NEW: RESEARCH POSITIONS AVAILABLE

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

 


Research Summary :

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.

 


Publications:


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
Submitted to 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:

B. Colbert, H. Mohammadi and M. Peet
Combining SOS and Moment Relaxations with Branch and Bound to Extract Solutions to Global Polynomial Optimization Problems
Submitted to 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
Submitted to 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
To Appear at the 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
To Appear at the 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
To appear at the 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]


Contact Information:

Home Phone: +1 480-699-4891

Mobile: +1 630-272-4451

Home Address:
23176 N. Dobson Rd.
Scottsdale, AZ 85255
USA

Work Phone: +1 480-965-5599

Mailing Address:
ERC 253
Arizona State University
School for Engineering of Matter, Transport and Energy
P.O. Box 876106
Tempe, AZ 85287-6106
USA

Shipping Address:
ECG337
501 E. Tyler Mall
Tempe, AZ 85287-6106
USA

Email 1 : mmpeet@gmail.com

Email 2 : mpeet@asu.edu