TY - JOUR
T1 - Data-driven model predictive control using interpolated koopman generators
AU - Peitz, Sebastian
AU - Otto, Samuel E.
AU - Rowley, Clarence W.
N1 - Funding Information:
\ast Received by the editors March 16, 2020; accepted for publication (in revised form) by J. Rubin July 2, 2020; published electronically September 24, 2020. https://doi.org/10.1137/20M1325678 Funding: The work of the first author was supported by the DFG Priority Programme 1962. \dagger Department of Mathematics, Paderborn University, Germany (speitz@math.upb.de). \ddagger Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544 (sotto@ princeton.edu, cwrowley@princeton.edu).
PY - 2020
Y1 - 2020
N2 - In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via dynamic mode decomposition, a quantization approach was recently proposed in [S. Peitz and S. Klus, Automatica J. IFAC, 106 (2019), pp. 184- 191]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls, as the data requirements increase only linearly with the input dimension. We show that when using the Koopman generator, this relaxation-realized by linear interpolation between two operators-does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.
AB - In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via dynamic mode decomposition, a quantization approach was recently proposed in [S. Peitz and S. Klus, Automatica J. IFAC, 106 (2019), pp. 184- 191]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls, as the data requirements increase only linearly with the input dimension. We show that when using the Koopman generator, this relaxation-realized by linear interpolation between two operators-does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.
KW - Dynamic mode decomposition
KW - Koopman operator
KW - Model predictive control
KW - Optimal control
KW - Reduced order modeling
UR - http://www.scopus.com/inward/record.url?scp=85094837223&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85094837223&partnerID=8YFLogxK
U2 - 10.1137/20M1325678
DO - 10.1137/20M1325678
M3 - Article
AN - SCOPUS:85094837223
VL - 19
SP - 2162
EP - 2193
JO - SIAM Journal on Applied Dynamical Systems
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
IS - 3
ER -