Input PRBS design for identification of multivariable systems NPCW21, Åbo, 2018-01-19 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

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

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

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

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

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

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

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

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

Determine time domain properties PRBS Determine time domain properties The clock period is determined as the largest 𝑡 𝑐𝑙 satisfying   𝑡 𝑐𝑙 ≤ 2.8 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

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 2 2 𝜎 𝑢 𝑒𝑥 2 =𝑛 𝐴 2 2 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 2 2 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

Amplitude in the multivariable case PRBS Amplitude in the multivariable case Consider 𝑝×𝑞 input sensitivity transfer function 𝜎 𝑦 1 2 ⋮ 𝜎 𝑦 𝑝 2 = 1 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 2 𝐴 1 2 ⋮ 𝐴 𝑞 2 On compact form 𝜆 1 ⋮ 𝜆 𝑝 = 1 2 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

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 1 0 0 0 ⋱ 0 0 0 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

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

Case Study Wood & Berry Simulated model with noise 𝐺 𝑠 = 12.8 16.7𝑠+1 𝑒 −𝑠 −18.9 21𝑠+1 𝑒 −3𝑠 6.6 10.9𝑠+1 𝑒 −7𝑠 −19.4 14.4𝑠+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

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

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