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DynamicSystems

  

NormH2

  

Compute the H2 norm of a linear system

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

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

• 

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

• 

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.

Continuous-time

• 

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

  

‖H‖2=Hjω2ⅆω2π 

• 

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

  

‖H‖2=TraceHjω?·Hjωⅆω2π

  

where A? 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 Hs has a state-space representation:

  

x.=Ax+Bw

  

y=Cx

  

so that Hs=YsWs and Hs=C. sIA1. 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 (D0), 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:

  

‖H‖2=TraceC·P·CT

  

where the Matrix P0 is calculated by solving a continuous Lyapunov equation:

  

A·P+P·AT+B·BT=0 

Discrete-time

• 

In the frequency domain, the H2 norm of a discrete-time LTI system is defined by:

  

‖H‖2=TraceHⅇjω?·Hⅇjωⅆω2π

  

where A? 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 Hz has a state-space representation:

  

xk+1=Axk+Bwk

  

yk=Cxk+Dwk

  

so that Hz=C. zIA1. B+D.

  

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

  

‖H‖2=TraceC·P·CT+D·DT

  

where the Matrix P0 is calculated by solving a discrete Lyapunov equation:

  

A·P·ATP+B·BT=0

• 

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 wt has an expected or mean value 𝔼wt=0 and covariance matrix 𝔼wt·wt+τT=𝕀·δτ, where 𝕀 is the Identity Matrix and δ is the Dirac delta function. It follows that the H2 norm is equivalent to: ‖H‖2=TraceCovariancesys,𝕀 from the interpretation above and DynamicSystems[Covariance].

Examples

with( DynamicSystems ):

Example 1 : Find the H2 norm of a system with discrete-time transfer function shown below.

sys1 := TransferFunction(10*(2*z+1)/(10*z^2 + 2*z + 5), discrete, sampletime = 0.1):

PrintSystem(sys1);

Transfer Functiondiscrete; sampletime = .11 output(s); 1 input(s)inputvariable=u1zoutputvariable=y1ztf1,1=20z+1010z2+2z+5

(1)

h2norm1 := NormH2(sys1);

h2norm12.46238673166698

(2)

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>|<1/2,1>>, <<0,0>|<0,0>> ):

PrintSystem(sys2);

State Spacecontinuous2 output(s); 2 input(s); 2 state(s)inputvariable&equals;u1t&comma;u2toutputvariable&equals;y1t&comma;y2tstatevariable&equals;x1t&comma;x2ta&equals;−533−4b&equals;2131c&equals;112−21d&equals;0000

(3)

h2norm2 := NormH2(sys2);

h2norm22.52637601270590

(4)

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):

PrintSystem(sys3);

Coefficientsdiscrete; sampletime = .11 output(s); 1 input(s)inputvariable&equals;u1zoutputvariable&equals;y1znum1,1&equals;1&comma;−2.841&comma;2.875&comma;−1.004den1,1&equals;1&comma;−2.417&comma;2.003&comma;−0.5488

(5)

h2norm3 := NormH2(sys3);

h2norm31.24382062647607

(6)

Example 4: Find the H2 norm of the system given by the following differential equation.

sys4 := DiffEquation(diff(diff(x(t),t),t) = -10*x(t) - diff(x(t),t) + w(t), [w(t)], [x(t)]):

PrintSystem(sys4);

Diff. Equationcontinuous1 output(s); 1 input(s)inputvariable&equals;wtoutputvariable&equals;xtde&equals;&DifferentialD;2&DifferentialD;t2xt=10xt&DifferentialD;&DifferentialD;txt+wt

(7)

h2norm4 := NormH2(sys4);

h2norm40.223606797749979

(8)

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>|<1/2,1>>, <<2,1>|<3,7>> ):

PrintSystem(sys5);

State Spacecontinuous2 output(s); 2 input(s); 2 state(s)inputvariable&equals;u1t&comma;u2toutputvariable&equals;y1t&comma;y2tstatevariable&equals;x1t&comma;x2ta&equals;−533−4b&equals;2131c&equals;112−21d&equals;2317

(9)

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+3)/(5*s^4 + 7*s^3 + 4*s^2 + 3*s + 1)):

PrintSystem(sys6);

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable&equals;u1soutputvariable&equals;y1stf1,1&equals;4s+35s4+7s3+4s2+3s+1

(10)

Since the H2 norm is infinite, an error message is displayed.

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

Compatibility

• 

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

• 

For more information on Maple 18 changes, see Updates in Maple 18.

See Also

DynamicSystems

DynamicSystems[Covariance]

DynamicSystems[Grammians]

LinearAlgebra[HermitianTranspose]

LinearAlgebra[LyapunovSolve]

LinearAlgebra[SylvesterSolve]