# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 5 Tracking with Perturbations
### Rejection of Measurable Perturbations  
Now, consider that perturbations are measurable. This case is of particular importance because, in many situations, it is possible to consider some external effects as measurable perturbations. For example, the variations in load or the DC input voltage can be available if some low-cost sensors are implemented. If \(y_{ref}(t)\) is the associated output reference, we must have  
\[y_{ref}(t)=C_{0}x_{ref}(t)+D_{0}(x_{ref}(t))\lambda_{ref}(t)+F_{0}(w(t)) \lambda_{ref}(t)\]  
where the state reference \(x_{ref}(t)\) is admissible. As in the previous paragraph, the reference signal is generated from an exosystem (defined in exactly the same way), whose state belongs to a compact set \(\mathbb{R}^{s}\) and is used to generate the state reference. However, the effects of the perturbation must be taken into account. This is why the state reference must include a term accounting for the perturbation. The state reference signal will be expressed as  
\[x_{ref}(t)=\Pi_{1}(z)(t)+\Pi_{p}(w)(t)\]  
where \(\Pi_{1}\) and \(\Pi_{p}\) are appropriate maps, \(\Pi_{1}\) being defined as in the previous paragraph, and \(\Pi_{p}(0)=0\). The reference belongs to the following set:  
\[X_{pert}=\left\{x_{ref}:\mathbb{R}_{+}\rightarrow\mathbb{R}^{n}:x_{ref}(t)= \Pi_{1}(z)(t)+\Pi_{p}(w)(t)\text{ and }\dot{z}(t)=\Theta z(t),\ z(0)=z_{0}\in\mathbb{R}^{s}\right\}\]  
A solution of the associated tracking control problem is only possible for admissible reference signals, the notion of admissibility being an extension of the one introduced in Section 3 and closely related to the signals that the converter is able to track in the presence of the measured perturbations. For the system described by (13), a reference signal is said to be admissible if it belongs to the set  
\[X_{Ad_{pert}}=\left\{x_{ref}\in X_{pert}\text{ and }w\in L_{2}:\exists \lambda_{ref}(t):\mathbb{R}\rightarrow\Lambda_{\mathbb{R}}:\Pi_{2}(z,w)(t) =B_{0}(x_{ref}(t))\lambda_{ref}(t)\right.\] \[\left.+E_{0}(w(t))\lambda_{ref}(t)\right\}\]  
where \(\Pi_{2}(z,w)=-A_{0}\Pi_{1}(z)-A_{0}\Pi_{p}(w)+\frac{\partial\Pi_{1}}{ \partial z}\Theta(z)+\frac{\partial\Pi_{p}}{\partial w}\frac{dw}{dt}\). By the definitions of \(\Pi_{2}(z,w)\) and the set \(X_{Ad_{pert}}\), we have  
\[\frac{\partial\Pi_{1}}{\partial z}\Theta(z)+\frac{\partial\Pi_{p}}{\partial w }\frac{dw}{dt}=A_{0}\left[\Pi_{1}(z)+\Pi_{p}(w)\right]+\Pi_{2}(z,w)\]  
From the admissibility of \(x_{ref}\), we deduce that \(x_{ref}\) satisfies the following differential system, which can be seen as an extension of the regulator equation when measured perturbations are present. \[\dot{x}_{ref}=A_{0}x_{ref}+B_{0}(x_{ref})\lambda_{ref}+E_{0}(x_{ref})\lambda_{ ref}\]  
If \(x_{ref}(t)\in X_{Ad_{pert}}\) and the error signal is defined as  
\[e(t)=x(t)-x_{ref}(t)\]we have  
\[\frac{de(t)}{dt} =A_{0}x(t)+B_{0}(x(t))\lambda(t)+E_{0}(w(t))\lambda(t)-\frac{ \partial\Pi_{1}}{\partial z}\Theta(z)-\frac{\partial\Pi_{p}}{\partial w}\frac{ dw}{dt}\] \[=A_{0}e(t)+B_{0}(x(t))\lambda(t)+E_{0}(w(t))\lambda(t)-\Pi_{2}(z,w )(t) \tag{20}\]  
Now, the problem can be precisely stated. **Problem 3**. : _Design a control law \(\lambda:\mathbb{R}\rightarrow\Lambda_{S}\), such that, for any initial condition, \(e(0)\in\mathbb{R}^{n}\), system (20) is globally asymptotically stable, ensuring that_  
\[x_{ref}(t)=\Pi_{1}(z)(t)+\Pi_{p}(w)(t)\]0  
**Assumption 3**. : _Given \(x_{ref}\in X_{Ad_{port}}\), \(x(t)\in\mathbb{R}^{n}\) and \(w(t)\in\mathbb{R}^{r}\), there exist \(\lambda(t)\in\Lambda_{R}\) and a symmetric matrix \(Q_{1}\), \(0\leq Q_{1}\leq Q\) such that, for all \(t\geq 0\)_  
\[x_{ref}(t)=\Pi_{1}(z)(t)+\Pi_{p}(w)(t)\]1  
It is important to note that if the conditions of Lemma 1 are satisfied, and this is a case for a large family of converters, for an admissible reference signal, Assumption 3 is met with \(P=S\). **Lemma 5**. : _Consider \(x_{ref}\in X_{Ad_{port}}\). Suppose that Assumption 3 is satisfied. Then,_  
\[x_{ref}(t)=\Pi_{1}(z)(t)+\Pi_{p}(w)(t)\]2  
Proof. : Similar as for Lemma 2. **Theorem 3**.