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state transition matrix

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niga

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Hi

I understand how to get to the state equations and state transition matrix.But ....i just cant figure out what is the practical aspect to it,the state transition matrix.
Can someone help?Thanx always.
 
It allows easy conversion between the standard canonical forms (control,observer and modal) such that control coefficients could be easily extracted. It's way simpler than finding the inverse or eigenvectors of a large matrix.
 
thanx for ur help checkmate.
I guess it will be clearer to me as i move ahead,eh?
cud u suggest some books i cud get hold of to understand all this stuff.
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Oh!Oh! But I am still an Electronics Newbie!!!
 
x'=ax+bu

which gives

x(t)=exp(at) x(0) + integration exp a(t-tau) bu(tau) dtau

what does this tau refer to? :?

the state transition matrix is just exp(At) right?


niga the newbie
 
Yes, the state transition matrix is simple exp(at).
The RHS term is a convolution, and tau is the variable of convolution. You do know convolution right, where you multiply 2 functions, find the area, then shift one function and repeat the process. tau is the variable that represents how much shifting one function is shifted wrt the other.

The 2 equations are equivalent.
You take the LT of x'=ax+bu
giving X(s)=x(0)/(s-a)+bU(s)/(s-a)
Take inverse LT and you get the 2nd equation.
The RHS term is evaluated by virtue that a product in the s-domain is equivalent to a convolution in the t-domain. If you see the LHS term as the response provided by an impulse of magnitude x(0) at t=0, then the response by the u(t) term should naturally be the convolution of the impulse response.
 
On a side note, my first response was totally crap. I was confused with state transformation matrix. I've never really used much of the state transition matrix, and I always feel that the beauty of using state equations are the manipulative power of matrices and the neater expressions of using differentials over exponentials.
 
Okay checkmate
i think i will move ahead and find out on transformation matrix etc.
thankyou. :)
 
but why shud one use convolution inorder to get the state transition matrix
from
x'=Ax+Bu

x(t)=exp(at)x(0) + integral[exp(a(t-tau)) bu(tau) dtau

where tau is the variable of convolution.what is the value of tau....
i have read that convolution is used when integration is reqd,for instance,in capacitors and inductors which dont respond to an input immediately.....but can someone PLEASEEEEEEEEEE help me understand how convolution really works in this case?i understand wat convolution is but to get state transition...... :?

Thanks for any help always!
 
The convolution term does not form part of the state transition matrix. The state transition matrix comes about purely by the initial response x(0), and not the input u(t).
Next, as I have already mentioned, tau is how much one function has been shifted in relation to the other during convolution. In this case, the two functions are u(t) and exp(at) respectively. In convolution, the limits of the integral are +-infinity, so tau will assume all these values during integration.
I guess many texts have taught you how to do convolution, but the principles behind it is really quite intuitive. You jab a billard ball, it responds by rolling, EVEN AFTER the jabbing has stopped. If you see the jabbing as an impulse input, so now imagine that instead of jabbing, you give a continuous input. If the system is linear, the law of superposition applies and the continuous input can be seen as a summation of impulse inputs. If each impulse input gives you the impulse response, you just have to sum the impulse response of all the impulse inputs, ie integrate and shift, that is your convolution.
In the derivation of the state transition matrix, the impulse response is exp(at). The continuous input is u(t). So the response due to u(t) is the convolution of u(t) and exp(at).
 
Very kind of u to have explained this stuff so clearly to me.
Thanx again and again,Checkmate.
:)
 
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