State-space models of linear dynamical systems

L. El Ghaoui

State-space models of linear dynamical systems

Definition

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

$x(t+1)=Ax(t)+Bu(t), y(t)=Cx(t)+Du(t), t=0,1,2,\dots$

where $x(t) \in \mathbb{R}^{n}$
is the state, which encapsulates the state of the system at time $t, u(t) \in \mathbb{R}^{p}$ contains control variables, $y(t) \in \mathbb{R}^{k}$ 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

$\dfrac{d}{dt}x(t)=Ax(t)+Bu(t), y(t)=Cx(t)+Du(t), t \ge 0.$

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 $u$ the external force acting on the mass, and $y$ the vertical position. (Here $\dot{y}$ and $\ddot{y}$ are the first and second derivatives of $y$ , 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

$x(t) := \begin{pmatrix} y(t) \\ \dot{y}(t) \end{pmatrix}.$

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

$\dot{x}(t) = \begin{pmatrix} 0 & 1 \\ -\dfrac{c}{m} & -\dfrac{k}{m} \end{pmatrix}x(t)+ \begin{pmatrix} 0 \\ 1 \end{pmatrix}u(t).$

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

$\dot{x}(t) = f(x(t))$

where $f: \mathbb{R}^{n+p} \rightarrow \mathbb{R}^{n}$ 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 $f$ , we can write a linear approximation to the above model:

$\dfrac{d}{dt}x(t)=Ax(t), t \ge 0.$

where

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

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

$\ddot{\theta} = - \sin(\theta).$

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

State-space models of linear dynamical systems

Definition

Motivation

A nonlinear system

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