I have been an associate professor in Electrical and Computer Engineering (ECE) at Johns Hopkins University (JHU) since July 2022. I earned my Ph.D. in ECE with a minor in Applied Mathematics from Cornell University in Jan 2014 under the supervision of an awesome advisor and person, Prof. A. Kevin Tang. Before joining JHU as an assistant professor in Jan 2016, I was a postdoctoral scholar at the Center for the Mathematics of Information (CMI) in the Computational and Mathematical Sciences (CMS) department at Caltech from 2013 to 2015, where I had the pleasure to be mentored by Prof. Steven Low and Prof. Adam Wierman.
Research Interests
- Networked Systems: coupled oscillators, clock synchronization, saddle-flows, network coherence, distributed coordination, consensus
- Power Systems: frequency control, inverter-based control, real-time congestion management, electricity markets, reduced-order models
- Optimization: time-varying optimization, primal-dual algorithms, semidefinite programming, sum-of-squares optimization
- Machine Learning: reinforcement learning, sparse recovery, subspace preserving recovery, network tomography, multi-armed bandits
Recent Talks
A complete list of talks can be found here.
- 2024-11-19: Nonparametric Analysis of Dynamical Systems: From Recurrent Sets to Generalized Lyapunov and Barrier Conditions, ESE Fall Colloquium.
[BibTeX] [Download PDF]@talk{upenn24, date = {11/20/2024}, day = {19}, event = {ESE Fall Colloquium}, host = {Rene Vidal}, month = {11}, role = {Speaker}, title = {Nonparametric Analysis of Dynamical Systems: From Recurrent Sets to Generalized Lyapunov and Barrier Conditions}, url = {https://mallada.ece.jhu.edu/talks/202411-UPenn.pdf}, year = {2024} } - 2024-09-25: Generalized Barrier Functions: Integral Conditions and Recurrent Relaxations, 60th Allerton Conference on 60th Allerton Conference on Communication, Control, and Computing.
[BibTeX] [Abstract] [Download PDF]
Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL’s suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
@talk{allerton24, abstract = {Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL's suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.}, date = {06/12/2024}, day = {25}, event = {60th Allerton Conference on 60th Allerton Conference on Communication, Control, and Computing}, host = {N/A}, month = {09}, role = {Speaker}, title = {Generalized Barrier Functions: Integral Conditions and Recurrent Relaxations}, url = {https://mallada.ece.jhu.edu/talks/202409-Allerton.pdf}, year = {2024} } - 2024-06-12: Reinforcement Learning for Safety Critical Applications, Tercera Conferencia Colombiana de Matematicas Aplicadas e Industriales.
[BibTeX] [Abstract] [Download PDF]
Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL’s suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
@talk{mapi24, abstract = {Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL's suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.}, date = {06/12/2024}, day = {12}, event = {Tercera Conferencia Colombiana de Matematicas Aplicadas e Industriales}, host = {Javier Peña (CMU), Mateo Diaz (JHU)}, month = {06}, role = {Speaker}, title = {Reinforcement Learning for Safety Critical Applications}, url = {https://mallada.ece.jhu.edu/talks/202406-MAPI.pdf}, year = {2024} } - 2024-06-18: Data-driven Analysis of Dynamical Systems Using Recurrent Sets, INFORMS International Conference.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.
@talk{informs2024, abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.}, date = {06/18/2024}, day = {18}, event = {INFORMS International Conference}, host = {Luis Zuluaga (Lehigh), Mateo Diaz (JHU)}, month = {06}, role = {Speaker}, title = {Data-driven Analysis of Dynamical Systems Using Recurrent Sets}, url = {https://mallada.ece.jhu.edu/talks/202406-Informs.pdf}, year = {2024} } - 2024-06-05: Data-driven Analysis of Dynamical Systems Using Recurrent Sets, Department of Automatic Control, Lund University.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.
@talk{lund2024, abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.}, date = {06/05/2024}, day = {05}, event = {Department of Automatic Control, Lund University}, host = {Richard Pates (Lund)}, month = {06}, role = {Lecture}, title = {Data-driven Analysis of Dynamical Systems Using Recurrent Sets}, url = {https://mallada.ece.jhu.edu/talks/202406-Lund.pdf}, year = {2024} } - 2024-06-06: Data-driven Analysis of Dynamical Systems Using Recurrent Sets, Cyber-Physical Systems Lab, Université catholique de Louvain.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.
@talk{ucl2024, abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.}, date = {06/06/2024}, day = {06}, event = {Cyber-Physical Systems Lab, Université catholique de Louvain}, host = {Raphael Jungers (UCL)}, month = {06}, role = {Lecture}, title = {Data-driven Analysis of Dynamical Systems Using Recurrent Sets}, url = {https://mallada.ece.jhu.edu/talks/202406-UCL.pdf}, year = {2024} } - 2024-05-16: Recurrence of Nonlinear Control Systems: Entropy and Bit Rates, Hybrid Systems: Computation and Control (HSCC).
[BibTeX] [Abstract] [Download PDF]
In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.
@talk{hscc2024, abstract = {In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.}, date = {05/16/2024}, day = {16}, event = {Hybrid Systems: Computation and Control (HSCC)}, month = {05}, role = {Lecture}, title = {Recurrence of Nonlinear Control Systems: Entropy and Bit Rates}, url = {https://mallada.ece.jhu.edu/talks/202405-HSCC.pdf}, year = {2024} } - 2024-03-28: Options for Mitigation Measures: Avenues for new Research, ESIG/G-PST Special Topic Workshop on Oscillations.
[BibTeX] [Download PDF]@talk{esig24, date = {03/28/2024}, day = {28}, event = {ESIG/G-PST Special Topic Workshop on Oscillations}, host = {Mark O'Malley (Imperial)}, month = {03}, role = {Lecture}, title = {Options for Mitigation Measures: Avenues for new Research}, url = {https://mallada.ece.jhu.edu/talks/202403-ESIG.pdf}, year = {2024} }
Recent News
A complete list of updates can be found here.
- Yue defended his dissertation
- 1 paper accepted to Allerton
- Yue to join Apple
- 3 papers accepted to CDC
- 1 paper accepted to L-CSS
- Roy to do an internship at Amazon
- 2 papers accepted to PES General Meeting
- 1 paper accepted to ICLR
- 1 paper presented at Asilomar
- DJ to join U. S. Joint Office of Energy and Transportation
- 1 paper accepted to IEEE TEMPR
- Tianqi defended his dissertation
- Rajni defended his dissertation
- Hancheng defended his dissertation
- Tianqi to join Amazon
- 2 papers accpeted to CDC
- Rajni to join UCSD as Postdoc
- Hancheng to join UPenn as Postdoc
- 1 paper accepted to ICML
- Tianqi passed his thesis proposal defense
- Chengda defended his dissertation
- Tianqi received MINDS Fellowship
- 1 paper accepted to AISTATS
- 2 papers accepted to ACC
- 1 paper accepted to TAC
Publications Snapshot
A complete list of publications can be found here or on my Google Scholar profile.
Preprints
- R. K. Bansal, E. Mallada, and P. Hidalgo-Gonzalez, A Market Mechanism for a Two-stage Settlement Electricity Market with Energy Storage, 2024, submitted.
[BibTeX] [Abstract] [Download PDF]
Electricity markets typically clear in two stages: a day-ahead market and a real-time market. In this paper, we propose market mechanisms for a two-stage multi-interval electricity market with energy storage, generators, and demand uncertainties. We consider two possible mixed bidding strategies: storage first bids cycle depths in the day ahead and then charge-discharge power bids in the real-time market for any last-minute adjustments. While the first strategy only considers day-ahead decisions from an individual participant’s perspective as part of their individual optimization formulation, the second strategy accounts for both the market operator’s and participants’ perspectives. We demonstrate that the competitive equilibrium exists uniquely for both mechanisms. However, accounting for the day-ahead decisions in the bidding function has several advantages. Numerical experiments using New York ISO data provide bounds on the proposed market mechanism.
@unpublished{bmh2024a-preprint, abstract = {Electricity markets typically clear in two stages: a day-ahead market and a real-time market. In this paper, we propose market mechanisms for a two-stage multi-interval electricity market with energy storage, generators, and demand uncertainties. We consider two possible mixed bidding strategies: storage first bids cycle depths in the day ahead and then charge-discharge power bids in the real-time market for any last-minute adjustments. While the first strategy only considers day-ahead decisions from an individual participant's perspective as part of their individual optimization formulation, the second strategy accounts for both the market operator's and participants' perspectives. We demonstrate that the competitive equilibrium exists uniquely for both mechanisms. However, accounting for the day-ahead decisions in the bidding function has several advantages. Numerical experiments using New York ISO data provide bounds on the proposed market mechanism.}, author = {Bansal, Rajni Kant and Mallada, Enrique and Hidalgo-Gonzalez, Patricia}, grants = {CPS-2136324, Global-Centers-2330450}, month = {03}, title = {A Market Mechanism for a Two-stage Settlement Electricity Market with Energy Storage}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-BMH.pdf}, year = {2024, submitted} } - P. You, Y. Liu, and E. Mallada, A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design, 2024, submitted.
[BibTeX] [Abstract] [Download PDF]
This work examines the conditions for asymptotic and exponential convergence of saddle flow dynamics of convex-concave functions. First, we propose an observability-based certificate for asymptotic convergence, directly bridging the gap between the invariant set in a LaSalle argument and the equilibrium set of saddle flows. This certificate generalizes conventional conditions for convergence, e.g., strict convexity-concavity, and leads to a novel state-augmentation method that requires minimal assumptions for asymptotic convergence. We also show that global exponential stability follows from strong convexity-strong concavity, providing a lower-bound estimate of the convergence rate. This insight also explains the convergence of proximal saddle flows for strongly convex-concave objective functions. Our results generalize to dynamics with projections on the vector field and have applications in solving constrained convex optimization via primal-dual methods. Based on these insights, we study four algorithms built upon different Lagrangian function transformations. We validate our work by applying these methods to solve a network flow optimization and a Lasso regression problem.
@unpublished{ylm2024a-preprint, abstract = {This work examines the conditions for asymptotic and exponential convergence of saddle flow dynamics of convex-concave functions. First, we propose an observability-based certificate for asymptotic convergence, directly bridging the gap between the invariant set in a LaSalle argument and the equilibrium set of saddle flows. This certificate generalizes conventional conditions for convergence, e.g., strict convexity-concavity, and leads to a novel state-augmentation method that requires minimal assumptions for asymptotic convergence. We also show that global exponential stability follows from strong convexity-strong concavity, providing a lower-bound estimate of the convergence rate. This insight also explains the convergence of proximal saddle flows for strongly convex-concave objective functions. Our results generalize to dynamics with projections on the vector field and have applications in solving constrained convex optimization via primal-dual methods. Based on these insights, we study four algorithms built upon different Lagrangian function transformations. We validate our work by applying these methods to solve a network flow optimization and a Lasso regression problem.}, author = {You, Pengcheng and Liu, Yingzhu and Mallada, Enrique}, grants = {CAREER-1752362, CPS-2136324, Global-Centers-2330450}, month = {9}, pages = {1-16}, title = {A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-YLM.pdf}, year = {2024, submitted} } - P. You, M. Fernandez, D. F. Gayme, and E. Mallada, Mixed Supply Function and Quantity Bidding in Two-Stage Settlement Markets, 2023, under revision, submitted Mar 2023.
[BibTeX] [Abstract] [Download PDF]
Motivated by electricity markets, we study the incentives of heterogeneous participants (firms and consumers) in a two-stage settlement market with a mixed bidding mechanism, in which firms participate using supply function bids and consumers use quantity bids. We carry out an equilibrium analysis of the market outcome and obtain closed-form solutions. The characterization of the equilibria allows us to gain insights into the market-power implications of mixed bidding and uncover the importance of accounting for consumers’ strategic behavior in a two-stage market, even when their demand is completely inelastic with respect to price. We show that strategic consumers are able to exploit firms’ strategic behavior to maintain a systematic difference between the forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payment and undermines the supply-side market power. However, it is only effective when firms are behaving strategically. We also observe situations where firms lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Our results further suggest that market competition has a heterogeneous impact across consumer sizes, particularly benefiting small consumers. Our analysis can accommodate other market policies, and we demonstrate this versatility by examining the impact of some example policies, including virtual bidding, on the market outcome.
@unpublished{yfgm2023a-preprint, abstract = {Motivated by electricity markets, we study the incentives of heterogeneous participants (firms and consumers) in a two-stage settlement market with a mixed bidding mechanism, in which firms participate using supply function bids and consumers use quantity bids. We carry out an equilibrium analysis of the market outcome and obtain closed-form solutions. The characterization of the equilibria allows us to gain insights into the market-power implications of mixed bidding and uncover the importance of accounting for consumers' strategic behavior in a two-stage market, even when their demand is completely inelastic with respect to price. We show that strategic consumers are able to exploit firms' strategic behavior to maintain a systematic difference between the forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payment and undermines the supply-side market power. However, it is only effective when firms are behaving strategically. We also observe situations where firms lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Our results further suggest that market competition has a heterogeneous impact across consumer sizes, particularly benefiting small consumers. Our analysis can accommodate other market policies, and we demonstrate this versatility by examining the impact of some example policies, including virtual bidding, on the market outcome.}, author = {You, Pengcheng and Fernandez, Marcelo and Gayme, Dennice F. and Mallada, Enrique}, grants = {CAREER-1752362;TRIPODS-1934979;CPS-2136324}, month = {8}, pages = {1-45}, title = {Mixed Supply Function and Quantity Bidding in Two-Stage Settlement Markets}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-YFGM.pdf}, year = {2023, under revision, submitted Mar 2023} } - R. K. Bansal, P. You, Y. Chen, and E. Mallada, Intercept Function and Quantity Bidding in Two-stage Electricity Market with Market Power Mitigation, 2023, under revision, submitted Aug 2023.
[BibTeX] [Abstract] [Download PDF]
Electricity markets typically operate in two stages, day-ahead and real-time. Despite best efforts striving efficiency, evidence of price manipulation has called for system-level market power mitigation (MPM) initiatives that substitute noncompetitive bids with default bids. Implementing these policies with a limited understanding of participant behavior may lead to unintended economic losses. In this paper, we model the competition between generators and inelastic loads in a two-stage market with stage-wise MPM policies. The loss of Nash equilibrium and lack of guarantee of stable market outcome in the case of conventional supply function bidding motivates the use of an alternative market mechanism where generators bid an intercept function. A Nash equilibrium analysis for a day-ahead MPM policy leads to a Stackelberg-Nash game with loads exercising market power at the expense of generators. A comparison of the resulting equilibrium with the standard market (not implementing any MPM policy) shows that a day-ahead policy completely mitigates the market power of generators. On the other hand, the real-time MPM policy increases demand allocation to real-time, contrary to current market practice with most electricity trades in the day-ahead market. Numerical studies illustrate the impact of the slope of the intercept function on the standard market.
@unpublished{bcym2023a-preprint, abstract = {Electricity markets typically operate in two stages, day-ahead and real-time. Despite best efforts striving efficiency, evidence of price manipulation has called for system-level market power mitigation (MPM) initiatives that substitute noncompetitive bids with default bids. Implementing these policies with a limited understanding of participant behavior may lead to unintended economic losses. In this paper, we model the competition between generators and inelastic loads in a two-stage market with stage-wise MPM policies. The loss of Nash equilibrium and lack of guarantee of stable market outcome in the case of conventional supply function bidding motivates the use of an alternative market mechanism where generators bid an intercept function. A Nash equilibrium analysis for a day-ahead MPM policy leads to a Stackelberg-Nash game with loads exercising market power at the expense of generators. A comparison of the resulting equilibrium with the standard market (not implementing any MPM policy) shows that a day-ahead policy completely mitigates the market power of generators. On the other hand, the real-time MPM policy increases demand allocation to real-time, contrary to current market practice with most electricity trades in the day-ahead market. Numerical studies illustrate the impact of the slope of the intercept function on the standard market.}, author = {Bansal, Rajni Kant and You, Pengcheng and Chen, Yue and Mallada, Enrique}, month = {11}, pages = {1-14}, title = {Intercept Function and Quantity Bidding in Two-stage Electricity Market with Market Power Mitigation}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-BCYMb.pdf}, year = {2023, under revision, submitted Aug 2023} } - T. Zheng, J. W. Simpson-Porco, and E. Mallada, Closed-Loop Motion Planning for Differentially Flat Systems: A Time-Varying Optimization Framework, 2023, submitted.
[BibTeX] [Abstract] [Download PDF]
Motion planning and control are two core components of the robotic systems autonomy stack. The standard approach to combine these methodologies comprises an offline/open-loop stage, \textitplanning, that designs a feasible and safe trajectory to follow, and an online/closed-loop stage, \textittracking, that corrects for unmodeled dynamics and disturbances. Such an approach generally introduces conservativeness into the planning stage, which becomes difficult to overcome as the model complexity increases and real-time decisions need to be made in a changing environment. This work addresses these challenges for the class of differentially flat nonlinear systems by integrating planning and control into a cohesive closed-loop task. Precisely, we develop an optimization-based framework that aims to steer a differentially flat system to a trajectory implicitly defined via a constrained \textittime-varying optimization problem. To that end, we generalize the notion of feedback linearization, which makes non-linear systems behave as linear systems, and develop controllers that effectively transform a differentially flat system into an optimization algorithm that seeks to find the optimal solution of a (possibly time-varying) optimization problem. Under sufficient regularity assumptions, we prove global asymptotic convergence for the optimization dynamics to the minimizer of the time-varying optimization problem. We illustrate the effectiveness of our method with two numerical examples: a multi-robot tracking problem and a robot obstacle avoidance problem.
@unpublished{zsm2023a-preprint, abstract = {Motion planning and control are two core components of the robotic systems autonomy stack. The standard approach to combine these methodologies comprises an offline/open-loop stage, \textitplanning, that designs a feasible and safe trajectory to follow, and an online/closed-loop stage, \textittracking, that corrects for unmodeled dynamics and disturbances. Such an approach generally introduces conservativeness into the planning stage, which becomes difficult to overcome as the model complexity increases and real-time decisions need to be made in a changing environment. This work addresses these challenges for the class of differentially flat nonlinear systems by integrating planning and control into a cohesive closed-loop task. Precisely, we develop an optimization-based framework that aims to steer a differentially flat system to a trajectory implicitly defined via a constrained \textittime-varying optimization problem. To that end, we generalize the notion of feedback linearization, which makes non-linear systems behave as linear systems, and develop controllers that effectively transform a differentially flat system into an optimization algorithm that seeks to find the optimal solution of a (possibly time-varying) optimization problem. Under sufficient regularity assumptions, we prove global asymptotic convergence for the optimization dynamics to the minimizer of the time-varying optimization problem. We illustrate the effectiveness of our method with two numerical examples: a multi-robot tracking problem and a robot obstacle avoidance problem.}, author = {Zheng, Tianqi and Simpson-Porco, John W. and Mallada, Enrique}, grants = {CAREER-1752362,CPS-2136324,EPICS-2330450}, month = {10}, pages = {1-14}, title = {Closed-Loop Motion Planning for Differentially Flat Systems: A Time-Varying Optimization Framework}, url = {https://mallada.ece.jhu.edu/pubs/2023-Preprint-ZSM.pdf}, year = {2023, submitted} } - H. Min, S. Tarmoun, R. Vidal, and E. Mallada, Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks, 2023, submitted.
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Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.
@unpublished{mtvm2023a-preprint, abstract = {Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution. }, author = {Min, Hancheng and Tarmoun, Salma and Vidal, Rene and Mallada, Enrique}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, month = {02}, title = {Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks}, url = {https://mallada.ece.jhu.edu/pubs/2022-Preprint-MTVM.pdf}, year = {2023, submitted} }
Recent Journals
- H. Min, R. Pates, and E. Mallada, “A Frequency Domain Analysis of Slow Coherency in Networked Systems,” Automatica, pp. 1-15, 2024.
[BibTeX] [Abstract] [Download PDF]
Network coherence generally refers to the emergence of simple aggregated dynamical behaviors, despite heterogeneity in the dynamics of the network’s subsystems. In this paper, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system’s input-output response. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network’s effective algebraic connectivity grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.
@article{mpm2024automatica, abstract = {Network coherence generally refers to the emergence of simple aggregated dynamical behaviors, despite heterogeneity in the dynamics of the network's subsystems. In this paper, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system's input-output response. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network's effective algebraic connectivity grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.}, author = {Min, Hancheng and Pates, Richard and Mallada, Enrique}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, journal = {Automatica}, month = {12}, pages = {1-15}, pubstate = {available online}, record = {accepted Oct 2024, revised Feb 2024, submitted Feb 2022}, title = {A Frequency Domain Analysis of Slow Coherency in Networked Systems}, url = {https://mallada.ece.jhu.edu/pubs/2024-Automatica-MPM.pdf}, year = {2024} } - P. You, Y. Jiang, E. Yeung, D. Gayme, and E. Mallada, “On the Stability, Economic Efficiency and Incentive Compatibility of Electricity Market Dynamics,” IEEE Transactions on Automatic Control, pp. 1-16, 2024.
[BibTeX] [Abstract] [Download PDF]
This paper focuses on the operation of an electricity market that accounts for participants that bid at a sub-minute timescale. To that end, we model the market-clearing process as a dynamical system, called market dynamics, which is temporally coupled with the grid frequency dynamics and is thus required to guarantee system-wide stability while meeting the system operational constraints. We characterize participants as price-takers who rationally update their bids to maximize their utility in response to real-time schedules of prices and dispatch. For two common bidding mechanisms, based on quantity and price, we identify a notion of alignment between participants’ behavior and planners’ goals that leads to a saddle-based design of the market that guarantees convergence to a point meeting all operational constraints. We further explore cases where this alignment property does not hold and observe that misaligned participants’ bidding can destabilize the closed-loop system. We thus design a regularized version of the market dynamics that recovers all the desirable stability and steady-state performance guarantees. Numerical tests validate our results on the IEEE 39-bus system.
@article{yjygm2024tac, abstract = {This paper focuses on the operation of an electricity market that accounts for participants that bid at a sub-minute timescale. To that end, we model the market-clearing process as a dynamical system, called market dynamics, which is temporally coupled with the grid frequency dynamics and is thus required to guarantee system-wide stability while meeting the system operational constraints. We characterize participants as price-takers who rationally update their bids to maximize their utility in response to real-time schedules of prices and dispatch. For two common bidding mechanisms, based on quantity and price, we identify a notion of alignment between participants' behavior and planners' goals that leads to a saddle-based design of the market that guarantees convergence to a point meeting all operational constraints. We further explore cases where this alignment property does not hold and observe that misaligned participants' bidding can destabilize the closed-loop system. We thus design a regularized version of the market dynamics that recovers all the desirable stability and steady-state performance guarantees. Numerical tests validate our results on the IEEE 39-bus system.}, author = {You, Pengcheng and Jiang, Yan and Yeung, Enoch and Gayme, Dennice and Mallada, Enrique}, grants = {CAREER-1752362, CPS-2136324, Global Centers-2330450}, journal = {IEEE Transactions on Automatic Control}, month = {12}, pages = {1-16}, pubstate = {available online}, record = {accepted Aug 2024, revised Dec 2023, submitted Dec 2021}, title = {On the Stability, Economic Efficiency and Incentive Compatibility of Electricity Market Dynamics}, url = {https://mallada.ece.jhu.edu/pubs/2024-TAC-YJYGM.pdf}, year = {2024} } - T. Zheng, N. Loizou, P. You, and E. Mallada, “Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization,” IEEE Control Systems Letters (L-CSS), vol. 8, pp. 2009-2014, 2024. doi:10.1109/LCSYS.2024.3413004
[BibTeX] [Abstract] [Download PDF]
Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA’s effectiveness in solving saddle point problems.
@article{zlym2024lcss, abstract = {Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA's effectiveness in solving saddle point problems.}, author = {Zheng, Tianqi and Loizou, Nicolas and You, Pengcheng and Mallada, Enrique}, doi = {10.1109/LCSYS.2024.3413004}, grants = {CPS-2136324, Global-Centers-2330450}, journal = {IEEE Control Systems Letters (L-CSS)}, month = {06}, pages = {2009-2014}, record = {published, accepted May 2024, submitted Mar 2024}, title = {Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization}, url = {https://mallada.ece.jhu.edu/pubs/2024-LCSS-ZLYM.pdf}, volume = {8}, year = {2024} } - R. K. Bansal, Y. Chen, P. You, and E. Mallada, “Market Power Mitigation in Two-stage Electricity Market with Supply Function and Quantity Bidding,” IEEE Transactions on Energy Markets, Policy and Regulation, vol. 1, iss. 4, pp. 512-522, 2023. doi:10.1109/TEMPR.2023.3318149
[BibTeX] [Abstract] [Download PDF]
The main goal of a sequential two-stage electricity market—e.g., day-ahead and real-time markets—is to operate efficiently. However, the price difference across stages due to inadequate competition and unforeseen circumstances leads to undesirable price manipulation. To mitigate this, some Inde- pendent System Operators (ISOs) proposed system-level market power mitigation (MPM) policies in addition to existing local policies. These policies aim to substitute noncompetitive bids with a default bid based on estimated generator costs. However, these policies may lead to unintended consequences when implemented without accounting for the conflicting interest of participants. In this paper, we model the competition between generators (bidding supply functions) and loads (bidding quantity) in a two-stage market with a stage-wise MPM policy. An equilibrium analysis shows that a real-time MPM policy leads to equilibrium loss, meaning no stable market outcome (Nash equilibrium) exists. A day-ahead MPM policy, besides, leads to a Stackelberg-Nash game with loads acting as leaders and generators as followers. In this setting, loads become winners, i.e., their aggregate payment is always less than competitive payments. Moreover, comparison with standard market equilibrium highlights that markets are better off without such policies. Finally, numerical studies highlight the impact of heterogeneity and load size on market equilibrium.
@article{bcym2023tempr, abstract = {The main goal of a sequential two-stage electricity market---e.g., day-ahead and real-time markets---is to operate efficiently. However, the price difference across stages due to inadequate competition and unforeseen circumstances leads to undesirable price manipulation. To mitigate this, some Inde- pendent System Operators (ISOs) proposed system-level market power mitigation (MPM) policies in addition to existing local policies. These policies aim to substitute noncompetitive bids with a default bid based on estimated generator costs. However, these policies may lead to unintended consequences when implemented without accounting for the conflicting interest of participants. In this paper, we model the competition between generators (bidding supply functions) and loads (bidding quantity) in a two-stage market with a stage-wise MPM policy. An equilibrium analysis shows that a real-time MPM policy leads to equilibrium loss, meaning no stable market outcome (Nash equilibrium) exists. A day-ahead MPM policy, besides, leads to a Stackelberg-Nash game with loads acting as leaders and generators as followers. In this setting, loads become winners, i.e., their aggregate payment is always less than competitive payments. Moreover, comparison with standard market equilibrium highlights that markets are better off without such policies. Finally, numerical studies highlight the impact of heterogeneity and load size on market equilibrium.}, author = {Bansal, Rajni Kant and Chen, Yue and You, Pengcheng and Mallada, Enrique}, doi = {10.1109/TEMPR.2023.3318149}, grants = {CAREER-1752362, CPS-2136324, EPICS-2330450}, journal = {IEEE Transactions on Energy Markets, Policy and Regulation}, month = {12}, number = {4}, pages = {512-522}, record = {published, online Sep 2023, revised July 2023, under revision May 2023, submitted Jan 2023}, title = {Market Power Mitigation in Two-stage Electricity Market with Supply Function and Quantity Bidding}, url = {https://mallada.ece.jhu.edu/pubs/2023-TEMPR-BCYM.pdf}, volume = {1}, year = {2023} } - A. Castellano, H. Min, J. Bazerque, and E. Mallada, “Learning to Act Safely with Limited Exposure and Almost Sure Certainty,” IEEE Transactions on Automatic Control, vol. 68, iss. 5, pp. 2979-2994, 2023. doi:10.1109/TAC.2023.3240925
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This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and seek to study the intrinsic trade-offs of learning safety in the presence of uncertainty. Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. We show that the (action) value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can further speed up the learning process.
@article{cmbm2023tac, abstract = {This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and seek to study the intrinsic trade-offs of learning safety in the presence of uncertainty. Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. We show that the (action) value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can further speed up the learning process.}, author = {Castellano, Agustin and Min, Hancheng and Bazerque, Juan and Mallada, Enrique}, doi = {10.1109/TAC.2023.3240925}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, journal = {IEEE Transactions on Automatic Control}, month = {5}, number = {5}, pages = {2979-2994}, record = {published, online May 2023, accepted Jan 2023, revised Oct 2022, submitted May 2021}, title = {Learning to Act Safely with Limited Exposure and Almost Sure Certainty}, url = {https://mallada.ece.jhu.edu/pubs/2023-TAC-CMBM.pdf}, volume = {68}, year = {2023} } - B. K. Poolla, Y. Lin, A. Bernstein, E. Mallada, and D. Groß, “Frequency shaping control for weakly-coupled grid-forming IBRs,” IEEE Control Systems Letters (L-CSS), pp. 937-942, 2022. doi:10.1109/LCSYS.2022.3228855
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We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.
@article{plbmg2023lcss, abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.}, author = {Poolla, Bala Kameshwar and Lin, Yashen and Bernstein, Andrey and Mallada, Enrique and Groß, Dominic}, doi = {10.1109/LCSYS.2022.3228855}, grants = {CAREER-1752362, CPS-2136324}, journal = {IEEE Control Systems Letters (L-CSS)}, month = {12}, pages = {937-942}, record = {published, online Dec 2022, accepted Nov 2022, submitted Sep 2022.}, title = {Frequency shaping control for weakly-coupled grid-forming IBRs}, url = {https://mallada.ece.jhu.edu/pubs/2022-LCSS-PLBMG.pdf}, year = {2022} } - G. H. Oral, E. Mallada, and D. Gayme, “On the Role of Interconnection Directionality in the Quadratic Performance of Double-Integrator Networks,” IEEE Transactions on Automatic Control, vol. 67, iss. 11, pp. 6211-6218, 2022. doi:10.1109/TAC.2021.3135358
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This paper provides a framework to evaluate the performance of single and double integrator networks over arbitrary directed graphs. Adopting vehicular network terminology, we consider quadratic performance metrics defined by the L2-norm of position and velocity based response functions given impulsive inputs to each vehicle. We exploit the spectral properties of weighted graph Laplacians and output performance matrices to derive a novel method of computing the closed-form solutions for this general class of performance metrics, which include H2-norm based quantities as special cases. We then explore the effect of the interplay between network properties (such as edge directionality and connectivity) and the control strategy on the overall network performance. More precisely, for systems whose interconnection is described by graphs with normal Laplacian L, we characterize the role of directionality by comparing their performance with that of their undirected counterparts, represented by the Hermitian part of L. We show that, for single-integrator networks, directed and undirected graphs perform identically. However, for double-integrator networks, graph directionality -expressed by the eigenvalues of L with nonzero imaginary part- can significantly degrade performance. Interestingly, in many cases, well-designed feedback can also exploit directionality to mitigate degradation or even improve the performance to exceed that of the undirected case. Finally we focus on a system coherence metric -aggregate deviation from the state average- to investigate the relationship between performance and degree of connectivity, leading to somewhat surprising findings. For example, increasing the number of neighbors on a ω-nearest neighbor directed graph does not necessarily improve performance. Similarly, we demonstrate equivalence in performance between all-to-one and all-to-all communication graphs.
@article{omg2022tac, abstract = {This paper provides a framework to evaluate the performance of single and double integrator networks over arbitrary directed graphs. Adopting vehicular network terminology, we consider quadratic performance metrics defined by the L2-norm of position and velocity based response functions given impulsive inputs to each vehicle. We exploit the spectral properties of weighted graph Laplacians and output performance matrices to derive a novel method of computing the closed-form solutions for this general class of performance metrics, which include H2-norm based quantities as special cases. We then explore the effect of the interplay between network properties (such as edge directionality and connectivity) and the control strategy on the overall network performance. More precisely, for systems whose interconnection is described by graphs with normal Laplacian L, we characterize the role of directionality by comparing their performance with that of their undirected counterparts, represented by the Hermitian part of L. We show that, for single-integrator networks, directed and undirected graphs perform identically. However, for double-integrator networks, graph directionality -expressed by the eigenvalues of L with nonzero imaginary part- can significantly degrade performance. Interestingly, in many cases, well-designed feedback can also exploit directionality to mitigate degradation or even improve the performance to exceed that of the undirected case. Finally we focus on a system coherence metric -aggregate deviation from the state average- to investigate the relationship between performance and degree of connectivity, leading to somewhat surprising findings. For example, increasing the number of neighbors on a ω-nearest neighbor directed graph does not necessarily improve performance. Similarly, we demonstrate equivalence in performance between all-to-one and all-to-all communication graphs.}, author = {Oral, H. Giray and Mallada, Enrique and Gayme, Dennice}, doi = {10.1109/TAC.2021.3135358}, grants = {ENERGISE-DE-EE0008006, EPCN-1711188,AMPS-1736448, CPS-1544771, CAREER-1752362, AMPS-1736448, ARO-W911NF-17-1-0092}, journal = {IEEE Transactions on Automatic Control}, month = {11}, number = {11}, pages = {6211-6218}, record = {published, online Dec. 2021, accepted Nov. 2021, conditionally accepted Apr 2021, revised Nov. 2020, submitted Nov. 2019}, title = {On the Role of Interconnection Directionality in the Quadratic Performance of Double-Integrator Networks}, url = {https://mallada.ece.jhu.edu/pubs/2019-Preprint-OMG.pdf}, volume = {67}, year = {2022} } - R. K. Bansal, P. You, D. F. Gayme, and E. Mallada, “A Market Mechanism for Truthful Bidding with Energy Storage,” Electric Power Systems Research, vol. 211, iss. 108284, pp. 1-7, 2022. doi:https://doi.org/10.1016/j.epsr.2022.108284
[BibTeX] [Abstract] [Download PDF]
This paper proposes a market mechanism for multiinterval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energycycling functions that map prices to cycle depths. The resulting market-clearing process–implemented via convex programming–yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumerbased approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful cost, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.
@article{bygm2022epsr, abstract = {This paper proposes a market mechanism for multiinterval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energycycling functions that map prices to cycle depths. The resulting market-clearing process--implemented via convex programming--yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. We illustrate the benefits of our solution by comparing the competitive equilibrium of the resulting mechanism to that of an alternative solution that uses prosumer-based bids. Our solution shows several advantages over the prosumerbased approach. It does not require a priori price estimation. It also incentivizes participants to reveal their truthful cost, thus leading to an efficient, competitive equilibrium. Numerical experiments using New York Independent System Operator (NYISO) data validate our findings.}, author = {Bansal, Rajni Kant and You, Pengcheng and Gayme, Dennice F. and Mallada, Enrique}, doi = {https://doi.org/10.1016/j.epsr.2022.108284}, grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324}, issn = {0378-7796}, journal = {Electric Power Systems Research}, month = {10}, note = {also in PSCC 2022}, number = {108284}, pages = {1-7}, record = {published, accepted Feb. 2022, submitted Oct. 2021}, title = {A Market Mechanism for Truthful Bidding with Energy Storage}, url = {https://mallada.ece.jhu.edu/pubs/2022-EPSR-BYGM.pdf}, volume = {211}, year = {2022} }
Recent Conference Papers
- K. Poe, E. Mallada, and R. Vidal, “Invertibility of Discrete-Time Linear Systems with Sparse Inputs,” in 63rd IEEE Conference on Decision and Control (CDC), 2024.
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One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics, and spectral structure of a system have been used to characterize the well-posedness of this problem, without assumptions on the inputs. However, certain structural assumptions, such as input sparsity, have been shown to translate to practical gains in the performance of inversion algorithms, surpassing classical guarantees. Establishing necessary and sufficient conditions for left invertibility of systems with sparse inputs is therefore a crucial step toward understanding the performance limits of system inversion under structured input assumptions. In this work, we provide the first necessary and sufficient characterizations of left invertibility for linear systems with sparse inputs, echoing classic characterizations for standard linear systems. The key insight in deriving these results is in establishing the existence of two novel geometric invariants unique to the sparse-input setting, the weakly unobservable and strongly reachable subspace arrangements. By means of a concrete example, we demonstrate the utility of these characterizations. We conclude by discussing extensions and applications of this framework to several related problems in sparse control.
@inproceedings{pmv2024cdc, abstract = {One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics, and spectral structure of a system have been used to characterize the well-posedness of this problem, without assumptions on the inputs. However, certain structural assumptions, such as input sparsity, have been shown to translate to practical gains in the performance of inversion algorithms, surpassing classical guarantees. Establishing necessary and sufficient conditions for left invertibility of systems with sparse inputs is therefore a crucial step toward understanding the performance limits of system inversion under structured input assumptions. In this work, we provide the first necessary and sufficient characterizations of left invertibility for linear systems with sparse inputs, echoing classic characterizations for standard linear systems. The key insight in deriving these results is in establishing the existence of two novel geometric invariants unique to the sparse-input setting, the weakly unobservable and strongly reachable subspace arrangements. By means of a concrete example, we demonstrate the utility of these characterizations. We conclude by discussing extensions and applications of this framework to several related problems in sparse control.}, author = {Poe, Kyle and Mallada, Enrique and Vidal, Rene}, booktitle = {63rd IEEE Conference on Decision and Control (CDC)}, grants = {CPS-2136324, Global-Centers-2330450}, month = {12}, pubstate = {presented}, record = {accepted Jul 2024, submitted Mar 2024}, title = {Invertibility of Discrete-Time Linear Systems with Sparse Inputs}, url = {https://mallada.ece.jhu.edu/pubs/2024-CDC-PMV.pdf}, year = {2024} } - T. Zheng, N. Loizou, P. You, and E. Mallada, “Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization,” in 63rd IEEE Conference on Decision and Control (CDC), 2024.
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Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA’s effectiveness in solving saddle point problems.
@inproceedings{zlym2024cdc, abstract = {Gradient Descent Ascent (GDA) methods for min-max optimization problems typically produce oscillatory behavior that can lead to instability, e.g., in bilinear settings. To address this problem, we introduce a dissipation term into the GDA updates to dampen these oscillations. The proposed Dissipative GDA (DGDA) method can be seen as performing standard GDA on a state-augmented and regularized saddle function that does not strictly introduce additional convexity/concavity. We theoretically show the linear convergence of DGDA in the bilinear and strongly convex-strongly concave settings and assess its performance by comparing DGDA with other methods such as GDA, Extra-Gradient (EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these methods, achieving superior convergence rates. We support our claims with two numerical examples that showcase DGDA's effectiveness in solving saddle point problems.}, author = {Zheng, Tianqi and Loizou, Nicolas and You, Pengcheng and Mallada, Enrique}, booktitle = {63rd IEEE Conference on Decision and Control (CDC)}, grants = {CPS-2136324, Global-Centers-2330450}, month = {12}, note = {also in L-CSS}, pubstate = {presented}, record = {accepted Jul 2024, submitted Mar 2024}, title = {Dissipative Gradient Descent Ascent Method: A Control Theory Inspired Algorithm for Min-max Optimization}, url = {https://mallada.ece.jhu.edu/pubs/2024-CDC-ZLYM.pdf}, year = {2024} } - Y. Shen, H. Sibai, and E. Mallada, “Generalized Barrier Functions: Integral Conditions & Recurrent Relaxations,” in 60th Allerton Conference on Communication, Control, and Computing, 2024, pp. 1-8.
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Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function $h$ that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set $h_\ge 0$. This methodology, however, does not prescribe a general method for finding the function $h$ that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for $h$ easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of $h$. Subsequently, we further replace the stringent invariance requirement on $h≥0$ with a more flexible concept known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ($τ$-)recurrence of the super-level set $h_≥ 0$ is sufficient to guarantee the system’s safety.
@inproceedings{ssm2024allerton, abstract = {Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function $h$ that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set $h_\ge 0$. This methodology, however, does not prescribe a general method for finding the function $h$ that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for $h$ easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of $h$. Subsequently, we further replace the stringent invariance requirement on $h≥0$ with a more flexible concept known as recurrence. A set is ($τ$-)recurrent if every trajectory that starts in the set returns to it (within $τ$ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the ($τ$-)recurrence of the super-level set $h_≥ 0$ is sufficient to guarantee the system's safety.}, author = {Shen, Yue and Sibai, Hussein and Mallada, Enrique}, booktitle = {60th Allerton Conference on Communication, Control, and Computing}, grants = {CPS-2136324, Global-Centers-2330450}, keywords = {Barrier Functions}, month = {09}, pages = {1-8}, pubstate = {presented}, record = {accepted Jul 2024, submitted Jul 2024}, title = {Generalized Barrier Functions: Integral Conditions & Recurrent Relaxations}, url = {https://mallada.ece.jhu.edu/pubs/2024-Allerton-SSM.pdf}, year = {2024} } - Y. Liu, E. Mallada, Z. Li, and P. You, “Accelerated Saddle Flow Dynamics for Bilinearly Coupled Minimax Problems,” in 63rd IEEE Conference on Decision and Control (CDC), 2024.
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Minimax problems have attracted much attention due to various applications in constrained optimization problems and zero-sum games. Identifying saddle points within these problems is crucial, and saddle flow dynamics offer a straightforward yet useful approach. This study focuses on a class of bilinearly coupled minimax problems and designs an accelerated algorithm based on saddle flow dynamics that achieves a convergence rate beyond stereotype limits. The algorithm is derived based on a sequential two-step transformation of the objective function. First, a change of variable is aimed at a better-conditioned saddle function. Second, a proximal regularization, when staggered with the first step, guarantees strong convexity-strong concavity of the objective function that can be tuned for accelerated exponential convergence. Besides, such an approach can be extended to a class of weakly convex-weakly concave functions and still achieve exponential convergence to one stationary point. The theory is verified by a numerical test on an affine equality-constrained convex optimization problem.
@inproceedings{lmly2024cdc, abstract = {Minimax problems have attracted much attention due to various applications in constrained optimization problems and zero-sum games. Identifying saddle points within these problems is crucial, and saddle flow dynamics offer a straightforward yet useful approach. This study focuses on a class of bilinearly coupled minimax problems and designs an accelerated algorithm based on saddle flow dynamics that achieves a convergence rate beyond stereotype limits. The algorithm is derived based on a sequential two-step transformation of the objective function. First, a change of variable is aimed at a better-conditioned saddle function. Second, a proximal regularization, when staggered with the first step, guarantees strong convexity-strong concavity of the objective function that can be tuned for accelerated exponential convergence. Besides, such an approach can be extended to a class of weakly convex-weakly concave functions and still achieve exponential convergence to one stationary point. The theory is verified by a numerical test on an affine equality-constrained convex optimization problem.}, author = {Liu, Yingzhu and Mallada, Enrique and Li, Zhongkui and You, Pengcheng}, booktitle = {63rd IEEE Conference on Decision and Control (CDC)}, grants = {CPS-2136324, Global-Centers-2330450}, month = {07}, pubstate = {accepted, submitted Mar 2024}, title = {Accelerated Saddle Flow Dynamics for Bilinearly Coupled Minimax Problems}, url = {https://mallada.ece.jhu.edu/pubs/2024-Preprint-LMLY.pdf}, year = {2024} } - B. K. Poolla, Y. Lin, A. Bernstein, E. Mallada, and D. Groß, “Dynamic Shaping of Grid Response of Multi-Machine Multi-Inverter Systems Through Grid-Forming IBRs,” in PES General Meeting, 2024, pp. 1-5. doi:10.1109/PESGM51994.2024.10688717
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We consider the problem of controlling the frequency response of weakly-coupled multi-machine multi-inverter low-inertia power systems via grid-forming inverter-based resources (IBRs). In contrast to existing methods, our approach relies on dividing the larger system into multiple strongly-coupled subsystems, without ignoring either the underlying network or approximating the subsystem response as an aggregate harmonic mean model. Rather, through a structured clustering and recursive dynamic shaping approach, the frequency response of the overall system to load perturbations is shaped appropriately. We demonstrate the proposed approach for a three-node triangular configuration and a small-scale radial network. Furthermore, previous synchronization analysis for heterogeneous systems requires the machines to satisfy certain proportionality property. In our approach, the effective transfer functions for each cluster can be tuned by the IBRs to satisfy such property, enabling us to apply the shaping control to systems with a wider range of heterogeneous machines.
@inproceedings{plbmg2024pesgm, abstract = {We consider the problem of controlling the frequency response of weakly-coupled multi-machine multi-inverter low-inertia power systems via grid-forming inverter-based resources (IBRs). In contrast to existing methods, our approach relies on dividing the larger system into multiple strongly-coupled subsystems, without ignoring either the underlying network or approximating the subsystem response as an aggregate harmonic mean model. Rather, through a structured clustering and recursive dynamic shaping approach, the frequency response of the overall system to load perturbations is shaped appropriately. We demonstrate the proposed approach for a three-node triangular configuration and a small-scale radial network. Furthermore, previous synchronization analysis for heterogeneous systems requires the machines to satisfy certain proportionality property. In our approach, the effective transfer functions for each cluster can be tuned by the IBRs to satisfy such property, enabling us to apply the shaping control to systems with a wider range of heterogeneous machines.}, author = {Poolla, Bala Kameshwar and Lin, Yashen and Bernstein, Andrey and Mallada, Enrique and Groß, Dominic}, booktitle = {PES General Meeting}, doi = {10.1109/PESGM51994.2024.10688717}, grants = {CPS-2136324, CAREER-1752362, Global Centers-2330450}, month = {06}, pages = {1-5}, record = {presented Jun. 2024, accepted Mar. 2024, submitted Nov. 2023}, title = {Dynamic Shaping of Grid Response of Multi-Machine Multi-Inverter Systems Through Grid-Forming IBRs}, url = {https://mallada.ece.jhu.edu/pubs/2024-PESGM-PLBMG.pdf}, year = {2024} } - Z. Siahaan, E. Mallada, and S. Geng, “Decentralized Stability Criteria for Grid-Forming Control in Inverter-Based Power Systems,” in PES General Meeting, 2024, pp. 1-5. doi:10.1109/PESGM51994.2024.10689037
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This paper presents a decentralized stability analysis of power systems comprising grid-forming (GFM) inverters. We leverage a decentralized stability framework capable of ensuring the stability of the entire interconnection through individual assessments at each bus. The key novelty lies in incorporating voltage dynamics and their coupling with reactive power, in addition to the angle dynamics and their coupling with active power. We perform loop transformation to address the challenge posed by the non-Laplacian nature of the network Jacobian matrix in this case. This methodology is applied to characterize conditions on the droop gains of GFM controllers that can preserve system-wide stability. Our proposed stability criteria exhibit scalability and robustness, and can be extended to accommodate delays, variations in network conditions, and plug-and-play of new components in the network.
@inproceedings{smg2024pesgm, abstract = {This paper presents a decentralized stability analysis of power systems comprising grid-forming (GFM) inverters. We leverage a decentralized stability framework capable of ensuring the stability of the entire interconnection through individual assessments at each bus. The key novelty lies in incorporating voltage dynamics and their coupling with reactive power, in addition to the angle dynamics and their coupling with active power. We perform loop transformation to address the challenge posed by the non-Laplacian nature of the network Jacobian matrix in this case. This methodology is applied to characterize conditions on the droop gains of GFM controllers that can preserve system-wide stability. Our proposed stability criteria exhibit scalability and robustness, and can be extended to accommodate delays, variations in network conditions, and plug-and-play of new components in the network.}, author = {Siahaan, Zudika and Mallada, Enrique and Geng, Sijia}, booktitle = {PES General Meeting}, doi = {10.1109/PESGM51994.2024.10689037}, grants = {CPS-2136324, CAREER-1752362, Global Centers-2330450}, month = {06}, pages = {1-5}, record = {presented Jun. 2024, accepted Mar. 2024, submitted Nov. 2023}, title = {Decentralized Stability Criteria for Grid-Forming Control in Inverter-Based Power Systems}, url = {https://mallada.ece.jhu.edu/pubs/2024-PESGM-SMG.pdf}, year = {2024} } - H. Min, R. Vidal, and E. Mallada, “Early Neuron Alignment in Two-layer ReLU Networks with Small Initialization,” in International Conference on Representation Learning (ICLR), 2024.
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This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons’ directional dynamics allows us to provide an $\mathcalO(\fracłog n\sqrtμ)$ upper bound on the time it takes for all neurons to achieve good alignment with the input data, where $n$ is the number of data points and $μ$ measures how well the data are separated. After the early alignment phase, the loss converges to zero at a $\mathcalO(\frac1t)$ rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.
@inproceedings{mvm2024iclr, abstract = {This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an $\mathcalO(\fracłog n\sqrtμ)$ upper bound on the time it takes for all neurons to achieve good alignment with the input data, where $n$ is the number of data points and $μ$ measures how well the data are separated. After the early alignment phase, the loss converges to zero at a $\mathcalO(\frac1t)$ rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.}, author = {Min, Hancheng and Vidal, Rene and Mallada, Enrique}, booktitle = {International Conference on Representation Learning (ICLR)}, grants = {CAREER-1752362}, month = {05}, record = {published, accepted Jan 2024, submitted Sep 2023}, title = {Early Neuron Alignment in Two-layer ReLU Networks with Small Initialization}, url = {https://mallada.ece.jhu.edu/pubs/2024-ICLR-MVM.pdf}, year = {2024} } - H. Sibai and E. Mallada, “Recurrence of Nonlinear Control Systems: Entropy and Bit Rates,” in Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control (HSCC), New York, NY, USA, 2024, pp. 1-9. doi:https://doi.org/10.1145/3641513.3650121
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In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.
@inproceedings{sm2024hscc, abstract = {In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.}, address = {New York, NY, USA}, author = {Sibai, Hussein and Mallada, Enrique}, booktitle = {Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control (HSCC)}, doi = {https://doi.org/10.1145/3641513.3650121}, grants = {CPS-2136324, Global-Centers-2330450}, month = {05}, number = {23}, pages = {1--9}, publisher = {Association for Computing Machinery}, record = {accepted Jan 2024, submitted Nov 2023}, series = {HSCC '24}, title = {Recurrence of Nonlinear Control Systems: Entropy and Bit Rates}, url = {https://mallada.ece.jhu.edu/pubs/2024-HSCC-SM.pdf}, year = {2024} }