You are here : Control System Design - Index | Book Contents | Appendix A

A. Notation, Symbols, and Acronyms

To download a PDF version of this document for printing, right mouse click here.


Notation Meaning
$t$ Continuous-time variable
$f(t)$ Continuous-time signal
$k$ Discrete-time variable
$\{f[k]\}$ Discrete-time sequence
$\Delta$ Sampling period
$f(k\Delta)$ Sampled version of $f(t)$
$\delta $ Delta operator
$q$ forward shift operator
$\delta_K(k)$ Kronecker delta
$\delta(t)$ Dirac delta
$\ensuremath{{\cal{E}}}\{...\}$ Expected value of ...
$\Gamma_c$ Controllability matrix in state space description
$\Gamma_o$ Observability matrix in state space description
$\Lambda\{...\}$ Set of eigenvalues of matrix ...
$\mu (t-t_o)$ unit step (continuous time) at time $t=to$
$\mu [k-k_o]$ unit step (discrete time) at time $k=k_o$
$f^s(t)$ Dirac impulse-sampled version of $f(t)$
$\ensuremath{{\cal{F}}\left[...\right]} $ Fourier transform of ...
$\ensuremath{{\cal{L}}\left[...\right]} $ Laplace transform of ...
$\ensuremath{{\cal{D}}\left[...\right]} $ Delta-transform of ...
$\ensuremath{{\cal{Z}}\left[...\right]} $ Z-transform of ...
$\ensuremath{{\cal{F}}^{-1}\left[...\right]} $ inverse Fourier transform of ...
$\ensuremath{{\cal{L}}^{-1}\left[...\right]} $ inverse Laplace transform of ...
$\ensuremath{{\cal{D}}^{-1}\left[...\right]} $ inverse Delta-transform of ...
$\ensuremath{{\cal{Z}}^{-1}\left[...\right]} $ inverse Z-transform of ...
$s$ Laplace-transform complex variable
$\omega$ angular frequency
$\gamma$ Delta-transform complex variable
$z$ Z-transform complex variable
$F(j\omega)$ Fourier transform of $f(t)$
$F(s)$ Laplace transform of $f(t)$
$F_\delta(\gamma)$ Delta-transform of $\{f[k]\}$
$F_q(z)$ Z-transform of $\{f[k]\}$
$f_1(t)*f_2(t)$ Time convolution of $f_1(t)$ and $f_2(t)$
$F_1(s)*F_2(s)$ Complex convolution of $F_1(s)$ and $F_2(s)$
$\Re \{...\}$ real part of ...
$\Im \{...\}$ imaginary part of ...
$\mathbb{C} ^{m\times n}$ set of all $m\times n$ matrices with complex entries
${\cal{H}}_2$ Hilbert space of those functions square-integrable along the imaginary axis and analytic in the right-half plane
${\cal{L}}_1$ Hilbert space of those functions absolutely integrable along the imaginary axis
${\cal{L}}_2$ Hilbert space of those functions square-integrable along the imaginary axis.
${\cal{H}}_\infty$ Hilbert space of those functions bounded along the imaginary axis and analytic in the right-half plane
${\cal{RH}}_\infty$ Hilbert space of those rational functions bounded along the imaginary axis and analytic in the right-half plane
${\cal{L}}_\infty$ Hilbert space of those functions bounded along the imaginary axis.
$\mathbb{N} $ set of all natural numbers
$\mathbb{R} ^{+}$ set of real numbers larger than zero
$\mathbb{R} ^{-}$ set of real numbers smaller than zero
$\mathbb{R} ^{m\times n}$ set of all $m\times n$ matrices with real entries
$\ensuremath{\boldsymbol{{\cal{S}}}} $ set of all real rational functions with (finite) poles strictly inside the LHP
$\mathbb{Z} $ set of all integer numbers
$[\alpha_{ik}]$ Matrix where the element in the $i^{th}$ row and $k^{th}$ column is denoted by $\alpha_{ik}$
$[\ensuremath{\mathbf{A}} ]_{ik}$ element in the $i^{th}$ row and $k^{th}$ of matrix \ensuremath{\mathbf{A}}
$[\ensuremath{\mathbf{A}} ]_{i*}$ $i^{th}$ row of matrix \ensuremath{\mathbf{A}}
$ [\ensuremath{\mathbf{A}} ]_{*k}$ $k^{th}$ columm of matrix \ensuremath{\mathbf{A}}
$(...)^*$ complex conjugate of ...
$G_{h0}(s)$ transfer function of a zero-order hold
$G_{h1}(s)$ transfer function of a first-order hold
$H\langle ...\rangle$ operator notation, i.e. $H$ operates on ...
$H_1\otimes H_2\langle... \rangle$ composite operators, i.e., $H_1\langle H_2\langle ...\rangle \rangle$
$\ensuremath{\mathbf{I_k}} $ identity matrix in $\mathbb{R} ^{k\times k}$
d.c. direct current, i.e., zero-frequency signal
d.o.f. degrees of freedom
CTARE Continuous-Time Algebraic Riccati Equation
DTARE Discrete-Time Algebraic Riccati Equation
CTDRE Continuous-Time Dynamic Riccati Equation
DTDRE Discrete-Time Dynamic Riccati Equation
IMC Internal Model Control
IMP Internal Model Principle
LHP left half-plane
OLHP open left half-plane
RHP right half- plane
ORHP open right-half plane
NMP nonminimum phase
MFD Matrix fraction description
LMFD Left matrix fraction description
RMFD Right matrix fraction description
LTI Linear time invariant
LQR Linear quadratic regulator
w.r.t with respect to ...
   

 

<-
Previous		 ^
Up - Book Contents
Next