Commanded Filter-Based Robust Model Reference Adaptive Control for Quadrotor UAV with State Estimation Subject to Disturbances

Define $\mathbf{p}={[x,y,z]}^T$ and $\mathrm{\Theta }={[\varphi ,\theta ,\psi ]}^T$ to represent position and attitude quadrotor, respectively. The rotational and translational velocities by $\mathrm{\Omega }={[p,q,r]}^T$ and $v={[v_x,v_y,v_z]}^T$, respectively. Thus, the quadrotor motion can be given by the following [2]: where $z_e$ is the unit vector along the z-axis in the earth-fixed inertial frame, $S\left(\mathrm{\Omega }\right)$ is a skew-symmetric matrix, and $\mathbf{R}$ is the rotational matrix given as follows [2,47]:${\tau }_a={[{\tau }_1,{\tau }_2,{\tau }_3]}^T$ is the vector of torques generated by the rotors, where:

The four control inputs of the quadrotor, combined with the total thrust, are given as follows:After mathematical derivation and simplification, the dynamical model of the quadrotor can be expressed as follows: where ${\mathrm{\Omega }}_r={\omega }_1-{\omega }_2+{\omega }_3-{\omega }_4$. Next, for the state-space model of the quadrotor UAV, define $\varphi =x_1\left(t\right)$, $\dot{\varphi }=x_2\left(t\right)$, $\theta =x_3\left(t\right)$, $\dot{\theta }=x_4\left(t\right)$, $\psi =x_5\left(t\right)$, $\dot{\psi }=x_6\left(t\right)$, $x=x_7\left(t\right)$, $\dot{x}=x_8\left(t\right)$, $y=x_9\left(t\right)$, $\dot{y}=x_{10}\left(t\right)$, $z=x_{11}\left(t\right)$ and $\dot{z}=x_{12}\left(t\right)$. Simplification yields where $a_{\varphi }=\left(I_y-I_z\right)/I_x$, $b_{\varphi }=I_r/I_x$, $c_{\varphi }=l/I_x$, $a_{\theta }=\left(I_z-I_x\right)/I_y$, $b_{\theta }=I_r/I_y$, $c_{\theta }=l/I_y$, $a_{\psi }=\left(I_x-I_y\right)/I_z$, $c_{\psi }=1/I_z$.

This section presents the key lemmas and underlying assumptions that serve as the foundation for the proposed control design methodology.

Without loss of generality, consider the decoupled roll model in the attitude state-space representation, subject to nonlinear disturbances. The controller is designed to achieve tracking control performance, and the state-space model is given as follows: where $d_{\varphi }\left(t\right)$ represents the bounded nonlinear disturbance estimated using the technique given by ${\widehat{d}}_{\varphi }\left(t\right)=z_{\varphi }\left(t\right)+w_{\varphi }\left(t\right)$, where where ${\mu }_{\varphi }$ denotes a positive DO control gain. Next, the error dynamics can be analyses by defining ${\tilde{d}}_{\varphi }\left(t\right)=d_{\varphi }\left(t\right)-{\widehat{d}}_{\varphi }\left(t\right)$. Thus, it can be derived thatIntroducing Young’s inequality and simplify where ${\overline{d}}_{\varphi }\ge \max \{1+{\dot{d}}_{\varphi }^2\}$ is a maximum bound. Since, ${\overline{d}}_{\varphi }>0\forall t\ge 0$, therefore, with ${\mu }_{\varphi }>0$ sufficiently large, the nonlinear DO will converge and estimate the disturbance. Next, define where $f_{\varphi }\left(t\right)=a_{\varphi }{x}_4\left(t\right)x_6\left(t\right)+b_{\varphi }{x}_4\left(t\right){\mathrm{\Omega }}_r$. Hence, it can be written thatThe MRAC control technique developed in this research requires the matching conditions to be satisfied between the real and reference models. Therefore, the roll model must be transformed into the desired model in matrix form. Based on this, the roll control input, $U_{\varphi }\left(t\right)$, for feedback linearization is chosen as follows: where $v_{\varphi }$ is a virtual control input to be designed using the MRAC-SMC technique combined with nonlinear DO. $A_1=\left[\begin{array}{ccc}0& 0;k_1& k_2\end{array}\right]$ and $B_1={\left[\begin{array}{cc}0& k_3\end{array}\right]}^T$, with $k_1>0$, $k_2>0$ and $k_3>0$.

Next, substitute (13) into (12) where where $B_{\varphi }\left({\displaystyle \frac{1}{B_{\varphi }^T{B}_{\varphi }}}\right)B_{\varphi }^T{B}_1=k_3$. Now, the virtual control input for the design of adaptive laws using MRAC criteria is chosen as follows: where $r_{\varphi }\left(t\right)$ is the reference input, and ${\widehat{K}}_{\varphi }\left(t\right)$, ${\widehat{K}}_{r_{\varphi }}\left(t\right)$, and ${\widehat{K}}_{v_{\varphi }}\left(t\right)$ are adaptive gains to be designed. ${\overline{u}}_{\varphi }\left(t\right)$ is a chattering-free robust SMC for trajectory tracking. In next step, the reference model is chosen according to the second-order dynamical system, given as follows: where $x_m$ is the state variable of the reference model. ${\zeta }_m$ and ${\omega }_m$ are the damping ratio and natural frequency, respectively. They should be chosen such that the reference model satisfies the Hurwitz criteria. Furthermore, $b_m$ is a constant and $r_{\varphi }\left(t\right)$ is the known reference input. Simplification is given as follows:Error between real and reference model is defined as follows:Time derivative is given bySubstituting (16) into (19)Matching conditions for designing adaptive gains for MRAC technique should satisfyNext, (21) can be simplified as follows: where where $K_{\varphi }$, $K_{r_{\varphi }}$, and $K_{v_{\varphi }}$ are unknown positive ideal gains. Hence, it can be derived thatNext, the Lyapunov function is chosen as follows: where $P_{\varphi }{A}_{\varphi }+A_{\varphi }^T{P}_{\varphi }=-Q_{\varphi }$, $P_{\varphi }=P_{\varphi }^T>0$ and $Q_{\varphi }=Q_{\varphi }^T$. Furthermore, ${\mathrm{\Gamma }}_{\varphi }={\mathrm{\Gamma }}_{\varphi }^T\in R^{2\times 2}$, ${\alpha }_{\varphi }$ and ${\beta }_{\varphi }$ are adaptive control tuning gains. Next, time derivative is derived as follows:Substituting (25) into (27)SimplifyingHence, it can be obtained where ${\overline{P}}_2$ is the second column of $P_{\varphi }$. Furthermore, $2e_{\varphi }^T{P}_{\varphi }{B}_{\varphi }=2k_3{e}_{\varphi }^T{\overline{P}}_2\in R$ and $2e_{\varphi }^T{P}_{\varphi }{D}_{\varphi }=2e_{\varphi }^T{\overline{P}}_2\in R$. Choose adaptive laws as follows:Substituting (31) into (30) yields where ${\gamma }_{\varphi }=2e_{\varphi }^T{\overline{P}}_2\left(k_3^2+1\right)$ is a known constant, and ${\overline{b}}_{\varphi }\ge {\displaystyle \frac{1}{2}}max\{{\overline{u}}_{\varphi }^2\left(t\right)+d_{\varphi }^2\left(t\right)\}$ represents an upper bound on the SMC input and disturbance. Therefore, the uniformly ultimately bounded condition can be achieved, provided that $P_{\varphi }$ is appropriately designed.

It is important to note that since $d_{\varphi }\left(t\right)$ is bounded, its square is also bounded. Furthermore, ${\overline{u}}_{\varphi }\left(t\right)$ is a control input designed using SMC principles, which inherently ensures boundedness if the control gains are chosen to be positive. Consequently, its squared value is also bounded. The design of ${\overline{u}}_{\varphi }\left(t\right)$ will be presented next after the model is redesigned as follows:

Next, for the design of ${\overline{u}}_{\varphi }\left(t\right)$, firstly the commanded-filter is introduced as ${\dot{\mathbb{Z}}}_{\varphi }={\mathbb{P}}_{\varphi }{Z}_{\varphi }$, written as follows: where ${\varphi }_r$ is the desired roll angle to be differentiated. $p_1>0$ and $p_2>0$ are design constants. Furthermore, $z_1=x^c$ and $z_2={\dot{x}}^c={\dot{\varphi }}_r$ represent the output of the commanded filter and the derivative of the input signal, respectively. To handle the numerical differentiation errors, a compensation system is designed as ${\dot{\mathbb{E}}}_{\varphi }={\mathfrak{R}}_{\varphi }{\mathbb{E}}_{\varphi }+{\mathbb{X}}_{\varphi }$, written as follows: where $r_1>0$ and $r_2>0$ are design constants. Next, the sliding manifold is chosen as follows: where $C_{\varphi }=[c_1;1]$, with $c_1>0$. Next, the sliding mode surface dynamics are obtained as follows:Hence, the control input is chosen as follows:Since ${\overline{u}}_{\varphi }\in R$, it can be written as follows: where adaptive laws are governed as follows: where ${\gamma }_{\varphi }>0$ and ${\eta }_{\varphi }>0$ are control gains for adaptive laws.

Next, define Lyapunov function as follows: where ${\tilde{L}}_{\varphi }=L-{\widehat{L}}_{\varphi }$ and ${\tilde{K}}_{\varphi }=K_{\varphi }-{\widehat{K}}_{\varphi }$. Now, time derivative is obtained asNext, by using (36) and (39), the time derivative can be obtained as follows:Introducing (40) yieldsHence, the controller is stable if the adaptive control gains, i.e., ${\gamma }_{\varphi }$ and ${\eta }_{\varphi }$, are designed positive and sufficiently large. Since ${\widehat{L}}_{\varphi }\left(t\right)$ and ${\widehat{K}}_{\varphi }\left(t\right)$ remain non-negative if the adaptive control gains satisfy ${\gamma }_{\varphi }>0$ and ${\eta }_{\varphi }>0$, the positivity of these terms is evident from the adaptive law design. This is because the right-hand side of both adaptive laws is always non-negative. However, it should be noted that these terms will not increase indefinitely over time, as the sliding mode surface eventually converges to zero, leading to a steady-state condition for the adaptive laws.

Consequently, ${\widehat{L}}_{\varphi }\left(t\right)\to L_{\varphi }$ and ${\widehat{K}}_{\varphi }\left(t\right)\to K_{\varphi }$, where $L_{\varphi }$ and $K_{\varphi }$ are unknown positive constants, and the adaptive law ensures a dynamic convergence to these values.

Next, for obtaining the estimated roll and its rate, following the HGO technique is designed ${\dot{X}}_{\varphi }=F_{\varphi }\left(t\right)+B_{\varphi }{U}_{\varphi }+{\mathbb{K}}_{\varphi }(y_{\varphi }-{\widehat{x}}_1)\left(t\right)$, where $y_{\varphi }=x_1$ is the real output of the roll model and where $\epsilon =(0,1]$, ${\hslash }_{{\varphi }_1}>0$ and ${\hslash }_{{\varphi }_2}>0$ are observer gains. ${\widehat{f}}_1\left(t\right)=a_{\varphi }{\widehat{x}}_4\left(t\right){\widehat{x}}_6\left(t\right)+b_{\varphi }{\widehat{x}}_4\left(t\right){\mathrm{\Omega }}_r$.

In order to analyses error dynamics, define estimation errors as follows:Let ${\chi }_{\varphi }={[{\chi }_1,{\chi }_2]}^T$; then, taking the derivative and simplifying yields $\epsilon {\dot{\chi }}_{\varphi }=\mathbb{A}{\chi }_{\varphi }+\epsilon \delta$, where matrices $\delta$ and $\mathbb{A}$ are written as follows: where $\mathbb{A}$ is Hurwitz due to the characteristic polynomial gains ${\hslash }_{{\varphi }_1}$ and ${\hslash }_{{\varphi }_2}$. Now, choose a Lyapunov candidate function as follows: where $\mathbb{P}=[p_{11},p_{12};p_{21},p_{22}]$ with positive constants such that $\mathbb{P}={\mathbb{P}}^T>0$ and $p_{12}=p_{21}$. In addition, $\mathbb{P}$ is obtained by solving $\mathbb{P}\mathbb{A}+{\mathbb{A}}^T\mathbb{P}=-\mathbb{I}$. With simple mathematical derivations, the elements of the matrix $\mathbb{P}$ can be determined as $p_{11}={\displaystyle \frac{1}{2{\hslash }_{{\varphi }_1}}}\left(1+{\hslash }_{{\varphi }_2}\right)$, $p_{12}=p_{21}=-{\displaystyle \frac{1}{2}}$, and $p_{22}={\displaystyle \frac{1}{2{\hslash }_{{\varphi }_2}}}\left({\displaystyle \frac{1}{{\hslash }_{{\varphi }_1}}}\left(1+{\hslash }_{{\varphi }_2}\right)+{\hslash }_{{\varphi }_1}\right)$. Next, taking the derivative of $V_H$ and simplifying it along with using Young’s inequality yields:Hence, stable dynamics can be achieved by an appropriate choice of ${\hslash }_{{\varphi }_1}>0$, ${\hslash }_{{\varphi }_2}>0$ and by designing $\epsilon \in (0,1]$ to be sufficiently small.

With the similar control design, the controllers for pitch and yaw can be constructed. The complete algorithm of attitude control is given in Algorithm 1.

Algorithm 1 Attitude Control

Require:  Quadrotor parameters and desired attitude: ${\varphi }_{des}\left(t\right),{\theta }_{des}\left(t\right),{\psi }_{des}\left(t\right)$ Ensure:  $\mathrm{Tracking}\mathrm{error}\to 0:$${\varphi }_{des}\left(t\right)\leftarrow \varphi \left(t\right),{\theta }_{des}\left(t\right)\leftarrow \theta \left(t\right),{\psi }_{des}\left(t\right)\leftarrow \psi \left(t\right)$ 1: while$\mathrm{Attitude}\mathrm{error}\ne 0$do 2:     for $i\in (\varphi ,\theta ,\psi )$, $j\in (1,3,5)$ and $m\in (1,2,3)$ do 3:          ${\widehat{d}}_i=z_i+w_i;{\dot{z}}_i={\mu }_i\left(f_i+c_i{U}_i+w_i+z_i\right);w_i={\mu }_i{x}_{j+1}\left(t\right)$ 4:          $X_i={\left[x_j{x}_{j+1}\right]}^T,F_i={\left[x_{j+1}{f}_i\right]}^T,B_j={\left[0c_i\right]}^T,D_i={\left[01\right]}^T$ 5:          $A_m=[01;k_m{k}_{m+1}],B_m={\left[0k_{m+2}\right]}^T$ 6:          $$\begin{gathered} {\dot{\widehat{K}}}_i^T\left(t\right)=-{\mathrm{\Gamma }}_i{X}_i{e}_i^T{\overline{P}}_{j+1};{\dot{\widehat{K}}}_{r_i}\left(t\right)=-{\alpha }_i{k}_{m+2}{r}_i\left(t\right)e_i^T{\overline{P}}_{j+1}, \\ {\dot{\widehat{K}}}_{v_i}\left(t\right)=-{\beta }_i{k}_{m+2}{\overline{u}}_i\left(t\right)e_i^T{\overline{P}}_{j+2} \end{gathered}$$ 7:          ${\dot{\mathbb{Z}}}_i={\mathbb{P}}_i{Z}_i$, ${\dot{\mathbb{E}}}_i={\mathfrak{R}}_i{\mathbb{E}}_i+{\mathbb{X}}_i$ 8:          ${\dot{\widehat{X}}}_{\varphi }={\widehat{F}}_{\varphi }\left(t\right)+B_{\varphi }{U}_{\varphi }+\mathbb{K}(y_{\varphi }-{\widehat{x}}_1)\left(t\right)$ 9:          ${\overline{u}}_i=-{\left(C_i^T{B}_i\right)}^{-1}(C_i^T\left(A_i{X}_i+D_i{\widehat{d}}_i\left(t\right)-{\mathbb{P}}_i{Z}_i-{\mathfrak{R}}_i{\mathbb{E}}_i-{\mathbb{X}}_i\right)$        $+{\widehat{L}}_i\mathrm{sgn}\left(S_i\right)+{\widehat{K}}_i{S}_i)$, ${\dot{\widehat{K}}}_i={\beta }_i{S}_i^2$, ${\dot{\widehat{L}}}_i\left(t\right)={\alpha }_i\left|S_i\right|$ 10:          $v_i\left(t\right)={\widehat{K}}_i\left(t\right)X_i+{\widehat{K}}_{r_i}\left(t\right)r_i\left(t\right)-{\widehat{K}}_{v_i}\left(t\right){\overline{u}}_i\left(t\right)$ 11:          $U_i\left(t\right)=\left({\displaystyle \frac{1}{B_i^T{B}_i}}\right)B_i^T\{A_m{X}_i-F_i\left(t\right)+B_m{v}_i\}$ 12:     if Attitude error $\to 0$ then 13:         Stop 14:     else 15:         Redesign control gains and reference model 16:         Repeat