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Unformatted text preview: 13.4). I f w is t he t ransformed-state vector given by K ailath, T homas, L inear Systems, P rentice-Hall, E nglewood Cliffs, N.J., 1980. 2. 1
s) - 8z y[k] = [ -8(3)-k P roblems Z adeh, L., a nd C. Desoer, L inear S ystem Theory, M cGraw-Hill, New York,
1 3.1-1 Convert each of the following second-order differential equations into a set of two
first-order differential equations (state equations). S tate which of the sets represent
( a) i i+1O!i+2Y=f
( b) i i+2e Y !i+logy=f ( c) ii + <P1(y)!i + <P2(y)y = f w =Px 112 F t hen
w [k + IJ = P A P-1w[kJ + P Bf a nd
y[kJ = ( Cp-l)w + D f
C ontrollability a nd observability may be investigated by diagonalizing t he m atrix. 1 3.7 30 f Summary
F ig. P I3.2-1 An n th-order s ystem c an b e described in t erms o f n key variables~the s tate
v ariables of t he system. T he s tate variables a re n ot u nique, b ut c an b e selected
in a variety of ways. Every possible system o utput c an b e e xpressed as a linear
c ombination o f t he s tate variables a nd t he i nputs. Therefore t he s tate variables
describe t he e ntire system, n ot merely t he r elationship between c ertain i nput(s) a nd
o utput(s). F or t his reason, t he s tate variable description is a n i nternal d escription
o f t he s ystem. Such a d escription is therefore t he m ost g eneral s ystem d escription,
a nd i t c ontains t he i nformation of t he e xternal descriptions, such as t he impulse 13.2-1 Write the state equations for the R LC network in Fig. PI3.2-1.
13.2-2 Write the state and output equations for t he network in Fig. PI3.2-2.
13.2-3 Write the state and output equations for the network in Fig. PI3.2-3.
13.2-4 Write the state and output equations for the electrical network in Fig. PI3.2-4.
13.2-5 Write the state and output equations for the network in Fig. PI3.2-5. 8 32 13 State-Space Analysis
1130 Problems 833 10 + +
1 I2F I IH Y YI II F ig. P13.2-7
F ig. P 13.2-2
12 + 20 + w here f A=
1 3.3-2 Fig. P13.2-4 Fig. P13.2-3 [_~ _ :] a nd B= [~] x(O) = [ :] R epeat P rob. 13.3-1 if f (t) a nd
Y + 1 3.3-3 a nd
1 3.3-4 Fig. P13.2-5 W rite t he s tate a nd o utput e quations of t he s ystem shown i n Fig. PI3.2-7. 1 3.2-8 f= W rite t he different sets of s tate e quations (two canonical, series, a nd p arallel forms)
a nd t he o utput e quation for a s ystem having a transfer function H (8)- 1 3.2-9 = 1 3.3-5 Using t he L aplace t ransform m ethod, find t he r esponse y i f x= 38+10 A x + B f(t) y =Cx+Df(t) + 7 8 + 12
w here R epeat P roblem 13.2-8 if
( a) H(8) 1 3.3-:1 82 [U(t)]
6(t) W rite t he s tate a nd o utput e quations of the s ystem s hown i n Fig. PI3.2-6. 1 3.2-7 48
(8 + 1)(8 + 2)2 ( b) H(8) = 3 8 (s 2 + 78 + 128
+ 1)3(s + 2) F ind t he s tate v ector x (t) u sing t he Laplace t ransform m ethod if x= A x+Bf lOOt f (t) = u (t) R epeat P rob. 13.3-1 if Fig. P13.2-6 1 3.2-6 = sin R epeat P rob. 13.3-1 if 20 Y f f (t) = 0 a nd C =[O
a nd f (t) = u (t) x(O) = [ :] 1) D =O 834
1 3.3-6 13 S tat...
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