NONLINEAR MULTIVARIABLE ROBUST INTERNAL MODEL CONTROL OF A TWO-STAGE TURBOCHARGED DIESEL ENGINE

NONLINEAR MULTIVARIABLE ROBUST INTERNAL MODEL CONTROL OF A TWO-STAGE TURBOCHARGED DIESEL ENGINE Dieter Schwarzmann ∗ Rainer Nitsche ∗ Jan Lunze ∗∗ Marco Schmidt ∗ ∗

Robert Bosch GmbH Corporate Sector Research and Advance Engineering Stuttgart, Germany ∗∗

Institute of Automation and Computer Control Ruhr-Universität Bochum, Bochum, Germany

Abstract: This paper proposes a new approach to the design of Internal Model Controllers (IMC) for nonlinear systems with emphasis on robustness issues. A nonlinear IMC controller is used which is based on the right inverse of the plant model. As the right inverse is, in general, not realisable, it is combined with a linear filter. The right inverse is constructed using nonlinear geometric control methods for input-affine systems. Robust stability of the closed-loop is investigated for unstructured (multivariable) output uncertainties. The industrial feasibility of the controller and the robustness analysis is shown for the nonlinear IMC control loop of a two-stage turbocharged diesel engine. Keywords: Internal Model Control, Flatness, Robust Tracking Control

1. INTRODUCTION Many control problems in the automotive industry deal with nonlinear plants that have to be controlled with nonlinear controllers in order to reach the desired closed-loop performance. Additionally, robustness with respect to manufacturing tolerances and ageing have to be accounted for. A rather new problem in automotive applications is the pressure control of a two-stage turbocharged diesel engine. A solution to this control problem is discussed in (Schwarzmann et al., 2006b). Here, the controller is reviewed and robust stability is proven. To this end, a novel nonlinear controller design is presented which combines methods from geo-

metric nonlinear control (see e. g. (Isidori, 1995; Kraviris and Kantor, 1990a; Kraviris and Kantor, 1990b)) with the Internal Model Control (IMC) (Morari and Zafiriou, 1989; Economou and Morari, 1986) loop. Uncertainties are introduced for this control structure with which a robust stability analysis can be performed using methods from linear control theory. This paper consists of two main parts: In Section 2 the IMC structure and some of its properties are reviewed. Then, a generalised IMC controller structure is introduced consisting of the series of IMC filter and model’s right inverse. An IMC filter and its implementation for nonlinear systems is proposed. With this IMC structure, it is suggested to omit the internal model and to introduce uncertainties such that the closed-loop

behaviour results in a linear control loop for which robustness can be verified easily. Section 3 applies the presented theory to show robustness of the pressure controller of a two-stage turbocharged diesel engine of (Schwarzmann et al., 2006b). To this end, the plant and control problem is introduced and the right inverse of the model is developed. Finally, uncertainties are introduced for which robust stability is proven.

2. INTERNAL MODEL CONTROL (IMC) 2.1 Plant Model Some mathematical background is introduced. e is defined by The plant model Σ e : x˙ =f (x) + G(x)u, x(0) = x0 , x ∈ X (1a) Σ e u∈U ˜ =h (x) , y ˜ ∈ Y, y (1b) ˜ = m and consists with dim x = n, dim u = dim y of a nonempty open set X called the state space of e a nonempty set U called the domain of Σ e and Σ, e e a set Y called the range of Σ. The elements of the vector f , matrix G and the vector h are assumed to be analytic 1 . As a result of this assumption, the solution x(t) of (1a) exists and is unique. e is described by The behaviour of the system Σ regarding the input/output (i/o) behaviour. In e : U → Ye is written by short, the i/o map Σ e for u ∈ U. Thus, within this text, the ˜ = Σu y concatenation operator ‘◦’ is omitted. e ∞ implies the i/o behaviour at steadyThe term Σ state. It is defined using the steady-state signals e ∞ u∞ < ∞ ˜∞ = Σ limt→∞ u(t) = u∞ < ∞ and y as e∞ : Σ

0 =f (x, u∞ ) ˜ ∞ =h (x, u∞ ) . y

(2)

Definition 1. (Right Inverse). The right inverse e r : Ye → U Σ

(3)

e maps a given output signal y ˜ ∈ Ye of a system Σ into the therefore necessary input signal u ∈ U, such that eΣ e ry ˜=y ˜ Σ (4) holds. Remark 1. According to (3) the right inverse is e Thus, it can not be only defined for signals in Y. e ˜ ∈ applied to signals w / Y. 1

A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Taylor series in the neighbourhood of every point.

d

~ w

w −

Q

u

y

Σ ~ Σ

~ y −

Fig. 1. IMC Structure 2.2 IMC Structure and Properties Figure 1 shows the IMC structure with IMC controller Q some nonlinear plant Σ with disture bances d and nonlinear plant model Σ. e = Σ and the absence Assuming a perfect model Σ of disturbances d = 0 the feedback signal in Fig. 1 vanishes and Q acts as a feedforward controller. If an IMC controller Q is thought of as a feedforward e then the feedback controller for the model Σ ˜ structure will generate the reference signal w for Q depending on the modelling errors and disturbances and the original reference signal w by ˜ = w − (y − y ˜ ). w Thereby, the feedback structure attenuates the effects of model uncertainties and disturbances. The IMC structure itself has valuable properties which hold independently of the methods used to obtain the IMC controller Q. The following reviews these properties. The block diagram in Fig. 1 yields ˜ =w − y + y ˜ w e Q w. ˜ =Σ ˜ y

(5) (6)

From (5) and (6) the following three properties, which also hold for the nonlinear multivariable case, can be derived (Economou and Morari, 1986): Property 1. (Nominal Stability). Assume a pere and the absence of disturfect model (Σ = Σ) bances d = 0. Then, the closed-loop system in Fig. 1 is internally stable if the controller Q and e are stable. the plant Σ Property 2. (Perfect Control). Assume that the e r exists and that the right inverse of the model Σ closed-loop system is input-output-stable with the e r . Then the control will be perfect controller Q = Σ (i. e. y(t) = w(t)) for arbitrary disturbances d. Property 3. (Zero Offset). Assume that the right e r∞ exists inverse of the model in steady-state Σ and that the IMC controller has the same steadye r∞ , and that the closedstate gain Q∞ = Σ loop system is input-output-stable with this controller. Then, offset-free control y ∞ = w∞ is attained for asymptotically constant reference sig-

Q w

~ w −

d F

~ Σr

u

Q

y

Σ

~ Σ

w

~ w −

u F

~ Σr

d

y

Σ

~ y

~ y −

Fig. 2. Generalised IMC Structure nals limt→∞ w = w∞ and constant disturbances d. In summary, an IMC control loop possesses important properties like the stability of the nominal close-loop system and setpoint following. In the following, a design procedure for this control loop is suggested and a robustness analysis is introduced.

−

Fig. 3. Generalised IMC controller with no internal model. ˜ according to (4) and (6),(7) the model output y is equivalent to the output of the IMC filter F . Thus, the model can be substituted as shown in Fig. 3. Both structures (Fig. 3 and Fig. 2) have the same ˜ (t). behaviour, since both generate the same y The benefit of an IMC which is implemented as in Fig. 3 is a significant reduction of the overall controller complexity.

2.3 Generalised IMC Controller Structure Within this paper, the model class will be limited to stable systems which can be represented by (1). Additionally, the existence and stability of the model’s right inverse is required which further limits the system class to minimum phase quadratic systems. The restriction to stable systems is inherent to the IMC structure (see Property 1). The restriction to minimum phase systems allows the introduction of a simplified IMC controller design procedure. Finally, an input-affine structure of the model (1) allows to use established methods of geometric nonlinear control. According to Property 2, the desired IMC cone r. troller would be the model’s right inverse Q = Σ f ˜ ∈W In the IMC structure in Fig. 1 the signal w ˜ is the input signal for Q. In general, the signal w ˜. can not be produced as output from the model y Thus, it is assumed that f Ye = 6 W holds. Hence, according to Remark 1, a pure right inverse as IMC controller is not realisable! To this end, an operator is introduced, called IMC filter f into the domain ˜ ∈W F , which maps the signal w e Y f → Y. e F :W Thus, a generalised realisable IMC controller Q is introduced as er F Q=Σ (7) which is in agreement with the classic linear IMC controller. The resulting generalised IMC structure is shown in Fig. 2. From the relationships above, the following significant simplification of the IMC loop is proposed: e is simulated on-line in In Fig. 2 the plant model Σ ˜ (t). However, order to generate the model output y

2.4 IMC Filter F The role of the IMC filter F is to map its input f into the domain of the right inverse ˜ ∈W signals w e e r is only realisable in Y. Thus, the right inverse Σ combination with the filter F . ˜ should follow its The filter’s output signal y d = y ˜ “closely”. If zero steady-state offset input signal w is desired and Q is designed as in (7), the IMC filter F should fulfill (Property 3) F∞ = I

(8)

e has a non-zero finite steady-state if the model Σ 2 gain . Here, I is the unity operator. Since this paper is concerned with the multivariable case, one filter for each reference signal wi is proposed: F = diag(F1 , . . . , Fm ).

(9)

It is assumed, that the input-affine system (1) has a (vector) relative degree (Isidori, 1995) of r = {r1 , . . . , rm }. The notion of relative degree ri implies that an input uj is in an algebraic (r ) relationship to y˜i i . Thus, each output y˜i with i = 1, . . . , m is ri −1-times continuously differentiable, i. e. y˜i ∈ C ri −1 for limited input signals. e it is proposed to design With the knowledge of Y, the filter Fi as a linear filter with the following fi (s) to Y d i (s) transfer function from W 1 Y d i (s) . = r i fi (s) kri s + kri −1 sri −1 + · · · + 1 W (10) Thus, this filter fulfills (8). Fi (s) =

2

For models with zero or infinite steady-state gain the reader is referred to (Morari and Zafiriou, 1989) for the construction of the IMC filter F .

yd( ri )

Fi ~ w i −

³

1 / k ri +

k ri −1

+



³



³



AHPC

HPC Bypass Chargecooler

yd(ri −1)

yd( ri −2 )

V2

HPC V3

k ri − 2

V1

LPC

ydi nE m f



HP Shaft

LP Shaft

Engine

+ V4

Fig. 4. IMC filter Fi w

~ w −

F

HPT

∆

y − ~y

Exhaust aftertreatment

V5

LPT HPT Bypass

V6

AHPT

LPT Wastegate

V7

ALPT

Fig. 5. IMC control loop with multiplicative uncertainties ∆

Fig. 6. Air-system of a two-stage turbocharged diesel engine

Further, it is proposed to implement each IMC filter Fi as a state-variable filter (SVF) (Young, 1981) as shown in Fig. 4. Each filter Fi guarantees for any limited input signal w ˜i that the output y d i is at least (ri − 1)-times continuously differentiable. Additionally, it computes ri derivatives of the output y di which can be supplied to the model e r , if necessary. inverse Σ

Proof 1. The proof follows from the Small-GainTheorem. See e. g. (Lunze, 1997).

2.5 Robust Stability for Unstructured Uncertainties Suppose that the plant Σ can be represented using unstructured multiplicative output uncertainties ∆ with e Σ = (I + ∆)Σ. (11) Then, with (4) the nonlinear IMC structure from Fig. 2 is equal to the structure shown in Fig. 5. ¯ for the uncerAssume, that an upper bound ∆ tainties ∆ is given, such that ¯ g(∆) ≤ ∆

(12)

holds, where g(·) implies the gain of its argument (see e. g. (Khalil, 2000)). ¯ It is proposed to choose the upper bound ∆ ¯ as a linear multivariable transfer function ∆(s). Since the gain of the original uncertainties g(∆) is finite (otherwise the plant would be unstable) ¯ it is always possible to find a linear system ∆(s) with a larger gain such that (12) holds. Since F can also be represented by a linear transfer function (see (10)), well known methods from linear control theory can be employed to verify robustness: Theorem 1. (Robust Stability of IMC). An IMC loop is robustly stable for the plant given in (11) ¯ with the upper bound ∆(s) on the uncertainties of (12) if ª © ¯ λP ∆(jω) |F (jω)| < 1; ∀ ω (13) holds, where λP is the Perron root.

2.6 Conclusion The generalised IMC structure from Fig. 2 or its proposed realisation without the internal model from Fig. 3 is a suitable control structure for the control of nonlinear systems if a stable right e r of the model can be determined. Then, inverse Σ the proposed multivariable IMC filter F (9) in the SVF implementation in Fig. 4 can be employed in e r. combination with the right inverse Σ The resulting nonlinear control loop will be robustly stable if (13) holds and will have zero steady-state offset (see Property 3). This nonlinear IMC control loop does not rely on statee or the plant Σ. feedback from either the model Σ It is an output feedback controller. In the following, a right inverse and an IMC filter are developed for a two-stage turbocharged engine and linear uncertainties are chosen for which robust stability is proven.

3. IMC OF A TWO-STAGE TURBOCHARGED ENGINE The plant and control problem is discussed in detail in (Schwarzmann et al., 2006a; Schwarzmann et al., 2006b). Here, only a short review of the plant itself and the chosen IMC controller is given. The focus lies on the robustness analysis.

3.1 Description of the Plant and Control Problem The plant in consideration is the two-stage turbocharged diesel engine shown in Fig. 6. Fresh air is aspirated from the environment V1 and is compressed by a low-pressure compressor (LPC) and

a high-pressure compressor (HPC). The HPC can be bypassed through an orifice with variable cross section AHPC . Before the compressed air enters the intake manifold V3 , it is cooled down by the chargecooler. After combustion, the hot exhaust gas is pushed into the exhaust manifold V4 . The exhaust is first led over the high-pressure turbine (HPT) through the pipe V5 and then over the low-pressure turbine (LPT). Both turbines can be bypassed by setting the cross sections AHPT and ALPT accordingly. Finally, the exhaust gas flows through the exhaust pipe V6 and back into the environment V7 . The control problem can be formulated as: With the control input u as the bypass cross sections and the measured outputs y as the pressures in the pipes V3 , V4 and V5 (see Fig. 6), find a controller which sets the input u such that the output y follows the reference signal w. 3.2 Plant Model e has the structure of (1). It has six The model Σ states and three inputs and three outputs. Tab. 1 explains the states of the model. The matrix G is Table 1. States of the plant x ˜1 : x ˜2 : x ˜3 : x ˜4 : x ˜5 : x ˜6 :

given by

f1 : f2 : f3 : f4 : f5 : f6 : g11 : g22 : g23 : g33 : g24 : g34 :

Enthalpy flow through V3 with closed HPC bypass Enthalpy flow through V4 with closed HPT bypass Enthalpy flow through V5 with closed HPT, LPT bypass Mass flow through V5 with closed HPT and LPT bypass Resulting torque on LP Shaft Resulting torque on HP Shaft Enthalpy flow over HPC bypass with unitary open cross section Enthalpy flow over HPT bypass with unitary open cross section −g22 Enthalpy flow over LPT bypass with unitary open cross section Mass flow over HPT bypass with unitary open cross section Mass flow over LPT bypass with unitary open cross section

Transformation into Input/Output Normal Form In (Schwarzmann et al., 2006b) it is shown, that e from (1) with (14) is already given in the model Σ the i/o normal form. More specifically, one finds with the transformation Φ · ¸ £ ¤T ˜ y ˜1 , x ˜2 , x ˜3 , x ˜4 , x ˜5 , x ˜6 = Φ(˜ x) = x (17) η the following representation of the plant:

Boost pressure p3 in pipe V3 Exhaust back pressure p4 in pipe V4 Pressure between turbines p5 in pipe V5 Fluid mass m5 contained in pipe V5 Speed ω LP of LP Shaft Speed ω HP of HP Shaft

  g11 (˜ x) 0 0  0 g22 (˜ x) 0     0 g23 (˜ x) g33 (˜ x)  , G(˜ x) =  x) g34 (˜ x)  0 g24 (˜   0 0 0  0 0 0

Table 2. Components of f and G

(14)

˜ is defined as the model output y £ ¤T £ ¤T ˜ = h(˜ ˜1 , x ˜2 , x ˜3 = y˜1 , y˜2 , y˜3 , (15) y x) = x and the model input is £ ¤T £ ¤T u = AHPC , AHPT , ALPT = u1 , u2 , u3 . (16) The components of the vector field f = [f1 , . . . , f6 ]T and the matrix G are explained in Tab. 2.

a(y˜ ,η) z }| { f1 (˜ y , η) y , η) y ˜˙ = f2 (˜ f3 (˜ y , η)   (18) g11 (˜ y , η) 0 0  u, 0 g22 (˜ y , η) 0 + 0 g23 (˜ y , η) g33 (˜ y , η) | {z } B (y˜ ,η)     f4 (˜ y , η) 0 g24 (˜ y , η) g34 (˜ y , η)  u. y , η) + 0 0 0 η˙ = f5 (˜ f6 (˜ y , η) 0 0 0 (19) £ ¤T ˜= x ˜1 , x ˜2 , x ˜3 and The transformed states are y £ ¤T ˜ are the controlled ˜4 , x ˜5 , x ˜6 . The states y η= x outputs whereas the states η are the states of the internal dynamics (Isidori, 1995).

3.3 IMC Controller Q

e r Solving (18) for the input u Right Inverse Σ with (23) one gets ¡ ¡ ¢¢ u = B −1 (y d , η) · y˙ d − a y d , η . (20)

The model (1) is input affine and allows to cree r using the property of inate a right inverse Σ put/output linearisability (Isidori, 1995). Here, tools from geometric nonlinear control are used but no state- or i/o-linearisation is necessary for the IMC controller design.

Suppose, that a desired value y d i and its corresponding derivative y˙ d i is given for each output component y˜i . Then, with £ ¤T (21) yd = yd 1 , yd 2 , yd 3 , £ ¤T (22) y˙ d = y˙ d 1 , y˙ d 2 , y˙ d 3

~r Σ

³

Eq. (19)

yd

F

Ș u

Eq. (20)

regarded as mathematically sufficient. Nevertheless, in the following, the obtained upper bound ¯ ∆(s) is assumed to be given for the robustness analysis.

Fig. 7. Structure of the right inverse Σr

−

³

λi

ydi ydi

Fig. 8. Linear IMC filter Fi , i = 1, 2, 3 e r computes the control input u the right inverse Σ such that the model output y˜ follows the demand y d exactly: !

˜ = yd . y

Σ

Fig. 9. Open-loop of an IMC controller Q = Σr F

y d

wi

~ Σr

(23)

The states of the internal model dynamics η in (20) are obtained by integrating (19) with the ini£ ¤T ˜4 (0), x ˜5 (0), x ˜6 (0) . The tial condition η(0) = x stability of the solution of the internal dynamics is assumed which implies a minimum phase model. e r is defined and Now, the model’s right inverse Σ can be implemented as shown in Fig. 7. IMC filter F The right inverse in Fig. 7 is not realisable, since it needs differentiations y˙ d i of the desired signal, which cannot be assumed to exist. The relative degree of the model (see (18)) is r = {1, 1, 1}. Thus the IMC filter Fi from Fig. 4 for ri = 1 leads to the filter structure shown in Fig. 8. This IMC filter computes the desired trajectory y d i and its derivative y˙ d i with i = 1, . . . , 3. The series of the SVFs with the right inverse shown in Fig. 7 yields the multivariable IMC controller Q (see (7)). Simulation results of this nonlinear IMC control loop are shown in (Schwarzmann et al., 2006b).

First, a set of models M is generated from the e by varying significant paramenominal model Σ e within certain bounds: ters p of Σ n o e p | pmin ≤ p ≤ pmax M= Σ (24) e p signifies a model Σ e with the The expression Σ parametrisation p. Thus, M represents a set of nonlinear models of two-stage turbocharged engines. The open-loop behaviour G0 of the generalised IMC structure in Fig. 2 can be written as (with (4) and (11)) e Σ e r F = (ΣΣ e r − I)F G0 = (Σ − Σ) = ∆F

and is shown in Fig. 9. Now, G0 is linearised about suitable operating conditions where Σ is substie tuted by the corresponding models Σ → Σ(p) ∈ M. This is a required intermediate step to obtain ¯ ∆(s). With (25) and   1 0 0  s/λ1 + 1    1   0 0 F (s) =   (26) s/λ2 + 1     1 0 0 s/λ3 + 1 each linearisation with model parametrisation p and operating point o will give linear multivariable uncertainties ∆p,o (s) by ∆p,o (s) = G0 p,o (s)F −1 (s).

3.4 Robust Stability Analysis A major issue in a stability analysis for industrial ¯ of unstrucproblems is that an upper bound ∆ tured uncertainties ∆ is almost never given a priori. Thus, the first part of this analysis is fo¯ for cused on determining such an upper bound ∆ the two-stage turbocharged engine. In the second part, the actual analysis will take place. ¯ The procedure as Finding an Upper Bound ∆ performed in the following should be interpreted as a “hands-on” approach which can be performed in industry and should give the engineer a rough guess where the uncertainties lie. It should not be

(25)

(27)

¯ Finally, a linear upper bound ∆(s) can be found by ¯ ∆(jω) = max |∆p,o (jω)| ;

∀ p, o;

∀ω

(28)

where max | · | means an element-wise maximum of the amplitude of the transfer function. Thus, ¯ ∆(s) is a multivariable transfer function, whose amplitude of any element at any frequency is equal to the largest amplitude of that element at that frequency of all obtained uncertainties ∆(s). The computed linear upper bounds on the mul¯ tivariable output uncertainties ∆(s) are shown in ¯ Fig. 10. An evaluation of the upper bound ∆(s) in Fig. 10 shows that some of its elements have a significant high frequency gain of about 20.

boost pressure p3

0

w1

−2

y =p 1

3

dB

exhaust back pressure p4 w2 y2=p4

0

pressure p5

−2 −1

10

1

10

3

−1

rad/s

10

10

1

10

3

w3

10

y =p 3

¯ Fig. 10. Upper bound ∆(jω) 0

u1=AHPC

control inputs

10

5

u2=AHPT u3=ALPT

−1

10

λP

t1

time

t2

Fig. 12. Controller performance for the control variables with the control inputs

−2

10

stage turbocharged diesel engine despite manufacturing tolerances and ageing. −3

10

−2

10

0

10

rad/s

2

10

4

10

¯ Fig. 11. Perron root λP (jω) of ∆(jω) |F (jω)| over all frequencies ω Stability Analysis It is assumed, that (12) holds; that is: The gain of the actual nonlinear uncertainties ∆ is smaller or equal to the gain of the linear ¯ upper bound ∆(s). Additionally, the determination of the upper ¯ bound ∆(s) can in reality only be performed for a finite number of parameter variations and finite number of operating points. Nevertheless, it is assumed that such an approximation will give a ¯ representative upper bound ∆(s). Finally, the Perron root λP (Lunze, 1997) can be ¯ computed for ∆(jω) |F (jω)| for all frequencies ω and is displayed in Fig. 11. A visual inspection of the λP in Fig. 11 shows that it is smaller than one for all frequencies ω. According to (13), this implies robust stability of the closed-loop. The high frequency gain of the uncertainties is compensated by the filter F which can now be interpreted as ensuring robust stability at those frequencies. Since model parameters were varied within manufacturing tolerances and the model represents the plant well, this presented approach yields sufficient confidence for the control engineer, that the nonlinear IMC will successfully control the two-

However, it is very important to note that this approach is conservative! Simulations have shown stability for the closed-loop even for norms well beyond one. In simulations, instability could only be achieved with unrealistic parameter variations.

Controller Performance Simulation results of the closed loop shown in Fig. 1 with IMC controller Q shown in Fig. 9 are presented in the following. The plant which is controlled is the original model with twelve states. Thus, modelling errors are present. The results are shown in Fig. 12 and Fig. 13. Engine speed nE and fuel mass flow m ˙ f change during the simulation run. The reference signals wi have been chosen to portray the abilities of the controller. As can be seen in Fig. 12 excellent trajectory tracking is achieved. The reference signal w forces the system through the singularity y1 = p3 = p2 at time t1 and t2 , as marked in Fig. 12 with the dashed vertical lines. For sake of clearness, the pressure p2 is not plotted in Fig. 12. At the singularity the input u1 instantaneously switches from zero to its maximal permissible value. In other words, the HPC bypass opens immediately when the LPC is generating more pressure than the HPC is able to. Therefore, the developed controller automatically acts in full accordance with the strategy that experts on turbocharging suggest.

boost pressure p3 w1

y1= p3

white noise

time Fig. 13. Controller performance with noisy disturbances on the measurement signals Fig. 13 shows the answer of the controlled system with modelling errors under the influence of measurement noise. Noise attenuation and performance can be calibrated online at the engine test bed through the IMC filter pole −λ. In summary, the results show that the theoretically developed novel nonlinear IMC control approach based on the input-affine model for which a feedforward controller was developed successfully solved the extremely difficult control problem. This strongly suggests that the nonlinear IMC controller as presented here is an excellent solution to the control problem of a two-stage turbocharged engine.

4. CONCLUSIONS It was shown, that the IMC structure is valid for nonlinear systems. Constructing the IMC controller Q as the series of IMC filter F and the e r yields an IMC control model’s right inverse Σ loop with zero steady-state offset and nominal stability. However, the system class is limited to stable systems with stable inverses. The existence of an inverse further limits the system class to quadratic systems. A linear IMC filter was proposed whose order is the order of the model’s relative degree. This approach is identical to the linear IMC design. However, it was proposed to implement the IMC filter in a SVF structure such that it also yields the derivatives of its output as information. Through the proposed structure of the IMC controller Q some extensions are possible. First, it is not necessary to simulate the model on-line as part of the control loop which permits to implement the IMC loop more efficiently. Second, if the plant is constructed with the model and multiplicative output uncertainties, the IMC loop can be further simplified to the IMC filter and the uncertainties in a standard feedback structure. Thus, the closed-loop stability is only dependent on the filter and the uncertainties but is independent of the actual model.

Finally, robust stability was shown for the pressure control of a two-stage turbocharged engine. Furthermore, the developed IMC controller eliminates the effect of measured disturbances, respects the input limitations and singularity by driving the system through its utmost physically possible trajectory. Other than common geometric control approaches, only output feedback was used and no linearisation was performed. Control results in simulations with both measurement noise and parameter variations are outstanding.

REFERENCES Economou, C.G. and M. Morari (1986). Internal Model Control. 5. Extension to Nonlinear Systems. Ind. Eng. Chem. Process Des. Dev. 25, 403–411. Isidori, A. (1995). Nonlinear Control Systems. 3rd ed.. Springer. New York. Khalil, H.K. (2000). Nonlinear Systems. 3rd ed.. Prentice-Hall. New Jersey. Kraviris, C. and J.C. Kantor (1990a). Geometric methods for nonlinear process control. 1. background. Ind. Eng. Chem. Res. 29, 2295– 2310. Kraviris, C. and J.C. Kantor (1990b). Geometric methods for nonlinear process control. 2. controller synthesis. Ind. Eng. Chem. Res. 29, 2310–2323. Lunze, J. (1997). Regelungstechnik 2. SpringerVerlag. Berlin. Morari, M. and E. Zafiriou (1989). Robust Process Control. Prentice Hall. Englewood Cliffs, New Jersey. Schwarzmann, D., R. Nitsche and J. Lunze (2006a). Modelling of the airsystem of a twostage turbocharged diesel engine. In: MATHMOD. Schwarzmann, D., R. Nitsche, J. Lunze and A. Schanz (2006b). Pressure control of a twostage turbocharged diesel engine using a novel nonlinear imc approach. In: Proceedings of the Conference on Control Applications. Young, P.C. (1981). Parameter estimation for continuous-time models - a survey. Automatica 17, 23–39.