Closed-Loop Model Order Reduction and MPC for Diesel Engine ...

2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA

Closed-Loop Model Order Reduction and MPC for Diesel Engine Airpath Control R.B. Choroszucha1 , J. Sun1 , and K. Butts2 Abstract— This paper explores MPC for an 8th order air path dynamic model for an automotive diesel engine system. The relatively high order of the system makes the MPC realtime implementation on a practical engine control unit (ECU) infeasible, thereby motivating the model order reduction effort. For the MPC problem with terminal state penalty, Riccatibalanced truncation is used to find a reduced order model for the design of MPC. The presence of a direct feedthrough term from the input to the output in the model requires special treatment and is addressed. We present the analysis of the MPC designs for different reduced order models, compare the reduced order models through open-loop and closed-loop reduction methods, and demonstrate the performance of the resulting linear MPC on a proprietary nonlinear diesel model.

I. I NTRODUCTION

Fig. 1.

Diesel engines have a great fuel efficiency advantage for automotive applications, compared to their gasoline counterparts [1]. They, however, impose a special set of emission control challenges, particularly for nitrous oxide (N Ox ) emissions [2]. One critical task, which has significant impact on diesel emission as well as driveability performance, is the airpath control [3]. The main objective of the airpath control for diesel engines is to deliver air to meet drivers’ demands, and at the same time to provide desired EGR (exhaust gas recirculation) to meet emission control requirements. The diesel airpath (DAP) control system is illustrated in Fig. 1. The system under consideration has two control inputs. One is the variable geometry turbine (VGT), which dictates the speed of the turbine, and hence of the compressor, to regulate the airflow into the intake manifold. The other is the EGR, which controls the flow from the exhaust manifold to the intake manifold for effective N Ox treatment. The control objectives are often translated into desired intake manifold pressure and desired EGR by a high level controller. The airpath control problem can therefore be treated as a tracking problem. In achieving the desired intake manifold pressure and desired EGR, the airpath controller also has to consider several physical constraints on the inputs and outputs. This makes the model predictive control (MPC) framework a natural choice for the airpath control design. The efficacy of MPC on an ECU to the DAP problem was shown in [4]. Recently, MPC has been applied to an 8th order DAP model in [5] and [6]. To reduce computational complexity, an explicit model predictive controller (eMPC) 1 R.B. Choroszucha and J. Sun are with The University of Michigan, Ann Arbor, Michigan. E-mail: {riboch,jingsun}@umich.edu 2 K. Butts is with Toyota Technical Center, Ann Arbor, Michigan. E-mail: [email protected]

978-1-4799-8684-2/$31.00 ©2015 AACC

Diesel engine airpath diagram.

using open-loop model order reduction [5] was demonstrated. The follow-up work [6] applied an explicit rate-based MPC controller [7] using a single reduced order linearization over the entire fuel/engine speed operating space. In this work, we consider the combined rate-based implicit MPC and state estimator design problem for the airpath system. The 8th order model for the diesel airpath represents a challenge for both design and real-time implementation, given the limited engine control unit (ECU) computational resource. Meanwhile, open-loop model order reduction using standard balanced realization and truncation has been shown to have performance issues. Closed-loop model order reduction is therefore adopted in this work. Closed-loop model order reduction for linear quadratic Gaussian (LQG) design has been addressed in [8], [9], where Riccati based balanced realization and truncation was proposed to perform model-order reduction and achieve closedloop stability. The work in [9], however, applies to systems whose transfer functions are strictly causal, i.e., the input affects the output through the state and there is no input term in the ouput equation (D = 0 in the standard (A, B, C, D) state space realization). For the DAP control system shown in Fig. 1, there is a direct feedthrough in the state space equation from the input (e.g., EGR flow) to the output (EGR rate). Therefore the transfer function is not strictly causal and the Riccati based balance and truncation algorithm presented in [9] for model order reduction cannot be directly applied. In this paper, we propose an extension of model order reduction (MOR) technique proposed in [9] to systems with direct feedthrough from input to output. We apply the proposed algorithm to the DAP compensator, and demonstrate

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the closed-loop MPC and estimator design based on the reduced order model through nonlinear model simulations. The paper is structured as follows: Section II discusses the DAP model, the MPC control objective, and the link between LQR and MPC. Section III provides an introduction to Riccati balanced truncation and describes the extended MOR algorithm. Section IV compares performance of the reduced compensators using open-loop balanced truncation and Riccati balanced truncation on a nonlinear DAP model. Finally in Section V conclusions are presented. For the remainder of this paper, a controller, estimator, or compensator designed using a reduced order model will be referred to as a reduced controller, estimator, or compensator, respectively. II. DAP M ODEL AND C ONTROL O BJECTIVE A. DAP Model The derivation of the DAP model for control has been discussed extensively in [10], [11], and [12]. The dynamical equations for the engine components are derived through applications of the ideal gas law, conservation of mass, and conservation of energy in an adiabatic process. The equation for the turbine speed is a result from conservation of energy. The eight states present in this model are: pressures, p, in the intake manifold, the pre-throttle volume, and the exhaust manifold; densities, ρ, in the intake and exhaust manifold; burn gas fractions, F, in the intake and exhaust manifolds; and the VGT rotational speed, ω. Due to space limitations, the equations of the 8th order model are not included here, and the readers are referred to [10]. B. Control Objective The control objective of this paper is to track a set-point intake manifold pressure and EGR rate, defined as rn , subject to constraints. Therefore, the outputs of the system are intake manifold pressure and EGR rate, while the inputs are VGT duty cycle and EGR flow. To apply systematic linear model order reduction techniques to the DAP model, linearization is performed for the DAP model at the center of the fuel/engine speed operating range [6]. The model is discretized with a sampling rate of 0.016 seconds, and the discrete-time model Σ : (A, B, C, D) has the following generic form:  xn+1 = Axn + Bun Σ: (1) yn = Cxn + Dun , where the values of the A, B, C, D matrices, together with the definitions of inputs, outputs, and states are given in Table I. Remark 1: Note that EGR flow, instead of EGR valve position, was chosen as the input in (1) to avoid DC-gain reversals. This choice, however, has led to a direct feedthrough in the output equation of the state-space model [13]. Engine speed, fuel rate, and throttle position are considered as exogenous inputs to the airpath control problem.

The tracking objective, as well as the objective to reduce actuator input, can be captured by defining and minimizing the cost function:  min (xt+Np − x∞ )> P (xt+Np − x∞ )   t+Np −1  {ui }i=t    t+Np −1   X  > + (yn − rn ) Q(yn − rn ) (2)  n=t   t+Np −1  X    + (un − uref )> R(un − uref )   n=t

subject to dynamics and constraints. uref is a control required for y∞ = rNp , x∞ is the steady state when uref is applied to a stable system, Q ≥ 0 and R > 0 reflect different weightings on tracking performance and control effort, and Np is the prediction horizon. The control is calculated by reformulating (2) to a constrained convex quadratic programming problem. Note that with a properly defined final state penalty matrix P, the MPC problem defined in (2) can be recast as an infinite horizon LQR problem when there is no constraint [14]. This connection motivates the Riccati balanced truncation presented in the following sections. Several constraints for the nonlinear DAP control problem need to be enforced, as given in (3). The EGR flow constraint, f (·), is a predefined function of operating conditions that prevents too much EGR flow from being demanded.  max{rn,1 − c1 , 0} ≤ yn,1 ≤ 300 kP a    max{rn,2 − c2 , 0} ≤ yn,2 ≤ 50% (3) 40% ≤ un,1 ≤ 90%    kg 0 ≤ un,2 ≤ f (·) s where c1 and c2 are positive constants. The control objective (2) and the need to enforce constraints (3) naturally suggest MPC for DAP control. However, issues have been identified when the full order linear model is replaced by a nonlinear model and when a lower order model is used for compensator design. The model mismatch resulting from the linearization causes steady-state error and undesirable transient performance, motivating the use of rate-based MPC, which will be further developed in this paper. III. C LOSED -L OOP M ODEL O RDER R EDUCTION Given the high order of the DAP model and the limited onboard computational resources of the ECU, MPC based on the 8th order model, even for the linearized model, is not feasible. The quadratic cost of the MPC problem renders the Riccati-based closed loop model order reduction applicable, and therefore will be pursued. Nevertheless, the non-zero D in the output equation of the state space realization requires special treatment in applying closed-loop model order reduction to the DAP model. A. LQG Solution for Systems with Non-zero D The LQG compensator is a very well known problem, it consists of optimally controlling and estimating a system

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TABLE I D ISCRETE -T IME , L INEARIZED DAP MODEL AT THE CENTER OF FUEL / ENGINE SPEED OPERATING RANGE .



A

=

         

B

=

C

=

          

D

=

0.3892 0.4996 0.0008 −0.0063 0.0878 0.7681 0.0023 0.0042 0.5065 4.2151 0.8392 3.2266 0.0551 0.1096 −0.0019 0.4505 −0.0038 0.0041 0 0 −0.0005 0.0005 0 −0.0005 0 0 0 0 0.0001 −0.0002 0 0  −0.0085 0.0204 −0.0098 0.0015  −5.9338 −0.2074   1.4051 −0.0782   −0.0001 0.0001   0.0057 −0.0003   0 0 0 0  1 0 0 0 0 0 0 0 0 0.1943  −0.0037 0 0 0 0 0 0 0 0 0.092

subject to zero mean additive Gaussian process (vn ) and measurement (wn ) noises. In discrete-time, the LQG problem is defined as "∞ #  X  1  > > min E yn Qyn + un Run un 2 (4) n=0   subject to (1) where E is the expected value. When D = 0, the LQG problem has a guaranteed stabilizing solution if (A, B) is stabilizable, (Q1/2 C, A) is detectable, C > QC is positive semi-definite, and R is positive definite [15]. It is also well known that the compensator design can be separated into a linear quadratic regulator (LQR) and linear quadratic estimator (LQE) problem to obtain a controller and estimator, resp. The Riccati equations associated with the solution of LQR and LQE are in the “dual format” when D = 0 and the process noise is normalized, and the algorithm described in [9] can be applied to derive a balanced realization and perform truncation, resulting in a reduced order LQG design. When D 6= 0, the quadratic cost of (4) will consist of a > cross term 2x> n C QDun , when expressed in terms of xn and un , the corresponding LQR solution is given by [16] un = − (B > Pˆ B + R)−1 (B > Pˆ A + D> QC) xn , | {z }

(5)

K

where Pˆ solves the discrete control algebraic Riccati equation (CARE) Pˆ

=

ˆ + Aˆ> Pˆ Aˆ Q ˆ −1 (Aˆ> Pˆ B)> −(Aˆ> Pˆ B)(B > Pˆ B + R)

(6)

with ˆ = R Aˆ = ˆ = Q

 D> QD + R,  ˆ −1 D> QC, A − BR ˆ −1 D> QC.  C > QC − C > QDR

(7)

0.0022 0.0477 51.445 19.6934 0.8892 0.1884 −0.0002 −0.0526

x

u y

 0.753 −0.0057 0 0.1679 0.0019 0  95.4767 2.2906 0.0055   31.7711 0.8204 0.003    0.0054 0 0   0.7985 −0.0003 0  0.0001 0.8904 0.0057 0.0039 0.2999  0.6756 p  1.2404 in  0.26545 ppre    ω  0.1264      pex  0.014579  , µi =  =    Fin  0.0083883     0.00051878  ρex      7.8288 × 10−7 ρin F 6.0403 × 10−9 ex   VGT = EGR Flow  Intake Pressure = EGR Rate

          

On the other hand, the LQE problem needs to be redefined by defining the output as y˜n = yn − Dun .

(8)

Introducing the notation that x ˆa|b is the estimate of x at time a given information at time b, the optimal LQE solution with the redefined output becomes x ˆn|n−1 x ˆn|n

= Aˆ xn−1|n−1 + Bun−1 , = x ˆn|n−1 + L(˜ yn − C x ˆn|n−1 )

(9) (10)

with the estimation gain L being L = ΠC > (CΠC > + Λ)−1 ,

(11)

where Λ is the covariance matrix of the measurement noise, wn , Γ is the covariance matrix of the process noise, vn , and Π solves the discrete filter algebraic Riccati equation (FARE) Π

= BΓB > + AΠA> −(AΠC > )(CΠC > + Λ)−1 (AΠC > )> .

(12)

B. Riccati Balanced Truncation Linear MOR in its most popular form of “balanced realization and truncation” came into the controls community in [17] in the form of open-loop model order reduction. The next decade saw new model order reduction techniques including those that explicitly accounted for the closed-loop behavior [18]. There are several reasons why a closed-loop reduction is preferable to open-loop reductions. Perhaps most important is that although an open-loop reduction may approximate the full order model arbitrarily well in some norm, the plant can be destabilized by a reduced compensator when there are arbitrarily small uncertainties [19]. Because of theoretical issues of open-loop reduction, LQG balanced truncation arose for robust reduced compensator design [9]. The primary result of [9] was balancing and

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truncating based on the continuous dual algebraic Riccati equations (AREs) with D = 0 and a stability result in the single-input, single-output (SISO) case. In this section, we expand the LQG balanced truncation, which is referred to as Riccati balanced truncation, and propose and explain a closed-loop model order reduction. It should be noted that dual AREs are not required and therefore the results in [9] can be developed to account for the case when D 6= 0. Riccati balanced truncation can be interpreted as removing subspaces that are easy to control and easy to estimate so that the subspaces that contribute more to the cost functional may be used to design a reduced order compensator optimizing (4) over all possible reduced order compensators of a specified order. This is achieved through a process called balanced truncation [17]. The measure of how easy it is to control and estimate a subspace is given by the Riccati singular values (RSVs) defined as follows [9]. p Definition 1: The RSVs are defined as µi = λi (P Π). P and Π are the the solutions to (6) and (12). λi (·) denotes the eigenvalues of the matrix ordered from the largest to the smallest. A small RSV corresponds to a subspace that is easy to control and estimate, meaning it has little impact on the cost. State-space realizations that have certain properties are called balanced, given by the following definitions [17]: Definition 2: (A, B, C, D) is said to be input normal when P = I, Π = M 2 , output normal when P = M 2 , Π = I, and internally balanced when P = Π = M . I denotes the identity matrix, M = diag(µ1 , . . . , µm ) with µ1 ≥ µ2 ≥ . . . ≥ µm , where m is the order of the system. The problem of balancing is to calculate a similarity transformation T T : (A, B, C, D) → (T AT −1 , T B, CT −1 , D). such that the new realization is balanced in the sense of Definition 2. The similarity transformation that balances the system will be referred to as the balancing transformation. A reduced order model is obtained by conformably partitioning a balanced realization (T AT −1 , T B, CT −1 , D) into     A˜11 A˜12  −1 ˜ T AT = A=  ˜21 A˜22 ,   A      ˜ B 1 ˜= TB = B , (13) ˜2  B      CT −1 = C˜ = C˜1 C˜2 , and     ˜ =D D and truncating to ˜1 , C˜1 , D), ˜ Σr = (A˜11 , B

(14)

where r is the order of the reduced system. Only truncated internally balanced systems will be considered for the remainder of the paper. To perform Riccati balanced truncation, there are two steps: balancing (6) and (12), and then partitioning and

truncating as in (13) and (14). Again, this procedure differs from other balanced truncation techniques in the sense that the (A, B, C, D) used in (6) and (12) are different for D 6= 0. Consequently, two different reduced order models will result from the model order reduction. In the design of the ˜ˆ ˜ˆ ˜ˆ ˆ r = (A˜ˆ11 , B controller, Σ 1 , C1 , D), denoting the transformed and truncated system for (7), is used. To design the estimator ˜1 , C˜1 ), the reduction of the unmodified model Σr = (A˜11 , B without direct feedthrough, is used together with the newly defined output (8). Remark 2: A distinction must be made between Pˆr , the ˆ r and the truncated solution to the CARE defined with Σ ˜ˆ ˜ Pr = diag(µ1 , ..., µr ). In continuous-time Pˆr = Pˆr , however, in discrete-time this relation no longer holds. Instead ˜ Pˆr satisfies a result analogous to those presented in [20] for a truncated ARE. Assume solutions to (6) and (12) exist, we follow the procedure outlined in [21] which defines the balancing transformation: 1) Calculate the Cholesky factors of Pˆ = XX > , Π = Y Y >. 2) Calculate the singular value decomposition of Y > X = U M V ∗ , where M is a positive definite, diagonal matrix and U and V are orthogonal matrices. 3) Form the balancing transformation T = X −> V M 1/2 .

(15)

4) Define the contragredient transformation for the ARE solutions for the transformed system. ˜ Pˆ = T > Pˆ T, ˜ = T −1 ΠT −> . and Π

(16) (17)

To check that (15) internally balances the solutions to (6) and (12), substitute (15) into (16) and (17), and recall Y > X = ˜ ˜ = M . Further, the product of (16) U M V ∗ to obtain Pˆ = Π > and (17) results in T P ΠT −> = M 2 , showing the RSVs are invariant under similarity transformations. IV. R EDUCED O RDER R ATE -BASED M ODEL P REDICTIVE C ONTROL The control problem (2) can be reformulated as a ratebased (or velocity-form) MPC problem by defining the augmented state x ˜n , output zn , and input ∆un as     ∆xn xn − xn−1 x ˜n = = , (18) zn yn − rn and ∆un = un − un−1 . (19) With (18) and (19) the state-space equation becomes      A 0 B  x ˜n+1 = x ˜n + ∆un C I D    zn = 0 I x ˜n

(20)

where I is the identity matrix and has dimension equal to the number of outputs, for the linearized DAP model the dimension is 2 × 2.

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Time varying constraints can be reformulated as  ˜n, F˜n zn ≤ G ˜ n, V˜n ∆un ≤ W

TABLE II

(21)

˜ n ) and (V˜n , W ˜ n ) define the linear constraints where (F˜n , G along the prediction window. The rate-based receding horizon optimal control problem with prediction, control, and constraint horizon equal to Np subject to a quadratic cost is [22]:  t+Np −1 X   > > min x ˜t+Np P x ˜t+Np + zn Qzn    t+Np −1  {∆ui }i=t  n=t t+Np −1 (22) X   + ∆u>  n R∆un .    n=t  subject to (20) and (21) Remark 3: The MPC formulation given by (18)-(22) has integral action which mitigates the offset problem caused by the errors resulting from linearization [22]. Let the augmented system of (20) be denoted as (A, B, C). The rate-based MPC controller is designed by performing ˆ r , then Riccati balanced realization and truncation to obtain Σ ˆ ˆ ˆ using the augmented system (A, B, C) to form the predictions for a constrained convex quadratic programming problem to solve for ∆un . ∆un is added to un−1 to obtain un , the control at the specified time step. At the same time, a reduced order rate-based estimator is used. This is obtained by augmenting the Σr from the Riccati balanced realization and truncation, then solving (12) and forming (11) using (A, B, C). The rate-based estimator takes the same form as (10) and (9): ˆ x ˜k|k−1 ˆk|k x ˜

= =

A˜ xk−1|k−1 + B∆uk−1 , ˆk|k−1 + L(zk − Cx ˆ x ˜ ˜k|k−1 ).

E RROR DEFINED BY (25) USING A COMPENSATOR DESIGNED WITH A SYSTEM OF SPECIFIED ORDER (×106 ).

(23) (24)

Remark 4: One might wonder why the Riccati balanced truncation model order reduction is not applied directly to (18)-(22) which does not have a direct feedthrough term? With the addition of the “integrator” states in the augmented model, numerical issues arise in solving for the balancing transformation, often making it difficult and numerically unstable solving the Riccati equations associated with (18)(22). For performance evaluation in this paper, three different compensators are designed using the linear model given in Table I. The parameters used are: Np = 2, Ts = 0.016, Q = diag(1, 10), R = diag(10, 20), Λ = I2×2 , and Γ = I2×2 . 1) Compensator I: Designed with a reduced order model using open-loop balanced realization and truncation, 2) Compensator II: Designed with a reduced order model using Riccati balanced realization and truncation, and 3) Compensator III: Designed with the full order model of Table I. In Table I the RSVs, µi , are defined with parameters provided above. Plots of the output responses and inputs are provided in Fig. 2 for a 2nd order compensator applied to the nonlinear

Order 8 7 6 5 4 3 2 1

Compensator I 3.0174 3.0174 3.0174 3.0173 3.0259 3.0046 3.0366 -

Compensator II 2.8898 2.8898 2.8898 2.8898 2.8831 2.8842 2.8556 -

system when a reduced linear compensator is used. The following observations are noteworthy: • As seen in Fig. 2(a)-2(d), all compensators have similar performance, but Compensator II has smaller oscillations and converges to the reference faster when compared to Compensator I. • Fig. 2(e)-2(h) are the zoomed in plots for inputs and outputs over a small time window. To notice is that Compensator II has a slightly larger overshoot, but that it converges faster to the reference when compared to Compensator I. In Fig. 2(e) and 2(f) we see that that Compensator II is much closer to the performance of Compensator III. • Fig. 2(g) shows that VGT duty cycle varies considerably more for Compensator I, possibly causing excessive wear and tear on the VGT. To get a feeling for computational demand consider the operation count of forming the QP problem and calculating the observer which has a flop count of O(r2 ). When the augmented model is used, this results in r = 10 in the full order case to r = 4 for the 2nd order DAP model. Table II provides assessment of performance evaluated by (25), J =

N X

(yn − rn )> Q(yn − rn ) + ∆u> n R∆un ,

(25)

n=1

where N is the number of time steps used in the simulation. The mismatch between the two 8th order (full order) compensators is caused by accounting for the direct feedthrough in the balancing transformation. The non-monotonic behavior of the cost function is a result of constraints. Riccati balanced realization and truncation provides performance improvement, measured by reduced J of (25), for all reduced order models, compared to the open loop reduction approach. V. C ONCLUSIONS The paper provides an extension of the closed-loop model order reduction given by [9] to account for the direct feedthrough from input to output. The link between infinite time LQR and finite time MPC cost function was exploited to adapt closed-loop MOR to MPC through the use of a terminal state penalty. The developed procedure is applied for controlling a nonlinear DAP model using an MPC controller designed with

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160 140 120

0.01

80 VGT duty [%]

15 EGR rate [%]

Intake Pressure [kPa]

200 180

90

Compensator I Compensator II Compensator III Reference

10

0.008 EGR Flow [kg/s]

20

220

70

60

5 50

100 80 0

50

100 Time [s]

150

0 0

200

50

(a) Intake Manifold Pressure

100 Time [s ]

150

0.004

0.002

40 0

200

0.006

50

(b) EGR Rate

100 Time [s]

150

0 0

200

50

(c) VGT Duty

100 Time [s]

150

200

(d) EGR Flow −3

11

155 150

10 9 8 7

145

80 VGT duty [%]

160

EGR rate [%]

Intake Pressure [kPa]

Compensator I Compensator II Compensator III Reference

12

165

8.5 EGR Flow [kg/s]

170

x 10

82

78 76 74 72

8 7.5 7 6.5

6 140 50

52

(e) Intake (Zoomed)

54

56 Time [s ]

58

Manifold Fig. 2.

60

62

Pressure

50

52

54

56 Time [s ]

58

60

62

70 50

(f) EGR Rate (Zoomed)

52

54

56 Time [s]

58

60

(g) VGT Duty (Zoomed)

62

6 50

52

54

56 Time [s]

58

60

62

(h) EGR Flow (Zoomed)

Comparison of outputs and inputs when applying a 2nd order compensator to the nonlinear plant model.

a reduced order linearization of the model, demonstrating the benefit of using an estimator and MPC designed with the proposed Riccati balanced truncation over the open-loop reduction technique. VI. ACKNOWLEDGEMENTS The authors’ wish to thank Dr. Hayato Nakada of Toyota Motor Corporation for fruitful discussions. R EFERENCES [1] J. L. Sullivan, R. E. Baker, B. A. Boyer, R. H. Hammerle, T. E. Kenney, L. Muniz, and T. J. Wallington, “CO2 emission benefit of diesel (versus gasoline) powered vehicles,” Environmental Science & Technology, vol. 38, no. 12, pp. 3217–3223, 2004. [2] M. Zheng, G. T. Reader, and J. Hawley, “Diesel engine exhaust gas recirculationa review on advanced and novel concepts,” Energy Conversion and Management, vol. 45, no. 6, pp. 883 – 900, 2004. [3] P. Ortner, P. Langthaler, J. V. Garcia Ortiz, and L. del Re, “MPC for a diesel engine air path using an explicit approach for constraint systems,” in Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006 IEEE, Oct 2006, pp. 2760– 2765. [4] P. Ortner and L. del Re, “Predictive control of a diesel engine air path,” Control Systems Technology, IEEE Transactions on, vol. 15, no. 3, pp. 449–456, May 2007. [5] M. Huang, H. Nakada, S. Polavarapu, R. Choroszucha, K. Butts, and I. Kolmanovsky, “Towards combining nonlinear and predictive control of diesel engines,” in American Control Conference (ACC), 2013. IEEE, 2013, pp. 2846–2853. [6] M. Huang, H. Nakada, S. Polavarapu, K. R. Butts, and I. V. Kolmanovsky, “Rate-based model predictive control of diesel engines,” 7th IFAC Symposium on Advances in Automotive Control, vol. 7, pp. 177–182, 2013. [7] J. A. DeCastro, “Rate-based model predictive control of turbofan engine clearance,” Journal of propulsion and power, vol. 23, no. 4, pp. 804–813, 2007. [8] E. Verriest, “Suboptimal LQG-design via balanced realizations,” in Decision and Control including the Symposium on Adaptive Processes, 1981 20th IEEE Conference on, vol. 20. IEEE, 1981, pp. 686–687.

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