matlab與控制系統(tǒng)仿真課件-ch6stability

1、CHAPTER6Stability of LinearFeedback Systems6.16.2 BIroductiontability6.3 Robot-controller Motorcycle6.4 Stabilityysis6.5 Disturbance Response6.66.7 Summaryysis16.1roductionOf utmost importance systems is stabilityUnstable open loop system-stabilize using closed loop control.he design of controlStabl

2、e open loop system-do notroduceinstabilityo the closed loop system.Routh-Hurwitz method26.2 BItabilityA system is stable if the system response is bounded (有界) for every bounded input, referred to as bounded input bounded output or BIBO stability.Let h(t) represent impulse response of a system.t h (

3、 t) u ( ty(t)td0t(t )dt)u(yt()ht(u)h(tt)d00)tyt(M(t )M)h(dSupe input u(t) is boundedu0thdy(t) being bounded i(mpitsboundeds03Nesary and sufficient conditionA system to be BItable ist all of the polesof the system transfer function have negative realpartsThe system is stable only when all the closed-

4、looppoles are locatedhe lefnd half of the scomplex plane (LHP);The systeme unstable as soon as one closed-loop pole is locatedhe righnd half of the scomplex plane (RHP).The system is critical stable as soon as a pair ofimaginary poles and all other roots half of s complex plane (LHP).he lefnd46.3 Ro

5、bot controlled motorcycleGhost-rider56.3 Robot controlled motorcycleVhLcGhost-rider6Control DesignControl Goal: Control the motorcyclehe vertical(垂直)ition, and maahe prescribed.itionhepresence of disturbanVariable to be Controlled: motorcycle vertical Control Design SpecificationsClosed loop system

6、must be stableition from7Robot-controlled MotorcycleD(s)1Robot ControllerMotorcycle Dynamicss)G (s 2 (s)(s)B1dDesired bangle angle ss)G c(2s 132 Vg VLFeedback controller ,123hhchc) KGs(KsfPDSelect Key Tuning parameter :KP and KDWhat values of KP and KD lead to closed-loop stability?8Gf(s)G(s)Gc(s)6.

7、4 Stabilityysis2s 1 3s)s(2()1Closed-loop transfer functionCharacteristic equation( s) K(K)s123PDs 1)s(2()1 3( (K KsK021Ks)s)D3D2P 31P21D2KP31s3 s2s1(K1KP2)s091 KD31KD3K)(D2 PK3) 1 (KP2 11 KD3KP2 1 0KD 1/ 3 ( K ()K 0K 22)K23D2133PD32PKP 1 / 2 0.2; h 1.09;V 2.0; L 1.0; c 1.36 g 9,h1KD 0V 22hc 2.7,K 3.

8、33P VL 1.35hc310a1=9;a2=2.7;a3=1.35;tau=0.2; KP,KD=meshgrid(0:1:50,0:1:50);f=a2*a3*KD.2+(a2-tau*a1*a3+a32*KP).*KD+(a3-tau*a2)*KP;meshz(f) xlabel(KP)ylabel(KD)zlabel(f)grid 10, 5.KPSelectingKD11Open loop T.F. of a unity feedback controlsystem100(s 2)G(s) n1 =k=100;z=-2;p=0,-1,-20;s(s 1)(s 20)n1,d1=zp

9、2tf(z,p,k)0d1 =1P =010020021200P=n1+d1roots(P)12112020012Other methods to get the rootsG=tf(n1,d1) sys=feedback(G,1) Transfer function:100 s + 200-s3 + 21 s2 + 120 s + 200roots(sys.den1)sys1=zpk(sys);sys1.p1pole(sys)ans =-12.8990-5.0000-3.1010G=ss(sys); eig(G.a)13Exle:Mars Exploration Rover, MERSpir

10、it, MER-A14Opportunity, MER-B15車輪力矩右左C(s)漫游方向調速閥門兩組車輪的速度差R(s) +預期的轉動方向C(s)-Control goal: The velocity error for a ris tobe no moren 24% of the slope of the input。16R(s) +預期的轉動方向C(s)-Characteristic equation:K1(s a)1 0s(s 1)(s 2)(s 5)s4+8s3+17s2+(10+K1)s+aK1=017動力傳動系統(tǒng)與漫游車G(s)K1s(s 2)(s 5)控制器Gc(s)(s a)

11、(s 1)s4+8s3+17s2+(10+K1)s+aK1=0Stabilitys4117aK1rray：Rous3s28126 K110+K1aK18s11260 (116 64a) K K 211 126 K1aK1s0K1 126aK1 01260 (116 64a ) KK021118K aK 1260 (116064a)K1syms K1 af=1260+(116-64*a)*K1-K12ezplot(f,0,150,0,4)R(s) +預期的轉動方向C(s)-Control goal: The velocity error for a ris to be nomoren 24% o

12、f the slope of the input。K lim sG (s)G(s) aK1r(t)=Atc10s0A10 AA24e%ssaKK120Let aK1=42，ess 23.8%a and K1 can be selected as：K1=70，a=0.6； K1=50，a=0.84.ff=a*K1-42 ezplot(ff,0,150,0,4)a216.5 Disturbance responseD(s)Robot ControllerMotorcycle Dynamicsd(s)Desired bangle(s)B angleFeedback controllerDisturb

13、ance transfer functionG2223Robot-controlled Motorcycles) 2 3sD(s)G(Robot Controllercs 11Motorcycle Dynamicss)d(s)Desired bangle(s)BG (s21 angleG) KPsf(DKs2 VFeedback controllerg VL ,123hhchcIs itsible for the robot to control the motorcycle asthe velocity (V ) varies ?246.6ysisWhen v1.2, one pole is

14、itive, so motorcycle is unstable256.7 Summary1. 工程背景與指標要求機器人、機器人的要求最主要的是穩(wěn)定性要求，有一定斜坡輸入下穩(wěn)態(tài)誤差要求。2. 設計與設計過程判據(jù)確定K1與a的穩(wěn)定區(qū)域。由于有兩個控制參數(shù)，可用具置應按滿足斜坡作用下的穩(wěn)態(tài)誤差要求確定。滿足穩(wěn)態(tài)誤差要求的參數(shù)對（ K1，a ）設計斜坡輸入作用下的穩(wěn)態(tài)誤差ess Kv 關系；Kv與 K1與 a關系；263. 擴展與引伸（1）全方位性能要求：一個實際工程系統(tǒng)，不僅有穩(wěn)態(tài)要求，還應有動態(tài)要求與頻帶要求，試用車系統(tǒng)在不同（ K1，a）下的閉環(huán)零極點，環(huán)頻率響應及帶寬。a（2）參數(shù)魯棒域確定：利用理論推導或仿真方法，可否在容許參數(shù)穩(wěn)定區(qū)域內(nèi)，確定使斜坡作用下系統(tǒng)動態(tài)性能及頻寬較“好”的子區(qū)域？求出（繪出）漫游斜