PID Tuning Using Ziegler Nichols Method

 The distinguishing feature of the PID controller is the ability to use the three control terms of proportional, integral and derivative influence on the controller output to apply accurate and optimal control. The block diagram on the right shows the principles of how these terms are generated and applied. It shows a PID controller, which continuously calculates an error value as the difference between a

desired setpoint SP = r(t) and a measured process variable , and applies a correction based on proportional, integral, and derivative terms. The controller attempts to minimize the error over time by adjustment of a control variable u(t) , such as the opening of a control valve, to a new value determined by a weighted sum of the control terms.

In this model:

· Term P is proportional to the current value of the SP − PV error e( t

). For example, if the error is large and positive, the control output will be proportionately large and positive, taking into account the gain factor "K". Using proportional control alone will result in an error between the setpoint and the actual process value because it requires an error to generate the proportional response. If there is no error, there is no corrective response.

· Term I accounts for past values of the SP − PV error and integrates them over time to produce the I term. For example, if there is a residual SP − PV error after the application of proportional control, the integral term seeks to eliminate the residual error by adding a control effect due to the historic

cumulative value of the error. When the error is eliminated, the integral term will cease to grow. This will result in the proportional effect diminishing as the error decreases, but this is compensated for by the growing integral effect.

· Term D is a best estimate of the future trend of the SP − PV error, based on its current rate of change. It is sometimes called "anticipatory control", as it is effectively seeking to reduce the effect of the SP − PV error by exerting a control influence generated by the rate of error change. The more rapid the change, the greater the controlling or damping effect

 

Ziegler–Nichols method

 

Heuristic tuning method is known as the Ziegler–Nichols method, introduced by John G. Ziegler and Nathaniel B. Nichols in the 1940s. Ziegler–Nichols (ZN) rules are widely used to tune PID controllers for which the plant dynamics are precisely not known, it can also be applied to plants of known dynamics. Ziegler and Nichols proposed rules for determining values of proportional gain Kp, integral time Ti , and derivative time Td based on the transient response characteristic of a given plant .As in the method above, the Ki and Kd gains are first set to zero. The proportional gain is increased until it reaches the ultimate gain, Ku , at which the output of the loop starts to oscillate constantly. Ku and the oscillation period Tu are used to set the gains as follows:

 

The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining u(t) as the controller output, the final form of the PID algorithm is

where

Kp is the proportional gain, a tuning parameter, Ki is the integral gain, a tuning parameter,

Kd is the derivative gain, a tuning parameter,

e(t) = SP – PV(t) is the error (SP is the setpoint, and PV(t) is the process variable),

t is the time or instantaneous time (the present),

is the variable of integration (takes on values from time 0 to the present t ).

Equivalently, the transfer function in the Laplace domain of the PID controller is

L(S)    = Kp+ Ki/S +Kds,

 where s is the complex frequency.

The Ziegler-Nichols tuning method provides two different methods:

- the step response method and

 

- the frequency response method.

 

Ziegler-Nichols step response PID tuning method

 

The first tuning method is applied for plants that show an S-shaped characteristic for their step response. The S-shaped curve is characterized by two constants—the delay time, L and time constant, T. A tangent is drawn to the S-shaped curve at the point of inflection (as shown in Figure). The delay time, L is determined by the intersection of the tangent line with the time axis and time constant, T is determined by intersection of tangent line with final value of step response c(t)=K.

 

 

Ziegler and Nichols suggested setting the values of the parameters Kp, Ti , and Td according to Table below . The real usefulness of ZN tuning method is seen when the plant dynamics are unknown. The main advantage provided by ZN tuning rules is that they provide a starting The real usefulness of ZN tuning method is seen when the plant dynamics are unknown. The main advantage provided by ZN tuning rules is that they provide a starting point for the determination of the parameter values.

 

 

 

Ziegler-Nichols frequency response PID tuning method

 

The second design method is based on increasing Kp from 0 to a critical value Kcr at which the output first exhibits sustained oscillations. To do that, first set Ti=∞ and Td=0, then the critical gain Kcr and the corresponding period Pcr can be determined by using the Routh Criterion method. The Ziegler-Nichols formula used to set the values of the PID parameters is as shown in Table. This method can be applied to the unstable system to make it stable.

 

 

Frequency response method

 

 

 

The steps for tuning a PID controller via the Frequency Method method is as follows: Using only proportional feedback control:

 

· Reduce the integrator and derivative gains to 0.

· Increase Kp from 0 to some critical value Kp=Kcr at which sustained oscillations occur. If it does not occur then another method has to be applied.

· Note the value Kcr and the corresponding period of sustained oscillation, Pcr

 

Ziegler-Nichols method: the pros and cons

 

Using Ziegler-Nichols to tune PID loops has its advantages and disadvantages.

 

· Pro: It’s simple, intuitive and you get a reasonable performance for simple loops.

· Pro: Little process knowledge is required.

·

· Con: Originally designed for fast disturbance rejection and not for setpoint tracking.

· Con: Results in an oscillatory closed loop response (max overshoot at 25%).

· Con: Only suitable for small dead time processes (dead time is smaller than the process time constant)

· Con: High proportional gains (due to the 25% overshoot design specification), low integral action with too low damping of the closed loop system and too low robustness against changes in the process dynamics, including non- linearities.

· Con: It can’t define control objectives or closed loop performance requirements.

 

When to use which controller

 

· A Proportional controller is used to reduce the rise time and speed up the response. This controller makes no changes in the phase response of the plant.

· A Derivative controller is required to minimise the transient errors like overshoot and oscillations in the output of the plant. But this can create heavy instability in noisy environments. Be careful to use smaller gain with this controller. It provides a phase

lead to the output when compared with the input. , usually with no change in magnitude.

· An Integral controller corrects the time invariant errors. This provides a phase lag and no change in magnitude in the output.

· A PI controller helps in reducing both the rise time and the steady state errors of the system. To be useful whenever you need to change magnitude and lag the phase together.

· A PD controller reduces the transients like rise time, overshoot, and oscillations in the output. Useful for changing magnitude and want to add phase lead to the output.

· A PID controller is a general form of controller. The gains of the three control actions can be adjusted to achieve any controller. The change in magnitude along with either lead or lag in phase in the output can be made available through this general model of controller.

 

Applications of PID Controller

 

· Proportional-Integral-Derivative (PID) control is the most common control algorithm used in industry and has been universally accepted in industrial control. This is due to the fact that all design specifications of the system can be met through optimal tuning of constants Kp, Ki & Kd for maximum performance

· In the early history of automatic process control the PID controller was implemented as a mechanical device in steering system of Ships

· PID temperature controllers are applied in industrial ovens, plastics injection machinery, hot stamping machines and packing industry.

· Electronic analog PID control loops are often found within more complex electronic systems, for example, the head positioning of a disk drive, the power conditioning of a power supply, or even the movement-detection circuit of a modern seismometer.

· Most modern PID controllers in industry are implemented in programmable logic controllers (PLCs) or as a panel-mounted digital controller.

                                                     PID Control Simulator

 Remarks

· Ziegler-Nichols tuning typically yields an aggressive gain and overshoot, which may be unacceptable in some applications.

· However, it can serve as a starting point for finer tuning.

· For example, by increasing Ti and Td, we can expect the overshoot will be reduced.

· However, for certain applications where the measurement noise is significant, we need to be extra careful when increasing Td

· Ziegler-Nichols tuning rules have been widely used in process control where the plant dynamics are unknown. When the plant model is available, other controller design methods exists.

· Generally, to apply the step response method, one needs to obtain the S-shape response. Plants with complicated dynamics but no integrators are usually the cases.

· The frequency response method requires the plant to be forced into oscillation. This can be expensive and dangerous.

 

References https://www.mstarlabs.com/control/znrule.html https://yilinmo.github.io/EE3011/Lec9.html

https://en.wikipedia.org/wiki/Ziegler%E2%80%93Nichols_method https://www.google.com/url?sa=t&source=web&rct=j&url=https://pure.tue.nl/ws/files/ 4286492/625529.pdf&ved=2ahUKEwiQ4qbh_Z_1AhUTVpQKHSaJB0MQFnoECCIQ AQ&usg=AOvVaw3YEF-7gnCaZ1nASt74el_4