EECE 340 Homework, Fall 2007

Homework 2 Solution Problem 1 Determine and plot the convolution integral f1(t)*f2(t) for: a) 2 1 1 2 2 3 1 2 2 2 2 1 2 1 ( ) ( ) ( 2) ( ) ( ) ( 1) ( 1) ( ) ( ) ( ) 2 ( ) ( ) ( 1) ( 1) 2( 1) ( 1) ( 3) ( 3) s s s s s s e f t u t u t F s s s e e f t u t u t F s s s e e e F s F s s s s f t f t t u t t u t t u t − + − + − − = − − ⇒ = − = + − − ⇒ = − ⋅ = − + ⇒ ∗ = + + − − − + − − b) t 3 1 -1 f1(t)*f2(t) 1 1 2 2 2 2 2 2 2 3 1 2 3 3 3 3 2 2 2 2 1 2 ( ) ( 1) ( 1) ( ) 1 ( ) ( ) 2( 1) ( 1) ( 2) ( 2) ( ) 2 1 ( ) ( ) 2 2 1 1 ( ) ( ) ( 1) ( 1) ( ) ( 2) ( 2) ( 3) ( 3) 2 2 s s s s s s s e e f t u t u t F s s s e e f t tu t t u t t u t F s s s s e e e F s F s s s s s f t f t t u t t u t t u t t u t + − − − + − − = + − − ⇒ = − = − − − + − − ⇒ = − + ⋅ = − + − ⇒ ∗ = + + − + − − − − − -2 -1 0 1 2 3 4 5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 t Problem 2 Determine the transfer function C(s)/V(s) of the System below 1 2 1 2 2 2 1 1 2 1 2 1 2 2 2 1 1 2 1 2 2 2 1 ( ) ( ) ( ) ( ) : ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) 1 1 1 R s V s C s C s let G s R s G G G G H G H G G C s G s G G V s G s G G H G H G G G G H G H = − = ⇒ + − = = = + + − + + + − - + V(s) Problem 3 Consider the unit feedback system shown below: a) Determine the error signal E(s). b) Determine the range of K for the system to be stable. Problem 4 The transfer function of a linear control system is given by: 2 ( ) 1 ( ) 3 2 C s R s s s = + + a) Determine the state equation for this system. b) Determine the corresponding output equation. Problem 5 The input signal of the LTI system shown below (initially at rest) is x(t) = u(t)-u(t-3) The impulse responses of the subsystems are: ( ) 2 1( ) ( ) t t h t e e u t − − = + 3 2( ) t ( ) h t e = − u t a) Compute the impulse response h(t) of the overall system. 1 2 1 2 1 1 ( ) 1 2 1 ( ) 3 1 1 1 ( ) ( ) ( ) 1 2 3 1 1 ( 1)( 3) ( 2)( 3) 1 2 3 H s s s H s s H s H s H s s s s s s s s A B C s s s = + + + = + ⇒ ⎛ ⎞ = ⋅ = + ⋅ ⎜ ⎟ + + + ⎝ ⎠ = + + + + + = + + + + + Solving for A, B, and C we find: A = 1; B = 0; C = -1 � ( ) 3 1 1 ( ) 1 3 ( ) ( ) t t H s s s h t e e u t − − = − ⇒ + + = − b) Find an equivalent system (same impulse response) configured as a parallel interconnection of two LTI subsystems. c) Sketch the input signal x(t). Compute the output signal y(t). e -t e -3t + - x(t) y(t) 3 3 3 3 3 3 1 ( ) 1 1 ( ) 1 3 ( ) ( ) ( ) 1 1 ( 1) ( 1) ( 3) ( 3) 1 1 1 1 1 1 1 1 1 3 3 3 3 s s s s s s e X s s s H s s s Y s H s X s e e s s s s s s s s e e e s s s s s s s s − − − − − − = − ⎛ ⎞ = − ⎜ ⎟ + + ⎝ ⎠ = = − − + + + + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − − − − − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + + + + ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ( ) ( ) ( ) ( 3) 3 3( 3) 3 ( 3) 3( 3) 1 1 ( ) ( ) - ( ) ( 3) - ( 3) ( ) - ( ) ( 3) ( 3) 3 3 2 1 2 1 ( ) - ( ) ( ) ( 3) ( 3) ( 3) 3 3 3 3 t t t t t t t t y t u t e u t u t e u t u t e u t u t e u t u t e u t e u t u t e u t e u t − − − − − − − − − − − − ⇒ = − − − − + − − − = + − − + − − − t x(t) 1 3 MATLAB Solution Problem 1: Given that: 1 2 5 3 3 2 1 + + + + = s s s s G ; G2 = 4; 4 1 1 H = s + ; and 2 2 1 2 2 + + + = s s s H . % Matlab Solution G1 = tf([1 3 5],[1 0 2 1]); G2 = tf([4],[1]); H1 = tf([1],[1 4]); H2 = tf([1 1],[1 2 2]); H = G1*G2/(1+G1*G2*H2-G2*H1); Problem 2 Use Matlab to determine whether the system with the transfer function shown below is stable. Get a state space representation of the system. 2 6 5 3 2 3 4 ( ) 2 7 3 1 s s H s s s s s s + + = − + − + + % Enter the coefficients of the denominator A = [1 -2 0 7 -3 1 1]; % Compute the poles of the system. If not all of them lie in the left half %plane then the system is not stable roots(A) if real(roots(A) < 0) display ‘The system is stable’ else display ‘The system is not stable’ end % Get the State Space Representation [a, b, c, d] = tf2ss([1 3 4], [1 -2 0 7 -3 1 1]) ans = 1.6003 + 1.3159i 1.6003 - 1.3159i -1.5872 0.3678 + 0.5341i 0.3678 - 0.5341i -0.3490 The system is not stable a = 2 0 -7 3 -1 -1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 b = 1 0 0 0 0 0 c = 0 0 0 1 3 4 d = 0