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Linear Systems

! Textbook: Strum, “Contemporary Linear Systemsusing MATLAB.”

! Contents1. Basic Concepts2. Continuous Systems

a. Laplace Transforms and Applicationsb. Frequency Response of Continuous Systems c. Continuous-Time Fourier Series and

Transforms d. State-Space Topics for Continuous Systems

3. Discrete Systemsa. z Transforms and Applicationsb. Frequency Response of Discrete Systems c. Discrete Fourier Series and Transforms d. State-Space Topics for Discrete Systems

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Chap. 1 Basic Concepts

Continus V. S. Discrete

Continuous System

Discrete System

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Unit Impulse

1. A virtual function.2. Any arbitrary analog signal can be decomposed by impulses.3. The response of any analog signal can be decomposed by impulse

responses.

Properties:

(Sifting Property)

Example:

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Unit Step Function

Representing Signals

a.

b.

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Conversion between Continuous and Discrete Signals

Sampling Theorem: Let : the maximum frequency component of the signal,

: the sampling frequency. then if the signal can be uniquely represented by the

samples, or the signal can fully recovered from the samples.

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Chap. 2 Continuous Systems

Basic Concept

Linearity: A system is linear if and only if it satisfies the principle ofhomogeneity and additivity.1. Homogeneity2. Additivity3. Homogeneity and additivity

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Time Invariance: The same input applied at different times willproduce the same output except shifted in time. That is, for arbitrary

Linear Time-Invariant Systems (LTI): Linear + time-invariant.1. Can be analyzed by Laplace and Fourier transforms.2. In time domain, described by linear differential equations with

constant coefficients. 3. In transform domains, described by linear algebraic equation.

Causality: Output depends on only previous inputs. That is, onlydepends on .

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Stability: if the input is bounded, the output is also bounded(bounded-input-bounded-output BIBO).

Problems 2.1. Discuss the properties of the following continuoussystems.

a. . (i) Linear? (ii) Time invariant?

b. . Causal or noncausal?

c.

Nth-Order Differential Equation ModelIn general, a single-input-single-output LTI system can be modeledby Nth-order differential equation as follows,

Example:

Initial Condition Solution of a Differential Equation1. Find the homogeneous solution, i.e.

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Let , substitute this to the above equation. Then,

which is a polynomial called characteristic equation.Let be the roots of the characteristic equation, then

Note: if stable, the real parts of the roots must be negative.The initial condition solution is in the form

where are unknown coefficients to be determined by theinitial conditions

.

Problem 2.2Let

a. Find the system’s characteristicequation.b. Find the root.c. Is the system stable?d. Find the general form of the homogeneous solution.e. Find the initial solution for and .

The Unit Impulse Response Model

Let the input be , and output , then is called impulseresponse. For any input , we have

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Sifting property:

Stability Definition:

Problem 2.3

a. Find impulse response .b. Is the system stable?

Convolution

Let, , then

Let , then

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A continuous-time function can be broken up into a summation ofshifted impulse functions.

Suppose an LTI continous system has the response of impulse

1.Time Invariant

2.Homogeneity

3.Additivity

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4.Let , then

(Convolution Integral)

where “*” is the convolution symbol.

Alternative form:

Example:

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Example:

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Problem 2.4 , , find by MATLAB.

Analytic solution:

Problem 2.5 Convolution with Impulse functions.

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Sinusoidal Steady-State Response

What is the output when the input is an AC signal, that is, asinusoidal signal? Since

let,

then,

Let

,

then

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Conclusion: in general, when a sinusoidal signal

is applied to a stable LTI system, the output is.

That is, the output is also a sinusoidal signal.

Problem 2.6A highpass filter has impulse response

.If

Find .

First, compute .

The input has two frequency components1. : , no output.

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2. : , the output is

Alternative Path to

Problem 2.7A bandpass analog filter is described by

where is the output and is input.a. Determine the frequency response.b. Find the output of

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State-Space Model

If there is no derivative term in the input, that is,

we can define

.

Then we can turn the original equation to a set of first-orderdifferential equations:

We call state variables and the following statevector:

. Then, the set of equations become in matrix form

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A common definition for the state of a system is as follows:The state of a system is a minimum set of quantities

which if known at are uniquely determinedfor by specifying the inputs to the system for . Why?

The M ouputs of a system are related to the states and asingle input by the output equation

where is an M by N matrix and is an M by 1 vector.

Problem 2.8

a. Describe the system in state variable form with and.

b. Find the output equation if and .

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System Simulation (Numerical Solution Using MATLAB)Consider

Let ,

then

Case 1: initial condition . . .A=[0 1 ; -2 -3];B=[0;1/3];C=[1 0];D=0;v0=[1;0];tspan=[0 5];x=@(t) 0; df=@(t,v) A*v+B*x(t);[t vv]=ode45(df,tspan,v0);plot(t,vv(:,1));xlabel(‘t‘);ylabel(‘theta‘);

Case 2: initial condition . . .

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v0=[0;0];x=@(t) 1;df=@(t,v) A*v+B*x(t);[t vv]=ode45(df,tspan,v0);plot(t,vv(:,1));xlabel(‘t‘);ylabel(‘theta‘);

Example 2.1: An Oscillatory System

a. Find the characteristic equation and roots.b. Solve for initial conditions of and .c. Write the state-space equation and use MATLAB to solve the

equation. Assume . Plot the result with (b) to compare.

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Example 2.2: Second-Order Systems

where is the natural frequency.a. From

,

find

b. Find the characteristic equation in terms of RLC circuitparameters and .

c. Let L and C fixed and , find the range of R that will yield(i) Purely imaginary roots.(ii) Complex roots.(iii) Real roots.

, or

d. Find the state-space equation.

e. Plot for the following cases: , ,, , .

wn=10;zeta=2.3;K=100;A=[0 1 ; -wn^2 -2*zeta*wn];B=[0;K];v0=[0;0];tspan=[0 2];x=@(t) 1;

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df=@(t,v) A*v+B*x(t);[t vv]=ode45(df,tspan,v0);plot(t,vv(:,1));xlabel(‘t‘);ylabel(‘theta‘);

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Example 2.3 Unit Impulse Response of a Lowpass FilterA lowpass RC filter can be modeled by

a. Find the characteristic equation.

b. Find for and .

c. Find for and .

d. Find for and .e. Let , show that

delta=1;A=-1;B=1;v0=0;tspan=[0 4];x=@(t) (t<=delta && t>=0)/delta; df=@(t,v) A*v+B*x(t);[t vv]=ode45(df,tspan,v0);plot(t,vv);xlabel(‘t‘);ylabel(‘theta‘);

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Example 2.4 Convolutiona. An LTI causal system is modeled by unit impulse response .

Prove that for ,

b. A finite duration integrator can be modeled by the unit impulseresponse . If the input is

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,Find the output by graphic method.

c. Find the output by analytical method.d. If the hypothetical integrator is modeld by the noncausal unit

impulse response , find .