# PD Control for Vibration Attenuation in  a Physical Pendulum with Moving Mass

Authors: Oscar Octavio Gutiérrez-Frias; Juan Carlos Martínez-García; Rubén  A. Garrido Moctezuma
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

This paper proposes a Proportional Derivative controller plus gravity compensation to
damp out the oscillations of a frictionless physical pendulum with moving mass. A mass slides
along the pendulum main axis and operates as an active vibration-damping element. The Lyapunov
method together with the LaSalle's theorem allows concluding closed-loop asymptotic
stability. The proposed approach only uses measurements of the moving mass position and velocity
and it does not require synchronization of the pendulum and moving mass movements.
Numerical simulations assess the performance of the closed-loop system.


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## Body

## 1. Introduction

Vibrating mechanical systems are an important class of dynamic systems including buildings, bridges, car suspensions, pacemakers, wind generators, and hi-fi speakers [1]. In physical terms, the behavior of a vibrating system is describe by the interplay between an energy-storing component and an energy-carrying component. Thus, the system dynamics are described in terms of energy changes, that is, the motion of the system results from an energy exchange. The control of vibrating mechanical systems is an important area of research, which has provided technological solutions to several problems concerning oscillatory behaviors of some important classes of dynamic systems. For instance, active control of vibrations allows attenuating undesired oscillations in buildings affected by external forces such as strong winds and earthquakes (see, e.g., [2–9] and the references therein), and computer-based active suspension systems are now common in cars as a mean to improve road handling.As far as mathematical tools are concerned, the control of vibrations has mainly been tackled via frequency-domain techniques, which are essentially restricted to linear systems (see, e.g., [10, 11]). If the vibrating systems are nonlinear and if they oscillate too far away from their equilibrium points, then, frequency-domain techniques are not suitable. In the case of nonlinear systems characterized by small domains of attraction around their equilibrium points, the linear approach is not very effective. Hence, modern approaches, employing time-domain nonlinear control strategies, would yield better performance.This paper focuses on active control of a class of underdamped lumped nonlinear under-actuated vibrating mechanical systems following an energy-based approach, that is, the control of vibrations is tackled via the shaping of the energy flow that characterizes the system. The control of vibrations is considered in terms of the solution of a particular asymptotic stabilizing feedback control problem around a selected equilibrium point. A stabilizing controller is then obtained following an energy-based Lyapunov approach, which exploits the physical properties of the mechanical system. Intuitively speaking, the energy-based Lyapunov control shapes, using a feedback loop, the potential, and kinetic energies of the controlled system to ensure a motion guaranteeing the control objective (see, e.g., [8, 12, 13]). Moreover, this approach requires the total energy of the system to be a non-increasing function. The total energy function is also required to be at least locally positive definite around the selected equilibrium point (see, e.g., [6, 7, 9, 13]). In this way, the proposed approach avoids conservative control strategies based either on high gains or on canceling nonlinear terms [14, 15].The problem tackled in this work is the stabilization of a frictionless under-actuated physical pendulum with a radially moving mass. This system was studied in [16], where several control strategies solved the aforementioned stabilization problem. The proposed stabilizing strategies include a modified nonlinear Proportional Derivative (PD) controller and a neural network approach. Further works also studied this physical system [17, 18]. The proposed approach in these two references synchronizes the movement of the moving mass with the pendulum oscillation to damp out the pendulum oscillation. It is interesting to point out that the aforementioned approaches are mainly heuristic and they do not provide rigorous stability proofs. Reference [19] provides a control algorithm based on a switching strategy. The model of the pendulum includes damping friction, and the proposed control law needs measurement of the pendulum angle and the moving mass position. It is also worth remarking that, if not properly tuned, the switching strategy could introduce unbounded control signal chattering.This paper proposes a simple control law for damping out the oscillation of a frictionless pendulum through the movement of a mass sliding along the pendulum main axis. The control law is composed of two parts, a linear PD controller corresponds to the first part, and the second part is a constant term, that is, equal to the gravity force term associated to the moving mass. Compared with previous approaches, the proposed control law only needs measurements of the moving mass position and velocity and does not relies on the synchronization of the pendulum oscillation. Moreover, it is simple and it could be implemented using processors with limited computing capabilities.The contribution is organized as follows: Section2 presents the model of the physical pendulum with moving mass, as well as its main physical properties. Section 3 describes the proposed control law and the stability analysis of the closed loop system. Section 4 depicts some computer simulations. The paper ends with some final comments.


## 2. Physical Pendulum with Moving Mass

### 2.1. Lagrangian Modeling

Consider a mechanical system consisting of a physical frictionless pendulum of massM with its pivot at O and an auxiliary mass m able to slide to and from the pivot as depicted in Figure 1. The moment of inertia of the pendulum about the pivot is given by I0, and its center of mass C is located at a distance rc from the pivot. The forces acting on the mass m are the gravitational force mg and a force F parallel to the guide OC¯ and supplied by an actuator, that is, an electric motor, attached to the auxiliary mass. The pivot O is the origin of the reference frame x-y. The x-axis is set in the horizontal direction and the y-axis is set in the vertical direction. The set of generalized coordinates are the angle θ between OC¯ and the y-axis, and the radial displacement r of the mass m from the pivot O. It is easy to show that the total kinetic energy Kc and the total potential energy Kp for this system are given by
(2.1)Kc=12I0θ̇2+12mṙ2+12mr2θ̇2,Kp=-Mgrccosθ-mgrcosθ,
respectively. Note that the total kinetic energy comprises the rotational energy of the pendulum as well as the translational and rotational energy of the sliding mass. The above equations allow writing the Lagrangian function
(2.2)L(q,q̇)=Kc-Kp,
where q:=[r,θ]T. From the above, the corresponding Euler-Lagrange equations are given by
(2.3)mr̈-mrθ̇2-mgcosθ=F,(mr2+I0)θ̈+2mrṙθ̇+g(Mrc+mr)sinθ=0.Figure 1
Physical pendulum with moving mass.


### 2.2. Model Properties

Define the forceF as
(2.4)F=v-mg,
where variable v is a new input, and mg is a gravity compensation term. Substituting (2.4) into (2.3) leads to the following Euler-Lagrange system
(2.5)M(q)q̈+C(q,q)q̇+∇qKi(q)=Gv,
where G:=[1,0]T and Ki(q):=-Mgrccosθ+mgr(1-cosθ) and
(2.6)M(q)=[m00mr2+I0],C(q,q̇)=[0-mrθ̇mrθ̇mrṙ.].
System (2.5) satisfies the following properties:(P1)
M(q) is positive definite.(P2)
H:=Ṁ(q)-2C(q,q̇) is skew symmetric with
(2.7)H=[0-mrθ̇mrθ̇0].(P3)
The operatorv→ṙ is passive.Properties (P1) and (P2) are shared by any Euler-Lagrange mechanical system. In order to prove property (P3), define the following storage function:
(2.8)E(q,q̇)=12q̇TM(q)q̇+Ki(q).Taking the time derivative of (2.8) and using properties (P1) and (P2) yields
(2.9)Ė=vṙ.According to standard results [20, page 236] the operator v→ṙ is passive. Finally, the following remark concerns the local controllability of (2.5).Remark 2.1.
Definex=[q,q̇]T=[r,θ,ṙ,θ̇]T. Then, linearization of system (2.5) around  x¯=[r¯,0,0,0]T,   r̅>0 produces
(2.10)mr̈=v,(mr¯2+I0)θ̈+g(Mrc+mr¯)θ=0.
From the above, it is clear that (2.10) is not locally controllable since there is no way to affect the dynamics of θ. Further, x̅ is a stable equilibrium point of (2.5) if  v=0 and  -π/2<θ<π/2.


## 2.1. Lagrangian Modeling

Consider a mechanical system consisting of a physical frictionless pendulum of massM with its pivot at O and an auxiliary mass m able to slide to and from the pivot as depicted in Figure 1. The moment of inertia of the pendulum about the pivot is given by I0, and its center of mass C is located at a distance rc from the pivot. The forces acting on the mass m are the gravitational force mg and a force F parallel to the guide OC¯ and supplied by an actuator, that is, an electric motor, attached to the auxiliary mass. The pivot O is the origin of the reference frame x-y. The x-axis is set in the horizontal direction and the y-axis is set in the vertical direction. The set of generalized coordinates are the angle θ between OC¯ and the y-axis, and the radial displacement r of the mass m from the pivot O. It is easy to show that the total kinetic energy Kc and the total potential energy Kp for this system are given by
(2.1)Kc=12I0θ̇2+12mṙ2+12mr2θ̇2,Kp=-Mgrccosθ-mgrcosθ,
respectively. Note that the total kinetic energy comprises the rotational energy of the pendulum as well as the translational and rotational energy of the sliding mass. The above equations allow writing the Lagrangian function
(2.2)L(q,q̇)=Kc-Kp,
where q:=[r,θ]T. From the above, the corresponding Euler-Lagrange equations are given by
(2.3)mr̈-mrθ̇2-mgcosθ=F,(mr2+I0)θ̈+2mrṙθ̇+g(Mrc+mr)sinθ=0.Figure 1
Physical pendulum with moving mass.


## 2.2. Model Properties

Define the forceF as
(2.4)F=v-mg,
where variable v is a new input, and mg is a gravity compensation term. Substituting (2.4) into (2.3) leads to the following Euler-Lagrange system
(2.5)M(q)q̈+C(q,q)q̇+∇qKi(q)=Gv,
where G:=[1,0]T and Ki(q):=-Mgrccosθ+mgr(1-cosθ) and
(2.6)M(q)=[m00mr2+I0],C(q,q̇)=[0-mrθ̇mrθ̇mrṙ.].
System (2.5) satisfies the following properties:(P1)
M(q) is positive definite.(P2)
H:=Ṁ(q)-2C(q,q̇) is skew symmetric with
(2.7)H=[0-mrθ̇mrθ̇0].(P3)
The operatorv→ṙ is passive.Properties (P1) and (P2) are shared by any Euler-Lagrange mechanical system. In order to prove property (P3), define the following storage function:
(2.8)E(q,q̇)=12q̇TM(q)q̇+Ki(q).Taking the time derivative of (2.8) and using properties (P1) and (P2) yields
(2.9)Ė=vṙ.According to standard results [20, page 236] the operator v→ṙ is passive. Finally, the following remark concerns the local controllability of (2.5).Remark 2.1.
Definex=[q,q̇]T=[r,θ,ṙ,θ̇]T. Then, linearization of system (2.5) around  x¯=[r¯,0,0,0]T,   r̅>0 produces
(2.10)mr̈=v,(mr¯2+I0)θ̈+g(Mrc+mr¯)θ=0.
From the above, it is clear that (2.10) is not locally controllable since there is no way to affect the dynamics of θ. Further, x̅ is a stable equilibrium point of (2.5) if  v=0 and  -π/2<θ<π/2.


## 3. The Control Law

Before establishing the control objective of this work, we define the admissible setQ⊂R2 as
(3.1)Q={q=[r,θ]T:0<r̅-ε<r<r¯+ε,-π2<θ<π2,ε>0}.
The control objective is defined as follows.Problem 3.1.
Consider the physical pendulum with moving mass described in (2.5), under the assumption that the initial conditions satisfy q(0)∈Q-q¯. Then, the control objective is to bring asymptotically the rotating pendulum with moving mass to the equilibrium point x¯=[r¯,0,0,0]T, while q=[r,θ]T∈Q.To solve the aforementioned control problem, define the following Lyapunov function candidate:(3.2)ET(q,q̇)=12q̇TM(q)q̇+Km(q),
where Km(q) is the modified potential energy stated as
(3.3)Km(q)=kp2(r-r¯)2+Ki(q)+Mgrc
with kp>0.Remark 3.2.
The selected potential energyKm(q) has a minimum at q̅=[r̅,0]T  since
(3.4)Km(q¯)=0,∇qKm(q)q=q¯=0,∇q2Km(q)q=q¯=[kp00Mgrc+mgr̅]>0.
As a matter of fact, the above condition implies thatKm(q) is a convex function around q¯. In geometrical terms, the level curves of Km(q) consist of a set of closed curves around q¯. On the other hand, function Km(q) is positive definite as long as -π/2<θ<π/2.Taking into account property (P3), the first time derivative of ET along a trajectory of (2.5) is given by
(3.5)ĖT(q,q̇)=vṙ+kp(r-r¯)ṙ.Define the control inputv as a Proportional Derivative control law
(3.6)v=-kp(r-r¯)-kdṙ
with kp>0, kd>0. Therefore, substituting control law (3.6) into (3.5) yields
(3.7)ĖT(q,q̇)=-kdṙ2.As a consequence,ĖT≤0. Thus, this condition establishes stability of the equilibrium point x¯ in the Lyapunov sense. Moreover, it also shows that function ET(q,q̇) is not increasing so does q. Therefore, if q(0)=[r(0),θ(0)]T∈Q, then, q=[r,θ]T∈Q as t→∞. On the other hand, since ET(q,q̇) is not increasing, then, ET(q,q̇)≤ET(q(0),q̇(0)). The above result allows defining a compact invariant set Ω as follows:
(3.8)Ω={x=[q,q̇]T:ET(q,q̇)≤C¯},
where C¯=ET(q(0),q̇(0)). Therefore, if x(0)∈Ω, then, x∈Ω as t→∞.To end the stability proof, La Salle's Theorem [21] will allow concluding asymptotic stability. To this end, define the invariant set S as follows:
(3.9)S={[q,q̇]T∈Ω:ĖT(q,q̇)=0}.Clearly, in the setS, we have that ṙ=0 and as a consequence r̈=0 and r=r̲, where r̲>0 is a constant. Thus, substituting these quantities into (2.5) leads to
(3.10)-mr̲θ̇2-mg(cosθ-1)+kp(r̲-r¯)=0,(mr̲2+I0)θ̈+g(Mrc+mr̲)sinθ=0.The time derivative of the first equation in (3.10) yields
(3.11)(2r̲θ̈-gsinθ)θ̇=0.Two cases must then be analyzed.Case 1.
Ifθ̇=0 in the set S, it also follows that θ̈=0. From the second differential equation of (3.10), it is clear that sinθ=0 since Mrc+mr̲ is strictly positive. Hence, it follows that θ=0 in the set S. As a consequence, from the first equation of (3.10) r̲-r¯=0. Therefore, r=r¯ in the set S.Case 2.
Ifθ̇≠0 in the set S, then (3.11) implies that
(3.12)θ̈=gsinθ2r̲.
Since r̲>0, thus, θ̈ is well defined. Taking into account the second differential equation in (3.10) and (3.12) yields to the following algebraic equation for the variable θ:
(3.13)0=(2Mrcr̲+3mr̲2+I0)sinθ.This last equation implies thatθ=0 on the set S because 2Mrcr̲+3mr̲2+I0>0. This means that Case 2 is not possible since it assumes that θ̇≠0. Therefore, θ̇=0 and Case 1 is the only possibility.The above analysis allows concluding that the largest invariant set contained inS is given by x¯. According to the LaSalle's invariance theorem, all the trajectories starting in Ω asymptotically converge towards x¯=[r¯,0,0,0]T. The following proposition resumes the stability result previously presented.Proposition 3.3.
Consider the closed-loop dynamic system given by (2.5) in closed-loop with the control law
(3.14)F=-kp(r-r¯)-kdṙ-mg.
Then, all the trajectories starting in Ω asymptotically converge towards the equilibrium point x¯=[r¯,0,0,0]T.It is worth noting that the three terms defining (3.14) are the proportional part kp(r-r¯), the derivative part kdṙ, and a gravity compensation term mg. Moreover, (3.7) indicates that damping introduced by the derivative part kdṙ provides energy dissipation.


## 4. Numerical Simulations

To illustrate the performance of the proposed control law, a numerical simulation was carried out using the MATLAB program. The system physical parameters were set as follows:(4.1)m=1[Kg],M=2.5[Kg],rc=0.7[m],I0=1.22[Kg·m2].The radial displacement ofm was given by r¯=2.5[m]andε=0.75[m]. The initial conditions were chosen to be θ(0)=0.5[rad], r(0)=2.5[m], θ̇(0)=0[rad/s] and ṙ(0)=0[m/s]. The control gains, empirically proposed to increase the convergence rate of the closed-loop system, were set as kp=19.6 and kd=1.9. Figure 2 shows the angular displacement, Figure 3 depicts the displacement of the moving mass. Figure 4 displays the time evolution of both, the force applied to the moving mass and the Lyapunov function ET(q,q̇). From the above results, it is evident that the proposed control law attenuates the pendulum oscillations by a factor of five; the radial displacement remain bounded and converges to the value r¯=2.5[m]. On the other hand, the applied force stays also bounded and converges to -9.8[Nw], that is, the value given by the gravity compensation mg. Note also that the Lyapunov function ET(q,q̇), which accounts for the kinetic and potential energy of the closed-loop system, also decreases. Figure 5 displays the equivalent damping. This concept is introduced in [17] and is given by the following equation:
(4.2)ξEQ=12π∫0τ(ṙ/r)(θ̇2+(g/4r)θ2)dt[(1/2)θ̇2(0)+(g/2r(0))θ2(0)].Figure 2
Angular displacementθ.Figure 3
Moving mass displacementr.ForceF applied to the moving mass and time evolution of the Lyapunov function ET(q,q̇).
(a)(b)Figure 5
Time evolution of the equivalent dampingξEQ.Even if this measure was originally intended for evaluating the damping in one cycle of the pendulum oscillation, equation allows computingξEQ for any time τ provided that θ≪1. Note that the time evolution of r and ṙ determines the behaviour of ξEQ. The equivalent damping has final value of 2×10-4, which is indeed very small.It must be pointed out that the control strategy is effective in reducing the pendulum oscillations with reasonable control input effort for a large deviation from the equilibrium point. However, the damping injection capability of the proposed strategy is somewhat limited, that is, the system is brought to the desired equilibrium very slowly. This last observation is in agreement with the small value of the equivalent damping. Nevertheless, in real systems there always exists viscous friction, which helps to accelerate convergence to the equilibrium point. Also, Figure6 shows the level curves associated with the modified potential energy Km(q), for the two different values of the parameter kp=19.6 and kp=40 and, as we can see, the region of attraction of the specified equilibrium point can be increased by just augmenting the value of kp. However, it is not convenient to consider high values for the proportional gain kp, because it may generate high frequencies oscillations in the closed-loop system. Thus, it is better to consider small values for parameter kp in order to guarantee that |r-r¯|<ε holds.Level curves ofKm(q) around the origin for (a) kp=19.6 and (b) kp=40.
(a)(b)


## 5. Conclusions

This paper proposes a Proportional Derivative control law plus gravity compensation for active vibration damping in a frictionless physical pendulum with moving mass. The control law is able to damp out the oscillations of the pendulum by using the moving mass as active damper, and its design exploits the underlying physical properties of this system to shape a Lyapunov function candidate. LaSalle's theorem allows concluding asymptotic stability of the closed-loop system. Moreover, the control law only needs measurements of the position and velocity of the moving mass. Compared with previous approaches [16, 18], the proposed methodology does not need to synchronize the motion of the moving mass with the swings of the pendulum, then avoiding measurement of the pendulum position and velocity. The proposed strategy could be considered as a first step towards the reduction of undesirable oscillation in civil structures.



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*Source: 179724-2009-07-30.xml*