1. 디지털 제어 Sun Moon University 1 of 32 목 차 7. Stability Kyoung-Chul Koh(kckoh@sunmoon.ac.kr) DIGITAL CONTROL

2. 디지털 제어 Sun Moon University 2 of 32 강의진도 참고문헌 [1] 고경철, 이장명 저, “C 로 구현한 디지털 제어 시스템 ”, 홍릉과학서적, to be published 2004.4 [2] Ogata, Discrete-Time Control System, [3] Dorf, Modern Control Systems, Wesley, 1998 [1 주 ] 1 장 서론 [2 주 ] 2 장 Z 변환 [3 주 ] 3 장 역 Z 변환 [4 주 ] 4 장 이산 제어 시스템의 모델링 [5 주 ] 5 장 시간응답 [6 주 ] 6 장 정상 상태 오차 분석 [7 주 ] 7 장 안정도 해석

3. 디지털 제어 Sun Moon University 3 of 32 7.1 System stability mapping from s-plane to z-plane stability analysis of closed loop system in z-plane design approach (1) root locus method in z-plane (2) pulse transfer function design of continuous control system with root locus method method -> directly applied to digital control systems Negative half in s- plane Unit circle in z-plane

4. 디지털 제어 Sun Moon University 4 of 32 7.1 System stability locations of poles and zeros in discrete control system Stability condition of discrete system if For stability, Negative half in s- plane Unit circle in z-plane Mapping s-plane into z-plane by using

5. 디지털 제어 Sun Moon University 5 of 32 stability analysis of a closed loop system characteristic equation (1) the closed loop poles must lie within the unit circle in the z plane (2) if a simple pole or a single pair of conjugate complex poles lie at z=1, then the system becomes critically stable (3) closed loop zeros do not affect the absolute stability Stability conditions: + - Unit circle in z-plane Close loop poles in z-plane 7.1 System stability

6. 디지털 제어 Sun Moon University 6 of 32 7.1 System stability relative stability - linear time invariant continuous time closed loop system > locations of poles in s plane >> oscillatory behavior near imaginary axis >> exponential decay near real axis relationship between two complex variables z and s location of poles and zeros in z planes location of poles and zeros in s planes by using therefore, relative stability of linear time invariant discrete time closed loop system - can be analyzed by investigating locations of poles in z plane - also locating the poles at desired location > varying by controller gains

7. 디지털 제어 Sun Moon University 7 of 32 7.1 System stability relative stability - linear time invariant continuous time closed loop system > locations of poles in s plane >> oscillatory behavior near imaginary axis >> exponential decay near real axis relationship between two complex variables z and s Constant damping coefficient loci vs constant natural frequency loci

8. 디지털 제어 Sun Moon University 8 of 32 7.2 Stability of discrete 2 nd order system Example : stability analysis of a closed loop system with a open loop of z transform of G(s)

9. 디지털 제어 Sun Moon University 9 of 32 7.2 Stability of discrete 2 nd order system Example : stability analysis of a closed loop system Determine the stability of the closed loop system when K=1 Closed-loop pulse transfer function Characteristic equation which becomes or Since the system is stable !, Unit circle in z-plane

10. 디지털 제어 Sun Moon University 10 of 32 Example : stability analysis of a closed loop system System order n=2 System parameter a[0],a[1],a[2],b[0],b[1],b[2] Input type r(k) Controller gains Kp Controller u (k) Plant y(k+1) 7.2 Stability of discrete 2 nd order system + -

11. 디지털 제어 Sun Moon University 11 of 32 Example : stability analysis of a closed loop system 7.2 Stability of discrete 2 nd order system + -

12. 디지털 제어 Sun Moon University 12 of 32 Example : stability analysis of a closed loop system 7.2 Stability of discrete 2 nd order system double stability(double a1,double a2,double b1, double b2,double Kp) { double alpha,beta,a,b,magnitude; //characterist eq=z*z+alpha*z+beta=0 alpha=a1+Kp*b1; beta=a2+Kp*b2; // poles=a+bj and a-bj a=-alpha/2.0; b=sqrt(4.0*beta-alpha*alpha)/2.0; printf("z1=%f+%fj\n",a,b); printf("z2=%f-%fj\n",a,b); //stability condition magnitude=sqrt(a*a+b*b); return magnitude; }

13. 디지털 제어 Sun Moon University 13 of 32 Example : stability analysis of a closed loop system with Kp=1 7.2 Stability of discrete 2 nd order system + - Stable!

14. 디지털 제어 Sun Moon University 14 of 32 Example : stability analysis of a closed loop system with Kp=2 7.2 Stability of discrete 2 nd order system + - Relatively oscillatory, but still stable!

15. 디지털 제어 Sun Moon University 15 of 32 Example : stability analysis of a closed loop system with Kp=2.4 7.2 Stability of discrete 2 nd order system + - Unstable!

16. 디지털 제어 Sun Moon University 16 of 32 7.3 Root locus method Characteristic Eq. Root locus: location trajectory of system poles by varying gain Kp For higher order(n>2) systems: - difficult to draw the location trajectory of system poles since needing numerical solution of characteristic equation Commercialized design packages for control system - MATLAB, CemTools

17. 디지털 제어 Sun Moon University 17 of 32 7.3 Root locus method Root locus by MATLAB + - m-file for MATLAB

18. 디지털 제어 Sun Moon University 18 of 32 7.3 Root locus method : m-file for MATLAB

19. 디지털 제어 Sun Moon University 19 of 32 7.3 Root locus method : m-file for MATLAB

20. 디지털 제어 Sun Moon University 20 of 32 7.3 Root locus method: MATLAB practice(1) Ex. 7.3: Root locus by MATLAB with Coding m-file for drawing root locus Find maximum Kp guaranteeing the stability

21. 디지털 제어 Sun Moon University 21 of 32 7.3 Root locus method: MATLAB practice(1) Ex. 7.4: Root locus by MATLAB with Coding m-file for drawing root locus Find maximum Kp guaranteeing the stability + -

22. 디지털 제어 Sun Moon University 22 of 32 7.4 Summary - 시스템 안정도는 제어 시스템 설계에 있어 고려해야 할 가장 요소이다. 이 장에서는 이산 제어 시스템의 안정도를 판별하기 위해, 극점의 위치를 Z 평면 상에서 검사하는 방법을 소개하였다. 그리고 단위 시간 응답 시뮬레이션을 통해 결과를 비교하였다. 3 차 이상의 고차 시스템의 안정도 해석을 위해, 컴퓨터 기반 제어 시스템 설계용 상용 S/W 의 하나인 MATLAB 을 이용하는 방법을 간략히 소개하였다. 또한 예제로서, 산업용 제어기에서 가장 넓게 응용되는 PID 제어기의 설계를 안정도 해석을 통해 연습하였다.

23. 디지털 제어 Sun Moon University 23 of 32 HW#7 Solve Pr.7.1, 7.2, 7.3, 7.4, 7.5 Due: Next week this time

24. 디지털 제어 Sun Moon University 24 of 32 MATLAB: “ MATrix LABoratory” - Tool box >> Control system Tool box. 130 functions: Step, rlocus, bode, Nyquist, others >> Signal Processing Tool box >> Robust Control Tool box >> System Identification Tool box >> System Optimization Tool box >> Neural Network Tool box >> Fuzzy Tool box Matrix calculation >> a=[1 2 3]; b=[3 4 5]; >> c=a’*b >> c=a*b’ >> c=a.*b >> x=[0.0: 0.2: 0.5]; >> y=3*(1-x).^2.*exp(-x.^2)) >> c=[a b] >> c=[a; b] >> t=[0:10] 부록 :MATLAB 특강 (1):

25. 디지털 제어 Sun Moon University 25 of 32 Ex.10-1 %---- function calculation --------------- t=[0:0.1:1]; y=exp(-t).*sin(t); [t' y'] plot(t,y,'-w') ans = 0 0 0.1000 0.0903 0.2000 0.1627 0.3000 0.2189 0.4000 0.2610 0.5000 0.2908 0.6000 0.3099 0.7000 0.3199 0.8000 0.3223 0.9000 0.3185 1.0000 0.3096 00.20.40.60.81 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 부록 :MATLAB 특강 (2):

26. 디지털 제어 Sun Moon University 26 of 32 Matrix or vector dimension - length(a) ; vector dimension - size(a): matrix dimension >> [m, n]=size(a) who >> who Your variables are: a b t ans c y whos >> whos NameSizeElementsBytesDensityComplex a 1 by 3 3 24FullNo ans 11 by 2 22 176 Full No b 1 by 3 3 24 Full No c 1 by 3 3 24 Full No t 1 by 11 11 88 Full No y 1 by 11 11 88 Full No clear >> clear a >> who Your variables are: ans b c t y 부록 :MATLAB 특강 (2):

27. 디지털 제어 Sun Moon University 27 of 32 load d.dat; x=d(:,1); y=d(:,2); w=d(:,3); k=1; for i=1:1:51, for j=1:1:51, X(i,j)=x(k)*10.; Y(i,j)=y(k)*10.; Z(i,j)=(X(i,j)*X(i,j)-Y(i,j)*Y(i,j))*sin(0.5*X(i,j))*cos(0.5*Y(i,j)); k=k+1; end mesh(X,Y,Z,'g') view(-37,30) Ex.10-2 부록 :MATLAB 특강 (3):

28. 디지털 제어 Sun Moon University 28 of 32 %---- function calculation --------------- t0=0; tf=20; x0=[0 0.25]; [t, x]=ode45('mat141',t0,tf,x0); plot(t,x(:,1),'-w',t,x(:,2),'--w') %function name should be same as file name function xdot=mat141(t,x) xdot=zeros(2,1); xdot(1)=x(1).*(1-x(2).^2)-x(2); xdot(2)=x(1); Mat141.m Ex.10-3 Van der Pol 2nd order differential equation 부록 :MATLAB 특강 (4):

29. 디지털 제어 Sun Moon University 29 of 32 plot(a,b) : 2-dimensional graph loglog(a,b): logarithmic graph semilogx(a, b), semilogy(a,b) : semi-logarithm graphs plot3(a, b, c) subplt(mnp) 211 212 311 312 313 221 223 222 224 subplot(211) plot(t,x(:,1),'-w') subplot(212) plot(t,x(:,2),'--w') axis(‘square’) axis(‘equal’) input() printsys() Stiffness =1 Inerta =1 num/den = 1 ------- s^2 + 1 kp=input('Stiffness ='); J=input('Inerta ='); num = [0 0 kp]; den = [J 0 kp]; printsys(num, den) 부록 :MATLAB 특강 (5):

30. 디지털 제어 Sun Moon University 30 of 32 grid text(x,y,’message’) title(‘Caption title’) xlabel(‘ x-axis ‘) ylabel(‘ y-axis ‘) v = [ -x x -y y]; hold hold off 부록 :MATLAB 특강 (6):

31. 디지털 제어 Sun Moon University 31 of 32 conv() >> num =conv(num1, num2) >> den = conv(den1,den2) series() >> [num, den] = series(num1, den1, num2, den2) parallel() >> [num, den] = parallel(num1, den1, num2, den2) cloop() >> [num, den] = cloop(num1, den1,-1) feedback() >> [num, den] = feedback(num1, den1, num2, den2,-1) + - + - 부록 :MATLAB 특강 (7):

32. 디지털 제어 Sun Moon University 32 of 32 [r,p,k]=residue(num,den) /* pp.42 [y,x,t]= step(num, den,t) [y,x,t]=impulse(num, den, t) [r,K]=rlocus(num,den,K) [mag, phase, w] = bode(num, den, w) [re, im, w] = nyquist(num, den, w); z=re+i*im; theta=angle(z); r=abs(z); polar(theta, r) 부록 :MATLAB 특강 (8):