Tuning a Conventional and Non Conventional Pi Controller (Essay Sample)

Module: EAC4013-N-FJ1-2020
Title: In course Assessment (I.C.A)


Tuning of a PI Controller.
Teesside University
(School of Computing, Engineering and Digital Technologies)
Module name: Robust Control Systems
Abstract
Control systems engineering requires control engineers to at least have knowledge of the plant (mathematical model) and the desired output. However due to uncertainties, control engineers will always want to strike a good balance between performance of the system and the ability to meet the design requirement. The uncertainties arise due to inaccurate representation of the physical system due to factors not limited to unmodelled time delays, unpredicted disturbance and parameter changes over time. It is the responsibility of the engineers to retain the assurance of the system performance despite model inaccuracies and changes. A system is said to be robust if it displays acceptable changes in its performance due to these model inaccuracies. The question of robustness of a system surrounds its ability to perform steady state error cancellation, tracking of the set point and whether the system can achieve disturbance rejection by means of internal model principle. The controller can either be proportional (P), proportional integral (PI) or a proportional integral derivative (PID) controller. In this paper, an assessment of the design on both fuzzy and non-fuzzy PI controller is presented based on robustness measures.
Introduction
When the performance of a system is not satisfactory, the controller is usually the first component modified. This paper focuses on the design of a PI controller. The features of a PI controller include a proportional term that impacts the controller bias or null value based on the size of the controller error signal and an integral part that continually sums up the error. The controlled system response should have an overshoot of approximately 20% and a rise time of ˂0.2seconds.
Design of the PI controller
1 Design of conventional (non-fuzzy) controller
The open loop transfer function of the servo motor in question is approximated to be;
Ts=2.1739(s+6.75)s(s+4.348) Equation SEQ Equation \* ARABIC 1
The transfer function above describes the open loop transfer function that the design of the controller targets. The design of the PI controller can be achieved using several controller tuning methods. For this particular design, the controller is achieved by use of the SISO Design Tool on MATLAB. The design process is a matter of editing the position of poles and zeros to alter the closed loop step response. The SISO design tool can be started by typing the command sisotool(T) where T is the open loop transfer function desctibed in equation 1.
Upon firing up the SISO design tool, the root locus, the bode plot editor and the step response windows are visible but the root louc and the step response are of interest. It can be observed from figure (i) below that the system response has a rise time of 0.593 seconds and an overshoot 0.852%. This output characteristics are not desired and the specifications demand for 20±2% overshoot and a rise time ˂0.2 seconds. At this instance, the compensator C has a default gain value of 1. In this instance the effect of the controller on the process is null and void. The SISO design tool shows how the system will behave under closed loop conditions. Throughout the design process the tool gives an avenue to edit the compensator parameters.

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(a) Opening SISO tool on MATLAB

(b) SISO tool

Figure SEQ Figure \* roman i SISO tool on MATLAB

Since the controller behaves like a mere multiplier, an integrator is added to attain a PI configuration. By default, the integrator is placed at 0. Changing the position of the integrator takes the transient response further from achieving design requirements and the integrator better remain at its default position. The controller gives the process a PI controller effect with Kp=1 and Ki=1. The controller equation is now Gc=1+1s. From figure (ii) below, on addition of the integrator at 0, the oscillations grow with time. This is not a typical control scenario. Adding a real zero at say -2 causes the elimination of unwanted ringing in the transient response and steady state is achieved with time.
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* Adding an integrator to the Compensator

* Root locus and transient response after introduction of the integrator.

Figure SEQ Figure \* roman ii Addition of an integrator.


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* Adding a real zero to the compensator

* Transient response of compensator with an integrator at 0 and a real zero at -2

Figure SEQ Figure \* roman iii transient response after addition of a zero.

377824079213057658001173010Although the transient response upon adition of the zero is typical, an overshoot of 40.5% and a rise time of 0.526 go against the design demands.
The Root locus editor shows the location of the systems poles and zeros. The red circle is the zero that was added while the blue zero is system’s fixed zero from the plant’s open loop. On the other hand, the crosses are poles with the blue one fixed and the red ones movable. The red boxes are pole magneto blocks that can moved around the root locus to tune the compensator. The compensator gain can be adjusted as well. Say to a value 5.4. This affects the transient response in the following manner.

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* Tweaking C to 5.4

* Transient response under tuned compensator C.

Figure SEQ Figure \* roman iv Tuned controller C and corresponding transient response.

A percentage overshoot of 21% and a rise time of 0.184 seconds are in line with the design expectation. This way the design of the desired PI controller has been achieved. The values of the tuned controller are now exported to the MATLAB workspace and the controller parameters examined.



* Export button

* Export the tuned controller parameters to workspace

* Controller C on command window

Figure SEQ Figure \* roman v Exporting the tuned Controller parameters to MATLAB.

Obtaining the kp ki values for the PI controller using the command [kp ki kd]=piddata(C) with kp, ki and kd as place holders. The parameters Kp, Ki and Kd are observed to be 2.7, 5.4 and 0 respectively.
Figure SEQ Figure \* roman vi MATLAB command window showing the Kp ang Ki values.
Observing the closed loop transfer function with the servo motor and the PI controller under unity feedback. Observing the closed loop system response P using the MATLAB command step (P)

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* MATLAB command (closed loop TF)

* Closed loop transient response

Figure SEQ Figure \* roman vii Closed loop TF and transient response

Below is a closed loop system Simulink model and its transient response. The step time is set at 0.05 seconds.


* Closed loop Simulink model

* Closed loop transient response

Figure SEQ Figure \* roman viii system response observed from the Simulink model

2 Designing a Fuzzy PI controller
It allows for possible view in different angles. Fuzzy inference process involves, fuzzification (translate input into truth values), rule evaluation and defuzzification (transfer truth values into output). Fuzzification involves assigning degrees of membership to various inputs. Rule evaluation involves a set of if/then control rules. Therefore, fuzzy logic is an approach to computing based on degrees of truth. In this paper we will design a fuzzy PI controller using Fuzzy logic toolbox on MATLAB.


* Default fuzzy logic Designer

* Fuzzy logic designer with two inputs (sugeno)

Figure SEQ Figure \* roman ix Fuzzy Logic Designer

Starting the fuzzy logic designer is as simple as typing ‘fuzzylogicDesigner’ into the MATLAB command window. The default fuzzy logic has one input and one output with the FIS as mamndani. The FIS is changed from mamdani to sugeno and the set of input and output variables defined by adding any required and renaming them appropriately. The main difference between Mamdani and Sugeno is that Sugeno output membership functions are either linear or constant. The input variables are e and ce representing the error and the changes in that error. The output u responds to any changes in the values of both inputs.
The membership functions associated with both the inputs and the outputs are edited. Say the membership function associated with the inputs are N, Z, P and Nder, Zder and Pder for inputs e and ce respectively while those associated with the output u are HPOS, POS, ZERO, NEG and HNEG where POS is positive and NEG is negative.
Editing the range and the params on the designer for the input and output ...