LTISysteme

LTISysteme, or linear time-invariant systems, are a class of mathematical models used to describe dynamic processes in engineering and physics. They satisfy linear superposition: a scaled input produces a scaled output, and the response to a sum of inputs is the sum of the individual responses. Time invariance means the system's characteristics do not change over time; delaying the input by t0 delays the output by the same amount.

Common representations include continuous-time models, discrete-time models, and frequency-domain descriptions. In continuous time, the system can be described by a differential equation dx/dt = Ax + Bu, y = Cx + Du, where x is the state and u is the input. The transfer function G(s) = C(sI - A)^{-1}B + D relates input and output in the Laplace domain. In discrete time, x[k+1] = Ax[k] + Bu[k], y[k] = Cx[k] + Du[k], and G(z) = C(zI - A)^{-1}B + D. The impulse response h(t) or h[n] characterizes the system's output to a unit impulse, and for any input u, the output is the convolution y = h * u in continuous time or y[n] = ∑ h[k] u[n-k] in discrete time.

Important properties include BIBO stability (bounded input yields bounded output), causality, and realizability. State-space descriptions can be analyzed for controllability and observability, which determine whether the system can be controlled or observed from inputs and outputs. Realizations can be minimal, with the smallest possible state dimension.

LTISysteme are central to control theory, signal processing, and communications, serving as a foundation for model-based design, system identification, filter design, and stability analysis.