Discrete Sliding Mode Control Based on Equivalent Control

In control theory, Sliding Mode Control (SMC) is widely used in aerospace, robotics, power electronics, and other fields due to its excellent robustness and strong anti-interference capability against system uncertainties and external disturbances. Discrete Sliding Mode Control (DSMC) is a variant designed for digital control systems, suitable for computer-controlled implementations.

### Fundamental Concept of Discrete Sliding Mode Control The core idea of DSMC involves designing a sliding surface that enables the system state to slide along it, ultimately converging to the desired equilibrium point. Unlike traditional continuous SMC, DSMC accounts for sampling time effects, ensuring control signal switching within each sampling period while avoiding high-frequency chattering phenomena. In code implementation, this typically requires discretizing the system model using methods like Euler approximation or zero-order hold.

### Role of Equivalent Control Equivalent Control constitutes a critical component of DSMC, used to compute the ideal control input that maintains the system state on the sliding surface without deviation. The introduction of equivalent control reduces jitter caused by high-frequency switching, enhancing system stability and smoothness. Algorithmically, this involves solving for the control input that satisfies the sliding condition s(k+1)=s(k) in discrete-time, often derived through linearization or predictive approaches.

### Implementation Methodology Sliding Surface Design: Based on system dynamic equations, select appropriate sliding variables (e.g., linear combinations of state errors) ensuring surface convergence. Code implementation typically involves defining the sliding function as s(k)=C*x(k) where C is the weighting matrix. Equivalent Control Calculation: Derive equivalent control laws using discrete system models to maintain stable sliding on the surface. This can be implemented through matrix inversion operations when solving u_eq(k) = -(G*B)⁻¹*G*A*x(k) for linear systems. Robustness Enhancement: Incorporate switching terms (e.g., sign or saturation functions) into control laws to handle model uncertainties and disturbances. Practical implementations often use boundary layer techniques with saturation functions to minimize chattering. Stability Analysis: Prove convergence and robustness through Lyapunov functions or stability criteria, where discrete Lyapunov differences ΔV(k) = V(k+1)-V(k) must be negative definite.

### Application Prospects DSMC possesses inherent advantages in digital control systems, particularly suitable for embedded systems and real-time control scenarios (e.g., UAVs, satellite attitude control). As modern control increasingly relies on digital computation, research and optimization of DSMC will further improve system response speed and disturbance rejection capabilities. Code optimization techniques may include adaptive gain scheduling and quantization error compensation for practical deployments.