arXiv subjects

Matthew M. Peet

Publications and source records attributed to Matthew M. Peet.

62 records · Page 4Linked to original sources

arXiv2015

Optimal Thermostat Programming and Optimal Electricity Rates for Customers with Demand Charges

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.

math.OC↗

arXiv2015

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

In this paper we use SOS and SDP to design output feedback controllers for a class of one-dimensional parabolic partial differential equations with point measurements and point actuation. Our approach is based on the use of SOS to search for positive quadratic Lyapunov functions, controllers and observers. These Lyapunov functions, controllers and observers are parameterized by linear operators which are defined by SOS polynomials. The main result of the paper is the development of an improved class of observer-based controllers and evidence which indicates that when the system is controllable and observable, these methods will find a observer-based controller for sufficiently high polynomial degree (similar to well-known results from backstepping).

eess.SY↗

arXiv2014

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

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 primal-dual 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.

math.OC↗

arXiv2014

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

We consider the problem of global stability of nonlinear sampled-data systems. Sampled-data systems are a form of hybrid model which arises when discrete measurements and updates are used to control continuous-time plants. In this paper, we use a recently introduced Lyapunov approach to derive stability conditions for both the case of fixed sampling period (synchronous) and the case of a time-varying sampling period (asynchronous). This approach requires the existence of a Lyapunov function which decreases over each sampling interval. To enforce this constraint, we use a form of slack variable which exists over the sampling period, may depend on the sampling period, and allows the Lyapunov function to be temporarily increasing. The resulting conditions are enforced using a new method of convex optimization of polynomial variables known as Sum-of-Squares.We use several numerical examples to illustrate this approach.

math.OC↗

arXiv2012

A Converse Sum of Squares Lyapunov Result with a Degree Bound

Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

math.CA↗

arXiv2007

Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions

This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear ordinary differential equations on bounded sets. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of R^n, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, we show how to use polynomials to approximate a differentiable function in the Sobolev norm W^{1,\infty} to any desired accuracy. We combine this approximation result with the second-order Taylor series expansion to find that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. Our investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.

math.CA↗

arXiv2007

Positive Forms and Stability of Linear Time-Delay Systems

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 parametrization 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.

math.DS↗

arXiv2006

Stability and Control of Functional Differential Equations

This thesis addresses the question of stability of systems defined by differential equations which contain nonlinearity and delay. In particular, we analyze the stability of a well-known delayed nonlinear implementation of a certain Internet congestion control protocol. We also describe a generalized methodology for proving stability of time-delay systems through the use of semidefinite programming. In Chapters 4 and 5, we consider an Internet congestion control protocol based on the decentralized gradient projection algorithm. For a certain class of utility function, this algorithm was shown to be globally convergent for some sufficiently small value of a gain parameter. Later work gave an explicit bound on this gain for a linearized version of the system. This thesis proves that this bound also implies stability of the original system. In Chapter 7, we describe a general methodology for proving stability of linear time-delay systems by computing solutions to an operator-theoretic version of the Lyapunov inequality via semidefinite programming. The result is stated in terms of a nested sequence of sufficient conditions which are of increasing accuracy. This approach is generalized to the case of parametric uncertainty by considering parameter-dependent Lyapunov functionals. Numerical examples are given to demonstrate convergence of the algorithm. In Chapter 8, this approach is generalized to nonlinear time-delay systems through the use of non-quadratic Lyapunov functionals.

math.DS↗