# SSM

State-Space Model

State-Space Model

Mathematical frameworks used to model dynamic systems by describing their states in space and how these states evolve over time under the influence of inputs, disturbances, and noise.

State-space models are integral to control theory and signal processing, representing systems as a set of input, output, and state variables related by first-order differential equations. In these models, the system's current state is described by a set of state variables, and the evolution of these states is determined by linear or nonlinear equations. SSMs are particularly powerful for dealing with multi-variable systems where the interactions between variables may be complex and hidden. They are used extensively for system identification, time series analysis, forecasting, and control systems design, allowing for the accommodation of noise and other uncertainties in the modeling process.

The concept of state-space in control engineering was primarily developed in the late 1950s and early 1960s. It became a fundamental aspect of modern control theory, particularly through the work on optimal control and the development of the Kalman filter in the early 1960s. The state-space approach provided a unified framework that was applicable to both continuous and discrete systems, marking a significant shift from classical control methods that focused on transfer functions.

Rudolf E. Kalman was particularly influential in the development of state-space models through his work on the Kalman filter, which efficiently estimates the state of a linear dynamic system from a series of incomplete and noisy measurements. His contributions laid foundational principles for the use of SSMs in various applications, including aerospace and economics. Other notable contributors include Pierre Simon Laplace and Andrey Markov, who developed early concepts related to state estimation and stochastic processes, respectively.