norm

Norm of linear model

Syntax

n = norm(sys)

n = norm(sys,2)

n = norm(sys,Inf)

[n,fpeak] = norm(sys,Inf)

[n,fpeak] = norm(sys,Inf,tol)

Description

n = norm(sys) or n = norm(sys,2) returns the root-mean-squares of the impulse response of the linear dynamic system model sys. This value is equivalent to the H₂ norm of sys.

n = norm(sys,Inf) returns the L_∞ norm of sys, which is the peak gain of the frequency response of sys across frequencies. For MIMO systems, this quantity is the peak gain over all frequencies and all input directions, which corresponds to the peak value of the largest singular value of sys. For stable systems, the L_∞ norm is equivalent to the H_∞ norm. For more information, see hinfnorm (Robust Control Toolbox).

[n,fpeak] = norm(sys,Inf) also returns the frequency fpeak at which the gain reaches its peak value.

[n,fpeak] = norm(sys,Inf,tol) sets the relative accuracy of the L_∞ norm to tol.

Examples

Compute Norm of Discrete-Time Linear System

Compute the $$H_2$$ and $$L_{\infty }$$ norms of the following discrete-time transfer function, with sample time 0.1 second.

$$sys(z)=\frac{z^3-2.841z^2+2.875z-1.004}{z^3-2.417z^2+2.003z-0.5488}.$$

Compute the $$H_2$$ norm of the transfer function. The $$H_2$$ norm is the root-mean-square of the impulse response of sys.

sys = tf([1 -2.841 2.875 -1.004],[1 -2.417 2.003 -0.5488],0.1);
n2 = norm(sys)

n2 = 
1.2438

Compute the $$L_{\infty }$$ norm of the transfer function.

[ninf,fpeak] = norm(sys,Inf)

ninf = 
2.5721

fpeak = 
3.0178

Because sys is a stable system, ninf is the peak gain of the frequency response of sys, and fpeak is the frequency at which the peak gain occurs. Confirm these values using getPeakGain.

[gpeak,fpeak] = getPeakGain(sys)

gpeak = 
2.5721

fpeak = 
3.0178

Input Arguments

sys — Dynamic system

dynamic system model | model array

Input dynamic system, specified as any SISO or MIMO linear dynamic system model or model array. sys can be continuous-time or discrete-time.

tol — Relative accuracy

0.01 (default) | positive real scalar

Relative accuracy of the H_∞ norm, specified as a positive real scalar value.

Output Arguments

n — H₂ or L_∞ norm

scalar | array

H₂ norm or L_∞ norm of sys, returned as a scalar or an array.

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

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

More About

H2 norm

The H₂ norm of a stable system H is the root-mean-square of the impulse response of the system. The H₂ norm measures the steady-state covariance (or power) of the output response y = Hw to unit white noise inputs w:

$${\Vert H\Vert }_2^2=\underset{t\to \infty }{\lim E}\left\{y{(t)}^{T}y(t)\right\},E\left(w(t)w{(\tau )}^T\right)=\delta \left(t-\tau \right)I.$$

The H₂ norm of a continuous-time system with transfer function H(s) is given by:

$${\Vert H\Vert }_2=\sqrt{\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }\text{Trace}\left[H{(j\omega )}^{H}H(j\omega )\right]}d\omega }.$$

For a discrete-time system with transfer function H(z), the H₂ norm is given by:

$${\Vert H\Vert }_2=\sqrt{\frac{1}{2\pi }{\displaystyle {\int }_{-\pi }^{\pi }\text{Trace}\left[H{(e^{j\omega })}^{H}H(e^{j\omega })\right]}d\omega }.$$

The H₂ norm is infinite in the following cases:

• sys is unstable.

• sys is continuous and has a nonzero feedthrough (that is, nonzero gain at the frequency ω = ∞).

Using norm(sys) produces the same result as sqrt(trace(covar(sys,1))).

L-infinity norm

The L_∞ norm of a SISO linear system is the peak gain of the frequency response. For a MIMO system, the L_∞ norm is the peak gain across all input/output channels.

For a continuous-time system H(s), this definition means:

$$\begin{array}{l}{\Vert H\left(s\right)\Vert }_{L_{\infty }}=\underset{\omega \in R}{\max }\left|H\left(j\omega \right)\right|\text{ (SISO})\\ \\ {\Vert H\left(s\right)\Vert }_{L_{\infty }}=\underset{\omega \in R}{\max }{\sigma }_{\max }\left(H\left(j\omega \right)\right)\text{ (MIMO})\end{array}$$

where σ_max(·) denotes the largest singular value of a matrix.

For a discrete-time system H(z), the definition means:

$$\begin{array}{l}{\Vert H\left(z\right)\Vert }_{L_{\infty }}=\underset{\theta \in \left[0,2\pi \right]}{\max }\left|H\left(e^{j\theta }\right)\right|\text{(SISO})\\ {\Vert H\left(z\right)\Vert }_{L_{\infty }}=\underset{\theta \in \left[0,2\pi \right]}{\max }{\sigma }_{\max }\left(H\left(e^{j\theta }\right)\right)\text{(MIMO})\end{array}$$

For stable systems, the L_∞ norm is equivalent to the H_∞ norm. For more information, see hinfnorm (Robust Control Toolbox). For a system with unstable poles, the H_∞ norm is infinite. For all systems, norm returns the L_∞ norm, which is the peak gain without regard to system stability.

Algorithms

After converting sys to a state space model, norm uses the same algorithm as covar for the H₂ norm. For the L_∞ norm, norm uses the algorithm of [1]. norm computes the peak gain using 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 before R2006a