# Adaptive Optimal Control of UAV Brake Based on Parameter Estimation
## 3 Adaptive Optimal Control Design with Guaranteed Parameter Estimation
### Adaptive Control with Finite-Time Parameter Estimation  
For the position tracking control, we define filter tracking error variables as  
\[e_{2}=\dot{e}_{1}+k_{1}e_{1} \tag{13}\]  
\[r_{1}=\dot{e}_{2}+k_{2}e_{2} \tag{14}\]  
where \(k_{1}\) and \(k_{2}\) are positive constants, and \(r_{1}\) is the auxiliary filter error.  
From (4), (5), (6), (13), (14), one has  
\[r_{1}\ =\Theta^{T}\Psi(x_{1})+K_{1}K_{2}x_{3}+\Delta_{1}(t)-\ddot{x}_{1d}+k_{1} \dot{e}_{1}+k_{2}e_{2} \tag{15}\]  
\(x_{3}\) will be regarded as virtual input, the design virtual control rate is \(\alpha_{2}\), and we define the input error \(e_{3}\) as  
\[e_{3}=x_{3}-\alpha_{2} \tag{16}\]  
In order to reduce the impact of noise in measurement, the expected trajectory \(x_{1d}\) is used to replace \(x_{1}\), so \(r_{1}\) can be rewritten by using the expected trajectory as  
\[r_{1}=K_{1}K_{2}\alpha_{2}+\Theta^{T}\Psi(x_{1d})+S_{1}+\Delta_{1}(t)-\ddot{x }_{1d} \tag{17}\]where the auxiliary function \(S_{1}\) is  
\[S_{1}=k_{1}\dot{e}_{1}+k_{2}e_{2}+\Theta^{T}\Psi(x_{1})-\Theta^{T}\Psi(x_{1d})+K_ {1}K_{2}e_{3} \tag{18}\]  
The design virtual control rate is  
\[\alpha_{2}=\frac{1}{K_{1}K_{2}}\Big{(}-\hat{\Theta}^{T}\Psi(x_{1d})-k_{1}(e_{2} -k_{1}e_{1})-k_{2}e_{2}+\ddot{x}_{1d}+\mu_{1}\Big{)} \tag{19}\]  
where \(\mu_{1}\) is the feedback item of RISE and  
\[\mu_{1}=-k_{s1}e_{2}(t)+k_{s1}e_{2}(0)-\int_{0}^{t}[k_{s1}k_{2}e_{2}(\sigma)+ \beta_{1}\operatorname{sgn}(e_{2}(\sigma))]d\sigma \tag{20}\]  
The time derivative of \(\mu_{1}\) is  
\[\dot{\mu}_{1}=-k_{s1}r_{1}-\beta_{1}(e_{2}) \tag{21}\]  
where \(k_{s1}\) is a positive feedback gain, \(\beta_{1}\) is a positive constant, and \(\operatorname{sgn}(\bullet)\) is a function of signum. The extracted estimation error is added to the traditional adaptive law. This method makes the error of parameter estimation and system control converge eventually.  
The parameter-updating law of \(\Theta\) is  
\[\hat{\Theta}=\Gamma\big{(}\ddot{\Psi}_{d}r_{1}-M\big{)} \tag{22}\]  
with  
\[M=\sigma H+\sigma\frac{G^{T}H}{\|H\|} \tag{23}\]  
where \(\Psi_{d}\) is the regressor and \(\Gamma>0\), \(\sigma>0\) are the learning gains. Since \(r_{1}\) is not to directly implement online learning, inspired by [37], we write (22) in the following form:  
\[\hat{\Theta}(t)=\hat{\Theta}(0)+\sigma\int_{0}^{t}\bigg{(}H(\tau) +\frac{G^{T}(\tau)H(\tau)}{\|H(\tau)\|}\bigg{)}d\tau+\left.\Gamma\ddot{\Psi}_ {d}(\tau)e_{2}(\tau)\right|_{0}^{t}-\Gamma\] \[\int_{0}^{t}\big{[}\ddot{\Psi}_{d}e_{2}(\tau)-k_{2}\Psi_{d}e_{2}( \tau)\big{]}d\tau \tag{24}\]  
As follows from (24), the variables \(r_{1}\)can be replaced by \(\dot{e}_{2}+k_{2}e_{2}\), where the Newton-Leibniz formula is used to calculate \(\dot{e}_{2}\). Therefore, the parameters \(\Theta\) can be calculated according to (24). The auxiliary filtering signal is designed as  
\[r_{2}=\dot{e}_{3}+k_{3}e_{3} \tag{25}\]  
The error variable \(r_{2}\) combined with (4), (16), and (25) can be rewritten by using the desired trajectory \(x_{2d}\) and \(x_{3d}\) as  
\[r_{2}=-K_{4}x_{2d}-K_{5}x_{3d}+K_{3}u+S_{2}+\Delta_{2}(t)-\dot{x}_{2} \tag{26}\]  
where the auxiliary function \(S_{2}\) is  
\[S_{2}=-K_{4}(x_{2}-x_{2d})-K_{5}(x_{3}-x_{3d})+k_{3}e_{3} \tag{27}\]  
The brake control effect is also affected by brake performance. Thus, torque and energy consumption need to be considered in the brake control system, to realize efficient and stable control of the aircraft brake. According to (26), the controller is  
\[u=\frac{1}{K_{3}}(K_{4}x_{2d}+K_{5}x_{3d}+\dot{a}_{2}+\mu_{2}+u_{o}) \tag{28}\]where \(u_{o}\) is the auxiliary optimal controller for the given secondary performance index of the system and \(\mu_{2}\) is the feedback item of RISE with  
\[\mu_{2}=-k_{s2}e_{3}(t)+k_{s2}e_{3}(0)-\int\limits_{0}^{t}[k_{s2}k_{3}e_{3}( \sigma)+\beta_{2}sgn(e_{3}(\sigma))]d\sigma \tag{29}\]  
The time derivative of \(\mu_{2}\) is  
\[\dot{\mu}_{2}=-k_{s2}r_{2}-\beta_{2}sgn(e_{3}) \tag{30}\]  
where \(k_{s2}\), \(\beta_{2}\) are normal numbers.