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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 s 2 +3 s +2 (d) s s ( s 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 = 200 =25 . Hint: Do a partial fraction expansion to & nd the J term & rst. (c) F = 3 2 10 , G = 1 0 , H = £ 21 / =[0 ] . (d) F = 2 20 0 01 0 , G = 1 0 0 , H = £ 013 / =[0] . 443

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444 CHAPTER 7. STATE-SPACE DESIGN (e) F = 4 39 1 0 800 10 00 0 01 0 0 0 , G = 1 0 0 0 0 , H = £ 011 13 52 5 0 / ,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 g iv enb e lowandth enu s eth eMATLABcommand [F,G,H,J] = tf2ss(num,den). (a) num = £ / , den = £ 41 / . (b) num = £ 5 / 25 / , den = £ 1 / 10 1 / . (c) num = £ 021 / ,d en = £ 132 / . (d) num = £ 0013 / = £ 1220 / . (e) num = £ 0011 5 0 / = £ 1413 91 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 . (b) F = 10 = 200 =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).
445 F = 10 0 2 , G = 1 1 , H = £ 13 / ,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 = 00 0 0 2 2 01 0 , G = 1 1 0 , H = £ 3 2 3 2 2 / =[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 = 00 0 0 0 10 0 0 0 00 30 0 00 0 1 36 00 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 = 21 20 , G = 1 3 , H =[10 ] =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 = £ GF G / = 11 3 2 . t 2 = £ 01 / C 1 = 1 5 £ 3 1 / , t 1 = t 2 F = 1 5 £ 43 / .
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ch7soln - Chapter 7 State-Space Design Problems and...

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