meiling chensignals & systems1 Lecture #2 Introduction to Systems

meiling chensignals & systems2 system A system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

meiling chensignals & systems3 Example of system

meiling chensignals & systems4 System interconnection

meiling chensignals & systems5 System properties Causality Linearity Time invariance Invertibility

meiling chensignals & systems6 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.

meiling chensignals & systems7 Causal and noncausal system Example: distinguish between causal and noncausal systems in the following: (1) Case I Noncausal system

meiling chensignals & systems8 (2) Case II causal system Delay system (3) Case III causal system At present past

meiling chensignals & systems9 (4) Case IV noncausal system At presentfuture (5) Case V noncausal system

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meiling chensignals & systems11 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 ：

meiling chensignals & systems12 Example 1.19 linear system

meiling chensignals & systems13 Example 1.20 Non linear system

meiling chensignals & systems14 Properties of linear system : (1) (2)

meiling chensignals & systems15 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. Time invariant system

meiling chensignals & systems16 Example 1.18 Time varying system

meiling chensignals & systems17 Invertibility A system is said to be Invertible if the input of the system can be recovered from the output. HH inv

meiling chensignals & systems18 Example 1.15 Inverse system Example 1.16

meiling chensignals & systems19 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

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meiling chensignals & systems21 example The system is governed by a linear ordinary differential equation (ODE) Linear time invariant system linearity

meiling chensignals & systems22 LTI System representations 1.Order-N Ordinary Differential equation 2.Transfer function (Laplace transform) 3.State equation (Finite order-1 differential equations) ) 1.Ordinary Difference equation 2.Transfer function (Z transform) 3.State equation (Finite order-1 difference equations) Continuous-time LTI system Discrete-time LTI system

meiling chensignals & systems23 constants Order-2 ordinary differential equation Continuous-time LTI system Transfer function Linear system  initial rest

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meiling chensignals & systems25 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: Homogenous solutionParticular solution ＋ ＋ ＋ Zero-state responseZero-input responseNatural responseForced response Transient responseSteady state response

meiling chensignals & systems26 Example: solve the following O.D.E (1) Particular solution:

meiling chensignals & systems27 (2) Homogenous solution: have to satisfy I.C.

meiling chensignals & systems28 (3) zero-input response: consider the original differential equation with no input. zero-input response

meiling chensignals & systems29 (4) zero-state response: consider the original differential equation but set all I.C.=0. zero-state response

meiling chensignals & systems30 (5) Laplace Method:

meiling chensignals & systems31 Complex response Zero state responseZero input response Forced response (Particular solution) Natural response (Homogeneous solution) Steady state responseTransient response