R\'enyi entropy is an important measure in the context of information theory as a generalization of Shannon entropy. This information measure was often used for uncertainty quantification of dynamical behaviour of stochastic processes. In this paper, we study in detail this measure for multivariate controlled autoregressive moving average (MCARMA) systems. The characteristic function of output process is represented from the terms of its residual characteristic function. An explicit formula to compute the R\'enyi entropy for the output process of MCARMA system is derived. In addition, we investigate the covariance matrix to find the upper bound of R\'enyi entropy. We present three simulations that serve to illustrate the behavior of information in MCARMA system, where the control and noise follow the Gaussian, Cauchy and Laplace distributions. Finally, the behaviour of R\'enyi entropy is illustrated in two real-world applications: a paper-making process and an electric circuit system.

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