MATLAB: Conversion between state space and transfer function

state-spacetransfer function

these are the state space variables:

A=[-1 -1;6.5 0], B=[1 1;1 0], C=[1 0;0 1], D=[0 0;0 0]

when i convert them to transfer function, it works fine but, when i convert them back to state space then i dont get the same matrices. please help me out.

Best Answer

An infinite number of state space realizations correspond to a single transfer function. From a transfer function, the Control System Toolbox function ss() computes a realization that it believes has some desirable properties. You have some control, but not much, over what ss() returns. See the doc page for ss for more details.

Once you convert your original state space realization to a transfer function, CST doesn't really care, nor know, about the original state space realization from which you generated the transfer function. So, when it converts that tf back to ss, it just returns what it thinks is best.

However, the state space realization that you start with and the one that you end with after the two-step conversion will still have the same input/output dynamics, e.g., step responses and Bode plots should match (to within numerical precision).

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MATLAB: Do state space and its equivalent transfer function models yield different step responses

Hi there,

Examine the state-space model sys_ss:

>> sys_ss

Note that it has a non-unity E matrix. This happens because your H, a pure derivative, is improper. The connect command represents that block as a two-state ss with a non-unity E matrix - a descriptor ss. When connect builds the interconnected system, it preserves those states, so it yields a descriptor system.

And that means that your call to ss2tf, which only uses A,B,C, and D, is missing part of the dynamics of the system. sys_tf doesn't represent the same dynamics as sys_ss.

This is one of the reasons that ss2tf is not a great command to use when converting from ss to tf form. Try this instead:

>> sys_tf = tf(sys_ss)

This time when you plot the step responses, they match up. Also, tf removes those two extra states.

You can also reduce and simplify sys_ss into explicit form (E = 1) using the 'explicit' option of the ss command:

>> sys_ss_ex = ss(sys_ss,'explicit')

Now all of the system dynamics are described by A,B,C, and D.

For more information about explicit and descriptor state-space models, see the reference page for ss.

>> doc ss

Hope this helps!

MATLAB: Conversion from State Space to TF and then back to State Space

Some points:

Don't do "algebra" with tf variables, use the functions series, parallel and feedback.

sys_mimo1 = ss(sys_mimo) is going to convert the tf model to a ss model, I don't think there is a need for the extra step with ssdata.

This one here

sys_mimo_tf(1,2) = sys_mimo_tf(1,2)*feedback(sys_mimo_tf(1,2),C);

is probably not doing what you want it to, it will give you the transfer function

sys_mimo_tf(1,2)^2 /  1 + sys_mimo_tf(1,2)*C

so that's probably where many extra states come from. You should expect 2 extra states per added PID.

Lastly, look at the Hankel Singular Values or a similar measure to figure out whether there might be a lot of almost unobservable/almost uncontrollable states in your resulting model. That would indicate numerical issues.

Best Answer

Related Solutions

MATLAB: Do state space and its equivalent transfer function models yield different step responses

MATLAB: Conversion from State Space to TF and then back to State Space

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