Lecture Frequency response Luca Ferrarini - Basic Automatic Control
Response to a sinusoidal input u () t y( t) ( s) Similarly to the step response, we will analyze now how the system with transfer function (s) reacts to a sinusoidal input (and thus, more in general, to a periodic input, which can be decomposed into a sum of sinusoids). It regards a very practical problem, since often the input of a plant has a sinusoidal or periodic evolution, for example: Analysis of economics phenomena characterized by external inputs with cyclical evolution ( seasonal effects ). Analysis of communication systems, where sounds/voices are decomposed using harmonic analysis into a sum of sinusoids. Luca Ferrarini - Basic Automatic Control
Response to a sinusoidal input u () t y() t ( s) () s = μ g s i ( + st ) ( + sτi ) i i Attention: g must be lower or equal to 0 As. stable u () t = Asin( t) Y () s () s = U () s A = s + A = s + α α βs + γ = + + L+ + sτ + sτ s + Y (s) Y (s) = = πf π T Luca Ferrarini - Basic Automatic Control 3
Y(s) = Y (s) + Y (s) L y(t) = y (t) + y (t) Since the system is asymptotically stable For t y (t) 0 For t y(t) y (t) = L [ Y (s) ] asymptotically, only the forced response remains. Luca Ferrarini - Basic Automatic Control 4
Y L y βs + γ s + () s = = β + γ s + () t = βcost + sint = = Δsin( t + ϕ) K s γ s + To calculate the magnitude Δ and the phase ϕ : Xcosθ + Ysinθ = X Then we get + Y sin θ + atan X Y Δ = β γ + = β + γ ϕ = atan β γ y β () t = β + γ sin + atan γ t Luca Ferrarini - Basic Automatic Control 5
Remember the output transformation A βs + γ Y () s = () s = Y () s + s + s + Multiply both parts by s + () ( s A = s + ) Y () s + βs + γ Evaluate the equation fors = j and rewrite the magnitude and phase equations: ( j) A = jβ + γ Magnitude ( j) A = jβ + γ ( j) A = β + γ β j A ) = jβ + γ ( j) = atan Phase ( ( ) ) ( ) β + γ = ( j)a β ( j) = atan Luca Ferrarini - Basic Automatic Control 6 γ γ
Summarizing u () t A t A = sin U () s = s + A = s = Y s + Y s asymptotically s + () s () () () Y Δ = ϕ = atan γ β γ + β = = ( j) ( j)a y ( t) = Δsin( t + ϕ) L ( t) = ( j) Asin( t + ( j) ) The sinusoid y (t) is completely characterized by the magnitude and the phase of the complex number (j). y amplification displacement Luca Ferrarini - Basic Automatic Control 7
Theorem of the frequency response u () t = Asin( t) u ( t) y( t) ( s) As. stable Then, at steady-state condition (in practice for t > t a ) y ( t) Bsin( t + ϕ) where: B = ϕ = ( j) A ( j) independently of the system initial conditions. Luca Ferrarini - Basic Automatic Control 8
Example () s = + s () t ( 0. t) u = sin 5 = 0.5 rad/s () t ( t) u = sin 5 = 5 rad/s ( j0.5) = + j0.5 ( j0.5) = atan( 0.5) Attention: radian and degree are not the same unity! = 0.89 = 6.6 π y( t) 0.89sin(0.5t 6.6* ) 80 ( j5) = = 0.0 + j5 ( j5) = atan( 5) = 78.6 y( t) 0.0sin(5t π 78.6* ) 80 Luca Ferrarini - Basic Automatic Control 9
Frequency response: definition ( j) for all 0 Complex function of real variable ( j) Luca Ferrarini - Basic Automatic Control 0
Example + s + s + 3 () s =.5 5 s + j + j + 3 5 ( j) = j Imaginary Axis 0.5 0-0.5 ( j5) 0.44. 3 j - -.5 -.5 - -0.5 0 0.5.5 Real Axis Luca Ferrarini - Basic Automatic Control
Extension of the frequency response theorem u(t) u(t) u(t) sum of sinusoids periodic generic u ( t) y( t) ( s) Luca Ferrarini - Basic Automatic Control
Input composed of a sum of sinusoids u N () t = c sin( t + γ ) = At steady-state condition y N () t = c ( j ) sin( t + γ + ( j )) = Superposition (of effects) principle + frequency response theorem Luca Ferrarini - Basic Automatic Control 3
Periodic input u () t = c + c ( t + γ ) = 0 sin At steady-state condition y 0 () t = ( 0) c + c ( j ) ( t + γ + ( j )) μ 0 0 sin = Fourier series Superposition (of effects) principle + frequency response theorem 0 π o = T = 0 o Luca Ferrarini - Basic Automatic Control 4
eneric input u ( t) = C( ) sin( t + γ( ) ) d 0 magnitude spectrum phase spectrum Fourier integral (under wea hypothesis) Superposition (of effects) principle + frequency response theorem At steady-state condition y () t = C( ) ( j) sin( t + γ( ) + ( j) ) d 0 C '( ) γ' ( ) Luca Ferrarini - Basic Automatic Control 5