# σ-Stabilization of a Flexible Joint Robotic Arm via Delayed Controllers

Authors: G. Ochoa-Ortega; R. Villafuerte-Segura; M. Ramírez-Neria; L. Vite-Hernández
Journal: Complexity
 (2019)

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## Abstract

In the present contribution, the problem of establishing tuning rules to proportional retarded controller for LTI systems is addressed. Based on theD-decomposition methodology and σ-stability analysis, analytic conditions are determined on the parameters of a delayed controller that guarantee us that the system response reaches the maximal decay rate. The conditions presented in this paper are tested experimentally in tracking tasks of a flexible joint robotic arm.


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## Body

## 1. Introduction

Time delays naturally arise in many mathematical models from engineering, biology, and physics, among other science areas. A common belief is that the appearance of these phenomena can lead to detrimental effects, bad performance of the system, instability, or even damage to the system [1]. Nevertheless, many contributions have shown that the deliberate introduction of a time delay in a feedback control law can provide a stabilizing effect [2–7] and in some cases give or improve the robustness property of the system [8]. The stability analysis of linear time delay systems is studied in the framework of two main approaches: time domain and frequency domain. The former is based on the well-known Lyapunov’s criteria and its extensions (Razumikhin and Krasovskii) or by using linear matrix inequalities (LMIs) via convex optimization [9, 10]. However, only sufficient stability conditions are provided, which are generally very conservative or, in some cases, nonexistent, mainly because the feasibility of LMIs (whose parameters are adjusted by a frequency method) is usually nonexistent. The second approach is based on the analysis of the characteristic function of the system, where, unlike the temporal approach, it is possible to provide necessary and sufficient conditions that do not have a conservative nature.Even when in the recent decades there is a great development in the control theory, where many sophisticated control schemes have been developed, PID controllers remain as one of the most used control strategies in the industrial environment. According to [11, 12], approximately 90–95% of the industrial control loops still use PID-type controllers, many of which do not include the derivative term, mainly because this term induces a noise amplifying effect that can affect the system performance [12–14] or because the derivative of the state is not available for measurement. Some common strategies to avoid the use of the derivative term and, consequently, the noise amplification problem, are the use of filters, and observers or the implementation of estimation schemes.On the other hand, a common approach consists in replacing the derivative term in PID-type controllers by an approximation of the form(1)dxtdt≈xt−xt−ττ,and for an appropriate selection of the delay τ, the closed-loop stability is guaranteed [3, 14–16]. In this framework, in [17], a method for 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 presented. The idea of deliberately introducing time delays in closed-loop systems and considering it as a control parameter is not a novel approach, but it has been intensively studied in recent years, see [18–20] and the references therein. The analysis of such class of controllers focuses mainly on the following topics: characterization of the stability crossing curves [21], tuning of delayed controllers to stabilize second-order systems [18, 20, 22] (and its noise attenuation analysis [23]), the design of proportional integral controllers for second-order linear systems [19, 24], and design of maximum decay rate using elimination theory [25]. Particularly in [26], a Proportional Integral Retarded (PIR) controller to solve the regulation problem of a general class of stable second-order LTI systems is presented. This result ultimately guarantees a desired exponential decay rate σ. On the other hand, design of nonfragile controllers with a desired exponential decay rate is proposed in [20]. Here, the authors present conditions on the parameters kp,kδ such that the p−δ controller of the form kp+kδe−τs ensure the stability of the closed-loop system.Inspired by the previous contributions, in the present manuscript, following theD-decomposition methodology (see [27–30]), it allows us to delimit and decompose the space of control parameters, determining the stability boundaries by means of a set of parametric equations that depends explicitly on the control parameters, which play a key step to allow us to determine simple analytical expressions for tuning the control parameters of a Proportional Retarded (PR) control law to σ-stabilize a general class of SISO LTI systems. Under these conditions, three dominant real roots are placed in −σ, which guarantees to reach the maximal exponential decay rate in the system response.Theσ-stabilization of a system can be described as the design and tuning of a controller such that the corresponding characteristic equation of the closed-loop system has dominant roots with real part less than or equal to −σ, σ∈ℝ+. The σ-stabilization approach can be considered a robust scheme, since when taking parameters within a σ-stable region and by presenting variations on the parameters (that do not leave the σ-stable zone) the system stability is assured.Delay-based controllers have as their main advantages the simplicity with which control laws are designed and, as a consequence, their practical implementation facility. Common applications of such controllers focus mainly on second-order systems, e.g., regulation problems of DC servomotors [18, 31], haptic virtual systems [20], underactuated mechanical system [32], and numerous academic examples. In the present proposal, a more challenging implementation is addressed: a flexible joint robotic arm (fourth order system), where trajectory tracking tasks are addressed. The main challenges for the design of PR control law are the complexity of its dynamic equations (mainly by the presence of highly nonlinear elements) and the appearance of oscillations at the tip of the link [33, 34]. Thus, a first step in the design of the Proportional Retarded control law is to linearize via an exact linearization approach the dynamic equations of the flexible joint robotic arm [35, 36].The remainder of this manuscript is organized as follows. In Section2, the analyzed closed-loop system and the problem formulation are presented and the σ-stability boundaries are determined. In Section 3, analytic conditions to determine the maximal decay rate σ∗ are provided. Section 4 is devoted to both the dynamic model of the flexible joint robotic arm under the study and the design of the proportional plus delay controller. Also, a complementary tuning approach is presented. In Section 5, the experimental platform is described and the experimental results are presented. For comparison purposes, a classical feedback state controller law is also designed and implemented. The contributions end with some concluding remarks.Nomenclature. In the present contribution, ℝ denotes the real numbers, while ℂ the complex numbers, and Z+ stands for the positive integers. Given a vector xt∈ℝn, then xt⊤ denotes its transpose. Let s∈ℂ∣s=a+ib, then Res=a and Ims=b denote its real and imaginary parts of s, respectively, and i is the imaginary unit. For a function xt, x˙t denotes its time derivative and dkxt/dtj=xjt and j∈ℤ+ defines the jth time derivative.


## 2. Problem Formulation

Let us consider a LTI-SISO system of the form(2)x˙t=Axt+But,where A∈ℝn×n, B∈ℝn, xt=x1t,x2t,…,xnt⊤∈ℝn, and ut∈ℝ is the control input. Let us propose a state feedback a control law of the form(3)ut=k˜1x1t+k˜2x2t+⋯+k˜nxnt,where k˜j∈ℝ,j=1,2,…,n, denote the controller gains. This approach demands the knowledge of the complete state vector. Now, let us suppose that at least one of the velocity measurements is not available, and then the implementation of a state observer to estimate the unavailable states would be considered. Instead of this alternative, let us propose a delayed controller as follows:(4)ut=kpx1t−krx1t−τ+α1x3t+⋯+αn−2xnt,where kp, kr, α1…,αn−2 ∈ℝ are the controller gains, and τ>0 is the delay. Here, we assume that the state variable x2t is unable to measure. Then, the closed-loop system is now of form(5)x˙t=A0xt+A1xt−τ,where A0,A1∈ℝn×n. Then, the characteristic function of system (5) is(6)Qs,kp,kr,α1,…,αn−2=ps,kp,α1,…,αn−2+krqse−sτ,where p⋅ and q⋅ are polynomials and s∈ℂ.In the subsequent sections, analytic conditions will be developed on the control parameterskr and τ that guarantee a maximal exponential decay rate (denoted by σ∗) on the system response.


## 3. Decomposition of the Gain Controller Parametric Space

As discussed in the introductory section, a key step to determine the analytic conditions on the parameters to achieve the maximal decay rate is the decomposition of the space of parameters by means of theD-decomposition methodology. To this end, we first study the σ-stability or stability degree σ>0 of (6). Then, let us consider σ>0 and by proposing the change of variable, s⟶s−σ, quasi-polynomial (6) takes the form(7)Qσs,kp,kr,α1,…,αn−2≔Qs−σ,kp,kr,α1,…,αn−2=ps,kp,α1,…,αn−2,σ+kreστqs,σe−sτ.Following theD-decomposition methodology, we first compute Qσ⋅s=0=0:(8)p0,kp,α1,…,αn−2,σ+kreστq0,σ=0,since p0,kp,α1,…,αn−2,σ≠0 and q0,σ≠0, then(9)kr=−p0,kp,α1,…,αn−2,σeστq0,σ.Now, evaluatingQσ⋅ for s=iω, ω>0, results(10)piω,kp,α1,…,αn−2,σ+kreστqiω,σe−iωτ=0,or in an equivalent form:(11)−1kreστpiω,kp,α1,…,αn−2,σqiω,σ=e−iωτ.Then, from Euler’s identity, it follows that(12)cosωτ=−1kreστRepiω,kp,α1,…,αn−2,σqiω,σ,sinωτ=1kreστImpiω,kp,α1,…,αn−2,σqiω,σ,and from the former equations, an explicit expression for τ can be written as follows:(13)τ=1ωcot−1−Repiω,kp,α1,…,αn−2,σ/qiω,σImpiω,kp,α1,…,αn−2,σ/qiω,σ+jπω,for j=0,±1,±2,…. Finally from equation (12) yields(14)kr=1eστsinωτImpiω,kp,α1,…,αn−2,σqiω,σ.The above analysis is summarized in the following result.Proposition 1.
Consider a quasi-polynomial of form (6), and then the σ-stability regions are bounded by the following equations.Whens=0,(15)kr=−p0,kp,α1,…,αn−2,σeστq0,σ,τ∈ℝ+,and for s=iω(16)τ=1ωcot−1−Repiω,kp,α1,…,αn−2,σ/qiω,σImpiω,kp,α1,…,αn−2,σ/qiω,σ+jπω,(17)kr=1eστsinωτImpiω,kp,α1,…,αn−2,σqiω,σ,where, j=0,±1,±2,…. Expressions (15)–(17) of Proposition 1 define the σ-stability boundaries in the parametric space τ,kr. To exemplify these stability boundaries, in Figure 1, a generic parametric map is presented. The outer region denotes the stability boundary of the system, that is, it corresponds to the pair τ,kr for which the eigenvalues of the system lies on the imaginary axis, while each one of the inner contour curves define a region where the system is exponentially stable with a specific decay rate σ>0. The red point corresponds to the maximal reachable stability degree, denoted by σ∗, and it occurs when all the σ-stable regions collapse in a single point τ∗,kr∗ of the parametric space and it is characterized because the characteristic function present a root of multiplicity at least three at s=σ∗ (see [22, 26]).In the next section, analytic expressions to determine the maximal decay rateσ∗ and the associated parameters τ∗,kr∗ are presented.Figure 1
τ
,
k
r parametric space.


## 4. Tuning of the PR Control Law

In this section, analytic expressions for determining the controller parametersτ∗, kr∗, and σ∗ to reach the maximal decay rate are provided.Proposition 2.
Let us consider a quasi-polynomial of form (6), then it has a dominant root of multiplicity at least three at the point s=−σ∗, if σ∗ is the smallest positive real root of the polynomial:(18)fσ=1p0,kp,α1,…,αn−2,σ∂ps,kp,α1,…,αn−2,σ∂s−p0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂s2s=0+2q0,σ∂qs,σ∂s∂ps,kp,α1,…,αn−2,σ∂s−p0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂ss=0+p0,kp,α1,…,αn−2,σq0,σ∂2qs,σ∂s2s=0−∂2ps,kp,α1,…,αn−2,σ∂s2s=0=0,and the delay τ=τσ∗ and the controller gain kr=krσ∗,τ satisfy the following relations:(19)τ=−1p0,kp,α1,…,αn−2,σ∂ps,kp,α1,…,αn−2,σ∂s−p0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂ss=0,kr=−∂ps,kp,α1,…,αn−2,σ/∂ss=0eστ∂qs,σ/∂ss=0−τq0,σ.Proof 1.
If quasi-polynomial (7) has three dominant roots in s=0, it implies that quasi-polynomial (6) has three dominant roots at s=−σ. Thus, conditions Qσ⋅s=0=0, ∂/∂sQσ⋅s=0=0, and ∂2/∂s2Qσ⋅s=0=0 must hold, i.e.,(20)0=Qσ⋅s=0=p0,kp,α1,…,αn−2,σ+kreστq0,σ,(21)0=∂∂sQσ⋅s=0=∂ps,kp,α1,…,αn−2,σ∂s+kreστ∂qs,σ∂s−τqs,σe−sτs=0,(22)0=∂2∂s2Qσ⋅s=0=∂2ps,kp,α1,…,αn−2,σ∂s2+kreστ∂2qs,σ∂s2−2τ∂qs,σ∂s+τ2qs,σe−sτs=0.
Condition (19) follows directly from (21). Now, through direct computations, it can be verified that equations (20) and (21) lead us to(23)τ=−1p0,kp,α1,…,αn−2,σ∂ps,kp,α1,…,αn−2,σ∂s−p0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂ss=0.
On the other hand, equations (20) and (22), imply that(24)0=τ2p0,kp,α1,…,αn−2,σ−2τp0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂s−∂2ps,kp,α1,…,αn−2,σ∂s2+p0,kp,α1,…,αn−2,σq0,σ∂2qs,σ∂s2s=0,and finally, by substituting (23) in (24), yields(25)fσ=1p0,kp,α1,…,αn−2,σ∂ps,kp,α1,…,αn−2,σ∂s−p0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂s2s=0+2q0,σ∂qs,σ∂s∂ps,kp,α1,…,αn−2,σ∂s−p0,kp,α1,…,αn−2,σq0,σ∂qs,σ∂ss=0+p0,kp,α1,…,αn−2,σq0,σ∂2qs,σ∂s2s=0−∂2ps,kp,α1,…,αn−2,σ∂s2s=0=0,that completes the proof.


## 5. PR Controller Design for a Flexible Joint Arm

In this section, the conditions proposed to achieve the maximum decay rateσ∗ are considered for the design of a Proportional Retarded control law implemented on a flexible joint robotic arm experimental platform.


### 5.1. Robot Dynamic Model

The schematic representation of the flexible joint robotic arm is given in Figure2.Figure 2
Schematic of a flexible joint robot arm.The notation used is as follows (for the sake of simplicity of notation, in the following, we will omit the time dependence of the functions).θ1 and θ2 are the angular positions of the rotating base and of the arm, respectively. J1 is the moment of inertia of the rotating base and J2 is the moment of inertia of the rotating arm. m denotes the mass of the arm, while l represents the arm length. g is the gravity constant, ks is the spring stiffness, and M is the torque applied to the system. The motor torque Mm and the torque applied to the system M satisfy the relation M=NMm, where N denotes the mechanical advantage of the pulley system.Following the Euler–Lagrange formulation, the equations of motion of the flexible joint robotic arm can be derived as(26)M=J1θ¨1+J2θ¨1+θ¨2+mgl2sinθ1+θ2,(27)0=J2θ¨1+θ¨2+ksθ2+mgl2sinθ1+θ2.Besides, the control input voltage applied to the motorV and the torque Mm are related as follows:(28)Mm=kmRmV−km2NRmθ˙1,where Rm symbolizes the armature resistance and km the torque constant of the motor. Based on the above relations, (26) and (27) are now of the form(29)NkmRmV=J1+J2θ¨1+J2θ¨2+km2N2Rmθ˙1+mgl2sinθ1+θ2,0=J2θ¨1+θ¨2+ksθ2+mgl2sinθ1+θ2,or equivalently(30)θ¨1=κ2θ2−κ1θ˙1+δV,(31)θ¨2=−κ3θ2−κ4sinθ1+θ2+κ1θ˙1−δV,where(32)κ1=km2N2J1Rm,κ2=ksJ1,κ3=ksJ1+J2J1J2,κ4=mgl2J2,δ=kmNJ1Rm.


### 5.2. PR Controller Design

By performing the change of variablesx≔x1x2x3x4⊤=θ1θ˙1θ2θ˙2⊤, systems (30) and (31) can be rewritten as(33)x˙=fx+gxV,where(34)fx=x2−κ1x2+κ2x3x4κ1x2−κ3x3−κ4sinx1+x3,gx=0δ0−δ.The first stage in the design of the delayed controller is to transform nonlinear system (33) into a linear system via feedback linearization approach. Let us define(35)hx=x1+x3,as the output function. Denoting Lfhx and Lghx as the Lie derivatives of function hx with respect of the vector fields f and g, respectively (see [35]). Thus, differentiating the output function hx,(36)h˙x=∂hx∂xx˙=Lfhx+Lghx=x2+x4,and computing the higher order derivative of hx,(37)h¨x=Lf2hx,h3x=Lf3hx,h4x=Lf4hx+VLgLf3hx,where(38)Lf2hx=κ2−κ3x3−κ4sinx1+x3,Lf3hx=−κ4x2+x4cosx1+x3+κ2−κ3x4,Lf4hx=κ2−κ3κ1x2−κ3x3+κ4sinx1+x3x2+x42−κ2−κ3+κ4cosx1+x3+κ4κ3−κ2x3cosx1+x3,LgLf3hx=−κ2−κ3δ.Hence, the system has relative degree four. Now, let us define the state transformationz1=hx, z2=h˙x, z3=h2x, and z4=h3x, then it follows that(39)z˙1=Lfhx,z˙2=Lf2hx,z˙3=Lf3hx,z˙4=Lf4hx+VLgLf3hx.Thus, the linearizing control function is of the form(40)V=1LgLf3hxu−Lf4hx.By denotingh∗x=z1∗ as the desired trajectory and hj∗x=zj+1∗, j=1,2,3,4, as its time derivatives, the errors functions are defined as ezj=zj−zj∗, then it follows(41)e˙z1=ez2,e˙z2=ez3,e˙z3=ez4,e˙z4=u.Now, by proposing the delayed controller,(42)uPR=−kpez1+krez1,τ−α1ez3−α2ez4,where the error ez1,τ is defined as(43)ez1,τ=hxτ−h∗xτ=x1t−τ+x3t−τ−x1∗t−τ+x3∗t−τ.Thus, system (41) in closed-loop, with control law (42), can be expressed as(44)e˙z=A0ez+A1ez,τ,where ez=ez1ez2ez3ez4⊤ and(45)A0=010000100001−kp0−α1−α2,A1=000000000000kr000.The characteristic quasi-polynomial of system (44) is(46)Qs,kp,kr,α1,α2=ps,kp,α1,α2+krqse−sτ,where ps,kp,α1,α2=s4+α2s3+α1s2+kp and qs=−1. Now, using the change of variable s=s−σ, (46) takes the form(47)Qσs,kp,kr,α1,α2=ps,kp,α1,α2,σ+kreστqs,σe−sτ,where(48)ps,kp,α1,α2,σ=s4+−4σ+α2s3+6σ2−3α2σ+α1s2+−4σ3+3α2σ2−2α1σs+σ4−α2σ3+α1σ2+kp,qs,σ=−1.By Proposition 1, we have that whens=0,(49)kr=σ4−α2σ3+α1σ2+kpeστ,while for s=iω(50)τ=1ωcot−1−Repiω,kp,α1,α2,σ/qiω,σImpiω,kp,α1,α2,σ/qiω,σ+jπω,kr=1eστsinωτImpiω,kp,α1,α2,σqiω,σ,where j=0,±1,±2,… and(51)Repiω,kp,α1,α2,σqiω,σ=−σ4+α2σ3−α1−6ω2σ2−3α2ω2σ−ω4+α1ω2−kp,Impiω,kp,α1,α2,σqiω,σ=ω4σ3−3α2σ2+−4ω2+2α1σ+α2ω2.According to Proposition 2 we first computeQσs,kp,kr,α1,α2s=0, as well as its first and second partial derivatives, that is,(52)Qσ⋅s=0=eστkr=σ4−α2σ3+α1σ2+kp,(53)∂∂sQσ⋅s=0=τeστkr=4σ3−3α2σ2+2α1σ,(54)∂2∂s2Qσ⋅s=0=τ2eστkr=12σ2−6α2σ+2α1.From (52) and (53), we get(55)τ=4σ3−3σ2α2+2σα1σ4−σ3α2+σ2α1+kp,and from (52) and (54), it follows that(56)τ2=12σ2−6σα2+2α1σ4−σ3α2+σ2α1+kp.Substituting (55) into (56), we have(57)fσ=4σ6−6σ5α2+2α1+3α22σ4−4α2α1σ3+2α12−12kpσ2+6α2kpσ−2α1kp.Summarizing, for givenkp,α1,andα2, then if σ∗ is the minimum real solution of polynomial (57), then the gains of the controller that determine the σ∗ stability of the system with a root of multiplicity at least three are established as(58)τ∗=4σ∗3−3σ∗2α2+2σ∗α1σ∗4−σ∗3α2+σ∗2α1+kp,(59)kr∗=σ∗4σ∗2−3σ∗α2+2α1τ∗eσ∗τ∗.In the previous procedure, analytic expressions to determine the optimal values of parameterskr∗, τ∗, and σ∗ were determined, but there are no words about parameters α1, α2, and kp; thus, in the subsequent paragraphs, a simple methodology for preselection or approximation of these parameters is presented.First, by computing the Maclaurin seriese−sτ=1−sτ+τ2s2/2−τ3s3/6 and substituting it in (47), we obtain the expression(60)Qσs=s4+α2+τ3kr6s3+α1−τ2kr2s2+τkrs+kp−kr.Now, it is desirable to select the controller gainskp,α1,andα2 in such a way that the dynamics of the close-loop system follow those of a proposed Hurwitz polynomial of the form(61)pas=s2+2ξωns+ωn22=s4+4ξωns3+2ωn2+4ξ2ωn2s2+4ξωn3s+ωn4,where ξ,ωn>0. In order to match the dynamics of polynomials (60) and (61), the following parametrized conditions must be fulfilled:(62)kp>4ξωn3324ξωn+ωn4,(63)α1=2ωn2+4ξ2ωn2+4ξωn322kp−ωn4,(64)α2=4ξωn−4ξωn336kp−ωn2.Remark 1.
It is worth mentioning, that expressions (62)–(64) can be considered only as a starting point to set the controller gains kp,α1,andα2 and must be adapted according to the control designer expertise. On the other hand, expressions (57)–(59) guarantee to reach the maximal decay rate for given parameters kp,α1,andα2.


## 5.1. Robot Dynamic Model

The schematic representation of the flexible joint robotic arm is given in Figure2.Figure 2
Schematic of a flexible joint robot arm.The notation used is as follows (for the sake of simplicity of notation, in the following, we will omit the time dependence of the functions).θ1 and θ2 are the angular positions of the rotating base and of the arm, respectively. J1 is the moment of inertia of the rotating base and J2 is the moment of inertia of the rotating arm. m denotes the mass of the arm, while l represents the arm length. g is the gravity constant, ks is the spring stiffness, and M is the torque applied to the system. The motor torque Mm and the torque applied to the system M satisfy the relation M=NMm, where N denotes the mechanical advantage of the pulley system.Following the Euler–Lagrange formulation, the equations of motion of the flexible joint robotic arm can be derived as(26)M=J1θ¨1+J2θ¨1+θ¨2+mgl2sinθ1+θ2,(27)0=J2θ¨1+θ¨2+ksθ2+mgl2sinθ1+θ2.Besides, the control input voltage applied to the motorV and the torque Mm are related as follows:(28)Mm=kmRmV−km2NRmθ˙1,where Rm symbolizes the armature resistance and km the torque constant of the motor. Based on the above relations, (26) and (27) are now of the form(29)NkmRmV=J1+J2θ¨1+J2θ¨2+km2N2Rmθ˙1+mgl2sinθ1+θ2,0=J2θ¨1+θ¨2+ksθ2+mgl2sinθ1+θ2,or equivalently(30)θ¨1=κ2θ2−κ1θ˙1+δV,(31)θ¨2=−κ3θ2−κ4sinθ1+θ2+κ1θ˙1−δV,where(32)κ1=km2N2J1Rm,κ2=ksJ1,κ3=ksJ1+J2J1J2,κ4=mgl2J2,δ=kmNJ1Rm.


## 5.2. PR Controller Design

By performing the change of variablesx≔x1x2x3x4⊤=θ1θ˙1θ2θ˙2⊤, systems (30) and (31) can be rewritten as(33)x˙=fx+gxV,where(34)fx=x2−κ1x2+κ2x3x4κ1x2−κ3x3−κ4sinx1+x3,gx=0δ0−δ.The first stage in the design of the delayed controller is to transform nonlinear system (33) into a linear system via feedback linearization approach. Let us define(35)hx=x1+x3,as the output function. Denoting Lfhx and Lghx as the Lie derivatives of function hx with respect of the vector fields f and g, respectively (see [35]). Thus, differentiating the output function hx,(36)h˙x=∂hx∂xx˙=Lfhx+Lghx=x2+x4,and computing the higher order derivative of hx,(37)h¨x=Lf2hx,h3x=Lf3hx,h4x=Lf4hx+VLgLf3hx,where(38)Lf2hx=κ2−κ3x3−κ4sinx1+x3,Lf3hx=−κ4x2+x4cosx1+x3+κ2−κ3x4,Lf4hx=κ2−κ3κ1x2−κ3x3+κ4sinx1+x3x2+x42−κ2−κ3+κ4cosx1+x3+κ4κ3−κ2x3cosx1+x3,LgLf3hx=−κ2−κ3δ.Hence, the system has relative degree four. Now, let us define the state transformationz1=hx, z2=h˙x, z3=h2x, and z4=h3x, then it follows that(39)z˙1=Lfhx,z˙2=Lf2hx,z˙3=Lf3hx,z˙4=Lf4hx+VLgLf3hx.Thus, the linearizing control function is of the form(40)V=1LgLf3hxu−Lf4hx.By denotingh∗x=z1∗ as the desired trajectory and hj∗x=zj+1∗, j=1,2,3,4, as its time derivatives, the errors functions are defined as ezj=zj−zj∗, then it follows(41)e˙z1=ez2,e˙z2=ez3,e˙z3=ez4,e˙z4=u.Now, by proposing the delayed controller,(42)uPR=−kpez1+krez1,τ−α1ez3−α2ez4,where the error ez1,τ is defined as(43)ez1,τ=hxτ−h∗xτ=x1t−τ+x3t−τ−x1∗t−τ+x3∗t−τ.Thus, system (41) in closed-loop, with control law (42), can be expressed as(44)e˙z=A0ez+A1ez,τ,where ez=ez1ez2ez3ez4⊤ and(45)A0=010000100001−kp0−α1−α2,A1=000000000000kr000.The characteristic quasi-polynomial of system (44) is(46)Qs,kp,kr,α1,α2=ps,kp,α1,α2+krqse−sτ,where ps,kp,α1,α2=s4+α2s3+α1s2+kp and qs=−1. Now, using the change of variable s=s−σ, (46) takes the form(47)Qσs,kp,kr,α1,α2=ps,kp,α1,α2,σ+kreστqs,σe−sτ,where(48)ps,kp,α1,α2,σ=s4+−4σ+α2s3+6σ2−3α2σ+α1s2+−4σ3+3α2σ2−2α1σs+σ4−α2σ3+α1σ2+kp,qs,σ=−1.By Proposition 1, we have that whens=0,(49)kr=σ4−α2σ3+α1σ2+kpeστ,while for s=iω(50)τ=1ωcot−1−Repiω,kp,α1,α2,σ/qiω,σImpiω,kp,α1,α2,σ/qiω,σ+jπω,kr=1eστsinωτImpiω,kp,α1,α2,σqiω,σ,where j=0,±1,±2,… and(51)Repiω,kp,α1,α2,σqiω,σ=−σ4+α2σ3−α1−6ω2σ2−3α2ω2σ−ω4+α1ω2−kp,Impiω,kp,α1,α2,σqiω,σ=ω4σ3−3α2σ2+−4ω2+2α1σ+α2ω2.According to Proposition 2 we first computeQσs,kp,kr,α1,α2s=0, as well as its first and second partial derivatives, that is,(52)Qσ⋅s=0=eστkr=σ4−α2σ3+α1σ2+kp,(53)∂∂sQσ⋅s=0=τeστkr=4σ3−3α2σ2+2α1σ,(54)∂2∂s2Qσ⋅s=0=τ2eστkr=12σ2−6α2σ+2α1.From (52) and (53), we get(55)τ=4σ3−3σ2α2+2σα1σ4−σ3α2+σ2α1+kp,and from (52) and (54), it follows that(56)τ2=12σ2−6σα2+2α1σ4−σ3α2+σ2α1+kp.Substituting (55) into (56), we have(57)fσ=4σ6−6σ5α2+2α1+3α22σ4−4α2α1σ3+2α12−12kpσ2+6α2kpσ−2α1kp.Summarizing, for givenkp,α1,andα2, then if σ∗ is the minimum real solution of polynomial (57), then the gains of the controller that determine the σ∗ stability of the system with a root of multiplicity at least three are established as(58)τ∗=4σ∗3−3σ∗2α2+2σ∗α1σ∗4−σ∗3α2+σ∗2α1+kp,(59)kr∗=σ∗4σ∗2−3σ∗α2+2α1τ∗eσ∗τ∗.In the previous procedure, analytic expressions to determine the optimal values of parameterskr∗, τ∗, and σ∗ were determined, but there are no words about parameters α1, α2, and kp; thus, in the subsequent paragraphs, a simple methodology for preselection or approximation of these parameters is presented.First, by computing the Maclaurin seriese−sτ=1−sτ+τ2s2/2−τ3s3/6 and substituting it in (47), we obtain the expression(60)Qσs=s4+α2+τ3kr6s3+α1−τ2kr2s2+τkrs+kp−kr.Now, it is desirable to select the controller gainskp,α1,andα2 in such a way that the dynamics of the close-loop system follow those of a proposed Hurwitz polynomial of the form(61)pas=s2+2ξωns+ωn22=s4+4ξωns3+2ωn2+4ξ2ωn2s2+4ξωn3s+ωn4,where ξ,ωn>0. In order to match the dynamics of polynomials (60) and (61), the following parametrized conditions must be fulfilled:(62)kp>4ξωn3324ξωn+ωn4,(63)α1=2ωn2+4ξ2ωn2+4ξωn322kp−ωn4,(64)α2=4ξωn−4ξωn336kp−ωn2.Remark 1.
It is worth mentioning, that expressions (62)–(64) can be considered only as a starting point to set the controller gains kp,α1,andα2 and must be adapted according to the control designer expertise. On the other hand, expressions (57)–(59) guarantee to reach the maximal decay rate for given parameters kp,α1,andα2.


## 6. Practical Implementation

In this section, the designed PR control law (40) is implemented in a flexible joint arm experimental platform using Matlab-Simulink, through a data acquisition device. The effectiveness of the proposed control law is assessed by means of tracking a rest-to-rest angular position reference trajectory. For comparative purposes, a feedback state control is designed and implemented in the experimental platform.


### 6.1. Experimental Setup

A block diagram of the experimental prototype is presented in Figure3. The experimental platform incorporate a DC motor NISCA model NC5475, attached to a rotating base trough a belt pulley system with a 16 : 1 ratio. The flexible joint consists of a main arm attached to the rotating base by springs. The angular positions, of the rotating base and the arm, are measured by incremental encoders. The data acquisition task was performed by a Sensoray 626 card. The control law was implemented in a Matlab-Simulink real time model, where the sampling period was fixed to be 0.001 s. The values of the flexible joint robotic arm parameters are l=0.5m, m=0.1633kg, J1=0.0136kg⋅m2, J2=0.002405kg⋅m2, ks=4N⋅m/rad, and N=16, while the motor parameters are km=0.0724N⋅m/A and Rm=2.983Ω.Figure 3
Diagram of the experimental platform architecture.


### 6.2. Feedback State Control

In order to compare PR control law (42) with a classical controller, we propose a feedback state control (FS):(65)uFS=−β1ez1−β2ez2−β3ez3−β4ez4,where the set of gains β1,β2,β3,β4 are positive constants. Using (40) and (41), we obtain the close-loop dynamic error:(66)ez14+β4ez13+β3e¨z1+β2e˙z1+β1ez1=0.The characteristic polynomial is matched with a Hurwitz polynomial of the form(67)pFLs=s4+β4s3+β3s2+β2s+β1=s2+2ζϖs+ϖ22.The set of controller gains is chosen as follows:(68)β1=ϖ4,β2=4ζϖ3,β3=4ζ2ϖ2+2ϖ2,β4=4ζϖ.


### 6.3. Experimental Results

The PR controller gains were selected as follows. We first proposedξ=1.8 and ωn=49, and the gains kp, α1, and α2 were selected from equations (62)–(64); after a slight adjustment, the gains were setted as kp=4.5×107, α1=5.7×104, and α2=80. On the other hand, the minimum real solution of (57) was σ∗=28.5495, then using equations (58) and (59) we obtained the values τ∗=0.0349 and kr∗=3.3×107.In Figure4, the σ-stability boundaries are depicted. The red mark represents the maximal achievable decay rate and corresponds to the place where all the σ-stable regions collapsed in a single point. According to (68), the FS controller gains were selected as follows ζ=1.8 and ϖ=52.Figure 4
σ
∗-stability region.The initial conditions for both experiment PR and FL controllers are settingθ1=x10=0 and θ2=x30=0; this implies that output function (35) is hx=0. The desired trajectory h∗x starts at time t=0 in the position h∗x=0, when the time is t=1, and it moves in 0.6 seconds. So, when t=1.6 and h∗x=π/2, it remains in this position for 8 seconds and after this time the reference is moved to the initial location in 0.6 seconds, that is, h∗x=0 when t=10.6, and it remains in this position until the test is finished.Vibrations are a common phenomenon in this class of systems, mainly when rest-to-rest point task is required. Figure5(a) shows the PR controller performance, the output trajectory tracking hPRx depicted in red, and the desired trajectory h∗x in black. In the same way, in Figure 5(b), the FL controller output hFSx is depicted in blue. We can notice that both outputs hPRx and hFSx smoothly follows the desired trajectory avoiding overshoot, as well as undesirable oscillations. The tracking trajectory errors ePR (up) and eFS (bottom) are shown in Figure 6. PR error, denoted by ePR, is bounded in a neighborhood of −0.02,0.02rad, while FL error eFS is bounded in a neighborhood of −0.02,0.03rad, and it shows some oscillations and a mayor amplitude in comparison with the PR controller.Figure 5
Trajectory tracking performance.
(a)
(b)Figure 6
Tracking error.
(a)
(b)PR voltage inputVPR is depicted in Figure 7(a) and the maximum amplitude of the signal is approximately −15,15 Volts, and it does not present noise when the system is in the steady state, and FS voltage VFS is shown in Figure 7(b). Here, we can notice similar performance of both VPR and VFS in amplitude and form, but VFS shows a clear increase in noise and frequency: In order to clarify this statement, a Power Density Spectrum (PSD) of each voltage control is depicted in Figure 8. As we can observe, PRPSD exhibits low-frequency components with a small peak at 7.7 Hz. On the other hand, FSPSD shows more high-frequency components. Low-frequency components on the PR controller give us some advantages such as less wear on the actuators, less power consumption, and vibrations, among others. The above can be verified by observing Figures 9 and 10, where the performance of the system is evaluated by means of a quadratic index for both the error and the applied voltage.Figure 7
Control input voltage.
(a)
(b)Figure 8
Power spectrum density.
(a)
(b)Figure 9
Performance index of the error.Figure 10
Performance index of the controller.Now, it is desirable to evaluate the performance of the flexible joint robotic platform in other points of the parametric spaceτ,kr. Here, we consider points with similar control parameters but with different exponential decay rate σ and consequently a different delay τ. These points are depicted in Figure 4 and its control parameters are given in Table 1.Table 1
Parameterskr,τ and settling time.
Parameters
Settling time
Color
σ
k
r
τ (s)
Time (s)
Cyan
2
3.5e7
0.006767
1.128
Blue
6
3.5e7
0.01775
0.947
Magenta
12
3.5e7
0.02784
0.655
Red
28.5495
3.3e7
0.0349
0.321The system responses, at the selected points, are depicted in Figure11, where it can be seen that the control tasks were carried out satisfactorily. Since each of the points corresponds to different σ-stable regions with a specific decay rate, the zoom at the bottom of Figure 11 allows us to note that the transient response of the system vanishes according to Table 1, and as expected, the system achieves the maximal decay rate at the point σ∗=28.5495. A similar statement can be made, when the error signals are analyzed (see Figure 12).Figure 11
Trajectory tracking results.Figure 12
Tracking error signals.Remark 2.
Theσ-stability approach provides us some benefits from the robustness perspective. Since the parameters τ∗ and kr∗ are selected deep inside the stability region, the effect of parameter variations only affect the decay rate but not the stability property. This statement can be verified in Figures 11 and 12, where the performance of the system is evaluated for parameters τ,kr taken far from the point τ∗,kr∗. It can be appreciated that the further the control parameters are selected from the point τ∗,kr∗, the greater the decay rate will be, but the stability of the system remains.


## 6.1. Experimental Setup

A block diagram of the experimental prototype is presented in Figure3. The experimental platform incorporate a DC motor NISCA model NC5475, attached to a rotating base trough a belt pulley system with a 16 : 1 ratio. The flexible joint consists of a main arm attached to the rotating base by springs. The angular positions, of the rotating base and the arm, are measured by incremental encoders. The data acquisition task was performed by a Sensoray 626 card. The control law was implemented in a Matlab-Simulink real time model, where the sampling period was fixed to be 0.001 s. The values of the flexible joint robotic arm parameters are l=0.5m, m=0.1633kg, J1=0.0136kg⋅m2, J2=0.002405kg⋅m2, ks=4N⋅m/rad, and N=16, while the motor parameters are km=0.0724N⋅m/A and Rm=2.983Ω.Figure 3
Diagram of the experimental platform architecture.


## 6.2. Feedback State Control

In order to compare PR control law (42) with a classical controller, we propose a feedback state control (FS):(65)uFS=−β1ez1−β2ez2−β3ez3−β4ez4,where the set of gains β1,β2,β3,β4 are positive constants. Using (40) and (41), we obtain the close-loop dynamic error:(66)ez14+β4ez13+β3e¨z1+β2e˙z1+β1ez1=0.The characteristic polynomial is matched with a Hurwitz polynomial of the form(67)pFLs=s4+β4s3+β3s2+β2s+β1=s2+2ζϖs+ϖ22.The set of controller gains is chosen as follows:(68)β1=ϖ4,β2=4ζϖ3,β3=4ζ2ϖ2+2ϖ2,β4=4ζϖ.


## 6.3. Experimental Results

The PR controller gains were selected as follows. We first proposedξ=1.8 and ωn=49, and the gains kp, α1, and α2 were selected from equations (62)–(64); after a slight adjustment, the gains were setted as kp=4.5×107, α1=5.7×104, and α2=80. On the other hand, the minimum real solution of (57) was σ∗=28.5495, then using equations (58) and (59) we obtained the values τ∗=0.0349 and kr∗=3.3×107.In Figure4, the σ-stability boundaries are depicted. The red mark represents the maximal achievable decay rate and corresponds to the place where all the σ-stable regions collapsed in a single point. According to (68), the FS controller gains were selected as follows ζ=1.8 and ϖ=52.Figure 4
σ
∗-stability region.The initial conditions for both experiment PR and FL controllers are settingθ1=x10=0 and θ2=x30=0; this implies that output function (35) is hx=0. The desired trajectory h∗x starts at time t=0 in the position h∗x=0, when the time is t=1, and it moves in 0.6 seconds. So, when t=1.6 and h∗x=π/2, it remains in this position for 8 seconds and after this time the reference is moved to the initial location in 0.6 seconds, that is, h∗x=0 when t=10.6, and it remains in this position until the test is finished.Vibrations are a common phenomenon in this class of systems, mainly when rest-to-rest point task is required. Figure5(a) shows the PR controller performance, the output trajectory tracking hPRx depicted in red, and the desired trajectory h∗x in black. In the same way, in Figure 5(b), the FL controller output hFSx is depicted in blue. We can notice that both outputs hPRx and hFSx smoothly follows the desired trajectory avoiding overshoot, as well as undesirable oscillations. The tracking trajectory errors ePR (up) and eFS (bottom) are shown in Figure 6. PR error, denoted by ePR, is bounded in a neighborhood of −0.02,0.02rad, while FL error eFS is bounded in a neighborhood of −0.02,0.03rad, and it shows some oscillations and a mayor amplitude in comparison with the PR controller.Figure 5
Trajectory tracking performance.
(a)
(b)Figure 6
Tracking error.
(a)
(b)PR voltage inputVPR is depicted in Figure 7(a) and the maximum amplitude of the signal is approximately −15,15 Volts, and it does not present noise when the system is in the steady state, and FS voltage VFS is shown in Figure 7(b). Here, we can notice similar performance of both VPR and VFS in amplitude and form, but VFS shows a clear increase in noise and frequency: In order to clarify this statement, a Power Density Spectrum (PSD) of each voltage control is depicted in Figure 8. As we can observe, PRPSD exhibits low-frequency components with a small peak at 7.7 Hz. On the other hand, FSPSD shows more high-frequency components. Low-frequency components on the PR controller give us some advantages such as less wear on the actuators, less power consumption, and vibrations, among others. The above can be verified by observing Figures 9 and 10, where the performance of the system is evaluated by means of a quadratic index for both the error and the applied voltage.Figure 7
Control input voltage.
(a)
(b)Figure 8
Power spectrum density.
(a)
(b)Figure 9
Performance index of the error.Figure 10
Performance index of the controller.Now, it is desirable to evaluate the performance of the flexible joint robotic platform in other points of the parametric spaceτ,kr. Here, we consider points with similar control parameters but with different exponential decay rate σ and consequently a different delay τ. These points are depicted in Figure 4 and its control parameters are given in Table 1.Table 1
Parameterskr,τ and settling time.
Parameters
Settling time
Color
σ
k
r
τ (s)
Time (s)
Cyan
2
3.5e7
0.006767
1.128
Blue
6
3.5e7
0.01775
0.947
Magenta
12
3.5e7
0.02784
0.655
Red
28.5495
3.3e7
0.0349
0.321The system responses, at the selected points, are depicted in Figure11, where it can be seen that the control tasks were carried out satisfactorily. Since each of the points corresponds to different σ-stable regions with a specific decay rate, the zoom at the bottom of Figure 11 allows us to note that the transient response of the system vanishes according to Table 1, and as expected, the system achieves the maximal decay rate at the point σ∗=28.5495. A similar statement can be made, when the error signals are analyzed (see Figure 12).Figure 11
Trajectory tracking results.Figure 12
Tracking error signals.Remark 2.
Theσ-stability approach provides us some benefits from the robustness perspective. Since the parameters τ∗ and kr∗ are selected deep inside the stability region, the effect of parameter variations only affect the decay rate but not the stability property. This statement can be verified in Figures 11 and 12, where the performance of the system is evaluated for parameters τ,kr taken far from the point τ∗,kr∗. It can be appreciated that the further the control parameters are selected from the point τ∗,kr∗, the greater the decay rate will be, but the stability of the system remains.


## 7. Conclusions

Analytic conditions for the design of proportional retarded controllers that guarantee reaching the optimal decay rate for a general class of LTI systems are presented. A detailed design of a proportional retarded controller, which shows the ease with which the proposed conditions can be computed and implemented, is developed for the control of an experimental platform that consists of a flexible joint robotic arm. The experimental results illustrate the effectiveness of the conditions and show remarkable tracking of rest-to-rest trajectories avoiding oscillations, noise, and overshoot.



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*Source: 7289689-2019-12-23.xml*