Tele-Operation of Autonomous Vehicles Over Additive White Gaussian Noise Channel

Scientia Iranica D (2021) 28(3), 1592{1605

Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering http://scientiairanica.sharif.edu

Tele-operation of autonomous vehicles over additive white Gaussian noise channel

A. Parsa and A. Farhadi

Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.

Received 17 September 2019; received in revised form 21 January 2020; accepted 14 April 2020

KEYWORDS Abstract. This paper is concerned with the tele-operation of autonomous vehicles over analog Additive White Gaussian Noise (AWGN) channel, which is subject to transmission Networked control noise and power constraint. The nonlinear dynamic of autonomous vehicles is described system; by the unicycle model and is cascaded with a bandpass lter acting as encoder. Using the Tele-operation system; describing function method, the nonlinear dynamic of autonomous vehicles is represented The describing by an approximate linear system. Then, the available results for linear control over analog function; AWGN channel are extended to account for linear continuous time systems with non-real The unicycle model; valued and multiple real valued eigenvalues and for tracking a non-zero reference signal. AWGN channel. Subsequently, by applying the extended results on the describing function of autonomous vehicles, a mean square control technique including an encoder, a decoder, and a controller is presented for reference tracking of the tele-operation of autonomous vehicles over AWGN channel. The satisfactory performance of the proposed control technique is illustrated by computer simulations. © 2021 Sharif University of Technology. All rights reserved.

1. Introduction the tele-operation of system given in Figure 1. Very often, autonomous vehicles are battery powered and 1.1. Motivation and background hence, communication from these vehicles to the base Tele-operation of autonomous vehicles has become an station where the operator is located must be done with active research direction in recent years. In these minimum possible transmission power to increase the systems, the remote autonomous vehicle must track a on-board battery life time. Therefore, communication reference signal generated by a remote operator which from vehicle to remote controller is subject to noise is communicated to it via a wireless link. In this and power constraint. However, as the communication application, the measurements from on-board sensors from the base station to remote vehicle can be done are also communicated to remote operators to help op- with high transmission power, in the block diagram erator to generate a desired reference signal. One of the of Figure 1, the communication link from remote abstract models for wireless communication is Additive controller to the remote vehicle (dynamic system) can White Gaussian Noise (AWGN) channel. This channel be considered without imperfections and limitations. is an abstract model for satellite communication, deep The tele-operation of Figure 1 is an example space communication, and when the line of sight is of networked control systems. In networked control strong. Therefore, in these situations, we deal with systems, we deal with controlling dynamic systems over communication channels subject to imperfections and *. Corresponding author. limitations. Some results addressing basic problems in E-mail address: [email protected] (A. Farhadi) stability and/or state tracking of dynamic systems over communication channels subject to imperfections and doi: 10.24200/sci.2020.54447.3755 limitations can be found in [1{28]. Dynamic systems A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 1593

back systems, in which the Multi-Input Multi-Output (MIMO) communication link is modeled as a parallel noisy Additive White Noise (AWN) channel. The idea of using the describing function for controlling nonlinear dynamic systems over AWGN channel for the rst time was presented in [29]. In addition to the AWGN channel, other communication channels, which have caught the attention of networked control systems research community, are the real and the packet erasure channels that are abstract models for communication Figure 1. A dynamic system over Multi-Input via the Internet and WiFi links. Other researchers [30] Multi-Output (MIMO) parallel Additive White Gaussian addressed the problems of optimal control and stability Noise (AWGN) channel. of LTI systems over the real erasure channel, which is subject to random packet dropout, where both remote can be viewed as continuous alphabet information and local controllers operate on the system. Moreover, sources with memory. Therefore, many works in the the problem of optimal control of LTI systems by a literature (e.g., [1,12,13,18,19,22{28]) are dedicated to remote controller over the real erasure channel was the question of state tracking and/or stability over addressed in [31]. AWGN channel, which itself is naturally a continuous The above literature review reveals that the alphabet channel. Researchers [12,13] addressed the available results on the stability and state tracking problem of mean square stability and state tracking of of dynamic systems over AWGN channel are mainly linear Gaussian dynamic systems over AWGN channel concerned with linear dynamic systems. To the best when noiseless feedback channel was available full time of our knowledge, control of nonlinear dynamic sys- and the communication of control signal from remote tems over AWGN channel is limited to [23] and [29], controller to system was perfect (see Figure 1). In [12], which are not concerned with the tele-operation of the authors presented an optimal control technique for autonomous vehicles subject to communication imper- asymptotic bounded mean square stability of partially fections. In the tele-operation of autonomous vehicles, observed discrete time linear Gaussian systems over we deal with state tracking as well as reference tracking AWGN channel. In [13], the authors addressed a of nonlinear systems. The dynamic of autonomous continuous time version of the problem addressed in vehicles (autonomous underwater, unmanned aerial, [12]. In [22], the researchers considered a framework and autonomous road vehicles) is described by the for discussing control over a communication channel six-degree-of-freedom model. However, the vehicle based on Signal-to-Noise Ratio (SNR) constraints and dynamic is handled by an on-board control loop that focused particularly on the feedback stabilization of an results in a kinematic unicycle model, which is non- open loop unstable plant via a channel with a SNR linear [16]. In addition, in deep space tele-operation constraint. By examining the simple case of a Linear systems or when the line of sight is strong, AWGN Time Invariant (LTI) plant and an AWGN channel, channel is a suitable model for wireless communication. the authors in [22] derived necessary and sucient These issues motivate research on tele-operation of conditions on the SNR for feedback stabilization with the unicycle model over AWGN channel, which is the an LTI controller. In [23], the authors addressed the subject of this paper. problem of state tracking of nonlinear systems over AWGN channel. In [25], the authors presented a sub- 1.2. Paper contributions optimal decentralized control technique for bounded Key contributions of this paper compared to the afore- mean square stability of a large-scale system with mentioned literature are summarized as follows: cascaded clusters of sub-systems. Each sub-system 1. In the eld of nonlinear dynamic systems, the is linear and time-invariant and both sub-system and describing function is used for building oscillators its measurement are subject to Gaussian noise. The [32]. However, in this paper, for the rst time, it control signals are exchanged between sub-systems has been used to stabilize and control a practical without any imperfections, but the measurements are nonlinear dynamic system. That is, the unicycle exchanged via an AWGN communication network. In model is an abstract model for describing the [27], the author presented a suboptimal technique dynamics of autonomous underwater, unmanned for mean square stability of a distributed system aerial and autonomous road vehicles; with geographically separated Gaussian sub-systems interconnected by a AWGN communication network. 2. To the best of our knowledge, earlier works on In [28], the authors investigated stabilization and controlling dynamic systems over the AWGN chan- performance issues for MIMO LTI networked feed- nel are mainly concerned with the stability and 1594 A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605

state tracking of linear dynamic systems, e.g., where x(t) and y(t) are the position vectors, (t) [1,12,13,18,19,22,24,25,27,28]. Only the referenced is the heading angle, and the control inputs are the studies [24,29] addressed the problem of controlling vehicle forward velocity v(t) and the angular velocity nonlinear dynamic systems over the AWGN chan- u(t). Note that for the remote controller the initial nel. Nevertheless, this study [24] is just concerned conditions (x(0), y(0), (0)) are unknown and have a with the estimation problem; and [29] addressed the Gaussian distribution. problem of tracking and stability of those systems that have periodic outputs to sinusoidal inputs. Communication channel. Communication channel Therefore, another study [29] used a linear dynamic between system and controller is an MIMO AWGN system subject to saturated excitation for computer channel without interference (parallel) with n inputs simulation as this type of noncommon dynamic and n outputs. The output of the encoder (which systems was consistent with the theoretical devel- will be described shortly) is transmitted through the opment in [29]. However, by nding a describing MIMO channel and a white Gaussian noise vector function for the unicycle model, which also has peri- is added to it (as is shown in Figure 1), where 0 ~ odic outputs to periodic inputs, this paper presents N(t) = n1(t) : : : nn(t) i.i.d. N(0; R) a real application of the theoretical development in ~ (R = diagfr~1; :::; r~ng) is the MIMO channel noise and [29] by controlling a practical nonlinear dynamic ni(t) N(0; r~i) is the additive noise of the ith input system over the AWGN channel. - output path of the MIMO parallel AWGN channel. 1.3. Paper organization Also, the MIMO parallel AWGN channel is subject to The paper is organized as follows. In Section 2, the the channel input power constraints P oi, i = 1; 2; :::; n 2 problem formulation is presented. Section 3 describes as follows: E[fi (t)] P oi, i = 1; 2; :::; n, where the describing function method. Then, Section 4 de- fi(t) is the ith element of the encoder output vector 0 velops the theory of mean square reference tracking of F (t) = f1(t) : : : fn(t) , which is the input of linear dynamic systems with multiple real and non-real the channel. That is, Y (t) = F (t) + N(t), where 0 valued eigenvalues over MIMO AWGN channel. Sec- Y (t) = y1(t) : : : yn(t) is the channel output. tion 5 is devoted to the tele-operation of autonomous vehicles. Section 6 gives simulation results for the Encoder. The encoder is a bandpass lter cascaded unicycle model and Section 7 concludes this paper with a matrix gain. This bandpass lter saves only by summarizing the contributions of the paper and the fundamental frequency of the system outputs and directions for future research. omits the other harmonics which have less information to be sent. For this purpose, a high pass lter with a 2. Problem formulation relatively low cut-o frequency (e.g., 0:1 Hz) is used to omit the DC part. This lter is cascaded with a low Throughout, certain conventions are used: E[] denotes pass lter with a cut-o frequency of ! 2 0:1 the expected value, var[:] the variance, and j j the rad/s. For constructing such a lter, we can use the absolute value and V 0 the transpose of vector/matrix following transfer functions: 1 V . A denotes the inverse of a square matrix A and H (s) = H (s)H (s); (2) N(m; n) is the Gaussian distribution with mean m and bp hp lp covariance n. R denotes the set of real numbers and In s2 the identity matrix with dimension n by n. trac(A), Hhp(s) = ;!h = 2 0:1 rad/s; s2 + !h s + !2 denotes the trace of a square matrix A, diagf:g denotes Q h the diagonal matrix, [A]ij denotes the i; jth element of Q = damping factor = 0:707; (3) the matrix A and 0 denotes the zero vector/matrix. This paper is concerned with asymptotic mean !2 H (s) = : (4) square stability and reference tracking of autonomous lp s2 + ! s + !2 vehicles over AWGN communication channel, as is Q shown in the block diagram of Figure 1. The building It will be shown in the next section that the nonlinear blocks of Figure 1 are described below. dynamic system (1) together with the bandpass lter has an approximate linear dynamic system with n = 7 0 Dynamic system: The dynamic system is the follow- states X(t) = x1(t) : : : xn(t) , in which these ing time-invariant unicycle system [16]: states and the matrix gain 8 2 3 c11(t) c12(t) ::: c1n(t) <>x_(t) = v(t) cos((t)) 6 7 6 c21(t) c22(t) ::: c2n(t) 7 y_(t) = v(t) sin((t)) (1) C(t) = ; > 4 ::: 5 : _ (t) = u(t) cn1(t) cn2(t) ::: cnn(t) A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 1595 form the encoder output as F (t) = C(t)(X(t) X^(t)), of !, and all other harmonics are eliminated. In other ^ 0 where X(t) = x^1(t) ::: x^n(t) is the estima- words, the output of the bandpass lter is given in the tion of states. following: y (t) = a sin(!t) + b cos(!t): (6) Decoder: Decoder is the minimum mean square f 1 1 estimator or the Kalman lter, also known as Linear Having that, we call the nonlinear dynamic system Quadratic Estimator (LQE). At each time instant t, that is cascaded with the bandpass lter with the the Kalman lter generates an estimation X^(t) using high cut-o frequency of !, a quasi-linear system [32], the channel output Y (t). because we can nd a linear dynamic system with the input u(t) = cos(!t) and the output yf (t) with the Controller. Controller is a certainty equivalent con- following transfer function: troller of the following form: u(t) = l(t)X^(t) + (t). The objective of this paper is to nd the matrix H(j!) = jH(j!)j\H(j!); (7) gain C(t) and u(t) to force the positions x(t) and y(t) p a2 + b2 and the heading angle (t) follows the desired paths. jH(j!)j = 1 1 ; (8)

3. Implementation of the describing function a1 method \H(j!) = arctan rad: (9) b1 In this section, we rst present the idea of the describ- This means that the nonlinear dynamic system can ing function to obtain the approximate linear dynamic be represented by a linear dynamic system with the system from a nonlinear time-invariant dynamic sys- above transfer function called the describing function tem. Then, we obtain the describing function for the of the nonlinear dynamic system. Note that the nonlinear dynamic system of (1). describing function represents the transfer function 3.1. The Describing Function of the approximate linear dynamic system and the This subsection is borrowed from [29]. For all nonlinear transfer function represents the response of the time - invariant dynamic systems that respond period- dynamic system to the input signal when the initial ically to sinusoidal inputs, we can nd an approximate conditions are set to be zero. Hence, the describing linear dynamic system, as de ned below [32]. function is obtained for zero initial conditions. Consider an Single-Input Single-Output (SISO) 3.2. The approximate linear system for the nonlinear time-invariant dynamic system with periodic unicycle model outputs in response to periodic inputs. Suppose For a given xed forward velocity v(t), the nonlinear that this nonlinear dynamic system is excited by the dynamic system (1) has periodic outputs to periodic following input: u(t) = cos(!t), where > 0 is large inputs u(t) = cos(!t). Hence, in the block diagram of enough to excite all modes of the nonlinear system. Figure 1, as this dynamic is cascaded with the bandpass Then, as the output is a periodic signal, it has a Fourier lter, it has an approximate linear system description. series representation that includes all harmonics of the To obtain this describing function, we rst assume that input with frequency of !. That is: the input is u(t) = cos(!t); andx _(t) andy _(t) are the X1 outputs of System (1), hence the inputs of the bandpass y(t) = y + (a sin(i!t) + b cos(i!t)); f1 d i i lter. Therefore, we obtain Hx_ (s) = 2 and f2s +f3s+f4 i=1 c1 Hy_ (s) = 2 transfer functions as describing c2s +c32s+c4 Z 2 functions from the input to the outputsx _(t) andy _(t), ! ! respectively (f s and c s are real coecients). Then, yd = y(t)dt; i i 4 2 ! obviously for obtaining the describing functions from input u(t) to outputs x(t) and y(t), we must multiply Z 2 ! ! 1 s to these transfer functions. Note that as is clear ai = (y(t) sin(i!t))dt; 2 2 from Figure 1, the inputs to the bandpass lter are ! x(t) and y(t). However, we can assume that these Z 2 ! ! inputs are obtained by integration ofx _(t) andy _(t); bi = (y(t) cos(i!t))dt: (5) thus, for the simplicity of obtaining the describing 2 2 ! functions, by moving this integration operator after Now, if this nonlinear system is cascaded with a the transfer function of the bandpass lter, we can bandpass lter with a high cut-o frequency of !, then assume that the outputs of the nonlinear system and we have a periodic output at the end of the lter which the inputs to the lter arex _(t) andy _(t). Also, note consists of only the rst harmonic with the frequency that from Eq. (1), it follows that the transfer function 1596 A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605

1 between input u(t) and output (t) is H(s) = s . value of is unknown and hence N(X0;Q0) That is, the relation between u(t) and (t) is linear. (Q0 is diagonal) is treated as the Gaussian random The equivalent state space representation of the variable with known mean and variance. Note that 0 ~ ~ approximate linear system has seven states: X(t) = N(t) = n1(t) : : : nn(t) i:i:d: N(0; R)(R = 0 [x1(t) x2(t); : : : ; x7(t)] which correspond to d1x(t) + diagfr~1; :::; r~ng) is the additive noise of the MIMO d2x_(t) + x(t), d3x(t) + d4x_(t), d5x(t) + d6x_(t), d7y(t) + channel and (ni(t) N(0; r~i) is the additive noise of d8y_(t) + y(t), d9y(t) + d10y_(t), d11y(t) + d12y_(t) and the ith path of the MIMO parallel AWGN channel, (t), respectively (dis are real coecients). The input which can be treated as the measurement noise proof this representation is u(t) and the output vector is vided the channel input power constraint is met. The [x(t) y(t) (t)]0 with the system matrices: objective here is to achieve mean square asymptotic ref- 2 3 erence tracking for the linear system (10) with n states Ax 0 0 4 5 0 and n outputs over the MIMO parallel AWGN channel. A = 0 Ay 0 ;B = [Bx By B] 0 0 A Throughout, it is assumed that the system matrix A has real eigenvalues real multiple eigenvalues and 2 3 distinct complex conjugate eigenvalues as the system Cx 0 0 4 5 matrix A that corresponds to the describing function C = 0 Cy 0 ; of the nonlinear system (1) including these types 0 0 C of eigenvalues. According to the previous section, where: 2 3 2 3 for the unicycle model, the system matrix A of the 0 0 0 0 0 0 approximate linear system is in the real Jordan form. 4 5 4 5 Ax = 0 e1 e2 ;Ay = 0 e5 e6 ; Therefore, throughout this section, it is assumed that 0 e2 e1 0 e6 e5 System (10) can be decomposed to several decoupled sub-systems; hence, for each sub-system, an encoder A =0;Bx = [e9 e10 e11];By = [e12 e13 e14]; and a decoder are designed separately. Note that the Jordan block associated with a real eigenvalue i(A) B = 1;Cx = [e15 e16 0];Cy = [e17 e18 0]; with multiplicity 2 is the following matrix: (A) 1 C = 1: i ; 0 (A) Hence, we need a MIMO parallel AWGN channel with i and the Jordan block associated with the complex 7 inputs and 7 outputs for transmitting x1(t) to x7(t). p Note that the equivalent state space representation of conjugate pair of eigenvalues i(A) = 1w w the approximate linear system is in the real Jordan (w 6= 0) is . form. w 4.1. Mean square asymptotic state tracking 4. Reference tracking of linear systems with In this section, it is assumed that each of the Jordan multiple real and non-real valued block is at most a 2 by 2 matrix. Then, for all three eigenvalues over AWGN channel possible cases: w Now, we extend the results of [13] to account for a) A = , ; w 2 R; w 6= 0, reference tracking and hence, the stability of linear w dynamic systems with multiple real and non-real a 1 b) A = , a 2 R, valued eigenvalues over MIMO parallel AWGN 0 a channel. In the next section, by applying these a 0 extended results on the describing functions associated c) A = 1 , a ; a 2 R; a 6= a , 0 a 1 2 1 2 with the nonlinear system (1), we address the reference 2 tracking problem of tele-operation of autonomous we nd the matrix gain C(t) for mean square asymp- vehicles, as is shown in the block diagram of Figure 1. totic state tracking of system states at the decoder. Suppose that the dynamic system in Figure 1 is 4.1.1. Sub-systems with complex conjugate eigenvalues linear with n states. Hence, the system that is seen by Suppose that the system matrix A in the linear system the remote controller is as follows: ( of (10) has the following form: _ X(t) = AX(t) + Bu(t);X(0) = ; w (10) A = ; ; w 2 R; w 6= 0: (11) Y (t) = C(t)(X(t) X^(t)) + N(t) w where X(0) is the initial state that is known for Then, we have the following proposition for mean the encoder, but is unknown for the remote con- square asymptotic state tracking of system states at troller, that is, for the remote controller the exact the decoder. A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 1597

( 12)t Proposition 4.1. Consider the block diagram of Fig- p_22(t)=(2 1)p22(t) ) p22(t)=e p22(0): ure 1 described by the linear system (10) with the (17) w system matrix A = ; ; w 2 R; w 6= 0. w Thus, in the above selection, we have: Suppose that P o1 > 2 and P o2 > 2,[Q0]12 = 0, e( 12)tp (0) 0 [Q ] = [Q ] andr ~ =r ~ . Let = min(P o ; P o ). 11 0 11 0 22 1 2 1 1 2 P (t) = ( 2)t ; (18) Then, by choosing the encoder matrix gain as: 0 e 1 p22(0) 2q 3 when the decoder description is as follows: 1r~1 0 C(t) = 4 p11(qt) 5 ; 1r~2 ^_ ^ ^ 0 X(t) = AX(t) + Bu(t) + K(t)Y (t); X(0) = X0; p22(t) and the following decoder: K(t) = P (t)C 0 (t)R~1: (19) _ X^(t)=AX^(t)+Bu(t)+K(t)Y (t); X^(0) = X ; (12) 0 Now, assuming that 1 > 2, we have:

0 ~1 ~ 2 K(t) = P (t)C (t)R ; R = diagfr~1; r~2g; (13) lim E[(x1(t) x^1(t)) ] = lim p11(t) t!1 t!1

0 0 1 ( 12)t P_ (t) = A P (t) + P (t)A P (t)C (t)R~ C(t)P (t); = lim e p11(0) = 0: t!1 0 p11(t) p12(t) Similarly, we have: P (0)=Q0;P (t)=P (t)= 0; (14) p12(t) p22(t) 2 lim E[(x2(t) x^2(t)) ] = lim p22(t) = 0: we have mean square asymptotic state tracking of t!1 t!1 system states at the decoder. Hence: 0 0 Proof. The encoder matrix gain has the following P (t) ! P = : general form: 0 0 c (t) c (t) This completes the proof. C(t) = 11 12 ; c21(t) c22(t) subsequently, the decoder can be extracted by Eq. (15) 4.1.2. Sub-systems with real multiple eigenvalues as shown in Box I (for the simplicity of presentation, Suppose that the system matrix A in the linear system the dependency on the timeq index, t, is dropped).q Now, of (10) has the following form: 1r~1 1r~2 by substituting c11(t) = , c22(t) = and p11(t) p22(t) a 1 A = ; a 2 R: (20) c12(t) = c21(t) = 0, we havep _12(t) = 0 () p12(t) = 0 a p12(0) = [Q0]12 = 0) and: Then, we have the following proposition for mean ( 12)t p_11(t)=(2 1)p11(t) ) p11(t) = e p11(0): square asymptotic state tracking of system states at (16) the decoder:

1 1 2 2 1 2 2 1 p_11 = 2p11 2wp12 (2p11p12c11c12r~1 + 2p11p12c21c22r~2 + p11c11r~1 + p12c12r~1

2 2 1 2 2 1 +p11c21r~2 + p12c22r~2 )

1 1 2 2 1 2 2 1 p22_ = 2p22 + 2wp12 (2p22p12c22c21r~2 + 2p22p12c21c11r~1 + p22c22r~2 + p12c21r~2

2 2 1 2 2 1 +p22c12r~1 + p12c11r~1 )

1 2 2 2 p12_ = 2p12 wp22 + wp11 r~1 (p11p12c11 + p12c11c12 + p11p22c11c12 + p12p22c12)

1 2 2 2 r~2 (p11p12c21 + p12c22c21 + p11p22c22c21 + p12p22c22) (15)

Box I 1598 A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605

Proposition 4.2. Consider the block diagram of Fig- by substituting: ure 1 described by the linear system (10) with the s s a 1 r~1 r~1 system matrix A = , a 2 R. Suppose that c11(t) = ; c12(t) = ; 0 a 2p11(t) 2p22(t) P o > 2a and P o > 2a,[Q ] = 0, [Q ] = [Q ] , 1 2 0 12 0 11 0 22 s s andr ~1 =r ~2. Then, by choosing the encoder matrix r~2 r~2 gain as: c21(t) = ; c22(t) = ; 2p11(t) 2p22(t) 2q q 3 r~1 r~1 we havep _12(t) = 0 () p12(t) = p12(0) = [Q0]12 = 0) C(t) = 4q 2p11(t) q 2p22(t) 5 ; and: r~2 r~2 2p11(t) 2p22(t) 1 p_ (t) = 2a + p (t) ) p_ (t) p 11 2 2 11 11 where = 2 1 and = min(P o ; P o ), and 1 1 1 1 2 the following decoder: 1 + 2 = 2a p (t) ); (25) 2 11 ^_ ^ ^ X(t)=AX(t)+Bu(t)+K(t)Y (t); X(0)=X0; (21) ( 12a)t p_11(t)=(2a 1)p11(t) ) p11(t)=e p11(0); 0 ~1 ~ (26) K(t) = P (t)C (t)R ; R = diagfr~1; r~2g; (22) 1 _ 0 0 ~1 p_ (t) = 2a + p (t) ) p_ (t) P (t) = A P (t) + P (t)A P (t)C (t)R C(t)P (t); 22 2 2 22 22 2 0 p11(t) p12(t) 1 + P (0)=Q0;P (t)=P (t) = 0; (23) = 2a p22(t) ); (27) p12(t) p22(t) 2

( 12a)t we have mean square asymptotic state tracking of p_22(t)=(2a 1)p22(t) ) p22(t)=e p22(0): system states at the decoder. (28)

Proof. The encoder matrix gain has the following Thus, by the above selection, we have: general form: ( 2a)t e 1 p11(0) 0 P (t) = ( 2a)t ; (29) 0 e 1 p22(0) c11(t) c12(t) C(t) = ; when the decoder description is as follows: c21(t) c22(t) ^_ ^ ^ subsequently, the decoder can be extracted by Eq. (24) X(t) = AX(t) + Bu(t) + K(t)Y (t); X(0) = X0; as shown in Box II (for the simplicity of presentation, 0 ~1 the dependency on the time index, t, is omitted). Now, K(t) = P (t)C (t)R : (30)

1 1 2 2 1 2 2 1 p_11 = 2ap11 (2p11p12c11c12r~1 + 2p11p12c21c22r~2 + p11c11r~1 + p12c12r~1

2 2 1 2 2 1 +p11c21r~2 + p12c22r~2 )

1 1 2 2 1 2 2 1 p22_ = 2ap22 + 2p12 (2p22p12c22c21r~2 + 2p22p12c21c11r~1 + p22c22r~2 + p12c21r~2

2 2 1 2 2 1 +p22c12r~1 + p12c11r~1 )

1 2 2 2 p12_ = 2ap12 + p11 r~1 (p11p12c11 + p12c11c12 + p11p22c11c12 + p12p22c12)

1 2 2 2 r~2 (p11p12c21 + p12c22c21 + p11p22c22c21 + p12p22c22) (24)

Box II A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 1599

Now, assuming that P o1 > 2a and P o2 > 2a, we we have mean square asymptotic state tracking of 2 have limt!1 E[(x1(t) x^1(t)) ] = limt!1 p11(t) = system states at the decoder. ( 2a)t limt!1 e 1 p11(0) = 0. Similarly, we have 2 limt!1 E[(x2(t) x^2(t))] = limt!1 p22(t) = 0. Proof. The encoder matrix gain has the following 0 0 general form: Hence, P (t) ! P = : This completes the proof. 0 0 c (t) c (t) C(t) = 11 12 ; 4.1.3. Sub-systems with real distinct eigenvalues c21(t) c22(t) Suppose that the system matrix A in the linear system subsequently, the matrix P (t) which represents the co- of (10) has the following form: variance matrix of the decoding error can be extracted by Eq. (35) as shown in Box III (for the simplicity a1 0 A = ; a1; a2 2 R; a1 6= a2: (31) of presentation, the dependency on the time index t 0 a2 q P o1r~1 is dropped). Now, by substituting c11(t) = , Then, we have the following proposition for mean q p11(t) square asymptotic state tracking of system states at P o2r~2 c22(t) = p (t) and c12(t) = c21(t) = 0, we have the decoder: 22 p_12(t) = 0 and p12(t) = p12(0) = [Q0]12 = 0. Subsequently, we have: Proposition 4.3. Consider the block diagram of Fig- p_ (t) = (2a P o )p (t) ) p (t) ure 1 described by the linear system (10) with the 11 1 1 11 11 a1 0 (P o12a1)t system matrix A = , a1; a2 2 R; a1 6= a2. = e p11(0); (36) 0 a2 Suppose that [Q0]12 = 0, P o1 > 2a1 and P o2 > 2a2 p_22(t) = (2a2 P o2)p22(t) ) p22(t) (P o1 and P o2 are the channel input power constraint). Then, by choosing the encoder matrix gain as: = e(P o22a2)tp (0): (37) 2q 3 22 P o1r~1 0 Thus, by the above selection, we have: C(t) = 4 p11(qt) 5 ; P o r~ 0 2 2 e(P o12a1)tp (0) 0 p22(t) 11 P (t) = (P o 2a )t ; (38) 0 e 2 2 p22(0) and a decoder with the following description: when the decoder description is as follows: ^_ ^ X(t) = AX(t) + Bu(t) + K(t)Y (t); ^_ ^ ^ X(t) = AX(t) + Bu(t) + K(t)Y (t); X(0) = X0;

X^(0) = X ; (32) 0 0 K(t) = P (t)C (t)R~1: (39) 0 ~1 ~ K(t) = P (t)C (t)R ; R = diagfr~1; r~2g; (33) Now, assuming that P o1 > 2a1 and P o2 > 2a2, we have: 0 0 1 P_ (t) = A P (t) + P (t)A P (t)C (t)R~ C(t)P (t); 2 lim E[(x1(t) x^1(t)) ] = lim p11(t) t!1 t!1 0 p11(t) p12(t) P (0) = Q ;P (t) = P (t) = 0;(34) (P o12a1)t 0 = lim e p11(0) = 0: p12(t) p22(t) t!1

1 1 2 2 1 2 2 1 p_11 = 2a1p11 (2p11p12c11c12r~1 + 2p11p12c21c22r~2 + p11c11r~1 + p12c12r~1

2 2 1 2 2 1 +p11c21r~2 + p12c22r~2 )

1 1 2 2 1 2 2 1 p22_ = 2a2p22 (2p22p12c22c21r~2 + 2p22p12c21c11r~1 + p22c22r~2 + p12c21r~2

2 2 1 2 2 1 +p22c12r~1 + p12c11r~1 )

1 2 2 2 p12_ = a1p12 + a2p12 r~1 (p11p12c11 + p12c11c12 + p11p22c11c12 + p12p22c12)

1 2 2 2 r~2 (p11p12c21 + p12c22c21 + p11p22c22c21 + p12p22c22) (35)

Box III 1600 A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605

Similarly, we have: x(0) y(0) (0) 0 is unknown for the remote controller and has the Gaussian distribution with diagonal 2 lim E[(x2(t) x^2(t)) ] = lim p22(t) = 0: covariance matrix. Both positions x(t) and y(t) must t!1 t!1 be controlled by one input u(t) as it is assumed that Hence: the forward velocity v(t) is xed. In the following, r(t), rx(t), and ry(t) are the reference signals for (t), 0 0 0 P (t) ! P = : x(t), and y(t), respectively (R(t) = [rx(t) ry(t) r(t)] ), 0 0 andx ^(t),y ^(t) and ^(t) are mean square estimation of This completes the proof. x(t), y(t), and (t) at the decoder, respectively, which are available for the remote controller. Now, from 4.2. Asymptotic reference tracking Figure 2 it follows that to control both x(t) and y(t) Now, to obtain the control signal u(t) for tracking the by one input u(t), we must have the reference signal reference signal R(t), we consider the following cost ry (t)y^(t) r(t) = arctan( ) [34], which forces x(t) and functional: rx(t)x^(t) y(t) to follow rx(t) and ry(t), respectively, when (t) Z tracks r (t) andx ^(t) ! x(t),y ^(t) ! y(t). 1 t1 J = lim E[[X(t) R(t)]0Q[X(t) R(t)] To this end, we notice that the equivalent t !1 t 1 1 0 state space representation of the approximate linear system of the nonlinear unicycle model cascaded +ru2(t)]dt; Q = Q0 0; r > 0: (40) with the bandpass lter, has seven states: X(t) = 0 Subsequently, the control signal u(t) is obtained by [x1(t) x2(t) : : : x7(t)] ; hence, we need an MIMO minimizing the above cost functional subject to the parallel AWGN channel with 7 inputs and 7 outputs dynamic system (10). From the classical LQG results for transmitting x1(t) to x7(t). Now, we use the [33], it follows that u(t) = l(t)X^(t) + (t), where: encoder and decoder proposed in the previous section with matrices derived from A, B, and C, as given l(t) = r1B0P~(t); (41) in Section 3. Since the system matrix A is in the real Jordan form, the encoder and decoder proposed (t) = r1B0S(t); (42) in Section 4 can be used for di erent 2 by 2 sub- 0 0 systems for transmitting [x2(t) x3(t)] and [x5(t) x6(t)] and P~(t) and S(t) are the solutions of the following and their reconstruction at the decoder, and, also, for equations: the scalar sub-systems for transmitting x1(t), x4(t), and x7(t) and their reconstruction. Then, using the ^ P~_ (t)=P~(t)AA0P~(t)Q+P~(t)Br1B0P~(t); (43) controller u(t) = l(t)(t) + (t) with gains computed by Eqs. (41) to (44) with A = A = 0, B = B = 1, S_(t) = [A0 P~(t)Br1B0]S(t) + QR(t): (44) and R(t) = r(t), we have mean square asymptotic

Under the assumption that the pair (A; B) is sta- bilizable and the pair (In;A) is detectable, we have P~(t) ! P~ [33], where P~ is the unique symmetric non- negative de nite stabilizing solution of the corresponding Algebraic Riccati equation; hence, J ! J = 0. This indicates that E[X(t)R(t)]0Q[X(t)R(t)] ! 0; 2 and hence, E[(xi(t) ri(t)) ] ! 0, i = 1; 2; :::; n, 0 (R(t) = r1(t) : : : rn(t) ).

5. Reference tracking of the tele-operation of autonomous vehicles In this section, the results of previous sections are applied to address the reference tracking problem of the tele-operation system described by the block diagram of Figure 1. In this block diagram, the nonlinear dynamics of the miniature drones, autonomous road vehicles, and autonomous underwater vehicles Figure 2. An autonomous vehicle with current positions are described by the unicycle model of (1). In this x(t) and y(t) moving towards the desired positions rx(t) tele-operation system, the initial conditions vector and ry(t). A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 1601

2 3 tracking of the reference signal r(t) by (t), and, 0 0 0 4 5 hence, the desired reference tracking. Note that at Ay = 0 0:1 2:914 ;A = 0; the decoder,x ^(t),y ^(t), and ^(t) are obtained as 0 2:914 0:1 follows: [^x(t)y ^(t) ^(t)]0 = CX^(t), which converge to [x(t) y(t) (t)]0 = CX(t), as X^(t) ! X(t) in mean Bx = [0:1176 0:1176 0:0040]; square sense. By = [0:1176 0:1176 0:0040];B = 1;

6. Simulation results Cx = [1 1 0];Cy = [1 1 0];C = 1: For the purpose of illustration, consider the block Hence, we need a MIMO parallel AWGN channel diagram of Figure 1 with the nonlinear dynamic of with 7 inputs and 7 outputs for transmitting x1(t) to (1). For the remote controller, the initial conditions x7(t), which has the following speci cation: (x(0), y(0), (0)) are unknown and have the following 0 description [x(t) y(t) (t)] N(0; 3I3). The N(t) i:i:d: N(0; diagf0:5; 0:5; 0:5; 0:5; 0:5; 0:5; 0:5g) autonomous vehicle must track a circle with the with the power constraints: P oi = 1 for i 2 center located at (xr; yr) and the radius of with 0 f1; 2; 3; :::; 7g, which meet the requirements of the the angular velocity of !r. Therefore, [x(t) y(t) (t)] 0 propositions of Section 4. must track the reference signal [rx(t) ry(t) r(t)] , Figure 3 to Figure 5 illustrate that the system where rx(t) = xr + cos(!rt), ry(t) = yr + sin(!rt) ry (t)y^(t) outputs x(t), y(t), and (t) track the reference signals and r(t) = arctan( r (t)x^(t) ). Note that as x rx(t), ry(t), and r(t), respectively, for = 3, = 1, r_x(t) = !r sin(!rt) andr _y(t) = !r cos(!rt), !r = 1, and xr = yr = 0 using the technique proposed for the simplicity of design, we choose the forward in Section 5. Figure 6 illustrates the control signal u(t). velocity constant and equal to v(t) = !r m/s. Also, Figures 7 and 8 illustrate that the autonomous To obtain the describing function for a given !, vehicle tracks the reference circle. As is clear from these e.g., ! = 1 rad/s, we apply the inputs v(t) = !r m/s gures, the proposed tracking technique is able to force and u(t) = !r cos(!t) rad/s ( 1 is a gain to the nonlinear system asymptotically to track the refer- excite all modes of the nonlinear system) to System ence signals very well. The Root Cumulative Square (1). Then, from this input and the corresponding Error (RCSE) computed for the period of [20, 50] outputs of the bandpass lter, the describing func- qR second (i.e., 50[(x(t) r (t))2 + (y(t) r (t))2]dt) tions for = 3, !r = 1 rad/s and = 1 are 20 x y 1 for di erent choices of ! is shown in Table 1. computed as Hx(s) = s(s2+0:2s+8:5) and Hy(s) = 1 , respectively. The equivalent linear state This table indicates that in order to have a satis- s(s2+0:2s+8:5) factory reference tracking, the high cut o frequency of space representation of this system has seven states: the bandpass lter is better to be at least equal to ! . X(t) = [x (t); x (t); : : : ; x (t)]0 which correspond to r 1 2 7 To compare the performance of the proposed 0:1176x(t)+0:0235x _(t)+x(t), 0:1176x(t)0:0235x _(t), technique, we apply the proposed technique and the 0:0040x(t) 0:3424x _(t), 0:1176y(t) + 0:0235y _(t) + y(t), 0:1176y(t) 0:0235y _(t), 0:0040y(t) 0:3424y _(t) and (t), respectively. The input of this representation is u(t) and the output vector is [x(t) y(t) (t)]0 with the system matrices: 2 3 Ax 0 0 4 5 0 A = 0 Ay 0 ;B = [Bx By B] ; 0 0 A 2 3 Cx 0 0 4 5 C = 0 Cy 0 ; 0 0 C where: 2 3 0 0 0 Figure 3. x(t) position of the system and its reference A = 40 0:1 2:9145 ; x signal rx(t). It is observed that after less than 4 seconds 0 2:914 0:1 x(t) converges to rx(t). 1602 A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605

Figure 7. x(t) y(t) time diagram Figure 4. y(t) position of the system and its reference signal ry(t). It is observed that after less than 4 second y(t) converges to ry(t).

Figure 8. x(t) y(t) diagram

Figure 5. The heading angle of the system (t) and its Table 1. Root Cumulative Square Error (RCSE) vs !. reference signal r(t). It is observed that (t) converges to r(t) immediately. ! RCSE 0.2 7.91 1 5.53 10 5.59 100 5.48 1000 5.45

feedback linearization control technique of [35] (with the linearized system of (9) and (10) of [35]) to the block diagram of Figure 1 for the reference signals of rx(t) = 0:0005t and ry(t) = 0:0002t. The RCSE computed for the period of [20, 50] second when the proposed technique is used is 5:43 (see Figure 9) and the RCSE computed for the feedback linearization control technique of [35] is 17:36 (see Figure 10). From these gures, it is clear that the proposed technique has a better performance in the presence of communication Figure 6. The control signal input to the system, which imperfections. determines the angular velocity. Over the packet erasure channel, which is subject A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 1603

to random packet dropout and quantization imper- allowable powers, P oi (i = 1; 2), the encoder gain C(t) fections, [36] presented a novel technique, which is is computed in Propositions 4.1 { 4.3 so that: based on the linearization method, for the reference h i tracking of the unicycle model of (1). Comparison ^ var[C(t)(X(t) X(t))] = P oi: of the simulation results of this section with those of ii [36] for tracking a circle reveals that the approximate However, by using the proposed coding scheme and method presented in this paper for the unicycle model as P (t) ! 0, some elements of the matrix gain C(t) is as good as the linearization method used in [36]. tend to in nity. To avoid this situation, the following modi cations in the encoder matrix gain C(t) are Remark 6.1. i) Simulation studies illustrate that the suggested: linear approximation method proposed in this paper which is based on the describing function is good for the For the encoder matrix gain of Proposition 4.1 when particular nonlinear system considered in this paper. pii(t) < 0:01 1r~i, we replace pii(t) in the matrix gain Although the describing function is obtained for a C(t) by pii(t) + 0:01 1r~i; sinusoidal input, from Figure 6, it is clear that for the non-sinusoidal inputs, the approximation must be also For the encoder matrix gain of Proposition 4.3, when good. Otherwise, we did not get such a good reference pii(t) < 0:01P oir~i, we replace pii(t) in the matrix tracking as well as state tracking of the nonlinear gain C(t) by pii(t) + 0:01P oir~i; system (in Figures 3{5, 7, 8) for an estimator and For the encoder matrix gain of Proposition 4.2, let controller which are based on the approximate linear r~1 r~2 h = min( 2 ; 2 ); when pii(t) < 0:01h, we replace system. ii) In order to utilize the communication pii(t) in the matrix gain C(t) by pii(t) + 0:01h. channel and transmit a signal with the maximum Using the above modi cations, when P (t) ! 0, the elements of the matrix gain C(t) remain bounded. We repeated computer simulation for ! = 1, !r = 1 and tracking the circle using the modi ed encoder matrix gain; and we observed that given the expense of transmission with the power a bit less than the maximum allowable power, we get RCSE = 5.55 which is very close to the RCSE of the corresponding case with in nite gain.

7. Conclusion and direction for future research In this paper, a new technique for mean square asymptotic state tracking and reference tracking of the autonomous vehicles over analog Additive White Figure 9. x(t) y(t) time diagram for the proposed Gaussian Noise (AWGN) channel was presented. Au- technique tonomous vehicle was cascaded with a bandpass lter acting as encoder; and hence, an approximate linear dynamic system was extracted using the describing function method. Then, the results of [13] were extended to account for systems with multiple real and non- real valued eigenvalues over Multi-Input Multi-Output (MIMO) parallel AWGN channel. Subsequently, by applying the extended results on the describing function, a technique for mean square asymptotic state tracking and reference tracking of autonomous vehicles was presented. Finally, the satisfactory performance of the proposed technique was illustrated using computer simulations. In addition to the describing function method, there are other methods, e.g., based on the Volterra series theorem [37{39] for nonlinear analysis in the Figure 10. x(t) y(t) time diagram for the feedback frequency domain. Using the Volterra series expansion, linearization technique of [35]. the Generalized Frequency Response Function (GFRF) 1604 A. Parsa and A. Farhadi/Scientia Iranica, Transactions D: Computer Science & ... 28 (2021) 1592{1605 was de ned in [40], which is multivariate Fourier trans- 12. Charalambous, C.D. and Farhadi, A. \LQG optimality form of the Volterra kernels. This provides a useful and separation principle for general discrete time par- concept for approximation of the nonlinear systems. tially observed stochastic systems over nite capacity Therefore, for future, it is interesting to compare the communication channels", Automatica, 44(12), pp. performance of the proposed technique with those 3181{3188 (2008). of other approximation methods (e.g., GFRF). In 13. Charalambous, C.D., Farhadi, A., and Denic, S.Z. addition, it is interesting to consider a more realistic \Control of continuous-time linear Gaussian systems model for the dynamic system (1), e.g., a dynamic over additive Gaussian wireless fading channels: A system subject to measurement and process noises. separation principle", IEEE Trans. Automat. Contr., 53(4), pp. 1013{1019 (2008). This research direction is currently under investigation in our research team. 14. Farhadi, A. \Stability of linear dynamics systems over the packet erasure channel: A co-design approach", International Journal of Control, 88(12), pp. 2488{ References 2498 (2015). 15. Farhadi, A. \Feedback channel in linear noiseless 1. Elia, N. \When Bode meets Shannon: Control- dynamics systems controlled over the packet erasure oriented feedback communication schemes", IEEE network", International Journal of Control, 88(8), pp. Trans. Automat. Contr., 49(9), pp. 1477{1488 (2004). 1490{1503 (2015). 2. 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26. Farhadi, A. and Charalambous, C.D. \Stability and Biographies reliable data reconstruction of uncertain dynamics systems over nite capacity channels", Automatica, 46(5), pp. 889{896 (2010). Ali Parsa received his BSc and MSc degrees in 27. Farhadi, A. \Sub-optimal control over AWGN commu- Electrical Engineering from the University of Tehran, nication network", European Journal of Control, 37, Iran in 2011 and 2014, respectively. He designed and pp. 27{33 (2017). implemented several systems and devices in the eld 28. Li, Y., Chen, J., Tuncel, E., and Su, W. \MIMO of Smart Grid and Smart Home in the Mechatronics control over additive white noise channels: Stabiliza- Lab. of the University of Tehran. He received the PhD tion and tracking by LTI controllers", IEEE Trans. degree from the Department of Electrical Engineering Automat. Contr., 61(5), pp. 1281{1296 (2016). at Sharif University of Technology, Iran in 2018. His 29. Parsa, A. and Farhadi, A. \Reference tracking of PhD research was concerned with theoretical aspects nonlinear dynamic systems over AWGN channel us- of remote estimation and control of nonlinear systems ing describing function", Scientia Iranica, 26(3), pp. over communication channels subject to imperfections 1727{1735 (2019). (wireless, WiFi, and the Internet) with applications 30. Liang, X. and Xu, J. \Control for networked control in telemetry and tele-operation of autonomous vehi- systems with remote and local controllers over unre- cles. Also, he has worked at the Department of liable communication channel", Automatica, 98, pp. Research and Development of Crouse Co., which is 86{94 (2018). the largest automotive part manufacturer in Iran, for 3 31. Imer, O.C., Yuksel, S., and Basar, T. \Optimal control years. Currently, he is a postdoctoral researcher at the of LTI systems over unreliable communication link", HiPeRT Lab. of the University of Modena and Reggio Automatica, 42(9), pp. 1429{1439 (2006). Emilia (UNIMORE), Modena, Italy. His main research 32. Slotine, J.J. and Li, W., Applied Nonlinear Control, areas include networked control systems, autonomous Prentice-Hall (1991). vehicles, automotive, Internet of Things (IoT), and 33. Kwakernaak, H. and Sivan, P., Linear Optimal Control smart home. Systems, Wiley-Interscience (1972). 34. Canudas de Wit, C., Sicilian, B., and Bastin, G., Alireza Farhadi received PhD degree in Electrical Theory of Robot Control, Springer - Verlag (1996). Engineering from the University of Ottawa, Ontario, 35. Oriolo, G., De Luca, A., and Vendittelli, M. \WMR Canada in 2007. After receiving PhD degree, Dr. control via dynamic feedback linearization: Design, Farhadi worked as a Postdoctoral Fellow at the De- implementation, and experimental validation", IEEE partment of Electrical Engineering at the University Trans. on Control Systems Technology, 10(6), pp. 835{ of Ottawa (2008{2009) and the French National Insti- 852 (2002). tute for Research in Computer Science and Control 36. Parsa, A. and Farhadi, A. \Measurement and control (INRIA), Grenoble, France (2010{2011). He then of nonlinear dynamic systems over the Internet (IoT): Applications in remote control of autonomous vehi- worked as a Research Fellow (academic level B4) at the cles", Automatica, 95, pp. 93{103 (2018). Department of Electrical Engineering of the University 37. Jing, X. and Lang, Z., Frequency Domain Analysis and of Melbourne, Australia (2011{2013). In September Design of Nonlinear Systems Based on Volterra Series 2013, he joined the Department of Electrical Engineer- Wxpansion - A Parametric Characteristic Approach, ing of Sharif University of Technology as Assistant Springer International Publishing Switzerland (2015). Professor. Currently, he is an Associate Professor 38. Rugh, W.J., Nonlinear System Theory: The at the Department of Electrical Engineering of Sharif Volterra/Wiener Approach, Baltimore, MD: The Johns University of Technology. Dr. Farhadi specializes in Hopkins Univ. Press (1981). the design, optimization and control of large-scale and 39. Worden, K. and Tomlinson, G.R., Nonlinearity In distributed Mechatronics systems and networks with a Structural Dynamics: Detection, Identi cation And particular focus on automated irrigation network, in- Modeling, Bristol, U.K.: Institute of Physics (2001). telligent inspection and fault diagnosis systems, smart 40. George, D.A., Continuous Nonlinear Systems, MIT oil eld, eets of autonomous under water, unmanned Research Lab. Electronics, Cambridge, MA, Tech. aerial and autonomous road vehicles, smart building, Rep. 355 (1959). IIOT, industry 4.0 and intelligent embedded systems.