## 1 Introduction
### Solutions and Their Approximations of the Optimal Control Problems  
Assume that the target system is a continuous-time linear time-invariant (CT-LTI) system as follows:  
\[\hat{\mathbf{x}}=\mathbf{Ax}+\mathbf{Bu}, \tag{1}\]\[\mathbf{y}=\mathbf{C}_{y}\mathbf{x}, \tag{2}\]  
where \(\mathbf{x}(t)\) is the state of the system, \(\mathbf{u}(t)\) is the input of the system, and \(\mathbf{y}(t)\) is the output of the system. Let the tracking error \(\mathbf{e}(t)\) be  
\[\mathbf{e}=\mathbf{r}-\mathbf{y}, \tag{3}\]  
where \(\mathbf{r}(t)\) is the reference for the output. Then, the linear-quadratic tracking (LQT) problem can be expressed as follows [6; 7; 8]:  
\[ \begin{array}{l}\underset{\mathbf{e},\mathbf{u}}{\text{minimize}}\ J=\frac{1}{2} \int_{0}^{t_{f}}\Big{(}\mathbf{e}^{T}\mathbf{Q}\mathbf{e}+\mathbf{u}^{T} \mathbf{R}\mathbf{u}\Big{)}dt\\ \text{subject to}\ \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}-\dot{\mathbf{x}}=0, \end{array}  \tag{4}\]  
where \(\mathbf{Q}\) and \(\mathbf{R}\) are positive definite matrices weighting for the tracking error and the input, \(J\) is the cost function, \(t_{0}\) is the initial time, and \(t_{f}\) is the final time. Assume that pair \((\mathbf{A},\mathbf{B})\) is controllable and pair \(\Big{(}\mathbf{A},\mathbf{C}_{y}\Big{)}\) is observable. Then, by the Lagrange multiplier method [6; 7; 8] and the variational approach (Theorem A1), the LQT problem (4) can be transformed into a problem of finding solutions of the Riccati equation with boundary conditions and an auxiliary dynamic equation related to the reference. The Riccati equation and the auxiliary dynamic equation can be derived as follows:  
\[\dot{\mathbf{P}}=-\mathbf{P}\mathbf{A}-\mathbf{A}^{T}\mathbf{P}+\mathbf{P} \mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{P}-\mathbf{C}_{y}{}^{T} \mathbf{Q}\mathbf{C}_{y}, \tag{5}\]  
\[\dot{\mathbf{g}}=-\Big{(}\mathbf{A}^{T}-\mathbf{P}\mathbf{B}\mathbf{R}^{-1} \mathbf{B}^{T}\Big{)}\mathbf{g}-\mathbf{C}_{y}{}^{T}\mathbf{Q}\mathbf{r}, \tag{6}\]  
where \(\mathbf{P}(t)\) and \(\mathbf{g}(t)\) are the solutions to be determined. These solutions are usually calculated in the backward direction in time or by using iterations since \(\mathbf{A}-\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{P}\), which is the system matrix of the closed-loop system, must have all the negative real parts of the eigenvalues. However, if there are no constraints on the input and the state, it is well known in control engineering that the solutions of the equations can be approximated to their steady-state values, and the solutions become independent of the time. In the case of input constraints alone, the optimal input can be determined by [PERSON]'s minimum principle [6; 7; 8; 9]. In general, the solutions determined by [PERSON]'s minimum principle are closely related with the solutions of the Equations (5) and (6) and have a form simple enough to be implemented on modern microcontrollers. On the other hand, if there are inequality constraints on the state, the problem statement (4) is not valid, therefore it must be redefined as follows:  
\[ \begin{array}{l}\underset{\mathbf{e},\mathbf{u}}{\text{minimize}}\ J=\frac{1}{2}\int_{0}^{t_{f}}\Big{(}\mathbf{e}^{T} \mathbf{Q}\mathbf{e}+\mathbf{u}^{T}\mathbf{R}\mathbf{u}\Big{)}dt\\ \text{subject to}\ \left\{\begin{array}{l}\mathbf{A}\mathbf{x}+\mathbf{B} \mathbf{u}-\dot{\mathbf{x}}=0\\ \mathbf{C}_{h}\mathbf{x}+\mathbf{w}\leq 0\end{array}\right.,\end{array}  \tag{7}\]  
where \(\mathbf{C}_{h}\) and \(\mathbf{w}\) are time-invariant parameters of the inequality constraints. The vector inequality \(\mathbf{h}\leq 0\) means that \(h_{i}\leq 0\ (i=1,\cdots,n)\), where \(\mathbf{h}=\left[ \begin{array}{ccc}h_{1}&\cdots&h_{n}\end{array} \right]^{T}\). In this case, the problem related to the direction of the calculation is hard to avoid. Since the problem (7) is difficult to solve exactly, many kinds of research were proposed to solve this problem approximately.