## 4 GNC/FDIR System Design and Analysis
### Control  
The controller is another key element. Similar to navigation, it must be robust to factors that impact its performance and reliability, and guarantee a proper controllability of the pose.  
In CS-1, the controller is designed with the H-infinity synthesis [34]. Within the adopted robust control design paradigm, the model is obtained by using the so-called Linear Fractional Transformation (LFT) framework. These LFTs allow for the representation of the system to control by means of the feedback connection of the nominal plant, \(G(s)\), and a block-diagonal uncertainty, \(\Delta(s)\), that gathers all the uncertain/time-varying parameters of the system, as illustrated in Figure 9.  
This framework ensures robustness by design against a selected set of uncertainties and non-idealities, namely the following:  
* Actuation errors,
* Navigation errors,
* Servicer and stack Mass, Centre of mass, Inertia uncertainties,
* Disturbances like flexible modes and sloshing,
* Systems delays.  
Concerning the exogenous disturbance and sensor noise, these are weighted by means of frequency domain filters, for the controller to reject disturbances in the desired range of frequencies and to penalize the controller sensitivity to sensor noise. On the other hand, design weights at the control outputs are used to bound the energy of the control signal, constraining the commands to the operational region of the actuator. Finally, to enforce closed-loop tracking performances, error performance weights are added to the synthesis problem, and the frequency response of the closed-loop plant is shaped in order to fulfil the performance requirements. Once the problem is posed, it is solved by using the iterative H-infinity synthesis methodology. More details on the adopted framework can be found in [35].  
The effects of flexible modes can be considered with two main approaches. The first approach considers the flexible modes as a frequency domain disturbance, with a controller being synthesized for the rigid body. An analysis of the frequency response of the closed loop is performed, to guarantee that the sensitivity function from the flexible modes entry point to the attitude error indeed attenuates the effect of the disturbance at the impacted frequencies. A second approach can be adopted where the flexible modes are explicitly modelled, e.g., as described in [33], and incorporated in the plant adopted in the controller  
Figure 9: Control synthesis problem.  
synthesis. The control optimization problem will have a higher degree of complexity, which may impact the convergence of the H-infinity iteration, but the resulting closed loop will naturally yield the desired attenuation of the flexible modes.  
The resulting controller complies with the requirements formulated in [11], summarized as follows:  
* An ability to control the position and attitude at any point during the mission;
* During all mission phases, compensation must take place for endogenous and exogenous perturbation forces and torques to the servicer, namely sloshing, flexible appendages, misalignment of the thrusters, thrust magnitude errors, solar pressure force, and torque;
* During the Close Rendezvous Phase and inside the KOZ, the 6 DOF relative control ensures that the servicer stays within the Approach Corridor.  
These properties are obtained by design and analysis. The V&V by testing is presented for the full GNC chain in Section 5.