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 specification 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.