real time implementation and design of predictive...

International Journal of Science, Engineering and Technology Research (IJSETR), Volume 4, Issue 7, July 2015

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ISSN: 2278 – 7798 All Rights Reserved © 2015 IJSETR

.

Abstract—Generalized Predictive Controller is a one type of

Model Predictive Controller. Speed Control of DC motor is

taken as a Case study for validation of GPC controller

algorithm implementation on Real Time Hardware. There is a

much gap between theoretical solution and real time

implementation of MPC. Basically MPC works better for slow

dynamic system, so here efforts are made to Implement MPC

for Fast Dynamic System. System Identification of DC motor is

done using a LABVIEW and system identification tool of

MATLAB. First of all ARIX model based GPC controller is

designed in MATLAB for Model Obtained from System

Identification of DC motor. Then the control law of GPC

controller is implemented on ATMEGA328P and Results shows

that GPC controller gives better results than discrete time PID

controller for Set Point Tracking as well as for Disturbance

Rejection.

Index Terms—MPC (Model Predictive Controller), GPC

(Generalized Predictive Controller), ARIX Model (Auto

Regressive Integrated Exogenous Model, FDS (Fast Dynamic

System).

I. INTRODUCTION

Model Predictive Controller is basically suitable only for

Slow Dynamic Systems like Chemical Process, Oil

Refineries, etc. Design of MPC for Fast Dynamic system is

difficult because MPC takes long time to calculate the value

of control action and if the sampling time of control action is

larger than the FDS like DC motor will not reach at the

Desired Set Point. So MPC with small sampling time and less

computation time will be used for FDS.

There are many types of MPC control algorithms used.

Some of them are listed here: 1) GPC (Generalized Predictive

Controller), 2) DMC (Dynamic Matrix Controller), 3) Steady

State Weighted Generalized Predictive Control, etc.

Generally GPC controller is used for Fast Dynamic System.

Here GPC controller results are compared with PID

controller for DC motor Speed Controller. PID and PI

controller are also used for Fast Dynamic System.

There is much gap between real time implementation and

theoretical solution of MPC. So here, effort is made for real

time implementation of GPC for DC motor Speed Control.

More detail on MPC and GPC control algorithm is given in

[2] and [6]. Now a day’s many researchers tries to reduce the

computation time of MPC.

Manuscript received June, 2015.

Sumit G. Vyas1Student of M.E., Instrumentation and Control Engineering,

L.D. College of Engineering, Ahmedabad,Gujarat, India

Vinodkumar P. Patel2 Associate Professor, Instrumentation and Control

Engineering, L.D. College of Engineering, Gujarat, India

In [1] Generalized Predictive controller results for DC motor

is compared to Discrete time Predictive Controller. More

detail and survey on Model Predictive Controller is given in

[2]. In [3] MPC is implemented for PMSM motor,

Generalized Predictive Controller is applied to control the

Position of Induction Motor in [4]. Robust Model Predictive

Controller is implemented for Fast Dynamic Vehicle System

in [5]. Above all MPC is designed particularly for specific

application.

In Section II GPC control Strategy is given, In Section III

description of DC motor is given, Section IV Describes the

System Identification of the System. In Section V Simulation

result and effect of tuning parameter of GPC on System

Output is shown. Section VI describes the real time

implementation of GPC and PID controller for DC motor

Speed Control is given.

II. GENERALIZED PREDICTIVE CONTROLLER

GPC is based on minimizing a weighted sum of the Set

Point error and the control effort and it allows plant models to

be updated frequently.GPC makes use of the j-step ahead

prediction error model.

ARIX model based Generalized Predictive Control law is

given by the following equation,

𝑢 = 𝑘𝑟 − 𝑘𝐻2𝑦𝑜𝑙𝑑 − 𝑘𝐻1𝑢𝑜𝑙𝑑

Where G =

𝑔𝑘 ,0 0 … 0

𝑔𝑘+1,1 𝑔𝑘+1,0 ⋯ 0⋮

𝑔𝑘+𝑁,𝑁

⋮𝑔𝑘+𝑁 ,𝑁−1

⋮⋯𝑔𝑘+𝑁 ,0

,

𝐻1 =

𝑔𝑘 ,1 ⋯ 𝑔𝑘 ,𝑑𝐺𝑘

𝑔𝑘+1,2 ⋯ 𝑔𝑘+1,𝑑𝐺𝑘+1

⋮𝑔𝑘+𝑁 ,𝑁+1

⋮ ⋯

⋮𝑔𝑘+𝑁 ,𝑑𝐺𝑘+𝑁

,

𝐻2 =

𝑓𝑘 ,0 ⋯ 𝑓𝑘 ,𝑑𝐴

𝑓𝑘+1,0 ⋯ 𝑓𝑘+1,𝑑𝐴

⋮𝑓𝑘+𝑁,0

⋮ ⋯

⋮𝑓𝑘+𝑁,𝑑𝐴

,

𝑦𝑜𝑙𝑑 = [𝑦 𝑛 …………𝑦(𝑛 − 𝑑𝐴)]𝑇 ,

Real Time Implementation and Design of

Predictive Controller for Fast Dynamic System

Sumit G. Vyas1, Vinodkumar P. Patel

2


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𝑢𝑜𝑙𝑑 = ∆𝑢 𝑛 − 1 ………∆𝑢 𝑛 − 𝑘 + 1 − 𝑑𝐵 𝑇 ,

𝑢 = ∆𝑢 𝑛 ……………… . ∆𝑢 𝑛 + 𝑁 𝑇,

where𝑟 is a trajectory of reference signals.

𝑟 = [𝑟 𝑛 + 𝑘 ……………𝑟(𝑛 + 𝑘 + 𝑁)]𝑇 ,

𝑘 = (𝐺𝑇𝐺)−1𝐺𝑇 ,

In the above Generalized Predictive Control law, the error

signal and the control effort are weighted over the same

length of time.

Now, generalizing this situation by minimizing the error

from n+k+N1 to n+k+N2, N2≥N1 and the control effort from

n to n + Nu.

The performance index is given by

𝐽𝐺𝑃𝐶 = [𝑦 𝑛 + 𝑘 + 𝑁1 − 𝑟(𝑛 + 𝑘 + 𝑁1)]2 +

… … …+[𝑦 𝑛 + 𝑘 + 𝑁2 − 𝑟(𝑛 + 𝑘 + 𝑁2)]2

+𝜌[∆𝑢(𝑛)]2 + ……… + 𝜌[∆𝑢(𝑛 + 𝑁𝑢 )]2

As a result of this 𝑦 and 𝑢 given by

𝑦 = [𝑦 𝑛 + 𝑘 + 𝑁1 ………𝑦 (𝑛 + 𝑘 + 𝑁2) ]𝑇

𝑢 = ∆𝑢 𝑛 ……………… . ∆𝑢 𝑛 + 𝑁𝑢 𝑇

The performance of the GPC depends on the parameters

N1, N2, Nu and ρ. So the proper selection of these parameters

gives better results. Tuning of these parameters is necessary

which is given in the [6], [7] and [8]. More detail on GPC

controller is given in [6].

III. DC MOTOR

To validate the proposed MPC algorithm, Speed Control of

DC Motor is taken as a Case Study for Fast Dynamic System.

DC Motor Specification is given below. Generally the MPC

gives better results for Slow Dynamics System so here efforts

are made to Implement GPC control algorithm for DC motor

Speed Control.

A. DC Motor Specification

Fig (1) show the DC motor used to validate the GPC

algorithm. Rotating Magnet mechanism on the pulley is put

on motor to give the load on the DC motor to checks whether

the GPC controller can rejects the disturbance or not.

Specifications:

1. Maximum speed: 1500 Rpm

2. Maximum input Voltage: 10 V DC

3. Maximum input Current: 0.9Amp

4. F to V converter output: 0 to 2 volt for 0 to 1500 rpm.

Fig- 1: DC motor

IV. SYSTEM IDENTIFICATION

System Identification uses the different statistical methods

to generate the model of the System using real time data of

input and output of the system with respect to time measured

in LABVIEW. LABVIEW stores the input and output

measured data of the system with respect to time at 0.001

sampling time. This data are used in the system identification

tool of the MATLAB to generate the model of the system.

Figure (2) shows the Interfacing of DC Motor with

LABVIEW for the System Identification of the Motor.

Fig-2: Interfacing of DC motor with LABVIEW to store

the value of input and output values in a Measurement File

As shown in figure DC motor is interfaced with

LABVIEW to obtain the model of the DC Motor and stored

data of input and output of the Motor is used in the system


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identification tool of the MATLAB. Figure (3) show the

output and input waveform of the motor stored in the File

using LABVIEW. Figure (4) shows the estimated output of

the obtained model of the DC motor.

The Estimated model of the DC motor is obtained as below,

𝐺 𝑆 = 0.5957 𝑆 + 0.0683

𝑆2 + 3.248 𝑆 + 0.339

With one zero and two pole and the estimated transfer

function fits the data by 89.54%.

Transfer function of DC motor in Z domain is obtained as

below,

G z = 0.0321 z − 0.03187

z2 − 1.825 z + 0.8256

Fig- 3: Input and Output response of measured data of DC motor

Fig- 4: Response of Actual System and Simulated Model of

DC motor

V. SIMULATION RESULTS

This section shows the simulation Results of the DC motor model with GPC controller and its results are compared with

discrete time PID controller. Sampling time Ts = 0.059s

taken to implement the PID and GPC controller. PID tuning

is done using Ziegler Nichols closed loop Method.

Obtained value of PID tuning parameter are Kp = 6.02, Ti =

0.05 and Td = 0.019. Then this parameter are used to

implement the discrete time PID controller using following

Equation,

𝑆0 = 𝐾𝑝 1 + 𝑇𝑠

2𝑇𝑖

+ 𝑇𝑑

𝑇𝑠

𝑆1 = 𝐾𝑝 −1 + 𝑇𝑠

2𝑇𝑖

− 2𝑇𝑑

𝑇𝑠

𝑆2 = 𝐾𝑝

𝑇𝑑

𝑇𝑠

Discrete Time PID Controller is given by,

1 − 𝑍−1 𝑢 𝑛 = 𝑆0 + 𝑆1𝑍−1 + 𝑆2𝑍

−2 𝑒(𝑛)

Tuning parameter of GPC are N1, N2, Nu and rho. Values of

tuning parameter of GPC are taken are N1 = 0, N2 = 10, Nu =

5 and rho = 0.5.

N1 = Starting point of Prediction Horizon, N2 = End point of Prediction Horizon,

Nu = Control Horizon,

rho = Control Weighting parameter.

For all case Set Point of DC Motor taken is 700 rpm

A. DC Motor’s Response without Disturbance

Motor’s closed loop response with PID controller and GPC

controller are show in figure (5). In this response, no

disturbance is given.

Fig-5: System Response without Disturbance

Figure (5) shows the both GPC and PID controller output

response without Disturbance. From the simulation results of


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the GPC and PID Controller, it is concluded that settling time

of GPC controller is 4 seconds while PID Controller takes

3seconds. So from settling time perspective PID is better than

GPC controller but PID controller have large Oscillation then

GPC controller. GPC controller gives smooth tracking of the

Set Point as compared to PID controller.

Fig – 6: Control Action Responsewithout Disturbance

Figure (6) shows the Control action response of the PID and

GPC Controller.

B. DC Motor’s Response with Disturbance

In this case step disturbance is applied after 10 seconds.

Both the Controller takes almost same time to rejects the

disturbance. System Response with Disturbance is shown in

figure (7).

Fig –7: System Response with Disturbance

Figure (8) shows the Control action Response of PID and

GPC Controller. Because of Disturbance, Oscillation occurs

in both controllers at time of 10 seconds. It shows that both

controllers reject the Disturbance effectively. Time taken to

remove the disturbance is almost same for both controller and

it is approximately 2 to 2.5 seconds.

Fig –8: Control action Response with Disturbance

C. Set Point Tracking Results of DC Motor

In this case variation in the Set Point is given. For 25

seconds Set Point value is 700 rpm and for another 25

seconds Set Point value is taken be 0. Figure (9) shows Set

point tracking of GPC controller is very smooth while PID

controller oscillated Set Point Tracking.

Fig – 9: System Response for Set Point Tracking

Figure (10) shows the control action response of GPC and

PID controller for Set Point tracking case study.


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Fig – 10: Control Action Response for Set Point Tracking

D. Effect of GPC Tuning Parameter on the System

Output

In this section effect of GPC tuning parameters on the

System Output is shown. Figure (11) shows the effect of

control horizon on System Output. For in all case Control

Horizon Nu = 5 is taken. Decreasing the value of Nu gives oscillated output while increasing the value of Nu makes

system slow. For Nu = 5 it gives better results for system

output.

Fig – 11: Effect of Control Horizon (Nu) on System Output

Figure (12) shows the effect of Starting Point of Prediction

Horizon on System Output. For N1 = 0 it gives good results

but when increasing the value of N1 it makes system very

slow.

Fig – 12: Effect of Starting Point of Prediction Horizon (N1)

on System Output Figure (13) shows the effect of End Point of Prediction

Horizon. For all cases N2 = 10 is taken. For N2 = 6 system

output is very slow and it takes more time to reach at set

point. For N2 = 13 continuous oscillation occurs in the

system output. So Small value of N2 makes system very slow

and large value of N2 makes system oscillated. So for N2 =

10 it gives system output have neither slow response nor

oscillated response.

Fig – 13: Effect of End Point of Prediction Horizon (N2) on

System Output

Remaining tuning parameter of GPC is Control Weighting

Parameter (rho). In figure (14) effect of the value of rho on

system output is shown. So it effects almost same as control

horizon parameter on system output. For rho = 0.3 it gives

oscillated response on system output and for rho = 0.8 system output have slow response. So for small value of rho system

output have oscillated response and for rho near to 1 have


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slow response on system output. For rho = 0.5 it system

output have neither oscillated response nor slow response.

Fig – 14: Effect of Control Weighting Parameter on System

Output

VI. REAL TIME IMPLEMENTATION OF GPC AND PID

CONTROLLER

In Previous section all are simulation results of GPC and PID controller, here results are of real time data of GPC and

PID controller is shown. Fig (15) shows the interfacing of DC

motor with Arduino Uno board. In Arduino Uno board

recursive function of eq (1) is implemented in C++ language.

Fig – 15: Interfacing of DC Motor with Arduino Uno Board

A. GPC Control Algorithm

Figure (16) shows the real implementation of GPC

controller to check the whether GPC rejects the Disturbance

or not peak in circle shows the disturbance given on Motor

using Magnet. It is show that GPC rejects the disturbance

very well. Blue line shows the reference Trajectory. From

the results it is concluded that using GPC Controller Motor

takes approximately 3 to 4 second to reach at the desired Speed and it rejects the disturbance in approximately 1 to 2

seconds.

Fig – 16: Motor Response for Disturbance Rejection Using

Motor Response fGPC Controller

Motor Response for Set Point Tracking using GPC

Controller is shown in figure (17). There are two values of

Set Point taken, first one is 900 Rpm and another is 500 Rpm.

Each Set Point is given to Motor for 20 seconds. From the

results it is concluded that using GPC Controller DC Motor

tracks the Set Point Smoothly. Blue line in the Results

shows the Reference Trajectory. It follows the Set Point smoothly after 1 to 2 seconds. Set Point tracking and

Disturbance Rejection of the DC Motor using GPC

Controller give better results than the PID Controller.

Fig – 17: Motor Response for Set Point Tracking Using GPC

Controller


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B. Discrete Time PID Control Algorithm

DC Motor’s Response for Disturbance Rejection Using

Discrete time PID Controller is shown in figure (18). Set

Point is 800 RPM is taken. From the results it is concluded

that GPC Controller have better results than Discrete time

PID Controller. There is large oscillation occurs in Discrete

time PID Controller. DC Motor takes 15 seconds initially to

reach at the desired Speed using PID Controller. Disturbance

on DC motor generates large oscillation and it takes more

time to rejects the Disturbance than GPC Controller. Settling

time and Overshoot both are large in PID Controller than

GPC Controller.

Fig – 18: Motor Response for Disturbance Rejection Using

PID Controller

DC Motor’s Output Results for Set Point Tracking using

PID Controller is shown in figure (19). Same as GPC controller, here also two Set Point values are given to DC

Motor for Set Point Tracking. Results show that for Set Point

Tracking GPC controller gives better performance than PID

controller.

Fig – 19: Motor Response for Set Point Tracking Using PID

Controller

There is large oscillation in PID Controller for Set Point

Tracking. It takes 15 to 16 seconds to settle at a desired set

point which is larger than GPC controller. Overshoot is also

larger in Discrete time PID Controller than GPC controller.

There is also a difference between Simulation Results and

Real Time Implementation Results of GPC and PID Controller for DC Motor Speed Control.

VII. CONCLUSION

From the Results of Real Time Performance and

Simulation of the DC Motor, it is concluded that GPC

controller gives better results for both the Set Point Tracking

and Disturbance Rejection than Discrete Time PID

controller. DC Motor using GPC Control takes only 3 to 4

seconds to reach at the Desired Set Point while Discrete Time PID Controller takes approximately 15 seconds to reach at

the Desired Set Point which is larger than the GPC Control.

In Disturbance Rejection case study, Discrete Time PID

Controller have more effects of Disturbance and it takes

more time to reject that disturbance while GPC control rejects

Disturbance and Continuous Load effects in small time than

PID controller. GPC gives Smooth and Stable Performance

than Discrete Time PID Control Algorithm. GPC control

algorithm performance depends on the proper selection of

tuning parameters which are Control Horizon (Nu), Starting

and End point of Prediction Horizon, Control Weighting parameter (rho). As Shown in simulation results, for taking

Nu = 2 System have oscillated output while taking Nu = 7

settlingtime of System is increases. For Nu = 5, System has

smooth response. neither it has slow response nor has

oscillated Response. For End point of Prediction Horizon N2

= 13, system have continuous oscillation and for N2 = 6,

settling time of system is increases. So for N2 = 10, system

has smooth response neither it is slow nor oscillated

response. Generally Starting Point of Prediction Horizon N1

= 0 is taken. After proper selection of tuning parameters,

GPC control algorithm gives smooth and stable response for

DC Motor Speed Control.

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