Control synthesis for nonlinear optimal control via convex relaxations

Abstract

This paper addresses the problem of control synthesis for nonlinear optimal control problems in the presence of state and input constraints. The presented approach relies upon transforming the given problem into an infinite-dimensional linear program over the space of measures. To generate approximations to this infinite-dimensional program, a sequence of Semi-Definite Programs (SDP)s is formulated in the instance of polynomial cost and dynamics with semi-algebraic state and bounded input constraints. A method to extract a polynomial control function from each SDP is also given. This paper proves that the controller synthesized from each of these SDPs generates a sequence of values that converge from below to the value of the optimal control of the original optimal control problem. In contrast to existing approaches, the presented method does not assume that the optimal control is continuous while still proving that the sequence of approximations is optimal. Moreover, the sequence of controllers that are synthesized using the presented approach are proven to converge to the true optimal control. The performance of the presented method is demonstrated on three examples.

Safe Trajectory Synthesis for Autonomous Driving in Unforeseen Environments

“…The error function g can also be improved by considering time variation of trajectory parameters across planning steps by posing the FRS computation as a hybrid problem [22]. Finally, we plan to use convex optimization to find a global solution to the nonlinear trajectory optimization problem at each planning step [26], [27].…”

“…This constrained nonlinear optimal control problem can be solved in a variety of different ways via collocation, solving a variational equation, sampling, or using convex relaxations [26]. If this optimization program is unable to conclude within τ plan , then one can always apply a braking maneuver which always exists as described in Section III-B.…”

Optimal Control of Polynomial Hybrid Systems via Convex Relaxations

“…In fact, this bound converges to the true optimal cost as the moment sequence extends to infinity under the assumption that the incremental cost is convex in control. Recent work has also shown how the optimal control policy can be extracted for systems that are affine in control [24], [25]. Unfortunately this relaxed control formulation for controlled hybrid systems, the subsequent development of a numerically implementable convex relaxation, and optimal control synthesis have remained unaddressed.…”

Safety Verification of Nonlinear Polynomial System via Occupation Measures

“…Note that any family of solutions x(t) of (1) with an initial distribution µ 0 induces an occupation measure (14) and a final measure (15) satisfying (16). Conversely, for any tuple of measures (µ 0 , µ, µ T ) satisfying (16), one can identify a distribution on the admissible trajectories starting from µ 0 whose average occupation measure and average final measure coincide with µ and µ T , respectively (see Lemma 3 in [21] and Lemma 6 in [25] for more details).…”