Low-Complexity Decentralized Active Damping of One-Dimensional Structures

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.


Introduction
e 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][6][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.erefore, 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.
e 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.is kind of control can be applied for automated highway systems [9], car formations [10], and also flexible structures.e 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.e 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][26][27][28].Sliding mode control on seat vibration reduction problem was applied in [29].

Structured Control Laws for Smart Materials
e 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.
e 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][31][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.
roughout the paper, the superscript T denotes transpose, I n 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.

One-Dimensional Structure Longitudinal Model
Let us consider a one-dimensional structure composed from the masses m, 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: where where p i , i � 1, . . ., η, are the positions of the masses, η is the number of nodes, and u ∈ R η is the vector of the input forces [32].e matrices A ∈ R 2η×2η and B ∈ R 2η×η are given by with where e matrix L ∈ R η×η is the Laplacian of the graph corresponding to the structure which is in this case given as Since for practical reasons at least one of the nodes has to be fixed, the equation (1) becomes where x ∈ R 2n , u ∈ R n comes from x and u by omitting the entries corresponding to the fixed nodes, and n is the number of the nonfixed nodes.e matrices A ∈ R 2n×2n and B ∈ R 2n×n are then given by where the matrix L g ∈ R n×n is called the grounded Laplacian that results from Laplacian L (5) by omitting the rows and 2 Shock and Vibration 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 e eigenvalues of L g are all positive and are given by We will see later that from the eigenvalues (9), the minimum and maximum ones are of special interest.ose can be determined as 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., with p 1 and p n+2 being fixed (Figure 1).e control law (11) can be written as After substituting ( 12) into (6), one obtains the description of the closed-loop system: where with For determination of the eigenvalues of matrix A c , we will use the following lemma.
Using Lemma 1, we immediately obtain the following result: It turns out that the closed-loop eigenvalues are given as the roots of the characteristic equation: i.e.,

Control Strategies
ere 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.

Prescribed Damping Ratio.
A quite natural option is to force all modes to be damped with a prescribed minimum damping ratio ζ min ∈ [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 λ min .e ratio of imaginary and real part of the least damped mode that corresponds to λ min is given by Hence, the minimum value of b satisfying this condition is given by where k damp is set arbitrarily.Since the corresponding damping ratio is given as after substitution into (21), we obtain Let us define stability margin as Stability margin of such a control law is determined by the distance of the least and most damped closed-loop eigenvalues, s 1 and s + n , respectively, from the imaginary axis Shock and Vibration 3 for b � b damp , k � k damp .e distance of complex conjugate s ± 1 corresponding to λ min whose position is given by the prescribed damping ratio is given by whereas distance of real s + n corresponding to λ max can be obtained as Stability margin is then given as minimum of ( 25) and (26): which after some algebraic manipulations yields It should be noted that, for very small damping (ζ min < λ min /λ max ), the eigenvalue s n 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.
e dependence of stability margin on prescribed minimum damping ratio given by (28) for n � 48 and k 0 + k damp � 1 is depicted in Figure 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 λ max and stability margin of the eigenvalues lying out of real axis is determined by those corresponding to λ min .us, the maximum stability margin is achieved if where s 1 and s n are the eigenvalues corresponding to λ min and λ max , respectively.Condition (29) can be written as from which we obtain for arbitrarily chosen k marg .Shock and Vibration By substituting (31) in (29), we obtain the maximum stability margin as e ratio of imaginary and real part of the least damped mode that corresponds to λ min is given by that yields after substitution from (31) and some simpli cations From that, the damping ratio of the least damping mode follows as e achievable stability margins for di erent values of n for both approaches are shown in Figure 3, whereas the minimum damping ratio corresponding to maximum stability margin is depicted in Figure 4.Both gures are plotted for k 0 + k marg 1.

Example
Let us illustrate the results derived in the previous section on an example.We will consider the following parameters: corresponding to stability margin: Damping ratio of the least damped mode is given by e corresponding position of dominant open-and closed-loop eigenvalues is depicted in Figure 6.

Shock and Vibration
To demonstrate the presented design, we compare time and frequency responses of a point lying in the middle of the beam for di erent values of damping ratios.e open-loop responses to initial condition p i (0) 0.01 m, _ p i (0) 0, i 1, . . ., n are shown in Figure 7.
e initial condition response for di erent prescribed minimum damping ratios is depicted in Figure 8. e Bode plots are compared in Figure 9.
Let us compare the achieved results with other two standard design methods typically used by the control community.At rst, we design an LQ controller with relative positions and velocities considered as measurable state variables, i.e., e di erence 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 modi ed state vector:  (43) e criterion to be minimized is then given by By tuning the weighting matrices Q and R to guarantee minimum damping ratio ζ min 0.6 we obtained the corresponding stability margin δ damp 1.162, see dominant poles in Figure 10.Tuning the weighting matrices to maximize stability margin, we arrived to δ max 1.01 with corresponding ζ min 0.903 that can be seen from dominant poles in Figure 11.Hence, the dominant poles con guration is very similar to the proposed design (Figures 5 and 6).
e control gains k ij and b ij 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.
To compare our methodology with another control design approach, we formulated the task as an H ∞ design for the xed structure controller (11).It can be easily done with hinfstruct() function in Matlab Robust Control Toolbox. is tuning minimizes the H ∞ norm of the closed-loop transfer function modeled by the closed-loop control system with tunable components and weighting lters.In our case, the high-pass lter with cuto frequency 8 rad/s has been used to penalize all system modes.e H-in nity design methodology o ers e cient algorithms how to obtain multivariable control laws by specifying closed-loop frequency response requirements.is approach was used, for e.g., in [33], where authors compare classical single-input single-output controllers with H-in nity approaches in terms of robustness and performance.e order of the H-in nity control system is however equal to the so-called augmented plant containing the model of the controlled system along with the weighting lters de ning performance and robustness requirements.is leads to excessively high-order control laws typically, with strong negative impact on implementation and experimental ne-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 e ect associated with this approach.anks to recent structured H-in nity control synthesis results (see, e.g., [35,36]), it is possible to receive the parameters of such reduced-order controllers directly, minimizing the H-in nity norm under the controller complexity constraints.
e Bode plots of original system, LQ controller tuned for prescribed minimum damping ratio ζ min 0.6, the H ∞ controller, and the proposed design for ζ min 0.6 are compared in Figure 14. e plots con rm that all designs give very similar results.

Conclusion
In the paper, we presented an active approach of onedimensional structures with dense array of collocated e control law was formulated in a distributive manner, i.e., each actuator uses information from its closest neighbors only.e achievable stability margins and damping ratios were analyzed based on the properties of Laplacian matrix describing the corresponding information graph.Comparison with LQ controller and H ∞ designs shows that the presented approach achieves similar results yet with much lower computational and actuator complexity.Shock and Vibration

Figure 1 :Figure 2 :
Figure 1: Distributed control law of a one-dimensional structure.

m 5 •
10 −4 kg, b 3.3 • 10 −3 Ns/m; k 0.4 N/m, and n 48.We set the control parameter k k damp k marg 10 N/m. e minimum and maximum eigenvalues of the grounded Laplacian become λ min 2 − 2 cos π n of b b damp damping all the eigenvalues with minimum damping ratio ζ min 0.6 is given by b damp 2ζ min k 0 + k damp λ min − b 0 527 s −1 .(37) Such a control law guarantees the stability margin: δ damp min 1.521, 1.095 { } 1.095.(38) e position of dominant open-and closed-loop eigenvalues is plotted in Figure 5. e value of control parameter b guaranteeing maximum stability margin yields

Figure 3 :
Figure 3: Comparison of stability margins of both approaches depending on number of nodes.

Figure 4 :
Figure 4: Minimum damping ratio for maximum stability margin approach depending on number of nodes.