8. L-Norm and H-Norm

8.1 L-Norm of a Vector

Given \(e = \begin{bmatrix} e_1 & \dots & e_m \end{bmatrix}^T\), the L-norm of the vector is:

Description	Formulation
\(L_2\) norm	\(\left\lVert e \right\rVert_2 = \sqrt{e^Te} = \sqrt{\sum_{i=1}^m e_i^2}\)
\(L_n\) norm	\(\left\lVert e \right\rVert_n = (\sum_{i=1}^m\left\lvert e_i \right\rvert ^n)\)
\(L_\infty\) norm	\(\left\lVert e \right\rVert_\infty = \sup_i \left\lvert e_i \right\rvert\)

8.2 L2-Norm of a Matrix

Given \(m\times n\) matrix \(A\), the induced 2-norm of this matrix is:

\[ ||A||_{i2} = \sup_{x \neq 0} \frac{||Ax||_2}{||x||_2} =\underbrace{\sqrt{\lambda_{max}(A^TA)}}_{\text{max singular value of } A} \]

Notes

• Singular value:

  Given \(A \in C^{m,n}\), \(i = 1, 2, \dots \min{\{m,n\}}\), the singular values of \(A\) are the largest roots between \(\lambda_i(A^*A)\) and \(\lambda_i(AA^*)\):

  \[ \begin{aligned} \sigma_i (A) &= \sqrt{\lambda_i(A^*A)} = \sqrt{\lambda_i(AA^*)}, &m=n \\ \sigma_i (A) &= \sqrt{\lambda_i(A^*A)}, &m<n\\ \sigma_i (A) &= \sqrt{\lambda_i(AA^*)}, &m>n \end{aligned} \]

Notes

• SVD decomposition:

  Given the SVD decomposition: \(A = U\Sigma V^*\), where \(\Sigma \in \mathbb R^{m\times n}\), \(U \in C^{m,m}\), \(V\in C^{n,n}\), and \(U\), \(V\) are unitary matrices, thus, there have:

  \[ \begin{aligned} U^*U &= UU^* = I \\ V^*V &= VV^* = I \end{aligned} \quad \Rightarrow \quad \begin{aligned} \sigma_i(U) &= 1 \\ \sigma_i(V) &= 1 \end{aligned} \]

  Condition	Formulation
  \(m > n\)	\(\Sigma = \begin{bmatrix}\Sigma_1 \\ \mathbf 0\end{bmatrix}\)
  \(m < n\)	\(\Sigma = \begin{bmatrix}\Sigma_1 & \mathbf 0\end{bmatrix}\)
  \(m = n\)	\(\Sigma = \Sigma_1\)

  where \(\Sigma_1 \in \mathbb R^{k\times k}\), \(k = \min{\{m,n\}}\), it has:

  \[ \Sigma_1 = \begin{bmatrix} \sigma_1(A) && \\ &\ddots& \\ &&\sigma_k(A) \end{bmatrix} \]

  \(\sigma_1(A) \geq \sigma_2(A) \geq \dots \geq \sigma_k(A) \geq 0\) and we can know that:

  \[ \begin{aligned} \bar \sigma(A) &= \max_i \sigma_i (A) = \sigma_1(A) \\ \underline \sigma(A) &= \left\{\begin{aligned} \min_i \sigma_i &= \sigma_k &m>n \\ &0 &m<n\end{aligned}\right. \end{aligned} \]

Info

Proof:

\[ \begin{aligned} ||Ax||_2^2 &= x^TA^TAx \leq \lambda_{max}(A^TA)||x||_2^2 \\ ||Ax||_2 &\leq \sqrt{\lambda_{max}(A^TA)}||x||_2 \\ \frac{||Ax||_2}{||x||_2} &\leq \sqrt{\lambda_{max}(A^TA)} \end{aligned} \]

8.2 L-Norm of a Signal

8.2.1 L-norm for signal

Given \(e = \begin{bmatrix} e_1 & \dots & e_m \end{bmatrix}^T\), the L-norm of a signal is:

Description	Formulation
\(L_2\) norm	\(\left\lVert e \right\rVert_2 = \sqrt{\int_{-\infty}^{+\infty} e^T(t)e(t)}\)
\(L_\infty\) norm	\(\left\lVert e \right\rVert_\infty = \sup_t (\sup_i \left\lvert e_i(t) \right\rvert)\)

8.2.2 L2-gain for system

From \(L_2\) norm of the system, we can give the \(L_2\) Gain:

• Given a general system,

  

  We assume that \(u(t) = 0, t < 0\), and the input is boundary, which satisfy:

  \[ \int_{0}^{+\infty} u^T(t)u(t)dt < \infty \]

  if \(u\in L_2\),

  \[ ||u||_2 = \sqrt{\int_0^{+\infty} u^T(t)u(t)dt} \]

  Thus, we can get the \(L_2\) gain of the system:

  \[ \begin{aligned} \gamma_2 &= \sup_{u\in L_2, ||u||_2 \neq 0} \frac{||y||_2}{||u||_2} = \sup_{u\in L_2, ||u||_2 \neq 0} \frac{S(u(s))}{||u||_2} \\ \gamma_2||u||_2 &\geq ||y||_2, \forall u \in L_2 \end{aligned} \]

  The system is input output \(L_2\) stable if its \(L_2\) gain is finite.

Example

Given the system,

\[ \begin{aligned} y(t) &= \int_0^{+\infty} u(\tau) d\tau \\ u(t) &= \left\{\begin{aligned} 1,\quad 0<t<1\\ 0,\quad \text{otherwise} \end{aligned}\right. \end{aligned} \]

We can find the \(L_2\) norm and \(L_2\) gain of the system,

\[ \begin{aligned} ||u||_2 &= \sqrt{\int_0^{+\infty} u^2(t)dt} = 1 \Rightarrow u \in L_2 \\ ||y||_2 &= \sqrt{\int_0^{+\infty}y^2(t)dt} = +\infty \\ \gamma_{2} &= \sup_{u\in L_2, ||u||_2 \neq 0} \frac{||y||_2}{||u||_2} = +\infty \end{aligned} \]

The system is not \(L_2\) stable.

Example

Given a cascaded system,

\[ y(s) = G(s)U(s) \]

Where, \(u \in L_2\) \(\to\) \(U(j\omega)\), with the Fourier Transform (FT),

\[ y(j\omega) = G(j\omega)U(j\omega) \]

Within the Parseval Theorem, we have:

\[ \begin{aligned} ||y||_2^2 &= \int_{-\infty}^{+\infty}y^2(t)dt \\ &= \frac{1}{2\pi} \int_{-\infty}^{+\infty}|y(j\omega)|^2 d\omega \\ &= \frac{1}{2\pi} \int_{-\infty}^{+\infty}|G(j\omega)|^2|U(j\omega)|^2 d\omega \end{aligned} \]

if \(|G(j\omega)| \leq k\), with \(|G(j\bar \omega)| = k\),

\[ \begin{aligned} ||y||_2^2 &\leq k^2\frac{1}{2\pi} \int_{-\infty}^{+\infty}|U(j\omega)|^2 d\omega\\ ||y||_2 &\leq k ||u||_2 \\ \gamma_2 &= \sup_{u\in L_2, ||u||_2 \neq 0} \frac{||y||_2}{||u||_2} \leq k \\ &= \sup_\omega |G(j\omega)| \end{aligned} \]

8.3 H-Norm

8.3.1 H-infinity norm for SISO system

Given a SISO A.S. linear system and its \(H_\infty\) norm,

\[ \gamma_2 = \underbrace{||G||_\infty}_{H_\infty \text{ norm of } G} = \sup_\omega |G(j\omega)| \]

8.3.2 H-infinity norm for MIMO system

To extend this result to MIMO A.S. linear system,

\[ G(j\omega) = \begin{bmatrix} G_{11}(j\omega) & \cdots & G_{1n} \\ \vdots & & \vdots \\ G_{m1} & \cdots & G_{mn} \end{bmatrix} \]

if \(m=3\), \(n = 4\), there have \(k = \min(m,n) = 3\) singular values,

\[ \gamma_2 = ||G||_\infty = \sup_{\omega} \bar \sigma(G(j\omega)) \]

At the position when \(m=n=1\), we get the upper bound of the singular value,

\[ \bar \sigma(G(j\omega)) = \sqrt{G^*(j\omega)G(j\omega)} = |G(j\omega)| \]

Info

Proof:

\[ \begin{aligned} ||y||_2^2 &= \int_{-\infty}^{+\infty}y^T(t)y(t)dt \\ &= \frac{1}{2\pi} \int_{-\infty}^{+\infty}y(j\omega)^*y(j\omega) d\omega \\ &= \frac{1}{2\pi} \int_{-\infty}^{+\infty} U(j\omega)^*G(j\omega)^*G(j\omega)U(j\omega)d\omega \\ &\leq \underbrace{\lambda_{max}(G(j\omega)^*G(j\omega))}_{\bar \sigma^2(G(j\omega))} \frac{1}{2\pi} \int_{-\infty}^{+\infty} U(j\omega)^*U(j\omega) d\omega\\ &\leq \sup_\omega \bar \sigma^2(G(j\omega)) ||u||_2^2 \end{aligned} \]

\[ \underline{\sigma}(G(j\omega)) \leq \frac{||G(j\omega)U(j\omega)||_2}{||U(j\omega)||_2} \leq \bar \sigma(G(j\omega)) \]

8.3.3 H-2 norm for SISO system

If system \(S\) is an A.S. strictly proper system, we can define the \(H_2\) norm of \(G\),

If the system is SISO,

\[ ||G||_2 = \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty}\underbrace{|G(j\omega)|^2}_{G(j\omega)^*G(j\omega)}d\omega} \]

Here, \(||G||_2\) equals to the impulse response \(||\delta||_2\), to evaluate it in bode plot, we let:

\[ |G(j\omega)|_{\text{dB}}^2 = 20\log_{10}|G(j\omega)|^2 = 2|G(j\omega)|_{\text{dB}} \]

Example

\[ \begin{aligned} G(j\omega) &= \frac{1}{j\omega+a}, a > 0\\ ||G||_\infty &= \sup_\omega |G(j\omega)| = \frac{1}{a}\\ ||G||_2 &= \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty}|\frac{1}{j\omega+a}|^2d\omega} \\ &= \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{1}{\omega^2+a^2}d\omega} \\ &= \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty}\arctan\frac{\omega}{a}d\omega} \\ &= \sqrt{\frac{1}{2a}} \end{aligned} \]

8.3.4 H-2 norm for MIMO system

If the system is MIMO

\[ ||G||_2 = \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty}\underbrace{\text{tr}(G(j\omega)G^*(j\omega))}_{\sum_i \sigma_i^2(G(j\omega))}d\omega} \]

8.3.5 Conclusion

Description	Formulation
\(H_2\) norm for SISO	\(\left\lVert G \right\rVert_2 = \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty} \left\lvert G(j\omega) \right\rvert^2 d\omega}\)
\(H_2\) norm for MIMO	\(\left\lVert G \right\rVert_2 = \sqrt{\frac{1}{2\pi}\int_{-\infty}^{+\infty} \text{tr}(G(j\omega)G^*(j\omega)) d\omega}\)
\(H_\infty\) norm	\(\left\lVert G \right\rVert_\infty = \sup_\omega \left\lvert G(j\omega) \right\rvert\)

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