Article pubs.acs.org/IECR
Quantitative Analysis of Influences of Model Plant Mismatch on Control Loop Behavior Hong Wang,† Tore Hag̈ glund,‡ and Zhihuan Song*,† †
State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China Department of Automatic Control, Lund University, Lund 22100, Sweden
‡
ABSTRACT: There is normally a mismatch between the current model of the plant and the model that was used for design. This mismatch is normally acceptable when the controller is designed. But if the process dynamics changes, the mismatch may be so large that the control behavior degrades and may be unacceptable. This paper evaluates the influences of model plant mismatch on the control loop behavior quantitatively. A novel method is developed to calculate proposed performance and robustness indices based on the controller and the mismatch. Priorities for maintenance can be made using these indices. Case studies demonstrate the efficacy of the methodology. In addition, the effects of different process dynamics are studied by general analysis. loop avoided interrupting the process. Kano et al.9 proposed a stepwise method to select the explanatory variables in the residual models. A scoring scheme was introduced to measure the significance of the mismatch in each submodel. Selvanathan and Tangirala10 demonstrated an index named plant model ratio in the frequency domain. It can identify the mismatch in corresponding components of a single input single output (SISO) transfer function model, such as gain, time constant, and time delay. It is of great importance in relating the mismatch to physical problems in the plant. The following important step is to assess the influences of the mismatches. Only a few related methods are found in the literature. Badwe et al. studied the influences of MPM on the closed loop performance of IMC according to the relative and designed sensitivity functions, which can be extended to MPC loops.11 Boundaries of the consequences were given. The representations in H∞ as discussed in robust control, however, are conservative.12 Tighter relationships have not been presented in other literature yet. Robustness is another critical aspect when assessing the behavior of a control loop. It is closely related to the stability and safety of the whole system. Therefore robustness should be considered when designing control systems13 and also when studying mismatches. In addition, MPM may result in the improvement or degradation of the control loop behavior which deserves different reactions from engineers. Hence, not only the magnitude but also the direction of the influences should be pointed out in the evaluation of the significance. Evaluation results and prior knowledge are both decisive in the final disposal of mismatches. The counter measures include interrupting the normal production, identifying the model with plant tests, updating the model, retuning the controller, replacing the instruments, and even reconstructing the loop. These measures cost a lot of labor and money, especially when they are
1. INTRODUCTION System maintenance is critical to operations of industrial processes. Owing to fierce competitions and strict demands, the scale and complexity of industrial systems are growing dramatically. Accordingly the maintenance staff has to take care of hundreds of control loops with fewer personnel, as investigated by Kano and Ogawa.1 A system is usually under good control at the early stage after commission. The system dynamics, however, is often changing because of the differences among raw materials, load disturbances, equipment wear, nonlinearities, and so on. These changes can be accompanied by great risks to both profitability and safety of the system.2 Thus it is essential to monitor these changes and evaluate their potential consequences. Plant dynamics is one of the major sources of changes in control systems. Model plant mismatch (MPM) is the difference of dynamics between the model built for the design and that of the current plant. Models can be built by first principles, system identification, or a combination of both. Corresponding studies can be found in books cited.3,4 The real processes containing nonlinearities and high-order dynamics are normally so complicated that it is inevitable to have MPM in the control loops.3 Sometimes it is unnecessary to pursue the exact model because feedback may deal with certain unknown dynamics, as discussed by Åström and Murray.5 MPM, however, may vary so much that it can degrade the control performance6 and push the system to the unstable zone. To improve profitability and reduce the potential risks, these mismatches should be handled opportunely. Detection of the mismatch is the first step for mismatch monitoring. Some approaches have been proposed to detect MPM in recent years. Three specific signatures were utilized to detect mismatch in the system matrices which are used to design model prediction control (MPC) systems by Jiang et al.7 This methodology was first formulated under the framework of state space models. Badwe et al.8 developed a partial correlation approach to analyze the relationship between residuals and manipulated variables and then eliminate the correlations among them. The mismatches were located in specific input−output channels. The utilization of routine operation data under a closed © 2012 American Chemical Society
Received: Revised: Accepted: Published: 15997
March 29, 2012 September 12, 2012 November 7, 2012 November 7, 2012 dx.doi.org/10.1021/ie300834y | Ind. Eng. Chem. Res. 2012, 51, 15997−16006
Industrial & Engineering Chemistry Research
Article
carried out beyond the scheduled maintenance period. Different strategies are needed for different mismatches. These strategies depend on the evaluation results, and the engineers need to arrange the practical schedule. When it comes to hundreds of loops, priorities need to be sorted. In addition, industrial knowledge is also important in the decision making. This knowledge can help us trace the sources of problems in the physical plant. The purpose of this paper is to develop a methodology to evaluate the consequences of MPM and thus sort the priorities for the engineers. A successful maintenance job covers detection, evaluation, and counteractions. Here we focus on the evaluation part, which has not been thoroughly discussed yet. Control performance and loop robustness are two typical aspects considered in the design and operation.14 Here control loop behavior (CLB) including both of them is used to represent the influences of mismatches. To achieve the previous goal, the relationship between MPM and CLB needs to be derived: how much will CLB change when a certain amount of MPM occurs in the loop? First MPM and CLB should be quantified. Their relations are related by the derivation of the indices. Then significance of the mismatches can be compared. Priorities and threshold values for alarms are suggested for the maintenance. Therefore, maintenance is carried out according to the influences of mismatch on loop behavior instead of its magnitude. Examples are used to demonstrate the efficacy of the methodology. Different types of process dynamics are also studied and how the loop behavior is influenced by these differences. The rest of the paper is organized as follows. In section 2, the framework of the research is formulated. Then the normalized indices are presented in section 3. A detailed demonstration of the proposed procedure is given in section 4 followed by an extensive analysis in section 5. Then a general study on different types of processes are discussed in section 6, and some remarks are given in section 7. Section 8 concludes the article.
Figure 2. The schematic of the loop structure in the design stage (r(t), set point; e(t), control error; d(t), load disturbance; v(t), output noise; y(t), process output; Gc, controller; Gd, model of the process in the design stage).
In the monitoring stage, changes of the process may appear after an operation period. We assume that the model structure of the plant is sufficient to describe its dynamics. Then a new model of the current process Gm(s) can be developed. The control loop structure as shown in Figure 3 is used for monitoring. Let Gm(s) = f(θm,s) (θm = [θm1, θp2, ..., θmn]).
Figure 3. The schematic of the loop structure in the monitoring stage (r(t), set point; e(t), control error; d(t), load disturbance; v(t), output noise; y(t), process output; Gc, controller; Gm, model of the process in the monitoring stage).
For MPM, parametric mismatches in transfer functions are considered, such as changes of gain, time constant, time delay, or their combinations. Additive representations of the parametric mismatch is adopted here. They can be transformed into multiplicative mismatch as formulated by Badwe et al.8 As we assume that the model structure is fixed, parametric mismatch is a better choice. There are several other notations of parametric mismatches, such as area index based on Markov parameters15 and state space forms.7 But parametric mismatch in transfer functions is explicit in deriving the expressions for the following analysis, and mismatches based on these parameters are usually related to some physical parameters, e.g., the time delay with the length of the pipe. They are easier to understand and deal with for the engineers. A mismatch between Gm(s) and Gd(s) is denoted as MPM parameter vector ΔMPM = [θMPM1, θMPM2, ..., θMPMn], where ΔMPM = θm − θd. In the mismatch analysis, we assume the controllers are the same in Figures 2 and 3. It is reasonable because the controller is updated only when the loop is not able to keep certain behavior. For CLB, integral absolute error (IAE) and the stability margin (sm) are adopted to measure control performance and loop robustness, respectively. Integral of squared error (ISE), minimum variance (MV), and other time response measures including overshoot, settling time, and rise time, can be used to assess the control performance. IAE, however, is commonly used in controller tuning and when comparing the control performance under deterministic disturbance.16 Therefore, the method to be presented provides a direct derivation from the model and controller to IAE. The control performance measure IAE is defined as
2. PROBLEM FORMULATION AND PROPOSED METHODOLOGY Suppose we have a process P and design a control system as shown in Figure 1. r(t), e(t), d(t), v(t), and y(t) are the set point,
Figure 1. The schematic of the loop structure in an implemented plant (r(t), set point; e(t), control error; d(t), load disturbance; v(t), output noise; y(t), process output; Gc, controller; P, process).
control error, load disturbance, output noise, and process output signals at time t, respectively. The following suffixes “d”, “m”, “c”, and “MPM” indicate parameters or models related to the plant in design stage, the plant in monitoring stage, the controller, and the mismatch, respectively. In the design stage, an approximated model Gd(s) of P is built based on first-principles or system identification. Let Gd(s) = f(θd,s) (θd = [θd1, θd2, ..., θdn]) where θd is the parameter vector and n is the number of parameters in the model). On the basis of this model, a linear time invariant controller Gc(s) is developed. Then we are actually considering the loop shown in Figure 2 for the design. In this methodology, we say that the loop is in the design stage when a new controller is implemented.
IAE =
∫0
t
e(t ) dt
(1)
where e(t) = r(t) − y(t). Load disturbances are the major disturbances in process industries. Therefore, we consider load 15998
dx.doi.org/10.1021/ie300834y | Ind. Eng. Chem. Res. 2012, 51, 15997−16006
Industrial & Engineering Chemistry Research
Article
xth parameter in ΔMPM is defined as δθx: δθx = θmx − θdx. It is normalized as relative variation
disturbance and assume that r(t) is constant. To simplify the expression, the following signals are supposed to be deviations from stationary values. This assumption is true according to the principle of superposition. So r(t) = 0 and d(t) is chosen as a unit step signal. Then e(t) = −y(t) and the transfer function from d(t) to y(t) is Y (s ) =
Gm(s) D(s) 1 + Gm(s)Gc(s)
ηx =
(2)
(3)
By substitution of eq 3 into eq 1, IAE can be directly calculated from the controller and current plant Gm(s) IAE =
∫0
t
⎧ ⎫ Gm(s) ⎬ dt L−1⎨ ⎩ s(1 + Gm(s)Gc(s)) ⎭
ICPI =
(4)
(5)
The sensitivity function is denoted as: S(s) = (1 + L(s))−1 = (1 + Gm(s)Gc(s))−1. The maximum sensitivity is defined as MS = max |S(jω)| which is a single parametric criterion measuring the
0 ≤ ω
