Abstract
Many control objectives can be stated in terms of the size of some particular signals. Therefore, a quantitative treatment of the performance of control systems requires the introduction of appropriate norms, which give measurements of the sizes of the signals considered. Another concept closely related to the size of a signal, is the size of a LTI system. The latter concept is of great practical importance because it is the basis of technical control \(\mathbf{H}_{\infty}\) and the study of robustness (see Chap. 6). These different concepts are detailed in this chapter.
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Notes
- 1.
In the case of a stochastic signal, we will always assume that it is modeled as an ergodic stationary stochastic process. For a comprehensive description of stochastic signals see the references given in the section Notes and References.
- 2.
Parseval’s theorem states that for a causal signal u∈L 2, we have
$$\int_0^{+\infty}u(t)^2\,dt= \frac{1}{2\pi}\int_{-\infty}^{+\infty}U^*(j\omega)U(j \omega)\,d\omega=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\bigl|U(j \omega)\bigr|^2\,d\omega $$where U(jω) represents the Fourier transform of u(t)
$$U(j\omega)=\mathcal{F}\bigl(u(t)\bigr)=\int_{-\infty }^{+\infty}u(t)e^{-j\omega t}\,dt $$ - 3.
The PSD of a signal u(t) is defined as \(S_{u}(\omega)=\int_{-\infty}^{+\infty} r_{u}(\tau)e^{-j\omega\tau}\,d\tau\), where r u (τ) is the autocorrelation function of the signal u(t): \(r_{u}(\tau )=\lim_{T\rightarrow\infty}\frac{1}{T}\int _{0}^{T}u(t)u(t+\tau)\,dt\). Note that the square of the RMS-value of the signal u(t) is nothing but \(u_{\mathrm{rms}}^{2}=r_{u}(0)\).
- 4.
The PSD matrix of the vector signal u(t) is defined as \(S_{u}(\omega )=\int_{-\infty}^{+\infty} R_{u}(\tau)e^{-j\omega\tau }\,d\tau\), where R u (τ) is the correlation matrix of the signal vector u(t): \(R_{u}(\tau)=\lim_{T\rightarrow\infty }\frac{1}{T}\int_{0}^{T}u(t)u(t+\tau)^{T}\,dt\). Note that the square of the RMS-value of the signal vector u(t) is nothing but \(u_{\mathrm {rms}}^{2}=\mathrm{Trace}(R_{u}(0))\). The matrix R u (0) is often referred to as the covariance matrix of the signal vector u(t).
- 5.
Recall that the Dirac delta function (or Dirac impulse), denoted δ(t), is the neutral element of the convolution product. Therefore, when the input is a Dirac impulse u(t)=1 k δ(t), where 1 k is a unit vector (e.g., 1 3=(0,0,1,0,…,0)), the state response is given by e At B 1 k , this is why e At B is called input-to-state impulse matrix.
- 6.
Consider a nonlinear autonomous system described by \(\dot{x}(t)=f(x(t))\). A point x e is said to be an equilibrium point (or a stationary point) for this system if f(x e )=0. In other words the equilibrium points are those from which the system does not evolve anymore. In the case of a LTI system the equilibrium points are the solutions of the equation Ax e =0. If A is of full rank, then x e =0, we have a single equilibrium point which is the origin of the state space. Otherwise, the solutions lie in the null space of A.
- 7.
Recall that the autonomous system is defined by \(e^{A(t-t_{0})}x(t_{0})\) which represents the state evolution of the system for u=0, see relation (2.15).
- 8.
The origin represents the unique equilibrium point of a LTI autonomous system for which det(A)≠0. If det(A)=0, the system is necessarily unstable in the sense of (2.17).
- 9.
A matrix P is symmetric if P=P T. The eigenvalues of a symmetric matrix are real. A symmetric matrix P is said to be positive definite if the associated quadratic form is always positive: x T Px>0 for all \(x\in \mathbf{R}^{n_{x}}\). This last condition is satisfied if and only if all the eigenvalues of P are positive. We denote by P≻0 a positive definite matrix.
- 10.
The Laplace transform of a given signal u(t) is defined as \(U(s)=\mathcal{L}(u(t))=\int _{0}^{\infty}x(t)e^{-st}\,dt\). From this definition, it is easy to show that the Laplace transform of the derivative of a signal is given by \(\mathcal{L}(\dot{u}(t))=s\mathcal {L}(u(t))-u(0)\).
- 11.
A complex matrix is said to be Hermitian if it is equal to its conjugate transpose.
- 12.
Indeed, it can be shown that for a complex matrix A∈C p×m and a complex vector x∈C m, we have
$$\bar{\sigma}(A)=\max_{{x\in\mathbf {C}^m}\atop{\|x\|_2\neq0}}\frac{\|Ax\|_2}{\|x\|_2}\quad\mbox{and} \quad\underline{\sigma}(A)=\min_{{x\in \mathbf{C}^m}\atop{\|x\|_2\neq0}}\frac{\|Ax\|_2}{\|x\|_2} $$To observe this, consider the first-order optimality condition of \(\lambda=\|Ax\|_{2}^{2}/\|x\|_{2}^{2}=(x^{*}A^{*}Ax)/(x^{*}x)\). We have
$$\frac{\partial\lambda}{\partial x}=\bigl(A^*A-\lambda I\bigr)x=0 $$thus, λ represents the eigenvalues of the matrix A ∗ A. Therefore, since \(\lambda=\|Ax\|_{2}^{2}/\|x\|_{2}^{2}\), the maximum of ∥Ax∥2/∥x∥2 is given by the square root of the largest eigenvalue of A ∗ A i.e., \(\bar{\sigma}(A)=\sqrt{\bar{\lambda}(A^{*}A)}\), and the minimum of ∥Ax∥2/∥x∥2 is given by the square root of the smallest eigenvalue of A ∗ A i.e., \(\underline{\sigma}(A)=\sqrt {\underline{\lambda}(A^{*}A)}\). Note that the input vector for which the gain is maximal (respectively, minimal) is given by the eigenvector associated to the largest (respectively, smallest) eigenvalue of A ∗ A.
- 13.
A transfer matrix is called proper (respectively, strictly proper) if for each of the component transfer function matrix, the degree of the numerator is less than or equal (respectively, strictly less than) the degree of the denominator.
- 14.
From (2.13), the RMS-value of the system response is given by \(y_{\mathrm{rms}}= (\frac {1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{Trace}(S_{y}(\omega ))\,d\omega)^{1/2}\). Using (2.29) we deduce that \(y_{\mathrm {rms}}= (\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm {Trace}(G(j\omega)G(j\omega)^{*})\,d\omega)^{1/2}\), which is by definition the H 2-norm of the system.
- 15.
It can be shown that for a complex matrix \(M\in\mathbf{C}^{n_{y}\times n_{u}}\), we have \(\mathrm{Trace}(MM^{*})=\sum_{i=1}^{\min(n_{y},n_{u})}\sigma _{i}^{2}(M)\).
- 16.
The observability Gramian is related to the total output energy of the autonomous system when it evolves from a given initial state x 0, we have \(x_{0}^{T}G_{o}x_{0}=\int_{0}^{\infty}y(t)^{T}y(t)\,dt\).
- 17.
The controllability Gramian makes it possible to determine the set of the state-space points that can be reached with an input of unit-energy; Consider the system \(\dot{x}(t)=Ax(t)+Bu(t)\), x(0)=0. A point x d can be reached at time T with a unit-energy signal (i.e. \(\int_{0}^{T}u^{T}(t)u(t)\,dt\leq 1\)) if and only if \(x_{d}^{T}G_{c}^{-1}x_{d}\leq1\).
References
Boyd S, Barratt C (1991) Linear controller design: limits of performance. Prentice-Hall, New York
Chen C-T (1999) Linear systems theory and design. Oxford University Press, Oxford
Grant M, Boyd S (2010) CVX: Matlab software for disciplined convex programming, version 1.21. http://cvxr.com/cvx
Kwakernaak H, Sivan R (1991) Modern signals and systems. Prentice-Hall, Englewood Cliffs
Rugh WJ (1996) Linear system theory. Prentice-Hall, Upper Saddle River
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Toscano, R. (2013). Signal and System Norms. In: Structured Controllers for Uncertain Systems. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-5188-3_2
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