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Papers by Matthew Peet

SIAM Journal on Control and Optimization, Jan 9, 2009

We consider the problem of constructing Lyapunov functions for linear differential equations with... more We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.

SIAM Journal on Control and Optimization, Jan 9, 2009

We consider the problem of constructing Lyapunov functions for linear differential equations with... more We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.

IEEE Robotics and Automation Letters

52nd IEEE Conference on Decision and Control, 2013

In this paper, we introduce an algorithm to decen tralize the computation associated with the sta... more In this paper, we introduce an algorithm to decen tralize the computation associated with the stability analysis of systems of nonlinear differential equations with a large number of states. The algorithm applies to dynamical systems with polynomial vector fields and checks the local asymptotic stability on hypercubes. We perform the analysis in three steps. First, by applying a multi-simplex version of Polya's theorem to some Lyapunov inequalities, we derive a sequence of stability conditions of increasing accuracy in the form of structured linear matrix inequalities. Then, we design a setup algorithm to decentralize the computation of the coefficients of the LMIs, among the processing units of a parallel environment. Finally, we use a parallel primal-dual central path algorithm, specifically designed to solve the structured LMIs given by the setup algorithm. For a sufficiently large number of available processors, the per-core computational complexity of the re sulting algorithm is fixed with the accuracy. The algorithm demonstrates a near-linear speed-up in numerical experiments.

2015 American Control Conference (ACC), 2015

We consider the coupled problems of optimal thermostat programming and optimal pricing of electri... more We consider the coupled problems of optimal thermostat programming and optimal pricing of electricity. Our framework consists of a single user and a single provider (a regulated utility). The provider sets prices for the user, who pays for both total energy consumed ($/kWh, including peak and off-peak rates) and the peak rate of consumption in a month (a demand charge) ($/kW). The cost of electricity for the provider is based on a combination of capacity costs ($/kW) and fuel costs ($/kWh). In the optimal thermostat programming problem, the user minimizes the amount paid for electricity while staying within a pre-defined temperature range. The user has access to energy storage in the form of thermal capacitance of the interior structure of the building. The provider sets prices designed to minimize the total cost of producing electricity while meeting the needs of the user. To solve the user-problem, we use a variant of dynamic programming. To solve the provider-problem, we use a descent algorithm coupled with our dynamic programming code-yielding optimal on-peak, off-peak and demand prices. We show that thermal storage and optimal thermostat programming can reduce electricity bills using current utility prices from utilities Arizona Public Service (APS) and Salt River Project (SRP). Moreover, we obtain optimal utility prices which lead to significant reductions in the cost of generating electricity and electricity bills.

Proceedings of the 2010 American Control Conference, 2010

This paper considers the problem of reducing the computational complexity associated with the Sum... more This paper considers the problem of reducing the computational complexity associated with the Sum-of-Squares approach to stability analysis of time-delay systems. Specifically, this paper considers systems with a large state-space but with relatively few delays-the most common situation in practice. The paper uses the general framework of coupled differential-difference equations with delays in low-dimensional feedback channels. This framework includes both the standard delayed and neutral-type systems. The approach is based on recent results which introduced a new type of Lyapunov-Krasovskii form which was shown to be necessary and sufficient for stability of this class of systems. This paper shows how exploiting the structure of the new functional can yield dramatic improvements in computational complexity. Numerical examples are given to illustrate this improvement.

Lecture Notes in Control and Information Sciences, 2012

11th Workshop on Time-Delay Systems, 2013

In this paper, we show that the controller synthesis of delayed systems can be formulated and sol... more In this paper, we show that the controller synthesis of delayed systems can be formulated and solved in a convex manner through the use of a duality transformation, a structured class of operators, and the Sum-of-Squares (SOS) methodology. The contributions of this paper are as follows. We show that a dual stability condition can be formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators which can be parameterized using Sum-of-Squares. Next, we show how any operator in this class can be inverted using simple operations on the SOS variables which can be performed in Matlab. Next we use SOS and semidefinite programming to formulate a dual stability test for time-delay systems. Next, we use the dual stability results to formulate a convex test for stabilizability and show how SOS can be used to solve this test and recover the controller. Finally, we give a numerical example. The results of this paper are significant in that they open the way for dynamic output H ∞ optimal control of infinite-dimensional systems by giving the first truly convex, numerically realizable full-state feedback controller synthesis criterion.

2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012

ABSTRACT

IEEE Conference on Decision and Control and European Control Conference, 2011

ABSTRACT

We consider the coupled problems of optimal thermostat programming and optimal pricing of electri... more We consider the coupled problems of optimal thermostat programming and optimal pricing of electricity. Our framework consists of multiple users (solar and non-solar customers) and a single provider (a regulated utility). The cost of electricity for the provider is based on a combination of fixed costs ($/user), capacity costs ($/kW) and fuel costs ($/kWh). The provider sets prices for the users, who pay for both total energy consumed ($/kWh, including peak and off-peak rates) and for the maximum electricity demand (peak rate of consumption) ($/kW). In the optimal thermostat programming problem, the users minimize the amount paid for electricity while staying within a pre-defined temperature range. The users have access to energy storage in the form of thermal capacitance of interior structures. Meanwhile, the provider sets prices designed to minimize the total cost of producing electricity while meeting the needs of the users. To solve the users problem, we use a variant of dynamic programming. To solve the provider problem, we use a descent algorithm coupled with our dynamic programming code-yielding optimal on-peak, off-peak and demand prices. We then apply our algorithms to a variety of scenarios. We show that: 1-Thermal storage and optimal thermostat programming can reduce electricity bills by 12% using current rates from utilities Arizona Public Service (APS) and Salt River Project (SRP). 2-Optimal pricing can significantly reduce the cost of generating electricity. 3-In our framework, the impact of solar power on the electricity bills of the non-solar customers is not significant (< 2%). 4-A full-day-peak pricing strategy induces a substantial reduction in the peak consumption but does not significantly change production costs.

Proceedings of the 45th IEEE Conference on Decision and Control, 2006

We provide an algorithmic approach for the analysis of infinite dimensional systems described by ... more We provide an algorithmic approach for the analysis of infinite dimensional systems described by Partial Differential Equations. In particular, we look at the stability properties of a class of strongly continuous semigroups generated by nonlinear parabolic partial differential equations with appropriate boundary conditions. Our approach is based on the application of semidefinite programming to the computation of Lyapunov-type certificates defined by polynomial functions. An illustrative example is given.

Lecture Notes in Control and Information Sciences, 2009

In this chapter we illustrate how the Sum of Squares approach can be used for understanding the s... more In this chapter we illustrate how the Sum of Squares approach can be used for understanding the stability properties of models of networks of biological and communication systems. The models we consider are in the form of nonlinear delay differential equations with multiple, incommensurate delays. Using the sum of squares approach, appropriate Lyapunov-Krasovskii functionals can be constructed, both for testing delay-independent and delay-dependent stability. We illustrate the methodology using examples from congestion control for the Internet and gene regulatory networks.

IEEE Transactions on Automatic Control, 2000

In this paper, we propose a distributed computing approach to solving large-scale robust stabilit... more In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polya's theorem to convert the polynomial optimization problem to a set of highly structured Linear Matrix Inequalities (LMIs). We then use a slight modification of a common interior-point primaldual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis.

In this paper, conditions for the stability of Zeno executions in hybrid dynamical systems with n... more In this paper, conditions for the stability of Zeno executions in hybrid dynamical systems with nonlinear dynamics and nonlinear transitions are presented. Zeno executions are non-equilibrium trajectories which experience an infinite number of transitions within a bounded time interval. The main result of the paper shows how to use the Sum-of-Squares optimization method to construct Lyapunov functions proving that all trajectories with initial conditions within a subset of the state-space are Zeno executions. Examples illustrating the use of the proposed technique are also provided. Finally, we use Sum-of-Squares to show convergence to Zeno executions for systems with parametric uncertainty in the vector field and transition map, along with an illustrative example.

under the direction of Professor Sanjay Lall during the period from August of 2002 through Januar... more under the direction of Professor Sanjay Lall during the period from August of 2002 through January of 2006. With the exception of Chapter 8, almost all of the work presented here has been previously published or submitted to peer-reviewed academic journals or conferences. The chapters of this thesis can be divided into those containing original research and those containing an overview of existing results. Specifically, Chapters 5, 7 and 8 contain original results, while most of the rest of the chapters can be considered as survey material. Some attempt has been made in this thesis to keep the length of the chapters minimal in order to improve readability. The presentation style is mathematically oriented, with specifics of numerical implementation suppressed. Proofs, when especially long, have been moved to the appendices. To maintain focus on the theoretical contribution, in-depth treatment of various special cases has been omitted when such treatment can be inferred from previous exposition.

Lecture Notes in Control and Information Sciences, 2009

This paper gives a description of how "sum-of-squares" (SOS) techniques can be used to check freq... more This paper gives a description of how "sum-of-squares" (SOS) techniques can be used to check frequency-domain conditions for the stability of neutral differential systems. For delay-dependent stability, we adapt an approach of Zhang et al. [11] and show how the associated conditions can be expressed as the infeasibility of certain semialgebraic sets. For delay-independent stability, we propose an alternative method of reducing the problem to infeasibility of certain semialgebraic sets. Then, using Positivstellensatz results from semi-algebraic geometry, we convert these infeasibility conditions to feasibility problems using sum-of-squares variables. By bounding the degree of the variables and using the Matlab toolbox SOSTOOLS [7], these conditions can be checked using semidefinite programming

SIAM Journal on Control and Optimization, Jan 9, 2009

We consider the problem of constructing Lyapunov functions for linear differential equations with... more We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.

SIAM Journal on Control and Optimization, Jan 9, 2009

We consider the problem of constructing Lyapunov functions for linear differential equations with... more We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.

IEEE Robotics and Automation Letters

52nd IEEE Conference on Decision and Control, 2013

In this paper, we introduce an algorithm to decen tralize the computation associated with the sta... more In this paper, we introduce an algorithm to decen tralize the computation associated with the stability analysis of systems of nonlinear differential equations with a large number of states. The algorithm applies to dynamical systems with polynomial vector fields and checks the local asymptotic stability on hypercubes. We perform the analysis in three steps. First, by applying a multi-simplex version of Polya's theorem to some Lyapunov inequalities, we derive a sequence of stability conditions of increasing accuracy in the form of structured linear matrix inequalities. Then, we design a setup algorithm to decentralize the computation of the coefficients of the LMIs, among the processing units of a parallel environment. Finally, we use a parallel primal-dual central path algorithm, specifically designed to solve the structured LMIs given by the setup algorithm. For a sufficiently large number of available processors, the per-core computational complexity of the re sulting algorithm is fixed with the accuracy. The algorithm demonstrates a near-linear speed-up in numerical experiments.

2015 American Control Conference (ACC), 2015

We consider the coupled problems of optimal thermostat programming and optimal pricing of electri... more We consider the coupled problems of optimal thermostat programming and optimal pricing of electricity. Our framework consists of a single user and a single provider (a regulated utility). The provider sets prices for the user, who pays for both total energy consumed ($/kWh, including peak and off-peak rates) and the peak rate of consumption in a month (a demand charge) ($/kW). The cost of electricity for the provider is based on a combination of capacity costs ($/kW) and fuel costs ($/kWh). In the optimal thermostat programming problem, the user minimizes the amount paid for electricity while staying within a pre-defined temperature range. The user has access to energy storage in the form of thermal capacitance of the interior structure of the building. The provider sets prices designed to minimize the total cost of producing electricity while meeting the needs of the user. To solve the user-problem, we use a variant of dynamic programming. To solve the provider-problem, we use a descent algorithm coupled with our dynamic programming code-yielding optimal on-peak, off-peak and demand prices. We show that thermal storage and optimal thermostat programming can reduce electricity bills using current utility prices from utilities Arizona Public Service (APS) and Salt River Project (SRP). Moreover, we obtain optimal utility prices which lead to significant reductions in the cost of generating electricity and electricity bills.

Proceedings of the 2010 American Control Conference, 2010

This paper considers the problem of reducing the computational complexity associated with the Sum... more This paper considers the problem of reducing the computational complexity associated with the Sum-of-Squares approach to stability analysis of time-delay systems. Specifically, this paper considers systems with a large state-space but with relatively few delays-the most common situation in practice. The paper uses the general framework of coupled differential-difference equations with delays in low-dimensional feedback channels. This framework includes both the standard delayed and neutral-type systems. The approach is based on recent results which introduced a new type of Lyapunov-Krasovskii form which was shown to be necessary and sufficient for stability of this class of systems. This paper shows how exploiting the structure of the new functional can yield dramatic improvements in computational complexity. Numerical examples are given to illustrate this improvement.

Lecture Notes in Control and Information Sciences, 2012

11th Workshop on Time-Delay Systems, 2013

In this paper, we show that the controller synthesis of delayed systems can be formulated and sol... more In this paper, we show that the controller synthesis of delayed systems can be formulated and solved in a convex manner through the use of a duality transformation, a structured class of operators, and the Sum-of-Squares (SOS) methodology. The contributions of this paper are as follows. We show that a dual stability condition can be formulated in terms of Lyapunov operators which are positive, self-adjoint and preserve the structure of the state-space. Second, we provide a class of such operators which can be parameterized using Sum-of-Squares. Next, we show how any operator in this class can be inverted using simple operations on the SOS variables which can be performed in Matlab. Next we use SOS and semidefinite programming to formulate a dual stability test for time-delay systems. Next, we use the dual stability results to formulate a convex test for stabilizability and show how SOS can be used to solve this test and recover the controller. Finally, we give a numerical example. The results of this paper are significant in that they open the way for dynamic output H ∞ optimal control of infinite-dimensional systems by giving the first truly convex, numerically realizable full-state feedback controller synthesis criterion.

2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012

ABSTRACT

IEEE Conference on Decision and Control and European Control Conference, 2011

ABSTRACT

We consider the coupled problems of optimal thermostat programming and optimal pricing of electri... more We consider the coupled problems of optimal thermostat programming and optimal pricing of electricity. Our framework consists of multiple users (solar and non-solar customers) and a single provider (a regulated utility). The cost of electricity for the provider is based on a combination of fixed costs ($/user), capacity costs ($/kW) and fuel costs ($/kWh). The provider sets prices for the users, who pay for both total energy consumed ($/kWh, including peak and off-peak rates) and for the maximum electricity demand (peak rate of consumption) ($/kW). In the optimal thermostat programming problem, the users minimize the amount paid for electricity while staying within a pre-defined temperature range. The users have access to energy storage in the form of thermal capacitance of interior structures. Meanwhile, the provider sets prices designed to minimize the total cost of producing electricity while meeting the needs of the users. To solve the users problem, we use a variant of dynamic programming. To solve the provider problem, we use a descent algorithm coupled with our dynamic programming code-yielding optimal on-peak, off-peak and demand prices. We then apply our algorithms to a variety of scenarios. We show that: 1-Thermal storage and optimal thermostat programming can reduce electricity bills by 12% using current rates from utilities Arizona Public Service (APS) and Salt River Project (SRP). 2-Optimal pricing can significantly reduce the cost of generating electricity. 3-In our framework, the impact of solar power on the electricity bills of the non-solar customers is not significant (< 2%). 4-A full-day-peak pricing strategy induces a substantial reduction in the peak consumption but does not significantly change production costs.

Proceedings of the 45th IEEE Conference on Decision and Control, 2006

We provide an algorithmic approach for the analysis of infinite dimensional systems described by ... more We provide an algorithmic approach for the analysis of infinite dimensional systems described by Partial Differential Equations. In particular, we look at the stability properties of a class of strongly continuous semigroups generated by nonlinear parabolic partial differential equations with appropriate boundary conditions. Our approach is based on the application of semidefinite programming to the computation of Lyapunov-type certificates defined by polynomial functions. An illustrative example is given.

Lecture Notes in Control and Information Sciences, 2009

In this chapter we illustrate how the Sum of Squares approach can be used for understanding the s... more In this chapter we illustrate how the Sum of Squares approach can be used for understanding the stability properties of models of networks of biological and communication systems. The models we consider are in the form of nonlinear delay differential equations with multiple, incommensurate delays. Using the sum of squares approach, appropriate Lyapunov-Krasovskii functionals can be constructed, both for testing delay-independent and delay-dependent stability. We illustrate the methodology using examples from congestion control for the Internet and gene regulatory networks.

IEEE Transactions on Automatic Control, 2000

In this paper, we propose a distributed computing approach to solving large-scale robust stabilit... more In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polya's theorem to convert the polynomial optimization problem to a set of highly structured Linear Matrix Inequalities (LMIs). We then use a slight modification of a common interior-point primaldual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis.

In this paper, conditions for the stability of Zeno executions in hybrid dynamical systems with n... more In this paper, conditions for the stability of Zeno executions in hybrid dynamical systems with nonlinear dynamics and nonlinear transitions are presented. Zeno executions are non-equilibrium trajectories which experience an infinite number of transitions within a bounded time interval. The main result of the paper shows how to use the Sum-of-Squares optimization method to construct Lyapunov functions proving that all trajectories with initial conditions within a subset of the state-space are Zeno executions. Examples illustrating the use of the proposed technique are also provided. Finally, we use Sum-of-Squares to show convergence to Zeno executions for systems with parametric uncertainty in the vector field and transition map, along with an illustrative example.

under the direction of Professor Sanjay Lall during the period from August of 2002 through Januar... more under the direction of Professor Sanjay Lall during the period from August of 2002 through January of 2006. With the exception of Chapter 8, almost all of the work presented here has been previously published or submitted to peer-reviewed academic journals or conferences. The chapters of this thesis can be divided into those containing original research and those containing an overview of existing results. Specifically, Chapters 5, 7 and 8 contain original results, while most of the rest of the chapters can be considered as survey material. Some attempt has been made in this thesis to keep the length of the chapters minimal in order to improve readability. The presentation style is mathematically oriented, with specifics of numerical implementation suppressed. Proofs, when especially long, have been moved to the appendices. To maintain focus on the theoretical contribution, in-depth treatment of various special cases has been omitted when such treatment can be inferred from previous exposition.

Lecture Notes in Control and Information Sciences, 2009

This paper gives a description of how "sum-of-squares" (SOS) techniques can be used to check freq... more This paper gives a description of how "sum-of-squares" (SOS) techniques can be used to check frequency-domain conditions for the stability of neutral differential systems. For delay-dependent stability, we adapt an approach of Zhang et al. [11] and show how the associated conditions can be expressed as the infeasibility of certain semialgebraic sets. For delay-independent stability, we propose an alternative method of reducing the problem to infeasibility of certain semialgebraic sets. Then, using Positivstellensatz results from semi-algebraic geometry, we convert these infeasibility conditions to feasibility problems using sum-of-squares variables. By bounding the degree of the variables and using the Matlab toolbox SOSTOOLS [7], these conditions can be checked using semidefinite programming