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Lecture #2. Introduction to Systems. system. A system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. Example of system. System interconnection. System properties. Causality Linearity Time invariance Invertibility. Causality.
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Lecture #2 Introduction to Systems signals & systems
system A system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. signals & systems
Example of system signals & systems
System interconnection signals & systems
System properties • Causality • Linearity • Time invariance • Invertibility signals & systems
Causality A system is said to be causal if the present value of the output signal depends only on the present or past values of the input signal. signals & systems
Causal and noncausal system Example: distinguish between causal and noncausal systems in the following: (1) Case I Noncausal system signals & systems
(2) Case II Delay system causal system (3) Case III causal system At present past signals & systems
(4) Case IV noncausal system At present future (5) Case V noncausal system signals & systems
Linearity A system is said to be linear in terms of the system input x(t) and the system output y(t) if it satisfies the following two properties of superposition and homogeneity. Superposition： Homogeneity： signals & systems
Example 1.19 linear system signals & systems
Example 1.20 Non linear system signals & systems
Properties of linear system : (1) (2) signals & systems
Time invariant system Time invariance A system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal. signals & systems
Example 1.18 Time varying system signals & systems
Invertibility A system is said to be Invertible if the input of the system can be recovered from the output. H Hinv signals & systems
Example 1.15 Inverse system Example 1.16 signals & systems
LINEAR TIME-INVARIANT (LTI) SYSTEMS: A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs System identification signals & systems
example The system is governed by a linear ordinary differential equation (ODE) Linear time invariant system linearity signals & systems
LTI System representations Continuous-time LTI system • Order-N Ordinary Differential equation • Transfer function (Laplace transform) • State equation (Finite order-1 differential equations) ) Discrete-time LTI system • Ordinary Difference equation • Transfer function (Z transform) • State equation (Finite order-1 difference equations) signals & systems
Continuous-time LTI system Order-2 ordinary differential equation constants Linear system initial rest Transfer function signals & systems
Homogenous solution Particular solution Natural response Forced response Zero-input response Zero-state response System response:Output signals due to inputs and ICs. 1. The point of view of Mathematic: ＋ 2. The point of view of Engineer: ＋ 3. The point of view of control engineer: ＋ Transient response Steady state response signals & systems
(1) Particular solution: Example: solve the following O.D.E signals & systems
(2) Homogenous solution: have to satisfy I.C. signals & systems
(3) zero-input response: consider the original differential equation with no input. zero-input response signals & systems
(4) zero-state response: consider the original differential equation but set all I.C.=0. zero-state response signals & systems
(5) Laplace Method: signals & systems
Complex response Zero state response Zero input response Forced response (Particular solution) Natural response (Homogeneous solution) Steady state response Transient response signals & systems
