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Routine hinflb_real

Compute lower bounds on the worst-case $\mathcal{H}_\infty$ norm for real perturbations.


This routine computes lower bounds on the worst-case $\mathcal{H}_\infty$ norm 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. Lower bounds are first computed on a rough frequency grid. A series of quadratic programming problems is then solved using these results as an initialization.

Let $\mathcal{B}_{\bf\Delta}$ be the set of all admissible block-diagonal operators $\Delta$ such that $|\delta_i|<1$ for all $i=1,\dots,N$. Assume that the interconnection between $M(s)$ and $\Delta$ is stable for all $\Delta\in\mathcal{B}_{\bf\Delta}$. Let $\mathcal{F}_u(M(s),\Delta)$ denote the transfer matrix from $e$ to $y$, which must be square (see make_square). The worst-case $\mathcal{H}_\infty$ norm is defined by:
$$\max_{\Delta\in\mathcal{B}_{\bf\Delta}} \|\mathcal{F}_u(M(s),\Delta)\|_\infty$$

where the $\mathcal{H}_\infty$ norm can be computed either on the imaginary axis or on the boundary of a truncated sector.

Note: The aim is not to compute a lower bound as a function of frequency but to compute a lower bound over the whole frequency range. The frequency for which the largest singular value of the transfer matrix from $e$ to $y$ is maximum is left free during the optimization and can be very far from the initial one. Thus, a rough initial grid is usually sufficient to capture the most critical frequencies.



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)))].
grid Frequency grid in rad/s ($1\times m$ array). It can also be a negative integer -$m$, in which case a $m$-point grid is automatically generated from $\Omega$. The default value is options.grid=-10.
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 The default value is
trace Trace of execution. The default value is options.trace=1.
warn Warnings display. The default value is options.warn=1.
wpar If zero, frequency is treated as an additional parameter in $\Delta$ and the problem is solved on the intervals [grid(1) grid(2)],...,[grid(end-1) grid(end)]. The default value is options.wpar=0.
prec Level of precision of the final result (options.prec=2 $\Leftrightarrow$ good precision, options.prec=1 $\Leftrightarrow$ faster but potentially less precise). The default value is options.prec=1.
local If zero, the algorithm focuses on the global worst-case configuration. Otherwise, it also tries to detect local maxima, but the probability to determine the global worst-case configuration is lower. The default value is options.local=0.
method $1\times 2$ array defining the method used for the line search:

  • options.method=[1 tol] means that the optimum is computed from the real eigenvalues of a hamiltonian matrix with a precision of tol.
  • options.method=[2 m] means that the optimum is computed on a grid composed of $m$ points.

options.method=[1 tol] is often more accurate but CPU time is higher. The default value is options.method=[1 0.001] or [2 6] depending on the size of $\Delta$ (less or more than 50).

Output arguments

lbnd Best lower bound on the worst-case $\mathcal{H}_\infty$ norm on the considered frequency interval $\Omega$.
wc Frequency $\omega_c$ in rad/s for which lbnd has been computed.
pert Associated value of $\tilde{\Delta}$: unless lbnd=0, the largest singular value of $\mathcal{F}_u(M(\omega_c),\tilde{\Delta})$ is equal to lbnd, 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 lower bounds on the worst-case $\mathcal{H}_\infty$ norm ($m\times 1$ array).
  • wc: associated frequencies in rad/s ($m\times 1$ array).
  • pert: associated values of $\tilde{\Delta}$ ($n\times n\times m$ array, where $n=\displaystyle\sum_{i=1}^{N}{n_i}$).


System with 10 states and 2 uncertainties. The last two inputs and outputs correspond to the performance channel:
blk=[-2 0;-1 0];
options.sector=[-0.05 0];

See also



[1] C. Roos, "A practical approach to worst-case $\mathcal{H}_\infty$ performance computation", in Proceedings of the IEEE Multiconference on Systems and Control, Yokohama, Japan, September 2010, pp. 380-385.