Robust Discrete-Time Lateral Control of Racecars by Unknown Input Observers

This brief addresses the robust lateral control problem for self-driving racecars. It proposes a discrete-time estimation and control solution consisting of a delayed unknown input-state observer (UIO) and a robust tracking controller. Based on a nominal vehicle model, describing its motion with respect to a generic desired trajectory and requiring no information about the surrounding environment, the observer reconstructs the total force disturbance signal, resulting from imperfect knowledge of the time-varying tire-road interface characteristics, presence of other vehicles nearby, wind gusts, and other model uncertainty. Then, the controller actively compensates the estimated force and asymptotically steers the tracking error to zero. The brief also presents a closed-loop stability proof of the method, ensuring perfect asymptotic estimation and tracking by the controlled vehicle. The proposed solution advantageously needs no a-priori information about the total disturbance boundedness, additional variables to model uncertainty, or observer parameters to be tuned. Its effectiveness and superiority to existing methods are studied in theory and shown in simulations where a full racecar model, based on the vehicle dynamics blockset, is required to track aggressive maneuvers. Through a faster and more accurate disturbance estimation, the solution robustly ensures better dynamic responses even with measurement noise.


I. INTRODUCTION
T HE motion of a vehicle is governed by the traction force, generated at the wheels, and all resistance forces that apply to it [1]. The traction force results from the complex interaction between the tire contact patch and the road. It nonlinearly depends on the driving motor torque and a set of parameters that vary with time and can only be identified for typical road typologies (cf., Burckhardt's [2] and Pacejka's [3] models). Resistance forces include the aerodynamic load losses and rolling resistance which have known expressions under nominal conditions. Yet, they still depend on coefficients whose accurate knowledge requires perfect modeling of the vehicle profile. Other external forces are disturbance signals, not easy to predict, such as wind gusts (normally modeled via stochastic Dryden winds [4]), Manuscript  airflow-turbulence due to surrounding vehicles, and road bank angle changes.
To obtain precise motion regulation for a vehicle, robust control techniques are naturally advocated. The technique presented in [5] achieved improved performance on a fourwheel vehicle. A comparison of the immersion and invariance principle, sliding mode control, and passivity techniques is reported [6]. These methods require a-priori knowledge of the maximum disturbance magnitude, involve high control gains and integral terms, or discontinuous control signals, which may result in over-conservative strategies, needing bigger efforts to prevent instability. Another way to cope with uncertainty is to estimate and compensate for the actual disturbance. A cascaded backstepping control method with an augmented observer is proposed in [7] to control the lateral dynamics of an autonomous vehicle in the presence of disturbance. However, it assumes that the disturbance slowly changes, thus limiting its applicability and effectiveness in general. Traditional disturbance estimation approaches, based on extended Kalman filters (EKFs), suffer from known disadvantages due to the necessity to calibrate noise covariance matrices and introduce additional states, whose dynamics can only be based on the generic random walk [8]. Beyond the fact that a convergence proof in the general scenario cannot be found, they result in over-delayed estimation or, even worse, divergent estimation behaviors (see the discussion in [9]).
Other more effective solutions rely on disturbance observerbased (DOB) approaches [10], [11], [12], [13]. Built upon a nominal vehicle model, they reach great estimation performance when the system state is entirely measurable. In fact, when the state is not fully accessible, they require an additional state observer, introducing further complexity and reducing the solution's efficiency. To overcome this limitation and simultaneously estimate disturbance inputs and states, extended state observers (ESOs) come in handy, which, similar to EKFs, model the deviation with respect to a nominal behavior as additional states and only require tuning a set of control parameters [14], [15], [16]. Despite their simplicity, they involve again high-gain parameters that make them often too sensitive to measurement noise and lead to peaky estimations. Also, they assume a small or negligible change over time of the disturbance signals [17]. Closely related to the ESOs is the active disturbance rejection control (ADRC) technique [18], [19], [20], [21]. It is a robust control method assuming that parameter and model uncertainties are modeled as a disturbance input vector, estimated via an ESO, and finally compensated via a state-feedback control law [14]. Despite its simplicity, its closed-loop stability remains strongly linked to the underlying ESO's primary assumption that the disturbance acting on the system has a negligible rate of change, which leads to worse estimation performance in systems with fast varying disturbances.
Within this context, this brief proposes a novel discrete-time robust control solution for racecar lateral dynamics, which uses a linear unknown input-state observer (UIO) [22]. Built upon a double-track nominal model, the observer reconstructs the total force disturbance signal, resulting from imperfect knowledge of the time-varying tire-road interface characteristics, the presence of other nearby vehicles, wind gusts, model reduction, and uncertainty. Then, a controller actively compensates the estimated force and asymptotically steers the tracking error to zero. Among other advantages, the proposed solution needs no a-priori information about the total disturbance boundedness, additional variables to model uncertainty, or tuning of observer parameters. It allows faster and more accurate disturbance estimation and better dynamic response.
Contribution: The brief's contribution is at least fourfold. First, by starting from the so-called double-track system, the brief derives a nominal model of the lateral racecar dynamics, in a form where the input-disturbance and state estimation can be addressed by using delayed UIO theory; second, a robust control law is devised which uses state and disturbance estimates to ensure perfect asymptotic tracking of any desired trajectory, and the full state closed-loop asymptotic stability is formally proved with convergence speed guarantees; third, the superiority of the proposed method to existing state-of-theart solutions is shown; finally, the effectiveness, the robustness, and the real-time implementability of the proposed solution are tested by using the models of MATLAB/Simulink's Vehicle Dynamics Blockset [23] and a Raspberry PI 4 system.

II. MODEL FORMALIZATION AND PROBLEM STATEMENT
Consider a rear-wheel drive, front-steering racecar with mass m and inertia J , moving along a flat horizontal road, i.e., with zero bank angle. The in-plane lateral dynamics of the vehicle can be described, in a noninertial frame attached to it, by the double-track model [1] that reads where y is its lateral position, ψ is its heading angle, a 1 and a 2 are its wheelbases, Y 1 and Y 2 are the front and rear lateral forces applied at its center of mass, δ v is the front steering angle and system input, F w and χ w are the lateral wind force and moment, and u is its longitudinal speed, which can be considered a time-varying parameter, resulting from the control of its longitudinal dynamics. The lateral forces Y 1 and Y 2 can be decomposed as follows: where F y i j are the front (i = 1) and rear (i = 2) forces applied at the left ( j = 1) and right ( j = 2) tires, respectively, and are nonlinear functions of the front wheel steer angles δ 1 j . Moreover, refer to Fig. 1 and assume that a global positioning system (GPS) sensor is used to measure the (X, Y ) coordinates Indicating with a y d = u 2 /R = uψ d an approximated desired lateral acceleration in body frame, the corresponding lateral acceleration error reads [1] as follows: where the actual lateral acceleration a y has been expanded as a y =ÿ +ψ u and the first equation in (1) has been used.
Furthermore, achieving precise and complete characterization of the nonlinear and possibly time-varying functions Y 1 and Y 2 requires ad hoc identification procedures which also need to be repeated over time [1]. Beyond that, the wind force and moment signals are only predictable via statistical models and hence their actual values over time remain unknown. Therefore, it is convenient to obtain a nominal vehicle model based on quantities that can be easily identified. For this purpose, one recalls that the lateral forces at the tires depend on the respective wheel slip angles α i j , for i, j = 1, 2; such dependence can be approximated, for small α i j , as F y i j = C i j α i j , where C i j are the tires' cornering stiffness coefficients, which are known with good accuracy. Moreover, the wheel slip angles can be expressed by the following formulas: with t 1 and t 2 being the front and rear vehicle tracks, v = y and r =ψ are the lateral and yaw speeds. While the functions δ 1 j (δ v ) are highly nonlinear, for small values of their argument, the following second-order Taylor expansions can be used [1]: with δ 0 1 , l, τ , and β being the static toe angle, the total wheelbase, the steering gear ratio, and the Ackermann coefficient, respectively, and ν the approximation error signal. In light of the above reasoning, the following nominal model can be assumed, involving only the nominal values of the vehicle massm, the cornering coefficientC 1 =C 11 +C 12 , and the steering gear ratioτ as follows: where w is a disturbance signal lumping together the effects due to parameter and model uncertainties, and even exogenous unknown inputs. Moreover, given a sampling time λ and a discrete time step k and defined the sampled state vector and disturbance with Z k = (e 1k ,ė 1k ) T = (e 1 (kλ),ė 1 (kλ)) T and w k = w(kλ), respectively, the model in (3) can be discretized via Euler's method and finally written in state form as follows: where δ k = δ v (kλ) is the input sample signal, y k = y(kλ), and the involved matrices are It should be noted that, even though part of the expression of the disturbance w is known, the discrete-time signal w k is assumed to be fully unknown.
Within this setting, the following problem is addressed as follows.
Problem 1: Design a robust observer-based controller for the model in (2) ensuring the asymptotic tracking of desired accelerations a y d by using only information about the lateral position y and the nominal values of mass m, front cornering coefficients, C 11 and C 12 , and steering gear ratio τ .

III. DELAYED UNKNOWN-INPUT OBSERVER (DUIO)-BASED ROBUST LATERAL VEHICLE CONTROL
The here adopted strategy to solve Problem 1 is to devise an input-state observer of the system in (4) and to provide the estimated data to a suitable controller ensuring the overall system stability. This is formalized in the two steps described in the remainder of this section, after briefly recalling the DUIO theory, while a graphical depiction of it is illustrated in Fig. 2.

A. Mathematical Framework of DUIOs
Consider a discrete-time linear system of the form where Z k ∈ R n , U k ∈ R m , and k ∈ R p are state, input, and output vectors, respectively, and k is discrete time step. Without loss of generality, assume (B T , D T ) T to be a full-column rank matrix. Given a positive integer, L ∈ N + , the L-step invertibility and observability matrices are obtained via the following recursive definitions: 2. Schematic of the estimation and control strategy. The output y k = e 1k of the racecar is buffered in the output history vector Y k , which is used by the DUIO to estimate the racecar delayed state Z k−L and unknown input w k−L ; this information along with the desired acceleration a y d are finally used by the controller to asymptotically steer the vehicle on the desired trajectory.
Then, the following can be recalled from [22]. Proposition 1 (DUIO): Given a large enough delay L, the discrete-time linear system T , and E and F satisfy the conditions as follows: , having eigenvalues in the unit circle (free solution convergence); is a DUIO for the model in (5), i.e., it generates delayed state and input estimates,Ẑ k−L andÛ k−L , asymptotically tracking the ones of model in (5), or equivalently The existence of a DUIO is connected to the solvability of A1 for some L and F, for which the following can be stated as follow.
Proposition 2 (System Invertibility): Given the sequence of matrices {H L }, for L = 1, . . . , n, Condition A1) can be solved for some F if, and only if, there exists L such that

B. Discrete-Time UIO Design
Consider the racecar model in (4) with inputs given by the steering angle δ k and the unknown disturbance w k , generated by model deviations from the nominal behavior as well as external signals, and the output given by the lateral position error e k . The following first main result can be proved as follow.
Theorem 1 (UIO Design for Racecars): The discrete-time linear system described by the iterative rulê Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
That is, the filter in (8) Consequently, the output history can be chosen as Y 2 k = (e k , e k−1 , e k−2 ) T . Condition A1 in Section III-A implies that matrix F belong to the left-nullspace of the last columns of matrix H 2 , which are given by P = 0 2×2 H 1 . To determine F, consider first a matrixN whose rows form a basis for the left-nullspace of H 1 , so that is a matrix whose rows form a basis for the left-nullspace of P. Constant p equals the unity since the system output e k is scalar. Given the null value of H 1 , for this system, it suffices to chooseN = I 2×2 . Furthermore, for any invertible matrix W , we can define a matrix N = W 1 0 1×2 0 2×1N = W I 3×3 , whose rows also form a basis for the left nullspace of P. Therefore, given the one-step observability matrix (see Section III-A), Moreover, as Proposition 2 is satisfied for a delay L = 2, the first columns of H 2 are linearly independent, which implies that the matrix have rank equal to the unity. Direct computation of the above matrix shows indeed that Splitting the rows of the matrix NO 2 on the right-hand side into two sub-matrices S 1 and S 2 , that is, , allows further expanding the expression of matrix E as follows: To finally satisfy also Condition A3, one can impose matrix E to be given by a Schur diagonal matrix as in (9). This condition can be attained by choosing the remaining part of matrix F asF 1 = −σ 1 1 σ 2 /λ −(1+σ 2 )/λ . By the above choices, the solution of the dynamics for the estimation errorẽ which has a convergent behavior with a speed of convergence directly dependent on the free parameters σ 1 and σ 2 .
Having guaranteed the convergence of the state estimation error, the unknown input disturbance w k can be retrieved as follows. First, (4) can be rearranged as follows: and its both sides can be left-multiplied by the pseudo-inverse of (W T k , D T k ) T , i.e., matrix G in (9). Doing so and then replacing the state with its estimate yieldŝ Finally, writing explicitly the term y k−L −CẐ k−L allows reaching the formula in (8) and concludes the proof.

C. Design of the Robust Lateral Position Control
As a second step, leveraging on the data reconstructed by the above designed DUIO, a robust lateral position controller for the steering angle input δ k is derived, which ensures e k 's convergence independently of all model uncertainty and external disturbance. This is formalized in the following second main result.
Theorem 2: Given the dynamics of the racecar model in (4), the feedback steering angle control law whereẐ k−L andŵ k−L are the state and disturbance estimates from the DUIO in (8) and where K = (k 1 , k 2 ) is a free control gain matrix, ensures robust, and global asymptotic convergence of the state estimation errorẽ k and robust and global uniform bounded stability of the racecar state Z k around the origin, with a bound decreasing at least linearly with the sampling time λ. Proof: The full dynamics of the racecar in (4) and the DUIO in (8) is in which the state estimation error dynamics is independent of the controlled input δ k and the disturbance signal w k (cf., the last set of equations). This fact implies that the convergence ofẽ k in the closed-loop system is again ensured by E being Schur, and that, after convergence,Ẑ k−L Z k−L andŵ k−L w k−L . Consequently, the feedback law in (13) becomes δ k −m/(C 1τ )(K Z k−L + w k−L ). Hence, closing the loop on (14) with such a feedback law yields where allows rewriting (15) as follows: with and γ 1 = 1−λk 2 . It should be first noticed that the free solution of (16) can be made convergent, by properly allocating the eigenvalues of A c . In this respect, matrix A c is upper-block triangular and hence the set of its eigenvalues comprises those of the DUIO, σ 1 and σ 2 , and those of and has full rank, which ensures the existence of a matrix K so that A k − B v K has all eigenvalues within the unit circle and then generates asymptotically stable modes only. Furthermore, recall from [24] that for a small enough sampling time λ, it holds, for consecutive samples,Ẑ k Ẑ k−1 and w k w k−1 or equivalently (17) with p Z , p w ∈ R + diminishing with the decrease of λ and where ||·|| 2 is the Euclidean norm. From the property in (17), derived from [24], it also follows δ k δ k−1 δ k−L , which solves the algebraic dependence of w k from δ k . Analogously, it holds y k y k−1 y k−L . Moreover, recalling that L = 2, the term φ k can be rewritten as and thus can be upper bounded as follows: From the first set of equations in (16), we have Since A c is Schur, the first addend in the equation above is contracting, i.e., ||A c Z k || < ||Z k || and hence, after a transient, the norm of the racecar state is only excited by the forcing term φ k which decreases at least linearly with the decrease of λ.
Furthermore, as for the convergence of disturbance estimation error,w k , from its definition, one can write as follows: Again under the hypothesis of [24] for small delays, the signals ν k , η k , and ξ k are bounded, i.e. there exist constant upper-bounds, S, V, T ∈ R + , such that ||ν k || 2 ≤ S, ||η k || 2 ≤ V , and ||ξ k || 2 ≤ T , for all k, and also convergent due to (17), i.e., for increasing values of k the upper bounds can be chosen as S, V, T → 0. As a consequence, from (18), one can obtain indicates the spectral radius of a matrix. Therefore, the fact that S, V, T → 0 also implies ||w k || 2 → 0 and, in turn,w k → 0. Hence, the disturbance estimation errorw k asymptotically converges to zero with the same speed of the state estimation errorẽ k , which is specified by the free constants σ 1 and σ 2 of the DUIO. This concludes the proof.
Therefore, the DUIO-based control observes the system state and inputs at the time-step k − L, reconstructs the unknown disturbanceŵ k−L , and finally determines the control action for the time step k steering the closed-loop system in (14) and allowing the sought asymptotic convergence to zero.

IV. SIMULATION AND VALIDATION
The correctness and robustness of the proposed method are shown in this section, along with a comparison of its effectiveness for a benchmark using a standard disturbance rejection technique.

A. Vehicle Model Implementation
To show the effectiveness and robustness of the proposed method, the implementation of a real Robocar model is used by using the Vehicle Dynamics Blockset of the MATLAB/ Simulink environment. The geometric and inertial parameters of the vehicle are listed in Table I. The interaction with the road surface is modeled by generating all lateral wheel forces, F yi j , via the nonlinear Pacejka tire model, the so-called the magic formula [26], i.e., F yi j = F zi j μ i j , with μ i j = D sin(C arctan(Bα i j −E(Bα i j − arctan(Bα i j )))) (19) where B, C, D, and E are dimensionless coefficients whose values depend on the road surface (cf., Table II) for the typical values, also used here, to represent dry, wet, snow, and icy surfaces, and where are the vertical forces applied at each wheel. These last forces are given by for j = 1, 2, where g is the gravity acceleration, h is the height of the vehicle center of gravity with respect to the road and a x is the vehicle's longitudinal acceleration in body frame, and Z i are the lateral load transfers due to the suspensions, which are given by where l is the vehicle wheelbase, d 1 and d 2 are the front and rear no-roll center height, k φ = k φ 1 + k φ 2 , with k φ 1 and k φ 2 being the front and rear suspension roll stiffness, respectively, and finally d = (a 2 d 1 + a 1 d 2 It should be noted that the roll and suspension effects on vehicle dynamics are taken into account via the lateral load transfers described above. Numerical values for the front and rear no-roll center height are d 1 = 0.025 m and d 2 = 0.045 m, respectively, with k φ 1 = 21740.6 (N/rad) and k φ 2 = 22322.2 N/rad. Moreover, the wind force signal is obtained by modeling the wind speed u w according to the stochastic Dryden model [4]. Specifically, the wind speed signal u w (t) is chosen to replicate turbulence at low altitudes, characterized by a height from the sea level of h = 6 m, an airspeed of V = 50 m/s, and a turbulence level of W 2 0 = 15 kn. Moreover, to generate the wind moment signal, the lever arm x w of the wind force is assumed to be a stochastic process with uniform distribution over the length of the vehicle, i.e., −a 2 ≤ x w ≤ a 1 . Accordingly, the wind force F w and wind moment χ w are obtained via the expressions where ρ = 1.225 kg/m 3 is the air density at sea level, S = 2 m 2 the so-called vehicle's lateral wetted area, C y = 1.5 its lateral aerodynamic coefficient. Overall, the second equation of the lateral dynamics in 1 becomes

B. Derivation of the DESO-Based Benchmark
The following benchmark based on the well-established theory described in [14] has been developed to compare the performance of our method with a de-facto standard disturbance rejection technique. It consists of a discrete-time implementation of the system obtained by the application of such a method, according to which an ADRC, based on an ESO, can be obtained as follows. First, consider an augmented state k = (Z T k , w k ) T , including as an additional variable the disturbance w k for which a dynamics must also be introduced. Assuming, as for our method, that only lateral position error measures, y k = e k , are available, and that, for a small delay of one sample, it holds δ k−1 δ k [24], a discrete-time ESO (DESO) is described by the iterative rulê with where L ∈ R 3 is such that the closed-loop dynamic matrix A˜ = A −LC is Schur, ensures the asymptotic boundedness of the augmented state estimation error˜ k = k − k , if, and only if, the following conditions are met: Simulation scenario designed to test and assess the effectiveness and performance of the proposed methods. The longitudinal speed profile has alternating phases of acceleration and deceleration, and two plateau phases at quasi-constant velocities. The road surface ranges from dry, wet, and snow, while also highly varying wind gust force and moment have to be handled.
As a second step, once an estimateˆ k = (Ẑ T k ,ŵ k ) T of the augmented state k is retrieved via the DESO in (20), the steering control law with K = (k 1 , k 2 ) a free control gain, ensures the bounded stability of the full system with W = λ (0 1×4 , 1) T , if, and only if, the signal κ( k , w k , k) is bounded. The convergence proof straightforwardly follows from standard arguments typical of the ADRC technique, but it is omitted here for the sake of space. Finally, to obtain comparable behaviors for the proposed DUIO-based approach and the DESO-based one, the respective free control gains have been chosen so that the eigenvalues of the closed-loop matrix in (14) and that of (22) closed in the loop with (21) are in the similar locations. More specifically, without loss of generality, the speed of convergence of the lateral tracking errors has been tuned, via the control gain K , so that the eigenvalues of A k − B v K are in p 1 = (0.1, −0.1); simultaneously, the speed of convergence of the observers has been chosen to be ten times faster. In the DUIO case, this is obtained by placing the eigenvalues in p 2 = (−0.01, 0.01) (and hence choosing σ 1 = −0.01 and σ 2 = 0.01) and, for the DESO, this is obtained by placing the eigenvalues in p 3 = (−0.01, −0.01, 0.01) (and hence choosing l 1 , l 2 , and l 3 accordingly).

C. Simulation and Testing With Vehicle Dynamics Blockset and Raspberry PI Board
The testing and validation scenario is reported in Fig. 3. The Robocar system is required to track a trajectory with a time-varying longitudinal speed u(t) and curvature radius signal ρ(t), under the presence of sudden wind gusts. The longitudinal speed profile reproduces a typical telemetry profile with acceleration and braking phases [9]; the time-varying road friction is modeled via appropriate variation of the magic formula coefficients.
The goal of the testing is at least fourfold. First, it aims at showing the effectiveness of the proposed method as well as its robustness to unmodeled dynamics, parameter uncertainty, and measurement noise. For this purpose, the Robocar system is implemented as a double-track racecar by using the Vehicle Body 3-degree of freedom (DoF) block of vehicle dynamic blockset in MATLAB/Simulink [27]. Nominal values for the vehicle massm with a maximum variation of 45% from the real value are used in the numerical implementation of the estimators and controllers. Measurement noise is also added to the system outputs via the MATLAB/Simulink Random Source block, which generates pseudorandom Gaussian distributions [28]. Second, the testing aims at comparing the proposed approach with the above-described benchmark. Third, it aims at proving the real-time implementability of the solution and assessing the required computation time in terms of central processing unit (CPU) utilization, through a low-cost hardware setup, consisting of a Raspberry PI 4 Model B system. To achieve this, the proposed DUIO-based solution and the DESO-based benchmark are compiled for the Raspberry PI hardware, via the simulink real-time code generation, and built as standalone applications. The inclusion of both control methods represents a further computational load of the microcontroller, leading to an overestimate of the required CPU utilization and a further guarantee of the solution implementability. Finally, the testing intends to show if all the above-mentioned properties are maintained, even when enlarging the sampling time. For this reason, the scheduling times of the involved processes are chosen as λ = 10 −3 s and later as λ = 10 −2 s. Figs. 4 and 5 report the results of the testing withm/m = 1.45 and with scheduling times of λ = 10 −3 and λ = 10 −2 , respectively. More precisely, Fig. 4 shows the DUIO always better estimates the state and the disturbance, at least by an order of magnitude. The resulting control is also much less affected by the noise and can better cope with the disturbance. Numerically, the integral time absolute error (ITAE) index computed on the tracking error and the disturbance estimation   TABLE III   ITAE INDEX COMPARISON FOR DUIO AND DESO-BASED APPROACHES, RESPECTIVELY, WITH SAMPLING TIME λ = 10 −3 error shows a clear superiority of the DUIO-based solution over the benchmark one (cf., Table III). The total CPU utilization, during the testing, for the two estimation and control processes is always less than 12.7% with a mean value of about 6.9%. Finally, Fig. 5 shows that when the scheduling time is increased to λ = 10 −2 s, the DESO-based benchmark is unable to correctly estimate and cope with the disturbance, which leads the system to instability (see the diverging green line of the tracking error). On the contrary, our proposed DUIO-based solution allows still a nice transient behavior and maintains stability.
V. CONCLUSION A robust lateral controller for self-driving racecars was proposed using a delayed UIO. It showed robustness to time-varying tire-road interface characteristics, wind gusts, and model uncertainty. Its closed-loop asymptotic stability was proven and its performance was compared to that of a disturbance estimation and rejection technique, for which only asymptotic boundedness was obtained. The solution requires no a-priori knowledge of the boundedness or the statistical properties of the system and measurement noises. Testing results confirmed the superior performance of the DUIO over a DESO, as expected from the literature [29]. Simulations further highlighted its superiority by showing that the DUIO-based control generates smoother control signals for the steering angle, leading to smaller and less spiky tracking errors than those of the ADRC methodology.