# Low-Complexity Decentralized Active Damping of One-Dimensional Structures

Authors: Petr Hušek; Filip Svoboda; Martin Hromčík; Zbyněk Šika
Journal: Shock and Vibration
 (2018)

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## Abstract

In the paper, we propose distributed feedback control laws for active damping of one-dimensional mechanical structures equipped with dense arrays of force actuators and position and velocity sensors. We consider proportional position and velocity feedback from the neighboring nodes with symmetric gains. Achievable control performance with respect to stability margin and damping ratio is discussed. Compared to full-featured complex controllers obtained by modern design methods like LQG, H-infinity, or mu-synthesis, these simplistic controllers are more suitable for experimental fine tuning and are less case-dependent, and they shall be easier to implement on the target future smart-material platforms.


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## Body

## 1. Introduction

The established paradigm in past and current active damping projects is as follows: the mechanical object is defined first (plate, beam, car door, wing panel, etc.). Systems detailed design and modelling phases follow the methods in [1, 2] giving rise to very accurate FEM models with tens of thousands of degrees of freedom. Alternatively for existing prototypes, the experimental identification approach can be applied to get the mathematical models directly via experimental modal analysis [3]. Model order reduction [1, 4] then gives accurate enough yet tractable models for optimal actuators and sensors placement [5–7]. Finally, a very limited number of them are considered (say up to twenty) for the design of the control laws [1, 8]. Finally, validation and verification of the solutions by high-fidelity simulations is performed, followed by laboratory experiments and final deployment of the product. For any new project—or even a relatively mild modification of a previously accomplished project—all these steps must be performed (or re-visited) again. Implications towards requested research and development costs are significant and obvious.Therefore, there is a need to use other type of control methodologies, and recent advances in MEMS sensors and microactuators, ongoing intensive research on new smart materials, and progress in computational power pave the way to massive development of heavily distributed control in this context.Distributed control is now a very active field of research, thanks to potential applications which require high scalability and reliability. The main advantage of using distributed control is the locality of the necessary measurement and actuation—the measurements are collected and processed in a distributed manner. This kind of control can be applied for automated highway systems [9], car formations [10], and also flexible structures. The work in [11], for instance, studies a flexible beam model with bending and torsion motions, and a distributed arrangement with two force-actuators and three moment-actuators paired with rate gyros was elaborated. In [12], a dense network of piezoelectric patch actuators was proposed to realize the distributed actuation. In [13], a distributed piezoelectric actuation was involved and applied to the placement problem of patches so that the deformations are suppressed at preselected locations. Multipositive feedback approach for flexible structure control was presented in [14]. Since the flexible systems are passive by nature, one can also employ a lot of results available for distributed control of the passive system [15, 16]. Completely passive solutions can be obtained using piezostructures, as reported in [17].One of the natural goals when dealing with control of flexible mechanical structures is vibration suppression. One standard approach relies on application of a large number of neutralizers placed in prespecified locations along the structure composed from masses and springs. The goal is not only to design the neutralizers’ parameters but specify their locations as well since vibrations can be eliminated only at the attachment point of the vibrating beam while amplification of vibration may occur in other parts of the beam. Dynamic vibration absorbers using magnetorheological elastomers were used in [18]. In [19, 20], a set of optimum conditions for global control of the kinetic energy based on the fixed-points theory was proposed. Dynamic transfer matrices using mobility or impedance were used in [21]. In [22], an iterative procedure was developed to find the required resonance frequencies of variable stiffness neutralizers to create nodes at selected locations. Wide-band frequency passive vibration attenuation design for the absorbers was introduced in [23]. In [24], explicit model predictive vibration control was tested. A different approach consists in control and attenuation of multiple travelling waves propagated in a one-dimensional structure [25–28]. Sliding mode control on seat vibration reduction problem was applied in [29].


## 2. Structured Control Laws for Smart Materials

The paper presents an attempt to systematic proportional decentralized position-velocity feedback for active damping of mechanical structures equipped with dense arrays of force actuators and position and velocity sensors. Such a control law is characterized by a very small number of parameters and simple procedures for their tuning compared to centralized approach. Although the results are presented for a one-dimensional structure model only, it is believed that a generalization to two-dimensional mechanical structures will be possible.The research is motivated by vehicular platoon control where relative position and relative or absolute velocity feedback related to the preceding and succeeding vehicle is often considered [30–32]. Nevertheless, the measure of control performance in both applications is different. For vehicular platooning, the main goal consists in preserving a prescribed spacing between the vehicles and in keeping the leader’s velocity, whereas when dealing with mechanical structures, a fast and adequate damping of the oscillating modes is required. Hence, the presented control design is focused on investigation of feasible damping ratio of the least damped mode and achievable stability margin of all modes.Throughout the paper, the superscriptT denotes transpose, In stands for n×n identity matrix, Re⋅ and Im⋅ denotes real and imaginary part, respectively, ⊗ denotes the Kronecker product, and σ⋅ denotes the spectrum of a matrix.


## 3. One-Dimensional Structure Longitudinal Model

Let us consider a one-dimensional structure composed from the massesm, springs k¯, and dampings b¯, each of the same value. Let us assume that the input forces may act on each individual mass independently and we are able to measure positions and velocities of each mass, i.e., actuators and sensors are placed in the same positions. Longitudinal vibrations of such a structure can be described by a state-space model:(1)x¯˙=A¯x¯+B¯u¯,where(2)x¯=p1,p˙1,p2,p˙2,…,pη,p˙ηT∈ℜ2η,where pi,i=1,…,η, are the positions of the masses, η is the number of nodes, and u¯∈ℜη is the vector of the input forces [32].The matricesA¯∈ℜ2η×2η and B¯∈ℜ2η×η are given by(3)A¯=Iη⊗A1+L⊗A2,B¯=Iη⊗01T,with(4)A1=0100,A2=00−k0−b0,where k0=k¯/m, b0=b¯/m. The matrix L∈ℜη×η is the Laplacian of the graph corresponding to the structure which is in this case given as(5)L=1−1−12−1⋱⋱⋱−12−1−11.Since for practical reasons at least one of the nodes has to be fixed, the equation (1) becomes(6)x˙=Ax+Bu,where x∈ℜ2n,u∈ℜn comes from x¯ and u¯ by omitting the entries corresponding to the fixed nodes, and n is the number of the nonfixed nodes.The matricesA∈ℜ2n×2n and B∈ℜ2n×n are then given by(7)A=In⊗A1+Lg⊗A2,B=In⊗01T,where the matrix Lg∈ℜn×n is called the grounded Laplacian that results from Laplacian L (5) by omitting the rows and columns corresponding to the fixed nodes. In the sequel, we will assume without loss of generality that the fixed nodes are the first and last one, n=η−2, and(8)Lg=2−1−12−1⋱⋱⋱−12−1−12.The eigenvalues ofLg are all positive and are given by(9)λℓ=2−2cosℓπn+1,ℓ=1,…,n.We will see later that from the eigenvalues (9), the minimum and maximum ones are of special interest. Those can be determined as(10)λmin=minℓλℓ=λ1=2−2cosπn+1,λmax=maxℓλℓ=λn=2−2cosnπn+1.In this paper, we will use distributed control law where the control action applied to each node depends symmetrically on relative positions and velocities with respect to its neighbors, i.e.,(11)ui=−kpi+1−pi−kpi+1−pi+2−bp˙i+1−p˙i−bp˙i+1−p˙i+2,i=1,…,n,k,b≥0,with p1 and pn+2 being fixed (Figure 1).Figure 1
Distributed control law of a one-dimensional structure.The control law (11) can be written as(12)u=Kx=Lg⊗kbx.After substituting (12) into (6), one obtains the description of the closed-loop system:(13)x˙=Acx=A+BKx,where(14)A+BK=In⊗A1+Lg⊗A2+Lg⊗K2=In⊗A1+Lg⊗A2+K2,with(15)K2=00−k−b.For determination of the eigenvalues of matrixAc, we will use the following lemma.Lemma 1 [32].
(16)
σ
I
n
⊗
A
˜
1
+
L
˜
⊗
A
˜
2
=
∪
λ
ℓ
∈
σ
L
˜
σ
A
˜
1
+
λ
l
A
˜
2
.
Using Lemma 1, we immediately obtain the following result:(17)σAc=σIn⊗A1+Lg⊗A2+K2=∪λℓ∈σLgσ01−λℓk0+k−λℓb0+b.It turns out that the closed-loop eigenvalues are given as the roots of the characteristic equation:(18)s2+λℓb0+bs+λℓk0+k=0,ℓ=1,…,n,i.e.,(19)sℓ±=−λℓb0+b±λℓ2b0+b2−4λℓk0+k2,ℓ=1,…,n.


## 4. Control Strategies

There are many options where to place the closed-loop eigenvalues. Nevertheless, from the vibration suppression point of view, the following two are the most interesting ones.


### 4.1. Prescribed Damping Ratio

A quite natural option is to force all modes to be damped with a prescribed minimum damping ratioζmin∈0,1. From (19), one can see that if k is fixed then with increasing value of b, the least damped mode is that corresponding to λmin. The ratio of imaginary and real part of the least damped mode that corresponds to λmin is given by(20)ξmax=Ims1Res1=4λmink0+k−λmin2b0+b2λminb0+b.Hence, the minimum value of b satisfying this condition is given by(21)bdamp=4k0+kdampλminξmax2+1−b0,where kdamp is set arbitrarily.Since the corresponding damping ratio is given as(22)ζmin=11+ξmax2,after substitution into (21), we obtain(23)bdamp=2ζmink0+kdampλmin−b0.Let us define stability margin as(24)δ=minℓResℓ.Stability margin of such a control law is determined by the distance of the least and most damped closed-loop eigenvalues,s1 and sn+, respectively, from the imaginary axis for b=bdamp,k=kdamp. The distance of complex conjugate s1± corresponding to λmin whose position is given by the prescribed damping ratio is given by(25)ds1±=−Res1=λminb0+bdamp2,whereas distance of real sn+ corresponding to λmax can be obtained as(26)dsn+=λmaxb0+bdamp−λmax2b0+bdamp2−4λmaxk0+kdamp2.Stability margin is then given as minimum of (25) and (26):(27)ddamp=minds1±,dsn+,which after some algebraic manipulations yields(28)ddamp=minζminλmink0+kdamp,λmaxk0+kdampζminλmaxλmin−ζmin2λmaxλmin−1.It should be noted that, for very small dampingζmin<λmin/λmax, the eigenvalue sn is not real and the second term in (28) becomes complex and should not be considered. Nevertheless, considering such a small damping is highly impractical.The dependence of stability margin on prescribed minimum damping ratio given by (28) for n=48 and k0+kdamp=1 is depicted in Figure 2.Figure 2
Stability margin dependence on damping ratio.


### 4.2. Maximum Stability Margin

Another interesting option is to find the control parameters that maximize the stability margin (24). From (19), it follows that the stability margin of the eigenvalues lying on real axis is determined by that corresponding to λmax and stability margin of the eigenvalues lying out of real axis is determined by those corresponding to λmin. Thus, the maximum stability margin is achieved if(29)Res1±=Resn+,where s1 and sn are the eigenvalues corresponding to λmin and λmax, respectively.Condition (29) can be written as(30)−λminb0+b=−λmaxb0+b+λmax2b0+b2−4λmaxk0+k,from which we obtain(31)bmarg=4λmaxk0+kmargλmin2λmax−λmin−b0,for arbitrarily chosen kmarg.By substituting (31) in (29), we obtain the maximum stability margin as(32)δmax=k0+kmargλmaxλmin2λmax−λmin.The ratio of imaginary and real part of the least damped mode that corresponds toλmin is given by(33)ξmax=Ims1Res1=4λmink0+kmarg−λmin2b0+bmarg2λminb0+bmarg,that yields after substitution from (31) and some simplifications(34)ξmax=1−λminλmax.From that, the damping ratio of the least damping mode follows as(35)ζmin=11+ξmax2=λmax2λmax−λmin.The achievable stability margins for different values ofn for both approaches are shown in Figure 3, whereas the minimum damping ratio corresponding to maximum stability margin is depicted in Figure 4. Both figures are plotted for k0+kmarg=1.Figure 3
Comparison of stability margins of both approaches depending on number of nodes.Figure 4
Minimum damping ratio for maximum stability margin approach depending on number of nodes.


## 4.1. Prescribed Damping Ratio

A quite natural option is to force all modes to be damped with a prescribed minimum damping ratioζmin∈0,1. From (19), one can see that if k is fixed then with increasing value of b, the least damped mode is that corresponding to λmin. The ratio of imaginary and real part of the least damped mode that corresponds to λmin is given by(20)ξmax=Ims1Res1=4λmink0+k−λmin2b0+b2λminb0+b.Hence, the minimum value of b satisfying this condition is given by(21)bdamp=4k0+kdampλminξmax2+1−b0,where kdamp is set arbitrarily.Since the corresponding damping ratio is given as(22)ζmin=11+ξmax2,after substitution into (21), we obtain(23)bdamp=2ζmink0+kdampλmin−b0.Let us define stability margin as(24)δ=minℓResℓ.Stability margin of such a control law is determined by the distance of the least and most damped closed-loop eigenvalues,s1 and sn+, respectively, from the imaginary axis for b=bdamp,k=kdamp. The distance of complex conjugate s1± corresponding to λmin whose position is given by the prescribed damping ratio is given by(25)ds1±=−Res1=λminb0+bdamp2,whereas distance of real sn+ corresponding to λmax can be obtained as(26)dsn+=λmaxb0+bdamp−λmax2b0+bdamp2−4λmaxk0+kdamp2.Stability margin is then given as minimum of (25) and (26):(27)ddamp=minds1±,dsn+,which after some algebraic manipulations yields(28)ddamp=minζminλmink0+kdamp,λmaxk0+kdampζminλmaxλmin−ζmin2λmaxλmin−1.It should be noted that, for very small dampingζmin<λmin/λmax, the eigenvalue sn is not real and the second term in (28) becomes complex and should not be considered. Nevertheless, considering such a small damping is highly impractical.The dependence of stability margin on prescribed minimum damping ratio given by (28) for n=48 and k0+kdamp=1 is depicted in Figure 2.Figure 2
Stability margin dependence on damping ratio.


## 4.2. Maximum Stability Margin

Another interesting option is to find the control parameters that maximize the stability margin (24). From (19), it follows that the stability margin of the eigenvalues lying on real axis is determined by that corresponding to λmax and stability margin of the eigenvalues lying out of real axis is determined by those corresponding to λmin. Thus, the maximum stability margin is achieved if(29)Res1±=Resn+,where s1 and sn are the eigenvalues corresponding to λmin and λmax, respectively.Condition (29) can be written as(30)−λminb0+b=−λmaxb0+b+λmax2b0+b2−4λmaxk0+k,from which we obtain(31)bmarg=4λmaxk0+kmargλmin2λmax−λmin−b0,for arbitrarily chosen kmarg.By substituting (31) in (29), we obtain the maximum stability margin as(32)δmax=k0+kmargλmaxλmin2λmax−λmin.The ratio of imaginary and real part of the least damped mode that corresponds toλmin is given by(33)ξmax=Ims1Res1=4λmink0+kmarg−λmin2b0+bmarg2λminb0+bmarg,that yields after substitution from (31) and some simplifications(34)ξmax=1−λminλmax.From that, the damping ratio of the least damping mode follows as(35)ζmin=11+ξmax2=λmax2λmax−λmin.The achievable stability margins for different values ofn for both approaches are shown in Figure 3, whereas the minimum damping ratio corresponding to maximum stability margin is depicted in Figure 4. Both figures are plotted for k0+kmarg=1.Figure 3
Comparison of stability margins of both approaches depending on number of nodes.Figure 4
Minimum damping ratio for maximum stability margin approach depending on number of nodes.


## 5. Example

Let us illustrate the results derived in the previous section on an example. We will consider the following parameters:m=5⋅10−4kg,b¯=3.3⋅10−3Ns/m;k¯=0.4N/m, and n=48. We set the control parameter k=kdamp=kmarg=10N/m.The minimum and maximum eigenvalues of the grounded Laplacian become(36)λmin=2−2cosπn+1=0.0041,λmax=2−2cosnπn+1=3.99.The minimum value ofb=bdamp damping all the eigenvalues with minimum damping ratio ζmin=0.6 is given by(37)bdamp=2ζmink0+kdampλmin−b0=527s−1.Such a control law guarantees the stability margin:(38)δdamp=min1.521,1.095=1.095.The position of dominant open- and closed-loop eigenvalues is plotted in Figure5.Figure 5
Dominant modes for minimum damping ratioζmin=0.6.The value of control parameterb guaranteeing maximum stability margin yields(39)bmarg=4λmaxk0+kmargλmin2λmax−λmin−b0=622s−1,corresponding to stability margin:(40)δmax=k0+kmargλmaxλmin2λmax−λmin=1.29.Damping ratio of the least damped mode is given by(41)ζmin=λmax2λmax−λmin=0.707.The corresponding position of dominant open- and closed-loop eigenvalues is depicted in Figure6.Figure 6
Dominant modes for maximum stability margin approach.To demonstrate the presented design, we compare time and frequency responses of a point lying in the middle of the beam for different values of damping ratios. The open-loop responses to initial conditionpi0=0.01m,p˙i0=0,i=1,…,n are shown in Figure 7.Figure 7
Initial condition displacement response of uncontrolled structure.The initial condition response for different prescribed minimum damping ratios is depicted in Figure8. The Bode plots are compared in Figure 9.Figure 8
Initial condition displacement response for different damping ratios.Figure 9
Bode plot for different prescribed minimum damping ratios.Let us compare the achieved results with other two standard design methods typically used by the control community. At first, we design an LQ controller with relative positions and velocities considered as measurable state variables, i.e.,(42)ui=∑j=1n−kijpj+1−pj−kijpj+1−pj+2−bijp˙j+1−p˙j−bijp˙j+1−p˙j+2,i=1,…,n.The difference between the LQ and presented control law is that the LQ control law uses relative positions and velocities between all neighboring nodes and not between the closest neighbors only as in (11). To force the LQ control to use relative positions and velocities, we introduce a modified state vector:(43)z=p2−p1+p2−p3,p˙2−p˙1+p˙2−p˙3,…,pn+1−pn+pn+1−pn+2,p˙n+1−p˙n+p˙n+1−p˙n+2T.The criterion to be minimized is then given by(44)J=∫0∞zTQz+uTRudt,resulting in LQ feedback control law(45)u=Kz,with(46)K=k22b22k23b23⋯k2n+1b2n+1k32b32k33b33⋯k3n+1b3n+1⋮⋮⋮⋮⋯⋮⋮kn+12bn+12kn+13bn+13⋯kn+1n+1bn+1n+1.By tuning the weighting matricesQ and R to guarantee minimum damping ratio ζmin=0.6 we obtained the corresponding stability margin δdamp=1.162, see dominant poles in Figure 10. Tuning the weighting matrices to maximize stability margin, we arrived to δmax=1.01 with corresponding ζmin=0.903 that can be seen from dominant poles in Figure 11. Hence, the dominant poles configuration is very similar to the proposed design (Figures 5 and 6).Figure 10
Dominant poles for LQ controller design,ζmin=0.6.Figure 11
Dominant poles for LQ controller design—maximum stability margin.The control gainskij and bij for the former case are depicted in Figures 12 and 13, respectively. One can see that the control law uses the relative positions and velocities to the closest neighbors only and that the gains are almost the same for all nodes.Figure 12
Control gainskij for LQ controller design, ζ=0.6.Figure 13
Control gainsbij for LQ controller design, ζ=0.6.To compare our methodology with another control design approach, we formulated the task as anH∞ design for the fixed structure controller (11). It can be easily done with hinfstruct() function in Matlab Robust Control Toolbox. This tuning minimizes the H∞ norm of the closed-loop transfer function modeled by the closed-loop control system with tunable components and weighting filters. In our case, the high-pass filter with cutoff frequency 8 rad/s has been used to penalize all system modes.TheH-infinity design methodology offers efficient algorithms how to obtain multivariable control laws by specifying closed-loop frequency response requirements. This approach was used, for e.g., in [33], where authors compare classical single-input single-output controllers with H-infinity approaches in terms of robustness and performance. The order of the H-infinity control system is however equal to the so-called augmented plant containing the model of the controlled system along with the weighting filters defining performance and robustness requirements. This leads to excessively high-order control laws typically, with strong negative impact on implementation and experimental fine-tuning. For this reason, e.g., in [34], there was a method presented for the controller order reduction which is one possible way how to get control laws with reasonable complexity. Nevertheless, loss or deterioration of closed-loop performance and/or stability is often an unwanted effect associated with this approach. Thanks to recent structured H-infinity control synthesis results (see, e.g., [35, 36]), it is possible to receive the parameters of such reduced-order controllers directly, minimizing the H-infinity norm under the controller complexity constraints.The Bode plots of original system, LQ controller tuned for prescribed minimum damping ratioζmin=0.6, the H∞ controller, and the proposed design for ζmin=0.6 are compared in Figure 14. The plots confirm that all designs give very similar results.Figure 14
Bode plot for displacement for different control strategies.


## 6. Conclusion

In the paper, we presented an active approach of one-dimensional structures with dense array of collocated sensors and actuators using proportional position and velocity feedback control laws. The control law was formulated in a distributive manner, i.e., each actuator uses information from its closest neighbors only. The achievable stability margins and damping ratios were analyzed based on the properties of Laplacian matrix describing the corresponding information graph. Comparison with LQ controller andH∞ designs shows that the presented approach achieves similar results yet with much lower computational and actuator complexity.



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