  We introduce a new approach for the robust control of quantum dynamics of
strongly interacting many-body systems. Our approach involves the design of
periodic global control pulse sequences to engineer desired target Hamiltonians
that are robust against disorder, unwanted interactions and pulse
imperfections. It utilizes a matrix representation of the Hamiltonian
engineering protocol based on time-domain transformations of the Pauli spin
operator along the quantization axis. This representation allows us to derive a
concise set of algebraic conditions on the sequence matrix to engineer robust
target Hamiltonians, enabling the simple yet systematic design of pulse
sequences. We show that this approach provides an efficient framework to (i)
treat any secular many-body Hamiltonian and engineer it into a desired form,
(ii) target dominant disorder and interaction characteristics of a given
system, (iii) achieve robustness against imperfections, (iv) provide optimal
sequence length within given constraints, and (v) substantially accelerate
numerical searches of pulse sequences. Using this systematic approach, we
develop novel sets of pulse sequences for the protection of quantum coherence,
optimal quantum sensing and quantum simulation. Finally, we experimentally
demonstrate the robust operation of these sequences in a system with the most
general interaction form.
