Accepted Manuscript. Particle filtering based parameter estimation for systems with output-error type model structures

Particle filtering based parameter estimation for systems with output-error type model structures

Accepted Manuscript

Particle filtering based parameter estimation for systems with output-error type model structures

Jie Ding, Jiazhong Chen, Jinxing Lin, Lijuan Wan

PII: S0016-0032(19)30309-6

DOI: https://doi.org/10.1016/j.jfranklin.2019.04.027

Reference: FI 3913

To appear in: Journal of the Franklin Institute

Received date: 16 September 2018 Revised date: 1 March 2019 Accepted date: 29 April 2019

Please cite this article as: Jie Ding, Jiazhong Chen, Jinxing Lin, Lijuan Wan, Particle filtering based parameter estimation for systems with output-error type model structures, Journal of the Franklin Institute (2019), doi:https://doi.org/10.1016/j.jfranklin.2019.04.027

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ACCEPTED MANUSCRIPT

Particle filtering based parameter estimation for systems with output-error type model structures

Jie Ding^a,∗, Jiazhong Chen^a, Jinxing Lin^a, Lijuan Wan^b

aSchool of Automation and Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

bCollege of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao 266061, PR China

Abstract

The output-error model structure is often used in practice and its identification is important for analysis of output-error type systems. This paper considers the parameter identification of linear and nonlinear output-error models. A particle filter which approximates the posterior probability density function with a weighted set of discrete random sampling points is utilized to estimate the unmeasurable true process outputs. To improve the convergence rate of the proposed algorithm, the scalar innovations are grouped into an innovation vector, thus more past information can be utilized. The convergence analysis shows that the parameter estimates can converge to their true values. Finally, both linear and nonlinear results are verified by numerical simulation and engineering.

Keywords: output-error model, particle filter, parameter estimation, convergence analysis

1. Introduction

Mathematical models and model parameter estimation are the basis of system analysis and con- troller design [1–4]. These involve scalar systems [5–7] and multivariable systems [8–12]. The output- error (OE) type systems have received considerable attention due to their wide application in engineer- ing. Some discussion includes but not limited to parameter estimation, stability analysis and control

5

of OE type systems [13, 14]. The parameter estimation for OE type systems is of prime importance [15, 16]. The key issue is how to estimate the true output of OE type systems with noise-contaminated data, so that the observation errors can reduce the influence on the accuracy of parameter estimation.

Some methods involve the estimation of noise-free outputs and the bias corrected parameter estima- tion. One of the techniques is the auxiliary model idea [17, 18], which replaces the unmeasurable

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output of the OE type system by the output of a properly designed or selected auxiliary model [19].

For example, for multivariate output-error autoregressive systems, an auxiliary model-based recursive generalized least squares algorithm was proposed by using the data filtering [20].

Recently, Jin et al. studied an auxiliary model based identification algorithm for multivariable OE- like systems with missing outputs [21]. For bias corrected parameter estimation, a bias correction or

15

bias compensation is devoted to eliminate estimation bias caused by colored noise [22]. For nonlinear systems with OE type model structures, Piga and T´oth presented a bias-corrected estimator [23].

A bias correction method was discussed by Zheng for the identification of linear dynamic errors-in- variables models [24]. Inspired by the auxiliary model idea [25], a particle filter is introduced to handle the noisy output data, which can improve the accuracy of the priori noise-free output estimates. The

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particle filter has been successfully applied to nonlinear systems. An EM-based particle filter algorithm was developed for nonlinear parameter varying systems in [26]. Chen et al. proposed a novel particle filter based gradient iterative algorithm for the identification of dual-rate nonlinear systems [27] and

✩This work was supported by the National Natural Science Foundation of China (Nos. 61203028 and 61473158) and the Natural Science Foundation of NJUPT (No. NY217063).

∗Corresponding author

Email address: [email protected] (Jie Ding)

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investigated a stochastic gradient based particle filter algorithm for an ARX model with nonlinear communication output [28].

25

In this paper, the benefits of the particle filter are taken into account to estimate the process output. The main contributions can be phrased as follows:

• Motivated by the auxiliary model idea, a particle filtering technique is introduced to estimate the true output of generalized OE type systems. It approximates the posterior probability density function with a weighted set of discrete random sampling points, yielding much better

30

identification performance in nonlinear OE models than linear ones. The convergence properties of the proposed algorithm is analyzed in detail.

• The multi-innovation identification theory is employed to improve the convergence rate of the proposed algorithm. By grouping a number of innovations into an innovation vector, a particle filter based multi-innovation identification algorithm is proposed.

35

Briefly, the rest of this paper is organized as follows. In Section 2, an auxiliary model based recursive least squares algorithm for OE type systems is introduced. In Section 3, a particle filter is employed to estimate the noise-free output, further the parameter estimation algorithm is presented.

In Section 4, the multi-innovation identification theory is introduced to improve the convergence rate of the proposed algorithm. In Section 5, the convergence result of the proposed algorithm is analyzed.

40

Both linear and nonlinear examples are given in Section 6. Finally, some conclusions and future topics are given in Section 7.

2. Problem formulation

Consider the generalized output-error type system with the following input-output relation:

x(t) = f (x(t− 1), · · · , x(t − nx), u(t),· · · , u(t − nu)), (1)

y(t) = x(t) + v(t), (2)

where u(t) and y(t) are the input and output sequence of the system, respectively, x(t) is the noise-free process output and is unknown, v(t) is a white Gaussian noise sequence with zero mean and variance σ², the function f (·) is a linear one with respect to x(t) and u(t), e.g.,

x(t) = B(z⁻¹)

A(z⁻¹)u(t) = [1− A(z⁻¹)]x(t) + B(z⁻¹)u(t) = f (·),

with A(z⁻¹) and B(z⁻¹) are polynomials in unit backward shift operator z⁻¹:

45

A(z) = 1 + a1z⁻¹+ a2z⁻²+· · · + anaz^−na, B(z) = b1z⁻¹+ b2z⁻²+ b3z⁻³+· · · + bn_bz^−nb,

where z⁻¹ represents the unit backward shift operator, i.e., z⁻¹y(t) = y(t− 1).

More generally, f (·) can be treated as a nonlinear function with a real-valued multivariate poly- nomial, which is parameterized as follows:

f (x(t−1), · · · , x(t − nx), u(t),· · · , u(t − nu))

=

nθ

X

i=1

θiψi(x(t− 1), · · · , x(t − nx), u(t),· · · , u(t − nu)),

where θ_i (i = 1, 2,· · · , nθ) are the unknown parameters to be estimated, while ψ_i (i = 1, 2,· · · , nθ) are a-priori chosen functions (e.g., the canonical polynomial basis) in the variables{x(t − 1), · · · , x(t − n_x), u(t),· · · , u(t − nu)}.

Define the parameter vector θ and the information vector ϕ(t) as

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θ := [θ₁, θ₂,· · · , θnθ]^T∈ R^nθ,

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- f(·) -+j? -

v(t)

u(t) x(t) y(t)

ppppp ppppp ppppp ppppp ppppp ppppp ppp p p p p p p p p p p p p p p p

1 N

PN j=1

ˆ xj(t)

ϕ^Ta(t)ˆθ(t) ppp ppp -ppp ppp -ppp ppp ppp ppp

ˆ xa(t)

ˆ x(t)

Original System

Auxiliary model Particle filter

Figure 1: The output-error type system with the auxiliary model or particle filter

ϕ(t) := [ψ₁(x(t− 1), · · · , x(t − nx), u(t),· · · , u(t − nu)),· · · , ψ_nθ(x(t− 1), · · · , x(t − nx), u(t),· · · , u(t − nu))]^T∈ R^nθ, Then Eqs. (1)–(2) can be rewritten as

x(t) = ϕ^T(t)θ,

y(t) = x(t) + v(t). (3)

In Eq. (3), the information vector ϕ(t) is unknown since it contains variables x(t− i). Thus the standard least squares cannot be applied directly. Based on the auxiliary model idea, the unknown variables x(t− i) in ϕ(t) can be replaced with the outputs ˆxa(t− i) of an auxiliary model as shown in Figure 1, where{ϕa(t), θa(t)} are the information vector and the parameter vector of the auxiliary

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model at time t that are defined as follows:

ϕ_a(t) := [ψ₁(ˆx_a(t− 1), · · · , ˆxa(t− nx), u(t),· · · , u(t − nu)),· · · ,

ψ_nθ(ˆx_a(t− 1), · · · , ˆxa(t− nx), u(t),· · · , u(t − nu))]^T∈ R^nθ. (4) Then the cost function can be formed as

J(θ) = Xt i=1

[y(i)− ϕ^Ta(i)θ]².

Minimizing J(θ) and defining the estimate of θ as ˆθ yield the following auxiliary model based recursive least squares (AM-RLS) algorithm [29]:

ˆθ(t) = ˆθ(t− 1) + P (t)ϕa(t)[y(t)− ϕ^Ta(t)ˆθ(t− 1)], (5)

P⁻¹(t) = P⁻¹(t− 1) + ϕa(t)ϕ^T_a(t), (6)

ˆ

xa(t) = ϕ^T_a(t)ˆθ(t). (7)

To initialize this algorithm, let ˆθ(0) = 1_n/p₀, where 1_n is an n-dimensional column vector whose elements are all 1 and p₀ is a large positive number (p₀= 10⁶), and P (0) = p₀I, where I denotes an

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identity matrix of appropriate dimensions.

Remark 1. The basic idea of the auxiliary model is to construct an auxiliary model, generally, an OE-like model is the top choice. Other auxiliary models can be designed in different cases, e.g., a finite impulse response model, which was proposed in [30] to predict the noise-free fast-rate output.

An over-parameterization identification model was presented for input nonlinear OE autoregressive

65

systems in [31]. A key term separation based identification model was developed for input nonlinear OE systems in [32]. A well-designed auxiliary model can improve the accuracy of the parameter estimation.

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3. The particle filtering based RLS algorithm

The output of the auxiliary model ˆx_a(t) is actually the estimate of the unmeasurable noise-free

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process outputs x(t), inspired by this idea, a particle filter is proposed to get the similar estimate by approximating the posterior probability density function (pdf) with a weighted set of discrete random sampling points as shown in Figure 2.

3.1. Particle filtering technique

To define the noise-free process outputs, consider the evolution of the process output sequence

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{x(t − i)} of a target given by Eq. (1). The noise-free process outputs x(t − i) can be recursively computed from noisy outputs Eq. (2). Actually, we seek the filtered estimates of x(t− i) based on the set of all available inputs and noisy outputs {u(j), y(j)} (j = 1, · · · , t − i) and estimated parameter ˆθ(t− i − 1) at time t − i.

From a Bayes perspective [33], the noise-free process outputs estimation problem is to recursively

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calculate some degree of belief in the process output x(t− i) at time t − i with {u(1), · · · , u(t − i), y(1),· · · , y(t − i), ˆθ(t − i − 1)}. Thus, it is required to construct the posterior pdf p(x(t − i)|y(t − i),· · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)). It is assumed that the posterior pdf at previous time t− i − 1 which is also known as the prior, is available. Then the posterior pdf at time t− i may be obtained recursively in two steps: prediction and update.

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The prediction step involves using the noise-free process output model Eq. (1) to obtain the prior pdf at time t− i via the Chapman-Kolmogorov equation

p(x(t− i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1))

= Z

p(x(t− i), x(t − i − 1)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), θ(tˆ − i − 1))dx(t − i − 1)

= Z

p(x(t− i)|x(t − i − 1), y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), θ(tˆ − i − 1))p(x(t − i − 1)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t− i − 1))dx(t − i − 1)

= Z

p(x(t− i)|x(t − i − 1))p(x(t − i − 1)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t− i), · · · , u(1), ˆθ(t − i − 1))dx(t − i − 1), (8) with the fact that p(x(t−i)|x(t−i−1), y(t−i−1), · · · , y(1), x(t−i−1), · · · , x(1), u(t−i), · · · , u(1), ˆθ(t−

i− 1)) = p(x(t − i)|x(t − i − 1)), where Eq. (1) is a Markov process of order one. The probabilistic model of the process output evolution p(x(t− i)|x(t − i − 1)) is defined by Eq. (1).

At time step t− i, a noisy output y(t − i) becomes available, and this may be used to update the prior (update step) via Bayes’ rule

p(x(t− i)|y(t − i), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1))

= p(y(t− i)|x(t − i), y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1))

×p(x(t− i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)) p(y(t− i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)),

= p(y(t− i)|x(t − i)) (9)

×p(x(t− i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)) p(y(t− i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)), where the normalizing constant

p(y(t− i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1))

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= Z

p(y(t− i)|x(t − i))p(x(t − i)|y(t − i − 1), · · · , y(1), x(t − i − 1), · · · , x(1), u(t− i), · · · , u(1), ˆθ(t − i − 1))dx(t − i),

depends on the likelihood function p(y(t−i)|x(t−i)) which can be expressed by the noisy output model in Eq. (2). In the update step Eq. (9), the noisy output y(t− i) is used to modify the prior density to obtain the required posterior density of the current process output. The recurrence relations (8) and (9) form the basis for the optimal Bayes solution. In general, due to the existence of integration, the optimal Bayes solution can be unavailable. Based on the Monte Carlo method [34], the posterior pdf can be expressed as

p(x(t− i)|y(t − i), · · · , y(1),x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1))

≈ XN j=1

ωjδ(x(t− i) − ˆxj(t− i)), (10)

where ω_j is the normalized weight associated with the jth particle, δ(·) represents the Dirac delta function and ˆx_j(t− i) denotes the jth particle sampled from the posterior pdf p(x(t − i)|y(t − i),· · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)) at time t − i. Once the particles and their weights have been determined, the noise-free process outputs can be computed as

ˆ

x(t− i) = XN j=1

ω_jxˆ_j(t− i),

where ω_j is the normalized weight, i.e., ω_j = ω_j/(P^N

j=1

ω_j).

However, it is difficult to sample particles from the posterior pdf directly. An alternative way is to sample from a reference distribution which is referred to as the importance density q(·). Then the regular choice of the importance density q(·) is the prior density:

q(x(t− i)|y(t − i), · · · , y(1), x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1))

= p(x(t− i)|x(t − i − 1), · · · , x(1), u(t − i), · · · , u(1), ˆθ(t − i − 1)). (11) Thus the new particles can be obtained from Eqs. (3) and (11). Next step is to update the weight associated with each new particle. Based on [26], one can get the updated weight ω_j(t) as follows:

ω_j(t) = p(y(t)|ˆxj(t))ω_j(t− 1). (12)

According to Eq. (3), the density function p(y(t)|ˆxj(t)) can be computed by p(y(t)|ˆxj(t)) = 1

√2πσexp



−(y(t)− ˆxj(t))² 2σ²



. (13)

Then, Eq. (12) can be written as ω_j(t) = 1

√2πσexp



−(y(t)− ˆxj(t))² 2σ²



ω_j(t− 1). (14)

Because the variance of the particle weight will increase with time iteration, degradation may occur in the particle filter algorithm, the weights of most particles will be small enough to be neglected except for some important particles. In order to reduce the impact of degradation, three choices can be taken: increasing the number of particles, re-sampling technique, and choosing a reasonable importance density. Here, the re-sampling technique is utilized, i.e., copying and eliminating the particles ˆx_j(t− i) according to the normalized weight ω_j(t), then resetting the weight ω_j(t) = ω_j(t) = _N¹ yields

ˆ

x(t− i) = 1 N

XN j=1

ˆ

xj(t− i). (15)

Then the noise-free process outputs can be estimated by the following particle filter:

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Step 1. Initialization: Suppose the initial process output x(0) is Gaussian, then sample N initial particles {ˆxj(0)}^Nj=1 from the prior density p(x(0)|ˆθ(t − 1)) and set the weight of each particle to _N¹.

Step 2. Importance sampling:

• Collect the input-output data {u(1), y(1), u(2), y(2), · · · , u(t), y(t)}, compute {ˆxj(t −

95

i)}^Nj=1 from Eq. (11).

• Recalculate weights for each particle by Eq. (14).

Step 3. Re-sampling:

• According to the normalized weight, copy and eliminate ˆxj(t− i).

• Reset the weight ωj(t) = ω_j(t) = _N¹.

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Step 4. Output: Compute{ˆx(t − i)}^Nj=1 by Eq. (15).

Step 5. Iteration: Let i = i− 1, if i ≥ 0 go to step 2.; otherwise, end the procedure.

3.2. Identification algorithm

In general, the estimate ˆθ(t) is considered to be much closer to the true value θ than previous estimates{ˆθ(t − 1), · · · , ˆθ(0)}. Thus, in order to estimate the noise-free process outputs at time t − i

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more accurately, parameter estimate ˆθ(t−i−1) is utilized, and corresponding {x(1), x(2), · · · , x(t−i)}

can be updated with Eqs. (11)–(15).

With process output estimate ˆx(t− i), a novel particle filtering based recursive least squares (PF- RLS) algorithm can be presented as follows:

θ(t) = ˆˆ θ(t− 1) + L(t)[y(t) − ˆϕ^T(t)ˆθ(t− 1)], (16) L(t) = P (t) ˆϕ(t) = P (t− 1) ˆϕ(t)[1 + ˆϕ^T(t)P (t− 1) ˆϕ(t)]⁻¹, (17)

P (t) = [I_n− L(t) ˆϕ^T(t)]P (t− 1), (18)

ˆ

ϕ(t) = [ψ₁(ˆx(t− 1), · · · , ˆx(t − nx), u(t),· · · , u(t − nu)),· · · ,

ψnθ(ˆx(t− 1), · · · , ˆx(t − nx), u(t),· · · , u(t − nu))]^T∈ R^nθ, (19) ˆ

x(t− i) = 1 N

XN j=1

ˆ

x_j(t− i), i = 1, 2, 3, · · · , nx. (20)

The initialization of the PF-RLS algorithm is similar to that of the AM-RLS algorithm. The pseudo

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code of the PF-RLS algorithm is described by Algorithm 1.

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Algorithm 1 The particle filtering based RLS algorithm Data: u, y, n_θ, length, N

Result: ˆθ

1: Initialization: ˆθ(0) = [1/p0,· · · , 1/p0]^T ∈ R^nθ, P (0) = p0In_θ, p0 = 10⁶, ˆx(0) = randn(N, 1), {ωj(0)}^Nj=1 = _N¹

2: for t = 1 : length do

3: function Particle Filter(u, y, ˆθ(t− 1), t) 4: for k = 1 : t do

5: for j = 1 : N do 6: Draw ˆx_j(k)∼ (11) 7: Calculate ω_j(k) by (14)

8: end for

9: Normalize ωj(k)

10: [ˆx_j(k), _N¹]=Re-sample [ˆx_j(k), ω_j(k)]

11: Compute ˆx(k) from (15)

12: end for

13: end function

14: Form ˆϕ(t) according to (19) 15: Calculate L(t) from (17) 16: Compute P (t) by (18) 17: Estimate ˆθ(t) by (16) 18: end for

4. The particle filtering based multi-innovation recursive least squares algorithm

The PF-RLS algorithm in Eqs. (16)–(20) can achieve good estimation accuracy but suffer long convergence rate, which needs more iterations. In order to improve its performance while keeping the same accuracy, the multi-innovation identification theory [35–38] is employed as shown in Figure 2,

115

where e(t) is stacked before parameter estimation.

- f(·) -+i? -

v(t)

u(t) x(t) y(t)

-

RLS Algorithm

Particle Filter -+i?

 ?

e(t),· · · , e(t − p + 1) ˆ

x(t)

-



>



Figure 2: The output-error type system with PF-MILS

Let p represent the innovation length. Notice that the innovation e(t) = y(t)− ˆϕ^T(t)ˆθ(t− 1) in Eq. (16) is a scalar quantity. In order to make good use of limited information, one can expand this scalar innovation e(t)∈ R¹ to an innovation vector

E(p, t) := [y(t)− ˆϕ^T(t)ˆθ(t− 1), y(t − 1) − ˆϕ^T(t− 1)ˆθ(t − 1), · · · , y(t− p + 1) − ˆϕ^T(t− p + 1)ˆθ(t − 1)]^T

= = Y (p, t)− ˆφ^T(p, t)ˆθ(t− 1) ∈ R^p.

Define the information matrix ˆφ(p, t) and the stacked output vector Y (p, t) as

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φ(p, t) := [ ˆˆ ϕ(t), ˆϕ(t− 1), · · · , ˆϕ(t− p + 1)] ∈ R^nθ×p,

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Y (p, t) := [y(t), y(t− 1), · · · , y(t − p + 1)]^T∈ R^p, the innovation vector E(p, t) can be expressed as

E(p, t) = Y (p, t)− ˆφ^T(p, t)ˆθ(t− 1).

An improved particle filtering based multi-innovation least squares (PF-MILS) algorithm is derived as θ(t) = ˆˆ θ(t− 1) + L(t)[Y (p, t) − ˆφ^T(p, t)ˆθ(t− 1)], (21) L(t) = P (t) ˆφ(p, t) = P (t− 1)ˆφ(p, t)[I_p+ ˆφ^T(p, t)P (t− 1)ˆφ(p, t)]⁻¹, (22)

P (t) = [I_nθ− L(t)ˆφ^T(p, t)]P (t− 1), (23)

φ(p, t) = [ ˆˆ ϕ(t), ˆϕ(t− 1), · · · , ˆϕ(t− p + 1)], (24) ˆ

ϕ(t) = [ψ₁(ˆx(t− 1), · · · , ˆx(t − nx), u(t),· · · , u(t − nu)),· · · ,

ψ_nθ(ˆx(t− 1), · · · , ˆx(t − nx), u(t),· · · , u(t − nu))]^T∈ R^nθ, (25) ˆ

x(t− i) = 1 N

XN j=1

ˆ

x_j(t− i), i = 1, 2, 3, · · · , nx. (26)

When the innovation length p = 1, E(1, t) = e(t), ˆφ(1, t) = ˆϕ(t), Y (1, t) = y(t), the PF-MILS algorithm in Eqs. (21)–(26) degrades to the PF-RLS algorithm in Eqs. (16)–(20). The pseudo code of the PF-MILS algorithm is described by Algorithm 2.

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Algorithm 2 The particle filtering based MILS algorithm Data: u, y, nθ, length, N , p

Result: ˆθ

1: Initialization: ˆθ(0) = [1/p₀,· · · , 1/p0]^T ∈ R^nθ, P (0) = p₀I_nθ, p₀ = 10⁶, ˆx(0) = randn(N, 1), {ωj(0)}^N_j=1 = _N¹

2: for t = 1 : length do

3: function Particle Filter(u, y, ˆθ(t− 1), t) 4: for k = 1 : t do

5: for j = 1 : N do 6: Draw ˆxj(k)∼ (11) 7: Calculate ω_j(k) by (14)

8: end for

9: Normalize ω_j(k)

10: [ˆxj(k), _N¹]=Re-sample [ˆxj(k), ωj(k)]

11: Compute ˆx(k) from (15)

12: end for

13: end function

14: Form ˆϕ(t) according to (25) 15: Form ˆφ(p, t) by (24)

16: Calculate L(t) from (22) 17: Compute P (t) by (23) 18: Estimate ˆθ(t) by (21) 19: end for

5. Convergence analysis

In this section, we present and prove the convergence property of the proposed algorithm in Eqs.

(16)–(20) according to the method in [39, 40].

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Theorem 1. For the output-error type system in Eq. (3) and the PF-RLS algorithm in Eqs. (16)–

(20),{v(t)} is a white noise sequence with zero mean and variance σ², that is

130

(A1) E[v(t)] = 0, (A2) E[v²(t)] = σ²,

and that there exist positive constants α, β and integer t₀such that for t≥ t0, the following generalized persistent excitation condition holds:

(A3) αIn6 1 t

Xt j=1

ˆ

ϕ(j) ˆϕ^T(j)6 βIn, a.s.

Then, the parameter estimates ˆθ(t) given by the algorithm Eqs. (16)–(20) converge to teir true value θ.

Proof of Theorem 1. Define the parameter estimation error vector

135

θ(t) = ˆ˜ θ(t)− θ. (27)

Substituting Eqs. (3) and (5) into Eq. (27) gives

θ(t) = ˆ˜ θ(t− 1) + P (t) ˆϕ(t)[y(t)− ˆϕ^T(t)ˆθ(t− 1)] − θ

= ˜θ(t− 1) + P (t) ˆϕ(t)[−˜y(t) + ∆(t) + v(t)], (28)

where

˜

y(t) : = ˆϕ^T(t)˜θ(t− 1) ∈ R,

∆(t) : = [ϕ(t)− ˆϕ(t)]^Tθ∈ R.

Using Eqs. (6) and (28), we have

˜θ^T(t)P⁻¹(t)˜θ(t) ={˜θ(t − 1) + P (t) ˆϕ(t)[−˜y(t) + ∆(t) + v(t)]}^TP⁻¹(t){˜θ(t − 1) +P (t) ˆϕ(t)[−˜y(t) + ∆(t) + v(t)]}

= ˜θ^T(t− 1)P⁻¹(t− 1)˜θ(t − 1) − [1 − ˆϕ^T(t)P (t) ˆϕ(t)]˜y²(t)

+2[1− ˆϕ^T(t)P (t) ˆϕ(t)]˜y(t)[∆(t) + v(t)] + ˆϕ^T(t)P (t) ˆϕ(t)[v²(t) + ∆²(t) + 2∆(t)v(t)]

= ˜θ^T(t− 1)P⁻¹(t− 1)˜θ(t − 1) + 2[1 − ˆϕ^T(t)P (t) ˆϕ(t)]˜y(t)[∆(t) + v(t)]

+ ˆϕ^T(t)P (t) ˆϕ(t)[v²(t) + ∆²(t) + 2∆(t)v(t)]. (29) Note that 1− ˆϕ^T(t)P (t) ˆϕ(t) = [1 + ˆϕ^T(t)P (t) ˆϕ(t)]⁻¹≥ 0, and {v(t)} is a white Gaussian noise with zero mean independent of the input signal{u(t)}. Referring to the methods in [39, 40], we can know

140

that the particle filter algorithm is convergent, i.e., when t → ∞, ˆx(t) → x(t). Then we can obtain that when t→ ∞, ˆϕ(t) → ϕ(t), i.e., ∆(t) is bounded with ∆²(t) ≤ ε < ∞. Define a non-negative definite function

V (t) := E[˜θ^T(t)P⁻¹(t)˜θ(t)].

Since ˜θ^T(t− 1)P⁻¹(t− 1)˜θ(t − 1), ˜y(t), ˆϕ^T(t)P (t) ˆϕ(t) and ∆(t) are uncorrelated with v(t), taking the expectation of both sides of Eq. (29) and using (A1) and (A2) get

145

V (t)≤ V (t − 1) + 0 + E{ ˆϕ^T(t)P (t) ˆϕ(t)[v²(t) + ∆²(t)]}

≤ V (t − 1) + E[ ˆϕ^T(t)P (t) ˆϕ(t)](σ²+ ε)

≤ V (0) + E[

Xt j=1

ϕˆ^T(j)P (j) ˆϕ(j)](σ²+ ε). (30)

From Eq. (6), we can obtain

P⁻¹(t− 1) = P⁻¹(t)− ˆϕ(t) ˆϕ^T(t)

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= P⁻¹(t)[In− P (t) ˆϕ(t) ˆϕ^T(t)].

Taking the determinant of both sides of above equation gives:

|P⁻¹(t− 1)| = |P⁻¹(t)||In− P (t) ˆϕ(t) ˆϕ^T(t)|

=|P⁻¹(t)|[1 − ˆϕ^T(t)P (t) ˆϕ(t)].

or ˆ

ϕ^T(t)P (t) ˆϕ(t) = |P⁻¹(t)| − |P⁻¹(t− 1)|

|P⁻¹(t)| .

Replacing t with j and summing for j from j = 1 to t yield Xt

j=1

ϕˆ^T(j)P (j) ˆϕ(j) = Xt j=1

|P⁻¹(j)| − |P⁻¹(j− 1)|

|P⁻¹(j)|

≤ ln |P⁻¹(t)| − ln |1

p₀I_n| = ln |P⁻¹(t)| + n ln p0. (31) Using Eq. (6) and (A3), we get

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P⁻¹(t) = Xt j=1

ˆ

ϕ(j) ˆϕ^T(j) + P⁻¹(0)≤ βtIn+ I_n/p₀= (βt + 1/p₀)I_n, P⁻¹(t)≥ αtIn+ In/p0 = (αt + 1/p0)I,

and

(αt + 1/p0)ⁿ ≤ |P⁻¹(t)| ≤ (βt + 1/p0)ⁿ. Substituting above condition into Eq. (31) gives

Xt j=1

ϕˆ^T(j)P (j) ˆϕ(j) = ln|P⁻¹(t)| + n ln p0 ≤ n ln(βt + 1/p0) + n ln p0.

According to the definition of V (t), we have

V (t) = E[˜θ^T(t)P⁻¹(t)˜θ(t)]≥ (αt + 1/p0)E[k˜θ(t)k²].

Since V (0) = E[˜θ^T(0)P⁻¹(0)˜θ(0)] = n/p²₀, from Eq. (30), we have

(αt + 1/p0)E[k˜θ(t)k²]≤ V (t) ≤ V (0) + E[

Xt j=1

ˆ

ϕ^T(j)P (j) ˆϕ(j)](σ²+ ε)

≤ n/p²0+ [n ln(βt + 1/p0) + n ln p0](σ²+ ε).

Taking the limit of both sides of the above equation gives

155

tlim→∞E[k˜θ(t)k²]≤ lim_t

→∞

n/p²₀+ [nln(βt + 1/p₀) + nlnp₀](σ²+ ε)

(αt + 1/p₀) = 0.

This means that the parameter estimation errorkˆθ(t) − θk converges to zero with the increase of t.

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6. Case studies

Case study 1. Consider the following linear two-order output-error system:

y(t) = X4 i=1

θiψi(x(t− 1), x(t − 2), u(t), u(t − 1), u(t − 2)) + v(t), ϕ(t) = [−x(t − 1), −x(t − 2), u(t − 1), u(t − 2)]^T,

θ = [θ1, θ2, θ3, θ4]^T= [0.412, 0.309, 0.6804, 0.6303]^T.

Here, {u(t)} is taken as a persistent excitation signal sequence with zero mean and unit variance, and{v(t)} as a white noise sequence with zero mean and variance σ² = 0.50² with the corresponding

160

noise-to-signal ratios (NSR): δns= 67.86%. In simulation, we choose 50 particles.

The AM-RLS algorithm in Eqs. (4)–(7) and the PF-RLS algorithm in Eqs. (16)–(20) are applied to estimate the parameters θ_i of this system. Further, the PF-MILS algorithm in Eqs. (21)–(26) with the innovation length p = 2 and 3 is applied to estimate the parameters of this system. The parameter estimates and their errors are shown in Tables 1 to 3 and the estimation errors δ :=kˆθ(t) − θk/kθk

165

versus t are shown in Figures 3 to 4.

Table 1: The AM-RLS estimates and errors (σ²= 0.50², δns= 67.86%)

t θ₁ θ₂ θ₃ θ₄ δ (%)

100 0.23728 0.34316 0.67433 0.41086 26.64187

200 0.37396 0.32041 0.69209 0.53047 10.18674

500 0.40668 0.35113 0.69840 0.55136 8.61788

1000 0.40755 0.34643 0.69120 0.60374 4.46389

2000 0.40395 0.32467 0.69038 0.61313 2.50211

3000 0.40711 0.31837 0.68724 0.61693 1.73159

True values 0.41200 0.30900 0.68040 0.63030

Table 2: The PF-RLS estimates and errors (σ²= 0.50², δns= 67.86%)

t θ1 θ2 θ3 θ4 δ (%)

100 0.21370 0.30012 0.70090 0.41462 27.69758

200 0.39000 0.29028 0.70566 0.55396 8.05393

500 0.41494 0.33589 0.70509 0.56456 7.09380

1000 0.41183 0.33827 0.69393 0.61096 3.54478

2000 0.40604 0.31989 0.69183 0.61716 2.01602

3000 0.40861 0.31474 0.68824 0.61995 1.37588

True values 0.41200 0.30900 0.68040 0.63030

0 500 1000 1500 2000 2500 3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t

δ

PF−RLS AM−RLS

Figure 3: The AM-RLS and PF-RLS estimation errors with σ²= 0.50²

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Table 3: The PF-MILS estimates and errors (σ²= 0.50², δns= 67.86%)

p t θ₁ θ₂ θ₃ θ₄ δ (%)

1 100 0.21370 0.30012 0.70090 0.41462 27.69758

200 0.39000 0.29028 0.70566 0.55396 8.05393

500 0.41494 0.33589 0.70509 0.56456 7.09380

800 0.41672 0.33558 0.69159 0.60154 3.86475

2 100 0.24936 0.30815 0.68445 0.45138 22.79525

200 0.40375 0.29457 0.69600 0.56782 6.26967

500 0.41746 0.34262 0.70448 0.56640 7.19303

800 0.41184 0.33439 0.68374 0.62011 2.59784

3 100 0.30909 0.26896 0.72887 0.55025 13.64422

200 0.43326 0.27162 0.73050 0.62831 6.22629

500 0.42389 0.32863 0.71296 0.56153 7.49176

800 0.40247 0.32123 0.68433 0.63107 1.50952

True values 0.41200 0.30900 0.68040 0.63030

0 100 200 300 400 500 600 700 800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PF−RLS ( MILS,p=1)

PF−MILS,p=2 PF−MILS,p=3

t

δ

Figure 4: The PF-MILS estimation errors with different innovation length (σ²= 0.50²)

c

g

(((((((((((((

((((((((((((( c

DD DD

DD DD DD

DD DD

DD

"!

# h

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pα Ball- _Beam

_{Level Arm}

Servo Motor

Figure 5: A ball and beam system

Case study 2. Consider an experiment setup of a ball and beam system in Figure 6, where the input u(t) is angle of the beam α and the output y(t) is position of the ball. The ball and beam system consists of a horizontal beam which can pivot about one end, a servo motor whose shaft is attached to the other end of the beam and a ball which can freely roll on top of the beam [41]. Due to the

170

connection of the beam with a level arm, the beam can move up and down according to the change of the position angle of the motor. Thus the system can be considered as the following linear third-order OE system:

y(t) = X6 i=1

θ_iψ_i(x(t− 1), x(t − 2), x(t − 3), u(t), u(t − 1), u(t − 2), u(t − 3)) + v(t),

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ϕ(t) = [−x(t − 1), −x(t − 2), −x(t − 3), u(t − 1), u(t − 2), u(t − 3)]^T,

where {u(t), y(t)} are the input and output data downloaded from DaISy [42]. The first 900 data is employed to estimate the parameters θ_i of this system with σ² = 0.50² by applying PF-RLS and

175

PF-MILS algorithms, while the last 100 data can be used to test and verify. Substituting the estimate parameters and the last 100 input data into above OE system, the estimate of the last 100 output data can be obtained. Then the estimation errors are shown in Figures 6–7.

910 920 930 940 950 960 970 980 990 1000

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

t

δ

True value Prediction Estimation errors

Figure 6: The PF-RLS estimation errors (σ²= 0.50²)

910 920 930 940 950 960 970 980 990 1000

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

t

δ

True value Prediction Estimation errors

Figure 7: The PF-MILS estimation errors with innovation length p = 3 (σ²= 0.50²)

Case study 3. Consider the following nonlinear output-error system:

y(t) = X5 i=1

θ_iψ_i(x(t− 1), x(t − 2), u(t), u(t − 1)) + v(t), ϕ(t) = [x(t− 1), x(t − 2), x²(t− 1), u(t − 1), u²(t− 1)]^T,

θ = [θ₁, θ₂, θ₃, θ₄, θ₅]^T= [−0.5, 0.3, 0.2, 0.15, −1.4]^T,

where the initialization of simulation is the same as that of Case study 1. Applying the AM-RLS

180

algorithm, PF-RLS algorithm, and PF-MILS algorithm with the innovation length p = 2 and 3 to estimate the parameters θ_i of this system. The parameter estimates and their errors are shown in Tables 4 to 6 and the estimation errors δ :=kˆθ(t) − θk/kθk versus t are shown in Figures 8 to 9.

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Table 4: The AM-RLS estimates and errors (σ²= 0.50², δns= 37.17%)

t θ₁ θ₂ θ₃ θ₄ θ₅ δ (%)

100 -0.22690 0.35098 0.27919 0.21101 -1.14691 25.30094

200 -0.49324 0.19968 0.18242 0.25654 -1.17595 17.45332

500 -0.51464 0.29607 0.23221 0.18240 -1.25324 10.04853

1000 -0.56426 0.27038 0.21346 0.15035 -1.30870 7.56562

2000 -0.56155 0.28366 0.20970 0.13949 -1.34659 5.48694

3000 -0.55420 0.28657 0.20407 0.13662 -1.36205 4.48587

True values -0.50000 0.30000 0.20000 0.15000 -1.40000

Table 5: The PF-RLS estimates and errors (σ²= 0.50², δns= 37.17%)

t θ₁ θ₂ θ₃ θ₄ θ₅ δ (%)

100 -0.37668 0.34197 0.19781 0.16892 -1.21374 14.83952

200 -0.47321 0.31026 0.20143 0.18435 -1.30245 6.98325

500 -0.52697 0.28898 0.19662 0.15502 -1.31754 5.70338

1000 -0.50050 0.30593 0.20022 0.14267 -1.35316 3.10884

2000 -0.50764 0.30282 0.19782 0.13835 -1.36573 2.41781

3000 -0.49841 0.30274 0.19669 0.13676 -1.36790 2.27899

True values -0.50000 0.30000 0.20000 0.15000 -1.40000

0 500 1000 1500 2000 2500 3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

t

δ

PF−RLS AM−RLS

Figure 8: The AM-RLS and PF-RLS estimation errors with σ²= 0.50²

Table 6: The PF-MILS estimates and errors (σ²= 0.50², δns= 37.17%)

p t θ1 θ2 θ3 θ4 θ5 δ (%)

1 100 -0.37668 0.34197 0.19781 0.16892 -1.21374 14.83952

200 -0.47321 0.31026 0.20143 0.18435 -1.30245 6.98325

500 -0.52697 0.28898 0.19662 0.15502 -1.31754 5.70338

800 -0.50186 0.30598 0.19837 0.15646 -1.35503 2.98583

2 100 -0.48167 0.35824 0.20636 0.13164 -1.27974 8.86523

200 -0.47763 0.32396 0.19821 0.18846 -1.34415 4.90178

500 -0.50411 0.30290 0.19392 0.18663 -1.42355 2.87956

800 -0.48745 0.31068 0.19316 0.18217 -1.40813 2.45120

3 100 -0.48952 0.31011 0.19160 0.11431 -1.40197 2.57014

200 -0.45495 0.30202 0.19441 0.16137 -1.40670 3.07860

500 -0.47557 0.29433 0.19168 0.16144 -1.44023 3.21900

800 -0.47751 0.29145 0.19571 0.15538 -1.41487 1.89389

True values -0.50000 0.30000 0.20000 0.15000 -1.40000

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0 100 200 300 400 500 600 700 800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PF−RLS ( MILS,p=1) PF−MILS,p=2

PF−MILS,p=3

t

δ

Figure 9: The PF-MILS estimation errors with different innovation length (σ²= 0.50²)

Table 7: The iterations of PF-MILS with different innovation length at the same estimation error

Case studies Case study 1 Case study 3

δ (%) 2.6 2.46

p 1 2 3 1 2 3

Iterations 1319 794 744 1805 760 632

Remark 2: At the same estimation error, the PF-RLS needs much more iterations than PF-MILS.

Therefore, the PF-MILS can improve convergence rate at the same convergence accuracy. More

185

detailed results are as follows.

• For output-error type systems, the parameter estimates of the AM-RLS and PF-RLS algorithms can converge to their true values very fast as shown in Tables 1–2 and 4–5. The PF-RLS algorithm performs slightly better than AM-RLS does in the linear output-error system as shown in Figure 3. The main advantages of the particle filter are in the application of strong nonlinear and

190

non-Gaussian systems, so the PF-RLS performs obviously better than AM-RLS in the nonlinear output-error system as shown in Figure 8.

• With different innovation length p, the PF-MILS algorithm with larger p can lead to better convergence rate and accuracy as shown in Figures 4 and 9. It is easy to find that the PF-MILS needs fewer iterations than the PF-RLS to achieve the same convergence accuracy as shown in

195

Table 7.

• The proposed algorithms have very small estimation errors as shown in Figures 6 and 7. Thus they are widely provided with practical value in engineering.

7. Conclusions

By employing the particle filtering technique and the multi-innovation identification theory, two

200

particle filter based algorithms are proposed for output-error type systems. The limitation exists in multi-input and multi-output (MIMO) systems, since the variables of the MIMO system will be of high dimension, yielding to heavy computation burden. But it is still a good choice for single-input and single-output systems to get accurate models. From simulation results, we can find that the convergence of the particle filter is slightly better than the auxiliary model method in linear systems,

205

showing the obvious advantages in nonlinear system. Furthermore, the convergence analysis shows that the parameter estimation error consistently converges to zero. The simulation results verify that the proposed algorithms have effective convergence rate and high accuracy. The propoased algorithm can coombine the hierarchical principle [43, 44] and the iterative methods [45, 46] to study the parameter identificatin problems of multivariable systems [47, 48], bilinear systems [49–51] and nonlinear systems

210

[52], and can be applied to otther fields [53–57].

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