Aircraft Landing Control Using the H-inf Control and the Dynamic Inversion Technique

Abstract

The chapter presents the automatic control of aircraft during landing, taking into account the sensor errors and the wind shears. Both planes—longitudinal and lateral-directional—are treated; the new obtained automatic landing system (ALS) will consists of two subsystems—the first one controls aircraft motion in longitudinal plane, while the second one is for the control of aircraft motion in lateral-directional plane. These two systems can be treated separately, but in the same time, these can be put together to control all the parameters which interfere in the dynamics of aircraft landing. The two new ALSs are designed by using the H-inf control, the dynamic inversion, optimal observers, and reference models. To validate the new obtained ALS, one uses the dynamics associated to the landing of a Boeing 747, software implements the theoretical results and analyzes the accuracy of the results and the precision standards' achievement with respect to the requirements of the Federal Aviation Administration (FAA).

1. Introduction

Landing is one of the most critical stages of flight; the aircraft has to perform a precise maneuver in the proximity of the ground to land safely at a suitable touch point with acceptable sink rate, speed, and attitude. During aircraft landing, the presence of different unknown or partially known disturbances in aircraft dynamics leads to the necessity of using modern automatic control systems. Sometimes, the conventional controllers are difficult to use due to the drastically changing of the atmospheric conditions and the dynamics of aircraft [1, 2]. In order to control aircraft landing, the feedback linearization has been used in [3], but the drawback of this method is that all the parametric plant uncertainties must appear in the same equation of the state-space representation. Other automatic landing systems (ALSs) use feed-forward neural networks based on the back propagation learning algorithms [4]; the main disadvantage is that the neural networks require a priori training on normal and faulty operating data. From the optimal synthesis' point of view, a mixed technique for the H₂/H_∞ control of landing has been introduced by Shue and Agarwal [5], while Ochi and Kanai [6] have used the H-inf technique for the same purpose; the negative point of these papers is the robustness of the controllers since the sensor errors and other external disturbances are not considered. The fuzzy logic has been also used to imitate the pilot's experience in compromising between trajectory tracking and touchdown safety [7]. In other studies [8], it has been proved that an intelligent on-line-learning controller is helpful in assisting different baseline controllers in tolerating a stuck control surface in the presence of strong wind.

The main drawback of all the papers dealing with aircraft landing is that the designed ALSs are designed either for the longitudinal plane or for the lateral-directional plane. Our work focuses on aircraft automatic control in the two planes, during landing, by using the linearized dynamics of aircraft, the H-inf control, and the dynamic inversion concept, taking into consideration the wind shears, the crosswind, and the errors of the sensors. Our aim is to design a new landing control system (both planes) which cancels the negative effect of wind shears, the crosswind, and the errors of the sensors. According to this work's authors, little progress has been reported for the landing flight control systems (using the H-inf control and the dynamic inversion) handling all the above presented problems.

The three phases of a typical landing procedure are: the initial approach, the glide slope, and the flare [1, 9]. The initial approach involves a descend of the aircraft from the cruise altitude to approximately 420 m (heavy aircraft). Aircraft pitch, attitude, and speed must be controlled during the glide slope path; its speed should be constant during this stage of landing. For a Boeing 747, the pitch should be between −5 and 5 degrees, while the sink rate should be 3 m/s. For the same type of aircraft, when the altitude is 20–30 m above the ground, a flare maneuver should be accomplished and, therefore, the slope angle control system is disengaged; during the flare, aircraft pitch angle is adjusted (between 0 and 5 degrees) for a soft touchdown of the runway. These issues will be achieved by the first system presented in this chapter—the one for the control of aircraft trajectory in the longitudinal plane. The motion of aircraft in lateral plane should be done without errors, this meaning the cancel of aircraft deviation with respect to the runway direction; for this purpose, flight direction automatic control systems are necessary; this issue will be achieved by the second system presented in this chapter—the one for the control of aircraft trajectory in the lateral-directional plane.

2. Design the first subsystem of the ALS (longitudinal plane)

2.1. Aircraft dynamics in longitudinal plane

The linearized dynamics of aircraft in longitudinal plane is described by the state equation [10]: ${\dot{\mathit{x}}}_{long}=A_{long}{\mathit{x}}_{long}+B_{long}{\mathit{u}}_{long}+G_{long}{\tilde{\mathit{u}}}_{long},$ with ${\mathit{x}}_{\mathit{long}}={\left[\begin{array}{ccccc}\hfill u\hfill & \hfill w\hfill & \hfill q\hfill & \hfill \theta \hfill & \hfill H\hfill \end{array}\right]}^T-$ the state vector, ${\mathit{u}}_{\mathit{long}}={\left[\begin{array}{cc}\hfill {\delta }_e\hfill & \hfill {\delta }_T\hfill \end{array}\right]}^T-$ the command vector, while ${\tilde{\mathit{u}}}_{\mathit{long}}={\left[\begin{array}{cc}\hfill V_{vx}\hfill & \hfill V_{vz}\hfill \end{array}\right]}^T$ is the vector of disturbances – V_vx and V_vz (the components of the wind velocity along the longitudinal and vertical axes of the aircraft [11]); in the above equations, u is the longitudinal velocity of aircraft, w − the vertical velocity, q − the pitch angular rate, θ − the pitch angle, H − aircraft altitude, while δ_e and δ_T are the elevator deflection and the engine command, respectively. Because the vertical velocity w is much smaller than u , one can consider the velocity in longitudinal plane to be $V=\sqrt{u^2+w^2}\cong u$ [10]; thus, the nominal value of V is considered to be V₀ ≅ u(0) = u₀ . The equations of the actuators are ${\dot{\delta }}_e=-\frac{1}{T_e}{\delta }_e+\frac{1}{T_e}{\delta }_{ec},{\dot{\delta }}_T=-\frac{1}{T_T}{\delta }_T+\frac{1}{T_t}{\delta }_{Tc};$ δ_ec and δ_Tc are the commands applied to elevator engine, respectively. Considering δ_e and δ_T as new states, x_long and u_long become ${\mathit{x}}_{\mathit{long}}={\left[\begin{array}{ccccccc}\hfill u\hfill & \hfill w\hfill & \hfill q\hfill & \hfill \theta \hfill & \hfill H\hfill & \hfill {\delta }_e\hfill & \hfill {\delta }_T\hfill \end{array}\right]}^T,{\mathit{u}}_{\mathit{long}}={\left[\begin{array}{cc}\hfill {\delta }_{ec}\hfill & \hfill {\delta }_{Tc}\hfill \end{array}\right]}^T\text{,}$ while the new matrices A_long ∈ R^7 × 7, B_long ∈ R^7 × 2, and G_long ∈ R^7 × 2 are [10].

$A_{\mathit{long}}=\left[\begin{array}{ccccccc}\hfill a_{11}\hfill & \hfill a_{12}\hfill & \hfill 0\hfill & \hfill a_{14}\hfill & \hfill 0\hfill & \hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \hfill a_{21}\hfill & \hfill a_{22}\hfill & \hfill a_{23}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill b_{21}\hfill & \hfill b_{22}\hfill \\ \hfill a_{31}\hfill & \hfill a_{32}\hfill & \hfill a_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill b_{31}\hfill & \hfill b_{32}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill a_{52}\hfill & \hfill 0\hfill & \hfill a_{54}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1/T_e\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1/T_T\hfill \end{array}\right],B_{\mathit{long}}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1/T_e\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1/T_T\hfill \end{array}\right],G_{\mathit{long}}=\left[\begin{array}{cc}\hfill g_{11}\hfill & \hfill g_{12}\hfill \\ \hfill g_{21}\hfill & \hfill g_{22}\hfill \\ \hfill g_{31}\hfill & \hfill g_{32}\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{.}$ E1

2.2. Wind shears' model

By using the velocities' spectrum and generator filters having as inputs white noises, one can define the vector ũ_long which corresponds to a stochastic process. In this work, the disturbances are considered to be the wind shears, the equations associated to them being [10, 11]: $V_{vx}=-V_{v{x}_0}sin\left({\omega }_{0}t\right),V_{vz}=-V_{v{z}_0}\left[1-cos\left({\omega }_{0}t\right)\right],{\omega }_0=2\pi /T_0\text{,}$ where T₀ is the flight time period inside the wind shear, while $V_{v{x}_0}$ and $V_{v{z}_0}$ are the maximum absolute values of the wind velocities with respect to aircraft longitudinal and vertical axes, respectively.

In order to calculate the matrix G_long , one replaces ũ_long = 0 in the aircraft dynamics and, after that, one replaces u with (u − V_vx) and w with (w − V_vz) ; the coefficients of the velocities V_vx , V_vz and, thus, the elements of the matrix G_long are obtained as follows: $g_{11}=-a_{11},$ $g_{12}=-a_{12},$ $g_{21}=-a_{21},$ $g_{22}=-a_{22},$ $g_{31}=-a_{31},$ $g_{32}=-a_{32}$ .

2.3. The general form of the control law (longitudinal plane)

One considers the vector ${\mathit{z}}_{\mathit{long}}={\left[\begin{array}{cc}\hfill H\hfill & \hfill u\hfill \end{array}\right]}^T=C_{\mathit{long}}^{\prime }{\mathit{x}}_{\mathit{long}}$ that contains the system-controllable output variables, while the vector ${\overline{\mathit{z}}}_{\mathit{long}}={\left[\begin{array}{cc}\hfill \overline{H}\hfill & \hfill \overline{u}\hfill \end{array}\right]}^T$ contains the reference variables (the imposed values of the flight altitude and velocity). The system output vector is y_long , chosen of the following form: ${\mathit{y}}_{long}={\left[\begin{array}{cccccc}H& \dot{H}& u& \dot{u}& \theta & q\end{array}\right]}^T=C_{long}{\mathit{x}}_{long}.$ Taking into account the differential equations of the states H and u, obtained from aircraft dynamics, by using x_long and u_long the matrices A_long , B_long , G_long , one yields:

$C_{\mathit{long}}^{\prime }={\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^T,C_{\mathit{long}}=\left[\begin{array}{ccccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill a_{52}\hfill & \hfill 0\hfill & \hfill a_{54}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{11}\hfill & \hfill a_{12}\hfill & \hfill 0\hfill & \hfill a_{14}\hfill & \hfill 0\hfill & \hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{.}$ E2

Now, by using ${\overline{\mathit{z}}}_{\mathit{long}}$ and the dynamic inversion principle, ${\overline{\mathit{x}}}_{\mathit{long}}$ and ū_long are calculated with respect to ${\overline{\mathit{z}}}_{\mathit{long}}$ and, after that, the vector ${\overline{\mathit{y}}}_{\mathit{long}}$ is obtained by means of the equations [11]: ${\dot{\overline{\mathit{x}}}}_{long}=A_{\mathit{long}}{\overline{\mathit{x}}}_{\mathit{long}}+B_{\mathit{long}}{\overline{\mathit{u}}}_{\mathit{long}},{\overline{\mathit{z}}}_{\mathit{long}}=C_{\mathit{long}}^{\prime }{\overline{\mathit{x}}}_{\mathit{long}},{\overline{\mathit{y}}}_{\mathit{long}}=C_{\mathit{long}}{\overline{\mathit{x}}}_{\mathit{long}}\text{.}$ The command law is calculated with the formula [10]:

${\mathit{u}}_{\mathit{long}}={\overline{\mathit{u}}}_{\mathit{long}}+{{\mathit{u}}_{\infty }}_{\mathit{long}}\text{,}$ E3

where u_∞long is the optimal command that is calculated by means of the H-inf method [12, 13], while the component ū_long is calculated by using the dynamic inversion.

2.4. Design of the control law's first component (longitudinal plane)

A coordinates’ change is achieved by means of the transformation matrix T ∈ R ^7 × 7 [11]:

$\left[\begin{array}{l}\xi \\ \eta \end{array}\right]=T{x}_{long},x_{long}=T^{-1}\left[\begin{array}{l}\xi \\ \eta \end{array}\right],$ E4

where ξ is a state vector consisting of the controlled variables and their derivatives, i.e. [11]: $\xi ={\left[\begin{array}{ccccccccccccc}{z}_1& {\dot{z}}_1& \cdots & z_1^{\left({\text{r}}_1-1\right)}& z_2& {\dot{z}}_2& \cdots & z_2^{\left({\text{r}}_2-1\right)}& \cdots & z_p& {\dot{z}}_p& \cdots & z_p^{\left({\text{r}}_p-1\right)}\end{array}\right]}^T,$ with $z_i^{\left({\mathrm{r}}_i-1\right)}-$ the (r_i − 1) order derivatives of z_i ; for the aircraft dynamics in longitudinal plane, z₁ = H , z₂ = z_p = u . The state vector η consists of all the state variables not included in the vector ξ; considering n to be the dimension of the square matrix T, one can easily deduce the dimension of the vector η as $n-\mathrm{r}=n-\sum _{i=1}^p{\mathrm{r}}_i\text{,}$ where the values of the relative degrees ${\mathrm{r}}_i,i=\overline{1,2}\text{,}$ are to be deduced later.

For the obtaining of the relative degrees r₁ and r₂ , the equations of $\stackrel{\text{.}}{u}$ and $\stackrel{\text{.}}{w}$ are differentiated until terms containing the two components of the control law (δ_ec , δ_Tc) appears in the expressions of the variables ü and $\stackrel{..}{w}\text{;}$ by time derivation of the variables $\stackrel{\text{.}}{u}$ and $\stackrel{\text{.}}{w}$ (expressed by using aircraft dynamics), it results some terms containing the variables ${\stackrel{.}{\delta }}_{ec}$ and ${\stackrel{.}{\delta }}_{Tc}\text{;}$ these can be expressed by means of equations ${\dot{\delta }}_e=-\frac{1}{T_e}{\delta }_e+\frac{1}{T_e}{\delta }_{ec},{\dot{\delta }}_T=-\frac{1}{T_T}{\delta }_T+\frac{1}{T_T}{\delta }_{Tc};$ one obtains ü and $\ddot{w}$ as functions of δ_ec , δ_Tc , and other states. Thus, the relative degree of the state u is r₂ = 2 . In order to obtain the relative degree of the altitude (H), one derivates the differential equation associated to H, i.e. $\dot{H}=a_{52}w+a_{54}\theta ,$ and obtains $\ddot{H}=a_{52}\dot{w}+a_{54}\dot{\theta }.$ is obtained. Therefore, the relative degree of the altitude is r₁ = 3 . The following equations result:

$\begin{array}{l}\ddot{u}={a^{\prime }}_{11}u+{a^{\prime }}_{12}w+{a^{\prime }}_{13}q+{a^{\prime }}_{14}\theta +{a^{\prime }}_{16}{\delta }_e+{a^{\prime }}_{17}{\delta }_T+\frac{b_{11}}{T_e}{\delta }_{ec}\\ +\frac{b_{12}}{T_T}{\delta }_{Tc}+{g^{\prime }}_{11}{V}_{vx}+{g^{\prime }}_{12}{V}_{vz}+g_{11}{\dot{V}}_{vx}+g_{12}{\dot{V}}_{vz},\\ \stackrel{⃛}{H}={a^{\prime }}_{51}u+{a^{\prime }}_{52}w+{a^{\prime }}_{53}q+{a^{\prime }}_{54}\theta +{a^{\prime }}_{56}{\delta }_e+{a^{\prime }}_{57}{\delta }_T+\frac{a_{52}{b}_{21}}{T_e}{\delta }_{ec}\\ +\frac{a_{52}{b}_{22}}{T_T}{\delta }_{Tc}+{g^{\prime }}_{51}{V}_{vx}+{g^{\prime }}_{52}{V}_{vz}+{g^{″}}_{51}{\dot{V}}_{vx}+{g^{″}}_{52}{\dot{V}}_{vz},\end{array}$E5

with $a_{11}^{\prime }=a_{11}^2+a_{12}{a}_{21},$ $a_{12}^{\prime }=a_{11}{a}_{12}+a_{12}{a}_{22},$ $a_{13}^{\prime }=a_{12}{a}_{23}+a_{14},$ $a_{14}^{\prime }=a_{11}{a}_{14},$ $a_{16}^{\prime }=a_{11}{b}_{11}+a_{12}{b}_{21}-\frac{b_{11}}{T_e}\text{,}$ $a_{17}^{\prime }=a_{11}{b}_{12}+a_{12}{b}_{21}-\frac{b_{12}}{T_T},$ $g_{11}^{\prime }=a_{11}{g}_{11}+a_{12}{g}_{21},$ $g_{12}^{\prime }=a_{11}{g}_{12}+a_{12}{g}_{22},$ $a_{51}^{\prime }=a_{54}{a}_{31}+a_{52}\left(a_{21}{a}_{11}+a_{21}{a}_{12}+a_{23}{a}_{31}\right)\text{,}$ $a_{52}^{\prime }=a_{54}{a}_{32}+a_{52}\left(a_{21}{a}_{12}+a_{22}^2+a_{23}{a}_{32}\right),$ $a_{53}^{\prime }=a_{54}{a}_{33},a_{54}^{\prime }=a_{52}\left(a_{22}{a}_{23}+a_{22}{a}_{33}+a_{21}{a}_{14}\right),a_{56}^{\prime }=a_{54}{b}_{31}+$ $+a_{52}\left(a_{21}{b}_{11}+a_{22}{b}_{21}+a_{23}{b}_{31}-\frac{1}{T_e}{b}_{21}\right),$ $a_{57}^{\prime }=a_{54}{b}_{32}+a_{52}\left(a_{21}{b}_{12}+a_{22}{b}_{22}+a_{23}{b}_{32}-\frac{1}{T_T}{b}_{22}\right),g_{51}^{\prime }=a_{54}{g}_{31}+$ $a_{52}\left(a_{21}{g}_{11}+a_{22}{g}_{21}+a_{23}{g}_{31}\right),$ $g_{52}^{\prime }=a_{54}{g}_{32}+a_{52}\left(a_{21}{g}_{12}+a_{22}{g}_{22}+a_{23}{g}_{33}\right),g_{51}^{″}=a_{52}{g}_{21},g_{52}^{″}=a_{52}{g}_{22}\text{.}$

Thus, the state vectors ξ and η are $\xi ={\left[\begin{array}{ccccc}H& \dot{H}& \ddot{H}& u& \dot{u}\end{array}\right]}^T,\eta ={\left[\begin{array}{cc}\theta & q\end{array}\right]}^T.$ By means of the coordinates' change (4), considering ũ_long = 0 and u_long = ū_long , aircraft dynamics gets the form [11]: $\left[\begin{array}{c}\dot{\xi }\\ \dot{\eta }\end{array}\right]={\widehat{A}}_{long}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]+{\widehat{B}}_{long}{\overline{\mathit{u}}}_{long};{\widehat{A}}_{long}=T{A}_{long}{T}^{-1},{\widehat{B}}_{long}=T{B}_{long}.$ If the matrices Â_long and ${\widehat{B}}_{\mathit{long}}$ are partitioned with respect to the dimensions of the vectors ξ and η , it results [4]: $\left[\begin{array}{c}\dot{\xi }\\ \dot{\eta }\end{array}\right]=\left[\begin{array}{cc}{\widehat{A}}_{lon{g}_{11}}& {\widehat{A}}_{lon{g}_{12}}\\ {\widehat{A}}_{lon{g}_{21}}& {\widehat{A}}_{lon{g}_{22}}\end{array}\right]\left[\begin{array}{c}\xi \\ \eta \end{array}\right]+\left[\begin{array}{c}{\widehat{B}}_{lon{g}_1}\\ {\widehat{B}}_{lon{g}_2}\end{array}\right]{\overline{u}}_{long}.$ Imposing $\xi =\overline{\xi },\dot{\xi }=\dot{\overline{\xi }},$ with $\overline{\xi }={\left[\begin{array}{ccccc}\overline{H}& \overline{\dot{H}}& \overline{\ddot{H}}& \overline{u}& \overline{\dot{u}}\end{array}\right]}^T,$ $\overline{\dot{\xi }}={\left[\begin{array}{ccccc}\overline{\dot{H}}& \overline{\ddot{H}}& \overline{\stackrel{⃛}{H}}& \overline{\dot{u}}& \overline{\ddot{u}}\end{array}\right]}^T,$ the vector ū_long is obtained as: ${\overline{\mathit{u}}}_{\mathit{long}}={\widehat{B}}_{lon{g}_1}^{+}\left(\overline{\stackrel{.}{\xi }}-{\widehat{A}}_{lon{g}_{11}}\overline{\xi }-{\widehat{A}}_{lon{g}_{12}}\eta \right)\text{,}$ with ${\widehat{B}}_{lon{g}_1}^{+}-$ pseudo-inverse of the matrix ${\widehat{B}}_{lon{g}_1}\text{.}$

For the obtaining of the matrix T, the vectors ξ and are replaced in (4) and the following differential equations result: $\dot{H}=a_{52}u+a_{54}\theta ,\ddot{H}=a_{52}\dot{w}+a_{54}\dot{\theta },\dot{\theta }=q,\dot{u}=a_{11}u+a_{12}w+$ $a_{14}\theta +b_{11}{\delta }_e+b_{12}{\delta }_T,\dot{w}=a_{21}u+a_{22}w+a_{23}q+b_{21}{\delta }_e+b_{22}{\delta }_T;$ one yields:

$\left[\begin{array}{c}H\\ \dot{H}\\ \ddot{H}\\ u\\ \dot{u}\\ \theta \\ q\end{array}\right]=\underset{T}{\underbrace{\left[\begin{array}{ccccccc}0& 0& 0& 0& 1& 0& 0\\ 0& a_{52}& 0& a_{54}& 0& 0& 0\\ a_{52}{a}_{21}& a_{52}{a}_{22}& \left(a_{54}+a_{52}{a}_{23}\right)& 0& 0& a_{52}{b}_{21}& a_{52}{b}_{22}\\ 1& 0& 0& 0& 0& 0& 0\\ a_{11}& a_{12}& 0& a_{14}& 0& b_{11}& b_{12}\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\end{array}\right]}}\left[\begin{array}{c}u\\ w\\ q\\ \theta \\ H\\ {\delta }_e\\ {\delta }_T\end{array}\right].$E6

Replacing the vectors $\overline{\xi }={\left[\begin{array}{ccccc}\overline{H}& \overline{\dot{H}}& \overline{\ddot{H}}& \overline{u}& \overline{\dot{u}}\end{array}\right]}^T$ and $\overline{\dot{\xi }}={\left[\begin{array}{ccccc}\overline{\dot{H}}& \overline{\ddot{H}}& \overline{\stackrel{⃛}{H}}& \overline{\dot{u}}& \overline{\ddot{u}}\end{array}\right]}^T$ into equation ${\overline{\mathit{u}}}_{\mathit{long}}={\widehat{B}}_{lon{g}_1}^{+}\left(\overline{\stackrel{.}{\xi }}-{\widehat{A}}_{lon{g}_{11}}\overline{\xi }-{\widehat{A}}_{lon{g}_{12}}\eta \right)\text{,}$ one obtains

${\overline{\mathit{u}}}_{\mathit{long}}={\widehat{B}}_{lon{g}_1}^{+}\left\{{\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \overline{\stackrel{⃛}{H}}\hfill & \hfill 0\hfill & \hfill \overline{\ddot{u}}\hfill \end{array}\right]}^T-{\widehat{A}}_{lon{g}_{11}}^{\prime }\overline{\xi }-{\widehat{A}}_{lon{g}_{12}}\eta \right\}\text{,}$ E7

where ${\widehat{A}}_{lon{g}_{11}}^{\prime }$ is calculated from ${\widehat{A}}_{lon{g}_{11}}$ making the substitutions: ${\widehat{a}}_{12}^{\prime }={\widehat{a}}_{12}-1,$ ${\widehat{a}}_{23}^{\prime }={\widehat{a}}_{23}-1\text{,}$ ${\widehat{a}}_{45}^{\prime }={\widehat{a}}_{45}-1\text{,}$ the other elements of the matrices ${\widehat{A}}_{lon{g}_{11}}$ and ${\widehat{A}}_{lon{g}_{11}}^{\prime }$ being the same; ${\widehat{a}}_{ij},i,j=\overline{1,5}$ are the elements of the matrix ${\widehat{A}}_{lon{g}_{11}}\text{.}$

Replacing (7) in $\stackrel{.}{\eta }={\widehat{A}}_{lon{g}_{21}}\xi +{\widehat{A}}_{lon{g}_{22}}\eta +{\widehat{B}}_{lon{g}_2}{\overline{\mathit{u}}}_{\mathit{long}}\text{,}$ with $\xi =\overline{\xi }\text{,}$ one obtains:

$\stackrel{.}{\eta }={\widehat{A}}_{\eta }\eta +{\widehat{B}}_y\overline{\mathit{Z}}\text{,}$ E8

where ${\widehat{A}}_{\eta }={\widehat{A}}_{lon{g}_{22}}-{\widehat{B}}_{lon{g}_2}{\widehat{B}}_{lon{g}_1}^{+}{\widehat{A}}_{lon{g}_{12}},$ ${\widehat{A}}_{\xi }={\widehat{A}}_{lon{g}_{21}}-{\widehat{B}}_{lon{g}_2}{\widehat{B}}_{lon{g}_1}^{+}{\widehat{A}}_{lon{g}_{11}}^{\prime },$ ${\widehat{B}}_z{\overline{\mathit{z}}}_{\mathit{long}}^{\left(\mathrm{r}\right)}={\widehat{B}}_{lon{g}_2}{\widehat{B}}_{lon{g}_1}^{+}{\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \overline{\stackrel{⃛}{H}}\hfill & \hfill 0\hfill & \hfill \overline{\ddot{u}}\hfill \end{array}\right]}^T\text{,}$ ${\overline{z}}_{long}^{\left(\text{r}\right)}={\left[\begin{array}{cc}{\overline{z}}_1^{\left({\text{r}}_1\right)}& {\overline{z}}_2^{\left({\text{r}}_2\right)}\end{array}\right]}^T={\left[\begin{array}{cc}\overline{\stackrel{⃛}{H}}& \overline{\ddot{u}}\end{array}\right]}^T,$ ${\widehat{B}}_y=\left[\begin{array}{cc}{\widehat{B}}_z& {\widehat{A}}_{\xi }\end{array}\right],$ $\overline{\mathit{Z}}={\left[\begin{array}{cc}{\overline{\mathit{z}}}_{long}^{\left(\text{r}\right)}& \overline{\xi }\end{array}\right]}^T={\left[\begin{array}{ccccccc}\overline{\stackrel{⃛}{H}}& \overline{\ddot{u}}& \overline{H}& \overline{\dot{H}}& \overline{\ddot{H}}& \overline{u}& \overline{\dot{u}}\end{array}\right]}^T.$ If one considers ${\widehat{B}}_{lon{g}_2}{\widehat{B}}^{+}{}_{lon{g}_1}=\left[\begin{array}{l}{\widehat{b}}_{11}{\widehat{b}}_{12}{\widehat{b}}_{13}{\widehat{b}}_{14}{\widehat{b}}_{15}\\ {\widehat{b}}_{21}{\widehat{b}}_{22}{\widehat{b}}_{23}{\widehat{b}}_{24}{\widehat{b}}_{25}\end{array}\right]$ then ${\widehat{B}}_z=\left[\begin{array}{cc}\hfill {\widehat{b}}_{13}\hfill & \hfill {\widehat{b}}_{15}\hfill \\ \hfill {\widehat{b}}_{23}\hfill & \hfill {\widehat{b}}_{25}\hfill \end{array}\right]\text{.}$ Thus, for the calculation of the command vector ū_long , one solves equation (8) and obtains the vector η and then uses the equation (7). From the expression of ${\widehat{B}}_z{\overline{\mathit{z}}}_{\mathit{long}}^{\left(\mathrm{r}\right)}\text{,}$ it results: ${\widehat{B}}_{lon{g}_1}^{+}{\left[\begin{array}{ccccc}0& 0& \overline{\stackrel{⃛}{H}}& 0& \overline{\ddot{u}}\end{array}\right]}^T={\widehat{B}}_{lon{g}_2}^{+}{\widehat{B}}_z{\overline{z}}_{long}^{\left(\text{r}\right)},$ which, replaced in (7), leads to

${\overline{\mathit{u}}}_{\mathit{long}}={\widehat{B}}_u^{-1}\left({\overline{\mathit{z}}}_{\mathit{long}}^{\left(\mathrm{r}\right)}-{\widehat{B}}_{\xi }\overline{\xi }-{\widehat{B}}_{\eta }\eta \right)\text{,}$ E9

with ${\widehat{B}}_u^{-1}={\widehat{B}}_{lon{g}_2}^{+}{\widehat{B}}_z,{\widehat{B}}_{\xi }={\widehat{B}}_u{\widehat{B}}_{lon{g}_1}^{+}{\widehat{A}}_{lon{g}_{11}}^{\prime },{\widehat{B}}_{\eta }={\widehat{B}}_u{\widehat{B}}_{lon{g}_1}^{+}{\widehat{A}}_{lon{g}_{12}}\text{.}$ Therefore, ū_long can be obtained by means of equation (7) or by using equation (9).

2.5. Design of the control law’s second component (longitudinal plane

To calculate the second component of the control law u_long , one combines the aircraft dynamics, the equations associated to z₁ = H and z₂ = u , as well as the equation of the output vector ${\mathit{y}}_{long}$:

$\left[\begin{array}{c}\hfill {\stackrel{.}{\mathit{x}}}_{\mathit{long}}\hfill \\ \hfill z_1\hfill \\ \hfill z_2\hfill \\ \hfill {\mathit{y}}_{\mathit{long}}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill A_{\mathit{long}}\hfill & \hfill B_{\mathit{long}}\hfill & \hfill G_{\mathit{long}}\hfill & \hfill 0_{\left(7\times 6\right)}\hfill \\ \hfill C_{0\mathit{long}}\hfill & \hfill D_{01\mathit{long}}\hfill & \hfill 0_{\left(1\times 2\right)}\hfill & \hfill 0_{\left(1\times 6\right)}\hfill \\ \hfill C_{1\mathit{long}}\hfill & \hfill D_{11\mathit{long}}\hfill & \hfill 0_{\left(1\times 2\right)}\hfill & \hfill 0_{\left(1\times 6\right)}\hfill \\ \hfill C_{\mathit{long}}\hfill & \hfill 0_{\left(6\times 2\right)}\hfill & \hfill 0_{\left(6\times 2\right)}\hfill & \hfill D_{22\mathit{long}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\mathit{x}}_{\mathit{long}}\hfill \\ \hfill {\mathit{u}}_{\mathit{long}}\hfill \\ \hfill {\tilde{\mathit{u}}}_{\mathit{long}}\hfill \\ \hfill {\mathit{e}}_{\mathit{long}}\hfill \end{array}\right]\text{,}$ E10

with $C_{0\mathit{long}}=\left[\begin{array}{ccccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],C_{1\mathit{long}}=\left[\begin{array}{ccccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],D_{01\mathit{long}}=\left[\begin{array}{cc}\hfill c_1\hfill & \hfill 0\hfill \end{array}\right],D_{11\mathit{long}}=\left[\begin{array}{cc}\hfill c_2\hfill & \hfill 0\hfill \end{array}\right]\text{,}$ the matrix C_long has the form (2), while D_22 long = I₆ for the vector containing the sensor errors: $${\mathit{e}}_{long}={\left[\begin{array}{cccccc}{e}_H& e_{\dot{H}}& e_u& e_{\dot{u}}& e_{\text{θ}}& e_q\end{array}\right]}^T.$$ The optimal control law has the form [10, 14]: $u_{\infty }{}_{long}=-K_{\infty }{}_{long}({\widehat{x}}_{{}_{long}}-{\overline{x}}_{{}_{long}}),K_{\infty }{}_{long}=R_1{}^{-1}{B}_{long}{}^T{P}_{\infty },R_1=D_{11long}^T{D}_{11long}$; u _∞long minimizes the cost functional:

. The symmetric and positive defined matrix P_∞ is the stabilizing solution of the Riccati equation [15]:

$A_{\mathit{long}}^T{P}_{\infty }+P_{\infty }{A}_{\mathit{long}}-P_{\infty }\left(B_{\mathit{long}}{R}_1^{-1}{B}_{\mathit{long}}^T-{\mu }_1^{-2}{G}_{\mathit{long}}{G}_{\mathit{long}}^T\right)P_{\infty }+Q_1=0\text{.}$ E11

Q₁ and R₁ are positive defined matrices, while μ₁ is a small enough positive scalar such that the Riccati equation (11) has a stabilizing solution [10]. To solve the H-inf control problem means to calculate the controller gain matrix (K_∞ long) ; the system has control inputs as well as disturbances; the control input (u_long) is the controller's output, this becoming the actuators' input. The disturbances of the system (ũ_long and e_long) are the exogenous inputs [10, 12, 13]. The H-inf control problem means to find a controller for the generalized plant such that the infinity norm of the transfer function relating exogenous inputs to performance outputs is minimum. The controller gain matrix (K_∞ long) has the general form $K_{\infty \mathit{long}}=R_1^{-1}{B}_{\mathit{long}}^T{P}_{\infty }$ [10, 12, 13], the optimal control law $u_{\infty long}$ depending on $\Delta {\widehat{\mathit{x}}}_{\mathit{long}}={\widehat{\mathit{x}}}_{\mathit{long}}-{\overline{\mathit{x}}}_{\mathit{long}}\text{;}$ to estimate this signal and the state ${\widehat{\mathit{x}}}_{\mathit{long}}\text{,}$ one borrowed the observer presented in [14], i.e.: $\Delta {\stackrel{.}{\widehat{\mathit{x}}}}_{\mathit{long}}=A_{\mathit{long}}\Delta {\widehat{\mathit{x}}}_{\mathit{long}}+B_{\mathit{long}}{\mathit{u}}_{\mathit{long}}+$ $+G_{\mathit{long}}{\tilde{\mathit{u}}}_{\mathit{long}}+L_{\infty \mathit{long}}\left(\Delta {\mathit{y}}_{\mathit{long}}-C_{\mathit{long}}\Delta {\widehat{\mathit{x}}}_{\mathit{long}}\right)\text{.}$ The observer gain matrix L_∞ long is calculated with the formula: $L_{\infty \mathit{long}}=P_{\infty }^{*}{C}_{\mathit{long}}^T{\left(D_{22\mathit{long}}^T{D}_{22\mathit{long}}\right)}^{-1}\text{,}$ with $P_{\infty }^{*}-$ the stabilizing solution of the Riccati equation [15]:

$A_{\mathit{long}}{P}_{\infty }^{*}+P_{\infty }^{*}{A}_{\mathit{long}}^T-P_{\infty }^{*}\left(C_{\mathit{long}}^T{C}_{\mathit{long}}-{\mu }_2^{-2}{Q}_1\right)P_{\infty }^{*}+G_{\mathit{long}}{G}_{\mathit{long}}^T=0\text{;}$ E12

μ₂ is a small positive scalar for which the Riccati equation (12) has a stabilizing solution.

3. Design the second subsystem of the ALS (lateral-directional plane)

3.1. Aircraft dynamics in lateral-directional plane

Before the start of the landing, two main stages in longitudinal plane (glide slope and flare), the pilot must cancel the aircraft lateral deviation with respect to the runway. This can be achieved by means of the second subsystem of the ALS designed in this chapter or by using other control systems for the flight direction control with radio navigation subsystem and equipment for the measurement of the distance between the aircraft and the radio markers. The linear model of the aircraft motion, in lateral-directional plane, can be described again by the state equation: ${\stackrel{.}{\mathit{x}}}_{lat}=A_{lat}{\mathit{x}}_{lat}+B_{lat}{\mathit{u}}_{lat}+G_{lat}{\tilde{\mathit{u}}}_{lat}\text{,}$ where x_lat = [β p r ϕ ψ Y δ_a δ_r]^T, ${\mathit{u}}_{lat}={\left[\begin{array}{cc}\hfill {\delta }_{a_c}\hfill & \hfill {\delta }_{r_c}\hfill \end{array}\right]}^T,{\tilde{\mathit{u}}}_{lat}=V_{vy}\text{,}$ with β − aircraft sideslip angle, ϕ and ψ are the roll angle and the yaw angle, respectively, p − aircraft roll angular rate, r − the aircraft yaw angular rate, Y − aircraft lateral error, δ_a and δ_r − the ailerons and rudder’s deflection angles, ${\delta }_{a_c}$ and ${\delta }_{r_c}-$ the roll and yaw commands (commands applied to the actuators), V_vy − the wind component having as direction—the aircraft the lateral axis, T_a and T_r − the effectors' time delay constants of the ailerons and rudder, respectively. The matrices A_lat, B_lat, and G_lat are [1]:

$A_{lat}=\left[\begin{array}{cccccccc}\hfill a_{11}\hfill & \hfill a_{12}\hfill & \hfill a_{13}\hfill & \hfill a_{14}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \hfill a_{21}\hfill & \hfill a_{22}\hfill & \hfill a_{23}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill b_{21}\hfill & \hfill b_{22}\hfill \\ \hfill a_{31}\hfill & \hfill a_{32}\hfill & \hfill a_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill b_{31}\hfill & \hfill b_{32}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -V_0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill V_0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{T_a}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{1}{T_r}\hfill \end{array}\right],B_{lat}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{1}{T_a}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{T_r}\hfill \end{array}\right],G_{lat}=\left[\begin{array}{c}\hfill \frac{a_{11}}{V_0}\hfill \\ \hfill \frac{a_{21}}{V_0}\hfill \\ \hfill \frac{a_{31}}{V_0}\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]\text{;}$E13

For the design of the second optimal subsystem of the ALS, let us consider the vector ${\mathit{z}}_{lat}={\left[\begin{array}{cc}\hfill Y\hfill & \hfill \beta \hfill \end{array}\right]}^T=C_{lat}^{\prime }{\mathit{x}}_{lat}$ —the vector of the system's controllable output variables and the vector ${\overline{\mathit{z}}}_{lat}={\left[\begin{array}{cc}\hfill \overline{Y}\hfill & \hfill \overline{\beta }\hfill \end{array}\right]}^T$ —the reference variables' vector, i.e. the desired values for the lateral deviation and the sideslip angle of the aircraft. The system's output vector is y_lat = $${\left[\begin{array}{ccccccc}Y& \dot{Y}& \beta & \phi & p& \psi & r\end{array}\right]}^T=C_{lat}{x}_{lat};$$ sensors' errors have been not taken into account here; knowing the forms of z_lat and y_lat , the matrices C_lat and $C_{lat}^{\mathit{\prime }}$ can be easily deduced.

3.2. The general form of the control law (lateral-directional plane)

The command law is similar to the one for longitudinal plane; it is calculated with the formula:

${\mathit{u}}_{lat}={\overline{\mathit{u}}}_{lat}+{{\mathit{u}}_{\infty }}_{lat}\text{,}$ E14

where u_∞lat is the optimal command calculated by means of the H-inf method, while the component ū_lat is calculated by using the dynamic inversion [3, 12, 13]. For the design of the signal ū_lat there are used the dynamic inversion principle and the vectors ${\overline{\mathit{z}}}_{lat}=C_{lat}^{\prime }{\overline{\mathit{x}}}_{lat},{\overline{\mathit{y}}}_{lat}=C_{lat}{\overline{\mathit{x}}}_{lat}\text{,}$ where ${\overline{\mathit{x}}}_{lat}$ is determined from the equation ${\stackrel{.}{\overline{\mathit{x}}}}_{lat}=A{\overline{\mathit{x}}}_{lat}+B_{lat}{\overline{\mathit{u}}}_{lat}\text{.}$

3.3. Design of the control law's first component (lateral-directional plane)

First, one obtains the relative degrees of the variables z₁ = Y and z₂ = β ; these relative degrees are denoted here with r₁ and r₂ , respectively. One derivates with respect to time the equations associated to Y and β (r₁ times and r₂ times, respectively) until the components of the command vector u_lat , i.e. ${\delta }_{a_c}$ and ${\delta }_{r_c}\text{,}$ are obtained; the following equations have resulted:

$\begin{array}{l}\stackrel{⃛}{Y}=a_{61}^{\prime }\beta +a_{62}^{\prime }p+a_{63}^{\prime }r+a_{64}^{\prime }\phi +a_{67}^{\prime }{\delta }_a+a_{68}^{\prime }{\delta }_r-\frac{V_0{b}_{11}}{T_a}{\delta }_{a_c}-\frac{V_0{b}_{12}}{T_r}{\delta }_{r_c}+\left(a_{31}-a_{11}^{\prime }\right)V_{vy}-a_{11}{\stackrel{.}{V}}_{vy},\\ \ddot{\beta }=a_{11}^{\prime }\beta +a_{12}^{\prime }p+a_{13}^{\prime }r+a_{14}^{\prime }\phi +a_{17}^{\prime }{\delta }_a+a_{18}^{\prime }{\delta }_r+\frac{b_{11}}{T_a}{\delta }_{a_c}+\frac{b_{12}}{T_r}{\delta }_{r_c}+\frac{a_{11}^{\prime }}{V_0}{V}_{vy}+\frac{a_{11}}{V_0}{\stackrel{.}{V}}_{vy}\text{,}\end{array}$ E15

where ${a^{\prime }}_{11}=a_{11}^2+a_{12}{a}_{21}+a_{13}{a}_{31},$ ${a^{\prime }}_{12}=a_{11}{a}_{12}+a_{12}{a}_{22}+a_{13}{a}_{32}+a_{14},$ ${a^{\prime }}_{13}=a_{11}{a}_{13}+a_{12}{a}_{23}+a_{13}{a}_{33},{a^{\prime }}_{14}=a_{11}{a}_{14},$ ${a^{\prime }}_{17}=a_{11}{b}_{11}+a_{12}{b}_{21}+a_{13}{b}_{31}-b_{11}/T_a,$ ${a^{\prime }}_{18}=a_{11}{b}_{12}+a_{12}{b}_{22}+a_{13}{b}_{32}-b_{12}/T_r,$ ${a^{\prime }}_{61}=V_0\left(a_{31}-{a^{\prime }}_{11}\right),$ ${a^{\prime }}_{62}=V_0\left(a_{32}-{a^{\prime }}_{12}\right),{a^{\prime }}_{63}=V_0\left(a_{33}-{a^{\prime }}_{13}\right),$ ${a^{\prime }}_{64}=-V_0{a^{\prime }}_{14},{a^{\prime }}_{67}=V_0(b_{31}-{a^{\prime }}_{17}),{a^{\prime }}_{68}=V_0(b_{32}-{a^{\prime }}_{18}),{a^{\prime }}_{31}=a_{31}-{a^{\prime }}_{11}.$

Thus, according to equations (15), the relative degrees are r₁ = 3 and r₂ = 2 . The equations (15) may be combined in the equation of the vector ${\mathit{z}}_{lat}^{\left(\text{r}\right)}={\left[\begin{array}{cc}\stackrel{⃛}{Y}& \ddot{\beta }\end{array}\right]}^T,$ i.e.: ${\mathit{z}}_{lat}^{\left(\mathrm{r}\right)}=A_x{\mathit{x}}_{lat}+B_u{\overline{\mathit{u}}}_{lat}+G_{lat}^{\prime }{\tilde{\mathit{u}}}_{lat}\text{,}$ where ${\overline{\mathit{u}}}_{lat}={\left[\begin{array}{cc}\hfill {\overline{\delta }}_{a_c}\hfill & \hfill {\overline{\delta }}_{r_c}\hfill \end{array}\right]}^T,$ ${\tilde{\mathit{u}}}_{lat}={\left[\begin{array}{cc}\hfill V_{vy}\hfill & \hfill {\stackrel{.}{V}}_{vy}\hfill \end{array}\right]}^T,$ $A_x=\left[\begin{array}{cccccccc}\hfill a_{61}^{\prime }\hfill & \hfill a_{62}^{\prime }\hfill & \hfill a_{63}^{\prime }\hfill & \hfill a_{64}^{\prime }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill a_{67}^{\prime }\hfill & \hfill a_{68}^{\prime }\hfill \\ \hfill a_{11}^{\prime }\hfill & \hfill a_{12}^{\prime }\hfill & \hfill a_{13}^{\prime }\hfill & \hfill a_{14}^{\prime }\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill a_{17}^{\prime }\hfill & \hfill a_{18}^{\prime }\hfill \end{array}\right],$ $B_u=\left[\begin{array}{cc}\hfill -\frac{V_0{b}_{11}}{T_a}\hfill & \hfill -\frac{V_0{b}_{12}}{T_r}\hfill \\ \hfill \frac{b_{11}}{T_a}\hfill & \hfill \frac{b_{12}}{T_r}\hfill \end{array}\right]\text{,}$ $G_{lat}^{\prime }=\left[\begin{array}{cc}\hfill a_{31}^{\prime }\hfill & \hfill -a_{11}\hfill \\ \hfill -\frac{a_{11}^{\prime }}{V_0}\hfill & \hfill \frac{a_{11}}{V_0}\hfill \end{array}\right]\text{.}$ The form of the control law ū_lat results from ${\mathit{z}}_{lat}^{\left(\mathrm{r}\right)}=A_x{\mathit{x}}_{lat}+B_u{\overline{\mathit{u}}}_{lat}+G_{lat}^{\prime }{\tilde{\mathit{u}}}_{lat}\text{,}$ if one imposes the convergence of $z_{lat}^{\left(\text{r}\right)}={\left[\begin{array}{cc}\stackrel{⃛}{Y}& \ddot{\beta }\end{array}\right]}^T$ to ${\overline{z}}_{lat}^{\left(\text{r}\right)}={\left[\begin{array}{cc}\overline{\stackrel{⃛}{Y}}& \overline{\ddot{\beta }}\end{array}\right]}^T$ and the convergence of the system estimated state $\left({\widehat{\mathit{x}}}_{lat}\right)$ to x_lat ; in these conditions, one gets [1]:

${\overline{\mathit{u}}}_{lat}=B_u^{-1}\left({\overline{\mathit{z}}}_{lat}^{\left(\mathrm{r}\right)}-A_x{\widehat{\mathit{x}}}_{lat}-G_{lat}^{\prime }{\tilde{\mathit{u}}}_{lat}\right)\text{.}$ E16

3.4. Design of the control law's second component (lateral-directional plane)

To obtain the second component of the command law u_lat , the H-inf control is used; the state equation associated to aircraft dynamics in lateral-directional plane, the equations associated to z₁ = Y and z₂ = β , as well as the equation of the output vector y_lat , may be combined into the following equation:

$\left[\begin{array}{c}\hfill {\stackrel{.}{\mathit{x}}}_{lat}\hfill \\ \hfill z_1\hfill \\ \hfill z_2\hfill \\ \hfill {\mathit{y}}_{lat}\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill A_{lat}\hfill & \hfill B_{lat}\hfill & \hfill G_{lat}\hfill & \hfill 0_{\left(8\times 7\right)}\hfill \\ \hfill C_{0lat}\hfill & \hfill D_{01lat}\hfill & \hfill 0_{\left(1\times 1\right)}\hfill & \hfill 0_{\left(1\times 7\right)}\hfill \\ \hfill C_{1lat}\hfill & \hfill D_{11lat}\hfill & \hfill 0_{\left(1\times 1\right)}\hfill & \hfill 0_{\left(1\times 7\right)}\hfill \\ \hfill C_{lat}\hfill & \hfill 0_{\left(7\times 2\right)}\hfill & \hfill 0_{\left(7\times 1\right)}\hfill & \hfill D_{22lat}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\mathit{x}}_{lat}\hfill \\ \hfill {\mathit{u}}_{lat}\hfill \\ \hfill {\tilde{\mathit{u}}}_{lat}\hfill \\ \hfill {\mathit{e}}_{lat}\hfill \end{array}\right]\text{;}$ E17

the matrices A_lat , B_lat , G_lat have the forms (13) and $C_{0lat}=\left[\begin{array}{cccccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{,}$ $C_{1lat}=\left[\begin{array}{cccccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],D_{01lat}=\left[\begin{array}{cc}\hfill c_1\hfill & \hfill 0\hfill \end{array}\right],D_{11lat}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill c_2\hfill \end{array}\right]\text{;}$ the matrix D_22lat has the form D_22 lat = I₇ for the vector containing the sensor errors: ${\mathit{e}}_{lat}={\left[\begin{array}{ccccccc}\hfill e_Y\hfill & \hfill e_{\stackrel{.}{Y}}\hfill & \hfill e_{\mathrm{\beta }}\hfill & \hfill e_{\phi }\hfill & \hfill e_p\hfill & \hfill e_{\mathrm{\psi }}\hfill & \hfill e_r\hfill \end{array}\right]}^T$. It is known that the sensors (used to measure some important variables) have sometimes errors; for example, the most important errors of a gyro sensor are [1]: 1) the bias; 2) the scale factor; 3) the calibration error of the scale factor; 4) the noise of the sensor; 5) the sensibility to an acceleration applied along an arbitrary direction. The bias and the noise are the most severe for the control performance during landing. Usually, on aircraft, there are used gyros to measure the angular rates (e.g. p and r); by integration of these angular rates, one obtains the roll and yaw angles. Because on aircraft there are also transducers (sensors) for the attack angle and for the sideslip angle (β), in this chapter, one considered sensor errors for β, p, and r. Similar remarks can be done for the automatic landing subsystem in longitudinal plane. In the software validation of the two automatic landing subsystems, the authors will use some information from [1], but the values of the sensors' errors will be increased to analyze the robustness of the two ALS subsystems. Also, it is interesting to proof that, for the steady regime, the forms of z₁ = Y and z₂ = β are the same with the expressions in (17); the expansion of ${\mathit{z}}_{lat}={\left[\begin{array}{cc}\hfill z_1\hfill & \hfill z_2\hfill \end{array}\right]}^T$ as function of state (x_lat) and of the system command vector (u_lat) leads to the equations:

$+\left[\begin{array}{c}\hfill D_{01lat}\hfill \\ \hfill D_{11lat}\hfill \end{array}\right]\Delta {\mathit{u}}_{lat}\iff {\mathit{z}}_{lat}\cong \left[\begin{array}{c}\hfill C_{0lat}\hfill \\ \hfill C_{1lat}\hfill \end{array}\right]{\mathit{x}}_{lat}+\left[\begin{array}{c}\hfill D_{01lat}\hfill \\ \hfill D_{11lat}\hfill \end{array}\right]{\mathit{u}}_{lat}\cong {\overline{C}}_{lat}{\mathit{x}}_{lat}+{\overline{D}}_{lat}{\mathit{u}}_{lat}\text{,}$ with
$C_{0lat}={\left[\begin{array}{ccc}\hfill \frac{\partial z_1}{\partial x_1}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial z_1}{\partial x_n}\hfill \end{array}\right]}_{\left({\mathit{x}}_{la{t}_0},0\right)}\text{,}$ $C_{1lat}={\left[\begin{array}{ccc}\hfill \frac{\partial z_2}{\partial x_1}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial z_2}{\partial x_n}\hfill \end{array}\right]}_{\left({\mathit{x}}_{la{t}_0},0\right)};x_i\left(i=\overline{1,n}\right)$
are the system’s states (n = 8), $D_{01lat}={\left[\begin{array}{cc}\hfill \frac{\partial Y}{\partial {\delta }_{a_c}}\hfill & \hfill \frac{\partial Y}{\partial {\delta }_{r_c}}\hfill \end{array}\right]}_{\left({\mathit{x}}_{la{t}_0},0\right)}=$ $=\left[\begin{array}{cc}\hfill c_1\hfill & \hfill 0\hfill \end{array}\right],D_{11lat}={\left[\begin{array}{cc}\hfill \frac{\partial z_2}{\partial u_1}\hfill & \hfill \frac{\partial z_2}{\partial u_2}\hfill \end{array}\right]}_{\left({\mathit{x}}_{la{t}_0},0\right)}={\left[\begin{array}{cc}\hfill \frac{\partial \beta }{\partial {\delta }_{a_c}}\hfill & \hfill \frac{\partial \beta }{\partial {\delta }_{r_c}}\hfill \end{array}\right]}_{\left({\mathit{x}}_{la{t}_0},0\right)}=\left[\begin{array}{cc}\hfill 0\hfill & \hfill c_2\hfill \end{array}\right],{\overline{C}}_{lat}=\left[\begin{array}{c}\hfill C_{0lat}\hfill \\ \hfill C_{1lat}\hfill \end{array}\right]\text{,}$ ${\overline{D}}_{lat}=\left[\begin{array}{c}\hfill D_{01lat}\hfill \\ \hfill D_{11lat}\hfill \end{array}\right]$. c₁ and c₂ have small positive values; in steady regime (u_lat = 0) , one gets z₁ = Y and z₂ = β .

The optimal control law in lateral-directional plane has the form [14]: ${\mathit{u}}_{\infty lat}=-K_{\infty lat}\left({\widehat{\mathit{x}}}_{lat}-{\overline{\mathit{x}}}_{lat}\right)\text{,}$ $K_{\infty lat}=R_1^{-1}{B_{lat}}^T{P}_{\infty },R_1={{\overline{D}}_{lat}}^T{\overline{D}}_{lat}\text{;}$ u_∞ lat must minimize the cost functional $J_{lat}=\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{{\mathit{z}}_{lat}}^T{\mathit{z}}_{lat}\mathrm{d}t}=$

. The symmetric and positive defined matrix P_∞ is the stabilizing solution of the Riccati matriceal equation [1]:

$A_{lat}^T{P}_{\infty }+P_{\infty }{A}_{lat}-P_{\infty }\left(B_{lat}{R}_1^{-1}{B}_{lat}^T-{\mu }_1^{-2}{G}_{lat}{G}_{lat}^T\right)P_{\infty }+Q_1=0\text{.}$ E18

The remarks regarding Q₁ , R₁ , and μ₁ remain the same; the controller gain matrix (K_∞ lat) has the general form $K_{\infty lat}=R_1^{-1}{B}_{lat}^T{P}_{\infty }$ which is typical for the optimal control theory. The optimal control law u_∞ lat depends on $\Delta {\widehat{\mathit{x}}}_{lat}={\widehat{\mathit{x}}}_{lat}-{\overline{\mathit{x}}}_{lat}\text{,}$ as one can see above. To obtain this signal, the observer presented above is again used; this time, one obtains the estimated state vector $\left({\widehat{\mathit{x}}}_{lat}\right)$ and the signal $\Delta {\widehat{\mathit{x}}}_{lat}={\widehat{\mathit{x}}}_{lat}-{\overline{\mathit{x}}}_{lat}\text{;}$ the equation of the observer is

$\Delta {\stackrel{.}{\widehat{\mathit{x}}}}_{lat}=A_{lat}\Delta {\widehat{\mathit{x}}}_{lat}+B_{lat}{\mathit{u}}_{lat}+G_{lat}{\tilde{\mathit{u}}}_{lat}+L_{\infty lat}\left(\Delta y_{lat}-C_{lat}\Delta {\widehat{\mathit{x}}}_{lat}\right)\text{.}$ E19

The observer gain matrix L_∞lat is calculated by using the formula: $L_{\infty lat}=P_{\infty }^{*}{C}_{lat}^T{\left(D_{22lat}^T{D}_{22lat}\right)}^{-1}\text{,}$ with $P_{\infty }^{*}-$ the stabilizing solution of the Riccati matriceal equation [15]:

$A_{lat}{P}_{\infty }^{*}+P_{\infty }^{*}{A}_{lat}^T-P_{\infty }^{*}\left(C_{lat}^T{C}_{lat}-{\mu }_2^{-2}{\overline{C}}_{lat}^T{\overline{C}}_{lat}\right)P_{\infty }^{*}+G_{lat}{G}_{lat}^T=0\text{;}$ E20

μ₂ is a small positive scalar for which the Riccati equation (20) has a stabilizing solution; one used the same notations for the matrices Q₁ , R₁ , P_∞ , and $P_{\infty }^{*}$ but their values are completely different from the ones in the case of aircraft motion in longitudinal plane.

4. Structure of the complete automatic landing system

To control all the variables in longitudinal plane, one also uses two reference models (Figure 1a) providing the desired altitude, velocity on the landing curve, and their derivatives up to relative degrees of the system [16]. Aircraft desired state $\left({\overline{\mathit{x}}}_{\mathit{long}}\right)$ and the desired output vector $\left({\overline{\mathit{y}}}_{\mathit{long}}\right)$ are obtained by using the states of the reference models. The optimal control is calculated on-line by means of the error $\Delta x_{long}={\widehat{x}}_{long}-{\overline{x}}_{long}$ .

Figure 1.

Block diagrams of the reference models.

4.1. Block diagrams of the reference models

The two reference models (the former being a three order reference model, while the latter is a second order reference model) are also used for the calculation of the vector $\overline{\mathit{Z}}\text{.}$ The two reference models receive information from a block which models the geometry of landing in longitudinal plane; this block uses two equations—one for the glide slope phase and one for the flare phase. The equation associated to the glide slope phase ($H\ge {\overline{H}}_0\text{,}$ ${\overline{H}}_0-$ the altitude at which the glide slope phase ends and the second landing phase begins) is [10]:

$\overline{H}=\left(X-X_{p_0}\right)tan\left({\gamma }_c\right)\text{,}$ E21

where X is the covered distance, horizontally measured, $X_{p_0}-$ describes the point where the horizontal axis intersects the glide slope, $\overline{H}-$ the desired altitude, while γ_c is the aircraft desired slope angle. The seven components of the imposed vector $\overline{\mathit{Z}}$ are: $\overline{H},\overline{\stackrel{.}{H}}=u\cdot {\gamma }_c,$ $\overline{\ddot{H}}=0,\overline{\stackrel{⃛}{H}}=0,\overline{u}=u_0\cong V_0,\overline{\dot{u}}=0,$ and $\overline{\ddot{u}}=0$. The equation associated to the flare phase $\left(H<{\overline{H}}_0\right)$ is [10]:

$\overline{H}={\overline{H}}_{0}exp\left(-t/\tau \right)\text{,}$ E22

with τ − the time constant that defines the exponential curvature (the flare phase); as a consequence, $\overline{\dot{H}}=-\frac{1}{\tau }\overline{H},\overline{\ddot{H}}=\frac{1}{{\tau }^2}\overline{H},\overline{\stackrel{⃛}{H}}=-\frac{1}{{\tau }^3}\overline{H},\overline{u}=u_0\cong V_0,\overline{\dot{u}}=0,\overline{\ddot{u}}=0.$ . If the reference models in Figure 1a are used, the roles of the variables $\overline{H}$ and ū from (21) and (22) are played by the variables H_c and u_c , respectively; the variables $\overline{\stackrel{⃛}{H}},\overline{\ddot{H}},\overline{\dot{H}},\overline{H},\overline{\ddot{u}},\overline{\dot{u}},$ and ū are provided by the reference models.

Similar approach is used for aircraft motion in lateral-directional plane; the vectors $\overline{\mathit{z}}$ and ${\mathit{z}}_{lat}^{\left(\mathrm{r}\right)}$ are calculated by means of other two reference models, the former being a three-order reference model (associated to Y), while the latter is a second-order reference model (associated to β) (Figure 1b) [1].

4.2. The block diagram of the new automatic landing system

The structure of the new ALS, using dynamic inversion and H-inf method, is presented in Figure 2; it consists of two subsystems—the first one controls aircraft motion in longitudinal plane, while the second one is for the control of aircraft motion in lateral-directional plane.

In longitudinal plane, to track the desired trajectory, one must control the aircraft speed (u) and its altitude (H). The landing requirements involve the aircraft descend to an altitude of 420 m above the ground in the same time with the reduction of its speed from the cruise value to the landing value (70 m/s); then, the speed should be maintained constant. In lateral-directional plane, the desired landing trajectory of aircraft mainly involves two variables' control: the aircraft lateral deviation with respect to the runway (Y) and the sideslip angle (β) .

In longitudinal plane, the dynamic inversion and H-inf method must assure the convergences: $\Delta {\mathit{y}}_{\mathit{long}}\to 0\left({\mathit{y}}_{\mathit{long}}=C_{\mathit{long}}{\mathit{x}}_{\mathit{long}}\to {\overline{\mathit{y}}}_{\mathit{long}}=C_{\mathit{long}}{\overline{\mathit{x}}}_{\mathit{long}},\right)$ ${\mathit{x}}_{\mathit{long}}\to {\overline{\mathit{x}}}_{\mathit{long}},\Delta {\mathit{z}}_{\mathit{long}}\to 0\left(\mathit{z}=C_{\mathit{long}}^{\prime }\mathit{x}\to \overline{\mathit{z}}=C_{\mathit{long}}^{\prime }\overline{\mathit{x}}\right),$ $\Delta {\widehat{\mathit{x}}}_{\mathit{long}}\to 0$ $\left({\widehat{\mathit{x}}}_{\mathit{long}}\to {\mathit{x}}_{\mathit{long}}\to {\overline{\mathit{x}}}_{\mathit{long}}\right),$ ${\mathit{u}}_{\infty \mathit{long}}\to 0,$ ${\overline{\mathit{u}}}_{\mathit{long}}\to 0,{\mathit{u}}_{\mathit{long}}\to 0\text{;}$ in lateral-directional plane, the following convergences should be assured: $\Delta {\mathit{y}}_{lat}\to 0\left({\mathit{y}}_{lat}=C_{lat}{\mathit{x}}_{lat}\to {\overline{\mathit{y}}}_{lat}=C_{lat}{\overline{\mathit{x}}}_{lat},{\mathit{x}}_{lat}\to {\overline{\mathit{x}}}_{lat}\right),$ $\Delta {\mathit{z}}_{lat}\to 0$ $\left(\mathit{z}=C_{lat}^{\prime }\mathit{x}\to \overline{\mathit{z}}=C_{lat}^{\prime }\overline{\mathit{x}}\right),\Delta {\widehat{\mathit{x}}}_{lat}\to 0\left({\widehat{\mathit{x}}}_{lat}\to {\mathit{x}}_{lat}\to {\overline{\mathit{x}}}_{lat}\right),{\mathit{u}}_{\infty lat}\to 0,{\overline{\mathit{u}}}_{lat}\to 0,{\mathit{u}}_{lat}\to 0\text{.}$

Figure 2.

Structure of the new architecture for aircraft automatic control during landing.

5. Numerical simulation results

5.1. Numerical simulation setup

In order to analyze the behaviour and the performances of the designed ALS, one considers a numerical example associated to the flight of a Boeing 747. The Matlab/Simulink environment is used for complex simulations; to obtain the time histories of the main variables describing the aircraft motion in longitudinal and lateral-directional planes, one software implemented the two optimal observers, the four reference models, and the two H-inf controllers.

For aircraft dynamics in lateral-directional plane, the values of the coefficients are [11]: a₁₁ = − 0.0013 , a₁₂ = 0 , a₁₃ = − 1 , a₁₄ = 0.15 , a₂₁ = − 1.33 , a₂₂ = − 0.98 , a₂₃ = 0.33 , a₃₁ = 0.17 , a₃₂ = − 0.17 , a₃₃ = − 0.217 , b₁₁ = 0.001 , b₁₂ = 0.015 , b₂₁ = 0.23 , b₂₂ = 0.06 , b₃₁ = 0.026 , b₃₂ = − 0.15 , V₀ = 67 m/s , T_a = 0.7 s , $T_r=0.1\mathrm{s},{\mu }_1=1,{\mu }_2=1,c_1=c_2=0.01,\overline{Y}=0\mathrm{m},\overline{\beta }=0deg\text{;}$ the vector of sensor errors has been chosen as: e_lat = [0 m 0 m/s 1 deg 0 deg 1 deg/s 0 deg 1 deg/s]  ^T , the matrix G_lat has been obtained by means of the equation (13), while the system's initial state is ${\mathit{x}}_{lat}\left(0\right)={\left[\begin{array}{cccccccc}\hfill 0.1deg\hfill & \hfill 0deg/\mathrm{s}\hfill & \hfill -2deg/\mathrm{s}\hfill & \hfill 0deg\hfill & \hfill 0.1deg\hfill & \hfill 25\mathrm{m}\hfill & \hfill 0deg\hfill & \hfill 0\hfill \end{array}deg\right]}^T\text{;}$ for the reference models, one has chosen: p₂ = 25 , ξ₃ = ξ₄ = 0.7 , ω₃ = ω₄ = 2 rad/s . To test the robustness of the new ALS (lateral-directional plane) with respect to the crosswind (lateral wind—V_vy), in simulations different values for ${\tilde{\mathit{u}}}_{lat}=V_{vy}$ between 2 and 10 m/s are considered. The values considered here for the sensors' errors (both planes—longitudinal and lateral-directional) are chosen very large because it is important to use strong disturbances instead of small ones when designing a robust ALS.

For aircraft dynamics in longitudinal plane, the values of the coefficients for Boeing 747 have been borrowed from [11]: a₁₁ = − 0.021 , a₁₂ = 0.122 , a₁₄ = − 0.322 , a₂₁ = − 0.209 , a₂₂ = − 0.53 , a₂₃ = 2.21 , a₃₁ = 0.017 , a₃₂ = − 0.164 , a₃₃ = − 0.412 , a₅₂ = − 1 , a₅₄ = V₀ = 70 m/s , b₁₁ = 0.01 , b₁₂ = 1 , b₂₁ = − 0.064 , b₂₂ = − 0.044 , b₃₁ = − 0.378 , b₃₂ = 0.544 , T_e = 0.3 s , T_T = 2 s , ū = V₀ , V_vx0 = 1 m/s , V_vz0 = 1 m/s , T₀ = 30 s . The vector of sensor errors is ${\mathit{e}}_{\mathit{long}}={\left[\begin{array}{cccccc}\hfill 0.2\mathrm{m}\hfill & \hfill 0.2deg/\mathrm{s}\hfill & \hfill 0.2\mathrm{m}\hfill & \hfill 0\mathrm{m}/{\mathrm{s}}^2\hfill & \hfill 0.2deg\hfill & \hfill 0.1deg/\mathrm{s}\hfill \end{array}\right]}^T\text{,}$ while, for the reference models, one has chosen: p₁ = 25 , ξ₁ = ξ₂ = 0.7 , ω₁ = ω₂ = 2 rad/s . For the first landing phase in longitudinal plane, the following values have been used: ${\overline{H}}_p=\overline{H}\left(0\right)=420\mathrm{m},X\left(0\right)=0\text{,}$ $X_{p_0}=-{\overline{H}}_p/tan\left({\gamma }_c\right),{\gamma }_c=-2.5deg\text{.}$ The other components of vector $\overline{\xi }$ are provided by the reference models, $\eta \left(0\right)={\left[\begin{array}{cc}\hfill -4.2deg\hfill & \hfill -0.46deg/\mathrm{s}\hfill \end{array}\right]}^T,{\mu }_1=50,{\mu }_2=1\text{,}$ while the initial value of the state is ${\mathit{x}}_{\mathit{long}}\left(0\right)={\left[\begin{array}{ccccccc}\hfill 72\mathrm{m}/\mathrm{s}\hfill & \hfill -3.1\mathrm{m}/\mathrm{s}\hfill & \hfill 0\mathrm{g}\mathrm{r}\mathrm{d}/\mathrm{s}\hfill & \hfill -0.6deg\hfill & \hfill 420\mathrm{m}\hfill & \hfill 0deg\hfill & \hfill 0deg\hfill \end{array}\right]}^T\text{.}$

Figure 3.

Time characteristics of the lateral-directional control subsystem.

5.2. Results and discussion

In Figure 3, one represents the time characteristics for the flight direction control system (the second subsystem of the complete ALS in Figure 2); before the start of the two landing main stages in longitudinal plane, the pilot must cancel aircraft's lateral deviation with respect to the runway. The characteristics have been represented for the first ALS affected by crosswind (V_vy = 2 m/s) in the presence or in the absence of sensor errors (the sensors are used for the measurement of the states). The presence of the sensor errors is not visible—the curves with solid line (obtained in the absence of sensor errors) overlap almost perfectly over the curves plotted with dashed line (obtained in the presence of sensor errors).

Figure 4.

Time characteristics of ALS, for the glide slope phase, with or without sensor errors.

The landing approach (the only landing phase which takes place in the lateral-directional plane) begins at the nominal speed of 67 m/s; the speed should be maintained constant. To test the robustness of the first designed ALS, in the simulations for lateral-directional plane, one has taken into consideration the crosswind, because low-altitude crosswind can be a serious threat to the safety of aircraft in landing. From sixth mini-graphic in Figure 3 (achieved for V_vy = 2 m/s), one can see that the stationary value of Y (aircraft lateral deviation) is very close to zero; analyzing the Aviation Administration (FAA) accuracy requirements for Category III (best category) [17], one can remark that this error is very good; for best category, it must less than 4.1 m, for Category II precision standards, the error must be between 4.1 and 4.6 m, while Category I precision standards involve an error between 4.6 and 9.1 m. The H-inf control technique is ideal for handling plants affected by sensor errors, measurement noise, or other disturbances (e.g. crosswind). Because the convergence error of the sideslip angle is less than 0.01 deg, one can notice the convergence β → β_c = 0 deg .

Figure 5.

Time characteristics of ALS, for the flare phase, with or without sensor errors.

In Figures 4 and 5, there are represented the time characteristics for the glide slope landing phase and flare landing phase, respectively; the characteristics have been represented for the ALS affected by wind shears in the presence or in the absence of sensor errors. The last four mini-graphics in Figures 4 and 5 represent the differences between the real values of the speed (u), sink rate $\left(\stackrel{.}{H}\right)\text{,}$ slope angle (γ) , altitude (H) and the desired values of these variables: $u-\overline{u},\stackrel{.}{H}-\overline{\stackrel{.}{H}},\gamma -{\gamma }_c,H-\overline{H}\text{.}$ As in the case of aircraft motion in lateral-directional plane, the sensor errors do not affect the landing. One may also remark in Figures 4 and 5 that the slope angle is in perfect accord with its desired values: −2.5 degrees during glide slope phase and 0 degrees during flare, respectively. During the glide slope phase, the aircraft describes a linear descendent trajectory (eighth graphic in Figure 4), while in the flare phase, it describes a parabolic trajectory (eighth graphic in Figure 5) with a null slope angle.

In longitudinal plane, to test the robustness of the new ALS, in all simulations, one has taken into consideration the wind shears. Figures 4 and 5 prove that the altitude error (the difference between the desired path and the actual path) is less than 0.3 m during the first landing stage (longitudinal plane) and less than 0.2 m during the second landing stage (longitudinal plane). According to the Federal Aviation Administration (FAA) accuracy requirements for Category III [17], the resulted errors are very small; thus, according to FAA Category III accuracy requirements, the vertical error (altitude deviation with respect to its nominal value) must be less than 0.5 m, while the final altitude at the end of flare must be 0 m. The ALS designed in this chapter meets the requirement because the H-inf robust control technique has been used; this method can handle the plant with measurement noise (sensor errors) and wind shears.

5.3. Comparison with other works

The ALS designed in this chapter represents an improved version of the ALS designed in [11] and it differs from other similar ALSs from the specialty literature; first of all, our ALS is not designed only for the longitudinal (vertical) plane but also for the lateral-directional plane; two subsystems have resulted, the new ALS being the mixture of these two automatic landing subsystems. Our new ALS has some additional elements with respect to the one presented in [11]: two optimal observers and four reference models which provide the desired altitude, velocity on the landing curve, their derivatives up to relative degrees of the system, the desired lateral deviation with respect to the runway, and the desired sideslip angle.

The results in this work have been compared to the ones obtained in [2] where the authors have designed a system which controls the lateral angular deviation of aircraft longitudinal axis with respect to the runway, by using a classical controller, a radio-navigation system, a system for the calculation of the distances between aircraft and the runway radio-markers, and an adaptive controller mainly used for the control of aircraft roll angle and its deviation with respect to the runway; the adaptive control system uses the dynamic inversion concept, a dynamic compensator, a neural network trained by the system's estimated error vector (signal provided by a linear observer), and a Pseudo Control Hedging block. The time regime period is better in our work (almost 15 seconds) but the lateral deviation's overshoot is larger; therefore, one can conclude that the neural networks-based adaptive controllers are more efficient than the conventional ones for aircraft landing in lateral-directional plane but their main disadvantage is that the neural networks require a priori training on normal and faulty operating data and these are enable only under limited conditions; on the other hand, the usage of Pseudo Control Hedging blocks (when the actuators are nonlinear) does not modify the final values of the variables. For the same aircraft type, same direction controller, and radio-navigation system but with a proportional-derivative type control after the roll angle and a proportional type control after Δψ [18], there have been obtained performance inferior to those obtained here by using the H-inf control, the dynamic inversion, optimal observers, and reference models.

The problem of landing in longitudinal plane has been also discussed in other papers, different types of ALSs being designed [9, 10]. If one makes a brief comparison between our ALS (longitudinal plane) and the ones based on an Instrumental Landing System or conventional/ fuzzy control of flight altitude by using the system's state [9], one remarks that from the system transient regime period and overshoot's points of view, the ALS based on the H-inf technique and dynamic inversion works slightly better. Improvement of the performance was obtained by replacing the conventional controllers with fuzzy controllers [9], but those ALSs cannot be used for no-bounded exogenous signals or strongly nonlinear aircraft dynamics. Our new ALS uses the H-inf technique, this having the advantage over classical control techniques in that it has applicability to problems involving multivariate systems with cross-coupling between channels; the only disadvantage is related to the non-linear constraints which are generally not well-handled.

5.4. Current and future work

This chapter presents some of the work that has been carried out at Laboratory of Aerospace Engineering, University of Craiova. Till now, there have been designed and software implemented: 1) two new ALSs (longitudinal plane) using the Instrumental Landing System and the flight altitude's control by means of the state vector (the controllers of the ALSs are designed both with classical and fuzzy logic approaches); 2) a new ALS (longitudinal plane) using the dynamic inversion concept and PID controllers in conventional and fuzzy variants, taking into consideration the wind shears and sensor errors; 3) a new ALS (lateral-directional plane) which controls the lateral angular deviation of aircraft longitudinal axis with respect to the runway, by using a classical controller, a radio-navigation system, a system for the calculation of the distances between aircraft and the runway radio-markers, and an adaptive controller mainly used for the control of aircraft roll angle and its deviation with respect to the runway. Our future work will focus on the design of ALSs (mixtures between subsystems designed for the longitudinal and lateral-directional planes) putting together the dynamic inversion technique, dynamics compensators, feed-forward neural networks, and Pseudo Control Hedging blocks.

6. Conclusions

The purpose of this study was to design a robust ALS by using the H-inf and dynamic inversion techniques taking into consideration the sensor errors and other different disturbances; two landing subsystems have been designed, software implemented and validated; the first subsystem is useful for landing control in longitudinal plane, while the second one is used in lateral-directional plane. After the separate design of the two subsystems, these have been combined to obtain a complete landing auto-pilot. The H-inf control technique handles the plant with measurement noise (sensor errors) and wind shears; the use of the dynamic inversion makes our control system more general and, therefore, it can be used both for the case when aircraft dynamics is nonlinear and for the case when the aircraft dynamics is linear; thus, this technique increases the generality character of our new ALS. Promising results have been obtained; these prove the robustness of the designed ALS even in the presence of disturbances and sensor errors.

Acknowledgments

This work is supported by the grant no. 89/1.10.2015 of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, project code PN-II-RU-TE-2014-4-0849.