**3. Case of study mathematical model**

In this section, the internal combustion motor is modeled. The idle mode operation condition is considered, which means the vehicle engine is running without accelerating; this is because the original objective is to reduce the fuel consumption when the vehicle is on and stopped, such as waiting for the green of the traffic light when circulating in the city. This fuel economy can be achieved increasing the air ratio of the fuel mixture, but this action causes instability in engine operation, resulting in a variation in the angular speed of the motor shaft. The mathe‐ matical model was compiled from article [7] and it is divided into three parts for better comprehension: (a) manifold chamber, (b) internal combustion chamber, and (c) rotational motion system.

### **3.1. Manifold chamber**

The rate of chance of the pressure in the manifold chamber is affected by the current chamber pressure, the opening position of the throttle valve *d* (*t*), which controls the incoming air mass flow in a proportional way, and the outgoing flow that is proportional to the motor angular velocity *n* (*t*). The output of the chamber is the relative air pressure *p* (*t*). Then, the equation that gives the relationship between these two inputs and the output is given as the first-order differential equation as follows:

$$
\dot{p}\left(t\right) + k\_2 p\left(t\right) = k\_1 d\left(t\right) - k\_3 n\left(t\right) \tag{10}
$$

where *k*1, *k*2, *y k*<sup>3</sup> are the respective proportionality constants. Applying the Laplace transform to previous equation, the following transfer function corresponding to the manifold chamber is obtained:

$$p\left(s\right) = \frac{1}{s + k\_2} [k\_1 d\left(s\right) - k\_3 n\left(s\right)] \tag{11}$$

### **3.2. Internal combustion chamber**

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,

> 0 0 Ï "³ *VT* (w w

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.

In this section, the internal combustion motor is modeled. The idle mode operation condition is considered, which means the vehicle engine is running without accelerating; this is because the original objective is to reduce the fuel consumption when the vehicle is on and stopped, such as waiting for the green of the traffic light when circulating in the city. This fuel economy can be achieved increasing the air ratio of the fuel mixture, but this action causes instability in engine operation, resulting in a variation in the angular speed of the motor shaft. The mathe‐ matical model was compiled from article [7] and it is divided into three parts for better comprehension: (a) manifold chamber, (b) internal combustion chamber, and (c) rotational

The rate of chance of the pressure in the manifold chamber is affected by the current chamber pressure, the opening position of the throttle valve *d* (*t*), which controls the incoming air mass flow in a proportional way, and the outgoing flow that is proportional to the motor angular velocity *n* (*t*). The output of the chamber is the relative air pressure *p* (*t*). Then, the equation that gives the relationship between these two inputs and the output is given as the first-order

*p t k p t kd t kn t*

& ( ) + =- 2 13 ( ) ( ) ( ) (10)

) has at least one member stable for *τ* = 0, then *p*(*s, q, r, e*‒

) (9)

12]. This result will be applied to verify robust stability property.

**Theorem 1:** Suppose that *p*(*s, q, r, e*‒τ*<sup>s</sup>*

154 Robust Control - Theoretical Models and Case Studies

**Proof:** See reference [8]

motion system.

**3.1. Manifold chamber**

differential equation as follows:

) is robustly stable only if the value set satisfies:

**3. Case of study mathematical model**

τ*s*

The combustion chamber produces the necessary torque to move the motor shaft. The torque generation subsystem can be modeled in a simple way having as inputs the spark advance (forward position of the rotor) *a*(*t*), the relative air pressure on the chamber *p*(*t*), the motor angular velocity *n*(*t*), and the fuel flow *f* (*t*). These variables contribute to the torque generation in a linear form. There is a time delay *τd* called *induction-to-power-stroke delay*, which affects the fuel control and the chamber pressure variables; this delay *τd* depends on the motor speed and the number of cylinders activated independently (denoted by *nc*) as given by the following formula:

$$
\tau\_d = \frac{120}{n\_c n(t)} \tag{12}
$$

Thus, the Laplace transform of the engine torque delivery *Te* (*s*) generated for the combustion block is represented by the following equation:

$$T\_e(\mathbf{s}) = e^{-\tau\_d s} \left[ k\_4 p(\mathbf{s}) + k\_5 n(\mathbf{s}) + k\_f f(\mathbf{s}) \right] + k\_6 a(\mathbf{s}) \tag{13}$$

Most of the time the delay is neglected, but when parameter uncertainty is considered, this has an influence on the control system stability, which is the reason this has been taken into account for analysis in this work.

#### **3.3. Shaft rotational dynamics**

Finally, the equations that represent the rotational can be obtained using Newton's second law for rotational movement as follows:

$$J\dot{m}\left(t\right) = T\_e\left(t\right) - T\_L\left(t\right) - k\_7 n\left(t\right) \tag{14}$$

where *J* represents the rotational inertia, *TL* is the torque of external load including its distur‐ bances, and *k*<sup>7</sup> is an attenuation constant of the viscous friction that depends on the tempera‐ ture, type of lubricant, and the gear that the motor uses. Therefore, the Laplace transform of the engine speed *n* (*s*) can be described as follows:

$$m\left(s\right) = \frac{1}{Js + k\_7} \left[T\_e\left(t\right) - T\_L\left(s\right)\right] \tag{15}$$

Then, considering the transfer functions for each subsystem, the complete block diagram representing the mathematical model of the internal combustion motor can be obtained and implemented in Simulink MATLAB, for numerical simulation as shown in **Figure 3**.

**Figure 3.** System block diagram in Simulink MATLAB.

The interest in this case of study is the relation between the variables *d* (*s*) and *n* (*s*) taking into account the time delay; then using block diagram algebra, the corresponding transfer function is given as follows:

$$\log\left(\mathbf{s}\right) = \frac{k\_1 k\_4 e^{-\tau\_d s}}{J\mathbf{s}^2 + \left(k\_2 J + k\_7\right)\mathbf{s} + k\_2 k\_7 + \left[k\_3 k\_4 - k\_2 k\_5 - k\_5 s\right] \mathbf{e}^{-\tau\_d s}}\tag{16}$$

The previous transfer function will be considered as the mathematical model of the internal combustion motor.

### **4. Problem statement**

It is important to mention that the engine mathematical model is expressed in terms of the parameters *ki* and *J*. These parameters depend on the motor operating point; therefore, they cannot be considered as constant in the transfer function and they will be considered as uncertain terms that change depending on the operation point, but around the nominal parameters. The nominal values considered here are the same values as [7]: *k*1 = 3.4329, *k*2 = 0.1627, *k*3 = 0.1139, *k*4 =0.2539, *k*5 = 1.7993, *k*<sup>7</sup> =1.8201, and the inertia *J* = 1. Making a change on the variables, the transfer function on the mathematical model can be simplified as follows:

$$\log\left(s\right) = \frac{a\_1e^{-\tau\_d s}}{s^2 + a\_2s + a\_3 + \left(a\_4s + a\_5\right)e^{-\tau\_d s}}\tag{17}$$

where the uncertain parameters are the ones included in *ai* , the inertia *J* will be considered as a fixed number and equal to 1. Comparing the transfer functions (16) and (17), it can be observed that the new nominal parameters take the following values: *a*<sup>1</sup> = 0.8716, *a*2 = 1.9828, *a*3 = 0.2961, *a*4 = 1.7993, *a*5 = ‒0.2638.

A controller can be connected in the closed-loop system, as shown in **Figure 4**, with two objectives: the first one to regulate the angular speed and the second to improve the perform‐ ance of the process. This controller is given by the following function:

$$c(s) = \frac{50.0194s + 26.3065}{s^2 + 9.8165s + 33.1664} \tag{18}$$

which assigns the system poles of the feedback control, considering a unitary feedback [13], at the following position: {−1, −2, −3, −4} on the complex plane. The controller was designed with the nominal parameters and time delay equal to zero. The characteristic equation of the closed-loop control system considering the uncertain parameters and the time delay has the following structure:

$$d\left(\mathbf{s}, q, r, e^{-T\_d s}\right) = d\left(\mathbf{s}, r\right) + n(\mathbf{s}, q)e^{-\tau\_d s}$$

where

ture, type of lubricant, and the gear that the motor uses. Therefore, the Laplace transform of

Then, considering the transfer functions for each subsystem, the complete block diagram representing the mathematical model of the internal combustion motor can be obtained and

The interest in this case of study is the relation between the variables *d* (*s*) and *n* (*s*) taking into account the time delay; then using block diagram algebra, the corresponding transfer function

1 4


*Js k J k s k k k k k k k s e*

*kk e*

2 7 27 34 25 5 [ ] *d*

The previous transfer function will be considered as the mathematical model of the internal

It is important to mention that the engine mathematical model is expressed in terms of the

cannot be considered as constant in the transfer function and they will be considered as uncertain terms that change depending on the operation point, but around the nominal parameters. The nominal values considered here are the same values as [7]: *k*1 = 3.4329, *k*2 =

and *J*. These parameters depend on the motor operating point; therefore, they

t


*s*

*d*

t

*s*

(16)

*Js k* <sup>=</sup> é ù - ë û <sup>+</sup> (15)

( ) () ( ) 7

implemented in Simulink MATLAB, for numerical simulation as shown in **Figure 3**.

1 *e L ns T t T s*

the engine speed *n* (*s*) can be described as follows:

156 Robust Control - Theoretical Models and Case Studies

**Figure 3.** System block diagram in Simulink MATLAB.

*g s*

( ) ( )

2

is given as follows:

combustion motor.

parameters *ki*

**4. Problem statement**

$$\begin{aligned} d\left(s, r\right) &= s^4 + \left(a\_2 + 9.82\right)s^3 + a\_3 + 9.82a\_2 + \\ \text{(33.17)} &s^2 + \left(9.82a\_3 + 33.17a\_2\right)s + 33.17a\_3 \end{aligned} \tag{19}$$

$$\begin{aligned} m(s, q) &= -a\_4 s^3 + \left( a\_5 - 9.82a\_4 \right) s^2 + (9.82a\_5 - \\ 133.17a\_4 + 50.02a\_1)s + 33.17a\_5 + 26.31a\_1 \end{aligned} \tag{20}$$

The problem considered in this work is to determine the property of robust stability of the internal combustion motor when uncertainty is included in the new nominal parameters *ai* . This property is directly related to the characteristic equation.

**Figure 4.** Control for angular velocity of the internal combustion motor.

### **5. Robust stability verification**

First, the characteristic equation is considered without uncertainty, taking into account the nominal parameters and the time delay, which means:

$$p\left(\mathbf{s}, \mathbf{q}, \mathbf{e}^{-T\_d s}\right) = \mathbf{s}^4 + \eta \mathbf{s}^3 + r\_2 \mathbf{s}^2 + r\_3 \mathbf{s} + r\_4 + \left(q\_1 \mathbf{s}^3 + q\_2 \mathbf{s}^2 + q\_3 \mathbf{s} + q\_4\right) \mathbf{e}^{-\tau\_d s}$$

where *r*1 = 11.80, *r*2 = 52.93, *r*3 = 68.67, *r*4 = 9.82, *q*1 = −1.80, *q*2 = −17.93, *q*3 = −18.67, *q*4 = 14.18.

Using Lemma 1, the corresponding value set for *ω* ∈ 0, .5 y *τ<sup>d</sup>* ∈ 0, 9.9 is obtained, which is represented on the graphic shown in **Figure 5**.

**Figure 5.** Value set without uncertainty in the parameters.

From the previous graphic, it can be appreciated that the value set does not reach the zero of the complex plain, but is very close; thus, it can be determined that the maximum delay the control system supports to preserve the stability is 9.9 s. Several numerical simulations can be run for different time delay limits to appreciate more clearly if the value set reaches the origin, for instance, considering the time delays *τ*<sup>1</sup> =2.5, *τ*<sup>2</sup> =9.9, *τ*<sup>3</sup> =13, which correspond to the stable, oscillatory, and unstable system responses, respectively, as shown in **Figures 6**–**8**, respectively.

**Figure 6.** Transient response for stable behavior.

**Figure 4.** Control for angular velocity of the internal combustion motor.

nominal parameters and the time delay, which means:

represented on the graphic shown in **Figure 5**.

**Figure 5.** Value set without uncertainty in the parameters.

First, the characteristic equation is considered without uncertainty, taking into account the

( ) ( ) 43 2 3 2 1 2 34 1 2 3 4 , , *T s <sup>d</sup> <sup>d</sup> <sup>s</sup> p s q e s rs r s rs r qs q s q s q e* - -

where *r*1 = 11.80, *r*2 = 52.93, *r*3 = 68.67, *r*4 = 9.82, *q*1 = −1.80, *q*2 = −17.93, *q*3 = −18.67, *q*4 = 14.18.

Using Lemma 1, the corresponding value set for *ω* ∈ 0, .5 y *τ<sup>d</sup>* ∈ 0, 9.9 is obtained, which is

= + + + ++ + + +

t

**5. Robust stability verification**

158 Robust Control - Theoretical Models and Case Studies

**Figure 7.** Transient response for oscillatory behavior.

**Figure 8.** Transient response for unstable behavior.

Finally, considering an uncertainty of 10% in each of the parameters, the following character‐ istic equation is obtained, where an overestimation was made to obtain interval polynomials with time delay:

$$p\left(\mathbf{s}, \mathbf{q}, \mathbf{r}, \mathbf{e}^{-T\_4 s}\right) = \mathbf{s}^4 + r\_1 \mathbf{s}^3 + r\_2 \mathbf{s}^2 + r\_3 \mathbf{s} + r\_4 + \left(q\_1 \mathbf{s}^3 + q\_2 \mathbf{s}^2 + q\_3 \mathbf{s} + q\_4\right) \mathbf{e}^{-\mathbf{r}\_d s}$$

where *r*1 = [11.60,11.99],, *r*2 = [50.95, 54.90], *r*3 = [61.80,75.53], *r*4 = [8.84,10.80], *q*1 = −1.98, −1.62], *q*2 = [−19.72, −16.13], *q*3 = −29.25, −8.08], *q*4 = [11.01,17.35].

Once again, using the lemma, the value set of the characteristic equation is obtained now with uncertainty in the coefficients; This can be appreciated in the graphic of the **figure 9**.

**Figure 9.** Value set with uncertainty in the parameters.

The previous value set was obtained for a value range of *ω* ∈ 0, 1.5 , *τ<sup>d</sup>* ∈ 0, 2.8 and like the previous case it can be observed that it barely reaches the zero of the complex plain; therefore, it can be assumed that the maximum time delay the control system supports is 2.8 s. Therefore, it can be clearly appreciated that parameter uncertainty decreases the delay margin that the control system supports and it is important to take this into account when the robust stability property is being verified. It is necessary to note that due to the overestimation about the parameters to represent the characteristic equation as a polynomial delay interval, the results obtained only warranty enough robust stability conditions; nevertheless, the next simulation shows how instability is presented for each of the values contained in the uncertainty, on the control system, when having a delay of *τd* = 4.1 s (see **Figure 10**).

**Figure 10.** Step response with uncertainty in both parameters and time delay.
