Quy dao nghiem

Kho st ng dng MATLAB trong iu khin t ng

NHM LNH V QU O NGHIM (Roots Locus)1. Lnh PZMAP a) Cng dng: V biu cc-zero ca h thng. b) C php: [p,z]= pzmap(num,den) [p,z]= pzmap(a,b,c,d) [p,z]= pzmap(a,b,c,d) c) Gii thch: Lnh pzmap v biu cc-zero ca h LTI. i vi h SISO th cc cc v zero ca hmtruyn c v. Nu b qua cc i s ng ra th lnh pzmap s v ra biu cc-zero trn mn hnh. pzmap l phng tin tm ra cc cc v zero tuyn t ca h MIMO. pzmap(a,b,c,d) v cc cc v zero ca h khng gian trng thi trong mt phng phc. i vi cc h thng MIMO, lnh s v tt c cc zero truyn t t tt c cc ng vo ti tt c cc ng ra. Trong mt phng phc, cc cc c biu din bng du cn cc zero c biu din bng du o. pzmap(num,den) v cc cc v zero ca hm truyn trong mt phng phc. Vector num v den cha cc h s t s v mu s theo chiu gim dn s m ca s. pzmap(p,z) v cc cc v zero trong mt phng phc. Vector ct p cha ta cc cc v vector ct z cha ta cc zero trong mt phng phc. Lnh ny v cc cc v zero c tnh sn trong mt phng phc. Nu gi li cc i s ng ra th : [p,z]= pzmap(num,den) [p,z]= pzmap(a,b,c,d) [p,z]= pzmap(a,b,c,d) to ra cc ma trn p v z trong p cha cc cc cn z cha cc zero. d) V d: (Trch trang 11-174 sch Control system Toolbox) V cc cc v zero ca h lin tc c hm truyn : 2 s 2 + 5s + 1 H (s) = 2 s + 2s + 3 num = [2 5 1]; den = [1 2 3]; pzmap(num,den) title(Bieu do cuc-zero)


2. Lnh RLOCFIND a) Cng dng: Tm li qu o nghim vi tp hp nghim cho trc. b) C php: [k,poles]= rlocfind(a,b,c,d) [k,poles]= rlocfind(num,den) [k,poles]= rlocfind(a,b,c,d,p) [k,poles]= rlocfind(num,den,p) c) Gii thch: Lnh rlocfind to ra li qu o nghim kt hp vi cc cc trn quo nghim. Lnh rlocfind c dng cho h SISO lin tc v gin on. [k,poles]= rlocfind(a,b,c,d) to ra du x trong ca s ha m ta dng chn mt im trn qu o nghim c sn. li ca im ny c to ra trong k v cc cc ng vi li ny nm trong poles. s dng lnh ny th qu o nghim phi c sn trong ca s ha. [k,poles]= rlocfind(num,den) to ra du x trong ca s ha m ta dng chn mt im trn qu o nghim ca h thng c hm truyn G = num/den trong c num v den cha cc h s a thc theo chiu gim dn s m ca s hoc z. [k,poles]= rlocfind(a,b,c,d,p) hoc [k,poles]= rlocfind(num,den,p) to ra vector li k v vector cc cc kt hp pole vi mi thnh phn trong mi vector ng vi mi nghim trong p. d) V d: (Trch t trang 11-180 sch Control System Toolbox) Xc nh li hi tip cc cc vng kn ca h thng c h s tt dn = 0.707 v c hm truyn :

Kho st ng dng MATLAB trong iu khin t ng 2 s 2 + 5s + 1 H (s) = 2 s + 2s + 3 num = [2 5 1]; den = [1 2 3]; % V qu o nghim: rlocus(num,den);title(Do loi quy dao nghiem); % Tm li ti im c chn: rlocfind(num,den); Sau khi nhp xong lnh, trn mn hnh ca Matlab s xut hin dng ch: Select a point in the graphics window v trn hnh v c thc ta ko chut v chn im ta c qu o nghim:

3. Lnh RLOCUS a) Cng dng: Tm qu o nghim Evans. b) C php: r = rlocus(num,den) r = rlocus(num,den,k) r = rlocus(a,b,c,d) r = rlocus(a,b,c,d,k) c) Gii thch:

Kho st ng dng MATLAB trong iu khin t ng Lnh rlocus tm qu o nghim Evans ca h SISO. Qu o nghim c dng nghin cu nh hng ca vic thay i li hi tip ln v tr cc ca h thng, cung cp cc thng tin v p ng thi gian v p ng tn s. i vi i tng iu khin c hm truyn G(s) v khu b chnh hi tip k*f(s), hm truyn vng kn l : g ( s) g ( s) h( s ) = = 1 + kg ( s ) f ( s ) q ( s ) Nu b qua cc i s ng ra th lnh rlocus s v ra qu o trn mn hnh. Lnh rlocus dng cho c h lin tc v gin on. r = rlocus(num,den) v qu o nghim ca hm truyn : num( s) q(s) = 1 + k =0 den( s) vi vector li k c xc nh t ng. Vector num v den ch ra h t s v mu s theo chiu gim dn s ca s hoc z. num( s ) num(1) s nn 1 + num(2) s nn 2 + ...... + num(nn) = den( s ) den(1) s nd 1 + den(2) s nd 2 + ...... + den(nd ) r = rlocus(a,b,c,d) v ra qu o nghim ca h khng gian trng ti SISO lin tc v gin on vi vector li c xc nh t ng r = rlocus(num,den,k) hoc r = rlocus(a,b,c,d,k) v ra qu o nghim vi vector li k do ngi s dng xc nh. Vector k cha cc gi tr v li m nghim h vng kn c tnh. Nu s dng cc i s ng ra th : [r,k] = rlocus(num,den) [r,k] = rlocus(num,den,k) [r,k] = rlocus(a,b,c,d) [r,k] = rlocus(a,b,c,d,k) to ra ma trn ng ra cha cc nghim v vector li k. Ma trn r c length(k) hng v (length(den) 1) ct, ng ra cha v tr cc nghim phc. Mi hng trong ma trn tng ng vi mt li trong vector k. Qu o nghim c th c v bng lnh plot(r,x). d) V d: (Trch t trang 11-183 sch Control System Toolbox) Tm v v qu o nghim ca h thng c hm truyn : 2 s 2 + 5s + 1 H (s) = 2 s + 2s + 3 % Xc nh hm truyn : num = [2 5 1]; den = [1 2 3]; % V qu o nghim : rlocus(num,den) title(Quy dao nghiem)


4. Lnh SGRID a) Cng dng: To li cho qu o nghim v biu cc-zero lin tc. b) C php: sgrid sgrid(new) sgrid(z,wn) sgrid(z,wn,new) c) Gii thch: Lnh sgrid to li cho qu o nghim v biu cc-zero lin tc trong mt phng s. ng li v l cc ng hng s t s tt dn () v tn s t nhin ( n). ng t s tt dn c v t 0 ti 1 theo tng nc l 0.1. sgrid(new) xa mn hnh ha trc khi v v thit lp trng thi hold on qu o nghim hay biu cc-zero c v ln li bng cc lnh : sgrid(new) rlocus(num,den) hoc pzmap(num,den) sgrid(z,wn) v cc ng hng s t l tt dn c ch nh trong vector z v v ng tn s t nhin c ch nh trong vector wn. sgrid(z,wn,new) xa mn hnh ha trc khi v cc ng t s tt dn v tn s t nhin c ch nh trong vector z v wn. Trng thi hold on c thit lp. d) V d: Trch t trang 11-200 sch Control System Toolbox V li trong mt phng s trn qu o nghim ca h thng c hm truyn : num = [2 5 1]; % ta c th thay i 2 dng num=, den= thnh dng lnh sau:

Kho st ng dng MATLAB trong iu khin t ng den = [1 2 3]; % H(s)=tf([2 5 1],[1 2 3]); rlocus(num,den) title(Quy dao nghiem) sgrid

5. Lnh ZGRID a) Cng dng: V li t l tt dn v tn s t nhin cho qu o nghim gin on. b) C php: zgrid zgrid(new) zgrid(z,wn) zgrid(z,wn,new) c) Gii thch: Lnh zgrid to li qu o cho nghim hoc biu cc-zero trong mt phng z. Cc ng hng s t l tt dn () v tn s t nhin chun ha s c v. c thay i t 0 ti 1 theo tng nc thay i l 0.1 v tn s t nhin c v t 0 ti vi tng nc thay i l /. zgrid(new) xa mn hnh ha trc khi v li v thit lp trng thi hold on qu o nghim hoc biu cc-zero c v ln li s dng cc lnh : zgrid('new') rlocus(num,den) hoc pzmap(num,den)

Kho st ng dng MATLAB trong iu khin t ng zgrid(z,wn) v hng s tt dn c ch nh trong vector z v v hng s tn s t nhin cho cc tn s chun ha c ch nh trong vector wn. Cc tn s chun ha c th c v bng lnh zgrid(z,wn/Ts) vi tn s l thi gian ly mu. zgrid(z,wn,new) xa mn hnh ha trc khi v t s tt dn v tn s t nhin c ch nh trong vector z v wn. Trng thi hold on c thit lp. zgrid([ ],[ ]) s v ra vng trn n v. d) V d: Trch t 11-236 sch Control System Toolbox V li trong mt phng cho qu o nghim ca h thng c hm truyn : 2 z 2 3.4 z + 1.5 H ( z) = 2 z 1.6 s + 0.8 num = [2 -3.4 1.5]; den = [1 -1.6 0.8]; axis(square) zgrid(new) rlocus(num,den) title(Ve luoi cho quy dao nghiem)


CC BI TPV QU O NGHIMBi 1: KGH =

k vi k = 2 s(s + 4)(s + 5)

num = 2; den = [1 9 20 0]; rlocus(num,den)

10 8 6 4 2 0 -2 -4 -6 -8 -10 -7

Imag Ax is

-6

-5

-4

-3 -2 Real Axis

-1

0

1

2


T th cho ta: 1. im cc: 0 ,-4,-5. 2. Qu o nghim c 3 nhnh. 3. im zero v cng ( ). 4. im tch c xc nh bng cch t ca s MATLAB ta nhp: num = 2; den = [1 9 20 0]; rlocus(num,den); rlocfind(num,den) Sau khi nhp lnh th trn ca s lnh s xut hin hng ch: Select a point in the graphics window (hy chn 1 im trn th minh ha). Trn th s c thc cho ta chn im ko r chut chn im cn chn. selected_point = -1.4516 im tch c gi tr: -1.4516 Giao im ca qu o nghim vi trc o (tng t nh tm im tch): +4.472j, -4.472j. T gi tr ti giao im ca qu o nghim vi trc o ta th vo phng trnh c trng: F(s) = s3+9s2+20s+k = 0 F(jw) = -jw3-9w2+20jw+k = 0 kgh = 180 Kt lun: h thng s n nh khi 0 < k < 180 Bi 2: KGH =

k s(s + 4)(s + 4s + 20 )2

(k = 2)

num = 2; den = [1 8 36 80 0]; rlocus(num,den)


5 4 3 2 1 0 -1 -2 -3 -4 -5 -6

Imag Ax is

-5

-4

-3

-2 Real Axis

-1

0

1

2

T th cho ta: 1. im cc: 0,-4,-2+4j,-2-4j; 2. Qu o nghim c 4 nhnh 3. im zero v hn ( ) 4. im tch c xc nh bng cch t ca s MATLAB ta nhp: num = 2; den = [1 8 36 80 0]; rlocus(num,den); rlocfind(num,den); Sau khi nhp lnh th trn ca s lnh s xut hin hng ch: Select a point in the graphics window (hy chn 1 im trn th minh ha). Trn th s c thc cho ta chn im ko r chut chn im cn chn. selected_point = -2, -2.0184 + 2.4561j, -2.0184 - 2.4561j im tch c gi tr: -2, -2.0184 + 2.4561j, -2.0184 - 2.4561j Giao im ca qu o nghim vi trc o (tng t nh tm im tch): +3.16j, -3.16j T gi tr ti giao im ca qu o nghim vi trc honh ta th vo phng trnh c trng: F(jw) = w4-8jw3-36w2+80jw+k kgh = 260 Kt lun : h thng s n nh khi 0 < k < 260 Bi 3: KGH =

num = 2; den =[1 0 0 ]; rlocus(num,den)

k (k = 2) s2


1.5

1

0.5 Imag Ax is

0

-0.5

-1

-1.5 -1

-0.8

-0.6

-0.4

-0.2

0 0.2 Real Ax is

0.4

0.6

0.8

1

T th ta c: 1. im cc : 0 2. Qu o nghim c 2 nhnh 3. im zero v hn ( ) im tch c gi tr: 0 Kt lun: h thng khng n nh. Bi 4: KGH =

num = 2; den =[1 0 0 0]; rlocus(num,den)

k s3


1 0.8 0.6 0.4 0.2 Imag Ax is 0 -0.2 -0.4 -0.6 -0.8 -1 -1

-0.8

-0.6

-0.4

-0.2

0 0.2 Real Ax is

0.4

0.6

0.8

1

T th ta c: 1. im cc: 0. 2. Qu o nghim c 3 nhnh. 3. im zero v hn (). 4. im tch c gi tr: 0 Kt lun: h thng khng n nh (v hai nhnh ca qu o nghim s lun nm na phi mt phng phc). Bi 5: KGH =

k (k = 1, t = 2) s(ts+ 1)

num = 1; den = [2 1 0]; rlocus(num,den)


2 1.5 1 0.5 Imag Ax is 0 -0.5 -1 -1.5 -2 -1

-0.8

-0.6

-0.4

-0.2

0 0.2 Real Ax is

0.4

0.6

0.8

1

1. im cc : 0,-0.5 2. Qu o nghim c 2 nhnh 3. im zero v hn ( ) 4. im tch c c xc nh bng cch t ca s MATLAB ta nhp: num = 1; den = [2 1 0 ]; rlocus(num,den); rlocfind(num,den) Sau khi nhp lnh th trn ca s lnh s xut hin hng ch: Select a point in the graphics window (hy chn 1 im trn th minh ha). Trn th s c thc cho ta chn im ko r chut chn im cn chn. selected_point = -0.253 im tch c gi tr: -0.253Kt lun: h thng lun n nh (v qu o nghim lun nm na tri mt phng phc). Bi 6: KGH =

num = [1 1]; den = [1 0 0]; rlocus(num,den)

k(ts+ 1 ) (k = 1, t = 1) 2 s


1 0.8 0.6 0.4 0.2 Imag Ax is 0 -0.2 -0.4 -0.6 -0.8 -1 -3

-2.5

-2

-1.5

-1 -0.5 Real Axis

0

0.5

1

1. im cc: 0 2. Qu o nghim c 2 nhnh 3. im zero , -1 4. im tch c c xc nh bng cch t ca s MATLAB ta nhp: num = [1 1]; den = [1 0 0]; rlocus(num,den); rlocfind(num,den) Sau khi nhp lnh th trn ca s lnh s xut hin hng ch: Select a point in the graphics window (hy chn 1 im trn th minh ha). Trn th s c thc cho ta chn im ko r chut chn im cn chn. selected_point = -2 im tch c gi tr: -2. Kt lun: h thng bin n nh. Bi 7: KGH =

k (k = 1, t1 = 2, t2 = 1) (t1s + 1)(t2s + 1)

num = 1; den = [2 3 1]; rlocus(num,den)


2 1.5 1 0.5 Im ag Ax is 0 -0.5 -1 -1.5 -2 -2

-1.5

-1

-0.5 Re a l A x i s

0

0.5

1

1. im cc: -0.5, -1. 2. Qu o nghim c 2 nhnh 3. im zero v hn ( ) 4. im tch c c xc nh bng cch t ca s MATLAB ta nhp: num = 1; den = [2 3 1]; rlocus(num,den); rlocfind(num,den) Sau khi nhp lnh th trn ca s lnh s xut hin hng ch: Select a point in the graphics window (hy chn 1 im trn th minh ha). Trn th s c thc cho ta chn im ko r chut chn im cn chn. selected_point = -0.75. im tch c gi tr: -0.75 Kt lun: h thng lun n nh. Bi 8: KGH =

( s + 1)( s 1)( s+ 4) 2

k

(k = 10)

num =10; den = [ 1 8 15 -8 -16]; rlocus(num,den)


5 4 3 2 1 Im ag Ax is 0 -1 -2 -3 -4 -5 -6

-5

-4

-3

-2 -1 Real Ax is

0

1

2

3

1. im cc : 1, -1 v 1 cc kp -4. 2. Qu o nghim c 4 nhnh. 3. im zero: c 4 zero v cng ( ). 4. im tch c c xc nh bng cch t ca s MATLAB ta nhp: num =10; den = [ 1 8 15 -8 -16]; rlocus(num,den); rlocfind(num,den) Sau khi nhp lnh th trn ca s lnh s xut hin hng ch: Select a point in the graphics window (hy chn 1 im trn th minh ha). Trn th s c thc cho ta chn im ko r chut chn im cn chn. selected_point = 0.2308, -4 im tch c gi tr: 0.2308, -4 Kt lun: H thng lun khng n nh v tn ti 1 nghim ca phng trnh c trng nm bn phi mt phng phc. Bi 9: Trch t trang 5-19 sch Control System Toolbox Bi ny tng hp cc lnh: h=tf([4 8.4 30.8 60],[1 4.12 17.4 30.8 60]); subplot(221) bode(h) subplot(222) step(h) subplot(223) pzmap(h) subplot(224)

Kho st ng dng MATLAB trong iu khin t ng plot(rand(1,100) plot(rand(1,100))B o d e D ia g ra m s P h a s e (d e g ); M a g n it u d e (d B )From : U(1) 20 1 .5

S te p R e s p o n s eFrom: U(1 )

0

A m p lit u d e

1 To: Y (1)

-20 100 To: Y (1)

0 .5

0

-100 1 0 -1 100 101

0 0 2 4 6 8 10 12

F re q u e n c y (ra d / s e c ) P o le -z e ro m a p 14

T im e (s e c . ) b a i to n g h o p

0 .82

Im a g A x i s

0 .60

0 .4-2

0 .2-4 -3 -2 -1 0 1

0 0 50 100

R e a l A x is

Bi 10: Cho hm truyn nh sau: s +4 G s) = ( (s + 1)(s + 2) Vit theo cu trc sau ta c c th biu din qu o nghim: num=[1 4]; den=conv([1 1],[1 2]) rlocus(num,den) Kt qu nh hnh sau:


Documents

Quy dao nghiem