## 6 Robustness Considerations
### Uncertain Model  
Sometimes it is desirable not to include integral action in the inner loops of cascaded control systems since this introduces limitations on achievable bandwidth in the inner loop [1]. Therefore, we focus on a version of the controller that uses a fixed--possibly time- and state-varying, but bounded--disturbance estimate. This estimate does not necessarily equal the true value of \(\Delta\). Besides, for some hierarchical flight control loops, the angular velocity reference is not made available for the inner loop. We thus remove the assumption that \(\omega_{d}^{\perp}\) is known in the control design, and redefine the backstepping variable as \(\hat{z}=\omega-\hat{\omega}_{d}\), and \(\hat{\omega}_{d}=\omega_{d}^{\parallel}-\kappa\varepsilon_{\eta}\). The backstepping procedure in the previous section has provided us with a strict Lyapunov function that can be used to show uniform ultimate boundedness of the solutions of the closed-loop system. To account for model uncertainties, consider again the control law (34)  
\[u=u_{\text{trim}}+\frac{1}{V_{\hat{u}}^{2}}\hat{B}^{-1}\Big{[}u_{\text{pd}}+u _{\text{ff}}-\hat{\Delta}(v_{r},t)\Big{]} \tag{45}\]  
where  
\[u_{\text{pd}} =-k_{1}e_{\eta}-K_{2}\hat{z} \tag{46}\] \[u_{\text{ff}} =\hat{J}\hat{\omega}_{d}^{\prime}-S(\hat{J}\hat{\omega}_{d})\hat {\omega}_{d}-V_{a}\hat{D}\hat{\omega}_{d}, \tag{47}\]  
and \(\hat{\Delta}(v_{r},t)\), \(\hat{B},\hat{J},\hat{D}\) are estimates of \(\Delta(v_{r},t)\), \(B,J,D\), respectively. The term \(\hat{\omega}_{d}^{\prime}\) is defined as the parts of \(\hat{\omega}_{d}\) that do not require reference angular velocities or accelerations:  
\[\hat{\omega}_{d}^{\prime}=\bigg{(}\frac{g}{V_{a}}\tan\phi_{d}\bigg{)}\hat{\eta }-\frac{g}{V_{d}^{2}}\tan(\phi_{d})\hat{V}_{a}\eta-\kappa(\hat{\eta}\times \eta_{d}). \tag{48}\]  
The next proposition states that, for sufficiently small model uncertainties, and sufficiently small \(\omega_{d}^{\perp}\), the solutions are ultimately bounded. This is essentially a local input-to-state stability (ISS) property [85]. To parametrize the model uncertainty, let \(\delta B\triangleq B\hat{B}^{-1}\), \(E\triangleq I_{3}-\delta B\), \(\hat{J}\triangleq\hat{J}-J\), and \(\hat{D}\triangleq\hat{D}-D\). For compactness, we define \(c_{I}=\|\hat{J}\|+\|E\|\|\hat{J}\|\) and \(c_{D}=\|\hat{D}\|+\|E\|\|\hat{D}\|\). **Proposition 4**. : _Consider the tracking error dynamics (18) and the perturbed controller (45). Assume that \(\delta B\) satisfies \(x^{\top}\delta Bx>0,\forall x\
eq 0\). Then, there exists some gain matrix \(K_{2}=K_{2}^{\top}>0\) such that the matrix \(\operatorname{sym}(\delta BK_{2})\) is positive definite. If the matrix \(K_{2}\) is chosen such that_  
\[\lambda_{\min}^{\operatorname{sym}(\delta BK_{2})}>\frac{a^{2}}{4\kappa k_{1} }+\lambda_{\max}^{I}(c_{\omega_{d}^{\perp}}+\kappa)+b, \tag{49}\]  
_where \(a=k_{1}\|E\|+k_{1}c_{\omega_{d}^{\perp}}+\kappa c_{D}V_{\max}+c_{I}\kappa\bigg{(} 2\kappa+2c_{\omega_{d}^{\perp}}+b_{\hat{\omega}_{d}^{\perp}}\bigg{)}+\lambda_{ \max}^{I}\|\delta B\|\kappa c_{\hat{\theta}_{d}}\sin c_{\theta_{d}}\) and \(b=c_{I}(x+b_{\omega_{d}^{\perp}})+\lambda_{\max}^{I}\|\delta B\|c_{\hat{\phi}_{ d}}\sin c_{\theta_{d}}\), and if \(c\) as defined by Equation (A11) is sufficiently small, then the solutions of the closed-loop system are uniformly ultimately bounded, with an ultimate bound that depends on the controller parameters, the model estimation errors, and the reference velocity bounds._  
Proof. : See Appendix E.  
In essence, the matrix \(K_{2}\) can be chosen such that the controller is robust to model uncertainties, even when the derivatives of the reduced-attitude reference are not available. However, a necessary condition is that the uncertainty in the input matrix is not too large. The condition \(x^{\top}\delta Bx>0\) implies that the control direction is known up to an error of 90 degrees. **Remark 10**. : _Global stability of the nominal system is a necessary condition for (global) ISS.