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7. Design of Exitation Signals for Identification


  • Excitation of all relevant frequencies and amplitudes.
  • Uniform coverage of the input space with data.

Approach: OMNIPUS − Optimal Excitation Signal Generator

  • Building a signal generator, not restricted to a special signal type like APRBS, chirp, etc.
  • Model-free (only rough assumption about dominate pole required).


  • Universally applicable.
  • Constraints can be incorporated easily (e.g. input or input rate limits).
  • Can be utilized for optimizing the data distribution of already existing measurements.
  • Can be extended to the multivariate case (multiple inputs).
  • Can be tailored to specific dynamic realizations (NARX, NOBF, NFIR).

Popular Excitation Signals

Interpretation of an Input Space in NARX Configuration


  • Classification of regions in the input space:
    - Quasi-static region along the diagonal
    - More dynamic excitation towards upper left and lower right corner
  • A good approximation of the system in high dynamic regions is necessary for high performance control (e.g.: MPC based)


Optimized Excitation Signal

Optimization of Sequences

  • Instead of N-dimensional optimization of u(1), u(2), …, u(N):
  • Iterative optimization:
    - Subsequent 2-dimensional optimizations
    - Amplitude and sequence length as optimization parameters
    ⇒ Increased robustness
  • Scalable to multiple inputs

Optimization result:
OptiMized Nonlinear InPUt Signal (OMNIPUS)


Optimization of Sequences

  • Sequence with highest quality is appended to the signal
  • Quality function:


  • Iterative optimization
  • Taking the “old” signal into account
  • Future sequence are not considered in the optimizatio

Implementation of Constraints


  • Most processes have input constraints:
    - Amplitude constraints
    - Velocity/Rate constraints
    - Higher order constraints (acceleration constraints)
  • Gray shaded area highlights constraint handling
  • Checking of feasibility
  • Increasing restrictions:
    ⇒ Decreasing feasible regions of (pseudo) input space


  • Increasing restrictions:
    ⇒ Lower frequent signals

Example: The High Pressure Fuel Supply Systemf79_b01


Qualitative Analysis – Local Models Networks

  • Ramp-chirp significantly worse in steady-state regions
    - Model simulates negative rail pressure occasionally!
  • OMNIPUS model fits the test data best
  • Significant difference between OMNIPUS model and ramp-chirp mode
  • High differences in error values
  • OMNIPUS data mostly significantly better


Popular Signal Types

  • Popular excitation signals for nonlinear identification are analyzed
  • Input space coverage is discussed


  • Objective is similar to static DoE strategies: Uniform data distribution
  • Input space established with linear model
  • Iterative optimization of excitation signal
    - Increased robustness due to subsequent 2-dimensional optimizations
  • Easy implementation of constraints

Results on the High Pressure Fuel Supply System

  • OMNIPUS covers extreme regions of the input space with data
  • OMNIPUS reveals infeasible regions of the input space
  • Ramp-chirp shows dramatically wrong extrapolation behavior


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