iqc_multi_harmonic

Purpose

Defines IQCs for operators defined as a multiplication with a harmonic multiplier in the time domain, i.e., the input output relation

$$ w(t)=\Delta_{H}(t)v(t),$$

where

$$ \Delta_{H}(t)v=\begin{bmatrix} \cos(k_{1}\omega_{0}t)\\ \sin(k_{1}\omega_{0}t)\\ \vdots\\ \cos(k_{N}\omega_{0}t)\\ \sin(k_{N}\omega_{0}t)\end{bmatrix},$$

and where \( 0 < k_{1} < k_{2} < \cdots < k_{N}\)  are integers, \(w_{0}\)  is any real number, and I is the n × n identity matrix, where n is the dimension of v.

Synopsis

w===iqc_multi_harmonic(v,K,a)

[w,Qr,Pr,Qi,Pi]=iqc_multi_harmonic(v,K,a)

Description

Let \(w(t) = \Delta_{H}v(t)\). The IQCs generated by the command iqc_multi_harmonic have the form

$$ \int_{0}^{\infty} \begin{bmatrix} v\\ \Delta_{H}(v)\end{bmatrix}^{T} M \begin{bmatrix} v\\ \Delta_{H}(v)\end{bmatrix} dt \geq 0,$$

where \( M = \textrm{Re}(U^{*}QU)\),

$$ U=\begin{bmatrix} I & 0 & 0 & 0 & \cdots & 0 & 0 & 0\\0 & I & iI & 0 & \cdots & 0 & 0 & 0\\ & & & & \cdots & & & \\0 & 0 & 0 & 0 &\cdots & 0 & I & iI\end{bmatrix},$$

and \( Q=Q^{*} \in \textbf{C}^{N+1}n\times(N+1)n\) satisfies the LMI

$$ \begin{bmatrix} A^{T}PA-P & A^{T}PB\\ B^{T}PA & B^{T}PB\end{bmatrix}+\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}Q\begin{bmatrix}C & D\end{bmatrix} \geq 0$$

 for some \(P=P^{*} \in \textbf{C}^{nK_{N}\times nK_{N}}\). Here A,B,C,D is any controllable realization of

$$ H(z)=\frac{1}{(z+a)^{k_{N}}}\begin{bmatrix}I\\z^{k_{1}}I\\\vdots\\z^{k_{N}}I\end{bmatrix}.$$

In order to use the parametrization \( \tau\Delta_{H},\ \tau \in [0,\ 1]\), we also impose the constraints \(M_{11} \geq 0\) and \(M_{22} \leq 0\), where \(M_{11} \in \textbf{R}^{n\times n}\) and \(M_{22} \in \textbf{R}^{2nN\times 2nN}\) are the submatrices of M defined by

$$ M=\begin{bmatrix}M_{11} & M_{12}\\M_{12}^{T} & M_{22}\end{bmatrix}.$$

Inputs/Outputs

Inputs:

v   Input to the harmonic multiplier.

K   Defines the harmonic frequencies \(K = [k_{1} \ \cdots \ k_{N}]\), where \(0 < k_{1}  < \cdots < k_{N}\) . Default K = 1.

a   The pole location of H, a > 1. Default a = 2.

Outputs:

w   Output of the harmonic multiplier.

Qr  Real part of Q (optional).

Pr  Real part of P (optional).

Qi  Imaginary part of Q (optional).

Pi  Imaginary part of P (optional).

Example

Let us consider the robustness problem of finding a bound on \(| \Delta(t)|\) , such that the system

$$ \dot{x}=(A(t)+\Delta(t))x,\quad x(0)=x_{0},$$

is exponentially stable. We assume that

$$ A(t)=A_{0}+A_{1}\cos(t)+B_{1}\sin(t)+A_{2}\cos(2t)+B_{2}\sin(2t),$$

where

 $$ A_{0}=\begin{bmatrix} -3 & 0\\ 0 & -6\end{bmatrix},\quad A_{1}=\begin{bmatrix} 0 & 1\\2 & 0\end{bmatrix},\quad B_{1}=\begin{bmatrix} 1 & 0\\0 & -1\end{bmatrix}$$

$$A_{2}=\begin{bmatrix} 0 & 2\\ 1 & 0\end{bmatrix},\quad B_{2}=\begin{bmatrix} -1 & 1\\ 1 & 1\end{bmatrix},$$

Figure 1: Equivalent representation of the system.

The system can equivalently be represented as in Figure 1, where

$$ G(s)=(sI-A_{0})^{-1}\begin{bmatrix} A_{1} & B_{1} & A_{2} & B_{2} & I \end{bmatrix}$$

$$\Delta_{H}(t)=\begin{bmatrix} \cos(t)I\\ \sin(t)I\\ \cos(2t)I\\ \sin(2t)I\end{bmatrix},$$

and where f corresponds to the initial condition. The following sequence of commands estimates the gain from f to v to be gain=38.6516. This shows that the system is stable when \(|\Delta(t) \leq 0.4\).

>>n=2

>>I=eye(n)

>>O=zeros(n,n)

>>A0=[-3 0;0 -6];

>>A1=[0 1;2 0];

>>B1=[1 0;0 -1];

>>A2=[0 2;1 0];

>>B2=[-1 1;1 1];

>>G=ss(A0,[A1 B1 A2 B2 I],I,[O O O O O]);

>>abst_init_iqc;

>>w1=signal(4*n);

>>w2=signal(n);

>>f=signal(n);

>>v=G*[w1;w2]+f;

>>w1==iqc_multi_harmonic(v,[1 2],2);

>>w2==iqc_ltvnorm(v,n,0.4);

>>gain=iqc_gain_tbx(f,v)

References

U. J¨onsson and A. Megretski. Some new IQCs and their application. Under Preparation.