A model-based virtual sensing approach for the injection molding process.
The injection molding industry has continuously sought precision and quality molding. Many efforts have been devoted to understanding how process variables determine part qualities, notably pvT data for part quality (1-4). Once important variables were identified, the regulation of process variables became the next crucial factor for ensuring production of consistently good-quality parts. However, instead of regulating process variables directly, all existing machine controls are designed to regulate machine variables, which in most situations behave quite differently from the corresponding process variables (5, 6). Examples include the utilization of barrel temperature for melt-temperature regulation, screw velocity for melt front velocity control, and hydraulic pressure for nozzle and/or cavity pressure tracking. Such apparent discrepancies between the theory and practice can be attributed to a lack of proper sensing mechanisms, due to cost, inaccessibility, and sometimes, appearance concerns. As a result, developing solutions for process sensing constitutes a new research focus in the molding control community.
Both hardware and software solutions have been reported. The most frequently discussed hardware solution is ultrasonic sensing. Variables subject to ultrasonic sensing include mold filling, flow front speed, temperature, shrinkage, solidification, etc. (7). Fluorescent sensing of the viscoelastic state of polymers was proposed by Rose et al. (8). Leveraging the advances in MEMS (Micro Electro-Mechanical Systems) technologies, thin film sensor arrays for cavity pressure and temperature have been reported (9-11). Further extending the frontier of hardware sensing, indirect sensing via software algorithms represents an exciting research direction. A concept of inferring melt front position and velocity using the ratio of nozzle and hydraulic pressures, screw position, and rate of change of nozzle pressure was proposed (6). Estimation of melt front velocity was also studied (12), based on nozzle pressure and screw position and velocity.
One deficiency in existing software sensing approaches is their reliance on black-box models empirically relating the behaviors of desired variables to data of directly measurable variables. In this study, a new software solution--a model-based virtual sensing approach--is proposed, in which estimation of process behavior is achieved by exploiting the dynamic interaction between the process and machine variables using the nonlinear observer theory (13). In particular, a virtual sensor of nozzle pressure (a process variable) utilizing information of the screw position (a machine variable) is considered. The design adopts the so-called "high gain observer" method (14).
In addition to its usefulness to molding control as discussed above, the proposed virtual sensing scheme has the potential to greatly enhance existing process-monitoring packages. One example is Fanuc's Resin Evaluation System (15), in which injection pressure data are utilized to identify changes in the resin via a database look-up. Incorporation of the proposed nozzle pressure virtual sensor is expected to provide a direct and potentially more effective means for resin evaluation. Another example is the Moldflow Plastics Xpert (16), which achieves process automation and assists in molding control by exploitation and fusion of machine transducer outputs, sensor readouts on cavity pressure, and the operator's quality reports. The proposed virtual sensors and the physical model-based design philosophy may not only provide a direct inference on hard-to-access process variables such as cavity pressure and melt flow rate but also offer an efficient avenue for data fusion and interpretation of various sensor outputs and operator's inputs.
The paper is organized as follows. Section 2 reviews the dynamic modeling of the filling process. Section 3 starts with an overview of the proposed virtual sensing approach. Application to on-line estimation of nozzle pressure during the so-called "nozzle resistance test" is then illustrated. Results of experimental evaluation on a commercial injection molding machine are reported in Section 4. Section 5 concludes the paper.
MODELING OF FILLING DYNAMICS
By applying the mass and momentum conservation and Reynolds transport theorem, a flow dynamic model of the filling process can be obtained. In this study, we follow the formulation described in Chiu et al. (17). The filling process is approximated by the transient phenomenon of non-Newtonian fluid flow. Some assumptions are made:
1. The filling process is isothermal.
2. The polymer viscosity is a function of temperature and shear rate only.
3. The polymer is treated as a compressible fluid in the continuity equation.
4. The polymer is treated as an incompressible fluid in the momentum equation.
With variables defined in the Nomenclature at the end of the paper, the resultant dynamic model of the filling process becomes
[[d.sup.2]x]/[d[t.sup.2]] = [1/M] ([P.sub.1][A.sub.1] - [P.sub.2][A.sub.2] - [C.sub.f] * sgn ([dx]/[dt]) - [B.sub.f] * [dx]/[dt]) (1a)
[d[P.sub.2]]/[dt] = [K.sub.p]/[V.sub.2] * ([A.sub.2] [[dx]/[dt]] - Q - [Q.sub.LP]), [V.sub.2] = [A.sub.2] ([L.sub.2] - x) (1b)
[dQ]/[dt] = [[P.sub.2] - [P.sub.p] - [[summation].sub.i=1.sup.N] ([F.sub.si]/[A.sub.ct])]/[[rho] * [[summation].sub.i=1.sup.N] ([H.sub.ci]/[A.sub.ci])] (1c)
Equation 1a is a result of force balancing for screw dynamics. Application of mass conservation and the Reynolds transport theorem results in Eq 1b for nozzle pressure. Equation 1c is obtained by applying the Reynolds transport theorem and momentum conservation with the polymer in the conduit being divided into N sections. The value of shear force [F.sub.si] can be calculated by
[F.sub.si] = [C.sub.ci] * [H.sub.ci] * [[tau].sub.i] (2)
where [C.sub.ci] and [H.sub.ci] are the equivalent circumference and length of the ith section of the conduit, respectively, and [[tau].sub.i] the shear stress. The shear stress is formulated by a power law model modified for non-Newtonian polymer flow,
[[tau].sub.i] = [[mu].sub.i][[GAMMA].sub.i] [equivalent to] sgn ([[GAMMA].sub.i]) * [K.sub.i] * |[[GAMMA].sub.i]|[.sup.v], 0 < v < 1 (3)
and [[mu].sub.i] = [[mu].sub.0]/[1 + [C.sub.0]([[mu].sub.0]|[[GAMMA].sub.i]|)[.sup.1-v]] (4)
where [[GAMMA].sub.i] represents the polymer shear rate, [[mu].sub.i] the shear-rate dependent viscosity, [[mu].sub.0] the zero shear-rate viscosity, v the exponent of the power law, and [C.sub.0] a constant. The polymer shear rate is geometry dependent. For example, we have for a tubular conduit
[[GAMMA].sub.tube] = [32Q]/[[pi] * [D.sub.h.sup.3]] (5)
where [D.sub.h] is the equivalent hydraulic diameter of the conduit.
From the point of view of system theory, the filling dynamics (1) can be considered as a nonlinear dynamic system with the injection pressure [P.sub.1] as its input and screw velocity dx/dt as its output. For hydraulic molding machines, [P.sub.1] refers to the hydraulic actuation pressure, while for all electric machines, it is proportional to the output of the load cell placed at the back of the screw assembly.
VIRTUAL SENSING APPROACH WITH APPLICATION TO NOZZLE PRESSURE ESTIMATION
Step-by-Step Design Procedure
The proposed approach is based upon recent developments in advanced control theory, in particular, the nonlinear observer theory (13). Specifically, construction of virtual sensors adopts the so-called "high gain observer" design (14), the design procedure of which is summarized in the following.
Step 1. Modeling Process Dynamics Based on Physics
Availability of a dynamic model of the process of interest is a prerequisite of the model-based observer design. The model is usually presented as a set of simultaneous ordinary or/and partial differential equations. When presented in partial differential equations, model reduction to meaningful finite dimensional ordinary differential equations is needed.
Step 2. Representing Dynamics in State Space Form
Most nonlinear control theory, including the nonlinear observer theory, adopts the state space description of dynamic systems (13, 18). For the sake of convenience, instead of considering the most general nonlinear system, we consider systems in the following special form.
[dot.x] = F(x) + G(x)u, x [member of] [R.sup.n], u [member of] R (6)
y = h(x), y [member of] R
where x denotes the vector of state variables of the system, u the input, y the output, and F(*), G(*), and h(*) are nonlinear functions of appropriate dimensions. Note that throughout the presentation, boldface is used to symbolize vectors and matrices, and scalar variables are written without boldface.
Step 3. Conducting Nonlinear Coordinate Transformation
For given input u(*), assume that the following nonlinear mapping constitutes a legitimate coordinate transformation, i.e., being invertible for all x,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
with [h.sub.i](*) defined recursively as follows,
[h.sub.0](x) = h(x), [h.sub.i](x) = [[partial derivative][h.sub.i-1](x)]/[[partial derivative]x] F(x), 1 [less than or equal to] i [less than or equal to] (n - 1) (8)
where [partial derivative]/[partial derivative]x represents the gradient of a scalar function. The gradient is presented as a row vector here. To understand the coordinate transformation (Eq 7), note that [h.sub.0](x) = y(t), and when u(*) = 0, [h.sub.i](x) is simply the i-th time derivative of the output, i.e., [h.sub.i](x) = [d.sup.i]y(t)/d[t.sup.i]. We then have model representation in a new coordinate, [xi], as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where [xi] = ([[xi].sub.1], [[xi].sub.2],..., [[xi].sub.n])[.sup.T] with superscript T representing vector transposition. Direct verification gives
[psi]([xi]) = [[[partial derivative][h.sub.n-1](x)]/[[partial derivative]x]] F(x)|[.sub.x=[[PHI].sup.-1]([xi])] (10)
[bar.g.sub.i]([xi]) = [[[partial derivative][h.sub.i-1](x)]/[[partial derivative]x]] G(x)|[.sub.x=[[PHI].sup.-1]([xi])], 1 [less than or equal to] i [less than or equal to] n
Step 4. Verifying Observability
Before designing an observer, one must verify if the system under investigation satisfies certain "observability" conditions. It was stated in (14) that the system satisfies the so-called "uniform observability for any input" if [bar.G]([xi]) assumes the following special structure,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
and an additional requirement, the so-called "global Lipschitz" of [psi](*) is satisfied. Furthermore, the global Lipschitz of [bar.g.sub.i](*), i = 1,..., n, is necessary for successful development of an asymptotically convergent observer, to be discussed in Step 5 below. A couple of facts on Lipschitz are useful for our application in the next section and are recalled here for reference. They are (i) linear functions are globally Lipschitz and (ii) a continuously differentiable function over a bounded domain is Lipschitz.
Step 5. Designing Virtual Sensor
Once the system is verified to be uniformly observable for any input and with the global Lipschitz condition on [bar.g.sub.i](*) satisfied, an asymptotically convergent observer, or in other words, a virtual sensor, can be designed based on Theorem 3 of (14). Let [^.[xi]] denote the estimate of the state variable [xi]. We have the following observer,
[??] = [bar.F]([^.[xi]]) + [bar.G]([^.[xi]])u + [S.sup.-1][bar.c.sup.T] (y - [bar.c][^.[xi]]) (12)
where the gain matrix S is the positive definite solution of the following equation,
[A.sup.T]S + SA + [theta]S = [bar.c.sup.T][bar.c] (13)
with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and [theta] a positive design parameter. The observer (Eq 12) can be viewed as an open-loop prediction, [??] = [bar.F]([^.[xi]]) + [bar.G]([^.[xi]])u, plus an on-line correction, [S.sup.-1] [c.sup.-T] (y - c[bar.[xi]]), based upon the estimation error, (y - c[bar.[xi]]).
The theorem states that with the design parameter [theta] "large enough," the above observer is asymptotically convergent, i.e.,
|[^.[xi]](t) - [xi](t)| [right arrow] 0 as t [right arrow] [infinity] (14)
One must wonder "How large of [theta] is enough?" This question was answered in the proof of the theorem in (14). Or it can be experimentally determined with the understanding that as long as we keep increasing the parameter [theta], the estimator will converge eventually.
Application to Nozzle Pressure Virtual Sensing
For illustration and evaluation purposes, in the following we apply the proposed approach to designing a nozzle pressure virtual sensor during the nozzle resistance test. In the nozzle resistance test, no mold is installed and the melt is shot freely into air. The virtual sensor takes information of a machine variable, screw position x, to estimate the behavior of an unavailable process variable, nozzle pressure [P.sub.2] (Fig. 1). In the following, we describe design of the nozzle pressure virtual sensor step by step.
[FIGURE 1 OMITTED]
Step 1. Modeling Process Dynamics
Treating the dissipation [P.sub.p] and leakage [Q.sub.LP] as unmodeled dynamics, the molding dynamics (Eq 1) during the nozzle resistance test becomes
[[d.sup.2]x]/[d[t.sup.2]] = [1/M] ([P.sub.1][A.sub.1] - [P.sub.2][A.sub.2] - [C.sub.f] - [B.sub.f] * [dx]/[dt]) (15a)
[d[P.sub.2]]/[dt] = [[K.sub.P]/[[A.sub.2]([L.sub.2] - x)]] * ([A.sub.2] [[dx]/[dt]] - Q) (15b)
[dQ]/[dt] = [[A.sub.c1]/[rho]] [[P.sub.2]/[H.sub.c1]] - [[C.sub.c1]/[rho]] * [/[[pi][D.sub.h.sup.3]]] * [[mu].sub.1](Q) * Q (15c)
Note that [[mu].sub.1] is Q dependent viscosity (Eq 4) and we have made use of the understanding that the screw velocity is always positive during filling when dealing with the Coulomb friction.
Step 2. Representing Dynamics in State Space Form
With the state vector x defined as
x = [[x.sub.1] [x.sub.2] [x.sub.3] [x.sub.4]][.sup.T] = [x dx/dt [P.sub.2] Q][.sup.T] (16)
u for the combined effect of the injection pressure [P.sub.1] and the Coulomb friction [C.sub.f], and y for measurement of screw position x, the state model of the molding dynamics becomes
[dot.x] = F(x) + bu (17)
y = cx
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[a.sub.1] = - [[B.sub.f]/M], [a.sub.2] = - [[A.sub.2]/M], [a.sub.3] = [[A.sub.1]/M], [a.sub.4] = [K.sub.p], [a.sub.5] = - [[K.sub.p]/[A.sub.2]] [a.sub.6] = [[A.sub.c1]/[rho][H.sub.c1]], [a.sub.7] = - [[C.sub.c1]/[rho]] [/[[pi][D.sub.h.sup.3]]], [[mu].sub.1]([x.sub.4]) = [[mu].sub.0]/[1 + [C.sub.0] ([32[[mu].sub.0]]/[[pi][D.sub.h.sup.3]] [x.sub.4])[.sup.1-v]]
Step 3. Conducting Nonlinear
Accordingly, the following nonlinear coordinate transformation is chosen,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
The transformed state model becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
[psi]([xi]) = [[alpha].sub.1][x.sub.2] + [[alpha].sub.2][x.sub.3] + [[alpha].sub.3] [[x.sub.2]/[[L.sub.2] - [x.sub.1]]] + [[alpha].sub.4] [[x.sub.3]/[[L.sub.2] - [x.sub.1]]] + [[alpha].sub.5] [[x.sub.4]/[[L.sub.2] - [x.sub.1]]] + [[alpha].sub.8] [[[x.sub.4][[mu].sub.1]([x.sub.4])]/[[L.sub.2] - [x.sub.1]]] + [[alpha].sub.6] [[x.sub.2.sup.2]/[([L.sub.2] - [x.sub.1])[.sup.2]]] + [[alpha].sub.7] [[[x.sub.2][x.sub.4]]/[([L.sub.2] - [x.sub.1])[.sup.2]]]|[.sub.x=[[PHI].sup.-1]([xi])]
[bar.g.sub.4]([xi]) = [[alpha].sub.9] + [[alpha].sub.10] [1/[[L.sub.2] - [x.sub.1]]]|[.sub.x=[[PHI].sup.-1]([xi])]
with [[alpha].sub.1] = [a.sub.1.sup.3], [[alpha].sub.2] = [a.sub.1.sup.2][a.sub.2], [[alpha].sub.3] = 2[a.sub.1][a.sub.2][a.sub.4], [[alpha].sub.4] = [a.sub.2.sup.2][a.sub.4] + [a.sub.2][a.sub.5][a.sub.6], [[alpha].sub.5] = [a.sub.1][a.sub.2][a.sub.5], [[alpha].sub.6] = [a.sub.2][a.sub.4], [[alpha].sub.7] = [a.sub.2][a.sub.5], [[alpha].sub.8] = [a.sub.2][a.sub.5][a.sub.7], [[alpha].sub.9] = [a.sub.1.sup.2][a.sub.3], [[alpha].sub.10] = [a.sub.2][a.sub.3][a.sub.4].
Step 4. Verifying Observability
[bar.G] ([xi]) in Eq 19 trivially satisfies the structural requirement stated in Eq 11. The global Lipschitzness of [psi](*) is satisfied by noting that d([x.sub.4][[mu].sub.1]([x.sub.4]))/d[x.sub.4] and d([L.sub.2] - [x.sub.1])[.sup.-1]/d[x.sub.1] are bounded for all [x.sub.4] [greater than or equal to] 0 and 0 [less than or equal to] [x.sub.1] [less than or equal to] [L.sub.2] - [delta] where [delta] is a small fixed positive number and [x.sub.1] [less than or equal to] [L.sub.2] - [delta] prevents singularity in 1/([L.sub.2] - [x.sub.1]). Thus the filling dynamics during the nozzle resistance test is uniformly observable for any input.
Step 5. Designing Virtual Sensor
Designing the virtual sensor becomes straightforward at this point. By solving Eq 13 for gain matrix S with a large enough parameter [theta], we have a virtual sensor for nozzle pressure as described in Eq 12. As discussed in Step 4 on boundedness of d([L.sub.2] - [x.sub.1])[.sup.-1]/d[x.sub.1] for all 0 [less than or equal to] [x.sub.1] [less than or equal to] [L.sub.2] - [delta], [bar.g.sub.4](*) is globally Lipschitz, guaranteeing asymptotic convergence of the designed observer.
EXPERIMENT AND DISCUSSION
The injection molding machine for the experiment is an electrohydraulic machine, Model VS-50 by Victor Taichung Machinery. Three sensors were available: a potentiometer for screw position sensing, pressure readout of the hydraulic actuation system, and a Kistler 4085A nozzle pressure sensor. The potentiometer and hydraulic pressure sensor are standard devices on the machine, while the nozzle pressure sensor was installed externally. A data-acquisition board. Advantech PCI-1710HG, was used to interface with the sensors and existing machine controller. The data-acquisition board and the virtual sensor software reside on a 533 MHz Intel Celeron PC.
The resin used in this study was PS, Type PG-80, by Chi-Mei Corporation.
Experiment and Results
The experiment consists of two parts, one for system identification of the filling dynamics and another for performance evaluation of the proposed virtual sensor.
Part 1. System Identification
Instead of identifying the filling dynamics equation (Eq 15) as a whole, its three equations (15a, 15b, 15c) were identified separately. Not all parameters were identified experimentally. Some were assumed to be available and will be pointed out during the discussion.
Identification of Eq 15a. In this experiment, the machine operated without any mold installed. Polymers were shot into air through the nozzle. During the experiment, the hydraulic flow control valve command was varied while the hydraulic pressure relief valve command was held at a constant 5 volts. The flow control valve command of each injection was composed of 10 randomly selected sinusoids with frequencies varying between 0 and 4 Hz and magnitudes between 3 and 4 volts. Two hundred injections were conducted. Data of injection pressure [P.sub.1], screw position x, and nozzle pressure [P.sub.2] were collected from the beginning of clamping to the end of the filling. To attenuate the effect of measurement noises, [P.sub.1], [P.sub.2] and x were processed with zero-phase smoothing (19) based on second order Butter-worth filters with bandwidths 120 rad/s, 120 rad/s, and 60 rad/s, respectively. Furthermore, to avoid direct differentiation, a low pass filter with transfer function
[G.sub.L1](s) = [1/[[s.sup.2] + 2[[zeta].sub.1][[omega].sub.n1]s + [[omega].sub.n1.sup.2]]], [[zeta].sub.1] = 0.707 and [[omega].sub.n1] = 60
rad/s is applied to Eq 15a, resulting in the following simple regression model.
[MATHEMATICAL EXPRESSON NOT REPRODUCIBLE IN ASCII] (20)
where [bar.x], [bar.v], [bar.P.sub.1], [bar.P.sub.2] represent the low-pass filtered x, dx/dt, [P.sub.1], and [P.sub.2], respectively, and machine parameters [A.sub.1] = 1.13 X [10.sup.4] [mm.sup.2], [A.sub.2] = 804.25 [mm.sup.2], and M = 140 kg were given by direct measurement. Recursive least square (RLS) estimation resulted in
[MATHEMATICAL EXPRESSON NOT REPRODUCIBLE IN ASCII]
where P denotes the covariance matrix of the estimated [B.sub.f] and [C.sub.f]. Square roots of the diagonal terms of P provide estimates on the standard deviations of the identified [B.sub.f] and [C.sub.f]. With a standard deviation being 0.26% of the last identified value and steady state covnergence observed in the evolution of identified [B.sub.f] (not shown here), [B.sub.f] = 59.6 can be treated as the true value. In [C.sub.f] case, the standard deviation stands at 0.90% of the last identified value and the evolution of identified [C.sub.f] (not shown here) still varied. Thus, more data are needed to claim [C.sub.f] = 961.3 the true value.
Identification of Eq 15b. In this experiment, we sealed the nozzle but still applied hydraulic flow control valve command, as previously. Since the nozzle was sealed, no polymer flowed out of the nozzle, creating zero polymer flow rate Q = 0. As above, [P.sub.2] and x were first processed by zero-phase smoothing. A low-pass filter [G.sub.L2] = [1/[s + [[omega].sub.n2]]] with [[omega].sub.n2] = 60 rad/s was applied to Eq 15b to avoid direct differentiation of [P.sub.2]. However, screw velocity v (= dx/dt) was obtained by directly differentiating the smoothed x. The following regression model was obtained.
[P.sub.2] - [[omega].sub.n2][~.P.sub.2]] = [[v.sub.f]][[K.sub.p]] (21)
where [~.P.sub.2]] and [v.sub.f] denote the filtered signals of [P.sub.2] and [v/[[L.sub.2] - x]] by [G.sub.L2]. [L.sub.2] = 40 mm was set in experiments. RLS estimation gave [K.sub.p] = 400.2 MPa and a variance P = 1.6. The facts that estimated standard deviation runs at 0.32% of the last identified value and that the evolution of identified [K.sub.p] (not shown here) did not reach steady state suggest that more data be collected before the true value of [K.sub.p] can be conclusively determined.
Identification of Eq 15c. In this experiment, several long-duration injections were conducted until the process reached its steady state. At steady state when dQ/dt = 0, Eq 15c gives
0 = [[A.sub.c1]/[rho]] [[P.sub.2]/[H.sub.c1]] - [[C.sub.c1]/[rho]] * [K.sub.1] * ([32Q]/[[pi][D.sub.h.sup.3]])[.sup.v] (22)
In the above equation, approximate viscosity [[mu].sub.1] [approximately equal to] [[[mu].sub.0.sup.v][[GAMMA].sub.1.sup.v-1]]/[C.sub.0], consequently [K.sub.1] [approximately equal to] [[mu].sub.0.sup.v]/[C.sub.0], was used because of the large Q at steady state. Taking the logarithm gives the following regression model.
[MATHEMATICAL EXPRESSON NOT REPRODUCIBLE IN ASCII] (23)
Cross-sectional area [A.sub.c1] = 176.71 [mm.sup.2], length [H.sub.c1] = 110 mm, diameter [D.sub.h] = 15 mm and circumference [C.sub.c1] = 47.12 mm of the tubular nozzle conduit were available from direct measurement. Before application of RLS, [P.sub.2] and x were processed by zero-phase smoothing as described above and the flow rate Q = [A.sub.2]*v obtained by direct differentiation of x. RLS estimation resulted in
[MATHEMATICAL EXPRESSON NOT REPRODUCIBLE IN ASCII]
with P being the corresponding covariance matrix. The identified log([K.sub.1]) = 12.1 is equivalent to an estimate on [K.sub.1] = 1.78 X [10.sup.5]. As indicated by the estimated standard deviations, which stand at about 11% of the final identified values, more experiments are required for better identification of v and [K.sub.1].
The above described identification approach does not provide information on the zero shear rate viscosity [[mu].sub.0]. Instead, [[mu].sub.0] = 3.2 X [10.sup.4] Pa*s reported in (20) was adopted in this study.
Model validation. The injection pressure [P.sub.1] of the 200 injections used in identification of Eq 15a was applied to the identified model to evaluate its predictive capacity. Two performance measures, the normalized 2-norm and infinite-norm, were applied to the model prediction errors in nozzle pressure [P.sub.2] and screw position x. Let [e.sub.p2,2] and [e.sub.p2,[infinity]] denote the normalized 2-norm and infinite-norm of prediction error ([P.sub.2] - [P.sub.2pred]) and [e.sub.x,2] and [e.sub.x,[infinity]] the corresponding measures of prediction error (x - [x.sub.pred]). Note [P.sub.2pred] and [x.sub.pred] denote the predicted nozzle pressure and screw position. Specifically, we have
[e.sub.p2,2] [equivalent to] [[square root of ([[summation].sub.t]([P.sub.2](t) - [P.sub.2pred](t))[.sup.2])]/[square root of ([[summation].sub.t][P.sub.2.sup.2](t))]], [e.sub.p2,[infinity]] [equivalent to] [[max.t]|[P.sub.2](t) - [P.sub.2pred](t)|]/[[max.t]|[P.sub.2](t)|] (24), (25)
[e.sub.x,2] [equivalent to] [[square root of ([[summation].sub.t](x(t) - [x.sub.pred](t))[.sup.2])]/[square root of ([[summation].sub.t][x.sup.2](t))]], [e.sub.x,[infinity]] [equivalent to] [[max.t]|x(t) - [x.sub.pred](t)|]/[[max.t]|x(t)|] (26), (27)
Average prediction errors of the 200 injections are mean ([e.sub.p2,2]) = 10.7%, mean ([e.sub.p2,[infinity]]) = 19.4%, mean ([e.sub.x,2]) = 18.3%, and mean ([e.sub.x,[infinity]]) = 22.4%. Table 1 summarizes other statistics, including the standard deviation and the best and worst results with injection identification numbers indicated. Figures 2 and 3 present the best and worst model prediction results in [e.sub.p2,2], occurring at the 169th and 151st injections, respectively. The results indicate an acceptable predictive capability of the identified filling dynamic model.
Part 2. Evaluation of Proposed Virtual Sensor
Design of virtual sensor. Determination of the observer gain [S.sup.-1][bar.c.sup.T] was done by first solving Eq 12 for S.
[MATHEMATICAL EXPRESSON NOT REPRODUCIBLE IN ASCII] (28), (29)
Let the overhead cap (^) indicate estimate of the variable of interest. [^.P.sub.2], the estimate of [P.sub.2], was obtained using the following virtual sensor.
[MATHEMATICAL EXPRESSON NOT REPRODUCIBLE IN ASCII] (30)
where [psi] and [bar.g.sub.4] were defined in Eq 19 and the expression of [^.P.sub.2]] obtained by solving Eq 18. Because of the high noise content, before being submitted to Eq 30, the measured screw position x was first processed using zero-phase smoothing based on second order Butter-worth filter with bandwidth 60 rad/s, as we did in system identification. The design parameter [theta] of the virtual sensor was set to 1400 in the data presented below.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Performance of designed virtual sensor. Let [^.e.sub.p2,2], [^.e.sub.x,2], [^.e.sub.p2,[infinity]] and [^.e.sub.x,[infinity]] denote the normalized 2-norms and infinite-norms of the estimation errors ([P.sub.2] - [^.P.sub.2]) and (x - [^.x]). They are defined similarly as those for [e.sub.p2,2], [e.sub.x,2], [e.sub.p2,[infinity]] and [e.sub.x,[infinity]] with [P.sub.2pred] and [x.sub.pred] replaced with [^.P.sub.2] and [^.x], respectively. Error statistics of the designed virtual sensor are summarized in Table 2 with mean ([^.e.sub.p2,2]) = 7.1%, mean ([^.e.sub.p2,[infinity]]) = 19.3%, mean ([^.e.sub.x,2]) = 0.7%, and mean ([^.e.sub.x,[infinity]]) = 0.6%, representing 34%, 0.6%, 96% and 97% improvement over the corresponding model prediction measures [e.sub.p2,2], [e.sub.p2,[infinity]], [e.sub.x,2] and [e.sub.x,[infinity]]. Figures 4 and 5 depict the best and worst virtual sensing performance in [^.e.sub.p2,2]. occurring at the 155th and 151st injections, respectively.
Like a feedback control, the observer-based virtual sensor inevitably went through a transient before it attained the expected performance. It is interesting to note that the averages become mean ([^.e.sub.p2,2]) = 6.4%, mean ([^.e.sub.p2,[infinity]]) = 14.5%, mean ([^.e.sub.x,2]) = 0.7%, and mean ([^.e.sub.x,[infinity]]) = 0.6% if the first 30 ms data were excluded from the calculation of normalized error norms (Table 3). Compared with [e.sub.p2,2], [e.sub.p2,[infinity]], [e.sub.x,2] and [e.sub.x,[infinity]] whose first 30 ms data were also excluded (Table 4), the virtual sensor performance now correspondingly reached 39%, 25%, 96% and 97% improvement. These results positively validate the effectiveness of the designed virtual sensor.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Can model prediction, consequently virtual sensing, performance be further improved? From the covariance analysis conducted in system identification. we know that the adopted nominal values of v and [K.sub.1] are most likely erroneous and those of [K.sub.p] and [C.sub.f] may vary a bit. Moreover, our zero shear rate viscosity [[mu].sub.o] was borrowed from the literature. Thus, it will be worthwhile to see how model prediction and virtual sensing perform with perturbed parameters. Are they robust against parametric perturbations?
A parametric sensitivity study has been conducted in which each parameter was varied individually around its nominal value. Figures 6 and 7 record variations in average square-norms of [P.sub.2] prediction and virtual sensing errors, while Fig. 8 contains variations in x prediction and virtual sensing errors. To discount the initial transient, the first 30 ms data were excluded from error calculation. The figures reveal that improvement of model prediction is possible with better identification of more sensitive parameters. In order from high to low sensitivity, we have
[FIGURE 6 OMITTED]
Sensitivity order (v [approximately equal to] [K.sub.1] > [B.sub.f] > [[mu].sub.o] [approximately equal to] [C.sub.f] > [K.sub.p]) F(31)
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
with [K.sub.p] playing a negligible effect. The same sensitivity order was observed in virtual sensing errors as in model prediction errors.
The sensitivity results also emphasize the advantages of closed-loop virtual sensing over open-loop model prediction. First, because of the screw position error feedback correction as evidenced in the x prediction and virtual sensing errors compared in Fig. 8, the virtual sensor delivered almost consistently better performance than open-loop prediction (Figs. 6 and 7). The one exception occurs in the case when [K.sub.1] was perturbed by +30% with mean ([e.sub.p2.2]) = 6.67% and mean ([^.e.sub.p2.2]) = 6.84%, a negligible difference. In the case when v was reduced by 10%, such performance improvement reached a level as high as 52%. Second, the screw position error feedback helps reduce the effect of parametric modeling uncertainties. In another words, the virtual sensor is much more robust against parameter variations than simple model prediction.
There is room for improvement. (a) As one may already notice, in order to remove the significant noise content from screw position measurements, a data treatment of zero-phase smoothing (also called smoothing filter) was applied. Because a zero-phase filter is non-causal, the virtual sensor reported above is suitable only for cycle-to-cycle process monitoring. For real-time within-cycle applications, causal filtering must be adopted. (b) One deficiency in the proposed virtual sensing approach is its "high-gain" design. Like any high-gain feedback control, a virtual sensor with very high gain will suffer performance deterioration, sometimes even instability, when it indiscriminately processes very noisy measurements. It is more desirable to have virtual sensing approaches without resorting to high-gain designs.
A new virtual sensing approach based on physical model and nonlinear observer theory was proposed for the injection molding process. A step-by-step design procedure of the approach was discussed. Application to virtual sensing of nozzle pressure during the so-called "nozzle resistance test" was carried out as an illustrative example. On-machine evaluation successfully established the feasibility of the proposed virtual sensing approach. A parametric sensitivity study was carried out to investigate the effect of modeling uncertainties on model prediction and virtual sensing performance. Through screw position error feedback correction, the proposed model-based virtual sensor delivered consistently better performance and was more robust against parametric variations than simple open-loop model prediction. Without considering the transient responses in the first 0.03 second, the designed nozzle pressure virtual sensor achieved, on average, a normalized 2-norm of the estimation error at 6.4% and an infinite-norm at 14.5%, which represent 39% and 25% improvement over corresponding performance measures of model prediction. Room for further improvement of the proposed virtual sensing approach was identified.
[A.sub.1]: Cross-sectional area of the injection cylinder
[A.sub.2]: Cross-sectional area of the barrel
[A.sub.ci]: Cross-sectional area of conduit at ith section
[B.sub.f]: Viscous damping coefficient
[C.sub.o]: Constant in power law model of non-Newtonian viscosity
[C.sub.f]: Coulomb friction force
[C.sub.ci]: Circumference of the conduit at ith section with i = 1 for the nozzle section
[D.sub.h]: Equivalent hydraulic diameter of a tubular conduit
[F.sub.si]: Shear force between the polymer and wall of the conduit at ith section
[H.sub.ci]: Length of the conduit at ith section
[K.sub.p]: Bulk modulus of polymer
[L.sub.2]: Length of the barrel section
M: Mass of the screw assembly
[P.sub.1]: Injection pressure
[P.sub.2]: Nozzle pressure
[P.sub.p]: Polymer dissipation pressure
Q: Volumetric melt flow rate
[Q.sub.LP]: Leakage melt flow rate in the barrel section
[V.sub.2]: Volume of the polymer in the barrel section
x: Screw Position
[[GAMMA].sub.i]: Shear rate of the polymer in ith section
[[GAMMA].sub.tube]: Shear rate of the polymer in a tubular conduit
[[mu].sub.o]: Zero-shear-rate viscosity of the polymer
[[mu].sub.i]: Viscosity of the polymer in ith section
v: Exponent of power law model of non-Newtonian polymer flow
[rho]: Density of polymer
[[tau].sub.i]: Shear stress between the polymer and wall of the conduit in ith section
Table 1. Performance Statistics of Model Prediction. Normalized Standard Error Mean Deviation [P.sub.2] prediction [e.sub.p2,2] 10.73% 1.48% [e.sub.p2,[infinity]] 19.38% 2.31% x prediction [e.sub.x,2] 18.25% 3.55% [e.sub.x,[infinity]] 22.35% 3.89% Min Max (Injection ID) (Injection ID) [P.sub.2] prediction 7.24% (169) 19.33% (151) 14.13% (66) 29.29% (151) x prediction 7.91% (143) 29.44% (151) 10.46% (143) 33.98% (152) Table 2. Performance Statistics of Virtual Sensing. Normalized Standard Error Mean Deviation [P.sub.2] estimation [^.e.sub.p2,2] 7.05% 0.81% [^.e.sub.p2,[infinity]] 19.26% 4.36% x estimation [^.e.sub.x,2] 0.68% 0.10% [^.e.sub.x,[infinity]] 0.62% 0.10% Min Max (Injection ID) (Injection ID) [P.sub.2] estimation 5.28% (155) 10.74% (151) 11.16% (115) 30.28% (117) x estimation 0.40% (179) 0.93% (88) 0.34% (169) 1.25% (92) Table 3. Performance Statistics of Virtual Sensing Without Considering Transient Data in First 0.03 Second. Normalized Standard Error Mean Deviation [P.sub.2] estimation [^.e.sub.p2.2] 6.35% 0.78% [^.e.sub.p2,[infinity]] 14.47% 2.38% x estimation [^.e.sub.x,2] 0.67% 0.10% [^.e.sub.x,[infinity]] 0.60% 0.08% Min Max (Injection ID) (Injection ID) [P.sub.2] estimation 4.52% (155) 9.60% (151) 9.22% (16) 23.95% (1) x estimation 0.40% (179) 0.88% (88) 0.31% (200) 0.81% (123) Table 4. Performance Statistics of Model Prediction Without Considering Transient Data in First 0.03 Second. Normalized Standare Error Mean Deviation [P.sub.2] prediction [e.sub.p2,2] 10.45% 1.48% [e.sub.p2,[infinity]] 19.22% 2.32% x prediction [e.sub.x,2] 18.27% 3.57% [e.sub.x,[infinity]] 22.35% 3.89% Min Max (Injection ID) (Injection ID) [P.sub.2] prediction 7.08% (169) 19.04% (151) 13.85% (137) 29.29% (151) x prediction 7.87% (143) 29.53% (151) 10.46% (143) 33.98% (152)
This research was supported by the National Science Council in Taiwan through Grant NSC 89-2622-E-194-009. Technical support from Victor Taichung Machinery, especially, Mr. F.-S. Liao and H.-A. Kao, is greatly appreciated. The authors express their gratitude to Professor M.-S. Tsai for his suggestion of zero-phase filtering and to the reviewers for their constructive comments.
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J.-W. JOHN CHENG*, TZU-CHING CHAO, LIH-HARNG CHANG, and BO-FENG HUANG
Department of Mechanical Engineering
National Chung Cheng University
Chia-Yi, Taiwan, R.O.C.
*To whom correspondence should be addressed.
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|Author:||Cheng, J.-W. John; Chao, Tzu-Ching; Chang, Lih-Harng; Huang, Bo-Feng|
|Publication:||Polymer Engineering and Science|
|Date:||Sep 1, 2004|
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