to distinguish them from the variables of the equations satisfied by the physical system. The observer is called asymptotically stable if the observer error converges to zero when . For a Luenberger observer, the observer error satisfies . The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix has all the eigenvalues inside the unit circle. For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix . The observer equations then become: or, more simply, Due to the separation principle we know that we can choose and independently without harm to the overall stability of the systems. As a rule of thumb, the poles of the observer are usually chosen to converge 10 times faster than the poles of the system . Continuous-time case The previous example was for an observer implemented in a discrete-time LTI system. However, the process is similar for the continuous-time case; the observer gains are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when is a Hurwitz matrix). For a continuous-time linear system where , the observer looks similar to discrete-time case described above: . The observer error satisfies the equation . The eigenvalues of the matrix can be chosen arbitrarily by appropriate choice of the observer gain when the pair is observable, i.e. observability condition holds. In particular, it can be made Hurwitz, so the observer error when . Peaking and other observer methods When the observer gain is high, the linear Luenberger observer converges to the system states very quickly. However, high observer gain leads to a peaking phenomenon in which initial estimator error can be prohibitively large (i.e., impractical or unsafe to use). As a consequence, nonlinear high-gain observer methods are available that converge quickly without the peaking phenomenon. For example, sliding mode control can be used to design an observer that brings one estimated state's error to zero in finite time even in the presence of measurement error; the other states have error that behaves similarly to the error in a Luenberger observer after peaking has subsided. Sliding mode observers also have attractive noise resilience properties that are similar to a Kalman filter. Another approach is to apply multi observer, that significantly improves transients and reduces observer overshoot. Multi-observer can be adapted to every system where high-gain