## 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).

### Keywords

- Landing
- H-inf control
- Dynamic inversion
- Observer
- Reference model

## 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 H_{2}/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]: *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* is considered to be *V*_{0} ≅ *u*(0) = *u*_{0} . The equations of the actuators are*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 *A*_{long} ∈ *R*^{7 × 7}, *B*_{long} ∈ *R*^{7 × 2}, and *G*_{long} ∈ *R*^{7 × 2} are [10].

### 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]: *T*_{0} is the flight time period inside the wind shear, while

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:

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

One considers the vector **y**_{long} , chosen of the following form: *H* and *u*, obtained from aircraft dynamics, by using **x**_{long} and **u**_{long} the matrices *A*_{long} , *B*_{long} , *G*_{long} , one yields:

Now, by using **ū**_{long} are calculated with respect to

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]:

where ξ is a state vector consisting of the controlled variables and their derivatives, i.e. [11]: _{i} − 1) order derivatives of *z*_{i} ; for the aircraft dynamics in longitudinal plane, *z*_{1} = *H* , *z*_{2} = *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

For the obtaining of the relative degrees r_{1} and r_{2} , the equations of *δ*_{ec} , *δ*_{Tc}) appears in the expressions of the variables *ü* and *ü* and *δ*_{ec} , *δ*_{Tc} , and other states. Thus, the relative degree of the state *u* is r_{2} = 2 . In order to obtain the relative degree of the altitude (*H*), one derivates the differential equation associated to *H*, i.e. _{1} = 3 . The following equations result:

with

Thus, the state vectors *ξ* and *η* are **ũ**_{long} = 0 and **u**_{long} = **ū**_{long} , aircraft dynamics gets the form [11]: *Â*_{long} and *ξ* and *η* , it results [4]: **ū**_{long} is obtained as:

For the obtaining of the matrix *T*, the vectors *ξ* and are replaced in (4) and the following differential equations result:

Replacing the vectors

where

Replacing (7) in

where **ū**_{long} , one solves equation (8) and obtains the vector *η* and then uses the equation (7). From the expression of

**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 *z*_{1} = *H* and *z*_{2} = *u* , as well as the equation of the output vector

with *C*_{long} has the form (2), while *D*_{22 long} = *I*_{6} for the vector containing the sensor errors: **u **_{∞long} minimizes the cost functional:*P*_{∞} is the stabilizing solution of the Riccati equation [15]:

*Q*_{1} and *R*_{1} are positive defined matrices, while *μ*_{1} 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 *L*_{∞ long} is calculated with the formula:

*μ*_{2} 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: *xlat* = [*β p r ϕ ψ Y δa δr*]^{T}, *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, *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]:

For the design of the second optimal subsystem of the ALS, let us consider the vector *ylat* = **z**_{lat} and **y**_{lat} , the matrices *C*_{lat} and

### 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:

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

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

First, one obtains the relative degrees of the variables *z*_{1} = *Y* and *z*_{2} = *β* ; these relative degrees are denoted here with r_{1} and r_{2} , respectively. One derivates with respect to time the equations associated to *Y* and *β* (r_{1} times and r_{2} times, respectively) until the components of the command vector **u**_{lat} , i.e.

where

Thus, according to equations (15), the relative degrees are r_{1} = 3 and r_{2} = 2 . The equations (15) may be combined in the equation of the vector **ū**_{lat} results from **x**_{lat} ; in these conditions, one gets [1]:

### 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 *z*_{1} = *Y* and *z*_{2} = *β* , as well as the equation of the output vector **y**_{lat} , may be combined into the following equation:

the matrices *A*_{lat} , *B*_{lat} , *G*_{lat} have the forms (13) and *D*_{22lat} has the form *D*_{22 lat} = *I*_{7} for the vector containing the sensor errors: *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 *z*_{1} = *Y* and *z*_{2} = *β* are the same with the expressions in (17); the expansion of **x**_{lat}) and of the system command vector (**u**_{lat}) leads to the equations:

are the system’s states (*n* = 8), *c*_{1} and *c*_{2} have small positive values; in steady regime (**u**_{lat} = 0) , one gets *z*_{1} = *Y* and *z*_{2} = *β* .

The optimal control law in lateral-directional plane has the form [14]: **u**_{∞ lat} must minimize the cost functional *P*_{∞} is the stabilizing solution of the Riccati matriceal equation [1]:

The remarks regarding *Q*_{1} , *R*_{1} , and *μ*_{1} remain the same; the controller gain matrix (*K*_{∞ lat}) has the general form **u**_{∞ lat} depends on

The observer gain matrix *L*_{∞lat} is calculated by using the formula:

*μ*_{2} is a small positive scalar for which the Riccati equation (20) has a stabilizing solution; one used the same notations for the matrices *Q*_{1} , *R*_{1} , *P*_{∞} , and

## 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

### 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

where *X* is the covered distance, horizontally measured, *γ*_{c} is the aircraft desired slope angle. The seven components of the imposed vector

with *τ* − the time constant that defines the exponential curvature (the flare phase); as a consequence, *ū* from (21) and (22) are played by the variables *H*_{c} and *u*_{c} , respectively; the variables *ū* are provided by the reference models.

Similar approach is used for aircraft motion in lateral-directional plane; the vectors *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:

## 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]: *a*_{11} = − 0.0013 , *a*_{12} = 0 , *a*_{13} = − 1 , *a*_{14} = 0.15 , *a*_{21} = − 1.33 , *a*_{22} = − 0.98 , *a*_{23} = 0.33 , *a*_{31} = 0.17 , *a*_{32} = − 0.17 , *a*_{33} = − 0.217 , *b*_{11} = 0.001 , *b*_{12} = 0.015 , *b*_{21} = 0.23 , *b*_{22} = 0.06 , *b*_{31} = 0.026 , *b*_{32} = − 0.15 , *V*_{0} = 67 m/s , *T*_{a} = 0.7 s , **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 _{2} = 25 , *ξ*_{3} = *ξ*_{4} = 0.7 , *ω*_{3} = *ω*_{4} = 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

For aircraft dynamics in longitudinal plane, the values of the coefficients for Boeing 747 have been borrowed from [11]: *a*_{11} = − 0.021 , *a*_{12} = 0.122 , *a*_{14} = − 0.322 , *a*_{21} = − 0.209 , *a*_{22} = − 0.53 , *a*_{23} = 2.21 , *a*_{31} = 0.017 , *a*_{32} = − 0.164 , *a*_{33} = − 0.412 , *a*_{52} = − 1 , *a*_{54} = *V*_{0} = 70 m/s , *b*_{11} = 0.01 , *b*_{12} = 1 , *b*_{21} = − 0.064 , *b*_{22} = − 0.044 , *b*_{31} = − 0.378 , *b*_{32} = 0.544 , *T*_{e} = 0.3 s , *T*_{T} = 2 s , *ū* = *V*_{0} , *V*_{vx0} = 1 m/s , *V*_{vz0} = 1 m/s , *T*_{0} = 30 s . The vector of sensor errors is _{1} = 25 , *ξ*_{1} = *ξ*_{2} = 0.7 , *ω*_{1} = *ω*_{2} = 2 rad/s . For the first landing phase in longitudinal plane, the following values have been used:

### 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).

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 .

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 *γ*) , altitude (*H*) and the desired values of these variables:

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.