# Robust Speed Control of a Multi-Mass System: Analytical Tuning and Sensitivity Analysis
## 1 Introduction  
The powertrain test benches apply the desired driving resistances directly to the wheel-hub of the test vehicle. So, the loading machines will be coupled to the vehicle's powertrain [1, 2]. This setup establishes high stability and allows the powertrain to operate without restraint while the wheel-hubs are attached to electrical load machines [3]. Therefore, powertrain test benches are practical tools for highly dynamic driving condition simulation and advanced driving assistance systems applications. For instance, in [1, 2], a control concept for powertrain test benches was proposed to simulate different street conditions, such as dry, wet, or icy. The reliability of this concept was proved by successfully simulating the anti-lock braking system performance on the road with a side ice patch. Nevertheless, accurate tire modeling becomes indispensable to compensate for the absence of the actual tire characteristics. For example, an approach was proposed to establish a correlation between the road and the test bench measurements in [4]. The developed method is based on empirically optimizing the Pacejka tire model parameters. The complete vehicle test benches are developed to allow the steering of the front wheels and even vertical excitation. These assets contribute to bridging the gap between real-world driving and vehicle-in-the-loop (VHiL) simulation. In addition, the complete vehicle test benches took the VHiL-testing to a new level by enabling more degrees of freedom in automotive testing. In addition to the four electrical loading machines used in the powertrain test benches, the complete vehicle test benches have actuators on the steering wheels to simulate the self-aligning torque phenomenon [5]. Many researchers used a proportional-integral-controller (PI) to reduce the unwanted dynamics in mechanical torsional systems [6, 7]. The widely spread utilization of the PI controller is due to its simple design and tuning techniques, its ability to be adapted to manyapplications, and its lack of an SS model [8; 9]. However, the performance of the PI control is inferior to the other control algorithms, such as SS feedback control [8; 9]. Furthermore, only a single design goal can be optimally solved. For instance, a PI controller can be tuned for high robustness or quick signal tracking performance, but not both [8; 10; 11]. In addition, many studies proved that PI control could not effectively dampen the torsional vibrations [10; 12; 13]. Moreover, since it has only two tunable parameters (i.e., proportional gain (\(k_{p}\)) and integral gain (\(k_{i}\))), it has limited performance with high-order systems [8; 10]. In contrast, SS feedback control allows placing the closed-loop poles anywhere in the space [10]. The limitations of the PI controller were further investigated in [10; 14]. One of the techniques to compensate for the limited number of closed-loop poles of the PI controller is implemented by adding more internal feedback to the system control loop. The authors of [15] conducted a study for employing a double feedback coefficient pole placement to improve two-inertia system performance. Based on a proper choice for the location of the closed-loop system poles, different cases were investigated for the poles of a cascade control structure with a PI speed controller. System dynamics were investigated for different natural angular frequencies and damping coefficients (i.e., different closed-loop poles). Likewise, two additional feedback gains are employed in [16]: one is for the shaft moment, and the second is for the angular speed difference between the motor and the wheel. A more efficient approach for vibration suppression is achieved with feedback gains from all system state variables, namely, shafts' torsional moment and the corresponding rotational speed of each inertia. The SS control realizes this concept [17; 18; 19]. SS control gained its advantage from its features that support the control process.