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DynamicSystems

  

NormHinf

  

Compute the H norm of a linear system

 

Calling Sequence

Parameters

Options

Description

Examples

References

Compatibility

Calling Sequence

NormHinf(sys)

NormHinf(sys, eps)

Parameters

sys

-

System; system object

eps

-

(optional) nonnegative; relative accuracy. The default value is 10^(-6).

opts

-

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

Options

• 

output = norm or peakfreq or list of these names.

Specifies the returned values. By default, only the H norm is returned. If peakfreq is specified, the angular frequency (rad/s) at which the peak gain of sys occurs is returned.

• 

checkstability = truefalse

True means check whether the system is stable; if it is not stable, raise a warning. False means skip the check. The default is true.

Description

• 

The NormHinf command computes the H norm of a linear system sys, with relative accuracy eps. 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 Gs, the H norm is defined in the frequency domain as:

  

‖G‖ = supωGjω

• 

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

  

‖G‖ = supωσ[max]Gjω

  

where σmax is the maximum singular value.

• 

In the time domain, the H norm of a transfer function is calculated assuming that the stable transfer function Gs has a state-space representation:

  

x.=Ax+Bw

  

y=Cx+Dw

  

where: A ∈ ℝn×n, B ∈ ℝn×m, C ∈ ℝp×n, and D ∈ ℝp×m, and n, m, and p are the number of states, inputs and outputs of the linear system respectively.

  

Gs=YsWs and Gs=C. sIA1. B+D, with A stable (all eigenvalues of A have a negative real part).

  

Then the H norm of the transfer function Matrix Gs is ‖G‖<γ for some 0<γ, not equal to a singular value of Matrix D, if and only if σ&lsqb;max&rsqb;H<γ has no eigenvalues on the imaginary axis. The Matrix H is defined as:

  

Hγ = ABR1DTC&gamma;BR1BT&gamma;CTS1CAT+CTDR1BT

  

where R&equals;DT&period;Dγ2Im and S&equals;D&period;DTγ2Ip (subscripts m and p indicate the dimensions of the respective identity Matrices).

Discrete-time

• 

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

  

‖G‖ = sup0ω<2πG&ExponentialE;jω

• 

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

  

‖G‖ = sup0ω<2πσ&lsqb;max&rsqb;G&ExponentialE;jω

  

where σmax is the maximum singular value.

• 

In the time domain, the H norm of a transfer function is calculated assuming that the stable transfer function Gz has a state-space representation:

  

xk&plus;1&equals;Axk&plus;Bwk

  

yk&equals;Cxk&plus;Dwk

  

so that Gz&equals;YzWz and Gz&equals;C. zIA1. B&plus;D, with A stable (all eigenvalues of A have a magnitude less than 1).

• 

The H norm of the transfer function Matrix Gz is calculated using the bilinear transformation, since the H norm for a discrete-time LTI system is preserved in the continuous-time domain under such transformation.

• 

The H norm provides a measure of the worst-case system gain, i.e., the largest factor by which any sinusoidal input is magnified by the system. For instance, the H norm of the transfer function G from w (disturbance input) to y (output) provides a measure of the worst-case influence of the noise w on the output y of an LTI system.

• 

For a SISO linear system, the H norm is the maximum gain of the frequency response of the system. In an analogous way, for a MIMO linear system, the H norm is the maximum gain across all inputs and outputs of the system.

• 

The H norm of G equals the peak value on the Bode magnitude plot of G. It also equals the distance from the origin to the farthest point on the Nyquist plot of G.

• 

The H norm is finite if and only if the transfer function G is proper (degree of denominator greater than or equal to degree of numerator) and has no poles on the imaginary axis (continuous-time) or on the unit circle (discrete-time).

Examples

withDynamicSystems&colon;

Example 1 : Find the H norm of a continuous-time system.

sys1TransferFunction100s&plus;5&colon;

PrintSystemsys1

Transfer Functioncontinuous1 output(s); 1 input(s)inputvariable&equals;u1soutputvariable&equals;y1stf1,1&equals;100s+5

(1)

hinfnorm1NormHinfsys1&comma;1010

hinfnorm1:=20.00000000

(2)

MagnitudePlotsys1&comma;decibels&equals;false&comma;range&equals;0.001..100

magMagnitudePlotsys1&comma;decibels&equals;false&comma;range&equals;0.001..100&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=19.9999996000000

(3)

Example 2: Find the H norm of the system given by the following differential equation. Show the peak frequency and the norm in that order.

sys2DiffEquation&DifferentialD;&DifferentialD;t&DifferentialD;&DifferentialD;txt&equals;10xt&DifferentialD;&DifferentialD;txt&plus;wt&comma;wt&comma;xt&colon;

PrintSystemsys2

Diff. Equationcontinuous1 output(s); 1 input(s)inputvariable&equals;wtoutputvariable&equals;xtde&equals;x..t=10xtx.t+wt

(4)

hinfnorm2NormHinfsys2&comma;&apos;output&apos;&equals;peakfreq&comma;norm

hinfnorm2:=3.08220698298668&comma;0.3202566278

(5)

MagnitudePlotsys2&comma;decibels&equals;false

magMagnitudePlotsys2&comma;decibels&equals;false&comma;output&equals;data&colon;

membermaxmag1..1&comma;2..2&comma;mag1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmagp

fHinf:=3.079785057

(6)

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=0.320252649756933

(7)

Example 3 : Find the H norm of a continuous state-space MIMO system.

sys3StateSpace0&comma;0&comma;3&verbar;1&comma;0&comma;4&verbar;0&comma;1&comma;7&comma;0&comma;0&comma;1&comma;1&verbar;0&verbar;0&comma;Matrix1&comma;1&colon;

PrintSystemsys3

State Spacecontinuous1 output(s); 1 input(s); 3 state(s)inputvariable&equals;u1toutputvariable&equals;y1tstatevariable&equals;x1t&comma;x2t&comma;x3ta&equals;010001−3−4−7b&equals;001c&equals;100d&equals;0

(8)

hinfnorm3NormHinfsys3&comma;&apos;output&apos;&equals;norm&comma;peakfreq

hinfnorm3:=0.4513222615&comma;0.559605319143178

(9)

MagnitudePlotsys3&comma;decibels&equals;false

magMagnitudePlotsys3&comma;decibels&equals;false&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=0.451320291397442

(10)

memberHinfgraph&comma;mag1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmagp

fHinf:=0.5590478459

(11)

Example 4: Find the H norm of a continuous transfer function G(s) with .1% of tolerance.

sys4TransferFunctionMatrix1s3&plus;s2&plus;5s&plus;2&comma;ss3&plus;s2&plus;5s&plus;2&comma;s2s3&plus;s2&plus;5s&plus;2&colon;

PrintSystemsys4

Transfer Functioncontinuous3 output(s); 1 input(s)inputvariable&equals;u1soutputvariable&equals;y1s&comma;y2s&comma;y3stf1,1&equals;1s3+s2+5s+2tf2,1&equals;ss3+s2+5s+2tf3,1&equals;s2s3+s2+5s+2

(12)

hinfnorm4NormHinfsys4&comma;0.001&comma;&apos;output&apos;&equals;norm&comma;peakfreq

hinfnorm4:=1.899661306&comma;2.180899209

(13)

MagnitudePlotsys4&comma;decibels&equals;false

magMagnitudePlotsys4&comma;decibels&equals;false&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=1.69438112239631

(14)

memberHinfgraph&comma;mag&lsqb;3&rsqb;1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmag&lsqb;3&rsqb;p

fHinf:=2.184166359

(15)

Example 5: Find the H norm of a continuous transfer function matrix.

sys5TransferFunctionMatrix1s2&plus;s&plus;4&comma;0&comma;0&comma;1s2&plus;s&plus;4&colon;

PrintSystemsys5

Transfer Functioncontinuous2 output(s); 2 input(s)inputvariable&equals;u1s&comma;u2soutputvariable&equals;y1s&comma;y2stf1,1&equals;1s2+s+4tf2,1&equals;0tf1,2&equals;0tf2,2&equals;1s2+s+4

(16)

hinfnorm5NormHinfsys5&comma;&apos;output&apos;&equals;norm&comma;peakfreq

hinfnorm5:=0.5163982960&comma;1.87082283019111

(17)

MagnitudePlotsys5&comma;decibels&equals;false

magMagnitudePlotsys5&comma;decibels&equals;false&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=0.516350854134402

(18)

memberHinfgraph&comma;mag&lsqb;1&rsqb;1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmag&lsqb;1&rsqb;p

fHinf:=1.863838004

(19)

Example 6: Find the H norm of a continuous state-space SISO system.

sys6StateSpaceMatrix0&comma;1&comma;25&comma;0.1&comma;Matrix0&comma;1&comma;Matrix1&comma;0&comma;Matrix0&colon;

PrintSystemsys6

State Spacecontinuous1 output(s); 1 input(s); 2 state(s)inputvariable&equals;u1toutputvariable&equals;y1tstatevariable&equals;x1t&comma;x2ta&equals;01−25−0.1b&equals;01c&equals;10d&equals;0

(20)

hinfnorm6NormHinfsys6&comma;&apos;output&apos;&equals;norm&comma;peakfreq

hinfnorm6:=2.000102008&comma;4.99949995099029

(21)

MagnitudePlotsys6&comma;decibels&equals;false

magMagnitudePlotsys6&comma;decibels&equals;false&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=1.99995097863956

(22)

memberHinfgraph&comma;mag1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmagp

fHinf:=5.000110374

(23)

Example 7 : Find the H norm of a system with discrete-time transfer function shown below.

sys7TransferFunction102z&plus;110z2&plus;2z&plus;5&comma;discrete&comma;sampletime&equals;0.1&colon;

PrintSystemsys7

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

(24)

hinfnorm7NormHinfsys7&comma;108&comma;&apos;output&apos;&equals;norm&comma;peakfreq

hinfnorm7:=4.264978970&comma;17.0452791622670

(25)

MagnitudePlotsys7&comma;decibels&equals;false&comma;range&equals;0.01..&pi;sys7:-sampletime

magMagnitudePlotsys7&comma;decibels&equals;false&comma;range&equals;0.01..&pi;sys7:-sampletime&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=4.289843575

(26)

memberHinfgraph&comma;mag1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmagp

fHinf:=16.66285136

(27)

Example 8 : Find the H norm of a system with discrete-time transfer function shown below.

sys8TransferFunction5&comma;14.2&comma;14.4&comma;5&comma;5&comma;12.1&comma;10&comma;2.7&comma;discrete&comma;sampletime&equals;0.5&colon;

PrintSystemsys8

Transfer Functiondiscrete; sampletime = .51 output(s); 1 input(s)inputvariable&equals;u1zoutputvariable&equals;y1ztf1,1&equals;5.z314.20000000z2+14.40000000z5.5.z312.10000000z2+10.z2.700000000

(28)

hinfnorm8NormHinfsys8&comma;&apos;output&apos;&equals;norm&comma;peakfreq

hinfnorm8:=4.635710037&comma;0.615651253990522

(29)

MagnitudePlotsys8&comma;decibels&equals;false&comma;range&equals;0.01..&pi;sys8:-sampletime

magMagnitudePlotsys8&comma;decibels&equals;false&comma;range&equals;0.01..&pi;sys8:-sampletime&comma;output&equals;data&colon;

Hinfgraphmaxmag1..1&comma;2..2

Hinfgraph:=4.635634989

(30)

memberHinfgraph&comma;mag1..1&comma;2..2&comma;&apos;p&apos;&colon;fHinfmagp

fHinf:=0.6152990634

(31)

References

  

S. Boyd, V. Balakrishnan, P. Kabamba, On computing the H norm of a transfer matrix, 1988.

  

N. A. Bruinsma, M. Steinbuch, A fast algortihm to compute the H-norm of a transfer function matrix, 1990.

Compatibility

• 

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

• 

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

See Also

DynamicSystems

DynamicSystems[MagnitudePlot]

DynamicSystems[NormH2]

DynamicSystems[ToContinuous]

 


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