A Contraction Theory Approach to Singularly Perturbed Systems with Application to Retroactivity Attenuation

In this paper, we revisit standard results for singularly perturbed systems on the infinite time interval by employing tools from nonlinear contraction theory. This allows us to determine explicit bounds both on the rate of convergence of trajectories to the slow manifold, and on the distance between these trajectories and those of the reduced system. We illustrate the application of the proposed technique to the problem of retroactivity attenuation in biomolecular systems, that is, to the problem of attenuating the effects of output loading due to interconnection to downstream systems. By virtue of the explicit bounds, we can single out the key biochemical parameters to tune in order to enhance retroactivity attenuation. This provides design guidelines for synthetic biology devices that are robust to loading and can function as insulation devices just like insulating amplifiers work in electronics.National Science Foundation (U.S.). Division of Computing and Communication Foundations (NSF-CCF Grant 1058127

Optimal design of phosphorylation-based insulation devices

We seek to minimize both the retroactivity to the output and the retroactivity to the input of a phosphorylation-based insulation device by finding an optimal substrate concentration. Characterizing and improving the performance of insulation devices brings us a step closer to their successful implementation in biological circuits, and thus to modularity. Previous works have mainly focused on attenuating retroactivity effects to the output using high substrate concentrations. This, however, worsens the retroactivity to the input, creating an error that propagates back to the output. Employing singular perturbation and contraction theory tools, this work provides a framework to determine an optimal substrate concentration to reach a tradeoff between the retroactivity to the input and the retroactivity to the output.Grant FA9550-12-1-021

Singular Perturbation via Contraction Theory

In this paper, we provide a novel contraction-theoretic approach to analyze two-time scale systems. In our proposed framework, systems enjoy several robustness properties, which can lead to a more complete characterization of their behaviors. Key assumptions are the contractivity of the fast sub-system and of the reduced model, combined with an explicit upper bound on the time-scale parameter. For two-time scale systems subject to disturbances, we show that the distance between solutions of the nominal system and solutions of its reduced model is uniformly upper bounded by a function of contraction rates, Lipschitz constants, the time-scale parameter, and the time variability of the disturbances. We also show local contractivity of the two-time scale system and give sufficient conditions for global contractivity. We then consider two special cases: for autonomous nonlinear systems we obtain sharper bounds than our general results and for linear time-invariant systems we present novel bounds based upon log norms and induced norms. Finally, we apply our theory to two application areas -- online feedback optimization and Stackelberg games -- and obtain new individual tracking error bounds showing that solutions converge to their (time-varying) optimizer and computing overall contraction rates.Comment: This paper has been submitted to IEEE Transactions on Automatic Contro

Retroactivity to the Output of Transcription Devices: Quantification and Insulation.

Traditional engineering often relies on hierarchical design techniques to build com- plex systems from simpler subsystems. This technique requires modularity, a prop- erty that states that the input/output characteristics of a system are not affected by interconnections. In this work we investigate retroactivity, an impedance-like effect in biomolecular systems that makes the behavior of a system change upon intercon- nection. We show, through analysis and experiments, that retroactivity in synthetic biology circuits is responsible for substantial changes in a system dynamic response. In order to construct circuits modularly, we propose the design of insulation de- vices, which, similar to insulating amplifiers in electronics, attenuate retroactivity effects and recover modular behavior. Our technique is based on a novel disturbance attenuation approach based on singular perturbation theory.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/95939/1/jayanthi_1.pd

Robust stability assessment for future power systems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2018.Cataloged from PDF version of thesis. "Due to the condition of the original material, there are unavoidable flaws in this reproduction. Some pages in the original document contain text that is illegible"--Disclaimer Notice page.Includes bibliographical references (pages 119-128).Loss of stability in electrical power systems may eventually lead to blackouts which, despite being rare, are extremely costly. However, ensuring system stability is a non-trivial task for several reasons. First, power grids, by nature, are complex nonlinear dynamical systems, so assessing and maintaining system stability is challenging mainly due to the co-existence of multiple equilibria and the lack of global stability. Second, the systems are subject to various sources of uncertainties. For example, the renewable energy injections may vary depending on the weather conditions. Unfortunately, existing security assessment may not be sufficient to verify system stability in the presence of such uncertainties. This thesis focuses on new scalable approaches for robust stability assessment applicable to three main types of stability, i.e., long-term voltage, transient, and small-signal stability. In the first part of this thesis, I develop a novel computationally tractable technique for constructing Optimal Power Flow (OPF) feasibility (convex) subsets. For any inner point of the subset, the power flow problem is guaranteed to have a feasible solution which satisfies all the operational constraints considered in the corresponding OPF. This inner approximation technique is developed based on Brouwer's fixed point theorem as the existence of a solution can be verified through a self-mapping condition. The self-mapping condition along with other operational constraints are incorporated in an optimization problem to find the largest feasible subsets. Such an optimization problem is nonlinear, but any feasible solution will correspond to a valid OPF feasibility estimation. Simulation results tested on several IEEE test cases up to 300 buses show that the estimation covers a substantial fraction of the true feasible set. Next, I introduce another inner approximation technique for estimating an attraction domain of a post-fault equilibrium based on contraction analysis. In particular, I construct a contraction region where the initial conditions are "forgotten", i.e., all trajectories starting from inside this region will exponentially converge to each other. An attraction basin is constructed by inscribing the largest ball in the contraction region. To verify contraction of a Differential-Algebraic Equation (DAE) system, I also show that one can rely on the analysis of extended virtual systems which are reducible to the original one. Moreover, the Jacobians of the synthetic systems can always be expressed in a linear form of state variables because any polynomial system has a quadratic representation. This makes the synthetic system analysis more appropriate for contraction region estimation in a large scale. In the final part of the thesis, I focus on small-signal stability assessment under load dynamic uncertainties. After introducing a generic impedance-based load model which can capture the uncertainty, I propose a new robust small signal (RSS) stability criterion. Semidefinite programming is used to find a structured Lyapunov matrix, and if it exists, the system is provably RSS stable. An important application of the criterion is to characterize operating regions which are safe from Hopf bifurcations. The robust stability assessment techniques developed in this thesis primarily address the needs of a system operator in electrical power systems. The results, however, can be naturally extended to other nonlinear dynamical systems that arise in different fields such as biology, biomedicine, economics, neuron networks, and optimization. As the robust assessment is based on sufficient conditions for stability, there is still room for development on reducing the inevitable conservatism. For example, for OPF feasibility region estimation, an important open question considers what tighter bounds on the nonlinear residual terms one can use instead of box type bounds. Also, for attraction basin problem, finding the optimal norms and metrics which result in the largest contraction domain is an interesting potential research question.by Hung Dinh Nguyen.Ph. D

Proceedings of the 7th Sound and Music Computing Conference

Proceedings of the SMC2010 - 7th Sound and Music Computing Conference, July 21st - July 24th 2010