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COMPARISION OF DIFFERENT TYPES OF PID CONTROL METHODS FOR OPERATION OF A SMART CAR

Presented by :Manisha Tripathy (904114)Manish Kumar Singh (904112)M Naga Praveen (904109)Indrajeet Upadhyay (904090)Himanshu (904086)

Under the guidance of Rajeev Kumar Mishra

TOPICS :

1. INTRODUCTION2. STRUCTURE OF GENERAL CONTROLLER3. MODES OF CONTROLLER4. INTRODUCTION TO SIMULINK5. ANALYSIS OF LINEAR PID CONTROLLER6. ANALYSIS OF NON-LINEAR PID

CONTROLLER7. ANALYSIS OF NON-LINEAR FEED

FORWARD PID CONTROLLER

INTRODUCTION : A control system is a device, or set of

devices to manage,command,direct or regulate the behaviour of other device(s) or system(s).Almost all appliances that we use have a control system to monitor their functioning and to ensure that the desired result is obtained.A traditional PID controller is one of the earliesr developed control strategies.These controllers have been proven to be robust and extremely beneficial in the control of many important applications.

A PID controller is generic control loop feedback mechanism widely used in industrial control systems.

However in certain applications such as smart car which is vulnerable to outside disturbance.The traditional (linear) PID system cannot meet the requirements for a high performance and it is difficult to gain good results.

Hence non-linear feed-forward PID is adopted which achieves satisfactory results in time regulation and speed response.

Structure of a general controller :• A controller is a device

that generates an o/p based on the i/p signal it receives.

• The i/p signal is an error signal,which is the difference between the measured variable and the desired value.

• The controller provides an o/p signal to the final control element, which adjusts the process system to reduce this deviation.

Modes of Controllers :Proportional (P)Proportional plus Reset (PI)Proportional plus Rate (PD)Proportional plus Reset plus Rate

(PID)Mathematically,these are stated

as:

Propotional controllers:Proportional controllers also known as

throttling control.In the proportional mode,there is a contionus linear relation between value of the controlled variable and the position of the final control element.

Proportional band is the change in the value of the controlled variable that causes full travel of the final control element.

Gain,also called sensitivity,compares the ratio of amount of change in the final control element to amount of change in the controlled variable.

Offset,also called droop,is deviation that remains after a process has stabilized.

Proportional plus integral control :It is also called proportional plus reset

control.Integral control describes a controller

in which the output ate of change is dependent on the magnitude of the input.Specifically,a smaller amplitude i/p causes a slower rate of change of the o/p.

This controller is called an integral controller because it approximates the mathematical function of integration.

PI characteristics :

Disadvantages of PI :An inherent disadvantage to proportional

plus reset controllers is the possible adverse effects caused by large error signals.

The large error can be caused by a large demand deviation or when initially starting up the system

This is a problem because a large sustained error signal will eventually cause the controller to drice to its limit and the result is called “reset windup”.

Because of this,this control mode is not well-suited for processes that frequently shutdown and started up.

Proportional plus derivative control :

It is a control mode in which a derivative section is added to the proportional controller.

This derivative section responds to the rate of change of the error signal, not the amplitude; this derivative action responds to the rate of change the instant it starts. This causes the controller output to be initially larger in direct relation with the error signal rate of change.

A device that produces a derivative signal is called a differentiator.

The derivative constant is expressed in units of seconds and defines the differential controller output.

Derivative cannot be used alone as a control mode.

This is because a steady-state input produces a zero output in a differentiator.

If the differentiator were used as a controller, the input signal it would receive is the error signal.A steady-state error signal corresponds to any number of necessary output signals for the positioning of the final control element.

Therefore, derivative action is combined with proportional action in a manner such that the proportional section output serves as the derivative section input.

PD Characteristics :

Disadvantages of PD controller :

Rate action cannot be employed with fast responding processes such as flow control or noisy processes because derivative action responds to any rate of change in the error signal, including the noise.

Proportional plus integral plus derivative control (PID) :Proportional plus reset plus rate controllers

combine proportional control actions with integral and derivative actions.

There are some processes that cannot tolerate offset error, yet need good stability. The logical solution is to use a control mode that combines the advantages of proportional, reset, and rate action.

When an error is introduced to a PID controller, the controller’s response is a combination of the proportional, integral, and derivative actions.

PID controller response curves :

The proportional action of the controller stabilizes the process.The reset action combined with the proportional action causes the measured variable to return to the set point.

The rate action combined with the proportional action reduces the initial overshoot and cyclic period.

Introduction to simulink :MATLAB is a high-level language and

interactive environment for numerical computation, visualization, and programming.

For multi domain and Model based design, Matlab has a block diagram environment called Simulink, which supports system-level design, simulation, automatic code generation, and continuous test and verification of embedded systems.

Simulink provides a graphical editor, customizable block libraries, and solvers for modelling and simulating dynamic systems.

The Simulink Library Browser has many commonly used blocks and toolbars which can be used to design any desired model. New blocks and models can also be added to the library browser. These can further be used for designing other complex models. Every model must have a signal source , block(s) representing the desired system and a sink for displaying the result obtained. The results can be analysed graphically which helps us to obtain a satisfactory response of the given design model or system.

Analysis of linear PID controller :

A PID controller calculates an "error" value as the difference between a measured process variable and a desired setpoint.The controller attempts to minimize the error by adjusting the process control inputs.

The PID controller calculation (algorithm) involves three separate constant parameters, and is accordingly sometimes called three-term control: the proportional, the integral and derivative values, denoted P, I, and D. Heuristically, these values can be interpreted in terms of time: P depends on the present error, I on the accumulation of past errors, and D is a prediction of future errors, based on current rate of change. The weighted sum of these three actions is used to adjust the process via a control element.

The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining as the controller output, the final form of the PID algorithm is:

Where ◦Kp : Proportional gain T :variable of

integration◦Ki : Integral gain t : instantaneous

time◦e : Error=SP-PV

This control signal (u) is sent to the plant, and the new output (y) is obtained. The new output (y) is then fed back and compared to the reference to find the new error signal (e).

Proportional term :The proportional term produces an output

value that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp.

Proportional term is : Pout = Kp e(t)

Integral term :

Plot of PV vs time,for three values of Ki (Kp and Kd held constant).

The integral term is given by :

Derivative term :

Plot of PV vs time, for three values of Kd (Kp and Ki held constant).

The derivative term is given by :

Effects of increasing a parameter independently :

Parameter

Rise time

Overshoot

Settling time

Steady state error

Stability

Kp Decrease Increase Small change

Decrease Degrade

Ki Decrease Increase Increase Eliminate Degrade

Kd Minor change

Decrease Decrease No effect in theory

Improve if Kd small

Simulation of linear PID controller :Using Simulink, a linear pid controller is

designed with reference to the control system of a smart car. By choosing suitable parameters, the transfer function of the object (controlled motor) is given as 2/(S2 +S+1).

Block diagram of linear PID controller :

This block implements continuous-and discrete-time PID control algorithms and includes advanced features such as anti-windup, external reset, and signal tracking. We can tune the PID gains automatically using the 'Tune...' button (requires Simulink Control Design).

By choosing different values of KP,Ki and Kd,the output of pid controller can be tuned by the pid tuner.

All the characteristics of step response are plotted. They are: peak response: time required for the response to reach the first peak

of the o overshoot. settling time : time required for the response curve to reach and stay

within 2% of the final value rise time : time required for the response curve to rise from 10% to

90% of its final value. steady state response : time after which the response becomes

steady and transient effects no longer occur.

Block diagram of linear PID (without using the pid controller block) :

The controlled output is as follows:

Defects of linear PID controller : u=kpe+ ki∫edt+ kd de/dt

where e(t) = r (t) - y(t) . This form of pure linear is very effective in relatively simple practical process. However, it is not effective to control the scope of the changes in the parameters of the larger or non-linear object.

Above all, r(t), as the reference input, is often not smooth or continuous, and y(t), as the system output, has to be smooth. With y(t) as the direct target of the output, the inertia effect of the controlled object is not considered, which easily causes overshoot oscillation in practical applications.

Lastly, there is conflict between the high speed and the overshoot variable caused by the "linear combination" of the traditional PID. In actual application, we hope three parameters will adjust in response to the response process of the system. In the initial phase, for example, a larger Kp can increase the system’s response speed, but with the decrease of the e(t), we hope Kp can be reduced accordingly and thus reduce the overshoot. When e(t)<0 and de (t)/dt> 0, we hope K will gradually increase in Kp

in order to reduce the overshoot through the increase of the reverse effect. When e(t)<0 and de (t)/dt<0, we hope Kp will gradually decrease, so that there is no big overshoot when the system is back to the balanced state. This is however impossible in case of linear PID.

Analysis of non-linear PID controller It is the linear combination that causes

the contradiction between overshoot and high speed. In the non-linear control, for the purpose of solving this contradiction, it is necessary to shake off the restraint of the mathematical model. The function of a nonlinear module is shown as:

f(e,α,δ)=IeIα sign(e), > δ =e/ δ1- α < δ

where sign(e)=1 ,e>0 =-1 ,e<0

The specific form of the control law of the non-linear PID is :

U = kp.e. fp(ep,α0,δ0) + ki. fi(ei,α1,δ1)∫edt + kd . fd(ed,α2,δ2)de/dt

∫

d/dt

fi

fd

fp

Linear combination

object

r(t) e

y(t)u(t)

Simulation of non-linear PID controller :

The parameters of the non-linear PID will be Kp =8 , Ki =1.3, Kd =2.1.The Simulink integration tools of the Matlab simulation environment is used in the simulation control system.The block diagram of non-linear pid using Simulink is as follows :

The simulation result of non-linear PID for simulation step size of t=0.01s is shown below :

Defects of non-linear PID controller The non-linear PID is more robust and

adaptable than the linear PID, but it should be noted that the entire system remains a closed-loop control system. As for some special objects such as the delay control systems, high-end systems, non-minimum phase systems, especially for smart car system, the system's response time will be affected, showing a longer time to adjust.

If the feed-forward is joined, that is, to introduce an open-loop, the system's dynamic performance and steady-state performance will be further improved.

Analysis of non-linear feed-forward PID controller :

Control object

Feed-forward control

Non-linear PID

r(t) y(t)

The specific form of non-linear feed forward controller is as follows:

u = kp.e. fp(ep,α0,δ0) + ki. fi(ei,α1,δ1)∫edt + kd . fd(ed,α2,δ2)de/dt + kf.r(t)

where kf is the gain constant of

feed-forward control. Thus the above equation has an additional term( kf.r(t)) as compared to the non-linear PID.

SIMULATION OF NON-LINEAR FEED-FORWARD PID CONTROLLER :

In order to get a satisfactory response, the values of the gain constants are chosen after testing the response with a trial and error method.

The parameters of the non-linear feed-forward PID will be Kp=8, Ki =1.1, Kd=2.6, Kf=0.6.

The control system performance can be improved by combining the feedback (or closed-loop) control of a PID controller with feed-forward (or open-loop) control. Knowledge about the system (such as the desired acceleration and inertia) can be fed forward and combined with the PID output to improve the overall system performance. The feed-forward value alone can often provide the major portion of the controller output.

The response curve of non-linear feed forward pid whose reference signal was a simulation step with step size of t=0.01s is given below :

Conclusion :In this report, with reference to the

design of the controller of a smart car system, we discuss the contradiction which the linear PID controller itself cannot overcome. By introducing the non-linear PID, we effectively solve the contradiction between speed and stability of the linear PID controller. The final simulation results show that the nonlinear PID, compared with the linear PID, dramatically enhances both the dynamic performance and stability.

On the basis of this, we add

the feed-forward control system so that the dynamic performance and stability is further enhanced. As for some special systems, the non-linear feed-forward PID controller reduces the system's response time, allowing the system to further accelerate the response speed.

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