**Robust Control Design for Automotive Applications: A Variable Structure Control Approach**

Benedikt Alt and Ferdinand Svaricek *University of the German Armed Forces Munich Germany*

## **1. Introduction**

24 Will-be-set-by-IN-TECH

72 Challenges and Paradigms in Applied Robust Control

Hoerner, S. F. (1993). *Fluid dynamics drag: theoretical, experimental and statistical information*,

Khalil, H. (2002). *Nonlinear Systems*, 3 edn, Prentice Hall, Michigan State University, East

Matos, A. & Cruz, N. (2009). MARES - navigation, control and on-board software, *in* A. V.

Seshagiri, S. & Khalil, H. K. (2005). Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators, *Automatica* 41(1): 43–54. Singh, A. & Khalil, H. K. (2005). Regulation of nonlinear systems using conditional integrators, *International Journal of Robust and Nonlinear Control* 15(8): 339–362. 930VR Teixeira, F. C., Aguiar, A. P. & Pascoal, A. (2010). Nonlinear adaptive control of an underwater

Triantafyllou, M. S. & Hover, F. S. (2002). Maneuvering and control of marine vehicles,

Wu, N. E., Zhang, Y. M. & Zhou, K. M. (2000). Detection, estimation, and accommodation

Zhang, F. Z. (1997). Quaternions and matrices of quaternions, *Linear Algebra and Its Applications*

Zhang, Y. M. & Jiang, J. (2002). Active fault-tolerant control system against partial actuator failures, *IEEE Proceedings-Control Theory and Applications* 149(1): 95–104.

of loss of control effectiveness, *International Journal of Adaptive Control and Signal*

Gelb, e. A. (1974). *Optimal applied estimation*, MIT Press.

Lansing.

Published by the author, Bakerfield, CA.

Inzartsev (ed.), *Underwater Vehicles*, InTech.

*Processing* 14(7): 775–795.

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towed vehicle, *Ocean Engineering* 37(13): 1193–1220.

*Technical report*, Massachussets Institute of Technology. White, F. M. (2008). *Fluid Mechanics*, 5 edn, McGraw Hill, Boston.

> The steady rise in fuel prices and the increased awareness on climate issues led and still lead to considerable efforts in the development of automotive engines and drivetrains (Guzzella & Sciarretta (2005)). Thus, fuel savings and emission reduction are of general interest and obviously as important as improved riding comfort or driveability.

Fig. 1. Evolution of memory size, computing power and number of calibration parameters of an automotive engine control unit from 1996 to 2009 (ETAS GmbH (2010))

However, it is hard to find a suitable trade-off between all of these requirements and many resulting solutions lead to increased complexity of the vehicle systems. This is in particular true for common automotive combustion engines where the number of free calibration parameters of the corresponding electronic control unit (ECU) software has been increased up to five times during the last fifteen years (see Figure 1). From today's state of the art it takes up to five calibration engineers one whole year to finish all the calibration work on a series-production engine (Reif (2007)). Consequently, this time consuming calibration results in considerable development cost. Since the complexity of future drivetrains (e.g. battery electric vehicles or hybrid electric vehicles) will be drastically increased (Ehsani et al. (2010)) an ongoing rise on development cost is inevitable. However, with this effect cars may become

Torque [Nm]

(right)

design requirements.

system through the control channel.

Speed [1/min]

**2. Sliding mode control and second order sliding mode control**

affine nonlinear single input single output (SISO) systems

Torque [Nm]

Fig. 3. Operating ranges of heuristically tuned PI or PID controller (left) and SMC controller

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 75

et al. (2009b)). For this purpose a short introduction on SMCs and SOSM based controllers will be given. Here, the robustness properties will be analyzed and it will be shown how chattering effects can be alleviated efficiently. Then, the ISC control design task will be outlined and a corresponding simulation model will be introduced and validated on a research vehicle. Finally, the SOSM based control design approach will be applied to the ISC problem. Since the control parameters remain fixed no gain scheduling technique is necessary. Thus, the overall design and calibration efforts are considerably reduced compared to the series-production solution. However, representative nonlinear simulation and experimental results show impressively that the proposed controller is still able to satisfy all current ISC

Sliding mode control theory has attracted great interest among scientists and control engineers within the last decades. The resulting control laws can be applied but are not restricted to

**x**˙(*t*) = **f**(**x**(*t*)) + **g** (**x**(*t*)) *u*(*t*) + **z**(**x**(*t*))

where **<sup>x</sup>** <sup>=</sup> **<sup>x</sup>**(*t*) <sup>∈</sup> **<sup>R</sup>***n*, *<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>** and *<sup>y</sup>* <sup>=</sup> *<sup>y</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>**. The system nonlinearities **<sup>f</sup>** <sup>∈</sup> **<sup>R</sup>***n*, **<sup>g</sup>** <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* and *<sup>h</sup>* <sup>∈</sup> **<sup>R</sup>** are considered to be sufficiently smooth (Bartolini et al. (1998)). The discontinuous structure of these sliding mode controllers allows to switch between different system structures (or components) such that a new type of system motion, called sliding mode, exists in a dedicated manifold *σ* (**x**) = 0. In particular the corresponding system trajectory moves onto this sliding manifold in finite time which leads to better system performance than the asymptotic behaviour of e.g. linear control systems. After reaching the manifold *σ* (**x**) = 0 the system motion is uniquely characterized from the design of the sliding manifold and independent to any of the corresponding subsystems. Thus, once the system trajectory reached the sliding manifold its motion is insensitive to model uncertainties and disturbances that satisfy the so-called matching conditions (see Drazenovic (1969)). Here, the term matching conditions means that all these model uncertainties and disturbances enter the

Regarding the overall control gain of the sliding mode control law the aforementioned robustness properties are easy to understand. As soon as the system trajectory reaches the sliding manifold the corresponding sliding variable *σ* (**x**) is equal to zero. Since *σ* (**x**) appears

*<sup>y</sup>*(*t*) = *<sup>h</sup>* (**x**(*t*)) (1)

Speed [1/min]

unaffordable to many customers in the near future. Thus, novel control design strategies have to be introduced such that today's and future calibration work is minimized.

Fig. 2. Comparison of black box approach (left) and model based approach (right) for control design in ECU software development

Since several years common black box control design approaches are more and more replaced with model based design strategies (see Figure 2). Here, the corresponding control parameters are referred to a single subsystem of the plant and no longer to the entire process. Thus, each parameter has a clear physical meaning and any model uncertainties or unknown load torque disturbances can be systematically incorporated within the control design process. With this strategy a considerable reduction of calibration efforts can be achieved (Schopp et al. (2010)). However, the efforts for the design of a suitable process model have to be taken into account as well, since it is not easy to find a trade-off between model accuracy and complexity. Thus, it becomes clear that model based control design strategies are not the unique solution to minimize the development cost on ECU software. Often the desired reduction of efforts is less than expected. To overcome this major drawback a combination of model based and robust control design strategies is proposed since it is the best way to reduce the modeling and calibration efforts similarly (Alt (2010)).

Among robust control design methods the class of variable structure controllers (especially sliding mode controllers (SMCs)) is well known for their low burden on model accuracy. Regarding the operating range of a common combustion engine it is well known that the operating range of sliding mode control is enlarged compared to conventional solutions with gain scheduling techniques and heuristically tuned PI or PID controllers even if simple linear system models are used for control design (Edwards & Spurgeon (1998)). Hence, the total number of required operating points can be considerably reduced thus leading to less calibration efforts (see Figure 3). Moreover, sliding mode control shows good robustness properties against a wide class of model uncertainties and external disturbances including environmental influences, aging and tolerance effects (Hung et al. (1993); Utkin (1977)).

Due to its discontinuous nature a high frequency oscillation may arise and deteriorate the performance of closed-loop systems with SMCs (Utkin et al. (2009)). These so called chattering effects take usually place if the plant includes actuator dynamics which cannot be neglected (e.g. electromechanical actuators) or if the discretization effects affect the overall system behaviour. To alleviate the chattering phenomenon several control design approaches have been investigated. Among these control design methods second order sliding modes (SOSM) controllers attract great attention since they guarantee excellent robustness properties and even better accuracy compared to conventional SMCs (Alt et al. (2009a); Bartolini et al. (1998); Butt & Bhatti (2009); Khan et al. (2001); Levant (1993)).

In this contribution a SOSM based control strategy will be applied to a typical automotive control design task, namely the idle speed control (ISC) of a spark ignition (SI) engine (Alt 2 Robust Control

unaffordable to many customers in the near future. Thus, novel control design strategies have

**<sup>y</sup>**

**<sup>y</sup>** 1*y*

**<sup>y</sup>**

*y*

**<sup>y</sup>** 2

*u y*

**<sup>y</sup>**

Fig. 2. Comparison of black box approach (left) and model based approach (right) for control

Since several years common black box control design approaches are more and more replaced with model based design strategies (see Figure 2). Here, the corresponding control parameters are referred to a single subsystem of the plant and no longer to the entire process. Thus, each parameter has a clear physical meaning and any model uncertainties or unknown load torque disturbances can be systematically incorporated within the control design process. With this strategy a considerable reduction of calibration efforts can be achieved (Schopp et al. (2010)). However, the efforts for the design of a suitable process model have to be taken into account as well, since it is not easy to find a trade-off between model accuracy and complexity. Thus, it becomes clear that model based control design strategies are not the unique solution to minimize the development cost on ECU software. Often the desired reduction of efforts is less than expected. To overcome this major drawback a combination of model based and robust control design strategies is proposed since it is the best way to reduce the modeling

Among robust control design methods the class of variable structure controllers (especially sliding mode controllers (SMCs)) is well known for their low burden on model accuracy. Regarding the operating range of a common combustion engine it is well known that the operating range of sliding mode control is enlarged compared to conventional solutions with gain scheduling techniques and heuristically tuned PI or PID controllers even if simple linear system models are used for control design (Edwards & Spurgeon (1998)). Hence, the total number of required operating points can be considerably reduced thus leading to less calibration efforts (see Figure 3). Moreover, sliding mode control shows good robustness properties against a wide class of model uncertainties and external disturbances including environmental influences, aging and tolerance effects (Hung et al. (1993); Utkin (1977)). Due to its discontinuous nature a high frequency oscillation may arise and deteriorate the performance of closed-loop systems with SMCs (Utkin et al. (2009)). These so called chattering effects take usually place if the plant includes actuator dynamics which cannot be neglected (e.g. electromechanical actuators) or if the discretization effects affect the overall system behaviour. To alleviate the chattering phenomenon several control design approaches have been investigated. Among these control design methods second order sliding modes (SOSM) controllers attract great attention since they guarantee excellent robustness properties and even better accuracy compared to conventional SMCs (Alt et al. (2009a); Bartolini et al. (1998);

In this contribution a SOSM based control strategy will be applied to a typical automotive control design task, namely the idle speed control (ISC) of a spark ignition (SI) engine (Alt

3 **<sup>y</sup>**

*y*

*u* **<sup>f</sup>**

**f**

**<sup>y</sup> <sup>y</sup>** 2 *f*

**<sup>y</sup>**

**<sup>y</sup>** 1*f*

to be introduced such that today's and future calibration work is minimized.

**<sup>y</sup>** 2

**<sup>y</sup>** 1*y*

3 **<sup>y</sup>**

design in ECU software development

and calibration efforts similarly (Alt (2010)).

Butt & Bhatti (2009); Khan et al. (2001); Levant (1993)).

*y*

Fig. 3. Operating ranges of heuristically tuned PI or PID controller (left) and SMC controller (right)

et al. (2009b)). For this purpose a short introduction on SMCs and SOSM based controllers will be given. Here, the robustness properties will be analyzed and it will be shown how chattering effects can be alleviated efficiently. Then, the ISC control design task will be outlined and a corresponding simulation model will be introduced and validated on a research vehicle. Finally, the SOSM based control design approach will be applied to the ISC problem. Since the control parameters remain fixed no gain scheduling technique is necessary. Thus, the overall design and calibration efforts are considerably reduced compared to the series-production solution. However, representative nonlinear simulation and experimental results show impressively that the proposed controller is still able to satisfy all current ISC design requirements.

#### **2. Sliding mode control and second order sliding mode control**

Sliding mode control theory has attracted great interest among scientists and control engineers within the last decades. The resulting control laws can be applied but are not restricted to affine nonlinear single input single output (SISO) systems

$$\begin{aligned} \dot{\mathbf{x}}(t) &= \mathbf{f}\left(\mathbf{x}(t)\right) + \mathbf{g}\left(\mathbf{x}(t)\right)u(t) + \mathbf{z}(\mathbf{x}(t))\\ y(t) &= h\left(\mathbf{x}(t)\right) \end{aligned} \tag{1}$$

where **<sup>x</sup>** <sup>=</sup> **<sup>x</sup>**(*t*) <sup>∈</sup> **<sup>R</sup>***n*, *<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>** and *<sup>y</sup>* <sup>=</sup> *<sup>y</sup>*(*t*) <sup>∈</sup> **<sup>R</sup>**. The system nonlinearities **<sup>f</sup>** <sup>∈</sup> **<sup>R</sup>***n*, **<sup>g</sup>** <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* and *<sup>h</sup>* <sup>∈</sup> **<sup>R</sup>** are considered to be sufficiently smooth (Bartolini et al. (1998)). The discontinuous structure of these sliding mode controllers allows to switch between different system structures (or components) such that a new type of system motion, called sliding mode, exists in a dedicated manifold *σ* (**x**) = 0. In particular the corresponding system trajectory moves onto this sliding manifold in finite time which leads to better system performance than the asymptotic behaviour of e.g. linear control systems. After reaching the manifold *σ* (**x**) = 0 the system motion is uniquely characterized from the design of the sliding manifold and independent to any of the corresponding subsystems. Thus, once the system trajectory reached the sliding manifold its motion is insensitive to model uncertainties and disturbances that satisfy the so-called matching conditions (see Drazenovic (1969)). Here, the term matching conditions means that all these model uncertainties and disturbances enter the system through the control channel.

Regarding the overall control gain of the sliding mode control law the aforementioned robustness properties are easy to understand. As soon as the system trajectory reaches the sliding manifold the corresponding sliding variable *σ* (**x**) is equal to zero. Since *σ* (**x**) appears

−10 −5 <sup>0</sup> <sup>5</sup> <sup>10</sup> −10

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −10

*<sup>σ</sup>*˙ <sup>=</sup> *<sup>∂</sup> ∂t σ* + *∂ ∂***x**

*<sup>σ</sup>*¨ <sup>=</sup> *<sup>∂</sup> ∂t σ*˙ + *∂ ∂***x**

time [s]

x 1 −10 −5 0

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −20

time [s]

x 1

0

u [

Fig. 5. Nonlinear simulation results for example in (2) with first order sliding mode control law in (3), sampling time *ts* = 20 ms, phase portrait of closed-loop system (up, left), zoom-in of phase portrait (up, right), system states *x*<sup>1</sup> (blue) and *x*<sup>2</sup> (red), (low, left) and control input

Utkin et al. (2009)). The corresponding design of this specific control law which is referred to super twisting algorithm (STA) is briefly discussed in the remainder of this section. For this purpose the class of affine nonlinear SISO systems is considered as already introduced in (1). Additionally, it is assumed that the system trajectory should reach the sliding manifold *σ*˙ = *σ* = 0 in finite time and that the relative degree of this system is one, i.e. the control input appears in the first time derivative of the sliding variable *σ* (**x**). Although this assumption looks restrictive it has been shown in Alt (2010) that many systems in the field of automotive, electric drive or robotic systems fulfill this requirement. Finally, the first and second order

time derivatives of *σ* (**x**) have to be calculated for the following control design steps:

 *φ*(**x**)

From *σ*˙ and *σ*¨ it can be clearly seen that the lumped model uncertainties and external disturbances **z** (**x**) appear within *φ* (**x**) and *γ* (**x**). However, no detailed knowledge of these nonlinear relationships is required for the following control design steps. Instead it turned out to be sufficient to introduce dedicated lower and upper bounds |*φ* (**x**)| < Φ and 0 < Γ*<sup>m</sup>* < *γ* (**x**) < Γ*<sup>M</sup>* on *φ* (**x**) and *γ* (**x**), respectively to cope with the matched model uncertainties and external disturbances where <sup>Φ</sup>, <sup>Γ</sup>*m*, <sup>Γ</sup>*<sup>M</sup>* <sup>∈</sup> **<sup>R</sup>**+. Thus, the robustness properties are considered to be similar to those of a closed-loop system with first order sliding mode control law.

*σ* (**f**(**x**) + **g** (**x**) *u* + **z** (**x**)) ,

*σ*˙ (**f**(**x**) + **g** (**x**) *u* + **z** (**x**))

+ *∂ ∂u σ*˙ *γ*(**x**)

*<sup>u</sup>*˙ . (4)

−]

20

x2

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 77

−5

0

x

*u* (low, right)

1, x2 [

−]

10

0

x2

5

10

in the denominator of the overall control gain *k* = *<sup>u</sup> <sup>σ</sup>* this variable is drastically increased. In practice that means that the discontinuous control law acts directly with its maximum but finite control input if the system motion on the sliding manifold is affected. Due to that high gain effect the robustness properties of the sliding mode control system are similar to a closed-loop system with high-gain control law (Khalil (1996)). On the contrary to this class of nonlinear controllers the corresponding sliding mode control input doesn't suffer from unrealistic large control efforts. Instead it is well known that this control input is bounded by a finite value as shown in Figure 4. In the remainder of this section the following second

Fig. 4. Nonlinear single input single output (SISO) system with sliding mode controller, high gain effect with good robustness properties against matched model uncertainties and external disturbances after the system trajectory has reached the sliding manifold

order system

$$
\begin{array}{l}
\dot{\mathfrak{x}}\_1 = \mathfrak{x}\_2 \\
\dot{\mathfrak{x}}\_2 = a^2 \underline{u}
\end{array}
\tag{2}
$$

with *a* > 0 is considered to explain the design of first order and second order sliding mode control laws. First, a so-called first order sliding mode control law (Perruquetti & Barbot (2002)) is given that guarantees the existence and the reachability (Edwards & Spurgeon (1998)) of the sliding motion in the entire state space:

$$\mu\_{\text{smc}} = -\delta \left| \mathbf{x}\_1 \right| \text{sgn} \left( \sigma \left( \mathbf{x}\_1, \mathbf{x}\_2 \right) \right) = \begin{cases} \delta \left| \mathbf{x}\_1 \right| & \text{for} \quad \sigma \left( \mathbf{x} \right) < 0 \\ -\delta \left| \mathbf{x}\_1 \right| & \text{for} \quad \sigma \left( \mathbf{x} \right) > 0 \end{cases}. \tag{3}$$

As soon as the system trajectory reaches the sliding manifold *σ* (**x**) = 0 the control input *usmc* shows a switching effect with infinite frequency. Of course, this infinite fast switching effect cannot occur in practical applications since each actuator has a limited bandwith and the corresponding control laws are calculated with finite sampling rates. Thus, the intended ideal sliding motion is also not realizable and the system trajectory oscillates around the given manifold as shown in Figure 5. These so-called chattering effects have to be alleviated in practical applications since chattering may lead to high power loss or even damages on the actuators or the overall system (Utkin et al. (2009)). Thus, the alleviation of chattering effects has been also intensively studied in the last decades (Bartolini et al. (1998); Hung et al. (1993); Utkin (1977); Utkin et al. (2009); Young et al. (1999)). Here, the so-called boundary layer approach (Edwards & Spurgeon (1998)) represents an efficient solution for many practical applications. However, it is well known that this alleviation approach suffers from reduced robustness properties since the system trajectory is no longer able to reach the sliding manifold exactly. Instead it can only be guaranteed that the trajectory moves within a dedicated boundary layer around the sliding manifold.

Another interesting approach for the alleviation of chattering effects can be found within the class of second order sliding mode (SOSM) controllers (Bartolini et al. (1998); Levant (1993); 4 Robust Control

In practice that means that the discontinuous control law acts directly with its maximum but finite control input if the system motion on the sliding manifold is affected. Due to that high gain effect the robustness properties of the sliding mode control system are similar to a closed-loop system with high-gain control law (Khalil (1996)). On the contrary to this class of nonlinear controllers the corresponding sliding mode control input doesn't suffer from unrealistic large control efforts. Instead it is well known that this control input is bounded by a finite value as shown in Figure 4. In the remainder of this section the following second

*<sup>u</sup>* **<sup>x</sup>**(*t*) *<sup>y</sup>*(*t*) *<sup>h</sup>*(**x**(*t*)) *<sup>σ</sup> <sup>σ</sup>* <sup>=</sup> *<sup>σ</sup>*(**x**)

*<sup>i</sup> <sup>σ</sup>* <sup>≈</sup> <sup>0</sup> **<sup>x</sup>**˙ <sup>=</sup> **<sup>f</sup>**(**x**) + **<sup>g</sup>**(**x**)*<sup>u</sup>* <sup>+</sup> **<sup>z</sup>**(**x**) *<sup>u</sup>*(*t*)

Fig. 4. Nonlinear single input single output (SISO) system with sliding mode controller, high gain effect with good robustness properties against matched model uncertainties and external disturbances after the system trajectory has reached the sliding manifold

*x*˙1 = *x*<sup>2</sup> ,

with *a* > 0 is considered to explain the design of first order and second order sliding mode control laws. First, a so-called first order sliding mode control law (Perruquetti & Barbot (2002)) is given that guarantees the existence and the reachability (Edwards & Spurgeon

As soon as the system trajectory reaches the sliding manifold *σ* (**x**) = 0 the control input *usmc* shows a switching effect with infinite frequency. Of course, this infinite fast switching effect cannot occur in practical applications since each actuator has a limited bandwith and the corresponding control laws are calculated with finite sampling rates. Thus, the intended ideal sliding motion is also not realizable and the system trajectory oscillates around the given manifold as shown in Figure 5. These so-called chattering effects have to be alleviated in practical applications since chattering may lead to high power loss or even damages on the actuators or the overall system (Utkin et al. (2009)). Thus, the alleviation of chattering effects has been also intensively studied in the last decades (Bartolini et al. (1998); Hung et al. (1993); Utkin (1977); Utkin et al. (2009); Young et al. (1999)). Here, the so-called boundary layer approach (Edwards & Spurgeon (1998)) represents an efficient solution for many practical applications. However, it is well known that this alleviation approach suffers from reduced robustness properties since the system trajectory is no longer able to reach the sliding manifold exactly. Instead it can only be guaranteed that the trajectory moves within a dedicated

Another interesting approach for the alleviation of chattering effects can be found within the class of second order sliding mode (SOSM) controllers (Bartolini et al. (1998); Levant (1993);

**z**(**x**)

*<sup>x</sup>*˙2 <sup>=</sup> *<sup>a</sup>*2*<sup>u</sup>* (2)

<sup>−</sup>*<sup>δ</sup>* <sup>|</sup>*x*1<sup>|</sup> for *<sup>σ</sup>* (**x**) <sup>&</sup>gt; <sup>0</sup> . (3)

*<sup>δ</sup>* <sup>|</sup>*x*1<sup>|</sup> for *<sup>σ</sup>* (**x**) <sup>&</sup>lt; <sup>0</sup>

*<sup>σ</sup>* this variable is drastically increased.

in the denominator of the overall control gain *k* = *<sup>u</sup>*

*u*− *i*

(1998)) of the sliding motion in the entire state space:

boundary layer around the sliding manifold.

*usmc* = −*δ* |*x*1| sgn (*σ* (*x*1, *x*2)) =

order system

*u*+

Fig. 5. Nonlinear simulation results for example in (2) with first order sliding mode control law in (3), sampling time *ts* = 20 ms, phase portrait of closed-loop system (up, left), zoom-in of phase portrait (up, right), system states *x*<sup>1</sup> (blue) and *x*<sup>2</sup> (red), (low, left) and control input *u* (low, right)

Utkin et al. (2009)). The corresponding design of this specific control law which is referred to super twisting algorithm (STA) is briefly discussed in the remainder of this section. For this purpose the class of affine nonlinear SISO systems is considered as already introduced in (1). Additionally, it is assumed that the system trajectory should reach the sliding manifold *σ*˙ = *σ* = 0 in finite time and that the relative degree of this system is one, i.e. the control input appears in the first time derivative of the sliding variable *σ* (**x**). Although this assumption looks restrictive it has been shown in Alt (2010) that many systems in the field of automotive, electric drive or robotic systems fulfill this requirement. Finally, the first and second order time derivatives of *σ* (**x**) have to be calculated for the following control design steps:

$$\begin{split} \boldsymbol{\sigma} &= \frac{\partial}{\partial t} \boldsymbol{\sigma} + \frac{\partial}{\partial \mathbf{x}} \boldsymbol{\sigma} \left( \mathbf{f} \left( \mathbf{x} \right) + \mathbf{g} \left( \mathbf{x} \right) \boldsymbol{u} + \mathbf{z} \left( \mathbf{x} \right) \right) \,, \\ \ddot{\boldsymbol{\sigma}} &= \underbrace{\frac{\partial}{\partial t} \boldsymbol{\dot{\sigma}} + \frac{\partial}{\partial \mathbf{x}} \boldsymbol{\dot{\sigma}} \left( \mathbf{f} \left( \mathbf{x} \right) + \mathbf{g} \left( \mathbf{x} \right) \boldsymbol{u} + \mathbf{z} \left( \mathbf{x} \right) \right)}\_{\boldsymbol{\phi}(\mathbf{x})} + \underbrace{\frac{\partial}{\partial \boldsymbol{u}} \boldsymbol{\dot{\sigma}} \boldsymbol{u} \boldsymbol{u}}\_{\boldsymbol{\gamma}(\mathbf{x})} \,. \end{split} \tag{4}$$

From *σ*˙ and *σ*¨ it can be clearly seen that the lumped model uncertainties and external disturbances **z** (**x**) appear within *φ* (**x**) and *γ* (**x**). However, no detailed knowledge of these nonlinear relationships is required for the following control design steps. Instead it turned out to be sufficient to introduce dedicated lower and upper bounds |*φ* (**x**)| < Φ and 0 < Γ*<sup>m</sup>* < *γ* (**x**) < Γ*<sup>M</sup>* on *φ* (**x**) and *γ* (**x**), respectively to cope with the matched model uncertainties and external disturbances where <sup>Φ</sup>, <sup>Γ</sup>*m*, <sup>Γ</sup>*<sup>M</sup>* <sup>∈</sup> **<sup>R</sup>**+. Thus, the robustness properties are considered to be similar to those of a closed-loop system with first order sliding mode control law.

−10 −5 <sup>0</sup> <sup>5</sup> <sup>10</sup> −10

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −10

time [s]

x 1 −10 −5 0

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −20

time [s]

(9)

x 1

0

u [

Fig. 6. Nonlinear simulation results for example in (2) with second order sliding mode control law in (5), sampling time *ts* = 20 ms, phase portrait of closed-loop system (up, left), zoom-in of phase portrait (up, right), system states *x*<sup>1</sup> (blue) and *x*<sup>2</sup> (red), (low, left) and

In this section a mathematical model of the spark ignition (SI) engine is briefly discussed. In the remainder of this contribution this engine model will basically be used as a nonlinear simulation model and thus as virtual engine test rig. It incorporates both the overall system dynamics of the engine and the torque structure of current engine management systems. For modeling purposes of the engine a continuous time mean value modeling approach turned out to be sufficient at idle condition (Guzzella & Sciarretta (2005)). This means, that all internal processes of the engine are spread out over one combustion period and differences from cylinder to cylinder are neglected. Thus, it is sufficient to take only the electronic throttle with its position controller, the intake manifold and the rotational dynamics of the crankshaft

*<sup>α</sup>*˙ *thr* <sup>=</sup> <sup>−</sup> <sup>1</sup>

*<sup>p</sup>*˙*im* <sup>=</sup> *<sup>R</sup>θim Vim*

> *<sup>N</sup>*˙ <sup>=</sup> <sup>30</sup> *πJ*

*τthr*

*αthr* +

where *τthr* represents the time constant of the closed loop behaviour of the electronic throttle. The variables *m*˙ *thr* = *m*˙ *thr*(*pim*, *αthr*,*u*) and *m*˙ *cc* = *m*˙ *cc*(*pim*, *N*) denote the air mass flow rates into the intake manifold and the combustion chamber, respectively. For the calculation of

1 *τthr*

(*m*˙ *thr* − *m*˙ *cc*) ,

(*Tind* − *Tloss* − *Tload*) ,

*αthr*,*<sup>u</sup>* ,

*cc*, *Tign*,*u*(*t* − *τd*)) per combustion cycle the air mass flow

−]

20

x2

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 79

−5

0

control input *u* (low, right)

**3. Nonlinear engine model**

the indicated torque *Tind* = *Tind*(*m*˙ �

x

into account:

1, x2 [

−]

10

0

x2

5

10

For a better general understanding the reduction of the chattering effects can be related to the additional integrator within the well-known form of the super twisting algorithm (Fridman & Levant (2002)) control law

$$\begin{aligned} u\_{\text{sta}} &= u\_{\text{sta},1} + u\_{\text{sta},2} \\ \dot{u}\_{\text{sta},1} &= \begin{cases} -u\_{\text{sta}} & \text{for } |u\_{\text{sta}}| > 1 \\ -\mathcal{W}\text{sgn}\left(\sigma\right) & \text{for } |u\_{\text{sta}}| \le 1 \end{cases} \end{aligned} \tag{5}$$

$$u\_{\text{sta},2} = \begin{cases} -\lambda |\sigma\_0|^\rho \text{sgn}\left(\sigma\right) & \text{for } |\sigma| > \sigma\_0 \\ -\lambda |\sigma|^\rho \text{sgn}\left(\sigma\right) & \text{for } |\sigma| \le \sigma\_0 \end{cases} \tag{6}$$

Thus, the discontinuous first order sliding mode control law in (3) is replaced by a continuous alternative. However, the resulting implementation and calibration efforts are increased with regards to practical applications.

For the calculation of the control gains *W*, *λ* and *ρ* the first order time derivative *u*˙*sta* of the control variable from (5) has to be inserted in the right hand side of the second order time derivative *σ*¨ in (4) where |*usta*| ≤ 1 and |*σ*| ≤ *σ*0:

$$\vartheta = \phi\left(\mathbf{x}\right) - \gamma\left(\mathbf{x}\right) \left(\mathrm{Wsgn}\left(\sigma\right) + \rho\lambda \frac{\vartheta}{\left|\sigma\right|^{1-\rho}}\right) \,. \tag{6}$$

Considering the lower and upper bounds Φ, Γ*<sup>m</sup>* and Γ*<sup>M</sup>* of *φ* (**x**) and *γ* (**x**), the right hand side of *σ*¨ turns from an ordinary differential equation into a differential inclusion (Emelyanov et al. (1996); Levant (1993)):

$$\forall \varepsilon \in [\Gamma\_m \mathcal{W} - \Phi \iota \Gamma\_M \mathcal{W} - \Phi] - [\Gamma\_{m\prime} \Gamma\_M] + \rho \lambda \frac{\dot{\sigma}}{|\sigma|^{1-\rho}} \,. \tag{7}$$

With regards to the calibration of the control gains *W*, *λ* and *ρ* it can be clearly seen from (7) that no unique bounds can be given such that the system trajectory reaches *σ*˙ = *σ* = 0 in finite time. However, with some further dedicated assumptions some more conservative bounds (Fridman & Levant (2002)) on *W*, *λ* and *ρ* can be introduced to satisfy this stringent condition:

$$\begin{aligned} \mathcal{W} &> \frac{\Phi}{\Gamma\_m} \\ \lambda^2 &\ge \frac{4\Phi}{\Gamma\_m^2} \frac{\Gamma\_M (\mathcal{W} + \Phi)}{\Gamma\_m (\mathcal{W} - \Phi)} \\ &0 < \rho \le 0.5 \end{aligned} \tag{8}$$

Here, it has to be noted that the assumptions on these conservative bounds for deriving *W*, *λ* and *ρ* may vary from reference to reference (see Levant (1993; 1998)). In practice, these sufficient conditions on *W*, *λ* and *ρ* are often used to simplify the heuristic calibration process (Bartolini et al. (1999)).

Finally, the introductionary example in (2) is considered to show the efficiency of the super twisting algorithm in terms of chattering alleviation purposes. The corresponding simulation results are depicted in Figure 6 and it can be clearly seen that the system trajectories reach the sliding manifold *σ*˙ = *σ* = 0 in finite time. Additionally, the chattering effects are considerably reduced.

Fig. 6. Nonlinear simulation results for example in (2) with second order sliding mode control law in (5), sampling time *ts* = 20 ms, phase portrait of closed-loop system (up, left), zoom-in of phase portrait (up, right), system states *x*<sup>1</sup> (blue) and *x*<sup>2</sup> (red), (low, left) and control input *u* (low, right)

#### **3. Nonlinear engine model**

6 Robust Control

For a better general understanding the reduction of the chattering effects can be related to the additional integrator within the well-known form of the super twisting algorithm (Fridman

> <sup>−</sup>*usta* for|*usta*<sup>|</sup> <sup>&</sup>gt; <sup>1</sup> −*W*sgn (*σ*) for|*usta*| ≤ 1 ,

Thus, the discontinuous first order sliding mode control law in (3) is replaced by a continuous alternative. However, the resulting implementation and calibration efforts are increased with

For the calculation of the control gains *W*, *λ* and *ρ* the first order time derivative *u*˙*sta* of the control variable from (5) has to be inserted in the right hand side of the second order time

Considering the lower and upper bounds Φ, Γ*<sup>m</sup>* and Γ*<sup>M</sup>* of *φ* (**x**) and *γ* (**x**), the right hand side of *σ*¨ turns from an ordinary differential equation into a differential inclusion (Emelyanov et al.

*<sup>σ</sup>*¨ <sup>∈</sup> [Γ*mW* <sup>−</sup> <sup>Φ</sup>, <sup>Γ</sup>*MW* <sup>−</sup> <sup>Φ</sup>] <sup>−</sup> [Γ*m*, <sup>Γ</sup>*M*] <sup>+</sup> *ρλ <sup>σ</sup>*˙

*W* >

*<sup>λ</sup>*<sup>2</sup> <sup>≥</sup>

Φ Γ*<sup>m</sup>* ,

4Φ Γ2 *m*

0 < *ρ* ≤ 0.5 .

Here, it has to be noted that the assumptions on these conservative bounds for deriving *W*, *λ* and *ρ* may vary from reference to reference (see Levant (1993; 1998)). In practice, these sufficient conditions on *W*, *λ* and *ρ* are often used to simplify the heuristic calibration process

Finally, the introductionary example in (2) is considered to show the efficiency of the super twisting algorithm in terms of chattering alleviation purposes. The corresponding simulation results are depicted in Figure 6 and it can be clearly seen that the system trajectories reach the sliding manifold *σ*˙ = *σ* = 0 in finite time. Additionally, the chattering effects are considerably

With regards to the calibration of the control gains *W*, *λ* and *ρ* it can be clearly seen from (7) that no unique bounds can be given such that the system trajectory reaches *σ*˙ = *σ* = 0 in finite time. However, with some further dedicated assumptions some more conservative bounds (Fridman & Levant (2002)) on *W*, *λ* and *ρ* can be introduced to satisfy this stringent condition:

> Γ*<sup>M</sup>* (*W* + Φ) <sup>Γ</sup>*<sup>m</sup>* (*<sup>W</sup>* <sup>−</sup> <sup>Φ</sup>) ,

*<sup>ρ</sup>* sgn (*σ*) for|*σ*<sup>|</sup> <sup>&</sup>gt; *<sup>σ</sup>*<sup>0</sup>

*<sup>W</sup>*sgn (*σ*) <sup>+</sup> *ρλ <sup>σ</sup>*˙


. (6)

<sup>1</sup>−*<sup>ρ</sup>* . (7)

*<sup>ρ</sup>* sgn (*σ*) for|*σ*<sup>|</sup> <sup>≤</sup> *<sup>σ</sup>*<sup>0</sup> . .

(5)

(8)

*usta* = *usta*,1 + *usta*,2 ,

−*λ* |*σ*|

 −*λ* |*σ*0|

*σ*¨ = *φ* (**x**) − *γ* (**x**)

*u*˙*sta*,1 =

*usta*,2 =

& Levant (2002)) control law

regards to practical applications.

(1996); Levant (1993)):

(Bartolini et al. (1999)).

reduced.

derivative *σ*¨ in (4) where |*usta*| ≤ 1 and |*σ*| ≤ *σ*0:

In this section a mathematical model of the spark ignition (SI) engine is briefly discussed. In the remainder of this contribution this engine model will basically be used as a nonlinear simulation model and thus as virtual engine test rig. It incorporates both the overall system dynamics of the engine and the torque structure of current engine management systems. For modeling purposes of the engine a continuous time mean value modeling approach turned out to be sufficient at idle condition (Guzzella & Sciarretta (2005)). This means, that all internal processes of the engine are spread out over one combustion period and differences from cylinder to cylinder are neglected. Thus, it is sufficient to take only the electronic throttle with its position controller, the intake manifold and the rotational dynamics of the crankshaft into account:

*<sup>α</sup>*˙ *thr* <sup>=</sup> <sup>−</sup> <sup>1</sup> *τthr αthr* + 1 *τthr αthr*,*<sup>u</sup>* , *<sup>p</sup>*˙*im* <sup>=</sup> *<sup>R</sup>θim Vim* (*m*˙ *thr* − *m*˙ *cc*) , *<sup>N</sup>*˙ <sup>=</sup> <sup>30</sup> *πJ* (*Tind* − *Tloss* − *Tload*) , (9)

where *τthr* represents the time constant of the closed loop behaviour of the electronic throttle. The variables *m*˙ *thr* = *m*˙ *thr*(*pim*, *αthr*,*u*) and *m*˙ *cc* = *m*˙ *cc*(*pim*, *N*) denote the air mass flow rates into the intake manifold and the combustion chamber, respectively. For the calculation of the indicated torque *Tind* = *Tind*(*m*˙ � *cc*, *Tign*,*u*(*t* − *τd*)) per combustion cycle the air mass flow rate into the combustion chamber has to be related to the crank-angle domain based software features of the electronic control unit:

$$
\dot{m}\_{\rm cc}^{\prime} = \frac{120}{N\_{\rm cc}N} \dot{m}\_{\rm cc} \,. \tag{10}
$$

Additionally, the physical actuator inputs (throttle position *αthr*,*<sup>u</sup>* and ignition setting *αign*,*SP*) are transformed into torque demands *Tair*,*<sup>u</sup>* and *Tign*,*<sup>u</sup>* on the air path and on the ignition path, respectively. In general the torque demand *Tign*,*<sup>u</sup>* is considered as only control input acting directly on the indicated torque *Tind* and hence on the engine speed *N*. The remaining control input *Tair*,*<sup>u</sup>* on the air path influences however the maximum brake torque *Tbas* = *Tbas*(*m*˙ *cc*, *N*). Thus both control inputs affect also the torque reserve

$$T\_{res} = T\_{bas} - T\_{ind} \,. \tag{11}$$

*u*<sup>1</sup> = *Tign*,*<sup>u</sup>*

*x*<sup>3</sup> = *N*

*x*˙3 *y*<sup>1</sup> = *N*

*h*21(*x*2, *x*3, *u*2)

*h*22(*x*2, *x*3, *u*1)

*y*<sup>2</sup> = *Tres*

−

*x*<sup>2</sup> = *pim*

*x*<sup>1</sup> = *αthr x*˙1

*x*˙2

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 81

virtual test rig for the simulation studies. To show the performance of the proposed modeling approach a validation process has been carried out on a series-production vehicle with a 2.0l SI engine and a common rapid control prototyping system. Since the validation should cover the whole idle operating range different engine speed setpoints have to be considered. In Figure 9 and 10 two representative examples are shown where the corresponding engine speed setpoint *NSP* = 800 1/min is situated in the middle of the idle operating range. For identification purposes a step in the torque demand *Tair*,*<sup>u</sup>* on the air path and a step in the torque demand *Tign*,*<sup>u</sup>* on the ignition path are applied to the system. In the first case the maximum torque *Tbas* of the engine is increased while the indicated torque *Tind* remains nearly the same. Due to the unidirectional coupling the engine speed *N* is not affected. In the second case the engine speed *N* and the torque reserve are both affected due to the step demand on the control input *Tign*,*u*. From both Figures it can be also seen that there exists a good matching

In this section a decoupling controller is proposed that will be able to hold the engine speed *N* and the torque reserve *Tres* at their reference values *NSP* and *Tres*,*SP*, respectively. Whenever the engine runs at idle condition and the reference value of the torque reserve *Tres*,*SP* is greater than zero, this ISC controller will be active. The corresponding control structure is shown in Figure 11. Here, it can be seen that the novel ISC controller includes two individual feedback

First, the design of the decoupling compensation is shown which will improve the driver's impression on the engine quality. In particular he should not registrate any influence on the engine speed *N* when changes in the reference value of the torque reserve *Tres*,*SP* occur. As seen in (12) the unilateral coupling between the control inputs *Tign*,*u*, *Tair*,*<sup>u</sup>* and the outputs *N* and *Tres* has to be taken into account such that any influence on the engine speed *N* vanishes. This decoupling compensation is based on a linear time invariant (LTI) model that can either

between the outputs of the simulation model and the real plant measurements.

*f*<sup>1</sup> (*x*1, *x*2, *x*3, *u*2)

*f*<sup>21</sup> (*x*2, *x*3)

*f*<sup>31</sup> (*x*2, *x*3)

*f*<sup>32</sup> (*u*1)

Fig. 8. Structure of nonlinear engine model

**4. Idle speed control design**

controllers and a decoupling compensation.

*f*<sup>22</sup> (*x*1)

*u*<sup>2</sup> = *Tair*,*<sup>u</sup>*

Fig. 7. Engine torque over spark ignition setting *αign* with fixed intake manifold mass flow *m*˙ *thr* = *m*˙ *thr*(*pim*, *αthr*,*u*), this characteristic is also known as spark sweep

As seen in Figure 7 the torque reserve *Tres* represents the amount of torque that is available on the ignition path. Hence there exists a unidirectional coupling between the torque demands on the air and the ignition path and the system outputs because the air path is able to adjust the dynamic actuator constraints on the ignition path. With equations (9), (10), (11) and the ECU related software structure from Alt (2010) a nonlinear state space representation can be

derived, where **x** = [*αthr pim N*] *<sup>T</sup>*, **u** = � *Tign*,*<sup>u</sup> Tair*,*<sup>u</sup>* �*T* and **y** = [*N Tres*] *T*:

$$
\begin{bmatrix}
\dot{\mathbf{x}}\_1 \\
\dot{\mathbf{x}}\_2 \\
\dot{\mathbf{x}}\_3
\end{bmatrix} = \begin{bmatrix}
f\_1\left(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{u}\_2\right) \\
f\_{21}\left(\mathbf{x}\_2, \mathbf{x}\_3\right) + f\_{22}\left(\mathbf{x}\_1\right) \\
f\_{31}\left(\mathbf{x}\_2, \mathbf{x}\_3\right) + f\_{32}\left(\mathbf{u}\_1\right)
\end{bmatrix},
$$

$$
\begin{bmatrix}
y\_1 \\ y\_2
\end{bmatrix} = \begin{bmatrix}
\mathbf{x}\_3 \\
h\_{21}\left(\mathbf{x}\_2, \mathbf{x}\_3, \mathbf{u}\_2\right) - h\_{22}\left(\mathbf{x}\_2, \mathbf{x}\_3, \mathbf{u}\_1\right)
\end{bmatrix}.
$$

The structure of the overall nonlinear engine model is shown in Figure 8. Here, it can be clearly seen that there exists a unidirectional coupling between the control inputs *Tign*,*u*, *Tair*,*<sup>u</sup>* and the outputs *N* and *Tres*. In the remainder of this paper the nonlinear model (12) is used as a

Fig. 8. Structure of nonlinear engine model

8 Robust Control

rate into the combustion chamber has to be related to the crank-angle domain based software

Additionally, the physical actuator inputs (throttle position *αthr*,*<sup>u</sup>* and ignition setting *αign*,*SP*) are transformed into torque demands *Tair*,*<sup>u</sup>* and *Tign*,*<sup>u</sup>* on the air path and on the ignition path, respectively. In general the torque demand *Tign*,*<sup>u</sup>* is considered as only control input acting directly on the indicated torque *Tind* and hence on the engine speed *N*. The remaining control input *Tair*,*<sup>u</sup>* on the air path influences however the maximum brake torque *Tbas* = *Tbas*(*m*˙ *cc*, *N*).

*NccN <sup>m</sup>*˙ *cc* . (10)

*Tres* = *Tbas* − *Tind* . (11)

*αign*,*SP αign*,*bas αign*

Fig. 7. Engine torque over spark ignition setting *αign* with fixed intake manifold mass flow

As seen in Figure 7 the torque reserve *Tres* represents the amount of torque that is available on the ignition path. Hence there exists a unidirectional coupling between the torque demands on the air and the ignition path and the system outputs because the air path is able to adjust the dynamic actuator constraints on the ignition path. With equations (9), (10), (11) and the ECU related software structure from Alt (2010) a nonlinear state space representation can be

*Tign*,*<sup>u</sup> Tair*,*<sup>u</sup>*

*f*<sup>1</sup> (*x*1, *x*2, *x*3, *u*2) *f*<sup>21</sup> (*x*2, *x*3) + *f*<sup>22</sup> (*x*1) *f*<sup>31</sup> (*x*2, *x*3) + *f*<sup>32</sup> (*u*1)

� *x*<sup>3</sup>

�*T*

*h*<sup>21</sup> (*x*2, *x*3, *u*2) − *h*<sup>22</sup> (*x*2, *x*3, *u*1)

The structure of the overall nonlinear engine model is shown in Figure 8. Here, it can be clearly seen that there exists a unidirectional coupling between the control inputs *Tign*,*u*, *Tair*,*<sup>u</sup>* and the outputs *N* and *Tres*. In the remainder of this paper the nonlinear model (12) is used as a

and **y** = [*N Tres*]

� .

⎤ ⎦ , *T*:

(12)

*cc* <sup>=</sup> <sup>120</sup>

*m*˙ �

features of the electronic control unit:

Thus both control inputs affect also the torque reserve

*Tres*

*m*˙ *thr* = *m*˙ *thr*(*pim*, *αthr*,*u*), this characteristic is also known as spark sweep

*<sup>T</sup>*, **u** = �

> ⎡ ⎣

⎡ ⎣ *x*˙1 *x*˙2 *x*˙3

> � *y*1 *y*2 � =

⎤ ⎦ =

*T*

*Tbas Tind*

derived, where **x** = [*αthr pim N*]

virtual test rig for the simulation studies. To show the performance of the proposed modeling approach a validation process has been carried out on a series-production vehicle with a 2.0l SI engine and a common rapid control prototyping system. Since the validation should cover the whole idle operating range different engine speed setpoints have to be considered. In Figure 9 and 10 two representative examples are shown where the corresponding engine speed setpoint *NSP* = 800 1/min is situated in the middle of the idle operating range. For identification purposes a step in the torque demand *Tair*,*<sup>u</sup>* on the air path and a step in the torque demand *Tign*,*<sup>u</sup>* on the ignition path are applied to the system. In the first case the maximum torque *Tbas* of the engine is increased while the indicated torque *Tind* remains nearly the same. Due to the unidirectional coupling the engine speed *N* is not affected. In the second case the engine speed *N* and the torque reserve are both affected due to the step demand on the control input *Tign*,*u*. From both Figures it can be also seen that there exists a good matching between the outputs of the simulation model and the real plant measurements.

## **4. Idle speed control design**

In this section a decoupling controller is proposed that will be able to hold the engine speed *N* and the torque reserve *Tres* at their reference values *NSP* and *Tres*,*SP*, respectively. Whenever the engine runs at idle condition and the reference value of the torque reserve *Tres*,*SP* is greater than zero, this ISC controller will be active. The corresponding control structure is shown in Figure 11. Here, it can be seen that the novel ISC controller includes two individual feedback controllers and a decoupling compensation.

First, the design of the decoupling compensation is shown which will improve the driver's impression on the engine quality. In particular he should not registrate any influence on the engine speed *N* when changes in the reference value of the torque reserve *Tres*,*SP* occur. As seen in (12) the unilateral coupling between the control inputs *Tign*,*u*, *Tair*,*<sup>u</sup>* and the outputs *N* and *Tres* has to be taken into account such that any influence on the engine speed *N* vanishes. This decoupling compensation is based on a linear time invariant (LTI) model that can either

*Tign*,*<sup>u</sup>*

*N*

−

Engine speed controller

> *G*22(*s*) *G*21(*s*)

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 83

of the range at idle condition. If the behaviour of the nonlinear engine model at this operating point has to be described with a LTI model it is clear that the unidirectional coupling structure

*Tres*(*s*) = *G*21(*s*)*Tair*,*u*(*s*) + *G*22(*s*)*Tign*,*u*(*s*) .

The operating point dependent continuous time transfer functions *G*12(*s*), *G*21(*s*) and *G*22(*s*) are calculated from various step responses using MATLAB's System Identication Toolbox

*<sup>s</sup>* <sup>+</sup> 2.235 ,

*<sup>s</sup>* <sup>+</sup> 4.625 ,

*<sup>s</sup>*<sup>2</sup> <sup>+</sup> 45.03*<sup>s</sup>* <sup>+</sup> 90.9 .

*<sup>G</sup>*22(*s*) = <sup>−</sup>26.14*<sup>s</sup>* <sup>−</sup> 91.07

The parameters of *G*12(*s*), *G*21(*s*) and *G*22(*s*) are calculated numerically using a maximum likelihood criterion. That means the underlying identification algorithm is based on continuous time low order transfer functions and it includes an iterative estimation method that minimizes the prediction errors. From the LTI model in (13) it can be seen that the transfer

*GDs*(*s*) = *<sup>G</sup>*22(*s*)

helps to compensate the influence of the torque demand *Tign*,*<sup>u</sup>* on the torque reserve *Tres* efficiently. Hence, the decoupled system with its inputs *Tign*,*<sup>u</sup>* and *Tair*,*<sup>u</sup>* can be controlled by two feedback controllers which are designed independently of each other. Since the dynamics of the air path are generally much slower than the dynamics on the ignition path a second order lag is additionally introduced to smooth the transient behaviour of the decoupling compensation in (15), see Figure 11. The corresponding damping of this filter and its natural

Second order lag

Torque reserve controller

Fig. 11. Block diagram of the decoupling controller at idle condition

*N*(*s*) = *G*12(*s*)*Tign*,*u*(*s*) ,

*<sup>G</sup>*12(*s*) = 246.4

*<sup>G</sup>*21(*s*) = 4.618

is still conserved. Hence, the LTI model can be written as

frequency have to be determined experimentally.

*NSP*

*Tres*,*SP*

(Ljung (2006)):

function

*Tres*

−

*Tair*,*<sup>u</sup>*

(13)

(14)

−

*<sup>G</sup>*21(*s*) (15)

Fig. 9. Experimental results for validation of the nonlinear engine model, step on the air path torque demand: Control input *Tign*,*<sup>u</sup>* (up, left), control input *Tair*,*<sup>u</sup>* (up, right), engine speed *N* (low, left), torque reserve *Tres* (low, right), experimental results (blue), simulation results (red)

Fig. 10. Experimental results for validation of the nonlinear engine model, step on the ignition path torque demand: Control input *Tign*,*<sup>u</sup>* (up, left), control input *Tair*,*<sup>u</sup>* (up, right), engine speed *N* (low, left), torque reserve *Tres* (low, right), experimental results (blue), simulation results (red)

be derived using analytical linearization or by system identification methods (Ljung (1999)). In many automotive control problems the latter techniques are more common since often no detailed nonlinear mathematical model is available. Instead test rig measurements are easily accessible. For this reason the remainder of the work is also based on identification methods. The resulting LTI models are generally valid in the neighbourhood of given operating points. Here, the required test rig measurements are taken from the validated nonlinear simulation model of (12) for the sake of simplicity. The aforementioned operating point with its reference values for the engine speed *NSP*,0 = 800 1/min and the torque reserve *Tres*,*SP*,0 = 8 Nm represents a good choice for the following control design steps since it is situated in the middle 10 Robust Control

Tair,u [Nm]

T

Fig. 9. Experimental results for validation of the nonlinear engine model, step on the air path torque demand: Control input *Tign*,*<sup>u</sup>* (up, left), control input *Tair*,*<sup>u</sup>* (up, right), engine speed *N* (low, left), torque reserve *Tres* (low, right), experimental results (blue), simulation results (red)

Tair,u [Nm]

T

Fig. 10. Experimental results for validation of the nonlinear engine model, step on the ignition path torque demand: Control input *Tign*,*<sup>u</sup>* (up, left), control input *Tair*,*<sup>u</sup>* (up, right), engine speed *N* (low, left), torque reserve *Tres* (low, right), experimental results (blue),

be derived using analytical linearization or by system identification methods (Ljung (1999)). In many automotive control problems the latter techniques are more common since often no detailed nonlinear mathematical model is available. Instead test rig measurements are easily accessible. For this reason the remainder of the work is also based on identification methods. The resulting LTI models are generally valid in the neighbourhood of given operating points. Here, the required test rig measurements are taken from the validated nonlinear simulation model of (12) for the sake of simplicity. The aforementioned operating point with its reference values for the engine speed *NSP*,0 = 800 1/min and the torque reserve *Tres*,*SP*,0 = 8 Nm represents a good choice for the following control design steps since it is situated in the middle

res [Nm]

res [Nm]

0

0

5

10

5

10

0 5 10

time [s]

0 5 10

time [s]

0 10 20 30

time [s]

0 10 20 30

time [s]

0 5 10

time [s]

0 5 10

time [s]

0 10 20 30

time [s]

0 10 20 30

time [s]

0

0

simulation results (red)

5

10

5

Tign,u [Nm]

N [1/min]

Tign,u [Nm]

N [1/min]

10

Fig. 11. Block diagram of the decoupling controller at idle condition

of the range at idle condition. If the behaviour of the nonlinear engine model at this operating point has to be described with a LTI model it is clear that the unidirectional coupling structure is still conserved. Hence, the LTI model can be written as

$$\begin{aligned} N(\mathbf{s}) &= \mathbf{G}\_{12}(\mathbf{s}) T\_{\mathbf{i}\mathbf{g}\mathbf{n},\mu}(\mathbf{s}) \ . \\\\ T\_{\text{res}}(\mathbf{s}) &= \mathbf{G}\_{21}(\mathbf{s}) T\_{\text{air},\mu}(\mathbf{s}) + \mathbf{G}\_{22}(\mathbf{s}) T\_{\mathbf{i}\mathbf{g}\mathbf{n},\mu}(\mathbf{s}) \ . \end{aligned} \tag{13}$$

The operating point dependent continuous time transfer functions *G*12(*s*), *G*21(*s*) and *G*22(*s*) are calculated from various step responses using MATLAB's System Identication Toolbox (Ljung (2006)):

$$G\_{12}(s) = \frac{246.4}{s + 2.235} \,\text{\,\,s$$

$$G\_{21}(s) = \frac{4.618}{s + 4.625} \,\text{\,\,s}\,\text{\,\,}\,\text{\,}\,\text{\,\,}\,\tag{14}$$

$$G\_{22}(s) = \frac{-26.14s - 91.07}{s^2 + 45.03s + 90.9} \,\text{\,\,s}\,\text{\,\,}\,\text{\,\,}\,\text{\,\,}\,\text{\,\,}\,\,\text{\,\,}\,\,\,$$

The parameters of *G*12(*s*), *G*21(*s*) and *G*22(*s*) are calculated numerically using a maximum likelihood criterion. That means the underlying identification algorithm is based on continuous time low order transfer functions and it includes an iterative estimation method that minimizes the prediction errors. From the LTI model in (13) it can be seen that the transfer function

$$\mathbf{G}\_{\rm Ds}(s) = \frac{\mathbf{G}\_{\rm 22}(s)}{\mathbf{G}\_{\rm 21}(s)} \tag{15}$$

helps to compensate the influence of the torque demand *Tign*,*<sup>u</sup>* on the torque reserve *Tres* efficiently. Hence, the decoupled system with its inputs *Tign*,*<sup>u</sup>* and *Tair*,*<sup>u</sup>* can be controlled by two feedback controllers which are designed independently of each other. Since the dynamics of the air path are generally much slower than the dynamics on the ignition path a second order lag is additionally introduced to smooth the transient behaviour of the decoupling compensation in (15), see Figure 11. The corresponding damping of this filter and its natural frequency have to be determined experimentally.

Here, the variables Γ*N*,*m*<sup>1</sup> and Γ*N*,*M*<sup>1</sup> denote lower and upper limitations of the nonlinear

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 85

Additionally, the variable Φ*N*,1 represents an upper bound for all effects which appear in case

<sup>32</sup>(*u*1) denotes the nominal value of *f*32(*u*1). Hence, the design of the engine speed controller is complete. The design of the torque reserve controller runs similarly to (16). The

> <sup>−</sup>*uTres*,1 for <sup>|</sup>*uTres*,1<sup>|</sup> <sup>&</sup>gt; <sup>1</sup> −*WTres*,1sgn(*σTres*) for |*uTres*,1| ≤ 1 ,

where *σTres* = 0 with *σTres* = *Tres* − *Tres*,*SP* represents the torque reserve related sliding

For the application of the super twisting algorithm it has to be again guaranteed that the

is calculated using the nonlinear relationship from (12) while the corresponding time

*res* <sup>≈</sup> *<sup>∂</sup>h*<sup>2</sup> *∂x*<sup>2</sup>

From (22) it can be clearly seen that the state *x*<sup>1</sup> appears in the nonlinear relationship

*<sup>∂</sup>x*<sup>2</sup> *f*22(*x*1) and thus in the first time derivative of *σTres*. However, to satisfy the relative degree one condition the dynamics of the subordinated electronic throttle control loop *x*˙1 = *f*1(*x*1, *x*2, *x*3, *u*2) in (12) have to be neglected for the following control design steps. This assumption is justified since the time lag of the subordinated throttle control loop is ten times smaller than the remaining ones of the SI engine model. With this simplification the state *x*<sup>1</sup> = *αthr* is assumed to be equal to the control input *αthr*,*SP* of the subordinated closed-loop

Under these conditions the time derivative of the torque reserve related sliding surface is

*∂x*<sup>2</sup>

With this assumption the corresponding system fulfills the relative degree one condition. Thus, the super twisting algorithm can be also applied to the torque reserve controller. Regarding the control gains *WTres*,1, *λTres*,1 und *ρTres*,1 it has to be guaranteed similar to the engine speed controller that starting from an initial value *σTres*,0 at an arbritary time instant

*f*22(*f*

∗(−1) <sup>22</sup> (*u*2)) <sup>−</sup> *<sup>T</sup>*˙

*∂x*<sup>2</sup>

*<sup>f</sup>*22(*x*1) <sup>−</sup> *<sup>T</sup>*˙

*<sup>f</sup>*21(*x*2, *<sup>x</sup>*3) + *<sup>∂</sup>h*<sup>2</sup>

considered system has relative degree one. For this purpose the time derivative

*T*˙

*<sup>f</sup>*21(*x*2, *<sup>x</sup>*3) + *<sup>∂</sup>h*<sup>2</sup>

*<sup>ρ</sup>Tres*,1 sgn(*σTres*) .


corresponding control law includes also an integral and a nonlinear part:

*uTres* = *uTres*,1 + *uTres*,2 ,

*uTres*,2 = −*λTres*,1 |*σTres*|

<sup>0</sup> <sup>&</sup>lt; <sup>Γ</sup>*N*,*<sup>m</sup>* <sup>≤</sup> *<sup>f</sup>*31(*x*2, *<sup>x</sup>*3) <sup>−</sup> *<sup>N</sup>*˙ *SP* <sup>≤</sup> <sup>Γ</sup>*N*,*<sup>M</sup>* . (19)

<sup>32</sup>(*u*1))| ≤ Φ*N*,1 . (20)

(21)

*res*,*SP* . (22)

*res*,*SP* . (24)

*x*˙2 . (23)

relationship *<sup>f</sup>*31(*x*2, *<sup>x</sup>*3) <sup>−</sup> *<sup>N</sup>*˙ *SP*, where

Here, *f* ∗

manifold.

*∂h*<sup>2</sup>

system.

given with

of model uncertainties due to the inversion of *f*32(*u*1):

*u*˙*Tres*,1 =

*<sup>σ</sup>*˙ *Tres* <sup>=</sup> *<sup>∂</sup>h*<sup>2</sup> *∂x*<sup>2</sup>

*<sup>σ</sup>*˙ *Tres* <sup>=</sup> *<sup>∂</sup>h*<sup>2</sup> *∂x*<sup>2</sup>

derivative of *Tres* is simplified to

For the design of both feedback controllers linear control theory would be generally sufficient as shown in current series-production applications or even in Kiencke & Nielsen (2005). Nevertheless, it is well known that classical linear controllers often do their job only in the neighbourhood of an operating point and the control parameters have to be scheduled over the entire operating range. This leads to time-consuming calibration efforts. In this work the potential of sliding mode control theory will be particularly analyzed with regards to reduced calibration efforts. Hence, both feedback controllers are designed using a second order sliding modes (SOSM) control design approach that has already been introduced in Section 2. This so-called super twisting algorithm (STA) has been developed to control systems with relative degree one in order to avoid chattering effects. Furthermore, it does not need any information on the time derivative of the sliding variable. For these reasons the super twisting algorithm has become very popular in recent years and it has been adopted to many real world control applications so far (Alt et al. (2009a); Butt & Bhatti (2009); Perruquetti & Barbot (2002)). In the following steps the control law for the engine speed *N* is derived while the engine runs at idle and the condition *Tres* > 0 holds true. This control law includes two major parts:

$$\begin{aligned} \boldsymbol{u}\_{N} &= \boldsymbol{u}\_{N,1} + \boldsymbol{u}\_{N,2} \\\\ \dot{\boldsymbol{u}}\_{N,1} &= \begin{cases} -\boldsymbol{u}\_{N,1} & \text{for } |\boldsymbol{u}\_{N,1}| > 1 \\ -\mathcal{W}\_{N,1} \text{sgn}(\sigma\_{N}) & \text{for } |\boldsymbol{u}\_{N,1}| \le 1 \end{cases} \end{aligned} \tag{16}$$
 
$$\boldsymbol{u}\_{N,2} = -\boldsymbol{\lambda}\_{N,1} |\sigma\_{N}|^{\rho\_{N,1}} \text{sgn}(\sigma\_{N}) \ . $$

where *σ<sup>N</sup>* = 0 with *σ<sup>N</sup>* = *N* − *NSP* represents the engine speed related sliding manifold.

For the application of the super twisting algorithm it has to be guaranteed that the considered system has relative degree one. For this purpose the time derivative

$$
\dot{\sigma}\_N = f\_{31}(\mathbf{x}\_{2\prime}\mathbf{x}\_3) + f\_{32}(\boldsymbol{\mu}\_1) - \dot{\mathbf{N}}\_{\rm SP} \tag{17}
$$

is calculated using the nonlinear model in (12). Here, it can be clearly seen that the control input *u*<sup>1</sup> appears in *f*32(*u*1) and thus in the first time derivative of *σN*. Thus, the aforementioned relative degree one condition is fulfilled for this case and the super twisting algorithm can be applied. For the calibration of the control gains *WN*,1, *λN*,1 and *ρN*,1 sufficient conditions for finite time convergence to the sliding surface *σ<sup>N</sup>* = 0 are derived in Levant (1993). Here, it is shown that starting from an initial value *σN*,0 at an arbritary time instant *tN*,0 the variable *σ<sup>N</sup>* converges to *σ<sup>N</sup>* = 0 if the following sufficient conditions (Fridman & Levant (2002); Levant (1993; 1998)) on *WN*,1, *λN*,1 and *ρN*,1 are satisfied:

$$\begin{aligned} \mathcal{W}\_{N,1} &> \frac{\Phi\_{N,1}}{\Gamma\_{N,m1}}, \\\\ \lambda\_{N,1}^2 &\ge \frac{4\Phi\_{N,1}}{\Gamma\_{N,m1}^2} \frac{\Gamma\_{N,M1}(\mathcal{W}\_{N,1} + \Phi\_{N,1})}{\Gamma\_{N,m1}(\mathcal{W}\_{N,1} - \Phi\_{N,1})}, \\\\ &0 < \rho\_{N,1} \le 0.5 \end{aligned} \tag{18}$$

12 Robust Control

For the design of both feedback controllers linear control theory would be generally sufficient as shown in current series-production applications or even in Kiencke & Nielsen (2005). Nevertheless, it is well known that classical linear controllers often do their job only in the neighbourhood of an operating point and the control parameters have to be scheduled over the entire operating range. This leads to time-consuming calibration efforts. In this work the potential of sliding mode control theory will be particularly analyzed with regards to reduced calibration efforts. Hence, both feedback controllers are designed using a second order sliding modes (SOSM) control design approach that has already been introduced in Section 2. This so-called super twisting algorithm (STA) has been developed to control systems with relative degree one in order to avoid chattering effects. Furthermore, it does not need any information on the time derivative of the sliding variable. For these reasons the super twisting algorithm has become very popular in recent years and it has been adopted to many real world control applications so far (Alt et al. (2009a); Butt & Bhatti (2009); Perruquetti & Barbot (2002)). In the following steps the control law for the engine speed *N* is derived while the engine runs at idle

and the condition *Tres* > 0 holds true. This control law includes two major parts:

<sup>−</sup>*uN*,1 for <sup>|</sup>*uN*,1<sup>|</sup> <sup>&</sup>gt; <sup>1</sup>

−*WN*,1sgn(*σN*) for |*uN*,1| ≤ 1 ,

(16)

(18)

*<sup>ρ</sup>N*,1 sgn(*σN*) ,

*<sup>σ</sup>*˙ *<sup>N</sup>* <sup>=</sup> *<sup>f</sup>*31(*x*2, *<sup>x</sup>*3) + *<sup>f</sup>*32(*u*1) <sup>−</sup> *<sup>N</sup>*˙ *SP* (17)

where *σ<sup>N</sup>* = 0 with *σ<sup>N</sup>* = *N* − *NSP* represents the engine speed related sliding manifold. For the application of the super twisting algorithm it has to be guaranteed that the considered

is calculated using the nonlinear model in (12). Here, it can be clearly seen that the control input *u*<sup>1</sup> appears in *f*32(*u*1) and thus in the first time derivative of *σN*. Thus, the aforementioned relative degree one condition is fulfilled for this case and the super twisting algorithm can be applied. For the calibration of the control gains *WN*,1, *λN*,1 and *ρN*,1 sufficient conditions for finite time convergence to the sliding surface *σ<sup>N</sup>* = 0 are derived in Levant (1993). Here, it is shown that starting from an initial value *σN*,0 at an arbritary time instant *tN*,0 the variable *σ<sup>N</sup>* converges to *σ<sup>N</sup>* = 0 if the following sufficient conditions (Fridman &

> Γ*N*,*M*1(*WN*,1 + Φ*N*,1) <sup>Γ</sup>*N*,*m*1(*WN*,1 <sup>−</sup> <sup>Φ</sup>*N*,1) ,

*uN* = *uN*,1 + *uN*,2 ,

*uN*,2 = −*λN*,1 |*σN*|

system has relative degree one. For this purpose the time derivative

Levant (2002); Levant (1993; 1998)) on *WN*,1, *λN*,1 and *ρN*,1 are satisfied:

Φ*N*,1 Γ*N*,*m*<sup>1</sup> ,

4Φ*N*,1 Γ2 *N*,*m*1

0 < *ρN*,1 ≤ 0.5 .

*WN*,1 >

*λ*2 *N*,1 ≥

*u*˙ *<sup>N</sup>*,1 =

Here, the variables Γ*N*,*m*<sup>1</sup> and Γ*N*,*M*<sup>1</sup> denote lower and upper limitations of the nonlinear relationship *<sup>f</sup>*31(*x*2, *<sup>x</sup>*3) <sup>−</sup> *<sup>N</sup>*˙ *SP*, where

$$0 < \Gamma\_{N,m} \le f\_{31}(\mathbf{x}\_2, \mathbf{x}\_3) - \dot{N}\_{SP} \le \Gamma\_{N,M} \,. \tag{19}$$

Additionally, the variable Φ*N*,1 represents an upper bound for all effects which appear in case of model uncertainties due to the inversion of *f*32(*u*1):

$$|f\_{32}(f\_{32}^\*(
u\_1))| \le \Phi\_{N,1} \,. \tag{20}$$

Here, *f* ∗ <sup>32</sup>(*u*1) denotes the nominal value of *f*32(*u*1). Hence, the design of the engine speed controller is complete. The design of the torque reserve controller runs similarly to (16). The corresponding control law includes also an integral and a nonlinear part:

$$\begin{aligned} \mu\_{\text{Tres}} &= u\_{\text{Tres},1} + u\_{\text{Tres},2} \\ \dot{\mu}\_{\text{Tres},1} &= \begin{cases} -u\_{\text{Tres},1} & \text{for } |u\_{\text{Tres},1}| > 1 \\ -W\_{\text{Tres},1} \text{sgn}(\sigma\_{\text{Tres}}) & \text{for } |u\_{\text{Tres},1}| \le 1 \end{cases} \\ \mu\_{\text{Tres},2} &= -\lambda\_{\text{Tres},1} \left| \sigma\_{\text{Tres}} \right|^{\rho\_{\text{Tres}}} \text{sgn}(\sigma\_{\text{Tres}}) \ . \end{aligned} \tag{21}$$

where *σTres* = 0 with *σTres* = *Tres* − *Tres*,*SP* represents the torque reserve related sliding manifold.

For the application of the super twisting algorithm it has to be again guaranteed that the considered system has relative degree one. For this purpose the time derivative

$$
\dot{\sigma}\_{\text{Tres}} = \frac{\partial h\_2}{\partial \mathbf{x}\_2} f\_{21}(\mathbf{x}\_2, \mathbf{x}\_3) + \frac{\partial h\_2}{\partial \mathbf{x}\_2} f\_{22}(\mathbf{x}\_1) - \dot{T}\_{\text{res},SP} \,. \tag{22}
$$

is calculated using the nonlinear relationship from (12) while the corresponding time derivative of *Tres* is simplified to

$$
\dot{T}\_{\rm res} \approx \frac{\partial h\_2}{\partial \mathbf{x}\_2} \dot{\mathbf{x}}\_2 \,. \tag{23}
$$

From (22) it can be clearly seen that the state *x*<sup>1</sup> appears in the nonlinear relationship *∂h*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> *f*22(*x*1) and thus in the first time derivative of *σTres*. However, to satisfy the relative degree one condition the dynamics of the subordinated electronic throttle control loop *x*˙1 = *f*1(*x*1, *x*2, *x*3, *u*2) in (12) have to be neglected for the following control design steps. This assumption is justified since the time lag of the subordinated throttle control loop is ten times smaller than the remaining ones of the SI engine model. With this simplification the state *x*<sup>1</sup> = *αthr* is assumed to be equal to the control input *αthr*,*SP* of the subordinated closed-loop system.

Under these conditions the time derivative of the torque reserve related sliding surface is given with

$$\psi\_{\rm Tres} = \frac{\partial h\_2}{\partial \mathbf{x}\_2} f\_{21}(\mathbf{x}\_2, \mathbf{x}\_3) + \frac{\partial h\_2}{\partial \mathbf{x}\_2} f\_{22}(f\_{22}^{\*(-1)}(\mathbf{u}\_2)) - \dot{T}\_{\rm res, SP} \,. \tag{24}$$

With this assumption the corresponding system fulfills the relative degree one condition. Thus, the super twisting algorithm can be also applied to the torque reserve controller.

Regarding the control gains *WTres*,1, *λTres*,1 und *ρTres*,1 it has to be guaranteed similar to the engine speed controller that starting from an initial value *σTres*,0 at an arbritary time instant

0 5 10 15

0 5 10 15

time [s]

0 5 10 15 20

time [s]

0 5 10 15

time [s]

0

5

10

15

T

Fig. 13. Nonlinear simulation and experimental results for super twisting algorithm based decoupling controller, tracking of an engine speed reference step profile: Engine speed *N* (left), torque reserve *Tres* (right), experimental results (blue), simulation results (red)

*NSP* is increased at *t*<sup>1</sup> = 4 s and lowered again at *t*<sup>2</sup> = 14 s. The corresponding simulation results are shown in Figure 13. Regarding the step response of the engine speed *N* it can be clearly seen that no overshoot occurs and the settling times are within less than 2 s and thus reasonable small. Additionally, the torque reserve *Tres* shows only small deviations due to the

T

Fig. 14. Nonlinear simulation and experimental results for super twisting algorithm based decoupling controller, tracking of a torque reserve reference step profile: Engine speed *N* (left), torque reserve *Tres* (right), experimental results (blue), simulation results (red)

res [Nm]

res [Nm]

10

T

Fig. 12. Nonlinear simulation and experimental results for super twisting algorithm based decoupling controller, disturbance rejection properties: Engine speed *N* (left), torque reserve

When disabling the load torque similar effects take place. Considering the engine speed *N* it can also clearly be seen that there exists a good matching between the nonlinear simulation data and the experimental measurements. For the torque reserve *Tres* this matching is less perfect since this variable is much more prone to unmodelled dynamics and tolerance effects that have not been considered in the nonlinear simulation model. This effect will be further evaluated in Section 6. In a second representative scenario the engine speed reference value

res [Nm]

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 87

20

time [s]

*Tres* (right), experimental results (blue), simulation results (red)

0 5 10 15 20

time [s]

0 5 10 15

time [s]

600

700

800

N [1/min]

900

N [1/min]

800

N [1/min]

1000

*tTres*,0 the sliding variable *σTres* converges to *σTres* = 0 in finite time. For this purpose the following sufficient conditions (Fridman & Levant (2002); Levant (1993; 1998)) have to be fulfilled:

$$\begin{aligned} \mathcal{W}\_{\text{Tres},1} &> \frac{\Phi\_{\text{Tres},1}}{\Gamma\_{\text{Tres},m1}},\\ \lambda\_{\text{Tres},1}^2 &\ge \frac{4\Phi\_{\text{Tres},1}}{\Gamma\_{\text{Tres},m1}^2} \frac{\Gamma\_{\text{Tres},M1}(\mathcal{W}\_{\text{Tres},1} + \Phi\_{\text{Tres},1})}{\Gamma\_{\text{Tres},m1}(\mathcal{W}\_{\text{Tres},1} - \Phi\_{\text{Tres},1})},\\ &0 < \rho\_{\text{Tres},1} \le 0.5 \,\text{.} \end{aligned} \tag{25}$$

Here, the variables Γ*Tres*,*m*<sup>1</sup> and Γ*Tres*,*M*<sup>1</sup> denote lower and upper limitations of the nonlinear relationship *<sup>∂</sup>h*<sup>2</sup> *<sup>∂</sup>x*<sup>2</sup> *<sup>f</sup>*21(*x*2, *<sup>x</sup>*3) <sup>−</sup> *<sup>T</sup>*˙ *res*,*SP*:

$$0 < \Gamma\_{\rm Tres,m} \le \frac{\partial h\_2}{\partial \mathbf{x}\_2} f\_{21}(\mathbf{x}\_2, \mathbf{x}\_3) - \mathcal{T}\_{\rm rres,SP} \le \Gamma\_{\rm Tres,M} \,. \tag{26}$$

The variable Φ*Tres*,1 represents similar to Φ*N*,1 an upper bound for all effects which appear due to possible model uncertainties that are related to the inversion of *f*22(*u*2):

$$\left| \frac{\partial \hbar\_2}{\partial \mathbf{x}\_2} f\_{22}(f\_{22}^{\*(-1)}(T\_{air,\mu})) \right| \le \Phi\_{\text{Tres},1} \,\,\,\,\,\,\tag{27}$$

Here, *f* ∗ <sup>22</sup>(*u*2) denotes the nominal value of *f*22(*u*2). Note that, in practice the engine speed control and the torque control loops are affected by model uncertainties and external disturbances leading to imperfect decoupling properties of the multivariable system. Nevertheless, it is well known from literature (Alt et al. (2009a); Bartolini et al. (1999); Levant (1993; 1998)) that the sliding surfaces *σ<sup>N</sup>* = 0 and *σTres* = 0 can still be reached in this case. Thus, the engine speed control and the torque reserve control loops are supposed to be robust against any disturbances due to improper decoupling. Finally, it has been shown in Alt (2010) that this multivariable control design approach leads to better performance and less calibration efforts than a similar approach without decoupling compensation.

#### **5. Nonlinear simulation and experimental results**

This section illustrates the efficiency and the robustness properties of the proposed decoupling controller. For this purpose some representative nonlinear simulation and experimental results are shown. All the simulations are based on the nonlinear engine model of Alt (2010) with a controller sampling time of *ts* = 10 ms. The experimental results include representative field test data with a 2.0l series-production vehicle and a common rapid control prototyping system.

In the first scenario the disturbance rejection properties of the closed-loop system are evaluated. For this purpose an additional load torque of *Tload* = 8 Nm (e.g. power steering) is applied to the engine at *t*<sup>1</sup> = 4 s and removed again at *t*<sup>2</sup> = 9 s. From Figure 12 it can be seen that due to this load torque the engine speed *N* and the torque reserve *Tres* drop below their reference values while the corresponding transients stay below Δ*N* = 40 1/min and Δ*Tres* = 8 Nm, respectively. However, the proposed idle speed controller steers both variables back to their reference values *NSP* = 800 1/min and *Tres*,*SP* = 8 Nm within less than 2 s. 14 Robust Control

*tTres*,0 the sliding variable *σTres* converges to *σTres* = 0 in finite time. For this purpose the following sufficient conditions (Fridman & Levant (2002); Levant (1993; 1998)) have to be

Here, the variables Γ*Tres*,*m*<sup>1</sup> and Γ*Tres*,*M*<sup>1</sup> denote lower and upper limitations of the nonlinear

*<sup>f</sup>*21(*x*2, *<sup>x</sup>*3) <sup>−</sup> *<sup>T</sup>*˙

The variable Φ*Tres*,1 represents similar to Φ*N*,1 an upper bound for all effects which appear

 

<sup>22</sup>(*u*2) denotes the nominal value of *f*22(*u*2). Note that, in practice the engine speed control and the torque control loops are affected by model uncertainties and external disturbances leading to imperfect decoupling properties of the multivariable system. Nevertheless, it is well known from literature (Alt et al. (2009a); Bartolini et al. (1999); Levant (1993; 1998)) that the sliding surfaces *σ<sup>N</sup>* = 0 and *σTres* = 0 can still be reached in this case. Thus, the engine speed control and the torque reserve control loops are supposed to be robust against any disturbances due to improper decoupling. Finally, it has been shown in Alt (2010) that this multivariable control design approach leads to better performance and

Γ*Tres*,*M*1(*WTres*,1 + Φ*Tres*,1) <sup>Γ</sup>*Tres*,*m*1(*WTres*,1 <sup>−</sup> <sup>Φ</sup>*Tres*,1) ,

(25)

,*res*,*SP* ≤ Γ*Tres*,*<sup>M</sup>* . (26)

≤ Φ*Tres*,1 . (27)

*WTres*,1 >

*λ*2 *Tres*,1 ≥

*<sup>∂</sup>x*<sup>2</sup> *<sup>f</sup>*21(*x*2, *<sup>x</sup>*3) <sup>−</sup> *<sup>T</sup>*˙

Φ*Tres*,1 Γ*Tres*,*m*<sup>1</sup>

4Φ*Tres*,1 Γ2 *Tres*,*m*1

0 < *ρTres*,1 ≤ 0.5 .

*∂h*<sup>2</sup> *∂x*<sup>2</sup>

due to possible model uncertainties that are related to the inversion of *f*22(*u*2):

∗(−1) <sup>22</sup> (*Tair*,*u*))

less calibration efforts than a similar approach without decoupling compensation.

This section illustrates the efficiency and the robustness properties of the proposed decoupling controller. For this purpose some representative nonlinear simulation and experimental results are shown. All the simulations are based on the nonlinear engine model of Alt (2010) with a controller sampling time of *ts* = 10 ms. The experimental results include representative field test data with a 2.0l series-production vehicle and a common rapid control prototyping

In the first scenario the disturbance rejection properties of the closed-loop system are evaluated. For this purpose an additional load torque of *Tload* = 8 Nm (e.g. power steering) is applied to the engine at *t*<sup>1</sup> = 4 s and removed again at *t*<sup>2</sup> = 9 s. From Figure 12 it can be seen that due to this load torque the engine speed *N* and the torque reserve *Tres* drop below their reference values while the corresponding transients stay below Δ*N* = 40 1/min and Δ*Tres* = 8 Nm, respectively. However, the proposed idle speed controller steers both variables back to their reference values *NSP* = 800 1/min and *Tres*,*SP* = 8 Nm within less than 2 s.

*f*22(*f*

*res*,*SP*:

0 < Γ*Tres*,*<sup>m</sup>* ≤

 *∂h*<sup>2</sup> *∂x*<sup>2</sup>

**5. Nonlinear simulation and experimental results**

,

fulfilled:

relationship *<sup>∂</sup>h*<sup>2</sup>

Here, *f* ∗

system.

Fig. 12. Nonlinear simulation and experimental results for super twisting algorithm based decoupling controller, disturbance rejection properties: Engine speed *N* (left), torque reserve *Tres* (right), experimental results (blue), simulation results (red)

When disabling the load torque similar effects take place. Considering the engine speed *N* it can also clearly be seen that there exists a good matching between the nonlinear simulation data and the experimental measurements. For the torque reserve *Tres* this matching is less perfect since this variable is much more prone to unmodelled dynamics and tolerance effects that have not been considered in the nonlinear simulation model. This effect will be further evaluated in Section 6. In a second representative scenario the engine speed reference value

Fig. 13. Nonlinear simulation and experimental results for super twisting algorithm based decoupling controller, tracking of an engine speed reference step profile: Engine speed *N* (left), torque reserve *Tres* (right), experimental results (blue), simulation results (red)

*NSP* is increased at *t*<sup>1</sup> = 4 s and lowered again at *t*<sup>2</sup> = 14 s. The corresponding simulation results are shown in Figure 13. Regarding the step response of the engine speed *N* it can be clearly seen that no overshoot occurs and the settling times are within less than 2 s and thus reasonable small. Additionally, the torque reserve *Tres* shows only small deviations due to the

Fig. 14. Nonlinear simulation and experimental results for super twisting algorithm based decoupling controller, tracking of a torque reserve reference step profile: Engine speed *N* (left), torque reserve *Tres* (right), experimental results (blue), simulation results (red)

0 10 20 30

nonlinearities and the intake-to-torque production delay

time [s]

max(Δ*Tres*,±(*t*)) = |*Tres*,*SP*(*t*) − *Tres*,±(*t*)| on the closed-loop system.

f1 f21 f22 f31 f32 h2 del

f1 f21 f22 f31 f32 h2 del

0

0

10

dev(Tres) [%]

20

1

dev(N) [%]

2

0 10 20 30

time [s]

0

10

T

Fig. 15. Simulation results of engine speed *N* (left) and torque reserve *Tres* (right) for load torque disturbance rejection scenario with multiplicative variations on the system

operates in nominal condition. In Figure 16 the calculated deviations dev(*N*) and dev(*Tres*) are shown for all 20 variations with strongest impact max(Δ*N*±(*t*)) = |*NSP*(*t*) − *N*±(*t*)| and

0

0

Fig. 16. Simulation results of engine speed deviations dev(*N*) (up) and torque reserve deviations dev(*Tres*) (low) for load torque disturbance rejection scenario with multiplicative

variations on the system nonlinearities and the intake-to-torque production delay

10

10

time [s]

time [s]

20

20

30

30

res [Nm]

Robust Control Design for Automotive Applications: A Variable Structure Control Approach 89

20

600

800

N [1/min]

1000

step changes on the engine speed *N* and it returns to its reference value *Tres*,*SP* within a short settling time.

Similar results can be seen from Figure 14 where the torque reserve reference value *Tres*,*SP* is increased at *t*<sup>1</sup> = 3 s and lowered again at *t*<sup>2</sup> = 14 s. During these changes on the torque reserve *Tres* the minimization of any effects on the engine speed *N* is considered as most important design criteria since this behaviour would affect the driver's comfort. From Figure 14 it can be clearly seen that the proposed idle speed controller is able to fulfill this requirement as specified. As known from existing series-production ISC controllers this overall performance can not be achieved using classical linear control design approaches without gain scheduling. Finally, the step response of the torque reserve *Tres* is also without any overshoot and faster than that for the engine speed *N*.

## **6. Robustness analysis**

After the first experimental studies the robustness properties of the closed-loop system have to be analyzed in detail. For the sake of simplicity this analysis will be performed using the validated nonlinear simulation model from Alt (2010). Here, a representative disturbance rejection scenario is used to illustrate the major effects of model uncertainties on the closed-loop system performance. This simulation scenario includes an external load torque disturbance of *Tload* = 10 Nm which is applied to the engine at *t*<sup>1</sup> = 10 s and removed again at *t*<sup>2</sup> = 20 s. The overall robustness analysis covers variations of ± 10 % in up to 19 different characteristic maps of the nonlinear simulation model. In particular, the system nonlinearities *f*1, *f*21, *f*22, *f*31, *f*<sup>32</sup> and *h*<sup>2</sup> = *h*<sup>2</sup> (*h*21, *hh*22) are varied one after another using multiplicative uncertainty functions:

$$\begin{array}{lcl}f\_{1} = & d\_{1,\pm} \cdot f\_{1,\text{non}} & \text{mit} \ d\_{1,\pm} \in [0.9, 1.1] \\ f\_{21} = d\_{21,\pm} \cdot f\_{21,\text{non}} & \text{mit} \ d\_{21,\pm} \in [0.9, 1.1] \\ f\_{22} = d\_{22,\pm} \cdot f\_{22,\text{non}} & \text{mit} \ d\_{22,\pm} \in [0.9, 1.1] \\ f\_{31} = d\_{31,\pm} \cdot f\_{31,\text{non}} & \text{mit} \ d\_{31,\pm} \in [0.9, 1.1] \\ f\_{32} = d\_{32,\pm} \cdot f\_{32,\text{non}} & \text{mit} \ d\_{32,\pm} \in [0.9, 1.1] \\ h\_{2} = d\_{2,\pm} \cdot h\_{2,\text{non}} & \text{mit} \ d\_{2,\pm} \in [0.9, 1.1] \end{array} \tag{28}$$

Furthermore, the intake-to-torque-production delay *τ<sup>d</sup>* has been increased up to 4 times to cope with any signal communication problems

$$
\tau\_d = \tau\_{d,nom} + \Delta \tau\_d \text{ with } \Delta \tau\_d = 20 \text{ ms.} \tag{29}
$$

All these nonlinear simulation results are depicted in Figure 15. In a second step all resulting deviations dev(*N*) and dev(*Tres*) on the nominal behaviour of the engine speed and the torque reserve are scaled with the reference values of the operating point (*NSP*,0 = 800 1/min, *Tres*,*SP*,0 = 8 Nm):

$$\text{dev}(N(t)) = \frac{|\max(\Delta N\_{\pm}(t)) - \Delta N\_{\text{nom}}(t)|}{N\_{SP}} \cdot 100 \,\text{\AA} \tag{30}$$

$$\text{dev}(T\_{\text{res}}(t)) = \frac{|\max(\Delta T\_{\text{res}\pm}(t)) - \Delta T\_{\text{res},\text{norm}}(t)|}{T\_{\text{res},\text{SP}}} \cdot 100 \,, \tag{31}$$

where Δ*Nnom*(*t*) = |*NSP*(*t*) − *Nnom*(*t*)| and Δ*Tres*,*nom*(*t*) = |*Tres*,*SP*(*t*) − *Tres*,*nom*(*t*)| represent the resulting errors to the corresponding reference values *NSP* and *Tres*,*SP* while the engine 16 Robust Control

step changes on the engine speed *N* and it returns to its reference value *Tres*,*SP* within a short

Similar results can be seen from Figure 14 where the torque reserve reference value *Tres*,*SP* is increased at *t*<sup>1</sup> = 3 s and lowered again at *t*<sup>2</sup> = 14 s. During these changes on the torque reserve *Tres* the minimization of any effects on the engine speed *N* is considered as most important design criteria since this behaviour would affect the driver's comfort. From Figure 14 it can be clearly seen that the proposed idle speed controller is able to fulfill this requirement as specified. As known from existing series-production ISC controllers this overall performance can not be achieved using classical linear control design approaches without gain scheduling. Finally, the step response of the torque reserve *Tres* is also without

After the first experimental studies the robustness properties of the closed-loop system have to be analyzed in detail. For the sake of simplicity this analysis will be performed using the validated nonlinear simulation model from Alt (2010). Here, a representative disturbance rejection scenario is used to illustrate the major effects of model uncertainties on the closed-loop system performance. This simulation scenario includes an external load torque disturbance of *Tload* = 10 Nm which is applied to the engine at *t*<sup>1</sup> = 10 s and removed again at *t*<sup>2</sup> = 20 s. The overall robustness analysis covers variations of ± 10 % in up to 19 different characteristic maps of the nonlinear simulation model. In particular, the system nonlinearities *f*1, *f*21, *f*22, *f*31, *f*<sup>32</sup> and *h*<sup>2</sup> = *h*<sup>2</sup> (*h*21, *hh*22) are varied one after another using

> *<sup>f</sup>*<sup>1</sup> = *<sup>d</sup>*1,<sup>±</sup> · *<sup>f</sup>*1,*nom* mit *<sup>d</sup>*1,<sup>±</sup> ∈ [0.9, 1.1] , *<sup>f</sup>*<sup>21</sup> = *<sup>d</sup>*21,<sup>±</sup> · *<sup>f</sup>*21,*nom* mit *<sup>d</sup>*21,<sup>±</sup> ∈ [0.9, 1.1] , *f*<sup>22</sup> = *d*22,<sup>±</sup> · *f*22,*nom* mit *d*22,<sup>±</sup> ∈ [0.9, 1.1] , *<sup>f</sup>*<sup>31</sup> = *<sup>d</sup>*31,<sup>±</sup> · *<sup>f</sup>*31,*nom* mit *<sup>d</sup>*31,<sup>±</sup> ∈ [0.9, 1.1] , *f*<sup>32</sup> = *d*32,<sup>±</sup> · *f*32,*nom* mit *d*32,<sup>±</sup> ∈ [0.9, 1.1] , *h*<sup>2</sup> = *d*2,<sup>±</sup> · *h*2,*nom* mit *d*2,<sup>±</sup> ∈ [0.9, 1.1] .

Furthermore, the intake-to-torque-production delay *τ<sup>d</sup>* has been increased up to 4 times to

In a second step all resulting deviations dev(*N*) and dev(*Tres*) on the nominal behaviour of the engine speed and the torque reserve are scaled with the reference values of the operating

*NSP*

*Tres*,*SP*

dev(*N*(*t*)) = <sup>|</sup>max(Δ*N*±(*t*)) <sup>−</sup> <sup>Δ</sup>*Nnom*(*t*)<sup>|</sup>

dev(*Tres*(*t*)) = <sup>|</sup>max(Δ*Tres*±(*t*)) <sup>−</sup> <sup>Δ</sup>*Tres*,*nom*(*t*)<sup>|</sup>

where Δ*Nnom*(*t*) = |*NSP*(*t*) − *Nnom*(*t*)| and Δ*Tres*,*nom*(*t*) = |*Tres*,*SP*(*t*) − *Tres*,*nom*(*t*)| represent the resulting errors to the corresponding reference values *NSP* and *Tres*,*SP* while the engine

*τ<sup>d</sup>* = *τd*,*nom* + Δ*τ<sup>d</sup>* with Δ*τ<sup>d</sup>* = 20 ms. (29)

(28)

· 100 , (30)

· 100 , (31)

any overshoot and faster than that for the engine speed *N*.

settling time.

**6. Robustness analysis**

multiplicative uncertainty functions:

cope with any signal communication problems

point (*NSP*,0 = 800 1/min, *Tres*,*SP*,0 = 8 Nm):

All these nonlinear simulation results are depicted in Figure 15.

Fig. 15. Simulation results of engine speed *N* (left) and torque reserve *Tres* (right) for load torque disturbance rejection scenario with multiplicative variations on the system nonlinearities and the intake-to-torque production delay

operates in nominal condition. In Figure 16 the calculated deviations dev(*N*) and dev(*Tres*) are shown for all 20 variations with strongest impact max(Δ*N*±(*t*)) = |*NSP*(*t*) − *N*±(*t*)| and max(Δ*Tres*,±(*t*)) = |*Tres*,*SP*(*t*) − *Tres*,±(*t*)| on the closed-loop system.

Fig. 16. Simulation results of engine speed deviations dev(*N*) (up) and torque reserve deviations dev(*Tres*) (low) for load torque disturbance rejection scenario with multiplicative variations on the system nonlinearities and the intake-to-torque production delay

**8. Acknowledgment**

Germany.

3898.

ETAS GmbH (2010).

*Porto, Portugal*.

Vol. 5, No. 3: pp. 287 – 295.

Francis Ltd. London, UK.

Press, Boca Raton, London, New York.

**9. References**

This work has been supported by IAV GmbH Gifhorn in Germany. The authors express their

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Alt, B. (2010). *Modellbasierte Regelung ausgewählter Komponenten im Antriebsstrang eines*

Alt, B., Blath, J., Svaricek, F. & Schultalbers, M. (2009a). Control of idle engine speed and

Alt, B., Blath, J., Svaricek, F. & Schultalbers, M. (2009b). Multiple sliding surface control of idle

Bartolini, G., Ferrara, A., Levant, A. & Usai, E. (1999). On second order sliding mode

Bartolini, G., Ferrara, A. & Usai, E. (1998). Chattering avoidance by second order sliding mode control, *IEEE Transactions on Automatic Control,* Vol. 43, No. 2: 241 – 246. Butt, Q. & Bhatti, A. (2009). Estimation of gasoline-engine parameters using higher order

Drazenovic, B. (1969). The invariance conditions in variable structure systems, *Automatica,*

Edwards, C. & Spurgeon, S. (1998). *Sliding mode control, theory and applications*, Taylor and

Ehsani, M., Gao, Y. & Emadi, A. (2010). *Modern electric, hybrid electric and fuel cell vehicles*, CRC

Emelyanov, S., Korovin, S. & Levant, A. (1996). Higher-order sliding modes in control systems,

Hung, J., Gao, W. & Hung, J. (1993). Variable structure control: a survey, *IEEE Transactions on*

Khalil, H. (1996). *Nonlinear Systems, 3rd edition*, Prentice Hall, Upper Saddle River, NJ, USA. Khan, M., Spurgeon, S. & Bhatti, A. (2001). Robust speed control of an automotive engine

Kiencke, U. & Nielsen, L. (2005). *Automotive control systems for engine, driveline and vehicle - 2nd*

Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control, *International*

URL: *http://www.etas.com/de/products/applications calibrating automotive electronics.php* Fridman, L. & Levant, A. (2002). Higher-order sliding modes, *in* W. Perruquetti & J. Barbot (eds), *Sliding mode control in engineering*, Marcel Dekker Inc., New York, Basel. Guzzella, L. & Sciarretta, A. (2005). *Vehicle propulsion systems, Introduction to modeling and*

using second order sliding modes, *Proceedings of the European Control Conference 2001,*

*Computational Mathematics and Modeling,* Vol. 7: pp. 294 – 318.

*optimization*, Springer, Berlin, Heidelberg, New York.

*Industrial Electronics,* Vol. 40, No. 1: 2 – 22.

*edition*, Springer, Berlin, Heidelberg, New York.

*Journal of Automatic Control,* Vol. 58, No. 6: 1247 – 1263.

*and nonlinear control*, Springer, London, Berlin, Heidelberg.

*Kraftfahrzeugs mit Ottomotor*, PhD thesis, Universität der Bundeswehr München,

torque reserve with higher order sliding modes, *Proceedings of the Multi-Conference on*

engine speed and torque reserve with dead start assist control, *IEEE Transactions on*

controllers, *in* K. Young & U. Özgüner (eds), *Variable structure systems, sliding mode*

sliding mode, *IEEE Transactions on Industrial Electronics,* Vol. 55, No. 11: pp. 3891 –

gratitude to Jan Peter Blath and Matthias Schultalbers for their support.

*Systems and Control, Saint Petersburg, Russia*.

*Industrial Electronics,* Vol. 56: 3580 – 3592.

From Figure 16 it can be seen that the engine speed deviation dev(*N*) is bounded with about 1 % while the deviation dev(*Tres*) on the torque reserve is bounded with about 15 %. This large peak deviation on the torque reserve seems to be not reasonable since the impact on the system parameters is bounded with only 10 %. However, it has to be noted that the calibration of the controllers allows to find a trade off between the accuracy on *N* and *Tres* and thus to penalize the engine speed error more than the torque reserve error. Since the comfort and the driver's impression on the engine quality are mainly affected by deviations on the engine speed it becomes clear that large control errors on *N* should be more penalized than deviations on *Tres*. Keeping this effect in mind it can be anyhow summarized that the proposed control framework shows good robustness properties despite any uncertainties in the system parameters, e.g. aging, tolerance effects or environmental influences.

## **7. Conclusion and future work**

The paper deals with the idle speed control problem which represents an interesting multivariable control design application in the field of modern automotive spark ignition engines. In idle condition the engine speed and the torque reserve should be held at their reference values. The key design requirements include the decoupling of the underlying multivariable system and the improvement of the robustness properties against unknown load torque disturbances and tolerance effects. In the first step a nonlinear engine model is introduced that includes both the main dynamics of the engine internal processes and also the major parts of the torque structure of current engine management systems. The resulting nonlinear simulation model is validated on a series-production vehicle and it is used as a virtual engine test rig. Then, a decoupling control framework is introduced that is able to hold the idle engine speed and the torque reserve at their reference values despite external load torque disturbances or even uncertainites in the system parameters or the intake-to-torque-production delay.

The multivariable control framework consists of two independent feedback controllers and a decoupling compensation. Each of these two controllers is based on a second order sliding modes control design method that is also known as super twisting algorithm. The decoupling compensation is based on an identified linear time invariant model of the plant that is valid around a given operating point which is situated in the middle of the idle operating range. Here, the required LTI model is deduced from test rig measurements using system identification methods. The efficiency of the proposed control framework is shown by nonlinear simulation results. It can be seen that the controller shows good performance for the large signal behaviour although it is only designed for the neigbourhood of the given operating point. Nonlinear simulation and experimental results show as well that the proposed controller is able to handle a wide operating range at idle condition while the control gains remain unchanged. Hence, the proposed control framework is easier to calibrate since the number of control parameters is severly reduced compared to classical series-production control design methods using gain scheduling techniques. The efficiency and the robustness properties against system uncertainties and variations in the intake-to-torque production delay are evaluated by extended simulation studies. Current research includes the application of this second order sliding modes based multivariable design approach in the field of other automotive control design tasks (i.e. hybrid electric vehicles) and aerospace applications (i.e. smart aeroengines).

## **8. Acknowledgment**

This work has been supported by IAV GmbH Gifhorn in Germany. The authors express their gratitude to Jan Peter Blath and Matthias Schultalbers for their support.

## **9. References**

18 Robust Control

From Figure 16 it can be seen that the engine speed deviation dev(*N*) is bounded with about 1 % while the deviation dev(*Tres*) on the torque reserve is bounded with about 15 %. This large peak deviation on the torque reserve seems to be not reasonable since the impact on the system parameters is bounded with only 10 %. However, it has to be noted that the calibration of the controllers allows to find a trade off between the accuracy on *N* and *Tres* and thus to penalize the engine speed error more than the torque reserve error. Since the comfort and the driver's impression on the engine quality are mainly affected by deviations on the engine speed it becomes clear that large control errors on *N* should be more penalized than deviations on *Tres*. Keeping this effect in mind it can be anyhow summarized that the proposed control framework shows good robustness properties despite any uncertainties in

The paper deals with the idle speed control problem which represents an interesting multivariable control design application in the field of modern automotive spark ignition engines. In idle condition the engine speed and the torque reserve should be held at their reference values. The key design requirements include the decoupling of the underlying multivariable system and the improvement of the robustness properties against unknown load torque disturbances and tolerance effects. In the first step a nonlinear engine model is introduced that includes both the main dynamics of the engine internal processes and also the major parts of the torque structure of current engine management systems. The resulting nonlinear simulation model is validated on a series-production vehicle and it is used as a virtual engine test rig. Then, a decoupling control framework is introduced that is able to hold the idle engine speed and the torque reserve at their reference values despite external load torque disturbances or even uncertainites in the system parameters or the

The multivariable control framework consists of two independent feedback controllers and a decoupling compensation. Each of these two controllers is based on a second order sliding modes control design method that is also known as super twisting algorithm. The decoupling compensation is based on an identified linear time invariant model of the plant that is valid around a given operating point which is situated in the middle of the idle operating range. Here, the required LTI model is deduced from test rig measurements using system identification methods. The efficiency of the proposed control framework is shown by nonlinear simulation results. It can be seen that the controller shows good performance for the large signal behaviour although it is only designed for the neigbourhood of the given operating point. Nonlinear simulation and experimental results show as well that the proposed controller is able to handle a wide operating range at idle condition while the control gains remain unchanged. Hence, the proposed control framework is easier to calibrate since the number of control parameters is severly reduced compared to classical series-production control design methods using gain scheduling techniques. The efficiency and the robustness properties against system uncertainties and variations in the intake-to-torque production delay are evaluated by extended simulation studies. Current research includes the application of this second order sliding modes based multivariable design approach in the field of other automotive control design tasks (i.e. hybrid electric vehicles) and aerospace applications (i.e.

the system parameters, e.g. aging, tolerance effects or environmental influences.

**7. Conclusion and future work**

intake-to-torque-production delay.

smart aeroengines).


#### ETAS GmbH (2010).

URL: *http://www.etas.com/de/products/applications calibrating automotive electronics.php*


*Keio University* 

*Japan* 

**Robust Active Suspension Control for** 

Takuma Suzuki and Masaki Takahashi

**Vibration Reduction of Passenger's Body** 

An automotive performance has improved from the demand of ride comfort and driving stability. Many research have proposed various control system design methods for active and semi-active suspension systems. These studies evaluated the amount of reduced vibration in the vehicle body, i.e., the vertical acceleration in the center-of-gravity (CoG) of the vehicle's body (Ikeda *et al.,* 1999; Kosemura *et al.,* 2008; Itagaki *et al.,* 2008). However, any passengers always do not sit in the CoG of the vehicle body. In the seated position that is not the CoG of the vehicle body, vertical acceleration is caused by vertical, roll and pitch motion of the vehicle. In nearly the resonance frequency of the seated human, the passenger's vibration becomes larger than the seated position's vibration of the vehicle body due to the

The seated human dynamics and human sensibility of vibration are cleared by many researchers. So far some human dynamics model has been proposed (Tamaoki *et al.,* 1996, 1998, 2002; Koizumi *et al.,* 2000). Moreover, some of them are standardized in ISO (ISO-2631- 1*,* 1997; ISO-5982, 2001). At the research as for automotive comfort with the passengervehicle system, M.Oya et al. proposed the suspension control method considering the passenger seated position in the half vehicle model (Oya *et al.,* 2008). G.J. Stein et al. evaluated passenger's head acceleration at some vehicle velocities and some road profiles (Guglielmino *et al.,* 2008). There are few active suspension control design methods which are positively based on a passenger's dynamics and the seating position. These methods can be

In this paper, new active suspension control method is developed to reduce the passenger's vibration. Firstly, a vehicle and passenger model including those dynamics at seated position is constructed. Next, a generalized plant that uses the vertical acceleration of the passenger's head as one of the controlled output is constructed to design the linear *H*<sup>∞</sup> controller. In this paper, this proposed method defines as "Passenger Control". "Passenger Control" means passenger's vibration control. Moreover, in an active suspension control, it is very important to reduce the vibration at the condition of the limited actuating force. Then, we design two methods which are "Vehicle CoG Control", and "Seat Position Control", and compare the proposed method with two methods. "Vehicle CoG Control" means vibration control of vehicle. "Seat Position Control" means vibration control of seat position. Finally, several simulations are carried out by using a full vehicle model which has active suspension system. From the result, it was confirmed that in nearly the resonance

**1. Introduction** 

seated human dynamics.

expected to improve the control performance.

