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We present a novel approach to analyze the performance of linear dynamical systems in the presence of disturbances with bounds on their magnitudes and bounds on their rates of change. The performance considered is the maximum magnitude of the outputs of linear systems driven by such disturbances. First, the basic properties of this performance are given. Then, the performance computation is formulated as an optimal control problem. Applying the Pontryagin's maximum principle, we obtain necessary conditions, and systematically derive the numerical procedure to obtain the worst-case disturbance and its corresponding output. To show the effectiveness of the performance analysis, the worst-case performance is compared with widely used upper bounds in the numerical example. The comparison indicates that the new performance is significantly less conservative than the upper bounds. Therefore, this performance analysis is practical for system analysis and deemed to provide a viable means to improve the capabilities of control synthesis.