Discrete-Time Control Design for Setpoint Tracking of a Combustion ...
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007
ThB18.4
Discrete-time control design for setpoint tracking of a combustion engine test bench Dina Shona Laila and Engelbert Gr¨unbacher Abstract— A sampled-data control redesign for a combustion engine test bench is studied. The design applies a model reference approach, based on a continuous-time controller that has been designed using a robust optimal control design. The goal of the redesign is to make the discrete-time controller less sensitive to sampling and to keep the sampled-data tracking performance as close as possible to the continuous-time tracking performance. The performance of the redesigned controller is tested for a setpoint tracking of the speed and the torque of the test bench. Some numerical simulations are performed, comparing the performance of the discrete-time redesigned controller with a discrete-time emulation controller obtained by sampling and zero order hold of the continuous-time controller. Keywords: Combustion engine test bench control; Controller redesign; Discrete-time systems; Extended Hammerstein systems; Nonlinear systems; Setpoint tracking.
I. I NTRODUCTION Nonlinear phenomena are inevitably found in various mechanical systems in which a combustion engine belongs to. Although in many control applications linearization approach has been used for such systems, with the requirement for more sophisticated system performance and the prevalence of computer controlled systems, the application of nonlinear control designs for engines has become a better option. In this paper, we study a discrete-time nonlinear control design for a combustion engine test bench. As combustion engines are widely used in automotive as well as industrial applications, the topic has attracted a lot of researchers to study the control problems of the engine as well as the engine test bench (see [1]–[3] and references therein). In this study, we extend our previous result on a discretetime regulation design [4] to a tracking problem. We base our work on the recent work [5], where a continuous-time robust optimal controller has been designed for a combustion engine test bench, using the extension of the nonlinear robust optimal control design tools introduced in [6], [7]. In the application, the controller needs to be implemented digitally using computer. However, it is known that discretization would deteriorate the controller performance and due to the requirement to solve the Hamilton-Jacobi partial differential equation, it is not easy to construct a discrete-time counterpart of the available nonlinear continuous-time robust optimal control design tools. On the other hand, it is known that direct discrete-time design and discrete-time redesign D. S. Laila is with the Electrical & Electronic Engineering Department, Imperial College London, Exhibition Road, London SW7 2AZ, UK. Email: [email protected]. E. Gruenbacher is with the Institute f¨ur Design und Regelung Mechatronischer Systeme, Johannes Kepler Universit¨at Linz, Altenberger Strasse 69, A-4040 Linz, Austria. Email: [email protected].
1-4244-1498-9/07/$25.00 ©2007 IEEE.
usually yield better performance than discrete-time emulation [8]–[10] and this has been the main motivation for this work. In [4], a model reference controller redesign proposed in [11] has been applied, extending the original single input setup to a multi input setup which is required to apply the result to the test bench that is a multi input system. The construction makes use of the Fliess series expansion of the Lyapunov difference of the sampled-data system consisting of the continuous-time plant and the discrete-time controller to be designed. The design aims at obtaining a discretetime controller for sampled-data implementation that is less sensitive to sampling and preserves as much as possible the performance of the continuous-time controller. It has been shown in [4] that a discrete-time controller design using this approach yields the improvement to the closed-loop response of the system, compared to a discretetime controller that is obtained by sample and hold of the continuous-time controller obtained in [5]. In this paper, the design is extended to apply to a tracking problem rather than a regulation problem. The robustness of the controller with respect to a constant disturbance is also tested. The paper is constructed as follows. In Section 2 we present the technical tools needed to construct our main results. Section 3 is dedicated to present the model of the combustion engine test bench and the controller redesign. Simulation results are presented in Section 4 and we close with summary in Section 5. II. N OTATION AND T OOLS The set of real and natural numbers (including 0) are denoted respectively by R and N. A function γ : R≥0 → R≥0 is of class K if it is continuous, strictly increasing and zero at zero. It is of class K∞ if it is of class K and unbounded. Functions of class K∞ are invertible. A function β : R≥0 × R≥0 → R≥0 is of class KL if β(·, t) is of class K for each t ≥ 0 and β(s, ·) is decreasing to zero for each s > 0. Note that we often drop the time arguments t, k (or kT ) whenever they are clear from the context. Consider a general input affine nonlinear system x˙ = f (x) + g(x)u , y = h(x) ,
(1)
where x ∈ Rn is the state, u ∈ Rm is the control input and y ∈ Rp is the output. The functions f , g and h are smooth and f is zero at zero. Suppose that the family of discrete-time models of the system (1) is
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xk+1 = FT (xk , uk ) , yk = h(xk ) ,
(2)
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 where T > 0 is a constant sampling period. We assume that the control u = uk is constant during sampling intervals [kT, (k + 1)T ), k ≥ 0 and the output yk := y(kT ) is measured at every sampling instant kT . If the input u is a state feedback controller, we write the closed loop system of (1) and (2) respectively as x˙ = f˜(x)
(3)
xk+1 = F˜T (xk ) .
(4)
ThB18.4 linear part replaced by nonlinear dynamics of the process, and with the additional state feedback on its configuration. A schematic diagram of the extended Hammerstein system is illustrated in Figure 1.
and We use the following definitions throughout the paper. Definition 2.1 (Asymptotic stability): The system (3) is said to be asymptotically stable if there exists a class KL function β(·, ·) such that for any x◦ ∈ Rn , the solution x(t) of (3) with initial condition x(0) = x◦ satisfies an estimate of the form |x(t)| ≤ β(|x◦ | , t) (5) for all t ≥ 0. Definition 2.2 (AS Lyapunov function): A continuous and differentiable function V : Rn → R is called an asymptotic stability (AS) Lyapunov function for the continuous-time system (3) if there exist class K∞ functions α1 (·), α2 (·) and α3 (·) such that the following holds α1 (|x|) ≤ V (x) ≤ α2 (|x|) , ∂V ˜ f (x) ≤ −α3 (|x|) , ∂x
(6) (7)
for all x ∈ Rn . Definition 2.3 (SPAS Lyapunov function): A continuous and differentiable function VT : Rn → R is called a semiglobal practical asymptotic stability (SPAS) Lyapunov function for the discrete-time system (4) if there exist class K∞ functions α1 (·), α2 (·) and α3 (·) such that for any strictly positive numbers ∆ and ν there exists T ∗ > 0 such that the following holds α1 (|x|) ≤ VT (x) ≤ α2 (|x|) , VT (F˜T (x)) − VT (x) ≤ −T α3 (|x|) + T ν , for all T ∈ (0, T ∗ ) and all x ≤ ∆.
(8) (9)
A. Extended Hammerstein systems Motivated by practical experiences in the engine control field [12], the engine model in this paper is represented by a special cascaded class called an extended Hammerstein system. Hammerstein systems arise quite naturally in practice, considering that many technical systems are indeed a chain of subsystems and that a cascaded representation can offer the advantage of simplicity while retaining the essential characteristics of the plant [13]. A standard Hammerstein model assumes the process can be decomposed into two parts, a static part which contains all the nonlinearities followed by a linear part which contains all the dynamics of the process. This class of systems is commonly used to model systems in process control. The extended Hammerstein model used in this paper allows the
Fig. 1.
Structure of the extended Hammerstein system.
It can be seen from this figure that the state space representation of the system can then be written as x˙ = f (x) + g(x)m(x, u ¯) , y = h(x) .
(10)
Comparing with the system (1), the nonlinear static map m(x, u ¯) can be seen as the input to the system if it is invertible. We will use this approach to build the model of the combustion engine test bench. B. Model reference based redesign This model reference based controller redesign is proposed in [11] for single input systems, which can be extended directly for multi inputs systems. The goal is to make the sampled-data Lyapunov difference as close as possible to the continuous-time Lyapunov difference, in order to keep the sampled-data response to be as close as possible to the ideal continuous-time response. The following result is a slight generalization of Theorem 4.11 of [11] to multi inputs systems. First we state the assumption needed for this result to hold. Assumption 2.1: Suppose that a continuous-time static state feedback controller u = v0 (x) has been designed for system (1) so that the following holds. 1) There exits a Lyapunov function V (·), the functions α1 , α2 , α3 ∈ K∞ satisfying the Lyapunov characterization of asymptotic stability in Definition 2.2. 2) There exist a KL function β as in Definition 2.1 such that the system has satisfied all performance specifications in terms of overshoot and speed convergence. 3) The controller is to be implemented digitally using a sampler and zero order hold, that is for a given sampling period T > 0, the states xk := x(kT ) are available through measurement and the input u(t) = u(kT ) is constant for t ∈ [kT, (k + 1)T ), k ∈ N0 . The extension of [11, Theorem 4.11] to multi input systems is stated below, where the following notations are used ∆Vdt (T, x, u) := V (FT (x, u)) − V (x) , ∆Vct (T, x) := V (φ(T, x)) − V (x) . Theorem 2.1: [11] Consider a nonlinear input affine system (1) with multiple inputs (m ≥ 1). Suppose that Assumption 2.1 holds. Then we have
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∆Vct (T, x) − ∆Vdt (T, x, v0 (x)) = O(T 2 ) .
(11)
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Defining the redesigned controller by udt (x) = v0 (x) + T v1 (x) with v1 (x) =
1 ∂v0 (x) [f (x) + g(x)v0 (x)] , 2 ∂x
(12) (13)
then we have ∆Vct (T, x) − ∆Vdt (T, x, udt (x)) = O(T 3 ) .
(14)
Moreover, the controller (12) is a SPAS controller for system (1) in a sampled-data configuration. Sketch of the proof of Theorem 2.1: The proof for showing that (14) holds follows closely the proof of [11, Theorem 4.11], by adapting the dimension of the inputs from 1 (single input) to m (multi inputs) as u ∈ Rm . From Assumption 2.1(1) we have that v0 (x(t)) is an AS controller for plant (1). From [14], it is known that asymptotic stability is preserved in a semiglobal sense under sampling. Therefore, v0 (x(k)) which is the sampled and zero order hold version of v0 (x(t)) and satisfies (11) is a SPAS controller for the discrete-time model (2). Since the redesign controller (12) satisfies (14), it is obvious that it is also a SPAS controller for (2). Moreover, we utilize the framework [8] (see [9, Theorems III.1 and III.2] for a simpler formulation of the framework) and [10] to show that the discrete-time controller (12) is a SPAS controller for the continuous-time plant system (1) in a sampled-data configuration. Note that in the construction of the controller (12) we are not using a discrete-time model of the plant, but using the Fliess series expansion of the Lyapunov difference, which is one step consistent [8], [15] with the exact discretization of the Lyapunov derivative and serve as the replacement of the approximate model of the plant (see also comments in Section 3 of [11]). Remark 2.1: It has been commented in [11] that the extra term (13) is in fact the first derivative of the original controller v0 . Therefore by construction, the controller (12) is a PD type of the controller v0 (x) with the differential control gain equal to, or in a more general setting depending on, the sampling period T . C. Robustness to constant disturbances It is implied by Theorem 2.1 that the Lyapunov difference of the closed-loop system with the controller (12) is negative definite with a small constant offset (negative definite in a practical sense). This negative definiteness implies that the controller possess a robustness property to an exogenous disturbance in an input to state stability sense [9]. However, this does not imply the compensation to the disturbance, particularly to a constant disturbance. Since constant disturbances often appear in practice due to modeling uncertainty, robustness to constant disturbances is very important in the combustion engine test bench control applications. To compensate a constant disturbance, a classical approach of using an integral control is used. In this case, the controller can be seen as a nonlinear PI controller that takes form Z ∞ uP I (x) = udt (x) + kI udt (x(τ ))dτ , (15) 0
ThB18.4 where kI > 0 is the integral controller gain. Remark 2.2: From Remark 2.1, we can view the controller (12) as a PD type controller with the extra term (13) providing the derivative control action. It is then clear that including the extra term (13) in the construction of the integral control is like canceling the derivative action to the controller. From this point of view, in the implementation of the PI controller, it is better to exclude the extra term from the integration. In this way, the redesign controller plus the integral action takes form Z ∞ uP I,dt (x) = v0 (x) + T v1 (x) + kI v0 (x(τ ))dτ , (16) 0
which can be seen as a nonlinear PID type controller.
III. C ONTROL DESIGN FOR THE ENGINE TEST BENCH A. Engine test bench model A simple schematic diagram of the combustion engine test bench is illustrated in Figure 2. The main parts of such
Fig. 2.
The combustion engine test bench system
a dynamical engine test bench are the dynamometer, the connection shaft and the combustion engine itself. One of the control design objectives for a dynamical engine test bench control is to stabilize the engine torque and the engine speed. Considering the torque of the combustion engine and the air gap torque of the dynamometer as the inputs to the mechanical part of the engine test bench system, the dynamical model of the engine can be represented by a two mass oscillator ∆ψ˙ = ωE − ωD (17) 1 (TE − c∆ψ − d(ωE − ωD )) (18) ω˙ E = θE 1 (c∆ψ + d(ωE − ωD ) − TDSet ) , (19) ω˙ D = θD where ∆ψ is the rotation angle, ωE and ωD are respectively the engine and the dynamometer angular velocity, TE is the engine’s torque, TDSet is the air gap torque of the dynamometer, θE and θD are the inertia of the engine and the dynamometer, respectively. The dynamical model of the combustion engine test bench is described by T˙E = −ρ(TEstat , ωE )TE + ρ(TEstat , ωE )TEstat , where TEstat is the output of the static engine map and ρ(TEstat , ωE ) is the nonlinear state and input depending
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
ThB18.4
eigenvalue. Under some assumptions this dynamical model is approximated by the class of the extended Hammerstein systems (see [12] for more details) 2 T˙E = −(c0 + c1 ωE + c2 ωE )TE + m(ωE , TE , α) .
For the engine system (22), we have −(˜ c0 + c˜1 x3 + c˜2 x23 )x1 − γ1 x3 − γ2 x23 β(x3 − x4 ) c c 1 f = 1 − β x2 − d(x3 − x4 ) θE βx c 1 x + d(x − x ) 2 3 4 θD β
(20)
We define the state normalization as follows [5], TE − TE0 , ∆TE ωE − ωE0 x3 = , ∆ωE x1 =
∆ψ − ∆ψ0 , max(∆ψ) ωD − ωD0 x4 = , ∆ωD
and
x2 =
(21)
with TE0 , ∆ψ0 , ωE0 and ωD0 defines the operating point and ∆TE , max(∆ψ), ∆ωE and ∆ωD the maximum expected distance from the equilibrium point. With this scaling and taking c max(∆ψ) = ∆TE , the system (17)-(20) can now be represented as an extended Hammerstein system as follows x˙ 1 = − (˜ c0 + c˜1 x3 + c˜2 x23 )x1 − γ1 x3 − γ2 x23 + u1 x˙ 2 =β(x3 − x4 ) c c 1 x1 − x2 − d(x3 − x4 ) x˙ 3 = θE β β 1 c x˙ 4 = x2 + d(x3 − x4 ) + u2 , θD β
and c˜0 , c˜1 , c˜2 , β, γ1 , γ2 are positive constants. Note that the system has two inputs, and hence it is a multi inputs system. B. Discrete-time controller redesign In the same way as [5], a continuous-time controller has been constructed to satisfy some robust optimal design criteria. The control Lyapunov function used for designing the controller is V (x1 , x2 , x3 , x4 ) =
k1 x21
+ k2 x22 + k3 x23 +
k4 x24 + k5 x2 x4 ,
(23)
with ki ∈ R+ , i = 1 · · · 4 and k5 ∈ R − {0}. The positive definiteness of V (·) is guaranteed for some k5 with |k5 | sufficiently small. The controller takes form ∂V (x) u = −R(x)g(x)⊤ ∂x v 2r1 k1 x1 := 0,1 = v0 , =− v0,2 r2 (2k4 x4 + k5 x2 )
(24)
taking the positive diagonal matrix R = diag[r1 , r2 ]. Note that as the controller is to be implemented digitally, it is obvious that sampling will degrade the performance of the overall system [14]. For that, to retain as much as possible the continuous-time performance, we apply Theorem 2.1 to redesign a discrete-time controller based on the continuoustime controller (24).
(26)
1 ∂v0 (x) v1 = [f (x) + g(x)v0 (x)] 2 ∂x r1 k1 2r1 k1 x1 + (˜ c0 + c˜1 x3 + c˜2 x23 )x1 +γ1 x3 + γ2 x23 = 1 − 2 r2 k5 β(x3 − x4 ) cx2 +d(x −x ) 3 4 +r2 k4 r2 (2k4 x4 + k5 x2 ) − β θD v1,1 := . v1,2
with the inputs m(x1 , x3 , α) − m(0, 0, α0 ) , ∆TE TDSet − TD0 u2 = − , θD ∆ωD
0 0 . 0 1
By using this information, we can compute the extra terms of the redesigned controller (12), i.e.
(22)
u1 =
1 0 g= 0 0
(25)
Hence we have that the discrete-time redesigned controller takes form udt,1 v0,1 + T v1,1 udt = = = v0 + T v 1 . (27) udt,2 v0,2 + T v1,2 Remark 3.1: It has been shown in [4] that the redesign controller (27) outperforms the emulation controller. As this redesign approach depends on the choice of the control Lyapunov function used for the design, one may argue that using the control Lyapunov function V (x1 , x2 , x3 , x4 ) = k1 x21 + k2 x22 + k3 x23 + k4 x24 + k5 x2 x4 + k6 x21 x23 + k7 x1 x23 , as the one used in [5] might yield an emulation controller that performs as good as the redesign controller as in this case v0 will also contains the nonlinear terms appearing on the dynamics of x1 . However, this is not the case as those terms have only a minor effect as k6 and k7 have to be sufficiently small to retain the positive definiteness of the control Lyapunov function. Note that the controller (27) is designed to asymptotically stabilize the normalized model (22) of the engine. As our main objective is to apply the controller to the engine test bench, we need to transform back the normalized model of the test bench and test the stability of tracking of the original system. From the state transformation (21), we have the following relations 2 m(ωE , TE , α) = u1 ∆TE + TE0 (c0 + c1 ωE0 + c2 ωE0 ) (28) TDSet = −u2 θD ∆ωD + TD0 ,
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 where we have chosen ∆ψ0 = TE0 c , TD0 = TE0 and ωE0 = ωD0 . The setpoint tracking aims to follow the changing of operating points (TE0 , ωE0 ) of the engine.
ThB18.4 (b) − Engine Speed (rpm)
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IV. S IMULATION RESULTS In [4], the performance of the discrete-time redesigned controller (27) with the discrete-time emulation of the controller (24) has been compared, using the continuous-time controller (24) as the reference. The comparison is done by applying the controllers to stabilize the normalized model (22). In this setting, it has been shown that the controller (27) outperforms the emulation of the controller (24). In this section, the controllers are applied to control the original continuous-time model of the engine test bench (20), (17)-(19). The performance of the redesign controller is again compared to the emulation controller for a setpoint tracking assignment in a disturbance free case, as well as in the case when a constant disturbance is affecting the dynamic of the engine torque. TABLE I T HE ENGINE ’ S PARAMETERS Parameter Engine inertia (θE ) Dynamometer inertia (θD ) Damping constant (d) Stiffness of the shaft (c)
Value 0.32 0.28 3.5505 1.7441 × 103
Unit kgm2 kgm2 Nms/rad Nm/rad
The parameters of the test bench and the controllers are given on Table I and Table II, respectively. The engine parameters are based on a dynamic test bench with a production BMW M47D diesel engine. The coefficients of the approximate dynamic model after scaling are c˜0 = 6.3466, c˜1 = 3.2096, c˜2 = 2.7744, β = 1.8264 × 103 , γ1 = 4.8143 and γ2 = 4.1616. TABLE II T HE CONTROLLERS ’ PARAMETERS Parameter k1 k3 k5 r1
Value 1.56860 0.88000 −0.01450 1
Parameter k2 k4 − r2
Value 0.00174 1.05000 − 0.6
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points to the reference signal. This is consistent with the simulation results shown in [4] where the response of the engine test bench with the emulation controller is more oscillating than the response with the redesigned controller. It is seen that the extra terms of the redesigned controller reduces the oscillation which is a typical effect of a derivative controller. Note that the response of the engine speed ωE seems slower than the response of the engine torque TE . This is due to the fact that the dynamical model of the engine speed has larger time constant. Also, although theoretically it is possible to increase the magnitude of TDSet to make the engine speed responses faster, it is impossible in practice as in reality TDSet cannot be larger than 300 Nm. The control signals m(ωE , TE , α) and TDSet (without saturation) are plotted in Figures 3(c) and 3(d) respectively. Observing the control laws (24) and (27), it can be seen that (24) is in fact a linear controller whereas (27) is a nonlinear controller that captures more the nonlinearity of the test bench model, especially the dynamic of the states that are affected directly by the control action. This explain why the redesign controller performs better than the emulation controller, particularly in keeping the state TE close to the continuous-time response. B. Setpoint tracking and robustness to constant disturbance
A. Stability test with respect to step reference inputs Some simulations have been performed using the given parameters, starting from the initial condition [TE0 , ∆ψ0 , ωE0 , ωD0 ]⊤ = [50, TE0 /c, 3000, ωE0 ]⊤ . The control objective is to stabilize the engine with respect to the change of the operating point of the engine from ωE = 3000rpm and TE = 50Nm to ωE = 2500rpm and TE = 150Nm at time t = 2 sec. Figure 3 shows the responses of the engine when applying the discrete-time controllers. It is shown particularly on Figure 3(a) that with the redesign controller the engine torque increases more smoothly than with the emulation controller. Note that the arrow on the figure (and also other figures)
The stability test done in the previous subsection has shown that the discrete-time redesign controller performs better than the emulation controller. In this subsection, we apply the controller for a setpoint tracking, when we change the operating point (TE , ωE ) of the engine each following a square sequence reference signal. 1) Disturbance free setpoint tracking: In Figure 4 , the tracking performance in the case without disturbance is shown. The reference signals are pointed by the arrow in the figures. It is again shown by Figures 4(a) and 4(b) that the redesign controller keep the tracking behavior more similar to the continuous-time controller. The extra term that provide derivative action yields smoother transient tracking response.
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 In fact, Figures 4(c) and 4(d) show that the redesign control signals are lower than the emulation control signals, which is preferable in practice especially due to the limitation on TDSet . (a) − Engine Torque (Nm)
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R EFERENCES 2) Robustness under constant disturbance: Using the same reference signal, we assume that a constant disturbance w = 100 is affecting the dynamic of the engine torque due to the model uncertainty of the input m(ωE , TE , α). The response of the closed-loop systems with respect to the disturbance is shown in Figure 5. The left hand side figure shows the response with the controller without integral action, and the right hand side figure shows the response when integral action is added to the controller. It can be seen that the tracking error that appears when control without integral action is compensated by the integral action. Note that the curve that occurs at t = 8 sec is due the change of setpoint of the engine speed. The control signals are shown in Figure 6. The left hand side figure shows the control signal without integral action and the right hand side figure shows the nonlinear PI type control action. Note that we implement the controller (16) rather than (15) for the discrete-time redesign controller. Engine Torque − TE (Nm)
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Fig. 5. Tracking a square sequence under constant disturbance with sampling period T = 0.3 sec.
V. S UMMARY
[1] D. Carlucci, F. Donati, and M. Peisino, “Digital control of a test bench for a diesel engine,” in Modelling, Identification and Control. Proc. IASTED Int. Symp., 1984, pp. 248–251. [2] L. Guzzella and A. Amstutz, “Control of diesel engines,” Control Syst. Mag., vol. 18, pp. 53–71, 1998. [3] R. Outbib, X. Dovifaaz, A. Rachid, and M. Ouladsine, “A theoretical control strategy for a diesel engine,” Trans. ASME Dynamic Systems, Meas. and Control, vol. 128, pp. 453–457, 2006. [4] D. S. Laila, E. Gr¨unbacher, and L. del Re, “Discrete-time model reference controller design for a combustion engine test bench,” in Proc. IFAC Workshop on Nonlinear Control (NOLCOS), 2007. [5] E. Gruenbacher, P. Colaneri, and L. del Re, “Guaranteed robustness bounds for L2 and H∞ controlled matched-disturbance nonlinear systems,” Automatica, under revision, 2007. [6] A. Isidori and A. Astolfi, “Disturbance attenuation and H∞ control via measurement feedback in nonlinear systems,” IEEE Trans. on Automatic Control, vol. 37, pp. 1283–1293, 1992. [7] A. van der Schaft, “L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control,” IEEE Trans. on Automatic Control, vol. 37, pp. 770–784, 1992. [8] D. Neˇsi´c and A. R. Teel, “A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models,” IEEE TAC, vol. 49, pp. 1103–1122, 2004. [9] D. Neˇsi´c and D. S. Laila, “A note on input-to-state stabilization for nonlinear sampled-data systems,” IEEE Trans. Auto. Contr., vol. 47, pp. 1153–1158, 2002. [10] D. Neˇsi´c, A. R. Teel, and E. Sontag, “Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems,” Syst. Contr. Lett., vol. 38, pp. 49–60, 1999. [11] D. Neˇsi´c and L. Grune, “Lyapunov based continuous-time nonlinear controller redesign for sampled-data implementation,” Automatica, vol. 41, pp. 1143–1156, 2005. [12] E. Gruenbacher, Robust Inverse Control of A Class of Nonlinear Systems. PhD Thesis. Johannes Kepler Universit¨at, 2005. [13] M. Jankovi´c, R. Sepulchre, and P. V. Kokotovi´c, “CLF based design with robustness to dynamic input uncertainties,” Systems and Control Letters, vol. 37, pp. 45–54, 1999. [14] D. S. Laila, D. Neˇsi´c, and A. R. Teel, “Open and closed loop dissipation inequalities under sampling and controller emulation,” European Journal of Control, vol. 8, pp. 109–125, 2002. [15] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis. New York: Cambridge Univ. Press, 1996.
In this paper we have presented a discrete-time controller redesign for a combustion engine test bench system. The
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Discrete-time control design for setpoint tracking of a combustion engine test bench Dina Shona Laila and Engelbert Gr¨unbacher Abstract— A sampled-data control redesign for a combustion engine test bench is studied. The design applies a model reference approach, based on a continuous-time controller that has been designed using a robust optimal control design. The goal of the redesign is to make the discrete-time controller less sensitive to sampling and to keep the sampled-data tracking performance as close as possible to the continuous-time tracking performance. The performance of the redesigned controller is tested for a setpoint tracking of the speed and the torque of the test bench. Some numerical simulations are performed, comparing the performance of the discrete-time redesigned controller with a discrete-time emulation controller obtained by sampling and zero order hold of the continuous-time controller. Keywords: Combustion engine test bench control; Controller redesign; Discrete-time systems; Extended Hammerstein systems; Nonlinear systems; Setpoint tracking.
I. I NTRODUCTION Nonlinear phenomena are inevitably found in various mechanical systems in which a combustion engine belongs to. Although in many control applications linearization approach has been used for such systems, with the requirement for more sophisticated system performance and the prevalence of computer controlled systems, the application of nonlinear control designs for engines has become a better option. In this paper, we study a discrete-time nonlinear control design for a combustion engine test bench. As combustion engines are widely used in automotive as well as industrial applications, the topic has attracted a lot of researchers to study the control problems of the engine as well as the engine test bench (see [1]–[3] and references therein). In this study, we extend our previous result on a discretetime regulation design [4] to a tracking problem. We base our work on the recent work [5], where a continuous-time robust optimal controller has been designed for a combustion engine test bench, using the extension of the nonlinear robust optimal control design tools introduced in [6], [7]. In the application, the controller needs to be implemented digitally using computer. However, it is known that discretization would deteriorate the controller performance and due to the requirement to solve the Hamilton-Jacobi partial differential equation, it is not easy to construct a discrete-time counterpart of the available nonlinear continuous-time robust optimal control design tools. On the other hand, it is known that direct discrete-time design and discrete-time redesign D. S. Laila is with the Electrical & Electronic Engineering Department, Imperial College London, Exhibition Road, London SW7 2AZ, UK. Email: [email protected]. E. Gruenbacher is with the Institute f¨ur Design und Regelung Mechatronischer Systeme, Johannes Kepler Universit¨at Linz, Altenberger Strasse 69, A-4040 Linz, Austria. Email: [email protected].
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usually yield better performance than discrete-time emulation [8]–[10] and this has been the main motivation for this work. In [4], a model reference controller redesign proposed in [11] has been applied, extending the original single input setup to a multi input setup which is required to apply the result to the test bench that is a multi input system. The construction makes use of the Fliess series expansion of the Lyapunov difference of the sampled-data system consisting of the continuous-time plant and the discrete-time controller to be designed. The design aims at obtaining a discretetime controller for sampled-data implementation that is less sensitive to sampling and preserves as much as possible the performance of the continuous-time controller. It has been shown in [4] that a discrete-time controller design using this approach yields the improvement to the closed-loop response of the system, compared to a discretetime controller that is obtained by sample and hold of the continuous-time controller obtained in [5]. In this paper, the design is extended to apply to a tracking problem rather than a regulation problem. The robustness of the controller with respect to a constant disturbance is also tested. The paper is constructed as follows. In Section 2 we present the technical tools needed to construct our main results. Section 3 is dedicated to present the model of the combustion engine test bench and the controller redesign. Simulation results are presented in Section 4 and we close with summary in Section 5. II. N OTATION AND T OOLS The set of real and natural numbers (including 0) are denoted respectively by R and N. A function γ : R≥0 → R≥0 is of class K if it is continuous, strictly increasing and zero at zero. It is of class K∞ if it is of class K and unbounded. Functions of class K∞ are invertible. A function β : R≥0 × R≥0 → R≥0 is of class KL if β(·, t) is of class K for each t ≥ 0 and β(s, ·) is decreasing to zero for each s > 0. Note that we often drop the time arguments t, k (or kT ) whenever they are clear from the context. Consider a general input affine nonlinear system x˙ = f (x) + g(x)u , y = h(x) ,
(1)
where x ∈ Rn is the state, u ∈ Rm is the control input and y ∈ Rp is the output. The functions f , g and h are smooth and f is zero at zero. Suppose that the family of discrete-time models of the system (1) is
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xk+1 = FT (xk , uk ) , yk = h(xk ) ,
(2)
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 where T > 0 is a constant sampling period. We assume that the control u = uk is constant during sampling intervals [kT, (k + 1)T ), k ≥ 0 and the output yk := y(kT ) is measured at every sampling instant kT . If the input u is a state feedback controller, we write the closed loop system of (1) and (2) respectively as x˙ = f˜(x)
(3)
xk+1 = F˜T (xk ) .
(4)
ThB18.4 linear part replaced by nonlinear dynamics of the process, and with the additional state feedback on its configuration. A schematic diagram of the extended Hammerstein system is illustrated in Figure 1.
and We use the following definitions throughout the paper. Definition 2.1 (Asymptotic stability): The system (3) is said to be asymptotically stable if there exists a class KL function β(·, ·) such that for any x◦ ∈ Rn , the solution x(t) of (3) with initial condition x(0) = x◦ satisfies an estimate of the form |x(t)| ≤ β(|x◦ | , t) (5) for all t ≥ 0. Definition 2.2 (AS Lyapunov function): A continuous and differentiable function V : Rn → R is called an asymptotic stability (AS) Lyapunov function for the continuous-time system (3) if there exist class K∞ functions α1 (·), α2 (·) and α3 (·) such that the following holds α1 (|x|) ≤ V (x) ≤ α2 (|x|) , ∂V ˜ f (x) ≤ −α3 (|x|) , ∂x
(6) (7)
for all x ∈ Rn . Definition 2.3 (SPAS Lyapunov function): A continuous and differentiable function VT : Rn → R is called a semiglobal practical asymptotic stability (SPAS) Lyapunov function for the discrete-time system (4) if there exist class K∞ functions α1 (·), α2 (·) and α3 (·) such that for any strictly positive numbers ∆ and ν there exists T ∗ > 0 such that the following holds α1 (|x|) ≤ VT (x) ≤ α2 (|x|) , VT (F˜T (x)) − VT (x) ≤ −T α3 (|x|) + T ν , for all T ∈ (0, T ∗ ) and all x ≤ ∆.
(8) (9)
A. Extended Hammerstein systems Motivated by practical experiences in the engine control field [12], the engine model in this paper is represented by a special cascaded class called an extended Hammerstein system. Hammerstein systems arise quite naturally in practice, considering that many technical systems are indeed a chain of subsystems and that a cascaded representation can offer the advantage of simplicity while retaining the essential characteristics of the plant [13]. A standard Hammerstein model assumes the process can be decomposed into two parts, a static part which contains all the nonlinearities followed by a linear part which contains all the dynamics of the process. This class of systems is commonly used to model systems in process control. The extended Hammerstein model used in this paper allows the
Fig. 1.
Structure of the extended Hammerstein system.
It can be seen from this figure that the state space representation of the system can then be written as x˙ = f (x) + g(x)m(x, u ¯) , y = h(x) .
(10)
Comparing with the system (1), the nonlinear static map m(x, u ¯) can be seen as the input to the system if it is invertible. We will use this approach to build the model of the combustion engine test bench. B. Model reference based redesign This model reference based controller redesign is proposed in [11] for single input systems, which can be extended directly for multi inputs systems. The goal is to make the sampled-data Lyapunov difference as close as possible to the continuous-time Lyapunov difference, in order to keep the sampled-data response to be as close as possible to the ideal continuous-time response. The following result is a slight generalization of Theorem 4.11 of [11] to multi inputs systems. First we state the assumption needed for this result to hold. Assumption 2.1: Suppose that a continuous-time static state feedback controller u = v0 (x) has been designed for system (1) so that the following holds. 1) There exits a Lyapunov function V (·), the functions α1 , α2 , α3 ∈ K∞ satisfying the Lyapunov characterization of asymptotic stability in Definition 2.2. 2) There exist a KL function β as in Definition 2.1 such that the system has satisfied all performance specifications in terms of overshoot and speed convergence. 3) The controller is to be implemented digitally using a sampler and zero order hold, that is for a given sampling period T > 0, the states xk := x(kT ) are available through measurement and the input u(t) = u(kT ) is constant for t ∈ [kT, (k + 1)T ), k ∈ N0 . The extension of [11, Theorem 4.11] to multi input systems is stated below, where the following notations are used ∆Vdt (T, x, u) := V (FT (x, u)) − V (x) , ∆Vct (T, x) := V (φ(T, x)) − V (x) . Theorem 2.1: [11] Consider a nonlinear input affine system (1) with multiple inputs (m ≥ 1). Suppose that Assumption 2.1 holds. Then we have
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Defining the redesigned controller by udt (x) = v0 (x) + T v1 (x) with v1 (x) =
1 ∂v0 (x) [f (x) + g(x)v0 (x)] , 2 ∂x
(12) (13)
then we have ∆Vct (T, x) − ∆Vdt (T, x, udt (x)) = O(T 3 ) .
(14)
Moreover, the controller (12) is a SPAS controller for system (1) in a sampled-data configuration. Sketch of the proof of Theorem 2.1: The proof for showing that (14) holds follows closely the proof of [11, Theorem 4.11], by adapting the dimension of the inputs from 1 (single input) to m (multi inputs) as u ∈ Rm . From Assumption 2.1(1) we have that v0 (x(t)) is an AS controller for plant (1). From [14], it is known that asymptotic stability is preserved in a semiglobal sense under sampling. Therefore, v0 (x(k)) which is the sampled and zero order hold version of v0 (x(t)) and satisfies (11) is a SPAS controller for the discrete-time model (2). Since the redesign controller (12) satisfies (14), it is obvious that it is also a SPAS controller for (2). Moreover, we utilize the framework [8] (see [9, Theorems III.1 and III.2] for a simpler formulation of the framework) and [10] to show that the discrete-time controller (12) is a SPAS controller for the continuous-time plant system (1) in a sampled-data configuration. Note that in the construction of the controller (12) we are not using a discrete-time model of the plant, but using the Fliess series expansion of the Lyapunov difference, which is one step consistent [8], [15] with the exact discretization of the Lyapunov derivative and serve as the replacement of the approximate model of the plant (see also comments in Section 3 of [11]). Remark 2.1: It has been commented in [11] that the extra term (13) is in fact the first derivative of the original controller v0 . Therefore by construction, the controller (12) is a PD type of the controller v0 (x) with the differential control gain equal to, or in a more general setting depending on, the sampling period T . C. Robustness to constant disturbances It is implied by Theorem 2.1 that the Lyapunov difference of the closed-loop system with the controller (12) is negative definite with a small constant offset (negative definite in a practical sense). This negative definiteness implies that the controller possess a robustness property to an exogenous disturbance in an input to state stability sense [9]. However, this does not imply the compensation to the disturbance, particularly to a constant disturbance. Since constant disturbances often appear in practice due to modeling uncertainty, robustness to constant disturbances is very important in the combustion engine test bench control applications. To compensate a constant disturbance, a classical approach of using an integral control is used. In this case, the controller can be seen as a nonlinear PI controller that takes form Z ∞ uP I (x) = udt (x) + kI udt (x(τ ))dτ , (15) 0
ThB18.4 where kI > 0 is the integral controller gain. Remark 2.2: From Remark 2.1, we can view the controller (12) as a PD type controller with the extra term (13) providing the derivative control action. It is then clear that including the extra term (13) in the construction of the integral control is like canceling the derivative action to the controller. From this point of view, in the implementation of the PI controller, it is better to exclude the extra term from the integration. In this way, the redesign controller plus the integral action takes form Z ∞ uP I,dt (x) = v0 (x) + T v1 (x) + kI v0 (x(τ ))dτ , (16) 0
which can be seen as a nonlinear PID type controller.
III. C ONTROL DESIGN FOR THE ENGINE TEST BENCH A. Engine test bench model A simple schematic diagram of the combustion engine test bench is illustrated in Figure 2. The main parts of such
Fig. 2.
The combustion engine test bench system
a dynamical engine test bench are the dynamometer, the connection shaft and the combustion engine itself. One of the control design objectives for a dynamical engine test bench control is to stabilize the engine torque and the engine speed. Considering the torque of the combustion engine and the air gap torque of the dynamometer as the inputs to the mechanical part of the engine test bench system, the dynamical model of the engine can be represented by a two mass oscillator ∆ψ˙ = ωE − ωD (17) 1 (TE − c∆ψ − d(ωE − ωD )) (18) ω˙ E = θE 1 (c∆ψ + d(ωE − ωD ) − TDSet ) , (19) ω˙ D = θD where ∆ψ is the rotation angle, ωE and ωD are respectively the engine and the dynamometer angular velocity, TE is the engine’s torque, TDSet is the air gap torque of the dynamometer, θE and θD are the inertia of the engine and the dynamometer, respectively. The dynamical model of the combustion engine test bench is described by T˙E = −ρ(TEstat , ωE )TE + ρ(TEstat , ωE )TEstat , where TEstat is the output of the static engine map and ρ(TEstat , ωE ) is the nonlinear state and input depending
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eigenvalue. Under some assumptions this dynamical model is approximated by the class of the extended Hammerstein systems (see [12] for more details) 2 T˙E = −(c0 + c1 ωE + c2 ωE )TE + m(ωE , TE , α) .
For the engine system (22), we have −(˜ c0 + c˜1 x3 + c˜2 x23 )x1 − γ1 x3 − γ2 x23 β(x3 − x4 ) c c 1 f = 1 − β x2 − d(x3 − x4 ) θE βx c 1 x + d(x − x ) 2 3 4 θD β
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We define the state normalization as follows [5], TE − TE0 , ∆TE ωE − ωE0 x3 = , ∆ωE x1 =
∆ψ − ∆ψ0 , max(∆ψ) ωD − ωD0 x4 = , ∆ωD
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with TE0 , ∆ψ0 , ωE0 and ωD0 defines the operating point and ∆TE , max(∆ψ), ∆ωE and ∆ωD the maximum expected distance from the equilibrium point. With this scaling and taking c max(∆ψ) = ∆TE , the system (17)-(20) can now be represented as an extended Hammerstein system as follows x˙ 1 = − (˜ c0 + c˜1 x3 + c˜2 x23 )x1 − γ1 x3 − γ2 x23 + u1 x˙ 2 =β(x3 − x4 ) c c 1 x1 − x2 − d(x3 − x4 ) x˙ 3 = θE β β 1 c x˙ 4 = x2 + d(x3 − x4 ) + u2 , θD β
and c˜0 , c˜1 , c˜2 , β, γ1 , γ2 are positive constants. Note that the system has two inputs, and hence it is a multi inputs system. B. Discrete-time controller redesign In the same way as [5], a continuous-time controller has been constructed to satisfy some robust optimal design criteria. The control Lyapunov function used for designing the controller is V (x1 , x2 , x3 , x4 ) =
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(23)
with ki ∈ R+ , i = 1 · · · 4 and k5 ∈ R − {0}. The positive definiteness of V (·) is guaranteed for some k5 with |k5 | sufficiently small. The controller takes form ∂V (x) u = −R(x)g(x)⊤ ∂x v 2r1 k1 x1 := 0,1 = v0 , =− v0,2 r2 (2k4 x4 + k5 x2 )
(24)
taking the positive diagonal matrix R = diag[r1 , r2 ]. Note that as the controller is to be implemented digitally, it is obvious that sampling will degrade the performance of the overall system [14]. For that, to retain as much as possible the continuous-time performance, we apply Theorem 2.1 to redesign a discrete-time controller based on the continuoustime controller (24).
(26)
1 ∂v0 (x) v1 = [f (x) + g(x)v0 (x)] 2 ∂x r1 k1 2r1 k1 x1 + (˜ c0 + c˜1 x3 + c˜2 x23 )x1 +γ1 x3 + γ2 x23 = 1 − 2 r2 k5 β(x3 − x4 ) cx2 +d(x −x ) 3 4 +r2 k4 r2 (2k4 x4 + k5 x2 ) − β θD v1,1 := . v1,2
with the inputs m(x1 , x3 , α) − m(0, 0, α0 ) , ∆TE TDSet − TD0 u2 = − , θD ∆ωD
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By using this information, we can compute the extra terms of the redesigned controller (12), i.e.
(22)
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(25)
Hence we have that the discrete-time redesigned controller takes form udt,1 v0,1 + T v1,1 udt = = = v0 + T v 1 . (27) udt,2 v0,2 + T v1,2 Remark 3.1: It has been shown in [4] that the redesign controller (27) outperforms the emulation controller. As this redesign approach depends on the choice of the control Lyapunov function used for the design, one may argue that using the control Lyapunov function V (x1 , x2 , x3 , x4 ) = k1 x21 + k2 x22 + k3 x23 + k4 x24 + k5 x2 x4 + k6 x21 x23 + k7 x1 x23 , as the one used in [5] might yield an emulation controller that performs as good as the redesign controller as in this case v0 will also contains the nonlinear terms appearing on the dynamics of x1 . However, this is not the case as those terms have only a minor effect as k6 and k7 have to be sufficiently small to retain the positive definiteness of the control Lyapunov function. Note that the controller (27) is designed to asymptotically stabilize the normalized model (22) of the engine. As our main objective is to apply the controller to the engine test bench, we need to transform back the normalized model of the test bench and test the stability of tracking of the original system. From the state transformation (21), we have the following relations 2 m(ωE , TE , α) = u1 ∆TE + TE0 (c0 + c1 ωE0 + c2 ωE0 ) (28) TDSet = −u2 θD ∆ωD + TD0 ,
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 where we have chosen ∆ψ0 = TE0 c , TD0 = TE0 and ωE0 = ωD0 . The setpoint tracking aims to follow the changing of operating points (TE0 , ωE0 ) of the engine.
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IV. S IMULATION RESULTS In [4], the performance of the discrete-time redesigned controller (27) with the discrete-time emulation of the controller (24) has been compared, using the continuous-time controller (24) as the reference. The comparison is done by applying the controllers to stabilize the normalized model (22). In this setting, it has been shown that the controller (27) outperforms the emulation of the controller (24). In this section, the controllers are applied to control the original continuous-time model of the engine test bench (20), (17)-(19). The performance of the redesign controller is again compared to the emulation controller for a setpoint tracking assignment in a disturbance free case, as well as in the case when a constant disturbance is affecting the dynamic of the engine torque. TABLE I T HE ENGINE ’ S PARAMETERS Parameter Engine inertia (θE ) Dynamometer inertia (θD ) Damping constant (d) Stiffness of the shaft (c)
Value 0.32 0.28 3.5505 1.7441 × 103
Unit kgm2 kgm2 Nms/rad Nm/rad
The parameters of the test bench and the controllers are given on Table I and Table II, respectively. The engine parameters are based on a dynamic test bench with a production BMW M47D diesel engine. The coefficients of the approximate dynamic model after scaling are c˜0 = 6.3466, c˜1 = 3.2096, c˜2 = 2.7744, β = 1.8264 × 103 , γ1 = 4.8143 and γ2 = 4.1616. TABLE II T HE CONTROLLERS ’ PARAMETERS Parameter k1 k3 k5 r1
Value 1.56860 0.88000 −0.01450 1
Parameter k2 k4 − r2
Value 0.00174 1.05000 − 0.6
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points to the reference signal. This is consistent with the simulation results shown in [4] where the response of the engine test bench with the emulation controller is more oscillating than the response with the redesigned controller. It is seen that the extra terms of the redesigned controller reduces the oscillation which is a typical effect of a derivative controller. Note that the response of the engine speed ωE seems slower than the response of the engine torque TE . This is due to the fact that the dynamical model of the engine speed has larger time constant. Also, although theoretically it is possible to increase the magnitude of TDSet to make the engine speed responses faster, it is impossible in practice as in reality TDSet cannot be larger than 300 Nm. The control signals m(ωE , TE , α) and TDSet (without saturation) are plotted in Figures 3(c) and 3(d) respectively. Observing the control laws (24) and (27), it can be seen that (24) is in fact a linear controller whereas (27) is a nonlinear controller that captures more the nonlinearity of the test bench model, especially the dynamic of the states that are affected directly by the control action. This explain why the redesign controller performs better than the emulation controller, particularly in keeping the state TE close to the continuous-time response. B. Setpoint tracking and robustness to constant disturbance
A. Stability test with respect to step reference inputs Some simulations have been performed using the given parameters, starting from the initial condition [TE0 , ∆ψ0 , ωE0 , ωD0 ]⊤ = [50, TE0 /c, 3000, ωE0 ]⊤ . The control objective is to stabilize the engine with respect to the change of the operating point of the engine from ωE = 3000rpm and TE = 50Nm to ωE = 2500rpm and TE = 150Nm at time t = 2 sec. Figure 3 shows the responses of the engine when applying the discrete-time controllers. It is shown particularly on Figure 3(a) that with the redesign controller the engine torque increases more smoothly than with the emulation controller. Note that the arrow on the figure (and also other figures)
The stability test done in the previous subsection has shown that the discrete-time redesign controller performs better than the emulation controller. In this subsection, we apply the controller for a setpoint tracking, when we change the operating point (TE , ωE ) of the engine each following a square sequence reference signal. 1) Disturbance free setpoint tracking: In Figure 4 , the tracking performance in the case without disturbance is shown. The reference signals are pointed by the arrow in the figures. It is again shown by Figures 4(a) and 4(b) that the redesign controller keep the tracking behavior more similar to the continuous-time controller. The extra term that provide derivative action yields smoother transient tracking response.
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 In fact, Figures 4(c) and 4(d) show that the redesign control signals are lower than the emulation control signals, which is preferable in practice especially due to the limitation on TDSet . (a) − Engine Torque (Nm)
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R EFERENCES 2) Robustness under constant disturbance: Using the same reference signal, we assume that a constant disturbance w = 100 is affecting the dynamic of the engine torque due to the model uncertainty of the input m(ωE , TE , α). The response of the closed-loop systems with respect to the disturbance is shown in Figure 5. The left hand side figure shows the response with the controller without integral action, and the right hand side figure shows the response when integral action is added to the controller. It can be seen that the tracking error that appears when control without integral action is compensated by the integral action. Note that the curve that occurs at t = 8 sec is due the change of setpoint of the engine speed. The control signals are shown in Figure 6. The left hand side figure shows the control signal without integral action and the right hand side figure shows the nonlinear PI type control action. Note that we implement the controller (16) rather than (15) for the discrete-time redesign controller. Engine Torque − TE (Nm)
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Fig. 5. Tracking a square sequence under constant disturbance with sampling period T = 0.3 sec.
V. S UMMARY
[1] D. Carlucci, F. Donati, and M. Peisino, “Digital control of a test bench for a diesel engine,” in Modelling, Identification and Control. Proc. IASTED Int. Symp., 1984, pp. 248–251. [2] L. Guzzella and A. Amstutz, “Control of diesel engines,” Control Syst. Mag., vol. 18, pp. 53–71, 1998. [3] R. Outbib, X. Dovifaaz, A. Rachid, and M. Ouladsine, “A theoretical control strategy for a diesel engine,” Trans. ASME Dynamic Systems, Meas. and Control, vol. 128, pp. 453–457, 2006. [4] D. S. Laila, E. Gr¨unbacher, and L. del Re, “Discrete-time model reference controller design for a combustion engine test bench,” in Proc. IFAC Workshop on Nonlinear Control (NOLCOS), 2007. [5] E. Gruenbacher, P. Colaneri, and L. del Re, “Guaranteed robustness bounds for L2 and H∞ controlled matched-disturbance nonlinear systems,” Automatica, under revision, 2007. [6] A. Isidori and A. Astolfi, “Disturbance attenuation and H∞ control via measurement feedback in nonlinear systems,” IEEE Trans. on Automatic Control, vol. 37, pp. 1283–1293, 1992. [7] A. van der Schaft, “L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control,” IEEE Trans. on Automatic Control, vol. 37, pp. 770–784, 1992. [8] D. Neˇsi´c and A. R. Teel, “A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models,” IEEE TAC, vol. 49, pp. 1103–1122, 2004. [9] D. Neˇsi´c and D. S. Laila, “A note on input-to-state stabilization for nonlinear sampled-data systems,” IEEE Trans. Auto. Contr., vol. 47, pp. 1153–1158, 2002. [10] D. Neˇsi´c, A. R. Teel, and E. Sontag, “Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems,” Syst. Contr. Lett., vol. 38, pp. 49–60, 1999. [11] D. Neˇsi´c and L. Grune, “Lyapunov based continuous-time nonlinear controller redesign for sampled-data implementation,” Automatica, vol. 41, pp. 1143–1156, 2005. [12] E. Gruenbacher, Robust Inverse Control of A Class of Nonlinear Systems. PhD Thesis. Johannes Kepler Universit¨at, 2005. [13] M. Jankovi´c, R. Sepulchre, and P. V. Kokotovi´c, “CLF based design with robustness to dynamic input uncertainties,” Systems and Control Letters, vol. 37, pp. 45–54, 1999. [14] D. S. Laila, D. Neˇsi´c, and A. R. Teel, “Open and closed loop dissipation inequalities under sampling and controller emulation,” European Journal of Control, vol. 8, pp. 109–125, 2002. [15] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis. New York: Cambridge Univ. Press, 1996.
In this paper we have presented a discrete-time controller redesign for a combustion engine test bench system. The
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