## 3 Mathematical Modeling
### Suboptimal Control in Nonlinear Dynamics
#### 3.2.1 State-Dependent (SD) Formulation for Suboptimal Nonlinear Control  
The nonlinear techniques are very sensitive to the dynamics, leading to a more complex overall control design when compared to linear (ized) ones. The adoption of LQR (or LQT) by means of linearization relies on the use of rather simplified models for the dynamics, which may not be a great limitation when dealing with small deviations from the given references. Nonetheless, as the model errors tend to be instantaneously compensated for by control actions, fuel consumption will increase in the case of large deviations, which makes linear(ized) models not prone for applications, for example, in a rendezvous mission to a NEO, as usual uncertainties concerning the target body's environment or from reference orbits will induce greater errors on the dynamics. Additionally, if the system is highly nonlinear, the linear(ized) optimal controller does not provide adequate performance as the process of linearization renders rather local validity in some models, and the performance may degrade rapidly as the operation departs from the reference point or reference trajectory, as opposed to global stability/optimality of the autonomous linear models [99; 100; 101].  
On the other hand, LTI models are much easier to handle and allow linear optimal control theory to produce easily computable and globally valid solutions for a large class of problems. It looks extremely attractive and promising to extend to nonlinear problems this linear simplicity/effectivity in determining the laws of optimal control, for instance, by means of model linearization around a stationary point (Linear Quadratic Regulation-LQR) or a reference trajectory (Linear Quadratic Tracking-LQT) [87; 89; 91; 102; 103; 104; 105].  
There begins the search for trade-offs between achieving as easy and inexpensive implementability as occurs in linear/linearized models while keeping as much as possible the real (nonlinear) characteristic of the plants, allowing greater robustness when compared to Taylor or Jacobi linearization, albeit sacrificing some performance (error reduction), as compared to the full nonlinear models. Ref. [106] gave a seminal step in the direction of such an approach by suggesting to approximate the nonlinear optimal control problem by a linear time-varying one, i.e., at fixed instants, the nonlinear system is assumed as linear, enjoying all the mentioned benefits of the linear structure. Much better performance is then achieved from a linear-like construct at the computational cost of repeatedly resolving momentaneously frozen (time-invariant) linear state-dependent problems as the system model evolves in time with the states [107], somehow mimicking a state-related gain-scheduling. The approach relies then on recurrently resolving an Algebraic Riccati Equation (ARE) at the momentaneously fixed points in the state-space, so it was named the Frozen Riccati Equation (FRE) or State-Dependent Riccati Equation (SDRE) technique [100; 108; 100]. Some alternative approaches, along with their costs and benefits, are presented in [103; 101].  
The standard SDRE technique is based on representing an input affine nonlinear model in a way that resembles a linear overall structure by adopting state-dependent coefficient (SDC) matrices in a process called pseudo-linearization or extended linearization. Thus, the SDRE design allows the flexibility of LQR/LQT formulation to be directly translated to the control of nonlinear systems. Basically, in the SDRE approach, once the pseudo-linearization of the dynamics in (10) is established via SDC matrices, the resolution of the infinite-horizon feedback regulation/tracking with PI given by (11) is reduced to solving an ARE for the equivalent autonomous LQR/LQT problem at given instants (frozen points) along the trajectory. For the finite-horizon control (9), the solution is time-dependent and leads to the resolution of a Differential Riccati Equation (DRE), instead of an algebraic one, for establishing the control law [101].