Talk:State-space representation

==issue in the moving object example, if m tends to zero, and k1 (or k2) tends to infinity, then the system wont be controlable or observable, it is important to write this as it is the basis of nuclear physics

Definition of a state
Somebody seriously needs to start at the beginning, and define/explain the meaning of a 'state'. That is, what does a 'state of a system' actually mean? Does it mean actual set of values (1V, 5 degree C, 0.2 metres/sec) of state variables at a particular time? Or does it mean just the set of state variables (without their actual quantities/values) at a particular time? Does 'state' mean 'state variable quantities/values at time t'? It will be good for the article to define the meaning of a 'state' clearly - with very clear examples. Like, if 'state' means the same thing as 'state vector', then it needs to be indicated. KorgBoy (talk) 07:19, 10 July 2017 (UTC)

Small error in the transfer function derivation
The Laplace transform of x'(t) is not sX(s), but sX(s)-x(0). I believe that the initial conditions term is usually assumed to be zero anyhow, to indicate no initial energy. I'm new to both controls and editing wikipedia, so I'll leave the actual editing to someone better equipped. :)

—The preceding unsigned comment was added by 70.171.42.102 (talk) 06:15, 21 March 2007 (UTC).

The transfer function, by definition, is the representation of the system in the s-domain with zero initial conditions. Perhaps the article could specify this? JMatopos (talk) 10:50, 23 March 2011 (UTC)

What else does this article need?
Definitely some diagrams of the systems.

I think another article on equilibrium is needed (phase diagrams, stabile points, unstable points, etc.). Cburnett 19:06, 13 Dec 2004 (UTC)

Also, state feedback was left out of the feedback section... might be good to include those tidbits. not a far stretch from what is already there, but it is entirely neglected.

Formulas to determine state-space represetnation of interconected systems: +--|S1|--+ |       v --+--|S2|--+-- -->--|S1|--|S2|-->-
 * parallel connection
 * serial connection

--+--|S1|--+-- ^       |  +<-|S2|--+ Where both S1 and S2 are given in state space representation.
 * feedback connection

--- I'm trying to learn some of the math from this article, so if I don't know what I'm talking about, that's why. That being said, I think there is some sort of error in the description of variables in the Moving Object Example. In the second breakout "acceleration of the objection" it explains a variable not used in the equation.

Section blanking
An anonymous user blanked content back in July 2006. . I reverted and attempted to restore edits made since then. Sorry if I missed some, but the amount deleted was wholly worth any loss since then. :/ Cburnett 01:10, 22 November 2006 (UTC)


 * I don't know if you want to restore more, so I prefer to ask you first. Section "State variables" is neither correct nor self-consistent, imo. State variables can be redundant (dependent); a state-space model with the number of state variables is a minimal realization, which is a special (yet important) case. One should define "realization" first, noting that it isn't unique (see some of my previous contributions). "System variables" aren't defined (I'd just write "values" instead). In "linearly independent", linearly is superfluous, because linearity isn't required here. Finally, in "the minimum number of state variables is equal to the transfer function's denominator", it's equal to the denominator's degree, not to the denominator itself. Engelec 11:42, 22 November 2006 (UTC)

Feedback with reference input
If $$r(t)$$ is the reference input, or setpoint, the correct equation for $$u(t)$$ is $$u(t) = K(r(t) - y(t))$$ (compare to other articles on control, e.g. PID controller). $$D$$ is usually not left out just for the sake of simplification, but because it is zero in strictly causal systems. Engelec 10:03, 14 December 2006 (UTC)

There are two issues, which I would like to raise: I think it's incorrect to call $$r(t)$$ the reference or setpoint. That would need to be at a summing junction before the feedback - surely we are trying to control $$y(t)$$. What is represented in the article is the "disturbance" - that which interferes with our ability to control to the set point. I will let this sit for a few weeks before I edit the article.

I disagree that the inclusion of $$D$$ makes the system acausal. This represents feed-through - for example, in a mechanical system if the feedback is by means of a magnetic actuator and the sensor is an inductive pick-up, it is possible for the sensor signal to contain a component due to the actuator field. This is causal and is described by $$D$$. In practical designs, one tries to avoid this by (for example) using a capacitative sensor, where the cross-talk is much smaller (perhaps even zero). It is true that with the inclusion of $$D$$ the transfer function is proper (and not strictly proper. This is actually a deficiency with the formalism and presented. In reality, $$D$$ itself will have a frequency dependence, which is strictly proper but there seems to be no way to include this in the formalism. It would be possible to incorporate this into $$A$$ but then the feed-through acts on $$x(t)$$ rather than $$y(t)$$ . That seems a preferable way to deal with the feed-through, with the behaviour of the transducers incorporated into the state space model $$A$$. In that case, $$D$$ is not needed. Frank van kann (talk) 08:51, 3 April 2014 (UTC)

Frank, it's debatable whether feed-throughs are causal or not since the information is output instantaneously with the input, that is $$y(t)=u(t) \; (1)$$, and this sparks controversies on the definitions of causality $$(t < t_0 \; vs \; t \leq t_0)$$. A much more precise and definitive argument is that feed-throughs are not realizable since information is moved instantaneously and this is not possible in physical systems, think of a mass which moves instantaneously and thus requires infinite energy. In fact we can demonstrate the problem by deriving (1) and applying limit to an instant $$\lim_{dt\rightarrow 0} (dy(t)-du(t))/dt=Inf \; (2)$$. This is expected since at $$t_0$$ systems do exhibit instantaneous variation, it is pretty much what the Initial Value Theorem tells us. The problem arises with the definition of our feed-through in (1). Replacing (1) in (2) gives $$\lim_{dt\rightarrow 0} (dy(t)-dy(t))/dt=0/0 \; (3)$$ and thus we obtain an undetermined result. There is a good debate about this on the talk page of proper transfer functions. --Nihilscientia (talk) 16:04, 25 August 2014 (UTC)

why no state feedback?
it is not obvious for me why normal feedback is presented in detail here whereas state feedback, which is one of the advantages of a statespace model, is not even mentioned. State feedback is misused in the second feedback figure. this is not state feedback but ouput feedback (corrected)! —Preceding unsigned comment added by 134.130.44.166 (talk) 15:48, August 30, 2007 (UTC)

Image copyright issues
I noticed that the block diagrams on this page are created with Simulink, which constitutes screenshots of a copyrighted program. I think we should create block diagrams of our own and replace the Simulink screenshots. --Jiuguang Wang (talk) 15:55, 4 July 2008 (UTC)
 * The program is copyright, not the images created by it. Otherwise one could never publish any picture created with Paint, Photoshop etc. Mesdale (talk) 20:39, 11 January 2011 (UTC)

Discrete time
There also needs to either be a discussion about discrete time in this article or a link to another page which discusses that topic.

S243a (talk) 19:28, 16 July 2009 (UTC)John Creighton

General linear formulation is time-variant
"Notice that in this general formulation, all matrices are supposed to be time-invariant, i.e. none of their elements can depend on time."

This is backwards; in the formulation shown the matrices do carry a time dependence. I corrected this. 95.222.120.117 (talk) 21:50, 22 July 2009 (UTC)
 * Your fix made it sound like the matrices *MUST* have a time dependence. They need not. I've fixed your fix. &mdash;TedPavlic (talk/contrib/@) 12:14, 23 July 2009 (UTC)

Other Edits
I removed CitationNeeded from "The state variables defined must be linearly independent; no state variable can be written as a linear combination of the other state variables or the system will not be able to be solved." because linear independence is a fundamental property of linear algebra necessary for fully specified systems. I am not sure that it entire applies in this situation, so as opposed to a Citation, perhaps it warrants removal for applicability. J.H. Gorse (talk) 18:08, 12 May 2013 (UTC)

Alternative Representation
There seems to exist an alternative representation of the canonical realizations: [| link] (or at least another definition). Unfortunately I don't understand it completely. Alorgen (talk) 18:38, 15 August 2013 (UTC)
 * It looks like the canonical form on that page is the same as the one on this page (just with ordering reversed). Zfeinst (talk) 22:59, 15

August 2013 (UTC) The output equation in the observable canonical realization is wrong; It should be [0 0 0 1] (as an illustration, the first state only depends on the input and itself; with [1 0 0 0], the output will be that from a 1st order system

MatLab/Octave
These programs can convert between transfer functions and state-space so I tried one to see how it matched to the page discussion. The coefficients are purely made up The result was (: used as separators): A = B = C = [0 : 0 : 0 : 0.20 ] D = 0
 * num=[1 2 3 4];  den=[5 4 3 2 1];
 * [A,B,C,D]= tf2ss (num,den); % from tf to ss
 * 0 : 0 :  0 : 0.20
 * 1 : 0 :  0 :  0.40
 * 0 : -1 : 0 : -0.60
 * 0 : 0 :  1 : -0.80
 * -4.00
 * -3.00
 * 2.00
 * 1.00

As far as I know this does not mean the transfer function and State space model are not related as the transfer function consists of 9 parameters whereas the state space representation consists of 25 parameters and actually can hold more information. So when converting from state space to transfer function information gets lost and when converting from transfer function to state space there are many valid solutions. If the poles and zeros of the state space and transfer functon are the same the state space system below is one of these valid solutions Maartenvaandrager (talk) 20:32, 19 January 2017 (UTC)

Dr. Snyder's comment on this article
Dr. Snyder has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

{{quote|text=State space representations originated in engineering and so an article like this is warranted in Wikipedia. However, they are now also applied in other areas such as economic and business forecasting, where they often contain unit roots to cope with the non-stationary features of time series typically found in these disciplines. This gives rise to quite different classes of linear state space representations and to the need for more complex estimation methods than those based the traditional Kalman filter. My view is that a separate article is required, titled say "State Space Models in Economics and Business", to cover the issues missing from the current Wikipedia article. There is now quite a large literature in this area, but two references cover many of its features:

Durbin, J., & Koopman, S. J. (2012). Time series analysis by state space methods (No. 38). Oxford University Press.

Hyndman, R. J., Koehler, A. B., Ord, J. K. & Snyder, R. D. (2008), Forecasting with Exponential Smoothing: The State Space Approach, Springer-Verlag.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Snyder has expertise on the topic of this article, since he has published relevant scholarly research:


 * Reference : Ralph D. Snyder & Anne B. Koehler, 2008. "A View of Damped Trend as Incorporating a Tracking Signal into a State Space Model," Monash Econometrics and Business Statistics Working Papers 7/08, Monash University, Department of Econometrics and Business Statistics.

ExpertIdeasBot (talk) 17:02, 27 July 2016 (UTC)

not all state spaces are Euclidean
The 6-dof position and orientation (and corresponding rates) of an aircraft is not a euclidean space, but it is still the state space. — Preceding unsigned comment added by Intellec7 (talk • contribs) 19:34, 11 September 2019 (UTC)

Z-transform: error?
I have an inkling that the formula:

$$z \mathbf{X}(z) - z \mathbf{x}(0) = \mathbf{A} \mathbf{X}(z) + \mathbf{B} \mathbf{U}(z)$$

may be slightly off. Inspecting its structure in comparison with the continuous counterpart the Laplace transform, and studying https://en.wikipedia.org/wiki/Z-transform for the first difference forward, I cannot help but think that it should possibly read:

$$(z - 1) \mathbf{X}(z) - z \mathbf{x}(0) = \mathbf{A} \mathbf{X}(z) + \mathbf{B} \mathbf{U}(z)$$

Am I right on this?Redav (talk) 10:40, 22 June 2020 (UTC)

Adding Observer Canonical Matrix
I think, it would be good if someone add the Observer Canonical Matrix just after the Controllable Canonical Matrix. That would make the article more complete (I do not have the time to do it anytime soon).

Remove  template (or clarify)
Can we remove this More footnotes template?

Or make a prioritized plan for what needs to be cited inline? Mcint (talk) 22:28, 13 December 2023 (UTC)
 * What are technical pages that do this well?
 * I'm tempted to put the template down in particular later subsections, the first section seems thoroughly covered.