# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 4 Main Result  
We suppose that \(v_{in}(t)\) is constant and equal to \(V_{in}\). Because the matrix \(A_{0}\) is Hurwitz, there exist two positive definite matrices \(P\in\mathbb{R}^{n\times n}\) and \(Q\in\mathbb{R}^{n\times n}\) satisfying  
\[A_{0}^{T}P+PA_{0}+2Q<0 \tag{8}\]  
To solve the previous problem, we introduce an assumption that is quite similar to the one adopted in the context of the stabilization problem addressed in [19]. **Assumption 1**. : _Given an admissible reference signal \(x_{ref}\in X_{\text{Ad}}\) and \(x(t)\in\mathbb{R}^{n}\), there exist \(\lambda(t)\in\Lambda_{R}\) and a positive definite symmetric matrix \(Q_{1},\,0<Q_{1}\leq Q\) such that, for all \(t\geq 0\),_  
\[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]  
This assumption is central and it is important to note that it is satisfied for a large class of converters if matrices \(P\) and \(Q\) verify a condition stronger than (8), as stated in the following lemma. **Lemma 1**. : _Suppose that there exist positive definite symmetric matrices \(Q\in\mathbb{R}^{n\times n}\) and \(S\in\mathbb{R}^{n\times n}\)such that_  
\[(A_{0}+A_{i})^{T}S+S(A_{0}+A_{i})+2Q<0,\ \ \ i=1,\ldots,M \tag{9}\]  
_Then, Assumption 1 is satisfied, with \(P\) replaced by \(S\)._  
Proof. : Suppose that (9) satisfied. Then, for an admissible reference signal, there exists \(\lambda_{ref}(t)\) such that \(0\leq\lambda_{ref}(t)\leq 1\) and  
\[\Pi_{2}(z)(t)=B_{0}(x_{ref}(t),\lambda_{ref}(t))\text{ for all }t\geq 0\]  
We have (time is omitted for convenience)  
\[e^{T}S[B_{0}(x)\lambda-\Pi_{2}(z)]= e^{T}S[B_{0}(x)\lambda-B_{0}(x_{ref})\lambda_{ref}]\] \[=e^{T}S[B_{0}(x)(\lambda-\lambda_{ref})+\Big{[}B_{0}(x)-B_{0}(x_ {ref})\Big{]}\lambda_{ref}]\] \[=e^{T}S[B_{0}(x)(\lambda-\lambda_{ref})+\big{[}\ A_{1}e\ \ \cdots\ \ A_{M}e\ \big{]}\lambda_{ref}]\] \[=e^{T}SB_{0}(x)(\lambda-\lambda_{ref})+e^{T}S\sum_{i=1}^{M} \lambda_{ref_{i}}(A_{0}+A_{i})e-e^{T}SA_{0}e\]  
For \(\lambda=\lambda_{ref}\), we have  
\[e^{T}SA_{0}e+e^{T}S[B_{0}(x)\lambda_{ref}-B_{0}(x_{e})\lambda_{ref}])=e^{T}S \sum_{i=1}^{M}\lambda_{ref_{i}}(A_{0}+A_{i})e<-e^{T}Qe\]  
from (9). By continuity arguments, for all \(t>0\) fixed, there always exists a neighborhood of \(\lambda_{ref}(t)\) such that  
\[e^{T}SA_{0}e+e^{T}S[B_{0}(x)\lambda-B_{0}(x_{ref})\lambda_{ref}])=e^{T}S\sum_ {i=1}^{M}\lambda_{ref_{i}}(A_{0}+A_{i})e+e^{T}SB_{0}(x)(\lambda-\lambda_{ref}) <-e^{T}Qe\]  
Then, Assumption 1 is satisfied for \(P\) replaced with \(S\) and the proof is complete. If the conditions of Lemma 1 are satisfied, this means that the set of matrices \(A_{0}+A_{i}\) is a set of quadratically stable matrices [49]. From the above assumption, another important lemma can be deduced. **Lemma 2**. : _If Assumption 1 is satisfied, we have_  
\[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]0  
Proof. : Suppose that the reference signal is admissible; we have by convexity  
\[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]1 \[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]2 \[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]3 \[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]4  
The main result of this section can now be stated. **Theorem 1**. : _Consider system (3) with an admissible reference signal described by the exosystem (4) and suppose that Assumption 1 is satisfied. Then, the control \(\lambda(t)\) defined by_  
\[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]5  
_solves Problem 1. In addition, the tracking error satisfies_  
\[e^{T}(t)P[B_{0}(x(t))\lambda(t)-\Pi_{2}(z)(t)]\leq-e^{T}(t)PA_{0}e(t)-e^{T}(t)Q _{1}e(t)\]6  
Proof. : Consider the Lyapunov function \(V(e(t))=\frac{1}{2}e^{T}(t)Pe(t)\).