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Calling Sequence
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NormHinf(sys)
NormHinf(sys, eps)
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Parameters
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sys
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System; system object
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eps
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(optional) nonnegative; relative accuracy. The default value is 10^(-6).
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opts
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(optional) equation(s) of the form option = value; specify options for the NormHinf command
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Options
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output = norm or peakfreq or list of these names.
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Specifies the returned values. By default, only the norm is returned. If peakfreq is specified, the angular frequency (rad/s) at which the peak gain of sys occurs is returned.
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checkstability = truefalse
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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.
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Description
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The NormHinf command computes the 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.
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Continuous-time
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For a stable SISO linear system with transfer function , the norm is defined in the frequency domain as:
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For a MIMO linear system with transfer function Matrix , the definition of norm in the frequency domain is generalized to:
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where is the maximum singular value.
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In the time domain, the norm of a transfer function is calculated assuming that the stable transfer function has a state-space representation:
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where: , , , and , and , , and are the number of states, inputs and outputs of the linear system respectively.
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and . . , with stable (all eigenvalues of have a negative real part).
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Then the norm of the transfer function Matrix is for some , not equal to a singular value of Matrix , if and only if has no eigenvalues on the imaginary axis. The Matrix is defined as:
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=
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where and (subscripts and indicate the dimensions of the respective identity Matrices).
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Discrete-time
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For a stable SISO linear system with transfer function , the norm is defined in the frequency domain as:
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=
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For a MIMO linear system with transfer function Matrix , the definition of norm in the frequency domain is generalized to:
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=
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where is the maximum singular value.
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In the time domain, the norm of a transfer function is calculated assuming that the stable transfer function has a state-space representation:
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so that and . . , with stable (all eigenvalues of have a magnitude less than 1).
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The norm of the transfer function Matrix is calculated using the bilinear transformation, since the norm for a discrete-time LTI system is preserved in the continuous-time domain under such transformation.
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The 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 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.
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For a SISO linear system, the norm is the maximum gain of the frequency response of the system. In an analogous way, for a MIMO linear system, the norm is the maximum gain across all inputs and outputs of the system.
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The norm of equals the peak value on the Bode magnitude plot of . It also equals the distance from the origin to the farthest point on the Nyquist plot of .
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The norm is finite if and only if the transfer function 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).
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Examples
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Example 1 : Find the norm of a continuous-time system.
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Example 2: Find the norm of the system given by the following differential equation. Show the peak frequency and the norm in that order.
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Example 3 : Find the norm of a continuous state-space MIMO system.
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Example 4: Find the norm of a continuous transfer function G(s) with .1% of tolerance.
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Example 5: Find the norm of a continuous transfer function matrix.
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Example 6: Find the norm of a continuous state-space SISO system.
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Example 7 : Find the norm of a system with discrete-time transfer function shown below.
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Example 8 : Find the norm of a system with discrete-time transfer function shown below.
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References
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S. Boyd, V. Balakrishnan, P. Kabamba, On computing the norm of a transfer matrix, 1988.
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N. A. Bruinsma, M. Steinbuch, A fast algortihm to compute the -norm of a transfer function matrix, 1990.
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Compatibility
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The DynamicSystems[NormHinf] command was introduced in Maple 18.
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