Cascade Delayed Controller Design for a Class of Underactuated Systems

Universidad Politécnica del Valle de México, Av. Mexiquense s/n Colonia Villa Esmeralda, Tultitlán 54910, Mexico Universidad Autónoma del Estado de Hidalgo, CITIS, Carretera Pachuca-Tulancingo km. 4.5, Col. Carboneras, Mineral de la Reforma 42184, Mexico Instituto Politécnico Nacional-UPIITA, Av. IPN 2580 Colonia Barrio la Laguna Ticomán, Ciudad de México 07340, Mexico InIAT Universidad Iberoamericana Ciudad de México, Prolongación Paseo de la Reforma 880, Colonia Lomas de Santa Fé, Ciudad de México CP 01219, Mexico


Introduction
e control of underactuated systems has attracted some attention from the academic community, as noticed in the growing list of articles and new approaches to solve the problems of estimation, regulation, and trajectory tracking involving both linear and nonlinear underactuated systems. In the literature, there are a variety of control strategies, including controlled Lyapunov functions [1,2], energybased control [3][4][5], passivity approaches [6,7], active disturbance rejection [8][9][10], planning algorithms, and feedback stabilization schemes [11].
One of the most important problems of nonlinear underactuated systems is the fact that the controllability property may be subject to singularities (ill-conditioned relative degree [12]), which conditions the controllable (normal) forms of the models to a certain class of systems [13]. An alternative to locally overcome the aforementioned problem is the use of the tangent linearization technique, since the tangent linearized models of an important class of underactuated systems are differentially flat [9]. e last fact means that the linearized systems are controllable and there exists a set of variables, named flat outputs, which can characterize them [14]. Besides the flatness of the linearized system, another important advantage for this class of systems is the called cascade form [15], which allows finding a relation between 2n − th-order time derivatives (fourthorder, sixth-order, and so on) of the flat output and a measurable variable of the system. is form can reduce the complexity of the global control design in which, instead of designing a global controller of a 2n − th-order system, the task can be the cascade connection of second-order controllers, which is especially important for sensitive systems or noise amplification effects due to high-order observer designs, among others. In particular, for the case of fourth-order systems, this scheme allows, instead of stabilizing a fourth-order integrator, implementing a cascade control arrangement of two second-order systems, for instance, classic controllers with derivative action. e implementation of the derivative action has at least three natural disadvantages. Firstly, the estimation algorithms typically increase the controller design. Secondly, the application of filters or compensators often increases the order of the closed-loop system. irdly, the use of measurement tools (encoders) is usually very noisy. An alternative is the use of delayed controllers, which means to deliberately include retarded actions (time delays) in the controller. Delayed controllers have better performance in practical applications compared to controllers with derivative actions [16]. Since these controllers type noise attenuation, they do not require estimators or filters to approximate the time derivative, providing soft control signals which do not damage actuators, and their numerical implementation is computationally more efficient than other low-order controllers. Moreover, a delayed controller has a simple structure which is easy to implement, like classic controllers. e deliberate use of retarded actions to stabilize a system is a topic that has been investigated in recent decades, among which are the pioneering contributions of [17][18][19][20][21][22][23][24][25][26][27][28]. Concerning derivative-free control based schemes, such as the proportional retarded (PR) scheme [19,25,[29][30][31], which consists in using the relation between the time derivative approximations in terms of a differential difference equation, involving a time delay.
is approximation is taken as a baseline to avoid using a derivative compensation, but a proportional retarded one. e use of delayed controllers has been previously studied for the stabilization of chains of integrators. It has been proven [32][33][34] that a chain of n integrators cannot be stabilized with less than n delay blocks, which increases significantly the complexity of the closed-loop system. e migration of a double imaginary characteristic root to the left half-plane or the right half-plane under the variation of two parameters of a quasi-polynomial is given in [35]. However, in none of the above proposals, there are criteria or methodologies to explicitly tune the controller gains.
For the class of underactuated systems satisfying the cascade property, the use of a cascade control structure of PR controllers (delayed controller) can be a derivative-free alternative of stabilizing control, including the complete dynamics (some reported schemes based on the linearization deal with a part of the dynamics such as the switched control [36]). However, the existing analyses are restricted to a class of second-order LTI systems, and the system to analyze becomes a cascade of two-order systems, whose analysis cannot be considering the isolated dynamics but the cascade structure.
e stability analysis of delayed controllers can be commonly addressed on the study of the relative stability of degree σ of the characteristic function, also known as σ-stability.
is approach in conjunction with the D-decomposition methodology (see [37][38][39][40]) can lead us to obtain conditions, under which the system response can reach the maximal exponential decay rate. us, the contributions of this manuscript are listed as follows: (1) It analyzed a class of underactuated systems, especially those whose tangent linearization is controllable (flat). Moreover, it developed a controller through the use of the cascade property in the flat linearized system, which allows reducing the complexity of the control loops. (2) e control approach consists in using a cascade structure of proportional retarded controllers in each of the second-order structures derived from the cascade system. In this proposal, the fourth-order system is controlled by a tandem array of PR controllers in which two delays are used for the main structure.
(3) e stability analysis considers the complete fourthorder structure and provides the tuning conditions for the controller design and, moreover, provides competitive experimental results, which denote the importance of using a more complex controller with the aim of obtaining appropriate results for noisy measurements, in which additional filtering schemes are demanded. In addition, the tuning procedure is reported and the involved numerical algorithms are provided for practitioners.
(4) e combination of the aforementioned concepts provides an alternative control approach for a class of fourth-order underactuated systems subject to noisy measurements, with a possible extension to the highest-order systems.
(5) e analysis and synthesis of a cascade proportional retarded controllers for a class of underactuated systems in cascade structure are proposed ( Figure 1). Also, analytic conditions, on the parameters of the controller, to reach the maximal exponential decay rate of the system response are proposed.
(6) Experimental implementations of the proposal, which validate its practical effectiveness, are also proposed. Two classic challenging systems were used for the assessment: a rotatory flexible link and a pendubot.
e remaining of the contribution is organized as follows. In Section 2, the basic concepts related to time-delay systems and differentially flatness are presented. In Section 3, a methodology based on the differential flatness to represent a class of fourth-order underactuated systems as a cascade arrangement of two-order systems is presented. e main contributions of the present manuscript are introduced in Section 4, where the σ-stability of the system is analyzed and analytic conditions to reach the maximal exponential decay rate of the system response are provided. Section 5 is devoted to the implementation results in trajectory tracking tasks. e article ends with some concluding remarks.

Complexity
Notation. Given a vector x(t) ∈ R n , x ⊺ (t) denotes its transpose and ‖x(t)‖ stands for the Euclidean norm. Let s ∈ C; then, Re(s) and Im(s) denote its real and imaginary parts, respectively. For a function x(t), x (i) (t), i ∈ Z + denotes the i-th time derivative of the function.

Mathematical Preliminaries
In this section, we introduce some basic concepts, concerning both time-delay systems and differentially flat systems, necessary for the development of this contribution.

Differentially Flat Systems.
In this subsection, the flatness concept for single-input single-output (SISO) systems of nonlinear and linear types is taken. Further comprehensive information can be taken from [14,41,42] and references therein. Definition 1. Let us consider a nonlinear system of the following form: where x(t) ∈ R n , u(t) ∈ R, and f(·) � f 1 (·) f 2 (·) · · · f n (·)] ⊺ is a smooth function of x(t), u(t), and the rank of the Jacobian matrix, and (zf(·))/(zu(t)) is equal to 1. System (1) is said to be differentially flat if there exists a differential function, denoted as y(t) ∈ R, given by which completely differentially parametrizes all the variables of the system. is means that states, inputs, and original output variables in the system can be written as differential functions of the flat output y(t), that is, Flatness is a structural property of the system that trivializes the exact linearization procedure of a nonlinear system even when it is not expressed in an affine form. It also allows an easier manner of designing control laws in trajectory tracking tasks, by taking advantage of the differential parametrization. For the case of linear SISO systems, the flatness property can be directly related to the controllability of the system.
with A ∈ R n×n , b ∈ R n , x(t) ∈ R n , and u(t) ∈ R. Let the pair (A, b) be controllable, that is, the controllability matrix: which has full rank n. en, it is said that system (4) is controllable; hence, it is differentially flat with the flat output given by

Time-Delay Systems.
In the present section, some definitions and general results concerning linear time-delay systems are introduced; for these purposes, it is considered a basic system of the following form: Here, A 0 , A 1 ∈ R n×n , and τ, h ∈ R + are time delays. Now, some stability concepts related to system (7) are presented.
e solution x(t, φ) of a system of the form (7) is said to be exponentially stable, if there exist L, σ ∈ R + such that the following inequality holds: Here Definition 4 (see [43]). A s 0 ∈ C is said to be an eigenvalue of system (7) if it is a root of the characteristic function; that is, e set of all the distinct eigenvalues of the system is called the spectrum of system (7) and this is denoted as Λ(Q).
An equivalent definition of exponential stability is given hereinafter.
Definition 5 (see [43]). System (7) is exponentially stable if and only if the spectrum of the system lies in the open left half-plane of the complex plane; that is, Re(s j ) < 0, ∀s j ∈ Λ(Q).
From these definitions, note that σ-stability ensures exponential stability with the decay rate σ in the system response (7). Figure 1: Schematic of the PR controller for a cascade structure.

Complexity
In the following section, a class of fourth-order underactuated systems is described.

Preliminary Results
Euler-Lagrange underactuated systems of the fourth order can generally be described by where q(t) ≔ q 1 (t) q 2 (t) ⊺ represents the generalized coordinates, M(q(t)) ∈ R 2×2 is the inertia matrix, and C(q(t), _ q(t)) ∈ R 2×2 describes the Coriolis matrix and centrifugal forces. Let g(q(t)) ∈ R 2 be the vector of gravitational forces and P(q(t)) ∈ R 2 is the vector mapping the external forces. Finally, u(t) ∈ R denotes the control input.
Systems of the form (11) can be rewritten in a state-space representation as follows: where x a (t) stands for the vector of joint positions After a slight rearrangement of the state vector as x(t) � q 1 (t) _ q 1 (t) q 2 (t) _ q 2 (t) ⊺ and by denoting x(t) as an equilibrium point of the nonlinear representation (12), a tangent linearization around the equilibrium point x(t) is given by Here, . e controllability matrix of system (14) is given by Under the assumption that det(K c ) ≠ 0, the system is controllable and hence, according to [14,42], differentially flat. e flat output can be obtained as follows: where C f ∈ R n and defined as Here, ϵ ∈ R/ 0 { } is an arbitrary constant. In order to determine the relative degree of system (14), we first need to compute the high-order time derivatives. To this end, notice that the following relations hold: and the time derivatives of the flat output are given by Since C f A 3 b ≠ 0, then, the relative degree of the system is four. It is worth noting that the even time derivatives of y f (t) can be expressed as a linear combination of the flat output, which correspond to those variables that can be measured and coincide with the vector of generalized positions q(t).
Now, let us define the output estimation error as Here, y * f (t) denotes the desired output. en, the error dynamics is governed by by proposing the auxiliary control and then (21) can be expressed as a fourth-order integration chain; that is, Now, a delayed controller u(t) that fulfills the cascade control structure proposed in Figure 1 is given as follows: where τ, h ∈ R + are the delays and k p 1,2 , k r 1,2 ∈ R + are the control gains. us, the closed-loop representation of system

Complexity
Here, and us, the characteristic function or characteristic quasi-polynomial of system (25) is given by

Main Results
In the present section, analytic conditions on the parameters of the delayed controller (24) are presented. Here, a general form of quasi-polynomial (27) is considered: where P i (·), i � 1, 2, 3, are polynomials with real coefficients which satisfy the following degree condition: deg(P 1 (·)) > deg(P 2,3 (·)). In order to analyze the σ-stability of the quasi-polynomial (28), the change of variable s ⟶ (s − σ) is considered; thus, (28) is now in the following form: Based on the D-decomposition methodology [44], the σ-stability regions of the modified characteristic function (29) can be now obtained. is procedure will play a key role in determining the analytic conditions to obtain the maximal decay rate σ * . Proposition 1. Let us consider a quasi-polynomial of the form (28), with given control parameters k p 1 , k p 2 , k r 1 , and τ. Let σ ∈ R + be fixed; then, by defining the σ-stability regions on the (h, k r ) parametric space are bounded by the following conditions.
When s � 0, Proof. It should be noted that the σ crossing boundaries of the quasi-polynomial (28), in the parametric space (h, k r ), are given by the critical crossings roots of quasi-polynomial (29), which occur when s � 0 and s � ± iω. In this regard, on the one hand, the solutions of quasi-polynomial (29) when s � 0 are given by or equivalently On the other hand, the solutions of the quasi-polynomial (29) when s � iω are given by from which

Complexity
From the above equation, it follows that en, the solutions of (39) (with respect to h) are of the following form: Finally, solving k r 2 from (38), equation (33) follows. In the following corollary, the σ-stability boundaries in the (h, k r 2 ) parametric space are written particularly for the quasi-polynomial (23), where P 1 (s, k p 1,2 ) � s 4 + k p 2 s 2 + k p 1 , P 2 (s, k r 1 ) � − k r 1 , and P 3 (s) � − s 2 . □ Corollary 1. Consider the quasipolynomial of the form (23). en, the σ-stability boundaries (31)- (33) look as follows. For s � 0, Here, n � 0, ± 1, ± 2, . . ., and Ψ and Φ are defined as e methodology to graphically determine the σ-stability boundaries (31)-(33) is as follows: (1) Propose an initial value of σ ∈ R + close to zero, denoted by σ 0 (2) For the critical frequencies s � 0: from (31), graph the parametric boundaries for k r 2 (h) on the parametric space (h, k r 2 ) considering h ∈ (0, h], h ∈ R + , and σ 0 (3) For the critical frequencies s � iω: (a) From (32), calculate h considering ω ∈ (0, ω], ω ∈ R + , and σ 0 (b) From (33), compute k r 2 considering ω ∈ (0, ω], σ 0 , and h (from (a)) (c) Plot the values of h and k r 2 , obtained in the previous steps, on the parametric space (h, k r 2 ) (4) Detect the closed regions/curves in the parametric plane (h, k r 2 ) and discard the unstable ones; the remaining regions are considered as the σ-stability regions of (28) (5) Select σ 1 > σ 0 (6) Repeat steps 2-5, considering σ 1 , until the σ-stability regions collapse in a single point Proposition 1 and Corollary 1 provide conditions to determine the σ-stability boundaries in the parametric plane (h, k r 2 ) of quasi-polynomial (28). Next, the main result of the contribution is presented, where analytic expressions relative to the tuning of the controller gains h * and k * r 2 to reach the maximal decay rate σ * are obtained. e result is based on the fact that when the maximal decay rate σ * is reached, there exist three dominant roots of the quasi-polynomial (28) in s � − σ; this phenomenon occurs when the concentric σ-stability regions collapse in a single point, denoted by (h * , k * r 2 ), as σ increases. e value of σ, when the collapse occurs, determines the maximal decay rate σ * ; see [25,30]. en, the analytical equations are determined to obtain the parameters h * and k * r 2 , considering that there are three dominant roots on σ * .
□ Corollary 2. Consider the quasi-polynomial of the form (27); then, the controller gains h * , k * r 2 and the function f(σ, ·) given by equations (44)-(46) look as f σ, k p 1,2 , k r 1 , τ � σ 4 2σ k p 2 − 2σ 2 − k r 1 τe στ 2 + σ 2 κ σ 2 2k p 2 + 12σ 2 − k r 1 τ 2 e στ 2σ 2 κ 2 , where κ � k p 1 − k r 1 e στ + σ 2 (σ 2 + k p 2 ) and h * σ, k p 1,2 , k r 1 , τ � 2σμ 1 − k r 1 τe στ with μ 1 � k p 2 + 2σ 2 and μ 2 � μ 1 − σ 2 . e conditions introduced in this section are only focused on the appropriate selection of the control parameters σ * , k * r 2 , and h * that guarantee reaching the maximal exponential decay rate, but there are no conditions involving the remaining control parameters. In the following, a simple approach for the approximate selection of the control parameters is presented. On the one hand, let us consider quasi-polynomial (27), where the exponential terms are estimated by its first-order Taylor series truncated expansion; that is, Now, let us propose a Hurwitz stable polynomial of the following form: where ξ c , ω c ∈ R + . us, the dynamics of the previous polynomial is matched with (57). en, simple calculations show that the controller gain parameters k p 1 , k p 2 , k r 1 , and τ can be chosen according to the following conditions (for α 1 > 1 and α 2 > 2): Remark 1. Due to the simple nature of conditions (54) and (60), they must be considered only as a starting point in the process of tuning the delayed control law (24) and subsequently adjusted according to the performance of the system.

Rotatory Flexible Link.
e rotatory flexible link platform consists of a rotating base driven by a belt-pulley mechanism actuated by a DC motor, a link is attached to the rotating base by two springs of identical nature, which results in a flexible joint. e schematic of such a manipulator is presented in Figure 2. e variables θ 1 (t) and θ 2 (t) denote Complexity the angular positions of the rotating base and the link, respectively, l represents the length of the link, J b stands for the inertia of the rotating base, and J l is the moment of inertia of the link. Finally, k s denotes the stiffness of the spring and u(t) is the torque input applied to the system. e problem formulation is stated as follows. For the rotatory flexible link, a smooth rest to rest angular position reference trajectory for the rotating base is demanded while the nonactuated link is desired to remain in a vicinity zero position, avoiding oscillations during the tracking maneuver.

5.1.1.
e Dynamic Model. Following the methodology presented in Section 3 the Euler-Lagrange formalism (11) is used to represent the mathematical model of the rotatory flexible link system, where the viscous friction is neglected and the unmodeled nonlinear terms are ignored: with or in an equivalent form: (63)

Cascade Structure Representation.
Let us introduce the state vector: en, the applied torque can be expressed as τ(t) � Nτ m , where τ m � (k τ /R m )V(t) is the torque in terms of the input voltage, N represents the gear ratio, k τ is the torque constant, and R m denotes the motor armature resistance. Since (63) is linear, it can be rewritten in space state representation (24) as follows: Let us define the Kalman controllability matrix as in (5). Direct calculations show that the pair (A, b) is controllable and consequently, by Definition 2, it is flat. e flat output can be computed according to (16), where by defining ε � (R m J b J l )/(k τ k s N), the flat output y f (t) is expressed as (67) e flat output time derivatives are Let us define the output error as in (20); then, its dynamics are _ e 1 (t) � e 2 (t), (70)

Delayed Control
Law. If the following delayed controller, u(t) � u (t) is considered: e chain of integrators representation (25) is obtained.

Remark 2.
It is worth noting that both the auxiliary control V(t) and the delayed controller u D (t) depend completely on the measurable variables that correspond to angular positions x 1 (t) � θ 1 (t) and x 3 (t) � θ 2 (t).
Remark 3. Now, the auxiliary control V(t) and the feedback state control law u FS (t) depend on the complete vector state x(t), due to the fact that the velocity states x 2 (t) � θ . 1 (t) and x 4 (t) � θ . 2 (t) are not available; then, a low pass filter with transfer function G(s) � (200s)/(s + 200) is used to estimate the velocity using the measurable variables x 1 (t) � θ 1 (t) and x 3 (t) � θ 2 (t); this methodology allows us to reduce the noise generated by the estimation of the so-called "dirty derivative" [45]. Figure 3, the rotatory flexible link experimental platform is presented. e prototype consists of a DC motor NISCA model NC5475 attached to the rotating base by means of a belt-pulley system with a 16 : 1 ratio. e angular position of both the rotating base and the link is measured by means of incremental encoders with a resolution of 1000 counts per revolution. e data acquisition is carried out with a data card Sensoray, model 626. e parameters of the controller were selected as follows. According to (38) and (39), the following values were proposed ω c � 52 and ξ c � 1.2; thus, k p 1 � 29246464, k p 2 � 33572.864, k r 1 � 21934848, and τ � 0.02. Now, the conditions presented in Corollary 4 lead to σ * � 66.65 and consequently k * r 2 � 18839.729 and h * � 0.00476. e σ-stability boundaries were calculated following Corollary 1, which is illustrated in Figure 4. e red mark symbolizes the maximal achievable decay rate σ * and, as it can be seen, represents the point where all the σ-stable regions collapse. Figure 5 depicts the rightmost root locus of the closedloop system, where it can be appreciated that a triple real dominant root is located at the point σ * � 66.65 as stated in Proposition 3. e set of gains for the feedback state controller were chosen as ϖ � 52 and ς � 1.2. e main challenge when controlling the rotatory flexible link platform is to avoid the oscillations that appear intrinsically due to the flexible nature of the system. In Figure 6, the rotatory flexible link tracking trajectory task under the control actions of the proposed cascade PR scheme and feedback state is presented. e desired trajectory y * f (t) is represented by a black line, while the flat output y fD (t) with the delayed controller is depicted in blue and the y fFS (t) with the feedback is represented by a red line. It can be seen that the tracking task is carried out satisfactorily by both controllers, avoiding oscillations and reaching the reference without overshooting. e tracking errors are detailed in Figure 7 Complexity 9

Experimental Results. In
when the system follows the reference trajectory. Figure 10 shows the control voltages of both schemes. Notice that the peak voltage V D does not exceed ±10[Volts] but V FS presents a larger amplitude. As a consequence of avoiding the use of the time derivatives in the proposed control scheme, the voltage V D signal appears relatively free of noise, as expected but V FS presents high-frequency components; it can be corroborated in Figure 11 where a Power Spectrum Density of control signals V D and V FS is presented. Here, it can be observed that FS PSD presents more frequency components in comparison with D PSD . Finally, the performance of the system is evaluated by means of a quadratic index of the applied voltage (see Figure 12) where it can be noticed that V FS consumes more energy.   10 Complexity

Pendubot System.
e pendubot consists of a planar double inverted pendulum, whose schematic is presented in Figure 13. e first link is driven by a DC motor while the second link is an underactuated simple pendulum. e variables θ 1 and θ 2 denote the angular positions of the links, u represents the control torque input, and the parameters m 1 and m 2 denote the masses of the links. e lengths of the links are represented as l 1 and l 2 , while l c 1 and l c 2 are the distances to the center of the masses. Finally, I 1 and I 2 denote the inertias of the links. e problem formulation is given as follows: a smooth rest to rest angular position reference trajectory for the first link is demanded, while the nonactuated second link is desired to remain around its unstable vertical position, without falling during the entire tracking maneuver or moving away from the equilibrium point.

Cascade Structure Representation.
In this section, the procedure presented in Section 3 is applied. e state vector is defined as e applied torque can be expressed as τ(t) � (k τ /R m )V(t) in terms of the input voltage, where k τ is the torque constant and R m represents the motor armature resistance. For V � 0, the considered equilibrium point x(t) is en, the tangent linearization looks as e parameters of the controller were chosen according to (59 and 60), selecting ω c � 14 and ξ c � 1.2 and then k p 1 � 460992, k p 2 � 4562.88, k r 1 � 422576, and τ � 0.0311. e remainder of the parameters are calculated following Corollary 2, leading to σ * � 8.8477, k r 2 � 2832.39, and h * � 0.021. e σ-stability boundaries are computed by means of Corollary 1, see Figure 16, where the maximal achievable decay rate σ * is marked as a red spot. Here, the zone outside the concentric regions corresponds to the unstable region. e set of gains for the feedback state controller were chosen as ϖ � 14 and ς � 1.2. Notice that the gain value coincides with the gain value used to select the proposed delayed control. Figure 17 shows the flat output trajectory tracking performance of both controllers, the reference trajectory y * f (t) is depicted in black line, the delayed controller response y D (t) is shown in the blue line, and the feedback controller response is shown in the red line y FS (t), where it can be seen that the pendubot tracks the desired trajectory with adequate results, even when it is far from the equilibrium point. Figure 18 shows that the tracking errors e D (t) and e FS (t) are bounded in an interval of approximately [− 0.025, 0.025] [rad]. Figures 19 and 20 show the variation of the positions of the links during the tracking trajectory task. Figure 21 exhibits the control input voltage V D calculated without using any time derivatives, which shows less noise θ 2 = 0 * θ 1 = π/2 * θ 1 = 2/3π * θ 2 = -π/6 * θ 2 = π/6 * θ 1 = π/3 *  and less amplitude in contrast with V FS . e performance of both controllers is similar to good results; the main difference is on the voltage control.
e Power Spectrum Density of V D and V FS is depicted in Figure 22. Notice that FS PSD shows high-frequency components with a peak on 50 [Hz]. Figure 23 shows an analysis using a quadratic index of the voltage, where V FS consumes more energy. us, the proposed delay controller represents an attractive alternative to control a set of underactuated mechanical systems. A sensitivity analysis with a quadratic error index of the   Complexity trajectory tasks was performed in simulation using a set of initial conditions for the angular positions θ 1 (0) and θ 2 (0). Figure 24 shows the first link performance; notice that the PR controller rank of initial conditions θ 1 D (0) remains on [− 8, 8] degrees, in comparison with the FS controller rank of initial conditions θ 1FS (0) that remains on the interval [− 7, 7] degrees. Similarly, Figure 25 shows the second link performance, the rank of the initial conditions for θ 2 D (0) is [− 12, 12] degrees, while θ 2FS (0) is [ − 11, 11]. Notice that the PR controller can control the pendubot system with further initial conditions from the equilibrium point than the PD controller.

Conclusion
In this paper, an alternative approach to control a class of underactuated systems of the fourth order through proportional retarded based controllers was proven and experimentally tested, leading to smooth tracking results thanks to the derivative-free control philosophy. e tuning procedure allows the practitioner to find appropriate control gains (the proposal is constructive). e cascade form of the tangent linearization of the underactuated system is an important design tool for the model simplification and the   16 Complexity local controllability property for the design of a wide variety of controllers. Future research concerning the topic deals with the extension of the procedure to larger order systems as well as a more comprehensive development of the conditions involving all the control parameters. e case of study can be also extended for disturbed dynamics, in which integral actions can be proposed and tested.

Data Availability
All the data generated or analyzed during this study are included in this paper.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.