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SIGNALS AND CONTROL SIGNALS AND CONTROL SIGNALS AND CONTROL SIGNALS AND CONTROL SYSTEMSSYSTEMSSYSTEMSSYSTEMS

Week 7_2

Differential Equations of Physical Systems

Instructor : Dr. Raouf Fareh

2014/2015

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Introduction to Linear Control

A control system is an interconnection of components forming a system configuration

that will provide a desired system response.

• The basis for analysis of a system is provided by linear system theory, which

assumes a cause-effect relationship for the components of a system.

• The process to be controlled can be represented by a block.

• The input/output relationship represent the cause and effect relationship of the

process

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Introduction to Linear Control

An open-loop control system utilizes an actuating device to control the process directly

without using feedback

A closed-loop control system uses a measurement of the output and feedback of

this signal to compare it with the desired output

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Multivariable control system

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Many complex engineering systems are equipped with several actuators that may influence theirstatic and dynamic behavior.

Systems with more than one actuating control input and more than one sensor output may beconsidered as multivariable systems or multi-input-multi-output (MIMO).

The control objective for multivariable systems is to obtain a desirable behavior of several outputvariables by simultaneously manipulating several input channels

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Six Step Approach to Dynamic System Problems

1. Define the system and its components

2. Formulate the mathematical model and list the

necessary assumptions

3. Write the differential equations describing the model

4. Solve the equations for the desired output variables

5. Examine the solutions and the assumptions

6. If necessary, reanalyze or redesign the system

Physical law of the process � Differential Equation

Mechanical system (Newton’s laws)

Electrical system (Kirchhoff’s laws)

How do we obtain the equations?

� Examples:

i.

ii.

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Differential Equation of Physical Systems

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Differential Equation of Physical Systems

v21 Ltid

d⋅ E

1

2L⋅ i

2⋅

v211

k tFd

d⋅ E

1

2

F2

k⋅

ω211

k tTd

d⋅ E

1

2

T2

k⋅

P21 ItQd

d⋅ E

1

2I⋅ Q

2⋅

Electrical Inductance

Translational Spring

Rotational Spring

Fluid Inertia

Describing Equation Energy or Power

Inductive storage

Differential Equation of Physical Systems

Capacitive storage

Electrical Capacitance

Translational Mass

Rotational Mass

Fluid Capacitance

Thermal Capacitance

i Ctv 21

d

d⋅ E

1

2M⋅ v 21

2⋅

F Mtv 2

d

d⋅ E

1

2M⋅ v 2

2⋅

T Jtω 2

d

d⋅ E

1

2J⋅ ω 2

2⋅

Q CftP21

d

d⋅ E

1

2Cf⋅ P21

2⋅

q CttT2

d

d⋅ E C t T2⋅

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Differential Equation of Physical Systems

Electrical Resistance

Translational Damper

Rotational Damper

Fluid Resistance

Thermal Resistance

F b v21⋅ P b v212⋅

i1

Rv 21⋅ P

1

Rv21

2⋅

T b ω21⋅ P b ω212⋅

Q1

Rf

P21⋅ P1

Rf

P212⋅

q1

Rt

T21⋅ P1

Rt

T21⋅

Energy dissipators

Mechanical system: Spring-mass-damper

M2

ty t( )d

d

2⋅ b

ty t( )d

d⋅+ k y t( )⋅+ r t( )

where k is the spring constant of the ideal spring and b is the friction constant. This Equationis a second-order linear constant-coefficient differential equation

The net force applied to the mass m is � − �� − �� ; V: is the velocity. The net force is the force applied to the mass to cause it to accelerate thus: Newton’s law: net force applied to mass =ma

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Example

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Electrical System Resistor–capacitor–inductor system

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Electrical and Mechanical Analogy

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Force-Current Analogy

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Example (Electrical)

The circuit shown in the figure consists of an inductor L, a capacitor C, and two resistors R and

Ro. The input is the voltage Vi(t) and the output is the voltage V0 across the resistor Ro.

Obtain two differential equations for the system in term of iL and Vc

For R = Ro = 1 Ω, L = 1 H, and C =1 F

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Solution

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Solution

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