## 2 Radar Position Servo System Modeling  
A precise radar servo system generally has two branches, namely azimuth and pitch, which are both driven by the motor. This paper takes the azimuth axis of the radar position servo system as an example to verify the performance of the composite controller. Taking the antenna as the total load in this model, the following dynamic equations of the radar servo system are established and the model block diagram of the radar position servo system is shown in Figure 1.  
Where \(u_{c}\) is the input voltage of the motor, \(R_{a}\) is the armature resistance, \(L_{a}\) is the armature inductance [31], \(K_{T}\) is the motor torque coefficient, \(B\) is the elastic coefficient, \(F\) is the viscous damping coefficient, \(K_{P}\) is the power magnification, \(T_{L}\) is the load torque, \(K_{E}\) is the back electromotive force of the motor, \(J\) is the system moment of inertia, \(\theta\) is the output pointing angle of the load antenna, \(S\) is the complex frequency domain, and \(\frac{1}{R_{a}+L_{aS}}\) and \(\frac{1}{Js+F}\) are the output formulas of motor voltage after Laplace transform.  
Ignoring the influence of the elastic moment on the system, the relationship between \(u_{c}\) and \(\theta\) can be obtained as follows:  
\[\theta(s)=\frac{K_{T}K_{P}U_{C}(s)-(L_{a}s+R_{a})T_{L}(s)}{s[(Js+F)(L_{a}s+R_ {a})+K_{E}K_{T}]} \tag{1}\]  
Since the induction \(L_{a}\) of the motor utilized in this system is very small, the influence of \(L_{a}\) on the system can be ignored and the system (1) can be rewritten as:  
\[J\mathrm{R_{a}}\ddot{\theta}+(FR_{a}+K_{E}K_{T})\dot{\theta}=K_{T}K_{P}u_{c}- R_{a}T_{L} \tag{2}\]  
When the servo system drives the antenna to track the target, it is mainly affected by the unbalanced torque and inertia moment; thus, the load torque equation can be obtained as follows:  
\[T_{L}=m\ddot{\theta}r+J_{L}\ddot{\theta} \tag{3}\]  
where \(m\) is the weight of the load antenna, \(r\) is the eccentric disturbance of the load antenna, and \(J_{L}\) is the load moment of inertia.  
Substitute (3) into (2):  
\[R_{a}(J+mr+J_{L})\ddot{\theta}+(FR_{a}+K_{E}K_{T})\dot{\theta}=K_{T}K_{P}u_{c} \tag{4}\]  
Then (4) can also be written in the following state space equation as  
\[\left\{\begin{matrix}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-\frac{FR_{a}+K_{E}K_{T}}{R_{a}(J+mr+l_{L})}x_{2}+\frac{K_{T}K_{P} }{R_{a}(J+mr+l_{L})}u\\ y=x_{1}\end{matrix}\right. \tag{5}\]  
where \(\theta=x_{1}\), \(\dot{\theta}=w=x_{2}\) and \(w\) is the mechanical angular velocity.  
Since the backlash-like nonlinearities such as dead zone and friction nonlinearity, etc., exist in practical servo systems, then a dead zone function with non-smooth nonlinearity and the Stribeck friction model [11; 32] are both introduced into (5). The dead zone \(\Phi(\gamma(t),d)\) function is assumed and defined as  
Figure 1: Block diagram of radar position servo system model.  
where \(v_{d}\) is the desired signal, optimizing two key parameters \(k_{p}\) and \(k_{d}\) can make the system more stable. In this paper, the neural network is introduced to optimize \(k_{p}\) and \(k_{d}\) in real time.  
**Assumption 1**.: _The expected signal \(v_{d}(t)\) is glossy and bounded, so that \(\|v_{d}(t)\|\leq V_{d}\) holds, is a known constant._  
**Assumption 2**.: _The total disturbance \(\Lambda(t,x_{2},d)\) is bounded, so that \(\|\Lambda(t,x_{2},d\|\leq\Lambda_{d}\) holds, \(\Lambda_{d}\) is a known constant._