• State-space models of linear dynamical systems.

16.1 State-space models of linear dynamical systems.

Definition

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

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

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. The matrices \[ A \in \mathbb{R}^{n \times n}, \quad B \in \mathbb{R}^{n \times p}, \quad C \in \mathbb{R}^{k \times n}, \quad D \in \mathbb{R}^{k \times p} \] are of appropriate dimensions to ensure compatibility of matrix multiplications.

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 ‘‘exogenous’’ 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

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

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 instance, the second-order differential equation:
	\[ m\ddot{y}(t) + c\dot{y}(t) + ky(t) = u(t) \]
	which captures the dynamics of a damped mass-spring system. In this equation:
	\( m \) : the mass of the object attached to the spring.
	\( c \) : the damping coefficient.
	\( k \) : the spring constant
	\( u(t) \) : any external force applied to the mass
	\( y(t) \) : the vertical displacement of the mass from its equilibrium position.

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:

\[\dot{x}(t) = \begin{pmatrix} \dot{y}(t) \\ \ddot{y}(t) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -\dfrac{k}{m} & -\dfrac{c}{m} \end{pmatrix}x(t) + \begin{pmatrix} 0 \\[0.5ex] \dfrac{1}{m} \end{pmatrix}u(t).\]

The position $y(t)$ is a linear function of the state by the relation \( y(t) = Cx(t) \) where \[ C = (1, 0). \]

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:

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

where

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

	The pendulum’s motion is governed by the nonlinear equation \(\ddot{\theta} = – \sin(\theta)\), where \(\theta\) is the angular displacement from the vertical position and the dot denotes time differentiation.
	To understand the dynamics near \(\theta = 0\) and \(\theta = \pi\), we linearize this equation using the first-order Taylor series expansion around a point \(a\):
	\[ f(x) \approx f(a) + f'(a)(x-a). \]
	For \(\theta = 0\), setting \(f(x) = \sin(x)\) and \(a = 0\), we find \(f'(x) = \cos(x)\) and \(f'(0) = 1\). This gives the approximation \(\sin(\theta) \approx \theta\), resulting in the simplified equation
	\[\ddot{\theta} = -\theta.\]
	Similarly, for \(\theta = \pi\), setting \(a = \pi\) gives \(f'(\pi) = -1\). The linear approximation here is \(\sin(\theta) \approx -(\theta – \pi)\), leading to
	\[\ddot{\theta} = \theta – \pi.\]
	This linearization elucidates the pendulum’s unstable dynamics at \(\theta = \pi\), assisting in predicting substantial reactions to minor disturbances.

Original Version:

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 $\theta = 1 – \theta$.