I have a set of time-series data that consists of inputs $u_k$ where $ u \in R $ and $k = 1 ... T$, and outputs $ y_k $ where $ y \in R^2 $ and also $k = 1 ... T$, from a given system. I believe this system can be modeled in discrete canonical state space form as $$ x_{k+1} = Fx_k + Gu_k $$ $$ y_{k+1} = Cx_{k+1} $$ In this case, $ y_{k} = x_{k} $ so the form becomes $$ y_{k+1} = Fy_k + Gu_k $$ Given all of the $ y_k $ and $ u_k $ I am fairly sure that I should be able to solve for $ F $ and $ G $, but how do I actually do this? Some work on paper got me nowhere and I can't seem to find anything on the internet.
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2$\begingroup$ The general topic is "System Identification", but I think for success you probably need some idea of what your system is. Me being me I'd switch to the frequency domain and get a transfer function. au.mathworks.com/help/ident/gs/about-system-identification.html $\endgroup$– Greg LocockJan 4 at 0:33
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$\begingroup$ 1 Is $F$ a $2\times 2$ matrix ? 2 are the measurements $y_k$ noisy ? i.e. $y_{k+1} = Cx_{k+1} + \mathbf{w_{k+1}}$ where $w$ is unknown noise. $\endgroup$– AJNJan 4 at 12:09
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$\begingroup$ @AJN 1) Yes 2) yes but negligibly so, which is why I excluded it. $\endgroup$– ian.cookeJan 4 at 15:26
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$\begingroup$ OK, so what are the dimensions of y and u? $\endgroup$– Greg LocockJan 4 at 17:25
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$\begingroup$ Have you seen papers using Hankel matrices and Singular value decomposition methods like these ? 1, 2, 3 $\endgroup$– AJNJan 5 at 15:47
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The answer to this turned out to be to frame the problem as a least-squares problem. Specifically I found & used State Space Model Identification by Least Squares. This allowed me to solve for the rows of $F$ and $G$ one-by-one.