Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations

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Terence Parrish
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1 ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses of system cn be ltered o tedy-stte error cn be reduced o Response of system to disturbnces (disturbnce response) cn be lowered he effects of control system on the overll response of dynmic system cn be positive or negtive. It is the responsibility of the nlyst to design the control system so it hs beneficil effects on system performnce. Prmetric ensitivity: Concept From mthemticl perspective, dynmic system will hve identicl responses to repeted pplictions of the sme input. he step response of system s clculted by MALAB, for exmple, is lwys the sme (unless the trnsfer function is ltered). For rel systems, however, this is not the cse. Ech time rel system is subjected to n input, its response will vry. he vritions my be smll rndom fluctutions producing the sme verge response, or they my be lrge fluctutions producing very different responses. For exmple, the effects of friction nd dmping cn esily vry during the dyto-dy opertion of system. o compre the effects of vritions on the response of open-loop nd closed-loop systems, consider the systems shown in Fig. 1. In ech system, the trnsfer function s () hs been chnged to ( s) ( s). Here, s () represents smll chnges to s. () Fig. 1. Open-Loop nd Closed-Loop ystems with Plnt Vritions mmn ME 3600 pge: 1/8

2 he lgebric eqution ssocited with the open-loop system is Y( s) Y ( s) ( s) ( s) R( s) ( s) R( s) ( s) R( s) o, chnges in the output cn be written s Y ( s) ( s) R( s) (1) he plnt chnges re clerly pssed directly to the system output. he lgebric eqution ssocited with the closed-loop system is ( s) ( s) Y ( s) Y ( s) R( s) 1 ( s) ( s) H ( s) ( s) ( s) R( s) R( s) 1 ( s) ( s) H ( s) 1 ( s) ( s) H( s) ( s) ( s) R( s) R( s) 1 H ( s) 1 H ( s) s () Y ( s) R( s) 1 H ( s) o, chnges in the output cn be pproximted s s () Y ( s) R( s) 1 H ( s) () From this result it cler tht the mount of s () tht is pssed to the output depends on the mgnitude of the loop (or open-loop) trnsfer function H () s. he lrger the mgnitude of H () s, the less chnges in s () will ffect the system response. ensitivity: Clcultion he clcultion of sensitivity is done more formlly using derivtives. pecificlly, the sensitivity of system with trnsfer function () s to chnges in prmeter is defined s follows. enerlly, the sensitivity is function of s, nd hence is function of frequency. (3) mmn ME 3600 pge: /8

3 If the mgnitude of is between zero nd one 0 1, then the effects of s () will be lowered (i.e. 10% chnges in s () will result in less thn 10% chnges in the response). However, if the mgnitude of is greter thn one 1, then the effects of s () will be mgnified (i.e. 10% chnges in s () will result in greter thn 10% chnges in the response). o gin some generl insight into the issue of sensitivity for simple closed-loop system (Fig. ), consider the sensitivity of the system trnsfer function s () to bulk 1 H chnges in the trnsfer functions s () or H() s. nd H H H Using these definitions nd the quotient rule for differentition gives Fig.. imple Closed Loop ystem H 1H 1 H H 11 H 1 1 H 1 H 1 H H 1H H H H 1 H 1 H 1 H (4) (5) Eq. (4) indictes tht s H () s is incresed, the effects on the response of the system to chnges in s () re lowered. his is the sme conclusion tht ws drwn from Eq. (). However, Eq. (5) indictes tht s H () s is incresed, the effects on the response of the system to chnges in H() s re pssed directly to the output, tht is, 1. Note lso tht for n open-loop system with trnsfer function ( s) ( s), H 1 mmn ME 3600 pge: 3/8

4 his gin indictes tht chnges in the plnt will be pssed directly to the output. Although Eqs. (4) nd (5) provide generl insights into the usefulness of closed-loop control, Eq. (3) is used to determine the sensitivity to specific system prmeters. For this reson, Eq. (3) provides more detiled informtion bout the system t hnd. For exmple, in previous notes, the closed-loop trnsfer function for proportionl position control of spring-mss-dmper system ws found to be s ms bs ( k ) he sensitivity of this system to chnges in the dmping nd spring stiffness prmeters cn be clculted s follows. b k b ms bs ( k ) b b b ms bs ( k ) b ms bs ( k ) ms bs ( k ) b bs ( ) ms bs k ms bs ( k ) s ms bs ( k ) k ms bs ( k ) k k k ms bs ( k ) k ms bs ( k ) k k ( ) ms bs k ms bs ( k ) Control of rnsient Response In previous notes the block digrm for the open-loop response of n rmturecontrolled DC motor ws given. If the electricl response of the motor is much fster thn the mechnicl speed chnges, then the time dependence of the circuitry cn be ignored. Under these conditions, the block digrm reduces to tht shown in Fig 3. mmn ME 3600 pge: 4/8

5 Fig. 3. Block Digrm of n Armture-Controlled DC Motor As before, the input to the motor is the rmture voltge V () s nd the output is the ngulr velocity (speed) of the motor. Using block digrm reduction, the open-loop trnsfer function for this system is found to be () s m R J V () s s b m R c R J (6) he prmeters nd re constnts tht depend on the motor chrcteristics, inertil lod, nd dmping coefficient. Note tht the vlue of responds when step increse in voltge is pplied to the motor. determines how quickly the motor o study the effects of feedbck on trnsient response, consider proportionl, closedloop control of the DC motor s shown in Fig 4. he input to the system is the desired ngulr velocity () s nd the output of the system is the ctul ngulr velocity () s. he prmeter d t is the clibrtion constnt of the tchometer tht reltes chnges in ngulr velocity to chnges in voltge. he signl Es () represents tchometer voltge error, nd the prmeter is the proportionl gin. he trnsfer function of the closed-loop system is found using block digrm reduction to be Fig. 4. Proportionl Control of DC Motor Using chometer mmn ME 3600 pge: 5/8

ˆ ˆ t s ˆ d s ˆ t (7) he prmeter â determines the speed of response of the closed-loop, speed control system. he vlue of â cn be incresed by incresing the proportionl gin however, tht if the vlue of.

6 ˆ ˆ t s ˆ d s ˆ t (7) he prmeter â determines the speed of response of the closed-loop, speed control system. he vlue of â cn be incresed by incresing the proportionl gin however, tht if the vlue of. Note, is incresed too much, the voltge input to the motor my become too lrge, potentilly dmging the motor. Physicl limittions such s these re often not prt of the mthemticl model, so the nlyst must be wre of them. Control of tedy-tte Error Consider gin the closed-loop speed control system of Fig. 4. o trck the error in the system s it responds to commnded speed chnge, the error signl Es () is tken to be the output of the system s shown in Fig. 5. Fig. 5. Error of DC Motor peed Control ystem Using block digrm reduction, the system error trnsfer function is found to be E s s t( ) t( ) s ˆ d s t s (8) Using the finl vlue theorem, the stedy-stte error to unit step, speed chnge commnd is found to be e ss E lim 1 s s0 s d t ˆ (stedy-stte error) (9) mmn ME 3600 pge: 6/8

shown in Fig. 6. It is ssumed tht the disturbnce torque reduces the torque D generted by the motor under idel conditions. Fig. 6. Block Digrm of n Armture-Controlled DC Motor with Disturbnce o study the effect of the disturbnce on the response of this system, the disturbnce trnsfer function must be found.

7 his result shows tht the proportionl control gin ffects the stedy-stte error vlue of decresed.. As the is incresed, the vlue of â is incresed nd e ss the stedy-stte error is Control of Disturbnce Response Consider the block digrm of n rmture-controlled DC motor with disturbnce torque () s s shown in Fig. 6. It is ssumed tht the disturbnce torque reduces the torque D generted by the motor under idel conditions. Fig. 6. Block Digrm of n Armture-Controlled DC Motor with Disturbnce o study the effect of the disturbnce on the response of this system, the disturbnce trnsfer function must be found. One wy to do this is to move the disturbnce to the leftmost summing block s shown in Fig. 7. Fig. 7. Block Digrm of DC Motor with Disturbnce Input Only he disturbnce trnsfer function for the open-loop system is then identified to be R s s D m (open loop system) (10) mmn ME 3600 pge: 7/8

8 Following this sme pproch, the disturbnce trnsfer function of the closed-loop, speed control system of Fig. 4 cn be found. In tht cse, to move the disturbnce to the leftmost summing block, the disturbnce must be dditionlly moved over the proportionl gin block s shown in Fig. 8. Fig. 8. Disturbnce Input in peed Control ystem of DC Motor Using block digrm reduction, the disturbnce trnsfer function for the closed-loop, speed control system is found to be R ˆ m 1 J s s ˆ s ˆ D (closed-loop, speed control system) (11) he stedy-stte ngulr velocity chnge of the motor to unit step disturbnce torque is found using the finl vlue theorem. ss lim D s0 1 1 s s ˆ D J (closed-loop, speed control system) (1) hese lst two results indicte tht s the proportionl gin is incresed, the disturbnce response decys fster nd the stedy-stte ngulr velocity chnge is decresed. Both re positive effects. mmn ME 3600 pge: 8/8

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response