Linear dynamical systems

Definitions
A continuous-time linear dynamical system (CT LDS) has the form

$$\frac{dx}{dt}=A(t) x(t) + B(t)u(t)$$, $$\,\,\,\,y(t) = C(t) x(t) + D(t)u(t)$$

where $$t \in \mathbb{R}$$ denotes time, $$x(t) \in \mathbb{R}^n$$ is the state (vector), $$u(t) \in \mathbb{R}^m$$ is the input or control and $$y(t) \in \mathbb{R}^p$$ is the output. Further, $$A(t) \in \mathbb{R}^{n\times n}$$ is the dynamics matrix, $$B(t) \in \mathbb{R}^{n\times m}$$ is the input matrix, $$C(t) \in \mathbb{R}^{p\times n}$$ is the output or sensor matrix and $$D(t) \in \mathbb{R}^{p\times m}$$ is the feedthrough matrix. The matrices $$A,B,C,D$$ are generally time-invariant.

For lighter appearance, the CT LDS can be written as: $$\dot{x} = Ax + Bu$$, $$y = Cx+Du$$.

A discrete-time linear dynamical system (DT LDS) has the form : $$x(t+1) = A(t)x(t) + B(t)u(t),$$ $$\,\,\,\, y(t) = C(t)x(t) + D(t)u(t)$$

where $$t \in \mathbb{Z} = \{0,\pm 1, \pm 2, \ldots\}$$ and the (vector) signals $$x, u, y$$ are sequences.

What's the point?
There are lots of reasons to study linear dynamical systems (LDSs). One can model a lot of engineering problems as an LDS; if one succeeds in modelling an engineering problem as an LDS, then solving it becomes very easy. The art is taking the real-life problem and transforming it into the form of $$\dot{x} = Ax + Bu$$, $$y = Cx+Du$$.

Categories of LDS problems

 * 1: Estimation or inversion. This happens when we can observe $$y$$ and want to infer $$x$$ (e.g., your weight on the scale: we can measure the output of a strain gauge and we want to infer the weight from it).
 * 2: Control or design: We can control $$x$$ and we want to achieve a desired $$y$$ (e.g., you control the input of the CD drive's motor to bring the drive's head to a desired position).
 * 3: Transformation: We want to find a mapping between $$x$$ and $$y$$ (e.g. discrete cosine transformation).

Those have lots of applications, such as modelling the reaction of a building to wind or earthquake, modelling the current in circuit, predicting the temperature on various points of a circuit, predicting production cost of a certain product from various production elements, controlling the illumination on a surface... One can also perform smoothing, interpolation of signals, encode/decode signals. More specific applications would be, e.g., estimating the runtime of a computer program from measurements.