Large-size cat states are of fundamental significance to explore the quantum-to-classical transition, and serve as promising resources for quantum metrology and fault-tolerant quantum computation. However, scaling up the amplitude of cat states remains challenging because of We propose to generate large cat states by using the dynamical invariant of hybrid qubit-resonator systems under either Hermitian or non-Hermitian time-dependent Hamiltonian. It is a study in the framework of universal quantum control (UQC), where the system dynamics is analyzed in the ancillary picture via a unitary transformation conditioned on the qubit state. The controlled dynamics of the composite system can be encoded in the evolution of the dynamical invariant via the Heisenberg equation, which yields constrains on the Hamiltonian. When the qubit is prepared in a balanced superposition state, the bosonic mode can be deterministically prepared from the vacuum state to the cat state of a mean photon number over $120$, . In the Hermitian case, the generation is perfect. Our protocol can also be applied to the generation of large intrinsic cat states and four-legged cat states. Through the preparation of macroscopic quantum states, our work substantially advances UQC to hybrid discrete-continuous variable systems.