identification schemes for modelling of dead time processes

Important limit cycle information at the process output is measured and then substituted into a derived set of mathematical expressions for the identification of time delay processes. Pandey S., Majhi S., "Limit cycle-based identification of second-order processes with time delay," in Proc.

System Identification

Relay Control Systems and System Identification

Frequency domain based identification algorithms require multiple relay feedback trials to identify different processes. 42] proposed a feedback relay test to estimate the model parameters of the integration process and dead time.

Figure 1.1: Various characteristics of relay

Motivation

Contributions of this Thesis

Therefore, the relay feedback scheme was extended for modeling and identification of non-minimum phase processes with time delay. From the obtained process output information, a general set of mathematical expressions is derived for identification of a class of time delay processes in terms of non-minimum phase-stable and unstable SOPTD, non-minimum phase-integrating FOPTD process models.

Thesis Organization

Finally, the simulation results are included and compared with the models reported in the literature to validate the proposed relay-based identification algorithms. Shenet et al.[21] proposed a dual-input describing function (DIDF) approach for estimating two points on the Nyquist curve.

Proposed Identification Scheme

Therefore, for a better identification accuracy, the dynamic real-time processes are generally modeled as a dead-time process model. In relay feedback theory, the dead time of the process has played a key role in the generation of persistent oscillations/limit cycle, especially for lower order dynamical systems.

Figure 2.1: A schematic representation of relay feedback test

Frequency Domain Based Mathematical Expressions

FOPTD process model
SOPTD process model
SOPTD process model with repeated poles
Integrating FOPTD process model
PIPTD process model
Non-minimum phase FOPTD process model
Non-minimum phase SOPTD process model with repeated poles
Non-minimum phase integrating FOPTD process model

The general transfer function for a non-minimum stage SOPTD process model with repetitive pole is written as. Using (2.12), the steady-state gain (k) of the non-minimum-phase repeated-pole SOPTD process model is derived.

Figure 2.4 represents the typical limit cycle for a general SOPTD process. The process output

Reconstruction of Limit Cycle in Presence of Noise

Results and Discussion

Example 1: Stable SOPTD process
Example 2: Unstable SOPTD process
Example 3: SOPTD process with repeated poles
Example 4: Integrating FOPTD process
Example 5: Non-minimum phase stable FOPTD process
Example 6: Non-minimum phase higher order process
Example 7: Integrating higher order process

In the presence of measurement noise, a Fourier series based curve fitting technique is implemented to achieve a clean limit cycle. By substituting the limit cycle information into (2.40) and (2.41) and further solving these expressions, Table 2.5: Comparison of identified process models for example 5. When a process output is subjected to measurement noise, a clean limit cycle is recovered using a Fourier series based curve fitting method.

Figure 2.6: Plots for reconstruction of SOPTD process output using curve fitting method: (i) noisy data (ii) fitted data

Summary

Therefore, researchers have investigated time domain analysis to bring better accuracy to the estimation of process model parameters. The proposed set of mathematical expressions includes the limit cycle information, which brings accuracy to the estimation of the model parameters of the assumed transfer function. Then, a recovered limit cycle and its slope parameters are substituted into the derived set of explicit expressions for estimating the parameters of the original process model, respectively.

Proposed Identification Scheme

Henceforth, the process identification is carried out using the proposed methodology, which provides an exact estimation of the installation parameters.

Reconstruction of Limit Cycle in Presence of Disturbances

Minimization of static load disturbance

Therefore, attempts have been made to recover the original limit cycle using a signal processing technique, Fast Fourier Transform (FFT). The relay settings ensure that the overall system provides limit cycling rather than forced oscillations. In addition, due to the addition of PI controller, the modified closed-loop transfer function changes the behavior of limit cycle oscillation.

Mitigation of measurement noise

The proposed methodology can be implemented in real-time plants in the presence of static load disturbance as the effect of such disturbances can be easily nullified and furthermore can also be estimated using the expression in (3.1). The proposed filtering method uses MATLAB [67] based data processing of sample points in limit cycle output. Next, a clean limit cycle is derived from Inverse Fast Fourier Transform (IFFT) of threshold signal that brings a bit.

Figure 3.2: Plots for influence of measurement noise in FOPTD process: (i) relay output (ii) process output

State Space Based Mathematical Expressions

FOPTD process model

Thus, the expression for the time delay parameter (θ) for the FOPTD process model can be obtained from (3.13) and (3.14) as. Considering the first time derivative of (3.16) and substituting expressions for λ1 and ˜yields. 3.18) Subsequently, the expression for the first time derivative term at t=t2 =T can be written as. Furthermore, the expression for the first derivative term att=t3 can be written using (3.8) as.

PIPTD process model

Design of Model Based Controllers

PI tuning rules for stable FOPTD process model

The closed-loop errors of the FOPTD process model from its step response are expressed as. The integrated gain of controller Ki is derived using inertia index [75] from the ratio of response time measures with open and. Finally, both the average residence time obtained from open and closed loop transfer functions are equated to give the integral gainKi as.

PI-P tuning rules for unstable FOPTD process model

While substituting the error expressions of (3.42) into (3.34), one of the controller parameters is derived in terms of normalized time delay (˜θn=θn/α˜1) as.

PI-P tuning rules for PIPTD process model

Results and Discussion

Example 1: Stable FOPTD process
Example 2: Stable FOPTD process
Example 3: Unstable FOPTD process
Example 4: Pure integrating process with large time delay
Example 5: Pure integrating process with small time delay
Example 6: Higher order process

Subsequently, the recovered limit cycle information is substituted into the derived set of explicit expressions and (3.22) for the evaluation of the unsteady process model parameters. An asymmetric relay with amplitudes (h1 =1 and gh2 =−1.2) is fed back to an integrating process to induce the persistent asymmetric limit cycle at the process output. The limit cycle at the process output is achieved using an asymmetric relay setting (h1 = 1 and h2 = -1.1).

Table 3.1: Comparison of identified process models for Example 1

Summary

Soon after, Majhi [40] proposed an ideal relay feedback experiment to model and identify a class of time-delayed processes using a state space approach. Padhy and Majhi [80] derived a set of analytical expressions to identify non-minimum phase processes with time delay using an ideal relay. Bajarangbali and Majhi [47] proposed another set of nonlinear equations to identify a class of time-delayed processes with non-minimum phase behavior using an ideal hysteresis relay to help minimize the effects of measurement noise during identification.

Proposed Identification Scheme

State Space Based Mathematical Expressions

Non-minimum phase SOPTD process model
Non-minimum phase integrating FOPTD process model
SOPTD process model
Integrating FOPTD process model

Repeating the above procedure, the expression for the peak amplitude y(t1) occurring at t=t1 is represented as. 4.18) Now, one of the unknown parameters of the non-minimum phase stable and unstable SOPTD process models is obtained in terms of the sum of eigenvalues as. Expressions for another unknown parameter of non-minimum stable and unstable SOPTD process models (α2) are derived in terms of the product of eigenvalues as. A general transfer function model for the SOPTD process is written from the non-minimum phase SOPTD.

Figure 4.3 shows a typical limit cycle for stable SOPTD process where t 0 , t 2 , t 4 represent time instants of zero crossings, t 1 , t 3 refer to peak instants in upper and lower direction and T, T p signify the time for which y(t) remains greater than s

Results and Discussion

Example 1: Non-minimum phase stable SOPTD process
Example 2: Non-minimum phase unstable SOPTD process
Example 3: Non-minimum phase integrating FOPTD process
Example 4: Stable SOPTD process
Example 5: Unstable SOPTD process
Example 6: Integrating FOPTD process

Again, the recovered limit cycle information is substituted into the derived set of explicit expressions to estimate the parameters of the process model. Further, the pure limit cycle is derived using a curve fitting method based on Fourier series and Table 4.3: Comparison of identified process models with/without noise for Case 3. The effects of parametric error in the estimated parameters of the process model are compared with models reported in the literature and measurements of noise in table 4.4 and table 4.5.

Table 4.1: Comparison of identified process models with/without noise for Example 1

Summary

Being a time-efficient method, the relay feedback experiment plays a key role in the modeling and identification of industrial processes. In this chapter, we have tried to propose a modified relay feedback test for modeling and identifying time lag processes. While considering the non-zero set point, an attempt is made to modify the relay feedback to induce sustained oscillations in a class of time delay processes around the set value.

Figure 5.1: Proposed relay feedback experiment for process identification

State Space Based Mathematical Expressions

FOPTD process model
FOPTD process model with zero
SOPTD process model with repeated poles
SOPTD process model with repeated poles and zero

By solving the state space equation, we write down the expression of the output of the process for the time range t0 ≤t < θ as. The explicit representation of one of the unknown parameters (αp) of the SOPTDZ process model is derived from the expression β1 as an angle. In addition, the expression for αz can also be obtained for the SOPTDZ process model using the maximum output time of the process as

Results and Discussion

Example 1: FOPTD process
Example 2: SOPTD process with repeated poles
Example 3: Non-minimum phase FOPTD process
Example 4: Higher order process

During identification, an asymmetric relay with hysteresis (ε=±0.55) causes a biased limit cycle at the output of the process around the set point R= 2. Finally, the given information about the limit cycle is replaced in a derived set of mathematical expressions to estimate the unknown process model parameters. During the relay-based identification test, a symmetrical limit cycle occurs at the process output around the setpoint (R = 1) using the relay settings (h = 2).

Table 5.1: Comparison of identified process models for Example 1

Summary

Applications to Level Control System

Physical description and working

Identification tests and model validation

Using the Fourier series-based curve fitting method, a clean limit cycle is recovered as shown in Figure 6.4. Then the frequency domain (FD) and state space (SS) based mathematical expressions derived in previous chapters 2-5 are used for modeling and identification of level control system in terms of FOPTD, SOPTD and non-minimum phase transfer function models. The ultimate frequency of identified process model lies close to the operating point in Figure 6.5 compared to ZN method.

Figure 6.1: Physical layout of YOKOGAWA level control system

Applications to Coupled Tanks System

Physical description and working

Now the physical modeling of the single and double tank model is carried out using the mass balance equation. Using linearization, the nonlinear models for one and two tanks of the coupled reservoir system are presented in linear transfer function form. Taking into account the initial working points, the expressions for the water levels in each reservoir are written as.

Figure 6.8: Experimental set-up of coupled tanks system by FEEDBACK Instruments

Identification tests and model validation

Using the frequency domain-based identification algorithm, the obtained limit cycle information is replaced into the derived set of mathematical expressions for modeling and identifying single and two coupled tank systems in terms of first- and second-order time delay models. Therefore, the limit cycle information is further replaced in the state space-based explicit expressions for the estimation of accurate process model parameters for each tank of a coupled tank system. Finally, the comparison between the identified process models using the state space approach and frequency domain-based identification algorithms is shown in Table 6.2.

Figure 6.9: Plots for coupled tanks system responses: (i) setpoint (ii) Tank 1 output (iii) relay output

Summary

Therefore, a state space approach is used for an accurate modeling and identification of a class of time-delay processes under relay-feedback control. The proposed set of expressions yields better accuracy in the identification of time delay processes. It was observed in Chapter 3 that the state space based identification method brings better accuracy in the estimation of process model parameters.

Suggestions for further work

Based on the process output, the relay switches twice in one full limit cycle period as shown in figure f-1. In general, processes are assumed to exhibit low-pass filter characteristics, therefore, the equivalent gain of an asymmetric relay is derived by considering the main harmonic components available in the relay output signal. Substituting the expressions ℵ1 from (A-10) and ℵ2 from (A-12) to (A-8) to obtain the equivalent gain of an asymmetric relay as.

Figure f-1: Plots for SOPTD process: (i) relay output (ii) relay input

Detailed derivation of the expressions (4.4), (4.5) and (4.6)

Various characteristics of relay
A schematic representation of relay feedback test
Nyquist plots for FOPTD process models: (i) without delay (ii) with delay
Plots for FOPTD process: (i) relay output (ii) process output
Plots for SOPTD process: (i) relay output (ii) process output (iii) second derivative of
Plots for non-minimum phase SOPTD process: (i) relay output (ii) process output
Plots for reconstruction of SOPTD process output using curve fitting method: (i) noisy
Nyquist plots for Example 1: (i) actual process (ii) proposed FOPTD model without
Nyquist plots for Example 1: (i) actual process (ii) proposed SOPTD model without
Plots for Example 2: (i) relay output (ii) process output (iii) second derivative of process
Plots for Example 5: (i) relay output (ii) process output
A schematic representation of relay feedback test
Plots for influence of measurement noise in FOPTD process: (i) relay output (ii) process
Plots for FOPTD process: (i) relay output (ii) process output (iii) first derivative of
Proposed identification and control schemes for stable and unstable processes
Comparison of identified process models for Example 1
Comparison of controller performances for Example 2
Comparison of identified process models with/without noise for Example 1
Comparison of the identified process models for level control system

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