DynamicSystems[NormH2]  Compute the H2 norm of a linear system

Calling Sequence


NormH2(sys)


Parameters


sys



System; system object

opts



(optional) equation(s) of the form option = value; specify options for the NormH2 command





Options


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checkstability = truefalse

True means check whether the system is stable; if it is not stable, an error occurs. False means skip the check. The default is true.


Description


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The NormH2 command computes the H2 norm of a linear system sys. Both continuoustime and discretetime systems, and both singleinput singleoutput (SISO) and multipleinput multipleoutput (MIMO) systems are supported.


Continuoustime


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For a stable SISO linear system with transfer function , the H2 norm is defined in the frequency domain as:

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For a MIMO linear system with transfer function Matrix , the definition of H2 norm in the frequency domain is generalized to:


where is the Hermitian transpose of Matrix A.

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In the time domain, the H2 norm of a transfer function is calculated assuming that the stable transfer function has a statespace representation:


where the feedforward matrix is necessary for the H2 norm to be finite. It follows that, for nonstrictlycausal continuoustime linear timeinvariant (LTI) systems (), the H2 norm is infinite.


From the above definitions, it can be demonstrated that the H2 norm of a continuoustime LTI is equivalent to:


where the Matrix is calculated by solving a continuous Lyapunov equation:



Discretetime


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In the frequency domain, the H2 norm of a discretetime LTI system is defined by:


where is the Hermitian transpose of Matrix A.

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In the time domain, the H2 norm of a transfer function is calculated assuming that the stable transfer function has a statespace representation:


From the above definitions, it can be demonstrated that the H2 norm of a discretetime LTI is equivalent to:


where the Matrix is calculated by solving a discrete Lyapunov equation:


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For both continuous and discretetime systems, the H2 norm is finite if the LTI system is asymptotically stable. It follows that for unstable systems, the H2 norm is infinite.

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A deterministic interpretation of the H2 norm is that it measures the energy of the impulse response of the LTI system.



Compatibility


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The DynamicSystems[NormH2] command was introduced in Maple 18.



Examples


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Example 1 : Find the H2 norm of a system with discretetime transfer function shown below.
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 (1) 
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 (2) 
Example 2 : Find the H2 norm of a continuous statespace MIMO system.
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 (3) 
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 (4) 
Example 3 : Find the H2 norm of the following discrete system.
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 (5) 
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 (6) 
Example 4: Find the H2 norm of the system given by the following differential equation.
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 (7) 
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 (8) 
Example 5 : Find the H2 norm of a nonstrictlycausal continuous statespace MIMO system.
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 (9) 
Since the H2 norm is infinite, an error message is displayed.
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Example 6: Find the H2 norm of an unstable system given by the continuous transfer function G(s).
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 (10) 
Since the H2 norm is infinite, an error message is displayed.
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