Consider a time-invariant discrete-time controllable and observable linear system with single input and single output, controlled by a memoryless feedback of the output; as depicted in the following diagram:
We aim at state nullification, namely at choosing K1,K2,...,Kl such that the state x becomes zero in a finite number of steps, regardless of the initial state. The main theorem is as follows:
Theorem: A necessary and sufficient condition for nullification is CAdj(A)B ≠ 0 where Adj(A)
denotes the classical adjoint of A.
As an interesting corollary, we get that a continuous-time linear controllable and observable system can be nullified by a feedback of a sampled output, with almost any sampling rate. This can be shown by verifying that the above condition is almost always true for the sampled-data system.
For more details see:
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State nullification by memoryless output feedback Math. Control Signals Systems, 17(1):38-56, 2005. by Zvi Artstein and Gera Weiss