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Models of Dynamic Processes

1.1. Introduction to dynamic processes

1.1.1. Definition, hypotheses and notations

From an input and output perspective, a dynamic process, as illustrated in Figure 1.1, represents a controllable physical system, whose law governing the joint evolution in time of characteristic input variables U(t) and output variables Y(t), for example, can be algebraically modeled by a differential equation [LUE 79].

Figure 1.1. Controllable dynamic process

According to the simplifying hypotheses adopted for the mathematical representation of the dynamic processes studied in this book, these processes have the following traits:

• a) They can be modeled using ordinary differential equations.
• b) They are univariate, or, in other words, U(t) and Y(t) are scalar quantities.
• c) They are of finite order, in which case the highest degree of the differential term (of the output quantity) contained in the characteristic dynamic equation is equal to n, with 1 ≤ n < ∞.
• d) They have constant parameters, and therefore, the response profile to an input signal does not depend on the instant when the input signal is applied.
• e) They are linear in the vicinity of the fixed operating point, which means that the dynamic behavior to be studied is additive and homogeneous in the space of small variations u(t) of U(t) and y(t) of Y(t).
• f) They are deterministic, in the sense that the characteristic quantities of the dynamic behavior are not probabilistic.
• g) They are slightly disturbed, in which case the quantitative effect of an unknown disturbance at the output to be controlled is sufficiently limited and it can consequently be compensated in closed-loop by a robust control strategy.
• h) They can be with or without pure input delay, in which case τ0 ≥ 0. If τ0 > 0, an input quantity applied to the process at instant t has no effect until the previous instant t + τ0. This type of pure input delay phenomenon (or dead time) is quite present in processes in biology, ecology, transportation, signal processing and transmission, etc.

The basic notations employed are defined as follows:

• – main scalar input;
• – scalar output;
• – : rated operating point;
• – u(t), y(t), w(t) : variations of U(t) and Y(t), respectively, around

The following relation can therefore be written:

1.1.2. Implications of hypotheses

Hypotheses (a), (b) and (c) defined in section 1.1.1 entail that the general structure of the differential equations of the dynamic processes considered can be written as:

with pure delay time or dead time of input U.

Therefore, the rated operating point defined by [1.1] is necessarily a solution of [1.2]. Consequently, relation [1.2] leads to relation [1.3]:

Hypothesis (e) stated in section 1.1.1 entails that expression [1.2] can be linearized around . Therefore, a first-order Taylor series expansion of

[1.2], followed by the simplification of the result by replacing the right term of [1.2] with the equivalent left term, leads to the linear differential equation [1.4]:

Hypothesis (e) also entails that the initial conditions of small variations u(t) and y(t) yielded by [1.1] are null, this being due to the additive property in the space of small variations.

1.1.3. Dynamic model: an automation perspective

From the automation perspective, a dynamic model is a new mathematical representation of [1.4], whose algebraic structure can be directly treated by tools available in automation. It is the case of Matlab® [ATH 13]. The two types of dynamic models commonly used are the transfer function and the state model.

1.2. Transfer functions

1.2.1. Existence conditions

The existence conditions of the transfer function of a dynamic process from [1.4] are dictated by hypothesis (d) defined in section 1.1.1. In other words, the parameters of [1.4] are necessarily constant.

1.2.2. Construction

Introducing the notations of small variations in [1.4] yields:

The application of Laplace transform to [1.5] yields:

Then the factorization of terms common to U(s) and Y(s) in [1.6] leads to relation [1.7]:

Thus, in the frequency domain, the linear structure obtained for the small variations in the vicinity of the rated point can be written:

Considering that:

the expression of the transfer function deduced from [1.9] can be written as:

with m ≤ n (feasibility condition).

1.2.3. General structure of a transfer function

The general structure of a transfer function [1.10] of a dynamic process, which is deduced from [1.8] and [1.9], is written as:

1.2.4. Tools for the analysis of the properties of transfer functions

The main specialized tools for the analysis of properties of transfer functions are:

• – step response diagrams, for the measurement of static gain, critical times (rise, response), overshoot, accuracy, etc.;
• – Bode gain and phase diagrams, for the observation and estimation of static gain, cut-off frequencies, resonance frequencies and gain, bandwidth, etc.;
• – Nyquist plot, for the observation and estimation of the gain margin (gain for which the phase is equal to – π), phase margin (phase for which the gain is equal to unity), resonance gain, etc.

These tools and many others are at present integrated into automatic CAD (Computer-Aided Design) tool ranges, such as Sisotool, LTIview and Matlab/Simulink, in order to reduce the computer-aided design and simulation efforts of automatic feedback control systems.

1.2.5. First- and second-order transfer functions

First- and second-order transfer functions have great practical value. Indeed, their characteristic properties are analytically known. Moreover, they can be combined for the synthesis of transfer functions of order higher than 2.

A first-order (without zero) transfer function can be written as:

with:

• – Ks: static gain;
• – τ: time constant.

Therefore, the response y(t) of [1.12] to an input step signal U(s) = E0/s, obtained from the inverse Laplace transform of Y(s) = Gc(s) U(s), can be written as:

For a second-order system, it can be written:

with:

• – Ks: static gain;
• – ωn : natural angular frequency;
• – ξ : damping coefficient.

In this case, the inverse Laplace transform of Y(s) leads to equation [1.15], y(t) for a step input U(s) = E0/s [KAT 90]:

Figure 1.2 presents the examples of graphic profiles of responses to a step E0 (with y(∞) = Ks E0 = 1) of simple dynamic systems. Then Figure 1.3 presents the corresponding Bode diagrams.

Figure 1.2. Step response to simple dynamic models. For a color version of this figure, see www.iste.co.uk/mbihi/automation.zip

Figure 1.3. Bode diagrams of simple dynamic models. For a color version of this figure, see www.iste.co.uk/mbihi/automation.zip

Moreover, Table 1.1 summarizes the basic characteristic properties that result in each case.

Finally, the damped model of n identical poles (without zero) can be written in the following form:

It is important to note that, in experimental modeling, the basis of well-known graphic profiles, summarized in Figure 1.2, is an important source of inspiration when choosing an appropriate structure of representation of a real dynamic process to be modeled. This important remark will be revisited in chapter 2.

Table 1.1. Basic properties of dynamic processes of first and second orders

1.3. State models

1.3.1. Definition

The state X(t) of a dynamic process designates a quantity of information that is sufficient to predict at each instant t its future behavior. Figure 1.4 presents the variables of a dynamic process described in the state...