chapter 2 systems defined by differential or difference equations
TRANSCRIPT
Chapter 2Systems Defined by
Differential or Difference Equations
Chapter 2Systems 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( ) ( ) ( )
0 0
( ) ( )N M
N i i
i ii i
y a y t b x t
,i ia b ( ) ( )( )
ii
i
d x tx t
dt( ) ( )
( )i
i
i
d y ty t
dt
( )y t( )x t System
• In order to solve the previous equation for
, we have to know the N initial conditions
Initial ConditionsInitial Conditions
(1) ( 1)(0), (0), , (0)Ny y y
0t
• 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’dInitial Conditions – Cont’d
( )k t
(1) ( 1)(0 ), (0 ), , (0 )Ny y y
0t
• Consider the following differential equation:
• Its solution is
First-Order CaseFirst-Order Case
( )( ) ( )
dy tay t bx t
dt
( )
0
( ) (0) ( ) , 0t
at a ty t y e e bx d t ( )
0
( ) (0 ) ( ) , 0t
at a ty t y e e bx d t
or
if the initial time is taken to be 0
• Consider the equation:
• Define
• Differentiating this equation, we obtain
Generalization of the First-Order CaseGeneralization of the First-Order Case
1 0
( ) ( )( ) ( )
dy t dx tay t b b x t
dt dt
1( ) ( ) ( )q t y t b x t
1
( ) ( ) ( )dq t dy t dx tb
dt dt dt
1 0
( ) ( )( ) ( )
dy t dx tay t b b x t
dt dt
1
( ) ( ) ( )dq t dy t dx tb
dt dt dt
Generalization of the First-Order Case – Cont’d
Generalization of the First-Order Case – Cont’d
0
( )( ) ( )
dq tay t b x t
dt
• Solving for it is
which, plugged into ,
yields
0
( )( ) ( )
dq tay 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 0
0 1
( )( ) ( ) ( )
( ) ( ) ( )
dq ta q t b x t b x t
dtaq t b ab x t
Generalization of the First-Order Case – Cont’d
Generalization of the First-Order Case – Cont’d
( )
0
( ) (0) ( ) , 0t
at a ty t y e e bx d t
( )( ) ( )
dy tay t bx t
dt
( )
0 1
0
( ) (0) ( ) ( ) , 0t
at a tq t q e e b ab x d t
0 1
( )( ) ( ) ( )
dq taq t b ab x t
dt
If the solution of
then the solution of
is
is
System Modeling – Electrical CircuitsSystem Modeling – Electrical Circuits
( ) ( )
( ) 1 1( ) ( ) ( )
( ) 1( ) ( ) ( )
t
t
v t Ri t
dv ti t or v t i d
dt C C
di tv t L or i t v d
dt L
resistorresistor
capacitorcapacitor
inductorinductor
Example: Bridged-T CircuitExample: Bridged-T Circuit
1 1 1 2
1 2 2 2
2 2
( ) ( ) ( ) ( )
( ) ( ) ( ) 0
( ) ( ) ( )
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’s voltage lawKirchhoff’s voltage law
Mechanical SystemsMechanical Systems
• Newton’s second Law of MotionNewton’s second Law of Motion:
• Viscous frictionViscous friction:
• Elastic force:Elastic force:
2
2
( )( )
d y tx t M
dt
( )( ) d
dy tx t k
dt
( ) ( )sx t k y t
Example: Automobile Suspension System
Example: Automobile Suspension System
2
1 2
2
2 2
( ) ( ) ( )[ ( ) ( )] [ ( ) ( )]
( ) ( ) ( )[ ( ) ( )] 0
t s d
s d
d q t dy t dq tM k q t x t k y t q t k
dt dt dt
d y t dy t dq tM k y t q t k
dt dt dt
Rotational Mechanical SystemsRotational Mechanical Systems
• Inertia torqueInertia torque:
• Damping torque:Damping torque:
• Spring torque:Spring torque:
2
2
( )( )
d tx t I
dt
( )( ) d
d tx t k
dt
( ) ( )sx t k t
• Consider the DT SISO system
described by
Linear I/O Difference Equation With Constant Coefficients
Linear I/O Difference Equation With Constant Coefficients
1 0
[ ] [ ] [ ]N M
i ii i
y n a y n i b x n i
,i ia b
[ ]y n[ ]x n System
N is the order or dimension of the system
Solution by RecursionSolution by Recursion
• Unlike linear I/O differential equations, linear I/O difference equations can be solved by direct numerical procedure (N-th order recursion)
1 0
[ ] [ ] [ ]N M
i ii i
y n a y n i b x n i
(recursive DT systemrecursive DT system or recursive digital filterrecursive 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’dSolution by Recursion – Cont’d
[ ], [ 1], , [ 1]y N y N y
[ ], [ 1], , [ 1]x M x M x
0n
• 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 SolutionAnalytical Solution
1 0
[ ] [ ] [ ]N M
i ii i
y n a y n i b x n i
[[ ]] [] zzi sy n yy n n (total response(total response == zero-input response zero-input response ++ zero-state responsezero-state response))