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Approximate Closed-Form Solution to a Linear Quadratic Optimal Control Problem with Disturbance
| Content Provider | Semantic Scholar |
|---|---|
| Author | Zidek, Robert A. E. Kolmanovsky, Ilya V. |
| Copyright Year | 2017 |
| Abstract | with Q QT ⪰ 0 and R RT ≻ 0. For notational convenience, we may omit indication of the explicit time dependence of time-dependent variables when it is clear from the context. Problems (1) and (2) are relevant to many real-time optimal control applications: in particular, those where a preview is available or needs to be incorporated [1–5]. According to [6], most prior applications in preview control used discrete-time-based formulations and either H∞ or linear quadratic regulator (LQR) methods [2,3,7–10]. In contrast, treating the problem in continuous-time (i.e., without resorting to discrete-time approximations) can provide higher solution accuracy. In the continuous-time formulation, optimal control problems (1) and (2) lead to a two-point boundary value problem (TPBVP) with mixed boundary conditions. A solution to this problem is based on solving the Riccati differential equation. In addition, an ordinary differential equation (ODE) that accounts for the disturbance has to be solved [11,12]. In general, there is no explicit solution to this ODE, and numerical and approximate approaches need to be developed. Numerical approaches to solve this ODE are based on integrating backward in time, which may become computationally impractical, especially when larger time horizons are considered. Therefore, we solve the problem by approximating the disturbance term d as a piecewise-linear function of time. Using Pontryagin’s maximumprinciple, linear systems theory, and analytical integration, we obtain a closed-form solution to the TPBVP. In addition, we derive an upper bound on the error between the optimal solution and the approximate solution when the piecewise-linear disturbance approximation is used. To the best of our knowledge, there has not been a closed-form solution to LQ optimal control problems with previewed disturbance d based on piecewise-linear approximation of d previously established or analyzed/investigated at a level of detail as in this Note. Such an approximation yields higher accuracy than piecewise-constant disturbance approximation that is common in sampled data/discrete-time treatments of the problem. The presented approach allows fast computation of the optimal control, which facilitates potential onboard/real-time implementation. In particular, this may be useful in applications of model predictive control (MPC) with previewed disturbance [13–16] or with disturbance scenarios [17], where an LQ problem with a disturbance term similar to Eqs. (1) and (2) has to be solved repeatedly over a receding time horizon. Future research to address the inclusion of constraints can further extend the use of our techniques for MPC with a preview to constrained problems. As our subsequent example demonstrates (Sec. IV), the proposed strategy can be effective in spacecraft orbital maneuvering problems to account for higher-order gravity perturbations and air drag. We note that, in this example, the disturbance is computed for the trajectory of the nominal/target orbit. Because, in the example, the spacecraft is relatively close to the known target orbit, the error is small and the simulation results show that our proposed approach is effective in the context of controlling a nonlinear system: in particular, when recomputing the control over a receding time horizon using MPC techniques to account for unmodeled effects (Sec. IV.C). At the same time, given that the focus of our theoretical analysis is an LQ problem with previewed disturbance, we also include simulation results for the linear model in Sec. IV.B because they illustrate the conclusions from the analysis in a setting consistent with the assumptions in this Note. The developments in this Note are further motivated by enhancing the implementation of a computational strategy to solve nonlinear optimal control problems, where one iterates between using d to approximate a nonlinear term d 1 φ x in the equations of motion evaluated on a current iteration i of the trajectory and solving the optimal control problem given by Eqs. (1) and (2) [18]. The structure of this Note is as follows. In Sec. II, the necessary conditions for optimality are presented and the closed-form solution to the TPBVP is derived. Section III includes an analysis of the error incurred by the piecewise-linear approximation of d. The method is demonstrated for orbital maneuvering in Sec. IV. Section V provides a conclusion of the work. |
| Starting Page | 475 |
| Ending Page | 481 |
| Page Count | 7 |
| File Format | PDF HTM / HTML |
| DOI | 10.2514/1.G001666 |
| Volume Number | 40 |
| Alternate Webpage(s) | https://deepblue.lib.umich.edu/bitstream/handle/2027.42/143067/1.G001666.pdf?isAllowed=y&sequence=1 |
| Alternate Webpage(s) | https://doi.org/10.2514/1.G001666 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
