Embedding properties of network realizations of dissipative reduced order models
Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati,
Vladimir Druskin, and Liliana Borcea
Mathematical Sciences Department, Worcester Polytechnic Institute
https://www.wpi.edu/people/vdruskin
Abstract
Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and
blocktridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations
can be interpreted as ladder resistorcapacitorinductor (RCL) networks. They gave rise to network syntheses in the
rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that
tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient
compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to
their compressing properties, network realizations can be used to embed the data back into the state space of the
underlying continuum problems.
In more recent works of the authors Krein's ideas gave rise to socalled nitedierence Gaussian quadrature rules
(FDGQR), allowing to approximately map the ROM statespace representation to its full order continuum counterpart
on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without
solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly
learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse
problems and unsupervised machine learning.
Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic
wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in
dispersive media, such as seismic exploration, radars and sonars.
To x the idea, we consider a passive irreducible SISO ROM
fn(s) = Xn
j=1
yi
s + σj
, (62)
assuming that all complex terms in (62) come in conjugate pairs.
We will seek ladder realization of (62) as
rjuj + vj − vj−1 = −shˆjuj ,
uj+1 − uj + ˆrj vj = −shj vj ,
(63)
for j = 0, . . . , n with boundary conditions
un+1 = 0, v1 = −1,
and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete
inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively
masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63)
into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a
nonsymmetric Lanczos algorithm written in Jsymmetric form and fn(s) can be equivalently computed as
fn(s) = u1.
The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both
cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real
positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nitedierence
approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich
datamanifolds), a nitedierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3].
The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer
applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63)
cannot always be obtained in portHamiltonian form, i.e., the equivalent primary and dual conductors may change
sign [1].
100
Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or
electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible nonpositivity of conductors for the
nonStieltjes case, we introduce an additional complex sdependent coordinate stretching, vanishing as s → ∞ [1].
This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and
leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps
discrete coecients onto the values of their continuum counterparts.
Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems
for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss
another approach to embedding, based on KreinNudelman theory [5], that results in special datadriven adaptive
grids.
References
[1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to
inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021
[2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the threepoint second dierences:
I. A twopoint positive denite problem in a semiinnite domain, SIAM Journal on Numerical Analysis, V. 37,
N 2, pp.403422, 1999
[3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model
order reduction of graphLaplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V
and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022
[4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm
for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021,
[5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934
Go back to Plenary Speakers Go back to Speakers Go back
more »
« less
Role of Agent Update Cycle in Stability and Robustness of SecondOrder Consensus Networks
We consider the problems of asymptotic stability and robustness in largescale secondorder consensus networks and vehicle platoons in the discretetime domain. First, we develop a graphtheoretic methodology to design the state feedback law for the secondorder consensus networks and vehicle platoons in a discretetime framework. We analyze the stability of such networks based on algebraic properties of the Laplacian matrices of underlying graphs and each vehicle’s update cycle (also known as the time step). We further provide a necessary and sufficient condition of stability of a linear secondorder consensus network in the discretetime domain. Moreover, we evaluate the robustness of the consensus networks by employing the expected value of the steadystate dispersion of the state of the entire network, also known as squared H2norm, as a performance measure. We show the connection between performance measures with respect to network size, connectivity, and the update cycle. The main contribution of this work is that we provide a formal framework to quantify the relation between scaling performance measures and restrictions of the vehicles’ update cycles. Specifically, we show that denser networks (i.e., networks with more communications/edges) require faster agents (i.e., smaller update cycles) to outperform or achieve the same level of robustness as sparse networks (i.e., networks with fewer communications/edges).
more »
« less
 Award ID(s):
 2121121
 NSFPAR ID:
 10352499
 Date Published:
 Journal Name:
 2022 30th Mediterranean Conference on Control and Automation (MED)
 Page Range / eLocation ID:
 643 to 648
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


This work investigates local and global measures of disorder in largescale directed networks of doubleintegrator systems connected over a multidimensional torus. We quantify these performance measures in systems subjected to distributed disturbances using an H2 norm with outputs corresponding to local state errors or deviations from the global average. We consider two directed unidirectional state feedback inter connections that correspond to relative position and relative velocity feedback in vehicle network applications. Our main result reveals that absolute state feedback plays a critical role in system robustness when local state measurements are uni directional. Specifically, if absolute measurements of either state variable are available, then systems with unidirectional relative feedback perform as well as their symmetric bidirectional counterparts but have the advantage of reduced communication requirements. However in the absence of absolute feedback their performance is worse; in fact, it is impossible to maintain stability (i.e. a finite H2 norm) with unidirectional state mea surements for arbitrarily large networks. Numerical examples illustrate the theory.more » « less

We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher (n>=2). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without retuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For highorder agents (n>=3), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest nonzero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For secondorder consensus (n=2) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh–Hurwitz criteria for complexvalued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sublinear scaling of nodal neighborhoods can suffice to overcome the issue.more » « less

This research explores the inherent vulnerability of nonlinear vehicle platoons characterized by the oscillatory behavior triggered by external perturbations. The perturbation exerted on the vehicle platoon is regarded as an external force on an object. Following the mechanical vibration analysis in mechanics, this research proposes a vibrationtheoretic approach that advances our understanding of platoon vulnerability from two aspects. First, the proposed approach introduces damping intensity to characterize vehicular platoon vulnerability, which divides platoon oscillations into two types, i.e., underdamped and overdamped. The damping intensity measures the platoon’s recovery strength in responding to perturbations. Second, the proposed approach can obtain the resonance frequency of a nonlinear vehicle platoon, where resonance amplifies platoon oscillation magnitude when the external perturbation frequency equals the platoon’s damping oscillation frequency. The main contribution of this research lies in the analytical derivation of the closedform formulas of damping intensity and resonance frequency. In particular, the proposed approach formulates platoon dynamics under perturbation as a secondorder nonhomogeneous ordinary differential equation, enabling rigorous derivations and analyses for platoons with complicated nonlinear carfollowing behaviors. Through simulations built on realworld data, this paper demonstrates that an overdamped vehicle platoon is more robust against perturbations, and an underdamped platoon can be destabilized easily by exerting a perturbation at the platoon’s resonance frequency. The theoretical derivations and simulation results shed light on the design of reliable platooning control, either for humandriven or automated vehicles, to suppress the adverse effects of oscillations.more » « less

Abstract Background Few studies have systematically investigated robust controllers for lower limb rehabilitation exoskeletons (LLREs) that can safely and effectively assist users with a variety of neuromuscular disorders to walk with full autonomy. One of the key challenges for developing such a robust controller is to handle different degrees of uncertain humanexoskeleton interaction forces from the patients. Consequently, conventional walking controllers either are patientcondition specific or involve tuning of many control parameters, which could behave unreliably and even fail to maintain balance. Methods We present a novel, deep neural network, reinforcement learningbased robust controller for a LLRE based on a decoupled offline humanexoskeleton simulation training with three independent networks, which aims to provide reliable walking assistance against various and uncertain humanexoskeleton interaction forces. The exoskeleton controller is driven by a neural network control policy that acts on a stream of the LLRE’s proprioceptive signals, including joint kinematic states, and subsequently predicts realtime position control targets for the actuated joints. To handle uncertain human interaction forces, the control policy is trained intentionally with an integrated human musculoskeletal model and realistic humanexoskeleton interaction forces. Two other neural networks are connected with the control policy network to predict the interaction forces and muscle coordination. To further increase the robustness of the control policy to different human conditions, we employ domain randomization during training that includes not only randomization of exoskeleton dynamics properties but, more importantly, randomization of human muscle strength to simulate the variability of the patient’s disability. Through this decoupled deep reinforcement learning framework, the trained controller of LLREs is able to provide reliable walking assistance to patients with different degrees of neuromuscular disorders without any control parameter tuning. Results and conclusion A universal, RLbased walking controller is trained and virtually tested on a LLRE system to verify its effectiveness and robustness in assisting users with different disabilities such as passive muscles (quadriplegic), muscle weakness, or hemiplegic conditions without any control parameter tuning. Analysis of the RMSE for joint tracking, CoPbased stability, and gait symmetry shows the effectiveness of the controller. An ablation study also demonstrates the strong robustness of the control policy under large exoskeleton dynamic property ranges and various humanexoskeleton interaction forces. The decoupled network structure allows us to isolate the LLRE control policy network for testing and simtoreal transfer since it uses only proprioception information of the LLRE (joint sensory state) as the input. Furthermore, the controller is shown to be able to handle different patient conditions without the need for patientspecific control parameter tuning.more » « less