This paper investigates the problem of secure state estimation for cyber-physical systems modeled by continuous or discrete-time linear systems when some sensors are corrupted by an attacker. A novel state observer is proposed with adaptive switching mechanism. Attack tolerance principle is established based on adaptively truncating the injection channels of attacks. To implement it, a switching function matrix is introduced into the observer design. Driven by a well-defined performance index, the switching function matrix automatically reaches and remains in the desired entry mode and turns off the input channels of attacks. Based on the equivalence between s-strong detectability of the observation error system and 2s-sparse detectability of the original system, the observation error system is proven to be asymptotically stable even under the cyber attacks. Compared with the existing complex static batch optimization algorithms, the proposed adaptive observer can be derived only by offline solving a set of simple linear matrix inequalities. Simulation examples are given to illustrate the estimation performance and the computational efficiency of the proposed method.