Forced Response Prof. Seungchul Lee Industrial AI Lab.
Outline • LTI Systems • Time Response to Constant Input • Time Response to Singularity Function Inputs • Response to General Inputs (in Time) • Response to Sinusoidal Input (in Frequency) • Response to Periodic Input (in Frequency) • Response to General Input (in Frequency) • Fourier Transform 2
Linear Time-Invariant (LTI) Systems 3
Systems • 𝐼 is a transformation (a rule or formula) that maps an input signal 𝑦(𝑢) into a time output signal 𝑧(𝑢) • System examples 4
Linear Systems • A system 𝐼 is linear if it satisfies the following two properties: • Scaling • Additivity 5
Time-Invariant Systems • A system 𝐼 processing infinite-length signals is time-invariant (shift-invariant) if a time shift of the input signal creates a corresponding time shift in the output signal 6
Linear Time-Invariant (LTI) Systems • We will only consider Linear Time-Invariant (LTI) systems • Examples 7
Time Response to Constant Input 8
Natural Response • So far, natural response of zero input with non-zero initial conditions are examined 9
Response to Non-Zero Constant Input • Assume all the systems are stable • Inhomogeneous ODE • Same dynamics, but it reaches different steady state • Good enough to sketch 10
Response to Non-Zero Constant Input • Dynamic system response = transient + steady state • Transient response is present in the short period of time immediately after the system is turned on – It will die out if the system is stable • The system response in the long run is determined by its steady state component only • In steady state, all the transient responses go to zero 11
Example • Example 12
Response to Non-Zero Constant Input • Think about mass-spring-damper system in horizontal setting 13
ሶ Response to Non-Zero Constant Input • Mass-spring-damper system in vertical setting • 𝑧 0 = 0 no initial displacement • 𝑧 0 = 0 initially at rest 14
Response to Non-Zero Constant Input • Shift the origin of 𝑧 axis to the static equilibrium point, then act like a natural response with 𝑛 • 𝑧 0 = − 𝑙 and ሶ 𝑧 0 = 0 as initial conditions 15
Time Response to Singularity Function Inputs 16
Time Response to General Inputs • We studied output response 𝑧(𝑢) when input 𝑦(𝑢) is constant • Ultimate Goal: output response of 𝑧(𝑢) to general input 𝑦(𝑢) • Consider singularity function inputs first – Step function – Impulse function (Delta Dirac function) 17
Step Function • Step function 18
Step Response • Start with a step response example • Or • The solution is given: 19
Step Response 20
Impulse • Impulse: difficult to image • The unit-impulse signal acts as a pulse with unit area but zero width • The unit-impulse function is represented by an arrow with the number 1, which represents its area • It has two seemingly contradictory properties : – It is nonzero only at 𝑢 = 0 and – Its definite integral (−∞, ∞) is 1 21
Properties of Dirac Delta Function • The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, and which is also constrained to satisfy the identity • Sifting property 22
Impulse Response • Impulse response: difficult to image • Question: how to realize initial velocity of 𝑤 0 ≠ 0 • Momentum and impulse in physics I – Consider an "impulse" which is a sudden increase in momentum 0 → 𝑛𝑤 of an object applied at time 0 – To model this, – where force 𝑔(𝑢) is strongly peaked at time 0 – Actually the details of the shape of the peak are not important, what is important is the area under the curve – This is the motivation that mathematician and physicist invented the delta Dirac function 23
Impulse Response 24
Impulse Response to LTI system • Later, we will discuss why the impulse response is so important to understand an LTI system 25
Impulse Response to LTI system • Example: now think about the impulse response • The solution is given: (why?) • Impulse input can be equivalently changed to zero input with non-zero initial condition 26
Step Response Again • Relationship between impulse response and unit-step response • Impulse response is the derivative of the step response 27
Response to General Inputs (in Time) 28
Response to a General Input (in Time) • Finally, think about response to a "general input" in time • The solution is given • If this is true, we can compute output response to any general input if an impulse response is given – Impulse response = LTI system 29
Convolution: Definition • 𝑧(𝑢) is the integral of the product of two functions after one is reversed and shifted by 𝑢 30
Easier Way to Understand Continuous Time Signal 31
Easier Way to Understand Continuous Time Signal 32
Structure of Superposition • If a system is linear and time-invariant (LTI) then its output is the integral of weighted and shifted unit- impulse responses. 33
Impulse Response to LTI System Time-invariant Linear (scaling) 34
Response to Arbitrary Input 𝒚(𝒖) (1/2) 35
Response to Arbitrary Input 𝒚(𝒖) (2/2) 36
Response to Arbitrary Input: MATLAB (1/2) • Example • The solution is given: 37
Response to Arbitrary Input: MATLAB (2/2) • Example • The solution is given: 38
Response to Sinusoidal Input (in Frequency) 39
Response to a Sinusoidal Input • When the input 𝑦 𝑢 = 𝑓 𝑘𝜕𝑢 to an LTI system 40
Fourier Transform • Definition: Fourier transform • 𝐼 𝑘𝜕 𝑓 𝑘𝜕𝑢 rotates the same angular velocity 𝜕 41
Response to a Sinusoidal Input: MATLAB 42
Response to a Sinusoidal Input: MATLAB 43
Response to Periodic Input (in Frequency) 44
Response to a Periodic Input (in Frequency Domain) • Periodic signal: Definition • Fourier series represent periodic signals in terms of sinusoids (or complex exponential of 𝑓 𝑘𝜕𝑢 ) • Fourier series represent periodic signals by their harmonic components 45
Response to a Periodic Input (in Frequency) • Fourier series represent periodic signals by their harmonic components 46
Response to a Periodic Input (in Frequency) • What signals can be represented by sums of harmonic components? 47
Harmonic Representations • It is possible to represent all periodic signals with harmonics • Question: how to separate harmonic components given a periodic signal • Underlying properties 48
Harmonic Representations 2𝜌 • Assume that 𝑦(𝑢) is periodic in 𝑈 and is composed of a weighted sum of harmonics of 𝜕 0 = 𝑈 • Then 49
Fourier Series • Fourier series: determine harmonic components of a periodic signal 50
Example: Triangle Waveform • One can visualize convergence of the Fourier Series by incrementally adding terms. 51
Example: Triangle Waveform 52
Example: Triangle Waveform 53
Example: Triangle Waveform 54
Example: Triangle Waveform 55
Example: Triangle Waveform 56
Example: Triangle Waveform 57
Example: Triangle Waveform 58
Example: Triangle Waveform 59
Example: Triangle Waveform: MATLAB 60
Example: Square Waveform 61
Example: Square Waveform 62
Example: Square Waveform 63
Example: Square Waveform 64
Example: Square Waveform 65
Example: Square Waveform 66
Example: Square Waveform 67
Example: Square Waveform 68
Example: Square Waveform 69
Example: Square Waveform: MATLAB 70
Response to a Periodic Input (Filtering) • Periodic input: Fourier series → sum of complex exponentials • Complex exponentials: eigenfunctions of LTI system • Output: same eigenfunctions, but amplitudes and phase are adjusted by the LTI system • The output of an LTI system is a “filtered” version of the input 71
Output is a “Filtered” Version of Input 72
Output is a “Filtered” Version of Input 73
Output is a “Filtered” Version of Input 74
Output is a “Filtered” Version of Input 75
Response to a Square Wave Input: MATLAB • Decompose a square wave to a linear combination of sinusoidal signals • The output response of LTI 76
Response to a Square Wave Input: MATLAB • Decompose a square wave to a linear combination of sinusoidal signals • The output response of LTI • Given input 𝑓 𝑘𝜕𝑢 • 𝑧 = 𝐵𝑓 𝑘(𝜕𝑢+𝜚) 77
Response to a Square Wave Input: MATLAB • Linearity: input σ 𝑏 𝑙 𝑦 𝑙 (𝑢) produces σ 𝑏 𝑙 𝑧 𝑙 (𝑢) 78
Response to a Square Wave Input: MATLAB • Linearity: input σ 𝑏 𝑙 𝑦 𝑙 (𝑢) produces σ 𝑏 𝑙 𝑧 𝑙 (𝑢) 79
Response to General Input (in Frequency) 80
Recommend
More recommend
