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INTRODUCTION TO CONTROL THEORY
PRELIMINARY MATRIX THEORY
TRIX SOLUTION OF LINEAR SYSTEMS
12 other sections not shown
adjoint equations 6.30 algebraic applied arbitrary assumed asymptotically stable Chapter characteristic polynomial characteristic roots characteristic vectors closed loop system coefﬁcients Consider control theory control variable convergent corresponding deﬁned denotes determine differential equation discrete-time system divisors easy to verify equilibrium point Example Exercise expression ﬁnal ﬁnd ﬁnite ﬁrst ﬁxed follows given gives Hence deduce holds inﬁnite initial conditions input invariant Jordan form Laplace transform Liapunov function linear feedback linear system linearly independent methods minimal realization minimum necessary and sufﬁcient necessary condition negative deﬁnite nonsingular Nyquist locus obtain optimal control origin output positive deﬁnite possible problem proof quadratic rank result Riccati equation satisﬁes Section shown in Fig similarity invariants solution speciﬁed stability matrix sufﬁcient condition Suppose symmetric system described system equations Theorem Theorem 4.1 trajectory transfer function transition matrix unstable varying systems velocity x(to z-transforms zero