How to transfer sensor function to state space model?

In my sensor implementation, I got a function:

Y(t)=cos(theta(t))*( U(t) + W(t)+ C )

Where t  denotes the sampling time, Y denotes the measured azimuth,
'theta'  denotes the angle of the hilly road, U denotes the real
azimuth(input), W denotes the zero mean Gaussian white noise, C denotes
calibration error(constant).

How to transfer this relation function to the state space model?

Thanks!

		
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gorlic wrote:
> In my sensor implementation, I got a function:
> 
> Y(t)=cos(theta(t))*( U(t) + W(t)+ C )
> 
> Where t  denotes the sampling time, Y denotes the measured azimuth,
> 'theta'  denotes the angle of the hilly road, U denotes the real
> azimuth(input), W denotes the zero mean Gaussian white noise, C denotes
> calibration error(constant).
> 
> How to transfer this relation function to the state space model?
> 
> Thanks!
> 
> 		
> 		This message was sent using the Comp.DSP web interface on DSPRelated.com

Using 'n' for time, to denote sampling:

x(n+1) = A x(n) + B_i U(n) + B_n W(n)
y(n) = C_m x(n) + D_i U(n) + D_n W(n)

A = 1, B_i = 0, B_n = 0, C_m = 1, D_i = cos(theta(n)), D_n = 
cos(theta(n)), x(0) = C.

This is saved from being a no-state model (A, B_i and B_n empty) by the 
constant error.  Otherwise you don't have any states, so it's a little 
silly modeling it with state-space.

What are you really trying to do, and why do you want to model stateless 
systems with state-space representations?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott wrote:

  ...

> What are you really trying to do, and why do you want to model stateless
> systems with state-space representations?

Homework, probably.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Tim Wescott wrote:
>
>  ...
>
>> What are you really trying to do, and why do you want to model
stateless
>> systems with state-space representations?
>
I want to use the kalman filter to detect the sensor fault. Any better
way?

Thanks!


		
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gorlic wrote:

> Tim Wescott wrote:
> 
>> ...
>>
>>
>>>What are you really trying to do, and why do you want to model
> 
> stateless
> 
>>>systems with state-space representations?
>>
> I want to use the kalman filter to detect the sensor fault. Any better
> way?
> 
> Thanks!
> 
> 
> 		
> 		This message was sent using the Comp.DSP web interface on DSPRelated.com

If you have a model of the expected sensor output in azimuth and you 
expect the sensor to behave in a markedly different way when it is 
broken then a kalman filter of some sort may be part of the solution -- 
your kalman filter structure would then have more to do with the vehicle 
dynamics than the (nonexistent) sensor dynamics.

Have fun -- BIT has always struck me as a good way to generate false alarms.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

>Using 'n' for time, to denote sampling:
>
>x(n+1) = A x(n) + B_i U(n) + B_n W(n)
>y(n) = C_m x(n) + D_i U(n) + D_n W(n)
>
>A = 1, B_i = 0, B_n = 0, C_m = 1, D_i = cos(theta(n)), D_n = 
>cos(theta(n)), x(0) = C.
>
>This is saved from being a no-state model (A, B_i and B_n empty) by the 
>constant error.  Otherwise you don't have any states, so it's a little 
>silly modeling it with state-space.
>

>Tim Wescott
>Wescott Design Services
>http://www.wescottdesign.com
>

First of all, thanks for help! But I still have some questions:

1)  To my knowledge, I think c_m = cos(theta(n)), so that it can go back
to the sensor relation function. 
2)  Why you say that A (=1) is empty?
3)  y(n) is not in the form of measurement model: z(k)= H(k)x(k)+ v(k).
How to deal with U(n) when a kalman filter is used to estimate the state
of the sensor?

Thanks!

		
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