2.1 Aerodynamic forces and moments

Aerodynamic forces and moments include the motion and interaction of the aircraft and airflow relative to the ground. Control inputs are the aircraft’s control surfaces, such as the aileron, rudder and elevator. Figure 2 shows the aerodynamic control surfaces on the aircraft. The displacement of the control surfaces generates lift and drag.

Figure 2.

Aerodynamic control surfaces.

Aerodynamic forces in the centre of the aircraft also cause aerodynamic moments. A change in the elevator surface causes a pitching moment change. Figure 3 defines the pitching moment in the wind axis with lift and drag forces.

Figure 3.

Lift, drag force and pitching moment.

While longitudinal motions occur according to the vertical axis of symmetry, lateral motions happen in the form of lateral force, roll moments or yaw moments resulting from the asymmetric air flow of the aircraft. These interactions may arise from lateral wind or rudder control surfaces.

Equations (1–6) define the parameters for longitudinal and lateral motions and their functional dependencies. In these equations, ${{\textrm{C}}_L}$ , ${{\textrm{C}}_D}$ , ${{\textrm{C}}_m}$ are the dimensionless aerodynamic coefficients, ${\textrm{S}}$ is the platform area of the wing surface, ${\textrm{c}}$ is the mean aerodynamic cord, ${\textrm{b}}$ is the span of the wing, $\rho $ is aerodynamic pressure and ${{\textrm{V}}_a}$ is airspeed. ${{\textrm{F}}_{lift}}$ , ${{\textrm{F}}_{drag}}$ , ${{\textrm{F}}_{lat}}$ represent the forces acting on the aircraft’s ${\textrm{x}},{\textrm{y}},{\rm{z}}$ axis, while ${\rm{L}}$ , ${\rm{M}}$ , ${\rm{N}}$ stand for the roll, pitch, yaw moments, respectively.

\begin{align*}\begin{array}{*{20}{l}}{{\textrm{Longitudinal\,plane}}}\end{array}\end{align*}

(1) \begin{align}{F_{lift}} = \frac{1}{2}\rho .V_a^2.S\left( {{C_L}_0 + {C_L}_\alpha {\rm{\;}}.\alpha + {C_L}_q{\rm{\;}}.\frac{c}{{2{V_a}}}{\rm{\;}}.q + {C_L}_{{\delta _e}}{\rm{\;}}.{\delta _e}} \right)\end{align}

(2) \begin{align}{F_{drag}} = \frac{1}{2}\rho .V_a^2.S\left( {{C_D}_0 + {C_D}_\alpha .\alpha + {C_D}_q.\frac{c}{{2{V_a}}}.q + {C_D}_{{\delta _e}}.{\delta _e}} \right)\end{align}

(3) \begin{align}M = \frac{1}{2}\rho .V_a^2.S.c\left( {{C_M}_0 + {C_M}_\alpha .\alpha + {C_M}_q.\frac{c}{{2{V_a}}}.q + {C_M}_{{\delta _e}}.{\delta _e}} \right)\end{align}

\begin{align*}\begin{array}{*{20}{l}}{{\textrm{Lateral\,plane}}}\\{}\end{array}\end{align*}

(4) \begin{align}{F_{lat}} = \frac{1}{2}\rho .V_a^2.S\left( {{C_Y}_0 + {C_Y}_\beta {\rm{\;}}.\beta + {C_Y}_p{\rm{\;}}.\frac{b}{{2{V_a}}}{\rm{\;}}.p + {C_Y}_r{\rm{\;}}.\frac{b}{{2{V_a}}}{\rm{\;}}.r + {C_Y}_{{\delta _r}}{\rm{\;}}.{\delta _r} + {C_Y}_{{\delta _{}}}{\rm{\;}}.{\delta _a}} \right)\end{align}

(5) \begin{align}L = \frac{1}{2}\rho .V_a^2.S.b\left( {{C_l}_0 + {C_l}_\beta {\rm{\;}}.\beta + {C_l}_p{\rm{\;}}.\frac{b}{{2{V_a}}}{\rm{\;}}.p + {C_l}_r{\rm{\;}}.\frac{b}{{2{V_a}}}{\rm{\;}}.r + {C_l}_{{\delta _r}}{\rm{\;}}.{\delta _r} + {C_l}_{{\delta _a}}{\rm{\;}}.{\delta _a}} \right)\end{align}

(6) \begin{align}N = \frac{1}{2}\rho .V_a^2.S.b\left( {{C_n}_0 + {C_n}_\beta {\rm{\;}}.\beta + {C_n}_p{\rm{\;}}.\frac{b}{{2{V_a}}}{\rm{\;}}.p + {C_n}_r{\rm{\;}}.\frac{b}{{2{V_a}}}{\rm{\;}}.r + {C_n}_{{\delta _r}}{\rm{\;}}.{\delta _r} + {C_n}_{{\delta _a}}{\rm{\;}}.{\delta _a}} \right)\end{align}

$\alpha $ , $\beta $ , ${\rm{p}}$ , ${\rm{q}}$ and ${\rm{r}}$ in Equations (1–6) are angle-of-attack, sideslip angle, roll rate, pitch rate and yaw rate, respectively. ${\delta _{\rm{a}}}$ , ${\delta _{\rm{e}}}$ and ${\delta _{\rm{r}}}$ are defined as control derivatives for aileron, elevator and rudder control surfaces. In the Appendix section, Table A1 shows the values of some geometrical and aerodynamic parameters of the test model Cessna-182 aircraft.

The fixed-wing UAV model must be able to maintain a horizontal position and constant altitude during a certain flight speed. The UAV model must provide combinations such as angle-of-attack ( $\alpha $ ), thrust adjustment and angular changes of aileron, elevator and rudder required at ${V_{T0}}$ flight speed. These values are referred to as the trim (equilibrium) conditions of flight and the stabilised conditions of motion, which are given in Equation (7).

(7) \begin{align}\begin{array}{*{20}{l}}{{U_0} = {V_{T0}}. \cos {\alpha _0},{W_0} = {V_{T0}}.\sin {\alpha _0},{Q_0} = 0,{\theta _0} = {\alpha _0},} {}{}\\{{V_0} = {P_0} = {R_0} = {\phi _0} = {\psi _0} = 0.} {}{}\end{array}\end{align}

The longitudinal EoMs are

(8) \begin{align}\begin{array}{*{20}{l}}{{F_x}} {}{ = m{\rm{\;}}\dot U = - m.g.\sin \alpha + q.S.\left( {{C_L}.\sin \alpha - {C_D}.\cos \alpha } \right) + T,}\\[3pt]{{F_z}} {}{ = m{\rm{\;}}\dot W = m.g.\cos \alpha + q.S.\left( {{C_L}.\cos \alpha + {C_D}.\sin \alpha } \right),}\\[3pt]M {}{ = \dot Q{\rm{\;}}{I_y} = q.S{\rm{\;}}.c{\rm{\;}}.{C_M} + T{\rm{\;}}.{e_T} + q.S\left( {{C_L}.\sin \alpha - {C_D}.\cos \alpha } \right).{h_0}.}\end{array}\end{align}

T is the thrust force and ${e_T}$ is the thrust line offset. The distance from the aerodynamic centre to the centre of gravity can be expressed as ${h_0}$ .

(9) \begin{align}{C_L} = {C_{L0}} + {C_L}_\alpha {\rm{\;}}.\alpha + {C_L}_q{\rm{\;}}.\frac{c}{{2{V_a}}}{\rm{\;}}.q + {C_L}_{{\delta _e}}{\rm{\;}}.{{\rm{\delta }}_e},\end{align}

(10) \begin{align}{C_D} = {C_0} + {C_D}_\alpha {\rm{\;}}.\alpha + {C_D}_q{\rm{\;}}.\frac{c}{{2{V_a}}}{\rm{\;}}.q + {C_D}_{{\delta _e}}{\rm{\;}}.{\delta _e},\end{align}

(11) \begin{align}{C_M} = {C_{M0}} + {C_M}_\alpha {\rm{\;}}.\alpha + {C_M}_q{\rm{\;}}.\frac{c}{{2{V_a}}}{\rm{\;}}.q + {C_M}_{{\delta _e}}{\rm{\;}}.{\delta _e}\end{align}

The control surface parameters and graphs are below for the moment balance on the aircraft axes at a ${V_{T0}}$ = 60 km/h flight speed. These results were obtained using the “fminsearch” optimisation algorithm in MATLAB, and the results are shown below and in Fig. 4:

Figure 4.

Equilibrium states (u, v, w, p, q, r, $\phi ,\theta ,\psi $ ) of the Cessna182.

2.2 The longitudinal equations of the Cessna-182 UAV

If the linearised longitudinal parameters previously mentioned in Equations (1–6) are considered and expressed in the state-space matrix, Equation (12) is obtained. State-space for both longitudinal and lateral motion details for parameter explanations can be seen in Refs (Reference Ulus and Eski42, Reference Oktay, Konar, Onay, Aydin and Mohamed43):

(12) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{\rm{\Delta }}\dot u}\\[2pt]{{\rm{\Delta }}\dot w}\\[2pt]{{\rm{\Delta }}\dot q}\\[2pt]{{\rm{\Delta }}\dot \theta }\\[2pt]{{\rm{\Delta }}\dot h}\end{array}} \right\} = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{{X_u}}& {}{{X_w}}& {}0& {}{ - g}& {}0\\[2pt]{{Z_u}}& {}{{Z_w}}& {}{{u_0}}& {}0& {}0\\[2pt]{{M_u} + {M_{\dot w}}{Z_u}}& {}{{M_w} + {M_{\dot w}}{Z_w}}& {}{{M_q} + {M_{\dot w}}{u_0}}& {}0& {}0\\[2pt]0& {}0& {}1& {}0& {}0\\[2pt]{ - \sin {\theta _0}}& {}{ - \cos {\theta _0}}& {}0& {}{ - {u_0}\cos {\theta _0}}& {}0\end{array}} \right]\\[5pt] \left\{ {\begin{array}{*{20}{l}}{{\rm{\Delta }}u}\\[2pt]{{\rm{\Delta }}w}\\[2pt]{{\rm{\Delta }}q}\\[2pt]{{\rm{\Delta }}\theta }\\[2pt]{{\rm{\Delta }}h}\end{array}} \right\} + \left[ {\begin{array}{c@{\quad}c}{{X_{{\delta _e}}}}& {}{{X_{{\delta _T}}}}\\[2pt]{{Z_{{\delta _e}}}}& {}{{Z_{{\delta _T}}}}\\[2pt]{{M_{{\delta _e}}} + {M_{\dot w}}{Z_{{\delta _e}}}}& {}{{M_{{\delta _T}}} + {M_{\dot w}}{Z_{{\delta _T}}}}\\[2pt]0& {}0\\[2pt]0& {}0\end{array}} \right]\left\{ {\begin{array}{*{20}{l}}{{\rm{\Delta }}{\delta _e}}\\[2pt]{{\rm{\Delta }}{\delta _T}}\end{array}} \right\}\end{align}

(13) \begin{align}\left[ {\begin{array}{*{20}{l}}{\delta u}\\[4pt]{\delta w}\\[4pt]{\delta q}\\[4pt]{\delta \theta }\\[4pt]{\delta h}\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{ - 0.10}& {}{0.23}& {}0& {}{ - 9.8}& {}0\\[4pt]{ - 0.76}& {}{ - 5.49}& {}{16.67}& {}0& {}0\\[4pt]{0.98}& {}{ - 1.85}& {}{ - 7.94}& {}0& {}0\\[4pt]0& {}0& {}1& {}0& {}0\\[4pt]{ - 0.05}& {}{ - 0.99}& {}0& {}{ - 16.7}& {}0\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\rm{\Delta }}u}\\[4pt]{{\rm{\Delta }}w}\\[4pt]{{\rm{\Delta }}q}\\[4pt]{{\rm{\Delta }}\theta }\\[4pt]{{\rm{\Delta }}h}\end{array}} \right] + \left[ {\begin{array}{c@{\quad}c}{6.81}& {}0\\[4pt]0& {}{ - 8.9}\\[4pt]{0.07}& {}{ - 86.6}\\[4pt]0& {}0\\[4pt]0& {}0\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\delta _e}}\\[4pt]{{\delta _t}}\end{array}} \right]\end{align}

where u and w are the velocities in the direction of the x and z axes, respectively, and q, $\theta $ , h are the parameters related to pitch rate, pitch angle and altitude. $\delta_e$ and ${\delta_T}$ are the changes in the elevator and the thrust.

2.3. The lateral equations of the Cessna-182 UAV

The UAV lateral state-space equations are obtained in Equation (14) and Equation (15) for parameter definitions (see Ref. (Reference Ulus and Eski42)):

(14) \begin{align}\left[ {\begin{array}{*{20}{l}}{{\rm{\Delta }}\dot v}\\[1pt]{{\rm{\Delta }}\dot p}\\[1pt]{{\rm{\Delta }}\dot r}\\[1pt]{{\rm{\Delta }}\dot \phi }\\[1pt]{{\rm{\Delta }}\dot \psi }\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{{Y_v}}& {}{{Y_p}}& {}{ - \left( {{u_0} - {Y_r}} \right)}& {}{g\cos {\theta _0}}& {}0\\[1pt]{{L_v}}& {}{{L_p}}& {}{{L_r}}& {}0& {}0\\[1pt]{{N_v}}& {}{{N_p}}& {}{{N_r}}& {}0& {}0\\[1pt]0& {}1& {}{\tan {\theta _0}}& {}0& {}0\\[1pt]0& {}0& {}{\sec {\theta _0}}& {}0& {}0\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\rm{\Delta }}v}\\[1pt]{{\rm{\Delta }}p}\\[1pt]{{\rm{\Delta }}r}\\[1pt]{{\rm{\Delta }}\phi }\\[1pt]{{\rm{\Delta }}\psi }\end{array}} \right] + \left[ {\begin{array}{c@{\quad}c}0& {}{{Y_{{\delta _r}}}}\\[1pt]{{L_{{\delta _a}}}}& {}{{L_{{\delta _r}}}}\\[1pt]{{N_{{\delta _a}}}}& {}{{N_{{\delta _r}}}}\\[1pt]0& {}0\\[1pt]0& {}0\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\rm{\Delta }}{\delta _a}}\\[1pt]{{\rm{\Delta }}{\delta _r}}\end{array}} \right]\end{align}

If the values are substituted in the equation:

(15) \begin{align}\begin{array}{*{20}{l}}{\left[ {\begin{array}{*{20}{l}}{\delta \dot v}\\[1pt]{\delta \dot p}\\[1pt]{\delta \dot r}\\[1pt]{\delta \dot \phi }\\[1pt]{\delta \dot \psi }\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{ - 8.1}& {}0& {}{ - 16.4}& {}{9.81}& {}0\\[1pt]{ - 76.3}& {}{ - 33.5}& {}{2.42}& {}0& {}0\\[1pt]{22.2}& {}{ - 1.75}& {}{ - 2.2}& {}0& {}0\\[1pt]0& {}1& {}0& {}0& {}0\\[1pt]0& {}0& {}1& {}0& {}0\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{\delta v}\\[1pt]{\delta p}\\[1pt]{\delta r}\\[1pt]{\delta \phi }\\[1pt]{\delta \psi }\end{array}} \right] + \left[ {\begin{array}{c@{\quad}c}0& {}{3.8}\\[1pt]{191}& {}{13.5}\\[1pt]{ - 4.4}& {}{ - 19.9}\\[1pt]0& {}0\\[1pt]0& {}0\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\delta _a}}\\[1pt]{{\delta _r}}\end{array}} \right]}\end{array}\end{align}

3.1 PID controller using genetic algorithm

In this study, a GA is used to search the gains of the PID controller [Reference Ulus and Eski42]. Equation (16) is given for PID cost function ( ${\rm{J}}$ ) according to performance criteria.

(16) \begin{align}J = {\textrm{parm}}\left( 1 \right) + {\textrm{parm}}\left( 2 \right)\frac{1}{s} + {\textrm{parm}}\left( 3 \right)\frac{s}{{0.001s + 1}}\end{align}

The expressions given as parm(1), parm(2) and parm(3) are used in the GA code as ${{\textrm{K}}_{\textrm{p}}}$ , ${{\textrm{K}}_{\textrm{i}}}$ and ${{\textrm{K}}_{\textrm{d}}}$ . Here, unnecessary signals are avoided by using a high-pass filter in the derivative block. Various performance criteria are given in Equations (17–21), respectively,

(17) \begin{align}ISE = {\rm{\;}}\mathop \sum \limits_{t = 0}^k {({x_{{{\textrm{m}}_{}}}} - \hat x)^2}\end{align}

(18) \begin{align}IAE = {\rm{\;}}\mathop \sum \limits_{t = 0}^k \left| {\left( {{x_{{{\textrm{m}}_{}}}} - \hat x} \right)} \right|\end{align}

(19) \begin{align}ITSE = {\rm{\;}}\mathop \sum \limits_{t = 0}^k t{\rm{'}}.{({x_{{{\textrm{m}}_{}}}} - \hat x)^2}\end{align}

(20) \begin{align}ITAE = {\rm{\;}}\mathop \sum \limits_{t = 0}^k t{\rm{'}}.\left| {\left( {{x_{{{\textrm{m}}_{}}}} - \hat x} \right)} \right|\end{align}

(21) \begin{align} LQR = {\rm{\;}}\mathop \sum \limits_{t = 0}^k (Q.{\left| {\left( {{x_{{{\textrm{m}}_{}}}}\left( t \right) - \hat x\left( t \right)} \right)} \right|^2} + R.u\left( {t{)^2}} \right)\end{align}

Here, t is the time and $\left( {{x_{{{\textrm{m}}_{}}}} - \hat x} \right)$ is the error signal. According to LQR performance criteria, Q = 10 and R = 0.001 are settled to regulate controller effort between error and controller in Equation (21). In this way, a tendency to rely more on the sensor measurement and less on the controller is aimed. The GA parameters are shown in Table 2.

Table 2.

Values for the GA parameters

3.2 Linear quadratic regulator (LQR) controller

The LQR controller is used as an optimisation technique involving a quadratic objective function corresponding to the weighted energy of the control variables. The performance criterion in LQR is to select signal u(t) that can minimise the control input u in the ${J_{LQR}}$ function given by Equation (22).

(22) \begin{align}{J_{LQR}} = \mathop {{\textrm{lim}}}\limits_{T \to \infty } {\rm{\;}}\mathop \smallint \limits_0^T \left( {{x^T}.Q.x + {u^T}.R.u} \right).dt\end{align}

The result of the states and control signals given by Equation (22), including the weighted energy terms, must be minimised. For this purpose, stabilising Q and R are constructed to meet the transient performance requirements quickly with minimum controller energy. According to R, higher Q values indicate higher system performance and the reverse. The Q matrix is a positive semi-definite matrix. Adjustments were performed according to each state variable’s feedback K gains and simulation results. Similarly, the R matrix is a positive diagonal matrix chosen to adjust the energy and value of the corresponding control signal. The assumptions are as follows:

Assumption 1.

\begin{align*}Q = {Q^T} \ge 0{\rm{\;\;}}and{\rm{\;\;}}R = {R^T} \gt 0\end{align*}

and (A, B) the pair must be stable.

According to Assumption 1, the optimum controller design is $U\left( t \right) = - {K_r}.x\left( t \right)$ , and ${K_r}$ denotes the optimum state feedback gain, as in Equation (23).

(23) \begin{align}{K_r} = {R^{ - 1}}.{B^T}.P\end{align}

P is obtained from the algebraic Riccati equation required for stability.

(24) \begin{align}{A^T}.P + P.A - P.B.{R^{ - 1}}.{B^T}.P + Q = 0\end{align}

Here, the stability measure refers to the case where the closed-loop matrix A is ${A_c} = A - B{R^{ - 1}}{B^T}P$ asymptotically stable. The P matrix is positive, semi-definite and symmetric. The gain and phase margins are improved with good determination of these matrices. LQR has a linear control law for the UAV and minimises the state variables with a second-order cost function.

3.3 Optimum state estimation with Kalman filter

KF is the most common state estimator because it optimally compensates for measurement noise, disturbances and modelling uncertainties. LQG-type control is obtained by combining the KF with LQR control. The disturbance effect ${{\textrm{w}}_{\textrm{d}}}$ and sensor noise ${{\textrm{w}}_{\textrm{n}}}$ are added to redefine the system as in Equations (24–25).

(25) \begin{align}\frac{d}{{dt}}x\left( t \right) = A.x\left( t \right) + B.u\left( t \right) + {w_d}\left( t \right)\end{align}

(26) \begin{align}y\left( t \right) = C.x\left( t \right) + D.u\left( t \right) + {w_n}\left( t \right)\end{align}

The KF is a Gaussian operation with zero mean at known disturbance and noise covariance values.

(27) \begin{align}{\textrm{E}}({w_d}\left( t \right).{w_d}\left( {\tau {)^T}} \right) = {V_d}.\delta \left( {t - \tau } \right)\end{align}

(28) \begin{align}{\textrm{E}}({w_n}\left( t \right).{w_n}\left( {\tau {)^T}} \right) = {V_n}.\delta \left( {t - \tau } \right)\end{align}

In Equations (27–28), E is the expected value and $\delta $ is the Dirac delta function. ${{\textrm{V}}_{\textrm{d}}}$ and ${{\textrm{V}}_{\textrm{n}}}$ are matrices that include positive semi-definite disturbance and noise covariances. The KF depends on unknown noise, disturbance and direct effects. It estimates the state ${\hat{\textrm{x}}}$ from the state measurements x according to the input u and output y measurements as in Equations (29–36).

(29) \begin{align}\frac{d}{{dt}}{\hat x} \left( t \right) = A.{\hat x} \left( t \right) + B.u\left( t \right) + {K_f}.\left( {y - {\hat y} } \right)\end{align}

(30) \begin{align}{\hat y} \left( t \right) = C.{\hat x} \left( t \right) + D.u\left( t \right)\end{align}

The matrices A, B, C and D reflect the UAV model, and the ${K_f}$ gain can be obtained similarly for LQR. y is the solution of algebraic Riccati equation,

(31) \begin{align}{A^T}.Y + Y.A - Y.C.{V_n}^{ - 1}.{C^T}.Y + {V_d} = 0\end{align}

The solution given by Equation (31) is called the KF.

(32) \begin{align}\frac{d}{{dt}}{\hat x} \left( t \right) = \left( {A - {K_f}.C} \right){\hat x} \left( t \right) + \left[ {{K_f},{\rm{\;\;}}\left( {B{K_f}.D} \right)} \right]\left[ {\begin{array}{*{20}{c}}y\\u\end{array}} \right]\end{align}

3.4 Linear quadratic Gaussian (LQG) controller

Since the assumption that all state variables can be measured in LQR control is not very realistic in terms of application, LQG control theory was preferred. For the LQG controller, disturbance effects are filtered Dryden wind turbulence signals. Leaving aside the measurement logic of all variables in LQR control, LQG control is based on noisy measurements. In this case, two different sub-problem solutions arise. One is the filter design in which the state variables can be optimally estimated and the other is to obtain the optimal controller setting. KF is essential here as the optimal observer that balances disturbance and noise signals. Here, LQR gain ${K_r}$ and KF gain ${K_f}$ are obtained separately. LQR and KF full-state observer form the LQG controller together. The LQG controller is the controller structure of the dynamic system which has input y, output u, internal state $\hat x$ .

(33) \begin{align}\frac{d}{{dt}}{\hat x} \left( t \right) = \left( {A - {K_f}.C - B.{K_r}} \right){\hat x} \left( t \right) + {K_f}.y\left( t \right)\end{align}

(34) \begin{align}u = - {K_r}.{\hat x} \end{align}

Equation (33) defines the newly derived matrix A and B for the LQG. Equation (34) is the negative feedback of the ${K_r}$ gains containing the LQR controller output. Equation (35) defines the LQG controller’s cost function.

(35) \begin{align} J\left( t \right) = \mathop \int \limits_0^t [x{(\tau )^T}.Q.x\left( \tau \right) + u\left( {\tau {)^T}.R.u\left( \tau \right)} \right].d\tau \end{align}

The controller $u = - {K_r}{\hat x} $ is a cost function related to the state estimation and requires averaging over many realisations of disturbance and noise. If we apply the LQR for $\hat x$

(36)

The final form for the closed-loop system is given by Equation (37).

(37) \begin{align}\frac{d}{{dt}}\left[ {\begin{array}{*{20}{l}}x {}\\[2pt]\varepsilon \end{array}} \right] = \left[ {\begin{array}{c@{\quad}c}{A - B{K_r}}& {}{B{K_r}}\\[2pt]0& {}{A - {K_f}C}\end{array}} \right]{\rm{\;}}\left[ {\begin{array}{*{20}{l}}x\\[2pt]\varepsilon {}\end{array}} \right] + \left[ {\begin{array}{c@{\quad}c}I& {}0\\[2pt]I& {}{ - {K_f}} {}\end{array}} \right]{\rm{\;}}\left[ {\begin{array}{*{20}{l}}{{w_d}}\\[2pt]{{w_n}}\end{array}} \right]\end{align}

The LQG block diagram can be found in Fig. 5. For the LQR and KF, the eigenvalues $\left( {A - B.{K_r}} \right)$ and $\left( {A - {K_f}.C} \right)$ , respectively, provide the closed-loop eigenvalues of the LQG controlled system.

Figure 5.

Block diagram of the LQG controller.

3.5 Model reference adaptive control based on MIT rule

In this paper, the dominant transfer function of the aircraft dynamics is controlled using an MRAC system based on the traditional Massachusetts Institute of Technology (MIT) rule as in Fig. 6. To provide transparency in robustness and disturbance-rejection analysis, the controller is designed inside a single-input, single-output (SISO) architecture.

Figure 6.

MIT rule-based MRAC structure with additive disturbance input.

The desired closed-loop system behaviour is given by a stable second-order reference model whose output represents the ideal system response. ${\omega _n}$ and ${\rm{\zeta }}$ are the natural frequency and damping ratio of the reference model, respectively. The reference model is defined as

(38) \begin{align}{G_m}\left( s \right) = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\end{align}

The plant dynamics are described by a linear transfer function obtained from the linearised aircraft model. The tracking error is defined as

(39) \begin{align}e\left( t \right) = y\left( t \right) - {y_m}\left( t \right)\end{align}

and is used for both control law and parameter adaptation. The adaptive controller input is constructed in a PID-type design,

(40) \begin{align}u\left( t \right) = {K_p}\left( t \right)e\left( t \right) + {K_i}\left( t \right)\mathop \smallint \nolimits_0^t e\left( \tau \right){\rm{\;}}d\tau + {K_d}\left( t \right){\dot e_f}\left( t \right)\end{align}

where the derivative term is used as a first-order filtered error

(41) \begin{align}{e_f}\left( s \right) = \frac{\lambda }{{s + \lambda }}e\left( s \right)\end{align}

The adaptation laws are obtained using the MIT rule by minimising the cost function J

(42) \begin{align}J = \frac{1}{2}{e^2}\left( t \right)\end{align}

leading to gradient-based update equations for the controller gains,

(43) \begin{align} {{{\dot K}_p}} &= -{\gamma_p}{e^2}\left( t \right),\nonumber\\[3pt]{{{\dot K}_i}} &= - {\gamma _i}e\left( t \right)\mathop \smallint \nolimits_0^t \left( \tau \right){\rm{\;}}d\tau , \nonumber\\[3pt]{{{\dot K}_d}} &= - {\gamma _d}e\left( t \right){{\dot e}_f}\left( t \right) \end{align}

where ${\gamma _p}$ , ${\gamma _i}$ , and ${\gamma _d}$ are positive adaptation gains. Dryden turbulence model for moderate level of external disturbances is applied at the plant input, while the reference model remains disturbance-free. This ensures that the adaptive mechanism compensates for model uncertainties and disturbances directly through parameter adjustment. The final MRAC design provides improved robustness compared to fixed-gain controllers and offers a clear benchmark for comparison with PID, LQR and LQG designs.

4.1 Cessna-182 longitudinal motion controller results

ZN, RL, GA and MT were used to determine the gains of the PID controller. The simulation results obtained with PID (ZN), PID (RL), PID (GA), PID (MT), MRAC are given. Figures 7–10 show the results of the PID (GA) controller for the flight speed (u), downward speed (w), pitch rate (q) and pitch angle $(\theta)$ dynamics. In the appendix, Tables A2–A3 show the gain parameters $K_p$ , $K_i$ and $K_d$ obtained from the GA optimisation. The J term, which is the cost function of each performance parameter, is also aimed to be minimal. Especially for the LQR criterion, Q = 10 and R = 0.001 were chosen to reduce the load on the controller and reduce overshoot. Considering the controller effort, GA responses for the LQR performance criterion were generally preferred. GA settings show much better results than the PID (ZN) method as seen in Tables 3–4.

Figure 7.

PID(GA) – flight speed (u) unit step.

Figure 7 Long description

A line graph titled du/dt - Unit Step Response - Genetic Algorithm Tuning - u equals 16 meters per second. The horizontal axis represents Time in seconds ranging from 0 to 5 seconds. The vertical axis represents Airspeed in meters per second ranging from 0 to 1.2 meters per second. The graph includes a legend with labels for Unit Step, ISE, IAE, ITSE, ITAE, and LQR. The Unit Step line is cyan and remains constant at 1 meter per second. The ISE line is red and shows a gradual increase. The IAE line is blue and shows a slower increase compared to ISE. The ITSE line is brown and shows a moderate increase. The ITAE line is green and shows the slowest increase. The LQR line is a dashed green and shows a rapid initial increase followed by a leveling off. An inset zooms in on the time range from 0 to 0.25 seconds and airspeed range from 0.98 to 1.0045 meters per second, highlighting the initial responses of the different parameters.

Figure 8.

PID(GA) – downward speed (w) unit step.

Figure 9.

PID(GA) – pitch rate (q) unit step.

Figure 9 shows that the GA performed flawlessly when parameters such as overshoot and settling time were analysed. In controlling the pitch rate, the integral of squared error (ISE) was preferred as it has a better rise time and settling time than other criteria. As seen in Fig. 9, there is no overshoot. The controller results show that the error increases slightly with time. The ISE and integral of absolute error (IAE) performance are very fast regarding rise time for the pitch angle. In general, the PID gains for the LQR criterion are taken as the best case since the LQR is achieved stably in a very short time in terms of steady-state error.

To show the optimum PID parameters obtained with GA more clearly, PID(ZN), PID(MT) and PID(RL) and MRAC controller results are compared in Figs 11–12 for flight speed and pitch angle. As seen in both figures, the MRAC controller performance indicated the best result.

Table 3.

Step result comparisons for GA criteria

Figure 10.

PID(GA) – pitch angle ( $\theta $ ) unit step.

In Fig. 12, the overshoot for the pitch angle is high for PID(ZN) and PID(RL), but PID(GA) is acceptable for settling time. Overall results in Table 5 show that the MRAC controller exhibited less overshoot, a fast rise time and a settling time, even in the presence of a moderate level of Dryden turbulence. The MRAC controller can be preferred in this situation.

Table 4.

Step result comparisons for GA criteria (contd.)

Figure 11.

PID results – flight speed (u) unit step.

Table 5.

Controller performance comparison for selected states

Figure 12.

PID results – pitch angle ( $\theta $ ) unit step.

4.2 Controller results of the lateral motion

The results of the lateral dynamics of roll rate (p), yaw rate (r), roll ( $\phi $ ) and yaw angle ( $\psi $ ) are presented. (p) has the simulation output shown in Fig. 13 among the lateral dynamics. All performance criteria responded quite fast. The ISE criterion has the fastest settling time.

Figure 14 shows that the integral of time-weighted squared error (ITSE) for (r) control stands out with the fastest settling and lowest overshoot. In the unit step response for roll angle ( $\phi $ ) control given in Fig. 15, the LQR criterion stands out regarding very low overshoot and acceptable settling speed. The LQR was preferred as the controller with low overshoot as seen in Table 4.

Figure 13.

PID(GA) – roll rate (p) unit step.

Figure 14.

PID(GA) – yaw rate (r) unit step.

Figure 15.

PID(GA) – roll angle( $\phi $ ) unit step.

Figure 16 and Table 4 show that the yaw angle responses are close to each other. However, the integral of time-weighted absolute error (ITAE) performance criterion regarding steady-state error and settling time is accepted.

Figure 16.

PID(GA) – yaw angle( $\psi $ ) unit step.

The lateral dynamic controller gains and J cost function values obtained by optimisation for different criteria using GA are given in Appendix Table A3. Figures 17–18 and Table 4 results, the MRAC controller represents much better settling time and less overshoot and the MRAC design exhibits a stable result under Dryden wind moderate turbulence conditons. It is acceptable to use the UAV at a moderate level of wind.

Figure 17.

Different PID tuning – roll angle ( $\phi $ ) unit step.

Figure 18.

PID results – roll angle ( $\psi $ ) unit step.

4.3 LQR, LQG and MRAC simulation results

4.3.1 LQR, LQG, MRAC longitudinal motion results

The longitudinal motion simulation results of the LQR controller are given in this section. As mentioned in the previous section explaining the LQR, the system’s feedback coefficients (K) are found, and the closed-loop simulation results are obtained. Defining the ${\textrm{Q}}$ and ${\textrm{R}}$ matrices is necessary to minimise this criterion according to the determined performance criterion.

The ratio between ${\textrm{Q}}$ and ${\textrm{R}}$ should be well-determined to obtain the best performance value by minimising ${{\textrm{J}}_{{\textrm{LQR}}}}$ defined by Equation (22). If the value of R matrix is larger than Q matrix, the settling time will increase. On the contrary, if Q matrix is larger than R matrix, the settling time will decrease.

For the state space longitudinal motion of the Cessna-182, the LQR feedback gains ${K_r}$ were obtained by selecting Q and R optimally in Equation (44) as a result of the trials. Here, the diagonal terms belong to the state variables u, w, q, $\theta $ , h, which are ${Q_{11}}$ , ${Q_{22}}$ , ${Q_{33}}$ , ${Q_{44}}$ and ${Q_{55}}$ , respectively. Here, ${\textrm{h}}$ is altitude and ${\textrm{w}}$ is emphasised more to provide altitude control.

(44) \begin{align}R = 1,Q = {\textrm{diag}}\left( {50,{\rm{\;\;}}100,{\rm{\;\;}}0.001,{\rm{\;\;}}0.001,{\rm{\;\;}}200} \right)\end{align}

The resulting ${K_r}$ feedback gains are:

(45) \begin{align}{K_r} = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}{7.07}& {}{ - 0.49}& {}{0.05}& {}{8.56}& {}{ - 1.16}\\{ - 0.01}& {}{ - 6.29}& {}{ - 1.51}& {}{ - 65.12}& {}{14.1}\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\delta _T}} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{}\\{{\delta _e}} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{}\end{array}} \right]\end{align}

The eigenvalues of the new closed loop system $\left( {{\textrm{A}} - {\textrm{B}}{{\textrm{K}}_{\textrm{r}}}} \right)$ formed with the found LQR gain are:

(46) \begin{align}{\textrm{Eigs}} = \left\{ { - 97.75 \pm 74.75i,{\rm{\;\;}} - 48.15,{\rm{\;\;}} - 2.35 \pm 1.34i} \right\}\end{align}

KF gains are required in addition to the LQR gains for the LQG control. If the altitude is monitored, the LQG can observe and control the system. The disturbance, noise covariances ${V_d}$ and ${V_n}$ for the KF given by Equation (47) were both selected as 0.1 to represent moderate uncertainty, providing a balanced trade-off between estimation responsiveness and noise amplification while avoiding overly aggressive or excessively sluggish Kalman filter behaviour.

(47) \begin{align}{V_d} = 0.1 \cdot {I_5},{\rm{\;\;\;\;}}{V_n} = 0.1\end{align}

Here, the new matrix B is obtained as ${B_{new}}$ = [B Vd 0 * B] and the matrix D is as ${D_{{\textrm{new}}}} = \left[ {0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}0{\rm{\;\;}}{V_n}} \right]$ and the ${{\textrm{K}}_{\textrm{f}}}$ gains, are given in Equation (48).

(48) \begin{align}{K_f} = \left[ {\begin{array}{*{20}{l}}{0.71}\quad {}{0.07}\quad {}{0.062}\quad {}{ - 0.074}\quad {}{1.51} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{}\end{array}} \right]\end{align}

The eigens of the state observer KF, i.e. poles and zeros, are obtained as in Equation (49). The poles are in the left half plane of the virtual axis, and the state observer KF is in steady-state.

(49) \begin{align}{\textrm{Eigs}}\left( {A - {K_f}C} \right) = \left\{ { - 0.1,{\rm{\;\;}} - 0.72 \pm 1.19i,{\rm{\;\;}} - 6.75 \pm 5.4i} \right\}\end{align}

According to the gains, the LQR noisy and noiseless state-space simulation results of the longitudinal model and the LQG noisy results are analysed. Since the KF required for using the LQG is a reasonable observer for noisy conditions, it is preferred for LQG design.

The results of the velocity and angle change in response to the altitude input to be controlled are shown in Fig. 19. The pitch angle changes by about 0.08 rad (4.58°) during the altitude increase and tries to settle around the equilibrium position (0°) when altitude is achieved. Here, the results of the LQR technique were realised relatively fast and without overshooting in the noiseless state; however, a Dryden wind moderate continuous turbulence was added to all state variables via Matlab Simulink by using Dryden block to model the external disturbance effect on the system. The noise variance was 0.1. In addition, to model the sensor noise in the system, a sensor noise with a variance of 0.01 was added to the sensor measurement sent to the KF, and the noisy state’s results were examined. The results clearly show that the LQR controller is more affected by noise and disturbance input. For this reason, it is seen that the system is less affected by disturbances with the LQG controller with the intuition of the noisy state of the examined KF. In addition to the LQG, the MRAC controller shows the best performance in terms of disturbance rejection and tracking accuracy.

Figure 19.

Altitude (h) control (LQR, LQG and MRAC) at Dryden turbulence.

In Fig. 20, a signal was input for altitude from 50 m to 100 m for ascent, constant altitude and descent, and the results were compared.

Figure 20.

(h,u, $\theta $ ) LQR and LQG results.

Fig. 20 clearly shows that the given reference altitude signal is followed by LQR and LQG controllers, but the LQR is much more affected by the noise and disturbance input signal. The ramp signal was defined for the UAV moving at a constant altitude, and the ascent was 100 m. Then, the wing-level flight was parallel to the ground, and the descent signal was followed at 50 m. During this time, the flight speed is 60 km/h. The pitch angle changes during ascent and descent show that the pitch angle remains constant during ascent and returns to the equilibrium angle when altitude is achieved. In general, the results show that the LQG controller is effective against noise, and therefore, the LQR + KF structure is valid and can be chosen at uncertain conditions. Beyond that the MRAC controller again demonstrates superior tracking performance and disturbance rejection compared to both LQR and LQG controllers under the same conditions.

4.3.2 LQR, LQG, MRAC lateral motion results

For the lateral dynamics of the Cessna-182 model, the weighting matrices Q and R were selected as shown in Equation (50), and the LQR feedback gains ${K_r}$ were obtained. Here, the diagonal terms ${Q_{11}}$ , ${Q_{22}}$ , ${Q_{33}}$ , ${Q_{44}}$ and ${Q_{55}}$ correspond to the state variables v, p, r, $\phi $ and $\psi $ , respectively. These state variables represent the sideslip velocity, roll angular velocity, yaw angular velocity, roll angle and yaw angle, respectively. In this case, greater weighting was applied to the roll angle $\phi $ and yaw angle $\psi $ to ensure effective control of these angles, and the necessary gain values were obtained.

(50) \begin{align}R = 0.1,{\rm{\;\;\;\;}}Q = {\textrm{diag}}\left( {0.1,0.1,0.1,100,100} \right)\end{align}

The obtained LQR feedback gains ( ${{\textrm{K}}_{\textrm{r}}}$ ) form a 2 $ \times $ 5 gain matrix, as there are two lateral control inputs: aileron ( $\delta a$ ) and rudder ( $\delta r$ ).

(51) \begin{align}{K_r} = \left[ {\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l}{-0.27}& {}{0.99}& {}{0.04}& {}{31.68}& {}{0.24}\\[5pt]-0.58 & {0.02} & {- 1.66}& {}{ - 0.88}& {}{ - 31.62}\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\delta _a}} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{}\\{{\delta _r}} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{}\end{array}} \right]\end{align}

The eigenvalues of the new closed-loop $\left( {{\textrm{A}} - {\textrm{B}}.{{\textrm{K}}_{\textrm{r}}}} \right)$ formed by the LQR gains:

(52) \begin{align}{\textrm{Eigs}} = \left\{ { - 191.5,{\rm{\;\;}} - 3.192,{\rm{\;\;}} - 31.46,{\rm{\;\;}} - 18.83 \pm 21.15i} \right\}\end{align}

As shown in Equation (52), the eigens are located in the left half of the complex plane, with real and complex conjugate roots. Since no poles are in the right half-plane, the system is controlled stably. For LQG control, the KF gains were obtained. As discussed in Section 4, the system is observable when the yaw angle $\psi $ is tracked, and it was explained that the system can be controlled using LQG. The disturbance and noise covariance matrices ${V_d}$ and ${V_n}$ for the KF are given in Equation (53) as explained in Equation (47).

(53) \begin{align}{V_d} = 0.1 \cdot {I_5},{\rm{\;\;\;\;}}{V_n} = 0.1\end{align}

The solution to KF gains is given in Equation (54):

(54) \begin{align}{K_f} = \left[ {\begin{array}{*{20}{l}}{6.9x{{10}^{ - 4}}}\quad {}{ - 1.01x{{10}^{ - 3}}} {}\quad {7.98x{{10}^{ - 3}}}\quad {}{1.4x{{10}^{ - 2}}} {}{0.13} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{} {}{}\end{array}} \right]\end{align}

The eigenvalues of the KF are given in Equation (55):

(55) \begin{align}{\textrm{Eigs}}\left( {A - {K_f}.} \right) = \left\{ { - 35.7,{\rm{\;\;}} - 4.05 \pm 2.1i,{\rm{\;\;}} - 0.1 \pm 0.08i} \right\}\end{align}

As seen in Equation (55), the poles are located in the left half-plane, which indicates that the KF is stable. Figure 21 shows the results for the other state variables in response to a 10° step input for roll angle control. The control surfaces (aileron and rudder) were limited to $ \mp 40^\circ $ using a saturation block in Simulink to reflect realistic conditions. The figure compares the noisy and noise-free responses of LQR with the noise of LQG for roll angle control.

Figure 21.

Roll angle ( $\phi $ ) control and other states.

The desired roll angle was achieved relatively quickly, while the other state variables exhibited changes until the roll angle settled, after which they stabilised. Both disturbance and sensor noise were introduced into the system to simulate the effects of noise. The variance of the randomly generated MATLAB Simulink noise was set to 0.01. In terms of noise, it was observed that the LQR feedback was more affected by noise. In contrast, the LQG, benefiting from the KF, effectively suppressed the noise. For the yaw angle ( $\psi $ ), a $10^\circ $ step input simulation results are shown in Fig. 22. The MRAC controller also demonstrates robust performance in tracking the yaw angle command under similar noisy conditions, outperforming both LQR and even LQG controllers.

Figure 22.

Yaw angle ( $\psi $ ) control and other states.

As in Fig. 22, the yaw angle settled within approximately 1 second, during which the sideslip and yaw angular velocity also exhibited changes. Once the system reached a steady state, the state variables converged to zero as expected. Additionally, the yaw condition only slightly affected the roll angular velocity and angle, settling to zero after minimal angular changes. When evaluating controller performance, it is evident that the LQG controller responded well to noise. In contrast, unlike its noise-free performance, the LQR controller exhibited significant oscillations in the presence of noise. But, the MRAC controller was preferred because of its fast and stable performance under same disturbance effects

As a result of the studies conducted, the proposed MRAC and GA-based PID parameter tuning, MATLAB PID tuning, RL design and ZN were implemented. Furthermore, full-state feedback control with LQR were explored to better demonstrate the interactions between all state-space variables and control inputs of the UAV. In noisy as well as noise-free conditions, the MRAC provides a flexible and adaptive response based on the transfer function. While LQG has an advantage over LQR due to its full-state design, MRAC offers more adaptable performance under varying conditions.

4.4 Frequency-domain robustness analysis

To analyse the robustness of the system dynamics, frequency-domain analyses were conducted using Bode diagrams. Figures 23–24 present the magnitude and phase responses of the altitude, pitch-angle, roll-angle and yaw-angle channels, respectively. The frequency-domain study shows that under open-loop operation, the longitudinal dynamics show marginal or even unstable behaviour, with poles near the imaginary axis and negative phase margins. The drastically decreased phase margins suggest poor robustness and significant sensitivity to parameter uncertainty and disturbances, even though some channels have reasonably substantial gain margins. On the other hand, even if some channels are operating near the stability boundary, the lateral dynamics show more favourable stability margins and remain open-loop stable with respectable gain and phase characteristics.

Figure 23.

Bode diagrams: (a) altitude response (b) pitch-angle response.

Figure 24.

Bode diagrams: (a) roll angle response (b) yaw-angle response.

Both longitudinal and lateral dynamics show notable increases in stability margins when closed-loop control, such as the MRAC and LQG controller, is used. Increased phase margins and decreased sensitivity to disturbances are evaluated by using the MRAC and LQG controllers. While the LQG controller successfully reduces the effects of noise and disturbances through optimal state estimation, the MRAC controller specifically improves robustness by adaptively responding to changing dynamics. Overall, the frequency-domain analysis verifies that the suggested control solutions significantly improve the stability and resilience of the UAV dynamics in both lateral and longitudinal motions.

4.5 Flight test results

Cessna-182 mini-UAV was used for flight tests. The aim was to compare the PID parameters obtained from simulation results. Several successful flights were carried out, and PID tuning was performed in real-time using the Mission Planner ground station software. The resulting PIDs are presented. The PIDs for roll and pitch angles were obtained during real-time flight using Mission Planner, and their proximity to the values derived from simulation results was examined.

During the flight, the UAV successfully executed roll, pitch, and yaw angle commands despite windy conditions, and the obtained PIDs were found suitable for controlling the aircraft. All hardware and the plan were prepared using the Pixhawk Cube controller board and the Mission Planner. The accelerometer, compass, radio controller, airspeed sensor and ESC motor were calibrated using ArduPilot instructions. Additionally, flight modes (manual, stabilise, mission, return-to-launch, FBWA) were configured on the controller. To ensure communication via telemetry, the RFD900x modem was used, operating in the 915–928 MHz frequency band with 21 channels. Necessary data were monitored through the autopilot during the flight.

In addition to previous flight tests, some information and the flight path obtained during this flight are shared. Figure 25 shows the GPS data for the flight during the PID auto-tuning attempt at the Talas model aircraft runway. The red trajectory represents the flight path during PID tuning, while the yellow trajectory represents the flight path in stabilise mode using the tuned PID parameters.

Figure 25.

(a) Auto-PID tuning; (b) second flight route at stabilise mode.

Figure 25(b) shows the automatic PID adjustment during real-time flight and the route during the adjustment with red lines. These adjustments continued for about 1.5 turns. Then, the UAV was put into stabilised mode, and the flight continued the route indicated by yellow lines. No abnormality was observed in the aircraft flight despite the windy conditions. Later, the landing was performed successfully, as there was no signal loss or deviation in the control. A flight was conducted for the next attempt to re-adjust the PIDs. Figure 25(b) shows the route of this flight.

The flight was experimented with, and all PID parameters were experimentally adjusted in real time. The results obtained are close to the theoretical results. Figure 26(a) shows the takeoff and Fig. 26(b) was captured while in flight phase. In Appendix Figs A1(b–c) indicate the desired roll and pitch angle versus the measured roll and pitch angle after the Mission Planner auto-tune. In general, desired and measured angle data follow each other. Airspeed data taken from GPS is depicted in Appendix Figure A1(a).

Figure 26.

(a) Take off; (b) flight.