90 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
The gap between a variational problem and its occupation measure relaxation
Recent works have proposed linear programming relaxations of variational
optimization problems subject to nonlinear PDE constraints based on the
occupation measure formalism. The main appeal of these methods is the fact that
they rely on convex optimization, typically semidefinite programming. In this
work we close an open question related to this approach. We prove that the
classical and relaxed minima coincide when the dimension of the codomain of the
unknown function equals one, both for calculus of variations and for optimal
control problems, thereby complementing analogous results that existed for the
case when the dimension of the domain equals one. In order to do so, we prove a
generalization of the Hardt-Pitts decomposition of normal currents applicable
in our setting. We also show by means of a counterexample that, if both the
dimensions of the domain and of the codomain are greater than one, there may be
a positive gap. The example we construct to show the latter serves also to show
that sometimes relaxed occupation measures may represent a more
conceptually-satisfactory "solution" than their classical counterparts, so that
-- even though they may not be equivalent -- algorithms rendering accessible
the minimum in the larger space of relaxed occupation measures remain extremely
valuable. Finally, we show that in the presence of integral constraints, a
positive gap may occur at any dimension of the domain and of the codomain.Comment: 46 pages, 10 figure
Moment-sum-of-squares hierarchies for set approximation and optimal control
This thesis uses the idea of lifting (or embedding) a nonlinear controlled dynamical system into an infinite-dimensional space of measures where this system is equivalently described by a linear equation. This equation and problems involving it are subsequently approximated using well-known moment-sum-of-squares hierarchies. First, we address the problems of region of attraction, reachable set and maximum controlled invariant set computation, where we provide a characterization of these sets as an infinite-dimensional linear program in the cone of nonnegative measures and we describe a hierarchy of finite-dimensional semidefinite-programming (SDP) hierarchies providing a converging sequence of outer approximations to these sets. Next, we treat the problem of optimal feedback controller design under state and input constraints. We provide a hierarchy of SDPs yielding an asymptotically optimal sequence of rational feedback controllers. In addition, we describe hierarchies of SDPs yielding approximations to the value function attained by any given rational controller, from below and from above, as well as a hierarchy of SDPs providing approximations from below to the optimal value function, hence obtaining performance certificates for the designed controllers as well as for any given rational controller. Finally, we describe a method to verify properties of a closed loop interconnection of a nonlinear dynamical system and an optimization-based controller (e.g., a model predictive controller) for deterministic and stochastic nonlinear dynamical systems. Properties such as global stability, the gain or performance with respect to a given infinite-horizon cost function can be certified. The methods presented are easy to implement using freely available software packages and are documented by a number of numerical examples
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