From the multivariable Nyquist plot, if 1/k < or 1/k > 5.36, or 0.19 < k < 2.7, then the closed loop system is stable.

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Marion Hardy
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1 1. (a) Multivariable Nyquist Plot (diagonal uncertainty case) dash=neg freq solid=pos freq Im Re From the multivariable Nyquist plot, if 1/k < -.37 or 1/k > 5.36, or.19 < k < 2.7, then the closed loop system is stable. (b)-(i) 6 singular value plot max singular value = bound on Delta =.1839 max (k 1, k 2 ) < 1/ =.184 (b)-(ii) bounds on (kl,k2): ++,+-,-+,-- ++: , +-: , -+: , --:

2 Plots of Hermitian part of diag(k 1 +/-,k 2 +/- ) G(jw). (k 1 >,k 2 >) (k 1 >,k 2 <) minimum eigenvalue of (G(jw)+G T (-jw))/2: k1>,k2> minimum eigenvalue of (G(jw)+G T (-jw))/2: k1>,k2< (k 1 <,k 2 >) (k 1 <,k 2 <) minimum eigenvalue of (G(jw)+G T (-jw))/2: k1<,k2> minimum eigenvalue of (G(jw)+G T (-jw))/2: k1<,k2< (b)-(iii) With k 1 in [-.18,.18] and k 2 in [-.18,.18], the following set of LMI s P (A+B diag(k 1 +,k 2 + )C) + (A+B diag(k 1 +,k 2 + )C) T P< P (A+B diag(k 1 +,k 2 - )C) + (A+B diag(k 1 +,k 2 - )C) T P< P (A+B diag(k 1 -,k 2 + )C) + (A+B diag(k 1 -,k 2 + )C) T P< P (A+B diag(k 1 -,k 2 - )C) + (A+B diag(k 1 -,k 2 - )C) T P< where (A,B,C,) is a realization of G(s), has a solution. Output of LMI solver: Iteration : Best value of t so far Result: best value of t: f-radius saturation:. of R = 1.e *** feasible! *** stability bound: k1: -.18,.18 k2: -.18,.18

3 (b)-(iv) ********* real mu *********... Mu upper bound (peak value): 5.372e+ Robust stability margin : 1.861e ********* complex mu *********... Mu upper bound (peak value): 5.372e+ Robust stability margin : 1.861e ********* Delta = unstructured (should be 1/H-infty norm of F) *********... Mu upper bound (peak value): 5.465e+ Robust stability margin : 1.83e maximum real mu = maximum real bound for diag(k1,k2) = maximum complex mu = maximum complex bound for diag(k1,k2) = maximum bound for unstructured Delta = maximum spectral radius of F = maximum bound for k when Delta = k I (k complex) = (b)-(v) Closed loop characteristic polynomial: 2 s + (2 + k1 + 4 k2) s + 4 k2 + k k1 k2 Assume -k1max < k1 < k1max, -k2max < k2 < k2max, then the coefficients can be bounded by: 2 - k1max - 4 k2max < (2 + k1 + 4 k2) < 2 + k1max + 4 k2max - 4 k2max - k1max + 1 < 4 k2 + k1 + 1 < 4 k2max + k1max k1max k2max < 1-2 k1 k2 < 1 + k1max k2max all Kharitonov polynomials are stable when k1 <.18 k2 <.18 Solution Script (fin1.m) solution of problem 1 (Multivariable Final 22) clear all; close all; open loop plant

4 G=[1 2;3 4]*tf(1,[1 1]); [aa,bb,cc,dd]=ssdata(ss(g)); multivariable nyquist w=logspace(-4,4,5); sys=ss(aa,bb,cc,dd); sigmah1=mvarnyq(sys,w); diagonal uncertainty case [k ; k] /k<-.88 /k> <k<11.4 figure(1); plot(real(sigmah1(1,:)),imag(sigmah1(1,:)),'-r',... real(sigmah1(1,:)),-imag(sigmah1(1,:)),'--r',... real(sigmah1(2,:)),imag(sigmah1(2,:)),'-b',... real(sigmah1(2,:)),-imag(sigmah1(2,:)),'--b'); title(['multivariable Nyquist Plot (diagonal uncertainty case)',... ' dash=neg freq solid=pos freq']) xlabel('re');ylabel('im');grid; axis equal small gain w=logspace(,2,2); sys=ss(aa,bb,cc,dd); sv=sigma(sys,w); figure(3);semilogx(w,sv);title('singular value plot'); xlabel(''); disp(['max singular value = ',num2str(max(sv(1,:)))]); disp(['bound on Delta = ',num2str(1/max(sv(1,:)))]); print -f3 -depsc smallgainexample.eps positive real realpartpp=pr(sys,w); figure(4);semilogx(w,realpartpp); title('minimum eigenvalue of (G(jw)+G^T(-jw))/2: k1>,k2>'); xlabel(''); realpartpn=pr([1 ; ]*sys,w); figure(5);semilogx(w,realpartpn); title('minimum eigenvalue of (G(jw)+G^T(-jw))/2: k1>,k2<'); xlabel(''); realpartnp=pr([ ; 1]*sys,w); figure(6);semilogx(w,realpartnp); title('minimum eigenvalue of (G(jw)+G^T(-jw))/2: k1<,k2>'); xlabel(''); realpartnn=pr([ ; ]*sys,w); figure(7);semilogx(w,realpartnn); title('minimum eigenvalue of (G(jw)+G^T(-jw))/2: k1<,k2<'); xlabel('');

5 print -f4 -depsc prexample1.eps print -f5 -depsc prexample2.eps print -f6 -depsc prexample3.eps print -f7 -depsc prexample4.eps bpp=min(realpartpp); bpn=min(realpartpn); bnp=min(realpartnp); bnn=min(realpartnn); disp(['bounds on (kl,k2): ++,+-,-+,--']); disp(sprintf('++:.5g, +-:.5g, -+:.5g, --:.5g\n',... 1/bpp,1/bpn,1/bnp,1/bnn)); LMI n=size(aa,1); L1=[1 ; ]; L2=[ ; 1]; a1=-bb*l1*cc; a2=-bb*l2*cc; define lmi variable to be solved setlmis([]); [P,ndec,Pdec]=lmivar(1,[n 1]); set up uncertain variable range k1=[,1]*.18 range of k1 k2=[,1]*.18; range of k2 k1a=k1(1);k1b=k1(2); k2a=k2(1);k2b=k2(2); set up LMI to be solved left hand side of first lmi: m1 [1,1]th block of the lmi: [l1,l2] first argument = [m1,l1,l2,p] second/third a,b 's' to specify a*p*b+b'*p*a' lmiterm([1,1,1,p],(aa+k1a*a1+k2a*a2)',1,'s'); left hand side of second lmi: m1 [1,1]th block of the lmi: [l1,l2] a*p*b first argument = [m1,l1,l2,p] second/third a,b lmiterm([2,1,1,p],(aa+k1b*a1+k2a*a2)',1,'s'); left hand side of third lmi: m1 [1,1]th block of the lmi: [l1,l2] a*p*b

6 first argument = [m1,l1,l2,p] second/third a,b lmiterm([3,1,1,p],(aa+k1a*a1+k2b*a2)',1,'s'); left hand side of fourth lmi: m1 [1,1]th block of the lmi: [l1,l2] a*p*b first argument = [m1,l1,l2,p] second/third a,b lmiterm([4,1,1,p],(aa+k1b*a1+k2b*a2)',1,'s'); example=getlmis; options=[,1,,,]; [tmin,pfeas]=feasp(example,options); disp(tmin) if tmin<; disp('*** feasible! ***'); disp('stability bound: '); disp(['k1: ',num2str(k1a),',',num2str(k1b)]); disp(['k2: ',num2str(k2a),',',num2str(k2b)]); else disp('*** no feasible solution (reduce uncertainty bound)! ***'); end mu sys1=ltisys(aa,bb,cc,dd,1); delta1=ublock(1,1,'ltisr'); delta2=ublock(1,1,'ltisr'); deltar=udiag(delta1,delta2); disp('********* real mu ********* '); [marginr,peakf,fs,ds,gs] = mustab(sys1,deltar); delta1=ublock(1,1,'ltisc'); delta2=ublock(1,1,'ltisc'); deltac=udiag(delta1,delta2); disp('********* complex mu ********* '); [marginc,peakf,fs,ds,gs] = mustab(sys1,deltac); deltaf=ublock(2,1,'ltifc'); disp(['********* Delta = unstructured ',... '(should be 1/H-infty norm of F) ********* ']); [marginf,peakf,fs,ds,gs] = mustab(sys1,deltaf); should be same as H- inf disp(['maximum real mu = ',num2str(1/marginr)]); disp(['maximum real bound for diag(k1,k2) = ',num2str(marginr)]); disp(['maximum complex mu = ',num2str(1/marginc)]); disp(['maximum complex bound for diag(k1,k2) = ',num2str(marginc)]); disp(['maximum bound for unstructured Delta = ',num2str(marginc)]); spectral radius rho_c=(max(abs(sigmah1))); disp(['maximum spectral radius of F = ',... num2str(max(rho_c))]); disp(['maximum bound for k when Delta = k I (k complex) = ',... num2str(1/max(rho_c))]);

7 Kharitonov syms s k1 k2 k1mx=.18;k2mx=.18; kk=[k1 ; k2]; charpoly=collect(simplify(det(s*eye(n,n)-(aa-bb*kk*cc))),s); pretty(charpoly); a1u=2+k1mx+4*k2mx; a1l=2-k1mx-4*k2mx; au=4*k2mx + k1mx *k1mx *k2mx; al=-4*k2mx - k1mx *k1mx *k2mx; poly1=[1 a1l au]; poly2=[1 a1u al]; poly3=[1 a1l al]; poly4=[1 a1u au]; r1=max(real(roots(poly1))); r2=max(real(roots(poly2))); r3=max(real(roots(poly3))); r4=max(real(roots(poly4))); if r1<&r2<&r3<&r4< disp('all Kharitonov polynomials are stable'); disp([' k1 <',num2str(k1mx),' ',' k2 <',num2str(k2mx)]); else disp('not all Kharitonov polynomials are stable, reduce bounds!'); end

From the multivariable Nyquist plot, if 1/k < or 1/k > 5.36, or 0.19 < k < 2.7, then the closed loop system is stable.

1. (a) Multivariable Nyquist Plot (diagonal uncertainty case) dash=neg freq solid=pos freq 2.5 2 1.5 1.5 Im -.5-1 -1.5-2 -2.5 1 2 3 4 5 Re From the multivariable Nyquist plot, if 1/k < -.37 or 1/k > 5.36,