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Identification of low-order output-error models
Mikael Manngård, Jari Böling, Hannu Toivonen Process Control Laboratory, Faculty of Science and Engineering, Åbo Akademi University. My research has mainly been in systems identification Today i’m presenting a method for identification of low order systems
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Motivation Why output-error models?
Describes the system dynamics separately from the noise dynamics. Performs well on ballisting simulations. Why constrain the model order? Acts as a regularizer. Identified models of the correct order tells us something about the true system (poles). Low-order models are suitable for synthesis of controllers. 12/3/2018 NPCW 2018
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Output-error model structure
Disturbances 𝑣 + 𝑢 𝐺 𝑦 𝑦 Inputs System Output Measured output Since we are developing general methods that can be applied to any system... 12/3/2018 NPCW 2018
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System representations
Any stable, rational system can be represented in terms of an orthonormal basis { 𝜓 𝑘 𝑧 } 𝑘=0,1,2,… in 𝐻 2 as 𝐺 𝑧 = 𝑘=0 ∞ ℎ 𝑘 𝜓 𝑘 𝑧 . A few examples of bases are: the natural basis { 𝑧 −𝑘 } the Laguerre basis the two-parameter Kautz basis 12/3/2018 NPCW 2018
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System representations
Any stable, rational system can be represented in terms of an orthonormal basis { 𝜓 𝑘 𝑧 } 𝑘=0,1,2,… in 𝐻 2 as 𝐺 𝑧 = 𝑘=0 ∞ ℎ 𝑘 𝜓 𝑘 𝑧 . A few examples of bases are: the natural basis { 𝑧 −𝑘 } the Laguerre basis the two-parameter Kautz basis Grossmann, C., Jones, C. N., & Morari, M. (2009). System identification via nuclear norm regularization for simulated moving bed processes from incomplete data sets. 12/3/2018
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System representations
Representing the system in terms of 𝑧 −𝑘 , the output 𝑦 𝑘 = 𝑡=0 ∞ ℎ 𝑡 𝑢 𝑘−𝑡 . It follows that there exists an operator 𝐻 that maps past inputs to future outputs 𝑦 0 𝑦 1 𝑦 2 ⋮ = ℎ 1 ℎ 2 ℎ 3 ⋯ ℎ 2 ℎ 3 ℎ 4 ⋯ ℎ 3 ℎ 4 ℎ 5 ⋯ ⋮ ⋮ ⋮ ⋱ 𝑢 −1 𝑢 −2 𝑢 −3 ⋮ ℎ 0 ⋯ 0 ℎ 1 ℎ 0 ⋮ ℎ 2 ℎ 1 ℎ 0 ⋮ ⋮ ⋮ ⋱ 𝑢 0 𝑢 1 𝑢 2 ⋮ 𝐻(ℎ) 12/3/2018 NPCW 2018
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The model order A well known result in systems theory is that the system order is given by the rank of the Hankel operator 𝐻(ℎ) = ℎ 1 ℎ 2 ℎ 3 ⋯ ℎ 2 ℎ 3 ℎ 4 ⋯ ℎ 3 ℎ 4 ℎ 5 ⋯ ⋮ ⋮ ⋮ ⋱ i.e. deg G =rank 𝐻 ℎ . 12/3/2018 NPCW 2018
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Problem statement P: Identify a system 𝐺 from input-output data {𝑢(𝑘),𝑦(𝑘)} that solves the output-error problem minimize ℎ∈ ℓ 2 𝑘=0 𝑁−1 𝑦 𝑘 − 𝑡=0 ∞ ℎ 𝑡 𝑢 𝑘−𝑡 2 subject to rank 𝐻 ℎ ≤𝑛 12/3/2018 NPCW 2018
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Problem statement P: Identify a system 𝐺 from input-output data {𝑢 𝑘 ,𝑦 𝑘 } that solves the output-error problem minimize ℎ∈ ℓ 2 𝑘=0 𝑁−1 𝑦 𝑘 − 𝑡=0 ∞ ℎ 𝑡 𝑢 𝑘−𝑡 2 subject to rank 𝐻 ℎ ≤𝑛 12/3/2018 NPCW 2018
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Problem statement P*: Identify a system 𝐺 from input-output data {𝑢 𝑘 ,𝑦 𝑘 } that solves the output-error problem minimize ℎ∈ ℝ 𝑀 𝑘=0 𝑁−1 𝑦 𝑘 − 𝑡=0 𝑀−1 ℎ 𝑡 𝑢 𝑘−𝑡 2 subject to rank 𝐻 ℎ ≤𝑛 12/3/2018 NPCW 2018
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Problem statement P*: Identify a system 𝐺 from input-output data {𝑢(𝑘),𝑦(𝑘)} that solves the output-error problem minimize ℎ∈ ℝ 𝑀 𝑘=0 𝑁−1 𝑦 𝑘 − 𝑡=0 𝑀−1 ℎ 𝑡 𝑢 𝑘−𝑡 2 subject to 𝐻(ℎ) ∗ ≤𝜂 Fazel, M. (2002). Matrix rank minimization with applications. Ph.D. Thesis, Stanford University. 12/3/2018
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Problem statement P*: Identify a system 𝐺 from input-output data {𝑢(𝑘),𝑦(𝑘)} that solves the output-error problem minimize ℎ∈ ℝ 𝑀 𝛾 𝑘=0 𝑁−1 𝑦 𝑘 − 𝑡=0 𝑀−1 ℎ 𝑡 𝑢 𝑘−𝑡 𝐻(ℎ) ∗ Fazel, M. (2002). Matrix rank minimization with applications. Ph.D. Thesis, Stanford University. 12/3/2018
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Simulation study Benchmark problem for output-error system identification (Van den Hof and Ninnes, 2005). 𝑦 𝑘 =𝐺 𝑞 𝑢 𝑘 +𝑣(𝑘) The process is described by a fifth order system 𝐺(𝑧)= 0.25 𝑧 −1 −0.97 𝑧 − 𝑧 −3 −0.95 𝑧 − 𝑧 −5 1−4.15 𝑧 − 𝑧 −2 −5.69 𝑧 − 𝑧 −4 −0.38 𝑧 −5 and noise process 𝑣 𝑘 =𝐹 𝑞 𝑒(𝑘)= 1−0.38 𝑞 − 𝑞 −2 1−1.90 𝑞 − 𝑞 −2 𝑒(𝑘) Van den Hof, P., and Ninness, B. (2005). System identification with generalized orthonormal basis functions. Automatica, 31, 12/3/2018
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Simulation study Compared to MATLAB OE routine (with correct model order). Assumed that the model order 𝑛<10. The weight parameter 𝛾 can be determined by cross-calidation. minimize ℎ∈ ℝ 𝑀 𝛾 𝑘=0 𝑁−1 𝑦 𝑘 − 𝑡=0 𝑀−1 ℎ 𝑡 𝑢 𝑘−𝑡 𝐻(ℎ) ∗ 12/3/2018 NPCW 2018
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Results Proposed method MATLAB: OE 12/3/2018 NPCW 2018
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Results Proposed method MATLAB: OE 12/3/2018 NPCW 2018
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Generalization The approach can be generalized to other bases. For example the Laguerre basis 𝜓 𝑘 𝑧 is defined as 𝜓 𝑘 𝑧 = 𝑧 1− 𝛼 2 𝑧−𝛼 1−𝛼𝑧 𝑧−𝛼 𝑘 , 𝑘=0, 1, 2,… where 𝛼 <1 corresponds to a real pole. The special case 𝛼=0 results in the natural basis functions 𝜓 𝑘 𝑧 = 𝑧 −𝑘 . 12/3/2018 NPCW 2018
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Generalization The system is then given by 𝐺(𝑧)= 𝑘=0 ∞ 𝑔 𝑘 𝜓 𝑘 𝑧 .
𝐺(𝑧)= 𝑘=0 ∞ 𝑔 𝑘 𝜓 𝑘 𝑧 . In general, it is not true that deg 𝐺 =rank 𝐻 𝑔 for other bases functions than 𝜓 𝑘 𝑧 = 𝑧 −𝑘 ! However, similar relations can be derived by utilizing properties of the signal and system transforms that the basis induce. 12/3/2018 NPCW 2018
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Summary Low-order output-error optimal models can be identified.
Combined classical results in systems theory with recent results in rank minimization. Future work Use generalized orthonormal basis functions. Identification of low-order MIMO systems. 12/3/2018 NPCW 2018
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Thank you “All models are wrong but some are useful” – Box 1978
12/3/2018 NPCW 2018
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12/3/2018 NPCW 2018
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