# Cascade Delayed Controller Design for a Class of Underactuated Systems

Authors: G. Ochoa-Ortega; R. Villafuerte-Segura; A. Luviano-Juárez; M. Ramírez-Neria; N. Lozada-Castillo
Journal: Complexity
 (2020)

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## Abstract

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 theσ-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 σ-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.


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## Body

## 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 [3–5], passivity approaches [6, 7], active disturbance rejection [8–10], 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]. 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 2n−th-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 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. 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 [17–28]. Concerning derivative-free control based schemes, such as the proportional retarded (PR) scheme [19, 25, 29–31], 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 [32–34] 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. 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 degreeσ of the characteristic function, also known as σ-stability. This approach in conjunction with the D-decomposition methodology (see [37–40]) 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 (Figure1). 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 Section2, 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 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. The article ends with some concluding remarks.Figure 1
Schematic of the PR controller for a cascade structure.Notation. Given a vector xt∈ℝn, x⊺t denotes its transpose and xt stands for the Euclidean norm. Let s∈ℂ; then, Res and Ims denote its real and imaginary parts, respectively. For a function xt, xit, i∈ℤ+ denotes the i-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:(1)x˙t=fxt,ut,where xt∈ℝn, ut∈ℝ, and f·=f1·f2·⋯fn·⊺ is a smooth function of xt,ut, and the rank of the Jacobian matrix, and ∂f·/∂ut is equal to 1. System (1) is said to be differentially flat if there exists a differential function, denoted as yt∈ℝ, given by(2)yt=rxt,ut,u˙t,u¨t,…,uαt,α∈ℤ+,which 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 yt, that is,(3)xt=ϕyt,y˙t,…,yγt,ut=ψyt,y˙t,…,yγ+1t,γ∈ℤ+.
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:(4)x˙t=Axt+but,with A∈ℝn×n, b∈ℝn,xt∈ℝn, and ut∈ℝ. Let the pair A,b be controllable, that is, the controllability matrix:(5)Kc=bAbA2b⋯An−1b,which has full rank n. Then, it is said that system (4) is controllable; hence, it is differentially flat with the flat output given by(6)yt=00⋯1Kc−1xt.


### 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:(7)x˙t=A0xt+A1xt−τ+A2xt−h,t≥0.Here,A0,A1∈ℝn×n, and τ,h∈ℝ+ are time delays. Now, some stability concepts related to system (7) are presented.Definition 3 (see [43]).
The solutionxt,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:(8)xt,φ≤Le−σtφH,t≥0.
Here,H=maxτ,h, φ:−H,0⟶ℭ denotes the initial function and ℭ≔C−H,0,ℝn is a Banach space of continuous functions in −H,0 with the norm φH=maxθ∈−H,0φθ.Definition 4 (see [43]).
As0∈ℂ is said to be an eigenvalue of system (7) if it is a root of the characteristic function; that is,(9)Qs0,τ,h=dets0I−A0−A1e−s0τ−A2e−s0h=0.
The set of all the distinct eigenvalues of the system is called the spectrum of system (7) and this is denoted as ΛQ.
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, Resj<0, ∀sj∈ΛQ.Definition 6.
A linear time-delay system of the form (7) is said to be σ-stable, if(10)s∗=−σ,for σ∈ℝ+ and s∗=maxj=1,…,∞Resj:sj∈ΛQ.
From these definitions, note thatσ-stability ensures exponential stability with the decay rate σ in the system response (7).
In the following section, a class of fourth-order underactuated systems is described.


## 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:(1)x˙t=fxt,ut,where xt∈ℝn, ut∈ℝ, and f·=f1·f2·⋯fn·⊺ is a smooth function of xt,ut, and the rank of the Jacobian matrix, and ∂f·/∂ut is equal to 1. System (1) is said to be differentially flat if there exists a differential function, denoted as yt∈ℝ, given by(2)yt=rxt,ut,u˙t,u¨t,…,uαt,α∈ℤ+,which 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 yt, that is,(3)xt=ϕyt,y˙t,…,yγt,ut=ψyt,y˙t,…,yγ+1t,γ∈ℤ+.
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:(4)x˙t=Axt+but,with A∈ℝn×n, b∈ℝn,xt∈ℝn, and ut∈ℝ. Let the pair A,b be controllable, that is, the controllability matrix:(5)Kc=bAbA2b⋯An−1b,which has full rank n. Then, it is said that system (4) is controllable; hence, it is differentially flat with the flat output given by(6)yt=00⋯1Kc−1xt.


## 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:(7)x˙t=A0xt+A1xt−τ+A2xt−h,t≥0.Here,A0,A1∈ℝn×n, and τ,h∈ℝ+ are time delays. Now, some stability concepts related to system (7) are presented.Definition 3 (see [43]).
The solutionxt,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:(8)xt,φ≤Le−σtφH,t≥0.
Here,H=maxτ,h, φ:−H,0⟶ℭ denotes the initial function and ℭ≔C−H,0,ℝn is a Banach space of continuous functions in −H,0 with the norm φH=maxθ∈−H,0φθ.Definition 4 (see [43]).
As0∈ℂ is said to be an eigenvalue of system (7) if it is a root of the characteristic function; that is,(9)Qs0,τ,h=dets0I−A0−A1e−s0τ−A2e−s0h=0.
The set of all the distinct eigenvalues of the system is called the spectrum of system (7) and this is denoted as ΛQ.
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, Resj<0, ∀sj∈ΛQ.Definition 6.
A linear time-delay system of the form (7) is said to be σ-stable, if(10)s∗=−σ,for σ∈ℝ+ and s∗=maxj=1,…,∞Resj:sj∈ΛQ.
From these definitions, note thatσ-stability ensures exponential stability with the decay rate σ 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 by(11)Pqtut=Mqtq¨t+Cqt,q˙tq˙t+gqt,where qt≔q1tq2t⊺ represents the generalized coordinates, Mqt∈ℝ2×2 is the inertia matrix, and Cqt,q˙t∈ℝ2×2 describes the Coriolis matrix and centrifugal forces. Let gqt∈ℝ2 be the vector of gravitational forces and Pqt∈ℝ2 is the vector mapping the external forces. Finally, ut∈ℝ denotes the control input.Systems of the form (11) can be rewritten in a state-space representation as follows:(12)x˙at=xbt,x˙bt=fxat,xbt+ζxatut,where xat stands for the vector of joint positions q1tq2t⊺, xbt denotes the articular velocities vector q˙1tq˙2t⊺, and(13)fxat,xbt=−Mqt−1Cqt,q˙tq˙t+gqt,ζxat=Mqt−1Pqt.After a slight rearrangement of the state vector asxt=q1tq˙1tq2tq˙2t⊺ 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(14)x˙δt=Axδt+Buδt.Here,xδt=xt−x¯t and uδt=ut−u¯t. The controllability matrix of system (14) is given by(15)Kc=bAbA2bA3b.Under the assumption that detKc≠0, the system is controllable and hence, according to [14, 42], differentially flat. The flat output can be obtained as follows:(16)yft=Cfxδt,where Cf∈ℝn and defined as(17)Cfϵ:=ϵ0001Kc−1.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:(18)Cfb=CfAb=CfA2b=0,CfA3b≠0,and the time derivatives of the flat output are given by(19)yft=Cfxδt,y˙ft=CfAxδt,y¨ft=CfA2xδt,yf3t=CfA3xδt,yf4t=CfA4xδt+CfA3buδt.SinceCfA3b≠0, then, the relative degree of the system is four. It is worth noting that the even time derivatives of yft 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 qt.Now, let us define the output estimation error as(20)eit≔yfi−1t−yf∗i−1t,i=1,…,4.Here,yf∗t denotes the desired output. Then, the error dynamics is governed by(21)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=yf4t−yf∗4t,by proposing the auxiliary control(22)uδt=CfA3b−1ut−CfA4xδt+yf∗4t,and then (21) can be expressed as a fourth-order integration chain; that is,(23)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ut.Now, a delayed controllerut that fulfills the cascade control structure proposed in Figure 1 is given as follows:(24)ut=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h,where τ,h∈ℝ+ are the delays and kp1,2,kr1,2∈ℝ+ are the control gains. Thus, the closed-loop representation of system (23) is now(25)e˙t=A0et+A1et−τ+A2et−h.Here,(26)A0=010000100001−kp10−kp20,A1=000000000000kr1000,A2=00000000000000kr20,and et=e1te2te3te4t⊺. Thus, the characteristic function or characteristic quasi-polynomial of system (25) is given by(27)Qs,kp1,2,kr1,2,τ,h=s4+kp2s2+kp1−kr1e−sτ−s2kr2e−sh.


## 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:(28)Qs,kp1,2,kr1,2,τ,h=P1s,kp1,2+P2s,kr1e−sτ+P3skr2e−sh,where Pi·, i=1,2,3, are polynomials with real coefficients which satisfy the following degree condition: degP1·>degP2,3·. 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:(29)Qs−σ,kp1,2,kr1,2,τ,h≔Qσs,kp1,2,kr1,2,τ,h=Pσ1s,kp1,2+Pσ2s,kr1e−τs−σ+Pσ3seσhkr2e−sh.Based on theD-decomposition methodology [44], the σ-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 σ∗.Proposition 1.
Let us consider a quasi-polynomial of the form (28), with given control parameters kp1,kp2,kr1, and τ. Let σ∈ℝ+ be fixed; then, by defining(30)Γs,σ=Pσ1s,kp1,2+Pσ2s,kr1e−s−στ,the σ-stability regions on the h,kr parametric space are bounded by the following conditions.Whens=0,(31)kr2h=−Γ0,σPσ30eσh,h∈ℝ+.Whens=iω, ω∈ℝ+,(32)hω=1ωcot−1−ReΓiω,σ/Pσ3iωImΓiω,σ/Pσ3iω+πnω,n=0,±1,±2,…,(33)kr2h,ω=1eσhsinωhImΓiω,σPσ3iω.Proof.
It should be noted that theσ crossing boundaries of the quasi-polynomial (28), in the parametric space h,kr, 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(34)0=Qσs,kp1,2,kr1,2,τ,hs=0=Pσ10,kp1,2+Pσ20,kr1eτσ+Pσ30eσhkr2,or equivalently(35)kr2=−Pσ10,kp1,2+Pσ20,kr1eτσPσ30eσh.
On the other hand, the solutions of the quasi-polynomial (29) when s=iω are given by(36)0=Qσs,kp1,2,kr1,2,τ,hs=iω=Pσ1iω,kp1,2+Pσ2iω,kr1e−iω−στ+Pσ3seσhkr2e−iωh.Thereby,(37)−1kr2eσhΓiω,σPσ3iω=e−iωh=cosωh−isinωh,from which(38)cosωh=−1kr2eσhReΓiω,σPσ3iω,sinωh=1kr2eσhImΓiω,σPσ3iω.
From the above equation, it follows that(39)cosωhsinωhImΓiω,σPσ3iω=−ReΓiω,σPσ3iω.
Then, the solutions of (39) (with respect to h) are of the following form:(40)h=1ωcot−1−ReΓiω,σ/Pσ3iωImΓiω,σ/Pσ3iω+πnω,n=0,±1,±2,…,andω∈ℝ+.
Finally, solvingkr2 from (38), equation (33) follows.
In the following corollary, theσ-stability boundaries in the h,kr2 parametric space are written particularly for the quasi-polynomial (23), where P1s,kp1,2=s4+kp2s2+kp1, P2s,kr1=−kr1, and P3s=−s2.Corollary 1.
Consider the quasipolynomial of the form (23). Then, the σ-stability boundaries (31)–(33) look as follows.
Fors=0,(41)kr2h=kp2σ4+kp2σ2+kp1−kr1eστσ2eσh.Fors=iω, ω∈ℝ+,(42)hω=1ωcot−1−ΦΨ+πnω,kr2h,ω=Ψeσhsinωhσ2+ω22.Here,n=0,±1,±2,…, and Ψ and Φ are defined as(43)Ψ=2ωσkp1−σ2+ω22−kr1eτσ2σωcosωτ−σ2−ω2sinωτ,Φ=σ2−ω2σ2ω2+kp1−kr1eτσcosωτ+kp2σ2+ω22−2kr1σωeτσsinωτ+σ6−ω6.The methodology to graphically determine theσ-stability boundaries (31)–(33) is as follows:(1)
Propose an initial value ofσ∈ℝ+ close to zero, denoted by σ0(2)
For the critical frequenciess=0: from (31), graph the parametric boundaries for kr2h on the parametric space h,kr2 considering h∈0,h¯, h¯∈ℝ+, and σ0(3)
For the critical frequenciess=iω:(a)
From (32), calculate h considering ω∈0,ω¯, ω¯∈ℝ+, and σ0(b)
From (33), compute kr2 considering ω∈0,ω¯, σ0, and h (from (a))(c)
Plot the values ofh and kr2, obtained in the previous steps, on the parametric space h,kr2(4)
Detect the closed regions/curves in the parametric planeh,kr2 and discard the unstable ones; the remaining regions are considered as the σ-stability regions of (28)(5)
Selectσ1>σ0(6)
Repeat steps 2–5, consideringσ1, until the σ-stability regions collapse in a single pointProposition1 and Corollary 1 provide conditions to determine the σ-stability boundaries in the parametric plane h,kr2 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 kr2∗ to reach the maximal decay rate σ∗ are obtained. The 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∗,kr2∗, as σ increases. The value of σ, when the collapse occurs, determines the maximal decay rate σ∗; see [25, 30]. Then, the analytical equations are determined to obtain the parameters h∗ and kr2∗, considering that there are three dominant roots on σ∗.Proposition 2.
Let us consider the closed-loop system (25). Then, the quasi-polynomial (28) has a root of multiplicity at least three on s=−σ∗ if kr2∗ and h∗ satisfy the following equations:(44)h∗=hσ∗=∂/∂sPσ3sPσ30−∂/∂sΓs,σ∗Γ0,σ∗s=0,(45)kr2∗=kr2σ∗,h∗=−∂/∂sΓs,σ∗s=0eσ∗h∗∂/∂sPσ3ss=0−h∗Pσ30,and σ∗ is the smallest positive real root of(46)fσ,kp1,kp2,kr1,τ=Pσ302Γ0,σ∂2∂s2Γs,σ−∂∂sΓs,σ2s=0+Γ0,σ2∂∂sPσ3s2−Pσ30∂2∂s2Pσ3ss=0.Proof.
Clearly, quasi-polynomial (28) presents three dominant roots at the point s=−σ if the quasi-polynomial (29) has three dominant roots at s=0. Thus, the conditions(47)Qσ·s=0=0,ddsQσ·s=0=0,d2ds2Qσ·s=0=0,must be satisfied; consequently,(48)0=Γ0,σ+Pσ30kr2eσh,(49)0=∂∂sΓs,σs=0+kr2eσh∂∂sPσ3se−shs=0,(50)0=∂2∂s2Γs,σs=0+kr2eσh∂2∂s2Pσ3se−shs=0.
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 that(51)0=∂∂sΓs,σs=0−Γ0,σPσ30∂∂sPσ3ss=0+hΓ0,σ,and we get (44):(52)h=∂/∂sPσ3sPσ30−∂/∂sΓs,σΓ0,σs=0.
To address (46), from (48) and (50), it follows that(53)0=∂2∂s2Γs,σs=0+kr2eσh∂2∂s2Pσ3se−shs=0=∂2∂s2Γs,σs=0−Γ0,σPσ30∂2∂s2Pσ3ss=0+2hΓ0,σPσ30∂∂sPσ3ss=0−h2Γ0,σ.
Substituting (52) in the above equation yields(54)0=Pσ302Γ0,σ∂2∂s2Γs,σ−∂∂sΓs,σ2s=0+Γ0,σ2∂∂sPσ3s2−Pσ30∂2∂s2Pσ3ss=0,
which ends proof.
In the following corollary, conditions (44)–(46) are written particularly for the quasi-polynomial (27), where P1s,kp1,2=s4+kp2s2+kp1, P2s,kr1=−kr1, and P3s=−s2.Corollary 2.
Consider the quasi-polynomial of the form (27); then, the controller gains h∗, kr2∗ and the function fσ,· given by equations (44)–(46) look as(55)fσ,kp1,2,kr1,τ=σ42σkp2−2σ2−kr1τeστ2+σ2κσ22kp2+12σ2−kr1τ2eστ2σ2κ2,where κ=kp1−kr1eστ+σ2σ2+kp2 and(56)h∗σ,kp1,2,kr1,τ=2σμ1−kr1τeστkp1−kr1eστ+σ2μ2−2σ,kr2∗σ,kp1,2,kr1,h,τ=eσh2σμ1−kr1τeστσ2+hσ,withμ1=kp2+2σ2 and μ2=μ1−σ2.The conditions introduced in this section are only focused on the appropriate selection of the control parametersσ∗, kr2∗, 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,(57)Qs,kp1,2,kr1,2,τ,h=s4+kp2s2+kp1−kr11−sτ−kr2s21−sh=s4+hkr2s3+kp2−kr2s2+τkr1s+kp1−kr1.Now, let us propose a Hurwitz stable polynomial of the following form:(58)ps=s2+2ξcωcs+ωc22,where ξc,ωc∈ℝ+. Thus, the dynamics of the previous polynomial is matched with (57). Then, simple calculations show that the controller gain parameters kp1,kp2,kr1, and τ can be chosen according to the following conditions (for α1>1 and α2>2):(59)kp1>α1ωc4,kp2>α2ωc21+2ξc2,(60)kr1=kp1−ωc4,τ=4ξcωc3kr1.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 Figure2. The variables θ1t and θ2t denote the angular positions of the rotating base and the link, respectively, l represents the length of the link, Jb stands for the inertia of the rotating base, and Jl is the moment of inertia of the link. Finally, ks denotes the stiffness of the spring and ut is the torque input applied to the system.Figure 2
Schematics of the rotatory flexible link.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 Section3 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:(61)Mqtq¨t+Cqt,q˙tq˙t+gqt=Pτt,with(62)Mq=Jb+JlJlJlJl,Cq,q˙=0000,gq=0ksθ2t,qt=θ1tθ2t⊺,P=10⊺,or in an equivalent form:(63)θ..1t=ksJbθ2t+τtJb,θ..2t=−ksJb+JlJbJlθ2t−τtJb.


#### 5.1.2. Cascade Structure Representation

Let us introduce the state vector:(64)xt=x1tx2tx3tx4t⊺=θ1tθ.1tθ2tθ.2t⊺.Then, the applied torque can be expressed asτt=Nτm, where τm=kτ/RmVt is the torque in terms of the input voltage, N represents the gear ratio, kτ is the torque constant, and Rm denotes the motor armature resistance. Since (63) is linear, it can be rewritten in space state representation (24) as follows:(65)x˙t=Axt+bVt,(66)A=010000ksJb0000100−Jl+JbJlJbks0,b=0NkτRmJb0−NkτRmJb.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. The flat output can be computed according to (16), where by defining ε=RmJbJl/kτksN, the flat output yft is expressed as(67)yft=x1t+x3t.The flat output time derivatives are(68)yft=x1t+x2t,y˙ft=x2t+x4t,y¨ft=−ksJlx3t,yf3t=−ksJlx4t,yf4t=ks2Jl+JbJl2Jbx3t+kτksNRmJbJlVt.Let us define the output error as in (20); then, its dynamics are(69)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ks2Jl+JbJl2Jbx3t+kτksNRmJbJlVt−yf∗4t,where the auxiliary control Vt is proposed as(70)Vt=a1ut−ks2Jl+JbJl2Jbx3t+yf∗4t,a1=RmJbJlkτksN.


#### 5.1.3. Delayed Control Law

If the following delayed controller,ut=ut is considered:(71)uDt=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h.The chain of integrators representation (25) is obtained.Remark 2.
It is worth noting that both the auxiliary controlVt and the delayed controller uDt depend completely on the measurable variables that correspond to angular positions x1t=θ1t and x3t=θ2t.


#### 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:(72)uFSt=−κ1e1t−κ2e2−κ3e3t−κ4e4t,where the set of gains κ1,κ2,κ3,κ4 is chosen as κ1=ϖ4, κ2=4ς¯ϖ3, κ3=4ς¯2ϖ2+2ϖ2, and κ4=4ς¯ϖ.Remark 3.
Now, the auxiliary controlVt and the feedback state control law uFSt depend on the complete vector state xt, due to the fact that the velocity states x2t=θ.1t and x4t=θ.2t are not available; then, a low pass filter with transfer function Gs=200s/s+200 is used to estimate the velocity using the measurable variables x1t=θ1t and x3t=θ2t; 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 Figure3, 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].Figure 3
Rotatory flexible link prototype.The rotatory flexible link parameters areInertias:Jb=0.0481 [Kg − m2] and Jl=0.0036 [Kg − m2].Length of the link:l=0.55 [m].Stiffness of the spring:ks=4 [N − m/rad].Armature resistance:kτ=0.0724Ω.Torque constant:τm=2.983Ω.Mechanical advantage:N=16.In this experiment, the tracking trajectory problem is addressed. The initial conditions arex0=0. The desired trajectory yf∗t consists of a path initialized at yf∗0=0[rad]. After two seconds in this position, the mechanism moves, in an interval of 0.7[s], to yf∗2=π/2[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 ωc=52 and ξc=1.2; thus, kp1=29246464, kp2=33572.864, kr1=21934848, and τ=0.02. Now, the conditions presented in Corollary 4 lead to σ∗=66.65 and consequently kr2∗=18839.729 and h∗=0.00476.Theσ-stability boundaries were calculated following Corollary 1, which is illustrated in Figure 4. The 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 4
σ-stability boundaries.Figure5 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 σ∗=66.65 as stated in Proposition 3.Figure 5
Root locus.The set of gains for the feedback state controller were chosen asϖ=52 and ς¯=1.2. 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 yf∗t is represented by a black line, while the flat output yfDt with the delayed controller is depicted in blue and the yfFSt 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 eD and eFS are restricted to the interval −0.075,0.075[rad] with similar performance. Figures 8 and 9 represent the evolution of the angular positions θ1 and θ2 when the system follows the reference trajectory. Figure 10 shows the control voltages of both schemes. Notice that the peak voltage VD does not exceed ±10[Volts] but VFS presents a larger amplitude. As a consequence of avoiding the use of the time derivatives in the proposed control scheme, the voltage VD signal appears relatively free of noise, as expected but VFS presents high-frequency components; it can be corroborated in Figure 11 where a Power Spectrum Density of control signals VD and VFS is presented. Here, it can be observed that FSPSD presents more frequency components in comparison with DPSD. 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 VFS consumes more energy.Figure 6
Trajectory tracking results.Figure 7
Tracking error.Figure 8
θ1 evolution.Figure 9
θ2 evolution.Figure 10
Control input voltage.Figure 11
Power spectrum density.Figure 12
Performance index of the controller.


### 5.2. Pendubot System

The pendubot consists of a planar double inverted pendulum, whose schematic is presented in Figure13. The first link is driven by a DC motor while the second link is an underactuated simple pendulum. The variables θ1 and θ2 denote the angular positions of the links, u represents the control torque input, and the parameters m1 and m2 denote the masses of the links. The lengths of the links are represented as l1 and l2, while lc1 and lc2 are the distances to the center of the masses. Finally, I1 and I2 denote the inertias of the links.Figure 13
Schematics of the pendubot system.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:(73)Mqtq¨t+Cqt,q˙tqt+gqt=Pτt,with qt=θ1tθ2t⊺ and P=10⊺. Here,(74)Mq=β1+β2+2β3cosθ2tM1,2θβ2+β3cosθ2tβ2,Cq,q˙=−β3θ.1tsinθ2tβ3C1,2θβ3θ.1tsinθ2t0,gq=β4gcosθ1t+β5gcosθ1t+θ2tβ5gcosθ1t+θ2t,where(75)β1=m1lc12+m2lc22+I1,β2=m2lc22+I2,β3=m2l1lc2,β4=m1lc1+m1l1,β5=m2lc2,M1,2θ=β2+β3cosθ2t,C1,2θ=θ.1t+θ.2tsinθ2t.


#### 5.2.2. Cascade Structure Representation

In this section, the procedure presented inSection 3 is applied. The state vector is defined as(76)xt=x1tx2tx3tx4t⊺=θ1tθ.1tθ2tθ.2t⊺.The applied torque can be expressed asτt=kτ/RmVt in terms of the input voltage, where kτ is the torque constant and Rm represents the motor armature resistance. For V=0, the considered equilibrium point x¯t is(77)x¯t=π2000⊺.Then, the tangent linearization looks as(78)x˙δt=Axδt+bVt,where xδt=x1t−π2x2tx3tx4t⊺,(79)A=0100β3β5−β2β4gβ32−β1β20β3β5gβ32−β1β200001β2+β3β4−β1+β3β5gβ32−β1β20−β1+β3β5gβ32−β1β20,b=0−kτβ2Rmβ32−β1β20kτβ2+β3Rmβ32−β1β2T.By defining the Kalman controllability matrix asKc=bAbA2bA3b, the pair A,b is controllable since Kc has full rank. Then, according to Definition 2, it implies that system (42) is flat, and the corresponding flat output is computed as(80)yft=ε0001Kc−1xδt=β2+β3β2xδ1t+xδ3t.Here,ε is selected as ε=Rmβ2β32−β1β2/kτβ3β5g. Now, by straightforward calculations, the flat output time derivatives (10) are given by(81)yft=β2+β3β2xδ1t+xδ3t,y˙ft=β2+β3β2xδ2t+xδ4t,y¨ft=gβ5β2xδ1t+xδ3t,yf3t=gβ5β2xδ2t+xδ4t,yf4t=ρ1xδ1t−ρ2xδ1t+xδ3t+ρ3Vt,with(82)ρ1=β3β4β5g2β2β32−β1β2,ρ2=β1β52g2β2β32−β1β2,ρ3=kτβ3β5gRmβ2β32−β1β2.By defining the output error as in (20), then, the set of error dynamics (21) are given as follows:(83)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ρ1xδ1t−ρ2xδ1t+xδ3t+ρ3uδ1t−yf∗4t.Now, by proposing the auxiliary control(84)Vt=ρ3−1ut−ρ1xδ1t+ρ2xδ1t+xδ3t+ρ3−1yf∗4t,and by considering the delayed control law(85)ut=uDt=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h,the desired chain of integrators representation (25) is obtained.Remark 4.
As in the previous experiment, the auxiliary controlVt and the delayed control law uDt 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:(86)uFSt=−κ1e1t−κ2e2−κ3e3t−κ4e4t,where the following set of gains κ1,κ2,κ3,κ4 is chosen as κ1=ϖ4, κ2=4ς¯ϖ3, κ3=4ς¯2ϖ2+2ϖ2, and κ4=4ς¯ϖ.Remark 5.
The velocity statesx2t and x4t are estimated using a low pass filter with transfer function Gs=200s/s+200.


#### 5.2.4. Experimental Results

Figure14 shows the experimental pendubot prototype. It consists of a DC motor NISCA model NC5475, which drives the first link. The angular position of both links is measured by means of incremental encoders with a resolution of 10000 counts per revolution. The same data acquisition model of the former example was used. The power amplifier consists of a Quanser amplifier model VoltPAQ-X2. The control strategy was implemented in the Matlab-Simulink platform, and the sampling time was set to be 0.001[s]. The pendubot parameters are as follows.Figure 14
Pendubot prototype.The test on the pendubot was carried out as follows:Links Inertias:I1=0.00053 [Kg − m2] and I2=0.00077 [Kg − m2].Mass of the links:m1=0.210 [Kg] and m2=0.1 [Kg].Length of the links:l1=0.15 [m] and l2=0.3 [m].Distance to the center of mass:lc1=0.12 [m] and lc2=0.15 [m].Armature resistance:kτ=0.0724 [Ω].Torque constant:τm=2.983 [Ω].The initial conditions are set asx10=π/2[rad] and x30=0[rad]. The desired trajectory consists of a rest to rest smooth trajectory, described as follows:(87)yf∗t=β2+β3β2xδ1∗t+xδ3∗t,yf∗t=β2+β3β2θ1∗t−π2+θ2∗t.Att=0, the trajectory is set at yf∗0=0; it implies that θ1∗t=π/2 and θ2∗t=0; when the time reaches t=4.5, the desired trajectory moves to yf∗6=β2+β3/β2π/6−π/6 with θ1∗t=2/3π and θ2∗t=−π/6 on a lapse of 2.5 seconds. Then, when time is t=9.5[s], it moves to yf∗13.5=−β2+β3/β2π/6+π/6 with θ1∗t=π/3 and θ2∗t=−π/6 in 4 seconds, and finally, when t=17.5[s], it returns to the initial position and remains in this position until the test is finished. Figure 15 shows the desired rest to rest positions of the pendubot system.Figure 15
Pendubot desired rest to rest positions.The parameters of the controller were chosen according to (59 and 60), selecting ωc=14 and ξc=1.2 and then kp1=460992, kp2=4562.88, kr1=422576, and τ=0.0311. The remainder of the parameters are calculated following Corollary 2, leading to σ∗=8.8477, kr2=2832.39, and h∗=0.021. The 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. The 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 16
σ-stability boundaries.Figure17 shows the flat output trajectory tracking performance of both controllers, the reference trajectory yf∗t is depicted in black line, the delayed controller response yDt is shown in the blue line, and the feedback controller response is shown in the red line yFSt, 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 eDt and eFSt 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 VD calculated without using any time derivatives, which shows less noise and less amplitude in contrast with VFS. The performance of both controllers is similar to good results; the main difference is on the voltage control. The Power Spectrum Density of VD and VFS is depicted in Figure 22. Notice that FSPSD shows high-frequency components with a peak on 50[Hz]. Figure 23 shows an analysis using a quadratic index of the voltage, where VFS consumes more energy. Thus, 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 trajectory tasks was performed in simulation using a set of initial conditions for the angular positions θ10 and θ20. Figure 24 shows the first link performance; notice that the PR controller rank of initial conditions θ1D0 remains on −8,8 degrees, in comparison with the FS controller rank of initial conditions θ1FS0 that remains on the interval −7,7 degrees. Similarly, Figure 25 shows the second link performance, the rank of the initial conditions for θ2D0 is −12,12 degrees, while θ2FS0 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.Figure 17
Flat output trajectory tracking.Figure 18
Tracking error.Figure 19
θ1 performance.Figure 20
θ2 performance.Figure 21
Control input voltage.Figure 22
Power spectrum density.Figure 23
Performance index of the controller.Figure 24
Assessment of the quadratic error index for the first link.Figure 25
Assessment of the quadratic error index for the second link.


## 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 Figure2. The variables θ1t and θ2t denote the angular positions of the rotating base and the link, respectively, l represents the length of the link, Jb stands for the inertia of the rotating base, and Jl is the moment of inertia of the link. Finally, ks denotes the stiffness of the spring and ut is the torque input applied to the system.Figure 2
Schematics of the rotatory flexible link.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 Section3 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:(61)Mqtq¨t+Cqt,q˙tq˙t+gqt=Pτt,with(62)Mq=Jb+JlJlJlJl,Cq,q˙=0000,gq=0ksθ2t,qt=θ1tθ2t⊺,P=10⊺,or in an equivalent form:(63)θ..1t=ksJbθ2t+τtJb,θ..2t=−ksJb+JlJbJlθ2t−τtJb.


### 5.1.2. Cascade Structure Representation

Let us introduce the state vector:(64)xt=x1tx2tx3tx4t⊺=θ1tθ.1tθ2tθ.2t⊺.Then, the applied torque can be expressed asτt=Nτm, where τm=kτ/RmVt is the torque in terms of the input voltage, N represents the gear ratio, kτ is the torque constant, and Rm denotes the motor armature resistance. Since (63) is linear, it can be rewritten in space state representation (24) as follows:(65)x˙t=Axt+bVt,(66)A=010000ksJb0000100−Jl+JbJlJbks0,b=0NkτRmJb0−NkτRmJb.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. The flat output can be computed according to (16), where by defining ε=RmJbJl/kτksN, the flat output yft is expressed as(67)yft=x1t+x3t.The flat output time derivatives are(68)yft=x1t+x2t,y˙ft=x2t+x4t,y¨ft=−ksJlx3t,yf3t=−ksJlx4t,yf4t=ks2Jl+JbJl2Jbx3t+kτksNRmJbJlVt.Let us define the output error as in (20); then, its dynamics are(69)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ks2Jl+JbJl2Jbx3t+kτksNRmJbJlVt−yf∗4t,where the auxiliary control Vt is proposed as(70)Vt=a1ut−ks2Jl+JbJl2Jbx3t+yf∗4t,a1=RmJbJlkτksN.


### 5.1.3. Delayed Control Law

If the following delayed controller,ut=ut is considered:(71)uDt=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h.The chain of integrators representation (25) is obtained.Remark 2.
It is worth noting that both the auxiliary controlVt and the delayed controller uDt depend completely on the measurable variables that correspond to angular positions x1t=θ1t and x3t=θ2t.


### 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:(72)uFSt=−κ1e1t−κ2e2−κ3e3t−κ4e4t,where the set of gains κ1,κ2,κ3,κ4 is chosen as κ1=ϖ4, κ2=4ς¯ϖ3, κ3=4ς¯2ϖ2+2ϖ2, and κ4=4ς¯ϖ.Remark 3.
Now, the auxiliary controlVt and the feedback state control law uFSt depend on the complete vector state xt, due to the fact that the velocity states x2t=θ.1t and x4t=θ.2t are not available; then, a low pass filter with transfer function Gs=200s/s+200 is used to estimate the velocity using the measurable variables x1t=θ1t and x3t=θ2t; 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 Figure3, 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].Figure 3
Rotatory flexible link prototype.The rotatory flexible link parameters areInertias:Jb=0.0481 [Kg − m2] and Jl=0.0036 [Kg − m2].Length of the link:l=0.55 [m].Stiffness of the spring:ks=4 [N − m/rad].Armature resistance:kτ=0.0724Ω.Torque constant:τm=2.983Ω.Mechanical advantage:N=16.In this experiment, the tracking trajectory problem is addressed. The initial conditions arex0=0. The desired trajectory yf∗t consists of a path initialized at yf∗0=0[rad]. After two seconds in this position, the mechanism moves, in an interval of 0.7[s], to yf∗2=π/2[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 ωc=52 and ξc=1.2; thus, kp1=29246464, kp2=33572.864, kr1=21934848, and τ=0.02. Now, the conditions presented in Corollary 4 lead to σ∗=66.65 and consequently kr2∗=18839.729 and h∗=0.00476.Theσ-stability boundaries were calculated following Corollary 1, which is illustrated in Figure 4. The 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 4
σ-stability boundaries.Figure5 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 σ∗=66.65 as stated in Proposition 3.Figure 5
Root locus.The set of gains for the feedback state controller were chosen asϖ=52 and ς¯=1.2. 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 yf∗t is represented by a black line, while the flat output yfDt with the delayed controller is depicted in blue and the yfFSt 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 eD and eFS are restricted to the interval −0.075,0.075[rad] with similar performance. Figures 8 and 9 represent the evolution of the angular positions θ1 and θ2 when the system follows the reference trajectory. Figure 10 shows the control voltages of both schemes. Notice that the peak voltage VD does not exceed ±10[Volts] but VFS presents a larger amplitude. As a consequence of avoiding the use of the time derivatives in the proposed control scheme, the voltage VD signal appears relatively free of noise, as expected but VFS presents high-frequency components; it can be corroborated in Figure 11 where a Power Spectrum Density of control signals VD and VFS is presented. Here, it can be observed that FSPSD presents more frequency components in comparison with DPSD. 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 VFS consumes more energy.Figure 6
Trajectory tracking results.Figure 7
Tracking error.Figure 8
θ1 evolution.Figure 9
θ2 evolution.Figure 10
Control input voltage.Figure 11
Power spectrum density.Figure 12
Performance index of the controller.


## 5.1.1. The Dynamic Model

Following the methodology presented in Section3 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:(61)Mqtq¨t+Cqt,q˙tq˙t+gqt=Pτt,with(62)Mq=Jb+JlJlJlJl,Cq,q˙=0000,gq=0ksθ2t,qt=θ1tθ2t⊺,P=10⊺,or in an equivalent form:(63)θ..1t=ksJbθ2t+τtJb,θ..2t=−ksJb+JlJbJlθ2t−τtJb.


## 5.1.2. Cascade Structure Representation

Let us introduce the state vector:(64)xt=x1tx2tx3tx4t⊺=θ1tθ.1tθ2tθ.2t⊺.Then, the applied torque can be expressed asτt=Nτm, where τm=kτ/RmVt is the torque in terms of the input voltage, N represents the gear ratio, kτ is the torque constant, and Rm denotes the motor armature resistance. Since (63) is linear, it can be rewritten in space state representation (24) as follows:(65)x˙t=Axt+bVt,(66)A=010000ksJb0000100−Jl+JbJlJbks0,b=0NkτRmJb0−NkτRmJb.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. The flat output can be computed according to (16), where by defining ε=RmJbJl/kτksN, the flat output yft is expressed as(67)yft=x1t+x3t.The flat output time derivatives are(68)yft=x1t+x2t,y˙ft=x2t+x4t,y¨ft=−ksJlx3t,yf3t=−ksJlx4t,yf4t=ks2Jl+JbJl2Jbx3t+kτksNRmJbJlVt.Let us define the output error as in (20); then, its dynamics are(69)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ks2Jl+JbJl2Jbx3t+kτksNRmJbJlVt−yf∗4t,where the auxiliary control Vt is proposed as(70)Vt=a1ut−ks2Jl+JbJl2Jbx3t+yf∗4t,a1=RmJbJlkτksN.


## 5.1.3. Delayed Control Law

If the following delayed controller,ut=ut is considered:(71)uDt=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h.The chain of integrators representation (25) is obtained.Remark 2.
It is worth noting that both the auxiliary controlVt and the delayed controller uDt depend completely on the measurable variables that correspond to angular positions x1t=θ1t and x3t=θ2t.


## 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:(72)uFSt=−κ1e1t−κ2e2−κ3e3t−κ4e4t,where the set of gains κ1,κ2,κ3,κ4 is chosen as κ1=ϖ4, κ2=4ς¯ϖ3, κ3=4ς¯2ϖ2+2ϖ2, and κ4=4ς¯ϖ.Remark 3.
Now, the auxiliary controlVt and the feedback state control law uFSt depend on the complete vector state xt, due to the fact that the velocity states x2t=θ.1t and x4t=θ.2t are not available; then, a low pass filter with transfer function Gs=200s/s+200 is used to estimate the velocity using the measurable variables x1t=θ1t and x3t=θ2t; 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 Figure3, 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].Figure 3
Rotatory flexible link prototype.The rotatory flexible link parameters areInertias:Jb=0.0481 [Kg − m2] and Jl=0.0036 [Kg − m2].Length of the link:l=0.55 [m].Stiffness of the spring:ks=4 [N − m/rad].Armature resistance:kτ=0.0724Ω.Torque constant:τm=2.983Ω.Mechanical advantage:N=16.In this experiment, the tracking trajectory problem is addressed. The initial conditions arex0=0. The desired trajectory yf∗t consists of a path initialized at yf∗0=0[rad]. After two seconds in this position, the mechanism moves, in an interval of 0.7[s], to yf∗2=π/2[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 ωc=52 and ξc=1.2; thus, kp1=29246464, kp2=33572.864, kr1=21934848, and τ=0.02. Now, the conditions presented in Corollary 4 lead to σ∗=66.65 and consequently kr2∗=18839.729 and h∗=0.00476.Theσ-stability boundaries were calculated following Corollary 1, which is illustrated in Figure 4. The 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 4
σ-stability boundaries.Figure5 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 σ∗=66.65 as stated in Proposition 3.Figure 5
Root locus.The set of gains for the feedback state controller were chosen asϖ=52 and ς¯=1.2. 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 yf∗t is represented by a black line, while the flat output yfDt with the delayed controller is depicted in blue and the yfFSt 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 eD and eFS are restricted to the interval −0.075,0.075[rad] with similar performance. Figures 8 and 9 represent the evolution of the angular positions θ1 and θ2 when the system follows the reference trajectory. Figure 10 shows the control voltages of both schemes. Notice that the peak voltage VD does not exceed ±10[Volts] but VFS presents a larger amplitude. As a consequence of avoiding the use of the time derivatives in the proposed control scheme, the voltage VD signal appears relatively free of noise, as expected but VFS presents high-frequency components; it can be corroborated in Figure 11 where a Power Spectrum Density of control signals VD and VFS is presented. Here, it can be observed that FSPSD presents more frequency components in comparison with DPSD. 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 VFS consumes more energy.Figure 6
Trajectory tracking results.Figure 7
Tracking error.Figure 8
θ1 evolution.Figure 9
θ2 evolution.Figure 10
Control input voltage.Figure 11
Power spectrum density.Figure 12
Performance index of the controller.


## 5.2. Pendubot System

The pendubot consists of a planar double inverted pendulum, whose schematic is presented in Figure13. The first link is driven by a DC motor while the second link is an underactuated simple pendulum. The variables θ1 and θ2 denote the angular positions of the links, u represents the control torque input, and the parameters m1 and m2 denote the masses of the links. The lengths of the links are represented as l1 and l2, while lc1 and lc2 are the distances to the center of the masses. Finally, I1 and I2 denote the inertias of the links.Figure 13
Schematics of the pendubot system.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:(73)Mqtq¨t+Cqt,q˙tqt+gqt=Pτt,with qt=θ1tθ2t⊺ and P=10⊺. Here,(74)Mq=β1+β2+2β3cosθ2tM1,2θβ2+β3cosθ2tβ2,Cq,q˙=−β3θ.1tsinθ2tβ3C1,2θβ3θ.1tsinθ2t0,gq=β4gcosθ1t+β5gcosθ1t+θ2tβ5gcosθ1t+θ2t,where(75)β1=m1lc12+m2lc22+I1,β2=m2lc22+I2,β3=m2l1lc2,β4=m1lc1+m1l1,β5=m2lc2,M1,2θ=β2+β3cosθ2t,C1,2θ=θ.1t+θ.2tsinθ2t.


### 5.2.2. Cascade Structure Representation

In this section, the procedure presented inSection 3 is applied. The state vector is defined as(76)xt=x1tx2tx3tx4t⊺=θ1tθ.1tθ2tθ.2t⊺.The applied torque can be expressed asτt=kτ/RmVt in terms of the input voltage, where kτ is the torque constant and Rm represents the motor armature resistance. For V=0, the considered equilibrium point x¯t is(77)x¯t=π2000⊺.Then, the tangent linearization looks as(78)x˙δt=Axδt+bVt,where xδt=x1t−π2x2tx3tx4t⊺,(79)A=0100β3β5−β2β4gβ32−β1β20β3β5gβ32−β1β200001β2+β3β4−β1+β3β5gβ32−β1β20−β1+β3β5gβ32−β1β20,b=0−kτβ2Rmβ32−β1β20kτβ2+β3Rmβ32−β1β2T.By defining the Kalman controllability matrix asKc=bAbA2bA3b, the pair A,b is controllable since Kc has full rank. Then, according to Definition 2, it implies that system (42) is flat, and the corresponding flat output is computed as(80)yft=ε0001Kc−1xδt=β2+β3β2xδ1t+xδ3t.Here,ε is selected as ε=Rmβ2β32−β1β2/kτβ3β5g. Now, by straightforward calculations, the flat output time derivatives (10) are given by(81)yft=β2+β3β2xδ1t+xδ3t,y˙ft=β2+β3β2xδ2t+xδ4t,y¨ft=gβ5β2xδ1t+xδ3t,yf3t=gβ5β2xδ2t+xδ4t,yf4t=ρ1xδ1t−ρ2xδ1t+xδ3t+ρ3Vt,with(82)ρ1=β3β4β5g2β2β32−β1β2,ρ2=β1β52g2β2β32−β1β2,ρ3=kτβ3β5gRmβ2β32−β1β2.By defining the output error as in (20), then, the set of error dynamics (21) are given as follows:(83)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ρ1xδ1t−ρ2xδ1t+xδ3t+ρ3uδ1t−yf∗4t.Now, by proposing the auxiliary control(84)Vt=ρ3−1ut−ρ1xδ1t+ρ2xδ1t+xδ3t+ρ3−1yf∗4t,and by considering the delayed control law(85)ut=uDt=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h,the desired chain of integrators representation (25) is obtained.Remark 4.
As in the previous experiment, the auxiliary controlVt and the delayed control law uDt 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:(86)uFSt=−κ1e1t−κ2e2−κ3e3t−κ4e4t,where the following set of gains κ1,κ2,κ3,κ4 is chosen as κ1=ϖ4, κ2=4ς¯ϖ3, κ3=4ς¯2ϖ2+2ϖ2, and κ4=4ς¯ϖ.Remark 5.
The velocity statesx2t and x4t are estimated using a low pass filter with transfer function Gs=200s/s+200.


### 5.2.4. Experimental Results

Figure14 shows the experimental pendubot prototype. It consists of a DC motor NISCA model NC5475, which drives the first link. The angular position of both links is measured by means of incremental encoders with a resolution of 10000 counts per revolution. The same data acquisition model of the former example was used. The power amplifier consists of a Quanser amplifier model VoltPAQ-X2. The control strategy was implemented in the Matlab-Simulink platform, and the sampling time was set to be 0.001[s]. The pendubot parameters are as follows.Figure 14
Pendubot prototype.The test on the pendubot was carried out as follows:Links Inertias:I1=0.00053 [Kg − m2] and I2=0.00077 [Kg − m2].Mass of the links:m1=0.210 [Kg] and m2=0.1 [Kg].Length of the links:l1=0.15 [m] and l2=0.3 [m].Distance to the center of mass:lc1=0.12 [m] and lc2=0.15 [m].Armature resistance:kτ=0.0724 [Ω].Torque constant:τm=2.983 [Ω].The initial conditions are set asx10=π/2[rad] and x30=0[rad]. The desired trajectory consists of a rest to rest smooth trajectory, described as follows:(87)yf∗t=β2+β3β2xδ1∗t+xδ3∗t,yf∗t=β2+β3β2θ1∗t−π2+θ2∗t.Att=0, the trajectory is set at yf∗0=0; it implies that θ1∗t=π/2 and θ2∗t=0; when the time reaches t=4.5, the desired trajectory moves to yf∗6=β2+β3/β2π/6−π/6 with θ1∗t=2/3π and θ2∗t=−π/6 on a lapse of 2.5 seconds. Then, when time is t=9.5[s], it moves to yf∗13.5=−β2+β3/β2π/6+π/6 with θ1∗t=π/3 and θ2∗t=−π/6 in 4 seconds, and finally, when t=17.5[s], it returns to the initial position and remains in this position until the test is finished. Figure 15 shows the desired rest to rest positions of the pendubot system.Figure 15
Pendubot desired rest to rest positions.The parameters of the controller were chosen according to (59 and 60), selecting ωc=14 and ξc=1.2 and then kp1=460992, kp2=4562.88, kr1=422576, and τ=0.0311. The remainder of the parameters are calculated following Corollary 2, leading to σ∗=8.8477, kr2=2832.39, and h∗=0.021. The 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. The 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 16
σ-stability boundaries.Figure17 shows the flat output trajectory tracking performance of both controllers, the reference trajectory yf∗t is depicted in black line, the delayed controller response yDt is shown in the blue line, and the feedback controller response is shown in the red line yFSt, 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 eDt and eFSt 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 VD calculated without using any time derivatives, which shows less noise and less amplitude in contrast with VFS. The performance of both controllers is similar to good results; the main difference is on the voltage control. The Power Spectrum Density of VD and VFS is depicted in Figure 22. Notice that FSPSD shows high-frequency components with a peak on 50[Hz]. Figure 23 shows an analysis using a quadratic index of the voltage, where VFS consumes more energy. Thus, 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 trajectory tasks was performed in simulation using a set of initial conditions for the angular positions θ10 and θ20. Figure 24 shows the first link performance; notice that the PR controller rank of initial conditions θ1D0 remains on −8,8 degrees, in comparison with the FS controller rank of initial conditions θ1FS0 that remains on the interval −7,7 degrees. Similarly, Figure 25 shows the second link performance, the rank of the initial conditions for θ2D0 is −12,12 degrees, while θ2FS0 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.Figure 17
Flat output trajectory tracking.Figure 18
Tracking error.Figure 19
θ1 performance.Figure 20
θ2 performance.Figure 21
Control input voltage.Figure 22
Power spectrum density.Figure 23
Performance index of the controller.Figure 24
Assessment of the quadratic error index for the first link.Figure 25
Assessment of the quadratic error index for the second link.


## 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:(73)Mqtq¨t+Cqt,q˙tqt+gqt=Pτt,with qt=θ1tθ2t⊺ and P=10⊺. Here,(74)Mq=β1+β2+2β3cosθ2tM1,2θβ2+β3cosθ2tβ2,Cq,q˙=−β3θ.1tsinθ2tβ3C1,2θβ3θ.1tsinθ2t0,gq=β4gcosθ1t+β5gcosθ1t+θ2tβ5gcosθ1t+θ2t,where(75)β1=m1lc12+m2lc22+I1,β2=m2lc22+I2,β3=m2l1lc2,β4=m1lc1+m1l1,β5=m2lc2,M1,2θ=β2+β3cosθ2t,C1,2θ=θ.1t+θ.2tsinθ2t.


## 5.2.2. Cascade Structure Representation

In this section, the procedure presented inSection 3 is applied. The state vector is defined as(76)xt=x1tx2tx3tx4t⊺=θ1tθ.1tθ2tθ.2t⊺.The applied torque can be expressed asτt=kτ/RmVt in terms of the input voltage, where kτ is the torque constant and Rm represents the motor armature resistance. For V=0, the considered equilibrium point x¯t is(77)x¯t=π2000⊺.Then, the tangent linearization looks as(78)x˙δt=Axδt+bVt,where xδt=x1t−π2x2tx3tx4t⊺,(79)A=0100β3β5−β2β4gβ32−β1β20β3β5gβ32−β1β200001β2+β3β4−β1+β3β5gβ32−β1β20−β1+β3β5gβ32−β1β20,b=0−kτβ2Rmβ32−β1β20kτβ2+β3Rmβ32−β1β2T.By defining the Kalman controllability matrix asKc=bAbA2bA3b, the pair A,b is controllable since Kc has full rank. Then, according to Definition 2, it implies that system (42) is flat, and the corresponding flat output is computed as(80)yft=ε0001Kc−1xδt=β2+β3β2xδ1t+xδ3t.Here,ε is selected as ε=Rmβ2β32−β1β2/kτβ3β5g. Now, by straightforward calculations, the flat output time derivatives (10) are given by(81)yft=β2+β3β2xδ1t+xδ3t,y˙ft=β2+β3β2xδ2t+xδ4t,y¨ft=gβ5β2xδ1t+xδ3t,yf3t=gβ5β2xδ2t+xδ4t,yf4t=ρ1xδ1t−ρ2xδ1t+xδ3t+ρ3Vt,with(82)ρ1=β3β4β5g2β2β32−β1β2,ρ2=β1β52g2β2β32−β1β2,ρ3=kτβ3β5gRmβ2β32−β1β2.By defining the output error as in (20), then, the set of error dynamics (21) are given as follows:(83)e˙1t=e2t,e˙2t=e3t,e˙3t=e4t,e˙4t=ρ1xδ1t−ρ2xδ1t+xδ3t+ρ3uδ1t−yf∗4t.Now, by proposing the auxiliary control(84)Vt=ρ3−1ut−ρ1xδ1t+ρ2xδ1t+xδ3t+ρ3−1yf∗4t,and by considering the delayed control law(85)ut=uDt=−kp1e1t+kr1e1t−τ−kp2e3t+kr2e3t−h,the desired chain of integrators representation (25) is obtained.Remark 4.
As in the previous experiment, the auxiliary controlVt and the delayed control law uDt 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:(86)uFSt=−κ1e1t−κ2e2−κ3e3t−κ4e4t,where the following set of gains κ1,κ2,κ3,κ4 is chosen as κ1=ϖ4, κ2=4ς¯ϖ3, κ3=4ς¯2ϖ2+2ϖ2, and κ4=4ς¯ϖ.Remark 5.
The velocity statesx2t and x4t are estimated using a low pass filter with transfer function Gs=200s/s+200.


## 5.2.4. Experimental Results

Figure14 shows the experimental pendubot prototype. It consists of a DC motor NISCA model NC5475, which drives the first link. The angular position of both links is measured by means of incremental encoders with a resolution of 10000 counts per revolution. The same data acquisition model of the former example was used. The power amplifier consists of a Quanser amplifier model VoltPAQ-X2. The control strategy was implemented in the Matlab-Simulink platform, and the sampling time was set to be 0.001[s]. The pendubot parameters are as follows.Figure 14
Pendubot prototype.The test on the pendubot was carried out as follows:Links Inertias:I1=0.00053 [Kg − m2] and I2=0.00077 [Kg − m2].Mass of the links:m1=0.210 [Kg] and m2=0.1 [Kg].Length of the links:l1=0.15 [m] and l2=0.3 [m].Distance to the center of mass:lc1=0.12 [m] and lc2=0.15 [m].Armature resistance:kτ=0.0724 [Ω].Torque constant:τm=2.983 [Ω].The initial conditions are set asx10=π/2[rad] and x30=0[rad]. The desired trajectory consists of a rest to rest smooth trajectory, described as follows:(87)yf∗t=β2+β3β2xδ1∗t+xδ3∗t,yf∗t=β2+β3β2θ1∗t−π2+θ2∗t.Att=0, the trajectory is set at yf∗0=0; it implies that θ1∗t=π/2 and θ2∗t=0; when the time reaches t=4.5, the desired trajectory moves to yf∗6=β2+β3/β2π/6−π/6 with θ1∗t=2/3π and θ2∗t=−π/6 on a lapse of 2.5 seconds. Then, when time is t=9.5[s], it moves to yf∗13.5=−β2+β3/β2π/6+π/6 with θ1∗t=π/3 and θ2∗t=−π/6 in 4 seconds, and finally, when t=17.5[s], it returns to the initial position and remains in this position until the test is finished. Figure 15 shows the desired rest to rest positions of the pendubot system.Figure 15
Pendubot desired rest to rest positions.The parameters of the controller were chosen according to (59 and 60), selecting ωc=14 and ξc=1.2 and then kp1=460992, kp2=4562.88, kr1=422576, and τ=0.0311. The remainder of the parameters are calculated following Corollary 2, leading to σ∗=8.8477, kr2=2832.39, and h∗=0.021. The 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. The 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 16
σ-stability boundaries.Figure17 shows the flat output trajectory tracking performance of both controllers, the reference trajectory yf∗t is depicted in black line, the delayed controller response yDt is shown in the blue line, and the feedback controller response is shown in the red line yFSt, 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 eDt and eFSt 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 VD calculated without using any time derivatives, which shows less noise and less amplitude in contrast with VFS. The performance of both controllers is similar to good results; the main difference is on the voltage control. The Power Spectrum Density of VD and VFS is depicted in Figure 22. Notice that FSPSD shows high-frequency components with a peak on 50[Hz]. Figure 23 shows an analysis using a quadratic index of the voltage, where VFS consumes more energy. Thus, 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 trajectory tasks was performed in simulation using a set of initial conditions for the angular positions θ10 and θ20. Figure 24 shows the first link performance; notice that the PR controller rank of initial conditions θ1D0 remains on −8,8 degrees, in comparison with the FS controller rank of initial conditions θ1FS0 that remains on the interval −7,7 degrees. Similarly, Figure 25 shows the second link performance, the rank of the initial conditions for θ2D0 is −12,12 degrees, while θ2FS0 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.Figure 17
Flat output trajectory tracking.Figure 18
Tracking error.Figure 19
θ1 performance.Figure 20
θ2 performance.Figure 21
Control input voltage.Figure 22
Power spectrum density.Figure 23
Performance index of the controller.Figure 24
Assessment of the quadratic error index for the first link.Figure 25
Assessment of the quadratic error index for the second link.


## 6. 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. The tuning procedure allows the practitioner to find appropriate control gains (the proposal is constructive). The cascade form of the tangent linearization of the underactuated system is an important design tool for the model simplification and the 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. The case of study can be also extended for disturbed dynamics, in which integral actions can be proposed and tested.



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