Behavioral system theory

In this post I introduce the mathematical language that will be frequently used to develop further concepts and topics in this blog.

Here I basically adopt the notation, ideas and principles of behavioral system theory. I include some references to complete material (including proofs) regarding this theory at the end of this post.

Notation

$\mathbb{R}^{n}$ denotes the space of $n$ dimensional real vectors.

$\mathbb{R}^{m\times n}$ denotes the space of $m\times n$ real matrices.

$\mathbb{R}^{\bullet\times m}$ denotes the space of real matrices with $m$ columns and an unspecified finite number of rows. 

Given matrices $A,B\in\mathbb{R}^{\bullet\times m}$, $\mbox{col}(A,B)$ denotes the matrix obtained by stacking $A$ over $B$.

$\mathbb{R}[s]$ denotes the ring of polynomials with real coefficients in the indeterminate $s$.

$\mathbb{R}^{m\times n}(s)$ denotes the set of rational $m\times n$ matrices.

$\mathfrak{C}^{\infty}(\mathbb{R},\mathbb{R}^{\tt w})$ denotes the set of infinitely differentiable functions from $\mathbb{R}$ to $\mathbb{R}^{{\tt w}}$.

Linear differential behaviors

Consider a linear time-invariant dynamical system whose physical laws are described by the following set of linear differential equations:

$R_0 w + R_1 \frac{d}{dt} w+ ... + R_L \frac{d^L}{dt^L} w =0$.

with $R_i\in\mathbb{R}^{\bullet\times \tt w}$, $i=0,...,L$, and $w=\mbox{col}(w_1,...,w_{\tt w})$ is the vector of external variables.

Such equations can be expressed in a compact way as 

$R\left(\frac{d}{dt}\right) w =0$,

where $R\in\mathbb{R}^{\bullet\times \tt w}[s]$. Therefore, $R(s)=R_0 + R_1 s + \cdots + R_L s^L$, represents the set of differential equations. 

If we adopt $\mathfrak{C}^{\infty}$ as solution space, we can define a linear differential behavior as

$\mathfrak{B}:=\left\{ w\in \mathfrak{C}^{\infty}(\mathbb{R},\mathbb{R}^{\tt w}) ~\mid~ R\left(\frac{d}{dt}\right) w =0 \right\}$.

In other words, the behavior is defined as the set of trajectories in the kernel of  $R$. For simplicity, we often refer to this set of trajectories as $\ker~R\left(\frac{d}{dt}\right)$. 

The set of linear differential behaviors taking their values in the signal space $\mathbb{R}^{\tt w}$ is denoted by $\mathfrak{L}^{\tt w}$.

  Controllability and Observability

Definition 1. A behavior $\mathfrak{B}\in \mathfrak{L}^{\tt w}$ is controllable if for all $w_1, w_2 \in \mathfrak{B}$ there exists a $t' \ge 0$ and $w \in \mathfrak{B}$, such that $w(t)=w_1(t)$ for $t < 0$ and $w(t + t')=w_2(t)$ for $t \ge 0$.

Controllability can be characterized algebraically as follows.

Proposition 1. Let $\mathfrak{B}:=\ker~R\left( \frac{d}{dt} \right)$, with $R\in\mathbb{R}^{\bullet \times \tt w}[s]$. $\mathfrak{B}$ is controllable iff $R(\lambda)$ is full row rank for all $\lambda \in \mathbb{C}$. 

Controllable behaviors admit a special representation called image representation.

Proposition 2. Let $\mathfrak{B}\in \mathfrak{L}^{\tt w}$. There exists $\tt z\in\mathbb{N}$ and $M \in  \mathbb{R} ^{\tt w \times z }[s]$ such that $\mathfrak{B}=\left\{ w ~\mid ~ \exists \,z \in\mathfrak{C}^{\infty}(\mathbb{R},\mathbb{R}^{\tt z})\, \, s.t. \, \, w=M \left( \frac{d}{dt} \right) z \right\}$, iff $\mathfrak{B}$ is controllable.

A behavior described by an image representation $w=M \left( \frac{d}{dt} \right) z$ as in Prop. 2, is  denoted by $\mbox{im}~M \left( \frac{d}{dt} \right)$. The auxiliary variable $z$ is called latent variable whose solution space is $\mathfrak{C}^{\infty}(\mathbb{R},\mathbb{R^{\tt z}})$.

Definition 2. Let $\mathfrak{B}\in\mathfrak{L}^{\tt w}$. Partition the external variable as $w=\mbox{col}(w_1,w_2)$. The variable $w_2$ is observable from $w_1$ if for all $\mbox{col}(w_1,w_2),\mbox{col}(w_1,w_2')\in\mathfrak{B}$, it follows that $w_2=w_2'$.

Proposition 3. Let $\mathfrak{B}\in\mathfrak{L}^{\tt w}$ be a controllable behavior described by $w=M \left( \frac{d}{dt} \right) z$. The latent variable $z$ is observable from $w$ iff $M(\lambda)$ is of full column rank for all $\lambda \in \mathbb{C}$.

References:

[1] Polderman, J. W., Willems, J. C., Introduction to mathematical systems theory: a behavioral approach, Springer, 1998.

[2] Willems, J. C., The behavioral approach to open and interconnected systems, IEEE Control Systems, no. 6, pp. 46-99, 2007.