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The control of high-dimensional nonlinear systems remains a challenge in control theory. We propose a framework to design controllers for high-dimensional nonlinear systems. The controller is designed by the use of the iterative linear quadratic regulator (ILQR) algorithm, an algorithm that computes control by iteratively applying the linear quadratic control (LQR) algorithm on the local linearizations of the system at each time step. The high dimensionality is addressed by model reduction, which constructs reduced-order models (ROMs) that approximate the dynamics of the original full-order models (FOMs). We apply balanced truncation (BT), a system-theoretic, trajectory-independent model reduction technique for open-loop systems, and its extension for closed-loop systems, linear quadratic Gaussian balanced truncation (LQG-BT). Numerical experiments of this framework are performed on an 1D Burgers’ equation, where the performances of ROMs of different orders, as well as the choice between BT and LQG-BT, are compared and discussed. We find that the ILQR algorithm produces good control on ROMs constructed either by BT or LQG-BT. While BT outperforms LQG-BT in terms of accuracy in open-loop simulations, LQG-BT results in better control in the closed-loop system when combined with ILQR.

The control of high-dimensional nonlinear systems remains a challenge in control theory. We propose a framework to design controllers for high-dimensional nonlinear systems. The controller is designed by the use of the iterative linear quadratic regulator (ILQR) algorithm, an algorithm that computes control by iteratively applying the linear quadratic control (LQR) algorithm on the local linearizations of the system at each time step. The high dimensionality is addressed by model reduction, which constructs reduced-order models (ROMs) that approximate the dynamics of the original full-order models (FOMs). We apply balanced truncation (BT), a system-theoretic, trajectory-independent model reduction technique for open-loop systems, and its extension for closed-loop systems, linear quadratic Gaussian balanced truncation (LQG-BT). Numerical experiments of this framework are performed on an 1D Burgers’ equation, where the performances of ROMs of different orders, as well as the choice between BT and LQG-BT, are compared and discussed. We find that the ILQR algorithm produces good control on ROMs constructed either by BT or LQG-BT. While BT outperforms LQG-BT in terms of accuracy in open-loop simulations, LQG-BT results in better control in the closed-loop system when combined with ILQR.