getPeakGain

Peak gain of dynamic system frequency response

Syntax

gpeak = getPeakGain(sys)

gpeak = getPeakGain(sys,tol)

gpeak = getPeakGain(sys,tol,fband)

[gpeak,fpeak] = getPeakGain(___)

Description

gpeak = getPeakGain(sys) returns the peak input/output gain in absolute units of the dynamic system model, sys.

• If sys is a SISO model, then the peak gain is the largest value of the frequency response magnitude.

• If sys is a MIMO model, then the peak gain is the largest value of the frequency response 2-norm (the largest singular value across frequency) of sys. This quantity is also called the L_∞ norm of sys, and coincides with the H_∞ norm for stable systems (see norm).

• If sys is a model that has tunable or uncertain parameters, getPeakGain evaluates the peak gain at the current or nominal value of sys.

• If sys is a model array, getPeakGain returns an array of the same size as sys, where gpeak(k) = getPeakGain(sys(:,:,k)).

gpeak = getPeakGain(sys,tol) returns the peak gain of sys with relative accuracy tol.

gpeak = getPeakGain(sys,tol,fband) returns the peak gain in the frequency interval fband = [fmin,fmax] with 0 ≤ fmin < fmax. This syntax also takes into account the negative frequencies in the band [–fmax,–fmin] for models with complex coefficients.

[gpeak,fpeak] = getPeakGain(___) also returns the frequency fpeak at which the gain achieves the peak value gpeak, and can include any of the input arguments in previous syntaxes. fpeak can be negative for systems with complex coefficients.

Examples

Peak Gain of Transfer Function

Compute the peak gain of the resonance in the following transfer function:

$$sys=\frac{90}{s^2+1.5s+90}.$$

sys = tf(90,[1,1.5,90]);
gpeak = getPeakGain(sys)

gpeak = 
6.3444

The getPeakGain command returns the peak gain in absolute units.

Peak Gain with Specified Accuracy

Compute the peak gain of the resonance in the transfer function with a relative accuracy of 0.01%.

$$sys=\frac{90}{s^2+1.5s+90}.$$

sys = tf(90,[1,1.5,90]);
gpeak = getPeakGain(sys,0.0001)

gpeak = 
6.3444

The second argument specifies a relative accuracy of 0.0001. The getPeakGain command returns a value that is within 0.0001 (0.01%) of the true peak gain of the transfer function. By default, the relative accuracy is 0.01 (1%).

Peak Gain Within Specified Band

Compute the peak gain of the higher-frequency resonance in the transfer function

$$sys=\left(\frac{1}{s^2+0.2s+1}\right)\left(\frac{100}{s^2+s+100}\right).$$

sys is the product of resonances at 1 rad/s and 10 rad/s.

sys = tf(1,[1,.2,1])*tf(100,[1,1,100]);
fband = [8,12];
gpeak = getPeakGain(sys,0.01,fband);

The fband argument causes getPeakGain to return the local peak gain between 8 and 12 rad/s.

Frequency of Peak Gain

Identify which of the two resonances has higher gain in the transfer function

$$sys=\left(\frac{1}{s^2+0.2s+1}\right)\left(\frac{100}{s^2+s+100}\right).$$

sys is the product of resonances at 1 rad/s and 10 rad/s.

sys = tf(1,[1,.2,1])*tf(100,[1,1,100]);
[gpeak,fpeak] = getPeakGain(sys)

gpeak = 
5.0747

fpeak = 
0.9902

fpeak is the frequency corresponding to the peak gain gpeak. The peak at 1 rad/s is the overall peak gain of sys.

Frequency of Peak Gain for System with Complex Coefficients

For systems with complex coefficients, getPeakGain can return a peak at a negative or positive frequency depending on the shape and fband you specify.

Generate a random state-space model with complex data.

rng(1)
A = complex(randn(10),randn(10));
B = complex(randn(10,3),randn(10,3));
C = complex(randn(2,10),randn(2,10));
D = complex(randn(2,3),randn(2,3));
sys = ss(A,B,C,D);

Compute the peak gain with a relative accuracy of 0.1%. Also, specify fband = [0,1] to compute the peak in the frequency interval [–1,1].

[gPeak,fPeak] = getPeakGain(sys,1e-3,[0,1])

gPeak = 
126.2396

fPeak = 
-0.4579

In this interval, sys attains a peak at a negative frequency value. Create a singular value plot in this range to confirm the result.

w = linspace(-1,1,100);
sp = sigmaplot(sys,w);
sp.FrequencyScale = 'linear';
sp.MagnitudeUnit = 'abs';

Now compute the peak gain in the frequency interval [–50,–1] ∪ [1,50]. To do so, specify fband = [1,50].

[gPeak,fPeak] = getPeakGain(sys,1e-3,[1,50])

gPeak = 
43.3303

fPeak = 
1.8097

In this interval, sys attains a peak at a positive frequency value.

Create a singular value plot in this range to confirm the result.

w = linspace(-50,50,5000);
sp = sigmaplot(sys,w);
sp.FrequencyScale = 'linear';
sp.MagnitudeUnit = 'abs';

For this interval, getPeakGain returns the magnitude and frequency value for the smaller peak shown in the plot.

Input Arguments

sys — Input dynamic system

dynamic system model | model array

Input dynamic system, specified as any dynamic system model or model array. sys can be SISO or MIMO.

tol — Relative accuracy

0.01 (default) | positive real scalar

Relative accuracy of the peak gain, specified as a positive real scalar value. getPeakGain calculates gpeak such that the fractional difference between gpeak and the true peak gain of sys is no greater than tol. The default value is 0.01, meaning that gpeak is within 1% of the true peak gain.

fband — Frequency interval

[0,Inf] (default) | 1-by-2 vector of positive real values

Frequency interval in which to calculate the peak gain, specified as a 1-by-2 vector of positive real values. Specify fband as a row vector of the form [fmin,fmax] with 0 ≤ fmin < fmax.

For models with complex coefficients, getPeakGain calculates the peak gain in the range [–fmax,–fmin]∪[fmin,fmax]. As a result, the function can return a peak at a negative frequency.

Output Arguments

gpeak — Peak gain of dynamic system

scalar | array

Peak gain of the dynamic system model or model array sys, returned as a scalar value or an array.

• If sys is a single model, then gpeak is a scalar value.

• If sys is a model array, then gpeak is an array of the same size as sys, where gpeak(k) = getPeakGain(sys(:,:,k)).

fpeak — Frequency of peak gain

real scalar | array of real values

Frequency at which the gain achieves the peak value gpeak, returned as a real scalar value or an array of real values. The frequency is expressed in units of rad/TimeUnit, relative to the TimeUnit property of sys.

• If sys is a single model, then fpeak is a scalar.

• If sys is a model array, then fpeak is an array of the same size as sys, where fpeak(k) is the peak gain frequency of sys(:,:,k).

fpeak can be negative for systems with complex coefficients.

Algorithms

getPeakGain uses the algorithm of [1]. All eigenvalue computations are performed using structure-preserving algorithms from the SLICOT library. For more information about the SLICOT library, see https://github.com/SLICOT.

References

[1] Bruinsma, N.A., and M. Steinbuch. "A Fast Algorithm to Compute the H_∞ Norm of a Transfer Function Matrix." Systems & Control Letters, 14, no.4 (April 1990): 287–93.

Version History

Introduced in R2012a