## 3 Mathematical Modeling
### Suboptimal Control in Nonlinear Dynamics
#### 3.2.4 Reference Trajectory Tracking Using SDRE Regulation  
The regulator problem described in the previous section is naturally adherent to the idea of some system's state or output being kept at a fixed distance from a given reference, as happens on an oil/gas production platform that keeps its position on the sea surface relative to a seabed oil well, or to a spacecraft hovering over the surface of a planet, or even keeping its position around an equilibrium point close to an asteroid [114; 115; 116]. Another category of problems concerns the accompaniment or tracking of a reference path, as happens in formation flying for airplanes or artificial satellites, mobile robots,synchronized chaotic systems, or even a space probe following a prescribed stable reference trajectory within the strongly perturbed environment around systems of Small Solar System Bodies [37; 61; 99; 117; 118; 119; 120; 121].  
The purpose of a tracking-control procedure is to guarantee that the plant's output will follow a prescribed reference trajectory \(\{\mathbf{r}(t)\in\mathcal{X}\subset\Omega,\ t\geq t_{{}_{0}}\}\), within assigned tolerances relative to output or state errors and control effort, analogous to the ones established in the regulation problem (20) and (21). However, there are some inherent difficulties related to numerical implementability due to the necessity of simultaneous resolution of the SDRE and feedforward differential equations [122] toward the eventual unboundedness of the cost function and toward the additional requirement that the reference trajectory be generated by (asymptotically) stable dynamics [98; 123].  
Measures simplifying these tasks can be adopted in order to circumvent some of the implementation difficulties. An effective and easier-to-implement solution is to use the additional formation of the tracking problem as a \"regulation with time-varying reference\", which permits the conversion of an SDRE-based LQT problem to a pseudo-LQR one by regulating the output error in SDC form [98; 124; 125; 126; 127; 128]. For this approach, assume the output of the plant with dynamics (20) must follow a reference-model trajectory driven by stable dynamics  
\[\dot{\mathbf{z}}=\ \  
\[A_{21}(X)=\omega^{2}-\frac{\mu}{r^{3}}+\frac{3\mu\,r_{0}^{2}\,C_{20}}{2r^{5}}+ \frac{6\mu\,r_{0}^{2}\,C_{22}}{r^{5}}-\frac{15\mu\,r_{0}^{2}\,C_{22}}{r^{7}}(x^{ 2}-y^{2})\, \tag{34}\]  
\[A_{43}(X)=\omega^{2}-\frac{\mu}{r^{3}}+\frac{3\mu\,r_{0}^{2}\,C_{20}}{2\,r^{5} }-\frac{6\mu\,r_{0}^{2}\,C_{22}}{r^{5}}-\frac{15\mu\,r_{0}^{2}\,C_{22}}{r^{7}}( x^{2}-y^{2})\,\]  
\(r\) is the distance of the spacecraft to the asteroid, \(\mu\) is the asteroid's gravity constant, \(C_{20}\) and \(C_{22}\) are the harmonic coefficients of the asteroid's gravity field [76; 78], and \(r_{0}\) is a reference distance, here assumed to be the synchronous radius defined in (2). We used the discount factor \(\alpha=-10^{-6}\), through the examples shown here. As can be seen in Figure 6, this value was enough to guarantee negative eigenvalues in the modified dynamics matrix in Equation (29a), thus avoiding numerical instabilities in the determination of control (30), while imparting marginal changes in the cost or performance index (29b).