## 2 Incremental State Model  
It is well established that discrete-time models are suitable for identifying continuous-time systems with multiple delays through the use of difference equations, as such models permit the integration of delays directly into the state formulation. In this context, the incremental state-space model offers distinct advantages over its non-incremental counterpart. Specifically, it simplifies the computation of the reference state, since a zero incremental state can be directly associated with a desired output, effectively eliminating the need for explicit reference state calculation. Additionally, the incremental formulation inherently introduces an integrative effect into the control law, thereby enabling zero steady-state error in response to constant references or disturbances [15]. We use a discrete-time multivariable system with \(p\) inputs, \(u\in\Re^{p}\), and \(q\) outputs, \(y\in\Re^{q}\), where \(q\leq p\)[35]:  
\[ \begin{split} y_{i}(k+1)&=a_{i0}+a_{1 }y_{i}(k)+\cdots+a_{in_{i}}y_{i}(k-n_{i}+1)+\\ &+\sum_{j=1}^{p}\left(b_{ij1}u_{j}(k)+b_{ij2}u_{j}(k-1)+\cdots+b_ {ijn_{i}}u_{j}(k-n_{i}+1)\right)\end{split}  \tag{1}\]  
The system has been identified using the least squares method with affine terms; therefore, some of the parameters \(a_{in_{i}}\), \(b_{ij1}\),..., \(b_{ipn_{i}}\) are non-zero. For each output, the difference equations can be transformed into a discrete-time state-space model with affine terms:  
\[ \begin{split} x_{i}(k)\in\Re^{n_{i}}\\ x_{i}(k+1)&=\left[\begin{array}{c}a_{i0}\cdot a_{i 1}\\ a_{i0}\cdot a_{i2}\\ \vdots\\ a_{i0}\cdot a_{in_{i}}\end{array}\right]+\left[\begin{array}{cccc}a_{i1}&1& \cdots&0\\ a_{i2}&0&\ddots&0\\ \vdots&\vdots&\ddots&1\\ a_{in_{i}}&0&\cdots&0\end{array}\right]x_{i}(k)\\ &+\left[\begin{array}{cccc}b_{i11}&b_{i21}&\cdots&b_{ip1}\\ b_{i12}&b_{i21}&\vdots&b_{ip2}\\ \vdots&\vdots&\vdots&\vdots\\ b_{11n_{i}}&b_{i1n_{i}}&\cdots&b_{ipn_{i}}\end{array}\right]\left[\begin{array}[] {c}u_{1}(k)\\ u_{2}(k)\\ \vdots\\ u_{p}(k)\end{array}\right]\\ y_{i}(k)&=a_{i0}+\left[\begin{array}{cccc}1&0&\cdots&0\end{array}\right]x_{ i}(k)\end{split}  \tag{2}\]  
In matrix notation,  
\[ \begin{split} x_{i}(k+1)&=a_{xi}+A_{i}x_{i}(k)+B_{i}u(k)\\ y_{i}(k)&=a_{i0}+C_{i}x_{i}(k)\end{split}  \tag{3}\]where \(a_{xi}\in\Re^{(n_{i}\times 1)}\), \(A_{i}\in\Re^{(n_{i}\times n_{i})}\), \(B_{i}\in\Re^{(n_{i}\times p)}\) y \(C_{i}\in\Re^{(1\times n_{i})}\). By grouping the entire multivariable system, the following is obtained:  
\[\left[ \begin{array}{c}x_{1}(k+1)\\ x_{2}(k+1)\\ \vdots\\ x_{q}(k+1)\end{array} \right]=\left[ \begin{array}{c}a_{x1}\\ a_{x2}\\ \vdots\\ a_{qq}\end{array} \right]+\left[ \begin{array}{cccc}A_{1}&0&\cdots&0\\ 0&A_{2}&\ddots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&A_{q}\end{array} \right]\left[ \begin{array}{c}x_{1}(k)\\ x_{2}(k)\\ \vdots\\ x_{q}(k)\end{array} \right]+\left[ \begin{array}{c}B_{1}\\ B_{2}\\ \vdots\\ B_{q}\end{array} \right]\left[ \begin{array}{c}u_{1}(k)\\ u_{2}(k)\\ \vdots\\ u_{p}(k)\end{array} \right] \tag{4}\]  
\[\left[ \begin{array}{c}y_{1}(k)\\ y_{2}(k)\\ \vdots\\ y_{q}(k)\end{array} \right]=\left[ \begin{array}{c}a_{10}\\ a_{20}\\ \vdots\\ a_{0}\end{array} \right]+\left[ \begin{array}{c}C_{1}&0&\cdots&0\\ 0&C_{2}&\ddots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&C_{q}\end{array} \right]\left[ \begin{array}{c}x_{1}(k)\\ x_{2}(k)\\ \vdots\\ x_{q}(k)\end{array} \right]\]  
This can be represented as follows:  
\[x(k+1) =a_{x}+Ax(k)+Bu(k) \tag{5}\] \[y(k) =a_{y}+Cx(k)\]  
\(a_{x}\in\Re^{(n\times 1)}\), \(A\in\Re^{(n\times n)}\), \(B\in\Re^{(n\times p)}\), \(a_{y}\in\Re^{(q\times 1)}\) y \(C\in\Re^{(q\times n)}\), where \(n=n_{1}+n_{2}+\cdots+n_{q}\). Equation (5) will be referred to as the discrete-time state-space model, which has been obtained through input-output identification; therefore, it is controllable and observable.