## 3 Digital Redesign of Full-Order Observer  
Consider a linear observable continuous-time system described as  
\[\dot{x}_{c}(t)=Ax_{c}(t)+Bu_{c}(t),\ x_{c}(t_{0})=x_{0}, \tag{24}\]  
\[y_{c}(t)=Cx_{c}(t), \tag{25}\]  
where \(x_{c}(t)\in\Re^{n}\), \(u_{c}(t)\in\Re^{m}\), and \(y_{c}(t)\in\Re^{p}\), \(A\), \(B\) and \(C\) are system matrices with appropriate dimensions. One method of estimating the unmeasured state is to construct a full-order model of the plant dynamics. The following derivation follows from [8]  
\[\dot{\hat{x}}_{c}(t)=A\hat{x}_{c}(t)+Bu_{c}(t), \tag{26}\]  
To derive the dynamics of the observer, we consider feeding back the difference between the actual output and the estimated output, and correcting the model continuously with this error signal. The equation is as follows  
\[\dot{\hat{x}}_{c}(t)=A\hat{x}_{c}(t)+Bu_{c}(t)+L_{c}[y_{c}(t)-C\hat{x}_{c}(t)], \tag{27}\]  
\[\dot{\hat{x}}_{c}(t)=[Ax_{c}(t)+Bu_{c}(t)]-\{A\hat{x}_{c}(t)+Bu_{c}(t)+L_{c}[y _{c}(t)-C\hat{x}_{c}(t)]\} \tag{28}\]  
Hence, the problem reduces designing an observer so that the error dynamics have their eigenvalues on the far left side of the complex s-plane, so the convergence to the original dynamics can be fast. Since the desirable observer design approach is to use the minimum error energy, an optimal design technique with a minimizing performance index is used. In general, an optimal design of controllers for a given controllable and observable linear system is defined as shown in Equations (24) and (25), and the optimal state-feedback control law minimizes the performance index  
\[J=\int_{0}^{\infty}\ e^{2 ht}[x_{c}(t)^{T}Qx_{c}(t)+u_{c}(t)^{T}Ru_{c}(t)]dt \tag{29}\]  
with \(Q\geq 0\), \(R>0\), and \(h\geq 0\) are obtained as  
\[u_{c}(t)=-K_{c}x_{c}(t) \tag{30}\]  
where the optimal feedback gain is \(K_{c}=R^{-1}B^{T}P\), with \(P\) being the positive definite and symmetric solution of the following Riccati equation [9]  
\[(A+hI_{n})^{T}P+P(A+hI_{n})-PBR^{-1}B^{T}P+Q=0. \tag{31}\]  
In a closed loop, the optimal control linear system has the form of  
\[\dot{x}_{c}(t)=(A-BK_{c})x_{c}(t),\ x_{c}(0)=x_{0}. \tag{32}\]  
Comparing Equations (32) and (28), we can see that  
\[(A-L_{c}C)^{T}=A^{T}-C^{T}L_{c}^{T}, \tag{33}\]which has the structure as a state-feedback controller. This is the dual property of linear systems, where the observer gain can be determined as the dual of the feedback controller gain. Thus, the optimal observer gain \(L_{c}\) can be found by designing the optimal control gain \(K_{c}\) for the dual system, via \(A=A^{T}\) and \(B=C^{T}\), so that \(L_{c}=K_{c}^{T}\); or equivalently, \(L_{c}=PC^{T}R^{-1}\), where \(P\) is the positive definite and symmetric solution of the Riccati equation  
\[y_{c}(t)=Cx_{c}(t), \tag{25}\]0  
Now, since a continuous-time observer for the system presented in Figure 1 is defined in Equation (27), the next step is to find a digital observer for the system. First, define the discrete-time observer error as  
\[y_{c}(t)=Cx_{c}(t), \tag{25}\]1  
Therefore, the discrete-time error dynamics match the continuous-time error dynamics at each sampling instant \(\widetilde{x}_{d}(kT)\approx\left.\widetilde{x}_{c}(t)\right|_{t=kT}\), or, equivalently, assuming that the continuous-time observer is asymptotically stable, and the original state and the digital state match \(\hat{x}_{d}(kT)\approx\left.\hat{x}_{c}(t)\right|_{t=kT}\approx\left.x_{c}(t) \right|_{t=kT}\).