In this paper, a delayed control strategy for a class of nonlinear underactuated fourth-order systems is developed. The proposal is based on the implementation of the tangent linearization technique, differential flatness, and a study of the -stabilization of the characteristic equation of the closed-loop system. The tangent linearization technique allows obtaining a local controllability property for the analyzed class of systems. Also, it can reduce the complexity of the global control design, through the use of a cascade connection of two second-order controllers instead of designing a global controller of the fourth-order system. The stabilizing behavior of the delayed controller design is supported by the -stability criterion, which provides the controller parameter selection to reach the maximum exponential decay rate on the system response. To illustrate the efficiency of the theoretical results, the proposal is experimentally assessed in two cases of study: a flexible joint system and a pendubot.

1. Introduction

The 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], energy-based control [35], passivity approaches [6, 7], active disturbance rejection [810], 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]. The 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 -order time derivatives (fourth-order, sixth-order, and so on) of the flat output and a measurable variable of the system. This form can reduce the complexity of the global control design in which, instead of designing a global controller of a -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. The 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. Thirdly, 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.

The 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 [1728]. Concerning derivative-free control based schemes, such as the proportional retarded (PR) scheme [19, 25, 2931], which consists in using the relation between the time derivative approximations in terms of a differential difference equation, involving a time delay. This approximation is taken as a baseline to avoid using a derivative compensation, but a proportional retarded one.

The use of delayed controllers has been previously studied for the stabilization of chains of integrators. It has been proven [3234] that a chain of integrators cannot be stabilized with less than delay blocks, which increases significantly the complexity of the closed-loop system. The 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.

The stability analysis of delayed controllers can be commonly addressed on the study of the relative stability of degree of the characteristic function, also known as -stability. This approach in conjunction with the -decomposition methodology (see [3740]) can lead us to obtain conditions, under which the system response can reach the maximal exponential decay rate.

Thus, 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)The 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)The stability analysis considers the complete fourth-order 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)The 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)The 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.

The 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. The main contributions of the present manuscript are introduced in Section 4, where the -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. The article ends with some concluding remarks.

Notation. Given a vector , denotes its transpose and stands for the Euclidean norm. Let ; then, and denote its real and imaginary parts, respectively. For a function , , denotes the -th time derivative of the function.

2. 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.

2.1. 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 , , and is a smooth function of , and the rank of the Jacobian matrix, and is equal to 1. System (1) is said to be differentially flat if there exists a differential function, denoted as , given bywhich completely differentially parametrizes all the variables of the system. This means that states, inputs, and original output variables in the system can be written as differential functions of the flat output , 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.

Definition 2. Consider the linear time-invariant SISO system:with , , and . Let the pair be controllable, that is, the controllability matrix:which has full rank . Then, it is said that system (4) is controllable; hence, it is differentially flat with the flat output given by

2.2. 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, , and are time delays. Now, some stability concepts related to system (7) are presented.

Definition 3 (see [43]). The solution of a system of the form (7) is said to be exponentially stable, if there exist such that the following inequality holds:Here, , denotes the initial function and is a Banach space of continuous functions in with the norm .

Definition 4 (see [43]). A is said to be an eigenvalue of system (7) if it is a root of the characteristic function; that is,The set of all the distinct eigenvalues of the system is called the spectrum of system (7) and this is denoted as .
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, , .

Definition 6. A linear time-delay system of the form (7) is said to be -stable, iffor and .
From these definitions, note that -stability ensures exponential stability with the decay rate in the system response (7).
In the following section, a class of fourth-order underactuated systems is described.

3. Preliminary Results

Euler-Lagrange underactuated systems of the fourth order can generally be described bywhere represents the generalized coordinates, is the inertia matrix, and describes the Coriolis matrix and centrifugal forces. Let be the vector of gravitational forces and is the vector mapping the external forces. Finally, denotes the control input.

Systems of the form (11) can be rewritten in a state-space representation as follows:where stands for the vector of joint positions , denotes the articular velocities vector , and

After a slight rearrangement of the state vector as and by denoting as an equilibrium point of the nonlinear representation (12), a tangent linearization around the equilibrium point is given by

Here, and . The controllability matrix of system (14) is given byUnder the assumption that , the system is controllable and hence, according to [14, 42], differentially flat. The flat output can be obtained as follows:where and defined as

Here, 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 , then, the relative degree of the system is four. It is worth noting that the even time derivatives of 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 .

Now, let us define the output estimation error as

Here, denotes the desired output. Then, the error dynamics is governed byby proposing the auxiliary controland then (21) can be expressed as a fourth-order integration chain; that is,

Now, a delayed controller that fulfills the cascade control structure proposed in Figure 1 is given as follows:where are the delays and are the control gains. Thus, the closed-loop representation of system (23) is now

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

4. 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 , , are polynomials with real coefficients which satisfy the following degree condition: . In order to analyze the -stability of the quasi-polynomial (28), the change of variable is considered; thus, (28) is now in the following form:

Based on the -decomposition methodology [44], the -stability regions of the modified characteristic function (29) can be now obtained. This procedure will play a key role in determining the analytic conditions to obtain the maximal decay rate .

Proposition 1. Let us consider a quasi-polynomial of the form (28), with given control parameters , and . Let be fixed; then, by definingthe -stability regions on the parametric space are bounded by the following conditions.

When ,

When , ,

Proof. It should be noted that the crossing boundaries of the quasi-polynomial (28), in the parametric space , are given by the critical crossings roots of quasi-polynomial (29), which occur when and . In this regard, on the one hand, the solutions of quasi-polynomial (29) when are given byor equivalentlyOn the other hand, the solutions of the quasi-polynomial (29) when are given byThereby,from whichFrom the above equation, it follows thatThen, the solutions of (39) (with respect to h) are of the following form:Finally, solving from (38), equation (33) follows.
In the following corollary, the -stability boundaries in the parametric space are written particularly for the quasi-polynomial (23), where , , and .

Corollary 1. Consider the quasipolynomial of the form (23). Then, the -stability boundaries (31)–(33) look as follows.
For ,

For , ,

Here, , and and are defined as

The methodology to graphically determine the -stability boundaries (31)–(33) is as follows:(1)Propose an initial value of close to zero, denoted by (2)For the critical frequencies : from (31), graph the parametric boundaries for on the parametric space considering , , and (3)For the critical frequencies :(a)From (32), calculate considering , , and (b)From (33), compute considering , , and (from (a))(c)Plot the values of and , obtained in the previous steps, on the parametric space (4)Detect the closed regions/curves in the parametric plane and discard the unstable ones; the remaining regions are considered as the -stability regions of (28)(5)Select (6)Repeat steps 2–5, considering , until the -stability regions collapse in a single point

Proposition 1 and Corollary 1 provide conditions to determine the -stability boundaries in the parametric plane of quasi-polynomial (28). Next, the main result of the contribution is presented, where analytic expressions relative to the tuning of the controller gains and to reach the maximal decay rate are obtained. The result is based on the fact that when the maximal decay rate is reached, there exist three dominant roots of the quasi-polynomial (28) in ; this phenomenon occurs when the concentric -stability regions collapse in a single point, denoted by , as increases. The value of , when the collapse occurs, determines the maximal decay rate ; see [25, 30]. Then, the analytical equations are determined to obtain the parameters and , considering that there are three dominant roots on .

Proposition 2. Let us consider the closed-loop system (25). Then, the quasi-polynomial (28) has a root of multiplicity at least three on if and satisfy the following equations:and is the smallest positive real root of

Proof. Clearly, quasi-polynomial (28) presents three dominant roots at the point if the quasi-polynomial (29) has three dominant roots at . Thus, the conditionsmust be satisfied; consequently,The first two (44) and (45) can be easily verified. First, expression (45) follows directly from (49). Now, to obtain (44), from (48) and (49), it follows thatand we get (44):To address (46), from (48) and (50), it follows thatSubstituting (52) in the above equation yieldswhich ends proof.
In the following corollary, conditions (44)–(46) are written particularly for the quasi-polynomial (27), where , , and .

Corollary 2. Consider the quasi-polynomial of the form (27); then, the controller gains , and the function given by equations (44)–(46) look aswhere and

with and .

The conditions introduced in this section are only focused on the appropriate selection of the control parameters , , and 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 . Thus, the dynamics of the previous polynomial is matched with (57). Then, simple calculations show that the controller gain parameters , and can be chosen according to the following conditions (for and ):

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.

5. Cases of Study

5.1. Rotatory Flexible Link

The 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. The schematic of such a manipulator is presented in Figure 2. The variables and denote the angular positions of the rotating base and the link, respectively, represents the length of the link, stands for the inertia of the rotating base, and is the moment of inertia of the link. Finally, denotes the stiffness of the spring and is the torque input applied to the system.

The 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. The 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:withor in an equivalent form:

5.1.2. Cascade Structure Representation

Let us introduce the state vector:

Then, the applied torque can be expressed as , where is the torque in terms of the input voltage, represents the gear ratio, is the torque constant, and 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 is controllable and consequently, by Definition 2, it is flat. The flat output can be computed according to (16), where by defining , the flat output is expressed as

The flat output time derivatives are

Let us define the output error as in (20); then, its dynamics arewhere the auxiliary control is proposed as

5.1.3. Delayed Control Law

If the following delayed controller, is considered:

The chain of integrators representation (25) is obtained.

Remark 2. It is worth noting that both the auxiliary control and the delayed controller depend completely on the measurable variables that correspond to angular positions and .

5.1.4. Feedback State Control Law

In order to compare the proposed delayed controller with a classical scheme, a feedback state control is designed as follows:where the set of gains is chosen as , , , and .

Remark 3. Now, the auxiliary control and the feedback state control law depend on the complete vector state , due to the fact that the velocity states and are not available; then, a low pass filter with transfer function is used to estimate the velocity using the measurable variables and ; this methodology allows us to reduce the noise generated by the estimation of the so-called “dirty derivative” [45].

5.1.5. Experimental Results

In Figure 3, the rotatory flexible link experimental platform is presented. The 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. The 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. The data acquisition is carried out with a data card Sensoray, model 626. The data card acquires the signals from the optical incremental encoders and supplies the control voltages to the power amplifiers (Quanser amplifier model VoltPAQ-X2). The control scheme is implemented in the Matlab-Simulink platform with a sampling time of 0.001[s].

The rotatory flexible link parameters areInertias: [Kg − ] and [Kg − ].Length of the link: [m].Stiffness of the spring: [N − m/rad].Armature resistance: .Torque constant: .Mechanical advantage: .

In this experiment, the tracking trajectory problem is addressed. The initial conditions are . The desired trajectory consists of a path initialized at [rad]. After two seconds in this position, the mechanism moves, in an interval of 0.7[s], to [rad] where it stands still for 7.3[s]. Finally, the reference path returns to its initial condition in an interval of 0.7[s].

The parameters of the controller were selected as follows. According to (38) and (39), the following values were proposed and ; thus, , , , and . Now, the conditions presented in Corollary 4 lead to and consequently and .

The -stability boundaries were calculated following Corollary 1, which is illustrated in Figure 4. The red mark symbolizes the maximal achievable decay rate and, as it can be seen, represents the point where all the -stable regions collapse.

Figure 5 depicts the rightmost root locus of the closed-loop system, where it can be appreciated that a triple real dominant root is located at the point as stated in Proposition 3.

The set of gains for the feedback state controller were chosen as and . The 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. The desired trajectory is represented by a black line, while the flat output with the delayed controller is depicted in blue and the 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. The tracking errors are detailed in Figure 7, noticing that both errors and are restricted to the interval [rad] with similar performance. Figures 8 and 9 represent the evolution of the angular positions and when the system follows the reference trajectory. Figure 10 shows the control voltages of both schemes. Notice that the peak voltage does not exceed [Volts] but presents a larger amplitude. As a consequence of avoiding the use of the time derivatives in the proposed control scheme, the voltage signal appears relatively free of noise, as expected but presents high-frequency components; it can be corroborated in Figure 11 where a Power Spectrum Density of control signals and is presented. Here, it can be observed that presents more frequency components in comparison with . 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 consumes more energy.

5.2. Pendubot System

The pendubot consists of a planar double inverted pendulum, whose schematic is presented in Figure 13. The first link is driven by a DC motor while the second link is an underactuated simple pendulum. The variables and denote the angular positions of the links, represents the control torque input, and the parameters and denote the masses of the links. The lengths of the links are represented as and , while and are the distances to the center of the masses. Finally, and denote the inertias of the links.

The 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.

5.2.1. The Nonlinear Dynamic Model

As carried out for the flexible link system, the Euler-Lagrange formalism is used to represent the dynamic model of the considered system:with and . Here,where

5.2.2. Cascade Structure Representation

In this section, the procedure presented in Section 3 is applied. The state vector is defined as

The applied torque can be expressed as in terms of the input voltage, where is the torque constant and represents the motor armature resistance. For , the considered equilibrium point is

Then, the tangent linearization looks aswhere ,

By defining the Kalman controllability matrix as , the pair is controllable since has full rank. Then, according to Definition 2, it implies that system (42) is flat, and the corresponding flat output is computed as

Here, is selected as . Now, by straightforward calculations, the flat output time derivatives (10) are given bywith

By defining the output error as in (20), then, the set of error dynamics (21) are given as follows:

Now, by proposing the auxiliary controland by considering the delayed control lawthe desired chain of integrators representation (25) is obtained.

Remark 4. As in the previous experiment, the auxiliary control and the delayed control law depend completely on the measurable variables that correspond to the angular positions.

5.2.3. Feedback State Control Law

As in the previous test, feedback state control is designed in order to compare its performance with the proposed delayed controller:where the following set of gains is chosen as , ,