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 that is not limited to a specific signal type such as APRBS, chirp, etc.
• Model-free (only a rough assumption about the dominant pole is required).

Advantages

• Universally applicable.
• Constraints can be easily incorporated (e.g., input or input rate limits).
• Can be used to optimize the data distribution of 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 toward the upper-left and lower-right corners
• A good approximation of the system in highly 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

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

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

Implementation of Constraints
 

• Most processes have input constraints:
  • Amplitude constraints
  • Velocity/Rate Constraints
• - Higher-order constraints (acceleration constraints)
• The gray-shaded area highlights constraint handling
• Feasibility check
   

• Increasing constraints:
  ⇒ Reducing the feasible regions of the (pseudo) input space

Increasing restrictions:
⇒ Less frequent signals

Example: The High-Pressure Fuel Supply System

Results

Qualitative Analysis – Local Model Networks
 

• Ramp-chirp is significantly worse in steady-state regions
  • The model occasionally simulates negative rail pressure!
• The OMNIPUS model best fits the test data
• Significant difference between the OMNIPUS model and ramp-chirp mode
   

• Large differences in error values
• OMNIPUS data is mostly significantly better

Conclusions

Popular Signal Types

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

OMNIPUS

• The objective is similar to that of static DoE strategies: uniform data distribution
• Input space established using a linear model
• Iterative optimization of the 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 identifies infeasible regions of the input space
• Ramp-chirp exhibits dramatically incorrect extrapolation behavior

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