Compute bounds on the (skewed) robust stability margin or on the worst-case $\mathcal{H}_\infty$ norm with desired accuracy.
Let us consider the following interconnections between a nominal LTI system $M(s)$ and a block-diagonal operator $\Delta(s)$.
If $M(s)$ has no exogenous input and output, a robust stability problem is considered: (skew-)$\mu$ upper and lower bounds, i.e. lower and upper bounds $k_{lb}$ and $k_{ub}$ on the robustness margin $k_{max}$, are computed (see problem 1 on the overview page of the toolbox for more details). The lower bound $k_{lb}$ is guaranteed on a whole frequency interval and satisfies $k_{ub}\le (1+\epsilon)k_{lb}$, where $\epsilon$ is a user-defined threshold. A destabilizing value $\tilde{\Delta}(s)$ of $\Delta(s)$ associated to the upper bound $k_{ub}$ is also determined, and the considered stability region can be bounded either by the imaginary axis or by a truncated sector.
If $M(s)$ has exogenous inputs $e$ and outputs $y$, a worst-case $\mathcal{H}_\infty$ performance problem is considered: lower and upper bounds $\gamma_{lb}$ and $\gamma_{ub}$ are computed on the highest $\mathcal{H}_\infty$ norm $\gamma_{max}$ of the transfer matrix between $e$ and $y$ when $\Delta(s)$ takes any admissible value (see problem 2 on the overview page of the toolbox for more details). The upper bound $\gamma_{ub}$ is guaranteed on a whole frequency interval and satisfies $\gamma_{ub}\le (1+\epsilon)\gamma_{lb}$, where $\epsilon$ is a user-defined threshold. A value $\tilde{\Delta}(s)$ of $\Delta(s)$ for which the lower bound $\gamma_{lb}$ is obtained is also determined, and the $\mathcal{H}_\infty$ norm can be computing either along the imaginary axis or along the boundary of a truncated sector.
Note: In both cases, the initial interconnection is normalized with convert_data
before the problem is solved.
[bnds,wc,pert,iodesc]=mubb(usys{,skew/perf,options})
[bnds,wc,pert,iodesc]=mubb(gsys{,skew/perf,options})
[bnds,wc,pert,iodesc]=mubb(lsys{,skew/perf,options})
[bnds,wc,pert,iodesc]=mubb(sys,blk{,skew/perf,options})
The interconnection between $M(s)$ and $\Delta(s)$ can be described by:
usys
,
gsys
,
lsys
,
sys
describing the LTI system $M(s)$ and a $N\times 2$ matrix blk
defining the structure of the block-diagonal operator $\Delta(s)=diag\left(\Delta_1(s),\dots,\Delta_N(s)\right)$:
blk(i,:)=[-ni 0]
$\Rightarrow$ $\Delta_i(s)=\delta_iI_{n_i}$ with $\delta_i$ real,
blk(i,:)=[ni 0]
$\Rightarrow$ $\Delta_i(s)=\delta_iI_{n_i}$ with $\delta_i$ complex,
blk(i,:)=[ni mi]
$\Rightarrow$ $\Delta_i(s)$ is a $n_i\times m_i$ LTI system.
For a robust stability problem, an additional input argument skew
can be defined, indicating whether the maximum admissible $\mathcal{H}_\infty$ norm of each $\Delta_i(s)$ is bounded (skew(i)=0
) or not (skew(i)=1
). If skew
is undefined, skew(i)
is set to 1 for all $i=1,\dots,N$, which means that a classical $\mu$ problem is considered.
For a worst-case $\mathcal{H}_\infty$ performance problem, an additional input argument perf
can be defined to specify how the transfer between $e$ and $y$ is structured. Each line of perf
corresponds to a performance channel. For example, if perf=[2 1;1 3]
, the transfer between inputs 1-2 and output 1 is considered independently of the one between input 3 and outputs 2-3-4. If perf
is undefined, it is set to [ne ny]
, where ne
and ny
denote the size of the signals $e$ and $y$.
The last input argument options
is an optional structured variable with fields:
maxgap |
Maximum gap between the bounds. The default value is options.maxgap=0.05 , i.e. 5%. In case of a $\mu$-test, (options.umax>0 ), options.maxgap is used to set options.tol in muub_mixed . |
maxcut |
Maximum number of times the real parametric domain can be bisected. The default value is options.maxcut=100 . |
maxtime |
Maximum computational time. The default value is options.maxtime=Inf . |
freq |
Frequency interval $\Omega$ in rad/s on which the bounds are to be computed. The default value is options.freq=[0 10*max(abs(eig(sys)))] . |
sector |
Vector $[\alpha\ \xi]$ or $[\alpha\ \xi\ \omega_c]$ characterizing the considered truncated sector (see display_sector for a complete description). The default value is options.sector=[0 0] , which means that the left half plane is considered. |
trace |
Trace of execution. The default value is options.trace=1 . |
warn |
Warnings display. The default value is options.warn=1 . |
umax |
If nonzero, the algorithm checks whether the (skew-)$\mu$ upper bound or the upper bound the worst-case $\mathcal{H}_\infty$ norm is less than options.umax ($\mu$-test). In this case, no lower bound is computed. The default value is options.umax=0 . |
Additional fields can be defined (see mubb_mixed
), but the ones listed above are usually sufficient for non-expert users.
bnds |
Bounds $k_{lb}$ and $k_{ub}$ on the robustness margin for the normalized interconnection, or bounds $\gamma_{lb}$ and $\gamma_{ub}$ on the worst-case $\mathcal{H}_\infty$ norm for each performance channel, on the considered frequency interval $\Omega$. |
wc |
Frequency $\omega_c$ in rad/s for which $k_{ub}$ or $\gamma_{lb}$ has been computed. |
pert |
The normalized interconnection is unstable (robust stability problem) or the $\mathcal{H}_\infty$ norm of the transfer between $e$ and $y$ is greater than or equal to $\gamma_{lb}$ (worst-case $\mathcal{H}_\infty$ performance problem) for every perturbation $\tilde{\Delta}(s)$ whose frequency response $\tilde{\Delta}(p(\omega_c))$ is equal to pert , where $p(\omega_c)$ denotes the point $j\omega_c$ on the imaginary axis or the corresponding point on the boundary of the truncated sector defined by options.sector . |
iodesc |
Cell array of structured variables:
In case a worst-case $\mathcal{H}_\infty$ performance problem is considered, the last cell of |
1. Robust stability problem
Description of the uncertain system by a USS object:
a1=ureal('a',5,'Percentage',20);
b1=ucomplex('b',4+3*sqrt(-1),'Percentage',40);
c1=ultidyn('c',[2 1],'Bound',2,'Type','GainBounded');
A1=[-3*a1*b1-2 1;a1 -b1^2-1]+c1*[a1 -b1];
usys=lft(tf({1 0;0 1},[1 0]),A1);
usys=simplify(usys,'full');
Computation of bounds $k_{lb}$ and $k_{ub}$ on the robustness margin $k_{max}$ for the normalized interconnection:
options.lmi=1;
options.maxgap=0.005;
[bnds1,wc1,pert1,iodesc1]=mubb(usys,options);
k_lb=1/bnds1(2),k_ub=1/bnds1(1)
Values of $\Delta(s)$ for which the stability of the initial interconnection is guaranteed and frequency response $\tilde{\Delta}(\omega_c)$ of a perturbation $\tilde{\Delta}(s)$ which makes the initial interconnection unstable:
iodesc1{:}
sys1=usubs(usys,'a',iodesc1{1}.value,'b',iodesc1{2}.value,'c',iodesc1{3}.value);
damp(sys1)
Description of the uncertain system by a GSS object:
a3=gss('a',5,[4 6]);
b3=gss('b',4+3i,[4 3 2]);
c3=gss('c','LTI',[2 1],0,2);
A3=[-3*a3*b3-2 1;a3 -b3^2-1]+c3*[a3 -b3];
gsys=abcd2gss(A3,2);
Computation of bounds $k_{lb}$ and $k_{ub}$ on the robustness margin $k_{max}$ for the normalized interconnection:
options.lmi=1;
options.maxgap=0.005;
[bnds3,wc3,pert3,iodesc3]=mubb(gsys,options);
k_lb=1/bnds3(2),k_ub=1/bnds3(1)
Values of $\Delta(s)$ for which the stability of the initial interconnection is guaranteed and frequency response $\tilde{\Delta}(\omega_c)$ of a perturbation $\tilde{\Delta}(s)$ which makes the initial interconnection unstable:
iodesc3{:}
sys3=eval(gsys,{'a' 'b' 'c'},{iodesc3{2}.value iodesc3{3}.value iodesc3{1}.value});
damp(sys3)
2. Worst-case $\mathcal{H}_\infty$ performance problem
Description of the uncertain system by a LFR object:
a2=lfr('a','ltisr',1,[4 6],'minmax');
b2=lfr('b','ltisc',1,[4+3*sqrt(-1) 2],'disc');
c2=lfr('c','ltifc',[2 1],ltisys([],[],[],2),'freq');
d2=lfr('d','ltifc',[2 2],ltisys(-1,1,3,2),'freq');
A2=[-3*a2*b2-2 1;a2 -b2^2-1]+c2*[a2 -b2];
B2=[1 3;2 a2];
C2=d2*[b2 1;2 -3];
D2=[0 2;0 1];
lsys=abcd2lfr([A2 B2;C2 D2],2);
lsys=minlfr(lsys);
size(lsys)
Computation of bounds $\gamma_{lb}$ and $\gamma_{ub}$ on the worst-case $\mathcal{H}_\infty$ norm $\gamma_{max}$:
[bnds2,wc2,pert2,iodesc2]=mubb(lsys,options);
gamma_lb=bnds2(1),gamma_ub=bnds2(2)
Values of $\Delta(s)$ for which the worst-case $\mathcal{H}_\infty$ norm is guaranteed to be less than $\gamma_{ub}$ for the initial interconnection and frequency response $\tilde{\Delta}(\omega_c)$ of a perturbation $\tilde{\Delta}(s)$ for which the $\mathcal{H}_\infty$ norm or the initial interconnection is equal to $\gamma_{lb}$:
iodesc2{:}
a=iodesc2{1}.value;b=iodesc2{2}.value;c=iodesc2{3}.value;d=iodesc2{4}.value;
sys2=eval(lsys);
svd(calc_freq_resp(ss(sys2.a,sys2.b,sys2.c,sys2.d),wc2,[0 0]))