Performance analysis of the generalised projection identification for time-varying systems

TLDR: In this paper, a generalised projection identification algorithm (or a finite data window stochastic gradient identification algorithm) for time-varying systems is presented and its convergence is analyzed by using the Stochastic Process Theory.

Abstract: The least mean square methods include two typical parameter estimation algorithms, which are the projection algorithm and the stochastic gradient algorithm, the former is sensitive to noise and the latter is not capable of tracking the time-varying parameters. On the basis of these two typical algorithms, this study presents a generalised projection identification algorithm (or a finite data window stochastic gradient identification algorithm) for time-varying systems and studies its convergence by using the stochastic process theory. The analysis indicates that the generalised projection algorithm can track the time-varying parameters and requires less computational effort compared with the forgetting factor recursive least squares algorithm. The way of choosing the data window length is stated so that the minimum parameter estimation error upper bound can be obtained. The numerical examples are provided.

Figures

Table 1: The SG estimates and errors

Figure 4: The estimation errors δ versus t for Example 1 (q = 1)

Figure 3: The estimation errors δ versus t for Example 1 (σ2 = 0.502)

Figure 2: The estimation errors δ versus t for Example 1 (σ2 = 0.202)

Table 4: The GPJ estimates and errors with q = 30

Figure 1: The estimation errors δ versus t for Example 1 (σ2 = 0.102)

Full text

Performance analysis of the generalized projection
identification for time-varying systems
Feng Ding
1∗
, Ling Xu
1,2
, Quanmin Zhu
3
1. Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education),
School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China
2. School of Internet of Things Technology, Wuxi Vocational Institute of Commerce, Wuxi 214153, PR China
3. Department of Engineering Design and Mathematics, Bristol BS16 1QY, University of the West of England
* Corresponding author: fding@jiangnan.edu.cn
February 23, 2016
Abstract: The least mean square methods include two typical parameter estimation algorithms, which are the
projection algorithm and the stochastic gradient algorithm, the former is sensitive to noise and the latter is
not capable of tracking the time-varying parameters. On the basis of these two typical algorithms, this paper
presents a generalized projection identification algorithm for time-varying systems and studies its convergence
by using the stochastic process theory. The analysis indicates that the generalized projection algorithm can
track the time-varying parameters and requires less computational effort compared with the forgetting factor
recursive least squares algorithm. The way of choosing the data window length is stated so that the minimum
parameter estimation error upper bound can be obtained. The numerical examples are provided.
Keywords: Parameter estimation; recursive identification; time-varying system
1 Introduction
Establishing the mathematical models of things or systems is the main task of natural sciences. Mathematical
models are very important in many areas such as controller design [1, 2], information filtering [3, 4], fault
detection and diagnosis [5, 6], and state filtering and estimation [7–9]. System identification is the theory and
methods of establishing the mathematical models of systems [10–12]. Parameter estimation methods are basic
for system identification. Recently, Ding and Gu analyzed the performance analysis of the auxiliary model-based
stochastic gradient parameter estimation algorithm for state space systems with one-step state delay [13]. This
paper considers the identification algorithm and its performance analysis for time-varying systems [14, 15],
A(t, z)y(t) = B(t, z)u(t) + v(t), (1)
where {u(t)} and {y(t)} are the input and output sequences of the system, respectively, {v(t)} is a stochastic
noise sequence with zero mean, and z
−1
is a unit backward shift operator: z
−1
y(t) = y(t−1), A(t, z) and B(t, z)
are time-varying coefficient polynomials in z
−1
, and
A(t, z) := 1 + a
1
(t)z
−1
+ a
2
(t)z
−2
+ · · · + a
n
a
(t)z
−n
a
,
∗
Corresponding author
E-mail: fding@jiangnan.edu.cn (F. Ding)
1

B(t, z) := b
1
(t)z
−1
+ b
2
(t)z
−2
+ · · · + b
n
b
(t)z
−n
b
.
Define the time-varying parameter vector ϑ(t − 1) ∈ R
n
to be identified and the regressive information vector
ψ(t) ∈ R
n
consisting of the observations up to and including time (t − 1),
ϑ(t − 1) := [a
1
(t), · · · , a
n
a
(t), b
1
(t), · · · , b
n
b
(t)]
T
∈ R
n
,
ψ(t) := [−y(t − 1), −y(t − 2), · · · , −y(t − n
a
), u(t − 1), u(t − 2), · · · , u(t − n
b
)]
T
∈ R
n
,
where the superscript T denotes a vector transpose. Equation (1) can be written in vector form
y(t) = ψ
T
(t)ϑ(t − 1) + v(t). (2)
The forgetting factor recursive least squares (FF-RLS) algorithm is effective for estimating the time-varying
parameter vector [16]. In this literature, Lozano [17], and Canetti and Espana [18] analyzed the performance
of the FF-RLS algorithms for time-invariant and time-varying systems, respectively. Unfortunately, as the
forgetting factor approaches unity, their results (i.e., the parameter estimation errors) goes to infinity even for
time-invariant systems whose parameters are constant [15]. Bittanti et al studied the convergence properties of
the directional FF-RLS algorithms for time-invariant deterministic systems [19]; for ergodic input-output data,
Ljung and Prioret [20, 21] and Guo et al [22] obtained a parameter estimation error (PEE) upper bound like
E[k
ˆ
ϑ(t) − ϑ(t)k
2
] 6 k
1
(1 − λ) sup E[v
2
(t)] +
k
2
1 − λ
sup E[kw(t)k
2
]
+O
³
(1 − λ)
3/2
+ c(1 − λ)
−1/2
´
(3)
for large enough t, where
ˆ
ϑ(t) is the estimate of ϑ(t), E represents the expectation operator, 0 < λ < 1 is
the forgetting factor, k
1
, k
2
and c are positive constants, w(t) is the parameter changing rate. However, for
deterministic time-varying systems, i.e., the observation noise v(t) ≡ 0, as λ → 0, the PEE upper bound [i.e.,
the expression on the right-hand side of (3)] is bounded. Unfortunately, this result is incompatible with the
existing ones because as λ → 0, the covariance matrix goes to infinity; it is impossible to obtain the bounded
PEE. This motivates us to present a novel generalized stochastic gradient algorithm.
Although the forgetting factor recursive least squares algorithm can estimate the time-varying parameter
vector ϑ(t), its computational load is heavy due to computing the covariance matrix [15,23]. From the perspec-
tive of decreasing computational complexity, the projection algorithm is sensitive to noise and the stochastic
gradient (SG) algorithm is not capable of tracking the time-varying parameters [23]. On the basis of the work
in [24], this paper combines the advantages of the projection algorithm and the SG algorithm to present a gen-
eralized projection identification algorithm for time-varying systems, and studies the convergence performance
of the proposed algorithm by using the stochastic process theory. The generalized projection algorithm can
track the time-varying parameters and requires less computational effort compared with the forgetting factor
recursive least squares algorithm.
This paper is organized as follows. Section 2 gives several time-varying parameter estimation algorithms and
derives the generalized projection algorithm. Section 3 provides several lemmas to prove the main convergence
results in Section 4. Section 5 provides two numerical examples and summarizes some conclusions.
2

2 The generalized projection algorithm
Let us introduce some notations. Let I
n
be an identity matrix of order n, tr[X] denote the trace of a square
matrix X, and the norm of X be kXk :=
p
tr[X
T
X] =
p
tr[XX
T
] and λ
max
[X] represent the maximum
eigenvalue of the positive definite matrix X.
The following discusses several time-varying parameter estimation algorithms to real-time identify the pa-
rameter vector ϑ(t) by using the input-output data, i.e., the observations {u(j), y(j), j 6 t}.
Using the Newton method and minimizing the cost function
J(ϑ(t)) :=
t
X
j=1
[y(j) − ψ
T
(j)ϑ(t)]
2
give the following recursive algorithm of estimating the parameters ϑ(t) [14]:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) + R
−1
(t)ψ(t)[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (4)
where
ˆ
ϑ(t) is the estimate of ϑ(t) at time t. The difference of the matrix R(t) ∈ R
n×n
will lead to different
identification algorithms, e.g., the forgetting factor recursive least squares algorithm, the projection algorithm,
the finite data window least squares algorithm, the stochastic gradient algorithm, and so on.
1. If we take R(t) := P
−1
(t), and
P
−1
(t) := λP
−1
(t − 1) + ψ(t)ψ
T
(t), 0 < λ < 1, (5)
then Equations (4) and (5) form the FF-RLS algorithm [15]:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) + P (t)ψ(t)[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (6)
P
−1
(t) = λP
−1
(t − 1) + ψ(t)ψ
T
(t), 0 < λ < 1. (7)
As the forgetting factor λ goes to unity, the FF-RLS algorithm in (6) and (7) reduces to the recursive least
squares (RLS) algorithm:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) + P (t)ψ(t)[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (8)
P
−1
(t) = P
−1
(t − 1) + ψ(t)ψ
T
(t), P (0) = p
0
I
n
. (9)
The number p
0
should be large enough if the regression variables take very small values.
2. If we take R(t) := P
−1
(t) and
P
−1
(t) =
q−1
X
i=0
ψ(t − i)ψ
T
(t − i) (10)
= P
−1
(t − 1) + ψ(t)ψ
T
(t) − ψ(t − q)ψ
T
(t − q), (11)
then Equations (4) and (11) form the finite data window recursive least squares (FDW-RLS) algorithm:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) + P (t)ψ(t)[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (12)
P
−1
(t) = P
−1
(t − 1) + ψ(t)ψ
T
(t) − ψ(t − q)ψ
T
(t − q), P (0) = p
0
I
n
, (13)
where q is the length of the data window.
3

3. If we take R(t) := r(t)I
n
and the trace of b oth sides of Equation (7) and define
r(t) := tr[P
−1
(t)] = r(t − 1) + kψ(t)k
2
, (14)
then Equations (4) and (14) form the forgetting factor stochastic gradient (FFSG) algorithm (the forgetting
gradient algorithm for short):
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) +
ψ(t)
r(t)
[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (15)
r(t) = λr(t − 1) + kψ(t)k
2
, 0 < λ < 1, r(0) = 1. (16)
As the forgetting factor λ goes to unity, the algorithm in (15)–(16) becomes the following stochastic gradient
(SG) algorithm [23]:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) +
ψ(t)
r(t)
[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (17)
r(t) = r(t − 1) + kψ(t)k
2
, r (0) = 1. (18)
Recently, a multi-innovation stochastic gradient algorithm was proposed to track time-varying parameters for
linear regression models [25].
4. If we take λ = 0 in (16), then the forgetting factor stochastic gradient algorithm in (15)–(16) becomes
the projection (PJ) identification algorithm:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) +
ψ(t)
kψ(t)k
2
[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)]. (19)
5. If we take R(t) := r(t)I
n
and the trace of b oth sides of Equation (10) and define
r(t) := tr[P
−1
(t)] =
q−1
X
i=0
kψ(t − i)k
2
, (20)
then Equations (4) and (20) form the generalized projection (GPJ) identification algorithm:
ˆ
ϑ(t) =
ˆ
ϑ(t − 1) +
ψ(t)
r(t)
[y(t) − ψ
T
(t)
ˆ
ϑ(t − 1)], (21)
r(t) = r(t − 1) + kψ(t)k
2
− kψ(t − q)k
2
, r (0) = 1. (22)
As the data window length q = 1, the GPJ algorithm is the projection algorithm in (19); as we take q = t, the
GPJ algorithm becomes the stochastic gradient algorithm in (17)–(18).
3 The basic lemmas
Define the transition matrix
Φ(t + 1, j) = [I
n
− ψ(t)ψ
T
(t)/r(t)] Φ(t, j), Φ(j, j) = I
n
and the maximum eigenvalue
ρ(t) := λ
max
[Φ
T
(t + s, t)Φ(t + s, t)].
It follows that
Φ(t + 1, t) = I
n
− ψ(t)ψ
T
(t)/r(t) ∈ R
n×n
.
4

Lemma 1 For the system in (2) and the GPJ algorithm in (21)–(22), assume that there exist positive constants
α and β and an integer s > n such that the following strong persistent excitation (SPE) condition holds,
(SPE) αI
n
6
1
s
s−1
X
j=0
ψ(t + j)ψ
T
(t + j) 6 βI
n
, a.s., t > 0.
Then the maximum eigenvalue rho(t) satisfies
ρ(t) 6 1 − α
2
{(q + s)(nβ)
2
(s + 1)
2
}
−1
=: ρ < 1, a.s.
Proof Here, we refer to the way in [25] for proving this lemma. Let ζ
0
∈ R
n
be the unit eigenvector correspond-
ing to the maximum eigenvalue ρ(t) of the matrix Φ
T
(t + s, t)Φ(t + s, t), i.e., Φ
T
(t + s, t)Φ(t + s, t)ζ
0
= ρ(t)ζ
0
.
Use the transition matrix Φ(t + 1, j) to construct the difference equation,
ξ(j + 1) = Φ(j + 1, j)ξ(j), ξ
t
= ζ
0
. (23)
Using the relation Φ(t, i)Φ(i, j) = Φ(t, j), we have
ξ(t + s) = Φ(t + s, t)ξ(t) = Φ(t + s, t)ζ
0
.
Taking the norm to both sides and using the definition of the eigenvalue, we have
kξ(t + s)k
2
= ζ
T
0
Φ
T
(t + s, t)Φ(t + s, t)ζ
0
= ζ
T
0
ρ(t)ζ
0
= ρ(t).
Taking the norm to both sides of (23) gives
kξ(j + 1)k
2
= ξ
T
(j + 1)ξ(j + 1)
= ξ
T
(j)Φ
T
(j + 1, j)Φ(j + 1, j)ξ(j)
= ξ
T
(j)[I
n
− ψ(j)ψ
T
(j)/r(j)]
2
ξ(j)
6 ξ
T
(j)[I
n
− ψ(j)ψ
T
(j)/r(j)]ξ(j)
= ξ
T
(j)ξ(j) − [ψ
T
(j)ξ(j)]
2
/r(j)
= kξ(j)k
2
− [ψ
T
(j)ξ(j)]
2
/r(j).
Thus we have
kψ
T
(j)ξ(j)k
2
/r(j) 6 kξ(j)k
2
− kξ(j + 1)k
2
.
Replacing t + j with j gives
kψ
T
(t + j)ξ(t + j)k
2
/r(t + j) 6 kξ(t + j)k
2
− kξ(t + j + 1)k
2
.
Summing for j from j = 0 to j = s − 1 gives
s−1
X
j=0
kψ
T
(t + j)ξ(t + j)k
2
/r(t + j) 6
s−1
X
j=0
kξ(t + j)k
2
− kξ(t + j + 1)k
2
= kξ(t)k
2
− kξ(t + s)k
2
= 1 − ρ(t). (24)
5

Frequently asked questions

Q1. What are the contributions in "Performance analysis of the generalized projection identification for time-varying systems" ?

The least mean square methods include two typical parameter estimation algorithms, which are the projection algorithm and the stochastic gradient algorithm, the former is sensitive to noise and the latter is not capable of tracking the time-varying parameters. On the basis of these two typical algorithms, this paper presents a generalized projection identification algorithm for time-varying systems and studies its convergence by using the stochastic process theory. The analysis indicates that the generalized projection algorithm can track the time-varying parameters and requires less computational effort compared with the forgetting factor recursive least squares algorithm. The numerical examples are provided. 

Q2. What future works have the authors mentioned in the paper "Performance analysis of the generalized projection identification for time-varying systems" ?

The proposed method in the paper can be extended to study identification problems of other time-varying or time-invariant scalar or multivariable systems [ ?, 28–31 ]. 

Q3. What is the disadvantage of the projection algorithm?

From the perspective of decreasing computational complexity, the projection algorithm is sensitive to noise and the stochastic gradient (SG) algorithm is not capable of tracking the time-varying parameters [23]. 

Q4. What is the disadvantage of the forgetting factor recursive least squares algorithm?

Although the forgetting factor recursive least squares algorithm can estimate the time-varying parameter vector ϑ(t), its computational load is heavy due to computing the covariance matrix [15,23]. 

Q5. What is the PEE upper bound for deterministic time-varying systems?

for deterministic time-varying systems, i.e., the observation noise v(t) ≡ 0, as λ → 0, the PEE upper bound [i.e., the expression on the right-hand side of (3)] is bounded. 

Q6. what is the recursive least squares algorithm?

If the authors take R(t) := P−1(t) andP−1(t) = q−1∑i=0ψ(t− i)ψT(t− i) (10)= P−1(t− 1) + ψ(t)ψT(t)−ψ(t− q)ψT(t− q), (11)then Equations (4) and (11) form the finite data window recursive least squares (FDW-RLS) algorithm:ϑ̂(t) = ϑ̂(t− 1) + P (t)ψ(t)[y(t)−ψT(t)ϑ̂(t− 1)], (12) P−1(t) = P−1(t− 1) + ψ(t)ψT(t)−ψ(t− q)ψT(t− q), P (0) = p0In, (13)where q is the length of the data window.3. 

Q7. What is the noise-to-signal ratio in the example?

In simulation, the input {u(t)} is taken as an uncorrelated stochastic sequence with zero mean and unit variance and {v(t)} as a white noise sequence with zero mean and variance σ2v = 0.202, the noise-to-signal ratio is δns = 24.51%. 

Q8. What is the simplest explanation of the PEE upper bound?

For practical identification problem, the task first is collecting the input-output data {u(t), y(t)} and uses them to construct the information vector ψ(t), and then Remark x A longer window permits better performance of the GPJ algorithm for slowly time varying systems. 

Q9. What is the value of r(t) in the FFSG algorithm?

If the information vector ψ(t) has the lower and upper bounds with 0 < α 6 ‖ψ(t)‖2 6 β, then r(t) in the FFSG algorithm (15)–(16) satisfiesα 

Q10. What is the generalized projection algorithm?

The generalized projection algorithm has better stability performance for a larger data window length and its performance is superior to the projection algorithm and stochastic gradient algorithm. 

Q11. What is the main purpose of this paper?

This paper considers the identification algorithm and its performance analysis for time-varying systems [14,15],A(t, z)y(t) = B(t, z)u(t) + v(t), (1)where {u(t)} and {y(t)} are the input and output sequences of the system, respectively, {v(t)} is a stochastic noise sequence with zero mean, and z−1 is a unit backward shift operator: z−1y(t) = y(t−1), A(t, z) and B(t, z) are time-varying coefficient polynomials in z−1, andA(t, z) := 1 + a1(t)z−1 + a2(t)z−2 + · · ·+ ana(t)z−na , ∗ Corresponding authorE-mail: fding@jiangnan.edu.cn (F. Ding)B(t, z) := b1(t)z−1 + b2(t)z−2 + · · ·+ bnb(t)z−nb . 

Q12. what is the parameter estimation error vector?

If the conditions in Lemma 1 hold, then the parameter estimation error vector ϑ̂(t)−ϑ(t) given by the generalized projection algorithm is mean square bounded, i.e.,E[‖ϑ̂(t)− ϑ(t)‖2 6 ρb ts cE[‖ϑ̂(0)− ϑ(0)‖2 + k1 (q + s)σ 2 v(q − s + 1)2 + k2(q + s)σ 2 w,where k1 = 2ns3(s + 1)2β3/α4, k2 = 2n2s2(s + 1)2β2/α2.Proof 

Q13. What is the corollary of the following study?

Let q = t in (22), i.e., r(t) = r(t− 1) + ‖ψ(t)‖2, the authors havelim t→∞ E[‖ϑ̂(t)− ϑ‖2] 6 lim t→∞k1 (t + s)σ2v(t− s + 1)2 = 0.Thus, for the time invariant stochastic systems, the parameter estimates given by the stochastic gradient algorithm converges to the true parameters – see Theorem 2.Corollary 2 

Q14. What is the main argument for the paper?

On the basis of the work in [24], this paper combines the advantages of the projection algorithm and the SG algorithm to present a generalized projection identification algorithm for time-varying systems, and studies the convergence performance of the proposed algorithm by using the stochastic process theory. 

Q15. What is the parameter estimation error vector?

From Theorem 1, the authors can see that the identification algorithms encounter difficulties for fast-changingparameter systems because the fast-changing parameters have large variance σ2w and the parameter estimation error upper bound become large. 

Q16. What is the simplest way to estimate the parameters of the SG algorithm?

when the generalized projection algorithm works at the beginning of operation, the authors choose a smaller data window length and then a large data window length as the time passes. 

Q17. what is the eigenvalue of the matrix?

Rn be the unit eigenvector corresponding to the maximum eigenvalue ρ(t) of the matrix ΦT(t + s, t)Φ(t + s, t), i.e., ΦT(t + s, t)Φ(t + s, t)ζ0 = ρ(t)ζ0.