Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Abstract

Motivated by uncertain parameters in nonlinear dynamic systems, we define a nonclassical optimal control problem where the cost functional is given by a Riemann-Stieltjes "functional of a functional." Using the properties of Riemann-Stieltjes sums, a minimum principle is generated from the limit of a semidiscretization. The optimal control minimizes a Riemann-Stieltjes integral of the Pontryagin Hamiltonian. The challenges associated with addressing the noncommutative operations of integration and minimization are addressed via cubature techniques leading to the concept of hyper-pseudospectral points. These ideas are then applied to address the practical uncertainties in control moment gyroscopes that drive an agile spacecraft. Ground test results conducted at Honeywell demonstrate the new principles. The Riemann-Stieltjes optimal control problem is a generalization of the unscented optimal control problem. It can be connected to many independently developed ideas across several disciplines: search theory, viability theory, quantum control and many other applications involving tychastic differential equations.

Robust Aircraft Trajectory Planning Under Wind Uncertainty Using Optimal Control

“…This is only appropriate for practical problems where the actual system has low-level controllers that 5 Note that this system is a special case of a stochastic differential equation in the sense of [19,Chapter 2] where the random parameters are not random processes, i.e., are constant. This motivated [16] to employ the term "tychastic", proposed in earlier work, to facilitate the distinction.…”

“…In previous literature employing this approach (see [16], [18] or [23]), the control law is considered as only dependant on time u = u F (t), thus leading to an "open-loop" control scheme where the controls are equal in each scenario (i.e. u(…”

“…This class of systems are called tychastic dynamical systems in [16] and related work. Uncertainty is described with the aid of a standard Kolmogorov probability space (, F, P ); it is composed by a sample space of possible outcomes , a σ-algebra of events F containing sets of outcomes, and the probability function P that assigns a probability to each of these events.…”

“…Then, an optimal control problem is formulated in order to search for the control sequence that minimizes a probabilistic functional that aggregates the cost among all of the trajectories (see Figure 1.b). Some examples of aerospace applications employing this technique are robust trajectory optimization for an aircraft based on dynamic soaring [17,Chapters 2,3], trajectory optimization for a supersonic aircraft with uncertainties in the aerodynamic data [18] and optimization of a spacecraft reorientation maneuver [16].…”

“…In a "robust" or "tychastic" methodology, a discrete set of values for the uncertain variables is again determined by a Polynomial Chaos rule (though other possibilities exist, such as the cubature approach of [16]). An augmented dynamical system is built by stacking one copy of the state variables for each computed value of the uncertain parameters.…”