements. In this case the control element can be made simpler, and a shorter settling time can be obtained. Consider the control system shown in Fig. 10, including both a reverse element and a compensator D(z) containing delay elements. It is assumed that r(t) + Reverse J c(tJ S(l+TS)(I+r'SJ Controlled element Fig. 10. Block diagram of a sampled-data control system when a reverse element is used along with a compensator D(z) containing delay elements. the polarity of the control signal is reversed when t= Ti in every sampling period, and that the system error is reduced to zero for t "2. 2 To. First, the function </J(t) is defined by </J(t) - g(t)-2g(t-T1) H(t-T1) + g(t- T 0 ) H(t- T 0 ) (16) where g(t) is the indicial response of the controlled element. The system output c(t) is written as c(t) = K</J(t) +Kh</JCt-To) HCt-To) where K and h are unknown constants. Design conditions are given by c' (2T0) = 0 l c"(2T0 ) = 0 C (2T0 ) = 1 Substituting Eq. (17) into Eq. (18) yields </J' (2 T 0 ) </J" ( To) = </J" (2 T 0 ) </J' ( T 0 ) D(z) = Kl-</J'(2To)z-1/r/J'(To) l 1+ {l-</J(T0)}z-1 K1i,o = </J(2To)-</J(To)</J'(2T0)/r/J'(T0 ) (17) (18) (19) Figure 11 gives the step response of the system output c(t) for a=2.0 and b= 1.0, and shows that two solutions exist. 348 ~ -~ lo <> ~ ~ '-5 Shigenori HAYASHI and Satoshi HOSHINO C 2.0 1/, ,;, Fig. 11. Step response of the control system shown in Fig. 10. (a=2.0, b=l.0). For the solution shown with a solid line, T1 = 0.72 KTo = 3.65 D(z) = 3.65. 1-0.36&-1 To 1 + 0.432z-1 For the solution shown with a dotted line, T1 = 0.62 KTo = 4.80 D(z) = ~.80. 1-0.138z-1 To 1 + 0.325z-1 l l IV. Extension to a General Order Controlled Element Assume the controlled element G(s) given as K r,, T,J r,, (20) (21) (22) -I( The polarity of the control signal is re-versed as shown in Fig. 12. The design condi-tions are written for a step input as Fig: 12. Reversal of the control signal in a saJ:?pling period. where 9t:tmPled-data Control System Design Using Reverse Element a,== To/'<; m1 = 1-T./To (k = 1, 2, ... , n-1) (i = 1, 2, ··· , n-1) (i = 1, 2, · · · , n-1) Similar conditions can also be obtained for a ramp input. V. The Effect of Saturation of the Controlled Element 349 (23) (24) (25) It was indicated in previous sections that the system error can be reduced to zero in one sampling period for either a step or ramp input. For a controlled element with a symmetric saturation characteristic, it is clear that the settling 2.0 ½. Fig. 13. Step resp<;mse of a control system with the controlled element given by Eq. (26) when the response speed is limited with saturation. time becomes longer with an increase in the input level. But when the working point of the controlled element enters a linear region, the system error is reduced to zero in one sampling period. Figure 13 shows the step response of the system output c(t) for a controlled element G(s) given by 1 G(s) = s(s+ 1) (26) 350 Shigenori HAYASHI and Satoshi HOSHINO VI. Response for a Random Input In this section, the response for a random input is analysed for the control system shown in Fig. 1. The variance a; of the system error at sampling instants is given by'l 2 _ 1 ,( I 1 12 dz a.-2nj :Y 1l+G(z) (f)rrCz)z (27) unit circle where G(z) is the open loop pulse transfer function, and 0rrCz) is the auto-correlation series of the system input. The polarity of the control signal is reversed when t= T,. Thus G(z) is represented as (28) Using tables of the z-transform and the modified z-transform, we obtain G(z) as where and therefore, where m-1-T,/To a= To/T 1 _ (z-e- 0 )(z-1) l+G(z) - z2 +C(m)z+D(m) C(m) =KTA(m)-(l+e-0 ) D(m) = KTB(m) + e-a Assume that (/)rrCz) is given by (/) ( ) - (1- q2 ) z rr Z - ( ) (l ) z-q -qz Substituting Eqs. (33) and (35) into Eq. (27) yields } (29) (30) (31) (32) (33) (34) (35) (36) where Sampled-data Control System Design Using Reverse Element C1 = 1-e-a C2 = l+e-a do= (1-q){l+C(m) +D(m)} di= (3-q) + (l+q)C(m)-(l-3q)D(m) d2 = (3+q)-(1-q)C(m)-(1+3q)D(m) d3 = (l+q){l-C(m)+D(m)} 351 (37) Figure 14 shows the variance a; of the system error as a function of KTo and m when a= 1 and q=0.9. If the control system discussed above is controlled with Normalized gain KT,, Fig. 14. Variance of the system error as a function of KT0 and m. (a=l.O). an ordinary compensator containing delay elements, the optimum over-all pulse transfer function K(z) is given by 1 ) K(z) = qz-1 (38) Therefore, the variance of the system error is equal to (l-q2) ( = 0.19 when q=0.9), which is shown with a dotted line in Fig. 14. Inspection shows that the optimum values of KT0 and m are approximately 4 and 0.4 respectively which agree well with the results obtained in section II. Reference 1) S. S. L. Chang: Trans. AIEE, 76, pt. III. 702 (1957) (Jan. 1958 section). 

Sampled-data Control System Design Using Reverse Element

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Sampled-data Control System Design Using Reverse Element

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