A multiple reference points linear model predictive control (MRP-LMPC) method was proposed to address the trade-off between accuracy and real-time performance in traditional path tracking control methods for car-like robots. The linearization expansion point was redefined, and the linear difference model was modified. A nonlinear compensation term was obtained through iterative nonlinear and linear prediction. Multiple reference points along the path were incorporated to construct an MRP-LMPC controller that adapted to abrupt changes in curvature. Co-simulation results showed that the proposed MRP-LMPC achieved a maximum lateral displacement error of 0.0971 m on a U-shaped path and 0.1185 m on a single-lane change path. Hardware-in-the-loop (HIL) experiments results showed that the maximum displacement error was 0.1897 m without localization error and 0.2486 m with localization error. Compared with nonlinear model predictive control (NMPC) under identical conditions, the accuracy loss of MRP-LMPC was limited, with a maximum error increase of less than 0.0949 m. Significant accuracy advantages were observed over single-reference-point linear model predictive control (SRP-LMPC) and proportional integral derivative (PID) controllers. In terms of real-time performance, the worst-case average solution time in co-simulation was 3.53 ms, and in the worst-case HIL test, the average was 5.59 ms. In all tests, the maximum computation time accounted for 44.15% of the control period. Under the same conditions, compared to NMPC, the average solution time was reduced by 39.14%. In summary, the results of co-simulation and hardware-in-the-loop tests demonstrate that the MRP-LMPC method effectively balances accuracy and real-time performance, achieving faster computational speed than NMPC, and maintaining higher accuracy than PID and SRP-LMPC.