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Parameter estimation for a dual-rate system with time delay$ Lei Chen a,b, Lili Han b, Biao Huang b,n, Fei Liu a a b
Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, PR China Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada AB T6G 2G6
art ic l e i nf o
a b s t r a c t
Article history: Received 23 August 2013 Received in revised form 20 December 2013 Accepted 6 January 2014 This paper was recommended for publication by Dr. Qing-Guo Wang
This paper investigates the parameter estimation problem of the dual-rate system with time delay. The slow-rate model of the dual-rate system with time delay is derived by using the discretization technique. The parameters and states of the system are simultaneously estimated. The states are estimated by using the Kalman filter, and the parameters are estimated based on the stochastic gradient algorithm or the recursive least squares algorithm. When concerning state estimate of the dual-rate system with time delay, the state augmentation method is employed with lower computational load than that of the conventional one. Simulation examples and an experimental study are given to illustrate the proposed algorithm. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Parameter estimation Dual-rate Time delay Stochastic gradient Least squares Kalman filter
1. Introduction In many industrial processes, modeling the process system is a fundamental problem for control applications, such as soft sensor and controller design [1–4]. The main obstacle of modeling the industrial process system is the lack of the measurable input– output data due to time delay and different sampling rates between the regularly sampled data in sensors and laboratory analyses [5,6]. As the input and the output are sampled in two different sampling periods, the system is often described by a dual-rate system. The challenge of modeling is how to deal with the parameter estimation problem based on the dual-rate input and output data with time delay. The methods for the parameter estimation of the multirate system have been widely developed. For the general multi-input, multi-output and multirate system, the system is divided into subsystems and then a least square method is used to estimate the system parameters [7]. For a fast input slow output dual-rate system, the polynomial transformation technique adopted to transform the dual-rate system into the one which can be identified by the measurable data, and the parameter and intersample output can be estimated by a stochastic gradient (SG) ☆
The short version of this paper was presented in ICCA 2013. Corresponding author. Tel.: þ 1 780 492 9016; fax: þ 1 780 492 2881. E-mail addresses: [email protected] (L. Chen), [email protected] (L. Han), [email protected] (B. Huang), fl[email protected] (F. Liu). n
algorithm [8] and a recursive least square (RLS) algorithm [9]. For a multi-input multirate system, the discretization technique employed to convert the system into a slow-rate system which can be directly identified based on the known data and the RLS algorithm is used to estimate the system parameters [10]. A common strategy of these methods is to convert the original multirate system into systems which are identifiable based on the traditional system identification methods. For a system with very slow output samples, the output error method has been used to directly identify the fast rate model from the known fast rate input and the slow rate output [11]. Some researchers have applied the expectation maximization method to deal with the parameter estimation problem of the system with missing data. The statespace model identification based on the expectation maximization (EM) algorithm and the Kalman filtering estimation method has been adopted to identify a chemical process system based on the irregular sampled output [12]. For example, the grey-box identification techniques have been applied to the bleaching operation in a pulp mill to identify a dynamic model with irregular outputs, and the EM algorithm in the sense of maximum likelihood estimation is used to estimate the system parameters [13]. Currently, for the single rate time delay systems, many identification methods exist [14–16]; however, these methods cannot be directly applied to multirate systems. This paper focuses on the parameter estimation problem for a single-input single-output dual-rate system with time delay, which has two different sampling periods. First, the discretization technique is used to derive
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Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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2
the slow-rate state-space model of the original dual-rate system which can be identified from the measurable input–output data. Second, inspired by [17–19], the observability canonical form of the obtained state-space model is derived. With some algebraic manipulation the model for the dual-rate system is obtained of which the information vector contains the unmeasurable states. Finally, the Kalman filtering algorithm is applied to estimate the model states, and the SG algorithm or the RLS algorithm is used to identify the model parameters. The parameters and states of the system are simultaneously estimated. The proposed state estimation method based on the state augmentation strategy has less computational load compared with the traditional one. This paper is an extended version of our conference paper [20]. The main extensions include an additional identification algorithm, a practical simulation example and a real experimental evaluation. The rest of the paper is organized as follows. Section 2 describes the dual-rate system and problem statements. Section 3 derives a discrete-time state-space model for the dual-rate systems. Section 4 discusses the Kalman filtering algorithm for state estimation, and the SG algorithm or the RLS algorithm for identification of the parameters. Section 5 and 6 provide two illustrative examples and an experimental study on a pilot-scale multiple tank system. Concluding remarks are given in Section 7.
Consider a continuous time process Pc in Fig. 1 with the statespace representation: ( _ ¼ Ac xðtÞ þ Bc uðtÞ; xðtÞ ð1Þ yðtÞ ¼ Cxðt τÞ þ vðtÞ; where xðtÞ A Rn is the state vector, uðtÞ A R1 is the control input which is sampled by a zero-order hold H T 1 with sampling period T 1 ¼ ph, yðtÞ A R1 is the output which is sampled by the sampler ST 2 with period T 2 ¼ qh, τ is the time delay, vðtÞ A R1 is a stochastic noise vector with zero mean, and Ac , Bc and C are matrices of appropriate dimensions. Here, h is the positive number called basic period, and p and q are the two positive co-prime integers. Because of the zero-order hold, we have for kT 1 r t o ðk þ 1ÞT 1 :
ð2Þ
For such a dual-rate system, the measurable input–output data is fuðkT 1 Þ, yðkT 2 Þ : k ¼ 0; 1; 2; …g. This means fuðkT 1 þ ihÞ, yðkT 2 þ jhÞ, i ¼ 1; 2; …; p 1, j ¼ 1; 2; …; q 1g are unknown. The main objective of this paper is to answer the following two questions:
How to derive the slow-rate state-space model for the dual
ð3Þ
where Ah ¼ eAc h ;
Z Bh ¼
h
0
eAc t dt Bc
d¼
and
hτ i h
:
Theorem 1. For the system (3), letting T≔pqh be the frame period and m≔pq d, and assuwming that pq b d, then the state-space model of the dual-rate system can be derived as 8 q1 p > pq jp i > > Bh uðkT þjT 1 Þ; > xðkT þ TÞ ¼ AxðkTÞ þ ∑ ∑ Ah > > j ¼ 0i ¼ 1 > > > > pq > > < xðkT þ T dhÞ ¼ AxðkT dhÞ þ ∑ Apq þ m j B uðkT þ ðj 1Þh TÞ j ¼ mþ1
h
h
> > > m > j > > Bh uðkT þ ðj 1ÞhÞ; þ ∑ Am > h > > j ¼ 1 > > > : yðkTÞ ¼ CxðkT dhÞ þ vðkTÞ; ð4Þ where A ¼ Apq . h Proof. Substituting k with kpq in (3), we obtain ( xðkT þ hÞ ¼ Ah xðkTÞ þ Bh uðkTÞ;
2. System description
uðtÞ ¼ uðkT 1 Þ
sampling time h to get Ph as follows: ( xðkh þ hÞ ¼ Ah xðkhÞ þ Bh uðkhÞ; yðkhÞ ¼ Cxðkh dhÞ þvðkhÞ;
rate system with a time delay so that it can be directly identified based on the measured input–output data? How to estimate the states and parameters of the system with the time delay based on the Kalman filtering and the SG algorithm or the RLS algorithm?
ð5Þ
yðkTÞ ¼ CxðkT dhÞ þvðkTÞ: This leads to q1
p
jp i xðkTÞ þ ∑ ∑ Apq Bh uðkT þjT 1 þ ði 1ÞhÞ: xðkT þ TÞ ¼ Apq h h
ð6Þ
j¼0i¼1
According to (2), we have q1
p
jp i xðkT þ TÞ ¼ AxðkTÞ þ ∑ ∑ Apq Bh uðkT þ jT 1 Þ: h
ð7Þ
j¼0i¼1
Combining pq d
d dj xðkTÞ þ ∑ Apq Bh uðkT þ ðj 1ÞhÞ xðkT þ T dhÞ ¼ Apq h h
ð8Þ
j¼1
and xðkTÞ ¼ Adh xðkT dhÞ þ
pq
∑
j ¼ pq d þ 1
j Apq Bh uðkT þ ðj 1Þh TÞ; h
ð9Þ
yields xðkT þ T dhÞ ¼ AxðkT dhÞ þ
pq
∑
j ¼ mþ1
þmj Apq Bh uðkT þ ðj 1Þh TÞ h
m
j Bh uðkT þ ðj 1ÞhÞ: þ ∑ Am h
ð10Þ
j¼1
From (10) and (7), the results of (4) can be obtained.
□
4. Parameter and state estimation 3. Model derivation In order to derive the state-space model for the dual-rate system, the continuous process Pc is discretized via the zero-order hold with
Fig. 1. The general dual-rate sampled-data system.
In this section, the observability canonical model of system (4) is given which is equivalent to the original controllable and observable system and has the least number of the parameters. Then the model ready for the identification of the system is derived. Based on the measurable input–output data, the state estimation can be obtained by using the Kalman filter, and the parameter estimation can be obtained by using the SG algorithm or the RLS algorithm. Due to the time delay, the states of the system cannot be directly estimated by the Kalman filter. A state
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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augmentation strategy is applied to the system to identify the system states. Assume that the state-space model in (1) is identifiable, which means that (3) is controllable and observable. The state-space model (4) is also controllable and observable under some mild conditions that are provided in [17]. Thus, the model (4) can be rewritten in the observability canonical form as follows: ( xðkT þ T dhÞ ¼ Ao xðkT dhÞ þ Bo uðkTÞ; ð11Þ yðkTÞ ¼ C o xðkT dhÞ þ vo ðkTÞ; where 2
0 1 0 60 0 1 6 6 ⋮ ⋮ Ao ¼ 6 6 ⋮ 60 0 0 4 a1 a2 a3 2 3 B1 6 B2 7 6 7 Bo ¼ 6 7 A Rnpq ; 4 ⋮ 5 Bn
3 0 07 7 7 nn 1 ⋮ 7 7 A R ; ai A R ; 17 5 an
⋯ ⋯ ⋮ ⋯ ⋯
Bi A R1pq ; C o ¼ ½1 0 ⋯ 0 A R1n ;
uðkTÞ ¼ ½uðkT dhÞ ⋯ uðkTÞ uðkT þhÞ⋯ uðkT þ ðpq d 1ÞhÞT A Rpq : Define the parameter vector θ and the information vector
φðkTÞ as
θ≕½αT B1 B2 ⋯ Bn T A Rn þ npq ; α≕½a1 a2 ⋯an T A Rn ; φðkTÞ ¼ ½ x T ðkT dhÞ uT ðkT þ nT TÞ uT ðkT þ nT 2TÞ ⋯ xðkTÞ ¼ ½x 1 ðkTÞ x 2 ðkTÞ ⋯ x n ðkTÞT A Rn : 2
x 1 ðkT dhÞ
Based on the least square (LS) method, the cost function J is N
J ¼ ∑ ‖yðkT þ nTÞ φT ðkTÞθ‖2 ;
ð15Þ
k¼1
where N is the number of the data point. Unfortunately, the parameter vector θ cannot be estimated by using the LS method. In fact, from (14), we can see that the information vector φðkTÞ contains unknown state xðkT dhÞ, so that the LS algorithm cannot be directly applied for the estimation of the parameter θ. The solution that we are taking is to replace the unknown state xðkT dhÞ with their estimates x^ ðkT dhÞ which is obtained by the Kalman filtering algorithm. Since the dual-rate system contains the time delay, the states of the system (11) can be estimated by using the state augmentation method which is the general method to deal with the problem of states estimation of a delayed system. Augment the state as 2 3 xðkT dhÞ 6 7 6 xðkT dh þ hÞ 7 7 A Rnd þ n : Z~ ðkTÞ ¼ 6 ð16Þ 6 7 ⋮ 4 5 xðkTÞ However, this approach of augmentation will introduce a heavy computational load in executing the Kalman filtering algorithm. In order to reduce the cost of the computational effort, the state can be extended as " # xðkT dhÞ ZðkTÞ ¼ A R2n : ð17Þ xðkTÞ Then the state-space model can be rewritten as ( ZðkT þ TÞ ¼ Az ZðkTÞ þ Bz U z ðkTÞ;
uT ðkTÞT A Rn þ npq ; Let
3
yðkTÞ ¼ C z ZðkTÞ þ vðkTÞ;
ð18Þ
where
3
Az ¼ A0 I; " B0 Bz ¼ 0
6 7 6 x 2 ðkT dhÞ 7 7 A Rn : xðkT dhÞ ¼ 6 6 7 ⋮ 4 5 x n ðkT dhÞ Then we have 8 x 1 ðkT þT dhÞ ¼ x 2 ðkT dhÞ þ B1 uðkTÞ; > > > > > > < x 2 ðkT þT dhÞ ¼ x 3 ðkT dhÞ þ B2 uðkTÞ; ⋮ > > > x n 1 ðkT þ T dhÞ ¼ x n ðkT dhÞ þBn 1 uðkTÞ; > > > : x ðkT þT dhÞ ¼ a x ðkT dhÞ þa x ðkT dhÞ þ ⋯ þ a x ðkT dhÞþ B uðkTÞ: n n n n 1 1 2 2
# 0 ; B0
C z ¼ ½1 0; U z ðkTÞ ¼ ½uðkT dhÞ uðkT dh þhÞ ⋯ uðkT hÞ uðkTÞ ⋯ uðkT þ ðpq d 1ÞhÞuðkT þ ðpq dÞhÞ ⋯ uðkT þ ðpq 1ÞhÞT :
When the system is one dimensional, Az and Bz can be expressed as
ð12Þ Az ¼ Apq I; 2h pq 1 Ah Bh Bz ¼ 4 0
2 Apq Bh h
⋯
d Apq Bh h
d1 Apq Bh h
⋯
Bh
⋯
0
0
⋯
0
1 Apq Bh h
⋯
d Apq Bh h
⋯
Bh
Let z be a forward shift operator, i.e., zx 1 ðkTÞ ¼ zx 1 ðkT þTÞ. Multiplying the jth equation above by zn j and summing the obtained equations, we can obtain x 1 ðkT þ nT dhÞ ¼ a1 x 1 ðkT dhÞ þ a2 x 2 ðkT dhÞ þ ⋯ þ an x n ðkT dhÞ
3 5:
Based on the Kalman filter, the state estimation can be obtained as follows: Z^ ðkT þ TÞ ¼ A^ z ðkTÞZ^ ðkTÞ þ B^ z ðkTÞU z ðkTÞ þ LðkTÞ½yðkTÞ C z Z^ ðkTÞ; ð19Þ
þBn uðkTÞ þ Bn 1 uðkT TÞ þ ⋯ þ B1 uðkT þ nT TÞ: ð13Þ
LðkTÞ ¼ A^ z ðkTÞP x ðkTÞC Tz ½I þ C z P x ðkTÞC Tz 1 ;
ð20Þ
Here we have adopted the approach of [17]. As a result, we have yðkT þ nTÞ ¼ φT ðkTÞθ þvo ðkT þ nTÞ:
T
ð14Þ
P x ðkTÞ ¼ ½A^ z ðkTÞ LðkTÞC z P x ðkTÞA^ z ðkTÞ;
ð21Þ
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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vector, e.g., θ^ ð0Þ ¼ 1n =p0 . The initial value for the parameter estimation algorithm is rð0Þ ¼ 1 (SG) or P p ð0Þ ¼ p0 I (RLS). Determine the data length Le. Step 2: Collect the dual-rate measurement data fuðkT 1 Þ, yðkT 2 Þg and form the information vector φðkTÞ. Step 3: By using (19)–(21) to calculate the state estimation ^ ðkTÞ by (22). Z^ ðkTÞ and form the estimated information vector φ Step 4: Calculate (24) to get r(kT) and update the parameter estimation by (23), or calculate (26) to get P p ðkTÞ and update the parameter estimation by (25). Step 5: If k ¼ Le , terminate the procedure and obtain the estimate θ^ ðLe TÞ of the parameter vector θ; otherwise, increment k by 1 and go to step 2.
5. Examples
Example 1 (Numerical example). Consider a single-input singleoutput dual-rate system with the input updating time T 1 ¼ h and the output sampling period T 2 ¼ 5h. Let h¼ 2; the corresponding discrete system is ( xðkh þ hÞ ¼ 0:7413xðkhÞ þ 0:4571uðkhÞ; ð28Þ yðkTÞ ¼ 1:00xðkT dhÞ þvðkTÞ; Fig. 2. The flowchart of the proposed algorithm for computing the parameter and state estimates.
where LðkTÞ is the Kalman gain and P x ðkTÞ is the covariance matrix of the state estimation error. A^ z ðkTÞ and B^ z ðkTÞ represent the estimates of Az and Bz at time kT, respectively. The parameter estimation can be derived by using the SG algorithm or the RLS algorithm. Define the new information vector φ^ ðkTÞ using the estimated state as T
φ^ ðkTÞ ¼ ½ x^ ðkT dhÞ uT ðkT þ nT TÞ uT ðkT þ nT 2TÞ ⋯ uT ðkTÞT : ð22Þ
where
The SG algorithm is as follows:
θ^ ðkT þ TÞ ¼ θ^ ðkTÞ þ
φ^ ðkTÞ rðkTÞ
^ ðkTÞθ^ ðkTÞ; ½yðkT þ nTÞ φ
^ ðkTÞ‖2 ; rðkT þ TÞ ¼ rðkTÞ þ ‖φ
T
rð0Þ ¼ 1;
ð23Þ ð24Þ
θ^ ðkT þ TÞ ¼ θ^ ðkTÞ þ Lp ðkTÞ½yðkT þ nTÞ φ^ T ðkTÞθ^ ðkTÞ; ^ ðkTÞ P p ðkTÞφ ^ ðkTÞ; ^ T ðkTÞP p ðkTÞφ 1þφ
^ T ðkTÞÞP p ðkTÞ: P p ðkT þ TÞ ¼ ðI Lp ðkTÞφ
A ¼ a1 I ¼
" B¼ ¼
and the RLS algorithm can be represented as
Lp ðkT þ TÞ ¼
where d ¼2 and the frame period T ¼ 5h. The input fuðkhÞg is taken as uncorrelated persistent excitation signal sequences with zero mean and unit variances, fvðkTÞg consists of two white-noise sequences with zero mean and variances s2 ¼ 0:52 and s2 ¼ 1:02 . By using the discretization technique, the state-space model with time T can be derived as " # 8 xðkT dhÞ > > > ; > < ZðkTÞ ¼ xðkTÞ ð29Þ > ZðkT þ TÞ ¼ AZðkTÞ þ BuðkTÞ; > > > : yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ;
ð25Þ ð26Þ ð27Þ
Here θ^ ðkTÞ represents the estimation of θðkTÞ at time kT. 1=rðKTÞ is
^ ðkTÞ is defined by the step-size and the norm of the matrix φ ^ ðkTÞ J 2 ¼ tr½φ ^ ðkTÞφ ^ T ðkTÞ. P p ðkTÞ is the covariance matrix of the Jφ parameter estimation error, and Lp ðkTÞ is the gain vector. The SG algorithm has a slower convergence rate compared with the RLS algorithm, but the SG algorithm requires lower computation load. From (19) to (21) and from (22) to (27), the Kalman filtering algorithm and the parameter estimation algorithm (SG or RLS) are applied to estimate the system parameters and states in an iterative way. The flowchart of the algorithm is given in Fig. 2 and the steps taken for the algorithm are summarized below. Step 1: Let k ¼1; the initial value P x ð0Þ is generally taken to be p0 I with p0 normally being a large positive number, e.g., p0 ¼ 106 , and the initial value θ^ ð0Þ and x^ ð0Þ are a zero vector or a small real
b1 0
0:2239
0
0
0:2239
b2 0
0:1380 0
b3 b1 0:1862 0
b4 b2
;
b5 b3 0:2512 0:1380
0 b4
0 b5
#
0:3388 0:1862
0:4571 0:2512
0 0:3388
0 ; 0:4571
uðkTÞ ¼ ½uðkT 2hÞ uðkT hÞ uðkTÞ uðkT þ hÞ uðkT þ 2hÞ uðkT þ 3hÞ uðkT þ 4hÞT :
Apply the Kalman filter and SG algorithm to estimate the parameters of this system, and the parameter estimation error is used to quantify the estimation accuracy [10]:
δ≔
J θ^ ðkTÞ θ J 100%; JθJ
ð30Þ
Table 1 The parameters and their estimates using KF and SG algorithm ðs2 ¼ 0:52 Þ. k
100
0.0886 a1 b1 0.1670 b2 0.2526 b3 0.1554 b4 0.2332 b5 0.3426 δ ð%Þ 34.0778
500
1000
2500
5000
0.1261 0.1316 0.1428 0.1489 0.1644 0.1578 0.1551 0.1552 0.2430 0.2402 0.2316 0.2256 0.1793 0.1935 0.2067 0.2131 0.2911 0.2970 0.3110 0.3176 0.4002 0.4074 0.4237 0.4331 22.1891 19.8576 16.1423 14.2382
10 000
True values
0.1546 0.1558 0.2230 0.2183 0.3226 0.4411 12.8241
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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where θ is the true parameters, and θ^ ðkTÞ is the estimated parameters at time KT. The parameter estimates and their errors when noise variance is 0.52 are shown in Table 1. Apply the Kalman filter and RLS algorithm to estimate the parameters of this system, the results of the parameter estimates and their errors with noise variances of 0.52, 1.02 and 1.52 are shown in Tables 2 and 4, respectively. The parameter estimation errors δ versus k with different noise variances are shown in Fig. 3. Figs. 4 and 5 compare the outputs of the actual slow-rate system and the estimated slow-rate system.
5
2 Outputs of actual system Outputs of estimated model
1.5
1
0.5
0
−0.5
Table 2 The parameters and their estimates using KF and RLS algorithms ðs2 ¼ 0:52 Þ. k
100
500
1000
2500
5000
10 000
True values
a1 b1 b2 b3 b4 b5 δ ð%Þ
0.1223 0.0947 0.3063 0.2232 0.3512 0.5812 29.5926
0.1849 0.1427 0.2035 0.2232 0.3932 0.5194 13.8785
0.2537 0.1134 0.1894 0.2392 0.3369 0.4493 5.9164
0.2374 0.1219 0.1803 0.2498 0.3464 0.4610 3.3641
0.2335 0.1311 0.1722 0.2461 0.3443 0.4639 2.9935
0.2199 0.1402 0.1816 0.2444 0.3446 0.4702 2.4459
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
−1
−1.5
0
20
40
60
80
100
k Fig. 4. Comparisons of the actual outputs and estimated outputs using RLS. 2 Outputs of actual system Outputs of estimated model
1.5
Table 3
1
The parameters and their estimates using KF and RLS algorithms ðs2 ¼ 1:02 Þ. 100
500
1000
2500
5000
10 000
True values
a1 b1 b2 b3 b4 b5 δ ð%Þ
0.1863 0.0325 0.1840 0.0340 0.3300 0.4422 34.9780
0.2947 0.1309 0.1374 0.2787 0.3836 0.3712 18.9390
0.2305 0.0873 0.1845 0.2296 0.3723 0.4826 9.9506
0.2816 0.1399 0.1942 0.2414 0.3394 0.4488 8.5318
0.1760 0.1422 0.1687 0.2418 0.3453 0.4620 7.5137
0.2634 0.1515 0.1729 0.2434 0.3256 0.4583 6.6334
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
0.5 y(k)
k
0 −0.5
−1
−1.5 −1.5
Table 4 The parameters and their estimates using KF and RLS algorithms ðs2 ¼ 1:52 Þ. k
100
500
1000
2500
5000
10 000
True values
a1 b1 b2 b3 b4 b5 δ ð%Þ
0.1261 -0.0260 0.5126 0.1201 0.2982 0.7243 68.9345
0.2003 0.1394 0.2168 0.1459 0.4672 0.5947 31.2697
0.2358 0.0738 0.2126 0.2330 0.3609 0.4727 11.0762
0.2327 0.0937 0.1747 0.2548 0.3726 0.4839 9.0760
0.2314 0.1201 0.1480 0.2413 0.3625 0.4872 8.3282
0.2179 0.1436 0.1711 0.2293 0.3540 0.4939 6.9273
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
−1
−0.5
0
0.5
1
1.5
2
y(k)
Fig. 5. The scatter plot comparisons of the actual outputs and estimated outputs using RLS.
1.4 1.2 1 0.8
2
2
δ
σ =0.5
Fig. 6. The schematic diagram of the single tank system.
0.6 2
2
σ =1.0
0.4
2
The estimated slow-rate model with s2 ¼ 0:52 using the Kalman filter and the RLS algorithm for the dual-rate system is given by
2
σ =1.5
0.2 0
0
500
1000
1500
2000
k Fig. 3. Parameter estimation error δ versus k using RLS.
2500
8 0:2199 0 > > ZðkTÞ ¼ ZðkTÞ > > > 0 0:2199 > < 0:1402 0:1816 0:2444 0:3446 þ > > > 0 0 0:1402 0:1816 > > > : yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ;
0:4702
0
0
0:2444
0:3446
0:4702
uðkTÞ;
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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6
which is close to the actual slow-rate dual-rate system (29). Then, the model for the dual-rate system can be derived as ( xðkh þ hÞ ¼ 0:7386xðkhÞ þ 0:4702uðkhÞ; ð31Þ yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ:
8.8
8.4 8.2
From Tables 1 to 3 and from Figs. 3 to 5, it is clear that the RLS algorithm has faster convergence rates than the SG algorithm, but the latter has less computation load. One can see that with the increase of time k, the parameter estimation error is reduced and outputs of the estimated slow-rate model approach to those of the actual slow-rate system. Example 2 (A holding tank). A process stream comprises a chemical species in dilute liquid solution. It is the effluent of some process, and it is used to feed a downstream process. One can moderate flow swings by holding up a volume in a tank. Intuitively we expect the changes in the outlet flow rate to be more moderate than those of the feed stream [21]. The schematic diagram of this process is displayed in Fig. 6. It is assumed that the tank operates in overflow, and the volume V is constant. Both streams are represented in terms of a single volumetric flow F, and CAi and CAo are the inlet and the outlet concentration, respectively. A component material balance over
8 0:3971 0 > > ZðkTÞ ¼ ZðkTÞ > > > 0 0:3971 > < 0:0658 0:0745 0:0871 þ > > > 0 0 0 > > > : yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ;
Outputs of actual system Outputs of estimated model (RLS) Outputs of estimated model (SG)
8.6
8 7.8 7.6 7.4
0
20
40
60
80
100
k Fig. 7. Comparisons of the actual outputs and estimated outputs (Example 2).
Apply the Kalman filter and the RLS algorithm to estimate the parameters of this system, the estimated slow-rate model using the Kalman filter and the RLS algorithm for the dual-rate system is given by
0:1077
0:1292
0:1460
0
0
0
0:0658
0:0745
0:0871
0:1077
0:1292
0:1460
uðkTÞ;
the solution is d ðVC Ao Þ ¼ FC Ai C Ao : dt
ð32Þ
The inlet concentration CAi and outlet concentration CAo are considered as the system input u and output y, respectively. Let the basic period h ¼1, and the input and output sampling time be T 1 ¼ h and T 2 ¼ 6h, respectively. Consider the time delay d ¼3, and the frame period T ¼ 6h. The corresponding discrete system can be derived as ( xðkh þ hÞ ¼ 0:85xðkhÞ þ 0:15uðkhÞ; ð33Þ yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ:
The input uðkhÞ is taken as uncorrelated persistent excitation signal sequences with unit variances, and vðkTÞ is the white-noise sequence with zero mean and variance s2 ¼ 0:22 . By using the proposed discretization technique, the state-space model with sampling time T can be derived as Eq. (29), where 0:3771 0 A ¼ a1 I ¼ ; 0 0:3771 " b2 b3 b4 b5 b1 B¼ b2 0 0 0 b1 0:0666 0:0783 0:0921 ¼ 0 0 0
b6
0
0
0
b3
b4
b5
b6
which is close to the actual slow-rate dual-rate system. From this model, the model for the dual-rate system can be derived as ( xðkh þ hÞ ¼ 0:8575xðkhÞ þ 0:14602uðkhÞ; ð34Þ yðkTÞ ¼ 1:00xðkT dhÞ þvðkTÞ:
Figs. 7 and 8 compare the outputs of the actual slow-rate system and the estimated slow-rate system using the Kalman filter with RLS and SG algorithms. The RLS has faster convergence. With a limited data, the estimated model parameters using RLS has higher accuracy than the SG algorithm. In spite of the slower convergence rate, parameter estimation using SG is also consistent. 6. Experiment An experimental study is performed on a pilot-scale multiple tank system to verify the effectiveness of the proposed approach. The
#
0:1084
0:1275
0:1500
0
0
0
0:0666
0:0783
0:0921
0:1084
0:1275
0:1500
;
uðkTÞ ¼ ½uðkT 3hÞ uðkT 2hÞ uðkT hÞ uðkTÞ uðkT þ hÞ uðkT þ2hÞuðkT þ 3hÞ uðkT þ 4hÞ uðkT þ 5hÞT : Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
L. Chen et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
dH 2 1 1 ¼ D H α1 D H α2 ; dt β2 ðH 2 Þ 1 1 β2 ðH2 Þ 2 2
8.8 8.6
7
Outputs of actual system Outputs of estimated model (RLS) Outputs of estimated model (SG)
ð35bÞ
8.4
y(k)
The inflow (q)
8.2 8 7.8
0.45 0.4 0.35
7.4 7.5
8
8.5
y(k)
Fig. 8. The scatter plot comparisons of the actual outputs and estimated outputs (Example 2).
Water level of Tank 2 (H2)
7.6
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
17 16 15 14 13 12
Time
Fig. 10. The input and output data of the multiple tank system.
16.5 True output Estimated output
16
Water level of Tank 2
15.5 15 14.5 14 13.5 13 12.5 12
0
100
200
300
400
500
600
700
800
900
Time
Fig. 11. Self-validation results of the identified model.
16.5 True output Estimated output
16
Fig. 9. Schematic diagram of the multitank system.
pilot-scale system is located in the Computer Process Control Laboratory in the Department of Chemical and Materials Engineering at the University of Alberta. The schematic diagram is displayed in Fig. 9. Based on the assumption that the laminar outflow is an ideal fluid, the dynamics of the three-tank system can be described as follows [3,22]
Water level of Tank 2
15.5 15 14.5 14 13.5 13 12.5 12 12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
Water level of Tank 2
dH 1 1 1 ¼ q D H α1 ; dt β1 ðH1 Þ β1 ðH1 Þ 1 1
ð35aÞ
Fig. 12. The scatter plot comparisons of self-validation results of the identified model.
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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8
dH 3 1 1 ¼ D H α2 D H α3 ; dt β3 ðH3 Þ 2 2 β3 ðH3 Þ 3 3
ð35cÞ
where Hi is the fluid levels in the ith tanks, q is inflow to the upper tank, Di is resistance of the output orifice of ith tank, βi is the cross sectional area of the ith tank, and αi is the flow coefficient for the ith tank, where i ¼ 1, 2, 3. In this experiment, the inflow q is selected as the input, and the level of the second tank H2 is considered to be the output. Let the system operate at a steady-state condition first, and then a random binary signal with level [ 0.05 0.05] is added to the input of the system. The observed input and output data are shown in Fig. 10. The input updating time T 1 ¼ h and the output sampling period T 2 ¼ 5h. Let h¼1, the corresponding discrete system is ( xðkh þ hÞ ¼ AxðkhÞ þ BuðkhÞ; ð36Þ yðkTÞ ¼ CxðkT dhÞ þ vðkTÞ; where 0:9922 0 A¼ ; 0 0:3698 0:1116 B¼ ; C ¼ ½0:06243; 0:04157: 0:06244
Water level of Tank 2
The parameter and state estimation problem of a single-input single-output dual-rate system is discussed in this paper. The slow-rate sate-space model of the dual-rate system with time delay is derived by using the discretization technique. A Kalman filtering approach based on the state augmentation strategy is developed to deal with the state estimation for the time delay system, and then system parameters are identified by using the SG algorithm or the RLS algorithm. The results are verified through simulations as well as a pilot-scale experiment. The proposed method has potential to solve the multi-input multi-output system with time delay, but heavy computational effort is expected.
Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Alberta Innovates Technology Futures (AITF), the Fundamental Research Funds for the Central Universities (JUDCF10064), Jiangsu Innovation Program for Graduates (CXZZ11_0464), China Scholarship Council (CSC), the Foundation of NSFC61134007 and 111 Project (B12018).
True output Estimated output
15.5 15
References
14.5 14 13.5 13 12.5 12
0
50
100
150
200
250
300
350
400
Time
Fig. 13. Cross-validation results of the identified model.
16.5 True output Estimated output
16 15.5 Water level of Tank 2
7. Conclusions
ð37Þ
16.5 16
The frame period T ¼ 5h. Using the proposed approach, the selfvalidation results using KF and RLS are shown in Figs. 11 and 12, and the cross-validation results are shown in Figs. 13 and 14, respectively. Therefore, it is concluded that the identified model can well capture the process dynamics using the sampled process data.
15 14.5 14 13.5 13 12.5 12 12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
Water level of Tank 2
Fig. 14. The scatter plot comparisons of cross-validation results of identified model.
the
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Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
L. Chen et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [19] Zhuang L, Pan F, Ding F. Parameter and state estimation algorithm for singleinput single-output linear systems using the canonical state space models. Appl Math Model 2012;36(8):3454–63. [20] Han L, Huang B. Parameter estimation for a dual rate system with time delay. In: The 10th IEEE international conference on control & automation, Hangzhou, China; June 12–14, 2013.
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[21] Johnston BS. Process dynamics, operations, and control. Spring; 2006 (MIT OpenCourseWare: Massachusetts Institute of Technology), 〈http://ocw.mit.edu/ courses/chemical-engineering/10-450-process-dynamics-operations-and-controlspring-2006〉. [22] Deng J, Huang B. Identification of nonlinear parameter varying systems with missing output data. AIChE J 2012;58(11):3454–67.
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Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Parameter estimation for a dual-rate system with time delay$ Lei Chen a,b, Lili Han b, Biao Huang b,n, Fei Liu a a b
Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, PR China Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada AB T6G 2G6
art ic l e i nf o
a b s t r a c t
Article history: Received 23 August 2013 Received in revised form 20 December 2013 Accepted 6 January 2014 This paper was recommended for publication by Dr. Qing-Guo Wang
This paper investigates the parameter estimation problem of the dual-rate system with time delay. The slow-rate model of the dual-rate system with time delay is derived by using the discretization technique. The parameters and states of the system are simultaneously estimated. The states are estimated by using the Kalman filter, and the parameters are estimated based on the stochastic gradient algorithm or the recursive least squares algorithm. When concerning state estimate of the dual-rate system with time delay, the state augmentation method is employed with lower computational load than that of the conventional one. Simulation examples and an experimental study are given to illustrate the proposed algorithm. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Parameter estimation Dual-rate Time delay Stochastic gradient Least squares Kalman filter
1. Introduction In many industrial processes, modeling the process system is a fundamental problem for control applications, such as soft sensor and controller design [1–4]. The main obstacle of modeling the industrial process system is the lack of the measurable input– output data due to time delay and different sampling rates between the regularly sampled data in sensors and laboratory analyses [5,6]. As the input and the output are sampled in two different sampling periods, the system is often described by a dual-rate system. The challenge of modeling is how to deal with the parameter estimation problem based on the dual-rate input and output data with time delay. The methods for the parameter estimation of the multirate system have been widely developed. For the general multi-input, multi-output and multirate system, the system is divided into subsystems and then a least square method is used to estimate the system parameters [7]. For a fast input slow output dual-rate system, the polynomial transformation technique adopted to transform the dual-rate system into the one which can be identified by the measurable data, and the parameter and intersample output can be estimated by a stochastic gradient (SG) ☆
The short version of this paper was presented in ICCA 2013. Corresponding author. Tel.: þ 1 780 492 9016; fax: þ 1 780 492 2881. E-mail addresses: [email protected] (L. Chen), [email protected] (L. Han), [email protected] (B. Huang), fl[email protected] (F. Liu). n
algorithm [8] and a recursive least square (RLS) algorithm [9]. For a multi-input multirate system, the discretization technique employed to convert the system into a slow-rate system which can be directly identified based on the known data and the RLS algorithm is used to estimate the system parameters [10]. A common strategy of these methods is to convert the original multirate system into systems which are identifiable based on the traditional system identification methods. For a system with very slow output samples, the output error method has been used to directly identify the fast rate model from the known fast rate input and the slow rate output [11]. Some researchers have applied the expectation maximization method to deal with the parameter estimation problem of the system with missing data. The statespace model identification based on the expectation maximization (EM) algorithm and the Kalman filtering estimation method has been adopted to identify a chemical process system based on the irregular sampled output [12]. For example, the grey-box identification techniques have been applied to the bleaching operation in a pulp mill to identify a dynamic model with irregular outputs, and the EM algorithm in the sense of maximum likelihood estimation is used to estimate the system parameters [13]. Currently, for the single rate time delay systems, many identification methods exist [14–16]; however, these methods cannot be directly applied to multirate systems. This paper focuses on the parameter estimation problem for a single-input single-output dual-rate system with time delay, which has two different sampling periods. First, the discretization technique is used to derive
0019-0578/$ - see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2014.01.001
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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2
the slow-rate state-space model of the original dual-rate system which can be identified from the measurable input–output data. Second, inspired by [17–19], the observability canonical form of the obtained state-space model is derived. With some algebraic manipulation the model for the dual-rate system is obtained of which the information vector contains the unmeasurable states. Finally, the Kalman filtering algorithm is applied to estimate the model states, and the SG algorithm or the RLS algorithm is used to identify the model parameters. The parameters and states of the system are simultaneously estimated. The proposed state estimation method based on the state augmentation strategy has less computational load compared with the traditional one. This paper is an extended version of our conference paper [20]. The main extensions include an additional identification algorithm, a practical simulation example and a real experimental evaluation. The rest of the paper is organized as follows. Section 2 describes the dual-rate system and problem statements. Section 3 derives a discrete-time state-space model for the dual-rate systems. Section 4 discusses the Kalman filtering algorithm for state estimation, and the SG algorithm or the RLS algorithm for identification of the parameters. Section 5 and 6 provide two illustrative examples and an experimental study on a pilot-scale multiple tank system. Concluding remarks are given in Section 7.
Consider a continuous time process Pc in Fig. 1 with the statespace representation: ( _ ¼ Ac xðtÞ þ Bc uðtÞ; xðtÞ ð1Þ yðtÞ ¼ Cxðt τÞ þ vðtÞ; where xðtÞ A Rn is the state vector, uðtÞ A R1 is the control input which is sampled by a zero-order hold H T 1 with sampling period T 1 ¼ ph, yðtÞ A R1 is the output which is sampled by the sampler ST 2 with period T 2 ¼ qh, τ is the time delay, vðtÞ A R1 is a stochastic noise vector with zero mean, and Ac , Bc and C are matrices of appropriate dimensions. Here, h is the positive number called basic period, and p and q are the two positive co-prime integers. Because of the zero-order hold, we have for kT 1 r t o ðk þ 1ÞT 1 :
ð2Þ
For such a dual-rate system, the measurable input–output data is fuðkT 1 Þ, yðkT 2 Þ : k ¼ 0; 1; 2; …g. This means fuðkT 1 þ ihÞ, yðkT 2 þ jhÞ, i ¼ 1; 2; …; p 1, j ¼ 1; 2; …; q 1g are unknown. The main objective of this paper is to answer the following two questions:
How to derive the slow-rate state-space model for the dual
ð3Þ
where Ah ¼ eAc h ;
Z Bh ¼
h
0
eAc t dt Bc
d¼
and
hτ i h
:
Theorem 1. For the system (3), letting T≔pqh be the frame period and m≔pq d, and assuwming that pq b d, then the state-space model of the dual-rate system can be derived as 8 q1 p > pq jp i > > Bh uðkT þjT 1 Þ; > xðkT þ TÞ ¼ AxðkTÞ þ ∑ ∑ Ah > > j ¼ 0i ¼ 1 > > > > pq > > < xðkT þ T dhÞ ¼ AxðkT dhÞ þ ∑ Apq þ m j B uðkT þ ðj 1Þh TÞ j ¼ mþ1
h
h
> > > m > j > > Bh uðkT þ ðj 1ÞhÞ; þ ∑ Am > h > > j ¼ 1 > > > : yðkTÞ ¼ CxðkT dhÞ þ vðkTÞ; ð4Þ where A ¼ Apq . h Proof. Substituting k with kpq in (3), we obtain ( xðkT þ hÞ ¼ Ah xðkTÞ þ Bh uðkTÞ;
2. System description
uðtÞ ¼ uðkT 1 Þ
sampling time h to get Ph as follows: ( xðkh þ hÞ ¼ Ah xðkhÞ þ Bh uðkhÞ; yðkhÞ ¼ Cxðkh dhÞ þvðkhÞ;
rate system with a time delay so that it can be directly identified based on the measured input–output data? How to estimate the states and parameters of the system with the time delay based on the Kalman filtering and the SG algorithm or the RLS algorithm?
ð5Þ
yðkTÞ ¼ CxðkT dhÞ þvðkTÞ: This leads to q1
p
jp i xðkTÞ þ ∑ ∑ Apq Bh uðkT þjT 1 þ ði 1ÞhÞ: xðkT þ TÞ ¼ Apq h h
ð6Þ
j¼0i¼1
According to (2), we have q1
p
jp i xðkT þ TÞ ¼ AxðkTÞ þ ∑ ∑ Apq Bh uðkT þ jT 1 Þ: h
ð7Þ
j¼0i¼1
Combining pq d
d dj xðkTÞ þ ∑ Apq Bh uðkT þ ðj 1ÞhÞ xðkT þ T dhÞ ¼ Apq h h
ð8Þ
j¼1
and xðkTÞ ¼ Adh xðkT dhÞ þ
pq
∑
j ¼ pq d þ 1
j Apq Bh uðkT þ ðj 1Þh TÞ; h
ð9Þ
yields xðkT þ T dhÞ ¼ AxðkT dhÞ þ
pq
∑
j ¼ mþ1
þmj Apq Bh uðkT þ ðj 1Þh TÞ h
m
j Bh uðkT þ ðj 1ÞhÞ: þ ∑ Am h
ð10Þ
j¼1
From (10) and (7), the results of (4) can be obtained.
□
4. Parameter and state estimation 3. Model derivation In order to derive the state-space model for the dual-rate system, the continuous process Pc is discretized via the zero-order hold with
Fig. 1. The general dual-rate sampled-data system.
In this section, the observability canonical model of system (4) is given which is equivalent to the original controllable and observable system and has the least number of the parameters. Then the model ready for the identification of the system is derived. Based on the measurable input–output data, the state estimation can be obtained by using the Kalman filter, and the parameter estimation can be obtained by using the SG algorithm or the RLS algorithm. Due to the time delay, the states of the system cannot be directly estimated by the Kalman filter. A state
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
L. Chen et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
augmentation strategy is applied to the system to identify the system states. Assume that the state-space model in (1) is identifiable, which means that (3) is controllable and observable. The state-space model (4) is also controllable and observable under some mild conditions that are provided in [17]. Thus, the model (4) can be rewritten in the observability canonical form as follows: ( xðkT þ T dhÞ ¼ Ao xðkT dhÞ þ Bo uðkTÞ; ð11Þ yðkTÞ ¼ C o xðkT dhÞ þ vo ðkTÞ; where 2
0 1 0 60 0 1 6 6 ⋮ ⋮ Ao ¼ 6 6 ⋮ 60 0 0 4 a1 a2 a3 2 3 B1 6 B2 7 6 7 Bo ¼ 6 7 A Rnpq ; 4 ⋮ 5 Bn
3 0 07 7 7 nn 1 ⋮ 7 7 A R ; ai A R ; 17 5 an
⋯ ⋯ ⋮ ⋯ ⋯
Bi A R1pq ; C o ¼ ½1 0 ⋯ 0 A R1n ;
uðkTÞ ¼ ½uðkT dhÞ ⋯ uðkTÞ uðkT þhÞ⋯ uðkT þ ðpq d 1ÞhÞT A Rpq : Define the parameter vector θ and the information vector
φðkTÞ as
θ≕½αT B1 B2 ⋯ Bn T A Rn þ npq ; α≕½a1 a2 ⋯an T A Rn ; φðkTÞ ¼ ½ x T ðkT dhÞ uT ðkT þ nT TÞ uT ðkT þ nT 2TÞ ⋯ xðkTÞ ¼ ½x 1 ðkTÞ x 2 ðkTÞ ⋯ x n ðkTÞT A Rn : 2
x 1 ðkT dhÞ
Based on the least square (LS) method, the cost function J is N
J ¼ ∑ ‖yðkT þ nTÞ φT ðkTÞθ‖2 ;
ð15Þ
k¼1
where N is the number of the data point. Unfortunately, the parameter vector θ cannot be estimated by using the LS method. In fact, from (14), we can see that the information vector φðkTÞ contains unknown state xðkT dhÞ, so that the LS algorithm cannot be directly applied for the estimation of the parameter θ. The solution that we are taking is to replace the unknown state xðkT dhÞ with their estimates x^ ðkT dhÞ which is obtained by the Kalman filtering algorithm. Since the dual-rate system contains the time delay, the states of the system (11) can be estimated by using the state augmentation method which is the general method to deal with the problem of states estimation of a delayed system. Augment the state as 2 3 xðkT dhÞ 6 7 6 xðkT dh þ hÞ 7 7 A Rnd þ n : Z~ ðkTÞ ¼ 6 ð16Þ 6 7 ⋮ 4 5 xðkTÞ However, this approach of augmentation will introduce a heavy computational load in executing the Kalman filtering algorithm. In order to reduce the cost of the computational effort, the state can be extended as " # xðkT dhÞ ZðkTÞ ¼ A R2n : ð17Þ xðkTÞ Then the state-space model can be rewritten as ( ZðkT þ TÞ ¼ Az ZðkTÞ þ Bz U z ðkTÞ;
uT ðkTÞT A Rn þ npq ; Let
3
yðkTÞ ¼ C z ZðkTÞ þ vðkTÞ;
ð18Þ
where
3
Az ¼ A0 I; " B0 Bz ¼ 0
6 7 6 x 2 ðkT dhÞ 7 7 A Rn : xðkT dhÞ ¼ 6 6 7 ⋮ 4 5 x n ðkT dhÞ Then we have 8 x 1 ðkT þT dhÞ ¼ x 2 ðkT dhÞ þ B1 uðkTÞ; > > > > > > < x 2 ðkT þT dhÞ ¼ x 3 ðkT dhÞ þ B2 uðkTÞ; ⋮ > > > x n 1 ðkT þ T dhÞ ¼ x n ðkT dhÞ þBn 1 uðkTÞ; > > > : x ðkT þT dhÞ ¼ a x ðkT dhÞ þa x ðkT dhÞ þ ⋯ þ a x ðkT dhÞþ B uðkTÞ: n n n n 1 1 2 2
# 0 ; B0
C z ¼ ½1 0; U z ðkTÞ ¼ ½uðkT dhÞ uðkT dh þhÞ ⋯ uðkT hÞ uðkTÞ ⋯ uðkT þ ðpq d 1ÞhÞuðkT þ ðpq dÞhÞ ⋯ uðkT þ ðpq 1ÞhÞT :
When the system is one dimensional, Az and Bz can be expressed as
ð12Þ Az ¼ Apq I; 2h pq 1 Ah Bh Bz ¼ 4 0
2 Apq Bh h
⋯
d Apq Bh h
d1 Apq Bh h
⋯
Bh
⋯
0
0
⋯
0
1 Apq Bh h
⋯
d Apq Bh h
⋯
Bh
Let z be a forward shift operator, i.e., zx 1 ðkTÞ ¼ zx 1 ðkT þTÞ. Multiplying the jth equation above by zn j and summing the obtained equations, we can obtain x 1 ðkT þ nT dhÞ ¼ a1 x 1 ðkT dhÞ þ a2 x 2 ðkT dhÞ þ ⋯ þ an x n ðkT dhÞ
3 5:
Based on the Kalman filter, the state estimation can be obtained as follows: Z^ ðkT þ TÞ ¼ A^ z ðkTÞZ^ ðkTÞ þ B^ z ðkTÞU z ðkTÞ þ LðkTÞ½yðkTÞ C z Z^ ðkTÞ; ð19Þ
þBn uðkTÞ þ Bn 1 uðkT TÞ þ ⋯ þ B1 uðkT þ nT TÞ: ð13Þ
LðkTÞ ¼ A^ z ðkTÞP x ðkTÞC Tz ½I þ C z P x ðkTÞC Tz 1 ;
ð20Þ
Here we have adopted the approach of [17]. As a result, we have yðkT þ nTÞ ¼ φT ðkTÞθ þvo ðkT þ nTÞ:
T
ð14Þ
P x ðkTÞ ¼ ½A^ z ðkTÞ LðkTÞC z P x ðkTÞA^ z ðkTÞ;
ð21Þ
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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4
vector, e.g., θ^ ð0Þ ¼ 1n =p0 . The initial value for the parameter estimation algorithm is rð0Þ ¼ 1 (SG) or P p ð0Þ ¼ p0 I (RLS). Determine the data length Le. Step 2: Collect the dual-rate measurement data fuðkT 1 Þ, yðkT 2 Þg and form the information vector φðkTÞ. Step 3: By using (19)–(21) to calculate the state estimation ^ ðkTÞ by (22). Z^ ðkTÞ and form the estimated information vector φ Step 4: Calculate (24) to get r(kT) and update the parameter estimation by (23), or calculate (26) to get P p ðkTÞ and update the parameter estimation by (25). Step 5: If k ¼ Le , terminate the procedure and obtain the estimate θ^ ðLe TÞ of the parameter vector θ; otherwise, increment k by 1 and go to step 2.
5. Examples
Example 1 (Numerical example). Consider a single-input singleoutput dual-rate system with the input updating time T 1 ¼ h and the output sampling period T 2 ¼ 5h. Let h¼ 2; the corresponding discrete system is ( xðkh þ hÞ ¼ 0:7413xðkhÞ þ 0:4571uðkhÞ; ð28Þ yðkTÞ ¼ 1:00xðkT dhÞ þvðkTÞ; Fig. 2. The flowchart of the proposed algorithm for computing the parameter and state estimates.
where LðkTÞ is the Kalman gain and P x ðkTÞ is the covariance matrix of the state estimation error. A^ z ðkTÞ and B^ z ðkTÞ represent the estimates of Az and Bz at time kT, respectively. The parameter estimation can be derived by using the SG algorithm or the RLS algorithm. Define the new information vector φ^ ðkTÞ using the estimated state as T
φ^ ðkTÞ ¼ ½ x^ ðkT dhÞ uT ðkT þ nT TÞ uT ðkT þ nT 2TÞ ⋯ uT ðkTÞT : ð22Þ
where
The SG algorithm is as follows:
θ^ ðkT þ TÞ ¼ θ^ ðkTÞ þ
φ^ ðkTÞ rðkTÞ
^ ðkTÞθ^ ðkTÞ; ½yðkT þ nTÞ φ
^ ðkTÞ‖2 ; rðkT þ TÞ ¼ rðkTÞ þ ‖φ
T
rð0Þ ¼ 1;
ð23Þ ð24Þ
θ^ ðkT þ TÞ ¼ θ^ ðkTÞ þ Lp ðkTÞ½yðkT þ nTÞ φ^ T ðkTÞθ^ ðkTÞ; ^ ðkTÞ P p ðkTÞφ ^ ðkTÞ; ^ T ðkTÞP p ðkTÞφ 1þφ
^ T ðkTÞÞP p ðkTÞ: P p ðkT þ TÞ ¼ ðI Lp ðkTÞφ
A ¼ a1 I ¼
" B¼ ¼
and the RLS algorithm can be represented as
Lp ðkT þ TÞ ¼
where d ¼2 and the frame period T ¼ 5h. The input fuðkhÞg is taken as uncorrelated persistent excitation signal sequences with zero mean and unit variances, fvðkTÞg consists of two white-noise sequences with zero mean and variances s2 ¼ 0:52 and s2 ¼ 1:02 . By using the discretization technique, the state-space model with time T can be derived as " # 8 xðkT dhÞ > > > ; > < ZðkTÞ ¼ xðkTÞ ð29Þ > ZðkT þ TÞ ¼ AZðkTÞ þ BuðkTÞ; > > > : yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ;
ð25Þ ð26Þ ð27Þ
Here θ^ ðkTÞ represents the estimation of θðkTÞ at time kT. 1=rðKTÞ is
^ ðkTÞ is defined by the step-size and the norm of the matrix φ ^ ðkTÞ J 2 ¼ tr½φ ^ ðkTÞφ ^ T ðkTÞ. P p ðkTÞ is the covariance matrix of the Jφ parameter estimation error, and Lp ðkTÞ is the gain vector. The SG algorithm has a slower convergence rate compared with the RLS algorithm, but the SG algorithm requires lower computation load. From (19) to (21) and from (22) to (27), the Kalman filtering algorithm and the parameter estimation algorithm (SG or RLS) are applied to estimate the system parameters and states in an iterative way. The flowchart of the algorithm is given in Fig. 2 and the steps taken for the algorithm are summarized below. Step 1: Let k ¼1; the initial value P x ð0Þ is generally taken to be p0 I with p0 normally being a large positive number, e.g., p0 ¼ 106 , and the initial value θ^ ð0Þ and x^ ð0Þ are a zero vector or a small real
b1 0
0:2239
0
0
0:2239
b2 0
0:1380 0
b3 b1 0:1862 0
b4 b2
;
b5 b3 0:2512 0:1380
0 b4
0 b5
#
0:3388 0:1862
0:4571 0:2512
0 0:3388
0 ; 0:4571
uðkTÞ ¼ ½uðkT 2hÞ uðkT hÞ uðkTÞ uðkT þ hÞ uðkT þ 2hÞ uðkT þ 3hÞ uðkT þ 4hÞT :
Apply the Kalman filter and SG algorithm to estimate the parameters of this system, and the parameter estimation error is used to quantify the estimation accuracy [10]:
δ≔
J θ^ ðkTÞ θ J 100%; JθJ
ð30Þ
Table 1 The parameters and their estimates using KF and SG algorithm ðs2 ¼ 0:52 Þ. k
100
0.0886 a1 b1 0.1670 b2 0.2526 b3 0.1554 b4 0.2332 b5 0.3426 δ ð%Þ 34.0778
500
1000
2500
5000
0.1261 0.1316 0.1428 0.1489 0.1644 0.1578 0.1551 0.1552 0.2430 0.2402 0.2316 0.2256 0.1793 0.1935 0.2067 0.2131 0.2911 0.2970 0.3110 0.3176 0.4002 0.4074 0.4237 0.4331 22.1891 19.8576 16.1423 14.2382
10 000
True values
0.1546 0.1558 0.2230 0.2183 0.3226 0.4411 12.8241
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
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where θ is the true parameters, and θ^ ðkTÞ is the estimated parameters at time KT. The parameter estimates and their errors when noise variance is 0.52 are shown in Table 1. Apply the Kalman filter and RLS algorithm to estimate the parameters of this system, the results of the parameter estimates and their errors with noise variances of 0.52, 1.02 and 1.52 are shown in Tables 2 and 4, respectively. The parameter estimation errors δ versus k with different noise variances are shown in Fig. 3. Figs. 4 and 5 compare the outputs of the actual slow-rate system and the estimated slow-rate system.
5
2 Outputs of actual system Outputs of estimated model
1.5
1
0.5
0
−0.5
Table 2 The parameters and their estimates using KF and RLS algorithms ðs2 ¼ 0:52 Þ. k
100
500
1000
2500
5000
10 000
True values
a1 b1 b2 b3 b4 b5 δ ð%Þ
0.1223 0.0947 0.3063 0.2232 0.3512 0.5812 29.5926
0.1849 0.1427 0.2035 0.2232 0.3932 0.5194 13.8785
0.2537 0.1134 0.1894 0.2392 0.3369 0.4493 5.9164
0.2374 0.1219 0.1803 0.2498 0.3464 0.4610 3.3641
0.2335 0.1311 0.1722 0.2461 0.3443 0.4639 2.9935
0.2199 0.1402 0.1816 0.2444 0.3446 0.4702 2.4459
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
−1
−1.5
0
20
40
60
80
100
k Fig. 4. Comparisons of the actual outputs and estimated outputs using RLS. 2 Outputs of actual system Outputs of estimated model
1.5
Table 3
1
The parameters and their estimates using KF and RLS algorithms ðs2 ¼ 1:02 Þ. 100
500
1000
2500
5000
10 000
True values
a1 b1 b2 b3 b4 b5 δ ð%Þ
0.1863 0.0325 0.1840 0.0340 0.3300 0.4422 34.9780
0.2947 0.1309 0.1374 0.2787 0.3836 0.3712 18.9390
0.2305 0.0873 0.1845 0.2296 0.3723 0.4826 9.9506
0.2816 0.1399 0.1942 0.2414 0.3394 0.4488 8.5318
0.1760 0.1422 0.1687 0.2418 0.3453 0.4620 7.5137
0.2634 0.1515 0.1729 0.2434 0.3256 0.4583 6.6334
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
0.5 y(k)
k
0 −0.5
−1
−1.5 −1.5
Table 4 The parameters and their estimates using KF and RLS algorithms ðs2 ¼ 1:52 Þ. k
100
500
1000
2500
5000
10 000
True values
a1 b1 b2 b3 b4 b5 δ ð%Þ
0.1261 -0.0260 0.5126 0.1201 0.2982 0.7243 68.9345
0.2003 0.1394 0.2168 0.1459 0.4672 0.5947 31.2697
0.2358 0.0738 0.2126 0.2330 0.3609 0.4727 11.0762
0.2327 0.0937 0.1747 0.2548 0.3726 0.4839 9.0760
0.2314 0.1201 0.1480 0.2413 0.3625 0.4872 8.3282
0.2179 0.1436 0.1711 0.2293 0.3540 0.4939 6.9273
0.2239 0.1380 0.1862 0.2512 0.3388 0.4571
−1
−0.5
0
0.5
1
1.5
2
y(k)
Fig. 5. The scatter plot comparisons of the actual outputs and estimated outputs using RLS.
1.4 1.2 1 0.8
2
2
δ
σ =0.5
Fig. 6. The schematic diagram of the single tank system.
0.6 2
2
σ =1.0
0.4
2
The estimated slow-rate model with s2 ¼ 0:52 using the Kalman filter and the RLS algorithm for the dual-rate system is given by
2
σ =1.5
0.2 0
0
500
1000
1500
2000
k Fig. 3. Parameter estimation error δ versus k using RLS.
2500
8 0:2199 0 > > ZðkTÞ ¼ ZðkTÞ > > > 0 0:2199 > < 0:1402 0:1816 0:2444 0:3446 þ > > > 0 0 0:1402 0:1816 > > > : yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ;
0:4702
0
0
0:2444
0:3446
0:4702
uðkTÞ;
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6
which is close to the actual slow-rate dual-rate system (29). Then, the model for the dual-rate system can be derived as ( xðkh þ hÞ ¼ 0:7386xðkhÞ þ 0:4702uðkhÞ; ð31Þ yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ:
8.8
8.4 8.2
From Tables 1 to 3 and from Figs. 3 to 5, it is clear that the RLS algorithm has faster convergence rates than the SG algorithm, but the latter has less computation load. One can see that with the increase of time k, the parameter estimation error is reduced and outputs of the estimated slow-rate model approach to those of the actual slow-rate system. Example 2 (A holding tank). A process stream comprises a chemical species in dilute liquid solution. It is the effluent of some process, and it is used to feed a downstream process. One can moderate flow swings by holding up a volume in a tank. Intuitively we expect the changes in the outlet flow rate to be more moderate than those of the feed stream [21]. The schematic diagram of this process is displayed in Fig. 6. It is assumed that the tank operates in overflow, and the volume V is constant. Both streams are represented in terms of a single volumetric flow F, and CAi and CAo are the inlet and the outlet concentration, respectively. A component material balance over
8 0:3971 0 > > ZðkTÞ ¼ ZðkTÞ > > > 0 0:3971 > < 0:0658 0:0745 0:0871 þ > > > 0 0 0 > > > : yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ;
Outputs of actual system Outputs of estimated model (RLS) Outputs of estimated model (SG)
8.6
8 7.8 7.6 7.4
0
20
40
60
80
100
k Fig. 7. Comparisons of the actual outputs and estimated outputs (Example 2).
Apply the Kalman filter and the RLS algorithm to estimate the parameters of this system, the estimated slow-rate model using the Kalman filter and the RLS algorithm for the dual-rate system is given by
0:1077
0:1292
0:1460
0
0
0
0:0658
0:0745
0:0871
0:1077
0:1292
0:1460
uðkTÞ;
the solution is d ðVC Ao Þ ¼ FC Ai C Ao : dt
ð32Þ
The inlet concentration CAi and outlet concentration CAo are considered as the system input u and output y, respectively. Let the basic period h ¼1, and the input and output sampling time be T 1 ¼ h and T 2 ¼ 6h, respectively. Consider the time delay d ¼3, and the frame period T ¼ 6h. The corresponding discrete system can be derived as ( xðkh þ hÞ ¼ 0:85xðkhÞ þ 0:15uðkhÞ; ð33Þ yðkTÞ ¼ 1:00xðkT dhÞ þ vðkTÞ:
The input uðkhÞ is taken as uncorrelated persistent excitation signal sequences with unit variances, and vðkTÞ is the white-noise sequence with zero mean and variance s2 ¼ 0:22 . By using the proposed discretization technique, the state-space model with sampling time T can be derived as Eq. (29), where 0:3771 0 A ¼ a1 I ¼ ; 0 0:3771 " b2 b3 b4 b5 b1 B¼ b2 0 0 0 b1 0:0666 0:0783 0:0921 ¼ 0 0 0
b6
0
0
0
b3
b4
b5
b6
which is close to the actual slow-rate dual-rate system. From this model, the model for the dual-rate system can be derived as ( xðkh þ hÞ ¼ 0:8575xðkhÞ þ 0:14602uðkhÞ; ð34Þ yðkTÞ ¼ 1:00xðkT dhÞ þvðkTÞ:
Figs. 7 and 8 compare the outputs of the actual slow-rate system and the estimated slow-rate system using the Kalman filter with RLS and SG algorithms. The RLS has faster convergence. With a limited data, the estimated model parameters using RLS has higher accuracy than the SG algorithm. In spite of the slower convergence rate, parameter estimation using SG is also consistent. 6. Experiment An experimental study is performed on a pilot-scale multiple tank system to verify the effectiveness of the proposed approach. The
#
0:1084
0:1275
0:1500
0
0
0
0:0666
0:0783
0:0921
0:1084
0:1275
0:1500
;
uðkTÞ ¼ ½uðkT 3hÞ uðkT 2hÞ uðkT hÞ uðkTÞ uðkT þ hÞ uðkT þ2hÞuðkT þ 3hÞ uðkT þ 4hÞ uðkT þ 5hÞT : Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
L. Chen et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
dH 2 1 1 ¼ D H α1 D H α2 ; dt β2 ðH 2 Þ 1 1 β2 ðH2 Þ 2 2
8.8 8.6
7
Outputs of actual system Outputs of estimated model (RLS) Outputs of estimated model (SG)
ð35bÞ
8.4
y(k)
The inflow (q)
8.2 8 7.8
0.45 0.4 0.35
7.4 7.5
8
8.5
y(k)
Fig. 8. The scatter plot comparisons of the actual outputs and estimated outputs (Example 2).
Water level of Tank 2 (H2)
7.6
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
17 16 15 14 13 12
Time
Fig. 10. The input and output data of the multiple tank system.
16.5 True output Estimated output
16
Water level of Tank 2
15.5 15 14.5 14 13.5 13 12.5 12
0
100
200
300
400
500
600
700
800
900
Time
Fig. 11. Self-validation results of the identified model.
16.5 True output Estimated output
16
Fig. 9. Schematic diagram of the multitank system.
pilot-scale system is located in the Computer Process Control Laboratory in the Department of Chemical and Materials Engineering at the University of Alberta. The schematic diagram is displayed in Fig. 9. Based on the assumption that the laminar outflow is an ideal fluid, the dynamics of the three-tank system can be described as follows [3,22]
Water level of Tank 2
15.5 15 14.5 14 13.5 13 12.5 12 12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
Water level of Tank 2
dH 1 1 1 ¼ q D H α1 ; dt β1 ðH1 Þ β1 ðH1 Þ 1 1
ð35aÞ
Fig. 12. The scatter plot comparisons of self-validation results of the identified model.
Please cite this article as: Chen L, et al. Parameter estimation for a dual-rate system with time delay. ISA Transactions (2014), http://dx. doi.org/10.1016/j.isatra.2014.01.001i
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8
dH 3 1 1 ¼ D H α2 D H α3 ; dt β3 ðH3 Þ 2 2 β3 ðH3 Þ 3 3
ð35cÞ
where Hi is the fluid levels in the ith tanks, q is inflow to the upper tank, Di is resistance of the output orifice of ith tank, βi is the cross sectional area of the ith tank, and αi is the flow coefficient for the ith tank, where i ¼ 1, 2, 3. In this experiment, the inflow q is selected as the input, and the level of the second tank H2 is considered to be the output. Let the system operate at a steady-state condition first, and then a random binary signal with level [ 0.05 0.05] is added to the input of the system. The observed input and output data are shown in Fig. 10. The input updating time T 1 ¼ h and the output sampling period T 2 ¼ 5h. Let h¼1, the corresponding discrete system is ( xðkh þ hÞ ¼ AxðkhÞ þ BuðkhÞ; ð36Þ yðkTÞ ¼ CxðkT dhÞ þ vðkTÞ; where 0:9922 0 A¼ ; 0 0:3698 0:1116 B¼ ; C ¼ ½0:06243; 0:04157: 0:06244
Water level of Tank 2
The parameter and state estimation problem of a single-input single-output dual-rate system is discussed in this paper. The slow-rate sate-space model of the dual-rate system with time delay is derived by using the discretization technique. A Kalman filtering approach based on the state augmentation strategy is developed to deal with the state estimation for the time delay system, and then system parameters are identified by using the SG algorithm or the RLS algorithm. The results are verified through simulations as well as a pilot-scale experiment. The proposed method has potential to solve the multi-input multi-output system with time delay, but heavy computational effort is expected.
Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Alberta Innovates Technology Futures (AITF), the Fundamental Research Funds for the Central Universities (JUDCF10064), Jiangsu Innovation Program for Graduates (CXZZ11_0464), China Scholarship Council (CSC), the Foundation of NSFC61134007 and 111 Project (B12018).
True output Estimated output
15.5 15
References
14.5 14 13.5 13 12.5 12
0
50
100
150
200
250
300
350
400
Time
Fig. 13. Cross-validation results of the identified model.
16.5 True output Estimated output
16 15.5 Water level of Tank 2
7. Conclusions
ð37Þ
16.5 16
The frame period T ¼ 5h. Using the proposed approach, the selfvalidation results using KF and RLS are shown in Figs. 11 and 12, and the cross-validation results are shown in Figs. 13 and 14, respectively. Therefore, it is concluded that the identified model can well capture the process dynamics using the sampled process data.
15 14.5 14 13.5 13 12.5 12 12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
Water level of Tank 2
Fig. 14. The scatter plot comparisons of cross-validation results of identified model.
the
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