Design of Feedback Control for Quadrotors Considering Signal Transmission Delays. Stephen K. Armah, Sun Yi, and Wonchang Choi

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1 Design of Feedback Control for Quadrotors Considering Signal Transmission Delays Stephen K. Armah, Sun Yi, and Wonchang Choi International Journal of Control, Automation and Systems 14(6) (2016) ISSN: (print version) eissn: (electronic version) To link to this article:

2 International Journal of Control, Automation and Systems 14(6) (2016) ISSN: eissn: Design of Feedback Control for Quadrotors Considering Signal Transmission Delays Stephen K. Armah*, Sun Yi, and Wonchang Choi* Abstract: For quadrotor types of unmanned aerial vehicles (UAVs), existence of transmission delays caused by wireless communication is one of the critical challenges. Estimation of the delays and analysis of their effects are not straightforward for designing controllers. An estimation method is introduced using experimental data and analytical solutions of delay differential equations (DDEs). Collected altitude responses in the time domain are compared to the predicted ones obtained from the analytical solutions. The Lambert W function-based approach for first-order DDEs is used for such analysis. The dominant characteristic roots among infinite number of roots are obtained in terms of coefficients and the delay. The effects of the time delay on the responses are analysed through the locations of the characteristic roots in the complex plane. Based on the estimation result, proportional plus velocity controllers are proposed to improve transient altitude responses. Keywords: Delay differential equation, delay estimation, Lambert W function, quadrotor. 1. INTRODUCTION Time delays are inherent typically in dynamic systems having signal transmission via Wi-Fi. Estimation of the delays and analysis of its effects on performance of dynamic systems is an important topic in many applications. Estimating delays is a challenging problem and has been an area of great research interest in fields as diverse as radar, sonar, seismology, geophysics, ultrasonic, controls, and communications [1, 2]. Although considerable efforts have been made on parameter estimation, there are still many open problems and there is no common approach in time-delay identification due to difficulty in formulation [3 5]. Development of effective control systems for quadrotor types of unmanned aerial vehicles (UAVs) has been the focus of active research during the past decades. One of the challenges in designing effective control systems for UAVs is existence of signal transmission delay, which has nonlinear effects on the flight performance of autonomously controlled UAVs. A controller designed using a non-delayed system model may result in disappointingly slow and oscillating responses due to the delays. For autonomous aerial robots, typical delay is around 0.4±0.2 s [6]. Also, for large delays (e.g., larger than 0.2 s) the system response might not be stabilized or converged due to the dramatic increase in torque. This poses a significant challenge [7]. Parrot AR.Drone 2.0 is a UAV controlled through Wi- Fi and, thus, its dynamics contains a time delay. Refer to Section 2 for the drone s control architecture. The overall time delay is attributed to: 1) the processing capability of the host computer, 2) the electronic devices processing the motion signals, 3) the measurement reading devices, e.g., the distance between the ultrasonic altitude sensor and the ground, and 4) the software on the host computer for implementing the controllers. For UAVs, wireless communication delays may not be critical when the controllers are on-board. However, delays have significant effects when the control software is run on an external computer and signals are transmitted wirelessly. For example, the experiments presented in this paper were conducted using MATLAB/Simulink on an external computer, and decoding process of the navigation data (yaw, pitch, roll, altitude, etc.) contributes to the delay. Also, the numerical solvers introduce additional delay. There are various existing methods for delay estimation. One of those is based on finite-dimensional Chebyshev spectral continuous time approximation (CTA) [8]. Manuscript received March 10, 2015; revised September 21, 2015; accepted November 13, Recommended by Associate Editor Changchun Hua under the direction of Editor Myo Taeg Lim. This material is based on research sponsored by Air Force Research Laboratory and OSD under agreement number FA The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Stephen K. Armah and Sun Yi are with the Department of Mechanical Engineering in North Carolina A & T State University, 1601 E Market St., Greensboro, NC 27411, USA ( s: skarmah@aggies.ncat.edu, syi@ncat.edu). Wonchang Choi is with the Department of Architectural Engineering in Gachon University, 1342 Seongnamdaero, Sujeong-gu, Seangnam-si, Gyeonggi-do, , Korea ( wchoi@gachon.ac.kr). * Corresponding authors. c ICROS, KIEE and Springer 2016

3 1396 Stephen K. Armah, Sun Yi, and Wonchang Choi The finite-dimensional CTA was used to approximately solve delay differential equations (DDEs) for estimation of constant and time-varying delays. The accuracy is dependent on the size of the Chebyshev spectral differentiation matrix. Also, techniques using graphical methods [9 11], an integral equation approach [12], discrete and continuous approaches [11], a cost function for a set of time delays in a certain range [13, 14], and a frequencydomain maximum likelihood [15] have been developed. Methods for transfer function identification ignore the delays and their effects, or just assume that the delay is known [16,17]. Moreover, none of the existing linear filter methods directly estimate the time delay, except for some very special type of excitation [11]. This paper presents a new method to estimate a constant time delay in the AR.Drone 2.0 altitude control system. In real applications, drones fly around and the time delay may vary. The altitude dynamics is assumed to be linear time-invariant (LTI) first-order and the time delay is incorporated into the model as an explicit parameter. Here, the delay is not restricted to be a multiple of the sampling interval. Experimental data and analytical solutions of infinite-dimensional continuous DDEs are used. The approach in this paper is inspired by the wellknown technique for LTI ordinary differential equations (ODEs), see [3, 18] for more details. Measured transient responses are compared to time-domain descriptions obtained by using the Lambert W function. Then, the dominant characteristic roots are obtained in terms of system parameters including the delay. Proportional (P) controllers are used to generate the responses for estimation. The effects of the time delay on the responses are analyzed. Then, proportional plus velocity (PV) control is designed to achieve better transient responses. This paper continues with the description of quadrotor s altitude model and the AR.Drone 2.0 altitude control system in Section 2. Section 3 presents the approaches used for estimating the system s time delay. In Section 4, the P and PV controllers are presented. In Section 5, results are summarized. Concluding remarks and future work is presented in Section ALTITUDE MODEL AND CONTROL SYSTEM The notation for the quadrotor model is shown in Fig. 1. Quadrotors has three main coordinate systems attached to it; the body-fixed frame, {B}, the vehicle frame, {V }, and the global inertial frame, {I}, which is not shown. There are two other coordinate systems, not shown, that are of interest, the vehicle-1 frame, {V 1 }, and vehicle-2 frame, {V 2 } [19]. The configuration of the quadrotor is represented by six degrees of freedom in terms of position, (x I,y I,z I ) T, and the attitude defined using the Euler angles, (ϕ V 2,θ V 1,φ V ) T, which gives a 12-state equation of motion [20]. Y B T 4 T Z 1 XB w Thrust, T d O B,V Y V w 1 T 3 T 2 w 2 w w Weight, mg 2 Fig. 1. Notation of coordinates for the quadrotor. The quadrotor has four rotors, labelled 1 to 4, mounted at the end of each cross arm. The rotors are driven by electric motors powered by electronic speed controllers. The vehicle s total mass is denoted by m and d is distance from the motors to the center of mass. The total upward thrust, T (t), on the vehicle is given by i=4 T (t) = T i (t), (1) i=1 and T i (t) = aω 2 i (t), i = 1, 2, 3, 4, where ω i(t) is the rotor s angular speed and a > 0 is the thrust constant [20]. The equation of motion in the z-direction can be obtain as [19, 21]. z(t) = 4aω2 (t) m g, (2) where ω(t) is the rotors average angular speed necessary to generate Tt. Thus, only the speed ω(t) needs to be controlled to regulate the altitude, z(t), of the quadrotor, since m, a, and g are constants. According to the AR.Drone 2.0 SDK documentation, z(t) is controlled by applying a reference vertical speed, ż re f (t), as control input; ż re f (t) has to be constrained to [ 1 1] ms 1, to prevent damage. The drone s flight management system sampling time, T s, is s, which is also the sampling time at which the control law is executed and the navigation data is received. The control diagram for the drone s altitude motion using MATLAB/Simulink program is shown in Fig. 2. The error between the desired reference input, z des (t), and the system altitude response, z(t), is denoted by e(t). The altitude motion dynamics is implemented using (2) to determine ω(t). The rotors then rotate with the calcuated ω(t), which will generate T (t) to produce z(t). These computation takes place on board the drone PID control engine program written in C. Thus, a cascade control is considered. The motor dynamics is assumed to be so fast that the altitude control system can be represented as a first-order system using an integrator (see Fig. 2). Cascade control X V

4 Design of Feedback Control for Quadrotors Considering Signal Transmission Delays Feedback loop for altitude regulation Fig. 2. Diagram for the drone altitude control system. is beneficial only if the dynamics of the inner is fast compared to those of the outer loop. Under such assumption, the control input, ż app (t), to the first-order system is approximated to be equal to the actual vertical speed, ż(t), of the drone. Thus, a first-order model is used for the analytical determination of the time delay and for obtaining the simulation altitude responses. The simulation setup used is shown in Fig. 3. The overall time delay, T d, in the system is represented as actuator time delay. The MATLAB/Simulink program setup developed for the experiments is shown in Fig. 4. The vertical speed control input constraints are applied using the saturation block. The experiments were conducted in an office environment with the drone s indoor hull attached. The drone is connected to the host PC through Wi-Fi. Data streaming, sending and receiving, are made possible using UDPs (user datagram protocols). UDP is a communication protocol, an alternative to TCP that offers a limited amount of service when messages are exchange between computers in a network that uses IP. The drone navigation data (from the sensors, cameras, battery, etc.) are received, and the control signals are sent, using AT commands. AT commands are combination of short text strings sent to the drone to control its actions. The drone has ultrasound sensor (at the bottom) for ground altitude measurement. It has 1 GHz 32 bit ARM Cortex A8 processor, 1GB DDR2 RAM at 200 MHz, and USB 2.0 high speed for extensions. Controller Actuator Time Delay Plant 3. TIME DELAYS ESTIMATION A continuous control system can be represented for active time-delay estimation (TDE) problem as [22] y(t) = G p u(t T d ) + n(t), (3) Fig. 3. Simulation control system setup. where G p is a single-input single-output (SISO) LTI dynamic system without time delay, y(t) is measured signal, u(t) is the control input signal, and n(t) is measurement noise (here, it is assumed that n(t) = 0). The time delay to be estimated is an explicit parameter in the model and it is not restricted to be a multiple of the sampling time. Then, the estimation problem can be formulated using analytical solutions to DDEs. Consider the first-order scalar homogenous DDE shown in (4). Unlike ODEs, two initial conditions need to be specified for DDEs: a preshape function, g(t), for T d t < 0, and initial point, z o, at t = 0. ż(t) a o z(t) a 1 z(t T d ) = 0. (4) The characteristic equation of (4) is given by Fig. 4. Simulink diagram for controlling the drone. s a o a 1 e st d = 0. (5) Then, the characteristic equation in (5) is solved as [3] s = 1 T d W(T d a 1 e a ot d ) + a o, (6) where the Lambert W function is defined as W (x)e W(x) = x [23]. As seen in (6), the characteristic root, s, is obtained analytically in terms of parameters, a o, a 1, and T d. The solution in (6) has an analytical form expressed in terms of the parameters of the DDE in (4). One can explicitly determine how the time delay is involved in the solution and, furthermore, how each parameter affects each characteristic root. That enables one to formulate estimation of time delays in an analytic way. Each eigenvalue is distinguished with the branches of the Lambert W function, which is already embedded in MATLAB [3]. For first-order scalar DDEs, it has been proved that the rightmost characteristic roots are always obtained by using the principal branch, k = 0, and/or k = 1 [24]. For

5 1398 Stephen K. Armah, Sun Yi, and Wonchang Choi x Fig. 6. Simulink block diagram for P-feedback control. Fig. 5. Stable dominant roots in the s-complex plane. the DDE in (4), one has to consider two possible cases for rightmost characteristic roots: characteristic equations of DDEs as in (5) can have one real dominant root or two complex conjugate dominant roots. Thus, when estimating time delays using characteristic roots, it is required to decide whether it is the former or the latter [3]. For ODEs, an estimation technique using the logarithmic decrement provides an effective way to estimate the damping ratio, ξ [?]. The technique makes use of the form s = ξ ω n ± jω n 1 ξ 2 (7) for obtaining s of second-order ODEs, where ω n is the system s natural frequency. The variables ξ and ω n are obtained from the response of the system, and different approaches can be applied depending on the nature of the response: oscillatory and nonoscillatory [3]. Here, the transient properties for oscillatory responses are used. Property such as the maximum overshoot, M o, is related to ξ, as shown below [25] M o = 100e ( ) ξ π 1 ξ 2. (8) Now, for a stable delayed control system, the dominant roots, s, lies in the left-hand complex plane (see Fig. 5), where is computed as ξ = cosθ = Re(s) (Re(s)) 2 + (Im(s)) 2. (9) Then, the drone control system with the unknown T d, is estimated by the following steps: Step 1: Calculate based on the system altitude response using (8), Step 2: Solve the nonlinear equation s = 1 T d W(T d a 1 e a ot d ) + a o for T d, since s is link to through (9). Fig. 7. Simulink block diagram for PV-feedback control. The equation in Step 2 can be solved using a nonlinear solver (e.g., fsolve in MATLAB). For comparison, a numerical approach using Simulink is also used. In this approach, M o of the drone s altitude responses are compared to those of simulation responses with various values of T d. 4. P AND PV CONTROLS The system has an integral term in the closed-loop transfer function and, thus, only P and PV feedback controllers are used to generate vertical speed signal. PV control, unlike PD control, does not yield numerator dynamics. The P-feedback controller is used to generate altitude responses for estimation of the time delay, and the PVfeedback controller is used to analyze the effect of the time delay on the drone s altitude response. Figs. 6 and 7 show the Simulink setups developed for conducting the simulations, and the controller gains were used in Fig. 4 for the experiments. The delay is applied using transport delay block. The transfer function of the time-delay closed-loop system for the P controller is given by Z (s) Z des (s) = K Pe std s + K P e st. (10) d This time-delay system is a retarded type. As expected the characteristic equation is transcendental, and therefore the number of the closed-loop poles are infinite. The exponential term in the characteristic equation introduces oscillations into the system. Comparing the characteristic equation of the closed-loop system in (10) to the firstorder system in (5), a o = 0 and a 1 = K P. The effect of T d on the drone s altitude response was studied using analytical, simulation, and experimental approaches with a PV controller. Suitable PV controller gains, K p and K v, are obtained to improve on the tran-

6 Design of Feedback Control for Quadrotors Considering Signal Transmission Delays 1399 Table 1. Simulated effect of control input saturation on the altitude response. M o (%) Without Saturation With Saturation K p = 1.00 & T d = 5T s K p = 3.00 & T d = 4T s K p = 1.31 & T d = 6T s Fig. 8. Experimented altitude responses: varying K p. sient behaviors. A high pass filter (HPF) with damping ratio, ξ f = 1.0 was used for the velocity controller. A suitable natural frequency, ω f, value was selected, by tuning and the use of the Bode plot, for the filter. The transfer function of the time-delay closed-loop system with the PV controller is given by Z (s) Z des (s) = K P e std s + (K P + K v s)e st. (11) d The time-delay system in (11) is a neutral type. 5. RESULTS AND DISCUSSION 5.1. Estimation of time delay Initially, the drone s altitude responses were obtained for several values of K p, as shown in Fig. 8. Note that if there is no delay (T d = 0), the characteristic root is K p (refer to (10)), which is a real number. Thus, there should be no overshoot. Also, increase in K p should not destabilize the system. However, as seen in Fig. 8, the delay introduces imaginary parts to the roots and, thus, oscillation in the responses. Therefore, the delay does have nontrivial effects and need be precisely estimated and considered in designing effective control. Note that for ease of analyzing the responses are shifted to start at (0 s, 0 m). The gain value, K p = 1.0 seems to be good for the controller since the response has no overshoot. However, the response is very slow. The results also show that increase in K p makes the response faster (the rise time becomes shorter) but introduces higher M o. This is partly due to the time delay in the system, which introduces nonlinearity on the dynamics. It was also observed that the saturation applied to the control input has a nonlinear effect on the system s response, especially as K p increases. The simulations summarized in Fig. 9 and Table 1 revealed that at a constant T d, as the K p value increases the control input saturation has a significant effect on the system response. At a constant T d and K p values, introducing the saturation decreases M o and increases t p. As K p increases the increase in M o becomes larger and the oscillations of the system increases. Fig. 9. Simulated effect of control input saturation on the altitude response. Also, it shows that for K p = 1.0 at T d = 5T s, there is no overshoot, as compared to K p = 3.0 at a lower T d = 4T s, with and without the saturation effect. Moreover, the experimented altitude responses show that at K p = 1.0, there is no or little overshoot, and the oscillations are very small (see Fig. 8). Thus, the system s time delay s effect is not shown clearly in the transient responses. The analysis via simulations shows that K p = 1.31 gives a response with a sufficient overshoot for estimation, and with minimum saturation effect Numerical method The drone s altitude response oscillates for high gains as shown in Fig. 8. Thus, the transient property M o is used for comparison of the simulation and experimental results. Table 2 shows the values of M o obtained from simulation by varying T d at K p = 1.31, where K is a real constant tuning parameter, a multiplier of T s. The drone s altitude responses with K p = 1.31 are shown in Fig. 10. Table 3 shows their corresponding M o values. The largest value, M o = 2.300%, gives the largest T d. Comparing the M o = 2.300% to the results in Table 2, T d is estimated as T s, which gives s. Note, the peak time, t p, was initially considered in the estimation of the delay, however, it was not used in the calculation of T d. There were differences in the simulation and the experimental values, see [18] and [26] for more details. This was largely due to the nonlinearity of the actuators saturation, and other disturbances such as the aerodynamics around the drone. Also, because there is one parameter, T d, to be estimated, only M o of the transient

7 1400 Stephen K. Armah, Sun Yi, and Wonchang Choi Table 2. Simulated altitude responses: K p = K T d = KT s (s) M o (%) * * 2.300* Table 4. Rightmost characteristic roots of the PV control system with K p = 2.0 and T d = 0.4 s. K v Rightmost Complex Roots ± 3.07 j ± 3.25 j 0.3* 3.25 ± j ± 6.98 j ± 7.47 j Table 3. Experimented altitude responses:k p = Flight M o (%) 2.300* * < 2 Fig. 12. PV control system characteristic roots spectrum distribution with K p = 2.0 and T d = 0.4 s. Fig. 10. Experimented altitude responses: K p = Fig. 11. Plot of the dominant roots as T d increases. properties was sufficient for the estimation Analytical approach using characteristic roots The drone altitude response oscillates (Figs. 8 and 10). Thus, the system has complex conjugate dominant roots. Therefore, M o is used to determine ξ. M o = % (see Section 5.1.1) and, thus, ξ is obtained as using (8). Applying the steps described in Section 3, T d is determined as s using the developed MATLAB program with initial guess value of 0.3 s. See Fig. 11 for the plot of the dominant roots as the delay increases PV controller: Design and implementation As shown above, the estimated time delay using both the numerical and the analytical methods is approximately 0.4 s. A MATLAB-based software package [27] was used to study the stability of the neutral type of time-delay systems by numerically solving the characteristic equation in (11). The closed-loop system characteristic roots within a specified region are then plotted for various K v values. Fig. 12 shows the spectrum distribution of the characteristic roots, and Table 4 shows a summary of the rightmost (i.e., dominant) roots for each system. The value K v = 0.3 yields the most suitable stable rightmost roots among them. The corresponding simulation altitude responses for the system were also obtained for the various K v values, see Fig. 13. It can be seen that as K v increases at K p = 2.0 and T d = 0.4 s, M o decreases and the rise time becomes longer. At higher values of K v, the response is oscillatory and the system becomes unstable. This is also observed in Fig. 12, that as K v increases the roots move to the right, increasing the instability in the system. When the rightmost roots are located farther from the imaginary axis (towards the left), the altitude responses are more stabilized. That is, it has reduced overshoots and faster convergence. Now, based on these analyses, a controller with K p = 2.0 and K v = 0.3 was selected as the most suitable, with closed-loop system response transient properties of M o = 0.44 %, t s = 1.52 s, and t p = 1.76 s. Using these controller gains, the HPF was included in the simulation control sys-

8 Design of Feedback Control for Quadrotors Considering Signal Transmission Delays 1401 Fig. 13. Simulated PV controller altitude response with K p = 2.0, T d = 0.4 s, without HPF. Table 5. Simulated effect of the HPF on the altitude response by varying ω f. ω f (rads 1 ) * M o (%) t p (s) t s (s) NaN Fig. 15. Bode plots of the PV-feedback close-loop system with and without the HPF. Table 6. PV controller altitude response transient properties, with K p = 2.0 and the HPF. K v * Simulation (T d = 0.4 s) M o (%) t p (s) t s (s) Experiment M o (%) t p (s) t s (s) Fig. 14. Simulated effect of the HPF on the altitude response by varying ω f. tem, and its effects on the altitude transient response was studied for different values of ω f, see Fig. 14 and Table 5. It is observed that at smaller ω f values the response oscillates, and at higher values the response distorts. The oscillations and the distortions effects were reduced by using the high-order solver, ode8 (Dormand-Prince). The HPF with ω f = 38 rads 1 and ξ f = 1.0 was then selected, with poles of 38 repeated. Now, looking at the poles distribution of the system in Fig. 12, it is observed that the poles of this filter is located to the left than the poles of the PV-feedback closed-loop system. This filter will respond faster, therefore, it will have smaller effect on the drone s altitude transient response. Bode plot (see Fig. 15) was used to determine the filter s cutoff frequency as 5.68 rads 1 (0.90 Hz). Fig. 16 and Table 6 shows the simulation altitude re- sponses and their corresponding transient properties, with the HPF and K p = 2.0, for different K v values. The results with K p = 2.0 and K v = 0.3 yields an improved transient response performance, which shows that the estimation of delay and analysis presented help. Fig. 17 and Table 6 also shows the experimented altitude responses and their corresponding transient properties, with the HPF and K p = 2.0, for several K v values. It demonstrates that as the K v value increases M o decreases and in general the responses becomes slower. The PV controller performed better for K v = 0.3, 0.5, and 0.7 at K p = 2.0. The simulations and experiments were conducted using the MATLAB/Simulink fixed-step solver, ode8 (Dormand-Prince). The solver calculates the states during simulation and code generation. Investigation revealed that the choice of a solver has a significant effect on the drone altitude response, refer to Fig. 18, and also see [28] and [29] for more details. The HPF performance is constrained by the type of solver used. The filter performs better at higher ω f when higher-order solvers are used for experiment. 6. CONCLUSIONS Estimation methods for the time delay in a quadrotor type of UAV, Parrot AR.Drone 2.0, are presented and

9 1402 Stephen K. Armah, Sun Yi, and Wonchang Choi Fig. 16. Simulated PV controller altitude responses with K p = 2.0 and T d = 0.4 s. using the MATLAB/Simulink high-order solver, ode8 (Dormand-Prince). Investigation through trials revealed that selection of the solvers has significant effects on the drone s altitude response. The HPF performance was constrained by the type of solver used and the filter performed better with the high-order solvers. Advanced control strategies (e.g., adaptive controllers considering varying payloads, robust controllers for the drone s attitude and x-y motions) are being developed by estimating and incorporating the time delay in the control systems. This problem is significantly more challenging, since the equation of motions are more complex compared to that of the altitude motion alone. Also, nonlinear control schemes (e.g., gain scheduling) can be designed taking into account the estimated time delay. Its performance will be compared to that of the linear PV controller. Furthermore, the presented time-delay estimating methods can be extended to general systems of DDEs (higher than first order), and be applied to delay problems in network systems and fault detection of actuators. REFERENCES Fig. 17. Experimented PV controller altitude responses with K p = 2.0. Fig. 18. Experimented effect of Simulink solvers on the PV controller altitude response. used for altitude regulation. Using the observed maximum overshoots and locations of the characteristic roots, the time delay was estimated as 0.4 s. For estimation of the time delay, an appropriate P controller was used and the gain that minimizes the effect of the applied control signal saturation on the system s response was selected. The effects of the time delay on the drone s altitude response were analyzed. The designed PV improve system performance in terms of stability and response speed than the P controller. Simulations and experiments data sets were collected [1] M. Azadegan, M.T. Beheshti, and B. Tavassoli, Using AQM for performance improvement of networked control systems, International Journal of Control, Automation and Systems, vol. 13, no. 3, pp , [2] M. Kchaou, F. Tadeo, M. Chaabane, and A. Toumi, Delaydependent robust observer-based control for discrete-time uncertain singular systems with interval time-varying state delay, Int. J. Control, Automation and Systems, vol. 12, no. 1, pp , February [click] [3] S. Yi, W. Choi, and T. Abu-Lebdeh, Time-delay estimation using the characteristic roots of delay differential equations, American Journal of Applied Sciences, vol. 9, no. 6, pp , [4] L. Belkoura, J.P. Richard, and M. Fliess, Parameters estimation of systems with delayed and structured entries, Automatica, vol. 45, no. 5, pp , [click] [5] J. P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica, vol. 39, no. 10, pp , [click] [6] G. Vásárhelyi, C. Virágh, G. Somorjai, N. Tarcai, T. Szorenyi, T. Nepusz, and Vicsek, T. Outdoor flocking and formation flight with autonomous aerial robots, Proc. of the 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2014), pp , [7] A. Ailon and S. Arogeti, Study on the effects of timedelays on quadrotor-type helicopter dynamics, Proc. of the 22nd Mediterranean Conference of Control and Automation (MED), pp , [8] S. Torkamani and E.A. Butcher, Delay, state, and parameter estimation in chaotic and hyperchaotic delayed systems with uncertainty and time-varying delay, International Journal of Dynamics and Control, vol. 1, no. 2, pp , 2013.

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