**1. Introduction**

The control of industrial processes requires efficient control loops. A majority of the control loops in various industries are implemented by the Proportional-Integrative-Derivative (PID) control algorithms. For efficient control, the PID controllers require proper tuning of the PID controller parameters. The parameters can be calculated to optimise various performance criteria such as integral of error (IE), integral of absolute error (IAE), integral of squared error (ISE) and similar [1–4]. However, the most important decision that should be made in advance is the choice of the main purpose of the closed-loop system. Namely, the user should choose between the optimal closed-loop responses to reference changes (so-called tracking responses) or the optimal response to process disturbances. While there are many industrial processes that require optimal reference tracking responses, such as robot manipulation, welding, and batch processes, the majority of industrial processes require optimal disturbance rejection.

The history of tuning rules is long, originating in the 1940s with the famous Ziegler-Nichols tuning rules. In the following decades, many other tuning rules have been developed [1, 2, 4–10]. The rules can be generally categorised according to the required data of the process. The process can be described either in parametric form, e.g., as a process model (transfer function), or in nonparametric form, e.g., as a process time-response.

represent the reference, the control error, the controller output, the process input

where *a*<sup>1</sup> to *a*<sup>n</sup> are the denominator coefficients, *b*<sup>1</sup> to *b*<sup>m</sup> are the numerator coefficients, *KPR* is the process gain, and *Tdel* is the process time delay. Note that *n* > *m* represents a strictly proper process transfer function and that the process is

where *KI* is the integrating gain, *KP* is the proportional gain, and *KD* is the derivative gain. Note that all three controller terms are filtered by the first-order

*GCL* <sup>¼</sup> *GCGP*

and disturbance rejection at the same time, the 2-degrees-of-freedom (2-DOF) controller can be used instead [1, 2, 4, 8, 16, 20], where *GCR* and *GCY* denote the controller transfer function from the reference and the process output, respectively:

*The 2-DOF PID controller and process in the closed-loop configuration.*

The closed-loop transfer function *GCL* between the reference (*r*) and the process

1 þ *GCGP*

Since the structure of a 1-DOF PID controller does not provide optimal tracking

*KPR* 1 þ *b*1*s* þ *b*2*s*

The PID controller is described by the following expression:

*GC*ðÞ¼ *s*

A process model (1) can be described by the following process transfer function:

*<sup>m</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*2*s*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *ansn <sup>e</sup>*

*KI* þ *KPs* þ *KDs*

*s*ð Þ 1 þ *sTF*

<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *bms*

2

�*sTdel* (1)

(2)

(3)

disturbance, and the process output, respectively.

*The 1-DOF PID controller and the process in the closed-loop configuration.*

*Improving Disturbance-Rejection by Using Disturbance Estimator*

*DOI: http://dx.doi.org/10.5772/intechopen.95615*

*GP*ðÞ¼ *s*

stable.

**Figure 2.**

**45**

**Figure 1.**

filter with time constant *TF*.

output (*y*) is as follows:

A relatively new tuning method that optimises either closed-loop tracking or disturbance rejection is the Magnitude-Optimum-Multiple-Integration (MOMI) method [7, 9, 11, 12]. The MOMI method is based on the Magnitude Optimum method, which aims to optimise the frequency response of the closed loop to achieve fast and stable closed loop time response [10, 13–15]. An interesting feature of the MOMI method is that it works either on the process given by its transfer function (of arbitrary order with time delay) or directly on the time response of the process during the steady state change. It is worth noting that both the parametric and non-parametric process data give exactly the same PID tuning results.

Many tuning methods for PID controllers provide different sets of controller parameters for tracking and disturbance rejection response. Similarly, the MOMI method primarily optimises the tracking response, while its modification, the Disturbance-Rejection-Magnitude-Optimum (DRMO) method, aims at optimising the disturbance rejection response. The latter significantly improves the disturbance rejection response, while the tracking response slows down due to the implemented reference-weighting gain or reference signal filter [9, 16, 17].

The main approach presented in this chapter is the alternative approach. First, the parameters of the PID controller are optimised for tracking performance. Then, a simple disturbance estimator is introduced to significantly increase the disturbance rejection performance [18, 19]. The advantages of the above approach are twofold. First, the disturbance rejection performance can significantly outperform that obtained by the DRMO method. Second, the parameters of the disturbance estimator can also be obtained directly from the non-parametric process data in the time domain. Therefore, the proposed approach can still be applied to the process data which is either in parametric or non-parametric form.

However, in practice, the process output noise is always present. If the controller or estimator gains are too high, the process input signals may be too noisy for practical applications. Therefore, noise attenuation should already be taken into account when calculating the controller and estimator parameters. This chapter shows how to achieve the best trade-off between performance and noise attenuation.
