% function [k,g,gfin,ax,hamx] = hinffi(p,ncon,gmin,gmax,tol,ricmethd,epr,epp) % % This function computes the FULL INFORMATION h-infinity n-state % controller, using Glover's and Doyle's 1988 result, % for a system P. The solution assumes that all the states % and disturbances can be measured. The P matrix has the form: % % p = | ap | b1 b2 | % | c1 | d11 d12 | % % Care must be taken by the user when closing the loop with the % full information controller. For feedback the interconnection % structure, P, must be augmented with state and disturbance % measurements. % % inputs: % P - interconnection matrix for control design % NCON - # controller outputs (nm2) % GMIN - lower bound on gamma % GMAX - upper bound on gamma % TOL - relative difference between final gamma values % RICMETHD - Ricatti solution via % 1 - eigenvalue reduction (balance) % -1 - eigenvalue reduction (no balancing) % 2 - real schur decomposition (balance,default) % -2 - real schur decomposition (no balancing) % EPR - measure of when real(eig) of Hamiltonian is zero % (default EPR = 1e-10) % EPP - positive definite determination of xinf solution % (default EPP = 1e-6) % % outputs: % K - H-infinity FULL INFORMATION controller % G - closed-loop system % GFIN - final gamma value used in control design % AX - Riccati solution as a VARYING matrix with independent % variable gamma (optional) % HAMX - Hamiltonian matrix as a VARYING matrix with independent % variable gamma (optional) % Copyright 1991-2004 MUSYN Inc. and The MathWorks, Inc. % $Revision: 1.1.6.1 $ function [k,g,gfin,ax,hamx] = hinffi(p,ncon,gmin,gmax,tol,imethd,epr,epp) % % The following assumptions are made: % (i) (a,b2,c2) is stabilizable and detectable % (ii) d12 has full rank % on return, the eigenvalues in egs must all be > 0 for the closed- % loop system to be stable and to have inf-norm less than gam. % this program provides a gamma iteration using a bisection method, % it iterates on the value of gam to approach the h-inf optimal % full information controller. % np nm1 nm2 % The system p is partitioned: | a b1 b2 | np % | c1 d11 d12 | np1 % % Reference Paper: % 'State-space formulae for all stabilizing controllers % that satisfy and h-infinity-norm bound and relations % to risk sensitivity,' by keith glover and john doyle, % systems & control letters 11, oct, 1988, pp 167-172. % storxinf = []; storhx = []; gammavecxi = []; gammavechx = []; if nargin == 0 disp('usage: [k,g,gfin,ax,hamx] = hinffi(p,ncon,gmin,gmax,tol,imethd,epr,epp)') return end if nargin < 5 disp('usage: [k,g,gfin,ax,hamx]=hinffi(p,ncon,gmin,gmax,tol,imethd,epr,epp)') error('incorrect number of input arguments') return end % setup default cases if nargin == 5 imethd = 2; epr = 1e-10; epp = 1e-6; elseif nargin == 6 epr = 1e-10; epp = 1e-6; else epp = 1e-10; end gam2=gmax; gam0=gmax; if gmin > gmax error(' gamma min is greater than gamma max') return end % save the input system to form the closed-loop system pnom = p; niter = 1; npass = 0; first_it = 1; totaliter = 0; % fprintf('checking rank conditions: '); [p,q12,r12,fail] = hinffi_t(p,ncon); if fail == 1, fprintf('\n') return end % gamma iteration fprintf('Test bounds: %10.4f < gamma <= %10.4f\n\n',gmin,gmax) fprintf(' gamma ham_eig x_eig pass/fail\n') while niter==1 totaliter = totaliter + 1; if first_it ==1 % first iteration checks gamma max gam = gmax; else gam=((gam2+gam0)/2.); gamall=[gam0 gam gam2]; end fprintf(' %9.3f ',gam) [xinf,f,fail,hmx,hxmin] = hinffi_g(p,ncon,epr,gam,imethd); if nargout >= 4 if isempty(xinf) % don't update the matrix else gammavecxi = [gammavecxi ; gam]; storxinf = [storxinf ; xinf]; end if nargout >= 5 if isempty(hmx) % don't update the matrix else gammavechx = [gammavechx ; gam]; storhx = [storhx ; hmx]; end end end % % check the conditions for a solution to the FULL INFORMATION % h-infinity control problem. % if fail == 0 xe = eig(xinf); xeig = min(real(xe)); mprintf(hxmin,'%9.2e ',' '); if xeig < -epp | xeig == nan % change to a stablility test mprintf(xeig,' %9.2e','#'); npass = 1; else mprintf(xeig,' %9.2e',' '); end else mprintf(hxmin,' %9.2e ','#'); fprintf(' ******** ') npass = 1; end if npass == 1 fprintf(' fail \n') else fprintf(' pass \n') end % % check the gamma iteration % if first_it == 1; if npass == 0; gam0 = gmin; gam2 = gmax; xinffin = xinf; fsave = f; gfin = gmax; first_it = 0; if gmin == gmax niter = 0; % stop since gmax = gmin end else niter = 0; fprintf('Gamma max, %10.4f, is too small !!\n',gmax); if nargout >= 4 ax = vpck(storxinf,gammavecxi); if nargout >= 5 hamx = vpck(storhx,gammavechx); end end return end else if abs(gam - gam0) < tol niter = 0; if npass == 0 gfin = gam; xinffin = xinf; fsave = f; else gfin = gam2; end else if npass == 0; gam2 = gam; xinffin = xinf; fsave = f; else gam0 = gam; end end npass = 0; end end xinf = xinffin; % finished with gamma iteration fprintf('\n Gamma value achieved: %10.4f\n',gfin) [k] = hinffi_c(p,ncon,fsave,r12); % % modify the interconnection structure to form the closed-loop plant % [ap,bp,cp,dp,b1,b2,c1,d11,d12,ndata] = hinffi_p(p,ncon); np = max(size(ap)); c21 = eye(np); [nr1,nc1] = size(b1); c22 = zeros(nc1,np); d211 = zeros(np,nc1); d212 = eye(nc1); [nr2,nc2] = size(b2); d221 = zeros(np,nc2); d222 = zeros(nc1,nc2); [a,b,c,d] = unpck(pnom); c = [c; c21; c22]; d = [d; d211 d221; d212 d222]; p_fi = pck(a,b,c,d); % % form closed-loop system % [g] = starp(p_fi,k,np+nc1,ncon); if nargout >= 4 ax = vpck(storxinf,gammavecxi); if nargout >= 5 hamx = vpck(storhx,gammavechx); end end % %