## 2 Design Considerations for the SUZI Prototype
### The Control Law
To counteract situations where the optimal control law produces a saturated control, which would cause vibrations/chattering/wind-up, a fuzzy supervised neuro-control, and thus a switching type structure, is considered as an anti-windup strategy in this study. The fuzzy logic control has three components: _the fuzzifier, the fuzzy reasoning,_ and _the defuzzifier_. Herein, _the fuzzifier_ component converts the crisp input signals:  
\[l_{2}(y_{1k}):=\sqrt{\sum_{j=k-2}^{k}y_{1j^{\prime}}^{2}},y_{1k},y_{2k},k=1,2,\ldots \tag{6}\]  
into fuzzy variables (membership functions), and the linguistic terms are chosen: zero (ZE), positive or negative small (PS, NS), positive or negative medium (PM, NM), and positive or negative big (PB, NB) (triangular and singleton type membership functions are chosen--see Figure 17. \(l_{2}\) is a norm that provides the maximum variation in the tracking error over a sliding window of three samples. Introducing the crisp signal in the fuzzifier will result in a reduction in the fuzzy control switches because of spurious noise signals. The fuzzy principle idea supposes a (direct) proportion between the error signal \(y_{1}\) and the required fuzzy control \(u_{f}\). Therefore, _fuzzy reasoning_ has \(n=4\times 7\times 7\) IF..., THEN \(\ldots\) rules, which are the number of elements comprising the Cartesian product \(A\times B\times C\), \(A\) := (ZE; PS; PM; PB), and \(B\) = \(C\) := (NB; NM; NS; ZE; PS; PM; PB). These sets are linked to the sets of linguistic terms chosen to define the membership functions for the fuzzy variables \(l_{2}\) (\(y_{1}\)), \(y_{1}\), and \(y_{2}\), respectively. The sequence of the \(n\) rules is the following:  
(1) IF \(l_{2}\) (\(y_{1}\)) is ZE and \(y_{2}\) is PB and \(y_{1}\) is PB, THEN \(u_{f}\) is PB (2) IF \(l_{2}\) (\(y_{1}\)) is ZE and \(y_{2}\) is PB and \(y_{1}\) is PM, THEN \(u_{f}\) is PM \(\ldots\) (7) IF \(l_{2}\) (\(y_{1}\)) is ZE and \(y_{2}\) is PB and \(y_{1}\) is NB, THEN \(u_{f}\) is NB 8) IF \(l_{2}\) (\(y_{1}\)) is ZE and \(y_{2}\) is PM and \(y_{1}\) is PB, THEN \(u_{f}\) is PB \(\ldots\) 196) IF \(l_{2}\) (\(y_{1}\)) is PB and \(y_{2}\) is NB and \(y_{1}\) is NB, THEN \(u_{f}\) is NB. \(l_{2}\) (\(y_{1k}\)), \(y_{1k}\), and \(y_{2k}\) are scaled variables at the time step \(t_{k}=k\tau\) (\(k\) = 1, 2, \(\ldots\) ), for which a number of \(M\leq 2^{3}\) IF..., THEN \(\ldots\) rules need to be evaluated:  
\[\text{IF }y_{1k}\text{ is }B_{i}\text{ and }y_{2k}\text{ is }C_{i}\text{ and }l_{2}(y_{1k})\text{ is }A_{i}\text{, }\text{ THEN }u_{fk}\text{ is }D_{i}\text{, }i=1,2,\ldots,M \tag{7}\]  
\(A_{i}\), \(B_{i}\), \(C_{i}\), and \(D_{i}\) belong to the linguistic terms groups \(A\), \(B\), \(C\), and \(D\) with \(D=B=C\), as seen in Figures 16 and 17. The _definzzifier_ component transforms these rules so as to produce the control variable \(u_{f}\) as output. A fuzzy set \(A_{i}\times B_{i}\times C_{i}\times D_{i}\) is defined by each rule (7) in the input-output Cartesian product space \(R_{+}\times R^{3}\), with the membership functions:  
\[\mu_{u_{i}}=min\big{[}\mu_{B_{i}}(y_{1k}),\mu_{C_{i}}(y_{2k}),\mu_{A_{i}}(l_{ 2}(y_{2k})),\mu_{D_{i}}(u)\big{]},i=1,\ldots,M,k=1,2,\ldots, \tag{8}\]  
The singleton-type membership function \(\mu_{D}(u)\) of control variable is considered for the sake of simplicity; thus, \(\mu_{D_{i}}(u)\) will be substituted by \(u_{i}^{0}\), the singleton abscissa. In conclusion, based on (1) the singleton fuzzifier for \(u_{f}\), (2) the center-average type defuzzifier, and (3) the min inference, the \(M\) IF..., THEN \(\ldots\) rules will produce the relation for crisp control \(u_{f}\) at each time step \(k\tau\)[45]  
\[u_{f}=\sum_{i=1}^{M}\mu_{u_{i}}u_{i}^{0}/\sum_{i=1}^{M}\mu_{u_{i}} \tag{9}\]  
The fuzzy supervised neurocontrol works as fuzzy logic control \(u_{f}\) when neurocontrol \(u_{n}\) saturates or when the \(l_{2}\)-norm of the tracking error \(y_{1}\) increases. If the fuzzy control \(u_{f}\) operates, the fuzzy neurocontrol \(u_{n}\) is updated as if the system were driven by the neurocontrol.