# Robust Speed Control of a Multi-Mass System: Analytical Tuning and Sensitivity Analysis
## 1 Introduction
According to [8], the fundamental feature of SS control is that it enables fast detection of deviations in all system states because it allows information from the internal states of the plant. In addition, it has a structured design procedure that makes high-control performance achievable. Furthermore, with SS representation, nonlinearities in the model become easier to handle. Nevertheless, it was also highlighted in [8; 13] that SS control has downsides. For instance, it does require thorough SS modeling. The second disadvantage is that the designing controller parameters are not directly interpretable. As a consequence, high implementation and tuning efforts are likely. Also, compared to the PI controller, it has a more complex design approach. Moreover, the steady-state accuracy is only apparent since the parameters are uncertain. Finally, and most importantly, it cannot regulate the output error. In order to overcome the inadequacy of SS control in compensating for the output error, a synthesized controller from SS control algorithms with the additional integral yield is suggested. The additional integral yields the closed-loop output to converge to zero steady-state error when the desired value is constant. This type of control is called state-space feedback with integral action (I-SS). More details about I-SS control can be found in [8; 10; 20]. It was also proven in [10] that the state-space feedback controller with the additional integral is a much better choice than the PI controller with additional feedback. Nevertheless, the modification of SS to I-SS control is made at the expense of increasing the settling time and reducing the phase margin compared to SS control. An investigation of the performance of different control algorithms, namely, the conventional PI control, I-SS control, and model predictive control (MPC), is performed [13]. The results showed that the PI ranked the lowest in the criterion of dynamic characteristics. In contrast, both I-SS and MPC achieved better results at the expense of higher complexity of implementation and tuning. Furthermore, regarding stability and robustness, I-SS was at the top of the list, followed by PI, while MPC came last. Finally, the MPC entailed the highest calculation effort in terms of processing time. It is concluded from this thorough study that the decision of the best speed control method depends on the user and the type of application. Among all the previously mentioned references, the authors of [7; 11] have developed an SS controller for a multi-mass system, while the rest had applications with a two-mass system at the most. Thus, an I-SS controller for a multi-mass system will be proposed in this work. Furthermore, an optimized PI and SS controller, similar to the control algorithm in [11],will be developed. Then, an extensive analysis of all controllers will be performed to define the limits of implementing each controller type. The analysis will comprise the theoretical equations, frequency response, time-domain response, and sensitivity analysis. An essential area that, unfortunately, did not receive enough attention from researchers is the need for a robust control algorithm for the combined system of the test bench and the test vehicle's powertrain. So, this research conducts an in-depth investigation of the development of a control algorithm for high-performance powertrain test benches'. In Section 2, a multi-mass model for the integrated system from the powertrain of the test vehicle and the mechanical part of a complete vehicle test bench is derived. The developed model will be employed in Section 3 to create different angular speed control algorithms. Frequency analysis and the measurement's noise-amplification are performed on all controllers to evaluate their robustness. A time-domain response investigation of the control alternatives is conducted in Section 4. The evaluation criteria of the different controllers are mainly based on performance, robustness, and sensitivity analysis.