# Optimal Robust LQI Controller Design for Z-Source Inverters
## 3 Robust LQI Controller Design  
Based on Lyapunov theory, for a linear time-invariant (LTI) system given by (15). \[\dot{x}=Ax \tag{15}\]  
the existence of a symmetrical positive definite matrix \(\mathbf{\Theta}\) such that the quadratic Lyapunov function given by (16). \[V(x)=x^{T}\mathbf{\Theta}x>0\ \ \ \forall x\
eq 0 \tag{16}\]  
Figure 4: Current source model of network load variations. satisfies (17), as  
\[\dot{V}(x)=x^{T}(A^{T}\mathbf{\Theta}+\mathbf{\Theta}A)x<0\ \ \ \forall x\
eq 0 \tag{17}\]  
is a necessary and sufficient condition to prove the quadratic stability of the system. Inequality given by (17) is satisfied if and only if \(A^{T}\mathbf{\Theta}+\mathbf{\Theta}A<0\). Given the system described by (13) with controllable pair \((A,B)\), the aim of the LQR control is to find an optimal SF controller \(u=-Kx\) that minimizes cost function expressed by (18). \[J=\int_{0}^{\infty}(x^{T}Qx+u^{T}\mathcal{R}u)dt \tag{18}\]  
where \(\mathcal{Q}\) and \(\mathcal{R}\) are symmetric semidefinite and definite positive matrices, respectively. The matrix \(\mathbf{\Theta}\) is the solution of the algebraic Riccati equation given by (19). \[A^{T}\mathbf{\Theta}+\mathbf{\Theta}A+\mathcal{Q}-\mathbf{\Theta}BR^{-1}B^{T}\mathbf{\Theta}=0 \tag{19}\]  
Then, the optimal SF gain can be calculated as \(K=R^{-1}B^{T}\mathbf{\Theta}\), which satisfies the so-called Return Difference Equality (RDE) given by (20) [32]. \[(I+\mathcal{L}(-s))^{T}\mathcal{R}(I+\mathcal{L}(s))=\mathcal{R}+G^{T}(-s)G(s) \tag{20}\]  
where \(\psi(s)=(sI-A)^{-1}\), \(\mathcal{L}(s)=K\psi(s)B\) is similar to the loop gain of the closed-loop system, \(G(s)=C\psi(s)B\) denotes the open-loop system, and \(I+\mathcal{L}(s)\) represents the Return Difference. On the basis of the Nyquist stability criterion, the closed-loop system is stable if the Nyquist plot of \(\mathcal{L}(j\omega)\) has the appropriate number of encirclement of the critical point \((-1+j0)\). Since the nominal linear quadratic system is stable, the Nyquist plot of \(\mathcal{L}(j\omega)\) reveals the robustness property of the closed-loop system. For the single input system, i.e., \(R=r\), the RDE yields (21). \[|1+\mathcal{L}(j\omega)|^{2}=1+\frac{1}{r}|G(j\omega)|^{2}\geq 1 \tag{21}\]  
Thus, \(\mathcal{L}(j\omega)\) is separated from \((-1+j0)\) by a disk of radius 1 centered at \((-1+j0)\), which guarantees the robustness of the closed-loop system. On the other hand, it was pointed out that sensitivity to model errors could be expressed as the largest value of the sensitivity function [33]. In this regard, the maximum sensitivity (\(M_{\rm s}\)) was defined as the inverse of the shortest distance from the Nyquist curve of the open-loop transfer function \(\mathcal{L}(j\omega)\) to the critical point \((-1+j0)\) in the complex plane [34], as given by (22). \[M_{\rm s}=\max_{\omega\rightarrow\infty}|\frac{1}{1+\mathcal{L}(j\omega)}|= \max_{\omega\rightarrow\infty}\frac{1}{\sqrt{1+\frac{1}{r}|G(j\omega)|^{2}}} \leq 1. \tag{22}\]  
The geometric interpretation of the maximum sensitivity is shown in Figure 5. As can be realized, \(M_{\rm s}\) is an effective way to measure the robustness of the system against modeling uncertainties, in which the lower the value of \(M_{\rm s}\) is, the more robust the system is. Additionally, there is a correlation between \(M_{\rm s}\) and the gain and phase margins, which guarantees the margins given by (23) based on (22) [34]. \[V(x)=x^{T}\mathbf{\Theta}x>0\ \ \ \forall x\
eq 0 \tag{16}\]0  
which means an infinite positive gain margin and 60 degree phase margin. Furthermore, it can be concluded that the Nyquist diagram of \(\mathcal{L}(j\omega)\) will never enter the unit circle N.  
As can be understood, the performance and optimality of the control system are contingent to a great extent on the choice of \(\mathcal{Q}\) and \(\mathcal{R}\)[35]. Generally, they are selected as diagonal matrices such that the quadratic cost function is a weighted integral of the squared error of the states and inputs.