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Y Fig. 9 Graph of f : U -+ R, with U C R2. Now, let us assume we are given the input-output relation, characterized by observations of the inputs and outputs. We then wish to study the internal properties of the system. More specifically, let the input-output relation be given by f .. ti U(k) + H 9 Y(W > where the system maps an input sequence u(k) u(k) t (UO,Ul,* Y(k) t (Yl,Yad%-) l l 7%) uj . 11) An internal model is then realized by the introduction of a state-space X2 where the transition map g: 17 -+ X (onto) and the output map h: 2 -+ Y 14 SYSTEM (one-to-one) ANALYSIS are given as: As the mapping refers to sequences, 6, 2, and ?

The matrix F represents the dynamics and is therefore frequently Matrix G describes the constraints on affecting the called system matrix. state of the system by the input, and H the constraints on observing the st,ate of the system from outputs. 15 show the general matrix block diagram of a state-space model for continuous- and discrete-time, respectively. 22). 24) should be read as describing the system evolving from the initial state x(to) at time to under the action of u(t). ZOWX! by certain other changes.

Inverse mapping f -1 : Iwm Y Fig. 13 The equation + lRr - x = F-‘y SYSTEMS x = F-ly 23 defines the . Composite output map. We should also note that the system is, in fact, a composition x:R t of two mappings + IEr e x(t) Such composite rn~~~~~g, applying is denoted as f 0 x, that is, (f ox) : R + t * f : lr x + lEtm t-) f( x > . 13. l commutative diagram and is X A system is called invertible, if the output uniquely determines the input. That is, if a system is represented by a mapping y = f(x), there must exist a map g such that x = g( f (x)).

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