Robust estimator synthesis

IQClab also provides the option to design robust estimators with general dynamic multipliers.

Note: It is assumed that the user is acquainted with the robust estimation literature. The reader is referred to [24] and references therein for further information.

The algorithm considered here corresponds to the one described in [24], which considers the problem of synthesizing robust estimators based on dynamic IQCs. This problem is convex and based on solving an LMI problem.

We consider the system interconnection shown in the following figure.

Here:

$G$ is the plant model
$\Delta$ is the uncertainty block, which is assumed to have a block diagonal structure $\Delta=\mathrm{diag}(\Delta_1,\ldots,\Delta_N)$ . The properties of the individual blocks should be compatible with those from the class iqcdelta (see details here)
$E$ is the to-be-designed robust estimator
the in- and output channels are given by
- $p$ and $z$ are the uncertainty in- and outputs
- $w$ is the disturbance input
- $z$ is the to-be-estimated output
- $y$ is the measurement output
- $e=z+z_E$ is the estimation error output

Some characteristics of the algorithm are:

The algorithms solves an LMI problem with the aim to synthesize a robust estimator, $z_E=Ey$ , which, for all $\Delta\in\mathbf{\Delta}$ , ensures that the interconnection in the figure above is robustly stable, while the induced $L_2$ -gain or $H_2$ -norm from $w\rightarrow e$ rendered less than $\gamma>0$ .
The $\Delta$ -block is compatible with the class iqcdelta (see details here)

Usage:

$[E,\gamma]=fRobsyn(P,\Delta,n_{out},n_{in})$
$[E,\gamma]=fRobsyn(P,\Delta,n_{out},n_{in},options)$

If feasible, the algorithm returns:

The estimator $E=ss(A_{E},B_{E},C_{E},D_{E})$ .
The (guaranteed) induced $L_2$ -gain or $H_2$ -norm, $\gamma>0$ , on the channel $w\rightarrow e$ .

The inputs should be provided as follows:

The (stable) LTI plant $G$ is assumed to admit the following state space description
$\left(\!\!\!\begin{array}{c}q\\z\\y\end{array}\!\!\!\right)\!=\!\left[\!\!\begin{array}{c|ccc}A&B_\mathrm{p}&B_\mathrm{w}\\ \hline C_\mathrm{q}&D_{\mathrm{qp}}&D_{\mathrm{qw}}\\ C_\mathrm{z}&D_{\mathrm{zp}}&D_{\mathrm{zw}}\\ C_\mathrm{y}&D_{\mathrm{yp}}&D_{\mathrm{yw}} \end{array}\!\!\right]\!\!\!\left(\!\!\begin{array}{c}p\\w\end{array}\!\!\!\right),$
where
$p(\cdot)\in\mathbb{R}^{n_p}$ and $w(\cdot)\in\mathbb{R}^{n_w}$ are the uncertainty and disturbance inputs, while $q(\cdot)\in\mathbb{R}^{n_q}$ , $z(\cdot)\in\mathbb{R}^{n_z}$ , and $y(\cdot)\in\mathbb{R}^{n_y}$ are the uncertainty, to-be-estimated, and measurement outputs respectively.
$A$ is Hurwitz.
The uncertainty block $\Delta=\{\Delta_1,\ldots,\Delta_N\}$ is an iqcdelta object, to which IQC-multipliers have been assigned with iqcassign (see details here).
The plant input and output dimension data and must be specified as follows:
- $n_\mathrm{in}=[n_{p};n_{w}]$
- $n_\mathrm{out}=[n_{q};n_{z};n_{y}]$
The last input, options, is a structure with various options as summarized in the following table.

Options	Description
options.perf	Selects the performance metric of interest (i.e., induced $L_2$ -gain or $H_2$ -norm): – options.perf = ‘L2’ (default) – options.perf = ‘H2’
options.subopt	If chosen larger than 1, the algorithm computes a suboptimal solution $options.subopt\cdot\gamma$ , while minimizing the norm on the LMI variables $\\| [A_E, B_E; C_E, D_E] \\|<\gamma$ . The default value is 1.01.
options.constants	options.constants= $[c_1,c_2,c_3,c_4]$ is a vector of (small) nonnegative constants, which perturb the LMIs as $LMI_i\prec-c_i$ , $i\in\{1,2,3\}$ . The constants are associated with the following LMI constraints – $c_1$ perturbs the main synthesis LMI as $LMI_\mathrm{main}\prec-c_1I$ – $c_2$ perturbs the coupling condition $X-\hat{Y}\succ c_2I$ – $c_3$ perturbs the coupling condition $T^T\hat{Y}T-\left(\!\!\begin{array}{cc}Y&0\\0&0\end{array} \!\!\right)\!\succ\!c_3I$ – $c_4$ perturbs the LMI corresponding to the $\Pi_{11}\succ c_4I$ The default value is $1e-9[1,1,1,1]$ .*
options.StrProp	Enforces the estimator, $E=ss(A_E,B_E,C_E,D_E)$ , to be strictly proper (i.e. $D_E=0$ ): – options.StrProp = ‘no’ (default) – options.StrProp = ‘yes’
options.Parser	The option options.Parser specifies which parser is used: – options.Parser=’LMIlab’ – options.Parser=’Yalmip’ The default options is ‘LMIlab’.
options.Solver	The option options.Solver specifies which solver is used when considering Yalmip as parser. See https://yalmip.github.io/ for further info. The default solver is ‘mincx’.
options.FeasbRad	This option allows setting the feasibility radius of the optimization problem (see MATLAB → help → mincx for further details). The default value is 1e9.
options.Terminate	This option can be used to change the LMI solver options (see MATLAB → help → mincx for further details). The default value is 0.
options.RelAcc	This option can be used to change the LMI solver options (see MATLAB → help → mincx for further details). The default value is 1e-4.