state-space representation
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From: fisico32 on 24 Jul 2010 10:38 hello forum, I know that a linear system can be represented in its state space representation. I am actually more used to the convolution integral, transfer function methods... What is the advantage of describing the dynamics of a system in state space representation? I know that control people used that a lot. Does a state space description always involve a feedback? What is exactly the state(s)? We have an input signal x(t), an output signal y(t)....The state variable u(t) represents the current state of the system....If a system has no memory (static system) it will simply mean that the state is the same at the output y(t).... Or is there always a difference between the output and state? Among the possible physical quantities that belong to a system, do we freely choose which ones will be the output and which the state variables? thanks! fisico32
From: Tim Wescott on 24 Jul 2010 13:15 On 07/24/2010 07:38 AM, fisico32 wrote: > hello forum, > > I know that a linear system can be represented in its state space > representation. I am actually more used to the convolution integral, > transfer function methods... > > What is the advantage of describing the dynamics of a system in state space > representation? First, when you're designing a control system you can preserve a lot of information about the structure of the system in the state-space form, which gets lost when everything is lumped into a transfer function. Second, if you're facing a nonlinear system, it is much easier to represent said nonlinear system in state space form -- there is no such thing as a 'nonlinear transfer function', but state-space is the generic form for a nonlinear dynamic system. Third, if you implement a system in state space on a processor, you can dodge numeric issues in ways that -- you can't with single transfer functions. The easiest to show in this group is the "cascade of 2nd-order biquads" structure that one traditionally needs in an IIR filter: x_n = A * x_{n-1} + B * u_n; [ A_1 C 0 ] A = [ 0 A_2 C ] [ 0 0 A_3 ] where A_k is a 2x2 matrix that makes a biquad, and C is either [ 0 0 ] [ 0 1 ] C = [ ] or C = [ ] [ 1 0 ] [ 0 0 ] depending on the choice of A_k. > I know that control people used that a lot. > Does a state space description always involve a feedback? No, it just gives a nice formal way to put feedback in. > What is exactly the state(s)? Subject to interpretation and traditional usage, if you dig deep enough. Basically, if you describe your system as an integral equation, the states are the things on the output side of the integral. > We have an input signal x(t), an output > signal y(t)....The state variable u(t) represents the current state of the > system....If a system has no memory (static system) it will simply mean > that the state is the same at the output y(t).... No, it means that there are no states at all, just some plain old (i.e. memoryless) function that connects input to output. > Or is there always a difference between the output and state? No, but it's not an "either/or" situation. In general the system output is a function of the states and (possibly) the input. > Among the possible physical quantities that belong to a system, do we > freely choose which ones will be the output and which the state variables? No, because you have to have enough states to fully express the system. If you've got a second order system and both states are "visible" then those states are also outputs -- but you can't arbitrarily choose to have just one state then. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
From: HardySpicer on 24 Jul 2010 20:50 On Jul 25, 2:38 am, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > hello forum, > > I know that a linear system can be represented in its state space > representation. I am actually more used to the convolution integral, > transfer function methods... > > What is the advantage of describing the dynamics of a system in state space > representation? I know that control people used that a lot. > Does a state space description always involve a feedback? > > What is exactly the state(s)? We have an input signal x(t), an output > signal y(t)....The state variable u(t) represents the current state of the > system....If a system has no memory (static system) it will simply mean > that the state is the same at the output y(t).... > Or is there always a difference between the output and state? > > Among the possible physical quantities that belong to a system, do we > freely choose which ones will be the output and which the state variables? > > thanks! > fisico32 All this stuff originated during the US missile program and for general rocket research. States can have physical meaning, position,velocity acceleration etc and it is form of simplification, splitting an nth order ODE in n first order ODE's. Yes it does fit in nicely with Kalman filters, state feedback and the like. There was a movement away from that in the mid-late 70s in europe,going back to polynomial methods and solving Diophantine equations instead of Ricatti equations. The two methods are equivalent. It is also possible to get descriptions that are mid-way bewteen polynomial and state-space. State-space has a lot going for it in filter design too as the previous poster points out. Hardy
From: Michael Plante on 26 Jul 2010 00:32 Tim wrote: >On 07/24/2010 07:38 AM, fisico32 wrote: >> hello forum, >> >> I know that a linear system can be represented in its state space >> representation. I am actually more used to the convolution integral, >> transfer function methods... >> >> What is the advantage of describing the dynamics of a system in state space >> representation? > >First, when you're designing a control system you can preserve a lot of >information about the structure of the system in the state-space form, >which gets lost when everything is lumped into a transfer function. To the OP: If you want to dig deeper into this area, I would suggest looking into the "Kalman decomposition", where you transform and partition the state space system into a canonical form so you can easily examine [un]controllable and [un]observable behaviors and states. Some information is lost in going to a transfer function. You might look at minimal realizations and pole/zero cancellation. All of this is covered in the overpriced but understandable book by Chi-Tsong Chen (0-19-511777-8) for both continuous and discrete-time cases. If the states represent physical variables that have limits (such as op-amp outputs, which can saturate at or near the rails), throwing away the states may prevent you from easily determining the situations where the physical variables leave the assumed-linear regime. Additionally, I don't commonly see MIMO systems described in transfer function form (though they certainly can be), partly because if it's all one intercoupled system, there's probably just a whole lot more to write down to describe the same thing. On the flip side, at least with *continuous* time, you might consider that arbitrary pure delays are more easily represented in a transfer function (but are not rational). You may still be able to use approximations (e.g., Pade) so that this is not a problem.
From: Rune Allnor on 26 Jul 2010 07:21 On 24 Jul, 16:38, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > hello forum, > > I know that a linear system can be represented in its state space > representation. I am actually more used to the convolution integral, > transfer function methods... > > What is the advantage of describing the dynamics of a system in state space > representation? The intro chapters in the book by Friedland contains a discussion about the pros and cons with state space representations: http://www.amazon.com/Control-System-Design-Introduction-State-Space/dp/0486442780/ref=sr_1_1?ie=UTF8&s=books&qid=1280143107&sr=8-1 It essentially says what others have already said; that the state space is a more general formulation that contains the usual transfer function formalism, but hat also can be extended in a number of directions with relative ease. Whatever 'relative' might mean... Rune
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