Compute the H2 norm of a linear system
System; system object
(optional) equation(s) of the form option = value; specify options for the NormH2 command
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.
The NormH2 command computes the H2 norm of a linear system sys. Both continuous-time and discrete-time systems, and both single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems are supported.
For a stable SISO linear system with transfer function H⁡s, the H2 norm is defined in the frequency domain as:
For a MIMO linear system with transfer function Matrix H⁡s, the definition of H2 norm in the frequency domain is generalized to:
where ATypesetting:-_Hold⁡%H is the Hermitian transpose of Matrix A.
In the time domain, the H2 norm of a transfer function is calculated assuming that the stable transfer function H⁡s has a state-space representation:
so that H⁡s=Y⁡sW⁡s and H⁡s=C. sI−A−1. B.
where the feedforward matrix D=0 is necessary for the H2 norm to be finite. It follows that, for non-strictly-causal continuous-time linear time-invariant (LTI) systems (D≠0), the H2 norm is infinite.
From the above definitions, it can be demonstrated that the H2 norm of a continuous-time LTI is equivalent to:
where the Matrix P≽0 is calculated by solving a continuous Lyapunov equation:
In the frequency domain, the H2 norm of a discrete-time LTI system is defined by:
In the time domain, the H2 norm of a transfer function is calculated assuming that the stable transfer function H⁡z has a state-space representation:
so that H⁡z=C. zI−A−1. B+D.
From the above definitions, it can be demonstrated that the H2 norm of a discrete-time LTI is equivalent to:
where the Matrix P≽0 is calculated by solving a discrete Lyapunov equation:
For both continuous and discrete-time systems, the H2 norm is finite if the LTI system is asymptotically stable. It follows that for unstable systems, the H2 norm is infinite.
A deterministic interpretation of the H2 norm is that it measures the energy of the impulse response of the LTI system.
A stochastic interpretation of the H2 norm is that it measures the energy of the output response to unit white Gaussian noise inputs. A white noise process w⁡t has an expected or mean value 𝔼⁡w⁡t=0 and covariance matrix 𝔼⁡w⁡t·w⁡t+τT=𝕀·δ⁡τ, where 𝕀 is the Identity Matrix and δ is the Dirac delta function. It follows that the H2 norm is equivalent to: ‖H‖2=Trace⁡Covariance⁡sys,𝕀 from the interpretation above and DynamicSystems[Covariance].
Example 1 : Find the H2 norm of a system with discrete-time transfer function shown below.
sys1 ≔ TransferFunction⁡10⁢2⁢z+110⁢z2+2⁢z+5,discrete,sampletime=0.1:
Transfer Functiondiscrete; sampletime = .11 output(s); 1 input(s)inputvariable=u1⁡zoutputvariable=y1⁡ztf1,1=20⁢z+1010⁢z2+2⁢z+5
h2norm1 ≔ NormH2⁡sys1
Example 2 : Find the H2 norm of a continuous state-space MIMO system.
sys2 ≔ StateSpace⁡−5,3|3,−4,2,3|1,1,1,−2|12,1,0,0|0,0:
State Spacecontinuous2 output(s); 2 input(s); 2 state(s)inputvariable=u1⁡t,u2⁡toutputvariable=y1⁡t,y2⁡tstatevariable=x1⁡t,x2⁡ta=−533−4b=2131c=112−21d=0000
h2norm2 ≔ NormH2⁡sys2
Example 3 : Find the H2 norm of the following discrete system.
sys3 ≔ Coefficients⁡1,−2.841,2.875,−1.004,1,−2.417,2.003,−0.5488,discrete,sampletime=0.1:
Coefficientsdiscrete; sampletime = .11 output(s); 1 input(s)inputvariable=u1⁡zoutputvariable=y1⁡znum1,1=1,−2.841,2.875,−1.004den1,1=1,−2.417,2.003,−0.5488
h2norm3 ≔ NormH2⁡sys3
Example 4: Find the H2 norm of the system given by the following differential equation.
sys4 ≔ DiffEquation⁡ⅆⅆt⁢ⅆⅆt⁢x⁡t=−10⁢x⁡t−ⅆⅆt⁢x⁡t+w⁡t,w⁡t,x⁡t:
Diff. Equationcontinuous1 output(s); 1 input(s)inputvariable=w⁡toutputvariable=x⁡tde=ⅆ2ⅆt2x⁡t=−10⁢x⁡t−ⅆⅆtx⁡t+w⁡t
h2norm4 ≔ NormH2⁡sys4
Example 5 : Find the H2 norm of a non-strictly-causal continuous state-space MIMO system.
sys5 ≔ StateSpace⁡−5,3|3,−4,2,3|1,1,1,−2|12,1,2,1|3,7:
State Spacecontinuous2 output(s); 2 input(s); 2 state(s)inputvariable=u1⁡t,u2⁡toutputvariable=y1⁡t,y2⁡tstatevariable=x1⁡t,x2⁡ta=−533−4b=2131c=112−21d=2317
Since the H2 norm is infinite, an error message is displayed.
h2norm5 ≔ NormH2⁡sys5
Error, (in DynamicSystems:-NormH2) H2 norm is infinite for continuous 'sys' with D<>0 (system is not strictly causal).
Example 6: Find the H2 norm of an unstable system given by the continuous transfer function G(s).
sys6 ≔ TransferFunction⁡4⁢s+35⁢s4+7⁢s3+4⁢s2+3⁢s+1:
Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable=u1⁡soutputvariable=y1⁡stf1,1=4⁢s+35⁢s4+7⁢s3+4⁢s2+3⁢s+1
h2norm6 ≔ NormH2⁡sys6
Error, (in DynamicSystems:-NormH2) H2 norm is infinite for unstable systems. Unstable eigenvalues of 'sys': .324596325e-1-.6550790710*I, .324596325e-1+.6550790710*I
The DynamicSystems[NormH2] command was introduced in Maple 18.
For more information on Maple 18 changes, see Updates in Maple 18.
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