Adaptive parameter identification of linear SISO systems with unknown ...
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Systems & Control Letters 66 (2014) 43–50
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Adaptive parameter identification of linear SISO systems with unknown time-delay✩ Jing Na a,∗ , Xuemei Ren b , Yuanqing Xia b a
Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, 650500, PR China
b
School of Automation, Beijing Institute of Technology, Beijing, 100081, PR China
article
info
Article history: Received 23 May 2013 Received in revised form 29 December 2013 Accepted 20 January 2014 Available online 14 February 2014 Keywords: System identification Parameter estimation Time-delay systems
abstract An adaptive online parameter identification is proposed for linear single-input-single-output (SISO) time-delay systems to simultaneously estimate the unknown time-delay and other parameters. After representing the system as a parameterized form, a novel adaptive law is developed, which is driven by appropriate parameter estimation error information. Consequently, the identification error convergence can be proved under the conventional persistent excitation (PE) condition, which can be online tested in this paper. A finite-time (FT) identification scheme is further studied by incorporating the sliding mode scheme into the adaptation to achieve FT error convergence. The previously imposed constraint on the system relative degree is removed and the derivatives of the input and output are not required. Comparative simulation examples are provided to demonstrate the validity and efficacy of the proposed algorithms. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Time-delay is usually unavoidable in control systems, which may lead to a sluggish response or even trigger instability [1]. The control design for time-delay systems mainly assumes a precise model (in particular for time-delay parameter). In this sense, online parameter identification of time-delay systems is an important and challenging topic [2]. An intuitive method is to approximate unknown time-delay by a rational transfer function (e.g. pade approximation) [3], where more parameters are introduced and the approximation error for a large time-delay may degrade the identification performance. In [4], a direct identification method has been developed based on step responses, and the relay-based method has been proposed for some low-order timedelay systems in [5]. These approaches are implemented offline and sensitive to noise. Another category has been rooted in adaptive identification: in [6] and the subsequent work [7], the identifiability analysis of SISO time-delay systems has been first studied, and then an adaptive identification was proposed based on the
✩ This work was supported by the National Natural Science Foundation of China under grant number: 61203066 and 61273150. ∗ Corresponding author. E-mail addresses: [email protected] (J. Na), [email protected] (X. Ren), [email protected] (Y. Xia).
http://dx.doi.org/10.1016/j.sysconle.2014.01.005 0167-6911/© 2014 Elsevier B.V. All rights reserved.
distributed delay flavored (DDF) procedure, where the time-delay parameter is determined in terms of an identifier that includes more fictitious delay terms than the actual system. Adaptive estimation for pure time-delay was also studied in [8], where the input signal and its derivative are all used. In [9], a nonlinear optimization was used in the adaptive identification, where the solution of the resulting min–max problem is not trivial. Recently, an algebraic identification method has been presented [10] using algebraic operations with annihilation and integration. Taking the merit of variable structure observers, several algorithms for delay identification were presented [11], where the derivatives of the input and/or output are assumed to be measurable and available. This assumption may be stringent for practical applications. An alternative idea, polynomial identification [12], was developed to identify the time-delay and rational transfer function parameters. However, the complexity of the practical implementation for high-order systems limits its application. Although the least-squares (LS) algorithm [13] has been well-recognized in the identification of linearly parametric systems, it cannot be directly used for time-delay systems because the time-delay usually imposes nonlinearity in the formulation and thus is difficult to be presented as an explicit parameter in a linear-in-parameterized form. To address this issue, in our previous work [14], a nonlinear least-squares algorithm was developed to simultaneously identify the time-delay and other parameters of linear SISO systems. This idea has also been extended to nonlinear time-delay systems by
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means of neural networks. [15]. However, the adaptive laws proposed in [14] are derived based on the gradient error embedded in a nonlinear optimization problem, and thus the parameter estimation convergence has not been proved. Moreover, a stringent assumption that the system relative degree should be larger than two is imposed in [14] so that the applicability of the method is limited. In this paper, we focus on the full parameter identification of linear continuous-time SISO systems with unknown time-delay and further improve our previous work [14] by proposing novel adaptive laws. It is assumed that the structure of the underlying system (e.g., system order) is known and only system parameters and time-delay are unknown. We first apply the Taylor series expansion to reformulate the time-delay system into a parameterized form, where the time-delay term can be explicitly indicated in a linear-in-parameterized formula. Then appropriate parameter estimation error information is derived, which is used in the design of adaptive laws. Consequently, the parameter convergence can be proved by means of Lyapunov theory under the conventional PE condition that can be online tested, and the improved performance (e.g., fast convergence) is achieved because of the inclusion of this parameter estimation error. In contrast to [12,14], the assumption on the system relative degree is successfully removed so that the identification for general high-order systems is straightforward. Finally, by incorporating the sliding mode technique into the design and synthesis of adaptive laws, an alternative parameter identification scheme is studied guaranteeing finite-time (FT) error convergence. Compared to other methods (e.g., [7,8,11]), the timedelay parameter can be directly estimated and the derivatives of the input and output are not required, i.e., only the input and output measurements are used. Comparative simulations are provided to verify the characteristics and the improved performance of the proposed algorithms. The paper is organized as follows. Section 2 presents the problem formulation and Section 3 introduces the adaptive parameter identification scheme. FT parameter identification is studied in Section 4. Simulation examples are provided in Section 5 and some conclusions are given in Section 6.
the input and output are assumed to be available (e.g., u˙ in [8] and y˙ in [11] are used) and thus the proposed schemes may be sensitive to noises. In this paper, only the input u and output y are required, while their derivatives are not utilized. Moreover, the unknown delay parameter τ can be directly identified. In this paper, without loss of generality, we assume that the structure of the underlying system (e.g., system order) is known and the input u and output y are available and bounded. This is a common assumption in system identification literature [3,14]. For the case with unbounded u and y, the normalization operation [13, 14] can be used accordingly to fulfill this condition. For the purpose of parameter identification, we filter (2) by a system1/Λ(s) with Λ(s) = sn + λ1 sn−1 + · · · + λn being a Hurwitz polynomial, and then rewrite (2) as sn
Λ(s)
[y] + a1
= b0
sn−1
Λ(s)
sm e−τ s
Λ(s)
+ bm
e−τ s
Λ(s)
[y] + · · · + an
[ u] + b 1 [ u] +
sm−1 e−τ s
Λ(s)
1
Λ(s)
1
Λ(s)
[y]
[u] + · · ·
[d]
(3)
i −τ s
i
where Λs(s) [y], s Λe(s) [u] and Λ1(s) [d] denote the filtered variables of the output y, input u(t − τ ) and the disturbance d by means of a i
linear proper system Λs(s) , i = 0, . . . , n. Adding and subtracting the term Λ(s)y/Λ(s) on both sides of (3), it can be rewritten as a parameterized form y = (λ1 − a1 )
sn−1
Λ(s)
+ (λn − an ) + b1
[y] + (λ2 − a2 )
1
Λ(s)
sm−1 e−τ s
Λ(s)
[y] + b0
sn−2
Λ(s)
sm e−τ s
Λ(s)
[ u] + · · · + b m
[y] + · · ·
[ u]
e−τ s
Λ(s)
[u] +
1
Λ(s)
[d ]
= θ T φ(τ ) + δ 2. Problem formulation
where
Consider the following linear continuous-time SISO system with an input time-delay y(n) (t ) + a1 y(n−1) (t ) + · · · + an y(t ) = b0 u(m) (t − τ )
+ b1 u(m−1) (t − τ ) + · · · + bm u(t − τ ) + d(t )
(1)
where u(t ), y(t ) are the system input and output, respectively, d(t ) is an unknown bounded disturbances, the system order (e.g.,m ≤ n) is a prior known, and a1 , . . . , an and b0 , . . . , bm are unknown parameters and τ is the unknown time-delay. The objective of this paper is to estimate a1 , . . . , an , b0 , . . . , bm and τ using only the input u and output y. Taking the Laplace transform of (1) under zero initial conditions, the following transfer function is obtained: G(s) =
Y (s) U (s)
m
=
b0 s + b1 s
(4)
m−1
+ · · · + bm
s n + a1 s n − 1 + · · · + an
e−τ s
(2)
where Y (s) and U (s) are the Laplace transforms of the output and input, respectively. Remark 1. The parameter identification of the linear SISO timedelay system (1) has been studied in the literature, e.g., [3–12]. In particular, the identifiability analysis of system (1) has been addressed in [6,7], where the unknown plant parameters including delays can be identified. In some of these results, the derivatives of
θ = [λ1 − a1 , λ2 − a2 , . . . , λn − an , b0 , . . . , bm ]T , n−1 s sn−2 φ(τ ) = [y], [y], . . . , Λ(s) Λ(s) T 1 sm e−τ s e−τ s 1 [y], [u], . . . , [ u] , δ= [d]. Λ(s) Λ(s) Λ(s) Λ(s) Denote θˆ as the estimation of the unknown parameter θ , and φ(τˆ ) as a vector that has the same form as φ(τ ) but uses the estimated delay τˆ , such that
θˆ = [λ1 − aˆ 1 , λ2 − aˆ 2 , . . . , λn − aˆ n , bˆ 0 , . . . , bˆ m ]T , φ(τˆ ) = sm e−τˆ s
Λ(s)
sn−1
Λ(s)
[y],
[u], . . . ,
sn−2
Λ(s) e−τˆ s
Λ(s)
[y], . . . ,
1
Λ(s)
[y],
T [ u]
,
where τˆ is the estimation of τ , aˆ i , i = 1 . . . n and bˆ i , i = 0 . . . m are the estimation of ai , i = 1 . . . n and bi , i = 0 . . . m, which will be determined in the following development. Then we reformulate (4) as y = θ T φ(τˆ ) + ζ
(5)
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J. Na et al. / Systems & Control Letters 66 (2014) 43–50
where the remainder term ζ = θ T [φ(τ ) − φ(τˆ )] + δ will be further analyzed. As the unknown time-delay τ should be identified together with the unknown parameter vector θ , we apply the Taylor series expansion for the continuous function e−τ s − e−τˆ s with respect to the estimated delay τˆ , such that
ζ = θ T [φ(τ ) − φ(τˆ )] + δ =
i =m sm−i [e−τ s − e−τˆ s ] bi [ u] + δ Λ(s) i=0
= −τ˜
i=m sm−i+1 e−τˆ s bi [u] + O(τ˜ ) + δ Λ(s) i=0
= −τ˜
i=m sm−i+1 e−τˆ s bˆ i [ u] + ρ Λ(s) i=0
(6)
where τ˜ = τ − τˆ is the time-delay estimation error, O(τ˜ ) represents high-order terms of the Taylor series expansion, and ρ = m−i+1 −τˆ s
b˜ i s Λ(s)e [u]+ O(τ˜ )+δ can be considered as a bounded disturbance [15]. Define an augmented parameter vector with unknown delay τ as
−τ˜
i=m i =0
Θ = [θ T , τ ]T = [λ1 − a1 , λ2 − a2 , . . . λn
− an , b0 , . . . , bm , τ ]T
(7)
and the augmented regressor vector as
i =m sm−i+1 e−τˆ s Φ = φ(τˆ ) , − bˆ i [u] Λ(s) i=0
T
s n −1
Λ(s)
[y],
s n −2
Λ(s)
[y], . . . ,
1
Λ(s)
[y],
i=m sm−i+1 e−τˆ s bˆ i [u], − [ u] Λ(s) Λ(s) i=0
e−τˆ s
sm e−τˆ s
Λ(s)
Remark 5. After representing time-delay system (1) as a linearly parameterized system (9), some well-known adaptive identification methods (e.g., gradient, LS and recursive LS [13]) may be adopted. However, they all depend on a predictor/observer to produce the predictor/observer error for driving the adaptive laws. In the following, inspired by the fact that the utilization of parameter estimation error in the adaptation may benefit the estimation performance [13,17], we will propose a novel adaptive law design such that the error convergence can easily be studied compared to [14]. Definition 1 ([18]). A vector or matrix function Φ is persistently t +T excited (PE) if there exist T > 0, ϖ > 0 such that t Φ (r )Φ T (r ) dr ≥ ϖ I , ∀t ≥ 0.
3. Adaptive parameter identification
[u], . . . ,
T .
(8)
To estimate the unknown parameter vector Θ , we define the auxiliary regressor matrix P ∈ R(m+n+2)×(m+n+2) and vector Q ∈ Rm+n+2 as
T ˙ P = −ℓP + ΦΦ ,
Then system (5) can be represented as i =m sm−i+1 e−τˆ s y = Θ T Φ + τˆ bˆ i [u] + ρ. Λ(s) i =0
Remark 4. In our previous work [14], a nonlinear LS identification has been proposed based on the gradient error, where the error convergence cannot be proved due to the induced nonlinearities of delay. Moreover, the system studied in [14] is assumed to be with a relative degree larger than 2 (i.e., m + 2 ≤ n). This paper is dedicated to further improve the result of [14] by proposing novel adaptive laws with guaranteed error convergence, to remove some assumptions on the system (e.g., input/output derivatives [7,8,11], relative degree [14]), and to avoid complex operations (e.g., pade approximation [3], annihilation and integration [10] and nonlinear optimization [9]).
In this paper, we define λmax (·), λmin (·) as the maximum and minimum eigenvalues of the corresponding matrices, respectively.
T
=
45
(9)
ˆ = The problem is now to find the parameter estimation Θ T T ˆ [θ , τˆ ] based on the measurements of input u and output y so that ˆ converges to Θ as close as possible. Θ Remark 2. In (6)–(9), the first order Taylor series expansion is applied for the function e−τ s − e−τˆ s to make delay parameter τ explicitly appear in the parameter vector Θ to be estimated, and the remainder term O(τ˜ ) depends on the delay error τ˜ = τ − τˆ , which will vanish as long as the parameter estimation convergence is achieved. Consequently, as suggested in [16], a ‘‘sufficiently near’’ initial delay value is necessary to improve the estimation performance. On the other hand, although the higher-order Taylor series expansion may be preferable [14] the resulting nonlinear formulation makes the delay identification nontrivial. With regard to this, the first order Taylor series expansion is used in this paper. Remark 3. To address the effect of measurement noises or disturbances, a bounded disturbance d ∈ L∞ is included in system (1), and thus an extra term 1/Λ(s)[d] is introduced in (3)–(8), which can be lumped into the auxiliary variable ρ in (9). In this case, we will show that the identification error converges to a bounded compact set around zero (see Theorems 1 and 2), i.e., the proposed methods are robust against to bounded noises and disturbances.
P (0) = 0
i=m sm−i+1 e−τˆ s bˆ i [ u] , Q˙ = −ℓQ + Φ y − τˆ Λ(s) i =0
Q (0) = 0
(10)
where ℓ > 0 is a design parameter. From (10), one may find that P and Q can be reformulated as the filtered variable of time-varying m−i+1 −τˆ s
e ˆ s system dynamics ΦΦ T and y − τˆ [u] in terms of i=0 bi Λ(s) a stable system 1/(s + ℓ), respectively. Subsequently, one can obtain the solution of (10) as
i=m
t P = e−ℓ(t −r ) Φ (r )Φ T (r ) dr 0 t Q = e−ℓ(t −r ) Φ (r ) 0 i=m sm−i+1 e−τˆ s ˆ [u(r )] dr . × y(r ) − τˆ (r ) bi ( r ) Λ(s) i=0
(11)
Moreover, define an auxiliary vector W ∈ Rm+n+2 as
ˆ − Q. W = PΘ
(12)
From (9), one can derive t
e−ℓ(t −r ) Φ (r ) Θ T Φ (r ) + ρ(r ) dr .
Q =
(13)
0
Consequently, based on (11)–(13), it follows that
ˆ − Q = −P Θ ˜ +ψ W = PΘ
(14)
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˜ ˆ is the parameter estimation error and ψ = where Θ Θ−Θ t −ℓ(t = −r ) − 0 e Φ (r )ρ(r ) dr is a bounded disturbance, e.g., ∥ψ∥ ≤ ε for a positive constant ε > 0. ˆ is given as The adaptive law for estimating Θ ˙ˆ = −Γ W Θ
(15)
where Γ > 0 is the adaptive learning gain. Remark 6. According (12), the variable W can be obtained based on P , Q defined in (10) or (11), which denotes the parameter ˜ and thus can be used to drive the estimation error information P Θ adaptation (15) so that the error convergence can be guaranteed. Moreover, by using the term W in adaptation (15), the imposed relative degree condition m + 2 ≤ n on system (1) in [14] is removed. In this paper, the positive definiteness condition of matrix P in (10) is crucial for the parameter estimation convergence. We first prove that this condition holds if the regressor vector Φ in (9) is PE. Lemma 1. The matrix P (t ) defined in (10) is positively definite (i.e., λmin (P (t )) > σ > 0 for a positive constant σ ) if the regressor matrix Φ in (9) is PE. Proof. Based on Definition 1 of if Φ is PE, there exist T > 0 and tPE, +T ε > 0 so that the inequality t Φ (r )Φ T (r ) dr > ϖ I is true for all t > 0. On the other hand, it can be shown that the inequality t
e−ℓ(t −r ) Φ (r )Φ T (r ) dr
0 t
e−ℓT Φ (r )Φ T (r ) dr > e−ℓT ϖ I
=
(16)
t −T
holds for all t > T , and then 0 e−ℓ(t −r ) Φ (r )Φ T (r ) dr > σ I is true for t > T and σ = e−ℓT ϖ . Consequently, P (t ) is positive definite (λmin (P (t )) > σ > 0) as long as the regressor vector Φ is PE.
t
Remark 7. Lemma 1 provides an intuitive way to online validate the well-known PE condition used for system identification [13,18], i.e., by calculating the minimum eigenvalue of matrix P (t ), which is numerically verifiable in the online implementation. To fulfill the PE condition, as suggested by [6], a ‘sufficiently nonsmooth’ control action can be used (e.g., sum of squares waves will be used as [12,14] for simulation). Now, the main results of this paper can be summarized as follows. Theorem 1. Consider the time-delay system (1) with an adaptive law (15), if the regressor vector Φ is PE, then all signals are bounded ˜ exponentially converges to a small and the estimation error Θ residual set around zero. Proof. Select the Lyapunov function as V =
1 2
˜ T Γ −1 Θ ˜. Θ
(17)
Then the derivative of (17) along (14)–(15) can be derived as
˙˜ = −Θ ˜ T Γ −1 Θ ˜ T PΘ ˜ +Θ ˜ T ψ. V˙ = Θ
(18)
According to Lemma 1, the condition λmin (P ) > σ > 0 holds provided that the regressor vector Φ is PE. Then by applying the ˜ T ψ , we inequality ab ≤ a2 /2k + kb2 /2 with k > 0 for the term Θ know
k ˙V ≤ − σ − 1 Θ ˜ TΘ ˜ + ε2 2k
≤ −α V + β
2
(19)
where α = 2(σ − 1/2k)/λmax (Γ −1 ), β = kε 2 /2 are all positive constant for k > 1/2σ . Since ρ and thus ε are all bounded, then according to the Lyapunov theorem [13], we know V and ˜ are uniformly ultimately bounded. Consequently, the estimated Θ ˆ and thus the auxiliary variables P , Q and W are all parameter Θ bounded. Moreover, to evaluate the error bound, we integrate both sides of (19) over [0, t ], then it follows V (t ) ≤ V (0) e−α t +
β β (1 − e−αt ) ≤ + V (0) e−αt . α α
(20)
Based on (17)–(20), we can obtain the bound of identification error as
˜ ≤ lim lim Θ
t →∞
t →∞
=
2 (β/α + V (0) e−α t ) /λmin (Γ −1 )
2β/αλmin (Γ −1 ).
(21)
˜ converges to a small This illustrates that the estimation error Θ compact set around zero in the exponential manner. Moreover, ˜ depends on the excitation level σ and from (21), the size of Θ learning gain Γ , i.e., for large σ and Γ , we can choose k small (so that α is large and β is small) to improve the parameter estimation convergence. Moreover, a ‘‘sufficiently near’’ initial value for the delay estimation τˆ (0) (as suggested by [16]) is necessary to reduce the residual error ρ in (9) and thus β in (19) to improve the convergence performance. This completes the proof. Remark 8. The presence of the term ρ in (9) is due to the Taylor series expansion on the unknown time-delay error e−τ s − e−τˆ s and the presence of disturbance d, which will vanish as long as the parameter estimation convergence is achieved. When the timedelay τ is precisely known, the parameters to be estimated are reduced to θ so that ζ = ρ = ψ = 0 and thus β = 0 in (19) hold under d = 0. In this case, the estimation error θ˜ can be proved to exponentially converge to zero. Hence the proposed method is also applicable for the identification of general systems without timedelay [17]. 4. Finite-time parameter identification In this section, the idea proposed in Section 3 is further improved to obtain finite-time parameter estimation. For this purˆ is updated by pose, the parameter estimation Θ T ˙ˆ = −Γ P W Θ ∥W ∥
(22)
where Γ > 0 is the adaptive learning gain. The identifier dynamics are governed by the functional differential equation (22) with a discontinuous right-hand side [19]. The meaning of this equation remains conventional beyond the discontinuity manifold W = 0, whereas along this manifold it is defined in the sense of sliding modes [20,21]. Then according to the results in [19] concerning discontinuous infinite-dimensional differential systems, the negativeness of a Lyapunov function derivative on the set {W ̸= 0} guarantees that the Filippov definition set on the right hand side of (22) is located on the negative part of the real axis, which further implies the error convergence everywhere including along the set {W = 0}, when in the sliding mode. Theorem 2. Consider the time-delay system (1) with an adaptive law (22), if the regressor vector Φ is PE, all signals are bounded and ˜ converges to a small residual set around zero the parameter error Θ ˜ = ψ. in finite-time and satisfies limt →∞ P Θ
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Proof. According to Lemma 1, we know λmin (P ) > σ > 0 holds if Φ is PE. For the Lyapunov functional analysis in this case, we need to first derive the derivative of P −1 W with respect to time tas [17]. ˜ + P −1 ψ , and then From (14), we have P −1 W = −Θ −1 ∂ P −1 W ˙˜ + ∂ P ψ + P −1 ψ˙ = −Θ ∂t ∂t ˙ ˙ˆ + ψ ′ − 1 ˙ −1 ψ + P −1 ψ˙ = Θ ˆ − P PP =Θ
(23)
˙ −1 ψ + P −1 ψ˙ . where ψ ′ is defined as ψ ′ = −P −1 PP Select the Lyapunov function as V =
1 2
W T P −1 P −1 W .
(24)
It follows from (14), (22) and the fact λmin (P ) > σ > 0 that
˙
ˆ + ψ ′ ) = −W T P −1 Γ V˙ = W T P −1 (Θ
PT W
∥W ∥ ≤ −λmin (Γ ) ∥W ∥ + P −1 ψ ′ ∥W ∥ ≤ −(λmin (Γ ) − ψ ′ /σ ) ∥W ∥ .
+ W T P −1 ψ ′
We now analyze the term ψ . According to ψ = −
Fig. 1. System input/output and minimum eigenvalue of P (t ).
(25)
t
−ℓ(t −r )
e Φ (r ) 0 ρ(r ) dr, it follows that ψ and ψ˙ are bounded as long as ρ is bounded. Moreover, the matricesP and ′ P˙ are also bounded. Thus, ψ exists and is bounded, i.e., ψ ′ ≤ ω with ω > 0. Thus, for large enough Γ > 0 such that λmin (Γ ) ≥ ω/σ , Eq. (25) can be reduced as V˙ ≤ 0, which implies limt →∞ V = 0. Consequently, ˜ and the estimated parameter Θ ˆ are all the parameter error W , Θ ′
bounded. To further prove the FT error convergence, we know that (25) can be represented as V˙ ≤ −(λmin (Γ ) − ω/σ ) ∥W ∥ ≤ −α
√
√ V
(26)
√
with α = (λmin (Γ ) − ω/σ )σ 2 = 2(σ λmin (Γ ) − ω) being a positive constant for large enough λmin (Γ ) ≥ ω/σ . In this case, using the results of [7,19] and according to (26), FT convergence of Lyapunov solution V and thus the parameter error W√ to zero is guaranteed, i.e., limt →∞ W = 0 in finite-time ts = 2 V (0)/α . Thus, starting from a finite time instant t ∈ [0, ts ], the system (22) evolves in a sliding mode along the surface W = 0. This further ˜ = ψ is true in finite-time, where the implies that limt →∞ P Θ convergence time depends on the excitation level σ and learning gain Γ and the error bound ω. This completes the proof. Remark 9. Compared to adaptation (15), a sliding mode term P T W / ∥W ∥ is employed in (22) and thus FT error convergence can be proved. In particular, we derive the error convergence bound ˜ = ψ for performance evaluation. Moreover, it should limt →∞ P Θ be noted that the switching in the implementation of adaptive law (22) is not as serious as in the controller implementation because the identifier input P T W / ∥W ∥ is not employed to drive an actuator [7]. 5. Simulation Example 1. Consider an SISO time-delay system studied in [14] as y(3) + 2.2y(2) + 0.5y˙ + y = 1.5u˙ (t − 1.3) − u(t − 1.3) + d
(27)
where a1 = 2.2, a2 = 0.5, a3 = 1, b0 = 1.5, b1 = −1 and τ = 1.3 are the parameters to be estimated. For fair comparison, the input signal u is taken as the sum of two square waves as [14]: one is with amplitude 3 and frequency 1 rad/s and the other one is with amplitude 1.5 and frequency 2 rad/s. The polynomial Λ(s) =
(s + 2.5)3 and the parameters Γ = 200, ℓ = 0.5 used in the simulation are determined via a trial-and-error method, and initial simulation conditions are set as zero. It should be noted that, as analyzed in the proof of theorems, alternative input signals with the higher excitation level (e.g., white noise) and ‘‘sufficiently near’’ initial value of time-delay estimation τˆ (0) may improve the convergence rate [14]. Moreover, an extra noise d = 1rand(0.1) with mean 0 and variance 0.1 is added to the system input measurement (see Fig. 1) to evaluate the robust effect against disturbances. Fig. 1 provides the input/output measurements of system (27), where the minimum eigenvalue of matrix P (t ) is also illustrated. It is shown that the condition λmin (P (t )) > σ > 0 is indeed true for t > 5 s and thus according to Lemma 1, we know that the regressor vector Φ is PE. Moreover, the PE condition can be online validated by calculating λmin (P (t )) in this paper, which is practically useful in the identification. Fig. 2(a)–(b) depict the profile of parameter estimations with an adaptive law (15) and FT adaptive law (22), respectively. As shown in Fig. 2, both the adaptive identification method and FT identification method obtain satisfactory performance. Moreover, FT identification (22) gives slightly faster transient, which, on the other hand, introduces minor oscillations in the steady-state due to the sliding-mode term in the adaptation. Comparison to other delay identification methods: the proposed adaptive laws (15) and (22) are compared to some available results for system (27): (1) direct identification [4]; (2) polynomial identification [12]; (3) least-squares identification [14]. The identified models T are summarized T in Table 1, where the error index E = 1/T 0 e(t ) dt = 1/T 0 (y − yˆ ) dt between the model output yˆ = θ T φ(τˆ ) and the system output y is also summarized. Moreover, the Nyquist curves of the identified models are given in Fig. 3. From Table 1 and Fig. 3, we can find that the proposed adaptive identification (15) obtained best model precision and error performance, and FT identification (22) performs similarly to the least-squares method [14], which all perform superior to the polynomial method [12]. The direct identification obtains a smaller error index than the polynomial identification but gives a model with a different relative degree and delay value. Comparison to gradient based method: finally, the efficacy of the proposed parameter estimation error information W in the adaptive law design is validated and compared to the conventional
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(a) Adaptive parameter identification (15).
(b) Finite-time parameter identification (22). Fig. 2. Parameter identification for system (27).
Table 1 Comparison to other methods. E=
Identification model 0.079s2 −0.4592s−0.0018 s3 +0.017s2 +0.4596s2 +0.0018
1 T
T 0
−3.0932s
0.0289
1.4742s−0.9833 s3 +2.1536s2 +0.5023s2 +0.9746 1.4960s−0.9961 s3 +2.1910s2 +0.4998s2 +0.9948
e−1.3020s
0.0789
e−1.3023s
0.0014
Adaptive identification (15)
1.5001s−1.0000 s3 +2.2001s2 +0.5000s2 +1.0000
e−1.3000s
0.000886
Finite-time identification (22)
1.4932s−1.0004 s3 +2.1908s2 +0.4945s2 +0.9989
−1.2937s
Direct identification [4] Polynomial identification [12] Least-squares identification [14]
e
e
e(t ) dt
0.0011
Fig. 3. Nyquist plots of the estimated models.
gradient-based method [13]. For this purpose, we represent system (27) into the linearly parameterized system (9) and then a predic-
ˆ T Φ (τˆ ) + τˆ tor/observer is developed as yˆ = Θ
i=m i=0
bˆ i s
m−i+1 e−τˆ s
Λ(s)
u
to yield an observer error e = y − yˆ , which is employed to design
˙
ˆ = Γ Φ T (τˆ )e. Comparative simulation results of the adaptive law Θ the proposed adaptive identification (15) and the gradient method are illustrated in Fig. 4. It is shown that faster parameter convergence speed is obtained with (15) than that of the gradient method. ˜ shown in This is due to that the parameter estimation error P Θ W is used in (15), while in the gradient-based scheme only the ob-
Fig. 4. Comparative parameter estimation performance to gradient method.
server error e is employed. This again validates the efficacy of the utilization of parameter estimation error W in the design of adaptive law [17].
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J. Na et al. / Systems & Control Letters 66 (2014) 43–50
49
Example 2. The air-to-fuel ratio (AFR) dynamics of an internal combustion engine (ICE) [7] described by a second-order SISO linear time-delay system is employed y(2) + a1 y˙ + a2 y = b0 u˙ (t − τ ) + b1 u(t − τ )
(28)
where y is the measured air-to-fuel ratio and u is the injected fuel over the air ratio, and τ denotes the delay between the injection event and the output event. The uncertain parameters to be estimated are a1 = 7.5, a2 = 12.5, b0 = 4, b1 = 12.5, τ = 0.4. The input u(t ) is designed as a ‘sufficiently nonsmooth’ signal [7] generated by u˙ (t ) = 0.03sign(sin(0.1 + 6.28t ))
+ 0.16sign(sin(0.1 + 28, 26t )). The initial conditions are aˆ 1 (0) = 6, aˆ 2 (0) = 10.5, bˆ 0 (0) = 2, bˆ 1 (0) = 13.6, τˆ (0) = 0 as [7] and the parameters Γ = 200, ℓ = 0.5 and Λ(s) = (s + 2.5)2 are used in the simulation. Fig. 5 provides the parameter identification performance of adaptive laws (15) and (22). It is shown that all parameters of system (28) are precisely identified, and the delay parameter τ is identified directly. In particular, FT identification (22) achieves faster convergence speed than adaptive identification (15) because of the employment of a sliding mode term in the adaptation. For comparison, the distributed delay flavored (DDF) identification [7] is also simulated. To account for the unknown delay, as shown in [7], five extra fictitious delay terms τi = ih, i = 1, . . . , 5, h = 0.1 are augmented into system as y(2) =
5
b0i u˙ (t − τi ) +
i =1
5
b1i u(t − τi ) − a1 y˙ − a2 y.
Fig. 5. Parameter identification with (15) and (22).
(29)
i =1
Then the following identifier [7] is proposed
˙ xˆ 1 = xˆ 2 + w 5 x˙ˆ 2 = γ w + bˆ 0i u˙ (t − τi ) i=1 5 + bˆ 1i u(t − τi ) − aˆ 1 (ˆx2 + w) − aˆ 2 xˆ 1
(30)
i =1
w = (L + xˆ 2 )sign(y − xˆ 1 ) ˙ ˙ bˆ 0i = λ1i u˙ (t − τi )w, bˆ 1i = λ2i u(t − τi )w, a˙ˆ 1 = −λ3 (ˆx2 + w)w, a˙ˆ 2 = −λ4 xˆ 1 w
Fig. 6. Parameter identification with (30).
i = 1, . . . , 5
where the parameters λ1i , L and γ in (30) are the same as [7]. As claimed in [7], bˆ 03 and bˆ 13 should converge to b0 and b1 , and bˆ 0i , bˆ 1i , i = 1, 2, 4, 5 converge to a small residual set around zero. Simulation results are given in Fig. 6, where the error convergence speed is fair. However, as shown in (30) more delay terms are included in the identifier and thus the time-delay τ = 0.4 cannot be directly estimated. In fact, the determination of delay τ in [7] depends on the convergence of bˆ 03 and bˆ 13 , and the derivative of the input signal u˙ (t − τi ) should be estimated online (see (30)), i.e., the estimation of bˆ 0i , i = 1, 2, 4, 5 may be sensitive to disturbances. From all aforementioned simulations, one may find that in the proposed identification methods (15) and (22) the delay τ can be explicitly identified. Moreover, no fictitious delay terms are augmented in the identifier. The improved performance can also be observed from Figs. 5–6.
6. Conclusion This paper is concerned with online parameter identification of linear SISO time-delay systems. All unknown parameters including time-delay are estimated simultaneously in terms of a novel design of adaptive laws based on the parameter estimation error, and finite-time identification error convergence can be achieved by introducing a sliding mode term in the adaptation. The structure of the underlying system (e.g., system order) is known, while the derivatives of input and output are not required and the previously imposed assumption on the system relative order is removed. The online implementation is thus simpler compared to some available results, in particular for high-order systems. The proposed methods can be taken as an alternative to gradient-based methods, where appropriate parameter estimation error information is derived and used for the design of adaptive laws to allow proving the error convergence. Future work will focus on extending the proposed idea for MIMO and nonlinear time-delay systems.
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Acknowledgments The author would like to thank the editors and anonymous reviewers for the constructive comments that helped to improve this paper. References [1] B. Zhou, Z.-Y. Li, W.X. Zheng, G.-R. Duan, Stabilization of some linear systems with both state and input delays, Syst. Control Lett. 61 (10) (2012) 989–998. [2] J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (10) (2003) 1667–1694. [3] P.J. Gawthrop, M.T. Nihtila, Identification of time delays using a polynomial identification method, Syst. Control Lett. 5 (4) (1985) 267–271. [4] Q.G. Wang, X. Guo, Y. Zhang, Direct identification of continuous time delay systems from step responses, J. Process Control 11 (5) (2001) 531–542. [5] S. Majhi, D.P. Atherton, Obtaining controller parameters for a new Smith predictor using autotuning, Automatica 36 (11) (2000) 1651–1658. [6] Y. Orlov, L. Belkoura, J.P. Richard, M. Dambrine, Adaptive identification of linear time-delay systems, Int. J. Robust Nonlinear Control 13 (9) (2003) 857–872. [7] O. Gomez, Y. Orlov, I.V. Kolmanovsky, On-line identification of SISO linear time-invariant delay systems from output measurements, Automatica 43 (12) (2007) 2060–2069. [8] S. Diop, I. Kolmanovsky, P.E. Moraal, M. Van Nieuwstadt, Preserving stability/performance when facing an unknown time-delay, Control Eng. Pract. 9 (12) (2001) 1319–1325. [9] M. Sugimoto, H. Ohmori, A. Sano, Continuous-time adaptive observer for linear system with unknown time delay, in: The 39th IEEE Conference on Decision and Control, IEEE, Sydney, NSW, 2000, pp. 1104–1109.
[10] L. Belkoura, J.P. Richard, M. Fliess, Parameters estimation of systems with delayed and structured entries, Automatica 45 (5) (2009) 1117–1125. [11] S.V. Drakunov, W. Perruquetti, J.P. Richard, L. Belkoura, Delay identification in time-delay systems using variable structure observers, Annu. Rev. Control 30 (2) (2006) 143–158. [12] P.J. Gawthrop, M.T. Nihtila, A.B. Rad, Recursive parameter estimation of continuous-time systems with unknown time delay, Control Theory Adv. Technol. 5 (3) (1989) 227–248. [13] P.A. Ioannou, J. Sun, Robust Adaptive Control, Prentice Hall, New Jersey, 1996. [14] X.M. Ren, A.B. Rad, P.T. Chan, W.L. Lo, Online identification of continuous-time systems with unknown time delay, IEEE Trans. Automat. Control 50 (9) (2005) 1418–1422. [15] X.M. Ren, A.B. Rad, Identification of nonlinear systems with unknown time delay based on time-delay neural networks, IEEE Trans. Neural Netw. 18 (5) (2007) 1536–1541. [16] R. Pintelon, J. Schoukens, System Identification: A Frequency Domain Approach, IEEE Press, Piscataway NJ, 2001. [17] J. Na, G. Herrmann, X.M. Ren, M.N. Mahyuddin, P. Barber, Robust adaptive finite-time parameter estimation and control of nonlinear systems, in: Proceeding of 2011 IEEE Multi-Conference on Systems and Control, IEEE, Denver, CO, USA, 2011, pp. 1014–1019. [18] S. Sastry, M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, Prentice Hall, New Jersey, 1989. [19] Y.V. Orlov, Discontinuous Systems: Lyapunov Analysis and Robust Synthesis Under Uncertainty Conditions, Springer-Verlag, London, 2009. [20] A.F. Filippov, Differential Equations with Discontinuous Right Hand Sides, Kluwer Academic Publishers, Dordrecht, 1988. [21] V.I. Utkin, Sliding Modes in Control and Optimization, Springer-Verlag, Berlin, 1992.
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Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Adaptive parameter identification of linear SISO systems with unknown time-delay✩ Jing Na a,∗ , Xuemei Ren b , Yuanqing Xia b a
Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, 650500, PR China
b
School of Automation, Beijing Institute of Technology, Beijing, 100081, PR China
article
info
Article history: Received 23 May 2013 Received in revised form 29 December 2013 Accepted 20 January 2014 Available online 14 February 2014 Keywords: System identification Parameter estimation Time-delay systems
abstract An adaptive online parameter identification is proposed for linear single-input-single-output (SISO) time-delay systems to simultaneously estimate the unknown time-delay and other parameters. After representing the system as a parameterized form, a novel adaptive law is developed, which is driven by appropriate parameter estimation error information. Consequently, the identification error convergence can be proved under the conventional persistent excitation (PE) condition, which can be online tested in this paper. A finite-time (FT) identification scheme is further studied by incorporating the sliding mode scheme into the adaptation to achieve FT error convergence. The previously imposed constraint on the system relative degree is removed and the derivatives of the input and output are not required. Comparative simulation examples are provided to demonstrate the validity and efficacy of the proposed algorithms. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Time-delay is usually unavoidable in control systems, which may lead to a sluggish response or even trigger instability [1]. The control design for time-delay systems mainly assumes a precise model (in particular for time-delay parameter). In this sense, online parameter identification of time-delay systems is an important and challenging topic [2]. An intuitive method is to approximate unknown time-delay by a rational transfer function (e.g. pade approximation) [3], where more parameters are introduced and the approximation error for a large time-delay may degrade the identification performance. In [4], a direct identification method has been developed based on step responses, and the relay-based method has been proposed for some low-order timedelay systems in [5]. These approaches are implemented offline and sensitive to noise. Another category has been rooted in adaptive identification: in [6] and the subsequent work [7], the identifiability analysis of SISO time-delay systems has been first studied, and then an adaptive identification was proposed based on the
✩ This work was supported by the National Natural Science Foundation of China under grant number: 61203066 and 61273150. ∗ Corresponding author. E-mail addresses: [email protected] (J. Na), [email protected] (X. Ren), [email protected] (Y. Xia).
http://dx.doi.org/10.1016/j.sysconle.2014.01.005 0167-6911/© 2014 Elsevier B.V. All rights reserved.
distributed delay flavored (DDF) procedure, where the time-delay parameter is determined in terms of an identifier that includes more fictitious delay terms than the actual system. Adaptive estimation for pure time-delay was also studied in [8], where the input signal and its derivative are all used. In [9], a nonlinear optimization was used in the adaptive identification, where the solution of the resulting min–max problem is not trivial. Recently, an algebraic identification method has been presented [10] using algebraic operations with annihilation and integration. Taking the merit of variable structure observers, several algorithms for delay identification were presented [11], where the derivatives of the input and/or output are assumed to be measurable and available. This assumption may be stringent for practical applications. An alternative idea, polynomial identification [12], was developed to identify the time-delay and rational transfer function parameters. However, the complexity of the practical implementation for high-order systems limits its application. Although the least-squares (LS) algorithm [13] has been well-recognized in the identification of linearly parametric systems, it cannot be directly used for time-delay systems because the time-delay usually imposes nonlinearity in the formulation and thus is difficult to be presented as an explicit parameter in a linear-in-parameterized form. To address this issue, in our previous work [14], a nonlinear least-squares algorithm was developed to simultaneously identify the time-delay and other parameters of linear SISO systems. This idea has also been extended to nonlinear time-delay systems by
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J. Na et al. / Systems & Control Letters 66 (2014) 43–50
means of neural networks. [15]. However, the adaptive laws proposed in [14] are derived based on the gradient error embedded in a nonlinear optimization problem, and thus the parameter estimation convergence has not been proved. Moreover, a stringent assumption that the system relative degree should be larger than two is imposed in [14] so that the applicability of the method is limited. In this paper, we focus on the full parameter identification of linear continuous-time SISO systems with unknown time-delay and further improve our previous work [14] by proposing novel adaptive laws. It is assumed that the structure of the underlying system (e.g., system order) is known and only system parameters and time-delay are unknown. We first apply the Taylor series expansion to reformulate the time-delay system into a parameterized form, where the time-delay term can be explicitly indicated in a linear-in-parameterized formula. Then appropriate parameter estimation error information is derived, which is used in the design of adaptive laws. Consequently, the parameter convergence can be proved by means of Lyapunov theory under the conventional PE condition that can be online tested, and the improved performance (e.g., fast convergence) is achieved because of the inclusion of this parameter estimation error. In contrast to [12,14], the assumption on the system relative degree is successfully removed so that the identification for general high-order systems is straightforward. Finally, by incorporating the sliding mode technique into the design and synthesis of adaptive laws, an alternative parameter identification scheme is studied guaranteeing finite-time (FT) error convergence. Compared to other methods (e.g., [7,8,11]), the timedelay parameter can be directly estimated and the derivatives of the input and output are not required, i.e., only the input and output measurements are used. Comparative simulations are provided to verify the characteristics and the improved performance of the proposed algorithms. The paper is organized as follows. Section 2 presents the problem formulation and Section 3 introduces the adaptive parameter identification scheme. FT parameter identification is studied in Section 4. Simulation examples are provided in Section 5 and some conclusions are given in Section 6.
the input and output are assumed to be available (e.g., u˙ in [8] and y˙ in [11] are used) and thus the proposed schemes may be sensitive to noises. In this paper, only the input u and output y are required, while their derivatives are not utilized. Moreover, the unknown delay parameter τ can be directly identified. In this paper, without loss of generality, we assume that the structure of the underlying system (e.g., system order) is known and the input u and output y are available and bounded. This is a common assumption in system identification literature [3,14]. For the case with unbounded u and y, the normalization operation [13, 14] can be used accordingly to fulfill this condition. For the purpose of parameter identification, we filter (2) by a system1/Λ(s) with Λ(s) = sn + λ1 sn−1 + · · · + λn being a Hurwitz polynomial, and then rewrite (2) as sn
Λ(s)
[y] + a1
= b0
sn−1
Λ(s)
sm e−τ s
Λ(s)
+ bm
e−τ s
Λ(s)
[y] + · · · + an
[ u] + b 1 [ u] +
sm−1 e−τ s
Λ(s)
1
Λ(s)
1
Λ(s)
[y]
[u] + · · ·
[d]
(3)
i −τ s
i
where Λs(s) [y], s Λe(s) [u] and Λ1(s) [d] denote the filtered variables of the output y, input u(t − τ ) and the disturbance d by means of a i
linear proper system Λs(s) , i = 0, . . . , n. Adding and subtracting the term Λ(s)y/Λ(s) on both sides of (3), it can be rewritten as a parameterized form y = (λ1 − a1 )
sn−1
Λ(s)
+ (λn − an ) + b1
[y] + (λ2 − a2 )
1
Λ(s)
sm−1 e−τ s
Λ(s)
[y] + b0
sn−2
Λ(s)
sm e−τ s
Λ(s)
[ u] + · · · + b m
[y] + · · ·
[ u]
e−τ s
Λ(s)
[u] +
1
Λ(s)
[d ]
= θ T φ(τ ) + δ 2. Problem formulation
where
Consider the following linear continuous-time SISO system with an input time-delay y(n) (t ) + a1 y(n−1) (t ) + · · · + an y(t ) = b0 u(m) (t − τ )
+ b1 u(m−1) (t − τ ) + · · · + bm u(t − τ ) + d(t )
(1)
where u(t ), y(t ) are the system input and output, respectively, d(t ) is an unknown bounded disturbances, the system order (e.g.,m ≤ n) is a prior known, and a1 , . . . , an and b0 , . . . , bm are unknown parameters and τ is the unknown time-delay. The objective of this paper is to estimate a1 , . . . , an , b0 , . . . , bm and τ using only the input u and output y. Taking the Laplace transform of (1) under zero initial conditions, the following transfer function is obtained: G(s) =
Y (s) U (s)
m
=
b0 s + b1 s
(4)
m−1
+ · · · + bm
s n + a1 s n − 1 + · · · + an
e−τ s
(2)
where Y (s) and U (s) are the Laplace transforms of the output and input, respectively. Remark 1. The parameter identification of the linear SISO timedelay system (1) has been studied in the literature, e.g., [3–12]. In particular, the identifiability analysis of system (1) has been addressed in [6,7], where the unknown plant parameters including delays can be identified. In some of these results, the derivatives of
θ = [λ1 − a1 , λ2 − a2 , . . . , λn − an , b0 , . . . , bm ]T , n−1 s sn−2 φ(τ ) = [y], [y], . . . , Λ(s) Λ(s) T 1 sm e−τ s e−τ s 1 [y], [u], . . . , [ u] , δ= [d]. Λ(s) Λ(s) Λ(s) Λ(s) Denote θˆ as the estimation of the unknown parameter θ , and φ(τˆ ) as a vector that has the same form as φ(τ ) but uses the estimated delay τˆ , such that
θˆ = [λ1 − aˆ 1 , λ2 − aˆ 2 , . . . , λn − aˆ n , bˆ 0 , . . . , bˆ m ]T , φ(τˆ ) = sm e−τˆ s
Λ(s)
sn−1
Λ(s)
[y],
[u], . . . ,
sn−2
Λ(s) e−τˆ s
Λ(s)
[y], . . . ,
1
Λ(s)
[y],
T [ u]
,
where τˆ is the estimation of τ , aˆ i , i = 1 . . . n and bˆ i , i = 0 . . . m are the estimation of ai , i = 1 . . . n and bi , i = 0 . . . m, which will be determined in the following development. Then we reformulate (4) as y = θ T φ(τˆ ) + ζ
(5)
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J. Na et al. / Systems & Control Letters 66 (2014) 43–50
where the remainder term ζ = θ T [φ(τ ) − φ(τˆ )] + δ will be further analyzed. As the unknown time-delay τ should be identified together with the unknown parameter vector θ , we apply the Taylor series expansion for the continuous function e−τ s − e−τˆ s with respect to the estimated delay τˆ , such that
ζ = θ T [φ(τ ) − φ(τˆ )] + δ =
i =m sm−i [e−τ s − e−τˆ s ] bi [ u] + δ Λ(s) i=0
= −τ˜
i=m sm−i+1 e−τˆ s bi [u] + O(τ˜ ) + δ Λ(s) i=0
= −τ˜
i=m sm−i+1 e−τˆ s bˆ i [ u] + ρ Λ(s) i=0
(6)
where τ˜ = τ − τˆ is the time-delay estimation error, O(τ˜ ) represents high-order terms of the Taylor series expansion, and ρ = m−i+1 −τˆ s
b˜ i s Λ(s)e [u]+ O(τ˜ )+δ can be considered as a bounded disturbance [15]. Define an augmented parameter vector with unknown delay τ as
−τ˜
i=m i =0
Θ = [θ T , τ ]T = [λ1 − a1 , λ2 − a2 , . . . λn
− an , b0 , . . . , bm , τ ]T
(7)
and the augmented regressor vector as
i =m sm−i+1 e−τˆ s Φ = φ(τˆ ) , − bˆ i [u] Λ(s) i=0
T
s n −1
Λ(s)
[y],
s n −2
Λ(s)
[y], . . . ,
1
Λ(s)
[y],
i=m sm−i+1 e−τˆ s bˆ i [u], − [ u] Λ(s) Λ(s) i=0
e−τˆ s
sm e−τˆ s
Λ(s)
Remark 5. After representing time-delay system (1) as a linearly parameterized system (9), some well-known adaptive identification methods (e.g., gradient, LS and recursive LS [13]) may be adopted. However, they all depend on a predictor/observer to produce the predictor/observer error for driving the adaptive laws. In the following, inspired by the fact that the utilization of parameter estimation error in the adaptation may benefit the estimation performance [13,17], we will propose a novel adaptive law design such that the error convergence can easily be studied compared to [14]. Definition 1 ([18]). A vector or matrix function Φ is persistently t +T excited (PE) if there exist T > 0, ϖ > 0 such that t Φ (r )Φ T (r ) dr ≥ ϖ I , ∀t ≥ 0.
3. Adaptive parameter identification
[u], . . . ,
T .
(8)
To estimate the unknown parameter vector Θ , we define the auxiliary regressor matrix P ∈ R(m+n+2)×(m+n+2) and vector Q ∈ Rm+n+2 as
T ˙ P = −ℓP + ΦΦ ,
Then system (5) can be represented as i =m sm−i+1 e−τˆ s y = Θ T Φ + τˆ bˆ i [u] + ρ. Λ(s) i =0
Remark 4. In our previous work [14], a nonlinear LS identification has been proposed based on the gradient error, where the error convergence cannot be proved due to the induced nonlinearities of delay. Moreover, the system studied in [14] is assumed to be with a relative degree larger than 2 (i.e., m + 2 ≤ n). This paper is dedicated to further improve the result of [14] by proposing novel adaptive laws with guaranteed error convergence, to remove some assumptions on the system (e.g., input/output derivatives [7,8,11], relative degree [14]), and to avoid complex operations (e.g., pade approximation [3], annihilation and integration [10] and nonlinear optimization [9]).
In this paper, we define λmax (·), λmin (·) as the maximum and minimum eigenvalues of the corresponding matrices, respectively.
T
=
45
(9)
ˆ = The problem is now to find the parameter estimation Θ T T ˆ [θ , τˆ ] based on the measurements of input u and output y so that ˆ converges to Θ as close as possible. Θ Remark 2. In (6)–(9), the first order Taylor series expansion is applied for the function e−τ s − e−τˆ s to make delay parameter τ explicitly appear in the parameter vector Θ to be estimated, and the remainder term O(τ˜ ) depends on the delay error τ˜ = τ − τˆ , which will vanish as long as the parameter estimation convergence is achieved. Consequently, as suggested in [16], a ‘‘sufficiently near’’ initial delay value is necessary to improve the estimation performance. On the other hand, although the higher-order Taylor series expansion may be preferable [14] the resulting nonlinear formulation makes the delay identification nontrivial. With regard to this, the first order Taylor series expansion is used in this paper. Remark 3. To address the effect of measurement noises or disturbances, a bounded disturbance d ∈ L∞ is included in system (1), and thus an extra term 1/Λ(s)[d] is introduced in (3)–(8), which can be lumped into the auxiliary variable ρ in (9). In this case, we will show that the identification error converges to a bounded compact set around zero (see Theorems 1 and 2), i.e., the proposed methods are robust against to bounded noises and disturbances.
P (0) = 0
i=m sm−i+1 e−τˆ s bˆ i [ u] , Q˙ = −ℓQ + Φ y − τˆ Λ(s) i =0
Q (0) = 0
(10)
where ℓ > 0 is a design parameter. From (10), one may find that P and Q can be reformulated as the filtered variable of time-varying m−i+1 −τˆ s
e ˆ s system dynamics ΦΦ T and y − τˆ [u] in terms of i=0 bi Λ(s) a stable system 1/(s + ℓ), respectively. Subsequently, one can obtain the solution of (10) as
i=m
t P = e−ℓ(t −r ) Φ (r )Φ T (r ) dr 0 t Q = e−ℓ(t −r ) Φ (r ) 0 i=m sm−i+1 e−τˆ s ˆ [u(r )] dr . × y(r ) − τˆ (r ) bi ( r ) Λ(s) i=0
(11)
Moreover, define an auxiliary vector W ∈ Rm+n+2 as
ˆ − Q. W = PΘ
(12)
From (9), one can derive t
e−ℓ(t −r ) Φ (r ) Θ T Φ (r ) + ρ(r ) dr .
Q =
(13)
0
Consequently, based on (11)–(13), it follows that
ˆ − Q = −P Θ ˜ +ψ W = PΘ
(14)
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˜ ˆ is the parameter estimation error and ψ = where Θ Θ−Θ t −ℓ(t = −r ) − 0 e Φ (r )ρ(r ) dr is a bounded disturbance, e.g., ∥ψ∥ ≤ ε for a positive constant ε > 0. ˆ is given as The adaptive law for estimating Θ ˙ˆ = −Γ W Θ
(15)
where Γ > 0 is the adaptive learning gain. Remark 6. According (12), the variable W can be obtained based on P , Q defined in (10) or (11), which denotes the parameter ˜ and thus can be used to drive the estimation error information P Θ adaptation (15) so that the error convergence can be guaranteed. Moreover, by using the term W in adaptation (15), the imposed relative degree condition m + 2 ≤ n on system (1) in [14] is removed. In this paper, the positive definiteness condition of matrix P in (10) is crucial for the parameter estimation convergence. We first prove that this condition holds if the regressor vector Φ in (9) is PE. Lemma 1. The matrix P (t ) defined in (10) is positively definite (i.e., λmin (P (t )) > σ > 0 for a positive constant σ ) if the regressor matrix Φ in (9) is PE. Proof. Based on Definition 1 of if Φ is PE, there exist T > 0 and tPE, +T ε > 0 so that the inequality t Φ (r )Φ T (r ) dr > ϖ I is true for all t > 0. On the other hand, it can be shown that the inequality t
e−ℓ(t −r ) Φ (r )Φ T (r ) dr
0 t
e−ℓT Φ (r )Φ T (r ) dr > e−ℓT ϖ I
=
(16)
t −T
holds for all t > T , and then 0 e−ℓ(t −r ) Φ (r )Φ T (r ) dr > σ I is true for t > T and σ = e−ℓT ϖ . Consequently, P (t ) is positive definite (λmin (P (t )) > σ > 0) as long as the regressor vector Φ is PE.
t
Remark 7. Lemma 1 provides an intuitive way to online validate the well-known PE condition used for system identification [13,18], i.e., by calculating the minimum eigenvalue of matrix P (t ), which is numerically verifiable in the online implementation. To fulfill the PE condition, as suggested by [6], a ‘sufficiently nonsmooth’ control action can be used (e.g., sum of squares waves will be used as [12,14] for simulation). Now, the main results of this paper can be summarized as follows. Theorem 1. Consider the time-delay system (1) with an adaptive law (15), if the regressor vector Φ is PE, then all signals are bounded ˜ exponentially converges to a small and the estimation error Θ residual set around zero. Proof. Select the Lyapunov function as V =
1 2
˜ T Γ −1 Θ ˜. Θ
(17)
Then the derivative of (17) along (14)–(15) can be derived as
˙˜ = −Θ ˜ T Γ −1 Θ ˜ T PΘ ˜ +Θ ˜ T ψ. V˙ = Θ
(18)
According to Lemma 1, the condition λmin (P ) > σ > 0 holds provided that the regressor vector Φ is PE. Then by applying the ˜ T ψ , we inequality ab ≤ a2 /2k + kb2 /2 with k > 0 for the term Θ know
k ˙V ≤ − σ − 1 Θ ˜ TΘ ˜ + ε2 2k
≤ −α V + β
2
(19)
where α = 2(σ − 1/2k)/λmax (Γ −1 ), β = kε 2 /2 are all positive constant for k > 1/2σ . Since ρ and thus ε are all bounded, then according to the Lyapunov theorem [13], we know V and ˜ are uniformly ultimately bounded. Consequently, the estimated Θ ˆ and thus the auxiliary variables P , Q and W are all parameter Θ bounded. Moreover, to evaluate the error bound, we integrate both sides of (19) over [0, t ], then it follows V (t ) ≤ V (0) e−α t +
β β (1 − e−αt ) ≤ + V (0) e−αt . α α
(20)
Based on (17)–(20), we can obtain the bound of identification error as
˜ ≤ lim lim Θ
t →∞
t →∞
=
2 (β/α + V (0) e−α t ) /λmin (Γ −1 )
2β/αλmin (Γ −1 ).
(21)
˜ converges to a small This illustrates that the estimation error Θ compact set around zero in the exponential manner. Moreover, ˜ depends on the excitation level σ and from (21), the size of Θ learning gain Γ , i.e., for large σ and Γ , we can choose k small (so that α is large and β is small) to improve the parameter estimation convergence. Moreover, a ‘‘sufficiently near’’ initial value for the delay estimation τˆ (0) (as suggested by [16]) is necessary to reduce the residual error ρ in (9) and thus β in (19) to improve the convergence performance. This completes the proof. Remark 8. The presence of the term ρ in (9) is due to the Taylor series expansion on the unknown time-delay error e−τ s − e−τˆ s and the presence of disturbance d, which will vanish as long as the parameter estimation convergence is achieved. When the timedelay τ is precisely known, the parameters to be estimated are reduced to θ so that ζ = ρ = ψ = 0 and thus β = 0 in (19) hold under d = 0. In this case, the estimation error θ˜ can be proved to exponentially converge to zero. Hence the proposed method is also applicable for the identification of general systems without timedelay [17]. 4. Finite-time parameter identification In this section, the idea proposed in Section 3 is further improved to obtain finite-time parameter estimation. For this purˆ is updated by pose, the parameter estimation Θ T ˙ˆ = −Γ P W Θ ∥W ∥
(22)
where Γ > 0 is the adaptive learning gain. The identifier dynamics are governed by the functional differential equation (22) with a discontinuous right-hand side [19]. The meaning of this equation remains conventional beyond the discontinuity manifold W = 0, whereas along this manifold it is defined in the sense of sliding modes [20,21]. Then according to the results in [19] concerning discontinuous infinite-dimensional differential systems, the negativeness of a Lyapunov function derivative on the set {W ̸= 0} guarantees that the Filippov definition set on the right hand side of (22) is located on the negative part of the real axis, which further implies the error convergence everywhere including along the set {W = 0}, when in the sliding mode. Theorem 2. Consider the time-delay system (1) with an adaptive law (22), if the regressor vector Φ is PE, all signals are bounded and ˜ converges to a small residual set around zero the parameter error Θ ˜ = ψ. in finite-time and satisfies limt →∞ P Θ
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Proof. According to Lemma 1, we know λmin (P ) > σ > 0 holds if Φ is PE. For the Lyapunov functional analysis in this case, we need to first derive the derivative of P −1 W with respect to time tas [17]. ˜ + P −1 ψ , and then From (14), we have P −1 W = −Θ −1 ∂ P −1 W ˙˜ + ∂ P ψ + P −1 ψ˙ = −Θ ∂t ∂t ˙ ˙ˆ + ψ ′ − 1 ˙ −1 ψ + P −1 ψ˙ = Θ ˆ − P PP =Θ
(23)
˙ −1 ψ + P −1 ψ˙ . where ψ ′ is defined as ψ ′ = −P −1 PP Select the Lyapunov function as V =
1 2
W T P −1 P −1 W .
(24)
It follows from (14), (22) and the fact λmin (P ) > σ > 0 that
˙
ˆ + ψ ′ ) = −W T P −1 Γ V˙ = W T P −1 (Θ
PT W
∥W ∥ ≤ −λmin (Γ ) ∥W ∥ + P −1 ψ ′ ∥W ∥ ≤ −(λmin (Γ ) − ψ ′ /σ ) ∥W ∥ .
+ W T P −1 ψ ′
We now analyze the term ψ . According to ψ = −
Fig. 1. System input/output and minimum eigenvalue of P (t ).
(25)
t
−ℓ(t −r )
e Φ (r ) 0 ρ(r ) dr, it follows that ψ and ψ˙ are bounded as long as ρ is bounded. Moreover, the matricesP and ′ P˙ are also bounded. Thus, ψ exists and is bounded, i.e., ψ ′ ≤ ω with ω > 0. Thus, for large enough Γ > 0 such that λmin (Γ ) ≥ ω/σ , Eq. (25) can be reduced as V˙ ≤ 0, which implies limt →∞ V = 0. Consequently, ˜ and the estimated parameter Θ ˆ are all the parameter error W , Θ ′
bounded. To further prove the FT error convergence, we know that (25) can be represented as V˙ ≤ −(λmin (Γ ) − ω/σ ) ∥W ∥ ≤ −α
√
√ V
(26)
√
with α = (λmin (Γ ) − ω/σ )σ 2 = 2(σ λmin (Γ ) − ω) being a positive constant for large enough λmin (Γ ) ≥ ω/σ . In this case, using the results of [7,19] and according to (26), FT convergence of Lyapunov solution V and thus the parameter error W√ to zero is guaranteed, i.e., limt →∞ W = 0 in finite-time ts = 2 V (0)/α . Thus, starting from a finite time instant t ∈ [0, ts ], the system (22) evolves in a sliding mode along the surface W = 0. This further ˜ = ψ is true in finite-time, where the implies that limt →∞ P Θ convergence time depends on the excitation level σ and learning gain Γ and the error bound ω. This completes the proof. Remark 9. Compared to adaptation (15), a sliding mode term P T W / ∥W ∥ is employed in (22) and thus FT error convergence can be proved. In particular, we derive the error convergence bound ˜ = ψ for performance evaluation. Moreover, it should limt →∞ P Θ be noted that the switching in the implementation of adaptive law (22) is not as serious as in the controller implementation because the identifier input P T W / ∥W ∥ is not employed to drive an actuator [7]. 5. Simulation Example 1. Consider an SISO time-delay system studied in [14] as y(3) + 2.2y(2) + 0.5y˙ + y = 1.5u˙ (t − 1.3) − u(t − 1.3) + d
(27)
where a1 = 2.2, a2 = 0.5, a3 = 1, b0 = 1.5, b1 = −1 and τ = 1.3 are the parameters to be estimated. For fair comparison, the input signal u is taken as the sum of two square waves as [14]: one is with amplitude 3 and frequency 1 rad/s and the other one is with amplitude 1.5 and frequency 2 rad/s. The polynomial Λ(s) =
(s + 2.5)3 and the parameters Γ = 200, ℓ = 0.5 used in the simulation are determined via a trial-and-error method, and initial simulation conditions are set as zero. It should be noted that, as analyzed in the proof of theorems, alternative input signals with the higher excitation level (e.g., white noise) and ‘‘sufficiently near’’ initial value of time-delay estimation τˆ (0) may improve the convergence rate [14]. Moreover, an extra noise d = 1rand(0.1) with mean 0 and variance 0.1 is added to the system input measurement (see Fig. 1) to evaluate the robust effect against disturbances. Fig. 1 provides the input/output measurements of system (27), where the minimum eigenvalue of matrix P (t ) is also illustrated. It is shown that the condition λmin (P (t )) > σ > 0 is indeed true for t > 5 s and thus according to Lemma 1, we know that the regressor vector Φ is PE. Moreover, the PE condition can be online validated by calculating λmin (P (t )) in this paper, which is practically useful in the identification. Fig. 2(a)–(b) depict the profile of parameter estimations with an adaptive law (15) and FT adaptive law (22), respectively. As shown in Fig. 2, both the adaptive identification method and FT identification method obtain satisfactory performance. Moreover, FT identification (22) gives slightly faster transient, which, on the other hand, introduces minor oscillations in the steady-state due to the sliding-mode term in the adaptation. Comparison to other delay identification methods: the proposed adaptive laws (15) and (22) are compared to some available results for system (27): (1) direct identification [4]; (2) polynomial identification [12]; (3) least-squares identification [14]. The identified models T are summarized T in Table 1, where the error index E = 1/T 0 e(t ) dt = 1/T 0 (y − yˆ ) dt between the model output yˆ = θ T φ(τˆ ) and the system output y is also summarized. Moreover, the Nyquist curves of the identified models are given in Fig. 3. From Table 1 and Fig. 3, we can find that the proposed adaptive identification (15) obtained best model precision and error performance, and FT identification (22) performs similarly to the least-squares method [14], which all perform superior to the polynomial method [12]. The direct identification obtains a smaller error index than the polynomial identification but gives a model with a different relative degree and delay value. Comparison to gradient based method: finally, the efficacy of the proposed parameter estimation error information W in the adaptive law design is validated and compared to the conventional
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J. Na et al. / Systems & Control Letters 66 (2014) 43–50
(a) Adaptive parameter identification (15).
(b) Finite-time parameter identification (22). Fig. 2. Parameter identification for system (27).
Table 1 Comparison to other methods. E=
Identification model 0.079s2 −0.4592s−0.0018 s3 +0.017s2 +0.4596s2 +0.0018
1 T
T 0
−3.0932s
0.0289
1.4742s−0.9833 s3 +2.1536s2 +0.5023s2 +0.9746 1.4960s−0.9961 s3 +2.1910s2 +0.4998s2 +0.9948
e−1.3020s
0.0789
e−1.3023s
0.0014
Adaptive identification (15)
1.5001s−1.0000 s3 +2.2001s2 +0.5000s2 +1.0000
e−1.3000s
0.000886
Finite-time identification (22)
1.4932s−1.0004 s3 +2.1908s2 +0.4945s2 +0.9989
−1.2937s
Direct identification [4] Polynomial identification [12] Least-squares identification [14]
e
e
e(t ) dt
0.0011
Fig. 3. Nyquist plots of the estimated models.
gradient-based method [13]. For this purpose, we represent system (27) into the linearly parameterized system (9) and then a predic-
ˆ T Φ (τˆ ) + τˆ tor/observer is developed as yˆ = Θ
i=m i=0
bˆ i s
m−i+1 e−τˆ s
Λ(s)
u
to yield an observer error e = y − yˆ , which is employed to design
˙
ˆ = Γ Φ T (τˆ )e. Comparative simulation results of the adaptive law Θ the proposed adaptive identification (15) and the gradient method are illustrated in Fig. 4. It is shown that faster parameter convergence speed is obtained with (15) than that of the gradient method. ˜ shown in This is due to that the parameter estimation error P Θ W is used in (15), while in the gradient-based scheme only the ob-
Fig. 4. Comparative parameter estimation performance to gradient method.
server error e is employed. This again validates the efficacy of the utilization of parameter estimation error W in the design of adaptive law [17].
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Example 2. The air-to-fuel ratio (AFR) dynamics of an internal combustion engine (ICE) [7] described by a second-order SISO linear time-delay system is employed y(2) + a1 y˙ + a2 y = b0 u˙ (t − τ ) + b1 u(t − τ )
(28)
where y is the measured air-to-fuel ratio and u is the injected fuel over the air ratio, and τ denotes the delay between the injection event and the output event. The uncertain parameters to be estimated are a1 = 7.5, a2 = 12.5, b0 = 4, b1 = 12.5, τ = 0.4. The input u(t ) is designed as a ‘sufficiently nonsmooth’ signal [7] generated by u˙ (t ) = 0.03sign(sin(0.1 + 6.28t ))
+ 0.16sign(sin(0.1 + 28, 26t )). The initial conditions are aˆ 1 (0) = 6, aˆ 2 (0) = 10.5, bˆ 0 (0) = 2, bˆ 1 (0) = 13.6, τˆ (0) = 0 as [7] and the parameters Γ = 200, ℓ = 0.5 and Λ(s) = (s + 2.5)2 are used in the simulation. Fig. 5 provides the parameter identification performance of adaptive laws (15) and (22). It is shown that all parameters of system (28) are precisely identified, and the delay parameter τ is identified directly. In particular, FT identification (22) achieves faster convergence speed than adaptive identification (15) because of the employment of a sliding mode term in the adaptation. For comparison, the distributed delay flavored (DDF) identification [7] is also simulated. To account for the unknown delay, as shown in [7], five extra fictitious delay terms τi = ih, i = 1, . . . , 5, h = 0.1 are augmented into system as y(2) =
5
b0i u˙ (t − τi ) +
i =1
5
b1i u(t − τi ) − a1 y˙ − a2 y.
Fig. 5. Parameter identification with (15) and (22).
(29)
i =1
Then the following identifier [7] is proposed
˙ xˆ 1 = xˆ 2 + w 5 x˙ˆ 2 = γ w + bˆ 0i u˙ (t − τi ) i=1 5 + bˆ 1i u(t − τi ) − aˆ 1 (ˆx2 + w) − aˆ 2 xˆ 1
(30)
i =1
w = (L + xˆ 2 )sign(y − xˆ 1 ) ˙ ˙ bˆ 0i = λ1i u˙ (t − τi )w, bˆ 1i = λ2i u(t − τi )w, a˙ˆ 1 = −λ3 (ˆx2 + w)w, a˙ˆ 2 = −λ4 xˆ 1 w
Fig. 6. Parameter identification with (30).
i = 1, . . . , 5
where the parameters λ1i , L and γ in (30) are the same as [7]. As claimed in [7], bˆ 03 and bˆ 13 should converge to b0 and b1 , and bˆ 0i , bˆ 1i , i = 1, 2, 4, 5 converge to a small residual set around zero. Simulation results are given in Fig. 6, where the error convergence speed is fair. However, as shown in (30) more delay terms are included in the identifier and thus the time-delay τ = 0.4 cannot be directly estimated. In fact, the determination of delay τ in [7] depends on the convergence of bˆ 03 and bˆ 13 , and the derivative of the input signal u˙ (t − τi ) should be estimated online (see (30)), i.e., the estimation of bˆ 0i , i = 1, 2, 4, 5 may be sensitive to disturbances. From all aforementioned simulations, one may find that in the proposed identification methods (15) and (22) the delay τ can be explicitly identified. Moreover, no fictitious delay terms are augmented in the identifier. The improved performance can also be observed from Figs. 5–6.
6. Conclusion This paper is concerned with online parameter identification of linear SISO time-delay systems. All unknown parameters including time-delay are estimated simultaneously in terms of a novel design of adaptive laws based on the parameter estimation error, and finite-time identification error convergence can be achieved by introducing a sliding mode term in the adaptation. The structure of the underlying system (e.g., system order) is known, while the derivatives of input and output are not required and the previously imposed assumption on the system relative order is removed. The online implementation is thus simpler compared to some available results, in particular for high-order systems. The proposed methods can be taken as an alternative to gradient-based methods, where appropriate parameter estimation error information is derived and used for the design of adaptive laws to allow proving the error convergence. Future work will focus on extending the proposed idea for MIMO and nonlinear time-delay systems.
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