the second row of the error dynamics, , will enter the sliding mode in finite time. Along the surface, the corresponding equivalent control will be equal to . Hence, so long as , the th row of the error dynamics, , will enter the sliding mode in finite time. So, for sufficiently large gains, all observer estimated states reach the actual states in finite time. In fact, increasing allows for convergence in any desired finite time so long as each function can be bounded with certainty. Hence, the requirement that the map is a diffeomorphism (i.e., that its Jacobian linearization is invertible) asserts that convergence of the estimated output implies convergence of the estimated state. That is, the requirement is an observability condition. In the case of the sliding mode observer for the system with the input, additional conditions are needed for the observation error to be independent of the input. For example, that does not depend on time. The observer is then Multi-observer Multi-observer extends the high-gain observer structure from single to multi observer, with many models working simultaneously. This has two layers: the first consists of multiple high-gain observers with different estimation states, and the second determines the importance weights of the first layer observers. The algorithm is simple to implement and does not contain any risky operations like differentiation. The idea of multiple models was previously applied to obtain information in adaptive control. Assuming that the number of high-gain observers equals , where is the observer index. The first layer observers consists of the same gain but they differ with the initial state . In the second layer all from observers are combined into one to obtain single state vector estimation where are weight factors. These factors are changed to provide the estimation in the second layer and to improve the observation process. Let assume that and where is some vector that depends on observer error . Some transformation yields to linear regression problem This formula gives possibility to estimate . To construct manifold we need mapping between and ensurance that is calculable relying on measurable signals. First thing is to eliminate parking phenomenon for from observer error . Calculate times derivative on to find mapping m lead to defined as where is some time constant. Note that relays on both and its integrals hence it is easily available in the control system. Further is specified by