Project - Position control of mass spring damper system

To design a closed loop control scheme for a DC motor the following changes need to be done in the model in the previously

created model.

Speed is the controllable parameter, so we will set the reference speed in step block as 10,20, 40 whichever you want. 

Subtract the actual speed from the reference speed to generate the error signal e(t)

Add Kp as K(1) and Kd as K(2) parameter determined from place command (K=place(A,B,P)). (use gain block to mention

values of Kp, Kd, and Ki)

The feedback for Kd is provided from the output current of the plant/model (figure 1). OR we can take feedback (error speed

signal as input to Kd shown by figure 2)

Add Ki manually with gain and integrator block and tune different values of Ki and check the impact of it on results. 

To check the location of poles on the s-plane from MATLAB the following command (‘pzmap ’-pole zero placement or

mapping) can be used in continuation to the above script. You can see two crosses in red indicate complex poles lying on the

left hand of the plane which indicate it is a stable system. 

As we know that given value of transfer function spring mass damper system as shown

in fig 

Block Used:

Pulse Generator Block

sum Block

Derivative Block

integrator block

Gain Block

Transfer Function Block

Scope

Pulse Generator:

Generate square wave pulses at regular intervals

Library:

Sources

Description:

The Pulse Generator block generates square wave pulses at regular intervals. The block's waveform

parameters, Amplitude, Pulse Width, Period, and Phase Delay, determine the shape of the output waveform. The following

diagram shows how each parameter affects the waveform.

The Pulse Generator can emit scalar, vector, or matrix signals of any real data type. To cause the block to emit a scalar signal,

use scalars to specify the waveform parameters. To cause the block to emit a vector or matrix signal, use vectors or

matrices, respectively, to specify the waveform parameters. Each element of the waveform parameters affects the

corresponding element of the output signal. For example, the first element of a vector amplitude parameter determines the

amplitude of the first element of a vector output pulse. All the waveform parameters must have the same dimensions after

scalar expansion. The data type of the output is the same as the data type of the Amplitude parameter

The block's Pulse type parameter allows you to specify whether the block's output is time-based or sample-based. If you

select sample-based, the block computes its outputs at fixed intervals that you specify. If you select time-based, Simulink

computes the block's outputs only at times when the output actually changes. This can result in fewer computations being

required to compute the block's output over the simulation time period.

Depending on the pulse's waveform characteristics, the intervals between changes in the block's output can vary. For this

reason, Simulink cannot use a fixed solver to compute the output of a time-based pulse generator. Simulink allows you to

specify a fixed-step solver for models that contain time-based pulse generators. However, in this case, Simulink computes a

fixed sample time for the time-based pulse generators. It then simulates the time-based pulse generators as sample-basedIf

you select time-based as the block's pulse type, you must specify the pulse's phase delay and period in units of seconds.

If you specify sample-based, you must specify the block's sample time in seconds, using the Sample Time parameter, then

specify the block's phase delay and period as integer multiples of the sample time. For example, suppose that you specify a

sample time of 0.5 second. And suppose you want the pulse to repeat every two seconds. In this case, you would specify 4

as the value of the block's Period parameter

Data Type Support:

A Pulse Generator block outputs real signals of any data type except int64 and uint64. The data type of the output signal is

the same as that of the Amplitude parameter.

 

Parameters and Dialog Box:

 

 

Pulse type:The pulse type for this block: time-based or sample-based. The default is time-based.

 

Amplitude:The pulse amplitude. The default is 1.

 

Period:The pulse period specified in seconds if the pulse type is time-based or as number of sample times if the pulse type is

sample-based. The default is 2.

 

Pulse width:The duty cycle specified as the percentage of the pulse period that the signal is on if time-based or as number of

sample times if sample-based. The default is 50 percent.

 

Phase delay:The delay before the pulse is generated specified in seconds if the pulse type is time-based or as number of

sample times if the pulse type is sample-based. The default is 0 seconds.

 

Sample Time:The length of the sample time for this block in seconds. This parameter appears only if the block's pulse type is

sample-based. See Specifying Sample Time  for more information.

 

Interpret vector parameters as 1-D:If this option is selected and the other parameters are one-row or one-column matrices,

after scalar expansion, the block outputs a 1-D signal (vector). Otherwise the output dimensionality is the same as that of

the other parameters.

Characteristics:

Sample Time	Inherited
Scalar Expansion	Of parameters
Dimensionalized	Yes
Zero Crossing	No

 

Sum:

Add or subtract inputs

Library:

Simulink Math Operations and Fixed-Point Blockset Math

Discription:

 

The Sum block performs addition or subtraction on its inputs. This block can add or subtract scalar, vector, or matrix inputs.

It can also collapse the elements of a single input vector.

You specify the operations of the block with the List of Signs parameter. Plus (+), minus (-), and spacer (|) characters

indicate the operations to be performed on the inputs:

If there are two or more inputs, then the number of characters must equal the number of inputs. For example, "+-+" requires

three inputs and configures the block to subtract the second (middle) input from the first (top) input, and then add the third

(bottom) input.

All nonscalar inputs must have the same dimensions. Scalar inputs will be expanded to have the same dimensions as the

other inputs.

 

A spacer character creates extra space between ports on the block's icon.

If only addition of all inputs is required, then a numeric parameter value equal to the number of inputs can be supplied

instead of "+" characters.

if only one vector is input, then a single "+" or "-" will collapse the vector using the specified operation.

When the Show additional parameters check box is selected, some of the parameters that become visible are common to

many blocks. For a detailed description of these parameters, refer to Block Parameters in the Fixed-Point Blockset

documentation.

Data Type Support:

The Sum block accepts signals of any complexity and data type, including fixed-point data types, except int64 and uint64.

The inputs may be of different data types unless the Require all inputs to have same data type parameter is selected.

Parametar And Dialog Box:

 

Show additional parameters:

If selected, additional parameters specific to implementation of the block become visible as shown.

 

Icon shape:Designate the icon shape of the block.

List of signs:Enter as many plus (+) and minus (-) characters as there are inputs. Addition is the default operation, so if you

only want to add the inputs, enter the number of input ports. For a single vector input, "+" or "-" will collapse the vector

using the specified operation.You can manipulate the positions of the input ports on the block icon by inserting spacers (|)

between the signs in the List of signs parameter. For example, "++|--" creates an extra space between the second and third

input ports.

Require all inputs to have same data type:Select this parameter to require that all inputs must have the same data type.

Output data type mode:Specify the output data type and scaling to be the same as the first input, or inherit the data type

and scaling from an internal rule or by backpropagation. You can also choose a built-in data type from the drop-down list.

Lastly, if you choose, the Output data type, Output scaling value, and Lock output scaling against changes by the autoscaling

tool parameters become visible.

Output data type:Specify any data type, including fixed-point data types. This parameter is only visible if is selected for

the Output data type mode parameter.

Output scaling value:Set the output scaling using radix point-only or [Slope Bias] scaling. This parameter is only visible if is

selected for the Output data type mode parameter.

Lock output scaling against changes by the autoscaling tool:If selected,Scalling of out-put is locked.This parameter is only

visible if is selected for the Output data type mode parameter.

Round integer calculations toward:Select the rounding mode for fixed-point output.

Saturate on integer overflow:If selected, overflows saturate.

Conversion and Operations:The Sum block first converts the input data type(s) to the output data type using the specified

rounding and overflow modes, and then performs the specified operations. Refer to Rules of Artimetic operation in the Fixed

Point Blockset documentation for more information about the rules that this block obeys when performing fixed-point

operations.

Characteristics:

Dimensionalized	Yes
Direct Feedthrough	Yes
Sample Time	Inherited from driving blocks
Scalar Expansion	Yes
States	0
Zero Crossing	No

Derivative:

Output the time derivative of the input

Library:

Continuous

Description:

The Derivative block approximates the derivative of its input by computin

where u is the change in input value and t is the change in time since the previous simulation time step. The block accepts

one input and generates one output. The value of the input signal before the start of the simulation is assumed to be zero.

The initial output for the block is zero.

The accuracy of the results depends on the size of the time steps taken in the simulation. Smaller steps allow a smoother and

more accurate output curve from this block. Unlike blocks that have continuous states, the solver does not take smaller steps

when the input changes rapidly.

When the input is a discrete signal, the continuous derivative of the input is an impulse when the value of the input changes,

otherwise it is 0. You can obtain the discrete derivative of a discrete signal using and taking z-tranfarm.

Using linmod to linearize a model that contains a Derivative block can be troublesome. For information about how to avoid

the problem, see Linearizing Models in Using Simulink

Data Type Support:

A Derivative block accepts and outputs a real signal of type double.

Dialog Box:

Characteristics:

Direct Feedthrough	Yes
Sample Time	Continuous
Scalar Expansion	N/A
States	0
Dimensionalized	Yes
Zero Crossing	No

 

Integrator:

Integrate a signal

Library:

Continuous

Description:

Integration deals with two essentially different types of problems.

In the first type, derivative of a function is given and we want to find the function. Therefore, we basically reverse the

process of differentiation. This reverse process is known as anti-differentiation, or finding the primitive function, or finding

an indefinite integral.

The second type of problems involve adding up a very large number of very small quantities and then taking a limit as the

size of the quantities approaches zero, while the number of terms tend to infinity. This process leads to the definition of

the definite integral.

Definite integrals are used for finding area, volume, center of gravity, moment of inertia, work done by a force, and in

numerous other applications.

By definition, if the derivative of a function f(x) is f'(x), then we say that an indefinite integral of f'(x) with respect to x is

f(x). For example, since the derivative (with respect to x) of x² is 2x, we can say that an indefinite integral of 2x is x².

In symbols −

f'(x²) = 2x, therefore,

∫ 2xdx = x².

Indefinite integral is not unique, because derivative of x² + c, for any value of a constant c, will also be 2x.

This is expressed in symbols as −

∫ 2xdx = x² + c.

Where, c is called an 'arbitrary constant'.

MATLAB provides an int command for calculating integral of an expression. To derive an expression for the indefinite integral

of a function, we write −

int(f);

Characteristics:

Direct Feedthrough	Yes, of the reset and external initial condition source ports
Sample Time	Continuous
Scalar Expansion	Of parameters
States	Inherited from driving block or parameter
Dimensionalized	Yes
Zero Crossing	If the Limit output option is selected, one for detecting reset, one each to detect upper and lower saturation limits, one when leaving saturation

 

Gain:

Multiply the input by a constant

Library:

Simulink Math Operations and Fixed-Point Blockset Math

Description:

 

The Gain block multiplies the input by a constant value (gain). The input and the gain can each be a scalar, vector, or matrix

You specify the value of the gain in the Gain parameter. The Multiplication parameter lets you specify element-wise or matrix

multiplication. For matrix multiplication, this parameter also lets you indicate the order of the multiplicands.

When the Show additional parameters check box is selected, some of the parameters that become visible are common to

many blocks. For a detailed description of these parameters, refer to Block Prameters in the Fixed-Point Blockset

documentation.

The Matrix gain block is an implementation of the Gain block with different default settings.

Data Type Support:

The input and gain of the Gain block can be a real or complex scalar, vector, or matrix of any data type except boolean,

int64,and uint64. If the input is real and the gain is complex, the output is complex.

Parametar And Dialog Box:

 

Gain:Specify the value by which to multiply the input. The gain may be a scalar, vector, or matrix.

• Multiplication:
• Specify the multiplication mode:
• Element-wise(k*u)--Each element of the input is multiplied by each element of the gain. The block performs expansions, if
• necessary, so that the input and gain have the same dimensions.
• Matrix(k*u)--The input and gain are matrix multiplied with the input as the second operand.
• Matrix(u*k)--The input and gain are matrix multiplied with the input as the first operand.
• Matrix(k*u)(u vector)--The input and gain are matrix multiplied with the input as the second operand. The input and the
• output are required to be vectors and their lengths are determined by the dimensions of the gain.
• Show additional parameters
• If selected, additional parameters specific to implementation of the block become visible as shown.

Parameter data type mode:

Set the data type and scaling of the gain to be the same as that of the input, or to be inherited via an internal rule.

Alternatively, choose to specify the data type and scaling of the gain through the Parameter data type, Parameter scaling

mode, and Parameter scaling parameters in the dialog.

Parameter data type:

Set the gain data type. This parameter is only visible if is selected for the Parameter data type mode parameter.

Parameter scaling mode:

Set the mode to determine the scaling of the gain.

 Use specified scaling --This mode allows you to set the scaling of the gain in the Parameter scaling parameter.

 

Best Precision: Element-wise--This mode sets radix points for the elements of the gain such that the precision of each elemnt is maximized.

Best Precision: Row-wise--This mode sets a common radix point within each row of the gain such that the largest element of

ach row has the best possible precision.

Best Precision: Column-wise--This mode sets a common radix point within each column of the gain such that the largest

element of each column has the best possible precision.

Best Precision: Matrix-wise--This mode sets a common radix point for all the elements of the gain such that the largest

element has the best possible precision.This parameter is only visible if  is selected for the Parameter data type

mode parameter.

Parameter scaling:Set the gain scaling using either radix point-only or [Slope Bias] scaling. This parameter is only visible if is

selected for the Parameter data type mode parameter, and if Use specified scaling is selected for the Parameter scaling

mode parameter.

Output data type mode:Set the data type and scaling of the output to be the same as that of the input, or to be inherited via

an internal rule or by backpropagation. Alternatively, choose to specify the data type and scaling of the output through

the Output data type and Output scaling value parameters in the dialog.

If you select internal rule for this parameter, Simulink chooses a combination of output scaling and data type that requires

the smallest amount of memory consistent with accommodating the output range and maintaining the output precision of the

block. If the Production hardware characteristics parameter on the Advanced pane of the Simulation Parameters dialog is set

to Unconstrained integer sizes, Simulink chooses the output data type without regard to hardware constraints. If the

parameter is set to Microprocessor, Simulink chooses the smallest available hardware data type capable of meeting the range

and precision constraints. For example, if the block multiplies an input of type int8 by a gain of int16 and Unconstrained

integer sizes is specified, the output data type is sfix24. If Microprocessor is specified and the microprocessor supports 8-bit,

16-bit, and 32-bit words, the output data type is int32. If none of the word lengths provided by the target microprocessorcan

accommodate the output range, Simulink displays an error message in the Simulink Diagnostic Viewer

 

Output data type:Set the output data type. This parameter is only visible if is selected for the Output data type

mode parameter.

Output scaling value:Set the output scaling using either radix point-only or [Slope Bias] scaling. This parameter is only visible

if is selected for the Output data type mode parameter.

Lock output scaling against changes by the autoscaling tool:If selected,scaling output is locked. This parameter is only visible

if  is selected for the Output data type mode parameter.

Round integer calculations toward:Select the rounding mode for integer output.

Saturate on integer overflow:If selected, overflows saturate.

 Conversion And Operation:

The gain is converted from doubles to the specified data type offline using round-to-nearest and saturation. Refer to

Parameter of conversion in the Fixed-Point Blockset documentation for more information about parameter conversions. The

input and gain are then multiplied, and the result is converted to the output data type using the specified rounding and

overflow modes. Refer to Rules of artimatic Operation in the Fixed-Point Blockset documentation for more information about

the rules that this block obeys when performing fixed-point operations.

Characteristics:

Dimensionalized	Yes
Direct Feedthrough	Yes
Sample Time	Inherited from driving block
Scalar Expansion	Of input and Gain parameter for Element wise multiplication
Zero Crossing	No

 

Transfer Function:

Implement a linear transfer function

Library:

Continuous

Description:

The Transfer Fcn block implements a transfer function where the input (u) and output (y) can be expressed in transfer

function form as the following equation

where nn and nd are the number of numerator and denominator coefficients, respectively. num and den contain the

coefficients of the numerator and denominator in descending powers of s. num can be a vector or matrix, den must be a

vector, and both are specified as parameters on the block dialog box. The order of the denominator must be greater than or

equal to the order of the numerator

A Transfer Fcn block takes a scalar input. If the numerator of the block's transfer function is a vector, the block's output is

also scalar. However, if the numerator is a matrix, the transfer function expands the input into an output vector equal in width

to the number of rows in the numerator. For example, a two-row numerator results in a block with scalar input and vector

output. The width of the output vector is two.

Initial conditions are preset to zero. If you need to specify initial conditions, convert to state-space form using tf2ss and use

the State-Space block. The tf2ss utility provides the A, B, C, and D matrices for the system. For more information, type help

tf2ss or consult the Control System Toolbox documentation.

Transfer Fcn Block Icon:

The numerator and denominator are displayed on the Transfer Fcn block icon depending on how they are specified:

If each is specified as an expression, a vector, or a variable enclosed in parentheses, the icon shows the transfer function with

the specified coefficients and powers of s. If you specify a variable in parentheses, the variable is evaluated. For example, if

you specify Numerator as and Denominator as (den) where den is, the block icon looks like this:

If each is specified as a variable, the icon shows the variable name followed by (s). For example, if you specify Numerator as

num and Denominator as den, the block icon looks like this:

Specifying the Absolute Tolerance for the Block's States:

By default Simulink uses the absolute tolerance value specified in the Simulation Parameters dialog box (see Error

Tolerances) to solve the states of the Transfer Fcn block. If this value does not provide sufficient error control, specify a more

appropriate value in the Absolute tolerance field of the Transfer Fcn block's dialog box. The value that you specify is used to

solve all the block's states.

Data Type Support:

A Transfer Fcn block accepts and outputs signals of type double.

 

Parameters and Dialog Box:

 

Numerator:The row vector of numerator coefficients. A matrix with multiple rows can be specified to generate multiple

output. The default is [1].

Denominator:The row vector of denominator coefficients. The default is [1 1].

Absolute tolerance:Absolute tolerance used to solve the block's states. You can enter auto or a numeric value. If you

enter auto, Simulink determines the absolute tolerance (see "Error Tolerances). If you enter a numeric value, Simulink uses

the specified value to solve the block's states. Note that a numeric value overrides the setting for the absolute tolerance in

the Simulation Parameters dialog box.

 

Characteristics:

Direct Feedthrough	Only if the lengths of the Numerator and Denominator parameters are equal
Sample Time	Continuous
Scalar Expansion	No
States	Length of Denominator -1
Dimensionalized	Yes, in the sense that the block expands scalar input into vector output when the transfer function numerator is a matrix. See the preceding block description.
Zero Crossing	No

 

 Scope: To visualize your simulation results over time, use a Scop block or Time Scope (DSP System Toolbox) block

To connect multiple signals to a scope, drag additional signals to the scope block. An additional port is created automatically.

To specify the number of input ports:

Open a scope window.

From the toolbar, select File > Number of Input Ports > More.

Enter the number of input ports, up to 96.

 

Program.m File:

m = 3.6;

k= 400;

c = 100;

%% Design of PID Controller

A =[0 1; -k/m -c/m];

b=[0 ; 0.01];

%% Placement of Pole

fn=1;

wn=sqrt(k/m);

zeta=c/(2*sqrt(m*k));

p1=-zeta*wn+wn*sqrt(zeta^2-1);

p2=-zeta*wn-wn*sqrt(zeta^2-1);

p=[p1;p2];

K=place(A,b,p);

kp=K(1);

kd=K(2);

 

Command Window:

Out-Put Graph: Here the (ki) value consider = 800 than out-put get stable response shown in fig.

CONCLUSION :

To complte the project of Position control of mass spring damper system transfer function of a mass spring damper system

and use it in the model. Add a PID controller to adjust the force on mass so that its position follows a reference signal to

using matlab and simulink wave from plot.