Optimal input design for multibody systems by using an extended adjoint approach
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Abstract
We present a method for optimizing inputs of multibody systems for a subsequently performed parameter identification. Herein, optimality with respect to identifiability is attained by maximizing the information content in measurements described by the Fisher information matrix. For solving the resulting optimization problem, the adjoint system of the sensitivity differential equation system is employed. The proposed approach combines these two wellestablished methods and can be applied to multibody systems in a systematic, automated manner. Furthermore, additional optimization goals can be added and used to find inputs satisfying, for example, end conditions or state constraints.
Keywords
Optimal input design Parameter identification Adjoint method Design of experiment Sensitivity analysis1 Introduction

The explicit use of the model equations (including any constraint) and current parameters to predict the “information content” of the next experiment (through the evaluation of some suitable objective function), and

the application of an optimization framework to find a numerical solution of the resulting problem.
The latter mentioned approaches cannot be applied directly to the model of mechanical systems. Therefore, in this paper, we show how the process of optimal input design can be systematically applied for mechanical systems. As the adjoint method provides outstanding performance in the field of optimal control, this method is used for computing the update direction during the optimal input iteration process. First, a proper performance measure or cost functional for determining optimal inputs is defined via statistical context. For further analysis, the system of sensitivity differential equations is derived, and also the adjoint system of the original system is extended by these new terms.
2 Theoretical background
2.1 System sensitivity analysis
2.2 Maximization of the information content in experimental measurement data
In [1], several norms or optimality metrics are suggested for optimal input design. Investigating the determinant (Doptimality) or eigenvalues (Eoptimality) of ℳ in a cost functional does not allow us to apply straightforward variational calculus. Hence, the so called Aoptimality is chosen, which incorporates the trace of ℳ. Moreover, common optimization algorithms search for the minimum of a cost functional \(J(\mathbf {u})\). Therefore, the maximization of the information content leads to a cost functional using the negative trace of ℳ.
2.3 The adjoint method
2.4 Considering model input constraints
3 Parameter identification
The purpose of optimal input design is to generate the excitation for a subsequent parameter estimation. Application of the computed optimal input onto the real system leads to an optimal desired trajectory. Therefore, in the following, an approach utilizing the system sensitivities derived in Sect. 2.1 is presented.
4 Numerical examples
4.1 Twomass oscillator
4.2 Cart pendulum system
The goal of the optimization is to find the excitation force \(F(t)\) that is best suited to generate measurements \(\varphi (t)\) allowing the identification of \(s_{p}\) and \(d_{p}\). Again, the force \(F(t)\) is constrained to the interval \([F_{\mathrm{max}},F_{\mathrm{max}}]\). Incorporating a scrap function using \(\varphi (t_{f}) = v_{c}(t_{f}) = \dot{\varphi }(t_{f}) = 0\) prevents from movements at \(t>t_{f}\).
5 Conclusion
The proposed method is mainly based on the assumption of optimality regarding the minimum standard deviation of identified parameters. It further allows us to set end conditions, which prescribe states at the end of an experiment. Looking at the example of the cart pendulum, this results in more robust measurement signals. Even when dealing with biased signals, more accurate parameter estimates are generated. Moreover, we assume that the proposed method can also handle different norms of the Fisher matrix ℳ in order to not only optimize the information content but also the condition of the optimization problem.
Notes
Acknowledgements
S. Oberpeilsteiner and T. Lauss acknowledge the support from the Austrian funding agency FFG in Coinproject ProtoFrame (project number 839074). K. Nachbagauer acknowledges support from the Austrian Science Fund (FWF): T733N30. Open access funding provided by the University of Applied Sciences Upper Austria.
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