# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 5 Tracking with Perturbations
### Rejection of Non-Measurable Perturbations  
We begin with the case in which the perturbations are not available for measurement. In this case, it is not possible to use the knowledge of the perturbation in the controller structure. A means of measuring the rejection level of the perturbation is to consider the \(L_{2}\) gain between the perturbation \(w\) and the signal of interest (here, the tracking error \(e\)) defined as [49]  
\[\sup_{\|w\|_{2}\
eq 0}\frac{\|e\|_{2}}{\|w\|_{2}}\text{ where }\|\bullet\|_{2}^{2}=\int_{0}^{\infty} \bullet^{T}\bullet dt\]  
We also recall that if there exists a function \(V(x)\) and \(\gamma\geq 0\), such that, for all \(t\),  
\[\frac{dV(x)}{dt}+e^{T}e-\gamma^{2}w^{T}w<0\]  
integrating the previous inequality between \(0\) and \(T>0\) with \(x(0)=0\), we can deduce that the \(L_{2}\) gain is lower than or equal to \(\gamma\)[49]. All the notations are in line with those in the previous section. The dynamic of the tracking error can be written as follows:  
\[\frac{de(t)}{dt} =\frac{dx(t)}{dt}-\frac{\partial\Pi_{1}}{\partial z}\Theta(z)(t) \tag{14}\] \[=A_{0}\;x(t)+B_{0}(x(t))\lambda(t)+E_{0}(w(t)\lambda(t)-\frac{ \partial\Pi_{1}}{\partial z}\Theta(z)(t)\] \[=A_{0}e(t)+B_{0}(x(t))\lambda(t)+E_{0}(w(t)\lambda(t)-\Pi_{2}(z)( t)\text{ from }\eqref{eq:det} \tag{15}\]  
The problem tackled here can be summarized as follows. **Problem 2**. : _Design a control law \(\lambda:\mathbb{R}\rightarrow\Lambda_{S}\), such that, for any initial condition, \(e(0)\in\mathbb{R}^{n}\), we have_  
_(i) If_ \(w(t)=0\)_, the tracking error is globally asymptotically stable, i.e.,_  
\[\lim_{t\rightarrow\infty}e(t)=0\]  
_(ii) If_ \(w(t)\
eq 0\)_, for all_ \(t>0\)_, the tracking error is bounded and the perturbation is rejected with a_ \(L_{2}\) _gain lower than_ \(\gamma>0\)_, i.e., we have, for_ \(e(0)=0\)_,_  
\[\|e\|_{2}^{2}\leq\gamma^{2}\|w\|_{2}^{2}\]  
To solve Problem 2, we suppose that Assumption 1 is met and recall that it is the case for a large class of converters if \(P\) is replaced by a matrix satisfying the conditions of Lemma 1. However, to deal with the disturbance rejection problem, Assumption 1 must be refined. **Assumption 2**. : _Given an admissible reference signal \(x_{ref}\in X_{\mathcal{A}d}\) and \(x(t)\in\mathbb{R}^{n}\), there exist \(\lambda(t)\in\Lambda_{R}\), a positive definite symmetric matrix \(Q_{1}\) and a positive number \(\gamma\) such that, for all \(\mu(t)\in\Lambda_{R}\) and \(t\geq 0\),_  
\[e^{T}(t)P[B_{0}(x(t))\lambda(t)+E_{0}(w(t))\mu(t)-\Pi_{2}(z)(t)]+ e(t)^{T}e(t)-\gamma^{2}w(t)^{T}w(t)\] \[\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q_{1}e(t)\]  
It is important to note that this assumption is always satisfied for the class of converters satisfying the conditions of Lemma 1, as shown in the following one. **Lemma 3**. : _Suppose that the conditions of Lemma 1 are met. Then, there exists a positive definite symmetric matrix \(W\in\mathbb{R}^{n\times n}\) and a positive scalar \(\delta\) such that_  
\[\left[ \begin{array}{cc}(A_{0}+A_{i})^{T}W+W(A_{0}+A_{i})+2I&WE_{j}\\ E_{j}^{T}W&-2\delta^{2}I\end{array} \right]<0,\ \ \ i=1,\ldots,M;\ j=1,\ldots,M \tag{16}\]  
_Then, Assumption 2 is satisfied for \(P=W\) and \(\gamma=\delta\)._  
Proof. : To show the first part of Lemma 3, suppose that the conditions of Lemma 1 are satisfied. Then, there exist positive definite symmetric matrices \(Q\in\mathbb{R}^{n\times n}\) and \(S\in\mathbb{R}^{n\times n}\)such that  
\[(A_{0}+A_{i})^{T}S+S(A_{0}+A_{i})+2Q<0,\ \ \ i=1,\ldots,M \tag{17}\]  
Multiplying the previous inequalities by a positive scalar \(\alpha>0\) such that \(\alpha Q<I\), we can see that the matrix \(W=\alpha S\) satisfies the inequalities  
\[(A_{0}+A_{i})^{T}W+W(A_{0}+A_{i})+2I<0,\ \ \ i=1,\ldots,M \tag{18}\]  
and then there exists a positive number \(\delta\) that is sufficiently large such that  
\[(A_{0}+A_{i})^{T}W+W(A_{0}+A_{i})+2I+\frac{1}{2}\delta^{-2}WE_{j}E_{j}^{T}W<0, \ \ \ i=1,\ldots,M,\ j=1,\ldots,M\]  
which, by Schur complement [49], can be transformed into (16).