71 research outputs found
Convex computation of the region of attraction of polynomial control systems
We address the long-standing problem of computing the region of attraction
(ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a
controlled nonlinear system with polynomial dynamics and semialgebraic state
and input constraints. We show that the ROA can be computed by solving an
infinite-dimensional convex linear programming (LP) problem over the space of
measures. In turn, this problem can be solved approximately via a classical
converging hierarchy of convex finite-dimensional linear matrix inequalities
(LMIs). Our approach is genuinely primal in the sense that convexity of the
problem of computing the ROA is an outcome of optimizing directly over system
trajectories. The dual infinite-dimensional LP on nonnegative continuous
functions (approximated by polynomial sum-of-squares) allows us to generate a
hierarchy of semialgebraic outer approximations of the ROA at the price of
solving a sequence of LMI problems with asymptotically vanishing conservatism.
This sharply contrasts with the existing literature which follows an
exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix
inequalities or conservative LMI conditions. The approach is simple and readily
applicable as the outer approximations are the outcome of a single semidefinite
program with no additional data required besides the problem description
Stability and Performance Verification of Optimization-based Controllers
This paper presents a method to verify closed-loop properties of
optimization-based controllers for deterministic and stochastic constrained
polynomial discrete-time dynamical systems. The closed-loop properties amenable
to the proposed technique include global and local stability, performance with
respect to a given cost function (both in a deterministic and stochastic
setting) and the gain. The method applies to a wide range of
practical control problems: For instance, a dynamical controller (e.g., a PID)
plus input saturation, model predictive control with state estimation, inexact
model and soft constraints, or a general optimization-based controller where
the underlying problem is solved with a fixed number of iterations of a
first-order method are all amenable to the proposed approach.
The approach is based on the observation that the control input generated by
an optimization-based controller satisfies the associated Karush-Kuhn-Tucker
(KKT) conditions which, provided all data is polynomial, are a system of
polynomial equalities and inequalities. The closed-loop properties can then be
analyzed using sum-of-squares (SOS) programming
Learning Koopman eigenfunctions for prediction and control: the transient case
This work presents a data-driven framework for learning eigenfunctions of the
Koopman operator geared toward prediction and control. The method relies on the
richness of the spectrum of the Koopman operator in the transient,
off-attractor, regime to construct a large number of eigenfunctions such that
the state (or any other observable quantity of interest) is in the span of
these eigenfunctions and hence predictable in a linear fashion. Once a
predictor for the uncontrolled part of the system is obtained in this way, the
incorporation of control is done through a multi-step prediction error
minimization, carried out by a simple linear least-squares regression. The
predictor so obtained is in the form of a linear controlled dynamical system
and can be readily applied within the Koopman model predictive control
framework of [11] to control nonlinear dynamical systems using linear model
predictive control tools. The method is entirely data-driven and based purely
on convex optimization, with no reliance on neural networks or other non-convex
machine learning tools. The novel eigenfunction construction method is also
analyzed theoretically, proving rigorously that the family of eigenfunctions
obtained is rich enough to span the space of all continuous functions. In
addition, the method is extended to construct generalized eigenfunctions that
also give rise Koopman invariant subspaces and hence can be used for linear
prediction. Detailed numerical examples demonstrate the approach, both for
prediction and feedback control
Dictionary-free Koopman model predictive control with nonlinear input transformation
This paper introduces a method for data-driven control based on the Koopman
operator model predictive control. Unlike exiting approaches, the method does
not require a dictionary and incorporates a nonlinear input transformation,
thereby allowing for more accurate predictions with less ad hoc tuning. In
addition to this, the method allows for input quantization and exploits
symmetries, thereby reducing computational cost, both offline and online.
Importantly, the method retains convexity of the optimization problem solved
within the model predictive control online. Numerical examples demonstrate
superior performance compared to existing methods as well as the capacity to
learn discontinuous lifting functions
Polynomial argmin for recovery and approximation of multivariate discontinuous functions
We propose to approximate a (possibly discontinuous) multivariate function f
(x) on a compact set by the partial minimizer arg miny p(x, y) of an
appropriate polynomial p whose construction can be cast in a univariate sum of
squares (SOS) framework, resulting in a highly structured convex semidefinite
program. In a number of non-trivial cases (e.g. when f is a piecewise
polynomial) we prove that the approximation is exact with a low-degree
polynomial p. Our approach has three distinguishing features: (i) It is
mesh-free and does not require the knowledge of the discontinuity locations.
(ii) It is model-free in the sense that we only assume that the function to be
approximated is available through samples (point evaluations). (iii) The size
of the semidefinite program is independent of the ambient dimension and depends
linearly on the number of samples. We also analyze the sample complexity of the
approach, proving a generalization error bound in a probabilistic setting. This
allows for a comparison with machine learning approaches
Nonquadratic Stochastic Model Predictive Control: A Tractable Approach
This paper deals with the nite horizon stochastic optimal control problem with the expectation of the p-norm as the objective function and jointly Gaussian, although not necessarily independent, additive disturbance process. We develop an approximation strategy that solves the problem in a certain class of nonlinear feedback policies while ensuring satisfaction of hard input constraints. A bound on suboptimality of the proposed strategy in this class of nonlinear feedback policies is given for the special case of p = 1. We also develop a recursively feasible receding horizon policy with respect to state chance constraints and/or hard control input constraints in the presence of bounded disturbances. The performance of the proposed policies is examined in two numerical examples
Peak Estimation of Time Delay Systems using Occupation Measures
This work proposes a method to compute the maximum value obtained by a state
function along trajectories of a Delay Differential Equation (DDE). An example
of this task is finding the maximum number of infected people in an epidemic
model with a nonzero incubation period. The variables of this peak estimation
problem include the stopping time and the original history (restricted to a
class of admissible histories). The original nonconvex DDE peak estimation
problem is approximated by an infinite-dimensional Linear Program (LP) in
occupation measures, inspired by existing measure-based methods in peak
estimation and optimal control. This LP is approximated from above by a
sequence of Semidefinite Programs (SDPs) through the moment-Sum of Squares
(SOS) hierarchy. Effectiveness of this scheme in providing peak estimates for
DDEs is demonstrated with provided examplesComment: 34 pages, 14 figures, 3 table
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