Using the limit cycle measurements, the control transfer function is extracted at the converter output. To demonstrate the effectiveness of the identified model, it is evaluated with a closed-loop response.
N OMENCLATURE
M ATHEMATICAL N OTATIONS
G(s) Transfer function of a process Gm(s) Transfer function of a process model Gc(s) Transfer function of a controller Gf(s) Setpoint filter.
C HAPTER 1
I NTRODUCTION
- Research Background
- Motivation
- Contributions of this Thesis
- Thesis Organization
In literature it is observed that many researchers have used the relay test for identifying the stable and the unstable processes. The performance of the identified converter model is compared with the model obtained using the DIDF method.
C HAPTER 2
I DENTIFICATION OF FOPDT AND
SOPDT PROCESSES USING DUAL - INPUT DESCRIBING FUNCTION (DIDF)
Introduction
The transfer function of the converter is obtained for nominal and long parameter variations. A converter model prototype is developed for real-time implementation of the identification method.
Relay Identification using DIDF method
The general limit cycle output with offset is considered to have a sinusoidal component and a bias component, which is shown in Fig. Here, to andt2 are zero crossings of limit cycle output and t1 is the zero crossing of delayed input.
Limit cycle corrupted with noise
Important information about the limit cycle, such as the peak amplitude and period of the signal, is restored with greater accuracy.
Identification of process models
- FOPDT processes
- SOPDT process model
- Critically damped process
- Underdamped SOPDT process
- Identification of DC-DC buck converter
- Transfer function model using the state-space average method
- Relay feedback test to identify DC-DC buck converter Model
The approximation coefficients during the derivation of the limit cycle with noise are shown in Fig. The derived limit cycle measurements are close to the actual limit cycle measurements. The limit cycle measurements and identified parameters for different SNR levels are given in table 2.12.
The limit cycle output of the relay feedback test is used to identify the dynamic model converter. The changes are due to the variation in peak value and bias in the limit cycle.
Summary
With the help of relay identification, real dynamics of the converter are identified, which further resulted in better response. If there is a degradation of the converter performance over time due to parametric changes, model identification using relay can be performed to identify the converter dynamics. The model validation test proves that the identified model can be used for the analysis of the buck converter.
The identification test gives accurate data about the converter in the limit cycle, which can be used to evaluate the real dynamics of the converter. The results show that the obtained model works well and can be used for converter analysis.
C HAPTER 3
S TATE -S PACE APPROACH FOR THE
SOPDT P ROCESSES
Introduction
Majhi and Atherton [23, 25] used the state-space approach, which requires boundary cycle conditions to derive the expressions for modeling FOPDT and SOPDT processes. In this chapter, the state-space method is used to identify FOPDT, SOPDT, and underdamped SOPDT process models. A general time domain solution is derived which can be used to identify the above processes with few modifications.
The effectiveness of the method is evaluated in simulation and in real time on the DC-DC buck converter.
Identification Structure
Mathematical modeling
During the relay test, the feedback response of the relay is assumed to be symmetrical with the half period Tp as shown in the figure. During the identification test, the relay switches from +hto-hatt0, the delayed relay output shown in the figure.
Identification of Process Models
- SOPDT Process
- FOPDT Process
- Underdamped SOPDT Process
Time delay is measured from limit cycle output as θ =t1−t0 and steady-state gain is assumed to be known. The extracted limit cycle output measurements using wavelet transform for different SNR levels are shown in Table 3.2. It is observed that the second derivative of the peak output (¨y(tp)) of the recovered limit cycle has some error.
The state-space matrices for the underdamped SOPDT model (3.35) are A=. The output equation for the underdamped SOPDT model will be similar to the stable SOPDT model. 3.39) results in the following two equations. Relay feedback test is performed by setting h=1 and a sustained limit cycle output is obtained.
Identification of Buck converter
- Validation of the identified model
There are small variations in the dynamics between the identified and the average model. The effectiveness of the relay feedback test is also verified experimentally, and the schematic diagram of the identification test is shown in Fig. The performance of the identified model from simulation (3.43) and experiment (3.45) using the relay is validated with the step response and compared with the response obtained from the average model (3.42).
The response of the identified model from the experiment and the average model are shown in the figure. The settling time of the identified model is 4.86 ms, and 6.64 ms for the average model.
Summary
C HAPTER 4
I DENTIFICATION AND C ONTROL OF
FOPDT AND SOPDT P ROCESSES
Introduction
A compromise is achieved by using two closed-loop controllers or a setpoint filter along with the controller. Three controllers were present in the 2-DoF structure, one to stabilize the process and the other two to track set points or reject load disturbances. Vijayan and Panda [53] proposed double feedback loops with a setpoint filter to achieve better closed-loop performance.
Here the internal feedback is used to stabilize the process, the outer loop for setpoint tracking and the setpoint filter for reducing peak overshoot. The above works do not directly consider the performance evaluation for the selection of the desired tuning parameter.
Proposed feedback structure
The desired tuning parameter required for the controller design is chosen depending on time delay or by curve fitting [79, 81]. An attempt is made in this chapter to determine the desired tuning parameter with the best performance evaluation. Using the limit cycle information, simple explicit expressions are derived to obtain accurate dynamics of the processes.
Once the process parameters are identified, the controller (PID) is designed using the direct synthesis method. The controller structure is chosen as Gc(s) =KP. 4.1) The setpoint filter with the setpoint coefficient (β) is given by.
Identification and control of FOPDT process
- Expressions for process parameters
- Controller Design
Using simple expressions, the unknown process parameters for stable and unstable FOPDT process are obtained. Substituting the process and controller transfer functions into (4.15) and comparing the denominator with (4.14), we obtain the expressions for controller parameters. The setpoint coefficient β is used to track the reference, its range is between 0 and 1.
By conducting extensive simulation studies, a set point coefficient with less peak overshoot is determined.
Identification and control of SOPDT process
- Expressions for process parameters
- Controller Design
Limit cycle data such as Ap, t1, tp andt2 are measured and equations and (4.43) are simultaneously solved to estimate the unknown parameters τ1,τ2 and K. A direct synthesis method is used to design the PID controller primarily for satisfactory load disturbance rejection. To achieve the required control objective, the PID parameters are tuned by the closed-loop transfer function derived with respect to the load disturbance.
Substituting the corresponding models G, Gc and comparing the parts of the denominator, we get the expression. As expected and verified by extensive simulation studies, β in (4.50) results in greater peak overshoot in the setpoint response.
Estimation of tuning parameter using PSO
The inertia weight factorω actually leads to local and global search point and is given. The constraint during the search process is to maintain the value of Me within the specified range.
Simulation results
- Example 4
- Example 5
- Example 6
During the load disturbance, the proposed method shows slightly more undershoot but settles faster than Padma Sree et al. The proposed method provides good damping of load disturbances and the performance indices are shown in Table 4.3. The controller responses for the proposed method and the Vijayan and Panda method [53] with a load perturbation in unit steps of magnitude -0.5 at timet = 40 sec.
The proposed method has good load disturbance rejection and the performance measures are given in Table 4.6. But during load disturbance, the proposed method has lower ISE value than that proposed by Vijayan and Panda [53].
Summary
It is clear that the proposed method yields improved responses during load disturbances apart from the satisfactory set point response. To check the robustness of the controller, the time constants of the process are varied up to +10% and the response is shown in Fig. The proposed method quickly follows the reference during perturbation compared to Nema and Padhy PSO PI-PD [78].
Some of the examples in the literature are considered to show the effectiveness of the proposed identification and control method. However, there is a slight degradation of the setpoint response because the purpose of the regulator design is to dampen the load disturbance.
C HAPTER 5
C ONCLUSIONS AND F UTURE W ORK
Conclusions
State-space approach for identification of FOPDT and SOPDT processes Models evaluated using the DIDF method give approximate results. Mathematical expressions to determine the process parameters of stable/unstable FOPDT and SOPDT processes are derived for the off-line method. Process models are identified using the state space method and using the identified process parameters a model based controller is designed.
The PID controller is designed using the direct synthesis method primarily to reject load disturbances. To achieve better setpoint tracking, the design of the setpoint coefficient can be modified.
Scope for further work
In addition, the estimated parametric DC-DC converter model has better closed-loop performance. The model verification test shows that the identified model can perform well even under parameter perturbations. Some standard well-known examples are considered to demonstrate the effectiveness of the proposed method.
It is clear from the performance with closed loop that there is good attenuation of load disturbances. The DIDF method can be extended to identify the process models in the presence of disturbance/offset with better accuracy.
L IST OF P UBLICATIONS
A PPENDIX A
S UPPLEMENTARY M ATERIALS
State-space average Modelling
Detailed derivation of the expressions (3.14) and (3.15)
Detailed derivation of the expressions (4.29) and (4.30)
R EFERENCES
Mandal, "Parameter estimation of integration and time-delay processes using single relay feedback test," ISA Transactions, vol. Wang, "A systematic approach for on-line identification of second-order process model from relay feedback test," AIChE Journal, vol. Gao, “Alternative identification algorithms for obtaining a first-order stable/unstable process model from a single relay feedback test,” Ind.
Majhi, “Identification of FOPDT and SOPDT Process Dynamics Using a Closed-Loop Test,” ISA Transactions, vol. Maksimovic, “System identification of power converters with digital control using cross-correlation methods,” IEEE Trans.
