# Routine mulb_1real

Compute $\mu$ lower bounds for a single real perturbation.

### Description

This routine computes $\mu$ lower bounds for an interconnection between a nominal LTI system $M(s)$ and a block-diagonal operator $\Delta=diag\left(\Delta_1,\dots,\Delta_N\right)$, where $\Delta_i=\delta_iI_{n_i}$ and $\delta_i$ is real. Only one uncertainty is allowed to vary at a time. $N$ bounds are thus computed and the ith one corresponds to the inverse of the smallest destabilizing value of $\delta_i$ when $\delta_k=0$ for all $k\ne i$. The considered stability region can be bounded either by the imaginary axis or by a truncated sector.

Note: The aim is not to compute a $\mu$ lower bound as a function of frequency, but to compute a lower bound over the whole frequency range.

### Syntax

[lbnd,wc,pert,tab]=mulb_1real(sys,blk{,options})

### Input arguments

The first two input arguments are mandatory:

 sys LTI object describing the nominal system $M(s)$. blk Matrix defining the structure of the block-diagonal operator $\Delta=diag\left(\Delta_1,...,\Delta_N\right)$. Its first 2 columns must be defined as follows for all $i=1,...,N$: blk(i,1:2)=[-ni 0] $\Rightarrow$ $\Delta_i=\delta_iI_{n_i}$ with $\delta_i$ real.

The third input argument options is an optional structured variable with fields:

 freq Frequency interval $\Omega$ in rad/s on which the lower bound is 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. target The algorithm is interrupted if a lower bound is found which is greater than options.target. The default value is options.target=Inf. epsilon A $\mu$ lower bound $\tilde{\mu}$ is acceptable if a perturbation $\tilde{\Delta}$ of norm 1/$\tilde{\mu}$ and a frequency $\omega_c$ are found, which satisfy $|\mbox{det}(I-M(j\omega_c)\tilde{\Delta})|<\epsilon$, where $M(j\omega_c)$ is the frequency response of $M(s)$ at $j\omega_c$ (or at the corresponding point on the boundary of the truncated sector defined by options.sector). The default value is options.epsilon=1e-8. trace Trace of execution. The default value is options.trace=1. warn Warnings display. The default value is options.warn=1. kmax Largest considered perturbation. The default value is options.kmax=100. npts Maximum number of equally-spaced values of each $\delta_i$ in [0 options.kmax] for which the stability of the interconnection is checked. The default value is options.npts=1000.

### Output arguments

 lbnd Best $\mu$ lower bound on the considered frequency interval $\Omega$. wc Frequency $\omega_c$ in rad/s for which lbnd has been computed. pert Associated destabilizing perturbation $\tilde{\Delta}$: unless lbnd=0, $\mbox{det}(I-M(\omega_c)\tilde{\Delta})\approx 0$, where $M(\omega_c)$ is the frequency response of $M(s)$ at $j\omega_c$ (or at the corresponding point on the boundary of the truncated sector defined by options.sector). tab Structured variable with fields: lbnd: other $\mu$ lower bounds ($N\times 1$ array). wc: associated frequencies in rad/s ($N\times 1$ array). pert: associated destabilizing perturbations ($n\times n\times N$ array, where $n=\displaystyle\sum_{i=1}^{N}{n_i}$).

### Examples

System with 10 states and 1 repeated real parametric uncertainty:
sys=rss(10,4,4); blk=[-4 0]; options.sector=[0.1 0]; [lbnd,wc,pert,tab]=mulb_1real(sys,blk,options);lbnd display_sector(options.sector,lft(sys,pert)); check_stab(lft(sys,pert),options.sector)
One of the eigenvalues of the interconnection lies just outside the considered sector.
abs(det(eye(4)-calc_freq_resp(sys,wc,options.sector)*pert))
The value of $\mbox{det}(I-M(\omega_c)\tilde{\Delta})$ is almost equal to 0.

System with 46 states and 20 non-repeated real parametric uncertainties:
load flexible_airplane options.sector=[-0.05 0.01]; [lbnd,wc,pert,tab]=mulb_1real(sys,blk,options);lbnd display_sector(options.sector,lft(sys,pert)); check_stab(lft(sys,pert),options.sector)
One of the eigenvalues of the interconnection lies just outside the considered sector.
abs(det(eye(20)-calc_freq_resp(sys,wc,options.sector)*pert))
The value of $\mbox{det}(I-M(\omega_c)\tilde{\Delta})$ is almost equal to 0.

mulb
mulb_nreal
mulb_mixed
hinflb_real
display_sector