ch7soln - Chapter 7 State-Space Design Problems and...

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Chapter 7 State-Space Design Problems and Solutions for Section 7.2 1. Give the state description matrices in control-canonical form for the following transfer functions: (a) 1 4 s + 1 (b) 5( s/ 2 + 1) ( s/ 10 + 1) (c) 2 s + 1 s 2 + 3 s + 2 (d) s + 3 s ( s 2 + 2 s + 2) (e) ( s + 10)( s 2 + s + 25) s 2 ( s + 3)( s 2 + s + 36) Solution: (a) F = 0 . 25 , G = 1 , H = 0 . 25 , J = 0. (b) F = 10 , G = 1 , H = 200 , J = 25 . Hint: Do a partial fraction expansion to ° nd the J term ° rst. (c) F = 3 2 1 0 , G = 1 0 , H = £ 2 1 / , J = [0] . (d) F = 2 2 0 1 0 0 0 1 0 , G = 1 0 0 , H = £ 0 1 3 / , J = [0] . 443
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444 CHAPTER 7. STATE-SPACE DESIGN (e) F = 4 39 108 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 , G = 1 0 0 0 0 , H = £ 0 1 11 35 250 / , J = [0] 2. Use the MATLAB function tf2ss to obtain the state matrices called for Problem 1. Solution: In all cases, simply form num and den given below and then use the MATLAB command [F,G,H,J] = tf2ss(num,den). (a) num = £ 0 1 / , den = £ 4 1 / . (b) num = £ 5 / 2 5 / , den = £ 1 / 10 1 / . (c) num = £ 0 2 1 / , den = £ 1 3 2 / . (d) num = £ 0 0 1 3 / , den = £ 1 2 2 0 / . (e) num = £ 0 0 1 11 35 250 / , den = £ 1 4 1 39 108 0 0 / . Note that the answers are the same as for Problem 7.1. Hint: The MATLAB function conv will save time when forming the numerator and denominator for part (e). 3. Give the state description matrices in normal-mode form for the transfer functions of Problem 1. Make sure that all entries in the state matrices are real-valued by keeping any pairs of complex conjugate poles together, and realize them as a separate subblock in control canonical form. Solution: (a) F = 0 . 25 , G = 1 , H = 0 . 25 , J = 0 . (b) F = 10 , G = 1 , H = 200 , J = 25 . (c) 2 s +1 s 2 +3 s +2 = 2 s +1 ( s +1)( s +2) = 1 s +1 + 3 s +2 , The computation can also be done using the residue command in MATLAB. Block diagram for Problem 7.3 (c).
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445 F = 1 0 0 2 , G = 1 1 , H = £ 1 3 / , J = [0] . (d) s +3 s ( s 2 +2 s +2) = 3 / 2 s 3 / 2 s +2 s 2 +2 s +2 , The computation can also be done using the residue command in MATLAB. Block diagram for Problem 7.3 (d). F = 0 0 0 0 2 2 0 1 0 , G = 1 1 0 , H = £ 3 2 3 2 2 / , J = [0] . (e) The hard part is getting the expansion, ( s +10)( s 2 + s +25) s 2 ( s +3)( s 2 + s +36) = 0 . 5118 s +2 . 3148 s 2 + 0 . 57407 s +3 + 0 . 0622 s +0 . 3452 s 2 + s +36 You can use the MATLAB function residue to obtain this. From the ° gure, we have, Block diagram for Problem 7.3(e).
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446 CHAPTER 7. STATE-SPACE DESIGN F = 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 1 36 0 0 0 1 0 , G = 1 0 1 1 0 , H = £ 0 . 5118 2 . 3148 0 . 57407 0 . 0622 0 . 3452 / , J = [0] . 4. A certain system with state x is described by the state matrices, F = 2 1 2 0 , G = 1 3 , H = [ 1 0 ] , J = 0 . Find the transformation T so that if x = Tz , the state matrices describing the dynamics of z are in control canonical form. Compute the new matrices A , B , C , and D . Solution: Following the procedure outlined in the chapter, we have, C = £ G FG / = 1 1 3 2 .
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