Chapter 2 Chapter 2 Systems Defined by Systems Defined by - PDF document

Chapter 2 Chapter 2 Systems Defined by Systems Defined by Differential or Difference Differential or Difference Equations Equations 1

Linear I/ O Differential Equations Linear I/ O Differential Equations with Constant Coefficients with Constant Coefficients • Consider the CT SISO system x t ( ) y t ( ) System described by − 1 N M ∑ ∑ + = ( N ) ( ) i ( ) i y a y ( ) t b x ( ) t i i = = i 0 i 0 i i d y t ( ) d x t ( ) ∈ ∈ � � � � ( ) i ( ) i ( ) x ( ) t y t a , b i i i i dt dt 2

Initial Conditions Initial Conditions • In order to solve the previous equation for t > 0 , we have to know the N initial conditions − … (1) ( N 1) y (0), y (0), , y (0) 3

Initial Conditions – Cont’d Initial Conditions – Cont’d • If the M -th derivative of the input x ( t ) δ k ( ) t contains an impulse or a derivative of an impulse, the N initial conditions must be − = t 0 taken at time , i.e., − − − − … (1) ( N 1) y (0 ), y (0 ), , y (0 ) 4

First-Order Case First-Order Case • Consider the following differential equation: dy t ( ) + = ay t ( ) bx t ( ) dt • Its solution is t ∫ = + τ τ ≥ − − − τ at a t ( ) y t ( ) y (0) e e bx ( ) d , t 0 0 or t ∫ = + τ τ ≥ − − − − τ at a t ( ) y t ( ) y (0 ) e e bx ( ) d , t 0 − 0 − if the initial time is taken to be 0 5

Generalization of the First-Order Case Generalization of the First-Order Case • Consider the equation: dy t ( ) dx t ( ) + = + ay t ( ) b b x t ( ) 1 0 dt dt = − q t ( ) y t ( ) b x t ( ) • Define 1 • Differentiating this equation, we obtain dq t ( ) dy t ( ) dx t ( ) = − b 1 dt dt dt 6

Generalization of the First-Order Generalization of the First-Order Case – Cont’d Case – Cont’d dy t ( ) dx t ( ) + = + ay t ( ) b b x t ( ) 1 0 dt dt + dq t ( ) dy t ( ) dx t ( ) = − b 1 dt dt dt = dq t ( ) = − + ay t ( ) b x t ( ) 0 dt 7

Generalization of the First-Order Generalization of the First-Order Case – Cont’d Case – Cont’d = − q t ( ) y t ( ) b x t ( ) y t ( ) • Solving for it is 1 = + y t ( ) q t ( ) b x t ( ) 1 dq t ( ) = − + which, plugged into , ay t ( ) b x t ( ) 0 dt yields dq t ( ) ( ) = − + + = a q t ( ) b x t ( ) b x t ( ) 1 0 dt = − + − aq t ( ) ( b ab x t ) ( ) 0 1 8

Generalization of the First-Order Generalization of the First-Order Case – Cont’d Case – Cont’d ( ) dy t + = If the solution of ay t ( ) bx t ( ) dt t ∫ = + τ τ ≥ − − − τ at a t ( ) y t ( ) y (0) e e bx ( ) d , t 0 is 0 dq t ( ) = − + − then the solution of aq t ( ) ( b ab x t ) ( ) 0 1 dt is t ∫ = + − τ τ ≥ − − − τ at a t ( ) q t ( ) q (0) e e ( b ab x ) ( ) d , t 0 0 1 0 9

System Modeling – Electrical Circuits System Modeling – Electrical Circuits = v t ( ) Ri t ( ) resistor resistor t dv t ( ) 1 1 ∫ = = τ τ i t ( ) or v t ( ) i ( ) d capacitor capacitor dt C C −∞ t di t ( ) 1 ∫ = = τ τ inductor v t ( ) L or i t ( ) v ( ) d inductor dt L −∞ 10

Example: Bridged-T Circuit Example: Bridged-T Circuit Kirchhoff’ Kirchhoff ’s s voltage law voltage law ( ) ⎧ + − = v t ( ) R i t ( ) i t ( ) x t ( ) loop (or mesh) ⎪ 1 1 1 2 equations + + = ⎨ v t ( ) v t ( ) R i t ( ) 0 1 2 2 2 ⎪ = + ⎩ y t ( ) x t ( ) R i t ( ) 2 2 11

Mechanical Systems Mechanical Systems • Newton • Newton’ ’s second Law of Motion s second Law of Motion: 2 ( ) d y t = x t ( ) M 2 dt • Viscous friction Viscous friction: • dy t ( ) = x t ( ) k d dt • Elastic force: • Elastic force: = ( ) ( ) x t k y t s 12

Example: Automobile Suspension Example: Automobile Suspension System System ⎧ ⎡ ⎤ 2 d q t ( ) dy t ( ) dq t ( ) + − = − + − M k q t [ ( ) x t ( )] k y t [ ( ) q t ( )] k ⎪ ⎢ ⎥ ⎪ 1 t s d ⎣ ⎦ 2 dt dt dt ⎨ ⎡ ⎤ 2 ⎪ d y t ( ) dy t ( ) dq t ( ) + − + − = M k y t [ ( ) q t ( )] k 0 ⎢ ⎥ ⎪ ⎣ ⎦ ⎩ 2 s d 2 dt dt dt 13

Rotational Mechanical Systems Rotational Mechanical Systems • Inertia torque Inertia torque: • θ 2 d ( ) t = ( ) x t I 2 dt • Damping torque: Damping torque: • θ d ( ) t = x t ( ) k d dt • Spring torque: Spring torque: • = θ x t ( ) k ( ) t s 14

Linear I/ O Difference Equation With Linear I/ O Difference Equation With Constant Coefficients Constant Coefficients • Consider the DT SISO system y n [ ] x n [ ] System described by N M ∑ ∑ + − = − y n [ ] a y n [ i ] b x n [ i ] i i = = i 1 i 0 ∈ ∈ � � a , b N is the order or dimension of the system i i 15

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 order recursion) N M ∑ ∑ = − − + − y n [ ] a y n [ i ] b x n [ i ] i i = = i 1 i 0 ( recursive DT system recursive DT system or recursive digital filter recursive digital filter ) 16

Solution by Recursion – Cont’d Solution by Recursion – Cont’d n ≥ 0 • The solution by recursion for requires the knowledge of the N initial conditions − − + − … y [ N ], [ y N 1], , [ 1] y and of the M initial input values − − + − … x [ M ], [ x M 1], , [ 1] x 17

Analytical Solution Analytical Solution • Like the solution of a constant-coefficient differential equation, the solution of N M ∑ ∑ = − − + − y n [ ] a y n [ i ] b x n [ i ] i i = = i 1 i 0 can be obtained analytically in a closed form and expressed as = + y n [ ] y [ n ] y [ n ] zi z s (total response = (total response = zero zero- -input response input response + + zero zero- -state response state response) ) • Solution method presented in ECE 464/564 18