1. Input PRBS design for identification of multivariable systems
NPCW21, Åbo,
Input PRBS design for identification of multivariable systems
Present some ideas from a project by Winston and me
Winston Garcia-Gabin & Michael Lundh
ABB AB, Corporate Research, Västerås, Sweden

2. Outline Introduction Two slides about ABB Problem description Solution
Results
[klar 1]
November 29, 2018

3. ABB: the pioneering technology leader
What
(Offering)
Pioneering technology
Products 59%
Systems 24%
Services & software 17%
For whom (Customers)
Utilities
Industry
Transport & Infrastructure
~35% of revenue
~40% of revenue
~25% of revenue
Where (Geographies)
Globally
Asia, Middle East, Africa 37%
Americas 30%
Europe 33%
Technology company
Manly product offerings but also systems and services and sw
~$35 bn revenue
~100 countries
~135,000 employees
November 29, 2018

4. ABB Corporate Research
Research and Technology for the future
ABB Corporate Research, Västerås
Facts
established 1916
250 Co-Workers (50 nationalities)
60% PhD
Intense Lab infrastructure
University collaborations
CTH, KTH, LiTH, MdH, UU …
MIT, Stanford, Imperial College, …
Västerås
Ladenburg
Krakow
Raleigh
Dättwil
Beijing / Shanghai
Bangalore
[klar 3]
I work within the research organization, located in seven countries, about 800 persons.
I am sitting in Västerås, the largest lab.
November 29, 2018

5. Introduction Problem Description We have an MPC controlling a process
It has been in operation for a while
It does not perform as good as before
The process has probably changed
The model is not accurate any longer
Need to update the empirical model
Experiments are needed
Discuss with process operators in control room
Convince them for how much we are allowed to disturb the process
Methodology oriented to practitioners
Input signal design for identification to update the current model in an MPC controller
Based on required increments of the output variances and existing model
[klar 5]
Convince operator
November 29, 2018

6. Spectrum Characteristics
PRBS
Spectrum Characteristics
Power spectrum of a PRBS signal
𝑠 𝜔 = 𝐴 2 (𝑁+1) 𝑡 𝑐𝑙 𝑁 sin⁡(𝜔 𝑡 𝑐𝑙 /2) 𝜔 𝑡 𝑐𝑙 /2 2
The power is reduced by half at
𝜔= 2.8 𝑡 𝑐𝑙
Then consider the frequency range [ 𝜔 𝑙𝑜𝑤 , 𝜔 ℎ𝑖𝑔ℎ ] to be useful for excitation, i.e.
2π 𝑁∙ 𝑡 𝑐𝑙 ≤𝜔≤ 2.8 𝑡 𝑐𝑙 [Rad/s]
Next we will examine different ways to find an appropriate frequency range
[klar 6]
November 29, 2018

7. Determine PRBS Frequency Range
Lower bound based on open loop time domain properties
Lower frequency bound
Obtain crude estimates of the time constants and the time delays of the open loop process 𝜏 𝑖𝑗 𝑜𝑙 ,𝑡𝑑 𝑖𝑗 𝑜𝑙 for all outputs for each inputs using the existing process model in the MPC
Approximate the settling times for all the input-output pairs 𝑡 𝑖𝑗 𝑜𝑙 =4 𝜏 𝑖𝑗 𝑜𝑙 + 𝑡𝑑 𝑖𝑗 𝑜𝑙
Calculate the lower value of frequency as follows
𝜔 𝑙𝑜𝑤 = 1 𝑆 𝑓 max 𝑡 𝑖𝑗 𝑜𝑙
[klar 7]
Sf is an uncertainty factor between 1 and 4
Also for multi sine
November 29, 2018

8. Determine PRBS Frequency Range
Upper bound based on time domain properties 𝛼
2.0
0.5
Using open loop properties
𝜔 𝑖𝑗 𝑐𝑙 = 𝛼 𝑆 𝑓 𝜏 𝑖𝑗 𝑜𝑙 where 𝛼= max − 𝑡𝑑 𝑖𝑗 𝑜𝑙 𝜏 𝑖𝑗 𝑜𝑙 +2 , 0.5
𝜔 ℎ𝑖𝑔ℎ = max 𝜔 𝑖𝑗 𝑐𝑙
Using closed loop properties
𝜔 ℎ𝑖𝑔ℎ = 4 𝑆 𝑓 min 𝑡 𝑖 𝑐𝑙
𝑡𝑑 𝑖𝑗 𝑜𝑙 𝜏 𝑖𝑗 𝑜𝑙
[klar 9]
Approximation of closed loop band using rules of thumb for adequate response time
In both cases wHigh must be lower than the Nyquist frequency
November 29, 2018

9. Determine PRBS Frequency Range
Based on frequency domain properties of the input sensitivity function
Based on
Frequency of peak of input sensitivity function, ω 𝑝
Width controlled by parameter β
Upper value of the range
𝜔 ℎ𝑖𝑔ℎ =min(β ω 𝑝 , 𝜔 𝑝 𝜔 𝑁 )
The lower value of frequency is as follows
𝜔 𝑙𝑜𝑤 = 1 𝛽 ω 𝑝
[klar 10]
November 29, 2018

10. Determine time domain properties
PRBS
Determine time domain properties
The clock period is determined as the largest 𝑡 𝑐𝑙 satisfying
𝑡 𝑐𝑙 ≤ 𝜔 ℎ𝑖𝑔ℎ
The number of shift register is determined as the smallest 𝑛 𝑟 satisfying
𝑁= 2 𝑛 𝑟 −1≥ 2𝜋 𝑡 𝑐𝑙 ∗ 𝜔 𝑙𝑜𝑤
[klar 10]
November 29, 2018

11. PRBS Amplitude Consider a multi-sine input signal
𝑢 𝑒𝑥 (𝑡)= 𝑘=1 𝑛 𝐴 𝑘 𝑠𝑖𝑛 𝜔 𝑘 𝑡+ φ 𝑘 𝑢
The variance is
𝜎 𝑢 𝑒𝑥 2 = 𝑘=1 𝑛 2 𝐴 𝑘
𝜎 𝑢 𝑒𝑥 2 =𝑛 𝐴 if all amplitudes 𝐴 𝑘 =𝐴
Then the output is
𝑦(𝑡)= 𝑘=1 𝑛 𝐺 𝑘 𝐴 𝑘 𝑠𝑖𝑛 𝜔 𝑘 𝑡+ φ 𝑘 𝑦
With variance
𝜎 𝑦 2 = 𝑘=1 𝑛 𝐺 𝑘 𝐴 2 if all amplitudes 𝐴 𝑘 =𝐴
[klar 12]
When the frequency interval is determined it remains ti determine the excitation amplitudes.
G is the input sensitivity function
November 29, 2018

12. Amplitude in the multivariable case
PRBS
Amplitude in the multivariable case
Consider 𝑝×𝑞 input sensitivity transfer function
𝜎 𝑦 ⋮ 𝜎 𝑦 𝑝 2 = 𝑘=1 𝑛 𝐺 11 𝑘 2 ⋯ 𝑘=1 𝑛 𝐺 1𝑞 𝑘 2 ⋮ ⋱ ⋮ 𝑘=1 𝑛 𝐺 𝑝1 𝑘 2 ⋯ 𝑘=1 𝑛 𝐺 𝑝𝑞 𝑘 𝐴 1 2 ⋮ 𝐴 𝑞 2
On compact form
𝜆 1 ⋮ 𝜆 𝑝 = 𝜓 11 ⋯ 𝜓 1𝑞 ⋮ ⋱ ⋮ 𝜓 𝑝1 ⋯ 𝜓 𝑝𝑞 𝜁 1 ⋮ 𝜁 𝑞 or Λ=ΨΖ
Notice that in the multivariable case each multi-sine excitation signal must have different frequency distribution sequences 𝜔 1 , 𝜔 2, … , 𝜔 𝑘
[klar 13]
November 29, 2018

13. Amplitude in the multivariable case
PRBS
Amplitude in the multivariable case
Find 𝑍 such that each input produces enough persistence of excitation in all the outputs
by minimizing
J= arg 𝑚𝑖𝑛 ζ Ζ 𝜁 𝑇 Ζ(𝜁)
Subject to:
−Ψ 𝐼 ⋱ 𝐼 𝑞 Ζ≤−Λ one for each input 𝑖:1≤𝑖≤𝑞
The amplitude 𝐴 𝑖 of the multi-sine is
𝐴 𝑖 = 𝜁 𝑖
With variance
𝜎 𝑢 𝑒𝑥 𝑖 2 =𝑛 𝜁 𝑖 2 =𝑛 𝐴 𝑖 2 2
[klar 14]
Secure excitation is sufficient from each input to each output.
The quadratic term in the loss function could be modified to a linear?
November 29, 2018

14. PRBS Subtitle Apply the multi-sine framework!
For a PRBS that amplitude must switch between two levels ± 𝐴 𝑃𝑅𝐵𝑆 , given by
𝐴 𝑃𝑅𝐵𝑆 𝑖 = 𝜎 𝑢 𝑒𝑥 𝑖 2
[klar 15
November 29, 2018

15. Case Study Wood & Berry Simulated model with noise
𝐺 𝑠 = 𝑠+1 𝑒 −𝑠 − 𝑠+1 𝑒 −3𝑠 𝑠+1 𝑒 −7𝑠 − 𝑠+1 𝑒 −3𝑠
Desired output Variance increment
𝜎 𝑦 2 =0.2
PRBS frequency range
CV1
CV2
MV1
MV2
𝜎 2
0.131
0.123
0.057
0.027
[klar 16]
𝜔 𝑙𝑜𝑤 [rad/s]
𝜔 ℎ𝑖𝑔ℎ [rad/s]
Open- loop
0.01
0.125
Closed-loop --
0.182
November 29, 2018

16. Case Study Result Obtained variances 𝑢 𝑒𝑥1 ≠0, 𝑢 𝑒𝑥2 =0 𝑢 𝑒𝑥1 =0,
𝑢 𝑒𝑥1 ≠0,
𝑢 𝑒𝑥2 =0
𝑢 𝑒𝑥1 =0,
𝑢 𝑒𝑥2 ≠0
Theoretical variance
𝜎 𝑦1 2 =0.20
𝜎 𝑦2 2 =0.206
𝜎 𝑦2 2 =0.693
𝜎 𝑦1 2 =0.40
𝜎 𝑦2 2 =0.900
Multi-sine simulation
𝜎 𝑦1 2 =0.206
𝜎 𝑦2 2 =0.20
𝜎 𝑀𝑉1 2 =0.009
𝜎 𝑀𝑉2 2 =0.001
𝜎 𝑦1 2 =0.204
𝜎 𝑦2 2 =0.845
𝜎 𝑀𝑉1 2 =0.004
𝜎 𝑀𝑉2 2 =0.010
𝜎 𝑦1 2 =0.435
𝜎 𝑦2 2 =1.174
𝜎 𝑀𝑉1 2 =0.016
𝜎 𝑀𝑉2 2 =0.028
PRBS simulation
𝜎 𝑦1 2 =0.266
𝜎 𝑦2 2 =0.241
𝜎 𝑀𝑉1 2 =0.034
𝜎 𝑀𝑉2 2 =0.002
𝜎 𝑦1 2 =0.239
𝜎 𝑦2 2 =1.056
𝜎 𝑀𝑉1 2 =0.008
𝜎 𝑀𝑉2 2 =0.0279
𝜎 𝑦1 2 =0.450
𝜎 𝑦2 2 =1.152
𝜎 𝑀𝑉1 2 =0.043
𝜎 𝑀𝑉2 2 =0.027
[klar 18]
Approximation!
November 29, 2018

17. Conclusions Subtitle Simple approach to determine PRBS excitation
Intended for update of existing, somewhat bad, model
Oriented for practitioners
[klar 19]
November 29, 2018