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In the proportional control algorithm, the controller output is proportional to the error signal, which is the difference between the setpoint and the process variable. In other words, the output of a proportional controller is the multiplication product of the error signal and the proportional gain.
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In proportional control, the power output is always proportional to the (actual versus target speed) error. If the car is at target speed and the speed increases slightly due to a falling gradient, the power is reduced slightly, or in proportion to the change in error, so that the car reduces speed
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tells us that the spring applies a corrective force that is proportional to the object's displacement. While this will tend to hold the object in a particular location, the absolute resting location of the object will vary if its mass is changed. This difference in resting location is the offset
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gradually and reaches the new target point with very little, if any, "overshoot", which is much smoother control than on–off control. In practice, PID controllers are used for this and the large number of other control processes that require more responsive control than using proportional alone.
2378:, full power would again be applied. It can be seen that this would obviously result in poor control and large variations in speed. The more powerful the engine, the greater the instability; the heavier the car, the greater the stability. Stability may be expressed as correlating to the
171:. On–off control will work where the overall system has a relatively long response time, but can result in instability if the system being controlled has a rapid response time. Proportional control overcomes this by modulating the output to the controlling device, such as a
2334:, the proportional gain, is very high, the proportional band is very small, which means that the band of controller output over which the final control element will go from minimum to maximum (or vice versa) is very small. This is the case with on–off controllers, where
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adjustment, if possible) to eliminate offset error for expected conditions, when a disturbance (deviation from existing state or setpoint adjustment) occurs in the process, corrective control action, based purely on proportional control, will result in an offset error.
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A drawback of proportional control is that it cannot eliminate the residual SP − PV error in processes with compensation e.g. temperature control, as it requires an error to generate a proportional output. To overcome this the
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error. Over a range of operating conditions, proportional control alone is unable to eliminate offset error, as it requires an error to generate an output adjustment. While a proportional controller may be tuned (via
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The proportional band is the band of controller output over which the final control element (a control valve, for instance) will move from one extreme to another. Mathematically, it can be expressed as:
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The clear advantage of proportional over on–off control can be demonstrated by car speed control. An analogy to on–off control is driving a car by applying either full power or no power and varying the
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Consider an object suspended by a spring as a simple proportional control. The spring will attempt to maintain the object in a certain location despite disturbances that may temporarily displace it.
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2374:, to control speed. The power would be on until the target speed is reached, and then the power would be removed, so the car reduces speed. When the speed falls below the target, with a certain
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1797:{\displaystyle \lim _{t\to \infty }y(t)=\lim _{s\,\searrow \,0}\left(s\times {\frac {k_{CL}}{\tau _{CL}s+1}}\times {\frac {\Delta R}{s}}\right)=k_{CL}\times \Delta R=y(t)|_{t=\infty }}
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2204:{\displaystyle \lim _{t\to \infty }y(t)=\lim _{s\,\searrow \,0}\left(s\times {\frac {k_{c}}{s(s+1)+k_{c}}}\times {\frac {\Delta R}{s}}\right)=\Delta R=y(t)|_{t=\infty }}
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in which a correction is applied to the controlled variable, and the size of the correction is proportional to the difference between the desired value (
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is an early example of proportional control. The balls rise as speed increases, which closes the valve, reducing speed until a balance is achieved.
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term (I) to eliminate the residual offset error by integrating the error over time to produce an "I" component for the controller output.
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at a level which avoids instability, but applies correction as fast as practicable by applying the optimum quantity of proportional gain.
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meaning there is no offset in this system. This is the only process that will not have any offset when using a proportional controller.
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as a ratio with the span of the instrument. This span is in the same units as error (e.g. C degrees) so the ratio has no units.
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Flow control loop. If only used as a proportional controller, then there's always an offset between SP and PV.
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is very high and hence, for even a small error, the controller output is driven from one extreme to another.
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Constraints: In a real plant, actuators have physical limitations that can be expressed as constraints on
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1139:{\displaystyle g_{CL}={\frac {\frac {k_{p}k_{c}}{\tau _{p}s+1}}{1+{\frac {k_{p}k_{c}}{\tau _{p}s+1}}}}}
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is process transfer function; the input into the block is flow rate and output is tank level.
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was devised, which uses a proportional term (P) to remove the gross error, and an
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892:{\displaystyle {\mathit {g_{cl}}}={\frac {\mathit {g_{p}g_{c}}}{1+g_{p}g_{c}}},}
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Offset error is the difference between the desired value and the actual value,
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may be bounded between −1 and +1 if those are the maximum output limits.
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2411:. Upper Saddle River, New Jersey: Prentice Hall PTR. pp. 165–168.
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1871:, which, when combined with the closed-loop transfer function, becomes
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1009:. Combining this with the closed-loop transfer function above returns
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Introducing a step change to the system gives the output response of
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Introducing a step change to the system gives the output response of
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which shows that there will always be an offset in the system.
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Proportional control compared to on–off or bang–bang control
1375:{\displaystyle \tau _{CL}={\frac {\tau _{p}}{1+k_{p}k_{c}}}}
144:, PV). Two classic mechanical examples are the toilet bowl
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For an integrating process, a general transfer function is
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For a first-order process, a general transfer function is
129:, in engineering and process control, is a type of linear
1300:{\displaystyle k_{CL}={\frac {k_{p}k_{c}}{1+k_{p}k_{c}}}}
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The proportional control concept is more complex than an
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2010:{\displaystyle y(s)=g_{CL}\times {\frac {\Delta R}{s}}}
1593:{\displaystyle y(s)=g_{CL}\times {\frac {\Delta R}{s}}}
1213:{\displaystyle g_{CL}={\frac {k_{CL}}{\tau _{CL}s+1}}}
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1944:{\displaystyle g_{CL}={\frac {k_{c}}{s(s+1)+k_{c}}}}
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control system used in something like an automobile
1002:{\displaystyle g_{p}={\frac {k_{p}}{\tau _{p}s+1}}}
936:are stable, then the closed-loop system is stable.
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may be too technical for most readers to understand
267:{\displaystyle P_{\mathrm {out} }=K_{p}\,{e(t)+p0}}
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2408:Process Control: Modeling, Design, and Simulation
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752:, the setpoint, is the flowrate into a tank and
577:Qualifications: It is preferable to express
748:. From the block diagram shown, assume that
51:introducing citations to additional sources
797:The output as a function of the setpoint,
2297:{\displaystyle PB={\frac {100}{K_{p}}}\ }
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1864:{\displaystyle g_{p}={\frac {1}{s(s+1)}}}
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684:Learn how and when to remove this message
668:, without removing the technical details.
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1491:(standard practice is to make sure that
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199:This can be mathematically expressed as
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41:Relevant discussion may be found on the
1146:. Simplifying this equation results in
741:{\displaystyle {\mathit {g_{c}=k_{c}}}}
340:: Output of the proportional controller
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1809:
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400:: Instantaneous process error at time
165:proportional–integral–derivative (PID)
159:system such as a bi-metallic domestic
666:make it understandable to non-experts
637:Development of control block diagrams
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303:: Controller output with zero error.
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929:{\displaystyle {\mathit {g_{cl}}},}
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787:{\displaystyle {\mathit {g_{p}}},}
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567:{\displaystyle P_{\mathrm {out} }}
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532:{\displaystyle P_{\mathrm {out} }}
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333:{\displaystyle P_{\mathrm {out} }}
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1603:Using the final-value theorem,
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1382:. For stability in this system,
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34:relies largely or entirely on a
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2020:Using the final-value theorem,
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16:Linear feedback control system
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803:closed-loop transfer function
700:Simple feedback control loop2
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2405:Bequette, B. Wayne (2003).
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444:{\displaystyle e(t)=SP-PV}
1438:{\displaystyle \tau _{p}}
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146:float proportioning valve
2450:Classical control theory
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62:"Proportional control"
2380:power-to-weight ratio
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2354:{\displaystyle K_{p}}
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127:Proportional control
47:improve this article
2460:Control engineering
1810:Integrating process
940:First-order process
369:: Proportional gain
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500:: Process variable
493:{\displaystyle PV}
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296:{\displaystyle p0}
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2418:978-0-13-353640-9
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2161:
2157:
2151:
2147:
2144:
2138:
2130:
2126:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2098:
2094:
2088:
2085:
2081:
2075:
2071:
2067:
2063:
2059:
2056:
2053:
2050:
2047:
2042:
2039:
2036:
2032:
2004:
2000:
1997:
1991:
1986:
1983:
1979:
1975:
1972:
1969:
1966:
1963:
1935:
1931:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1903:
1899:
1893:
1888:
1885:
1881:
1857:
1854:
1851:
1848:
1845:
1842:
1838:
1833:
1828:
1824:
1811:
1808:
1791:
1788:
1785:
1780:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1749:
1746:
1742:
1738:
1734:
1728:
1724:
1721:
1715:
1709:
1706:
1703:
1698:
1695:
1691:
1684:
1681:
1677:
1671:
1668:
1664:
1658:
1654:
1650:
1646:
1642:
1639:
1636:
1633:
1630:
1625:
1622:
1619:
1615:
1587:
1583:
1580:
1574:
1569:
1566:
1562:
1558:
1555:
1552:
1549:
1546:
1523:
1520:
1515:
1511:
1505:
1501:
1480:
1477:
1474:
1469:
1465:
1459:
1455:
1432:
1428:
1407:
1404:
1399:
1396:
1392:
1366:
1362:
1356:
1352:
1348:
1345:
1339:
1335:
1329:
1324:
1321:
1317:
1291:
1287:
1281:
1277:
1273:
1270:
1263:
1259:
1253:
1249:
1242:
1237:
1234:
1230:
1206:
1203:
1200:
1195:
1192:
1188:
1181:
1178:
1174:
1168:
1163:
1160:
1156:
1129:
1126:
1123:
1118:
1114:
1106:
1102:
1096:
1092:
1085:
1082:
1076:
1073:
1070:
1065:
1061:
1053:
1049:
1043:
1039:
1031:
1026:
1023:
1019:
995:
992:
989:
984:
980:
973:
969:
963:
958:
954:
941:
938:
925:
918:
915:
911:
888:
880:
876:
870:
866:
862:
859:
852:
848:
842:
838:
831:
824:
821:
817:
783:
776:
772:
733:
729:
725:
720:
716:
692:
691:
653:
651:
644:
638:
635:
622:
619:
616:
613:
591:
587:
560:
557:
554:
549:
525:
522:
519:
514:
502:
501:
489:
486:
476:
464:
461:
451:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
389:
386:
383:
380:
370:
356:
352:
341:
326:
323:
320:
315:
304:
292:
289:
275:
274:
262:
259:
256:
253:
250:
247:
244:
237:
233:
229:
223:
220:
217:
212:
192:
189:
169:cruise control
157:on–off control
134:control system
110:
109:
45:. Please help
31:
29:
22:
15:
9:
6:
4:
3:
2:
2472:
2461:
2458:
2456:
2453:
2451:
2448:
2447:
2445:
2436:
2433:
2432:
2420:
2414:
2410:
2409:
2401:
2399:
2394:
2387:
2383:
2381:
2377:
2373:
2362:
2346:
2342:
2319:
2315:
2305:
2284:
2280:
2276:
2271:
2268:
2265:
2257:
2248:
2245:
2240:
2223:
2214:
2211:
2193:
2190:
2177:
2171:
2168:
2165:
2159:
2155:
2149:
2145:
2136:
2128:
2124:
2120:
2114:
2111:
2108:
2102:
2096:
2092:
2086:
2083:
2079:
2073:
2065:
2057:
2051:
2045:
2034:
2021:
2018:
2002:
1998:
1989:
1984:
1981:
1977:
1973:
1967:
1961:
1952:
1933:
1929:
1925:
1919:
1916:
1913:
1907:
1901:
1897:
1891:
1886:
1883:
1879:
1852:
1849:
1846:
1840:
1836:
1831:
1826:
1822:
1807:
1804:
1786:
1783:
1770:
1764:
1761:
1758:
1752:
1747:
1744:
1740:
1736:
1732:
1726:
1722:
1713:
1707:
1704:
1701:
1696:
1693:
1689:
1682:
1679:
1675:
1669:
1666:
1662:
1656:
1648:
1640:
1634:
1628:
1617:
1604:
1601:
1585:
1581:
1572:
1567:
1564:
1560:
1556:
1550:
1544:
1535:
1521:
1518:
1513:
1509:
1503:
1499:
1478:
1475:
1472:
1467:
1463:
1457:
1453:
1430:
1426:
1418:; therefore,
1405:
1402:
1397:
1394:
1390:
1364:
1360:
1354:
1350:
1346:
1343:
1337:
1333:
1327:
1322:
1319:
1315:
1289:
1285:
1279:
1275:
1271:
1268:
1261:
1257:
1251:
1247:
1240:
1235:
1232:
1228:
1204:
1201:
1198:
1193:
1190:
1186:
1179:
1176:
1172:
1166:
1161:
1158:
1154:
1127:
1124:
1121:
1116:
1112:
1104:
1100:
1094:
1090:
1083:
1080:
1074:
1071:
1068:
1063:
1059:
1051:
1047:
1041:
1037:
1029:
1024:
1021:
1017:
993:
990:
987:
982:
978:
971:
967:
961:
956:
952:
937:
923:
886:
878:
874:
868:
864:
860:
857:
829:
804:
800:
795:
781:
759:
755:
751:
698:
688:
685:
677:
667:
663:
657:
654:This section
652:
643:
642:
634:
617:
611:
589:
585:
575:
547:
512:
487:
484:
477:
462:
459:
452:
438:
435:
432:
429:
426:
423:
417:
411:
403:
384:
378:
371:
354:
350:
342:
313:
305:
290:
287:
280:
279:
278:
260:
257:
254:
248:
242:
235:
231:
227:
210:
202:
201:
200:
197:
188:
186:
182:
181:PI controller
176:
174:
173:control valve
170:
166:
162:
158:
153:
151:
147:
143:
139:
135:
132:
128:
121:
116:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
2407:
2384:
2368:
2306:
2258:
2254:
2241:
2228:
2217:Offset error
2212:
2022:
2019:
1953:
1813:
1805:
1605:
1602:
1536:
943:
802:
798:
796:
757:
753:
749:
703:
680:
671:
655:
576:
503:
401:
276:
198:
194:
177:
154:
126:
125:
100:
90:
83:
76:
69:
57:
33:
2244:Hooke's law
475:: Set point
2444:Categories
2390:References
2376:hysteresis
2372:duty cycle
2365:Advantages
674:March 2024
161:thermostat
103:March 2021
73:newspapers
2197:∞
2163:Δ
2143:Δ
2137:×
2087:×
2070:↘
2041:∞
2038:→
1996:Δ
1990:×
1790:∞
1756:Δ
1753:×
1720:Δ
1714:×
1690:τ
1670:×
1653:↘
1624:∞
1621:→
1579:Δ
1573:×
1476:−
1427:τ
1391:τ
1334:τ
1316:τ
1187:τ
1113:τ
1060:τ
979:τ
433:−
43:talk page
185:integral
148:and the
138:setpoint
131:feedback
2247:error.
2231:SP − PV
660:Please
87:scholar
2415:
2307:So if
2292:
1220:where
277:where
191:Theory
89:
82:
75:
68:
60:
758:error
94:JSTOR
80:books
2413:ISBN
1534:).
1519:>
1473:>
1403:>
1307:and
118:The
66:news
2277:100
2062:lim
2031:lim
1951:.
1645:lim
1614:lim
1600:.
756:is
664:to
404:.
49:by
2446::
2397:^
2236:p0
2017:.
805:.
152:.
2421:.
2347:p
2343:K
2320:p
2316:K
2285:p
2281:K
2272:=
2269:B
2266:P
2194:=
2191:t
2186:|
2181:)
2178:t
2175:(
2172:y
2169:=
2166:R
2160:=
2156:)
2150:s
2146:R
2129:c
2125:k
2121:+
2118:)
2115:1
2112:+
2109:s
2106:(
2103:s
2097:c
2093:k
2084:s
2080:(
2074:0
2066:s
2058:=
2055:)
2052:t
2049:(
2046:y
2035:t
2003:s
1999:R
1985:L
1982:C
1978:g
1974:=
1971:)
1968:s
1965:(
1962:y
1934:c
1930:k
1926:+
1923:)
1920:1
1917:+
1914:s
1911:(
1908:s
1902:c
1898:k
1892:=
1887:L
1884:C
1880:g
1856:)
1853:1
1850:+
1847:s
1844:(
1841:s
1837:1
1832:=
1827:p
1823:g
1787:=
1784:t
1779:|
1774:)
1771:t
1768:(
1765:y
1762:=
1759:R
1748:L
1745:C
1741:k
1737:=
1733:)
1727:s
1723:R
1708:1
1705:+
1702:s
1697:L
1694:C
1683:L
1680:C
1676:k
1667:s
1663:(
1657:0
1649:s
1641:=
1638:)
1635:t
1632:(
1629:y
1618:t
1586:s
1582:R
1568:L
1565:C
1561:g
1557:=
1554:)
1551:s
1548:(
1545:y
1522:0
1514:c
1510:k
1504:p
1500:k
1479:1
1468:c
1464:k
1458:p
1454:k
1431:p
1406:0
1398:L
1395:C
1365:c
1361:k
1355:p
1351:k
1347:+
1344:1
1338:p
1328:=
1323:L
1320:C
1290:c
1286:k
1280:p
1276:k
1272:+
1269:1
1262:c
1258:k
1252:p
1248:k
1241:=
1236:L
1233:C
1229:k
1205:1
1202:+
1199:s
1194:L
1191:C
1180:L
1177:C
1173:k
1167:=
1162:L
1159:C
1155:g
1128:1
1125:+
1122:s
1117:p
1105:c
1101:k
1095:p
1091:k
1084:+
1081:1
1075:1
1072:+
1069:s
1064:p
1052:c
1048:k
1042:p
1038:k
1030:=
1025:L
1022:C
1018:g
994:1
991:+
988:s
983:p
972:p
968:k
962:=
957:p
953:g
924:,
917:l
914:c
910:g
887:,
879:c
875:g
869:p
865:g
861:+
858:1
851:c
847:g
841:p
837:g
830:=
823:l
820:c
816:g
799:r
782:,
775:p
771:g
754:e
750:r
732:c
728:k
724:=
719:c
715:g
687:)
681:(
676:)
672:(
658:.
621:)
618:t
615:(
612:e
590:p
586:K
559:t
556:u
553:o
548:P
524:t
521:u
518:o
513:P
488:V
485:P
463:P
460:S
439:V
436:P
430:P
427:S
424:=
421:)
418:t
415:(
412:e
402:t
388:)
385:t
382:(
379:e
355:p
351:K
325:t
322:u
319:o
314:P
291:0
288:p
261:0
258:p
255:+
252:)
249:t
246:(
243:e
236:p
232:K
228:=
222:t
219:u
216:o
211:P
105:)
101:(
91:·
84:·
77:·
70:·
53:.
39:.
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