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4 min readMany definitions of a system are available, ranging from loose descriptions to strict mathematical formulations. In what follows, a system is considered to be an object in which different variables interact at all kinds of time and space scales and that produces observable signals. These types of systems are also called open systems. A graphical representation of a general open system (S) with vector-valued input and output signals is represented in Fig. 11.2. Thus, multiple inputs or outputs are combined in one single arrow. So, the system variables may be scalars or vectors. In addition, they can be continuous or discrete functions of time. It is important to stress that the arrows in Fig. 11.2 represent signal flows and thus not necessarily physical flows.

It is also possible to connect systems into a network, as in an AP system, with parallel, feedback and feedforward paths. Figure 11.3 presents an example of such a network.

For controller/management analysis and synthesis, it is often convenient to connect the system (S) to the controller or management strategy (C), as in Fig. 11.4. Most often the input to the controller or management strategy is the external steering signal of the controlled system, and the output of the system is the observed system's behaviour.

Fig. 11.2 General open system representation

Fig. 11.3 Open system network representation

Fig. 11.4 Controlled system

Fig. 11.5 Model-based controlled system

Finally, to emphasize the incorporation of a mathematical model (__M__) into the controller structure or management strategy, the following model-based controlled system representation is introduced (Fig. 11.5).

For now, it suffices to present the block diagram representation. In subsequent sections, the modelling of AP systems will be worked out in more detail.

In systems theory the basic structure of a mathematical model (__M__) is schematically represented as in Fig. 11.6. In Fig. 11.6, x is the so-called state of the system, u the control input, y the output, w the disturbance input and v the output noise. In general, each of these variables is vector-valued.

Fig. 11.6 Basic structure of mathematical model (M)

In continuous time, the following set of equations describes a general dynamic model (M), with parameter vector p, in what is called state-space form:

$\frac{dx(t)}{dt}=f(t,x(t),u(t),w(t);p),\ \ \ x(0)=x_0$ (11.1)

$y(t)=g(t,x(t),u(t);p)+v(t),\ \ \ t\in \Re^+$ (11.1)

where the first equation describes the nonlinear and time-varying dynamics of the system in terms of state variables (x) and the second one expresses the algebraic relationship between __u__, __x__ and __y__. This state-space model representation has been a starting point for many software implementations for design, control and estimation. In what follows, however, only deterministic models, thus without the stochastic vectors v and w, are considered. Let us illustrate this theory on a fish tank system.

**Example: Fish Tank System**

Consider the following fish tank, which is a typical example of the general system presented in Fig. 11.7.

Let us start with specifying our prior knowledge of the internal system mechanisms. The following mass balance can be defined in terms of the volume of the storage tank (__V__), also called the state of the system, inflows __u(t)__ and outflows __y(t)__:

$\frac{dV(t)}{dt}=u(t)-y(t)$ (11.2)

Suppose there is a level controller (LC) that keeps the outflow proportional to the volume in the tank. This can be enforced by implementing the following proportional control law,

$Y(t)=KV(t)$ (11.3)

with __K__ a real, positive constant. Hence, after substituting Eq. (11.3) into (11.2), we obtain the following differential equation

$\frac{dV(t)}{dt}+KV(t)=n(t)$ (11.4)

**Fig. 11.7** Fish tank with volume-controlled flow using level controller (LC)

For this specific linear differential equation with constant coefficients, an analytical solution exists and is given by

$y(t)=y(0)e^{-Kt}+\int^t_0Ke^{-K(t-s)}u(s)ds$ (11.5)

under the assumption that u(t) = 0 for t \ 0. From this example it is clear that applying first principles — mass conservation in this case — directly leads to an ordinary differential equation. In state-space format, the model can be represented as

$\frac{dx(t)}{dt}=-Kx(t)+u(t)$ (11.6)

$\ y(t)=Kx(t)$ (11.6)

With $x$ volume, $u$ flow input and $K$ controller gain. Thus, in terms of the general state-space Eq. (11.1), $f(t,x(t),u(t);p) \equiv -Kx(t) + u(t)$ and $g(t,x(t),u(t);p) \equiv Kx(t)$.

For two volume-controlled fish tanks in series with volume V<sub1/sub and Vsub2/sub, and controller gain Ksub1/sub and Ksub2/sub, respectively, two mass balances can be formulated, i.e.

$\frac{dV__1(t)}{dt}=-K__1V_1(t)+u(t)$ (11.7)

$\frac{dV__2(t)}{dt}=K__1V__1(t)-K__2V_2(t)$ (11.7)

In vector-matrix form, and for physical outflow y(t), we can write:

$\frac{d}{dt}\begin{bmatrix} V__1(t) \ V__2(t) \end{bmatrix} = \begin{bmatrix} -K__1 & 0 \ K__1 & -K__2 \end{bmatrix}\begin{bmatrix} V__1(t) \ V_2(t) \end{bmatrix}+\begin{bmatrix} 1\0\end{bmatrix}u(t) $ (11.8)

$y(t)=K__2V__2(t)$ (11.8)

And thus, with $x__1=V__1,x__2=V__2:f(t,x(t),u(t);p)\equiv \begin{bmatrix} -K__1x__1(t)+u(t) \ K__1x__1(t)-K__2x__2(t) \end{bmatrix}$ and $g(t,x(t),u(t);p) \equiv K__2x__2(t)$.

In the next sections, each of the subsystems of the AP system (Fig. 11.1) will be described in more detail.