Chapter 2 Chapter 2 Systems Defined by Systems Defined by - - PowerPoint PPT Presentation

Chapter 2 Systems Defined by Differential or Difference Equations Chapter 2 Systems Defined by Differential or Difference Equations

• Consider the CT SISO system

described by Linear I/ O Differential Equations with Constant Coefficients Linear I/ O Differential Equations with Constant Coefficients

1 ( ) ( ) ( )

( ) ( )

N M N i i i i i i

y a y t b x t

− = =

+ =

∑ ∑

,

i i

a b ∈ ∈

• ( )

( ) ( )

i i i

d x t x t dt

• ( )

( ) ( )

i i i

d y t y t dt

• ( )

y t ( ) x t

System

• In order to solve the previous equation for

, we have to know the N initial conditions Initial Conditions Initial Conditions

(1) ( 1)

(0), (0), , (0)

N

y y y

−

… t >

• If the M-th derivative of the input x(t)

contains an impulse or a derivative of an impulse, the N initial conditions must be taken at time , i.e., Initial Conditions – Cont’d Initial Conditions – Cont’d

( ) k t δ

(1) ( 1)

(0 ), (0 ), , (0 )

N

y y y

− − − −

… t

−

=

• Consider the following differential equation:
• Its solution is

First-Order Case First-Order Case

( ) ( ) ( ) dy t ay t bx t dt + =

( )

( ) (0) ( ) ,

t at a t

y t y e e bx d t

τ

τ τ

− − −

= + ≥

∫

( )

( ) (0 ) ( ) ,

t at a t

y t y e e bx d t

τ

τ τ

−

− − − −

= + ≥

∫

• r

if the initial time is taken to be 0

−

• Consider the equation:
• Define
• Differentiating this equation, we obtain

Generalization of the First-Order Case Generalization of the First-Order Case

1

( ) ( ) ( ) ( ) dy t dx t ay t b b x t dt dt + = +

1

( ) ( ) ( ) q t y t b x t = −

1

( ) ( ) ( ) dq t dy t dx t b dt dt dt = −

1

( ) ( ) ( ) ( ) dy t dx t ay t b b x t dt dt + = +

1

( ) ( ) ( ) dq t dy t dx t b dt dt dt = −

Generalization of the First-Order Case – Cont’d Generalization of the First-Order Case – Cont’d

+ =

( ) ( ) ( ) dq t ay t b x t dt = − +

• Solving for it is

which, plugged into , yields

( ) ( ) ( ) dq t ay t b x t dt = − +

Generalization of the First-Order Case – Cont’d Generalization of the First-Order Case – Cont’d

1

( ) ( ) ( ) q t y t b x t = − ( ) y t

1

( ) ( ) ( ) y t q t b x t = +

( )

1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) dq t a q t b x t b x t dt aq t b ab x t = − + + = = − + −

Generalization of the First-Order Case – Cont’d Generalization of the First-Order Case – Cont’d

( )

( ) (0) ( ) ,

t at a t

y t y e e bx d t

τ

τ τ

− − −

= + ≥

∫

( ) ( ) ( ) dy t ay t bx t dt + =

( ) 1

( ) (0) ( ) ( ) ,

t at a t

q t q e e b ab x d t

τ

τ τ

− − −

= + − ≥

∫

1

( ) ( ) ( ) ( ) dq t aq t b ab x t dt = − + −

If the solution of then the solution of is is

System Modeling – Electrical Circuits System Modeling – Electrical Circuits

( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( )

t t

v t Ri t dv t i t

• r

v t i d dt C C di t v t L

• r

i t v d dt L τ τ τ τ

−∞ −∞

= = = = =

∫ ∫

resistor resistor capacitor capacitor inductor inductor

Example: Bridged-T Circuit Example: Bridged-T Circuit

( )

1 1 1 2 1 2 2 2 2 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) v t R i t i t x t v t v t R i t y t x t R i t ⎧ + − = ⎪ + + = ⎨ ⎪ = + ⎩

loop (or mesh) equations Kirchhoff Kirchhoff’ ’s s voltage law voltage law

Mechanical Systems Mechanical Systems

• Newton

Newton’ ’s second Law of Motion s second Law of Motion:

• Viscous friction

Viscous friction:

• Elastic force:

Elastic force:

2 2

( ) ( ) d y t x t M dt = ( ) ( )

d

dy t x t k dt = ( ) ( )

s

x t k y t =

Example: Automobile Suspension System Example: Automobile Suspension System

2 1 2 2 2 2

( ) ( ) ( ) [ ( ) ( )] [ ( ) ( )] ( ) ( ) ( ) [ ( ) ( )]

t s d s d

d q t dy t dq t M k q t x t k y t q t k dt dt dt d y t dy t dq t M k y t q t k dt dt dt ⎧ ⎡ ⎤ + − = − + − ⎪ ⎢ ⎥ ⎪ ⎣ ⎦ ⎨ ⎡ ⎤ ⎪ + − + − = ⎢ ⎥ ⎪ ⎣ ⎦ ⎩

Rotational Mechanical Systems Rotational Mechanical Systems

• Inertia torque

Inertia torque:

• Damping torque:

Damping torque:

• Spring torque:

Spring torque:

2 2

( ) ( ) d t x t I dt θ = ( ) ( )

d

d t x t k dt θ = ( ) ( )

s

x t k t θ =

• Consider the DT SISO system

described by Linear I/ O Difference Equation With Constant Coefficients Linear I/ O Difference Equation With Constant Coefficients

1

[ ] [ ] [ ]

N M i i i i

y n a y n i b x n i

= =

+ − = −

∑ ∑

,

i i

a b ∈ ∈

• [ ]

y n [ ] x n

System N is the order or dimension of the system

Solution by Recursion Solution by Recursion

• Unlike linear I/O differential equations,

linear I/O difference equations can be solved by direct numerical procedure (N-th

• rder recursion)

1

[ ] [ ] [ ]

N M i i i i

y n a y n i b x n i

= =

= − − + −

∑ ∑

(recursive DT system recursive DT system or recursive digital filter recursive digital filter)

• The solution by recursion for requires

the knowledge of the N initial conditions and of the M initial input values Solution by Recursion – Cont’d Solution by Recursion – Cont’d

[ ], [ 1], , [ 1] y N y N y − − + − … [ ], [ 1], , [ 1] x M x M x − − + − … n ≥

• Like the solution of a constant-coefficient

differential equation, the solution of can be obtained analytically in a closed form and expressed as

• Solution method presented in ECE 464/564

Analytical Solution Analytical Solution

1

[ ] [ ] [ ]

N M i i i i

y n a y n i b x n i

= =

= − − + −

∑ ∑

[ [ ] ] [ ]

z zi s

y n y y n n = +

(total response (total response = = zero zero-

• input response

input response + + zero zero-

• state response

state response) )