LMIs in Control/Click here to continue/Notation

Notations

Examples
Consider the square matrix $$ A \in \mathbb{R} ^{n\times n}$$. The eigenvalues of $$A$$ are denoted by $$ \lambda_i (A), i = 1, 2,. . ., n$$. The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane (i.e., Re $$ \lambda_i (A), i = 1, 2, . . ., n$$ ). A matrix is Schur if all of its eigenvalues are strictly within a unit disk centered at the origin of the complex plane (i.e., $$|\lambda_i (A)| < 1, i = 1, . . ., n)$$. If $$A \in \mathbb{S}^n $$, then the minimum eigenvalue of A is denoted by $$\lambda(A)$$ and its maximum eigenvalue is denoted by $$ \bar{\lambda} (A) $$.

Consider the matrix B $$\in \mathbb{R} ^{n\times m}$$. The minimum singular value of B is denoted by $$ \underline{\sigma} $$ (B) and its maximum singular value is denoted by $$ \bar{\sigma} $$(B). The range and nullspace of B are denoted by $$ \mathbb{R} $$(B) and $$ \mathbb{N} $$(B), respectively. The Frobenius norm of B is ||B|| = $$ \sqrt{tr(B ^H B)}$$.

A state-space realization of the continuous-time linear time-invariant (LTI) system

$$ \dot{x}(t) = Ax(t) + Bu(t)$$,

$$ y(t) = Cx(t) + Du(t),$$. is often written compactly as (A, B,C,D) in this document. The argument of time is often omitted in continuous-time state-space realizations, unless needed to prevent ambiguity. A state-space realization of the discrete-time LTI system

$$x_{k+1} = A_d x_k + B_d u_k,$$

$$y_k = C_d x_k + D_d u_k,$$

is often written compactly as $$(A_d, B_d,C_d,D_d)$$.

The $$\mathcal{H}$$∞ norm of the LTI system $$ \mathcal{G}$$ is denoted by ||$$ \mathcal{G} $$||∞ and the $$\mathcal{H}_2 $$ norm of $$\mathcal{G}$$ is denoted by
 * $$\mathcal{G}$$||$$_2$$.

The inner product spaces $$ \mathcal{L}_2 and \mathcal{L}_{2e}$$ for continuous-time signals are defined as follows.

$$ \{\mathcal{L}_2\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_2 = \int_{0}^\infty x^T(t)x(t)dt<\infty \right\}, $$

$$ \{\mathcal{L}_{2e}\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2T} = \int_{0}^T x^T(t)x(t)dt<\infty, \forall T \in \mathbb{R}_{\geq 0} \right\}. $$

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows. $$ \{\mathcal{l}_2\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2} = \sum_{k=0}^\infty x^T_k x_k <\infty, \right\}. $$ $$ \{\mathcal{l}_{2e}\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2N} = \sum_{k=0}^N x^T_k x_k <\infty, \forall N \in \mathbb{Z}_{\geq 0} \right\}. $$

Reference

 * LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
 * LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
 * LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.