# A Control-Oriented Linear Parameter-Varying Model of a Commercial Vehicle Air Brake System
## 4 Linearization Through System Identification
The identification input is the PRBS signal added to the input reference pressure; and the output is the system output pressure (brake chamber pressure) signal minus the steady-state pressure due to input reference signal without PRBS. The PRBS signal used in this paper is based on maximum length sequences (also called m-sequence) for which the length of PRBS signal is \(m=2^{n}-1\), where \(n\) is an integer (order of PRBS) [22], which contains \(m=2^{n-1}\) ones and \(m=2^{n-1}-1\) zeros. The PRBS signal is generated at the sample rate \(T_{p}\) and the output pressure is sampled at the sample rate \(T_{s}\). With the multirate q-Markov Cover identification [25], the sample rate ratio between the PRBS signal and the output pressure is defined as follows. \[n_{r}=\frac{T_{p}}{T_{s}} \tag{23}\]  
Let \(z^{-1}\) represent the delay operator, and define \(\hat{p}(z^{-1})\) and \(p(z^{-1})\) to be polynomials defined below  
\[p(z^{-1})=a_{n}z^{-n+1}\oplus a_{n-1}z^{-n+2}\ldots\oplus a_{2}z^{-1}\oplus a_ {1} \tag{24}\]  
\[p(z^{-1})=\hat{p}(z^{-1})z^{-1}\oplus 1 \tag{25}\]  
Figure 7: System identification framework. where \(a_{i}\) is either zero or one, which can be chosen according to Table 3 and also from [28], and \(\oplus\) obeys binary addition law. The coefficients of the irreducible polynomial with order between 1 and 34 are provided by [PERSON] [28]. The PRBS signal can be generated by the following equation  
\[\hat{u}(k+1)=\hat{p}(z^{-1})u(k) \tag{26}\]  
where \(k=0,1,2\dots,\)\(\hat{u}(0)=1,\hat{u}(-1)=\dots=\hat{u}(-n)=0\). Let  
\[s(k)=\left\{ \begin{array}{cc}a;&k\;\;even\\ -a;&k\;\;odd\end{array} \right.\]  
Then the signal  
\[u(k)=s(k)\odot(-a+2a\hat{u}(k)) \tag{27}\]  
is called the inversed PRBS, where operator \(\odot\) obeys \(a\odot a=-a\odot-a=-a\), \(-a\odot a=a\odot-a=a\). It's clear that \(u\) has a period of \(2m\) and \(u(k)=-u(k+m)\). The length of PRBS signal for one cycle is \(2\;mn_{r}\). In this paper, the notation PRBS is used to represent the inversed PRBS. Consider the discrete-time nonlinear air brake system  
\[\left\{ \begin{aligned} x(k+1)&=f(x(k),w(k))\\ y(k)&=g(x(k),w(k))\end{aligned} \right. \tag{28}\]  
where \(x(k)\), \(w(k)\), \(y(k)\) are state, input, and output vectors, respectively. For a given input sequence \(\{w(0)\), \(w(1)\), \(w(2)\)... \(\}\), the associated output sequence \(\{y(0)\), \(y(1)\), \(y(2)\)... \(\}\) can be obtained through simulation or experimentally. If a nonlinear system is \(q\)-identifiable, there exists the form of linear discrete-time asymptotically stable system in the following form  
\[\left\{ \begin{aligned} x(k+1)&=Ax(k)+Bw(k)\\ y(k)&=Cx(k)+Dw(k)\end{aligned} \right. \tag{29}\]  
that is able to generate the same output sequence \(\{y(0)\), \(y(1)\), \(y(2)\)... \(\}\) for the same given input sequence \(\{w(0)\), \(w(1)\), \(w(2)\)... \(\}\). In this paper, a 10 th order PRBS signal \(u(k)\) with magnitude of \(a=0.4\) or \(0.6\) is used to generate the output sequence \(y(k)\). Note that PRBS order is related to the lowest frequency of the signal covered, and at order 10, the lowest frequency covered is about 0.008 Hz when the PRBS sample period is chosen to be 0.12 s; and the PRBS magnitude is selected to maximize the signal to noise ratio. After the system response is generated, the PRBS input and output sequences are fed into the PRBS system identification \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order of Polynoma \(n\)** & **Period of Sequence \(m\)** & **Non-Zero Coefficients \(a_{i}\)** \\ \hline
2 & 3 & \(a_{1}\),\(a_{2}\) \\
3 & 7 & \(a_{2}\),\(a_{3}\) \\
4 & 15 & \(a_{3}\),\(a_{4}\) \\
5 & 31 & \(a_{3}\),\(a_{5}\) \\
6 & 63 & \(a_{5}\),\(a_{6}\) \\
7 & 127 & \(a_{4}\),\(a_{7}\) \\
8 & 255 & \(a_{2}\),\(a_{3}\),\(a_{4}\),\(a_{8}\) \\
9 & 511 & \(a_{5}\),\(a_{9}\) \\
10 & 1023 & \(a_{7}\),\(a_{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficients for m-sequences. GUI (graphic user interface) [22].