## 2 Design Considerations for the SUZI Prototype
### The Control Law  
It is known that the artificial intelligence paradigm in control problems tends to only consider the input-output behavior of the system. Herein, it assumes two components: the first one is neuro-control [42; 43; 44], a simple mono-layer perceptron, shown Figure 16a, as described by the following relationship:  
\[u:=u_{n}=\
u_{1}y_{1}+\
u_{2}y_{2}=:\
u_{1}\Big{(}T_{ref}-T\Big{)}+\
u_{2} \tilde{T} \tag{2}\]  
where \(T_{ref}\) is the input (reference temperature), and \(T\) is the measured temperature. The input to the manikin system is \(u_{n}\), and the output from the manikin system is \(\mathbf{y}=(y_{1},y_{2})\). The cost index \(J\) defined below evaluates the performance with which the control \(u\) regulates the temperature \(T\) at the manikin's surface and tends to minimize the deviation \(y_{1}\) from the reference temperature that is caused by environmental disturbances; obviously, this regulation must be balanced with the \"energy\" of the control \(u^{2}\); otherwise, there are risks of the control becoming saturated, resulting in negative consequences for the stability  
Figure 16: (**a**) Schema of elementary perceptron; (**b**) singleton membership function for scaled fuzzy control \(u_{f}\). of the system, which is a sine qua non condition of the regulation process (square rate of temperature variation \(T^{2}\) has also been included in the cost)  
\[J=\frac{1}{2n}\sum_{i=1}^{n}\Bigl{(}q_{1}y_{1}^{2}(i)+y_{2}^{2}(i)+q_{2}u_{n}^{ 2}(i)\Bigr{)}:=\frac{1}{2n}\sum_{i=1}^{n}J(i) \tag{3}\]  
The essence of the neural network principle consists of the correction (variation) \(\Delta\
u(n)\) of the weighting vector \(\
u=\left[ \begin{array}{cc}v_{1}&v_{2}\end{array} \right]^{T}\), which is based on the \"learning experience\", and this correction is performed to ensure a descending gradient to reduce the cost \(J\):  
\[\
u(n+1)=\
u(n)+\Delta\
u(n),\]  
\[\Delta\
u(n):=-diag(\delta_{1},\delta_{2})\frac{\partial J}{\partial\
u(n)}=- diag(\delta_{1},\delta_{2})\sum_{i=n-N}^{n}\biggl{(}\frac{\partial J(i)}{ \partial y(i)}\frac{\partial y(i)}{\partial u(i)}+\frac{\partial J(i)}{ \partial u(i)}\biggr{)}\frac{\partial u(i)}{\partial\
u(i)} \tag{4}\]  
The matrix \(diag(\delta_{1},\delta_{2})\) introduces a learning vector scale, \(\Delta\
u(n)\) is the vector of the improved weights, and \(N\) represents a back memory of the \(N\) time steps (herein, \(N=3\)). The novelty of the neural network paradigm is remarkable: it can be seen that three of the derivatives in (4) (first, vector; third, scalar; and fourth, vector) only require input-output information about the manikin system, thus always avoiding its mathematical model in situations where it can only be elaborated approximately. The derivative \(\frac{\partial y(i)}{\partial}u(i)\) can be approximated only through the relationship:  
\[(\mathbf{y}(i)-\mathbf{y}(i-1))/(u(i)-u(i-1)) \tag{5}\]  
For compliance, we write these derivatives from relationship (4):  
\[\Delta\mathbf{\
u}(n):=-\mathbf{diag}(\delta_{1},\delta_{2})M_{i;n,N}\] \[M_{i;n,N}=\sum_{i=n-N}^{n}\biggl{(}\frac{\partial J(i)}{ \partial y_{1}(i)}\frac{y_{1}(i)-y_{1}(i-1)}{u(i)-u(i-1)}+\frac{\partial J(i )}{\partial y_{2}(i)}\frac{y_{1}(i)-y_{1}(i-1)}{u(i)-u(i-1)}+\frac{\partial J (i)}{\partial u(i)}\biggr{)}\left[ \begin{array}{c}\frac{\partial u(i)}{ \partial y_{1}(i)}\\ \frac{\partial u(i)}{\partial u(i)}\\ \frac{\partial u(i)}{\partial u_{2}(i)}\end{array} \right]\]  
\[M_{i;n,N}=\sum_{i=n-N}^{n}\biggl{(}2q_{1}y_{1}(i)\frac{y_{1}(i)-y_{1}(i-1)}{ u(i)-u(i-1)}+2q_{2}y_{2}(i)\frac{y_{2}(i)-y_{2}(i-1)}{u(i)-u(i-1)}+2q_{2}u(i) \biggr{)}\left[ \begin{array}{c}y_{1}(i)\\ y_{2}(i)\end{array} \right]\]  
We mentioned the possibility of neural control saturation above. Neural control is optimal, and its synthesis is achieved without restrictions and is based on the solution of a Riccati equation.