List of Principal Notation
fs
Ts
ω
f
t
k
u(t), y(t)
y*(t+d+1)
r(t)
e(t)
q-1
s, z
A(q-1), B(q-1), C(q-1)
d
sampling frequency
sampling period
radian frequency (rad/s)
frequency (Hz) or normalized frequency(f /fs)
continuous time or normalized discrete-time (with
respect to the sampling period t /Ts)
normalized discrete time (t /Ts)
plant input and output
tracking reference
reference or external excitation
discrete time Gaussian white noise
shift (delay) operator (q-1 y(t) = y (t-1))
complex variables ( z = esTs)
polynomials in the variable q-1
delay of the discrete-time system (integer)
 (t,q-1), B̂ (t,q-1), Cˆ (t,q-1) estimation of polynomials A, B,C at instant t
â i(t),  b̂ i(t), ĉ i(t)
H(q-1)
H(z-1), H(z)
τ
R(q-1), S(q-1), T(q-1)
Sxy(s), Sxy(z-1)
P(z-1)
∆M
∆τ
θ
θˆ (t)
φ(t), Φ(τ)
F, F(t)
ε°(t), ε(t)
estimated coefficients of polynomials A,B,C
pulse transfer operator (discrete time systems)
discrete-time transfer functions
time delay of a continuous-time system
pulse transfer operators used in a RST digital
controller
sensitivity functions
closed loop characteristic polynomial
modulus margin
delay margin
parameter vector
estimated parameter vector
measurement / observation vector
adaptation gain
a priori /a posteriori prediction error
xxiii
xxiv List of Principal Notation
ε°CL(t), εCL(t)
υ°(t), υ(t)
A, F
F>0
tR
tS
M
ωo , ζ,
E{.}
MV
var.
σ
R(i)
RN(i)
tim
OL
CL
BP
AF-CLOE
ARMAX
CLIM
CLOE
CLOM
ELS
F-CLOE
GLS
IVAM
OEAFO
OEEPM
OEFC
OEFO
PAA
PID
PRBS
RLS
RML
RST
X-CLOE
closed loop a priori / a posteriori prediction error
a priori / a posteriori adaptation error
matrices
positive definite matrix
rise time
settling time
maximum overshoot
natural frequency and damping factor for a
continuous-time second-order system
expectation
mean value
variance
standard deviation
auto-correlation or cross-correlation
normalized auto-correlation or cross-correlation
maximum length of pulse in a PRBS
open loop
closed loop
bandwidth
adaptive filtered closed loop output error
Auto-Regressive Moving Average with eXogenous
input process
closed loop input matching
closed loop output error
closed loop output matching
extended least squares
filtered closed loop output error
generalized least squares
instrumental variable with auxiliary model
output error with adaptive filtered observations
output error with extended prediction model
output error with fixed compensator
output error with filtered observations
parameter adaptation algorithm
proportional + integral + derivative controller
pseudo random binary sequence
recursive least squares
recursive maximum likelihood
two degrees of freedom digital controller
extended closed loop output error
Warning
For sake of notation uniformity, we shall often use, in the case of linear systems
with constant coefficients, q-1 notation both for the delay operator and the complex
variable z-1. The z-1 notation will be especially employed when an interpretation in
the frequency domain is needed (in this case z = e − jωT ).
s