Abstract
We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A T P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB T PB>0 and that the determinant ofB T PW be Hurwitz, whereW=block diag[w 1,...,w m ], with elementw i =[s k i −1,s k i −2,...,s, 1]T; here, the symbolsk i ,i=1, 2, ...,m, denote the Kronecker invariants with respect to the pair {A, B}. This result has application in designing robust controllers for linear uncertain systems.
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Chao, C.H., Stalford, H.L. Necessary and sufficient condition in Lyapunov robust control: Multi-input case. J Optim Theory Appl 66, 1–21 (1990). https://doi.org/10.1007/BF00940529
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DOI: https://doi.org/10.1007/BF00940529