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Ind. Eng. Chem. Res. 2010, 49, 11572–11581
Step Response Identification under Inherent-Type Load Disturbance with Application to Injection Molding Tao Liu, Feng Zhou, Yi Yang, and Furong Gao* Department of Chemical and Biomolecular Engineering, Hong Kong UniVersity of Science & Technology, Clear Water Bay, Kowloon, Hong Kong
Motivated by autotuning of injection velocity in an industrial injection molding machine, a step response identification method is proposed for practical application subject to inherent-type load disturbance. Both models of the process and the inherent-type load disturbance can be simultaneously derived from a step test. Identification algorithms are detailed for obtaining the widely used low-order models of first-order plus dead time (FOPDT) and second-order plus dead time (SOPDT). Based on the identified process and disturbance models, a two-degree-of-freedom (2DOF) internal model control (IMC) structure plus feed-forward control is proposed for improving load disturbance rejection. Analytical controller design formulas and tuning guidelines are developed accordingly. A practical application to the velocity control of an injection molding machine is shown to illustrate the effectiveness and merits of the proposed identification method and control strategy. 1. Introduction Step response tests have been widely applied in the process industries for control-oriented model identification.1-3 Since the earlier developments of fitting several representative points in the process transient response to a step change of the set point4-7 or using the graphical area ratio of the output response to the step input,2 a number of enhanced identification methods have been reported in the recent literature. Bi et al.8 proposed a leastsquares- (LS-) based identification algorithm to improve fitting over the low-frequency range in terms of the most widely used low-order model structure of first-order-plus-dead-time (FOPDT) model, which was further extended to obtain a second-orderplus-dead-time (SOPDT) or higher-order model.9,10 An alternative step response identification method11 was developed in the frequency domain, by introducing a damping factor for the Laplace transform. Using polygonal curve approximation for data preprocessing, Piroddi and Leva12 proposed a step response classification method to determine the process dynamic characteristics and then choose a suitable model structure. A criterion named “stability rate” was suggested by Garnier et al.13 to evaluate the performance of an identification algorithm in terms of Monte Carlo tests. Recent surveys on the robust identification of process models from experimental data can be found in refs 14 and 15. Because step response tests in engineering practice are likely to be subject to load disturbance or unsteady initial process conditions, a number of robust step response identification methods have recently been developed to address this problem. By defining initial states of the process output and its derivatives as part of the parameters to be identified while assuming no presence of a load disturbance, Ahmed et al.16,17 developed two robust identification algorithms, one based on an iterative procedure combined with a linear filtering method and the other based on evolving multiple fitting conditions to establish a linear LS solution. Liu et al.18 suggested the use of multiple piecewise step tests for identification under nonzero initial process conditions or load disturbance with slow dynamics, and by comparison, Wang et al.19 developed an alternative algorithm to improve identification robustness in the presence of unknown * To whom correspondence should be addressed. Tel.: +852-23587139. Fax: +852-2358-0054. E-mail:
[email protected].
initial process conditions and static disturbances. By using the transient response data from adding and subsequently removing a step change to the process input, a robust identification algorithm20 was proposed to construct independent LS regression for unbiased parameter estimation against unexpected load disturbance. Note that, for the presence of a repetitive load disturbance, which is commonly encountered in various industrial batch process operations that are periodically initiated by a step change to the set point, the resulting step response can be viewed as a pure process response plus a load disturbance response, according to the linear superposition principle. Such a load disturbance is herein called an inherent-type load disturbance. Modeling only the pure process response might not be sufficient for describing the overall dynamic response characteristics for control system design or controller tuning. For instance, a water pump in an air conditioning system obviously gives different step responses under different loads. Modeling both the pure pump response to the set point without a load and the disturbance response of the load can facilitate control design for pump operation at a variety of load levels. For velocity control of an industrial injection molding machine,21 the open-loop response of the injection velocity during the filling process will gradually decrease after a step change of the valve opening, because of the presence of mold cavity pressure that gradually increases until the end of mold filling. Modeling the influence caused by the mold cavity pressure can facilitate advanced control design for maintaining the injection velocity during the filling process to guarantee product consistency and quality. To facilitate step response identification subject to an inherenttype load disturbance, as encountered in industrial injection molding processes,21,22 a piecewise model identification method is proposed herein for practical application. Both the process model and the inherent-type load disturbance model can be simultaneously derived from a step test. Correspondingly, a control scheme consisting of a two-degree-of-freedom (2DOF) internal model control (IMC) structure plus feed-forward control is proposed for improving process operation against such a load disturbance. A practical application to the injection velocity control of an industrial injection molding machine is presented to demonstrate the effectiveness of the proposed identification algorithms and control strategy. This article is organized as
10.1021/ie1015427 2010 American Chemical Society Published on Web 10/15/2010
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Figure 2. Block diagram of proposed control scheme for inherent-type disturbance rejection.
Figure 1. Illustration of step response test under nonzero initial conditions and load disturbance.
follows: Section 2 presents some guidelines on the choice of step response data for model identification. In section 3, identification algorithms are detailed for obtaining low-order process and disturbance models of FOPDT and SOPDT types. These algorithms can be applied with nonzero initial process conditions. Subsequently, a 2DOF IMC plus feed-forward control scheme is proposed in section 4 for practical applications subject to inherent-type load disturbance. Illustrative examples from the recent literature are presented in section 5 to demonstrate the effectiveness and merits of the proposed identification method and control scheme. A practical application to the injection velocity control of an injection molding machine is described in section 6. Finally, conclusions are drawn in section 7. 2. Step Response Test It is well-known that only the process transient response to a step change is useful for model identification. To effectively extract such data from a step test, it is suggested that the process output trend be monitored before the step test. If the initial process output, y(t0), has a decreasing trend, that is, y˙(t0) < 0, a positive step change to the input (u) is suggested, as shown in Figure 1, so that an obvious turning point in the output response can be observed and referenced to take the first point (t1) of the step response data used for identification. Accordingly, if the initial process output has an increasing trend, a negative step change should be introduced. If a clear trend in the initial process output cannot be observed, particularly for the presence of a high noise level, either a positive or a negative step change to the process input can be used based on observations/knowledge of the process response characteristics; the magnitude of this change should be set reasonably large to yield an admissible fluctuation range of the process output around its operating level. That is, the step response should be conducted within a workable range of the process output for model identification. For the presence of an inherent-type load disturbance, as illustrated in Figure 1, the corresponding turning point in the output response can be clearly observed, and it should therefore be referenced to take the starting point (td) of the load disturbance response for identification. Practically, it is suggested that td be taken slightly earlier than the time of the observed turning point to ensure identification effectiveness against measurement noise, in consideration of the time delay usually associated with process response. Hence, the step response data in the time interval [t1, td) can be used to identify a model of the pure process response without the influence of the inherenttype load disturbance, and correspondingly, the step response data in the time interval [td, tN] from which the obtained process model response has been subtracted can be used to identify the load disturbance model, where tN can be taken roughly after the output response has recovered to a steady state.
For the case where the inherent-type load disturbance occurs at a very early stage of a step test, a reasonable division of the observed transient response data is needed for separate identification of the pure process response model and the load disturbance model to facilitate control design. That is, the number of transient response data points (M) chosen in terms of the obvious turning point in the output response should be large enough for identification of the pure process model. In general, it is suggested that M be at least 2 times larger than the number of parameters to be estimated in such an LS fitting algorithm to guarantee effectiveness of the model identification, as will be illustrated by a later experimental example. To ensure a sufficient number of transient response data, a compromise can be made between the two segments of data chosen for identification of the pure process model and the inherent-type load disturbance model. For the worst case, where the obvious turning point appears at the very early stage of a step test, it has been revealed20 that the standard step test cannot allow for independent identification of the process model against the influence from load disturbance, and therefore, a modified step test can be performed to apply a robust identification algorithm20 for identification of the pure process model. Then, the inherenttype load disturbance model can be identified by using the piecewise identification method presented in the next section. For the cases where the overall transient process response to a step change can be effectively described by a single model structure or the modeling aims at controller tuning after the inherent-type disturbance response has become steady, the identification effort might be correspondingly reduced to determining only a single low-order model for describing the dynamic response characteristics of interest in control design. In other words, there is no need to choose td any longer for model identification. It should be noted that, to obtain a good tradeoff between identification accuracy and computation efficiency for the use of LS fitting for parameter estimation, the number of transient response data (M) for identification of the pure process model or the load disturbance model is generally suggested to be taken in the range of 50-200. This guideline will be illustrated in the later examples. 3. Identification Algorithms The process output response to the set point and the inherenttype load disturbance can be generally expressed as Y(s) ) G(s) U(s) + Gd(s) Ud(s)
(1)
where G(s) and Gd(s) denote the process model and the disturbance model, respectively, and U(s) and Ud(s) denote the process input and the inherent-type load disturbance, respectively. For convenience of model identification, Ud(s) is herein normalized as a unity step signal, as shown in Figure 2, and correspondingly, the gain of Gd(s) reflects the magnitude of the inherent-type load disturbance.
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In view of the fact that low-order FOPDT and SOPDT process models are most widely employed for control system design, the following model structures are studied here kp
G(s) )
a2s2 + a1s + 1
Gd(s) )
-θs
e
kd e-θds τd s + 1
∫∫ t
0
τm-1
∫
···
0
τ1
0
f(τ0) dτ0 · · · dτm-1 )
∫
(m)
[0,t]
f(t), m g 2 (9)
It can be derived for t > θ that
(2)
∫
(2)
[0,t]
∫
(3)
(3)
[0,t]
u(t) )
h + ∆h 2 ∆h 2 t - ∆htθ + θ 2 2
(10)
h + ∆h 3 ∆h 2 ∆h 2 ∆h 3 t tθ+ tθ θ 6 2 2 6
u(t) )
(11)
where kp is the process gain, θ is the process time delay, a1 and a2 are positive coefficients to reflect the process dynamic characteristics and kd is the magnitude of the inherent-type load disturbance, θd is the time delay of the disturbance response, and τd is the time constant of the disturbance response. Note that the time delay in eq 3 can be decomposed as θd ) θ˜ d + td, where td is taken in terms of the observed turning point of the process response caused by the inherent-type load disturbance. By using a time shift of td (i.e., letting td ) 0), θ˜ d can be separately derived for model fitting of the load disturbance response in the time interval [td, tN]. Using the linear superposition principle, the time-domain response of eq 1 can be decomposed as y ) y r + yd
(4)
a2y¨r(t) + a1y˙r(t) + yr(t) ) kpu(t - θ)
(5)
∫
(2)
[0,t]
∫
(3)
[0,t]
∫
(3)
[0,t]
y˙(t) )
t
(12)
0
∫
y˙(t) )
y¨(t) )
∫ y(t) - y(0)t (2)
[0,t]
1 y(t) - y(0)t2 2
∫ y(t) - y(0)t - 21 y˙(0)t t
(13) 2
(14)
0
Note that y(t) ) yr(t) for t < td, as shown in Figure 1. With the input description of eq 7, the time-domain response shown in eq 5 can be equivalently expressed as a2y¨r(t) + a1y˙r(t) + yr(t) ) kpu(t)
(15)
By triply integrating both sides of eq 15 and rearranging the resulting equation using eqs 11, 13, and 14, we obtain ψ(t) ) φT(t)γ
τdy˙d(t) + yd(t) ) kdud(t - θd)
(6)
where yr denotes the pure process response to the input change and yd is the inherent-type load disturbance response. Note that, by letting a2 ) 0, the expression of eq 2 is reduced to an FOPDT model similar to eq 3. Hence, the following development of identification algorithms is detailed for obtaining an SOPDT model and then is briefly summarized for obtaining an FOPDT model. To allow for practical identification with unsteady or nonzero initial process conditions, the initial process state for a step test is herein considered as u(t0) ) h, h ∈ R, and y(t0) * 0, which is illustrated in Figure 1, as encountered in injection molding processes.21,22 Note that, u(t0) ) 0 was assumed in the nonzero initial conditions studied in recent step response identification articles,17-19 which can be viewed as a zero-input (or normalized zero-input) case associated with a nonzero initial transient output response that might be unknown before a step test. By comparison, we change the initial process conditions to facilitate practical applications with raw step response data. 3.1. SOPDT Model. With initial process conditions as described above, by using a time shift of t0 (i.e., letting t0 ) 0), one can express the process input under a step test as u(t) )
{
h, 0et θ, it can be derived that
∫ u(t) dt ) ∫ t
0
θ
0
h dt +
∫ (h + ∆h) dt ) (h + ∆h)t - ∆hθ t
θ
(8) Denote multiple integrals for a time function of f(t) as
(16)
where
∫ yt φ(t) ) [- ∫ y t , - ∫
ψ(t) )
(3)
()
[0,t]
t
( 2)
()
0
[0,t]
T
]
y(t), (h + ∆h)t3 /6, t2 /2, t/2, -∆h/6
γ ) [a2, a1, kp, -∆hkpθ + a2y˙(0) + a1y(0), ∆hkpθ2 + 2a2y(0), kpθ3]T
(17) Hence, using the step response data in the time interval [t1, td), as shown in Figure 1 (i.e., t0 < t1 < t2 < · · · < tM < td) and letting Ψ ) [ψ(t1) ψ(t2) · · · ψ(tM) ]T and Φ ) [φ(t1) φ(t2) · · · φ(tM) ]T we obtain a linear LS algorithm for parameter estimation Ψ ) Φγ
(18)
Accordingly, the parameter vector can be solved as γ ) (ΦTΦ)-1ΦTΨ
(19)
Note that the first two columns of Φ, vectors of single and double integrals for the process output, are obviously independent of the other columns of Φ, and the third to fifth columns of Φ are all time vectors with exponents of different indexes, so they are linearly independent of each other. Therefore, Φ is guaranteed to be full column rank. Owing to the matrix property rank(ΦTΦ) ) rank(Φ), we ensure that ΦTΦ is invertible for the computation of eq 19, leading to a unique solution of γ. Subsequently, the model parameters can be retrieved from eq 19 as
[] [ ] γ(1) a2 γ(2) a1 ) γ(3) kp 3 θ √γ(6)/γ(3)
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lim (Z Φ)/M
Mf∞
(20)
Mf∞
where
γ(5) - ∆hkpθ2 2a2
(21)
υ ) [ν(t1) ν(t2) · · · ν(tM) ]T
1 [γ(4) + ∆hkpθ - a1y(0)] a2
(22)
denotes the sampled measurement noise matrix. Accordingly, a consistent parameter estimation can be obtained as
y(0) )
Therefore, the above estimation of the initial state can be used to represent the step response test for evaluating the fitting effect of the identified model under unsteady/nonzero initial process conditions. Note that, in the case of model mismatch for the identification of a higher-order process, an enhanced fitting effect can be obtained by using preliminary knowledge of the initial process conditions. That is, apart from the other model parameters, the process time delay can be alternatively estimated from the redundant fitting conditions in eq 17 as θ)
exists. lim (ZTυ)/M ) 0
Note that initial state of the process output can be estimated from eqs 17 and 19 as
y˙(0) )
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T
1 [a y˙(0) + a1y(0) - γ(4)] ∆hkp 2
(23)
γ ) (ZTΦ)-1ZTΨ
3.2. FOPDT Model. For identification of an FOPDT model for an inherent-type load disturbance in terms of the time interval (td,tN], as shown in Figure 1, the load disturbance response can be computed as yd ) y - yˆr
or
γ(5) - 2a2y(0) ∆hkp
(24)
or mean of the three values computed from eqs 20, 23, and 24. The best choice can be determined from model fitting of the step response. Remark 1. By doubly integrating both sides of eq 15, an identification algorithm with less computation effort can be obtained in a similar way, but its identification robustness against measurement noise is inferior to that of the above algorithm. It can be seen from eq 17 that, rather than using individual output response points measured from the step test, single to triple integrals of output response points are used for parameter estimation, which facilitates the reduction of the influence from measurement errors according to the statistical averaging principle. To guarantee parameter estimation consistency in the presence of measurement noise, the instrumental variable (IV) method23 can be used to circumvent this issue. There is, however, no uniform choice of the IV matrix for consistent estimation. A feasible choice is proposed as Z ) [z1 z2 · · · zM ]T
(27)
where yˆr denotes the above-identified process model response in terms of the step test. Because the inherent-type load disturbance is normalized as a unity step signal, by letting td ) 0 for θd ) θ˜ d + td, one obtains ud(t) )
θ)
(26)
{
0, 0 e t < θ˜ d 1, t g θ˜
(28)
d
With the disturbance description of eq 28, the time-domain response shown in eq 6 can be equivalently expressed as τdy˙d(t) + yd(t) ) kdud(t)
(29)
By doubly integrating both sides of eq 29 and rearranging the resulting equation using eqs 10, 12, and 28, we can formulate a linear LS fitting in the form of eq 16, for which ψ(t) )
∫
(2) y (t) [0,t] d t - 0 yd(t),
[ ∫
]
T
φ(t) ) t2 /2, t, 1/2 γ ) [τd, kd, -kdθ˜ d + τdyd(0), kdθ˜ 2d ]T
(30)
Accordingly, the model parameters can be derived using eqs 19 and 30 as
[] [ ] τd γ(1) kd ) γ(2) θ˜ d √γ(4)/γ(2)
(31)
To guarantee consistent estimation against measurement noise, the corresponding IV is suggested as
(25)
where zi ) [1/ti 1/ti2 ti3 ti2 ti 1 ]T It can be easily verified using the consistent estimation theorem given in ref 20 that the above IV satisfies the following two limiting conditions: The inverse of
zi ) [1/ti ti2 ti 1 ]T
(32)
which can also be verified as satisfying the aforementioned limiting conditions for consistent estimation. Remark 2. In case different step tests are used to verify the identification effectiveness of the load disturbance model, particularly for the presence of a high noise level, different choices of td will result in different values of θ˜ d. Therefore, verification of the time delay of the load disturbance model should be made in terms of θd ) θ˜ d + td. That is, model
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verification should be made in terms of a time-domain response fitting criterion Nd
∑ [yˆ (kT ) - y (kT )] /N 2
d
s
d
s
d
θ
(40)
which indicates that there is no overshoot for the nominal case and that the quantitative time-domain performance specification for set-point tracking can be easily satisfied by tuning the single adjustable parameter of λs. For instance, define the rising time, tr, as the time required to reach 95% of the set-point change (∆r), the tuning formula can be derived from eq 40 as tr ) 4.7439λs + θ. The closed-loop structure set between the process input and output is used for eliminating output error in the presence of model mismatch and other process uncertainties. Note that, if model mismatch exists for the feed-forward control, the redundant control signal (∆ud) can be viewed as a load disturbance (denoted as di) that enters into the process input. If G ) Gm, the transfer function from ud to uf can be derived as Hdi(s) ) G(s) Cf(s)
(41)
which is exactly equivalent to the nominal open-loop system transfer function for set-point tracking. Hence, using the control idea developed in ref 24 for optimal load disturbance rejection, the desired closed-loop transfer function is proposed as Tf(s) ) Hdi(s) )
e-θs (λfs + 1)2
(42)
where λf is an adjustable time constant for tuning the closedloop performance for disturbance rejection. Substituting eqs 2 and 41 into eq 42, we obtain Cf )
a2s2 + a1s + 1 kp(λfs + 1)2
(43)
which is similar to the form of Cs in eq 39. However, the tuning of Cf is subject to a stability constraint of the closedloop structure. According to IMC theory,25 tuning Cf aims
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Table 1. Step Response Identification for Example 1 under Different Measurement Noise Levels NSR (%)
process model
err
-6.04667s
load disturbance model
err
-1.0192s
0
1.2000e 8.9999s2 + 2.4001s + 1
1.2 × 10-5
0.1000e 1.0001s + 1
2.65 × 10-7
2
(1.2007 ( 0.013)e(-6.0458(0.26)s (8.9946 ( 0.35)s2 + (2.4212 ( 0.36)s + 1
5.52 × 10-5
(0.1019 ( 0.015)e(-1.0306(0.13)s (1.0011 ( 0.36)s + 1
2.25 × 10-6
10
(1.2184 ( 0.047)e(-6.2913(0.89)s (8.4433 ( 1.25)s2 + (2.9135 ( 1.38)s + 1
3.81 × 10-3
(0.1123 ( 0.046)e(-0.9539(0.24)s (1.1894 ( 0.79)s + 1
6.04 × 10-5
30
(1.2531 ( 0.055)e(-7.0594(0.75)s (8.4726 ( 2.09)s2 + (3.8558 ( 1.71)s + 1
4.02 × 10-2
(0.1405 ( 0.052)e(-0.8529(0.19)s (1.2489 ( 0.71)s + 1
9.01 × 10-4
at a compromise between achievable closed-loop performance for disturbance rejection and its robust stability, that is
fitting criterion of the step response error,27,28 which is defined by
|∆m(s) Tf(s)| + |W(s)[1 - Tf(s)]| < 1
1 err ) [y(kTs) - yˆ(kTs)]2 Ns k)1
(44)
where ∆m(s) )[G(s) - Gm(s)]/G(s) defines the process multiplicative uncertainty and W(s) is a weighting function of the closed-loop sensitivity function, Sf(s) ) 1 - Tf(s). For instance, W(s) ) 1/s can be taken for a step change in the load disturbance that enters into the process input. Decreasing λf can improve the disturbance rejection performance of the closed-loop structure, but degrades its robust stability in the presence of process uncertainties. In contrast, increasing λf can strengthen the robust stability of the closed-loop structure, but in exchange for a degradation in its disturbance rejection performance. According to the small gain theorem,26 the closed-loop structure for disturbance rejection holds robust stability if and only if |∆m(s) Tf(s)|∞ < 1
(45)
Substituting eq 42 into eq 45, we obtain the robust stability constraint for tuning λf
√λf2ω2 + 1 > ∆m(jω),
∀ω g 0
(46)
which can be intuitively checked by observing whether the magnitude plot of the left-hand side of eq 46 is larger than the right-hand side for ω ∈ [0,∞). Therefore, given an upper bound of ∆m as is usually specified in practice (e.g., the maximal range of the model parameters), the admissible tuning range of λf can be numerically ascertained from eq 46. 5. Simulation Results To demonstrate the effectiveness and accuracy of the proposed identification algorithms, illustrative examples from the recent literature are presented in this section. Example 1 is given to demonstrate the achievable accuracy of the proposed algorithms for identifying first- and second-order processes, together with measurement noise tests for demonstrating identification robustness. Example 2 is presented to show the effectiveness of the proposed algorithms for the identification of higher-order processes. For all cases, the number of transient response data is taken as M ) 100 for computation. For assessment of the model fitting error, we use a widely used
Ns
∑
where y(kTs) and yˆ(kTs) denote the process and model responses, respectively, to a unity step change and NsTs is the transient response time or the settling time. The sampling period is taken as Ts ) 0.01 s for all simulation tests. 5.1. Example 1. Consider the second-order process studied in ref 17 Y(s) )
e-s 1.2e-6s U(s) + U (s) s+1 d 9s + 2.4s + 1 2
Note that an FOPDT model studied by Bi et al.8 is used to represent the dynamics of an inherent-type load disturbance. For illustration, assume that the initial process conditions are y(t0) ) 1.2, y˙(t0) ) 0.01, and u(t0) ) 1.0. A step change of ∆h ) 0.2 is added to the set point for identification at t0 ) 3 s, and an inherent-type load disturbance with a magnitude of -0.1 is added through the above FOPDT model to the process at t ) 18 s. For illustration, td ) 19 s, corresponding to an obvious turning point of the process response as shown in Figure 1, is chosen for load disturbance identification. The process transient response in the time interval [10, 19) s is employed to derive the process model. The results are listed in Table 1, along with the fitting error for the transient response used for identification. It can be seen that good accuracy is obtained by the proposed algorithm. Subsequently, using the load disturbance response estimated in the time interval [19, 26] s, that is, subtracting the resulting SOPDT process model response from the real step response, an FOPDT disturbance model is therefore derived as listed in Table 1, which also indicates good accuracy. Note that the gain of the FOPDT disturbance model is reduced to one-tenth, as the magnitude of the load disturbance is normalized to unity for model identification. The fitting error for the load disturbance response is correspondingly evaluated in terms of the transient response in the time interval [19, 26] s. To demonstrate identification robustness in the presence of measurement noise, assume that a random noise of N(0, σζ2 ) 0.0012%), causing a noise-to-signal ratio (NSR, defined as {mean[abs(noise)]/mean[abs(signal)]}) of 2%, is added to the output measurement. By performing 100 Monte Carlo tests in terms of varying the “seed” of the noise generator from 1 to 100, the proposed algorithms based on the above time intervals
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Figure 3. Fitting effect of step response for example 2.
of the step response data for identification give the results listed in Table 1, where the model parameters are the mean values for the 100 Monte Carlo tests along with the sample standard deviations. The results for NSR ) 10% and 30% are also listed in Table 1 to demonstrate the achievable identification accuracy and robustness. Note that, for NSR ) 10% and the absence of a load disturbance but with nonzero initial conditions of y(t0) ) 0.2, y˙(t0) ) 0.01. and u(t0) ) 0, Ahmed et al.17 gave the identification results based on 100 Monte Carlo tests using a settling time length of tN ) 50 s and M ) 500 for computation Gm(s) )
(1.2 ( 0.007)e(-5.95(0.36)s (9.1 ( 0.8)s2 + (2.41 ( 0.16)s + 1
which shows obviously better accuracy compared to the results listed in Table 1. When the same time length of the transient response to the step change of the set point is used for identification of the process model, the proposed algorithm gives the following results using M ) 100 for computation Gm(s) )
(1.2 ( 0.0003)e(-6.065(0.22)s (8.98 ( 0.21)s2 + (2.39 ( 0.09)s + 1
It can be seen that the proposed method can also give very good accuracy in the absence of a load disturbance but with nonzero initial process conditions, thus demonstrating good applicability for various step tests in practice. 5.2. Example 2. Consider the high-order process studied in refs 9 and 10 Y(s) )
1 1 U(s) + Ud(s) 5 (s + 1) (s + 1)8
Note that the eighth-order process model studied by Bi et al.8 is used to describe the dynamics of an inherent-type load disturbance. Assume that the initial process conditions are y(t0) ) 1.0, y˙(t0) ) 0, and u(t0) ) 1.0. A step change of ∆h ) 0.2 is
added to the set point at t0 ) 0 s, and an inherent-type load disturbance with a magnitude of -0.1 is added to the process at t ) 10 s. The corresponding step response is shown in Figure 3. Using the process transient response in the time interval of [2, 12] s, the proposed identification formulas of eqs 17-20 give an SOPDT process model, Gm ) 0.9976e-2.06s/(3.4825s2 + 3.1803s + 1), corresponding to err ) 1.74 × 10-3 for the time interval [0, 12] s. Note that further enhanced fitting accuracy can be obtained using known initial process conditions and the corresponding identification formula of eq 23 as Gm ) 0.9976e-1.72s/(3.4825s2 + 3.1803s + 1), which corresponds to err ) 3.42 × 10-4. Then, using the load disturbance response estimated in the time interval [14, 30] s with a choice of td ) 12 s that is slightly ahead of the observed turning point at t ) 14 s (i.e., subtracting the above SOPDT model response from the real step response), an FOPDT disturbance model is therefore derived as Gd ) 0.1045e-3.03s/(3.7663s + 1), corresponding to err ) 2.14 × 10-4 for the transient response in the time interval [12, 30] s. The combined step response of the above SOPDT process model and the FOPDT disturbance model for representing the real step response are also plotted in Figure 3 for comparison, which demonstrates the good fitting accuracy. Note that the assumed inherent-type disturbance dynamics that was studied as a high-order process by Bi et al.8 was identified as Gm ) 1.06e-4.94s/(3.81s + 1), corresponding to err ) 7.7 × 10-4 in terms of an unity step test, and by comparison, the wellknown graphical area method2 gave Gm ) 1.0e-4.3s/(4.3s + 1), corresponding to err ) 2.2 × 10-3. To demonstrate identification robustness to different choices of the time length of the transient response associated with the turning point chosen for piecewise model identification, Table 2 lists the identification results for using different time lengths of transient response, which indicates that the proposed method is not sensitive to different choices of the turning point and transient response data for model fitting. To demonstrate the achievable control effect based on the identified models, the proposed control scheme shown in Figure 2 is applied in comparison with the standard 2DOF IMC control structure. For the above initial process conditions and the setpoint change with an inherent-type load disturbance as in the above step test, the control results are shown in Figure 4. It can be seen that using the same control parameters (i.e., λs ) 0.5 and λf ) 1.0), the standard 2DOF IMC control structure based on using the identified SOPDT process model gives similar setpoint tracking performance but with better load disturbance rejection, compared to using the real process model. To obtain the same disturbance rejection performance, the tuning parameter for Cf, λf, should be increased to 1.6 for use of the identified SOPDT process model, which, in fact, facilitates better closedloop stability according to the robust stability analysis in section
Table 2. Identification Results for Example 2 Using Transient Responses of Different Time Lengths time interval (s)
process model
err
time interval (s)
inherent disturbance model
err
[2, 12]
0.9976e-1.72s 3.4825s2 + 3.1803s + 1
3.42 × 10-4
[14, 30]
0.1045e-3.03s 3.7663s + 1
2.14 × 10-4
[2, 13]
0.9981e-1.74s 3.4021s2 + 3.1862s + 1
3.22 × 10-4
[15, 30]
0.1027e-3.39s 3.2801s + 1
2.01 × 10-4
[2, 11]
0.9963e-1.68s 3.6297s2 + 3.1551s + 1
4.28 × 10-4
[16, 30]
0.1015e-3.69s 2.9068s + 1
1.91 × 10-4
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Figure 6. Open-loop step test for injection velocity response.
Figure 4. Comparison of control effects for example 2.
shown in eq 17 that the number of parameters to be estimated [dim(γ)] is 6. To ensure identification effectiveness, the turning point is chosen at t ) 0.13 s for determining the starting point of the load disturbance response. Accordingly, the transient response in the time interval [0.05, 0.13] s is used to identify the dynamic response of injection velocity to the valve opening, corresponding to a value of M ) 17, which complies with the guideline given in section 2 for piecewise model identification. By using a low-pass third-order Butterworth filter with a cutoff frequency of fc ) 20 Hz for predenoising the measured data, the proposed identification method gives the SOPDT model Gm )
Figure 5. Schematic diagram of injection molding machine.
4. Note that, based on the identified FOPDT disturbance model, the proposed control scheme with λs ) 0.5, λf ) 1.6, and λd ) 2.0 apparently gives an improved load disturbance response, demonstrating well that identifying both models of the process and the inherent-type load disturbance from a step test facilitates advanced control design and performance. 6. Experimental Results Consider the injection velocity control of an industrial reciprocating screw injection molding machine (Chen-Hsong, model JM88-MKIII-C), a schematic diagram of which is shown in Figure 5. The injection velocity is regulated by a proportional valve (4WRP-10-63S-1X/G24Z24/W), denoted as PV1 in Figure 5, and is measured by an MTS Temposonics III displacement and velocity transducer (RH-N-0200M-RG0-1-V0-1). A 16-bit data acquisition card (PCL-816) from Advantech is used for analog-to-digital (A/D) and digital-to-analog (D/A) conversions. For illustration, a rectangular mold of length 150 mm, width 200 mm, and thickness 2 mm, corresponding to a weight of 28 g, is used for the injection molding experiments. The plastic material is high-density polyethylene (HDPE). For an open-loop step test through a sudden change of 40% in the valve opening of PV1, the injection velocity response measured over a sampling period of 0.005 s is plotted in Figure 6. It can be seen that the injection velocity response has an obvious overshoot for startup from an initial value of -2 m/s and then drops to a roughly steady value about 25 m/s in the time interval [0.1, 0.2] s. Because of the presence of mold cavity pressure that increases gradually during the filling process, the injection velocity decreases continuously until the end of the filling process. In view of the obvious overshoot in the step response, an SOPDT model structure is employed for identification. It can be seen from the corresponding LS fitting algorithm
71.1198e-0.05s 1.0792 × 10 s + 6.6425 × 10-3s + 1 -4 2
Then, by letting td ) 0.12 s in consideration of the time delay, the load disturbance response in the time interval [0.13, 2.5] s, estimated by subtracting the above SOPDT model response from the real step response, is used for modeling the influence caused by mold cavity pressure. An FOPDT disturbance model is therefore derived as Gd )
23.025e-0.13s 0.4726s + 1
For comparison, the SOPDT model response and the combined model response for representing the real step test are also plotted in Figure 6, which shows the good fitting effect. Based on the identified models, the proposed control scheme is applied for closed-loop control of the injection velocity at a desired constant value, IV ) 30 m/s, for the filling process of injection molding. For implementation, the control sampling period is taken as Ts ) 0.01 s, and the one-step-backward discretization operator, e˙(kTs) ) {e(kTs) - e[(k - 1)Ts]}/Ts, is used for computational simplicity. Experimental results based on the tuning parameters of λs ) 0.2, λf ) 0.5, and λd ) 0.1 are plotted in Figure 7. It can be seen from Figure 7a that fast set-point tracking without overshoot is obtained by the proposed control method. A slight drop of the injection velocity during the time interval [0.6, 1] s is due to model mismatch for describing the influence caused by the mold cavity pressure, which, however, is quickly compensated by the feedback controller (Cf). Figure 7b shows the valve opening (as a percentage) and the controller outputs. For comparison, the control result obtained using the standard 2DOF IMC scheme is also plotted in Figure 7a, which indicates that the set-point tracking is obviously slower and the filling time is longer, when the feed-forward control based on the identified disturbance model is not used. To further demonstrate the achievable control effect, assume that the injection velocity profile is prescribed for molding a
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identified time delay of the injection velocity response, and correspondingly, the tuning parameters of Cs and Cf are adjusted as λs ) 0.05 and λf ) 0.1 to deal with the step change, so that the implemental constraint of moderating a step change of the set point into a ramp type21,22 is no longer necessary. 7. Conclusions
Figure 7. Closed-loop control of injection velocity during the filling process.
For step response identification subject to inherent-type load disturbance in practical applications, we have proposed a piecewise model identification method that allows direct use of the raw step response data for simultaneously identifying the pure process model and the inherent-type load disturbance model from a step test, based on intuitive decomposition of the transient step response data. Identification algorithms have been detailed for obtaining the widely practiced low-order models of FOPDT and SOPDT types. Note that these identification algorithms can be transparently extended to obtain higher-order models for describing more complex dynamic response characteristics of the process and the inherent-type load disturbance. Illustrative examples have been used to demonstrate that good identification accuracy and robustness can be obtained by the proposed identification method. Accordingly, a 2DOF IMC plus feed-forward control scheme has been proposed for improving process operation against inherent-type load disturbances. The application to velocity control of injection molding has been used to demonstrate the effectiveness of the proposed identification and control method. Acknowledgment This work was supported in part by the Hong Kong Research Grants Council under Project 613107. Literature Cited
Figure 8. Tracking an injection velocity profile during the filling process.
product of convex shape, as shown in Figure 8a (dashed line). The proposed control method with the above tuning parameters gives the results shown in Figure 8a,b. Note that, for the setpoint change from 30 to 40 m/s or the inverse, the output of the set-point tracking controller Cs is implemented one sampling step ahead of the set-point change to compensate for the
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ReceiVed for reView July 19, 2010 ReVised manuscript receiVed August 30, 2010 Accepted September 26, 2010 IE1015427
