Few remarks about behaviors

A brief introduction about myself; I am an electrical engineer with a background in power electronics (topic that I will elaborate later). Due to a fortunate decision I ended up working on a Ph.D. thesis that relies on behavioral system theory. Although in this blog I am mostly interested in discussing engineering applications, I will do my best to maintain the rigorous mathematical exposition and principles of the behavioral setting.

The justification for the development of concepts in behavioral system theory has been extensively argued before in books, articles, magazines, etc., and certainly, in a better way than I could possibly attempt to elaborate here. Hence, what is written here must be considered only as my humble and personal point of view.

* My opinion and ideas are of course open to debate and in fact I will be quite happy to hear about any critique or rebuke to my arguments, so please feel free to drop a comment or contact me with your remarks.

About behaviors

The main idea in the behavioral setting (see my previous post) is to focus on the study of dynamics at the level of trajectories rather than representations. Then the behavior of the system enters into the picture not only as a figure of speech, but as a mathematical object, e.g. for a set of linear differential equations

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

with  $R\in\mathbb{R}^{\bullet \times \tt w}[s]$, we can define the behavior $\mathfrak{B}:=\ker~R\left(\frac{d}{dt}\right)$.

It may seem a bit confusing at this point to talk about a representation $R\left(\frac{d}{dt}\right)w=0$ as a sort of starting point, but let us remember that so far we have discussed systems whose physical laws are described by a set of linear differential equations. Taking this point into account, let us then make the following important remarks.

First off note that a kernel representation is very general: it admits zeroth order equations as well as higher-order ones. Moreover, many other representations can adopt such structure in a straightforward manner. Consider for instance the traditional state space representation:

$\frac{d}{dt}x=Ax+Bu$.

We can define $w:=\mbox{col}(x,u)$ and $R(s):=\begin{bmatrix} sI-A & -B \end{bmatrix}$.

However, it must be clear that we are not forced to use a kernel representation to define a behavior. For instance in the last example the behavior can be simply defined as

$\mathfrak{B}:=\left\{ \mbox{col}(x,u)\in\mathfrak{C}^{\infty}(\mathbb{R},\mathbb{R}^{\bullet})~\mid~ \frac{d}{dt}x=Ax+Bu \right\}$.

There are plenty more representations that can be used to describe the laws of physical systems, for instance the impedance of an n-port driven circuit is modeled as a matrix of rational functions, i.e. $Z(s):=P(s)^{-1}Q(s)$, with $P,Q\in\mathbb{R}^{n\times n}[s]$. In the time domain such an impedance corresponds to the input-output representation $Q\left(\frac{d}{dt}\right)I=P\left(\frac{d}{dt}\right) V$ where $I,V$ are the port- currents and voltages of the circuit. Then

$\mathfrak{B}:=\left\{\mbox{col}(I,V)\in\mathfrak{C}^{\infty}(\mathbb{R},\mathbb{R}^{2n})~\mid~ Q\left(\frac{d}{dt}\right)I=P\left(\frac{d}{dt}\right) V  \right\}$.

As a preliminary conclusion, we can say that the behavior can be defined on the basis of the type of models that is most natural for each application. Moreover, there is no compelling reason to force the use of a particular representation (e.g. input-output descriptions, state-space) if we can study the overall properties of the system directly in terms of trajectories.

There are other sensible reasons why we should consider trajectories as the central object of study rather than representations. For example, the physical laws of a system may be satisfied by different representations as we shall discuss now.

Equivalence of representations

"Appearances can be deceiving."

Let us consider two behaviors: $\mathfrak{B}_1:=\ker~R_1\left(\frac{d}{dt}\right)$ and $\mathfrak{B}_2:=\ker~R_2\left(\frac{d}{dt}\right)$, where $R_1,R_2\in\mathbb{R}^{q\times\tt w}[s]$ correspond to two different kernel representations. We are interested in knowing under which circumstances $\mathfrak{B}_1=\mathfrak{B}_2$. 

Consider $V\in\mathbb{R}^{q\times q}[s]$. Define $R_1(s):=V(s)R_2(s)$, then it is easy to see that all the trajectories in the kernel of $R_2\left(\frac{d}{dt}\right)$ are also trajectories in the kernel of $VR_2\left(\frac{d}{dt}\right)$. Then we conclude that $\mathfrak{B}_2\subseteq\mathfrak{B}_1$.

Now note that if $V$ is unimodular, i.e. $V^{-1}\in\mathbb{R}^{q\times q}[s]$, following the same argument as above, it follows that $\mathfrak{B}_1\subseteq\mathfrak{B}_2$, and consequently $\mathfrak{B}_1=\mathfrak{B}_2$.

We conclude that a kernel representation $R\left(\frac{d}{dt}\right)w=0$ is equivalent to $VR\left(\frac{d}{dt}\right)w=0$ when $V$ is unimodular.

* For an electrical engineer like me this result is striking. Personally, I was accustomed to associate the laws of a given system with a particular set of equations derived from physical principles. However, any set of equations that can be written down to describe the laws of the system, is only one of the many mathematical models that can be used. In other words, the result recalled here suggests that trajectories are indeed something more fundamental than representations.

For further elaboration please refer to:

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

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.