1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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,
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14. 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

15. Results
Proposed method
MATLAB: OE
12/3/2018
NPCW 2018

16. Results
Proposed method
MATLAB: OE
12/3/2018
NPCW 2018

17. 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

18. 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

19. 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

20. Thank you “All models are wrong but some are useful” – Box 1978
12/3/2018
NPCW 2018

21. 12/3/2018
NPCW 2018