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).
- H-inf control
- Dynamic inversion
- Reference model
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 , 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 ; 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 , while Ochi and Kanai  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 . In other studies , 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 : with the state vector, the command vector, while is the vector of disturbances –
2.2. Wind shears' model
By using the velocities' spectrum and generator filters having as inputs white noises, one can define the vector
In order to calculate the matrix
2.3. The general form of the control law (longitudinal plane)
One considers the vector that contains the system-controllable output variables, while the vector contains the reference variables (the imposed values of the flight altitude and velocity). The system output vector is
Now, by using and the dynamic inversion principle, and
2.4. Design of the control law's first component (longitudinal plane)
A coordinates’ change is achieved by means of the transformation matrix
where ξ is a state vector consisting of the controlled variables and their derivatives, i.e. : with the (r
For the obtaining of the relative degrees r1 and r2 , the equations of and are differentiated until terms containing the two components of the control law (
Thus, the state vectors
For the obtaining of the matrix
Replacing the vectors and into equation one obtains
where is calculated from making the substitutions: the other elements of the matrices and being the same; are the elements of the matrix
Replacing (7) in with one obtains:
where If one considers then Thus, for the calculation of the command vector
2.5. Design of the control law’s second component (longitudinal plane
To calculate the second component of the control law
with the matrix
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: where
For the design of the second optimal subsystem of the ALS, let us consider the vector —the vector of the system's controllable output variables and the vector —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
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:
3.3. Design of the control law's first component (lateral-directional plane)
First, one obtains the relative degrees of the variables
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 i.e.: where The form of the control law
3.4. Design of the control law's second component (lateral-directional plane)
To obtain the second component of the command law
are the system’s states (
The optimal control law in lateral-directional plane has the form :
The remarks regarding
The observer gain matrix
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 . Aircraft desired state and the desired output vector are obtained by using the states of the reference models. The optimal control is calculated on-line by means of the error .
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 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 (the altitude at which the glide slope phase ends and the second landing phase begins) is :
Similar approach is used for aircraft motion in lateral-directional plane; the vectors and are calculated by means of other two reference models, the former being a three-order reference model (associated to
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 (
In longitudinal plane, the dynamic inversion and H-inf method must assure the convergences: in lateral-directional plane, the following convergences should be assured:
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 :
For aircraft dynamics in longitudinal plane, the values of the coefficients for Boeing 747 have been borrowed from :
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 (
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
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 (
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 , 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  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 : 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  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
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 , 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 , 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.
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.
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.