## 2 Problem Description and System Modeling  
The system model of this paper is illustrated in Figure 1. The wireless communication system consists of a transmitter and a corresponding receiver, with adaptive capabilities for power and modulation coding adjustment. The communication transmission of this system is affected by a malicious jammer, and the jamming signal can effectively cover the receiver. Consider an anti-jamming control model for the wireless communication system, as shown in Figure 2. After perceiving the electromagnetic environment, the transmitter adjusts its transmission power and modulation coding based on feedback information to ensure that the receiver's bit error rate does not exceed the preset target error rate. Figure 1: System model. For the convenience of the study, the following assumptions are made in this paper:  
1. The transmitter's transmission power is \(P_{\mathrm{s}}(t)\), and the wireless communication system operates over an Additive White Gaussian Noise (AWGN) channel. 2. The receiver has spectrum sensing capability, allowing it to sense the power levels of jamming and noise in channel on a time-slot basis, but it cannot determine the behavior characteristics, patterns, or probability distribution of the jamming. 3. Ignoring free-space propagation losses, and with both the transmission power \(P_{\mathrm{s}}(t)\) and the power of jamming plus noise \(P_{\mathrm{j}}(t)\) in the channel expressed in dBm, the SJNR during transmission can be simply represented as: \(\mathrm{SJNR}(t)=P_{\mathrm{s}}(t)-P_{\mathrm{j}}(t)\). 4. The BER at the receiver under \(P_{\mathrm{s}}(t)\) is \(y_{\mathrm{BER}}(t)\), with the target BER for normal system operation being \(y_{\mathrm{r}}(t)\). 5. The system has M modulation and coding schemes, corresponding to M transmission rates, where \(i\in\{1,\cdots,M\}\) represents the system under the \(i\)-th modulation and coding combination. Figure 3 shows the schematic curves of system BER under different modulation and channel coding combinations. Figure 3: Bit error rate curves under several modulation schemes and LDPC codes. Figure 2: Anti-jamming control model for wireless communication systems. The purpose of system modeling is to describe the system using the following form of linear differential switching equations:  
\[\left\{ \begin{array}{l}\mathrm{d}x_{i}(t)/\mathrm{d}t=A_{i}x_{i}(t)+B_{i}u_{i}(t) \\ y_{\mathrm{BER}}(t)=C_{i}x_{i}(t)+D_{i}\end{array} \right. \tag{1}\]  
Here, \(A_{i},B_{i},C_{i},D_{i}\) are referred to as the dynamic characteristic coefficients, control coefficients, sensing coefficients, and direct terms of the \(i\)-th subsystem, respectively. Choose the transmission power as the control input variable, the SJNR at the receiver as the system state variable, and the BER as the system output variable. The curves of BER versus SJNR under different modulation and coding schemes are shown in Figure 3. It can be observed that when the BER \(P_{\mathrm{e}}\leq 10^{-3}\), the BER curve generally enters the \"waterfall area\" and can be approximated as a straight line. Therefore, a linear equation can be used to approximate the relationship between SJNR and BER for a given modulation and coding scheme. Consequently, the system under M modulation and coding combinations can be expressed in the following piecewise function form:  
\[y_{\mathrm{BER}}(t)=\left\{ \begin{array}{l}C_{1}x_{i}(t)+D_{1},\ X_{1}\leq x <X_{2}\\ C_{2}x_{i}(t)+D_{2},\ X_{2}\leq x<X_{3}\\ \vdots\\ C_{i}x_{i}(t)+D_{i},\ X_{i}\leq x<X_{i+1}\\ \vdots\\ C_{M}x_{M}(t)+D_{M},\ X_{M}\leq x\end{array} \right. \tag{2}\]  
In the formula, the sensing coefficient \(C_{i}\) and the direct term \(D_{i}\) are determined by the slope and intercept of the fitted line for a given modulation and coding scheme; \(i\) represents the currently active subsystem, and \(X_{i}\) is the SJNR threshold value for the \(i\)-th modulation and coding scheme.