State-Space Model for Discrete-Time Systems

The state-space model for discrete-time systems serves as a fundamental mathematical framework for describing dynamic system evolution over time. Unlike continuous systems, discrete systems update their states only at specific time instances (such as sampling instants), making them particularly suitable for digital control implementations and computer-based processing scenarios.

The core structure of state-space equations consists of two key components: the state update equation and the output equation. The state update equation defines how the current state transforms into the next time step's state, typically incorporating the influence of system inputs. The output equation specifies how system outputs are generated from the current state. In MATLAB implementation, these are commonly represented as x[k+1] = A*x[k] + B*u[k] for state update and y[k] = C*x[k] + D*u[k] for output calculation, where A, B, C, D are system matrices.

Key steps for constructing discrete state-space models include: Identifying system state variables by selecting a minimal set of variables that completely characterize system dynamics Deriving state transition relationships based on physical laws or system mechanisms Discretizing continuous-time models when necessary, using common methods such as forward Euler discretization or zero-order hold (ZOH) approximation. The forward Euler method can be implemented as A_d = I + A_c*Ts, where Ts is the sampling time, while ZOH discretization utilizes the matrix exponential function expm() in MATLAB.

This modeling approach finds widespread applications in digital control systems, signal processing, and computer simulations. Through state-space models, engineers can analyze system stability, design controllers using methods like pole placement or LQR, and predict future system behavior. The matrix representation form is particularly advantageous for numerical computation and analysis using modern computer systems, enabling efficient implementation through matrix operations and eigenvalue analysis.