arXiv subjects

Peter Seiler

Publications and source records attributed to Peter Seiler.

At least 73 records · Page 4Linked to original sources

arXiv2021

Control Barrier Functions With Unmodeled Dynamics Using Integral Quadratic Constraints

This paper presents a control design method that achieves safety for systems with unmodeled dynamics at the plant input. The proposed method combines control barrier functions (CBFs) and integral quadratic constraints (IQCs). Simplified, low-order models are often used in the design of the controller. Parasitic, unmodeled dynamics (e.g. actuator dynamics, time delays, etc) can lead to safety violations. The proposed method bounds the input-output behavior of these unmodeled dynamics in the time-domain using an alpha-IQC. The alpha-IQC is then incorporated into the CBF constraint to ensure safety. The approach is demonstrated with a simple example.

eess.SY↗

arXiv2021

Estimating Regions of Attraction for Transitional Flows using Quadratic Constraints

This letter describes a method for estimating regions of attraction and bounds on permissible perturbation amplitudes in nonlinear fluids systems. The proposed approach exploits quadratic constraints between the inputs and outputs of the nonlinearity on elliptical sets. This approach reduces conservatism and improves estimates for regions of attraction and bounds on permissible perturbation amplitudes over related methods that employ quadratic constraints on spherical sets. We present and investigate two algorithms for performing the analysis: an iterative method that refines the analysis by solving a sequence of semi-definite programs, and another based on solving a generalized eigenvalue problem with lower computational complexity, but at the cost of some precision in the final solution. The proposed algorithms are demonstrated on low-order mechanistic models of transitional flows. We further compare accuracy and computational complexity with analysis based on sum-of-squares optimization and direct-adjoint looping methods.

physics.flu-dyn↗

arXiv2021

Imitation Learning with Stability and Safety Guarantees

A method is presented to learn neural network (NN) controllers with stability and safety guarantees through imitation learning (IL). Convex stability and safety conditions are derived for linear time-invariant plant dynamics with NN controllers by merging Lyapunov theory with local quadratic constraints to bound the nonlinear activation functions in the NN. These conditions are incorporated in the IL process, which minimizes the IL loss, and maximizes the volume of the region of attraction associated with the NN controller simultaneously. An alternating direction method of multipliers based algorithm is proposed to solve the IL problem. The method is illustrated on an inverted pendulum system, aircraft longitudinal dynamics, and vehicle lateral dynamics.

eess.SY↗

arXiv2021

Nonlinear Stability Analysis of Transitional Flows using Quadratic Constraints

The dynamics of transitional flows are governed by an interplay between the non-normal linear dynamics and quadratic nonlinearity in the incompressible Navier-Stokes equations. In this work, we propose a framework for nonlinear stability analysis that exploits the fact that nonlinear flow interactions are constrained by the physics encoded in the nonlinearity. In particular, we show that nonlinear stability analysis problems can be posed as convex feasibility and optimization problems based on Lyapunov matrix inequalities, and a set of quadratic constraints that represent the nonlinear flow physics. The proposed framework can be used to conduct global stability, local stability, and transient energy growth analysis. The approach is demonstrated on the low-dimensional Waleffe-Kim-Hamilton model of transition and sustained turbulence. Our analysis correctly determines the critical Reynolds number for global instability. For local stability analysis, we show that the framework can estimate the size of the region of attraction as well as the amplitude of the largest permissible perturbation such that all trajectories converge back to the equilibrium point. Additionally, we show that the framework can predict bounds on the maximum transient energy growth. Finally, we show that careful analysis of the multipliers used to enforce the quadratic constraints can be used to extract dominant nonlinear flow interactions that drive the dynamics and associated instabilities.

physics.flu-dyn↗

arXiv2021

Finite Horizon Robust Synthesis Using Integral Quadratic Constraints

We present a robust synthesis algorithm for uncertain linear time-varying (LTV) systems on finite horizons. The uncertain system is described as an interconnection of a known LTV system and a perturbation. The input-output behavior of the perturbation is specified by time-domain Integral Quadratic Constraints (IQCs). The objective is to synthesize a controller to minimize the worst-case performance. This leads to a non-convex optimization. The proposed approach alternates between an LTV synthesis step and an IQC analysis step. Both induced $\mathcal{L}_2$ and terminal Euclidean norm penalties on output are considered for finite horizon performance. The proposed algorithm ensures that the robust performance is non-increasing at each iteration step. The effectiveness of this method is demonstrated using numerical examples.

eess.SY↗

arXiv2021

Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers

A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal inner-approximation to the region of attraction (ROA). The first theorem addresses linear time-invariant systems, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the system to include perturbations such as unmodeled dynamics, slope-restricted nonlinearities, and time delay, using integral quadratic constraint (IQCs) to capture their input/output behavior. This in turn allows for off-by-one IQCs to refine the description of activation functions by capturing their slope restrictions. Both results rely on semidefinite programming to approximate the ROA. The method is illustrated on systems with neural networks trained to stabilize a nonlinear inverted pendulum as well as vehicle lateral dynamics with actuator uncertainty.

eess.SY↗

arXiv2020

Iterative Best Response for Multi-Body Asset-Guarding Games

We present a numerical approach to finding optimal trajectories for players in a multi-body, asset-guarding game with nonlinear dynamics and non-convex constraints. Using the Iterative Best Response (IBR) scheme, we solve for each player's optimal strategy assuming the other players' trajectories are known and fixed. Leveraging recent advances in Sequential Convex Programming (SCP), we use SCP as a subroutine within the IBR algorithm to efficiently solve an approximation of each player's constrained trajectory optimization problem. We apply the approach to an asset-guarding game example involving multiple pursuers and a single evader (i.e., n-versus-1 engagements). Resulting evader trajectories are tested in simulation to verify successful evasion against pursuers using conventional intercept guidance laws.

eess.SY↗

arXiv2020

An Efficient Algorithm to Compute Norms for Finite Horizon, Linear Time-Varying Systems

We present an efficient algorithm to compute the induced norms of finite-horizon Linear Time-Varying (LTV) systems. The formulation includes both induced $\mathcal{L}_2$ and terminal Euclidean norm penalties. Existing computational approaches include the power iteration and bisection of a Riccati Differential Equation (RDE). The power iteration has low computation time per iteration but overall convergence can be slow. In contrast, the RDE condition provides guaranteed bounds on the induced gain but single RDE integration can be slow. The complementary features of these two algorithms are combined to develop a new algorithm that is both fast and provides provable upper and lower bounds on the induced norm within the desired tolerance. The algorithm also provides a worst-case disturbance input that achieves the lower bound on the norm. We also present a new proof which shows that the power iteration for this problem converges monotonically. Finally, we show a controllability Gramian based simpler computational method for induced $\mathcal{L}_2$-to-Euclidean norm. This can be used to compute the reachable set at any time on the horizon. Numerical examples are provided to demonstrate the proposed algorithm.

eess.SY↗

arXiv2020

Construction of Periodic Counterexamples to the Discrete-Time Kalman Conjecture

This paper considers the Lurye system of a discrete-time, linear time-invariant plant in negative feedback with a nonlinearity. Both monotone and slope-restricted nonlinearities are considered. The main result is a procedure to construct destabilizing nonlinearities for the Lurye system. If the plant satisfies a certain phase condition then a monotone nonlinearity can be constructed so that the Lurye system has a non-trivial periodic cycle. Several examples are provided to demonstrate the construction. This represents a contribution for absolute stability analysis since the constructed nonlinearity provides a less conservative upper bound than existing bounds in the literature.

math.OC↗

arXiv2020

Optimal assignment of collaborating agents in multi-body asset-guarding games

We study a multi-body asset-guarding game in missile defense where teams of interceptor missiles collaborate to defend a non-manuevering asset against a group of threat missiles. We approach the problem in two steps. We first formulate an assignment problem where we optimally assign subsets of collaborating interceptors to each threat so that all threats are intercepted as far away from the asset as possible. We assume that each interceptor is controlled by a collaborative guidance law derived from linear quadratic dynamic games. Our results include a 6-DOF simulation of a 5-interceptor versus 3-threat missile engagement where each agent is modeled as a missile airframe controlled by an autopilot. Despite the assumption of linear dynamics in our collaborative guidance law and the unmodeled dynamics in the simulation environment (e.g., varying density and gravity), we show that the simulated trajectories match well with those predicted by our approach. Furthermore, we show that a more agile threat, with greater speed and acceleration, can be intercepted by inferior interceptors when they collaborate. We believe the concepts introduced in this paper may be applied in asymmetric missile defense scenarios, including defense against advanced cruise missiles and hypersonic vehicles.

eess.SY↗

arXiv2020

Finite Step Performance of First-order Methods Using Interpolation Conditions Without Function Evaluations

We present a procedure to numerically compute finite step worst case performance guarantees on a given algorithm for the unconstrained optimization of strongly convex functions with Lipschitz continuous gradients. The solution method provided serves as an alternative approach to that derived by Taylor, Hendrickx, and Glineur in [Math. Prog. 161 (1-2), 2017]. The difference lies in the fact that our solution uses conditions for the interpolation of a set of points and gradient evaluations by the gradient of a function in the class of interest, whereas their solution uses conditions for the interpolation of a set of points, gradient evaluations, and function evaluations by a function in the class of interest. The motivation for this alternative solution is that, in many cases, neither the algorithm nor the performance metric of interest rely upon function evaluations. The primary development is a procedure to avoid suffering from the factorial growth in the number of these conditions with the size of the set to be interpolated when solving for the worst case performance.

eess.SY↗

arXiv2020

Reachability Analysis Using Dissipation Inequalities For Uncertain Nonlinear Systems

We propose a method to outer bound forward reachable sets on finite horizons for uncertain nonlinear systems with polynomial dynamics. This method makes use of time-dependent polynomial storage functions that satisfy appropriate dissipation inequalities that account for time-varying uncertain parameters, L2 disturbances, and perturbations characterized by integral quadratic constraints (IQCs) with both hard and soft factorizations. In fact, to our knowledge, this is the first result introducing IQCs to reachability analysis, thus allowing for various types of uncertainty, including unmodeled dynamics. The generalized S-procedure and Sum-of-Squares techniques are used to derive algorithms with the goal of finding the tightest outer bound with a desired shape. Both pedagogical and practically motivated examples are presented, including a 7-state F-18 aircraft model.

eess.SY↗

arXiv2020

An Introduction to Disk Margins

This paper provides a tutorial introduction to disk margins. These are robust stability measures that account for simultaneous gain and phase perturbations in a feedback system. The paper first reviews the classical (gain-only and phase-only) margins and their limitations. This motivates the use of disk margins which are defined using a set of perturbations that have simultaneous gain and phase variations. A necessary and sufficient condition is provided to compute the disk margin for a single-input, single-output feedback system. Frequency-dependent disk margins can also be computed yielding additional insight. The paper concludes with a discussion of stability margins for multiple-input, multiple output (MIMO) feedback systems. A typical approach is to assess robust stability "loop-at-a-time" with a perturbation introduced into a single channel and all other channels held at their nominal values. MIMO disk margins provide a useful extension to consider simultaneous variations in multiple channels. This multiple-loop analysis can provide a more accurate robustness assessment as compared to the loop-at-a-time approach.

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arXiv2020

Analysis of Biased Stochastic Gradient Descent Using Sequential Semidefinite Programs

We present a convergence rate analysis for biased stochastic gradient descent (SGD), where individual gradient updates are corrupted by computation errors. We develop stochastic quadratic constraints to formulate a small linear matrix inequality (LMI) whose feasible points lead to convergence bounds of biased SGD. Based on this LMI condition, we develop a sequential minimization approach to analyze the intricate trade-offs that couple stepsize selection, convergence rate, optimization accuracy, and robustness to gradient inaccuracy. We also provide feasible points for this LMI and obtain theoretical formulas that quantify the convergence properties of biased SGD under various assumptions on the loss functions.

math.OC↗

arXiv2020

Direct Synthesis of Iterative Algorithms With Bounds on Achievable Worst-Case Convergence Rate

Iterative first-order methods such as gradient descent and its variants are widely used for solving optimization and machine learning problems. There has been recent interest in analytic or numerically efficient methods for computing worst-case performance bounds for such algorithms, for example over the class of strongly convex loss functions. A popular approach is to assume the algorithm has a fixed size (fixed dimension, or memory) and that its structure is parameterized by one or two hyperparameters, for example a learning rate and a momentum parameter. Then, a Lyapunov function is sought to certify robust stability and subsequent optimization can be performed to find optimal hyperparameter tunings. In the present work, we instead fix the constraints that characterize the loss function and apply techniques from robust control synthesis to directly search over algorithms. This approach yields stronger results than those previously available, since the bounds produced hold over algorithms with an arbitrary, but finite, amount of memory rather than just holding for algorithms with a prescribed structure.

eess.SY↗

arXiv2020

Backward Reachability using Integral Quadratic Constraints for Uncertain Nonlinear Systems

A method is proposed to compute robust inner-approximations to the backward reachable set for uncertain nonlinear systems. It also produces a robust control law that drives trajectories starting in these sets to the target set. The method merges dissipation inequalities and integral quadratic constraints (IQCs) with both hard and soft IQC factorizations. Computational algorithms are presented using the generalized S-procedure and sum-of-squares techniques. The use of IQCs in backward reachability analysis allows for a variety of perturbations including parametric uncertainty, unmodeled dynamics, nonlinearities, and uncertain time delays. The method is demonstrated on two examples, including a 6-state quadrotor with actuator uncertainties.

eess.SY↗

arXiv2020

Tractable Reinforcement Learning of Signal Temporal Logic Objectives

Signal temporal logic (STL) is an expressive language to specify time-bound real-world robotic tasks and safety specifications. Recently, there has been an interest in learning optimal policies to satisfy STL specifications via reinforcement learning (RL). Learning to satisfy STL specifications often needs a sufficient length of state history to compute reward and the next action. The need for history results in exponential state-space growth for the learning problem. Thus the learning problem becomes computationally intractable for most real-world applications. In this paper, we propose a compact means to capture state history in a new augmented state-space representation. An approximation to the objective (maximizing probability of satisfaction) is proposed and solved for in the new augmented state-space. We show the performance bound of the approximate solution and compare it with the solution of an existing technique via simulations.

cs.LG↗

arXiv2019

Backward Reachability for Polynomial Systems on A Finite Horizon

A method is presented to obtain an inner-approximation of the backward reachable set (BRS) of a given target tube, along with an admissible controller that maintains trajectories inside this tube. The proposed optimization algorithms are formulated as nonlinear optimization problems, which are decoupled into tractable subproblems and solved by an iterative algorithm using the polynomial S-procedure and sum-of-squares techniques. This framework is also extended to uncertain nonlinear systems with L_2 disturbances and L_{\infty} parametric uncertainties. The effectiveness of the method is demonstrated on several nonlinear robotics and aircraft systems with control saturation.

eess.SY↗