**2.3.1 The Ziegler–Nichols tuning method**

The Ziegler–Nichols tuning method is a heuristic method of tuning a PID controller. It was proposed by John G. Ziegler and Nichols in the 1940's. It is performed by setting I (integral) and D (derivative) gains to zero. The P (proportional) gain, Kp is then increased (from zero) until it reaches the ultimate gain Ku, at which the output of the control loop oscillates with a constant amplitude. Ku and the oscillation period Tu are used to set the P, I, and D gains depending on the type of controller used [3,4]:


There are several prescriptive rules used in PID tuning. The most effective methods generally involve the development of some form of process model, and then choosing P, I,

Requires experienced

Some trial and error, process upset and very aggressive

Good only for first order

Some cost and training

personnel

Some math

processes

involved

Very slow

tuning

This can be very useful to remove unstable poles.

and D based on the dynamic model parameters.

Manual Online method

Ziegler-Nichols Online method

We present here four tuning methods for a PID controller [2,3].

Software tools Online or offline method,

Algorithmic Online or offline method,

**2.3.1 The Ziegler–Nichols tuning method** 

depending on the type of controller used [3,4]:

Method **Advantages Disadvantages**

Proven method

Cohen-Coon Good process models Offline method

Very precise

No math expression

consistent tuning, Support Non-Steady State tuning

Consistent tuning, Support Non-Steady State tuning,

The Ziegler–Nichols tuning method is a heuristic method of tuning a PID controller. It was proposed by John G. Ziegler and Nichols in the 1940's. It is performed by setting I (integral) and D (derivative) gains to zero. The P (proportional) gain, Kp is then increased (from zero) until it reaches the ultimate gain Ku, at which the output of the control loop oscillates with a constant amplitude. Ku and the oscillation period Tu are used to set the P, I, and D gains

**2.3 Tuning methods** 

We can realise a PID controller by two methods: First, an analog PID controller Second, a digital PID controller

1. Circuit diagram below (figure.5) shows an analog PID controller. In this figure, we present an analog PID controller with three simple op amp amplifier, integrator and differentiator circuits.

Fig. 5. Electronic circuit implementation of an analog PID controller


PID Control Theory 221

While PID controllers are applicable to many control problems, and often perform satisfactorily without any improvements or even tuning, they can perform poorly in some

Fractional order systems are characterized by fractional-order differential equations. Fractional calculus considers any real number for derivatives and integrals. The FOPID controller is the expansion of the conventional integer-order PID controller based on

The PIDs are linear and in particular symmetric and they have difficulties in the presence of non-linearities. We can solve this problem by using a fractional-order PID (FOPID)

( ) *I P ID*

*K KS K KSS Gs K K S S S*

> 

 (14)

 

*P D*

Figure.7 describes the possibilities a FOPID for the different controllers.

Fig. 7. Generalization of the FOPID controller: from point to plane.

to the Riemann-Liouville definition.

There are several methods to calculate the fractional order derivative and integrator of a fractional order PID controller. For this purpose we present a real order calculus according

applications, and do not in general provide optimal control.

**3.1 Fractional-order PID (FOPID) controller** 

controller. A FOPID controller is presented below [7-9]:

**3. Fractional systems** 

fractional calculus [7,8].

Finally, we need to add the three PID terms together. Again the summing amplifier OP4 serves us well. Because the error amp, PID and summing circuits are inverting types, we need to add a final op amp inverter OP5 to make the final output positive.

2. Today, digital controllers are being used in many large and small-scale control systems, replacing the analog controllers. It is now a common practice to implement PID controllers in its digital version, which means that they operate in discrete time domain and deal with analog signals quantized in a limited number of levels. Moreover, in such controller we do not need much space and they are not expensive. A digital version of the PID controller is shown in figure 6 [5,6].

Fig. 6. Digital PID Controller

In its digital version, the integral becomes a sum and the deferential a difference. The continuous time signal e(t) is sampled in fixed time intervals equals a determined sample period, here called Tc (in figure 6 Tc = 1). An A/D (analog to digital) converter interfaces the input and a D/A (digital to analog) converter interfaces the output. This sampled and digitalized input, called eD[j], exists only in time instants *<sup>C</sup> t kT* for all 0 *k Z* . A lower bound for the sample period is the computing time of a whole cycle of the digital PID (which includes the A/D and D/A conversion).

Finally, we need to add the three PID terms together. Again the summing amplifier OP4 serves us well. Because the error amp, PID and summing circuits are inverting types, we

2. Today, digital controllers are being used in many large and small-scale control systems, replacing the analog controllers. It is now a common practice to implement PID controllers in its digital version, which means that they operate in discrete time domain and deal with analog signals quantized in a limited number of levels. Moreover, in such controller we do not need much space and they are not expensive. A digital version of

In its digital version, the integral becomes a sum and the deferential a difference. The continuous time signal e(t) is sampled in fixed time intervals equals a determined sample period, here called Tc (in figure 6 Tc = 1). An A/D (analog to digital) converter interfaces the input and a D/A (digital to analog) converter interfaces the output. This sampled and digitalized input, called eD[j], exists only in time instants *<sup>C</sup> t kT* for all 0 *k Z* . A lower bound for the sample period is the computing time of a whole cycle of the digital PID

need to add a final op amp inverter OP5 to make the final output positive.

the PID controller is shown in figure 6 [5,6].

Fig. 6. Digital PID Controller

(which includes the A/D and D/A conversion).

While PID controllers are applicable to many control problems, and often perform satisfactorily without any improvements or even tuning, they can perform poorly in some applications, and do not in general provide optimal control.
