Modeling and Analysis of Systems Lecture #8 - Transfer Function Guillaume Drion Academic year 2015-2016 1
Input-output representation of LTI systems Can we mathematically describe a LTI system using the following relationship? We exploit the superposition principle (linear systems): Response to a pulse: impulse response time-domain. Response to an oscillatory signal at a specific frequency: frequency response frequency-domain. 2
The complex exponential The frequency response of a LTI system determines how it transmits oscillatory signals? We use the complex exponential:. The complex exponential has two components: an exponential growth/decay and oscillatory component: Special case:, giving. In discrete time:. 3
The complex exponential 4
Transmission of complex exponentials through LTI systems Continuous case: LTI system where is the transfer function of the LTI system. 5
Transmission of complex exponentials through LTI systems The transfer function is a transformation of the impulse response. Transfer function Impulse response This transformation is called the Laplace transform: the transfer function of a LTI system is the Laplace transform of its impulse response (continuous time). 6
The Laplace/Fourier transforms: from time domain to frequency domain 7
Transmission of complex exponentials through LTI systems Discrete case: LTI system where is the transfer function of the LTI system. 8
Transmission of complex exponentials through LTI systems The transfer function is a transformation of the impulse response. Transfer function Impulse response This transformation is called the z-transform: the transfer function of a LTI system is the z- transform of its impulse response (discrete time). 9
Important properties of transforms: linearity If with ROC and with ROC then with ROC 10
Important properties of transforms: convolution/multiplication Duality convolution/multiplication (continuous) Duality convolution/multiplication (discrete) 11
Important properties of transforms: differentiation and integration Continuous: Discrete: 12
Transform of decaying exponential/time shift Laplace transform of decaying exponential (continuous) z-transform of time-shift (discrete) 13
Outline Transfer function Response of LTI systems: zero-state (bilateral Laplace transform) Response of LTI systems: non-zero-state (unilateral Laplace transform) Stability of LTI systems 14
The transfer function Transfer function: LTI system LTI system In practice, analysis and design of LTI systems is done using the transfer function. 15
Transfer function of LTI systems (continuous case) Let s consider the discrete LTI system described by the ODE the Laplace transform gives The transfer function is therefore given by 16
Transfer function of LTI systems (discrete case) Let s consider the discrete LTI system described by the difference equation the Z-transform gives The transfer function is therefore given by 17
Transfer function of LTI systems The transfer function of LTI systems has a specific form: it is rational. The roots of The roots of are called the zeros of the transfer function. are called the poles of the transfer function. 18
Transfer function of LTI systems: relationship with state-space representation Transfer function from state-space representation: which gives (1) (1) (2) and therefore 19
Relationship between ROC of transfer function and causality Recall the example of last lecture: If was the impulse response of a system, it would correspond to a causal system, and to its anti-causal equivalent. The ROC of the transfer function of a causal system is an half-plane bounded on the left by the pole that has the biggest real part. 20
Relationship between ROC of transfer function and causality ROC of transfer function of causal systems (continuous and discrete): 21
Block diagrams and transfer function The duality convolution/multiplication makes it easy to connect LTI systems using the transfer function. Parallel: H = H 1 + H 2 H Feedback: H = H 1 /(1+ H 1 H 2 ) H U H 1 Y U - H 1 Y H 2 H 2 Series: H = H 1 H 2 H U H 1 H 2 Y 22
Outline Transfer function Response of LTI systems: zero-state (bilateral Laplace transform) Response of LTI systems: non-zero-state (unilateral Laplace transform) Stability of LTI systems 23
Response of LTI systems: zero-state Can we evaluate the response of a LTI system by simply looking at its transfer function? Example: consider the continous-time LTI system described by the ODE The transfer function writes We want to compute the response of this system in zero-state to a step of amplitude a that starts at t=0 (step response). 24
Response of LTI systems: zero-state The problem writes and the response can be derived using the transfer function: Y (s) =H(s)U(s) = 1 a s 2 +3s +2 s, <(s) > 0 The step response in the time domain can be easily obtained using a partial fraction decomposition of the transfer function: 25
Response of LTI systems: zero-state The step response in the time domain can be easily obtained using a partial fraction decomposition of the transfer function: Using the Laplace transform of an exponential, it yields The response of a LTI system is a combination of exponentials (possibly complex)! 26
Response of LTI systems: zero-state The response of a LTI system is determined by the poles of the transfer function. 27
Response of LTI systems: zero-state The response of a LTI system is determined by the poles of the transfer function. Problem? 28
Outline Transfer function Response of LTI systems: zero-state (bilateral Laplace transform) Response of LTI systems: non-zero-state (unilateral Laplace transform) Stability of LTI systems 29
The transfer function Transfer function: LTI system LTI system In real-life, the input is time-limited and the system has often a non-zero state! 30
The unilateral Laplace transform To cope with non-zero initial conditions, we define the unilateral Laplace transform: where 0 - means that we include the effect of a pulse at t=0. The unilateral Laplace transform of a signal transform of a signal. is the bilateral Laplace The ROC is always a half-place bounded on the left! For a causal LTI system, the unilateral and bilateral Laplace transforms of the impulse response both give the transfer function. 31
Unilateral Laplace transform and initial conditions Why do we define the unilateral Laplace transform? The main difference between the bilateral and unilateral Laplace transforms concerns the derivation: Initial state! Similarly, 32
Unilateral z-transform and initial conditions 33
Response of LTI systems: non-zero-state Can we evaluate the response of a LTI system by simply looking at its transfer function? Example: consider the continuous-time LTI system described by the ODE The transfer function writes We want to compute the response of this system in non-zero-state to a step of amplitude a that starts at t=0 (step response). 34
Response of LTI systems: non-zero-state The problem writes and the response can be derived using the unilateral Laplace transform: which gives 35
Response of LTI systems: non-zero-state The problem writes and the response can be derived using the unilateral Laplace transform: which gives zero-input response zero-state response 36
Response of LTI systems: modes zero-input response zero-state response The zero-input (i.e. autonomous) response of a LTI system is composed of (complex) exponentials determined by the poles of the transfer function. The poles of the transfer function define the modes of the systems response (i.e. natural response). If the transfer function possess a positive real pole, the modes contain a growing exponential! Stability of the system? 37
Outline Transfer function Response of LTI systems: zero-state (bilateral Laplace transform) Response of LTI systems: non-zero-state (unilateral Laplace transform) Stability of LTI systems 38
Response of LTI systems The response of a LTI system is determined by the poles of the transfer function. Stability? 39
Bounded input bounded output (BIBO) stability A system is BIBO stable of all input-output pairs satisfy where is often referred as the gain of the system. In practice, it means that in a stable system, a bounded input will always give a bounded output? Stability is critical in engineering! 40
How do we characterize BIBO stability? A LTI system is stable if the poles of the transfer function all have negative real parts, i.e. the imaginary axis is included in its ROC. 41
How do we characterize BIBO stability? In general, stability is ensured if continuous:. discrete:. In other words: continuous: the imaginary axis is included in the ROC of the transfer function. discrete: the unit circle is included in the ROC of the transfer function. Note that stability conditions imply that the Fourier transform exists! 42
Relationship between ROC of transfer function and stability ROC of transfer function of an unstable causal systems: K>0 K>1 43
Relationship between transfer function and systems response 44