# A 6 DOF Virtual Environment Space Docking Operation with Human Supervision
## 3 Adaptive Control  
Originally, the controller used for this level was a single PI (proportional-integral) controller with feedforward action. An integer PI-cascaded controller has been outperformed in terms of robustness for maintaining closed loop specifications by a fractional order PI controller while being validated on the real-life simulator for European Space Agency at QINETIQ nva Antwerp, Belgium [16]. A robust fractional order PI controller has been proposed and simulated in [6]. Here, we use the adaptive robust fractional order PI (FOPI) control design described in [5]. It has already been shown that fractional order control is a good candidate for the robust control of DC motor components [17; 18; 19]. The FOPI described hereafter has the form:  
\[C_{FOPI}(s)=K_{p}(1+\frac{K_{i}}{s^{\mu}}) \tag{7}\]  
where \(K_{p}\) and \(K_{i}\) are the proportional and integral gains and \(\mu\in(0,1)\) the fractional order of integration. Note that the traditional PI controller is obtained for the special case when \(\mu=1\). The fractional order PI controller in (7) is thus a generalization of the integer order PI controller. The advantages of the fractional order PI controller stem from the extra tuning parameter \(\mu\), which can be used to enhance the robustness of the controller. Since there are three tuning parameters in the fractional order PI controller, three performance specifications are used. These three performance criteria lead to the typical tuning rules for fractional order controllers. and they are based on the relationship between time-domain performance specifications and corresponding frequency domain specifications. We enumerate here the most commonly used specifications:  
* Gain cross over frequency \(\omega_{gc}\), which is related to the settling time of the closed loop system--large values will yield smaller settling times at the cost of higher control efforts;
* Phase margin (PM), which is stability related and an indicator of the closed loop overshoot percentage--usual values are within the 45-65\({}^{\circ}\) interval [20];
* The iso-damping property, which is a condition ensuring robustness to gain changes such that overshoot remains constant within a certain gain variation range. To ensure a constant overshoot, a constant phase margin needs to be maintained around the desired gain crossover frequency, which ultimately implies that the phase of the open-loop system must be kept constant around the specified gain cross-over frequency. In other words, the derivative of the phase with respect to frequency around the gain cross over frequency must be (almost) null. The three performance specifications mentioned above are mathematically expressed as  
\[|H_{open-loop}(j\omega_{gc})|=1 \tag{8}\] \[\angle H_{open-loop}(j\omega_{gc})=-\pi+PM\] (9) \[\frac{d(\angle H_{open-loop}(j\omega))}{d\omega}|_{\omega= \omega_{gc}} \tag{10}\]  
where \(H_{open-loop}(s)=C_{FOPI}(s)P(s)\) stands for the open loop transfer function and \(P(s)\) is the system to be controlled. Equations (8)-(10) require the magnitude, phase and derivative of the phase for both the FOPI controller and \(P(s)\). The modulus and phase of the FOPI controller are given by  
\[ \begin{split}&|C_{FOPI}(j\omega)|=K_{p}\sqrt{1+2K_{i}\omega^{- \mu}\cos\frac{\pi\mu}{2}+K_{i}^{2}\omega^{-2\mu}}\\ &\angle C_{FOPI}=-\arctan\left(\frac{K_{i}\omega^{-\mu}\sin \frac{\pi\mu}{2}}{1+K_{i}\omega^{-\mu}\cos\frac{\pi\mu}{2}}\right)\end{split}  \tag{11}\]  
To determine analytically the derivative of the phase of the FOPI controller, the phase equation in (11) is used, resulting in  
\[\frac{\mu K_{i}\omega_{gc}^{-\mu-1}\sin\frac{\pi\mu}{2}}{1+2K_{i}\omega_{gc} ^{-\mu}\cos\frac{\pi\mu}{2}+K_{i}^{2}\omega_{gc}^{-2\mu}}=0 \tag{12}\]  
Equations (8)-(10) also use the magnitude, phase and derivative of the phase of \(P(s)\). The values for the modulus, phase and phase slope of the process should be known a priori at the gain cross over frequency \(\omega_{gc}\). These can be obtained analytically based on a model of \(P(s)\).