## 3 Mathematical Modeling
### Suboptimal Control in Nonlinear Dynamics
#### 3.2.3 SDRE Infinite-Horizon Regulator  
Consider a continuous-time autonomous infinite-horizon nonlinear regulator problem, with full-state observable dynamics and affine(linear) unconstrained inputs, given by (20) now with a state-dependent performance index  
\[PI_{{}_{dd}}=\int\limits_{t_{0}}^{\infty}\Big{(}\,\mathbf{x}^{T}\,\mathbf{Q}( \mathbf{x})\,\mathbf{x}+\mathbf{u}^{T}\,\mathbf{R}(\mathbf{x})\,\mathbf{u}\, \Big{)}\,dt \tag{21}\]with a quadratic-like structure in \(\mathbf{x}\in\mathbf{\Omega}\subset\mathbb{R}^{n}\) (the reachable/accessible region), and quadratic in \(\mathbf{u}\in\mathcal{U}\subset\mathbb{R}^{m}\) (the admissible controls), now admitting also the weighting functions \(\mathbf{Q}:\Omega\to\mathbb{R}^{n\times n}\) and \(\mathbf{R}:\Omega\to\mathbb{R}^{m\times m}\) to be \(\mathcal{C}^{\lambda}\ (k\geq 0)\quad\) state-dependent functions. Still enjoying the heritage of the linear models, as the weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\) may also be state-dependent, the nonlinear performance index (21) may have now only a quadratic-_like_ structure regarding the states \((\mathbf{x}^{T}\mathbf{Q}(\mathbf{x})\,\mathbf{x})\), but still presents a quadratic structure regarding control \((\mathbf{u}^{T}\,\mathbf{R}(\mathbf{x})\,\mathbf{u})\). This state-dependence permits extra flexibility in the control design by allowing extra degrees of freedom in the trade-off between control effort and state errors via tuning \(\mathbf{R}(\mathbf{x}(\tau))\) and \(\mathbf{Q}(\mathbf{x}(\tau))\) at the frozen instants \(\tau\geq t_{0}\), in the search for better compromise between cost, performance, and robustness [39; 91; 96; 97; 108; 111]. Analogously to the LTI case, the SDRE approach is based on defining the triplet \(\{\mathbf{A}(\mathbf{x}),\ \mathbf{B}(\mathbf{x}),\ \mathbf{C}(\mathbf{x})\}\), as well as the weights \(\{\mathbf{Q}(\mathbf{x}),\ \mathbf{R}(\mathbf{x})\}\), so as to guarantee the symmetric positive-definite solution \(\mathbf{P}(\mathbf{x})\) for the state-dependent version of the ARE (17) [39]. Nonetheless, while \(\mathbf{Q}\) and \(\mathbf{R}\) are on-demand design parameters, the triplet \(\{\mathbf{A},\ \mathbf{B},\ \mathbf{C}\}\) must be chosen so as to attain the necessary canonical structure [94; 105; 108]. Results concerning local stability are established in  
**Theorem 1**. : _Given the system (10), in which \(\mathbf{f}\) satisfy Proposition 1, assume that a SDC parameterization (20) is chosen such that \(\mathbf{A},\ \mathbf{B},\ \mathbf{C}\) are continuous and the pairs \(\{\mathbf{A},\ \mathbf{B}\}\), and \(\{\mathbf{A},\ \mathbf{C}\}\) are, respectively, point-wise stabilizable and point-wise detectable, in the linear sense, in some nonempty neighborhood \(\mathcal{X}_{\lambda}\) of the origin. Then, the SDRE nonlinear regulator (11) and (20) produces a closed-loop solution for the suboptimal control (15), which makes the system locally asymptotically stable._  
Proof. : see [108]. This means the SDRE approach for nonlinear feedback control produces a closed-loop system matrix [112]  
\[\mathbf{A}_{{}_{CL}}(\mathbf{x})=\mathbf{A}(\mathbf{x})-\mathbf{B}(\mathbf{ x})\mathbf{K}(\mathbf{x}) \tag{22}\]  
which is point-wise Hurwitz (eigenvalues with negative real part), \(\forall\mathbf{x}\in\mathcal{X}_{\lambda}\). However, as shown by [112], differently from what happens in the LTI models, this Hurwitz property does not guarantee global asymptotic stability for a nonlinear system (when it is possible to regulate any \(\mathbf{x}\in\Omega\) to the origin). Additionally, Ref. [91] proved that, under the hypotheses of Theorem 1, the SDRE solution is globally asymptotically stable if the closed-loop coefficient matrix (22) for the problem (20) and (21) is symmetric for all \(\mathbf{x}\in\Omega\). However, there is no rule of thumb for obtaining such a particular stabilizing matrix.