Dynamic behavior of simple processes refers to how these processes respond to changes in input or disturbance variables over time. The dynamic behavior can be described mathematically using differential equations or transfer functions, and can be visualized using time-domain or frequency-domain analysis.

Some examples of simple processes with different dynamic behaviors are:

First-order systems: A first-order system responds to a change in input with an exponential curve. The output initially increases or decreases rapidly and then gradually approaches a steady-state value. Examples of first-order systems include tanks, pipes, and filters.
Second-order systems: A second-order system has a similar response to a first-order system, but the response is more complex due to the presence of oscillations. The output overshoots the steady-state value before approaching it, and the oscillations can decay slowly or quickly depending on the system’s damping coefficient. Examples of second-order systems include springs, dampers, and pendulums.
Integrating systems: An integrating system responds to a change in input with a linear increase or decrease in output over time. The output does not reach a steady-state value unless the input remains constant. Examples of integrating systems include level control systems and integrators in electrical circuits.
Dead-time systems: A dead-time system has a delay between a change in input and a change in output. The output starts changing only after a certain time delay, and this delay can cause instability or oscillations in the system. Examples of dead-time systems include transportation systems and communication networks.

Understanding the dynamic behavior of a process is essential for designing effective control strategies. It allows for the selection of appropriate controllers, tuning of controller parameters, and prediction of system responses to changes in input or disturbance variables.

Initially, simple processes without a controller are considered and their open-loop behaviour is studied. Let us consider the response of the system to two types of inputs $u(t)$ :

A unit step: $u = 1$ for $t > 0$ , $u = 0$ for $t \leq 0$ ; the response of the system to this input to as the step response.
A Dirac unit impulse: $u = \delta$ (theoretical Dirac); the response of the system to such an input is referred to as an impulse response.

Figure 1.1 [Open-loop block diagram of a process].

Let $G(s)$ be the system transfer function subject to an input $U(s)$ . Except for possible time delays, $G(s)$ is a rational fraction. For any physical input (impulse, step, ramp, sinusoidal, …), $U(s)$ can also be expressed as rational fraction, therefore

(1) $\begin{equation*} G(s) = \frac{N_{g}(s)}{D_{g}(s)} \hspace{2cm} U(s) = \frac{N_{u}(s)}{D_{u}(s)} \end{equation*}$

The Laplace transform $Y(s)$ of the output can be decomposed into

(2) $\begin{equation*} \begin{split} Y(s) = G(s)U(s) = \frac{N_{g}(s) N_{u}(s)}{D_{g}(s) D_{u}(s)} = \frac{N_{1}(s)}{D_{g}(s)} + \frac{N_{2}(s)}{D_{u}(s)} = Y_{n} + Y_{f} \iff \\ \textrm{Response = Natural response + Forced response} \end{split} \end{equation*}$

provided that the product $(G(s)U(s))$ is strictly proper and that denominators $D_{g}(s)$ and the $D_{u}(s)$ have no common roots.

The response $y_{n}(t)$ depends on the modes of $G(s)$ and is called the natural response of the system, while $y_{f}(t)$ depends on the modes of $U(s)$ (linked to the input type) and is referred to as the forced response of the system.

First-Order Systems

A first-order differential equation of the form

(3) $\begin{equation*} \tau \frac{dy}{dt} + y(t) = K_{p} u(t) \end{equation*}$

describes a first-order system. The steady-state gain, or asymptotic gain, of the process is denoted by $K_{p}$ and the time constant by $\tau$ . The corresponding transfer function is

(4) $\begin{equation*} G(s) = \frac{K_{p}}{\tau s + 1}. \end{equation*}$

The block diagram for a first-order system is shown in Figure 1.2.

Figure 1.2 [Block diagram of a first-order system].

When the input $u(t)$ is a step function with amplitude $A$ , its Laplace transform is

(5) $\begin{equation*} U(s) = \frac{A}{s}. \end{equation*}$

Using the transfer function, the Laplace transform of the output $y(t)$ is obtained as

(6) $\begin{equation*} Y(s) = G(s)U(s) = \frac{K_{p}}{\tau s + 1}\frac{A}{s} = \frac{AK_{p}}{s} - \frac{AK_{p}\tau}{\tau s + 1} = Y_{f}(s) + Y_{n}(s), \end{equation*}$

where $Y_{f}(s)$ and $Y_{n}(s)$ are the forced and natural parts of the response, respectively. The time domain response $y(t)$ is

(7) $\begin{equation*} y(t) = AK_{p}(1 - \textrm{exp}(-t/\tau)). \end{equation*}$

The forced and natural responses are given by

(8) $\begin{equation*} y_{f}(t) = AK_{p} \hspace{2cm} y_{n}(t) = -AK_{p}\textrm{exp}(-t/\tau). \end{equation*}$

The asymptotic output, when $t \to \infty$ , is multiplied by the gain of the process $K_{p}$ . A first-order system is also known as a “first-order lag”.

The time constant $\tau$ corresponds to the time necessary for the system response to reach 63.2% of its asymptotic value for a step input. After $2\tau$ , the response reaches 86.5%, and after $5\tau$ , it reaches 99.3% (Table 1.1).

Many real physical systems exhibit first-order dynamics, such as systems storing mass, energy or momentum, or systems exhibiting resistance to the flow of mass, energy or momentum.

Figure 1.3 [Response of a first-order system ( $K_{p} = 1, \tau = 2$ ) to a unit step function].

Table 1.1 [Response of a first-order system to a unit step function expressed in percentage of the asymptotic value].

Integrating Systems

Processes that only contain the first-order derivative of $y(t)$ are known as pure integrator or pure capacitive processes. The equation describing the dynamics of such processes is:

(9) $\begin{equation*} \frac{dy}{dt} = K_{p}u(t) \end{equation*}$

The corresponding transfer function is:

(10) $\begin{equation*} G(s) = \frac{K_{p}}{s} \end{equation*}$

When a step input of magnitude $A$ is applied, the Laplace transform of the output $y(t)$ is:

(11) $\begin{equation*} Y(s) = \frac{AK_{p}}{s^{2}} \end{equation*}$

In the time domain, the response of the system $y(t)$ is given by:

(12) $\begin{equation*} y(t) = AK_{p}t \end{equation*}$

Pure capacitive processes are so-called because they accumulate energy, mass, or electrical charge. A surge tank is an example of a pure capacitive process.

Figure 1.4 [Response of a capacitive system ( $K_{p} = 1$ ) to a unit step function].

Second-Order Systems

A second-order system is described by a second-order differential equation written in the classical form as

(13) $\begin{equation*} \tau^{2}\frac{d^{2}y(t)}{dt^{2}} + 2 \zeta \tau \frac{dy(t)}{dt} + y(t) = K_{p} u(t) \end{equation*}$

with the corresponding transfer function

(14) $\begin{equation*} G(s) = \frac{K_{p}}{\tau^{2}s^{2} + 2 \zeta \tau s + 1} \end{equation*}$

where $\tau$ is the natural period of oscillation of the system which determines the stabilization time of the system, $\zeta$ is the damping coefficient and $K_{p}$ is the steady-state gain of the system.

The notions of natural period of oscillation and of damping factor are related to the sampled or undamped oscillators. For $\zeta = 0$ , the expression (15) shows that the response to a step input oscillates continuously with a frequency $1/\tau$ in radians/time unit.

The transfer function of a second-order system is sometimes written as

(15) $\begin{equation*} G(s) = \frac{K_{p}\omega^{2}_{n}}{s^{2} + 2 \zeta \omega_{n}s + \omega^{2}_{n}} \end{equation*}$

where $\omega_{n} = 1/\tau$ is the natural undamped frequency and $\sigma = \zeta \omega_{n}$ is the damping parameter.

Several real physical processes exhibit second-order dynamics, among them are:

Two first-order systems in series.
Intrinsic second-order systems, e.g. mechanical systems having an acceleration.
Feedback or closed-loop transfer function of a first-order process with a PI controller.

Note that the transfer function $G(s)$ defined by Eq. (14) has two poles, roots of: $\tau^{2} s^{2} + 2 \zeta \tau s + 1 = 0$ , which are equal to

(16) $\begin{equation*} \begin{split} s_{i} = \frac{1}{\tau} (-\zeta \pm \sqrt{\zeta^{2} - 1}) \hspace{2cm} \textrm{if: } \zeta \geq 1 \\ s_{i} = \frac{1}{\tau} (-\zeta \pm \sqrt{1 - \zeta^{2}}) = \omega_{n}(-\zeta \pm j \sqrt{1 - \zeta^{2}}) = -\sigma \pm j \omega_{a} \hspace{2cm} \textrm{if: } 0 \leq \zeta \geq 1 \end{split} \end{equation*}$

If the natural period of oscillation $\tau$ is fixed, then the position of the poles depends only on the damping coefficient $\zeta$ . The shape of the open-loop response to a given input is determined by the location of these poles on the s-plane. For $0 \leq \zeta \geq 1$ , the natural frequency $\omega_{n}$ is equal to the distance of the poles from the origin, the damped frequency $\omega_{a}$ is equal to the distance of the poles from the real axis, and the damping parameter $\sigma$ is equal to the distance of the poles from the imaginary axis.

Figure 1.5 [Normalised response of a second-order system to a unit step function for different values of the damping coefficient $\zeta$ (= 0.25; 1; 1.3 resulting in oscillatory underdamped response to overdamped response) ( $K_{p} = 1, \tau = 1$ )

If the input is a step function with magnitude $A$ , the output Laplace transform is equal to

(17) $\begin{equation*} Y(s) = A \frac{K_{p}}{s(\tau^{2} s^{2} + 2 \zeta \tau s + 1)} \end{equation*}$

which can be decomposed into

(18) $\begin{equation*} Y(s) = \frac{AK_{p}}{s} - \frac{AK_{p}\tau^{2} s + 2 \zeta \tau}{\tau^{2} s^{2} + 2 \zeta \tau s + 1} = Y_{f}(s) + Y_{n}(s) \end{equation*}$

The overall response consists of the forced and the natural responses

(19) $\begin{equation*} y(t) = y_{f}(t) + y_{n}(t) \end{equation*}$

The forced response is equal to

(20) $\begin{equation*} y_{f}(t) = AK_{p} \end{equation*}$

and the overall response is

(21) $\begin{equation*} \begin{split} y(t) = AK_{p}{1 - \textrm{exp}(-\zeta t/\tau)[\textrm{cos}(\frac{\sqrt{1-\zeta^{2}}}{\tau}t) + \frac{\zeta}{\sqrt{1-\zeta^{2}}} \textrm{sin} (\frac{\sqrt{1-\zeta^{2}}}{\tau} t)} \hspace{0.5cm} \textrm{if: } 0 \leq \zeta \geq 1 \\ y(t) = AK_{p}[1 - (1 + \frac{t}{\tau})\textrm{exp}(-t/\tau)] \hspace{0.5cm} \textrm{if: } \zeta = 1 \\ y(t) = AK_{p}{1 - \textrm{exp}(-\zeta t/\tau)[\textrm{cosh}(\frac{\sqrt{\zeta^{2} - 1}}{\tau}t) + \frac{\zeta}{\sqrt{\zeta^{2} - 1}} \textrm{sinh}(\frac{\sqrt{\zeta^{2} - 1}}{\tau}t)} \hspace{0.5cm} \textrm{if: } 1 < \zeta \end{split} \end{equation*}$

The forced response is constant and equal to $AK_{p}$ while the natural response tends towards $0$ when $t \to \infty$ . The natural response takes into account the natural modes of the system and thus depends on the value of $\zeta$ (Figure 1.5):

For $\zeta > 1$ , there will be two real and distinct poles. The response is overdamped (multi-capacitive systems) with no overshoot.
For $\zeta = 1$ , there will be one multiple second-order pole. The response is critically damped, which corresponds to the faster overdamped response.
For $0 < \zeta < 1$ , there will be two complex conjugate poles with negative real art. The response is underdamped. This response is initially faster than the critically damped and overdamped responses, which are sluggish; the drawback is the resulting overshoot.

Figure 1.6 [Response of a second-order system to a unit step input].

With reference to the underdamped response of Fig. 1.6, the following terms are defined:

Overshoot

(22) $\begin{equation*} \textrm{overshoot} = \frac{A'}{B} = \textrm{exp}(\frac{-\pi \zeta}{\sqrt{1 - \zeta^{2}}}) \end{equation*}$

Decay ratio = $C/A'$ = (overshoot) $^{2}$
Natural period of oscillation is defined for a system with a damping coefficient $\zeta$ equal to zero. Such a system oscillates continuously with the natural period $T_{n} = 2 \pi \tau = 2 \pi / \omega_{n}$ and the undamped natural frequency $\omega_{n}$ .
Actual period of oscillation $T_{a}$ , which is the time between two successive peaks, characterised by its damped frequency $\omega_{a}$

(23) $\begin{equation*} \omega_{a} = \omega_{n} \sqrt{1 - \zeta^{2}} = \frac{\sqrt{1 - \zeta^{2}}}{\tau} = \frac{2 \pi}{T_{a}} \end{equation*}$

Rise time: this is the time necessary to reach the asymptotic value for the first time

(24) $\begin{equation*} t_{m} = \frac{1}{\omega_{n}\sqrt{1 - \zeta^{2}}} \textrm{arctg} (-\frac{\sqrt{1 - \zeta^{2}}}{\zeta}) \end{equation*}$

First peak reach time: the time necessary for the response to reach the first peak

(25) $\begin{equation*} t_{p} = \frac{\pi}{\omega_{d}} = \frac{\pi}{\omega_{n}\sqrt{1 - \zeta^{2}}} \end{equation*}$

Settling time: time necessary for the response to remain in an interval between $\pm \epsilon (\pm 5 \%$ , or $\pm 2 \%)$ of the asymptotic value. For $\pm \epsilon = \pm 1\%$ , according to Goodwin and Sin (1984), the settling time is

(26) $\begin{equation*} t_{s} \approx \frac{4.6}{\zeta \omega_{n}} \end{equation*}$

For $(0 < \zeta < 1)$ , the time domain response can be written as

(27) $\begin{equation*} y(t) = AK_{p} - AK_{p}\frac{\omega_{n}}{\omega_{a}}\textrm{exp}(-\sigma t)\textrm(sin)(\omega_{d}t + \theta) \end{equation*}$

with

(28) $\begin{equation*} \theta = \textrm{arccos}\zeta = \textrm{arctan}(\sqrt{\frac{1 - \zeta^{2}}{\zeta}}) = \textrm{arcsin}(\sqrt{1 - \zeta^{2}}) \end{equation*}$

and the envelope of the undamped sinusoidal response is

(29) $\begin{equation*} \textrm{exp}(-\sigma t)\textrm{sin}(\omega_{d} t + \theta) \end{equation*}$

Figure 1.7 [Normalised amplitude ratio for a sinusoidal input with varying damping coefficient $\zeta$ between 0 and 1 per increment of 0.1].

The time domain response of a second-order system subjected to a sinusoidal input: $u(t) = A \textrm{sin}(\omega t)$ , after the transient response decays, will take the form

(30) $\begin{equation*} y_{\infty}(t) = \frac{K_{p}A}{\sqrt{[1 - (\omega \tau)^{2}]^{2} + (2 \zeta \omega \tau)^{2}}} \textrm{sin}(\omega t - \textrm{arctang}[\frac{2\zeta \omega \tau}{1 - (\omega \tau)^{2}}]) \end{equation*}$

and the normalised amplitude ratio is equal to

(31) $\begin{equation*} RA_{n} = \frac{1}{\sqrt{[1 - (\omega \tau)^{2}]^{2} + (2 \zeta \omega \tau)^{2}}} \end{equation*}$

which is maximum at a frequency $\omega_{max}$ given by

(32) $\begin{equation*} \omega_{max} = \sqrt{(1 - 2 \zeta^{2})}/\tau \end{equation*}$

The normalised amplitude ratio has a maximum equal to $1/(2\zeta\sqrt{1-\zeta^{2}})$ for $0 \leq \zeta \leq 0.707$ . This maximum increases very quickly when $\zeta$ becomes small.

Large oscillations are not desired, therefore small damping coefficients $\zeta$ must be avoided. In controlled processes, a damping coefficient around $\zeta = 1$ or $\zeta = 0.7$ (low overshoot, fast response), is often recommended.

CT.03 / Dynamic Behaviour of Simple Processes +

First-Order Systems

Integrating Systems

Second-Order Systems