## A two-mass-one-spring system

A  two-mass-one-spring system: description of the problem

In this section, a simple example is used to illustrate how IQC$$\beta$$ can be used to estimate the upper bound of the L2-induced gain of a pair of signals in an uncertain system. Consider the controlled two-mass-one-spring system in Figure 2.1. This example is adopted from the manual of MATLAB LMI Control Toolbox. It was a benchmark problem proposed in [5]. The goal of the original problem is to design an output-feedback law $$u = G_{c} (s)x_{2}$$ that adequately rejects the disturbances $$w_{1}$$, $$w_{2}$$, and guarantees stability for values of the stiffness parameter $$k$$ ranging in a certain given range. Here the control design problem will not be taken into account. We will use a proposed controller and illustrate how to verify stability of the interconnected control system by using IQC$$\beta$$ .

Figure 2.1: Two-mass-one-spring system

For $$m_{1} = m_{2} = 1$$, the dynamics of the system can be described by a fourth order state-space model

$$\dot{x}=Ax+B_{1}(u+w_{1})+B_{2}w_{2}$$

where the state vector $$x$$ consists of the position and the velocity of masses $$m_{1}$$ and $$m_{2}$$. The matrices $$A$$, $$B_{1}$$ and $$B_{2}$$ are given as follows

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

Matrix $$A$$ depends affinely on $$k$$. By letting $$k = 1.25+\frac{3}{4}\bar{k}$$, one can be further expressed as $$A = A_{0} +A_{1}\bar{k}A_{2}^{T}$$, where $$A_{0}$$, $$A_{1}$$, $$A_{2}$$ are again constant matrices

$$A_{0}=\begin{bmatrix} 0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\ -1.25 & 1.25 & 0 & 0\\ 1.25 & -1.25 & 0 & 0\end{bmatrix},\quad A_{1}=\begin{bmatrix} 0\\0\\-\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{bmatrix},\quad A_{2}=\begin{bmatrix} \frac{3\sqrt{2}}{4}\\ -\frac{3\sqrt{2}}{4}\\0\\0\end{bmatrix},$$

and $$\bar{k}$$ is an unknown constant in the range of $$[−a,\ a]$$. When $$a = 1$$, the spring coefficient $$k$$ belongs to the range of $$[0.5,\ 2]$$, which is the control design speci&#64257;cation of the original problem in [5].

A fourth-order stabilizing controller $$u = C_{c}(sI-A_{c})^{-1}B_{c}x_{2}$$ was proposed in [5], where

$$A_{c}=\begin{bmatrix} 0 & -0.7195 & 1 & 0\\0 & -2.9732 & 0 & 1\\ -2.5133 & 4.8548 & -1.7287 & -0.9616\\ 1.0063 & -5.4097 & -0.0081 & 0.0304\end{bmatrix},\quad B_{c}=\begin{bmatrix} 0.720\\ 2.973\\ -3.37\\ 4.419\end{bmatrix},\quad C_{c}^{T}=\begin{bmatrix} -1.506\\ 0.494\\ -1.738\\ -0.932\end{bmatrix}$$

The mass-spring control system can be expressed as the block diagram shown in Figure 2.2. where $$B_{1}$$, $$B_{2}$$, $$C$$ are $$[0 \ 0 \ 1 \ 0]^{T}$$, $$[0 \ 0 \ 0 \ -1]^{T}$$, $$[0 \ 1 \ 0 \ 0]$$, respectively.

Figure 2.2: Block diagram of the two-mass-one-spring system

In the following, we will show how to estimate the gain from $$[w1,\ w2]$$ to $$x_{2}$$ by IQC$$\beta$$. This is done by computing an estimate of the energy gain from $$[w1,\ w2]$$ to $$x_{2}$$. At current stage, checking stability is always performed by estimating the gain from some external disturbance to any internal signal in the system. A finite gain implies stability. We assume that matrices $$A_{0}$$, $$A_{1}$$, $$A_{2}$$, $$B_{1}$$, $$B_{2}$$, and $$C$$, as well as the LTI systems $$G_{p}(s)$$ and $$G_{c}(s)$$, are already defined in the MATLAB workspace. We will also assume that the readers have experience of working with MATLAB Control Systems Toolbox.