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S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn Laplace Transform Math Review with Matlab: Application: Linear Time Invariant (LTI) Systems

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Laplace Transform: X(s) Linear Time Invariant Systems 2 Linear Time Invariant (LTI) Systems Definition of a Linear Time Invariant System Impulse Response Transfer Function Simple Systems Simple System Example Pulse Response Example Transient and Steady State Example

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Laplace Transform: X(s) Linear Time Invariant Systems 3 System Definition A system can be thought of as a black box with an input and an output The signal connected to the input is called the Excitation The system performs a Transformation, T, (function) on the input Given an input excitation, the output signal is called the Response ExcitationResponse Output Signal y(t) Input Signal x(t)

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Laplace Transform: X(s) Linear Time Invariant Systems 4 Differential Equations Time domain systems are often described using a Differential Equation Recall that time domain Differentiation corresponds to Laplace Transform domain Multiplication by s with subtraction of Initial Conditions Output Signal y(t) Input Signal x(t)

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Laplace Transform: X(s) Linear Time Invariant Systems 5 Linear Systems A system is Linear if it satisfies the Superposition Principle ( where and are constants ): This can be restated given the excitation and response relationships: Then an Excitation of: Results in a Response of:

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Laplace Transform: X(s) Linear Time Invariant Systems 6 Time Invariance A system is time-invariant if its input-output relationship does not change as time evolves 0 t 0 t 0 t 0 t

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Laplace Transform: X(s) Linear Time Invariant Systems 7 Impulse Response The Impulse Response signal, h(t), of a linear system is determined by applying an Impulse to the Input, x(t), and determining the output response, y(t) Due to the properties of a Linear Time Invariant System, the Impulse Response Completely Characterizes the relationship between x and y for all x such that: Where * denotes the Convolution operation

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Laplace Transform: X(s) Linear Time Invariant Systems 8 Laplace Transform Since Convolution may be Mathematically Intensive, the Laplace Transform is often used as an aid to analyze the Linear Time Invariant Systems. Recall the relationship between Convolution in the Time- Domain and Multiplication in the Laplace Transform- Domain LT

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Laplace Transform: X(s) Linear Time Invariant Systems 9 Transfer Function The Transfer Function, H(s), of a system is the Laplace Transform of the Impulse Response, h(t) The Transfer Function completely specifies the relationship between the excitation (input) and response (output) in the Laplace Transform-Domain

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Laplace Transform: X(s) Linear Time Invariant Systems 10 Simple Systems Most systems can be created by combining the following simple system building blocks: Linear Operations: u Multiplication by a Constant u Addition of Signals Time-Domain Differentiation Time-Domain Integration Time-Domain Delay

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Laplace Transform: X(s) Linear Time Invariant Systems 11 Linear Operations Linear operations have a direct correlation between the Time-Domain and Laplace Transform-Domain (s-domain) counterparts Time-Domain Laplace Transform-Domain Time-Domain Laplace Transform- Domain

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Laplace Transform: X(s) Linear Time Invariant Systems 12 Time-Domain Differentiation Time-Domain Differentiation Operation Equivalent Laplace Transform-Domain Operation

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Laplace Transform: X(s) Linear Time Invariant Systems 13 Time-Domain Integration Time-Domain Integration Operation (no initial conditions) Equivalent Laplace Transform-Domain Operation

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Laplace Transform: X(s) Linear Time Invariant Systems 14 Time-Domain Delay Time-Domain Delay Operation Equivalent Laplace Transform-Domain Operation

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Laplace Transform: X(s) Linear Time Invariant Systems 15 Automatic Gain Controls for a radio Car Mufflers (mechanical filter) Suspension Systems (mechanical low pass filter) Cruise Control (motor speed control) Examples of LTI Systems The building blocks described previously can be used to model and analyze real world systems such as: Audio Equalizers (band pass filters)

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Laplace Transform: X(s) Linear Time Invariant Systems 16 System Example Create a system to implement the differential equation: 1)Determine the Transfer Function directly from the Differential Equation 2)Draw the system in the Time-Domain 3)Draw the system in the Laplace Transform-Domain 4)Write the Transfer Function from the System Diagram 5)Determine the Impulse Response

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Laplace Transform: X(s) Linear Time Invariant Systems 17 Directly Determine H(s) The Transfer Function H(s) can be directly determined by taking the Laplace Transform of the differential equation and manipulating terms LT By definition, H(s) = Y(s) / X(s)

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Laplace Transform: X(s) Linear Time Invariant Systems 18 Time-Domain-System Draw time-domain system representation for: 2)Start by drawing Input and Output at far ends + - 3)Draw Differentiation Block connected to y(t) 4)Draw Summation Block and its connections 1)Reorder terms to create a Function for y(t)

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Laplace Transform: X(s) Linear Time Invariant Systems 19 Laplace Transform-Domain The Laplace Transform-Domain System can be drawn by leaving the linear summation block and replacing the differentiating block with a multiplication by s + - Time-Domain + - Laplace Transform Domain

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Laplace Transform: X(s) Linear Time Invariant Systems 20 Verify H(s) The Transfer Function H(s) can also be determined by writing an expression from the Laplace Transform-Domain System + - Reordering terms gives the same result as taking the Laplace Transform of the Differential Equation System directly yields:

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Laplace Transform: X(s) Linear Time Invariant Systems 21 Impulse Response The Impulse Response of the system, h(t), is simply the Inverse Laplace Transform of the Transfer Function, H(s)

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Laplace Transform: X(s) Linear Time Invariant Systems 22 Pulse Response Example Given a system with an Impulse Response, h(t)=e -2t u(t) 1) Find the Transfer Function for the system, H(s) 2) Find the General Pulse Response,y(t) 3) Plot the Pulse Response for T=1 sec and T=2 sec Impulse Response Pulse Response Output 0 t T Input Pulse

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Laplace Transform: X(s) Linear Time Invariant Systems 23 Transfer Function The transfer function of the system is simply the Laplace Transform of the Impulse Response: The Transfer Function can be used to find the Laplace Transform of the pulse response, Y(s), using: LT

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Laplace Transform: X(s) Linear Time Invariant Systems 24 Laplace Transform of Input Given the equation for a General Pulse of period T The general Laplace Transform is thus: Combining Terms: LT

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Laplace Transform: X(s) Linear Time Invariant Systems 25 Determine Y(s) Y(s) is found using: Substituting for H(s) and X(s) Distributing terms Rewrite in terms of a new Y 1 (t) where

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Laplace Transform: X(s) Linear Time Invariant Systems 26 Partial Fraction Expansion The Matlab function residue can be used to perform Partial Fraction Expansion on Y 1 (s) [R,P,K] = RESIDUE(B,A) B = Numerator polynomial Coefficient Vector A = Denominator Polynomial Coefficient Vector R = Residues Vector P = Poles Vector K = Direct Term Constant

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Laplace Transform: X(s) Linear Time Invariant Systems 27 » B=[0 0 1];A=[1 2 0]; Expand Y 1 (s) Use residue to perform partial fraction expansion » [R,P,K]=residue(B,A) R = P = -2 0 K = []

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Laplace Transform: X(s) Linear Time Invariant Systems 28 General Solution y(t) Find y(t) by taking Inverse Laplace Transforms and substituting y 1 (t) back into y(t) LT -1

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Laplace Transform: X(s) Linear Time Invariant Systems 29 Matlab Declarations The General Pulse Response can be verified using Matlab Variables must be carefully declared using proper syntax » syms h H t s » h=exp(-2*t) Assuming the system to be causal, T must be explicitly declared as a positive number The Heaviside function is equivalent to the unit-step » T=sym('T','positive') » x=sym('Heaviside(t)-Heaviside(t-T)')

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Laplace Transform: X(s) Linear Time Invariant Systems 30 Matlab Verification » H=laplace(h) H = 1/(s+2) » X=laplace(x) X = 1/s-exp(-T*s)/s » Y=H*X Y = 1/(s+2)*(1/s-exp(-T*s)/s) » y=ilaplace(Y) y = -1/2*exp(-2*t)+1/2+ 1/2*Heaviside(t-T)*exp(-2*t+2*T) -1/2*Heaviside(t-T)

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Laplace Transform: X(s) Linear Time Invariant Systems 31 The following code recreates the Pulse Response as vectors for T=1 sec and T=2 sec Matlab Vector Code NOTE as of Matlab 6, ezplot cannot plot functions containing declarations of Heaviside or Dirac (Impulse) t=[0:0.01:4];% Time Vector tmax=size(t,2);% Index to last Time Value T1=find(t==1);% Index to 1 second T2=find(t==2);% Index to 2 seconds yexp=0.5*(1-exp(-2*t));% Base exponential vector y1T=[zeros(1,T1),yexp(1:tmax-T1)]; y1=yexp-y1T;% Pulse Response T=1 y2T=[zeros(1,T2),yexp(1:tmax-T2)]; y2=yexp-y2T;% Pulse Response T=2

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Laplace Transform: X(s) Linear Time Invariant Systems 32 Matlab Plots The response for T=1 and T=2 is plotted subplot(2,1,1);plot(t,y1); title('Pulse Response T=1'); grid on; subplot(2,1,2);plot(t,y2); title('Pulse Response T=2'); xlabel('Time in seconds'); grid on;

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Laplace Transform: X(s) Linear Time Invariant Systems 33 Transient and Steady State Example Determine an equation for the output of a system, y(t), described by the transfer function H(s) and input x(t) From the output y(t): 1.Identify the Transient Response, y trans (t), of the system (portion that goes to zero as t increases) 2.Identify the Steady State Response, y ss (t), of the system (portion that repeats for all t)

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Laplace Transform: X(s) Linear Time Invariant Systems 34 Laplace Transform of Input Recall the Laplace Transform of a general sine signal with an angular frequency 0 Find the Laplace Transform of the input signal x(t)

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Laplace Transform: X(s) Linear Time Invariant Systems 35 Roots of Y(s) Determine an expression for output signal Y(s) Determine general form for roots (poles) of denominator of Y(s) Purely Imaginary RootsComplex Roots

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Laplace Transform: X(s) Linear Time Invariant Systems 36 Verify Poles in Matlab » poles=roots( conv( [1 0 2], [1 2 2]) ) poles = i i i i

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Laplace Transform: X(s) Linear Time Invariant Systems 37 Partial Fraction Expansion Note that since poles are complex conjugates, coefficients will also be complex conjugates

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Laplace Transform: X(s) Linear Time Invariant Systems 38 Find Coefficients in Matlab » syms s t » p1=j*2^0.5; p1c=conj(p1); p2=(-1+j); p2c=conj(p2); » c1=(2*2^0.5)/(s-p1c)/(s-p2)/(s-p2c); » c1=subs(c1,'s',p1) c1 = i » c2=(2*2^0.5)/(s-p1)/(s-p1c)/(s-p2c); » c2=subs(c2,'s',p2) c2 = i

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Laplace Transform: X(s) Linear Time Invariant Systems 39 Inverse Laplace Take Inverse Laplace Transform of Y(s) Reduce terms by combining complex conjugates

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Laplace Transform: X(s) Linear Time Invariant Systems 40 Substitute Values When substituting coefficients, it is useful to use the polar representation to simplify cosine conversions

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Laplace Transform: X(s) Linear Time Invariant Systems 41 Steady State and Transient Responses The complex signal can be converted into a function of cosines Transient Response (Goes to 0 at t increases) Steady State Response (Repeats as t increases)

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Laplace Transform: X(s) Linear Time Invariant Systems 42 Matlab Verification Matlab can be used to determine Inverse Laplace Transform Result will have transient and steady state component Result will appear different but be mathematically equivalent » X=(2^0.5)/(s^2+2); H=2/(s^2+2*s+2); » Y=X*H; y=ilaplace(Y); » y=simplify(y); pretty(y) 1/2 1/2 - 1/2 2 cos(2 t) + 1/2 1/2 1/2 2 exp(-t) cos(t) + 1/2 2 exp(-t) sin(t) Steady State Transient

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Laplace Transform: X(s) Linear Time Invariant Systems 43 Verify Equivalence The Hand and Matlab steady state results are equivalent because a phase shift of is the same as negating the cosine The Hand and Matlab transient results are equivalent by applying the relationship:

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Laplace Transform: X(s) Linear Time Invariant Systems 44 Summary Laplace Transform is a useful technique for analyzing Linear Time Invariant Systems Impulse Response and its Laplace Transform, the Transfer Function, are used to describe system characteristics Simple System Blocks for multiplication, addition, differentiation, integration, and time shifting can be used to describe many real world systems Matlab can be used to determine the Transient and Steady-State Responses of a complex system

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