By A.Nazli Gündes, Charles A. Desoer
The algebraic concept of linear, time-invariant, multiinput-multioutput (MIMO) suggestions platforms has constructed speedily prior to now decade. The factorization strategy is straightforward and chic; it really is appropriate for either continuous-time and discrete-time lumped-parameter method versions, and plenty of of its effects observe on to distributed-parameter structures. This quantity streamlines the algebreaic method of the research and synthesis of linear time-invariant MIMO suggestions systems.
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Extra info for Algebraic Theory of Linear Feedback Systems with Full and Decentralized Compensators
R. 8). 1 (vi), det(Vp - Q / V p ) ~ I. Conversely, ifQ ~ m ( H ) ischosen sothatdet(Vp - Q AT~ ) ~ I, then (Vt, - Q / q t , ) isa valid choice for the denominator matrix /)c. 4, for all Q ~ m ( H ) , det( Vp - Q/Vp ) ~ det( 17p - Np Q ). 39). r. fx. 43) Dc In o In o 55 for some Q2 e H h i × ' ° . r. r. 42), for some Q 1 (~ Hn° x ni and for some Q2 e Hn°x nl. , Q1 = Q2. We conclude that C 1 = C 2 S ( P ) if and only if Q1 = Q2- Consequently, there is a unique (matrix-) parameter Q e m(H) corresponding to each H-stabilizing compensator C ~ S ( P ) .
8) where Q ~ m ( H ) . 6) for ( ( N c ,D c ), ( D c ,lye ) ) are parametrized by the matrix Q. 8). r,of P . 12) hold; then Np = P Dp ~ m ( G s ) and ~Tp = 5 . e ~ m(Gs). 5, (V,-Q ~7, )-1 e m(G) (~7e - N p Q )-1 e m ( G ) foralla e m ( H ) . 8) have the property that det( Vp - Q ATe,) e I and deft 17t, - Nr Q) ~ I , forallQ ~ m(H). 8), c = O:'N~ = (v,, - e ~ . )-l(v,, + e 5. 10) c =uco;'=(6. c. c. factorizations of C ~ re(G). 29 (iii) Now suppose that P ~ m(G) but not in m(Gs); then and not in m(G ) either;, consequently, det( lie - Q Np ) and dot( ~Tp -Np Q ) are not necessarily in I for all Q ~ m ( H ) .
In the unity-feedback system S ( P , C ), the plant has only one (vector-)input e and one (vector)-output y ; this output is used in feedback to the compensator;, hence, the plant model considers only additive inputs or disturbances (say u , u" ), which all affect the plant through its actuators. More generally, however, there may be inputs (for example, disturbances, initial conditions, noise, manual commands) which are applied directly to the plant without going through the actuators; hence, the map from the directly applied inputs to the plant output may be different from the map from the additive inputs to the plant output.