A model is said to be represented in state space form if it can be written as

$$Y_t = G_t X_t + W_t$$

where $$X_t =F_t X_{t-1} + V_t $$

and $W_t,V_t$ are some kind of random noise.

The steady state instead is written as

$$Y_t = G_t X_{t|t-1} + W_t$$

where $$ X_{t+1|t} = F_t X_{t|t-1} + V_t $$

This should remind me that we are using the information up to time $t-1$ to find $Y_t$, but is this not true even for a model in state space form? What is the true difference here?


State space form originated in control theory. There, the usual representation is similar to that shown in your first two equations, with additional terms for controller outputs that are inputs to the process. The equation propagating the state X from the previous state is called the state equation (your second equation), and the relationship of the measurement Y to the state is called the measurement equation (your first equation). (There are differences from the above in how the equations are typically written. By convention, V is the measurement noise and W is the process noise, the opposite of what's shown above. And the matrix shown as G above is, by convention, called H. But none of that matters other than avoiding confusion in discussions with others.) X is an internal representation. Y is what is measured. One typical goal, as in the case of Kalman filtering, is to take the measurements Y, and estimate the state X. The emphasis on estimating the state X is because with the state equation, predictions about the future can be made, and hence predictions of Y follow as well.

The system representation does not change when the system happens to achieve a steady state.

At steady state, by definition, the state X is not changing over time. That implies that F cannot not be changing over time either, because if F were to change, then X would have to change by the state equation. Subscripts referencing time are not relevant when solving for steady state solutions. So, at steady state, the state equation equation just reduces to X = FX (At a steady state, the process noise would have to be zero as well, otherwise by the state equation, process noise would change X).

For a useful steady state solution, you normally have another set of inputs, as is typical for process control. Otherwise you're left with the only solutions being X = 0, unless F is the identity matrix.

Normal definitions of a steady state system would include that the entire system is not changing, so that the measurement matrix (G in your notation) should be constant as well. I suppose that if you're differentiating a steady state system from a steady state X, then you could allow the measurement matrix to change over time. But your first equation still wouldn't change.

If I'm understanding your third equation, it says that Y at time t depends on X at time t, given the X values up to time t-1. The reference to t-1 is not needed there, whether at steady state or not. (With a similar comment about t+1|t and t|t-1 in the fourth equation). In state space representation, X always incorporates all previously known information --- the "t-1" is implied. Also, the state equation also allows you to always predict forward in time as far as you want, in terms of expected value (that is, assuming no process noise). Similarly, if F is invertible, you can reconstruct previous states (assuming no process noise). These issues are central to state space representation and its applications, because it means that for instance, for estimation, you never need to look backward further in time once you have an estimate of X at some time t.

So, in short, the third and fourth equations aren't needed, and the t|t-1 references aren't needed, if you're just talking about state space representation.

Sometimes you see references like the t|t-1 when looking at particular estimation solutions. For instance, Kalman filtering has two steps: a prediction from the previous state just using the state equation (ignoring measurements), and then a correction step that accounts for the measurements, balancing the measurement noise against the process noise. When differentiating between those two steps, sometimes that sort of notation is used to distinguish the intermediate prediction from the final result.

But all of that is a separate issue from making a steady state assumption.

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  • $\begingroup$ Thank you! so basically what I have written are two state space models just utilizing different notation correct? and could you please expand on what "a correction step that accounts for measurments" means? my understanding is that we use $F_tX_t $ as a predictor for $X_{t+1}$ but then we also use $$ \hat{X}_t + K_t ( Y_t - F_t \hat{X}_t )$$ this confuses me a bit maybe I should write another question. $\endgroup$ – Monolite Nov 13 '17 at 11:44
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    $\begingroup$ Right, the second two equations seem like just another way of describing the same system. Perhaps I'm missing some context as to why there's that t|t-1 notation in there. I'm answering the question just in terms of the state space representation, but maybe the question context is really about an estimation solution such as a Kalman filter. A full description can be found in the book Applied Optimal Estimation by Gelb, from MIT press It's on Amazon, but you can also Google to find a free pdf online. See p. 111. The equation in the above comment isn't quite right...see next comment... $\endgroup$ – gms Nov 13 '17 at 16:09
  • $\begingroup$ Gelb uses notation (-) to indicate the prediction, and (+) to indicate the correction, the final result. With your matrices, prediction is $$\hat{X}_t (-) = F_t \hat{X}_{t-1} (+) $$ and the final result (correction step) is $$ \hat{X}_t (+) = \hat{X}_t (-) + K_t ( Y_t - G_t \hat{X}_t (-) )$$ You need that G in there to compare the predicted measurement to the actual measurement - getting units and dimensions right. Calculating K, the Kalman gain matrix, takes more equations, including calculating the covariance matrix of the estimate for (-) and (+). See Gelb $\endgroup$ – gms Nov 13 '17 at 16:40
  • $\begingroup$ Loosely speaking, K is a form of ratio of process noise to measurement noise. If you have large measurement noise, K approaches 0, and you believe the prediction, practically ignoring the measurement. If you have large process noise, K will work out so that you ignore the prediction and just use the measurements. $\endgroup$ – gms Nov 13 '17 at 16:45
  • $\begingroup$ See also the related question stats.stackexchange.com/questions/313478/… $\endgroup$ – gms Nov 13 '17 at 19:48

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