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Robust PID controller design for nonlinear processes using JITL technique
Chemical Engineering Science 63 (2008) 5141 -- 5148
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Robust PID controller design for nonlinear processes using JITL technique Cheng Cheng, Min-Sen Chiu ∗ Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576, Singapore
A R T I C L E
I N F O
Article history: Received 14 May 2007 Received in revised form 18 June 2008 Accepted 10 July 2008 Available online 15 July 2008 Keywords: Robust control Just-in-time learning Nonlinear system PID controller
A B S T R A C T
A robust PID controller design methodology for nonlinear processes is proposed based on the just-in-time learning (JITL) technique. To do so, a composite model consisting of a nominal ARX model and the JITL, where the former is used to capture linear process dynamics and the latter to approximate the inevitable modeling error caused by the process nonlinearity, is employed to model the process dynamics in the operating space of interest. The state space realizations of this composite model and PID controller are then reformulated as an uncertain closed-loop system, by which the corresponding robust stability condition is developed. Literature examples are employed to illustrate the proposed methodology and a comparison with the previous result is made. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction For most chemical processes, the first-principle models are usually unavailable because of the lacking of physicochemical knowledge. An attractive alternative for controller design is to rely on the models extracted from process input and output measurements. These models generally have varying degrees of accuracy. If the plant/model mismatch is not taken into account in controller design, the resulting control performance may become poor or even unstable in the presence of significant modeling error. This design problem has motivated the researchers to pursue various robust controller design methods in the last two decades (Morari and Zafiriou, 1989; Malan et al., 2004). The objective of robust controller design is to ensure closed-loop stability and maintain control performance not only for the nominal model but also for a set of possible process models that captures the actual process dynamics. For linear system, various robust control problems had been tackled by using the transfer function model approaches (Morari and Zafiriou, 1989; Packard and Doyle, 1993). Normally, a given set of process models is represented by a nominal model together with a suitable uncertainty description equation to account for the modeling error between the nominal model and actual process dynamics. The associated design issue of estimating the uncertainty models also attracted much research investigations (Gustafsson and Makila, 2001; Boling et al., 2004). Some robustness analysis results developed for linear systems have also been applied to the nonlinear systems. For example, Doyle et al. (1989) proposed a robust controller design method for a
∗
Corresponding author. Tel.: +65 6516 2223; fax: +65 6779 1936. E-mail address: [email protected] (M.-S. Chiu).
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nonlinear CSTR for which a first-principle model is assumed to be available. By assuming that the process/model mismatch is entirely due to the nonlinearities of the process, bounds on the conic sectors that describe the process nonlinearities were developed and used in the standard M − structure for robust stability analysis. However, the identification of the conic bounds is not trivial and the resulting robust stability analysis tends to give conservative result, not to mention that first-principle models are generally not available for many chemical processes. To lessen the modeling requirement, Knapp and Budman (2000) developed a robust PID controller design methodology for nonlinear processes using an empirical state affine model developed from the available process data, which can be readily transformed into a suitable form for the robust stability analysis. Although their result provides an attractive alternative to the Doyle's approach, the construction of the state affine models is rather tedious and computational demanding. According to Knapp and Budman (2000), in order to obtain a state affine model, a NARMA model is initially constructed from the available process input and output data. Subsequently, an algorithm developed by Diaz and Desrochers (1988) is employed to find the parameters for a truncated Volterra model based on the NARMA model identified previously. Once the Volterra kernels are obtained, a generalized Hankel matrix can be developed to find a state affine model (Sontag, 1979). Obviously, the modeling efforts required to identify a state affine model are extensive and thus hampers the application of robust controller design method developed based on such a model. To circumvent the aforementioned drawbacks by using the conic sectors and state affine models in robust controller designs for nonlinear systems, a robust PID controller design methodology using the just-in-time learning (JITL) technique (Cybenko, 1996; Atkeson et al., 1997; Bontempi et al., 2001) is developed in this paper. It is noted
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that robust PID controller design methods have been developed using different techniques in the literature, for example linear matrix inequalities (LMIs) approach in the multiple-model modeling framework (Ge et al., 2002; Toscano, 2007) and numerical optimization approach by solving the maximization of the shortest distance from the Nyquist curve of the open-loop transfer function to the critical point (Toscano, 2005). However, these methods require a priori process knowledge or ad-hoc procedure to address the associated design issue of partition of the operating space in the multiple-model approach and determine the worst-case model (or optimal model for that matter) in the operating space of interest in order to obtain the best viable PID design in the numerical optimization approach. In contrast, the proposed design method is a one-shot design approach in the sense that the PID parameters are obtained directly from a set of process data characterizing the process dynamics of interest. In this respect, the proposed method is more advantageous than the previous model-based robust PID design methods because the required a priori process information may not be readily available in practical applications, leading to the tedious trial and error design procedure. In the proposed method, it is assumed that the process nonlinearity is the only source of the model uncertainty. A composite model consisting of a nominal ARX model and the JITL, where the former is used to capture linear process dynamics and the latter is applied to approximate the inevitable modeling error caused by the process nonlinearity. The state space realization of this composite model is then reformulated as an uncertain system, by which the robust stability condition of this uncertain system under PID control is developed. Literature examples are used to illustrate the proposed method and a comparison with the previous result is made. 2. Modeling methodology In this paper, a composite model consisting of a nominal ARX model and JITL models is used to describe the nonlinear process in the operating range of interest, where the former can be identified by using the process input and output data around a nominal operating condition and the latter is used to capture the modeling error caused by the process nonlinearity, i.e. the difference between the predicted output of nominal ARX model and actual process output. Suppose that an input sequence {u(k)} is injected into the process and the corresponding output sequence {y(k)} is measured. The following composite model is then used to model the nonlinear dynamics between {u(k)} and {y(k)}: yˆ (k) = yl (k) + ynl (k)
(1)
where yˆ (k) is the output of the composite model, yl (k) is the predicted output of nominal ARX model, and ynl (k) is the effect of process nonlinearity, i.e. y(k)−yl (k), to be approximated by the JITL technique (see Appendix A). Because JITL normally employs a first-order or second-order ARX model, we shall use a second-order model structure for both nominal ARX model and JITL in the subsequent developments. Therefore, yl (k) and ynl (k) are represented as follows:
z−1 x(k + 1)
x(k)
u(k)
A0
B0
E1
C0
D0
E2
F1
F2
E3
y(k)
~ y(k)
~ u(k)
1 2 3 a
Fig. 1. The interconnection structure for the composite model described by Eqs. (8) and (10).
where x(k) = [x1 (k) x2 (k)]T and + 1 l,2 + 2 A = l,1 1 0 B0 = [1 0]T ;
C = [l,1 + 1 0];
D0 = 0
Given the process input and output data {u(k), y(k)}, the plant/model mismatch caused by the nonlinearity, i.e. y(k) − yl (k), can be calculated after the nominal ARX model given in Eq. (2) is identified. Subsequently, the JITL technique can be applied to model the dynamics between the input sequence {u(k)} and the sequence {y(k) − yl (k)} by using the reference dataset constructed from {u(k), y(k)}. Upon the successful implementation of JITL algorithm, the respective ranges of variation for model coefficients in Eq. (3) are denoted by 1 ∈ [1,min 1,max ], 2 ∈ [2,min 2,max ], and 1 ∈ [1,min 1,max ], which can be represented by the following equations:
1 = ¯ 1 (1 + r1 1 ); |1 | 1 2 = ¯ 2 (1 + r2 2 ); |2 | 1 1 = ¯ 1 (1 + r3 3 ); |3 | 1
(6)
where i (i = 1–3) is the uncertainty bounded by one and other parameters are defined as follows:
1,max − 1,min 1,min + 1,max 2,max − 2,min r2 = 2,min + 2,max + 1,max − 1,min 1,max ; r3 = ¯ 1 = 1,min 2 1,min + 1,max ¯ 1 =
1,min + 1,max
; 2 + 2,max ; ¯ 2 = 2,min 2
r1 =
yl (k) = l,1 y(k − 1) + l,2 y(k − 2) + l,1 u(k − 1)
(2)
By using Eqs. (6) and (7), Eqs. (4) and (5) can be rewritten as A1 A2 x(k + 1) = A0 + 1 + x(k) + B0 u(k) 01×2 01×2 2
ynl (k) = 1 y(k − 1) + 2 y(k − 2) + 1 u(k − 1)
(3)
yˆ (k) = (C0 + C1 3 )x(k) + D0 u(k)
Consider the following state space realization of the composite model given in Eqs. (1)–(3):
where 0n×m denote a n × m zero matrix and + ¯ 1 l,2 + ¯ 2 A0 = l,1 1 0
x(k + 1) = Ax(k) + B0 u(k)
(4)
A1 = [¯ 1 r1 0];
yˆ (k) = Cx(k) + D0 u(k)
(5)
C0 = [l,1 + ¯ 1 0];
A2 = [0 ¯ 2 r2 ] C1 = [¯ 1 r3 0]
(7)
(8) (9)
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z−1 x(k)
x(k + 1)
Ψ(k)
Ψ(k + 1) M11
M12
M21
M22
To account for the modeling error resulting from the approximation of the nonlinear process by the proposed composite model, an additive uncertainty a is appended to the model output yˆ (k) as follows: y(k) = yˆ (k) + la x1 (k)a ,
|a | 1
(10)
where la is the magnitude of the worst perturbation as calculated by y(k) − yˆ (k) (11) la = Max ˆ (k)/(l,1 + 1 ) k y
~ y(k)
~ (k) u
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1 2
3
1
a
0.9
M
x(k) Ψ(k) ~ u(k)
M12
M21
M22
0.7 y
M11
0.8
x(k + 1) Ψ(k + 1) ~ y (k)
0.6 0.5 0.4
zI4
Process output ARX model Composite model
0.3
1 2
0.2
3
0
50
100
a
150
Fig. 4. Validation result for the CSTR example.
Fig. 2. The interconnection structure for the uncertain closed-loop system.
y
1 0.5 0 0
50
100
150
200
250
300
0
50
100
150
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300
0
50
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0
50
100
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yl
1 0.5 0
y-yl
0.2 0 -0.2
30 u
200
Sample
15 0 Sample Fig. 3. Input–output data used for the JITL.
250
300
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It is noted from Eq. (9) that the state variable x2 (k) has no effect on the model output yˆ (k), which explains why the additive uncertainty a in Eq. (10) is only associated with the state variable x1 (k). After some algebraic manipulation, the composite model by including the uncertainty a , i.e. Eqs. (8) and (10), can be recast into the interconnection structure as depicted in Fig. 1, where ˜ x(k + 1) = A0 x(k) + B0 u(k) + E1 u(k)
(12)
˜ y(k) = C0 x(k) + D0 u(k) + E2 u(k)
(13)
˜ ˜ y(k) = F1 x(k) + F2 u(k) + E3 u(k)
(14)
1 1 0 0 E1 = ; 0 0 0 0 ⎡
¯ 1 r1 ⎢ 0 F1 = ⎢ ⎣ ¯ r
1 3 la
⎤ 0 ¯ 2 r2 ⎥ ⎥; 0 ⎦
E2 = [0 0 1 1];
E3 = 04×4
˜ y(k) = [¯ 1 r1 x1 (k) ¯ 2 r2 x2 (k) ¯ 1 r3 x1 (k) la x1 (k)]T ˜ u(k) = [¯ 1 r1 x1 (k)1 ¯ 2 r2 x2 (k)2 ¯ 1 r3 x1 (k)3 la x1 (k)a ]T Lastly, when a first-order ARX model is employed for both nominal model and the JITL, the corresponding interconnection structure is given by A0 = l,1 + ¯ 1 ;
B0 = 1
C0 = l,1 + ¯ 1 ;
D0 = 0
E1 = [1 0 0];
E2 = [0 1 1];
F1 = [¯ 1 r1 ¯ 1 r3 la ]T ;
E3 = 03×3
F2 = 03×1
˜ y(k) = [¯ 1 r1 x(k) ¯ 1 r3 x(k) la x(k)]T F2 = 04×1
˜ u(k) = [¯ 1 r1 x(k)1 ¯ 1 r3 x(k)3 la x(k)a ]T
0 3. Robust stability analysis Since PID controller is widely used in the process industries, it is considered in the ensuing robust stability analysis. To do so, PID controller is represented by the following state space equation:
W(k + 1) = Ac W(k) + Bc e(k)
(15)
u(k) = Cc W(k) + Dc e(k)
(16)
where W(k) is a 2 × 1 state variable vector of PID controller, e(k) is the tracking error, i.e. the difference between the set-point and process output, and other model parameters are given by
1 0 1 Ac = ; Bc = 1 0 0 Cc =
− kc D kc D ; I
kc
Dc = kc +
kc
I
+ kc D
where kc , I , and D are the PID parameters. By using Eqs. (12)–(16), the uncertain closed-loop system under PID control as depicted in Fig. 2(a) is represented by the following
Fig. 5. Robust stability region (shadow) for the CSTR example.
1 0.8 x1
0.6 0.4 0.2 0 0
20
40
60
80
100
120
1 Set-point Process output
0.8 x1
0.6 0.4 0.2 0 0
20
(17)
40
60 Sample
80
100
120
Fig. 6. Set-point responses for the PI designs with I = 1.1 and (1) kc = 32.4 (top); (2) kc = 50 (bottom).
C. Cheng, M.-S. Chiu / Chemical Engineering Science 63 (2008) 5141 -- 5148
augmented state space equation: x(k) x(k + 1) ˜ = M11 + M12 u(k) W(k + 1) W(k) x(k) ˜ ˜ y(k) = M21 + M22 u(k) W(k)
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0.57 (18)
0.56
(19)
0.55 x1
where the matrices M11 , M12 , M21 , and M22 are given by A − B0 Dc C0 B0 Cc M11 = 0 −Bc C0 Ac E − B0 Dc E2 M12 = 1 −Bc E2
0.54 0.53 Set-point Process output
0.52
M21 = [F1 04×2 ] M22 = 04×4
0.51 0
10
20
30
40 50 Sample
60
70
80
0
10
20
30
40 50 Sample
60
70
80
0
10
20
30
40 50 Sample
60
70
80
0.88 0.875 0.87 0.865 x1
In Fig. 2(a), the delay operator z−1 can be considered as a perturbation bounded by one and therefore can be treated as an artificial uncertainty z with |z | 1, resulting in a repeated perturbation z I4 (I4 standing for a 4 × 4 identity matrix) in Fig. 2(b), where the overall uncertainty block is given by = diag[z I4 1 2 3 a ]. Consequently, the robust stability condition of this uncertain closed-loop system can be obtained by using the structured singular value test (Packard and Doyle, 1993) as follows: M11 M12 <1 (20) M21 M22 where denotes the structured singular value with perturbation structure . Lastly, when the composite model is constructed by a first-order ARX model and a PI controller is desired, Eq. (20) can be applied to by using the relevant model parameters provided at the end of last section and the following controller parameters, Ac = 1, Bc = 1, Cc = kc / I , and Dc = kc + kc / I , along with the corresponding perturbation block given by diag[z I2 1 3 a ].
0.86 0.855 0.85 0.845 0.84 0.835
4. Examples Example 1. Consider a CSTR described by the following equations (Doyle et al., 1989):
0.208 0.206
(21)
0.204
x˙ 2 = −x2 − b(x2 − xc ) + BDa(1 − x1 ) ex2 /(1+x2 / )
(22)
0.202
where x1 and x2 are the dimensionless concentration and temperature of the reactor, and xc (=u) is the cooling temperature chosen as manipulated variable while x1 (=y) is the controlled variable. The process has one stable steady state when Da = 0.072, B = 1, b = 0.3, and = 20. The following operating space xc ∈ [5 23] and x1 ∈ [0.1969 0.8781] is considered in this example.
0.2 x1
x˙ 1 = −x1 + Da(1 − x1 ) ex2 /(1+x2 / )
0.198 0.196 0.194 0.192
To construct the composite model, a first-order ARX model is adopted. By using the process input and output data obtained around the nominal operating condition, the parameters of the nominal ARX model are obtained as l,1 = 0.7216 and l,1 = 0.1231. To model the process nonlinearity by the JITL, a different set of input and output data is generated within the operating space as illustrated in Fig. 3, where yl is the predicted output by the nominal ARX model and y − yl is the modeling error caused by the process nonlinearity. To model the nonlinearity effect by the JITL with = 0.75, kmin = 10, and kmax = 60, the resulting model parameters are obtained as 1 ∈ [−0.3363, 0.0191] and 1 ∈ [−0.0217, 0.1112]. In addition, to quantify the modeling error of the composite model, the worst perturbation is calculated as la = 0.0514. Fig. 4 illustrates that the
0.19 0.188
Fig. 7. Closed-loop responses for −20% step disturbance at three operating conditions.
resulting composite model gives reasonably good prediction in the validation test by using input and output data different from that used in constructing the composite model.
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y
0 -0.05 -0.1 0
100
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400
500
600
700
800
0
100
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400
500
600
700
800
0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
yl
0 -0.05
y-yl
-0.1
5 0 -5 -10
x 10-3
u
0.05 0 -0.05 Sample Fig. 8. Input–output data used for the JITL.
0.01 0 -0.01 -0.02 y
-0.03 -0.04 -0.05 -0.06 Process Output Nominal ARX Model Composite Model
-0.07 -0.08 0
50
100 150 200 250 300 350 400 450 500 Sample
Fig. 9. Validation results for the distillation process.
Fig. 10. Robust stability region (shadow) for the distillation process.
To proceed with the proposed robust PI design, Eq. (20) is applied to obtain the PI parameters that guarantee the robust stability over the concerned operating space. Fig. 5 shows the resulting robust stability region in the kc − I parameter space. To verify the analysis result thus obtained, set-point changes covering the entire operating space for the viable PI design with kc = 32.4 and I = 1.1 are conducted. As can be seen from Fig. 6, this PI controller is able to maintain closed-loop stability over the entire operating space. For PI design with I = 1.1, the actual maximum allowable controller gain is found to be kc = 50 through the trial and error procedure via dynamic simulation studies. The set-point responses of this controller are also shown in Fig. 6 for comparison purpose. To illustrate the disturbance rejection capability of the PI controllers thus obtained, −20% step disturbance in the parameter b is considered. The control performances of the above PI design (kc = 32.4 and I = 1.1) at different operating conditions are illustrated in Fig. 7. It is clear that
this PI controller yields stable response at any operating condition within the pre-specified operating space. Lastly, it is noted that this example was previously studied by Knapp and Budman (2000) who developed a robust PID design method based on the state affine model. Compared with the viable PI designs reported in their paper, the proposed method gives less conservative result, for example, the maximum allowable kc obtained for I = 1.1 is 18 in their result, not to mention that tedious modeling procedure is required to build a state affine model. Example 2. Consider a distillation process described by (Gao et al., 2000): y(k) = 0.757y(k − 1) + 0.243g(u(k − 1))
(23)
g(x) = 1.04x − 14.11x2 − 16.72x3 + 562.7x4
(24)
C. Cheng, M.-S. Chiu / Chemical Engineering Science 63 (2008) 5141 -- 5148
systems. It is also known as instance-based learning (Aha et al., 1991), local weighted model (Atkeson et al., 1997), lazy learning (Aha, 1997; Bontempi et al., 2001), or model-on-demand (Braun et al., 2001) in the literature. The JITL has no standard learning phase because it merely stores the data in the database and the computation is not performed until a query data arrival by constructing a local model to approximate the process dynamics characterized by the current query data. The JITL algorithm adopted in this paper is now briefly reviewed in the sequel. Suppose that a database consisting of N process data (yi , xi )i=1∼N , yi ∈ R, xi ∈ Rn , is collected. Given a specific query data xq ∈ Rn , the objective of JITL is to predict the model output yˆ q = f (xq ) according to the known database (yi , xi )i=1∼N . To do so, the relevant data are selected from the database first by using the following similarity measure si (Cheng and Chiu, 2004):
0.02 Set-point Process output
0
y
-0.02 -0.04 -0.06 -0.08 -0.1 0
20
40
60
80
5147
100
2 si = e−d (xq ,xi ) + (1 − ) cos( i ) if cos( i ) 0
(25)
Sample Fig. 11. Set-point response for the PI design with kc = 1.24 and I = 2.5.
where the top column composition y(%) is the process output and the reflux flow rate u (mol/min) is the process input. The operating space under consideration is given by u ∈ [−0.05 0.01] and y ∈ [−0.0817 0.009]. Again, input and output data around the nominal operating condition are used to identify the nominal ARX model with parameters obtained as l,1 = 0.7686 and l,1 = 0.2183. To apply the JITL to model the process nonlinearity, a different set of input and output data as depicted in Fig. 8 is generated within the operating space. With the parameters of JITL chosen as = 0.95, kmin = 8, and kmax = 60, the resulting model parameters are obtained as 1 ∈ [−0.3750 0.0480] and 1 ∈ [−0.0133 0.6614]. In addition, the worst modeling error of the composite model is calculated as la = 0.0543. To illustrate the predictive performance of nominal ARX model and composite model, Fig. 9 shows that the composite model gives better prediction than the nominal ARX model over the entire operating space. Fig. 10 gives the viable PI designs in the kc − I parameter space that meet the robust stability condition, Eq. (20). To verify the result thus obtained, Fig. 11 shows the set-point responses of the PI controller with kc = 1.24 and I = 2.5. It is evident that this controller gives stable response for the set-point changes covering the entire operating space. 5. Conclusion A new methodology for robust PID controller design of nonlinear processes is developed. In the proposed method, a composite model is first constructed to model the process dynamics for the operating space of interest. This composite model consists of a nominal ARX model to capture linear process dynamics and the JITL to approximate the modeling error caused by process nonlinearity. The state space realizations of this composite model and PID controller are then reformulated as an uncertain closed-loop system, by which the corresponding robust stability condition is developed. Simulation results show that the proposed method can be used to design robust PID controller to ensure the closed-loop stability for the concerned operating space. Appendix A. Just-in-time learning The just-in-time learning (JITL) (Cybenko, 1996) technique was developed as an attractive alternative for modeling the nonlinear
where is a weight parameter and is constrained between 0 and 1, and i is the angle between xq and xi , where xq = xq − xq−1 and xi = xi − xi−1 . To apply JITL for the modeling of dynamic systems, all si are computed by Eq. (25) first and for each l ∈ [kmin kmax ], where kmin and kmax are the pre-specified minimum and maximum number of relevant data, the relevant data set (yl , l ), where yl ∈ Rl×1 and l ∈ Rl×n , are constructed by selecting l most relevant data (yi , xi ) corresponding to the largest si to the l-th largest si . Denote Wl ∈ Rl×l , a diagonal weight matrix with diagonal elements being the first l largest values of si . Next, the leave-one-out cross-validation test is conducted and the validation error is calculated. Upon the completion of the above procedure, the optimal l, l∗ , is determined by that giving the smallest validation error. Subsequently, the predicted output for query data is calculated as xqT (PlT∗ Pl∗ )−1 PlT∗ Wl∗ yl∗ , where Pl∗ = Wl∗ l∗ and Wl∗ is a diagonal matrix with entries being the first Wl ∈ Rl×l largest si , provided this optimal model satisfies the stability constraint; otherwise, an optimization procedure is carried out to recompute the optimal model.
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Robust PID controller design for nonlinear processes using JITL technique Cheng Cheng, Min-Sen Chiu ∗ Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576, Singapore
A R T I C L E
I N F O
Article history: Received 14 May 2007 Received in revised form 18 June 2008 Accepted 10 July 2008 Available online 15 July 2008 Keywords: Robust control Just-in-time learning Nonlinear system PID controller
A B S T R A C T
A robust PID controller design methodology for nonlinear processes is proposed based on the just-in-time learning (JITL) technique. To do so, a composite model consisting of a nominal ARX model and the JITL, where the former is used to capture linear process dynamics and the latter to approximate the inevitable modeling error caused by the process nonlinearity, is employed to model the process dynamics in the operating space of interest. The state space realizations of this composite model and PID controller are then reformulated as an uncertain closed-loop system, by which the corresponding robust stability condition is developed. Literature examples are employed to illustrate the proposed methodology and a comparison with the previous result is made. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction For most chemical processes, the first-principle models are usually unavailable because of the lacking of physicochemical knowledge. An attractive alternative for controller design is to rely on the models extracted from process input and output measurements. These models generally have varying degrees of accuracy. If the plant/model mismatch is not taken into account in controller design, the resulting control performance may become poor or even unstable in the presence of significant modeling error. This design problem has motivated the researchers to pursue various robust controller design methods in the last two decades (Morari and Zafiriou, 1989; Malan et al., 2004). The objective of robust controller design is to ensure closed-loop stability and maintain control performance not only for the nominal model but also for a set of possible process models that captures the actual process dynamics. For linear system, various robust control problems had been tackled by using the transfer function model approaches (Morari and Zafiriou, 1989; Packard and Doyle, 1993). Normally, a given set of process models is represented by a nominal model together with a suitable uncertainty description equation to account for the modeling error between the nominal model and actual process dynamics. The associated design issue of estimating the uncertainty models also attracted much research investigations (Gustafsson and Makila, 2001; Boling et al., 2004). Some robustness analysis results developed for linear systems have also been applied to the nonlinear systems. For example, Doyle et al. (1989) proposed a robust controller design method for a
∗
Corresponding author. Tel.: +65 6516 2223; fax: +65 6779 1936. E-mail address: [email protected] (M.-S. Chiu).
0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.07.013
nonlinear CSTR for which a first-principle model is assumed to be available. By assuming that the process/model mismatch is entirely due to the nonlinearities of the process, bounds on the conic sectors that describe the process nonlinearities were developed and used in the standard M − structure for robust stability analysis. However, the identification of the conic bounds is not trivial and the resulting robust stability analysis tends to give conservative result, not to mention that first-principle models are generally not available for many chemical processes. To lessen the modeling requirement, Knapp and Budman (2000) developed a robust PID controller design methodology for nonlinear processes using an empirical state affine model developed from the available process data, which can be readily transformed into a suitable form for the robust stability analysis. Although their result provides an attractive alternative to the Doyle's approach, the construction of the state affine models is rather tedious and computational demanding. According to Knapp and Budman (2000), in order to obtain a state affine model, a NARMA model is initially constructed from the available process input and output data. Subsequently, an algorithm developed by Diaz and Desrochers (1988) is employed to find the parameters for a truncated Volterra model based on the NARMA model identified previously. Once the Volterra kernels are obtained, a generalized Hankel matrix can be developed to find a state affine model (Sontag, 1979). Obviously, the modeling efforts required to identify a state affine model are extensive and thus hampers the application of robust controller design method developed based on such a model. To circumvent the aforementioned drawbacks by using the conic sectors and state affine models in robust controller designs for nonlinear systems, a robust PID controller design methodology using the just-in-time learning (JITL) technique (Cybenko, 1996; Atkeson et al., 1997; Bontempi et al., 2001) is developed in this paper. It is noted
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that robust PID controller design methods have been developed using different techniques in the literature, for example linear matrix inequalities (LMIs) approach in the multiple-model modeling framework (Ge et al., 2002; Toscano, 2007) and numerical optimization approach by solving the maximization of the shortest distance from the Nyquist curve of the open-loop transfer function to the critical point (Toscano, 2005). However, these methods require a priori process knowledge or ad-hoc procedure to address the associated design issue of partition of the operating space in the multiple-model approach and determine the worst-case model (or optimal model for that matter) in the operating space of interest in order to obtain the best viable PID design in the numerical optimization approach. In contrast, the proposed design method is a one-shot design approach in the sense that the PID parameters are obtained directly from a set of process data characterizing the process dynamics of interest. In this respect, the proposed method is more advantageous than the previous model-based robust PID design methods because the required a priori process information may not be readily available in practical applications, leading to the tedious trial and error design procedure. In the proposed method, it is assumed that the process nonlinearity is the only source of the model uncertainty. A composite model consisting of a nominal ARX model and the JITL, where the former is used to capture linear process dynamics and the latter is applied to approximate the inevitable modeling error caused by the process nonlinearity. The state space realization of this composite model is then reformulated as an uncertain system, by which the robust stability condition of this uncertain system under PID control is developed. Literature examples are used to illustrate the proposed method and a comparison with the previous result is made. 2. Modeling methodology In this paper, a composite model consisting of a nominal ARX model and JITL models is used to describe the nonlinear process in the operating range of interest, where the former can be identified by using the process input and output data around a nominal operating condition and the latter is used to capture the modeling error caused by the process nonlinearity, i.e. the difference between the predicted output of nominal ARX model and actual process output. Suppose that an input sequence {u(k)} is injected into the process and the corresponding output sequence {y(k)} is measured. The following composite model is then used to model the nonlinear dynamics between {u(k)} and {y(k)}: yˆ (k) = yl (k) + ynl (k)
(1)
where yˆ (k) is the output of the composite model, yl (k) is the predicted output of nominal ARX model, and ynl (k) is the effect of process nonlinearity, i.e. y(k)−yl (k), to be approximated by the JITL technique (see Appendix A). Because JITL normally employs a first-order or second-order ARX model, we shall use a second-order model structure for both nominal ARX model and JITL in the subsequent developments. Therefore, yl (k) and ynl (k) are represented as follows:
z−1 x(k + 1)
x(k)
u(k)
A0
B0
E1
C0
D0
E2
F1
F2
E3
y(k)
~ y(k)
~ u(k)
1 2 3 a
Fig. 1. The interconnection structure for the composite model described by Eqs. (8) and (10).
where x(k) = [x1 (k) x2 (k)]T and + 1 l,2 + 2 A = l,1 1 0 B0 = [1 0]T ;
C = [l,1 + 1 0];
D0 = 0
Given the process input and output data {u(k), y(k)}, the plant/model mismatch caused by the nonlinearity, i.e. y(k) − yl (k), can be calculated after the nominal ARX model given in Eq. (2) is identified. Subsequently, the JITL technique can be applied to model the dynamics between the input sequence {u(k)} and the sequence {y(k) − yl (k)} by using the reference dataset constructed from {u(k), y(k)}. Upon the successful implementation of JITL algorithm, the respective ranges of variation for model coefficients in Eq. (3) are denoted by 1 ∈ [1,min 1,max ], 2 ∈ [2,min 2,max ], and 1 ∈ [1,min 1,max ], which can be represented by the following equations:
1 = ¯ 1 (1 + r1 1 ); |1 | 1 2 = ¯ 2 (1 + r2 2 ); |2 | 1 1 = ¯ 1 (1 + r3 3 ); |3 | 1
(6)
where i (i = 1–3) is the uncertainty bounded by one and other parameters are defined as follows:
1,max − 1,min 1,min + 1,max 2,max − 2,min r2 = 2,min + 2,max + 1,max − 1,min 1,max ; r3 = ¯ 1 = 1,min 2 1,min + 1,max ¯ 1 =
1,min + 1,max
; 2 + 2,max ; ¯ 2 = 2,min 2
r1 =
yl (k) = l,1 y(k − 1) + l,2 y(k − 2) + l,1 u(k − 1)
(2)
By using Eqs. (6) and (7), Eqs. (4) and (5) can be rewritten as A1 A2 x(k + 1) = A0 + 1 + x(k) + B0 u(k) 01×2 01×2 2
ynl (k) = 1 y(k − 1) + 2 y(k − 2) + 1 u(k − 1)
(3)
yˆ (k) = (C0 + C1 3 )x(k) + D0 u(k)
Consider the following state space realization of the composite model given in Eqs. (1)–(3):
where 0n×m denote a n × m zero matrix and + ¯ 1 l,2 + ¯ 2 A0 = l,1 1 0
x(k + 1) = Ax(k) + B0 u(k)
(4)
A1 = [¯ 1 r1 0];
yˆ (k) = Cx(k) + D0 u(k)
(5)
C0 = [l,1 + ¯ 1 0];
A2 = [0 ¯ 2 r2 ] C1 = [¯ 1 r3 0]
(7)
(8) (9)
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z−1 x(k)
x(k + 1)
Ψ(k)
Ψ(k + 1) M11
M12
M21
M22
To account for the modeling error resulting from the approximation of the nonlinear process by the proposed composite model, an additive uncertainty a is appended to the model output yˆ (k) as follows: y(k) = yˆ (k) + la x1 (k)a ,
|a | 1
(10)
where la is the magnitude of the worst perturbation as calculated by y(k) − yˆ (k) (11) la = Max ˆ (k)/(l,1 + 1 ) k y
~ y(k)
~ (k) u
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1 2
3
1
a
0.9
M
x(k) Ψ(k) ~ u(k)
M12
M21
M22
0.7 y
M11
0.8
x(k + 1) Ψ(k + 1) ~ y (k)
0.6 0.5 0.4
zI4
Process output ARX model Composite model
0.3
1 2
0.2
3
0
50
100
a
150
Fig. 4. Validation result for the CSTR example.
Fig. 2. The interconnection structure for the uncertain closed-loop system.
y
1 0.5 0 0
50
100
150
200
250
300
0
50
100
150
200
250
300
0
50
100
150
200
250
300
0
50
100
150
200
250
300
yl
1 0.5 0
y-yl
0.2 0 -0.2
30 u
200
Sample
15 0 Sample Fig. 3. Input–output data used for the JITL.
250
300
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It is noted from Eq. (9) that the state variable x2 (k) has no effect on the model output yˆ (k), which explains why the additive uncertainty a in Eq. (10) is only associated with the state variable x1 (k). After some algebraic manipulation, the composite model by including the uncertainty a , i.e. Eqs. (8) and (10), can be recast into the interconnection structure as depicted in Fig. 1, where ˜ x(k + 1) = A0 x(k) + B0 u(k) + E1 u(k)
(12)
˜ y(k) = C0 x(k) + D0 u(k) + E2 u(k)
(13)
˜ ˜ y(k) = F1 x(k) + F2 u(k) + E3 u(k)
(14)
1 1 0 0 E1 = ; 0 0 0 0 ⎡
¯ 1 r1 ⎢ 0 F1 = ⎢ ⎣ ¯ r
1 3 la
⎤ 0 ¯ 2 r2 ⎥ ⎥; 0 ⎦
E2 = [0 0 1 1];
E3 = 04×4
˜ y(k) = [¯ 1 r1 x1 (k) ¯ 2 r2 x2 (k) ¯ 1 r3 x1 (k) la x1 (k)]T ˜ u(k) = [¯ 1 r1 x1 (k)1 ¯ 2 r2 x2 (k)2 ¯ 1 r3 x1 (k)3 la x1 (k)a ]T Lastly, when a first-order ARX model is employed for both nominal model and the JITL, the corresponding interconnection structure is given by A0 = l,1 + ¯ 1 ;
B0 = 1
C0 = l,1 + ¯ 1 ;
D0 = 0
E1 = [1 0 0];
E2 = [0 1 1];
F1 = [¯ 1 r1 ¯ 1 r3 la ]T ;
E3 = 03×3
F2 = 03×1
˜ y(k) = [¯ 1 r1 x(k) ¯ 1 r3 x(k) la x(k)]T F2 = 04×1
˜ u(k) = [¯ 1 r1 x(k)1 ¯ 1 r3 x(k)3 la x(k)a ]T
0 3. Robust stability analysis Since PID controller is widely used in the process industries, it is considered in the ensuing robust stability analysis. To do so, PID controller is represented by the following state space equation:
W(k + 1) = Ac W(k) + Bc e(k)
(15)
u(k) = Cc W(k) + Dc e(k)
(16)
where W(k) is a 2 × 1 state variable vector of PID controller, e(k) is the tracking error, i.e. the difference between the set-point and process output, and other model parameters are given by
1 0 1 Ac = ; Bc = 1 0 0 Cc =
− kc D kc D ; I
kc
Dc = kc +
kc
I
+ kc D
where kc , I , and D are the PID parameters. By using Eqs. (12)–(16), the uncertain closed-loop system under PID control as depicted in Fig. 2(a) is represented by the following
Fig. 5. Robust stability region (shadow) for the CSTR example.
1 0.8 x1
0.6 0.4 0.2 0 0
20
40
60
80
100
120
1 Set-point Process output
0.8 x1
0.6 0.4 0.2 0 0
20
(17)
40
60 Sample
80
100
120
Fig. 6. Set-point responses for the PI designs with I = 1.1 and (1) kc = 32.4 (top); (2) kc = 50 (bottom).
C. Cheng, M.-S. Chiu / Chemical Engineering Science 63 (2008) 5141 -- 5148
augmented state space equation: x(k) x(k + 1) ˜ = M11 + M12 u(k) W(k + 1) W(k) x(k) ˜ ˜ y(k) = M21 + M22 u(k) W(k)
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0.57 (18)
0.56
(19)
0.55 x1
where the matrices M11 , M12 , M21 , and M22 are given by A − B0 Dc C0 B0 Cc M11 = 0 −Bc C0 Ac E − B0 Dc E2 M12 = 1 −Bc E2
0.54 0.53 Set-point Process output
0.52
M21 = [F1 04×2 ] M22 = 04×4
0.51 0
10
20
30
40 50 Sample
60
70
80
0
10
20
30
40 50 Sample
60
70
80
0
10
20
30
40 50 Sample
60
70
80
0.88 0.875 0.87 0.865 x1
In Fig. 2(a), the delay operator z−1 can be considered as a perturbation bounded by one and therefore can be treated as an artificial uncertainty z with |z | 1, resulting in a repeated perturbation z I4 (I4 standing for a 4 × 4 identity matrix) in Fig. 2(b), where the overall uncertainty block is given by = diag[z I4 1 2 3 a ]. Consequently, the robust stability condition of this uncertain closed-loop system can be obtained by using the structured singular value test (Packard and Doyle, 1993) as follows: M11 M12 <1 (20) M21 M22 where denotes the structured singular value with perturbation structure . Lastly, when the composite model is constructed by a first-order ARX model and a PI controller is desired, Eq. (20) can be applied to by using the relevant model parameters provided at the end of last section and the following controller parameters, Ac = 1, Bc = 1, Cc = kc / I , and Dc = kc + kc / I , along with the corresponding perturbation block given by diag[z I2 1 3 a ].
0.86 0.855 0.85 0.845 0.84 0.835
4. Examples Example 1. Consider a CSTR described by the following equations (Doyle et al., 1989):
0.208 0.206
(21)
0.204
x˙ 2 = −x2 − b(x2 − xc ) + BDa(1 − x1 ) ex2 /(1+x2 / )
(22)
0.202
where x1 and x2 are the dimensionless concentration and temperature of the reactor, and xc (=u) is the cooling temperature chosen as manipulated variable while x1 (=y) is the controlled variable. The process has one stable steady state when Da = 0.072, B = 1, b = 0.3, and = 20. The following operating space xc ∈ [5 23] and x1 ∈ [0.1969 0.8781] is considered in this example.
0.2 x1
x˙ 1 = −x1 + Da(1 − x1 ) ex2 /(1+x2 / )
0.198 0.196 0.194 0.192
To construct the composite model, a first-order ARX model is adopted. By using the process input and output data obtained around the nominal operating condition, the parameters of the nominal ARX model are obtained as l,1 = 0.7216 and l,1 = 0.1231. To model the process nonlinearity by the JITL, a different set of input and output data is generated within the operating space as illustrated in Fig. 3, where yl is the predicted output by the nominal ARX model and y − yl is the modeling error caused by the process nonlinearity. To model the nonlinearity effect by the JITL with = 0.75, kmin = 10, and kmax = 60, the resulting model parameters are obtained as 1 ∈ [−0.3363, 0.0191] and 1 ∈ [−0.0217, 0.1112]. In addition, to quantify the modeling error of the composite model, the worst perturbation is calculated as la = 0.0514. Fig. 4 illustrates that the
0.19 0.188
Fig. 7. Closed-loop responses for −20% step disturbance at three operating conditions.
resulting composite model gives reasonably good prediction in the validation test by using input and output data different from that used in constructing the composite model.
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y
0 -0.05 -0.1 0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
0
100
200
300
400
500
600
700
800
yl
0 -0.05
y-yl
-0.1
5 0 -5 -10
x 10-3
u
0.05 0 -0.05 Sample Fig. 8. Input–output data used for the JITL.
0.01 0 -0.01 -0.02 y
-0.03 -0.04 -0.05 -0.06 Process Output Nominal ARX Model Composite Model
-0.07 -0.08 0
50
100 150 200 250 300 350 400 450 500 Sample
Fig. 9. Validation results for the distillation process.
Fig. 10. Robust stability region (shadow) for the distillation process.
To proceed with the proposed robust PI design, Eq. (20) is applied to obtain the PI parameters that guarantee the robust stability over the concerned operating space. Fig. 5 shows the resulting robust stability region in the kc − I parameter space. To verify the analysis result thus obtained, set-point changes covering the entire operating space for the viable PI design with kc = 32.4 and I = 1.1 are conducted. As can be seen from Fig. 6, this PI controller is able to maintain closed-loop stability over the entire operating space. For PI design with I = 1.1, the actual maximum allowable controller gain is found to be kc = 50 through the trial and error procedure via dynamic simulation studies. The set-point responses of this controller are also shown in Fig. 6 for comparison purpose. To illustrate the disturbance rejection capability of the PI controllers thus obtained, −20% step disturbance in the parameter b is considered. The control performances of the above PI design (kc = 32.4 and I = 1.1) at different operating conditions are illustrated in Fig. 7. It is clear that
this PI controller yields stable response at any operating condition within the pre-specified operating space. Lastly, it is noted that this example was previously studied by Knapp and Budman (2000) who developed a robust PID design method based on the state affine model. Compared with the viable PI designs reported in their paper, the proposed method gives less conservative result, for example, the maximum allowable kc obtained for I = 1.1 is 18 in their result, not to mention that tedious modeling procedure is required to build a state affine model. Example 2. Consider a distillation process described by (Gao et al., 2000): y(k) = 0.757y(k − 1) + 0.243g(u(k − 1))
(23)
g(x) = 1.04x − 14.11x2 − 16.72x3 + 562.7x4
(24)
C. Cheng, M.-S. Chiu / Chemical Engineering Science 63 (2008) 5141 -- 5148
systems. It is also known as instance-based learning (Aha et al., 1991), local weighted model (Atkeson et al., 1997), lazy learning (Aha, 1997; Bontempi et al., 2001), or model-on-demand (Braun et al., 2001) in the literature. The JITL has no standard learning phase because it merely stores the data in the database and the computation is not performed until a query data arrival by constructing a local model to approximate the process dynamics characterized by the current query data. The JITL algorithm adopted in this paper is now briefly reviewed in the sequel. Suppose that a database consisting of N process data (yi , xi )i=1∼N , yi ∈ R, xi ∈ Rn , is collected. Given a specific query data xq ∈ Rn , the objective of JITL is to predict the model output yˆ q = f (xq ) according to the known database (yi , xi )i=1∼N . To do so, the relevant data are selected from the database first by using the following similarity measure si (Cheng and Chiu, 2004):
0.02 Set-point Process output
0
y
-0.02 -0.04 -0.06 -0.08 -0.1 0
20
40
60
80
5147
100
2 si = e−d (xq ,xi ) + (1 − ) cos( i ) if cos( i ) 0
(25)
Sample Fig. 11. Set-point response for the PI design with kc = 1.24 and I = 2.5.
where the top column composition y(%) is the process output and the reflux flow rate u (mol/min) is the process input. The operating space under consideration is given by u ∈ [−0.05 0.01] and y ∈ [−0.0817 0.009]. Again, input and output data around the nominal operating condition are used to identify the nominal ARX model with parameters obtained as l,1 = 0.7686 and l,1 = 0.2183. To apply the JITL to model the process nonlinearity, a different set of input and output data as depicted in Fig. 8 is generated within the operating space. With the parameters of JITL chosen as = 0.95, kmin = 8, and kmax = 60, the resulting model parameters are obtained as 1 ∈ [−0.3750 0.0480] and 1 ∈ [−0.0133 0.6614]. In addition, the worst modeling error of the composite model is calculated as la = 0.0543. To illustrate the predictive performance of nominal ARX model and composite model, Fig. 9 shows that the composite model gives better prediction than the nominal ARX model over the entire operating space. Fig. 10 gives the viable PI designs in the kc − I parameter space that meet the robust stability condition, Eq. (20). To verify the result thus obtained, Fig. 11 shows the set-point responses of the PI controller with kc = 1.24 and I = 2.5. It is evident that this controller gives stable response for the set-point changes covering the entire operating space. 5. Conclusion A new methodology for robust PID controller design of nonlinear processes is developed. In the proposed method, a composite model is first constructed to model the process dynamics for the operating space of interest. This composite model consists of a nominal ARX model to capture linear process dynamics and the JITL to approximate the modeling error caused by process nonlinearity. The state space realizations of this composite model and PID controller are then reformulated as an uncertain closed-loop system, by which the corresponding robust stability condition is developed. Simulation results show that the proposed method can be used to design robust PID controller to ensure the closed-loop stability for the concerned operating space. Appendix A. Just-in-time learning The just-in-time learning (JITL) (Cybenko, 1996) technique was developed as an attractive alternative for modeling the nonlinear
where is a weight parameter and is constrained between 0 and 1, and i is the angle between xq and xi , where xq = xq − xq−1 and xi = xi − xi−1 . To apply JITL for the modeling of dynamic systems, all si are computed by Eq. (25) first and for each l ∈ [kmin kmax ], where kmin and kmax are the pre-specified minimum and maximum number of relevant data, the relevant data set (yl , l ), where yl ∈ Rl×1 and l ∈ Rl×n , are constructed by selecting l most relevant data (yi , xi ) corresponding to the largest si to the l-th largest si . Denote Wl ∈ Rl×l , a diagonal weight matrix with diagonal elements being the first l largest values of si . Next, the leave-one-out cross-validation test is conducted and the validation error is calculated. Upon the completion of the above procedure, the optimal l, l∗ , is determined by that giving the smallest validation error. Subsequently, the predicted output for query data is calculated as xqT (PlT∗ Pl∗ )−1 PlT∗ Wl∗ yl∗ , where Pl∗ = Wl∗ l∗ and Wl∗ is a diagonal matrix with entries being the first Wl ∈ Rl×l largest si , provided this optimal model satisfies the stability constraint; otherwise, an optimization procedure is carried out to recompute the optimal model.
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