By Hans Blomberg and Raimo Ylinen (Eds.)
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Extra info for Algebraic Theory for Multivariable Linear Systems
30)). Note also that the present formulation implies that we are allowed to use feedback from both the output components of the given system. There is only one control input available. 19) we have (cf. 24)) 12 L(p) = and (cf. 26)) yl+ P+ O2I 13 14 =["i4 1 p2+3p+2 The characteristic polynomial of 15 S1 is ; i -2p -1 thus found to be (cf. (2)) t(2p + l)(p + 2)(p2 - 4)(p2 + 3p + 21, and det Al(p) det L(p) is equal to 16 (2P + l)(P + 2). S1 is accordingly not asymptotically stable because (15) has a root at +2.
Next we therefore apply Step 4. Step 4. According to (42) we get for T3(p) 60 T3(p) = T311 = [-I2 -461, 41 Algebraic Theory for Multivariable Linear Systems then we apply (43). This yields the new candidate (30) given by 1 61 and the new coefficient matrix (33) given by 62 -11 -50 i 0 Still det Yl = Y 1= 0 (Xl # 0), and (61) is consequently not a satisfactory candidate for (29). Therefore we start again from Step 1. The reader is advised to perform the various steps as outlined above over and over again, always choosing t(p) = p + 1 63 in Step 5.
19) we already constructed a unimodular matrix P @ ) such that [ A @ )i - B @ ) ] P ( p ) = [ I i 01 with [ A @ )i -B@)] as in (50) (cf. 20), . . 22). To get a suitable matrix (36) as a first candidate Z @ ) for (29) we have accordingly first of all to compute P@)-'. The lower part of P@)-' consists in this case of one single nonzero row-this part is therefore necessarily always row proper. P@)-' would thus as such qualify as the first candidate. For the sake of convenience, we shall here take P(P)-' with the bottom row multiplied by 47 as our first candidate Z @ ) for (29).
Algebraic Theory for Multivariable Linear Systems by Hans Blomberg and Raimo Ylinen (Eds.)