Goals

• 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).

Advantages

• 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 

• 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 optimization

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 System

Results

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

Conclusions

Popular Signal Types

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

OMNIPUS

• 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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