This document discusses various methods for parameter estimation of discrete linear systems, including deterministic and stochastic methods. Deterministic methods described include the simplified algorithm, Kaczmarz's algorithm, projection algorithm, batch least squares, weighted least squares, recursive least squares, and recursive least squares with exponential forgetting. Stochastic methods described include extended recursive least squares, extended recursive least squares with exponential forgetting, and modified extended recursive least squares. The goal of these methods is to estimate parameters of linear system transfer functions from input-output data in an online, recursive fashion.

1. Contents
1.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Deterministic Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Simplified Algorithm (SA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Kaczmarz’s Algorithm (KA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.3 Projection Algorithm (PA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.4 Batch Least Square (BLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.5 Weighted Least Square (WLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.6 Recursive Least Square (RLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.7 Recursive Least Square with Exponential Forgetting (RLS with λ) . . . . . . . . . . . . 4
1.2.8 Recursive Least Square with Varying Exponential Forgetting
(RLS with varying λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Stochastic Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Extended Recursive Least Square (ERLS) . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Extended Recursive Least Square with Exponential Forgetting
(ERLS with λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.3 Extended Recursive Least Square with Varying Exponential Forgetting
(ERLS with varying λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.4 Modified Extended Recursive Least Square (MERLS) . . . . . . . . . . . . . . . . . . . 6
1.3.5 Modified Extended Recursive Least Square with Exponential Forgetting
(MERLS with λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.6 Modified Extended Recursive Least Square with Varying Exponential
Forgetting (MERLS with varying λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1

2. 1.1. PARAMETER ESTIMATION Adaptive Control
1.1 Parameter Estimation
On-line determination of process parameters is a key element in adaptive control. A recursive parameter
estimator appears explicitly as a component of a self-tuning regulator. Parameter estimation also occurs
implicitly in a model-reference adaptive controller. This section presents some methods for real-time parameter
estimation. It is useful to view parameter estimation in the broader context of system identification. The key
elements of system identification are selection of model structure, experiment design, parameter estimation,
and validation. Since system identification is executed automatically in adaptive systems. it is essential to
have a good understanding of all aspects of the problem. Selection of model structure and parameterization are
fundamental issues. Simple transfer function models will be used in this chapter. The identification problems
are simplified significantly if the models are linear in the parameters.
This is about parameter estimation of discrete linear system transfer function using linear estimation methods.
1.2 Deterministic Parameter Estimation
G(z−1
) =
z−dB(z−1)
A(z−1)
B(z−1
) = b0 + b1z−1
+ ... + bnbz−nb
A(z−1
) = 1 + a1z−1
+ ... + anaz−na
y(k) = φT
(k)ˆθ(k − 1)
φT
(k) = [−y(k − 1) − y(k − 2) ... − y(k − na)
u(k − d)u(k − d − 1) ... u(k − d − nb)]
ˆθ(k − 1) = a1 a2 ... ana b0 b1 ... bnb
T
number of knowns = nu = na + nb + 1
n = Max(na, nb + d)
number of sampled data = N > nu + n − 1
NOTE : Estimation stats from time n + 1
1.2.1 Simplified Algorithm (SA)
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)]
ˆθ(k + 1) = ˆθ(k) +
φ(k + 1)
φT (k + 1)φ(k + 1)
y(k + 1) − φT
(k + 1)ˆθ(k)
1.2.2 Kaczmarz’s Algorithm (KA)
φT
(k + 1) = [ −y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)]
ˆθ(k + 1) = ˆθ(k) +
γφ(k + 1)
φT (k + 1)φ(k + 1)
y(k + 1) − φT
(k + 1)ˆθ(k)
α ≥ 0
0 < γ < 2
Mohamed Mohamed El-Sayed Atyya Page 2 of 8

3. 1.2. DETERMINISTIC PARAMETER ESTIMATION Adaptive Control
1.2.3 Projection Algorithm (PA)
φT
(k + 1) = [ −y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)]
ˆθ(k + 1) = ˆθ(k) +
γφ(k + 1)
α + φT (k + 1)φ(k + 1)
y(k + 1) − φT
(k + 1)ˆθ(k)
1.2.4 Batch Least Square (BLS)
ψ =




−y(k − 1) ... −y(k − na) u(k − d) ... u(k − d − nb)
−y(k) ... −y(k − na + 1) u(k − d + 1) ... u(k − d − nb + 1)
. ... . . ... .
−y(N − 1) ... −y(N − na) u(N − d) ... u(N − d − nb)




Y = y(k) y(k + 1) ... y(N)
T
∴ θ = ψT
ψ
−1
ψT
Y
1.2.5 Weighted Least Square (WLS)
θ = ψT
Wψ
−1
ψT
WY
Where W is a diagonal matrix with wii = γN−n−i−1 and γ < 1
1.2.6 Recursive Least Square (RLS)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)]
K(k + 1) = P(k)φ(k + 1) I + φT
(k + 1)P(k)φ(k + 1)
−1
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) (k + 1)
P(k + 1) = I − K(k + 1)φT
(k + 1) P(k)
P(k + 1) =
P(k + 1) + PT (k + 1)
2
Mohamed Mohamed El-Sayed Atyya Page 3 of 8

4. 1.3. STOCHASTIC PARAMETER ESTIMATION Adaptive Control
1.2.7 Recursive Least Square with Exponential Forgetting (RLS with λ)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)]
K(k + 1) = P(k)φ(k + 1) λ + φT
(k + 1)P(k)φ(k + 1)
−1
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) (k + 1)
P(k + 1) = I − K(k + 1)φT
(k + 1) P(k)/λ
P(k + 1) =
P(k + 1) + PT (k + 1)
2
1.2.8 Recursive Least Square with Varying Exponential Forgetting
(RLS with varying λ)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
λ(k) = 0.3 → 0.999
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)]
K(k + 1) = P(k)φ(k + 1) λ(k) + φT
(k + 1)P(k)φ(k + 1)
−1
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) (k + 1)
P(k + 1) = λ(k) − K(k + 1)φT
(k + 1) P(k)
P(k + 1) =
P(k + 1) + PT (k + 1)
2
λ(k + 1) = 1 −
1 − φT (k + 1)K(k + 1) 2(k + 1)
σ2( )µ( )
NOTE : Update λ after n times
1.3 Stochastic Parameter Estimation
y(k) =
z−dB(z−1)
A(z−1)
u +
C(z−1)
A(z−1)
= φT
(k)ˆθ(k − 1)
B(z−1
) = b0 + b1z−1
+ ... + bnbz−nb
A(z−1
) = 1 + a1z−1
+ ... + anaz−na
C(z−1
) = 1 + c1z−1
+ ... + cncz−nc
ˆθ(k − 1) = [a1 a2 ... ana b0 b1 ... bnb c1 c2 ... cnc]T
number of knowns = nu = na + nb + nc + 1
n = Max(na, nb + nc + d)
number of sampled data = N > nu + n − 1
NOTE : Estimation stats from time n + 1
Mohamed Mohamed El-Sayed Atyya Page 4 of 8

5. 1.3. STOCHASTIC PARAMETER ESTIMATION Adaptive Control
Figure 1.1: Noise is applied on the output signal
1.3.1 Extended Recursive Least Square (ERLS)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)
(k) (k − 1) ... (k − nc + 1)]
K(k + 1) = P(k)φ(k + 1) I + φT
(k + 1)P(k)φ(k + 1)
−1
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) (k + 1)
P(k + 1) = I − K(k + 1)φT
(k + 1) P(k)
P(k + 1) =
P(k + 1) + PT (k + 1)
2
1.3.2 Extended Recursive Least Square with Exponential Forgetting
(ERLS with λ)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)
(k) (k − 1) ... (k − nc + 1)]
K(k + 1) = P(k)φ(k + 1) λ + φT
(k + 1)P(k)φ(k + 1)
−1
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) (k + 1)
P(k + 1) = I − K(k + 1)φT
(k + 1) P(k)/λ
P(k + 1) =
P(k + 1) + PT (k + 1)
2
Mohamed Mohamed El-Sayed Atyya Page 5 of 8

6. 1.3. STOCHASTIC PARAMETER ESTIMATION Adaptive Control
1.3.3 Extended Recursive Least Square with Varying Exponential Forgetting
(ERLS with varying λ)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
λ(k) = 0.3 → 0.999
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)
(k) (k − 1) ... (k − nc + 1)]
K(k + 1) = P(k)φ(k + 1) λ(k) + φT
(k + 1)P(k)φ(k + 1)
−1
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) (k + 1)
P(k + 1) = λ(k) − K(k + 1)φT
(k + 1) P(k)
P(k + 1) =
P(k + 1) + PT (k + 1)
2
λ(k + 1) = 1 −
1 − φT (k + 1)K(k + 1) 2(k + 1)
σ2( )µ( )
NOTE : Update λ after n times
1.3.4 Modified Extended Recursive Least Square (MERLS)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)
(k) (k − 1) ... (k − nc + 1)]
K(k + 1) = P(k)φ(k + 1) I + φT
(k + 1)P(k)φ(k + 1)
−1
−
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) −
(k + 1)
+
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k + 1)
P(k + 1) = I − K(k + 1)φT
(k + 1) P(k)
P(k + 1) =
P(k + 1) + PT (k + 1)
2
Mohamed Mohamed El-Sayed Atyya Page 6 of 8

7. 1.3. STOCHASTIC PARAMETER ESTIMATION Adaptive Control
1.3.5 Modified Extended Recursive Least Square with Exponential Forgetting
(MERLS with λ)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)
(k) (k − 1) ... (k − nc + 1)]
K(k + 1) = P(k)φ(k + 1) λ + φT
(k + 1)P(k)φ(k + 1)
−1
−
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) −
(k + 1)
+
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k + 1)
P(k + 1) = I − K(k + 1)φT
(k + 1) P(k)/λ
P(k + 1) =
P(k + 1) + PT (k + 1)
2
1.3.6 Modified Extended Recursive Least Square with Varying Exponential
Forgetting (MERLS with varying λ)
P0 = 106
I , or P0 = ψT
ψ
−1
, size(P) = nu x nu
λ(k) = 0.3 → 0.999
φT
(k + 1) = [−y(k) − y(k − 1) ... − y(k − na + 1)
u(k − d + 1) u(k − d − 2) ... u(k − d − nb + 1)
(k) (k − 1) ... (k − nc + 1)]
K(k + 1) = P(k)φ(k + 1) λ(k) + φT
(k + 1)P(k)φ(k + 1)
−1
−
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k)
ˆθ(k + 1) = ˆθ(k) + K(k + 1) −
(k + 1)
+
(k + 1) = y(k + 1) − φT
(k + 1)ˆθ(k + 1)
P(k + 1) = λ(k) − K(k + 1)φT
(k + 1) P(k)
P(k + 1) =
P(k + 1) + PT (k + 1)
2
λ−
(k + 1) = 1 −
1 − φT (k + 1)K(k + 1) −2
(k + 1)
σ2( −)µ( −)
λ+
(k + 1) = 1 −
1 − φT (k + 1)K(k + 1) +2
(k + 1)
σ2( +)µ( +)
σ−
=
σ2( −)
σ2( −) + σ2( +)
σ+
=
σ2( +)
σ2( −) + σ2( +)
λ(k + 1) = σ−
λ−
(k + 1) + σ+
λ+
(k + 1)
NOTE : Update λ after n times
Mohamed Mohamed El-Sayed Atyya Page 7 of 8

8. 1.4. MATLAB CODES Adaptive Control
1.4 MATLAB Codes
1.2.1 http://goo.gl/Vddvtt
1.2.2 http://goo.gl/UwWFTW
1.2.3 http://goo.gl/tV4Ni6
1.2.4 http://goo.gl/rY2n7I
1.2.6 http://goo.gl/e7J2kq
1.2.7 http://goo.gl/3q6Yc6
1.2.8 http://goo.gl/SCPvEW
1.3.1 http://goo.gl/JnrdNh
1.3.2 http://goo.gl/xjpHha
1.3.3 http://goo.gl/6wVeuW
1.3.4 http://goo.gl/vuKeaL
1.3.5 http://goo.gl/mL0RCz
1.3.6 http://goo.gl/vzViYE
1.5 References
1. Karl Johan Astrom, Adaptive Control, 2nd Edition.
2. David I. Wilson, Advanced Control using MATLAB or Stabilising the unstabilisable, Auckland University
of Technology, New Zealand, May 15, 2015
1.6 Contacts
mohamed.atyya94@eng-st.cu.edu.eg
Mohamed Mohamed El-Sayed Atyya Page 8 of 8