State-space models of linear dynamical systems

Matrices > Applications > State-space models

Definition

Many discrete-time dynamical systems can be modeled via linear state-space equations, of the form

x(t+1) = Ax(t) + Bu(t), ;; y(t) = Cx(t) + Du(t), ;; t=0,1,2,ldots

where $x(t) in mathbf{R}^n$ is the state, which encapsulates the state of the system at time , $u(t) in mathbf{R}^p$ contains control variables, $y(t) in mathbf{R}^k$ contains specific outputs of interest, and A,B,C,D are matrices of appropriate size.

In effect, a linear dynamical model postulates that the state at the next instant is a linear function of the state at past instants, and possibly other ‘‘exogeneous’’ inputs; and that the output is a linear function of the state and input vectors.

A continuous-time model would take the form of a differential equation

$frac{d}{dt} x(t) = A x(t) + Bu(t), ;; y(t) = Cx(t) + Du(t), ;; tge0.$

Finally, the so-called time-varying models involve time-varying matrices A,B,C,D (see an example below).

Motivation

The main motivation for state-space models is to be able to model high-order derivatives in dynamical equations, using only first-order derivatives, but involving vectors instead of scalar quantities.

Consider for example the second-order differential equation

m ddot{y}(t) + c dot{y}(t) + ky(t) = u(t),

which describes the evolution of a damped mass-spring system, with the external force acting on the mass, and the vertical position. (Here dot{y} and ddot{y} are the first and second derivatives of , respectively.)

The above involves second-order derivatives of a scalar function y(cdot) . We can express it in an equivalent form involving only first-order derivatives, by defining the state vector to be

x(t) := left(begin{array}{c}y(t)dot{y}(t) end{array}right) .

The price we pay is that now we deal with a vector equation instead of a scalar equation:

$dot{x}(t) = left( begin{array}{cc}0&1-frac{c}{m}&-frac{k}{m} end{array}right) x(t) + left( begin{array}{c} 0 1end{array}right)u(t).$

The position y(t) is a linear function of the state:

with C = (1,0) .

A nonlinear system

In the case of non-linear systems, we can also use state-space representations. In the case of autonomous systems (no external input) for example, these come in the form

where $f : mathbf{R}^{n+p} rightarrow mathbf{R}^n$ is now a non-linear map. Now assume we want to model the behavior of the system near an equilibrium point x_0 (such that f(x_0) = 0 ). Let us assume for simplicity that x_0 = 0 .

Using the first-order approximation of the map , we can write a linear approximation to the above model:

$frac{d}{dt} x(t) = A x(t), ;; tge0.$

where

$A = frac{partial f}{partial x} (0).$

The motion of a pendulum can be described by the dimensionless nonlinear equation

The linearization around the equilibrium point theta = 0 yields ddot{theta} = - theta . The linearization around the equilibrium point theta = pi yields ddot{theta} = 1 - theta .