Ziegler Nichols Method

Four Types of Controllers

George Ellis , in Control System Design Guide (Fourth Edition), 2012

6.4.1.3 The Ziegler–Nichols Method

A popular method for tuning P, PI, and PID controllers is the Ziegler–Nichols method. This method starts by zeroing the integral and differential gains and then raising the proportional gain until the system is unstable. The value of KP at the point of instability is called K MAX; the frequency of oscillation is f 0. The method then backs off the proportional gain a predetermined amount and sets the integral and differential gains as a function of f 0. The P, I, and D gains are set according to Table 6.1 32 .

Table 6.1. Settings for P, I, and D Gains According to the Ziegler–Nichols Method

K P K I K D
P controller 0.5 K MAX 0 0
PI controller 0.45 K MAX 1.2 f 0 0
PID controller 0.6 K MAX 2.0 f 0 0.125/f 0

If a dynamic signal analyzer is available to measure the GM and phase crossover frequency, there is no need to raise the gain all the way to instability. Instead, raise the gain until the system is near instability, measure the GM, and add the GM to the gain to find K MAX. For example, if a gain of 2 had a GM of 12 dB (a factor of 4), K MAX would be 2 plus 12 dB, or 2 times 4, or 8. Use the phase crossover frequency for f 0. A flowchart for the Ziegler–Nichols method is shown in Figure 6.15.

Figure 6.15. Ziegler–Nichols method for tuning P, PI, and PID controllers.

Note that the form shown here assumes KP is in series with KI and KD . For cases where the three paths are in parallel, be sure to add a factor of KP to the formulas for KI and KD in Table 6.1 and Figure 6.15. Note, also, that these formulas make no assumption about the units of KP , but KI and KD must be in SI units (rad/sec and sec/rad, respectively). This is the case for the Visual ModelQ models of this chapter, but often is not the case for industrial controllers which may use units of Hz, seconds, or non-standard units. Finally, the Ziegler–Nichols method is frequently shown using T 0, period of oscillation, instead of the f 0 when KP   = K MAX; of course, T 0  =   1/f 0.

The Ziegler–Nichols method is too aggressive for many industrial control systems. For example, for a proportional controller, the method specifies a GM of just 6 dB, compared with the 12 dB in the P controller tuned earlier in this chapter (Figure 6.5). In general, the gains from Ziegler–Nichols will be higher than from the methods presented here. Table 6.2 shows a comparison of tuning the P, PI, and PID controllers according to the method in this chapter and the Ziegler–Nichols method. (The value K MAX  =   4.8 and f 0  =   311 Hz were found using Experiment 6A.) Both sets of gains are stable, but the Ziegler–Nichols method provides smaller stability margins.

Table 6.2. Comparison of Results from Tuning Method in this Chapter and the Ziegler–Nichols Method

Method of Chapter 6 Ziegler–Nichols Method KMAX  =   4.8 and f 0  =   311 Hz
P controller KP   =   1.2 KP   =   2.4
PI controller KP   =   1.2 KP   =   2.2
KI   =   100 KI   =   373
PID controller KP   =   1.7 KP   =   2.9
KI   =   120 KI   =   622
KD   =   0.0002 KD   =   0.0004

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123859204000060

28th European Symposium on Computer Aided Process Engineering

Paisan Kittisupakorn , ... Ajaree Suwatthikul , in Computer Aided Chemical Engineering, 2018

4.2 The response of the closed-loop system

Figure 3 and 4 show the response of the process control by PID and MPC controllers respectively. The PID tuning parameters are determined by a Ziegler-Nichols method and then fine-tuned by a trial and error approach. It can be seen that the PID controller can control the concentration of neodymium ions at the set point with the IAE of 2069.5. The manipulated variable, feed flowrate, is adjusted and reduced to a constant value.

Figure 3

Figure 3. The neodymium ions concentration under the PID controller in the nominal case

Figure 4

Figure 4. The neodymium ions concentration under the MPC controller in the nominal case

The MPC controller can control the concentration of the extracted neodymium ions at the set point with the IAE of 1928.2. and can provide tracking of the set point faster than the PID does.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444642356501704

21st European Symposium on Computer Aided Process Engineering

José Luis de la Mata , Manuel Rodríguez , in Computer Aided Chemical Engineering, 2011

3 Industrial example

3.1 Process description

The process consists of a jacketed reactor, a cooler and a storage tank. An exothermic reaction of isomerization in liquid phase is carried out in the reactor. Cooling water is poured through the jacket to keep reactor temperature. The cooler is a shell and tube heat exchanger with one shell pass and two tubes passes. There are also five control loops controlling the reactor level and temperature, cooler outlet temperature, tank level and recycle flow. The P&ID of the process is shown in Fig. 1.

Fig. 1. Process P&I diagram.

3.2 Model description

The reactor and the tank are modeled using mass and energy balances. In both cases perfect mixing is considered. The cooler has been modeled using a discretization of its internal flow dynamics. The five control loops are considered as PIs and they are tuned using the Ziegler-Nichols method. The overall mathematical system consists of 41 EDOs, 54 constitutive equations and 10 flow constraints. There are 43 variables of interest in the process concerning flows, molar fractions and temperatures.

3.3 Fault analysis

3.3.1 Fouling in the reactor jacket

This is a particularly common and severe problem because the heat transfer decreases and the reaction can run away. The time evolution of the residuals is depicted in Fig. 2. Notice that only the most relevant residuals are shown in the figure.

Fig. 2. Fouling in the reactor jacket residuals.

There are two plots, the first one (left) shows the residuals until they reach the steady state and the second one (right) shows the first three minutes of the simulation. The parameters of this fault are:

•

Higher maximum: Cooling water flow.

•

Lower maximum time: B molar fraction in the reactor.

•

Higher initial slope: Jacket temperature.

•

Higher steady-state value: Cooling water flow.

Note that in the steady state variables such as reactor temperature and molar fractions in the reactor go back to zero residual. This is because the control system keeps the reactor variables under control. However, the flow to the jacket and the jacket temperature have a non-zero residual in steady state. This information is consistent with the fault described.

3.3.2 Fault in the reactor level sensor

The faulty sensor provides a level lower than the real. The time evolution of the residuals is shown in Fig. 3 and its parameters are:

Fig. 3. Residuals of a fault in the reactor level sensor.
•

Higher maximum: Reactor outlet flow.

•

Lower maximum time: Reactor outlet flow.

•

Higher initial slope: Reactor outlet flow.

•

Higher steady-state value: Cooling water flow.

The residuals shown in Fig. 3 are consistent with the fault described. At first the outlet flow varies a lot to compensate the difference from the level set point (control system action) and then, the cooling water flow also increases to compensate the increment of volume in the reactor.

3.3.3 On-line detection

Comparing figures 2 and 3 is obvious that the residuals of both faults are different and the parameters used to characterize them also differ. So if during normal operation of the plant we detect the pattern of any of these faults, we can identify them and corrective actions can be performed.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444537119501772

29th European Symposium on Computer Aided Process Engineering

Tibor Nagy , ... Peter Mizsey , in Computer Aided Chemical Engineering, 2019

2 Methods and experimental Setup

In our work we define two cases:

•

case A: purifying chlorobenzene from impurities

•

case B :separating n-heptane and hexane

The column (see figure 1) for case A glass wall distillation column with the height of 30 cm height of Mellapak structured packing from Sulzer Chemtech and for case B it is built of two packed stages filled with random glass raschig ring type packings. In both cases the top of the column includes a gas vapor separator below the condenser. The condensed liquid is driven out of the column to a glass valve that is able vary the reflux ratio based on its on/off time ratio. The glass valve is driven by a 24 V solenoid valve and mosfet structure driven by the controller digital output. Preliminary tests showed, that the minimal time of stay/state should set to be at least 2 seconds. For normal operation we set the periodic time to be 10 seconds.

Figure 1

Figure 1. Distillation column and control scheme

The reboiler of the distillation column is a 750 ml triple neck type round-bottom flask filled to with 500 ml of the initial liquid mixture. The heat is applied at the bottom by a heating basket by Bovimex MBO that is powered by 50 Hz 230 V AC and actuated with a solid state relay (SSR) that is regulated by the digital output of the Arduino board with the periodic time of 2 seconds. As the solid state relay is only capable of cutting power at 0 V crossings of the power source, 2 seconds is chosen as time period so the minimal resolution of the power adjustment is 1%.

In case of distillation, the temperature measurement should be particularly precise as in such case it is the process variable (PV) that gives the basis of the quality of control. In the investigated scenario, the temperature range is set to 0-200 °C with a precision range of +/- 0.1 °C. In order to satisfy these, RTD PT100 temperature sensors are applied with two approaches. The measured signal is either applied to a MAX31865 temperature sensor amplifier DAC from Adafruit or is sent to a signal transmitter to form 4-20 mA analog signals. In case of the utilization of MAX31865, in order to receive the most accurate signals the measured resistance is continually compared to a lookup table including resistance and temperature value pairs of DIN 43760/IEC 751 standard. The thermometer verification is carried out by the boiling of penthane at atmospheric pressure (36.1 ° C) and as the steady state is achieved, the thermometer oscillates within   +/- 0.1 ° C range. In case of the analog signal processing two point temperature verification is carried out, by adjusting the range minimum and span of the transmitter.

The microcontrollers utilized are Arduino UNO and Due that are particularly popular for process control studies Rubio-Gomez et al. (2019). In case of analog signal input the transmitted signal is converted to 0-3 V to match the controller input range. These analog input reads are then refined with running average that takes 200 samples and gives two digital signals/second. Such frequency of data acquisition in such system is sufficient and will not cause stability problems. In case of digital input, SPI communication is utilized. Output signals are connected to the digital output pins with PWN. The Arduino is programmable by integrated development environment (IDE) with a PC by defining the variables of the different input and output pin-s, setpoints and generating the control algorithm etc. In other cases it is also possible to retransmit the Arduino's I/O and perform the data acquisition and control at a higher level.

As for controller tuning, different methods are available. In order to test the built system and closed loop a number of tuning methods and setups are investigated, these are:

•

PID applying step response (open Z-N tuning) tuning

•

PID applying Matlab PID tuner

•

Cohen Coon method

2.1 Open Loop Ziegler-Nichols Method

The Ziegler-Nichols Method is one of the best known tuning methods, that is based an open loop step-response function of time (Ziegler (1942)). The produced response function is approached by a first order transfer function with dead time. Applying the extracted parameters (K, τ and D s ) heuristic tables are utilized fortuning.

2.2 Cohen Coon method

The Cohen Coon tuning method also approximates the process with a first order element with dead time, but it is calculated differently (Cohen and Coon (1953)). Some literature favor the Cohen Coon method as it is more flexible with dead time compared to the Z-N method. To determine the tuning parameters just like with Z-N method, tuning tables are used.

2.3 Matlab PID tuner

Matlab PID tuner is an application of Matlab software that allows the instant tuning of a processes. The tuner application includes process identification in which the One pole + delay model is used. The application automatically calculates the linear model of the process and it provides interactive tuning where performance and robustness can be balanced. In our case the control loop is tuned for both performance and robustness to have only a moderate overshoot and after damping to settle on the new setpoint.

2.4 Model based control

A model based composition control is carried out with the model mixture of n-heptane and hexane. The internal model is divided to a reflux plan and the reflux control part. In order to achieve the planed product purity the number of theoretical plate (NTP) for the column is measured. In order to keep the composition of the distillate constant, the slope of the distillation working line must change in time and thus the reflux ratio changes as well. The reflux control part executes the process including startup, setting the correct reflux ratio to the corresponding reboiled temperature and the shutdown procedure.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128186343503040

Optimal fuzzy controller parameters using PSO for speed control of Quasi-Z Source DC/DC converter fed drive

M. Ranjani , P. Murugesan , in Applied Soft Computing, 2015

1 Introduction

In general the DC motor drives are good choice for high performance motion control systems requiring four-quadrant operation that includes field weakening, minimum torque ripple, rapid speed recovery under impact load torque and in addition to fast dynamic torque and speed responses. So far PI controller is most widely used controller for DC drives [1]. Over these years, many controlling techniques such as P, PI and PID controllers have been developed. One of the most popular controllers used in industrial processes is the PID controller [2]. The advantage of this controller is its simplicity to understand and to implement [3]. The main drawback of using PID controller is the effect acquired as a result of disturbances and environmental conditions on the structure of the system [4] that adds complexity to the controller design. Mainly adaptive PI controllers avoid time consuming manual tuning by providing optimal PI controller settings automatically as the system dynamics or the operating points change [5] . There are various methods for tuning of PI controllers namely Trial and error, Continuous cycling method (Ziegler Nichols method), Process Reaction Curve methods (Ziegler–Nichols and Cohen–Coon methods), Ziegler–Nichols method (both types of responses) and Cohen–Coon method (self-regulating response only). The above stated methods lead to time consumption if a large number of trials are required or if the process dynamics are slow. Again unstable operation or hazardous situation may result if any external disturbances or a change in the process occurs during tuning of the controller.

Fuzzy logic, the logic of approximate reasoning, continues to grow in importance, as it provides an inexpensive solution for controlling ill-known complex systems. Fuzzy controllers have received adequate attention in motion control systems as they possess non-linear characteristics and a precise model is most often unknown. Fuzzy controller is already applied to phased controlled converter fed DC drive, resonant converter fed DC drive and linear servo drive [6] and induction motor drive [7,8]. Generally the fuzzy logic employed in control system design where human expert knowledge rather than precise mathematical modeling of a process or plant is used to model or implements the required controller. Uncertainty and ambiguity are evident in many engineering problems. Fuzzy logic controller therefore provides a formal method of translating imprecise human knowledge into control strategies. Optimal design of FLC knowledge base is central to performance of FLC. In absence of such knowledge, a common approach is to optimize these FLC parameters through a process of trial and error with respect to performance of the system [9–11].

On the other hand the tuning of fuzzy controller is a heuristic work. To eliminate such problems, the evolutionary techniques have been applied in tuning of FLC parameters. Emerging Computational intelligence based techniques such as, genetic algorithm (GA) and Particle Swarm Optimization (PSO) can be the solutions to above stated problems. GA [35] is search technique used in computer science and engineering to find the approximate solutions to optimization problems [12–14]. GA incorporates a particular class of evolutionary algorithms that uses techniques inspired by evolutionary biology such as inheritance, mutation, natural selection and recombination (crossover) [15,16].

The performance of GF algorithm based controller for Quasi-resonant converter for drive applications presented by Ranjani and Murugesan reduces the transient response of speed, switching stresses and losses but whereas the converter performs efficiently for narrow bandwidth ranges [62]. Even it can rapidly locate good solutions for difficult search spaces, it has some drawbacks associated with it: unless the fitness function is defined properly, GA has a tendency to converge toward local optima rather than global optimum of the problem. In addition operating on dynamic data sets is difficult for specific optimization problems. In this paper, Z-Source and Quasi-Z-Source converters are considered for wide range of operating frequencies also the PSO outperforms the GAs performance by finding better solutions for same amount of computation time. The Takagi–Sugeno type fuzzy-genetic controller for boost PFC converter described by Dogan et al. regulates output DC voltage and a hysteresis current controller that force the input current to track a sinusoidal reference. The steady state and transient performance are presented in simulation results [63]. In this paper the performance of the DC drive for various ranges of load torque values fed by GA and PSO optimized fuzzy controller are analyzed by considering the simulation and experimental results.

PSO is another evolutionary computation technique developed by Eberhart and Kennedy [23,26] in 1995, which was inspired by the social behavior of bird flocking and fish schooling. PSO has its roots in artificial life and social psychology, as well as in engineering and computer science. It utilizes a "population" of particles that fly through the problem hyperspace with the given velocities. At each iteration, the velocities of the individual particles are stochastically adjusted according to the historical best position for the particle itself and the neighborhood best position. Both the particle best and the neighborhood best are derived according to a user defined fitness function [54–56]. The movement of each particle naturally evolves to an optimal or near-optimal solution. Clerk and Kennedy presented the particle swarm explosion, stability and convergence in a multidimensional complex space [24]. Here the word "Swarm" comes from the irregular movements of the particles in the problem space, now more similar to a swarm of mosquitoes rather than a flock of birds or a school of fish.

Extensively the Particle Swarm Optimization (PSO) has been applied to optimization of FLC [19–23]. The Particle Swarm Optimization for fuzzy membership functions was applied by Esmin et al. [21]. Also Ghoshal [18] presented the optimization of PID gains by Particle Swarm Optimizations in fuzzy-based automatic generation control. A particle swarm-optimized fuzzy-neural network for voice-controlled robot systems was discussed by Chatterjee et al. [19]. The comparison between PSO and GA for parameter optimization of PID controller by Ou and Lin [17] infers that PSO outperforms GA by providing global optimal solution. Del Valle et al. [65] applied the Particle Swarm Optimization with its basic concepts, variants in power systems. It provides the technical details such as its type, practical formulation and efficient fitness function required for application of PSO in power system to effectively solve large scale nonlinear optimization problems [65]. This paper invokes the basic concepts and technical details for application of PSO for Fuzzy PI tuning of DC–DC converter fed DC drive speed control. Also it is evident that heuristics based swarm intelligence serves as an efficient alternative for analytical methods which suffer from slow convergence and the curse of dimensionality.

A PSO-Lyapunov hybrid stable adaptive fuzzy tracking control approach for vision-based robot navigation was developed by Das Sharma (2012). He also discussed about a random spatial Ibest PSO-based hybrid strategy for designing adaptive fuzzy controllers for a class of nonlinear systems.

In recent years PSO has gained much popularity in different kinds of applications because of its simplicity, easy implementation and reliable convergence [17,18]. A robust control method for solving continuous non-linear optimization problems linear for uncertain chaotic systems via an optimal type-2 fuzzy proportional integral derivative controller was found by Khooban (2013). Parallel Particle Swarm Optimization with parameters adaptation using fuzzy logic explained by Fevrier Vaaldez (2013) is a computational intelligence-based technique that is not largely affected by the size and nonlinearity of the problem and can converge to the optimal solution in many problems where most analytical methods fail to converge. Hence it can be effectively applied to different optimization problems in power electronics. An optimal fuzzy logic guidance law using Particle Swarm Optimization presented by Labeed Hassan (2013) has some advantages over other similar optimization techniques such as GA. In PSO, every particle remembers its own previous best values as well as neighborhood best and therefore PSO has more effective memory capability than that of GA. In addition PSO is more efficient in maintaining the diversity of the swarm [57–61], since all the particles use the information related to the most successful particle in order to improve them, whereas in GA, the worse solutions are discarded and only the good ones are saved.

Sedaghati and Babei [64] presented the different states of operation of double input Z-Source DC–DC converters, the input DC voltage can be boosted and deliver the power to load individually or the input DC sources may be combination of new energy sources. Only the steady state operations of converter are analyzed through simulation results [64]. This paper focuses on the closed loop performance of the double input Z-Source DC–DC converters in drive application with the various combinations of intelligent controllers and the performance are evaluated through practical and simulation results.

The above literatures do not explain the simulink and practical models of optimal fuzzy controller for Z-Source DC/DC converter (ZSC) fed DC drive and Quasi Z-Source DC/DC converter (QZSC) fed DC drive. Hence in this present work a new simulink model of Particle Swarm Optimization based fuzzy controller (FPSO) incorporated in performance evaluation of DC drive for various loaded conditions is developed. In addition new simulink models and practical models are developed for conventional fuzzy controller (FLC) and genetic algorithm based fuzzy controller (GA-FLC) for the Z-Source DC/DC converter fed DC drive and Quasi Z-Source DC/DC converter fed DC drive. The results are to be compared with that of the proposed FPSO controller fed drive.

This paper provides a review of the PSO technique, as well as its applications to converter fed drives with controller optimization problems. A brief introduction has been provided in this section on the existing optimization techniques that have been applied to tune the controller problems. The rest of this paper is arranged as follows. In Section 2, the basic problem formulations are explained along with the original formulation of the algorithm in the real number space. The details regarding modeling of converter are described in Section 3. The most common variants of the PSO algorithm with fuzzy PSO (FPSO) are described in Section 4. Section 5 provides an extensive survey on the implementation of FPSO in converter fed drives both in simulation type and experimental type. Finally, the concluding remarks appear in Section 6.

Read full article

URL:

https://www.sciencedirect.com/science/article/pii/S1568494614005596