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Strong observability as a sufficient condition for non-singularity and lossless convexification in optimal control with mixed constraints

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This paper analyzes optimal control problems with linear time-varying dynamics defined on a smooth manifold in addition to mixed constraints and pure control constraints. The main contribution is the identification of sufficient conditions for the optimal controls to be non-singular, which enables exact (or lossless) convex relaxations of the control constraints. The problem is analyzed in a geometric framework using a recent maximum principle on manifolds, and it is shown that strong observability of the dual system on the cotangent space is the key condition. Two minimum time problems are analyzed and solved. A minimum fuel planetary descent problem is then analyzed and relaxed to a convex form. Convexity enables its efficient solution in less than one second without any initial guess.

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Correspondence to Matthew W. Harris.

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The second author was partially funded by ONR Grant N00014-22-1-2131.

A Numerical solution method

A Numerical solution method

To numerically solve the infinite-dimensional problem P0, it is converted to a finite-dimensional parameter optimization problem. This is done by discretizing the problem and enforcing the constraints at the nodes. Note that the constraints in all of the examples are first or second-order cone constraints. Therefore, the resulting problems after discretization are finite-dimensional second-order cone programs that can be solved to global optimality using the powerful interior-point methods in convex optimization. We use a simple discretization method which is summarized below.

The time domain \([0,t_f]\) is discretized into \(N+1\) nodes separated by \(\Delta t\).

$$\begin{aligned} t_i = (i-1)\Delta t, ~~ i=1,\dots ,N+1. \end{aligned}$$

The states at time \(t_i\) are denoted by x[i] and they exist at all nodes \(i=1,\dots , N+1\). The controls at time \(t_i\) are denoted by u[i] and they exist at nodes \(i = 1,\dots ,N\). The controls are held constant over every interval. The system dynamics are discretized using the fundamental matrix resulting in

$$\begin{aligned} x[i+1] =\varPhi _i x[i] + \varGamma _i u[i], ~~ i = 1,\dots ,N, \end{aligned}$$

where \(\varPhi _i\) and \(\varGamma _i\) are matrices given by

$$\begin{aligned} \varPhi _i = \mathrm{e}^{A (t_{i+1}-t_i)}, ~~ \varGamma _i = \int _{t_i}^{t_{i+1}} \mathrm{e}^{A(t_{i+1}-\tau )} B(\tau ) \mathrm{d}\tau . \end{aligned}$$

The integral cost can be approximated by using any numerical integration technique, e.g., trapezoidal integration:

$$\begin{aligned} \int _0^{t_f}\ell (w(t))\approx \frac{\varDelta }{2}\textstyle \sum \limits _{i=1}^N\Big ( \ell (w[i+1])+\ell (w[i]) \Big ). \end{aligned}$$

All other constraints are enforced at the nodes. For example, control constraints are written as

$$\begin{aligned}&\Vert u[i]\Vert \le w[i], ~~ \; \forall \; i = 1,\dots ,N, \\&\rho _1 \le w[i] \le \rho _2, ~~ \forall \;i = 1,\dots ,N. \end{aligned}$$

The result is a finite-dimensional approximation to the infinite-dimensional optimal control problem. For free final time problems, a line search is conducted to find the N giving the least objective.

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Kunhippurayil, S., Harris, M.W. Strong observability as a sufficient condition for non-singularity and lossless convexification in optimal control with mixed constraints. Control Theory Technol. 20, 475–487 (2022). https://doi.org/10.1007/s11768-022-00115-w

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