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The Behavioral Approach to Systems Theory and Control

The behavioral approach is a mathematical framework for discussing mathematical models, in particular dynamical systems. The central concept is the "behavior" which consists of all signals which, according to the model, can occur. Thus the behavior is the essence of what a model expresses. An important feature which distinguishes the behavioral approach from more conventional approaches is that it does not distinguish a priori between inputs and outputs. The behavior treats all system variables on the same footing. This is very

=Introduction=

Systems theory is a branch of applied mathematics that studies the mathematical and conceptual foundations for areas as control, signal processing, circuit theory, etc. It is one of the basic subjects taught in electrical and mechanical engineering and has strong ties with computer science. Historically, systems theory can be seen as an outgrowth of electrical circuit theory and control theory. Systems theory problems generally aim at dynamical systems, that is at problems in which the time evolution plays a central role.

A crucial aspect of modern systems and control theory is that these subjects study open systems, that is, systems that interact with their environment. Open systems are in contrast with closed systems. The theory of closed systems is a very developed part of mathematics with applications reaching from mechanics to chaos. The theory of open systems is neglected in mathematics, notwithstanding the fact that the basic laws of physics, as Newton’s second law, the gas law, and Maxwell’s equations are examples of open systems.

The first problem that one encounters in setting up a mathematical theory of open systems is to come up with a good and general mathematical framework that answers the question: What do we mean by a ‘dynamical system’? The standard models view the interaction of a system with its environment in terms of inputs and outputs. The inputs are the variables through which the environment influences the system variables, while the outputs are the variables through which the system influences the environment. Pictorially this can be illustrated by means of a black box black box with inputs driving the systems, and outputs emanating from the system. Mathematically, the relation between the input $$u$$ and the output $$y$$ is usually expressed by either a transfer function or a state space model.

=Transfer function models=

A transfer function originates from specifying the output $$y:\mathbb{R} \rightarrow \mathbb{R}^p$$ in terms of the input $$u:\mathbb{R} \rightarrow \mathbb{R}^m$$ by means of a convolution

$$y(t) = \int_{-\infty}^t G(t-t')u(t')\, dt'.$$

The impulse response $$G: \mathbb{R}:\mathbb{R}^m \times \mathbb{R}^p$$ specifies the system model. The transfer function is the Laplace transform of the impulse response, and the mathematical theory of systems defined in terms of their transfer function is very much tied in with the intricacies of Laplace transforms. Convolutions are linear operators, and the theory of systems defined in terms of convolution is therefore restricted to linear systems. Attempts to generalize convolutions to nonlinear systems using Volterra series not been particularly successful.

=State space models=

A typical continuous-time state space system takes the form

$$\frac{d}{dt} x = f(x,u), y=h(x,u),$$

with $$u:\mathbb{R} \rightarrow \mathbb{U} \subseteq \mathbb{R}^m$$, $$y:\mathbb{R} \rightarrow \mathbb{Y}\subseteq \mathbb{R}^p $$, and $$x:\mathbb{R} \rightarrow \mathbb{X}\subseteq \mathbb{R}^n $$ respectively the input, the output, and the state as functions of time. In a state space model, the maps $$f:\mathbb{X} \times \mathbb{U} \rightarrow \mathbb{R}^n$$ and $$h:\mathbb{X} \times \mathbb{U} \rightarrow \mathbb{Y}$$ specify the system model. The state $$x$$ is computed from the input $$u$$ by solving the differential equation

$$\frac{d}{dt} x = f(x,u), x(t_0) = x_0.$$

The output then follows from $$u$$ and $$x$$ by

$$y(t)=h(x(t),u(t)).$$

Thus the output $$y$$ depends not only on the input $$u$$, but also on the initial state $$x_0$$. The initial state serves to parametrize the memory of the system. The fact that the state space model class incorporates the initial state, the memory, combined with the fact that it readily deals with nonlinear models is a major advantage over the model class defined by the convolution integral. Under mild conditions it is possible to write a convolution as a state model. The algorithms that perform this transformation are known as realization algorithms.

Models of the form of convolution or state-space have dominated the fields of signal processing or control throughout the 20-th century. Important names in these development are Heaviside and Wiener for convolution and Kalman for state space.

=The behavioral approach=

A disadvantage of both model classes convolution and state space is that the modeler has to make a choice of what the inputs and the outputs are. In some applications, for example in signal processing, this choice may be quite evident. However, in many situations, especially in models of physical systems, the choice of input and output is rather arbitrary and can be awkward. Indeed, often the same electrical circuit (for example a simple resistor) can be viewed as voltage driven or as current driven. A mechanical system can be force driven or displacement driven. To describe the motion of a mass using Newton’s second law it is natural to view the force as input and the displacement as output, while for a mass subject to a gravitational field, it is more natural to view the displacement as input and the force as output. These shortcomings have motivated the behavioral approach to systems theory and control.

In the behavioral approach, a dynamical system is defined as a triple

$$\Sigma=(\mathbb{T},\mathbb{W},\mathcal{B}),$$

with $$\mathbb{T}\subseteq\mathbb{R}$$ the time set, $$\mathbb{W}$$ the signal space, and $$\mathcal{B}\subseteq \mathbb{W}^\mathbb{T}$$ the behavior. The time set consists of the time instances over which the system evolves. Often, $$\mathbb{T}$$ is taken to be $$\mathbb{R}$$ for continuous-time systems, or $$\mathbb{Z}$$ for discrete-time systems. The signal space $$\mathbb{W}$$ expresses in which set the variables whose time evolution is modeled take on their values. Often, for physical systems, $$\mathbb{W}$$ is a finite dimensional real vector space, or a subset of a finite dimensional real vector space, and for discrete event systems, $$\mathbb{W}$$ is a finite set. As the notation $$\mathcal{B}\subseteq\mathbb{W}^\mathbb{T}$$ indicates ($$\mathbb{W}^\mathbb{T}$$ denotes the set of maps from $$\mathbb{T}$$ into $$\mathbb{W}$$), the behavior is a set of time functions $$w:\mathbb{T}\rightarrow\mathbb{W}$$; $$w\in\mathcal{B}$$ means that the trajectory $$w$$ is compatible with the laws of the system, while $$w\notin\mathcal{B}$$ means that the laws of the system forbid $$w$$ to happen.

It is readily seen that both the convolution and state space models are special cases of a system as a behavior, by taking, for convolution, $$\mathbb{T}=\mathbb{R}, \mathbb{W}=\mathbb{R}^m\times \mathbb{R}^p$$ and

$$\mathcal{B}= \{ (u,y) \ | \ y(t) = \int_{-\infty}^t G(t-t')u(t')\, dt'\}.$$

For the state space model, take $$\mathbb{T}=\mathbb{R}$$, $$\mathbb{W} = \mathbb{U} \times \mathbb{Y}$$ and

$$\mathcal{B}= \{(u,y) \ | \ \text{there exists } x: \mathbb{R} \rightarrow \mathbb{X}: \frac{d}{dt} x = f(x,u), y=h(x,u)\}.$$

The notion of a behavior fits very well the general mathematical language for modeling in which a model consist of a pair $$(\mathcal{U},\mathcal{B})$$ with $$\mathcal{U}$$ the universum of possibilities and $$\mathcal{B}$$ a subset of $$\mathcal{U}$$ called the behavior. Thus before the phenomenon is modeled every outcome in $$\mathcal{U}$$ is deemed possible, while after modeling only the outcomes in $$\mathcal{B}$$ remain. For a dynamical systems, $$\mathcal{U} = \mathbb{W}^\mathbb{T}$$.

=System properties=

System properties are conveniently defined in terms of the behavior. $$\Sigma=(\mathbb{T},\mathbb{W},\mathcal{B})$$ is said to be linear if $$\mathbb{W}$$ is a vector space and $$\mathcal{B}$$ is a linear subspace of $$\mathbb{W}^\mathbb{T}$$. $$\Sigma=(\mathbb{T},\mathbb{W},\mathcal{B})$$ is said to be time-invariant if $$\mathbb{T}=\mathbb{R}, [0,\infty),\mathbb{Z}$$, or $$\mathbb{Z}_+$$ and if $$\sigma^t\mathcal{B} \subseteq\mathcal{B}$$ for all $$t\in\mathbb{T}$$. Here $$\sigma^t$$ denotes the $$t$$-shift, defined by $$\sigma^t(f)(t'):=f(t'+t)$$.

=Linear time-invariant differential systems (LTIDS)=

An especially important class of dynamical systems are the linear time-invariant differential systems. A linear time-invariant differential system (LTIDSs) is a dynamical system $$\Sigma=(\mathbb{R},\mathbb{R}^{\mathtt{w}},\mathcal{B})$$ whose behavior $$\mathcal{B}$$ is the solution set of a system of ordinary differential equations (ODE’s)

$$R\left(\frac{d}{dt}\right) w=0, \text{ with } R\in\mathbb{R}^{\bullet \times {\mathtt{w}}} [\xi].$$

$$R\in\mathbb{R}^{\bullet \times {\mathtt{w}}} [\xi]$$ means that $$R$$ is a matrix of polynomials with real coefficients, with $$ {\mathtt{w}}$$ rows and any finite number of columns. In order to define the corresponding behavior, we need to specify when we consider $$w:\mathbb{R}\rightarrow\mathbb{R}^{\mathtt{w}}$$ to be a solution of the ODE above. For ease of exposition, often infinite differentiable solutions are considered. There are other possibilities, as taking distributional solutions, or solutions in $$\mathcal{L}^{\text{local}}(\mathbb{R},\mathbb{R}^ {\mathtt{w}})$$, and with the ODE’s interpreted in the sense of distributions. The behavior defined by the ODE is

$$\mathcal{B} = \{ w\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R}^ {\mathtt{w}}) \ | \ R\left(\frac{d}{dt}\right) w(t)=0 \text{ for all } t\in\mathbb{R}\}.$$

=Kernel representations=

Obviously, $$\mathcal{B} = \mathtt{kernel}\left(R\left(\frac{d}{dt}\right)\right)$$, with $$R\left(\frac{d}{dt}\right) $$ viewed as a differential operator mapping $$\mathcal{C}^\infty(\mathbb{R},\mathbb{R}^ {\mathtt{w}})$$ to $$\mathcal{C}^\infty(\mathbb{R},\mathbb{R}^\bullet)$$. We call the representation of a system by $$R\left(\frac{d}{dt}\right) w(t)=0$$ a kernel representation of the corresponding dynamical system. There are many other useful representations of the same behavior.