Chapter 5 5 1 (a) ζ. ω 0 707 rad / ec (b) 0 ζ 0. 707 ω rad / ec (c) ζ 0. 5 1 ω 5 rad / ec (d) 0. 5 ζ 0. 707 ω 0. 5 rad / ec 5 (a) Type 0 (b) Type 0 (c) Type 1 (d) Type (e) Type 3 (f) Type 3 (g) type (h) type 1 5 3 (a) K G p lim ( ) 1000 K lim G( ) 0 K lim G( ) 0 0 v 0 a 0 (b) K p lim G( ) 0 K v lim G( ) 1 K lim G( ) 0 0 5 1 a 0
(c) K p lim G( ) 0 K lim G( ) K v 0 K a lim G( ) 0 0 (d) K p lim G( ) 0 K v lim G( ) 0 K a lim G( ) 1 0 (e) K p lim G( ) 0 K v lim G( ) 1 K lim G( ) 0 0 a 0 (f) K p lim G( ) 0 K v lim G( ) 0 K lim G( ) K a 0 5 4 (a) Iput Error Cotat Steady tate Error u t K 1000 p 1 1001 tu t K 0 v t u ()/ t K a 0 (b) Iput Error Cotat Steady tate Error u t K p 0 tu t K 1 v 1 t u ()/ t K a 0 (c) Iput Error Cotat Steady tate Error u () t K 0 p 5
tu () t K K 1/ K v t u ()/ t K a 0 The above reult are valid if the value of K correpod to a table cloed loop ytem. (d) The cloed loop ytem i utable. It i meaigle to coduct a teady tate error aalyi. (e) Iput Error Cotat Steady tate Error u t K p 0 tu t K 1 v 1 t u ()/ t K a 0 (f) Iput Error Cotat Steady tate Error u () t K 0 p tu () t K 0 v t u ()/ t K K 1/ K a The cloed loop ytem i table for all poitive value of K. Thu the above reult are valid. 5 5 (a) K H ( 0) 1 M G( ) ( ) H 1 + GH ( ) ( ) 3 + 1 + + 3+ 3 a 3, a 3, a, b 1, b 1. 0 1 0 1 Uit tep Iput: e 1 bk 0 H 1 K H a 0 3 5 3
Uit ramp iput: a b K 3 1 0. Thu e. 0 0 H Uit parabolic Iput: a b K 0 ad a b K 1 0. Thu e. 0 0 H 1 1 H (b) K H ( 0) 5 H G( ) 1 M( ) a 5, a 5, b 1, b 0. 0 1 0 1 1 + GH ( ) ( ) + 5+ 5 Uit tep Iput: e 1 bk 1 5 K a 5 5 H 0 0 H 1 1 0 Uit ramp Iput: i 0: a b K 0 i 1: a b K 5 0 0 0 H 1 1 H e a b K 1 1 ak 0 H H 5 1 5 5 Uit parabolic Iput: e (c) K H ( 0) 1/ 5 H G( ) M( ) 1 + GH ( ) ( ) + 5 The ytem i table. 4 3 + 15 + 50 + + 1 a 1, a 1, a 50, a 15, b 5, b 1 0 1 3 0 1 Uit tep Iput: e 1 bk 5/5 K a 1 H 0 0 H 1 5 1 0 Uit ramp Iput: 5 4
i 0: a b K 0 i 1: a b K 4/ 5 0 0 0 H 1 1 H e a b K 1 1 ak 0 H H 1 1/ 5 4 1/ 5 Uit parabolic Iput: e (d) K H ( 0) 10 H G( ) 1 M( ) The ytem i table. 3 1 + GH ( ) ( ) + 1 + 5+ 10 a 10, a 5, a 1, b 1, b 0, b 0 0 1 0 1 Uit tep Iput: e 1 bk 1 10 K a 10 10 H 0 0 H 1 1 0 Uit ramp Iput: i 0: a b K 0 i 1: a b K 5 0 0 0 H 1 1 H e a b K 1 1 ak 0 H H 5 100 005. Uit parabolic Iput: e 5 6 (a) M ( ) + 4 K 4 3 H 1 The ytem i table. + 16 + 48 + 4 + 4 Uit tep Iput: a 4, a 4, a 48, a 16, b 4, b 1, b 0, b 0 0 1 3 0 1 3 e 1 bk 4 K a 4 H 0 0 H 1 1 0 Uit ramp iput: i 0: a b K 0 i 1: a b K 4 1 3 0 0 0 H 1 1 H 5 5
e a b K 1 1 ak 0 H H 4 1 3 4 4 Uit parabolic Iput: e (b) M( ) K( + 3) K 3 H 1 The ytem i table for K > 0. + 3 + ( K+ ) + 3K a 3K, a K+, a 3, b 3K, b K 0 1 0 1 Uit tep Iput: e 1 bk 3K K a 3K H 0 0 H 1 1 0 Uit ramp Iput: i 0: a b K 0 i 1: a b K K + K 0 0 0 H 1 1 H e a b K 1 1 ak 0 H H K+ K 3K 3K Uit parabolic Iput: e The above reult are valid for K > 0. (c) M + 5 10 H ( ) H( ) K lim ( ) 4 3 H + 15 + 50 + 10 + 5 0 Uit tep Iput: a 0, a 10, a 50, a 15, b 5, b 1 0 1 3 0 1 e 1 a bk 1 50 1 1 H K H a 10 1.4 Uit ramp Iput: 5 6
Uit parabolic Iput: e e (d) M ( ) K( + 5) K H 1 The ytem i table for 0 < K < 04. 4 3 + 17 + 60 + 5K + 5K a 5K, a 5K, a 60, a 17, b 5K, b K 0 1 3 0 1 Uit tep Iput: e 1 bk 5K K a 5K H 0 0 H 1 1 0 Uit ramp Iput: i 0: a b K 0 i 1: a b K 5K K 4K 0 0 0 H 1 1 H e Uit parabolic Iput: a b K 1 1 ak 0 H H 5K K 5K e 4 5 The reult are valid for 0 < K < 04. 5 7
5-7) Type of the ytem i zero 1 3 5 1 1 3 5 1 5 11 5 Pole: -.013 + 1.8773i, -.013-1.8773i, ad -0.5974 Zero: -1 5-8) 5( + 1) ( + )( + 3) 5 1 3 a) Poitio error: lim G lim b) Velocity error: lim lim c) Acceleratio error: lim lim 0 5-9) a) Steady tate error for uit tep iput: Referrig to the reult of problem 5-8, 0 b) Steady tate error for ramp iput: 5 8
Regardig the reult of problem 5-8, c) Steady tate error for parabolic iput: Regardig the reult of problem 5-8, 0 5-10) 4( + 1) ( + ) a) Step error cotat: lim b) Ramp error cotat: lim c) Parabolic error cotat: lim 5-11) 3 4 where x 1 i a uit tep iput, x i a ramp iput, ad x 3 i a uit parabola iput. Sice the ytem i liear, the the effect of X() i the ummatio of effect of each idividual iput. That i: 3 4 So: 0 0 4 5 9
5-1) The tep iput repoe of the ytem i: 1 1 1 1 1 Therefore: 1 1 1 The rie time i the time that uit tep repoe value reache from 0.1 to 0.9. The: It i obviou that t r > 0, the: A k < 1, the 0 Therefore.. 0 or.. which yield: 0 1 1.. 1 1.. 0 5 13) Y( ) G ( ) E( ) KG ( ) 0 p 100K 1+ KG ( ) 0( 1+ 0. + 100K ) t p t Type 1 ytem. 5K Error cotat: K, K, K 0 p v a 1 + 100K t (a) rt () u(): t e (b) r() t tu (): t e (c) rt () tu ()/ t : e 1 K a 1 0 1 + K p 1 1 K + 100 K 5K v t 5 10
5 14 G p 100 Y( ) KG ( ) p ( ) G ( ) ( 1+ 01. )( 1+ 05. ) E( ) 0 1+ K G ( ) 100K G ( ) 0 ( 1+ 01. )( 1+ 0. 5) + 100K t t p 5K Error cotat: K, K, K 0 p v a 1 + 100K t (a) rt () u(): t e (b) r() t tu (): t e (c) rt () tu ()/ t : e 1 K a 1 0 1 + K p 1 1 K + 100 K 5K v t Sice the ytem i of the third order, the value of K ad K t mut be cotraied o that the ytem i table. The characteritic equatio i Routh Tabulatio: ( ) 3 K K t + 1 + 0 + 000 + 100 0 3 1 0 1 0 + 000K 1 100K 40 + 4000K 100K 1 100K t t Stability Coditio: K > 0 ( K ) 1+ 100K 1 t 1 1+ 100 5K > 0 or > t 5K 1 Thu, the miimum teady tate error that ca be obtaied with a uit ramp iput i 1/1. 5 11
5 15 (a) From Figure 3P 9, KK KK + KKKK 1 i b 1 i t 1 + + Θ () R + L ( R + L )( B + J ) o a a a a t t Θ () KK KK + KKKK KKKKN r 1 i b 1 i t 1 i 1 + + + R + L R + L B + J R + L B + J ( )( ) ( )( ) a a a a t t a a t t [ ( R + L)( B + J) + KK ( B + J) + KK + KKKK] a a t t 1 t t i b 1 i t ( ) ( ) Θ ( ) o Θ + + + + + + + + 3 ( ) L J L B R J K K J R B K K KK K K K K B KK K K N r a t a t a t 1 t a t i b i 1 t 1 t 1 i 1 () (), () lim Θ () 0 e θ r t u t Θ r 0 Provided that all the pole of Θ ( ) are all i the left half plae. e (b) For a uit ramp iput, Θ r ( ) 1/. e lim θ ( t) lim Θ ( ) t e 0 e RB+ KKB+ KK + KKKK a t 1 t i b 1 i t KK K K N 1 i if the limit i valid. 5 1
5 16 (a) Forward path trafer fuctio: [(t) 0]: K(1 + 0.0 ) Y( ) ( + 5) K(1 + 0.0 ) G () E ( ) KK t 5 KK 1+ ( + 5) ( + + t ) Type 1 ytem. 1 Error Cotat: K, K, K 0 p v a K 1 1 For a uit ramp iput, r() t tu (), t R(), e lim e() t lim E() K t 0 K t v t Routh Tabulatio: 3 1 KK + 0.0K t 1 0 5 K ( ) 5K K + 0.0 K t 5 K Stability Coditio: ( ) K > 0 5 K + 0.0 K > 0 or K > 0.0 t t (b) With r(t) 0, t u t N 1 () (), () /. Sytem Trafer Fuctio with N() a Iput: K ( ) ( 5) Y + K N K(1 + 0.0 ) KK + + K K + + K ( ) 3 ( ) t 5 0.0 t 1 + + Steady State Output due to (t): ( + 5) ( + 5) y lim y( t) lim Y( ) 1 if the limit i valid. t 0 5 13
5 17 You may ue MATLAB i all Routh Hurwitz calculatio. 1. Activate MATLAB. Go to the directory cotaiig the ACSYS oftware. 3. Type i Acy 4. The pre the trafer fuctio Symbolic ad eter the Characteritic equatio 5. The pre the Routh Hurwitz butto 6. For example look at below Figure 5 14
(a) () t 0, r() t tu (). t Forward path Trafer fuctio: Y() K( + α)( + 3) G () E () 1 0 ( ) Type 1 ytem. Ramp error cotat: K lim G( ) 3Kα v 0 Steady tate error: e 1 1 K 3K v v 3 Characteritic equatio: + K + [ K( 3+ α) 1] + 3α K 0 Routh Tabulatio: 1 3K+ αk 1 K 3αK 3 1 0 K( 3K+ αk 1) 3αK K 3αK Stability Coditio: 1 3K 3K+ αk 1 3α > 0 or K > + 3+ α αk > 0 (b) Whe r(t) 0, t u t N 1 () (), () /. Trafer Fuctio betwee (t) ad y(t): K( + 3) Y( ) 1 K( + 3) 3 N( ) K( + α)( + 3) + K + [ K( + α) 1] + 3α K r 0 1+ ( 1) Steady State Output due to (t): y lim y( t) lim Y( ) 0 if the limit i valid. t 0 5 15
5 18 Percet maximum overhoot 0.5 e πζ 1 ζ Thu ( ) πζ 1 ζ l 0.5 1.386 π ζ 1.9 1 ζ Solvig for ζ from the lat equatio, we have ζ 0.404. Peak Time t max ω π 1 ζ 001. ec. Thu, ω π 001. 1 ( 0.404) 343.4 rad / ec Trafer Fuctio of the Secod order Prototype Sytem: Y( ) ω 117916 R ( ) + ζω + ω + 77. 3+ 117916 5 16
Exteded MATLAB olutio of problem imilar to 5 19 5 7 appear later o e.g. 5 58 5 19 Cloed Loop Trafer Fuctio: Characteritic equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 For a ecod order prototype ytem, whe the maximum overhoot i 4.3%, ζ 0. 707. Rie Time: ω 5K, ζω 5+ 500K 1.414 5K t 1 0.4167 ζ +. 917ζ. 164 0. ec Thu ω 108. rad / ec t r ω ω Thu, K ω ( 10. 8) + K ω 10. 3 4.68 5 500 1.414 15. 3 Thu K t t 0. 006 5 5 500 With K 4.68 ad K t 0. 006, the ytem trafer fuctio i Y ( ) 117 R ( ) + 15. 3+ 117 Uit tep Repoe: y 0.1 at t 0.047 ec. y 0.9 at t 0.44 ec. t r 0.44 0. 047 0. 197 ec. y max 0. 043 ( 4.3% max. overhoot) 5 17
5 0 Cloed loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 πζ Whe Maximum overhoot 10%, l 0.1.3 πζ ( ) 5.3 1 ζ Solvig for ζ, we get ζ 0.59. 1 ζ The Natural udamped frequecy i ω 5K Thu, 5+ 500 K ζω 118. ω Rie Time: t 1 0.4167 ζ. 917ζ. t r. ω ω + 01 17696 ec. Thu ω 17. 7 rad / ec ω 1588. K 1. 58 Thu K 0. 0318 t 5 500 With K 1.58 ad K t 0. 0318, the ytem trafer fuctio i Y ( ) 313 R ( ) + 0. 88+ 314.5 Uit tep Repoe: y 0.1 whe t 0.08 ec. y 0.9 whe t 0.131 ec. t r 0. 131 0. 08 0. 103 ec. y max 11. ( 10% max. overhoot) 5 18
5 1 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y ( ) 5K R ( ) + ( 5+ 500K ) + 5K ( ) t t πζ + 5+ 500K + 5K 0 Whe Maximum overhoot 0%, l 0. 1.61 πζ ( ).59 1 ζ Solvig for ζ, we get ζ 0.456. 1 ζ The Natural udamped frequecy ω 5K 5+ 500K ζω 0. 91ω Rie Time: 1 0.4167 ζ. 917ζ t r 005. ω ω t + 1.4165 1.4165 ec. Thu, ω 8. 33 005. ω K 31. 5+ 500K 0. 91ω 584. Thu, K 0. 0417 t t 5 With K 3.1 ad K t 0. 0417, the ytem trafer fuctio i Y ( ) 80. 59 R ( ) + 584. + 80. 59 Uit tep Repoe: y 0.1 whe t 0.0178 ec. y 0.9 whe t 0.07 ec. t r 0. 07 0. 0178 0. 054 ec. y max 1. ( 0% max. overhoot) 5 19
5 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 Delay time t d 11. + 0. 15ζ + 0.469ζ ω 01. ec. 1.43 Whe Maximum overhoot 4.3%, ζ 0. 707. t d 01. ω ec. Thu ω 14.3 rad/ec. ω 14.3 K 8.1 5 + 500K t ζω 1.414ω 0.1 5 5 151. Thu K t 0. 030 500 With K 0.1 ad K t 0. 030, the ytem trafer fuctio i Uit Step Repoe: Y ( ) 0. 5 R ( ) + 01. + 0. 5 Whe y 0.5, t 0.1005 ec. Thu, t d 0. 1005 ec. y max 1043. ( 4.3% max. overhoot) 5 0
5 3 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 Delay time t d 11. + 0. 15ζ + 0.469ζ ω 1337. 005. ω 1337. Thu, ω 6. 74 005. ω 6.74 K 8.6 5 5 5 + 500K ζω 0. 59 6. 74 3155. Thu K 0. 0531 t t With K 8.6 ad K t 0. 0531, the ytem trafer fuctio i Y ( ) 715 R ( ) + 3155. + 715 Uit Step Repoe: y 0.5 whe t 0.0505 ec. Thu, t d 0. 0505 ec. y max 11007. ( 10. 07% max. overhoot) 5 1
5 4 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 For Maximum overhoot 0., ζ 0.456. Delay time t d 11. + 0. 15ζ + 0.469 ζ 1.545 001. ω ω ec. 1.545 Natural Udamped Frequecy ω 15.45 rad/ec. Thu, 001. ω 15737.7 K 69.5 5 5 5 + 500 ζω 0.456 15.45 114.41 Thu, K t 0.188 K t With K 69.5 ad K t 0.188, the ytem trafer fuctio i Y ( ) 15737. 7 R ( ) + 114.41+ 15737. 7 Uit tep Repoe: y 0.5 whe t 0.0101 ec. Thu, t d 0. 0101 ec. y max 1. ( 0% max. overhoot) 5
5 5 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5 + 5000 + 5 0 ζ 0. 6 ζω 5 + 500K 1.ω t 3. 3. Settlig time t 01. ζω 06. ω 3. ec. Thu, ω 53. 33 rad / ec 006. K t 1.ω 5 ω 0118. K 113. 76 500 5 Sytem Trafer Fuctio: Y ( ) 844 R ( ) + 64+ 844 Uit tep Repoe: y(t) reache 1.00 ad ever exceed thi value at t 0.098 ec. Thu, t 0. 098 ec. 5 3
5 6 (a) Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 For maximum overhoot 0.1, ζ 0. 59. 5 + 500 ζω 0. 59ω 118. ω K t Settlig time: t 3. 3. 005. ζω 059. ω 3. ec. ω 108.47 005. 059. K t 118. ω 5 ω 0.46 K 470. 63 500 5 Sytem Trafer Fuctio: Y ( ) 11765. 74 R ( ) + 18+ 11765. 74 Uit Step Repoe: y(t) reache 1.05 ad ever exceed thi value at t 0.048 ec. Thu, t 0. 048 ec. (b) For maximum overhoot 0., ζ 0.456. 5 + 500 ζω 0. 91ω K t 3. 3. 3. Settlig time t 001. ec. ω 70175. ζω 0.456ω 0.456 001. rad / ec K t 0. 91 ω 5 1.7 500 Sytem Trafer Fuctio: Y ( ) 49453 Uit Step Repoe: R ( ) + 640+ 49453 5 4
y(t) reache 1.05 ad ever exceed thi value at t 0.0074 ec. Thu, t 0. 0074 ec. Thi i le tha the calculated value of 0.01 ec. 5 5
5 7 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () + 5+ 500 + 5 t ( ) K K t + 5+ 500 + 5 0 Dampig ratio ζ 0. 707. Settlig time t Sytem Trafer Fuctio: 4.5ζ 31815. 01. ec. Thu, ω 31815. rad/ec. ω ω ω 5+ 500K ζω 44.986 Thu, K 0. 08 K 40.488 t t ζ Y ( ) 101. R ( ) + 44.986+ 101. Uit Step Repoe: The uit tep repoe reache 0.95 at t 0.09 ec. which i the meaured t. 5 6
5 8 (a) Whe ζ 05., the rie time i 1 0.4167 ζ +. 917ζ 151. 1 t r ω ω ec. Thu ω 151. rad/ec. The ecod order term of the characteritic equatio i writte + ζω + ω + 151. +. 313 0 3 The characteritic equatio of the ytem i + ( a+ 30) + 30a+ K 0 Dividig the characteritic equatio by + 151. +. 313, we have For zero remaider, 8.48 a 45. 63 Thu, a 16. K 65874. +. 313a 69. 58 Forward Path Trafer Fuctio: G ( ) 69. 58 ( + 16. )( + 30) Uit Step Repoe: y 0.1 whe t 0.355 ec. y 0.9 whe t 1.43 ec. Rie Time: t r 1.43 0. 355 1075. ec. 5 7
(b) The ytem i type 1. (i) For a uit tep iput, e 0. (ii) For a uit ramp iput, K G v K 60. 58 1 lim ( ) 1.45 e 069. 0 30a 30 16. K v 5 9 (a) Characteritic Equatio: 3 + 3 + ( + K) K 0 Apply the Routh Hurwitz criterio to fid the rage of K for tability. Routh Tabulatio: 3 1 + K 1 0 3 6+ 4K 3 K K Stability Coditio: 1.5 < K < 0 Thi implifie the earch for K for two equal root. Whe K 0.7806, the characteritic equatio root are: 0.347, 0.347, ad.3054. (b) Uit Step Repoe: (K 0.7806) 5 8
(c) Uit Step Repoe (K 1) The tep repoe i (a) ad (b) all have a egative uderhoot for mall value of t. Thi i due to the zero of G() that lie i the right half plae. 5 9
5 30 (a) The tate equatio of the cloed loop ytem are: dx dt 1 dx x + 5x 6x k x k x + r 1 1 1 1 dt The characteritic equatio of the cloed loop ytem i + 1 5 Δ + ( 1+ k ) + ( 30+ 5k + k ) 0 1 6 + k + k 1 For ω 10 rad / ec, 30 + 5k + k ω 100. Thu 5k + k 70 1 1 k (b) For ζ 0. 707, ζω 1+ k. Thu ω 1+. 1.414 ( 1+ k ) ω 30 + 5 k + k Thu k 59 + 10k 1 1 (c) For ω ζ 10 rad / ec ad 0. 707, 5k + k 100 ad 1 + k ζω 14.14 Thu k 1314. Solvig for k, we have k 1137.. 1 1 1 (d) The cloed loop trafer fuctio i Y( ) 5 5 R k k k ( ) ( ) ( ) + + 1 + 30+ 5 + + 14.14 + 100 1 5 For a uit tep iput, lim y( t ) lim Y ( ) 005. t 0 100 (e) For zero teady tate error due to a uit tep iput, 30 + 5k + k 5 Thu 5k + k 5 1 1 5 30
Parameter Plae k 1 veru k : 5 31 (a) Cloed Loop Trafer Fuctio (b) Characteritic Equatio: ( K + K ) Y( ) 100 P D R + K+ K ( ) 100 100 D P + 100K + 100K 0 D P The ytem i table for K P > 0 ad K > 0. D (b) For ζ, ζω ω 1 100K. D 10 K Thu ω 100K 0 K K 0. K P D P D P (c) See parameter plae i part (g). 5 31
(d) See parameter plae i part (g). (e) Parabolic error cotat K a 1000 ec ( ) K G K K K K lim ( ) lim100 + 100 1000 Thu 10 a P D P P 0 0 (f) Natural udamped frequecy ω 50 rad/ec. ω K P Thu K P 10 50 5 (g) Whe K P 0, K K D G ( ) 100 100 D (pole zero cacellatio) 5 3
5 3 (a) Forward path Trafer Fuctio: Y() KK 10K i G () E ( ) J(1 T) KK 0.001 0.01 10K [ + + ] i t ( + + t) K 1 Whe r() t tu (), t K lim G() e v 0 K K t v K t K (b) Whe r(t) 0 Y() 1+ T 1+ 0.1 T ( ) J(1 + T) + K K + KK 0.001 + 0.01+ 10K + 10K [ ] ( ) d i t i t For T d 1 1 ( ) lim y( t ) lim Y ( ) t 0 10K if the ytem i table. (c) The characteritic equatio of the cloed loop ytem i 3 0. 001 + 0. 01 + 0. 1 + 10K 0 The ytem i utable for K > 0.1. So we ca et K to jut le tha 0.1. The, the miimum value of the teady tate value of y(t) i 1 10 K K 01. + 1 However, with thi value of K, the ytem repoe will be very ocillatory. The maximum overhoot will be early 100%. (d) For K 0.1, the characteritic equatio i 3 3 4 t 0. 001 + 0. 01 + 10K + 1 0 or + 10 + 10 K + 1000 0 t 5 33
For the two complex root to have real part of /5. we let the characteritic equatio be writte a ( )( ) 3 a b a b ab + + 5 + 0 or + ( + 5) + (5 + ) + 0 The, a + 5 10 a 5 ab 1000 b 00 5a + b 10 K K 0. 05 The three root are: a 5 a 5. 5± j13. 9 4 t t 5-33) Rie time:..... 0.56 Peak time:.. 0.785 Maximum overhoot:. Settlig time:. 0 0.69.. 1.067. 0.095 5 34
5-34) K ( J + a + KK f ) exp 0. ξ 0.456 0.1 ω 0.353 0.15 5.4.. 5.49. 19.88 5 35
5-35) a) 6.5 6.5 1 1. 6.5 1.. Subtitutig ito equatio () give: 7.5 6.5 6.5 6.5 Sice the ytem i multi iput ad multi output, there are 4 trafer fuctio a:,,, To fid the uit tep repoe of the ytem, let coider 6.5 6.5 6.5 6.5 1 1 1 6.5 By lookig at the Laplace traform fuctio table: where co 1 1 1 i 1 b) 5 36
Therefore: A a reult: which mea: 1 1 The uit tep repoe i: 1 1 Therefore a a reult: 1 1 1 i 1 where ω 1 ad ξ ω 1 / c) 4 Therefore: 1 1 A a reult, the tep repoe of the ytem i: 1 1 By lookig up at the Laplace trafer fuctio table: 5 37
5-36) MATLAB CODE (a) where ω 1, ad ξ ω 1 ξ 1 / i 1 where θ co -1 (ξ -1) co -1 (0.5), therefore, clear all Amat[-1-1;6.5 0] Bmat[1 1;1 0] Cmat[1 0;0 1] Dmat[0 0;0 0] dip(' State-Space Model i:') Statemodel(Amat,Bmat,Cmat,Dmat) [ma,a]ize(amat); rakarak(amat); dip(' Characteritic Polyomial:') chareqpoly(amat); 1 3 i 3 % p poly(a) where A i a -by- matrix retur a +1 elemet %row vector whoe elemet are the coefficiet of the characteritic %polyomialdet(i-a). The coefficiet are ordered i decedig power. [mchareq,chareq]ize(chareq); ym ''; polyym(chareq,) dip(' Equivalet Trafer Fuctio Model i:') HmatCmat*iv(*eye()-Amat)*Bmat+Dmat Sice the ytem i multi iput ad multi output, there are 4 trafer fuctio a: To fid the uit tep repoe of the ytem, let coider,,, 6.5 6.5 5 38
Let obtai thi term ad fid Y () time repoe for a tep iput. HHmat(,) ilaplace(h/) Pretty(H) Hpolytf([13/],chareq) tep(hpoly) H 13/(*^+*+13) a 1-1/5*exp(-1/*t)*(5*co(5/*t)+i(5/*t)) Trafer fuctio: 6.5 ------------- ^ + + 6.5 13 --------------- + + 13 To fid the tep repoe H11, H1, ad H1 follow the ame procedure. Other part are the ame. 5 39
5-37) Impule repoe: a).. b) c) where α co -1 ξ ad 1, therefore, 1 i 1 1 i 1 1 1 1 i 1 5-38) Ue the approach i 5-36 except: HHmat(,) ilaplace(h) Pretty(H) Hpolytf([13/],chareq) impule(hpoly) H 13/(*^+*+13) a 13/5*exp(-1/*t)*i(5/*t) Trafer fuctio: 6.5 ------------- ^ + + 6.5 13 --------------- + + 13 5 40
Other part are the ame. 5-39) a) The diplacemet of the bar i: The the equatio of motio i: i 0 A x i a fuctio of θ ad chagig with time, the co If θ i mall eough, the i θ θ ad co θ 1. Therefore, the equatio of motio i rewritte a: 0 5 41
(b) To fid the uit tep repoe, you ca ue the ymbolic approach how i Toolbox -1-1: clear all %tf( ); ym B L K ThetaB*//L/(B*+K) ilaplace(theta) Theta B/L/(B*+K) a 1/L*exp(-K*t/B) Alteratively, aig value to B L K ad fid the tep repoe ee olutio to problem 5-36. 5 40 (a) K t 10000 oz - i / rad The Forward Path Trafer Fuctio: 1 9 10 K G () 4 3 7 9 1 ( + 5000 + 1.067 10 + 50.5 10 + 5.74 10 ) 1 9 10 K ( + 116)( + 4883)( + 41.68 + j3178.3)( + 41.68 j3178.3) Routh Tabulatio: 5 1 7 1 1.067 10 5.74 10 4 9 1 5000 50.5 10 9 10 K 3 5 1 9 5.7 10 5. 7 10 18. 10 K 0 8 7 1. 895 10 + 1579. 10 K 9 10 K 1 13 1 9 16. 6 10 + 8.473 10 K. 84 10 K 9 + 1579. K 0 1 9 10 K 5 4
From the 1 row, the coditio of tability i 165710 + 8473K. 84K > 0 or K 98114. K 58303.47 < 0 or ( K + 19.43)( K 3000. 57) < 0 Stability Coditio: 0 < K < 3000.56 The critical value of K for tability i 3000.56. With thi value of K, the root of the characteritic equatio are: 4916.9, 41.57 + j3113.3, 41.57 + j3113.3, j75.68, ad j75.68 (b) K L 1000 oz i/rad. The forward path trafer fuctio i 11 9 10 K G () 4 3 6 9 11 ( + 5000 + 1.58 10 + 5.05 10 + 5.74 10 ) 11 9 10 K (1 + 116.06)( + 488.8)( + 56.48 + j1005)( + 56.48 j1005) (c) Characteritic Equatio of the Cloed Loop Sytem: 5 4 6 3 9 11 11 + 5000 + 1. 58 10 + 5. 05 10 + 5. 74 10 + 9 10 K 0 Routh Tabulatio: 5 1 6 11 1.58 10 5.74 10 4 9 11 5000 5.05 10 9 10 K 3 5 11 8 5.7 10 5. 74 10 18. 10 K 0 7 6 11 4.6503 10 + 15734. 10 K 9 10 K 1 18 15 14 6. 618 10 + 377.43 10 K. 83 10 0 11 7 6 4.6503 10 + 15734. 10 9 10 K K K 5 43
4 From the 1 row, the coditio of tability i 6. 618 10 + 3774.3K. 83K > 0 Or, K 133. 73K 93990 < 0 or ( K 1400)( K + 67. 14) < 0 Stability Coditio: 0 < K < 1400 The critical value of K for tability i 1400. With thi value of K, the characteritic equatio root are: 4885.1, 57.465 + j676, 57.465 j676, j748.44, ad j748.44 (c) K L. Forward Path Trafer Fuctio: K K K i G () J J + J T m L ( ) L J + R J + R L + R B + K K a T a T m a a m i b 891100K 891100K + 5000+ 566700 ( + 116)( + 4884) ( ) Characteritic Equatio of the Cloed Loop Sytem: 3 + 5000 + 566700 + 891100 K 0 Routh Tabulatio: 3 1 0 1 566700 5000 891100K 566700 178.K 8991100K From the 1 row, the coditio of K for tability i 566700 178.K > 0. 5 44
Stability Coditio: 0 < K < 3179.78 The critical value of K for tability i 3179.78. With K 3179.78, the characteritic equatio root are 5000, j75.79, ad j75.79. Whe the motor haft i flexible, K L i fiite, two of the ope loop pole are complex. A the haft become tiffer, K L icreae, ad the imagiary part of the ope loop pole alo icreae. Whe K L, the haft i rigid, the pole of the forward path trafer fuctio are all real. Similar effect are oberved for the root of the characteritic equatio with repect to the value of K L. 5 41 (a) 100( + ) G ( ) 1 G( ) c 1 K p lim G( ) 00 0 Whe d(t) 0, the teady tate error due to a uit tep iput i e 1 1 1 0. 00505 1 + K 1 00 199 p (b) + α 100( + )( + α) G ( ) G( ) K e 0 c p ( 1) (c) α 5 maximum overhoot 5.6% α 50 maximum overhoot % α 500 maximum overhoot 54.6% 5 45
A the value of α icreae, the maximum overhoot icreae becaue the dampig effect of the zero at α become le effective. Uit Step Repoe: (d) rt () G (). dt () u () t D () 1 0 ad 1 c Sytem Trafer Fuctio: (r 0) Y( ) D ( ) r 0 3 100( + ) + 100 + ( 199 + 100α) + 00α Output Due to Uit Step Iput: Y( ) 3 100( + ) + 100 + ( 199 + 100α) + 00α 00 1 y lim y( t) lim Y( ) t 0 00α α (e) rt () 0, dt () u() t G ( ) c + α 5 46
Sytem Trafer Fuctio [r(t) 0] Y( ) D ( ) r 0 3 100 ( + 0 + 100 + ( 199 + 100α) + 00α 1 D ( ) y lim y( t) lim Y( ) 0 t 0 (f) Y( ) α 5 D ( ) r 0 Y( ) α 50 D ( ) r 0 Y( ) α 5000 D ( ) r 0 3 100 ( + ) + 100 + 699+ 1000 3 100 ( + ) + 100 + 5199+ 10000 3 100 ( + ) + 100 + 50199+ 100000 Uit Step Repoe: (g) A the value of α icreae, the output repoe y(t) due to r(t) become more ocillatory, ad the overhoot i larger. A the value of α icreae, the amplitude of the output repoe y(t) due to d(t) become maller ad more ocillatory. 5 47
5 4 (a) Forward Path Trafer fuctio: Characteritic Equatio: H( ) 10N N G ( ) E( ) ( + 1)( + 10) ( + 1) + + N 0 N1: Characteritic Equatio: + + 1 0 ζ 05. ω 1 rad/ec. Maximum overhoot e πζ 1 ζ 0. 163 (16.3%) Peak time t max ω π 1 ζ 3. 68 ec. N10: Characteritic Equatio: + 10 0 ζ 0. 158 ω 10 rad/ec. πζ 1 ζ Maximum overhoot e 0. 605 (60.5%) Peak time t (b) Uit Step Repoe: N 1 max ω π 1 ζ 1006. ec. 5 48
Secod order Sytem Third order Sytem Maximum overhoot 0.163 0.06 Peak time 3.68 ec. 3.68 ec. Uit Step Repoe: N 10 Secod order Sytem Third order Sytem Maximum overhoot 0.605 0.96 Peak time 1.006 ec. 1.13 ec. 5 49
5 43 Uit Step Repoe: Whe T z i mall, the effect i lower overhoot due to improved dampig. Whe T z i very large, the overhoot become very large due to the derivative effect. T z improve the rie time, ice 1+ T z derivative cotrol or a high pa filter. i a 5 50
5 44 Uit Step Repoe The effect of addig the pole at 1 to G() i to icreae the rie time ad the overhoot. The ytem i T p le table. Whe T p > 0. 707, the cloed loop ytem i table. 5 51
5-45) You may ue the ACSYS oftware developed for thi book. For decriptio refer to Chapter 9. We ue a MATLAB code imilar to toolbox --1 ad thoe i Chapter 5 to olve thi problem. (a) Uig Toolbox 5-9-3 clear all um []; de [0-0.55-1.5]; Gzpk(um,de,1) t0:0.001:15; tep(g,t); hold o; for Tz[1 5 0]; t0:0.001:15; um [-1/Tz]; de [0-0.55-1.5]; Gzpk(um,de,1) tep(g,t); hold o; ed xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Zero/pole/gai: 1 ------------------ (+0.55) (+1.5) Zero/pole/gai: (+1) ------------------ (+0.55) (+1.5) Zero/pole/gai: (+0.) ------------------ (+0.55) (+1.5) Zero/pole/gai: (+0.05) ------------------ (+0.55) (+1.5) 5 5
(b) clear all for Tz[0 1 5 0]; t0:0.001:15; um [Tz 1]; de [1 ]; Gtf(um,de) tep(g,t); hold o; ed xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Trafer fuctio: 1 ------------- ^ + + Trafer fuctio: + 1 ------------- ^ + + Trafer fuctio: 5 + 1 ------------- ^ + + 5 53
Trafer fuctio: 0 + 1 ------------- ^ + + Follow the ame procedure for other part. 5-46) Sice the ytem i liear we ue uperpoitio to fid Y, for iput X ad D Firt, coider D 0 k K J + a K f The 1 ; 5 54
Kk ( J + a + KK k) f Accordig to above block diagram: Now coider X 0, the: 1 1 1 k K J + a K f +1 Accordigly, ad 1 I thi cae, E() D-KY() 1 1 1 5 55
Now the teady tate error ca be eaily calculated by: 1 1/: lim 1 1 lim lim 1 0 1 1 : lim lim lim (c) The overall repoe i obtaied through uperpoitio Y() Y() + Y() D 0 X 0 yt () yt () + yt () d() t 0 x() t 0 1 1 1 MATLAB clear all ym K k J a Kf X1/; D1/^ YK*k*X/(*(J*+a+K*k*Kf)+K*k)+k*(Kf+)*D/(*(J*+a)+k*(Kf+)*K) ilaplace(y) D 1/^ Y K*k//(*(J*+a+K*k*Kf)+K*k)+k*(Kf+)/^/(*(J*+a)+k*(Kf+)*K) a 1+t/K+1/k/K^/Kf/(a^+*a*K*k+K^*k^- 4*J*K*k*Kf)^(1/)*ih(1/*t/J*(a^+*a*K*k+K^*k^-4*J*K*k*Kf)^(1/))*exp(- 1/*(a+K*k)/J*t)*(a^+a*K*k-*J*K*k*Kf)-coh(1/*t/J*(a^+*a*K*k*Kf+K^*k^*Kf^- 4*J*K*k)^(1/))*exp(-1/*(a+K*k*Kf)/J*t)-(a+K*k*Kf)/(a^+*a*K*k*Kf+K^*k^*Kf^- 4*J*K*k)^(1/)*ih(1/*t/J*(a^+*a*K*k*Kf+K^*k^*Kf^-4*J*K*k)^(1/))*exp(- 1/*(a+K*k*Kf)/J*t)+1/k/K^/Kf*a*(-1+exp(- 1/*(a+K*k)/J*t)*coh(1/*t/J*(a^+*a*K*k+K^*k^-4*J*K*k*Kf)^(1/))) 5 56
5-47) (a) Fid the e( t) dt whe e(t) i the error i the uit tep repoe. 0 A the ytem i table the e( t) dt will coverge to a cotat value: 0 0 E( ) e( t) dt lim lim E( ) 0 0 Y ( ) G( ) ( A1 + 1)( A X ( ) 1 + G( ) ( B + 1)( B 1 + 1)...( A + 1)...( B m + 1) + 1) m Try to relate thi to Equatio (5-40). G () 1 E ( ) X() Y() X() X() X() 1 + G ( ) 1 + G ( ) ( A+ 1)( A+ 1)...( A+ 1) ( B+ 1)( B + 1)...( B + 1) ( A+ 1)( A+ 1)...( A+ 1) X X 1 1 m 1 1 ( ) ( ) ( B 1 + 1)( B + 1)...( Bm+ 1) ( B 1 + 1)( B + 1)...( Bm+ 1) 1 ( B 1 + 1)( B + 1)...( Bm+ 1) ( A 1 + 1)( A + 1)...( A + 1) lim E ( ) lim 0 0 ( B 1 + 1)( B+ 1)...( Bm+ 1) B + B + L+ B A + A + L+ A ( ) ( ) 1 m 1 ( B+ 1)( B+ 1)...( B + 1) ( B 1 + 1)( B + 1)...( Bm+ 1) ( A 1 + 1)( A + 1)...( A + 1) 1 m () 1 G ( A 1 + 1)( A + 1)...( A + 1) G () ( B 1 + 1)( B + 1)...( Bm+ 1) ( A 1 + 1)( A + 1)...( A + 1) 1 ( A+ 1)( A+ 1)...( A+ 1) e B B B 1 lim 1 0 0 ( 1 + 1)( + 1)...( m + 1) 5 57
(b) Calculate Recall 1 K 1 lim G( ) 0 G () 1 E ( ) X() Y() X() X() X() 1 + G ( ) 1 + G ( ) Hece 1 1 1 1 lim E ( ) lim X( ) lim lim 0 01 + G ( ) 0 + G ( ) 0 G ( ) K Ramp Error Cotat v 5 58
5-48) 10 5 10 10 35 5 10 Comparig with the ecod order prototype ytem ad matchig deomiator: 5 10 35 exp 10.5 Let ξ 0.4 ad ω 80 The 1.386 0.10 1 3. 0.1 80, 0 0.69 35 9 56.5 clear all p9; K56.5; um [10 10*K]; de [1 35+p 5*p+10*K]; Gtf(um,de) tep(g); xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Trafer fuctio: 10 + 56.5 ----------------- ^ + 64 + 188 5 59
5-49) Accordig to the maximum overhoot: which hould be le tha t, the or 1 jω π / t ω 1 ξ σ π / t ω 1 ξ 5 60
5-50) Uig a d order prototype ytem format, from Figure 5-15, ω i the radial ditace from the complex cojugate root to the origi of the -plae, the ω with repect to the origi of the how regio i ω 3.6. Therefore the atural frequecy rage i the regio how i aroud.6 ω 4.6 O the other had, the dampig ratio ζ at the two dahed radial lie i obtaied from: ζ co/ i ζ co/ i The approximatio from the figure give: ζ 0.56 ζ 0.91 Therefore 0.56 ζ 0.91 b) A K p, the: 1 If the root of the characteritic equatio are aumed to be lied i the cetre of the how regio: Comparig with the characteritic equatio: c) The characteritic equatio 3 3 6130 1 6 13 3.5 0 i a ecod order polyomial with two root. Thee two root ca be determied by two term (p+kk p ) ad KK p K I which iclude four parameter. Regardle of the p ad K p value, we ca alway chooe K ad o that to place the root i a deired locatio. 5 61
5-51) a) By ubtitutig the value: b) Speed of the motor i lim 10. 19.3. 0. 0.109 0. 0.109 (c) d) 0. 0.109 0. 0.109 0. 0.109 A a reult: 0. 0.109 0. e) A exp 0., the, ξ 0.404 Accordig to the trafer fuctio, 0.109, the where 0. K<0.0845 f) A..., the 0.6, a 0., therefore; 1.. 0.14/ 5 6
g) MATLAB clear all for K[0.5 1 ]; t0:0.001:15; um [0.*K]; de [1 0.109 0.*K]; Gtf(um,de) tep(g,t); hold o; ed xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Trafer fuctio: 0.1 ------------------- ^ + 0.109 + 0.1 Trafer fuctio: 0. ------------------- ^ + 0.109 + 0. Trafer fuctio: 0.4 ------------------- ^ + 0.109 + 0.4 5 63
Rie time decreae with K icreaig. Overhoot icreae with K. 5 64
5-5) exp 0.1, therefore, ξ 0.59 A., the,... 3.6 1 3 3 3 Therefore: 3 3 If p i a o-domiat pole; therefore comparig both ide of above equatio ad: 3 3 If we coider p 10a (o-domiat pole), ξ 0.6 ad ω 4, the: 4.8 10 3 48 16 3 160 160 8.3 89.8 83 clear all K160; a8.3; b89.8; p83; um [K K*a]; de [1 3+b 3*b+K K*a]; Gtf(um,de) tep(g); xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Trafer fuctio: 160 + 138 ------------------------------- ^3 + 9.8 ^ + 49.4 + 138 5 65
Both Overhoot ad ettlig time value are met. No eed to adjut parameter. 5-53) For the cotroller ad the plat to be i erie ad uig a uity feedback loop we have: MATLAB USE toolbox 5-8-3 clear all um[ 1 3]; deom[ 3+qrt(9 40) 3 qrt(9 40) 0.0+qrt(.004.07) 0.0 qrt(.004.07) 10]; Gzpk(um,deom,60) rlocu(g) Zero/pole/gai: 60 (+1) (+) (+3) (+10) (^ + 0.04 + 0.0664) (^ + 6 + 40) 5 66
Note the ytem ha two domiat complex pole cloe to the imagiary axi. Let zoom i the root locu diagram ad ue the curor to fid the parameter value. 5 67
A how for K0.933 the domiat cloed loop pole are at 0.46±j 0.66 AND OVERSHOOT IS ALMOST 10%. Icreaig K will puh the pole cloer toward le domiat zero ad pole. A a proce the deig proce become le trivial ad more difficult. To cofirm ue Mfeedback(G*.933,1) %See toolbox 5 4 tep(m) Zero/pole/gai: 55.98 (+3) (+) (+1) (+7.048) (^ + 0.9195 + 0.603) (^ + 8.07 + 85.9) 5 68
To reduce rie time, the pole have to move to left to make the ecodary pole more domiat. A a reult the little bump i the left had ide of the above graph hould rie. Try K3: >> Mfeedback(G*3,1) Zero/pole/gai: 180 (+3) (+) (+1) (+5.01) (^ + 1.655 + 1.058) (^ + 9.375 + 08.9) >> tep(m) **Try a higher K value, but lookig at the root locu ad the time plot, it appear that the overhoot ad rie time criteria will ever be met imultaeouly. 5 69
K5 Mfeedback(G*5,1) %See toolbox 5-4- tep(m) Zero/pole/gai: 300 (+3) (+) (+1) --------------------------------------------------------- (+4.434) (^ + 1.958 + 1.5) (^ + 9.648 + 39.1) 5 70
5 71
5-54) Forward path Trafer Fuctio: M( ) K G ( ) 3 1 M( ) + ( 0 + a) + ( 00 + 0a) + 00a K For type 1 ytem, 00a K 0 Thu K 00a Ramp error cotat: K v K 00a lim G( ) 5 Thu a 10 K 000 0 00 + 0a 00 + 0a MATLAB Symbolic tool ca be ued to olve above. We ue it to fid the root for the ext part: >> ym a K >>olve(5*00+5*0*a 00a) a 10 >> D(^+0*+00)*+a)) D (^+0*+00)*(+a) >> expad(d) a ^3+^*a+0*^+0**a+00*+00*a >> olve(a,) a a 10+10*i 10 10*i The forward path trafer fuctio i The cotroller trafer fuctio i 5 7
G () 000 ( + 30+ 400) ( + + ) ( + + ) G () 0 10 100 G () c G () 30 400 p MATLAB The maximum overhoot of the uit tep repoe i 0 percet. clear all K000; a10; um []; de [-10+10i -10-10i -a]; Gzpk(um,de,K) tep(g); xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Zero/pole/gai: 000 ------------------------- (+10) (^ + 0 + 00) Clearly PO0. 5 73
5-55) Forward path Trafer Fuctio: M( ) K G ( ) 3 1 M( ) + ( 0 + a) + ( 00 + 0a) + 00a K For type 1 ytem, 00a K 0 Thu K 00a Ramp error cotat: K v K 00a lim G( ) 9 Thu a 90 K 18000 0 00 + 0a 00 + 0a MATLAB Symbolic tool ca be ued to olve above. We ue it to fid the root for the ext part: >> ym a K olve(9*00+9*0*a 00*a) a 90 >>D(^+0*+00)*+a)) D (^+0*+00)*(+a) >> expad(d) a ^3+^*a+0*^+0**a+00*+00*a >> olve(a,) a a 10+10*i 10 10*i The forward path trafer fuctio i The cotroller trafer fuctio i 5 74
G () 18000 ( + 110+ 000) ( + + ) ( + + ) G () 180 10 100 G () c G ( ) 110 000 p The maximum overhoot of the uit tep repoe i 4.3 percet. From the expreio for the ramp error cotat, we ee that a a or K goe to ifiity, K v approache 10. Thu the maximum value of K v that ca be realized i 10. The difficultie with very large value of K ad a are that a high gai amplifier i eeded ad urealitic circuit parameter are eeded for the cotroller. clear all K18000; a90; um []; de [-10+10i -10-10i -a]; Gzpk(um,de,K) tep(g); xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Zero/pole/gai: 18000 ------------------------- (+90) (^ + 0 + 00) PO i le tha 4. 5 75
5-56) (a) Ramp error Cotat: MATLAB clear all ym Kp Kd kv Gum(Kp+Kd*)*1000 Gde (*(+10)) GGum/Gde Kv*G 0 eval(kv) Gum 1000*Kp+1000*Kd* Gde *(+10) G (1000*Kp+1000*Kd*)//(+10) Kv (1000*Kp+1000*Kd*)/(+10) 0 a 100*Kp ( + ) 1000 K K 1000K P D P K lim 100K 1000 v P 0 ( + 10) 10 Thu K P 10 Kp10 clear ym Mum(Kp+Kd*)*1000//(+10) Mde1+(Kp+Kd*)*1000//(+10) 5 76
Kp 10 Mum (10000+1000*Kd*)//(+10) Mde 1+(10000+1000*Kd*)//(+10) a (^+10*+10000+1000*Kd*)//(+10) Characteritic Equatio: ( ) Match with a d order prototype ytem + 10 + 1000K + 1000K 0 D P ω 1000K P 10000 100 rad/ec ζω 10 + 1000K D 0. 5 100 100 olve(10+1000*kd 100) a 9/100 90 Thu K D 1000 009. Ue the ame procedure for other part. (b) For K v 1000 ad ζ 0. 707, ad from part (a), ω 100 rad/ec, 131.4 ζω 10 + 1000K D 0. 707 100 141.4 Thu K D 0. 1314 1000 (c) For K v 1000 ad ζ 1.0, ad from part (a), ω 100 rad/ec, 190 ζω 10 + 1000K D 1 100 00 Thu K D 1000 019. 5 77
5-57) The ramp error cotat: ( K + K ) 1000 P D K lim 100K 10, 000 Thu K 100 v P P 0 ( + 10) The forward path trafer fuctio i: ( + K ) 1000 100 G () ( + 10) D clear all for KD0.:0.:1.0; um [-100/KD]; de [0-10]; Gzpk(um,de,1000); Mfeedback(G,1) tep(m); hold o; ed xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Zero/pole/gai: 1000 (+500) (^ + 1010 + 5e005) Zero/pole/gai: 1000 (+50) (+434.1) (+575.9) Zero/pole/gai: 1000 (+166.7) (+07.7) (+80.3) 5 78
Zero/pole/gai: 1000 (+15) (+144.4) (+865.6) Zero/pole/gai: 1000 (+100) (+111.3) (+898.7) Ue the curor to obtai the PO ad tr value. For part b the maximum value of KD reult i the miimum overhoot. 5 79
5-58) (a) Forward path Trafer Fuctio: ( + ) 4500K K K P D G () G() G() c p ( + 361.) 4500KK P Ramp Error Cotat: K lim G( ) 1.458KK v 0 361. e 1 0. 080 0. 001 Thu KK P 80. Let K K P K KK 1 ad 80. v P clear all KP1; K80.; figure(1) um [-KP]; de [0-361.]; Gzpk(um,de,4500*K) Mfeedback(G,1) tep(m) hold o; for KD00.0005:0.0005:0.00; um [-KP/KD]; de [0-361.]; Gzpk(um,de,4500*K*KD) Mfeedback(G,1) tep(m) ed xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') P Zero/pole/gai: 360900 (+1) (+361.) Zero/pole/gai: 360900 (+1) (+0.999) (+3.613e005) Zero/pole/gai: 180.45 (+000) (+361.) Zero/pole/gai: 180.45 (+000) 5 80
(^ + 541.6 + 3.609e005) Zero/pole/gai: 360.9 (+1000) (+361.) Zero/pole/gai: 360.9 (+1000) (^ + 7.1 + 3.609e005) Zero/pole/gai: 541.35 (+666.7) (+361.) Zero/pole/gai: 541.35 (+666.7) (^ + 90.5 + 3.609e005) Zero/pole/gai: 71.8 (+500) (+361.) Zero/pole/gai: 71.8 (+500) (^ + 1083 + 3.609e005) 5 81
K D t r (ec) t (ec) Max Overhoot (%) 0 0.001 0.0166 37.1 0.0005 0.004 0.0081 1.5 0.0010 0.0045 0.00775 1. 0.0015 0.004 0.0065 6.4 0.0016 0.0039 0.00597 5.6 0.0017 0.0038 0.0087 4.8 0.0018 0.0036 0.009 4.0 0.000 0.0033 0.0083.8 5 8
5-59) The forward path Trafer Fuctio: N 0 00( K + K ) P D G () ( + 1)( + 10) To tabilize the ytem, we ca reduce the forward path gai. Sice the ytem i type 1, reducig the gai doe ot affect the teady tate liquid level to a tep iput. Let K P 005. ( + K ) 00 0.05 D G () ( + 1)( + 10) ALSO try other Kp value ad compare your reult. clear all figure(1) KD0 um []; de [0-1 -10]; Gzpk(um,de,00*0.05) Mfeedback(G,1) tep(m) hold o; for KD0.01:0.01:0.1; KD um [-0.05/KD]; Gzpk(um,de,00*KD) Mfeedback(G,1) tep(m) ed xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') KD 0 Zero/pole/gai: 10 (+1) (+10) Zero/pole/gai: 10 (+10.11) (^ + 0.8914 + 0.9893) KD 0.0100 5 83
Zero/pole/gai: (+5) (+1) (+10) Zero/pole/gai: (+5) (+9.889) (^ + 1.111 + 1.011) KD 0.000 Zero/pole/gai: 4 (+.5) (+1) (+10) Zero/pole/gai: 4 (+.5) (+9.658) (^ + 1.34 + 1.035) KD 0.0300 Zero/pole/gai: 6 (+1.667) (+1) (+10) Zero/pole/gai: 6 (+1.667) (+9.413) (^ + 1.587 + 1.06) KD 0.0400 Zero/pole/gai: 8 (+1.5) (+1) (+10) Zero/pole/gai: 8 (+1.5) (+9.153) (^ + 1.847 + 1.093) 5 84
KD 0.0500 Zero/pole/gai: 10 (+1) (+1) (+10) Zero/pole/gai: 10 (+1) (+8.873) (+1.17) (+1) KD 0.0600 Zero/pole/gai: 1 (+0.8333) (+1) (+10) Zero/pole/gai: 1 (+0.8333) (+8.569) (+1.773) (+0.658) KD 0.0700 Zero/pole/gai: 14 (+0.7143) (+1) (+10) Zero/pole/gai: 14 (+0.7143) (+8.3) (+.1) (+0.547) KD 0.0800 Zero/pole/gai: 16 (+0.65) (+1) (+10) Zero/pole/gai: 16 (+0.65) (+7.85) (+.673) (+0.4765) 5 85
KD 0.0900 Zero/pole/gai: 18 (+0.5556) (+1) (+10) Zero/pole/gai: 18 (+0.5556) (+7.398) (+3.177) (+0.455) KD 0.1000 Zero/pole/gai: 0 (+0.5) (+1) (+10) Zero/pole/gai: 0 (+0.5) (+0.3861) (+3.803) (+6.811) 5 86
Uit tep Repoe Attribute: K D t (ec) Max Overhoot (%) 0.01 5.159 1.7 0.0 4.57 7.1 0.03.35 3. 0.04.56 0.8 0.05.71 0 0.06 3.039 0 0.10 4.317 0 Whe K D 005. the rie time i.71 ec, ad the tep repoe ha o overhoot. 5 87
5-60) (a) For e 1, ( K + K ) 00 P D K lim G( ) lim 0K 1 v P 0 0 ( + 1)( + 10) Thu K P 005. Forward path Trafer Fuctio: ( + K ) 00 0.05 D G () ( + 1)( + 10) Becaue of the choice of Kp thi i the ame a previou part. 5-61) (a) Forward path Trafer Fuctio: K I 100 K + P G ( ) + 10+ 100 For ( K+ K) 100 P I K 10, K lim G( ) lim K 10 v v I 0 0 ( + 10+ 100) Thu K I 10. G cl ( ) ( ) ( ) P I P 3 3 ( ) ) 100 K + K 100 K + 10 + 10 + 100+ 100 K + K + 10 + 100(1 + K + 1000 P I P (b) Let the complex root of the characteritic equatio be writte a σ + j σ j The quadratic portio of the characteritic equatio i σ ( σ ) + + + 5 0 3 The characteritic equatio of the ytem i ( ) 15 ad 15. + 10 + 100 + 100K + 1000 0 The quadratic equatio mut atify the characteritic equatio. Uig log diviio ad olve for zero remaider coditio. P 5 88
+ (10 σ ) ( ) 3 σ σ K P + + + 5 + 10 + 100 + 100 + 1000 ( σ ) 3 + + + 5 σ ( σ P ) ( ) ( ) (10 σ) + 100K 15 + 1000 (10 σ) + 0σ 4 σ + (10 σ) + 5 ( KP ) 3 100 + 3σ 0σ 15 + σ 10σ + 450σ 150 3 For zero remaider, σ 10σ + 450σ 150 0 (1) ad 100K P + 3σ 0σ 15 0 () The real olutio of Eq. (1) i σ. 8555. From Eq. (), 15+ 0σ 3σ K P 100 15765. The characteritic equatio root are:. 8555+ j15,. 8555 j15, ad 10 + σ 4.89 (c) Root Cotour: 3 Dividig both ide of ( ) + 10 + 100 + 100K + 1000 0 by the term that do ot cotai K p we have: P 100K P 1+ 1+ G 3 + 10 + 100+ 1000 eq 100K 100K P P G () eq 3 + 10 + 100+ 1000 + 10 + 100 ( )( ) Root Cotour: See Chapter 9 toolbox 9 5 for more iformatio clear all Kp.001; um [100*Kp 0]; de [1 10 100 1000]; rlocu(um,de) 5 89
5 90
5-6) (a) Forward path Trafer Fuctio: K I 100 K + P G ( ) + 10+ 100 For ( K+ K) 100 P I K 10, K lim G( ) lim K 10 v v I 0 0 ( + 10+ 100) Thu the forward path trafer fuctio become 100 10 G () ( + K ) ( + 10+ 100) P G cl ( ) ( ) ( ) P I P 3 3 ( ) ) 100 K+ K 100 K+ 10 + 10 + 100+ 100 K + K + 10 + 100(1 + K + 1000 P I P clear all for Kp.4:0.4:; um [100*Kp 1000]; de [1 10 100 0]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL); tep(gcl) hold o; ed 5 91
Ue the curer to fid the maximum overhoot ad rie time. For example whe K p, PO43 ad tr100%0.15 ec. Trafer fuctio: 00 + 1000 ^3 + 10 ^ + 300 + 1000 5 9
5-63) (a) Forward path Trafer Fuctio: 100 G () ( K+ K) P ( + 10+ 100) I For K v 100, ( K+ K) 100 P I K lim G( ) lim K 100 v I 0 0 ( + 10+ 100) 3 (b) The characteritic equatio i ( ) Routh Tabulatio: + 10 + 100 + 100K + 100K 0 P I Thu K I 100. 3 1 100 + 100K P 1 10 100K 900 P 10,000 0 For tability, 100K 900 > 0 Thu K > 9 P P 0 10,000 7. Activate MATLAB 8. Go to the directory cotaiig the ACSYS oftware. 9. Type i Acy 10. The pre the trafer fuctio Symbolic ad eter the Characteritic equatio 11. The pre the Routh Hurwitz butto 5 93
RH [ 1, 100+100*kp] [ 10, 10000] [ 900+100*kp, 0] [ ( 9000000+1000000*kp)/( 900+100*kp), 0] 5 94
Root Cotour: G eq ( ) 100K 100K P P 3 + 10 + 100+ 10,000 ( + 3. 65)( 6. 85 + j19.4)( 6. 85 j19.4) Root Cotour: See Chapter 9 toolbox 9 5 for more iformatio clear all Kp.001; um [100*Kp 0]; de [1 10 100 10000]; rlocu(um,de) 5 95
(c) K I 100 ( K+ ) 100 100 P G () ( + 10+ 100) The followig maximum overhoot of the ytem are computed for variou value of K P. clear all Kp[15 0 4 5 6 30 40 100 1000]; [N,M]ize(Kp); for i1:m um [100*Kp(i) 10000]; de [1 10 100 0]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL); figure(i) tep(gcl) ed K P 15 0 4 5 6 30 40 100 1000 5 96
y max 1.794 1.779 1.7788 1.7785 1.7756 1.779 1.78 1.795 1.844 1.859 Whe K P 5, miimum y max 1.7756 Ue: cloe all to cloe all the figure widow. 5 97
5-64) MATLAB olutio i the ame a 5-63. (a) Forward path Trafer Fuctio: 100 G () ( K+ K) P ( + 10+ 100) I For K v 100K I 10, K 10 I 100 (b) 3 Characteritic Equatio: ( ) Routh Tabulatio: + 10 + 100 K + 1 + 1000 0 P 3 1 0 1 100 + 100K 10 1000 100K 1000 P 0 P For tability, K P > 0 Root Cotour: G eq ( ) 3 100K + 10 + 100 + 1000 P 5 98
(c) The maximum overhoot of the ytem for differet value of K P ragig from 0.5 to 0 are computed ad tabulated below. K P 0.5 1.0 1.6 1.7 1.8 1.9.0 3.0 5.0 10 0 y max 1.393 1.75 1.317 1.416 1.44 1.441 1.46 1.8 1.37 1.514 1.64 Whe K P 1.7, maximum y max 1.416 5-65) ( ) + + G ( ) K + K + 1+ K K + c P D D1 P where K K K K K I D P I I K K + K K K K K K K P P D1 I D D1 P I I Forward path Trafer Fuctio: 100 G () G() G() c p ( KD + KP+ KI) ( + 10+ 100) Thu Ad reame the ratio: K / K A, K / K B D P I P K K v I KI lim G( ) 100 100 0 100 100 5 99
ForK D beig ufficietly mall: Forward path Trafer Fuctio: ( K+ ) 100 100 P G () ( + 10+ 100) Characteritic Equatio: ( ) 3 K P + 10 + 100 + 100 + 10, 000 0 For tability, K p >9. Select K p 10 ad oberve the repoe. clear all Kp10; um [100*Kp 10000]; de [1 10 100 0]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL) tep(gcl) Trafer fuctio: 1000 + 10000 ----------------------------- ^3 + 10 ^ + 1100 + 10000 Obviouly by icreaig Kp more ocillatio will occur. Add K D to reduce ocillatio. clear all Kp10; Kd; um [100*Kd 100*Kp 10000]; de [1 10 100 0]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL) tep(gcl) 5 100
Trafer fuctio: 00 ^ + 1000 + 10000 ------------------------------ ^3 + 10 ^ + 1100 + 10000 Uit tep Repoe The rie time eem reaoable. But we eed to icreae Kp to improve approach to teady tate. Icreae Kp to Kp30. clear all Kp30; Kd1; um [100*Kd 100*Kp 10000]; de [1 10 100 0]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL) tep(gcl) Trafer fuctio: 100 ^ + 3000 + 10000 ------------------------------ ^3 + 110 ^ + 3100 + 10000 5 101
To obtai a better repoe cotiue adjutig KD ad KP. 5 10
5-66) Thi problem ha received exteded treatmet i Chapter 6, Cotrol Lab ee Sectio 6-6. For the ake implicity, thi problem we aume the cotrol force f(t) i applied i parallel to the prig K ad damper B. We will ot cocer the detail of what actuator or eor are ued. Let look at Figure 4-84 ad equatio 4-3 ad 4-33. m c x c m c k c k m w c x w m x m w k w c w k w c w y k c y (a) (b) (c) Figure 4-84 Quarter car model realizatio: (a) quarter car, (b) degree of freedom, ad (c) 1 degree of freedom model. The equatio of motio of the ytem i defied a follow: mx &&() t + cx& () t + kx() t cy& () t + ky() t (4-3) which ca be implified by ubtitutig the relatio z(t) x(t) y(t) ad o dimeioalizig the coefficiet to the form && zt + zt & + zt && yt (4-33) () ζω () ω () () The Laplace traform of Eq. (4 33) yield the iput output relatiohip 5 103
Z() 1 Y&& () + ζω+ ω (4-34) Now let apply cotrol ee ectio 6-6 for more detail. m x k k f(t) c y For implicity ad better preetatio, we have caled the cotrol force a kf(t) we rewrite (4-34) a: mx &&() t + cx& () t + kx() t cy& () t + ky() t + kf () t && zt ζω zt & ω zt && yt ω f t () + () + () () + () + ζω+ ω A( ) + ω F( ) A () Y&& () (4-34) Settig the cotroller tructure uch that the vehicle bouce Z () X() Y() i miimized: KI F () 0 KP + K D + Z () Z () 1 A () K + ζω+ ω 1+ KP + KD+ I Z () A () + ζω + ω 1+ K + K + K (( ) ) 3 P D I See Equatio (6-4). For proportioal cotrol K D K I 0. 5 104
Pick ς 0.707 ad ω 1 for implicity. Thi i ow a uderdamped ytem. Ue MATLAB to obtai repoe ow. clear all Kp1; Kd0; Ki0; um [-1 0]; de [1 *0.707+Kd 1+Kp Ki]; Gtf(um,de) tep(g) Trafer fuctio: - --------------------- ^3 + 1.414 ^ + Adjut parameter to get the deired repoe if eceary. The proce i the ame for part b, c ad d. 5 105
5-67) Replace F() with KI F () Xref KP + K D + X () B ζω M K ω M X() 1 Xref () KI + ζω+ ω + KP + KD+ Ue MATLAB to obtai repoe ow. clear all Kp1; Kd0; Ki0; B10; K0; M1; omegaqrt(k/m); zeta(b/m)//omega; um [1 0]; de [1 *zeta*omega+kd omega^+kp Ki]; Gtf(um,de) tep(g) rafer fuctio: ------------------- ^3 + 10 ^ + 1 T0 achieve the proper repoe, adjut cotroller gai accordigly. 5 106
5-68) From problem 4-3 a) Rotatioal kietic eergy: Tralatioal kietic eergy: Relatio betwee tralatioal diplacemet ad rotatioal diplacemet: 1 Potetial eergy: A we kow, the: 1 1 1 By differetiatig, we have: 0 0 Sice caot be zero, the 0 b) 0 c) 5 107
1 1 1 where at the maximum eergy. 1 1 The: 1 1 Or: J d) G () ( m + K) % elect value of m, J ad K K100; J5; m5; Gtf([J],[m 0 K]) Pole(G) impule(g,10) xlabel( 'Time(ec)'); ylabel('amplitude'); Trafer fuctio: 5 ------------ 5 ^ + 100 a 0 +.0000i 0 -.0000i 5 108
Ucotrolled With a proportioal cotroller oe ca adjut the ocillatio amplitude the trafer fuctio i rewritte a: G () cl JK p ( m + K + JK p ) % elect value of m, J ad K Kp0.1 K100; J5; m5; Gtf([J*Kp],[m 0 (K+J*Kp)]) Pole(G) impule(g,10) xlabel( 'Time(ec)'); ylabel('amplitude'); 5 109
Kp 0.1000 Trafer fuctio: 0.5 -------------- 5 ^ + 100.5 a 0 +.0050i 0 -.0050i A PD cotroller mut be ued to damp the ocillatio ad reduce overhoot. Ue Example 5-11-1 a a guide. 5 110
5-69) From Problem 4-6 we have: a) y K( y 1 y ) K( y 1 y ) y 1 μmg& y μmg& y 1 b) From Newto Law: c) If y 1 ad y are coidered a a poitio ad v 1 ad v a velocity variable The: The output equatio ca be the velocity of the egie, which mea Obtaiig require olvig above equatio with repect to Y () From the firt equatio: Subtitutig ito the ecod equatio: 5 111
By olvig above equatio: 1 Replace Force F with a proportioal cotroller o that FK(Z-Z ref ): K P Z ref _ F Z 1 1 1 5-70) Alo ee derivatio i 4-9. x(t) L M F L θ Here i a alterative repreetatio icludig frictio (dampig) μ. I thi cae the agle θ i meaured differetly. Let fid the dyamic model of the ytem: 5 11
1) ) Let. If Φ i mall eough the 1 ad, therefore which give: Φ F Igorig frictio 0. Φ F where ; Igorig actuator dyamic (DC motor equatio), we ca icorporate feedback cotrol uig a erie PD compeator ad uity feedback. Hece, K P +K D R _ F Φ ( ( ) ) ( ) () p D ( ) F K R Φ KR Φ The ytem trafer fuctio i: Φ R ( Cotrol i achieved by eurig tability (K p >B) A ( Kp + KD) D + ( p ) + K A K B 5 113
Ue Routh Hurwitz to etablih tability firt. Ue Acy to do that a demotrated i thi chapter problem. Alo Chapter ha may example. Ue MATLAB to imulate repoe: clear all Kp10; Kd5; A10; B8; um [A*Kd A*Kp]; de [1 Kd A*(Kp-B)]; Gtf(um,de) tep(g) Trafer fuctio: 50 + 100 -------------- ^ + 5 + 0 Adjut parameter to achieve deired repoe. Ue THE PROCEDURE i Example 5-11-1. You may look at the root locu of the forward path trafer fuctio to get a better perpective. 5 114
( K + K ) AK ( z+ ) Φ A E AB fix z ad vary K. p D D clear all z100; Kd0.01; A10; B8; um [A*Kd A*Kd*z]; de [1 0 -(A*B)]; Gtf(um,de) rlocu(g) Trafer fuctio: 0.1 + 10 ---------- ^ 80 D AB For z10, a large K D 0.805 reult i: 5 115
clear all Kd0.805; Kp10*Kd; A10; B8; um [A*Kd A*Kp]; de [1 Kd A*(Kp-B)]; Gtf(um,de) pole(g) zero(g) tep(g) Trafer fuctio: 8.05 + 80.5 ------------------- ^ + 0.805 + 0.5 a -0.405 + 0.5814i -0.405-0.5814i a -10 Lookig at domiat pole we expect to ee a ocillatory repoe with overhoot cloe to deired value. For a better deig, ad to meet rie time criterio, ue Example 5-11-1. 5 116