74 research outputs found
Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation
- In this paper we introduce a new method to solve fixed-delay optimal
control problems which exploits numerical homotopy procedures. It is known that
solving this kind of problems via indirect methods is complex and
computationally demanding because their implementation is faced with two
difficulties: the extremal equations are of mixed type, and besides, the
shooting method has to be carefully initialized. Here, starting from the
solution of the non-delayed version of the optimal control problem, the delay
is introduced by numerical homotopy methods. Convergence results, which ensure
the effectiveness of the whole procedure, are provided. The numerical
efficiency is illustrated on an example
Exact Characterization of the Convex Hulls of Reachable Sets
We study the convex hulls of reachable sets of nonlinear systems with bounded
disturbances. Reachable sets play a critical role in control, but remain
notoriously challenging to compute, and existing over-approximation tools tend
to be conservative or computationally expensive. In this work, we exactly
characterize the convex hulls of reachable sets as the convex hulls of
solutions of an ordinary differential equation from all possible initial values
of the disturbances. This finite-dimensional characterization unlocks a tight
estimation algorithm to over-approximate reachable sets that is significantly
faster and more accurate than existing methods. We present applications to
neural feedback loop analysis and robust model predictive control
First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems
In this paper, we derive a set of first-order Pontryagin optimality
conditions for a risk-averse stochastic optimal control problem subject to
final time inequality constraints, and whose cost is a general finite coherent
risk measure. Unlike previous contributions in the literature, our analysis
holds for classical stochastic differential equations driven by standard
Brownian motions. Moreover, it presents the advantages of neither involving
second-order adjoint equations, nor leading to the so-called weak version of
the PMP, in which the maximization condition with respect to the control
variable is replaced by the stationarity of the Hamiltonian
A Gradient Descent-Ascent Method for Continuous-Time Risk-Averse Optimal Control
In this paper, we consider continuous-time stochastic optimal control
problems where the cost is evaluated through a coherent risk measure. We
provide an explicit gradient descent-ascent algorithm which applies to problems
subject to non-linear stochastic differential equations. More specifically, we
leverage duality properties of coherent risk measures to relax the problem via
a smooth min-max reformulation which induces artificial strong concavity in the
max subproblem. We then formulate necessary conditions of optimality for this
relaxed problem which we leverage to prove convergence of the gradient
descent-ascent algorithm to candidate solutions of the original problem.
Finally, we showcase the efficiency of our algorithm through numerical
simulations involving trajectory tracking problems and highlight the benefit of
favoring risk measures over classical expectation
Risk-Averse Trajectory Optimization via Sample Average Approximation
Trajectory optimization under uncertainty underpins a wide range of
applications in robotics. However, existing methods are limited in terms of
reasoning about sources of epistemic and aleatoric uncertainty, space and time
correlations, nonlinear dynamics, and non-convex constraints. In this work, we
first introduce a continuous-time planning formulation with an
average-value-at-risk constraint over the entire planning horizon. Then, we
propose a sample-based approximation that unlocks an efficient,
general-purpose, and time-consistent algorithm for risk-averse trajectory
optimization. We prove that the method is asymptotically optimal and derive
finite-sample error bounds. Simulations demonstrate the high speed and
reliability of the approach on problems with stochasticity in nonlinear
dynamics, obstacle fields, interactions, and terrain parameters
Sequential Convex Programming For Non-Linear Stochastic Optimal Control
This work introduces a sequential convex programming framework to solve
general non-linear, finite-dimensional stochastic optimal control problems,
where uncertainties are modeled by a multidimensional Wiener process. We
provide sufficient conditions for the convergence of the method. Moreover, we
prove that when convergence is achieved, sequential convex programming finds a
candidate locally-optimal solution for the original problem in the sense of the
stochastic Pontryagin Maximum Principle. We then leverage these properties to
design a practical numerical method for solving non-linear stochastic optimal
control problems based on a deterministic transcription of stochastic
sequential convex programming.Comment: Free-final-time problems with stochastic controls are now discussed
in a separate sectio
Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control
Sequential Convex Programming (SCP) has recently gained significant
popularity as an effective method for solving optimal control problems and has
been successfully applied in several different domains. However, the
theoretical analysis of SCP has received comparatively limited attention, and
it is often restricted to discrete-time formulations. In this paper, we present
a unifying theoretical analysis of a fairly general class of SCP procedures for
continuous-time optimal control problems. In addition to the derivation of
convergence guarantees in a continuous-time setting, our analysis reveals two
new numerical and practical insights. First, we show how one can more easily
account for manifold-type constraints, which are a defining feature of optimal
control of mechanical systems. Second, we show how our theoretical analysis can
be leveraged to accelerate SCP-based optimal control methods by infusing
techniques from indirect optimal control
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