Chapter (a) ζ. ω. 5 2 (a) Type 0 (b) Type 0 (c) Type 1 (d) Type 2 (e) Type 3 (f) Type 3. (g) type 2 (h) type (a) K G s.

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1 Chapter (a) ζ. ω rad / ec (b) 0 ζ ω rad / ec (c) ζ ω 5 rad / ec (d) 0. 5 ζ 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 (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( ) a 0

2 (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 (a) Iput Error Cotat Steady tate Error u t K 1000 p 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

3 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 ( ) ( ) a 3, a 3, a, b 1, b Uit tep Iput: e 1 bk 0 H 1 K H a

4 Uit ramp iput: a b K 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 GH ( ) ( ) Uit tep Iput: e 1 bk 1 5 K a 5 5 H 0 0 H Uit ramp Iput: i 0: a b K 0 i 1: a b K H 1 1 H e a b K 1 1 ak 0 H H Uit parabolic Iput: e (c) K H ( 0) 1/ 5 H G( ) M( ) 1 + GH ( ) ( ) + 5 The ytem i table a 1, a 1, a 50, a 15, b 5, b Uit tep Iput: e 1 bk 5/5 K a 1 H 0 0 H Uit ramp Iput: 5 4

5 i 0: a b K 0 i 1: a b K 4/ 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 GH ( ) ( ) a 10, a 5, a 1, b 1, b 0, b Uit tep Iput: e 1 bk 1 10 K a H 0 0 H Uit ramp Iput: i 0: a b K 0 i 1: a b K H 1 1 H e a b K 1 1 ak 0 H H Uit parabolic Iput: e 5 6 (a) M ( ) + 4 K 4 3 H 1 The ytem i table Uit tep Iput: a 4, a 4, a 48, a 16, b 4, b 1, b 0, b e 1 bk 4 K a 4 H 0 0 H Uit ramp iput: i 0: a b K 0 i 1: a b K H 1 1 H 5 5

6 e a b K 1 1 ak 0 H H Uit parabolic Iput: e (b) M( ) K( + 3) K 3 H 1 The ytem i table for K > ( K+ ) + 3K a 3K, a K+, a 3, b 3K, b K Uit tep Iput: e 1 bk 3K K a 3K H 0 0 H Uit ramp Iput: i 0: a b K 0 i 1: a b K K + K 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 H ( ) H( ) K lim ( ) 4 3 H Uit tep Iput: a 0, a 10, a 50, a 15, b 5, b e 1 a bk H K H a Uit ramp Iput: 5 6

7 Uit parabolic Iput: e e (d) M ( ) K( + 5) K H 1 The ytem i table for 0 < K < K + 5K a 5K, a 5K, a 60, a 17, b 5K, b K Uit tep Iput: e 1 bk 5K K a 5K H 0 0 H Uit ramp Iput: i 0: a b K 0 i 1: a b K 5K K 4K 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 <

8 5-7) Type of the ytem i zero Pole: i, i, ad Zero: ) 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

9 Regardig the reult of problem 5-8, c) Steady tate error for parabolic iput: Regardig the reult of problem 5-8, ) 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:

10 5-1) The tep iput repoe of the ytem i: Therefore: 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: ) Y( ) G ( ) E( ) KG ( ) 0 p 100K 1+ KG ( ) 0( K ) t p t Type 1 ytem. 5K Error cotat: K, K, K 0 p v a K t (a) rt () u(): t e (b) r() t tu (): t e (c) rt () tu ()/ t : e 1 K a K p 1 1 K K 5K v t 5 10

11 5 14 G p 100 Y( ) KG ( ) p ( ) G ( ) ( )( ) E( ) 0 1+ K G ( ) 100K G ( ) 0 ( )( ) + 100K t t p 5K Error cotat: K, K, K 0 p v a K t (a) rt () u(): t e (b) r() t tu (): t e (c) rt () tu ()/ t : e 1 K a K p 1 1 K 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 K 1 100K K 100K 1 100K t t Stability Coditio: K > 0 ( K ) K 1 t K > 0 or > t 5K 1 Thu, the miimum teady tate error that ca be obtaied with a uit ramp iput i 1/

12 5 15 (a) From Figure 3P 9, KK KK + KKKK 1 i b 1 i t 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 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 ( ) 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

13 5 16 (a) Forward path trafer fuctio: [(t) 0]: K( ) Y( ) ( + 5) K( ) 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 K ( ) 5K K K t 5 K Stability Coditio: ( ) K > 0 5 K 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( ) KK + + K K + + K ( ) 3 ( ) t t Steady State Output due to (t): ( + 5) ( + 5) y lim y( t) lim Y( ) 1 if the limit i valid. t

14 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

15 (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 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

16 5 18 Percet maximum overhoot 0.5 e πζ 1 ζ Thu ( ) πζ 1 ζ l π ζ ζ Solvig for ζ from the lat equatio, we have ζ Peak Time t max ω π 1 ζ 001. ec. Thu, ω π ( 0.404) rad / ec Trafer Fuctio of the Secod order Prototype Sytem: Y( ) ω R ( ) + ζω + ω

17 Exteded MATLAB olutio of problem imilar to appear later o e.g Cloed Loop Trafer Fuctio: Characteritic equatio: Y( ) 5K R K K ( ) () t ( ) K K t For a ecod order prototype ytem, whe the maximum overhoot i 4.3%, ζ Rie Time: ω 5K, ζω K K t ζ ζ ec Thu ω 108. rad / ec t r ω ω Thu, K ω ( 10. 8) + K ω Thu K t t With K 4.68 ad K t , the ytem trafer fuctio i Y ( ) 117 R ( ) Uit tep Repoe: y 0.1 at t ec. y 0.9 at t 0.44 ec. t r ec. y max ( 4.3% max. overhoot) 5 17

18 5 0 Cloed loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t πζ Whe Maximum overhoot 10%, l πζ ( ) ζ Solvig for ζ, we get ζ ζ The Natural udamped frequecy i ω 5K Thu, K ζω 118. ω Rie Time: t ζ. 917ζ. t r. ω ω ec. Thu ω rad / ec ω K Thu K t With K 1.58 ad K t , the ytem trafer fuctio i Y ( ) 313 R ( ) Uit tep Repoe: y 0.1 whe t 0.08 ec. y 0.9 whe t ec. t r ec. y max 11. ( 10% max. overhoot) 5 18

19 5 1 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y ( ) 5K R ( ) + ( K ) + 5K ( ) t t πζ K + 5K 0 Whe Maximum overhoot 0%, l πζ ( ).59 1 ζ Solvig for ζ, we get ζ ζ The Natural udamped frequecy ω 5K K ζω 0. 91ω Rie Time: ζ. 917ζ t r 005. ω ω t ec. Thu, ω ω K K 0. 91ω 584. Thu, K t t 5 With K 3.1 ad K t , the ytem trafer fuctio i Y ( ) R ( ) Uit tep Repoe: y 0.1 whe t ec. y 0.9 whe t 0.07 ec. t r ec. y max 1. ( 0% max. overhoot) 5 19

20 5 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t Delay time t d ζ ζ ω 01. ec Whe Maximum overhoot 4.3%, ζ t d 01. ω ec. Thu ω 14.3 rad/ec. ω 14.3 K K t ζω 1.414ω Thu K t With K 0.1 ad K t , the ytem trafer fuctio i Uit Step Repoe: Y ( ) 0. 5 R ( ) Whe y 0.5, t ec. Thu, t d ec. y max ( 4.3% max. overhoot) 5 0

21 5 3 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t Delay time t d ζ ζ ω ω Thu, ω ω 6.74 K K ζω Thu K t t With K 8.6 ad K t , the ytem trafer fuctio i Y ( ) 715 R ( ) Uit Step Repoe: y 0.5 whe t ec. Thu, t d ec. y max ( % max. overhoot) 5 1

22 5 4 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t For Maximum overhoot 0., ζ Delay time t d ζ ζ ω ω ec Natural Udamped Frequecy ω rad/ec. Thu, 001. ω K ζω Thu, K t K t With K 69.5 ad K t 0.188, the ytem trafer fuctio i Y ( ) R ( ) Uit tep Repoe: y 0.5 whe t ec. Thu, t d ec. y max 1. ( 0% max. overhoot) 5

23 5 5 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t ζ 0. 6 ζω K 1.ω t Settlig time t 01. ζω 06. ω 3. ec. Thu, ω rad / ec 006. K t 1.ω 5 ω K Sytem Trafer Fuctio: Y ( ) 844 R ( ) Uit tep Repoe: y(t) reache 1.00 ad ever exceed thi value at t ec. Thu, t ec. 5 3

24 5 6 (a) Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t For maximum overhoot 0.1, ζ ζω 0. 59ω 118. ω K t Settlig time: t ζω 059. ω 3. ec. ω K t 118. ω 5 ω 0.46 K Sytem Trafer Fuctio: Y ( ) R ( ) Uit Step Repoe: y(t) reache 1.05 ad ever exceed thi value at t ec. Thu, t ec. (b) For maximum overhoot 0., ζ ζω 0. 91ω K t Settlig time t 001. ec. ω ζω 0.456ω rad / ec K t ω Sytem Trafer Fuctio: Y ( ) Uit Step Repoe: R ( )

25 y(t) reache 1.05 ad ever exceed thi value at t ec. Thu, t ec. Thi i le tha the calculated value of 0.01 ec. 5 5

26 5 7 Cloed Loop Trafer Fuctio: Characteritic Equatio: Y( ) 5K R K K ( ) () t ( ) K K t Dampig ratio ζ Settlig time t Sytem Trafer Fuctio: 4.5ζ ec. Thu, ω rad/ec. ω ω ω K ζω Thu, K K t t ζ Y ( ) 101. R ( ) Uit Step Repoe: The uit tep repoe reache 0.95 at t 0.09 ec. which i the meaured t. 5 6

27 5 8 (a) Whe ζ 05., the rie time i ζ ζ t r ω ω ec. Thu ω 151. rad/ec. The ecod order term of the characteritic equatio i writte + ζω + ω The characteritic equatio of the ytem i + ( a+ 30) + 30a+ K 0 Dividig the characteritic equatio by , we have For zero remaider, 8.48 a Thu, a 16. K a Forward Path Trafer Fuctio: G ( ) ( )( + 30) Uit Step Repoe: y 0.1 whe t ec. y 0.9 whe t 1.43 ec. Rie Time: t r ec. 5 7

28 (b) The ytem i type 1. (i) For a uit tep iput, e 0. (ii) For a uit ramp iput, K G v K lim ( ) 1.45 e a K v 5 9 (a) Characteritic Equatio: ( + K) K 0 Apply the Routh Hurwitz criterio to fid the rage of K for tability. Routh Tabulatio: K K 3 K K Stability Coditio: 1.5 < K < 0 Thi implifie the earch for K for two equal root. Whe K , the characteritic equatio root are: 0.347, 0.347, ad (b) Uit Step Repoe: (K ) 5 8

29 (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

30 5 30 (a) The tate equatio of the cloed loop ytem are: dx dt 1 dx x + 5x 6x k x k x + r dt The characteritic equatio of the cloed loop ytem i Δ + ( 1+ k ) + ( 30+ 5k + k ) k + k 1 For ω 10 rad / ec, k + k ω 100. Thu 5k + k k (b) For ζ , ζω 1+ k. Thu ω ( 1+ k ) ω k + k Thu k k 1 1 (c) For ω ζ 10 rad / ec ad , 5k + k 100 ad 1 + k ζω Thu k Solvig for k, we have k (d) The cloed loop trafer fuctio i Y( ) 5 5 R k k k ( ) ( ) ( ) For a uit tep iput, lim y( t ) lim Y ( ) 005. t (e) For zero teady tate error due to a uit tep iput, k + k 5 Thu 5k + k

31 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 ( ) 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

32 (d) See parameter plae i part (g). (e) Parabolic error cotat K a 1000 ec ( ) K G K K K K lim ( ) lim Thu 10 a P D P P 0 0 (f) Natural udamped frequecy ω 50 rad/ec. ω K P Thu K P (g) Whe K P 0, K K D G ( ) D (pole zero cacellatio) 5 3

33 5 3 (a) Forward path Trafer Fuctio: Y() KK 10K i G () E ( ) J(1 T) KK K [ + + ] 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 T ( ) J(1 + T) + K K + KK K + 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 K 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 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 t K or K t 5 33

34 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 or + ( + 5) + (5 + ) + 0 The, a a 5 ab 1000 b 00 5a + b 10 K K The three root are: a 5 a 5. 5± j t t 5-33) Rie time: Peak time: Maximum overhoot:. Settlig time:

35 5-34) K ( J + a + KK f ) exp 0. ξ ω

36 5-35) a) Subtitutig ito equatio () give: 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 By lookig at the Laplace traform fuctio table: where co i 1 b) 5 36

37 Therefore: A a reult: which mea: 1 1 The uit tep repoe i: 1 1 Therefore a a reult: 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

38 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,,,

39 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: ^ To fid the tep repoe H11, H1, ad H1 follow the ame procedure. Other part are the ame. 5 39

40 5-37) Impule repoe: a).. b) c) where α co -1 ξ ad 1, therefore, 1 i 1 1 i i ) 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: ^

41 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:

42 (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 (a) K t oz - i / rad The Forward Path Trafer Fuctio: K G () ( ) K ( + 116)( )( j3178.3)( j3178.3) Routh Tabulatio: K K K 9 10 K K K K K 5 4

43 From the 1 row, the coditio of tability i K. 84K > 0 or K K < 0 or ( K )( K ) < 0 Stability Coditio: 0 < K < The critical value of K for tability i With thi value of K, the root of the characteritic equatio are: , j3113.3, j3113.3, j75.68, ad j75.68 (b) K L 1000 oz i/rad. The forward path trafer fuctio i K G () ( ) K ( )( )( j1005)( j1005) (c) Characteritic Equatio of the Cloed Loop Sytem: K 0 Routh Tabulatio: K K K 9 10 K K K K K 5 43

44 4 From the 1 row, the coditio of tability i K. 83K > 0 Or, K K < 0 or ( K 1400)( K ) < 0 Stability Coditio: 0 < K < 1400 The critical value of K for tability i With thi value of K, the characteritic equatio root are: , j676, j676, j748.44, ad j (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 K K ( + 116)( ) ( ) Characteritic Equatio of the Cloed Loop Sytem: K 0 Routh Tabulatio: K K K From the 1 row, the coditio of K for tability i K >

45 Stability Coditio: 0 < K < The critical value of K for tability i With K , the characteritic equatio root are 5000, j75.79, ad j 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 (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 K 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

46 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 ( + ) ( α) + 00α Output Due to Uit Step Iput: Y( ) 3 100( + ) ( α) + 00α 00 1 y lim y( t) lim Y( ) t 0 00α α (e) rt () 0, dt () u() t G ( ) c + α 5 46

47 Sytem Trafer Fuctio [r(t) 0] Y( ) D ( ) r ( ( α) + 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 ( + ) ( + ) ( + ) 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

48 5 4 (a) Forward Path Trafer fuctio: Characteritic Equatio: H( ) 10N N G ( ) E( ) ( + 1)( + 10) ( + 1) + + N 0 N1: Characteritic Equatio: ζ 05. ω 1 rad/ec. Maximum overhoot e πζ 1 ζ (16.3%) Peak time t max ω π 1 ζ ec. N10: Characteritic Equatio: ζ ω 10 rad/ec. πζ 1 ζ Maximum overhoot e (60.5%) Peak time t (b) Uit Step Repoe: N 1 max ω π 1 ζ ec. 5 48

49 Secod order Sytem Third order Sytem Maximum overhoot Peak time 3.68 ec ec. Uit Step Repoe: N 10 Secod order Sytem Third order Sytem Maximum overhoot Peak time ec ec. 5 49

50 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

51 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 > , the cloed loop ytem i table. 5 51

52 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 clear all um []; de [ ]; 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 [ ]; 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: (+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

53 (b) clear all for Tz[ ]; 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: ^ + + Trafer fuctio: ^ + + Trafer fuctio: ^

54 Trafer fuctio: ^ + + 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

55 Kk ( J + a + KK k) f Accordig to above block diagram: Now coider X 0, the: k K J + a K f +1 Accordigly, ad 1 I thi cae, E() D-KY()

56 Now the teady tate error ca be eaily calculated by: 1 1/: lim 1 1 lim lim : 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 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

57 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 + 1)( + 1)...( m + 1) 5 57

58 (b) Calculate Recall 1 K 1 lim G( ) 0 G () 1 E ( ) X() Y() X() X() X() 1 + G ( ) 1 + G ( ) Hece lim E ( ) lim X( ) lim lim G ( ) 0 + G ( ) 0 G ( ) K Ramp Error Cotat v 5 58

59 5-48) Comparig with the ecod order prototype ytem ad matchig deomiator: exp 10.5 Let ξ 0.4 ad ω 80 The , 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: ^

60 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

61 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 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

62 5-51) a) By ubtitutig the value: b) Speed of the motor i lim (c) d) A a reult: e) A exp 0., the, ξ Accordig to the trafer fuctio, 0.109, the where 0. K< f) A..., the 0.6, a 0., therefore; / 5 6

63 g) MATLAB clear all for K[0.5 1 ]; t0:0.001:15; um [0.*K]; de [ *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: ^ Trafer fuctio: ^ Trafer fuctio: ^

64 Rie time decreae with K icreaig. Overhoot icreae with K. 5 64

65 5-5) exp 0.1, therefore, ξ 0.59 A., the, 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: 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: ^ ^

66 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 clear all um[ 1 3]; deom[ 3+qrt(9 40) 3 qrt(9 40) 0.0+qrt( ) 0.0 qrt( ) 10]; Gzpk(um,deom,60) rlocu(g) Zero/pole/gai: 60 (+1) (+) (+3) (+10) (^ ) (^ ) 5 66

67 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

68 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: (+3) (+) (+1) (+7.048) (^ ) (^ ) 5 68

69 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) (^ ) (^ ) >> 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

70 K5 Mfeedback(G*5,1) %See toolbox 5-4- tep(m) Zero/pole/gai: 300 (+3) (+) (+1) (+4.434) (^ ) (^ ) 5 70

71 5 71

72 5-54) Forward path Trafer Fuctio: M( ) K G ( ) 3 1 M( ) + ( 0 + a) + ( a) + 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 a a 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

73 G () 000 ( ) ( + + ) ( + + ) G () G () c G () p MATLAB The maximum overhoot of the uit tep repoe i 0 percet. clear all K000; a10; um []; de [-10+10i i -a]; Gzpk(um,de,K) tep(g); xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Zero/pole/gai: (+10) (^ ) Clearly PO

74 5-55) Forward path Trafer Fuctio: M( ) K G ( ) 3 1 M( ) + ( 0 + a) + ( a) + 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 a a 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

75 G () ( ) ( + + ) ( + + ) G () G () c G ( ) 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 i -a]; Gzpk(um,de,K) tep(g); xlabel('time(ec)') ylabel('y(t)') title('uit-tep repoe of the ytem') Zero/pole/gai: (+90) (^ ) PO i le tha

76 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

77 Kp 10 Mum ( *Kd*)//(+10) Mde 1+( *Kd*)//(+10) a (^+10* *Kd*)//(+10) Characteritic Equatio: ( ) Match with a d order prototype ytem K K 0 D P ω 1000K P rad/ec ζω K D olve( *kd 100) a 9/ Thu K D Ue the ame procedure for other part. (b) For K v 1000 ad ζ , ad from part (a), ω 100 rad/ec, ζω K D Thu K D (c) For K v 1000 ad ζ 1.0, ad from part (a), ω 100 rad/ec, 190 ζω K D Thu K D

78 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 ) 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) (^ e005) Zero/pole/gai: 1000 (+50) (+434.1) (+575.9) Zero/pole/gai: 1000 (+166.7) (+07.7) (+80.3) 5 78

79 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

80 5-58) (a) Forward path Trafer Fuctio: ( + ) 4500K K K P D G () G() G() c p ( ) 4500KK P Ramp Error Cotat: K lim G( ) 1.458KK v e 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 KD :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: (+1) (+361.) Zero/pole/gai: (+1) (+0.999) (+3.613e005) Zero/pole/gai: (+000) (+361.) Zero/pole/gai: (+000) 5 80

81 (^ e005) Zero/pole/gai: (+1000) (+361.) Zero/pole/gai: (+1000) (^ e005) Zero/pole/gai: (+666.7) (+361.) Zero/pole/gai: (+666.7) (^ e005) Zero/pole/gai: 71.8 (+500) (+361.) Zero/pole/gai: 71.8 (+500) (^ e005) 5 81

82 K D t r (ec) t (ec) Max Overhoot (%)

83 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 ) D G () ( + 1)( + 10) ALSO try other Kp value ad compare your reult. clear all figure(1) KD0 um []; de [ ]; 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) (^ ) KD

84 Zero/pole/gai: (+5) (+1) (+10) Zero/pole/gai: (+5) (+9.889) (^ ) KD Zero/pole/gai: 4 (+.5) (+1) (+10) Zero/pole/gai: 4 (+.5) (+9.658) (^ ) KD Zero/pole/gai: 6 (+1.667) (+1) (+10) Zero/pole/gai: 6 (+1.667) (+9.413) (^ ) KD Zero/pole/gai: 8 (+1.5) (+1) (+10) Zero/pole/gai: 8 (+1.5) (+9.153) (^ ) 5 84

85 KD Zero/pole/gai: 10 (+1) (+1) (+10) Zero/pole/gai: 10 (+1) (+8.873) (+1.17) (+1) KD Zero/pole/gai: 1 ( ) (+1) (+10) Zero/pole/gai: 1 ( ) (+8.569) (+1.773) (+0.658) KD Zero/pole/gai: 14 ( ) (+1) (+10) Zero/pole/gai: 14 ( ) (+8.3) (+.1) (+0.547) KD Zero/pole/gai: 16 (+0.65) (+1) (+10) Zero/pole/gai: 16 (+0.65) (+7.85) (+.673) ( ) 5 85

86 KD Zero/pole/gai: 18 ( ) (+1) (+10) Zero/pole/gai: 18 ( ) (+7.398) (+3.177) (+0.455) KD Zero/pole/gai: 0 (+0.5) (+1) (+10) Zero/pole/gai: 0 (+0.5) ( ) (+3.803) (+6.811) 5 86

87 Uit tep Repoe Attribute: K D t (ec) Max Overhoot (%) Whe K D 005. the rie time i.71 ec, ad the tep repoe ha o overhoot. 5 87

88 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 ) 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 ( ) For ( K+ K) 100 P I K 10, K lim G( ) lim K 10 v v I 0 0 ( ) Thu K I 10. G cl ( ) ( ) ( ) P I P 3 3 ( ) ) 100 K + K 100 K K + K (1 + K 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 ) The characteritic equatio of the ytem i ( ) 15 ad K The quadratic equatio mut atify the characteritic equatio. Uig log diviio ad olve for zero remaider coditio. P 5 88

89 + (10 σ ) ( ) 3 σ σ K P ( σ ) σ ( σ P ) ( ) ( ) (10 σ) + 100K (10 σ) + 0σ 4 σ + (10 σ) + 5 ( KP ) σ 0σ 15 + σ 10σ + 450σ For zero remaider, σ 10σ + 450σ (1) ad 100K P + 3σ 0σ 15 0 () The real olutio of Eq. (1) i σ From Eq. (), 15+ 0σ 3σ K P The characteritic equatio root are: j15, j15, ad 10 + σ 4.89 (c) Root Cotour: 3 Dividig both ide of ( ) K by the term that do ot cotai K p we have: P 100K P G eq 100K 100K P P G () eq ( )( ) Root Cotour: See Chapter 9 toolbox 9 5 for more iformatio clear all Kp.001; um [100*Kp 0]; de [ ]; rlocu(um,de) 5 89

90 5 90

91 5-6) (a) Forward path Trafer Fuctio: K I 100 K + P G ( ) For ( K+ K) 100 P I K 10, K lim G( ) lim K 10 v v I 0 0 ( ) Thu the forward path trafer fuctio become G () ( + K ) ( ) P G cl ( ) ( ) ( ) P I P 3 3 ( ) ) 100 K+ K 100 K K + K (1 + K P I P clear all for Kp.4:0.4:; um [100*Kp 1000]; de [ ]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL); tep(gcl) hold o; ed 5 91

92 Ue the curer to fid the maximum overhoot ad rie time. For example whe K p, PO43 ad tr100%0.15 ec. Trafer fuctio: ^ ^

93 5-63) (a) Forward path Trafer Fuctio: 100 G () ( K+ K) P ( ) I For K v 100, ( K+ K) 100 P I K lim G( ) lim K 100 v I 0 0 ( ) 3 (b) The characteritic equatio i ( ) Routh Tabulatio: K + 100K 0 P I Thu K I K P K 900 P 10,000 0 For tability, 100K 900 > 0 Thu K > 9 P P 0 10, 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

94 RH [ 1, *kp] [ 10, 10000] [ *kp, 0] [ ( *kp)/( *kp), 0] 5 94

95 Root Cotour: G eq ( ) 100K 100K P P ,000 ( )( j19.4)( j19.4) Root Cotour: See Chapter 9 toolbox 9 5 for more iformatio clear all Kp.001; um [100*Kp 0]; de [ ]; rlocu(um,de) 5 95

96 (c) K I 100 ( K+ ) P G () ( ) The followig maximum overhoot of the ytem are computed for variou value of K P. clear all Kp[ ]; [N,M]ize(Kp); for i1:m um [100*Kp(i) 10000]; de [ ]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL); figure(i) tep(gcl) ed K P

97 y max Whe K P 5, miimum y max Ue: cloe all to cloe all the figure widow. 5 97

98 5-64) MATLAB olutio i the ame a (a) Forward path Trafer Fuctio: 100 G () ( K+ K) P ( ) I For K v 100K I 10, K 10 I 100 (b) 3 Characteritic Equatio: ( ) Routh Tabulatio: K P K K 1000 P 0 P For tability, K P > 0 Root Cotour: G eq ( ) 3 100K P 5 98

99 (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 y max Whe K P 1.7, maximum y max ) ( ) + + 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) ( ) Thu Ad reame the ratio: K / K A, K / K B D P I P K K v I KI lim G( )

100 ForK D beig ufficietly mall: Forward path Trafer Fuctio: ( K+ ) P G () ( ) Characteritic Equatio: ( ) 3 K P , For tability, K p >9. Select K p 10 ad oberve the repoe. clear all Kp10; um [100*Kp 10000]; de [ ]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL) tep(gcl) Trafer fuctio: ^ ^ 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 [ ]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL) tep(gcl) 5 100

101 Trafer fuctio: 00 ^ ^ ^ 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 [ ]; [umcl,decl]cloop(um,de); GCLtf(umCL,deCL) tep(gcl) Trafer fuctio: 100 ^ ^ ^

102 To obtai a better repoe cotiue adjutig KD ad KP. 5 10

103 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 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

104 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

105 Pick ς 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: ^ ^ + Adjut parameter to get the deired repoe if eceary. The proce i the ame for part b, c ad d

106 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: ^ ^ + 1 T0 achieve the proper repoe, adjut cotroller gai accordigly

107 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: By differetiatig, we have: 0 0 Sice caot be zero, the 0 b) 0 c) 5 107

108 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: ^ a i i 5 108

109 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

110 Kp Trafer fuctio: ^ a i i A PD cotroller mut be ued to damp the ocillatio ad reduce overhoot. Ue Example a a guide

111 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

112 By olvig above equatio: 1 Replace Force F with a proportioal cotroller o that FK(Z-Z ref ): K P Z ref _ F Z ) 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

113 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

114 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: ^ Adjut parameter to achieve deired repoe. Ue THE PROCEDURE i Example You may look at the root locu of the forward path trafer fuctio to get a better perpective

115 ( 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: ^ 80 D AB For z10, a large K D reult i: 5 115

116 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: ^ a i i 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