## iqc_slowtv

Purpose

Deﬁne IQC for the relation

$$w(t) = \delta(t)v(t),$$

where $$\delta(t)$$ is a scalar function which satisfies $$|\delta(t)| \leq D$$ and $$|\dot{\delta}(t)| \leq d\quad \forall t$$.

Synopsis

w==iqc_slowtv(v,d,a,D);

[w,K1,M1,K2,M2]=iqc_slowtv(v,d,a,D);

Description

Let A be a diagonal Hurwitz matrix with dimension m × m, and B be a m × n constant matrix whose elements are all equal to 1, where m is an arbitrary constant and n is the size of the signal v. The IQC given by function iqc slowtv.mhas the form:

$$D^{2}\langle v_{ext},K_{1}v_{ext}\rangle - \langle w_{ext}, K_{1}w_{ext} \rangle + \langle v_{ext}, M_{1} w_{ext} \rangle + \langle w_{ext},M_{1}^{T}v_{ext} \rangle \geq 0,$$

$$d^{2} \langle y, K_{2}y \rangle - \langle u, K_{2}u \rangle + \langle y, M_{2}u \rangle + \langle u,M_{2}^{T}y \rangle \geq 0,$$

where $$K_{1},\ K_{2}$$ are positive symmetric matrices; $$M_{1},\ M_{2}$$ are skew symmetric matrices.  $$v_{ext},\ w_{ext},\ y,\ u$$ are deﬁned as following

$$v_{ext} = \begin{bmatrix} y^{T},v^{T} \end{bmatrix}^{T},\quad w_{ext}= \begin{bmatrix} z^{T}+x^{T},w^{T} \end{bmatrix}^{T},$$

\begin{align*} & \dot{y}(t)=Ay(t)+Bv(t),\quad y(0)=0\\ &\dot{z}(t)=Az(t)+Bw(t),\quad z(0)=0\\ &\dot{x}(t)=Ax(t)+u(t),\quad x(0)=0,\\ & u(t)=\dot{\delta}(t)y(t),\end{align*}

and

$$A= \begin{bmatrix} -a_{1} & & \\ & \ddots & \\ & & -a_{n} \end{bmatrix},\quad B = \begin{bmatrix} 1 & \cdots & 1\\ & \vdots & \\ 1 & \cdots & 1 \end{bmatrix}.$$

Inputs/Outputs

Inputs:

v    Signal with dimension n × 1.

d    Positive scalar.  Upper bound of the $$L_{\infty}$$-norm of $$\dot{\delta}(t)$$, i.e., $$|\dot{\delta}(t)| < d,\ \forall t$$. Default d=1.

a    Row vertor. All of the elements must be positive. The negative value of each element of agives one diagonal entry of the matrix A. Default a=1.

D   Positive scalar.  Upper bound of the $$L_{\infty}$$-norm of $$\delta(t)$$, i.e., $$| \delta(t) | < D,\ \forall t$$.  Default D=1.

Outputs:

w   Basic signal which has the same dimension as v.

K1  Multiplier K1.

M1 Multiplier M1.

K2  Multiplier K2.

M2 Multiplier M2.

Example

Consider the system in Figure 1. The transfer function is

$$G(s)=\frac{0.8}{s^{2}+0.21s+1},$$

and $$|\delta(t)| < 1,\ |\dot{\delta}(t)| < 0.1$$ are known to us.  We would like to check whether the system is stable. The commands below show that the gain from f to v is ﬁnite (the gain is computed, with a=[1, 3, 5], to be 25.7744).  Hence, the system is stable under the appearance of the slowly time-varying coefﬁcient.

Figure 1: System with slowly time-varying coefﬁcient

>> abst_init_iqc;

>> A=[-0.21,-1;1,0];

>> B=[0.8;0];

>> C=[0,1];

>> D=0;

>> G=ss(A,B,C,D);

>> w=signal;

>> f=signal;

>> v=G*(f+w);

>> w==iqc_slowtv(v,0.1,[1,3,5],1);

>> gain=iqc_gain_tbx(f,v)