Milan Korda

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 (typically 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 a 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 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.

CONVEX COMPUTATION OF THE MAXIMUM CONTROLLED INVARIANT SET FOR POLYNOMIAL CONTROL SYSTEMS

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

Inner approximations of the region of attraction for polynomial dynamical systems

In a previous work we developed a convex infinite dimensional linear programming (LP) approach to approximating the region of attraction (ROA) of polynomial dynamical systems subject to compact basic semialgebraic state constraints. Finite dimensional relaxations to the infinite-dimensional LP lead to a truncated moment problem in the primal and a polynomial sum-of-squares problem in the dual. This primal-dual linear matrix inequality (LMI) problem can be solved numerically with standard semidefinite programming solvers, producing a hierarchy of outer (i.e. exterior) approximations of the ROA by polynomial sublevel sets, with a guarantee of almost uniform and set-wise convergence. In this companion paper, we show that our approach is flexible enough to be modified so as to generate a hierarchy of polynomial inner (i.e.\,interior) approximations of the ROA with similar convergence guarantees.

Turnpike and dissipativity properties in dynamic real-time optimization and economic MPC

We investigate the turnpike and dissipativity properties of continuous-time optimal control problems. These properties play a key role in the analysis and design of schemes for dynamic real-time optimization and economic model predictive control. We show in a continuous-time setting that dissipativity of a system with respect to a steady state implies the existence of a turnpike at this steady state and optimal stationary operation at this steady state. Furthermore, we investigate the converse statements: We show that the existence of a turnpike at a steady state implies (a) that this steady state is the optimal steady state; and (b) that over an infinite horizon the system is optimally operated at this steady state. We draw upon a numerical example to illustrate our findings.

Stochastic MPC Framework for Controlling the Average Constraint Violation

This technical note considers linear discrete-time systems with additive, bounded, disturbances subject to hard control input bounds and a stochastic constraint on the amount of state-constraint violation averaged over time. The amount of violations is quantified by a loss function and the averaging can be weighted, corresponding to exponential forgetting of past violations. The freedom in the choice of the loss function makes this formulation highly flexible-for instance, probabilistic constraints, or integrated chance constraints, can be enforced by an appropriate choice of the loss function. For the type of constraint considered, we develop a recursively feasible receding horizon control scheme exploiting the averaged-over-time nature by explicitly taking into account the amount of past constraint violations when determining the current control input. This leads to a significant reduction in conservatism. As a simple extension of the proposed approach we show how time-varying state-constraints can be handled within our framework. The computational complexity (online as well as offline) is comparable to existing model predictive control schemes. The effectiveness of the proposed methodology is demonstrated by means of a numerical example from building climate control.

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control

This paper presents a class of linear predictors for nonlinear controlled dynamical systems. The basic idea is to lift (or embed) the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear. In an uncontrolled setting, this procedure amounts to numerical approximations of the Koopman operator associated to the nonlinear dynamics. In this work, we extend the Koopman operator to controlled dynamical systems and apply the Extended Dynamic Mode Decomposition (EDMD) to compute a finite-dimensional approximation of the operator in such a way that this approximation has the form of a linear controlled dynamical system. In numerical examples, the linear predictors obtained in this way exhibit a performance superior to existing linear predictors such as those based on local linearization or the so called Carleman linearization. Importantly, the procedure to construct these linear predictors is completely data-driven and extremely simple – it boils down to a nonlinear transformation of the data (the lifting) and a linear least squares problem in the lifted space that can be readily solved for large data sets. These linear predictors can be readily used to design controllers for the nonlinear dynamical system using linear controller design methodologies. We focus in particular on model predictive control (MPC) and show that MPC controllers designed in this way enjoy computational complexity of the underlying optimization problem comparable to that of MPC for a linear dynamical system with the same number of control inputs and the same dimension of the state-space. Importantly, linear inequality constraints on the state and control inputs as well as nonlinear constraints on the state can be imposed in a linear fashion in the proposed MPC scheme. Similarly, cost functions nonlinear in the state variable can be handled in a linear fashion. We treat both the full-state measurement case and the input–output case, as well as systems with disturbances/noise. Numerical examples demonstrate the approach.

Strongly feasible stochastic model predictive control

In this article we develop a systematic approach to enforce strong feasibility of probabilistically constrained stochastic model predictive control problems for linear discrete-time systems under affine disturbance feedback policies. Two approaches are presented, both of which capitalize and extend the machinery of invariant sets to a stochastic environment. The first approach employs an invariant set as a terminal constraint, whereas the second one constrains the first predicted state. Consequently, the second approach turns out to be completely independent of the policy in question and moreover it produces the largest feasible set amongst all admissible policies. As a result, a trade-off between computational complexity and performance can be found without compromising feasibility properties. Our results are demonstrated by means of two numerical examples.

On turnpike and dissipativity properties of continuous-time optimal control problems

This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e.,we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings.

Convex computation of the maximum controlled invariant set for discrete-time polynomial control systems

We characterize the maximum controlled invariant (MCI) set for discrete-time systems as the solution of an infinite-dimensional linear programming problem. In the case of systems with polynomial dynamics and semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality relaxations of this problem that provides outer approximations with guaranteed set-wise convergence to the MCI set. The approach is compact and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description.

On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

Extended dynamic mode decomposition (EDMD) (Williams et al. in J Nonlinear Sci 25(6):1307–1346, 2015) is an algorithm that approximates the action of the Koopman operator on an N-dimensional subspace of the space of observables by sampling at M points in the state space. Assuming that the samples are drawn either independently or ergodically from some measure