By Hans Blomberg and Raimo Ylinen (Eds.)
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Additional resources for Algebraic Theory for Multivariable Linear Systems
Let us finally study the above case when we are allowed to use feedback only from the first component of the output of the given system. The generator (1) for this case is obtained from (11) or (19) by interchanging the first two columns and then transforming the generator thus obtained to an upper right triangular form by means of elementary row operations. The significant part of the result obtained in this way reads: A1(p)= 1, L(p) = 2p2 + 5p + 2 = (2p + 1)(p [&(p) 23 i + 2), -B4i(p)] = [(p2- 4)(p2 + 3p + 2 ) i -2p3 - 6p2 - 4p + 11.
To emphasize the system aspect of row equivalence we shall also say that two row equivalent regular generators are input-output equiualent. Stability A regular differential input-output relation S generated by a regular generator [A(p) i - B ( p ) ] is said to be asymptotically stable if all the roots of detA(p) have negative real parts (cf. 5). If so, then all the solutions y to A(p)y = 0 asymptotically approach zero with increasing time. The monic polynomial corresponding to detA(p) is called the characteristic polynomial of S as well as of "P) i -B(p)l.
Go to Step 3. Step 3. We have tl = z2 = z, = 1 - 1 = 0. 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.
Algebraic Theory for Multivariable Linear Systems by Hans Blomberg and Raimo Ylinen (Eds.)