Optimal input design for parameter estimation of nonlinear systems: case study of an unstable delta wing

ABSTRACT

A closed-loop optimal experimental design for online parameter identification approach is developed for nonlinear dynamic systems. The goal of the observer and the nonlinear model predictive control theories is here to perform online computation of the optimal time-varying input and to estimate the unknown model parameters online. The main contribution consists in combining Lyapunov stability theory with an existing closed-loop identification approach, in order to maximise the information content in the experiment and meanwhile to asymptotically stabilise the closed-loop system. To illustrate the proposed approach, the case of an open-loop unstable aerodynamic mechanical system is discussed. The simulation results show that the proposed algorithm allows to estimate all unknown parameters, which was not possible according to previous work, while keeping the closed-loop system stable.

KEYWORDS:

• Closed-loop identification
• optimal experiment design
• model predictive control
• stability
• nonlinear observer
• persistent excitation, delta wing
• mechanical system

Acknowledgment

This work is done in the context of a CIFRE PhD thesis between the LAGEP and the french company Acsystème which is gratefully acknowledged for the funding.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Other constraints may be specified on the measured outputs or estimation of the process states (dealing with safety, set-point tracking within error bounds, production, ...). In order to handle the stability with the method detailed here, such constraints are not included.

2. To simplify the notation in the following discrete formulations, s(k) = s(t_k) (resp. s(l) = s(t_l)) represents the value of the signal s at the current (resp. future) discrete time t = k × T_s (resp. t = l × T_s), where T_s is the constant sampling time and k (resp. l) is a time index (i.e. an integer). For the input, a zero-order hold is used between two consecutive sampling time instants. The various models are still formulated in a continuous framework and are solved numerically. Hence, sampled values may be taken at any discrete time. It is assumed that process data may also be sampled at the same rate.

3. As usual, the same development can be done for any pair (x, u) ≠ (0, 0) that represents a steady state in (13(13) $$\begin{array}{c}\hfill \dot{x}=F(x,u)+d\left(t\right),x\left(0\right)=x_0,\end{array}$$(13) ).

4. As discussed previously, since it is a Lyapunov function and since the cost function J has to be maximised, one adds a minus sign before J_L.

5. More information on odoe4ope.univ-lyon1.fr.

6. The time, the input and the states are also dimensionless.

7. To use this software, please visit http://odoe4ope.univ-lyon1.fr.

Additional information

Funding

This work was supported by the French company Acsystème and the French Ministry of Higher Education and Research and CIFRE [grant number 2011/0876].