Discrete state-space representation in FPGA

Hello all,
My aim is to simulate a discrete state-space model is the cRIO FPGA (in order to use an observer).
I am currently trying to simulate it on my computer without using the discrete state-space VI from the Control Design and Simulation toolbox (because there is not state-space model VI for the FPGA). However my discrete state-space model representation does not work.
There is below the continuous state-space model:
Then I obtained the discretized model (using the zero-order-hold and t=0.01s):
Below is the discretized state-space model I designed (in order to design a similar model in the FPGA):
Here is the discrete state-space VI I'm using to compare the results:
The graph plots the state X1, which is infinitely increasing. However it should look like a first order, as you can see:
I understand why the state X1 is increasing like this, I know I'm missing something (integration saturation ?).
Thanks in advance
Regards!
PS: Does somebody know if implenting other controller than the PID is doable in the cRIO FPGA? I also asking myself about matrix inversion if I want to use the Kalman gain in my observer.
PS2: I apologize for my disorganized/unclear Labview files, I'm beginning with it.
Solved!
Go to Solution.
Attachments:
SS Discret Simulation.vi ‏544 KB
SS Discret Model.vi ‏21 KB

Hi,
I think the problem might be caused by rounding errors in your A and B matrices. Have you tried getting it to display the values with greater precision?
Matlab calculates those values as: A = [0.995, 0.009925; 0 0.99], B = [2.488e-5; 0.004975]
Regards,
Ian

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Hello, state space models in LabVIEW most of the time are being implemented by utilizing mathscript syntax
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There´s a couple of nice tutorials for control design with these tools which I´d like to point you to as a first step.
http://www.ni.com/white-paper/6368/en/
http://www.ni.com/white-paper/6368/en/
http://www.ni.com/white-paper/6435/en/
http://home.hit.no/~hansha/documents/lab/Lab%20Work/MathScript/MathScript%20Lab%20-%20Part%20II.pdf
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Marco Brauner AES NIG

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Solved!
Go to Solution.
Attachments:
Cavities.vi ‏30 KB
Matrixes.vi ‏27 KB

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Hi!
I'm currently doing a project and I'm stuck and am in need for some help.
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Solved!
Go to Solution.
Attachments:
State-Space Model.png ‏11 KB

We have so many different ways to do this. Try to look at the following shipping examples:
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FIFO emulation.GIF ‏10 KB

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I am trying to do a fairly simple state space task with Control Design and Simulation toolbox, however I am quite new at using this toolbox and need some help :-)
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Message Edited by SCMAJA on 10-16-2009 11:21 AM
LabVIEW 8.6 / 2009 / 2010
Vision Development Module 8.6 / 2009 / 2010
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2 DOF robot with ramp input, polar coordinates plot, state space.vi ‏49 KB

Hello Simon,
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Attachments:
time out lv11.vi ‏8 KB
time outlv12.vi ‏11 KB

I usually just use the Elapsed Time express VI. I have taken it and made my own version to use, but the concept is the same. But it does really depend on what you are doing. Most times, you have the option for the user to abort. So you will want to bounce between a Check Time and Wait For Event states. The Wait For Event state will have a timeout of its own (something like 100ms usually works well). Instead of using the Tick Count, use the timestamp instead. It will give you a larger range to work with and you won't have to worry about the overflow. It is also easier to debug since you can see the actual time that it starts the current time.
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Input for CD Linear Simulation.vi in a Simulation Loop

Hi I'm new to LabView and I was trying to implement a real time estimator for a linear system, so a steady
state gain is acceptable (estimator doesn't need to be adaptive). It will be
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Hi Mike,
Thanks for the reply.
I browsed through some examples and tried some things; here are the updates, maybe useful for others in the future.
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How to make 1D Array but with only one element filled in Real Time

Hi folks,
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Ok, I did it that way. But I am facing another problem right now...
At some point the Discrete Observer return a NAN array, you will see the code in the code snippet?
I get rid of that component by component, but the observer gets "stuck" in it. So my Control law is zero... but the state stimate is NAN.
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RT - Pole Placement + Complete Observer.vi ‏40 KB

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Dear Chin ho,
After going through your questions I found some documentation which will be useful for you. But I you still have more questions, you can reply me any time.
1- About the data and amount of it I couldn't find any info but maybe you can use the new functions in LV version 2009 and find your answer.
Estimating States of Nonlinear Stochastic State-Space Models with Extended Kalman Filters
In previous versions of the LabVIEW Control Design and Simulation Module, you can use the Discrete Kalman Filter function and the Continuous Kalman Filter function to estimate the states of a linear discrete or linear continuous stochastic state-space model, respectively. In the LabVIEW 2009 Control Design and Simulation Module, you can use the Continuous Extended Kalman Filter function and the Discrete Extended Kalman Filter function to estimate the states of a nonlinear continuous or a nonlinear discrete stochastic state-space model, respectively.
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The Continuous Extended Kalman Filter function and the Discrete Extended Kalman Filter function linearize the nonlinear system either by calculating a Jacobian matrix internally or by using an external Jacobian matrix that you define. Use the SIM Continuous Jacobians template VI or the SIM Discrete Jacobians template VI, located in the labview\templates\Control and Simulation directory, to define an external Jacobian matrix.
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2- I think that I found a pdf document about this part of your question which will be helpful. If you would like to give your email address, I can send it to you.
3- About this part you can check the help function in LabVIEW when you open the "CD Convert Transfer function To State-space" function in your front panel.
I pasted some info about the mathematic part below.
The LabVIEW Control Design and Simulation Module provides tools to study the dynamics of systems described by linear time-invariant (LTI) continuous and discrete models. You can create deterministic state-space, transfer function, and zero-pole-gain models. You also can create stochastic state-space models and the second-order statistics noise models. You can use these forms to describe both single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems.
Continuous transfer function and zero-pole-gain models use the s variable to define time, whereas discrete transfer function and zero-pole-gain models use the z variable to define time. Continuous state-space models use the t variable to define time, whereas discrete state-space models use the k variable to define time.
Deterministic State-Space Model
Continuous
x(t) = Ax("t) + Bu(t)
y(t) = Cx(t) + Du(t)
Discrete
x(k + 1) = Ax(k) + Bu(k)
y(k) = Cx(k) + Du(k)
Stochastic State-Space Model
Continuous
x(t) = Ax(t) + Bu(t) + Gw(t)
y(t) = Cx(t) + Du(t) + Hw(t) + v(t)
Discrete
x(k + 1) = Ax(k) + Bu(k) + Gw(k)
y(k) = Cx(k) + Du(k) + Hw(k) + v(k)
Second-Order Statistics Noise Model
Q = E{w . wT} – E{w} . ET{w}
R = E{v . vT} – E{v} . ET{v}
N = E{w . vT} – E{w} . ET{v}
where
t is continuous time.
k is the model sampling time multiplied by the discrete time step, where the discrete time step equals 0, 1, 2, …
x is the model state vector.
u is the model input vector.
y is the model output vector.
w is the process noise vector.
v is the measurement noise vector.
A is an n × n state matrix of the given model.
B is an n × m input matrix of the given model.
C is an r × n output matrix of the given model.
D is an r × m direct transmission matrix of the given model.
n is the number of model states.
m is the number of model inputs.
r is the number of model outputs.
G is a matrix relating w to the model states.
H is a matrix relating w to the model outputs.
Q is the auto-covariance matrix of w.
R is the auto-covariance matrix of v.
N is the cross-covariance matrix between w and v.
E{} denotes the expected value or the mean of the enclosed term(s).

Discrete state-space representation in FPGA

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