The integrated fault estimation and fault-tolerant control scheme is developed in this article for a series of interval type-2 polynomial fuzzy systems with both sensor faults and actuator faults, where the bidirectional influence between fault estimation unit and fault-tolerant control unit is investigated. Considering the existence of sensor faults, unmeasurable premise variables are investigated for more general situations and Class III state/fault estimation observers are established for the final fault estimation and fault-tolerant control purposes. To increase design flexibility and reduce physical implementation complexity, the proposed method allows the observer and original system to share asynchronous membership functions and a different number of fuzzy rules. $(\mathcal {Q,S,R})-\alpha$ dissipative performance index is also introduced to fulfill a wider vary of performance requirements. Membership function-dependent stability constraints are given in the format of bilinear polynomial matrix inequalities to obtain less conservative results, which are computed by a two-step path-following method. Superiority and validity are demonstrated by an inverted pendulum example in terms of estimation errors, fault-tolerant control performance, and control inputs.