A new interval observer (IO) for uncertain switched linear dynamical systems is derived in this paper. We show that estimation accuracy and robustness can be achieved by combining Input-to-State Stability (ISS) concept and common quadratic Lyapunov function. The stability and positivity conditions are expressed in terms of Linear Matrix Inequalities (LMIs). This new observer overcomes the drawbacks of existing IO's, which are designed based on similarity transformations. The strengths of the proposed observer structure lie in having additional decision variables. The design procedure is very simple and uniform, it does not require changes of coordinates, which are costly in terms of computational time. The increased robustness of the presented method, compared to other commonly used approaches, is demonstrated by accurately estimating, at each time instant, the interval including vehicle sideslip angle and yaw rate. The proposed algorithm shows high potential for practical implementation.