With the appearance of inerter, the ISD suspension composed of “Inerter-Spring-Damper” opens up a new direction for efficient coordination of ride comfort and road friendliness for heavy vehicles. In this paper, a reference model based on the hybrid-hook damping strategy is proposed, and the mechatronic inerter is used as an active suspension force generator in the actual controlled model. Then, the coordinated control of the vehicle ISD suspension model is realized by the model predictive control (MPC) approach. The test results show that, the RMS values of the body acceleration and dynamic tire load of the controllable ISD suspension is smaller than the traditional passive and passive ISD suspension under the sinusoidal road inputs. Under the random road input, compared with the traditional passive suspension, the controllable ISD suspension has improved the RMS value of the body acceleration by 26.61%, and the road damage coefficient has been improved by 20.13%. Experiments show that the controllable ISD suspension of heavy vehicles based on the hybrid-hook damping strategy has comprehensive improvement of vehicle ride comfort and road friendliness.