arXiv subjects

Matthew M. Peet

Publications and source records attributed to Matthew M. Peet.

At least 37 records · Page 2Linked to original sources

arXiv2023

Integral Quadratic Constraints with Infinite-Dimensional Channels

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

math.OC↗

arXiv2023

A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose 1-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

math.DS↗

arXiv2023

A New Treatment of Boundary Conditions in PDE Solution with Galerkin Methods via Partial Integral Equation Framework

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing equation. The framework is based on a Partial Integral Equation (PIE) representation of PDEs, where a PDE equation is transformed into an equivalent PIE formulation that does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized PIE-Galerkin approximation methodology for a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries. The significance of this result is that solution to almost any linear PDE can now be constructed in a form of an analytical approximation based on a series expansion using a suitable set of basis functions, such as, e.g., Chebyshev polynomials of the first kind, irrespective of the boundary conditions. In many cases involving homogeneous or simple time-dependent boundary inputs, an analytical integration in time is also possible. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

math.NA↗

arXiv2023

Polynomial Approximation of Value Functions and Nonlinear Controller Design with Performance Bounds

For any suitable Optimal Control Problem (OCP) there exists a value function, defined as the unique viscosity solution to the Hamilton-Jacobi-Bellman (HJB) Partial-Differential-Equation (PDE), and which can be used to design an optimal feedback controller for the given OCP. In this paper, we approximately solve the HJB-PDE by proposing a sequence of Sum-Of-Squares (SOS) problems, each of which yields a polynomial subsolution to the HJB-PDE. We show that the resulting sequence of polynomial sub-solutions converges to the value function of the OCP in the L1 norm. Furthermore, for each polynomial sub-solution in this sequence, we show that the associated sequence of sublevel sets converge to the sublevel set of the value function of the OCP in the volume metric. Next, for any approximate value function, obtained from an SOS program or any other method (e.g. discretization), we construct an associated feedback controller, and show that sub-optimality of this controller as applied to the OCP is bounded by the distance between the approximate and true value function of the OCP in the Sobolev norm. Finally, we demonstrate numerically that by solving our proposed SOS problem we are able to accurately approximate value functions, design controllers and estimate reachable sets.

math.OC↗

arXiv2022

Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions.

math.OC↗

arXiv2022

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

In this paper, we present a new method for estimating the $L_2$-gain of systems governed by 2nd order linear Partial Differential Equations (PDEs) in two spatial variables, using semidefinite programming. It has previously been shown that, for any such PDE, an equivalent Partial Integral Equation (PIE) can be derived. These PIEs are expressed in terms of Partial Integral (PI) operators mapping states in $L_2[Ω]$, and are free of the boundary and continuity constraints appearing in PDEs. In this paper, we extend the 2D PIE representation to include input and output signals in $\mathbb{R}^n$, deriving a bijective map between solutions of the PDE and the PIE, along with the necessary formulae to convert between the two representations. Next, using the algebraic properties of PI operators, we prove that an upper bound on the $L_2$-gain of PIEs can be verified by testing feasibility of a Linear PI Inequality (LPI), defined by a positivity constraint on a PI operator mapping $\mathbb{R}^n\times L_2[Ω]$. Finally, we use positive matrices to parameterize a cone of positive PI operators on $\mathbb{R}^n\times L_2[Ω]$, allowing feasibility of the $L_2$-gain LPI to be tested using semidefinite programming. We implement this test in the MATLAB toolbox PIETOOLS, and demonstrate that this approach allows an upper bound on the $L_2$-gain of PDEs to be estimated with little conservatism.

math.OC↗

arXiv2022

Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems

This paper considers the problem of approximating the "maximal" region of attraction (the set that contains all asymptotically stable sets) of any given set of locally exponentially stable nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given a locally exponential stable ODE with a differentiable vector field, we show that there exists a globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set is equal to the maximal region of attraction of the ODE. We then propose a sequence of d-degree Sum-of-Squares (SOS) programming problems that yields a sequence of polynomials that converges to our proposed converse Lyapunov function uniformly from above in the L1 norm. We show that each member of the sequence of 1-sublevel sets of the polynomial solutions to our proposed sequence of SOS programming problems are certifiably contained inside the maximal region of attraction of the ODE, and moreover, we show that this sequence of sublevel sets converges to the maximal region of attraction of the ODE with respect to the volume metric. We provide numerical examples of estimations of the maximal region of attraction for the Van der Pol oscillator and a three dimensional servomechanism.

math.OC↗

arXiv2022

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

We introduce a Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D PDEs and time-delay systems without continuity constraints or boundary conditions -- making these PIE representations amenable to stability analysis using convex optimization. To extend the PIE framework to 2D PDEs, we first construct an algebra of Partial Integral (PI) operators on the function space L_2[x,y], providing formulae for composition, adjoint, and inversion. We then extend this algebra to R^n x L_2[x] x L_2[y] x L_2[x,y] and demonstrate that, for any suitable coupled, linear PDE in 2 spatial variables, there exists an associated PIE whose solutions bijectively map to solutions of the original PDE -- providing conversion formulae between these representations. Next, we use positive matrices to parameterize the convex cone of 2D PI operators -- allowing us to optimize PI operators and solve Linear PI Inequality (LPI) feasibility problems. Finally, we use the 2D LPI framework to provide conditions for stability of 2D linear PDEs. We test these conditions on 2D heat and wave equations and demonstrate that the stability condition has little to no conservatism.

math.AP↗

arXiv2021

SOSTOOLS Version 4.00 Sum of Squares Optimization Toolbox for MATLAB

The release of SOSTOOLS v4.00 comes as we approach the 20th anniversary of the original release of SOSTOOLS v1.00 back in April, 2002. SOSTOOLS was originally envisioned as a flexible tool for parsing and solving polynomial optimization problems, using the SOS tightening of polynomial positivity constraints, and capable of adapting to the ever-evolving fauna of applications of SOS. There are now a variety of SOS programming parsers beyond SOSTOOLS, including YALMIP, Gloptipoly, SumOfSquares, and others. We hope SOSTOOLS remains the most intuitive, robust and adaptable toolbox for SOS programming. Recent progress in Semidefinite programming has opened up new possibilities for solving large Sum of Squares programming problems, and we hope the next decade will be one where SOS methods will find wide application in different areas. In SOSTOOLS v4.00, we implement a parsing approach that reduces the computational and memory requirements of the parser below that of the SDP solver itself. We have re-developed the internal structure of our polynomial decision variables. Specifically, polynomial and SOS variable declarations made using sossosvar, sospolyvar, sosmatrixvar, etc now return a new polynomial structure, dpvar. This new polynomial structure, is documented in the enclosed dpvar guide, and isolates the scalar SDP decision variables in the SOS program from the independent variables used to construct the SOS program. As a result, the complexity of the parser scales almost linearly in the number of decision variables. As a result of these changes, almost all users will notice a significant increase in speed, with large-scaleproblems experiencing the most dramatic speedups. Parsing time is now always less than 10% of time spent in the SDP solver. Finally, SOSTOOLS now provides support for the MOSEK solver interface as well as the SeDuMi, SDPT3, CSDP, SDPNAL, SDPNAL+, and SDPA solvers.

math.OC↗

arXiv2020

Representation of Networks and Systems with Delay: DDEs, DDFs, ODE-PDEs and PIEs

Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation is limited. In network models with delay, the delayed channels are low-dimensional and accounting for this heterogeneity is not possible in the DDE framework. In addition, DDEs cannot be used to model difference equations. Furthermore, estimation and control of systems in DDE format has proven challenging, despite decades of study. In this paper, we examine alternative representations for systems with delay and provide formulae for conversion between representations. First, we examine the Differential-Difference (DDF) formulation which allows us to represent the low-dimensional nature of delayed information. Next, we examine the coupled ODE-PDE formulation, for which backstepping methods have recently become available. Finally, we consider the algebraic Partial-Integral Equation (PIE) representation, which allows the optimal estimation and control problems to be solved efficiently through the use of recent software packages such as PIETOOLS. In each case, we consider a very general class of delay systems, specifically accounting for all four possible sources of delay - state delay, input delay, output delay, and process delay. We then apply these representations to 3 archetypical network models.

math.OC↗

arXiv2020

A Partial Integral Equation (PIE) Representation of Coupled Linear PDEs and Scalable Stability Analysis using LMIs

We present a new Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in which it is possible to use convex optimization to perform stability analysis with little or no conservatism. The first result gives a standardized representation for coupled linear PDEs in a single spatial variable and shows that any such PDE, suitably well-posed, admits an equivalent PIE representation, defined by the given conversion formulae. This leads to a new prima facie representation of the dynamics without the implicit constraints on system state imposed by boundary conditions. The second result is to show that for systems in this PIE representation, convex optimization may be used to verify stability without discretization. The resulting algorithms are implemented in the Matlab toolbox PIETOOLS, tested on several illustrative examples, compared with previous results, and the code has been posted on Code Ocean. Scalability testing indicates the algorithm can analyze systems of up to 40 coupled PDEs on a desktop computer.

math.AP↗

arXiv2020

The Orbital Mechanics of Space Elevator Launch Systems

The construction of a space elevator would be an inspiring feat of planetary engineering of immense cost and risk. But would the benefit outweigh the costs and risks? What, precisely, is the purpose for building such a structure? For example, what if the space elevator could provide propellant-free (free release) orbital transfer to every planet in the solar system and beyond on a daily basis? In our view, this benefit might outweigh the costs and risks. But can a space elevator provide such a service? In this manuscript, we examine 3 tiers of space elevator launch system design and provide a detailed mathematical analysis of the orbital mechanics of spacecraft utilizing such designs. We find the limiting factor in all designs is the problem of transition to the ecliptic plane. For Tiers 1 and 2, we find that free release transfers to all the outer planets is possible, achieving velocities far beyond the ability of current Earth-based rocket technology, but with significant gaps in coverage due to planetary alignment. For Tier 3 elevators, however, we find that fast free release transfers to all planets in the solar system is possible on a daily basis. Finally, we show that Tier 2 and 3 space elevators can potentially use counterweights to perform staged slingshot maneuvers, providing a velocity multiplier which could dramatically reduce transit times to outer planets and interstellar destinations.

astro-ph.IM↗

arXiv2020

Duality and $H_{\infty}$-Optimal Control Of Coupled ODE-PDE Systems

In this paper, we present a convex formulation of $H_{\infty}$-optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and $H_{\infty}$ performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and $H_{\infty}$ performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and $H_{\infty}$-performance problems using the Linear PI Inequality (LPI) framework. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Next, we use our duality results to formulate the stabilization and $H_{\infty}$-optimal state-feedback control problems as LPIs. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.

math.OC↗

arXiv2020

A New Algorithm for Tessellated Kernel Learning

The accuracy and complexity of machine learning algorithms based on kernel optimization are limited by the set of kernels over which they are able to optimize. An ideal set of kernels should: admit a linear parameterization (for tractability); be dense in the set of all kernels (for robustness); be universal (for accuracy). The recently proposed Tesselated Kernels (TKs) is currently the only known class which meets all three criteria. However, previous algorithms for optimizing TKs were limited to classification and relied on Semidefinite Programming (SDP) - limiting them to relatively small datasets. By contrast, the 2-step algorithm proposed here scales to 10,000 data points and extends to the regression problem. Furthermore, when applied to benchmark data, the algorithm demonstrates significant improvement in performance over Neural Nets and SimpleMKL with similar computation time.

cs.LG↗

arXiv2020

Extensions of the Dynamic Programming Framework: Battery Scheduling, Demand Charges, and Renewable Integration

We consider a general class of Dynamic Programming (DP) problems with non-separable objective functions. We show that for any problem in this class, there exists an augmented-state DP problem which satisfies the Principle of Optimality and the solutions to which yield solutions to the original problem. Furthermore, we identify a subclass of DP problems with Naturally Forward Separable (NFS) objective functions for which this state-augmentation scheme is tractable. We extend this framework to stochastic DP problems, proposing a suitable definition of the Principle of Optimality. We then apply the resulting algorithms to the problem of optimal battery scheduling with demand charges using a data-based stochastic model for electricity usage and solar generation by the consumer.

math.OC↗

arXiv2020

A Convex Parametrization of a New Class of Universal Kernel Functions

The accuracy and complexity of kernel learning algorithms is determined by the set of kernels over which it is able to optimize. An ideal set of kernels should: admit a linear parameterization (tractability); be dense in the set of all kernels (accuracy); and every member should be universal so that the hypothesis space is infinite-dimensional (scalability). Currently, there is no class of kernel that meets all three criteria - e.g. Gaussians are not tractable or accurate; polynomials are not scalable. We propose a new class that meet all three criteria - the Tessellated Kernel (TK) class. Specifically, the TK class: admits a linear parameterization using positive matrices; is dense in all kernels; and every element in the class is universal. This implies that the use of TK kernels for learning the kernel can obviate the need for selecting candidate kernels in algorithms such as SimpleMKL and parameters such as the bandwidth. Numerical testing on soft margin Support Vector Machine (SVM) problems show that algorithms using TK kernels outperform other kernel learning algorithms and neural networks. Furthermore, our results show that when the ratio of the number of training data to features is high, the improvement of TK over MKL increases significantly.

stat.ML↗

arXiv2020

Computing Input-Output Properties of Coupled PDE systems

In this paper, we propose an LMI-based approach to analyze input-output properties of coupled linear PDE systems. This work expands on a newly developed state-space theory for coupled PDEs and extends the positive-real and bounded-real lemmas to infinite dimensional systems. We show that conditions for passivity and bounded L2 gain can be expressed as linear operator inequalities on RxL2. A method to convert these operator inequalities to LMIs by using parameterization of the operator variables is proposed. This method does not rely on discretization and as such, the properties obtained are prima facie provable. We use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that the bounds are computable on desktop computers for systems consisting of up to 20 coupled PDEs.

math.OC↗

arXiv2020

A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs.

math.AP↗