Process and device for testing the powertrain of at least partially electrically driven vehicles

Abstract

In a process for testing the powertrain of vehicles that are at least in part electrically driven, the voltage supplied to the powertrain is controlled by a controller coupled with a simulation system for the energy storage system in a way that the voltage acts dynamically as for a real energy storage system. The controller is designed with a model based controller design method, with a load model of the powertrain being used in the model of the controlled system.

Claims

1. A method for testing a powertrain of vehicles that are at least in part electrically driven, comprising: controlling a voltage supplied to the powertrain by a controller coupled with a simulation system for an energy storage system in a manner that the voltage acts dynamically as for a real energy storage system, wherein the controller is designed with a model based controller design method as model predictive control; using a load model of the powertrain in a discrete time model of a controlled system, wherein the model of the controlled system has a parameter set with a parameter (g.sub.P) that is variable over an operating range of the controlled system; determining at each time instant (k) an optimal sequence of control moves (u.sub.k); calculating a number (i) of parameter sets with a different parameter (g.sub.P,i) over the operating range and choosing a parameter set with parameter (g.sub.P,i) which is closest to the actual parameter (g.sub.P); and determining the sequence of control moves (u.sub.k) with the chosen parameter set.

2. The method according to claim 1, wherein the voltage is measured, the method further comprising estimating a load power demand of the powertrain and modifying the parameter set of the model of the controller system in dependence of the voltage and the estimated load power demand, by switching between complete parameter sets.

3. The method according to claim 2, wherein the estimation of the load power demand is accomplished with an observer system, based on a measured load current.

4. A device for testing of a powertrain of vehicles that are at least in part electrically driven, comprising: a simulation system for an energy storage system, and a controller that is coupled with the simulation system; wherein the controller controls a voltage supplied to the powertrain in a manner that said voltage acts dynamically as for a real energy storage system; wherein a model based controller as a model predictive control is realized for a controlled system, and a load model of the powertrain is integrated in a model of the controlled system and at each time instant (k) an optimal sequence of control moves (u.sub.k) is determined by the model predictive control; wherein the model of the controlled system has a parameter set with a parameter (g.sub.P) that is variable over an operating range of the controlled system; and wherein the model predictive control chooses a parameter set with parameter (g.sub.P,i) which is closest to the actual parameter (g.sub.P) from a number (i) of calculated and over the operating range allocated parameter sets with different parameter (g.sub.P,i) for determining the sequence of control moves (u.sub.k).

5. The device according to claim 4, wherein a load power demand dependent model is integrated into the model of the controlled system.

6. The device according to claim 5, wherein the load model of the powertrain is integrated into the model of the controlled system that depends on an output voltage of the energy storage system.

7. The device according to claim 4, wherein a model for the model predictive control comprises a converter model including a constant power load with an input filter capacitance.

8. The device according to claim 7, wherein the converter model is based on a linearized negative impedance approximation of the constant power load which depends on an output voltage and a power demand of a load of the powertrain.

9. A device for testing of a powertrain of vehicles that are at least in part electrically driven, comprising: a simulation system for simulating an energy storage system; and a controller that is coupled with the simulation system, for controlling a voltage supplied to the powertrain in a manner that said voltage acts dynamically as for a real energy storage system; wherein the controller is designed as a model based controller as a model predictive control for a model of a controlled system that has integrated a load model of the powertrain; and wherein the model of the controlled system has a parameter set with a parameter that is variable over an operating range of the controlled system, one or more parameter sets are calculated for different parameters, an actual parameter is determined based on an estimate of a load power demand of the powertrain, and a parameter set closest to the actual parameter is selected from the calculated one or more parameter sets by the model predictive control to determine a next control move.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a powertrain testbed with battery emulator in schematic form, FIG. 2 shows a schematic diagram of the battery emulator output stage, FIG. 3 is an illustration of the load model. (a) Static characteristic of a constant power load. (b) Nonlinear load model. (c) Small signal model linearized around operating point, FIG. 4 is a block diagram of the battery emulator model, FIG. 5 shows a block diagram of the proposed controller structure, FIG. 6 is an illustration of the proposed algorithm applied to receding horizon control, where the upper row shows predicted output voltage trajectories and the bottom row shows corresponding control variable sequences, while the columns represent subsequent time samples, FIG. 7 shows a comparison of the proposed algorithm with an MPC using a generic QP solver, FIG. 8 is a block diagram of the robust controller concept, FIG. 9 shows a block diagram of the scheduling controller, FIG. 10 shows the simulation results for load power step changes from 0 kW to 60 kW with a constant power load, the plots on the top showing output voltage, inductor current and load current. The plots on the bottom show the applied duty cycle, FIG. 11 shows experimental results for reference step changes without a load connected to the battery emulator, FIG. 12 is a diagram of experimental results for a reference step change showing the effectiveness of the inductor current limiting, FIG. 13 shows experimental results for a sequence of reference step changes with a constant power load, with the plots at the bottom showing the scheduling variable for the scheduling controller and the duty cycle for the robust controller, and FIG. 14 depicts experimental results for load power step changes with a constant power load from P=0 kW to P=24 kW.

DETAILED DESCRIPTION

(2) An example of a typical testbed configuration is shown in FIG. 1. The plant consists of the power electronics of an HEV or EV powertrain that represent the load on one hand and the battery emulator that replaces the actual traction battery on the other hand. For controller design, the battery emulator and the load are modeled separately and then combined into one system model.

(3) A schematic diagram of the battery emulator is shown in FIG. 2. The main part is the output stage consisting of three interleaved step-down DC-DC converters with a common output capacitor C.sub.1 providing the output voltage v.sub.2 that is used to emulate the battery terminal voltage. The rectifier is not considered here and C.sub.0 is large enough such that the DC-link voltage V.sub.0 can be assumed to be constant.

(4) An analog modulator performs pulse width modulation (PWM) and inductor current balancing from a single duty cycle command d at its input. A simplified model suitable for real-time MPC is obtained by averaged switch modeling [R. W. Erickson and D. Maksimovi, Fundamentals of power electronics. Springer, 2001] and by paralleling the three inductors to one lumped inductor L.sub.1 as in [S. Mariethoz, A. Beccuti, and M. Morari, Model predictive control of multiphase interleaved dc-dc converters with sensorless current limitation and power balance, in Power Electronics Specialists Conference, 2008. PESC 2008. IEEE, 2008, pp. 1069-1074] or [H. Bae, J. Lee, J. Yang, and B. H. Cho, Digital resistive current (drc) control for the parallel interleaved dc-dc converters, Power Electronics, IEEE Transactions on, vol. 23, no. 5, pp. 2465-2476, 2008]. Then the sum of all three currents i.sub.1=i.sub.1a+i.sub.1b+i.sub.1c is chosen as the new inductor current. Ohmic resistances of inductors and semiconductor switches are approximated by R.sub.L1. The current drawn by the load is denoted by i.sub.2. With the state vector for the converter chosen as x.sub.c=[i.sub.1 v.sub.2].sup.T, control input u=d.Math.V.sub.0 and disturbance input i.sub.2, the system is described by

(5) $\begin{array}{cc}{\stackrel{.}{x}}_c=\underset{\underset{A_c}{}}{\left[\begin{array}{cc}-\frac{R_{L1}}{L_1}& -\frac{1}{L_1}\\ \frac{1}{C_1}& 0\end{array}\right]}{x}_c+\underset{\underset{B_c}{}}{\left[\begin{array}{c}\frac{1}{L_1}\\ 0\end{array}\right]}u+\underset{\underset{E_c}{}}{\left[\begin{array}{c}0\\ -\frac{1}{C_1}\end{array}\right]}{i}_2{v}_2=\underset{\underset{C_c}{}}{\left[\begin{array}{cc}0& 1\end{array}\right]}{x}_c.& \left(1\right)\end{array}$
The quantities i.sub.1, v.sub.2 and i.sub.2 can be measured.

(6) The HEV/EV electric motor inverter is a tightly regulated voltage source inverter with its DC-link connected to the BE. The inverter's power output P is independent of its supply voltage v.sub.2 as long as it is within a specified range. This configuration is modeled as a CPL for which the relation between the current .sub.2 drawn by the CPL and the supply voltage is found as

(7) $\begin{array}{cc}{\tilde{i}}_2=\frac{P}{v_2}.& \left(2\right)\end{array}$
Equation (2) closes a feedback loop from the output voltage to the load current disturbance input, which leads to the nonlinear state equation

(8) $\begin{array}{cc}{\stackrel{.}{x}}_c=A_c{x}_c+E_{c}f\left(x_c\right)+B_{c}u\mathrm{with}f\left(x_c\right)=\frac{P}{v_2\left(x_c\right)}.& \left(3\right)\end{array}$
Introducing the equivalent resistance R.sub.2

(9) $\begin{array}{cc}\frac{1}{R_2}=\frac{}{v_2}f\left(x_c\right){.Math.}_{v_2^0,P}=-\frac{P}{{\left(v_2^0\right)}^2}=-\frac{i_2^0}{v_2^0}& \left(4\right)\end{array}$
at the operating point v.sub.2.sup.0 and i.sub.2.sup.0=P/v.sub.2.sup.0, an operating point dependent linearized model of the plant is found as

(10) $\begin{array}{cc}{\stackrel{.}{x}}_p=\left(A_c+E_c\frac{1}{R_2}{C}_c\right)x_p+B_{c}u+E_c\left(i_0^2-\frac{1}{R_2}{v}_2^0\right)& \left(5\right)\end{array}$
For P>0 and v.sub.2>0, R.sub.2 is negative, such that the plant becomes unstable.

(11) FIG. 3 is an illustration of the load model. Item (a) depicts a static characteristic of a constant power load, item (b) a nonlinear load model and item (c) Small signal model linearized around the operating point v.sub.2.sup.0, i.sub.2.sup.0.

(12) A new output vector z=[i.sub.1 v.sub.2 i.sub.2].sup.T is introduced to represent all measurable quantities. Cable resistances are sufficiently small so that the load inverter's DC-link capacitance C.sub.2 can be added in parallel to C.sub.1. As a result, one obtains the model in (6). For ease of notation, the symbol g.sub.P=1/R.sub.2 will be used as a parameter. The variable w=i.sub.2.sup.0v.sub.2.sup.0.Math.g.sub.P denotes the operating point offset.

(13) $\begin{array}{cc}{\stackrel{.}{x}}_p=\underset{\underset{A_p\left(g_p\right)}{}}{\left[\begin{array}{cc}-\frac{R_{L1}}{L_1}& -\frac{1}{L_1}\\ \frac{1}{C_1+C_2}& -\frac{1}{C_1+C_2}{g}_p\end{array}\right]}{x}_p+\underset{\underset{B_p}{}}{\left[\begin{array}{c}\frac{1}{L_1}\\ 0\end{array}\right]}u+\underset{\underset{E_p}{}}{\left[\begin{array}{c}0\\ -\frac{1}{C_1+C_2}\end{array}\right]}wz=\underset{\underset{C_p\left(g_p\right)}{}}{\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1-\frac{C_1}{C_1+C_2}& \frac{C_1}{C_1+C_2}{g}_p\end{array}\right]}{x}_p+\underset{\underset{F_p}{}}{\left[\begin{array}{c}0\\ 0\\ \frac{C_1}{C_1+C_2}\end{array}\right]}w& \left(6\right)\end{array}$

(14) FIG. 4 shows a block diagram of the resulting model structure that will be used for control design, including I/O interfaces and sampling for digital control. The sampling is synchronized to the centers of the three interleaved, symmetric PWM carrier signals, such that the sampling rate f.sub.s is three times the switching frequency f.sub.sw of each phase. The continuous time system model (6) is converted to a discrete time model (7) using a zero order hold (ZOH) approximation. For compact notation, the operating point offset w.sub.k is added as a new state such that x.sub.dk=[i.sub.1,k v.sub.2,k w.sub.k].sup.T.
x.sub.dk+1=A.sub.d(g.sub.p).Math.x.sub.dk+B.sub.d(g.sub.p).Math.u.sub.k y.sub.k=C.sub.d.sup.y.Math.x.sub.dk, z.sub.k=C.sub.d(g.sub.p).Math.x.sub.dk (7)

(15) The control design is preferably chosen to be a model predictive control (MPC). For time on-line MPC formulation according to [J. Maciejowski, Predictive control; with constraints. Pearson education, 2002], an augmented discrete time system description is utilized
x.sub.k+1=Ax.sub.k+Bu.sub.k, y.sub.k=Cx.sub.k, (8)
with the state vector chosen as x.sub.k=[x.sub.dk.sup.T u.sub.k1].sup.T. This allows offset free tracking in combination with a state observer according to [U. Maeder, F. Borrelli, and M. Morari, Linear offset-free model predictive control, Automatica, vol. 45, no. 10, pp. 2214-2222, 2009] and it also contains a computational delay of one sample.

(16) At each time instant k an optimal sequence of control moves u.sub.k is determined for a control horizon N.sub.c and a prediction horizon N.sub.p such that the following criterion is minimized:
J.sub.k=(R.sub.s,kY.sub.k).sup.TQ(R.sub.s,kY.sub.k)+U.sub.k.sup.TRU.sub.k (9)
The symmetric and positive definite weighting matrices Q and R penalize (i) deviations of the predicted output trajectory Y.sub.k from the reference trajectory R.sub.s,k and (ii) the control effort U.sub.k, respectively. The decision variable is the sequence of future control moves
U.sub.k=[u.sub.k|k . . . u.sub.k+N.sub.c.sub.1|k].sup.T (10)
and the output trajectory is the sequence of predicted outputs
Y.sub.k=[y.sub.k+1|k . . . y.sub.k+N.sub.p.sub.|k].sup.T=Fx.sub.k+U.sub.k (11)
with the matrices F and defined as:

(17) $\begin{array}{cc}F=\left[\begin{array}{c}\mathrm{CA}\\ \mathrm{.Math.}\\ {\mathrm{CA}}^{N_p}\end{array}\right];=\left[\begin{array}{ccc}\mathrm{CB}& \mathrm{.Math.}& 0\\ \mathrm{CAB}& \mathrm{.Math.}& 0\\ \mathrm{.Math.}& & \mathrm{.Math.}\\ {\mathrm{CA}}^{N_p-1}B& \mathrm{.Math.}& {\mathrm{CA}}^{N_p-N_c}B\end{array}\right]& \left(12\right)\end{array}$

(18) The power of MPC lies in the ability to handle constraints explicitly. Thus, the limitation of the PWM duty cycle to 0d.sub.k1 and hence 0u.sub.kV.sub.0 is formulated as inequality constraints in the form of
M.sub.uU.sub.k.sub.u (13)
to the minimization problem such that for a constraint horizon of length N.sub.ccN.sub.c the control moves are limited to

(19) $\begin{array}{cc}0u_{k+i.Math.k}=u_{k-1.Math.k}+\underset{j=0}{\overset{i}{.Math.}}{u}_{k+j.Math.k}{V}_0& \left(14\right)\end{array}$
for all iN.sub.cc.

(20) In addition, the inductor currents can be limited for overcurrent protection of the IGBT switches and to avoid magnetic saturation of the inductors. This is achieved by stating inequality constraints M.sub.xU.sub.k.sub.x on the predicted states such that
i.sub.1.sup.maxi.sub.1,k+i|ki.sub.1.sup.max (15)
holds for i N.sub.cc. The input and state constraints are then combined to one set of inequality constraints:

(21) $\begin{array}{cc}M{U}_k\mathrm{with}M=\left[\begin{array}{c}{M}_u\\ M_x\end{array}\right],=\left[\begin{array}{c}{}_u\\ {}_x\end{array}\right]& \left(16\right)\end{array}$

(22) By implementing the receding horizon principle only the first control move u.sub.k=u.sub.k1+u.sub.k|k is applied at each time instant and the rest of U.sub.k is discarded. A block diagram of the resulting controller scheme is depicted in FIG. 5.

(23) The controller can be tuned via the weighting matrices Q and R. With only a single controlled output, Q is assigned an N.sub.pN.sub.p unity matrix and R is assigned an N.sub.cN.sub.c diagonal matrix such that R=diag([R R . . . R].sup.T). This simplifies the tuning to choosing a single scalar value R. Smaller values of R increase the bandwidth of the closed loop but also increase the sensitivity to measurement noise and plant-model mismatches.

(24) In the following the advantages of using a real-time constrained MPC with regard to the present invention is explained. An illustration in FIG. 6 of the proposed algorithm applied to receding horizon control shows how the predictions evolve over time. The upper row shows predicted output voltage trajectories and the bottom row shows corresponding control variable sequences. The columns represent subsequent time samples. The control moves are not just cut off to fit between the constraints but also the unconstrained moves are modified. Over time the solution gets more and more refined.

(25) The challenge in control of power electronics with constrained MPC is to solve the minimization problem fast enough in order to achieve sampling rates in the kHz-range. Here we propose a simple yet effective algorithm that exploits the structure of the given problem.

(26) If the system had no constraints, then the optimal control sequence U.sub.k.sup.0 is found by minimization of (9) with respect to U.sub.k:
U.sub.k.sup.0=(.sup.TQ.sub.y+R).sup.1.sup.TQ.sub.y(R.sub.s,kFx.sub.k) (17)

(27) For constraints in the form of (13), each row m.sub.j of M and the corresponding element .sub.j of express one constraint. Any combination of active constraints is expressed as an active set M.sub.act, .sub.act. With the Hessian matrix H (18) and the vector of Lagrange multipliers .sub.act (19):
H=2(.sup.TQ.sub.y+R) (18)
.sub.act=(M.sub.actH.sup.1M.sub.act.sup.T).sup.1 (.sub.actM.sub.actU.sup.0), (19)
the constrained solution U.sub.k is found by updating the unconstrained solution to
U.sub.k=U.sub.k.sup.0H.sup.1M.sub.act.sup.T.sub.act (20)

(28) The remaining task is to find the active set that minimizes J.sub.k. Active set methods usually require many iterations of adding and removing constraints until the optimal solution is found. In the worst case, all possible combinations of constraints have to be tested. Thus it is not possible to find a polynomial upper bound for the number of iterations with active set methods [H. J. Ferreau, H. G. Bock, and M. Diehl, An online active set strategy to overcome the limitations of explicit mpc, Int. J. Robust Nonlinear Control, vol. 18, no. 8, pp. 816-830, 2008]. Testing all possible active sets is avoided by (i) exploiting the problem structure to eliminate irrelevant combinations of constraints and (ii) stopping after a limited number of iterations and applying a non-optimal solution as in [H. J. Ferreau, H. G. Bock, and M. Diehl, An online active set strategy to overcome the limitations of explicit mpc, Int. J. Robust Nonlinear Control, vol. 18, no. 8, pp. 816-830, 2008] and [Y. Wang and S. Boyd, Fast model predictive control using online optimization, Control Systems Technology, IEEE Transactions on, vol. 18, no. 2, pp. 267-278, 2010].

(29) A heuristic approach that is applicable to the specific problem at hand is as follows. (i) Find the constraint with maximum violation. Any element .sub.j>0 of the vector =(MU.sub.k) indicates a constraint violation. In the case of several simultaneous violations, the largest element .sub.i with

(30) 0$\begin{array}{cc}i=\underset{j}{\max }\left(m_j^T{U}_k-{}_j\right)& \left(21\right)\end{array}$
can be taken as an indicator for the maximum violation, under the condition that all constraints are equally scaled. (ii) Add {m.sub.i.sup.T, .sub.i} to the active set and (iii) re-compute (19), (20). (iv) Repeat for a maximum of N.sub.cc iterations, as long as constraints are violated. Constraints are only added but never removed from the active set. This procedure is summarized in Algorithm 1.

(31) If only input constraints are considered, then the obtained solution is feasible but not necessarily optimal. Because of the receding horizon control, only the first control move of U.sub.k is applied. So it is not necessary to find the full solution but only one that approximates

(32) $\begin{array}{cc}M=\left[\begin{array}{c}{M}_u{}_u\\ M_x{}_x\end{array}\right],\left[\begin{array}{c}{}_u{}_u\\ {}_x{}_x\end{array}\right]& \left(22\right)\\ {}_u=1/\left(u^{\max }-u^{\min }\right)& \left(23\right)\\ {}_x=1/\left(y_x^{\max }-y_x^{\min }\right)& \left(24\right)\end{array}$
the first move of the optimal sequence close enough such that the desired trajectory can be achieved. The simulated example in FIG. 6 illustrates how the solutions evolve over time for N.sub.p=16, N.sub.c=8, N.sub.cc=5. FIG. 7 shows that the resulting trajectory is virtually identical to the exact solution obtained from quadratic programming, [The Mathworks Inc., Optimization toolbox 4.3, 2009].

(33) The comparison of the proposed algorithm with an MPC using a generic QP solver in FIG. 7 shows that there is almost no difference between the two trajectories, although the initial solution at time sample 1 is far from optimal.

(34) In order to be also able to handle state constraints at the same time, it is necessary to scale the rows of M and such that the elements of , i.e. the amount of constraint violation, are comparable. This is achieved by normalizing the input constraints and state constraints to their respective admissible range.

(35) TABLE-US-00001 Algorithm 1 Active set method with early stopping 1: Initialize with empty active set. 2: Compute unconstrained solution (17). 3: for N.sub.cc iterations do 4: if all constraints are satisfied then 5: Stop. 6: else 7: Find maximum violation .sub.i from (21). 8: Add constraint {m.sub.i.sup.T, .sub.i} to active set. 9: Compute solution for active set with (19), (20). 10: end if 11: end for
Because the proposed algorithm does not exactly solve the QP, the resulting control law may cause brief violations of the inductor current limit while saturating the control variable for optimal reference tracking. Therefore it may be necessary to give precedence to the state constraints by scaling them with a precedence factor .
.sub.x=/(y.sub.x.sup.maxy.sub.x.sup.min), >1 (25)
Simulations show that a value of =10 gives good results for the application at hand.

(36) With the addition of state constraints, infeasible combinations of constraints can occur. In such a case, the iteration is stopped and the solution from the last iteration is applied. The big advantage of the proposed approach is that it only requires a small and bounded number of iterations, which facilitates a real-time implementation. A similar approach is described in [Y. Wang and S. Boyd, Fast model predictive control using online optimization, Control Systems Technology, IEEE Transactions on, vol. 18, no. 2, pp. 267-278, 2010] where extensive numerical experiments show that a surprisingly good control law can be achieved by stopping after a few iterations. The computation time can be reduced by precomputing the inverse of the Hessian H and by using a rank-1-update for the matrix inversion in (19). A further reduction of the average computation time can be achieved by stopping the algorithm as soon as the first control variable increment is fixed by an equality constraint.

(37) The effects and advantages of a state observer arid reference filtering are explained now. Offset free tracking is possible with the chosen MPC formulation, despite plant-model mismatches or unmeasured disturbances. In [U. Maeder, F. Borrelli, and M. Morari, Linear offset-free model predictive control, Automatica, vol. 45, no. 10, pp. 2214-2222, 2009] it is shown that this is achieved by using an observer such that the current state vector contains an estimate .sub.k1 instead of the actual previous controller output u.sub.k1. Furthermore, the observer can provide the full state vector even if not all states can be measured directly. A reference prefilter is used to generate a feasible reference trajectory vector R.sub.s,k=[r.sub.s,k,1 . . . r.sub.s,k,N.sub.p] from the scalar reference r.sub.k at every sample. The filter delays the trajectory two samples in order to account for the computation delay and the system's low pass characteristic. It also limits the trajectory's rate of change to |r.sub.s,k,ir.sub.s,k,i1|r.sub.s.sup.max.

(38) The linear MPC described so far only applies to loads with constant parameters. The load's filter capacitance is known or can be measured and does not change during operation. However, with a CPL, the parameter R.sub.2 changes over v.sub.2 and P according to (2). As one possible approach, a robustness concept is chosen to solve this control problem. For the system model (6), two extremal cases can be identified. First, for P=0, the uncertain parameter becomes g.sub.P.sup.max=0. Second, for the highest power demand P.sup.max at the lowest input voltage v.sub.2.sup.min specified for the load inverter, the uncertain parameter takes the value g.sub.P.sup.min=P.sup.max/(v.sub.2.sup.min).sup.2. For the two extremal cases, one can set up two prediction models in the form of (7), which are denoted by {A.sub.d(0), B.sub.d(0), C.sub.d(0)} with the state vector x.sub.d1k and {A.sub.d(g.sub.P.sup.min), B.sub.d(g.sub.P.sup.min), C.sub.d(g.sub.P.sup.min)} with the state vector x.sub.d2k. The basic idea is to use both models for the prediction in order to find a sequence of control moves that properly controls the actual plant on the one hand and stabilizes both extremal plants on the other hand. This is achieved by applying the same control variable sequence to both models and by taking the sum of both outputs weighted with .sub.1 and .sub.2 as the controlled output as shown in FIG. 8.

(39) This can be implemented by using the MPC algorithm from above sections and appropriately setting up the augmented model:

(40) $\begin{array}{cc}{x}_{k+1}=\left[\begin{array}{ccc}{A}_d\left(0\right)& 0& B_d\left(0\right)\\ 0& A_d\left(g_p^{\min }\right)& B_d\left(g_p^{\min }\right)\\ 0& 0& 1\end{array}\right]x_k+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u_k{y}_k=\left[\begin{array}{ccc}{}_1{C}_d^y& {}_2{C}_d^y& 0\end{array}\right]x_k& \left(26\right)\end{array}$
With the state vector chosen as x.sub.k=[x.sub.d1k.sup.T x.sub.d2k.sup.T u.sub.k1].sup.T.
The observer design requires both models to be observable via an extended output vector:

(41) $\begin{array}{cc}{\tilde{z}}_k=\left[\begin{array}{ccc}{C}_d\left(0\right)& 0& 0\\ 0& C_d\left(g_p^{\min }\right)& 0\end{array}\right]x_k& \left(27\right)\end{array}$
In practice there are only measurements z.sub.k.sup.T from the real plant available. These have to be stacked such that {tilde over (z)}.sub.k=[z.sub.k.sup.T z.sub.k.sup.T].sup.T in order to obtain the full output vector for the observer.

(42) If the previously unknown parameter g.sub.P can be measured or estimated, then the performance can be improved with a scheduling controller. The system description (7) has one single parameter g.sub.P, which is chosen as the scheduling variable. For a representative set of values
g.sub.P.sup.ming.sub.P,ig.sub.P.sup.max, i N.sub.g (28)
that uniformly span the expected operating range, local MPC parameterizations are obtained.
The corresponding parameter sets {.sub.i, H.sub.i, F.sub.i} are computed offline. At runtime, a scheduler then only has to select the parameter set for which

(43) $i=\underset{j}{\min }.Math.g_{p,j}-g_p.Math..$
This controller scheme is depicted in FIG. 9. The higher the number N.sub.g of local parameter sets, the smoother the system response will be. The number of parameter sets only has an impact on the amount of memory necessary for parameter storage but does not increase the complexity of online computations. Care has to be taken when implementing this scheme in order to achieve bumpless transfer between the parameter sets. The initial system (7) contains the combined disturbance and offset state w.sub.k=i.sub.2,k.sup.0v.sub.2,k.sup.0g.sub.P such that the meaning of the state vector depends on the scheduling variable. The same physical state would result in a different value of w.sub.k for each local MPC. This problem is mitigated by choosing an extended discrete time model
{tilde over (x)}.sub.dk+1=.sub.d(g.sub.P).Math.{tilde over (x)}.sub.dk+{tilde over (B)}.sub.d(g.sub.P).Math.u.sub.k y.sub.k={tilde over (C)}.sub.d.sup.y.Math.x.sub.dk (29)
with the state vector
{tilde over (x)}.sub.dk=[i.sub.1,k v.sub.2,k i.sub.2,k.sup.0 v.sub.2,k.sup.0].sup.T (30)
for which the relation between the state variables and the physical state is independent of the scheduling variable. Then the augmented prediction model for the MPG is defined as

(44) $\begin{array}{cc}{\tilde{x}}_{k+1}=\underset{A\left(g_p\right)}{\underset{}{\left[\begin{array}{cc}{\tilde{A}}_d\left(g_p\right)& {\tilde{B}}_d\left(g_p\right)\\ 0& 1\end{array}\right]}}{\tilde{x}}_k+\underset{\underset{B}{}}{\left[\begin{array}{c}0\\ 1\end{array}\right]}{u}_k{y}_k=\underset{\underset{C}{}}{\left[\begin{array}{cc}{\tilde{C}}_d^y& 0\end{array}\right]}{\tilde{x}}_k& \left(31\right)\end{array}$
with the augmented state vector
{tilde over (x)}.sub.k=[{tilde over (x)}.sub.dk.sup.T u.sub.k1].sup.T. (32)

(45) The matrices A(g.sub.P,i), B, C are used to find the corresponding sets {.sub.i, H.sub.i, F.sub.i} from (12), (18). For each sample of the controller, the states v.sub.2,k and v.sub.2,k.sup.0 are equal. However, within the MPC's prediction horizon, only v.sub.2,k+i|k changes whereas v.sub.2,k+i|k.sup.0 remains constant for all i N.sub.p. Because of the choice of state vector made above, the state can be estimated independently of the parameter g.sub.P such that the same observer can be used for the entire operating range. By choosing g.sub.P=0, the influence of the CPL is treated as a disturbance w.sub.k=i.sub.2,k.sup.0v.sub.2,k.sup.0.Math.0=i.sub.2,k.sup.0. Hence, the state observer is designed for the nominal model:

(46) $\begin{array}{cc}{\hat{x}}_{k+1}=\left[\begin{array}{cc}{A}_d\left(0\right)& B_d\left(0\right)\\ 0& 1\end{array}\right]{\hat{x}}_k+B_d\left(0\right)u_k{z}_k=\left[\begin{array}{cc}{C}_d\left(0\right)& 0\end{array}\right]{\hat{x}}_k,{\hat{x}}_k=\left[\begin{array}{c}{\hat{x}}_{\mathrm{dk}}\\ {\hat{u}}_{k-1}\end{array}\right]& \left(33\right)\end{array}$
The estimated state vector {circumflex over (x)}.sub.k is extended with {circumflex over (v)}.sub.2,k.sup.0=C.sub.d.sup.y{circumflex over (x)}.sub.dk such that it can be used for the scheduling MPC:
{tilde over (x)}.sub.k=[{circumflex over (x)}.sub.dk.sup.T {circumflex over (v)}.sub.2,k.sup.0 .sub.k1].sup.T (34)
Because g.sub.P cannot be measured directly, the observer is also used to obtain an estimate .sub.Pk of the scheduling variable from (4) such that

(47) $\begin{array}{cc}{\hat{g}}_{\mathrm{pk}}=-\frac{{\hat{i}}_{2,k}^0}{{\hat{v}}_{2,k}^0}.& \left(35\right)\end{array}$

(48) The crisp parameter switching can cause unwanted excitation of the system during transients. Limit cycles can occur if the steady state operating point is right in the middle of two supporting points and the scheduler constantly switches between them. More advanced scheduling techniques such as parameter blending or controller output blending could bring an improvement in performance [G. Gregorcic and G. Lightbody, Nonlinear model-based control of highly nonlinear processes, Computers & Chemical Engineering, vol. 34, no. 8, pp. 1268-1281, 2010].

(49) The proposed controller designs according to the present invention have been verified with simulations as well as experimentally with a 60 kW battery emulator. The parameters of the test system are listed in table 1. The PWM modulator of the test system had a low-pass filter at its input, which had to be added to the converter model (1).

(50) TABLE-US-00002 TABLE 1 System parameters parameter nominal value description C.sub.0 20 000 F DC-link capacitance U.sub.0 620 V DC-link voltage L.sub.1 .Math. 1800 H lumped storage inductance R.sub.1 .Math. 8 m lumped inductor resistance C.sub.1 450 F filter capacitance C.sub.2 20 000 F load input capacitance f.sub.sw 2.5 kHz switching frequency f.sub.s 7.5 kHz sampling rate

(51) The simulations were carried out using a detailed model of the BE output stage with three interleaved switching phases. The simulated load was modeled as an ideal CPL for voltages greater than 150 V. For lower voltages the simulated load switches to constant current behavior.

(52) In the simulation model, C.sub.2=0 F was chosen for the filter capacitance in order to show the effectiveness of the proposed approaches even in worst-case situations. Simulations were carried out using an MPC without CPL model, the proposed scheduling controller and the proposed robust controller. The robust controller was designed for a maximum power demand of P.sup.max=60 kW at v.sub.2.sup.min=245 V such that g.sub.P.sup.max=0 and g.sub.P.sup.min=1.sup.1. Choosing a wider range would lead to increased robustness but also to a slower response. With the scheduling approach, a wider parameter range can be covered without affecting the closed loop performance. Only the increased memory demand for additional parameter sets has to be considered. With the notation of g.sub.P=1/R.sub.2, the system matrices depend linearly on g.sub.P. Consequently, 21 parameter sets were uniformly placed between g.sub.P.sup.max=0 and g.sub.P.sup.min=4.sup.1 such that the resulting local controllers are placed 0.05.sup.1 apart.

(53) FIG. 10 shows simulation results for startup without load from 0 V to a set point of 320 V and a sudden load step from 0 kW to 60 kW while the reference is kept constant. At zero load, both the nominal controller and the scheduling controller use the same set of parameters such that their startup is identical for the first 5 ms. After the load step at 5 ms, the nominal controller does not adapt to the changed plant dynamics and the closed loop becomes unstable, whereas the scheduling controller remains stable. Despite a first large dip in the output voltage, the controller is able to quickly restore the voltage with virtually no overshoot by optimally using the available control variable range. With the robust controller variant the closed loop also remains stable but it causes an overshoot and it takes longer to settle after the step disturbance, Because the robust controller uses the control variable more cautiously, it also takes slightly longer to reach the set point during the start up. After 10 ms, a reference step from 320 V to 270 V is applied in order to demonstrate reference tracking and the ability to handle changes in the operating point. Both proposed controllers show similar behavior with a short rise-time and fast settling.

(54) For the experimental tests, the control algorithms have been implemented on a dSpace MicroAutoBox using embedded Matlab for automatic code generation. The dSpace platform features an IBM PPC 750FX processor clocked at 800 MHz. Reference voltage step changes without load are shown in FIG. 11, demonstrating the effectiveness of the constrained optimization. For a small reference step, the set point can be reached very quickly. The controller applies a large control variable increment in the first time step but immediately reduces the control variable to its lower bound for the following samples. After less than 0.8 ms, the set point is reached and the duty cycle is set to its new steady state value. As can be seen, the inductor current follows a triangular trajectory. For a large reference step, the controller fully utilizes the upper bound of the duty cycle during the first two samples. Then the controller follows the reference trajectory until the new set point is reached.

(55) The same large reference step was repeated with a reduced inductor current limit of 200 A and a constant current load drawing 100 A as shown in FIG. 12 to demonstrate the activation of the inductor current constraint. The inductor current is limited to its upper bound during the transient such that the output voltage can only rise slowly.

(56) For testing purposes, a DC to three-phase AC UPS inverter with a maximum power of 24 kW and DC link capacitance of C.sub.2=20 000 F was connected to the BE. On the AC side it was set to regulate a constant voltage across a three-phase resistor such that it appeared as a CPL towards the BE. Results for a sequence of reference step changes are shown in FIG. 13, using the scheduling controller and the robust controller. It can be seen that the load current decreases for increasing output voltage and vice versa as it is expected for a CPL. During the transients, the charging and discharging of the inverter's large DC-link capacitor caused large current spikes. But these were limited to the inductor current constraint of 300 A. Only the scheduling controller slightly violates the lower limit for a short instant.

(57) Load disturbances were tested by abruptly switching on the resistors on the AC side of the load inverter. The results are shown in FIG. 14.