**2. Process and controller description**

The classic 1-degree-of-freedom (1-DOF) control loop configuration of the process and the controller is shown in **Figure 1**, where the signals *r*, *e*, *u*, *d* and *y* *Improving Disturbance-Rejection by Using Disturbance Estimator DOI: http://dx.doi.org/10.5772/intechopen.95615*

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

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

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,

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

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

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

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

The classic 1-degree-of-freedom (1-DOF) control loop configuration of the process and the controller is shown in **Figure 1**, where the signals *r*, *e*, *u*, *d* and *y*

reference-weighting gain or reference signal filter [9, 16, 17].

data which is either in parametric or non-parametric form.

**2. Process and controller description**

processes require optimal disturbance rejection.

e.g., as a process time-response.

tuning results.

noise attenuation.

**44**

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

represent the reference, the control error, the controller output, the process input disturbance, and the process output, respectively.

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

$$G\_P(s) = \frac{K\_{PR}(\mathbf{1} + b\_1s + b\_2s^2 + \dots + b\_ms^m)}{\mathbf{1} + a\_1s + a\_2s^2 + \dots + a\_ns^n} e^{-sT\_{dd}} \tag{1}$$

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 stable.

The PID controller is described by the following expression:

$$G\_C(s) = \frac{K\_I + K\_P s + K\_D s^2}{s(1 + sT\_F)} \tag{2}$$

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 filter with time constant *TF*.

The closed-loop transfer function *GCL* between the reference (*r*) and the process output (*y*) is as follows:

$$G\_{\rm CL} = \frac{G\_{\rm C} G\_{\rm P}}{1 + G\_{\rm C} G\_{\rm P}} \tag{3}$$

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

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

$$u = G\_{CR}(s)r - G\_{CY}(s)y$$

$$G\_{CR} = \frac{K\_I + bK\_{PS} + cK\_Ds^2}{s(\mathbf{1} + sT\_F)}$$

$$G\_{CY} = \frac{K\_I + K\_{PS} + K\_Ds^2}{s(\mathbf{1} + sT\_F)},\tag{4}$$

*GF* <sup>¼</sup> <sup>1</sup>

*Improving Disturbance-Rejection by Using Disturbance Estimator*

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

function.

transfer function.

where

DE-MOMI method.

**47**

**4. DE-MOMI tuning method**

ð Þ 1 þ *sTF*

by using expression (7) [9]. The aforementioned modification of the method, referred to as the MOMI method, allowed the controller parameters to be computed directly from the process time response [12, 21] or from the process transfer

Since the MOMI method aims at optimising the tracking performance, the disturbance rejection performance may be degraded for some types of processes. To improve the disturbance-rejection performance, the optimisation criteria of the MOMI method were modified accordingly. The new method, referred to as the DRMO (Disturbance-Rejection-Magnitude-Optimum) method, achieved

Similar to the MOMI method, the controller parameters in the DRMO method are also based on characteristic areas or moments. Therefore, the controller parameters can be calculated either from the process time-response or from the process

The PID controller parameters are calculated according to the following expres-

*KP* <sup>¼</sup> *<sup>β</sup>* � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*KI* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *KPA*<sup>∗</sup>

<sup>1</sup> <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> <sup>2</sup>

*DA*<sup>∗</sup> <sup>2</sup> <sup>0</sup> *A*<sup>∗</sup>

2 *KDA*<sup>∗</sup> <sup>2</sup>

<sup>0</sup> *A*<sup>∗</sup>

and the derivative gain *KD* is calculated directly from expression (6). The

The DRMO tuning method significantly improved the disturbance rejection performance, especially for the lower-order processes. However, the reference tracking becomes slower due to the reference-weighting factors *b* = *c* = 0 in the 2-DOF control structure (4). The problem can be circumvented by including a simple disturbance estimator in the control scheme. Such a solution is denoted as

In order to improve the disturbance rejection response, while retaining the tracking response obtained by the MOMI method, a disturbance estimator has been

The disturbance estimator consists of the process model *GM*, the inverse process model *GMI* and the filter *GFD*. In hypothetical case, when the process model is ideal representation of the bi-proper process without time-delay, and the filter *GFD* = 1:

added to the PID controller *GC*(*s*) (2), as depicted in **Figure 3**.

<sup>3</sup> <sup>þ</sup> <sup>2</sup>*KD <sup>A</sup>*<sup>∗</sup>

<sup>1</sup> <sup>þ</sup> *KD* <sup>2</sup>*A*<sup>∗</sup>

*<sup>β</sup>*<sup>2</sup> � *αγ* <sup>p</sup> *α*

0 � �<sup>2</sup>

� � (9)

<sup>0</sup> <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> 1

<sup>3</sup> � <sup>2</sup>*A*<sup>∗</sup>

<sup>0</sup> *A*<sup>∗</sup> <sup>1</sup> *A*<sup>∗</sup> 2

<sup>0</sup> *A*<sup>∗</sup> <sup>2</sup>

<sup>0</sup> *A*<sup>∗</sup>

<sup>1</sup> � *<sup>A</sup>*<sup>∗</sup> <sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup> <sup>2</sup> 1 � � <sup>þ</sup> *<sup>A</sup>*<sup>∗</sup>

� �

<sup>0</sup> *A*<sup>∗</sup> 2

<sup>3</sup> (10)

significantly improved disturbance rejection performance [9, 16, 17].

sions when using the DRMO method [9, 16, 17]:

*<sup>β</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>∗</sup> <sup>1</sup> *A*<sup>∗</sup> <sup>2</sup> � *<sup>A</sup>*<sup>∗</sup> <sup>0</sup> *A*<sup>∗</sup>

*DA*<sup>∗</sup> <sup>4</sup>

*<sup>γ</sup>* <sup>¼</sup> *<sup>K</sup>*<sup>3</sup>

reference-weighting factors are *b* = *c* = 0.

*<sup>α</sup>* <sup>¼</sup> *<sup>A</sup>*<sup>∗</sup> <sup>3</sup>

<sup>0</sup> <sup>þ</sup> <sup>3</sup>*K*<sup>2</sup>

(8)

as shown in **Figure 2**, where parameters *b* and *c* are reference-weighting parameters for the proportional and derivative terms, respectively.
