**NEW:** RESEARCH POSITIONS AVAILABLE

"ATTENTION: This website contains material supported by the National Science Foundation under Grants No. 1100376, 1151018, 1301851, 1301660, 1538374."

My research goal is to develop tools for the analysis and control of nonlinear ordinary equations and also some kinds of partial differential equations. I am interested in applications in communications networks with nonlinear and decentralized dynamics, sparse interconnection and delayed feedback. I am currently working on the use of semidefinite programming for stability analysis of Chronic Myelogenous Leukemia. I have developed some matlab toolboxes which can be used for analysis of differential equations with delays. For a detailed description of my research, please refer to my cv or extended research statement.

My Thesis:

Stanford University, Department of Aeronautics and Astronautics

Stability and Control of Functional Differential Equations

Defended March 15, 2006.

Thesis - [arXiv:math/0607144v1] [.pdf] [.ps]

Defense Talk - [.pdf] [.ps]

Student Theses:

Reza Kamyar (PhD), Arizona State University

Parallel Optimization of Polynomials for Large-Scale Problems in Stability and Control

Defended January, 2016.

Thesis - [arXiv:math] [.pdf] [.ps]

Defense Talk - [.pdf] [.ps]

Aditya Gahlawat (PhD), (double PhD) University Joseph Fourier and Illinois Institute of Technology

Analysis and Control of Parabolic Partial Differential Equations with Application to Tokamaks using Sum-of-Squares Polynomials

Defended October, 2015.

Thesis - [arXiv:math] [.pdf] [.ps]

Defense Talk - [.pdf] [.ps]

Chaitanya Murti (MS), Illinois Institute of Technology

Analysis of Zeno Stability in Hybrid Systems Using Sum-of-Squares Programming

Defended December, 2012.

Thesis - [arXiv:math] [.pdf] [.ps]

Defense Talk - [.pdf] [.ps]

Journal Articles:

M. M. Peet

SOS Methods for Multi-Delay Systems: A Dual Form of Lyapunov-Krasovskii Functional

IEEE Transactions on Automatic Control.

[arXiv] [.pdf] [.ps] [related talk]

Summary:A new class of Lyapunov stability condition for systems with delay is proposed. The new class has additional structure and can be combined with feedback to create a convex approach to controller synthesis.R. Kamyar and M. M. Peet

Optimal Thermostat Programming for Peak and Time-of-Use Pricing Plans with Thermal Energy Storage and Implications for Regulated Utilities

To Appear in IEEE Transactions on Power Systems, 2017.

[arXiv] [.pdf] [.ps]

Summary:Recently, utilities such as SRP and APS have begun to charge consumers partly based on the maximum rate of electricity consumption (demand charges). Because buildings absorb and release heat during the day, these demand charges make programming even simple 4-setpoint thermostats a complicated mathematical challenge. In this paper, we solve the math problem and post the code for programming your thermostat online.A. Gahlawat and M. M. Peet

A Convex Sum-of-Squares Approach to Analysis, State Feedback and Output Feedback Control of Parabolic PDEs

IEEE Transactions on Automatic Control, Vol. 62, No. 4, April. 2017, pp. 1636-1651.

[arXiv] [.pdf] [.ps]

Summary:We show that control of alinear PDE in a single variable is convex in operator variables. We parameterize these operators and solve the resulting optimization problem using semidefinite programming. We consider stability, stabilization by point feedback, estimation by point measurements, and estimation-based feedback control.R. Kamyar and M. M. Peet

Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum-of-Squares

Discrete and Continuous Dynamical Systems - Series B. Special Issue on "Constructive and computational methods in Lyapunov and stability theory". Vol. 20, No. 9, pp 2383--2417. Oct., 2015. (Survey Paper).

[arXiv] [.pdf] [.ps]

Summary:We overview several convex approaches to optimization of polynomials, highlighting the advantages and disadvantages of each. We show how each approach can be used for stability and control of both uncertain and nonlinear systems. The overview includes Quantifier Elimination, Reformulation, Blossoming, and Groebner bases, while focusing on the methods of Polya, Bernstein and Handelman.C. Murti and M. M. Peet

Using SOS for Analysis of Zeno Stability in Hybrid systems with Nonlinearity and Uncertainy

Submitted to SIAM Journal on Control and Optimization.

[arXiv] [.pdf][.ps]

Summary:Hybrid models can get "stuck" at points which are not equilibria - leading to infinite loops and other undesireable numerical and physical phenomena. We show how to use SOS and the Positivstellensatz to design algorithms to verify that a model doesn't contain any such "Zeno Equilibria".R. Kamyar, M. M. Peet and Y. Peet

Solving Large-Scale Robust Control Problems by Exploiting the Parallel Structure of Polya's Theorem

IEEE Transactions on Automatic Control, Vol. 58, No. 8, Aug. 2013, pp. 1931-1947.

[arXiv] [.pdf] [.ps]

Summary:We exploit the structure of Poya's test for polynomial positivity to create an efficient parallel-computing platform for optimization of polynomials. We apply this to large-scale robust stability problems with hundreds of states and several uncertain parameters. We demostrate the algorithm on a discretized model of plasma in a tokamak.A. Seuret and M. M. Peet

Stability Analysis of Sampled-Data Systems using Sum-of-Squares

IEEE Transactions on Automatic Control, Vol 58, No. 6, June 2013. pp. 1620-1625.

[arXiv] [.pdf] [.ps]

Summary:Sampled-Data Systems are modeled as continuous-time plants coupled with a controller which updates at discrete, unpredictable times. We develop Lyapunov theory for these mixed discrete-continuous dynamics and use Sum-of-Squares to construct proofs of stability and exponential convergence.M. M. Peet and A. Papachristodoulou

A Converse Sum-of-Squares Lyapunov Result with a Degree Bound

IEEE Transactions on Automatic Control, Vol 57, No. 9, Sept. 2012. pp. 2281-2293.

[arXiv.1201.2619v1] [.pdf] [.ps]

Summary:We introduce a form of converse Lyapunov function and use it to show that exponential stability of a nonlinear system implies the existence of a Lyapunov function which is SOS. We give a bound on the degree of the converse Lyapunov function.Y. Zhang, M. M. Peet and K. Gu

Reducing the Complexity of the Sum-of-Squares Test for Stability of Delayed Linear Systems

IEEE Transactions on Automatic Control, Vol 56, No. 1, Jan. 2011

[arXiv] [.pdf] [.ps]

Summary:We use recently developed converse Lyapunov theory to reduce the complexity of the SOS test by several orders of magnitude for systems with few delays.M. M. Peet and P.-A. Bliman

On the Conservatism of the Sum-of-Squares Method for Analysis of Time-Delayed Systems

Automatica, Vol. 47, No. 11, Nov. 2011

[arXiv] [.pdf] [.ps]

Summary:A converse Lyapunov result showing the existence of polynomial Lyapunov-Krasovskii functionals for stability of linear time-delay systems. Also a proof that the Weierstrass approximation theorem holds on linear varieties of continuous functions.M. M. Peet

Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions

IEEE Transactions on Automatic Control, Vol 52, No. 5, May 2009

[arXiv:0707.0218v1] [.pdf] [.ps]

Summary:A proof that one can use only polynomial Lyapunov functions to prove exponential stability of ordinary differential equations with no additional conservatism.M. M. Peet, A. Papachristodoulou and S. Lall

Positive Forms and Stability of Linear Time-Delay Systems

SIAM Journal on Control and Optimization, Vol 47, No. 6, 2009

[arXiv:0707.0230v1] [.pdf] [.ps]

Summary:A framework for using semidefinite programming to construct Lyapunov functions for infinite-dimensional systems (i.e. delay-differential and partial differential equations).A. Papachristodoulou, M. M. Peet and S. Lall

Analysis of Polynomial Systems with Time Delays via the Sum of Squares Decomposition

IEEE Transactions on Automatic Control, Vol 52, No. 5, May 2009

[arXiv] [.pdf] [.ps]

Summary:An overview of how the methods from `` Positive Forms and Stability of Linear Time-Delay Systems'' can be applied to nonlinear time-delay systems.M. M. Peet, P. S. Kim, S.-I. Niculescu and D. Levy

New Computational Tools for Modeling Chronic Myelogenous Leukemia

Mathematical Modelling of Natural Phenomena, Vol 4, No. 2, January, 2009

[arXiv] [.pdf] [.ps]

Summary:We use SOS to analyze a recently-proposed model of CML.M. M. Peet and S. Lall

Stability Analysis of a Nonlinear Model of Internet Congestion Control with Delay

IEEE Transactions on Automatic Control, Vol. 50, No. 3, March 2007

[arXiv] [.pdf] [.ps]

Technical Report with Detailed Proofs [arXiv] [.pdf] [.ps]

Summary:An exact characterization of the hybrid, nonlinear, and time-delayed model and region of stability for a popular internet congestion control protocol.

Book Chapters:

G. Maio and M. Peet and K. Gu

Inversion of Separable Kernel Operator and Its Application in Control Synthesis

in "Delays and Interconnections: Methodology, Algorithms and Applications", Springer, To Appear.

[arXiv] [.pdf] [.ps]M. M. Peet and A. Seuret

Global Stability Analysis of Nonlinear Sampled-Data Systems Using Convex Methods

in "Delay Systems: From Theory to Numerics and Applications", Springer, 2013.

[arXiv] [.pdf] [.ps]Y. Zhang, M. M. Peet and K. Gu

Accelerating Convergence of Sum-of-Squares Stability Analysis of Coupled Differential-Difference Equations

in "Time Delay Systems - Methods, Applications and New Trends'', Springer Lecture Notes in Control and Information Sciences.

[arXiv] [.pdf] [.ps]M. M. Peet, C. Bonnet, and H. Ozbay

SOS Methods for Stability Analysis of Neutral Differential Systems

in "Topics in Time Delay Systems: Analysis, Algorithms and Control'', Springer Lecture Notes in Control and Information Sciences.

Preliminary version appeared at MTNS, 2008.

[arXiv] [.pdf] [.ps]A. Papachristodoulou and M. M. Peet

SOS Methods for Nonlinear Delayed Models in Biology and Networking

in "Topics in Time Delay Systems: Analysis, Algorithms and Control', Springer Lecture Notes in Control and Information Sciences.

[arXiv] [.pdf] [.ps]

Conference Papers:

M. Peet

A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework

Submitted to the IEEE Conference on Decision and Control, 2018.

[arXiv:1803.07290] [.pdf] [.ps] [slides]B. Colbert and M. Peet

Estimating the Region of Attraction using Stable Trajectory Measurements

Submitted to the IEEE Conference on Decision and Control, 2018.

[arXiv:1609.01019] [.pdf] [.ps] [slides]A. Doroudchi and S. Shivakumar and S. Berman and M. Peet

Decentralized Control of Distributed Actuation in a Segmented Soft Robot Arm

Submitted to the IEEE Conference on Decision and Control, 2018.

[arXiv:1609.01019] [.pdf] [.ps] [slides]M. Peet and Keqin Gu

Synthesis of Full-State Observers for Time-delay Systems using SOS

Submitted to the Mathematical Theory of Networks and Systems, 2018.

[arXiv:1609.01019] [.pdf] [.ps] [slides]M. Jones and M. Peet

A Dynamic Programing Aproach to Evaluating Multivariate Gaussian Probabilities

Submitted to the Mathematical Theory of Networks and Systems, 2018.

[arXiv:1609.01019] [.pdf] [.ps] [slides]B. Colbert, H. Mohammadi and M. Peet

Combining SOS and Moment Relaxations with Branch and Bound to Extract Solutions to Global Polynomial Optimization Problems

To appear at the American Control Conference, 2018.

[arXiv:1609.01019] [.pdf] [.ps] [slides]M. Peet

A Convex Reformulation of the Controller Synthesis Problem for Infinite-Dimensional Systems using Linear Operator Inequalities (LOIs) with Application to MIMO Multi-Delay Systems

To Appear at the American Control Conference, 2018.

[arXiv] [.pdf] [.ps] [slides]E. Meyer and M. Peet

A Convex Approach for Stability Analysis of Coupled PDEs with Spatially Dependent Coefficients

To be Submitted.

[arXiv] [.pdf] [.ps] [slides]M. Jones and M. Peet

Solving Dynamic Programming with Supremum Terms in the Objective and Application to Optimal Battery Scheduling for Electricity Consumers Subject to Demand Charges

IEEE Conference on Decision and Control, 2017.

[arXiv] [.pdf] [.ps] [slides]M. Jones, H.. Mohammadi and M. Peet

Estimating the Region of Attraction Using Polynomial Optimization: a Converse Lyapunov Result

IEEE Conference on Decision and Control, 2017.

[arXiv:1704.06983] [.pdf] [.ps] [slides]G. Miao, M. Peet and K. Gu

Inversion of Separable Kernel Operators in Coupled Differential-Functional Equations and Application to Controller Synthesis

IFAC World Congress. Toulouse, France. July 9-14, 2017.

[arXiv:1703.10253]] [.pdf] [.ps] [slides]M. Peet

A Convex Reformulation of the Controller Synthesis Problem for MIMO Single-Delay Systems with Implementation in SOS

Proceedings of the American Control Conference. Seattle, WA. May 22-26, 2017..

[arXiv] [.pdf] [.ps] [slides]R. Kamyar and M. Peet

A Multi-objective Approach to Optimal Energy Storage for Residential Customers in The Presence of Demand Charges

IEEE Conference on Decision and Control. Las Vegas, NV. December 12-14, 2016.

[arXiv] [.pdf] [.ps] [slides]A. Gahlawat and M. Peet

Optimal State-Feedback Boundary Control of Parabolic PDEs Using SOS Polynomials

The American Control Conference. Boston, MA. July 6-8, 2016.

[arXiv] [.pdf] [.ps] [slides]E. Meyer and M. Peet

Stability Analysis and Control of Parabolic Linear PDEs with two Spatial Dimensions Using Lyapunov Methods and SOS

54th IEEE Conference on Decision and Control. Osaka, Japan. December 15-17, 2015.

[arXiv] [.pdf] [.ps] [slides]A. Gahlawat and M. Peet

Output Feedback Control of Inhomogeneous Parabolic PDEs with Point Actuation and Point Measurement Using SOS and Semi-Separable Kernels

54th IEEE Conference on Decision and Control. Osaka, Japan. December 15-17, 2015.

[arXiv] [.pdf] [.ps] [slides]R. Kamyar and M. Peet

The effect of Distributed Thermal Storage on Optimal Pricing and Optimal Thermostat Programming in a Regulated Smart Grid

American Control Conference. Chicago, IL. July 1-3, 2015.

[arXiv] [.pdf] [.ps] [slides]R. Kamyar and M. Peet

Constructing Piecewise Polynomial Lyapunov Functions Over Arbitrary Convex Polytopes Using Handelman's Basis

53rd IEEE Conference on Decision and Control. Los Angeles, CA. December 15-17, 2014.

[arXiv] [.pdf] [.ps] [slides]M. M. Peet

LMI Parameterization of Lyapunov Functions for Infinite-Dimensional Systems: A Toolbox

American Control Conference. Portland, OR. June 4-6, 2014.

[arXiv] [.pdf] [.ps] [slides]R. Kamyar and M. Peet

Decentralized Polya’s Algorithm for Stability Analysis of Large-scale Nonlinear Systems

52nd IEEE Conference on Decision and Control. Florence, IT. December 10-13, 2013.

[arXiv] [.pdf] [.ps] [slides]B. Li and M. M. Peet

Stability Analysis of State-Dependent Delay Systems using Sum-of-Squares

AIAA Conference on Guidance, Navigation and Control. Boston, MA. Aug. 19-22, 2013.

[arXiv] [.pdf] [.ps] [slides]C. Murti and M. M. Peet

A Sum-Of-Squares Approach to the Analysis of Zeno Stability in Polynomial Hybrid Systems

European Control Conference. Zurich, CH. July 17-19, 2013.

[arXiv] [.pdf] [.ps] [slides]M. M. Peet

Full-State Feedback of Delayed Systems using SOS: A New Theory of Duality

Proceedings of the 11th IFAC Workshop on Time-Delay Systems. February 4-6, 2013.

[arXiv] [.pdf] [.ps] [slides]R. Kamyar and M. Peet

Decentralized Computation for Robust Stability of Large-scale Systems with Parameters on the Hypercube

51st IEEE Conference on Decision and Control, Maui, HI. December 15-17, 2012. pp. 6529-6264..

[arXiv] [.pdf] [.ps]A. Gahlawat, E. Witrant, M. Peet and M. Alamir

Bootstrap Current Optimization in Tokamaks Using Sum-of-Squares Polynomials

50th IEEE Conference on Decision and Control, Orlando, FL. December 12-15, 2011. pp. 4359-4365.

[arXiv] [.pdf] [.ps]R. Kamyar and M. M. Peet

Solving Large-Scale Robust Control Problems by Exploiting the Parallel Structure of Polya's Theorem

American Control Conference, Montreal, CA. June 27-29, 2012. pp. 5948-5954.

[arXiv] [.pdf] [.ps]M. M. Peet, P. Kim and P. Lee

Biological Circuit Models of Immune Regulatory Response: A Decentralized Control System

50th IEEE Conference on Decision and Control, Orlando, FL. December 12-15, 2011. pp. 3020-3025.

[arXiv] [.pdf] [.ps] [talk] [slides]A. Gahlawat and M. M. Peet

Designing Observer-Based Controllers for PDE systems: A Heat-Conducting Rod With Point Observation and Boundary Control

50th IEEE Conference on Decision and Control, Orlando, FL. December 12-15, 2011. pp. 6985-6990.

[arXiv] [.pdf] [.ps] [slides]A. Seuret and M. M. Peet

SOS for Sampled Data Systems

18th IFAC World Congress, Milan, Italy. Aug. 28- Sept. 2, 2011.

[arXiv] [.pdf] [.ps]A. Gahlawat, M. M. Peet and E. Witrant

Control and Verification of the Safety-Factor Profile in Tokamaks Using Sum-of-Squares Polynomials

18th IFAC World Congress, Milan, Italy. Aug. 28- Sept. 2, 2011. pp. 12556-12561.

[arXiv] [.pdf] [.ps] [slides]M. M. Peet and A. Papachristodoulou

A Converse Sum-of-Squares Lyapunov Result: An Existence Proof Based on the Picard Iteration

49th IEEE Conference on Decision and Control, Atlanta, GA. December 15-17, 2010.

[arXiv] [.pdf] [.ps] [slides]Y. Zhang, M. M. Peet and K. Gu

Reducing the Computational Cost of the Sum-of-Squares Stability Test for Time-Delayed Systems

Proceedings of the American Control Conference, Baltimore, MD. June 30-July 2, 2010.

[arXiv] [.pdf] [.ps]M. M. Peet and Y. V. Peet

A Parallel-Computing Solution for Optimization of Polynomials

Proceedings of the American Control Conference, Baltimore, MD. June 30-July 2, 2010.

[arXiv] [.pdf] [.ps]Y. Zhang and M. M. Peet and K. Gu

Accelerating Convergence of Sum-of-Square Formulation for Lyapunov-Krasovskii Stability Analysis of Coupled Differential-Difference Equations

9th IFAC Workshop on Time-Delay Systems. Prague, Czech Republic. June 7-9, 2010.

[arXiv] [.pdf] [.ps]M. M. Peet

A Bound on the Continuity of Solutions and Converse Lyapunov Functions

48th IEEE Conference on Decision and Control, Shanghai, China. December 16-18, 2009.

[arXiv] [.pdf] [.ps]M. M. Peet and A. Papachristodoulou

Inverses of Positive Linear Operators and State Feedback Design for Time-Delay Systems

8th IFAC Workshop on Time-Delay Systems. Siniai, Romania. Sept. 1-3, 2009.

[arXiv] [.pdf] [.ps]M. M. Peet and A. Papachristodoulou

Using Polynomial Semi-Separable Kernels to Construct Infinite-Dimensional Lyapunov Functions

47th IEEE Conference on Decision and Control, Cancun, Mexico. December 9-11, 2008.

[arXiv] [.pdf] [.ps]M. M. Peet, H. Ozbay, and C. Bonnet

SOS for Delay-Dependent Stability of Neutral Differential Equations

Mathematical Theory of Networks and Systems, Blacksburg, VA. July 28-Aug. 1, 2008.

[arXiv] [.pdf] [.ps]M. M. Peet, and P.-A. Bliman

The Weierstrass Approximation Theorem on Linear Varieties: Polynomial Lyapunov Functionals for Delayed Systems

Mathematical Theory of Networks and Systems, Blacksburg, VA. July 28-Aug. 1, 2008.

Preliminary version appeared at the workshop TDS 2007.

[arXiv] [.pdf] [.ps]A. Papachristodoulou and M. M. Peet

Global Stability Analysis of Primal Internet Congestion Control Schemes with Heterogeneous Delays

IFAC World Congress. Seoul, South Korea. June 6-11, 2008.

[arXiv] [.pdf] [.ps]M. M. Peet, and P.-A. Bliman

Polynomial Lyapunov Functions for Exponential Stability of Nonlinear Systems on Bounded Regions

IFAC World Congress. Seoul, South Korea. June 6-11, 2008.

Preliminary version appeared at Allerton, 2007.

[arXiv] [.pdf] [.ps]A. Papachristodoulou, M. M. Peet, and S.-I. Niculescu

Stability Analysis of Linear Systems with Time-Varying Delays: Delay Uncertainty and Quenching

46th IEEE Conference on Decision and Control, New Orleans, LA. December 12-14, 2007.

[arXiv] [.pdf] [.ps]M. M. Peet and A. Papachristodoulou

Positivity of Kernel Functions for Systems with Communication Delay

46th IEEE Conference on Decision and Control, New Orleans, LA. December 12-14, 2007.

Preliminary version appeared at the Conference de la SMAI su l'optimisation et la decision, 2007

[arXiv] [.pdf] [.ps]M. M. Peet

Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions

45th annual Allerton Conference on Communication, Control, and Computing. Monticello, IL. Sept 26-28, 2007.

[arXiv] [.pdf] [.ps]C. Bonnet and M. M. Peet

Using the Positivstellensatz for Stability Analysis of Neutral Delay Systems in the Frequency Domain

7th IFAC Workshop on Time-Delay Systems. Nantes, France. Sept. 17-19, 2007.

[arXiv] [.pdf] [.ps]M. M. Peet and P.-A. Bliman

An Extension of the Weierstrass Theorem to Linear Varieties: Application to Delay Systems

7th IFAC Workshop on Time-Delay Systems. Nantes, France. Sept. 17-19, 2007.

[arXiv] [.pdf] [.ps]M. M. Peet and C. Bonnet

Stability and Computation of Roots in Delayed Systems of the Neutral Type

IFAC Workshop on Control of Distributed Parameter Systems. Namur, Belgium. July 22-27, 2007. pp. 49-50.

[arXiv] [.pdf] [.ps]M. M. Peet

On Positive Quadratic Forms and Stability of Linear Systems

Conference de la SMAI su l'optimisation et la decision, April 18-20, 2007.

[arXiv] [.pdf] [.ps]M. M. Peet, A. Papachristodoulou and S. Lall

Positive Forms and the Stability of Linear Time-Delay Systems

Proceedings of the 45th IEEE Conference on Decision and Control(CDC), December 13-15, 2006. pp. 187-193.

[arXiv] [.pdf] [.ps]A. Papachristodoulou and M. M. Peet

On the Analysis of Systems Described by Classes of Partial Differential Equations

Proceedings of the 45th IEEE Conference on Decision and Control(CDC), December 13-15, 2006. pp. 747-752.

[arXiv] [.pdf] [.ps]A. Papachristodoulou, M. M. Peet and S. Lall

Constructing Lyapunov-Krasovskii Functionals for Linear Time Delay Systems

Proceedings of the American Control Conference, pp. 2845-2850, June 2005.

[arXiv] [.pdf] [.ps]M. M. Peet and S. Lall

On Global Stability of Internet Congestion Control

Proceedings of the 43rd IEEE Conference on Decision and Control(CDC), pp. 1035-1041, December 2004.

[arXiv] [.pdf] [.ps]M. M. Peet and S. Lall

Constructing Lyapunov Functions for Nonlinear Delay-Differential Equations using Semidefinite Programming

Proceedings of the 6th IFAC Symposium on Nonlinear Control Systems(NOLCOS), pp. 381-385, August 2004.

[arXiv] [.pdf] [.ps]

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