ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 14. EE 350Continuous-Time Linear SystemsRecitation 141 Jeffrey Schiano 2014. All rights reserved. Rec 14. Recitation 13 Topics2• Solved Problems – Solution of ODEs using the Laplace transform– Transfer function representation of a LTI system– Circuit analysis using the Laplace transform• MATLAB Exercises– Transfer function representation of a system– SIMULINK graphical programming language Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 1• Using the Laplace transform pairs• Determine the signals yzi(t) and yzs(t) from their Laplace transforms32yyy f 221()164()(1)( 1)zizssYssssYsss s s2222cos(b ) ( )()sin(b ) ( )()atatsaetutsa bbetutsa b Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 2• The ODE representation of a certain LTI system with input f(t) and output y(t) is• Determine the zero-input response yzi(t), zero-state response yzs(t), and total response y(t) given the following initial conditions and input72yyy f initial conditions: (0 ) 1, (0 ) 2input : ( ) 2 ( )tyyft e ut Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 3• The ODE representation of another LTI system with input f(t) and output y(t) is• Determine the characteristic roots and state whether the system is unstable, marginally stable, or asymptotically stable• Determine the system transfer function H(s)• Determine the poles and zeros of H(s) and show their location in the s-plane using a pole-zero map• Determine the impulse response representation of the system• Determine the zero-state unit-step response using the Laplace transform method105 6 12 12yyy f f Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 3 Solution11 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 3 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 4• Derive the Laplace transform representation of the inductor and capacitor14Time Domain s Domain()vtLi( )tV(s)sLI(s)(0 )Li()vtCi( )tV(s)1sCI(s)v(0 )s Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 4 Solution15 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 4 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 5• Find the zero-input response, zero-state response, and total response of the following circuit17f(t) = u(t)100mHL y(t)1R1k2R1k(0 ) 1mAi Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 5 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 5 Solution19 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 5 Solution20 Jeffrey Schiano 2014. All rights reserved. Rec 14. MATLAB Laplace Transform Tools21Operation FunctionCreates/Displays a transfer function representation tfObtains the numerator and denominator polynomials fromthe transfer function representationtfdataDetermines the poles of dynamic system poleDetermines the zeros and DC gain of a dynamic system zeroDetermines the DC gain of a dynamic system dcgainGraphs the pole-zero map of a dynamic system pzmapGenerates the unit-step response stepSimulate time response to an arbitrary input lsim Jeffrey Schiano 2014. All rights reserved. Rec 14. MATLAB Laplace Transform Tools• Model Construction– The command tf generates a transfer function model of a dynamic system• Transfer Function Properties – The command pole determines the poles – The command zero determines the zeros and DC gain – The command dcgain determines the DC gain – The command pzmap generates the pole-zero map– The command tfdata returns the numerator and denominator polynomials22 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 6• Consider a system with input u(t), output y(t), and ODE representation• Write a MATLAB M-file that– Represents the system using a transfer function– Determines the DC gain, poles, and zeros– Plots the pole-zero map– Plots the zero-state unit-step response2310yyyu u Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 6 M-file24 Jeffrey Schiano 2014. All rights reserved. Rec 14. SIMULINK• A dataflow graphical programming language for modeling and simulating dynamic systems• Widely used in control systems engineering and signal processing• Tightly integrated with the MATLAB environment• Launch by entering Simulink at the command prompt25 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 7• Once again consider the system with input u(t), output y(t), and ODE representation• Generate a SIMULINK model that– Represents the system use the Transfer Fcn block in the Library: Simulink/Continuous– Uses the Step block in the Library: Simulink/Sources to generate the input u(t-1) – Sets the stop time to 12 second– Displays the results using the Scope block in the Library: Simulink/Sinks2610yyyu u Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 7 SIMULINK27 Jeffrey Schiano 2014. All rights reserved. Rec 14. Problem 8• Modify the SIMULINK model in Problem 7 so that– The input, out, and time vectors are sent to the MATLAB workspace as arrays using the To Workspace block in the Library: Simulink/Sinks– Obtain the time vector using the Clock block in the Library: Simulink/Sources• Write a MATLAB M-file that executes the SIMULINK model and plots the input f and zero-state
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