**2. Preliminary**

In this section, we give the value-set concept, which is widely used to verify robust stability (see references [5, 9, 10]). The methodology that will be used consists of obtaining the value set for the characteristic equation, which results from the interval plants including time delay that are defined using the following definition:

**Definition 1.** A transfer function type interval plant is composed in the following manner:

$$\log(s, q, r) = \frac{n(s, q)}{d(s, r)} = \frac{\sum\_{l=0}^{m} \left\lfloor q\_l^-, q\_l^+ \right\rfloor s^l}{s^n + \sum\_{l=0}^{n-1} \left\lceil r\_l^-, r\_l^+ \right\rceil s^l} \forall q \in \mathcal{Q}, r \in \mathcal{R} \tag{1}$$

Now, if the interval plant includes the time delay, then it can be denoted by the following expression:

on the mathematical model of the physical process, and this does not represent in an exact way its dynamic behavior. This problem has been of great interest for scientific researchers in the last years, because it is desirable to get the best results when the control system is implement‐ ed but depends on these properties. A way of solving this problem is through the considera‐ tion of uncertainty on the mathematical model of the physical process. This uncertainty can be dynamic (see references [1–3]) or parametric (see references [4–6]). This chapter describes the qualitative property analysis of robust stability of a system. A case of study of an internal combustion engine is analyzed considering parametric uncertainty in the mathematical model. This case of study process is taken from references [5, 7], where conditions are obtained to verify

It is important to mention that in these research works several simplifications of the mathe‐ matical model were made, which may affect the real behavior of the system; one simplification performed by the authors is the cancellation of time delay in a part of the mathematical model, which has an influence that affects the stability property. The main contribution presented here is the consideration of the time delay on the mathematical model of an internal combustion engine, which is taken into account to obtain the property of robust stability. The methodology used is based on the application of the value-set concept to the particular case of the internal combustion engine (see reference [5]); to be more precise, it consists of the characterization of the value set of the resulting characteristic equation of the closed-loop system when the controller is connected. This controller assigns the poles in a position previously defined. Using this characteristic and applying the zero exclusion principle [8], it is possible to obtain robust

stability conditions through a visual inspection of a graphic in the complex plane.

This chapter is organized as follows: Section 2 presents the elements used to verify the robust stability, Section 3 provides an abstract regarding obtaining the mathematical model of the internal combustion motor, in Section 4 the problem statement is set, in Section 5 the robust stability tools are used in simulation to verify robustness margin, and finally in Section 6 the

In this section, we give the value-set concept, which is widely used to verify robust stability (see references [5, 9, 10]). The methodology that will be used consists of obtaining the value set for the characteristic equation, which results from the interval plants including time delay

**Definition 1.** A transfer function type interval plant is composed in the following manner:

1 0


å é ù ë û = = "Î Î + å é ù ë û

*m i i ii nn i i ii*

Q R *<sup>q</sup> q r q r <sup>r</sup>* (1)

, (, ) g ,, , (,) ,

*qq s n s <sup>s</sup> d s s rr s*

the robust stability of the control system.

150 Robust Control - Theoretical Models and Case Studies

conclusions of this work are presented.

that are defined using the following definition:

( ) <sup>0</sup>

**2. Preliminary**

$$\mathbf{g}\left(\mathbf{s},\mathbf{q},\mathbf{r},e^{-\tau s}\right) = \mathbf{g}\left(\mathbf{s},\mathbf{q},\mathbf{r}\right)e^{-\tau s}\mathbf{r} \in \left[0,\tau\_{\text{max}}\right] \tag{2}$$

Notice that *m* < *n*, then g(*s*,*q, r*) is a set of strictly proper rational functions, *Q* and *R* are sets that represent the parametric uncertainty and are defined as follows:

$$\mathcal{R} \triangleq \left\{ \mathbf{r} = \left[ r\_1 \cdots r\_{n-1} \right]^T : r\_i^- \le r\_i \le r\_i^+ \right\}$$

$$\mathcal{Q} \triangleq \left\{ \mathbf{q} = \left[ q\_1 \cdots q\_{n-1} \right]^T : q\_i^- \le q\_i \le q\_i^+ \right\}$$

These kinds of sets are called as boxes. Equation (2) represents a type of systems known as interval plants. Both the numerator and the denominator of the transfer function have coefficients of uncertain values, which reside in a closed interval. The main interest in the study of this chapter is the robust stability analysis of feedback control systems, as the one depicted in **Figure 1**, whose main process is represented by an interval plant, with time delay and negative unitary gain feedback. The characteristic equation of this feedback system can be expressed in terms of the numerator, denominator, and time delay as follows:

$$\mathbf{p}(\mathbf{s}, \mathbf{q}, \mathbf{r}, e^{-\tau s}) = \mathbf{d}(\mathbf{s}, \mathbf{r}) + \mathbf{n}(\mathbf{s}, \mathbf{q})e^{-\tau s} \tag{3}$$

**Figure 1.** Uncertain interval plant with time delay in negative feedback.

The term quasi-polynomial is used to describe these types of functions. Note that the above characteristic Eq. (3) is not a single equation but rather a family of an infinite number of characteristic equations. Then, this whole family must be considered if robust stability verification is carried out. This family is defined as follows:

$$P\_{\tau} \triangleq \left\{ \mathbf{p}\left(\mathbf{s}, \mathbf{q}, \mathbf{r}, e^{-\tau s}\right) = \mathbf{q} \in \mathcal{Q}; \mathbf{r} \in \mathcal{R}; \tau \in \left[0, \tau\_{\text{max}}\right] \right\} \tag{4}$$

The robust stability property is guaranteed if and only if the following equation is satisfied:

$$\text{sp(s, q, r, e^{-\tau S})} \neq 0 \text{ } \forall \text{ s } \in \mathbb{C}\_{+} \tag{5}$$

where C+ is used to denote the set of complex numbers with real part positive or equal to zero. From Eq. (3), it can be comprehended that the robust stability property of the dynamic system is very hard to verify using analytical methods, because this equation contains an endless number of equations. The goal of this work is to show a simple method to validate the robust stability property of time delay systems. The contribution of this work is based on the valueset characterization of the family of characteristic equations *Pτ*. The value set is defined here as follows:

**Definition 2**. The value set, denoted as *Vτ* (*ω*), of the characteristic equation of an interval polynomial with time delay is the set of complex numbers obtained by substituting *s* = *jω* in the polynomial:

$$\mathcal{V}\_{\boldsymbol{\tau}}(\boldsymbol{\omega}) \triangleq \{ p \mid \mathbf{s}, \mathbf{q}, \boldsymbol{\tau} \, e^{-\boldsymbol{\tau} \mathbf{s}} \}; \mathbf{s} = \{ \boldsymbol{\omega}, \boldsymbol{\omega} \in \mathbb{R}, \mathbf{q} \in \mathbb{Q}; \boldsymbol{\tau} \in \mathcal{R}; \boldsymbol{\tau} \in [0, \boldsymbol{\tau}\_{\max}] \} \tag{6}$$

where **Q** and **R** represent the set containing all possible values of the uncertainty of the parameters *qi* and *ri* expressed in a vector form as elements of the vector *q* and *r*. It is clear that the value set of *P<sup>τ</sup>* is a set of complex numbers plotted on the complex plane for values of *qi* , *ri* , *ω*, and *τ* inside the defined boundaries. An important result that is applied in this chapter is the characterization of the value set for a characteristic equation as the one considered in the previous definition; this result is represented in references [11] and [12] with the following lemma:

**Lemma 1**. For each frequency *ω*, the value set *Vτ*(*ω*) is formed by octagons, where each one changes their geometry in function of the time delay *τ*. The coordinates in the complex plane of the vertices or corners of each octagon are given by the following formulas:

$$
\omega\_{l+1} = d\_{l+1} \left( jso \right) + n\_k \left( jso \right) e^{-j o \tau} \tag{7}
$$

$$d\nu\_{l+5} = d\_{l+1} \left( jo\nu \right) + \eta\_h \left( jo\nu \right) e^{-jo\sigma} \tag{8}$$

where

*i* = 0,1,2,3, ( )mod 1 <sup>4</sup> *k i* =+ + g

$$h = \left(\gamma + i + 1\right) \text{mod}\_4 + 1$$

The robust stability property is guaranteed if and only if the following equation is satisfied:

where C+ is used to denote the set of complex numbers with real part positive or equal to zero. From Eq. (3), it can be comprehended that the robust stability property of the dynamic system is very hard to verify using analytical methods, because this equation contains an endless number of equations. The goal of this work is to show a simple method to validate the robust stability property of time delay systems. The contribution of this work is based on the valueset characterization of the family of characteristic equations *Pτ*. The value set is defined here

**Definition 2**. The value set, denoted as *Vτ* (*ω*), of the characteristic equation of an interval polynomial with time delay is the set of complex numbers obtained by substituting *s* = *jω* in

where **Q** and **R** represent the set containing all possible values of the uncertainty of the

*ω*, and *τ* inside the defined boundaries. An important result that is applied in this chapter is the characterization of the value set for a characteristic equation as the one considered in the previous definition; this result is represented in references [11] and [12] with the following

**Lemma 1**. For each frequency *ω*, the value set *Vτ*(*ω*) is formed by octagons, where each one changes their geometry in function of the time delay *τ*. The coordinates in the complex plane

of the vertices or corners of each octagon are given by the following formulas:

u

u

1 1 () () *<sup>j</sup> ii k d j nje*

5 1 () () *<sup>j</sup> ii h d j nje*

ww

ww

*i* = 0,1,2,3,

( )mod 1 <sup>4</sup> *k i* =+ + g

the value set of *P<sup>τ</sup>* is a set of complex numbers plotted on the complex plane for values of *qi*

and *ri* expressed in a vector form as elements of the vector *q* and *r*. It is clear that

wt

wt



as follows:

the polynomial:

152 Robust Control - Theoretical Models and Case Studies

parameters *qi*

lemma:

where

(5)

(6)

, *ri* , The term mod4 represents the whole module base four operation, for example, mod4 (3). *γ* can take integer values 0, 1, 2, and 3 depending on *ωτ*, see reference [9]:

$$\gamma \triangleq \begin{cases} 0 & 2n\pi \le o\sigma\tau \le \frac{\pi}{2} + 2n\pi \\ 1 & \frac{\pi}{2} + 2n\pi \le o\tau \le +2n\pi \\ 2 & \pi + 2n\pi \le o\tau \le \frac{3\pi}{2} + 2n\pi \\ 3 & \frac{3\pi}{2} + 2n\pi \le o\tau \le 2\pi + 2n\pi \end{cases}$$

The corresponding Kharitonov polynomials for the numerator and denominator of the interval plant are denoted by *ni* (*s*) y *di* (*s*), respectively.

An example of a particular value set for fixed frequency and time delay is presented in **Figure 2**.

**Figure 2.** Value set for a fixed frequency and time delay.

From the definition given for the value set, one can conclude that it includes all the values that the infinite family *P<sup>τ</sup>* can take when it is evaluated in *s* = *jω*. Then, if the complex plane origin is contained in the value set *Vτ*(*ω*), this means that *Pτ* has roots located over the imaginary axis *jω* for some values of *ω* ∈R. This, in fact, causes instability in the time delay feedback system. Consequently, the value-set technique can be used as an instrument that serves to validate the robust stability property. A question that arises in this point is how to show that *Pτ* does not have roots on the right half plane when a sweep over *jω* is done. The answer to this question is sustained in the following result known as the zero exclusion principle, see references [5, 11, 12]. This result will be applied to verify robust stability property.

**Theorem 1:** Suppose that *p*(*s, q, r, e*‒τ*<sup>s</sup>* ) has at least one member stable for *τ* = 0, then *p*(*s, q, r, e*‒ τ*s* ) is robustly stable only if the value set satisfies:

$$0 \notin V\_T\left(\alpha\right) \forall \alpha \ge 0 \tag{9}$$

### **Proof:** See reference [8]

From the previous theorem, the robust stability problem is transformed into a problem where it only needs to verify that the value-set plot just avoids the zero of the complex plane.
