IQC-invariance analysis

Next to performing the more “conventional” IQC-based analyses with the function iqcanalysis, it is also possible to perform invariance analyses with dynamic IQCs as was recently pointed out in, for example, [21], [22], among others.

Along the lines of [21], IQClab now includes various tests that can be performed by means of the new function iqcinvariance.

There are several options that can be considered as further discussed next.

Energy to State (option $e2x$ )
As a first application, we consider one of the most common examples for regional analysis, namely the computation of invariant sets in the state-space under the assumption that the external disturbances $w$ are bounded in energy.

Given the uncertain plant

$q=M_{11}p+M_{12}w,\ \ p=\Delta(q),$

with realization

$\begin{array}{l}\dot{x}=Ax+B_1p+B_2w,\ \ \ \ x(0)=0\\q = C_1x+D_{11}p+D_{12}w\end{array}$

this option computes the smallest hyper ellipsoidal region

$x(t)\in\left\{x\in\mathbb{R}^{n_x}: x^THx\leq\alpha^2\|w\|_2^2\right\}\ \mathrm{for\ all}\ t\geq0,$

by minimizing the trace of $H^{-1}$ for a given $\alpha>0$ (default is 1).

This option can be selected by specifying the performance metric option ‘e2x’ (energy to state) to the performance block.

As output you obtain:
– $H$
– $gamma=\mathrm{trace}(H^{-1})$

Energy to Output (option $e2z$ )
Similarly, one might wish to compute invariant sets for the output $z$ , again under the assumption that the external disturbances $w$ are bounded in energy.

Given the uncertain plant

$\begin{array}{l}q=M_{11}p+M_{12}w,\ \ p=\Delta(q),\\z=M_{21}p+M_{22}w\end{array}$

with realization

$\begin{array}{l}\dot{x}=Ax+B_1p+B_2w,\ \ \ \ x(0)=0\\q = C_1x+D_{11}p+D_{12}w\\z = C_2x\end{array}$

and $M_{21}(\infty)=M_{22}(\infty)=0$ is strictly proper, this option computes the smallest hyper ellipsoidal region

$z(t)\in\left\{z\in\mathbb{R}^n_z: z^TH z\leq\alpha^2\|w\|_2^2\right\}\ \mathrm{for\ all}\ t\geq0,$

by minimizing the trace of $H^{-1}$ for a given $\alpha>0$ (default is 1).

This option can be selected by specifying the performance metric option ‘e2z’ (energy to output) to the performance block.

As output you obtain:
– $H$
– $gamma=\mathrm{trace}(H^{-1})$

Energy to Peak (option $e2p$ )
In practical applications, one might also be interested in bounds on the individual components of $z_j$ , $j\in\{1,\cdots,k\}$ , which can be interpreted as providing guaranteed bounds on the energy to-peak gain for the performance channels $w\rightarrow z_j$ , $j\in\{1,\cdots,k\}$ . This leads to the third test.

Given the uncertain plant

$\begin{array}{l}q=M_{11}p+M_{12}w,\ \ p=\Delta(q),\\z=M_{21}p+M_{22}w\end{array}$

with realization

$\begin{array}{l}\dot{x}=Ax+B_1p+B_2w,\ \ \ \ x(0)=0\\q = C_1x+D_{11}p+D_{12}w\\z = C_2x\end{array}$

and $M_{21}(\infty)=M_{22}(\infty)=0$ is strictly proper, this option computes the peak gains $\gamma_j$ on the performance output channels $z_j$ , $j\in\{1,\cdots,k\}$ such that

$|z_j(t)|\leq\sqrt{\gamma_j}\alpha\|w\|_2\ \ \forall t\geq0,\ j\in\{1,\cdots,k\},$

by minimizing $\sum_{j=1}^k\gamma_j$ for a given value of $\alpha$ (default is 1).

This option can be selected by specifying the performance metric option ‘e2p’ (energy to peak) to the performance block.

As output you obtain the peak gain (PeakGain):
– $\mathrm{PeakGain}= \mathrm{col}\left(\sqrt{\gamma_1},\cdots,\sqrt{\gamma_{n_z}}\right)\alpha$

State to Output (option $e2z$ )
Another useful test is the possibility to compute invariant sets for the output $z$ for a given non-zero initial condition $x(0)=x_0\neq0$ that may have both zero as well as non-zero elements

For example, $x_0$ might be a unit vector $x_0=\mathrm{col}(0,...,1,...,0)$ ) with the non-zero element being an initial position or velocity. More than one non-zero element is also possible.

Consider the uncertain plant

$\begin{array}{l}q=M_{11}p,\ \ p=\Delta(q),\\z=M_{21}p\end{array}$

with realization

$\begin{array}{l}\dot{x}=Ax+B_1p,\ \ \ \ x(0)=x_0\\q = C_1x+D_{11}p\\z = C_2x\end{array}$

and $M_{21}(\infty)=0$ is strictly proper.

Inputs to be provided:
– $T_{x0}$ is a row/column selection matrix with $n_x$ rows and no more than $n_x$ columns consisting of unit-vectors such that $T_{x0}=\mathrm{im}(x_0)$ . For example for $x_0=\mathrm{col}(0,x_{0,12},0)$ we would have $T_{x0}=\mathrm{col}(0,1,0)$ (default is $T_{x0}=I$ ).
– $W_{x0}\succ0$ is a weighting matrix with as many rows/colums as $x_0$ has non-zero elements (default is $W_{x0}=I$ ).

Note: These two matrices enforce the constraint $T_{x0}^TX_{22}T_{x0}\prec W_{x0}$ on the Lyapunov matrix (i.e., the part associated with the system state).

Given the plant and the matrices $T_{x0}$ and $W_{x0}$ , this options computes the smallest hyper ellipsoidal region

$z(t)\in\left\{z\in\mathbb{R}^n_z: z^TH z\leq \hat{x}_0^TW_{x0}\hat{x}_0\right\}\ \mathrm{for\ all}\ t\geq0,$

by minimizing the trace of $H^{-1}$ for a given $W_{x0}\succ0$ . Here $\hat{x}_0$ denotes the non-zero part of $x_0$ (in the same order).

This option can be selected by specifying the performance metric option ‘x2z’ (state to output) to the performance block.

As output you obtain:
– $H$
– $gamma=\mathrm{trace}(H^{-1})$

State to Peak (option $e2p$ )
Next, we bound the individual components of $z_j$ , $j\in\{1,\cdots,k\}$ for a given non-zero initial condition $x(0)=x_0\neq0$ that may have (as above) both zero as well as non-zero elements.

Consider the uncertain plant

$\begin{array}{l}q=M_{11}p,\ \ p=\Delta(q),\\z=M_{21}p\end{array}$

with realization

$\begin{array}{l}\dot{x}=Ax+B_1p,\ \ \ \ x(0)=x_0\\q = C_1x+D_{11}p\\z = C_2x\end{array}$

and $M_{21}(\infty)=0$ is strictly proper.

Inputs to be provided: $T_{x0}$ , $W_{x0}\succ0$ (see option $x2z$ for details).

Given the plant and the matrices $T_{x0}$ and $W_{x0}$ , this options computes
the peak gains $\gamma_j$ on the performance output channels $z_j$ , $j\in\{1,\cdots,k\}$ such that

$|z_j(t)|\leq\sqrt{\gamma_j\hat{x}_0^TW_{x0}\hat{x}_0}\ \ \forall t\geq0,\ j\in\{1,\cdots,k\},$

by minimizing $\sum_{j=1}^k\gamma_j$ . Here $\hat{x}_0$ denotes the non-zero part of $x_0$ (in the same order).

This option can be selected by specifying the performance metric option ‘x2p’ (state to peak) to the performance block.

As output you obtain:
– $\mathrm{PeakGain}= \mathrm{col}\left(\sqrt{\gamma_1},\cdots,\sqrt{\gamma_{n_z}}\right)$

Application
The application proceeds in exactly the same fashion as the function iqcanalysis. Once all uncertainty blocks have been defined and all IQC-multipliers have been assigned, one can proceed by performing the invariance analysis by means of the function iqcinvariance:

$prob=iqcinvariance(M,\Delta,varargin).$

For the options $e2x$ , $e2z$ , and $e2p$ , the value of $\alpha$ (default is 1.) can be specified as follows:

$prob=iqcinvariance(M,\Delta,'alp','value'),$

For the options $x2z$ and $x2p$ the matrices $T_{x0}$ (default is $I$ .), and $W_{x0}$ (default is $I$ .) can be specified as follows:

$prob=iqcinvariance(M,\Delta,'Tx0','value','Wx0','value').$