MOTION CONTROL IN MOTOR VEHICLES

Abstract

A method for controlling actuators acting on vehicle wheels of a motor vehicle comprises ascertaining a force to be brought about on a reference point of the motor vehicle on the basis of driver specifications, ascertaining wheel forces to be brought about on the vehicle wheels to implement the force to be brought about on the reference point of the motor vehicle by means of a first dynamic allocation by model-based predictive control (MPC), ascertaining setpoint values for wheel parameters from the ascertained wheel forces, and actuating the actuators of the motor vehicle so as to implement the setpoint values of the wheel parameters.

Claims

1. A method for controlling actuators acting on vehicle wheels of a motor vehicle comprising: ascertaining a force to be brought about on a reference point of the motor vehicle on the basis of driver specifications; ascertaining wheel forces to be brought about on the vehicle wheels to implement the force to be brought about on the reference point of the motor vehicle with a first dynamic allocation by model-based predictive control; ascertaining setpoint values for wheel parameters from the wheel forces; and actuating the actuators of the motor vehicle so as to implement the setpoint values of the wheel parameters.

2. The method as claimed in claim 1, wherein the driver specifications are made available by at least one of a virtual driver and an assistance system.

3. The method as claimed in one claim 1, wherein the driver specifications are at least one of an acceleration and at least one steering angle.

4. The method as claimed in claim 1, wherein the setpoint values for the wheel parameters are at least one of torques respectively acting on the wheels, slip values of the wheels, rotational speeds of the wheels, and steering angles of the wheels.

5. The method as claimed in claim 1, wherein implementing a setpoint value for a torque acting on a vehicle wheel further comprises a second dynamic allocation, which includes slip control.

6. The method as claimed claim 1, wherein the actuators are at least one of electric motors and friction brakes.

7. The method as claimed in further comprising adapting kinematic setpoint motion variables by way of a virtually controlled single-track model which are taken into account in the first dynamic allocation of the wheel forces.

8. The method as claimed in claim 1, wherein the dynamic allocation of the wheel forces includes driving dynamics control.

9. The method as claimed in claim 1, wherein ascertaining the wheel forces further comprises ascertaining the wheel forces from a static allocation and feeding the wheel forces to the dynamic allocation as input values.

10. The method as claimed in claim 1, wherein the first dynamic allocation comprises a saturation of an assigned actuator, a reconfiguration to ensure the error tolerance in the event of failure of the actuator, an increase in the energy efficiency and a minimization of wear.

11. The method as claimed in claim 1, further comprising converting wheel forces with an inverse tire-force model into setpoint variables for torques, wheel slips, rotational speeds and steering angles of the wheels.

12. The method as claimed in claim 9, further comprising arbitrating the wheels forces of the static and dynamic allocation.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Further features, advantages and possible applications of the invention also result from the following description of exemplary embodiments on the basis of calculations and the drawings. All of the features described and/or pictorially depicted belong to the subject matter of the invention both individually and in any combination, also independently of their summarization in the claims or the back-references thereof.

[0022] FIG. 1 shows a vehicle configuration with a motion control,

[0023] FIG. 2 shows the setup in parent-child systems,

[0024] FIG. 3 schematically shows the motion control concept,

[0025] FIG. 4 shows the forces of the motor vehicle,

[0026] FIG. 5 shows the forces on a tire,

[0027] FIG. 6 shows a first test maneuver,

[0028] FIG. 7 shows a second test maneuver,

[0029] FIG. 8 shows a third test maneuver.

DETAILED DESCRIPTION

[0030] FIG. 1 shows a vehicle configuration for the motion control (MC) of the vehicle. The motion control (MC) has access to all brakes (b), which may for example be electromechanically formed, and both electric motors (m) on the front axle, which are suitable for regenerative braking. With this configuration, improved torque vectoring and blending can be realized, which improves the energy efficiency, the stability and the handling of the vehicle. The steering (s) cannot be controlled by the motion control (MC).

[0031] The regenerative braking improves the comfort of the driver and the passengers during the journey and extends the range of the vehicle.

[0032] The motion control concept uses elements which are inspired by object-oriented design. The elements that are used here have a setup which replicates the overall system respectively in parent systems 13 and child systems 2 (parent/child system pairs) along various functional chains, as represented in FIG. 2. The functional chains are either set up for specific purposes or represent external influences such as waste heat. Examples of parent/child system pairs are vehicle/chassis, chassis/corner or corner/actuator. Each system distinguishes internally between information providers/observers 3 (observer functions) and managers 4. Observer functions 3 provide estimated or measured actual values 9 for their own and the parent level 13 and coordinate the boundary conditions 10, which they obtain from the child level 2. They may also provide estimated or measured disturbances (“counters”) 12, which are used by their own or subordinate systems for improving the quality of the control. Manager functions ascertain requirements 8 for controlling the respective child system. They may also receive desired commands 11 from child levels and adjust them with their own requirements 8.

[0033] At vehicle level, the longitudinal and transverse requirements are determined by dynamic pre-control with personalizable vehicle response behavior and damping properties. A central element at chassis level is a model-predictive controller (MPC), which requires the forces of the curve module while taking into account stability and energy limits. For describing the chassis dynamics, a two-track model is used. Time-variant system matrices are produced by linearization and discretization at each sampling time. Friction circles between the tires and the road, which restrict the admissible forces on the wheels, are approximated by polytopes. These measures are intended to ensure the applicability of the convex quadratic programming that is suitable for the embedded real-time optimization. At the corner level, the modules process the incoming stream of chassis requirements and produce a stream of corresponding requirements for the actuators. An MPC framework for corner modules is capable of optimally dividing the wheel braking torque among the redundant actuators on the front axle, while traction control and antilock functions are provided by wheel slip control on all wheels. This approach offers a rapid transient response, without impairing the energy recovery efficiency of the electric motors, with different dynamic authorities of the friction brake and the electric motor being taken into account.

[0034] FIG. 3 shows the overall structure of the motion control concept. The system comprises modern methods, such as electronic stability control (ESC), active yaw control (AYC), antilock braking system (ABS) and traction control system (TCS). The arbitrator blocks (Arb) and an inverse tire model (iTire) are used for this. The remaining blocks in the control chain are the human driver (Drv), a single-track model (STM) with a virtual controller (VC), a feedforward control (FF), a static control assignment (CA), observers (Obs) and the actuators (Act), i.e. two electric motors on the front axle and friction brakes on each wheel. The specific variables are explained below. A virtual driver (Virt. Drv) can be used in special situations, in order to overrule the driver, whereby autonomous driving (AD) can also be directly integrated.

[0035] There follows firstly a general overview and then an increasingly detailed discussion of the systematic derivation and the setup of the models and also the generic structure of the optimization problem, which represents the basis for the model-predictive control.

[0036] The model-predictive control is an advanced optimization-based control method. By contrast with linear-quadratic (LQ) control, MPC offers the explicit handling of process restrictions resulting from natural requirements, for example energy efficiency, safety, actuator limits and others. The control decisions in MPC are calculated online on the basis of an internal model of the system dynamics.

[0037] All models in the motion control concept are based generally on linear time-variant discrete models with n state variables in the vector x, m inputs or control actions in u and p outputs in y. Each time-discrete model is derived from a non-linear time-continuous model in state space form.

[00001]$\dot{x}=f\left(x,u\right),$

[00002]$x\left(t_0\right)=x_0$

[00003]$y=h\left(x,u\right)$

[0038] The non-linear state space model is linearized along a reference trajectory for the state x.sub.d and the control action u.sub.d, which leads to

[00004]$f\left(x,u\right)\approx f\left(x_d,u_d\right)+\underset{A}{\underbrace{{\left(\frac{\partial f}{\partial x^T}\right|}_{x_d,u_d}}}\left(x-x_d\right)+\underset{B}{\underbrace{{\left(\frac{\partial f}{\partial u^T}\right|}_{x_d,u_d}}}\left(u-u_d\right),$

[00005]$h\left(x,u\right)\approx h\left(x_d,u_d\right)+\underset{C}{\underbrace{{\left(\frac{\partial h}{\partial x^T}\right|}_{x_d,u_d}}}\left(x-x_d\right)+\underset{D}{\underbrace{{\left(\frac{\partial h}{\partial u^T}\right|}_{x_d,u_d}}}\left(u-u_d\right).$

[0039] The result is an affine linear approximation of the non-linear system with the system matrices A, B, C, D and the affine vectors h.sub.x and h.sub.y, which are generally time-variant

[00006]$\dot{x}=Ax+Bu+h_x,$

[00007]$h_x=f\left(x_d,u_d\right)-A{x}_d-B{u}_d,$

[00008]$y=Cx+Du+h_y,$

[00009]$h_y=h\left(x_d,u_d\right)-C{x}_d-D{u}_d.$

[0040] The affine approximation allows the system variables to be used directly in the optimization and the problematic mixing of absolute values and difference-from-setpoint values is avoided. In the motion control concept, two discretization methods are used: The low-cost, but less accurate Euler forward method and the more accurate, but also more expensive Tustin (trapezoidal) method. A discrete model is characterized in the following by an index k. With the sampling time T.sub.s, the Euler discretization for the affine system leads to

[00010]$A_k=I_n+A{T}_s,$

[00011]$B_k=B{T}_s,$

[00012]$C_k=C,$

[00013]$D_k=D,$

[00014]$h_{xk}=h_x{T}_s,$

[00015]$h_{yk}=h_y$

[0041] The Tustin discretization is only used for non-affine systems and is given by the following transformation. This generally results in a direct implementation matrix D.sub.k.

[00016]$A_k={\left(I_n-A\frac{T_s}{2}\right)}^{-1}\left(I_n+A\frac{T_s}{2}\right),$

[00017]$B_k={\left(I_n-A\frac{T_s}{2}\right)}^{-1}B{T}_s,$

[00018]$C_k=C{\left(I_n-A\frac{T_s}{2}\right)}^{-1},$

[00019]$D_k=D+C{\left(I_n-A\frac{T_s}{2}\right)}^{-1}B\frac{T_s}{2}$

[0042] It has previously been assumed that the predictive controller calculates the control input u.sub.k in its absolute form. However, this is not the only possibility for producing inputs for the system. In certain applications, changes of the input u.sub.k can be calculated. This formulation is also referred to as rate-based or integral action formulation. The reason for using a controller formulation with integral action is to obtain offset-free tracking. There are several possible ways of expressing the controller model in its incremental, integrating form. With the incremental vectors

[00020]$\text{Δ}{x}_k=x_k-x_{k-1},$

[00021]$\text{Δ}{u}_k=u_k-u_{k-1},$

[00022]$\text{Δ}{y}_k=y_k-y_{k-1}$

[0043] the integral mode of action of the state space system is specified below, with the assumption that the time-variant system matrices and affine vectors do not differ very much between two consecutive time samples.

[00023]$\text{Δ}{x}_{k+1}=A_k\text{Δ}{x}_k+B_k\text{Δ}{u}_k+\text{Δ}{h}_{xk},$

[00024]$y_k=y_{k-1}+C_k\text{Δ}{x}_k+D_k\text{Δ}{u}_k+\text{Δ}{h}_{yk}$

[0044] The system description above does not correspond to the generic form that we introduced further above. In order to restore the generic form also for integral actions, the system state is extended with the output of the previous sampling interval. The extended system is then as follows

[00025]$\left[\begin{array}{c}\text{Δ}{x}_{k+1}\\ y_k\end{array}\right]=\left[\begin{array}{cc}{A}_k& 0\\ C_k& I_p\end{array}\right]\left[\begin{array}{c}\text{Δ}{x}_k\\ y_{k-1}\end{array}\right]+\left[\begin{array}{c}{B}_k\\ D_k\end{array}\right]\text{Δ}{u}_k+\left[\begin{array}{c}\text{Δ}{h}_{xk}\\ \text{Δ}{h}_{yk}\end{array}\right],$

[00026]$y_k=\left[\begin{array}{cc}{C}_k& I_p\end{array}\right]\left[\begin{array}{c}\text{Δ}{x}_k\\ y_{k-1}\end{array}\right]+D_k\text{Δ}{u}_k+\text{Δ}{h}_{yk}$

[0045] A further use of the state space model is to provide a prediction of the system state and the output over a specific time period into the future. With the so-called “batch” approach, it is possible to express the dynamics of the state and output vector for a specified time horizon N by backward multiplication of the linear matrices and vectors. The batch form for discrete linear time-variant affine systems is denoted as follows and must be newly set up in each sampling interval

[00027]$\begin{array}{l}\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ .Math.\\ x_N\end{array}\right]=\overline{A}{x}_0+\overline{B}\left[\begin{array}{c}{u}_0\\ u_1\\ u_2\\ .Math.\\ u_{N-1}\end{array}\right]+{\overline{H}}_x\left[\begin{array}{c}{h}_{x0}\\ h_{x1}\\ h_{x2}\\ .Math.\\ h_{xN-1}\end{array}\right],\left[\begin{array}{c}{y}_0\\ y_1\\ y_2\\ .Math.\\ y_{N-1}\end{array}\right]\\ =\overline{C}{x}_0+\overline{D}\left[\begin{array}{c}{u}_0\\ u_1\\ u_2\\ .Math.\\ u_{N-1}\end{array}\right]+{\overline{H}}_{yx}\left[\begin{array}{c}{h}_{x0}\\ h_{x1}\\ h_{x2}\\ .Math.\\ h_{xN-1}\end{array}\right]+\left[\begin{array}{c}{h}_{y0}\\ h_{y1}\\ h_{y2}\\ .Math.\\ h_{yN-1}\end{array}\right]\end{array}$

[00028]$\begin{array}{l}\overline{A}=\left[\begin{array}{c}{A}_0\\ A_1{A}_0\\ A_2{A}_1{A}_0\\ .Math.\\ {\displaystyle \underset{j=0}{\overset{N-1}{.Math.}}{A}_{N-1-j}}\end{array}\right],\overline{C}=\left[\begin{array}{c}{C}_0\\ C_1{C}_0\\ C_2{C}_1{C}_0\\ .Math.\\ C_{N-1}{\displaystyle \underset{j=0}{\overset{N-1}{.Math.}}{A}_{N-1-j}}\end{array}\right],{\overline{H}}_x\\ =\left[\begin{array}{ccccc}{I}_n& 0& 0& .Math.& 0\\ A_1& I_n& 0& .Math.& 0\\ A_2{A}_1& A_2& I_n& .Math.& 0\\ .Math.& .Math.& \ddots & \ddots & .Math.\\ \left({\displaystyle \underset{j=1}{\overset{N-1}{.Math.}}{A}_{N-j}}\right)& \left({\displaystyle \underset{j=1}{\overset{N-2}{.Math.}}{A}_{N-j}}\right)& .Math.& A_{N-1}& I_n\end{array}\right]\end{array}$

[00029]$\overline{B}=\left[\begin{array}{ccccc}{B}_0& 0& 0& .Math.& 0\\ A_1{B}_0& B_1& 0& .Math.& 0\\ A_2{A}_1{B}_0& A_2{B}_1& B_2& .Math.& 0\\ .Math.& .Math.& \ddots & \ddots & .Math.\\ \left({\displaystyle \underset{j=1}{\overset{N-1}{.Math.}}{A}_{N-j}}\right)B_0& \left({\displaystyle \underset{j=1}{\overset{N-2}{.Math.}}{A}_{N-j}}\right)B_1& .Math.& A_{N-1}{B}_{N-2}& B_{N-1}\end{array}\right]$

[00030]$\begin{array}{l}\overline{D}=\\ \left[\begin{array}{ccccc}{D}_0& 0& 0& .Math.& 0\\ C_1{B}_0& D_1& 0& .Math.& 0\\ C_2{A}_1{B}_1& C_2{B}_1& D_2& .Math.& 0\\ .Math.& .Math.& \ddots & \ddots & .Math.\\ C_{N-1}\left({\displaystyle \underset{j=1}{\overset{N-2}{.Math.}}{A}_{N-1-j}}\right)B_{N-2}& C_{N-1}\left({\displaystyle \underset{j=1}{\overset{N-3}{.Math.}}{A}_{N-1-j}}\right)B_{N-2}& .Math.& C_{N-1}{B}_{N-2}& D_{N-1}\end{array}\right],\end{array}$

[00031]$\begin{array}{l}{\overline{H}}_{yx}=\\ \left[\begin{array}{ccccc}0& 0& 0& .Math.& 0\\ C_1& 0& 0& .Math.& 0\\ C_2{A}_1& C_2& 0& .Math.& 0\\ .Math.& .Math.& \ddots & \ddots & .Math.\\ C_{N-1}\left({\displaystyle \underset{j=1}{\overset{N-2}{.Math.}}{A}_{N-1-j}}\right)& C_{N-1}\left({\displaystyle \underset{j=1}{\overset{N-3}{.Math.}}{A}_{N-1-j}}\right)& .Math.& C_{N-1}& 0\end{array}\right]\end{array}$

[0046] Essentially, an MPC controller is based on an iterative optimization with a finite horizon (constrained) of a route model. At each discrete sampling time (k), the route is sampled and the actual state x.sub.k is measured or estimated with the help of observers. The performance of the controller is expressed by a so-called cost function. Based on a dynamic model of the route, this cost function is formulated in such a way that it expresses the behavior of the MPC controller in the future for a current route state x.sub.k and a series of future inputs u.sub.k. In other words, this predicted cost function gives a numerical indicator of the quality of the control, with the assumption that the current system state is influenced by a specific sequence of inputs from the past. The question is not how the controller will perform, but what is the sequence of inputs u.sub.k that produces the best performance. In order to compute the optimal sequence of inputs, the cost function must be minimized at each sampling interval by using a numerical optimization algorithm. As in the case of most actual systems, the inputs, outputs and states can be restricted by physical boundary conditions, which can easily be included in the numerical minimization task. From the sequence of future inputs u.sub.k, only the first one is applied, then the process is repeated on the basis of brand-new measured state information. This type of repeated measure-predict-optimize-apply cycle is referred to as receding horizon control.

[0047] The MPC optimization approach used in the present motion control concept comprises the following generic form, where N denotes the forecast horizon and M denotes the control horizon,

[00032]$\underset{u_{c0:M-1},\epsilon }{\min }{J}_T\left(x_k,u_k\right)+\gamma J_E\left(x_k,u_k\right)+P{\epsilon }^2$

[00033]$x_{k+1}=A_k{x}_k+B_k{u}_k+h_{xk}$

[00034]$y_k=C_k{x}_k+D_k{u}_k+h_{yk}$

[00035]$w_k=C_{wk}{x}_k+D_{wk}{u}_k+\epsilon V_w$

[00036]${\left[u_0^T,.Math.,u_{N-1}^T\right]}^T=\left(M.Math.I_m\right){\left[u_{c0}^T,.Math.,u_{cM-1}^T\right]}^T$

[00037]$x_0=x_k$

[00038]$u_{min}\le u_k\le u_{max}$

[00039]$y_{min}-\epsilon V_{y,min}\le y_k\le y_{max}+\epsilon V_{y,max}$

[00040]$\Delta u_{min}-\epsilon V_{\Delta u,min}\le \Delta u_k\le \Delta u_{max}+\epsilon V_{\Delta u,max}$

[00041]$H_{xk}{x}_k+H_{uk}{u}_k\le b_{Hk}+\epsilon V_H$

[00042]$0\le \epsilon$

[00043]$\forall k\in \left\{0,.Math.,N-1\right\}$

[0048] The overall cost function is divided into a trajectory tracking part J.sub.T, an energy consumption part J.sub.E and a slip variable ε, provided with the weighting P. The constraints of the MPC problem can be divided into equality and inequality constraints. The first three equality constraints are the state and output model of the vehicle dynamics in affine form and a general soft constraint as a linear combination of state and input, reduced by the slip variable. The target vector w.sub.k is either zero or must be specified externally. The next equality constraint is used to realize a control horizon that is shorter than the forecast horizon, specifically by motion blocking. The matrix M determines the blocking scheme and the operator symbol is the Kronecker product. The last equality constraint maps the initial state onto the current state of the sampling period. The inequality constraints determine the bounds for the input u.sub.k, the output y.sub.k and the input rate u.sub.k. The last two are reduced in order to ensure the feasibility of the optimization. A polytopic set constraint, which is optionally relaxed, and the non-negativity of the slip variables are the remaining inequality constraints. The cost function used for target tracking is quadratic and allows the output error, the input error and the input rate to be penalized by the symmetrical positively semi-definite weighting matrices Q, R and R.sub.Δ.

[00044]$\begin{array}{l}{J}_T={\displaystyle \underset{k=0}{\overset{N}{.Math.}}\left({\left(y_k-y_{dk}\right)}^{T}Q\left(y_k-y_{dk}\right)\right)+}\\ {\displaystyle \underset{k=0}{\overset{N-1}{.Math.}}\left({\left(u_k-u_{dk}\right)}^{T}R\left(u_k-u_{dk}\right)+\Delta u_k^T{R}_{\Delta }\Delta u_k\right)}\end{array}$

[0049] Any energy- or performance-related function can be used for the energy consumption part J.sub.E of the cost function. The only constraint is that the cost function must be quadratic and/or linear. The relative importance of the energy costs can be specified by the weighting factor. A typical power-related function is the power loss P.sub.loss of electric motors, which can be formulated on the basis of the electric motor torque T.sub.m, the wheel speed w and given efficiency maps for generation and drive,

[00045]$P_{loss}=\left\{\begin{array}{rr}\hfill \left|T_m\right|{\omega }_w-\left|T_m\right|{\omega }_w{\eta }_{gen}\left(T_m,{\omega }_w\right),& \hfill T_m<0\\ \hfill \frac{T_m{\omega }_w}{{\eta }_{mot}\left(T_m,{\omega }_w\right)}-T_m{\omega }_w,& \hfill T_m\ge 0\end{array}\right)$

[0050] The efficiency maps are generally not quadratic, so that an approximation at a specific operating point with linear and quadratic terms can be used in the following form

[00046]$J_E={\nabla }^2{P}_{loss}{\left|{}_{T_m=T_{mo}}\Delta T_m^2+\nabla P_{loss}\right|}_{T_m=T_{mo}}\Delta T_m$

[0051] The following convex quadratic program (QP), which is reliably and efficiently solvable, is used to solve the optimization problem online.

[00047]$\underset{z}{\min }\frac{1}{2}{z}^{T}Hz+f^{T}z$

[00048]$\text{subject to}\begin{array}{c}{A}_{eq}z=b_{eq}\\ A_{ineq}z\le b_{ineq}\end{array}$

[0052] The optimization vector z is given by

[00049]$z={\left[x_0^T,.Math.,x_N^T,u_0^T,.Math.,u_{N-1}^T,u_{c0}^T,.Math.,u_{cM-1}^T,\epsilon \right]}^T.$

[0053] The only free variables for the optimization are the control interventions u.sub.c0..M-1 and the slip ε. Only the first controlled variable u.sub.c0 is used for controlling the system. The other variables, in particular the state x, are also optimal in terms of QP optimization and can be used for other purposes, for example for the transformation of data in subsequent steps. With this optimization vector, the optimization is in a so-called “sparse” or “recursive” form, by contrast with the “dense” or “batch” formulation. In the case of the latter, the state in the optimization problem is eliminated, which leads to an alternative QP problem. The Hessian matrix H and the gradient vector f are given as

[00050]$H=\left[\begin{array}{cccc}{Q}_{CC}& Q_{CD}& 0& 0\\ Q_{DC}& Q_{DD}+R_u+A_{\Delta }^T{R}_{\Delta \text{u}}{A}_{\Delta }& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& P\end{array}\right],$

[00051]$f=\left[\begin{array}{cccc}{Q}_C& 0& 0& 0\\ Q_D& R_u& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{h}_{y,pred}-y_{d,pred}\\ -u_{d,pred}\\ 0\\ 0\end{array}\right]-\left[\begin{array}{c}0\\ A_{\Delta }^T{R}_{\Delta \text{u}}{A}_{\Delta }{u}_{-1}\\ 0\\ 0\end{array}\right]$

[0054] With the following vectors

[00052]$y_{d,pred}={\left[y_{d0}^T,.Math.,y_{dN}^T\right]}^T,$

[00053]$h_{y,pred}={\left[h_{y0}^T,.Math.,h_{yN}^T\right]}^T,$

[00054]$u_{d,pred}={\left[u_{d0}^T,.Math.,u_{dN-1}^T\right]}^T,$

[00055]$u_{pred}={\left[u_0^T,.Math.,u_{N-1}^T\right]}^T,$

[00056]$\Delta u_{pred}={\left[\Delta u_0^T,.Math.,\Delta u_{N-1}^T\right]}^T=A_{\Delta }{u}_{pred}-A_{\Delta }{u}_{-1},$

[00057]$u_{-1}=\underset{N-1\text{times}}{\underbrace{{\left[u_{k-1}^T,.Math.,u_{k-1}^T\right]}^T}}$

And matrices of the cost function

[00058]$Q_{CC}=\left[\begin{array}{ccc}{C}_0^{T}Q{C}_0& & \\ & \ddots & \\ & & C_N^{T}Q{C}_N\end{array}\right],$

[00059]$Q_{CC}=\left[\begin{array}{ccc}{C}_0^{T}Q{D}_0& & \\ & \ddots & \\ & & C_{N-1}^{T}Q{D}_{N-1}\\ 0& .Math.& 0\end{array}\right],$

[00060]$Q_{DC}=\left[\begin{array}{cccc}{D}_0^{T}Q{C}_0& & & 0\\ & \ddots & & .Math.\\ & & D_{N-1}^{T}Q{C}_{N-1}& 0\end{array}\right],$

[00061]$Q_{DD}=\left[\begin{array}{ccc}{D}_0^{T}Q{D}_0& & \\ & \ddots & \\ & & D_{N-1}^{T}Q{D}_{N-1}\end{array}\right],$

[00062]$R_u=\left[\begin{array}{ccc}R& & \\ & \ddots & \\ & & R\end{array}\right],$

[00063]$R_{\Delta u}=\left[\begin{array}{ccc}{R}_{\Delta }& & \\ & \ddots & \\ & & R_{\Delta }\end{array}\right],$

[00064]$A_{\Delta }={\left[\begin{array}{ccc}{I}_m& 0& .Math.\\ I_m& I_m& .Math.\\ .Math.& .Math.& \ddots \\ I_m& I_m& .Math.\end{array}\right]}^{-1},$

[00065]$Q_C=\left[\begin{array}{ccc}{C}_0^{T}Q& & \\ & \ddots & \\ & & C_N^{T}Q\end{array}\right],$

[00066]$Q_D=\left[\begin{array}{cccc}{D}_0^{T}Q& & & 0\\ & \ddots & & .Math.\\ & & D_{N-1}^{T}Q& 0\end{array}\right]$

The linear equality constraints are given by the matrix A.sub.eq and the vector b.sub.eq,

[00067]$A_{eq}=\left[\begin{array}{ccccccccccc}-I_n& & & & 0& .Math.& 0& 0& .Math.& 0& 0\\ A_0& -I_n& & & B_0& & & 0& .Math.& 0& 0\\ & \ddots & \ddots & & & \ddots & & .Math.& \ddots & .Math.& .Math.\\ & & A_{N-1}& -I_n& & & B_{N-1}& 0& .Math.& 0& 0\\ C_{w0}& & & 0& D_{w0}& & & 0& .Math.& 0& V_w\\ & \ddots & & .Math.& & \ddots & & .Math.& \ddots & .Math.& .Math.\\ & & C_{wN-1}& 0& & & D_{wN-1}& 0& .Math.& 0& V_w\\ 0& .Math.& 0& 0& I_m& & & -I_m& & & 0\\ & & & & & & & & \ddots & & .Math.\\ .Math.& \ddots & .Math.& .Math.& & \ddots & & & & -I_m& 0\\ & & & & & & & & & .Math.& .Math.\\ 0& .Math.& 0& 0& & & I_m& & & -I_m& 0\end{array}\right],$

[00068]$b_{eq}=\left[\begin{array}{c}{x}_k\\ -h_{x0}\\ .Math.\\ -h_{xN-1}\\ w_0\\ .Math.\\ w_{N-1}\\ 0\\ .Math.\\ 0\\ .Math.\\ 0\end{array}\right]$

[0055] The inequality constraints are collected in the matrix A.sub.ineq and the vector b.sub.ineq,

[00069]$A_{ineq}=\left[\begin{array}{ccccc}0& 0& \text{diag}\left(I_m\right)& 0& 0\\ 0& 0& \text{-diag}\left(I_m\right)& 0& 0\\ 0& \text{diag}\left(C_{k+1}\right)& \text{diag}\left(D_k\right)& 0& -{\overline{V}}_{y,max}\\ 0& \text{-diag}\left(C_{k+1}\right)& \text{-diag}\left(D_k\right)& 0& -{\overline{V}}_{y,min}\\ 0& 0& A_{\Delta }& 0& -{\overline{V}}_{\Delta u,max}\\ 0& 0& -A_{\Delta }& 0& -{\overline{V}}_{u,min}\\ 0& \text{diag}\left(H_{xk+1}\right)& \text{diag}\left(H_{uk}\right)& 0& -{\overline{V}}_H\\ 0& 0& 0& 0& -1\end{array}\right],$

[00070]$b_{ineq}=\left[\begin{array}{c}{\overline{u}}_{max}\\ -{\overline{u}}_{min}\\ {\overline{y}}_{max}-{\overline{h}}_{y,pred}\\ -{\overline{y}}_{min}+{\overline{h}}_{y,pred}\\ {\overline{\Delta u}}_{max}+A_{\Delta }{u}_{-1}\\ -{\overline{\Delta u}}_{min}-A_{\Delta }{u}_{-1}\\ {\overline{b}}_H\\ 0\end{array}\right]$

[0056] The index k ranges from horizon step 0 to N-1 and the overline symbol identifies the corresponding vectors. The min and max limits are repeated here for each step of the horizon, but if a prediction is available along the horizon, the limits may differ from step to step. It should be noted that enforcing output constraints in the initial step is only meaningful if the input directly influences the constrained outputs. On the other hand, enforcing output constraints in the final step is only meaningful if there is no direct flow from the input to the output. These restrictions also apply analogously to the cost function.

[0057] The motion control requirements are as follows:

[0058] In straight-ahead driving or braking situations, the energy efficiency should have priority. In such cases, the maximum regenerative braking torque should be provided.

[0059] In cornering situations, agility should be improved under normal driving conditions and, in the area at the limit of tire-road friction, vehicle stability should have absolute priority.

[0060] Stability has priority in the event of failure of a brake actuator and the drivetrain should assist the deceleration process (stabilization and max. decelerating power)

[0061] In the following sections, the tasks of all modules of the motion control concept are described and the models and all necessary data are explained in more detail.

[0062] The task of the vehicle manager is to ascertain setpoint values for the vehicle state and the global forces that act on the body at the center of gravity (CoG). The vehicle manager is designed as a model-based feedforward control. The reference vehicle state is ascertained with the aid of a flat, linear single-track model, in which the longitudinal and transverse motion are only coupled by the vehicle speed. The input into the single-track model is the front steering angle and the acceleration required by the driver. Optionally, a virtual driver may also require a rear steering angle. In this case, the virtual controller (VC) described below is deactivated.

[0063] The longitudinal state of the vehicle, i.e. the reference vehicle speed v.sub.d, is simply ascertained from the integration of the required driver acceleration a.sub.xd minus an estimated resistance, i.e. the disturbance variable a.sub.dis. If the disturbance variable were not taken into account, the reference vehicle speed would be less realistic and would not be achievable by a tracking controller. The longitudinal reference model and its state space representation are given by

[00071]${\dot{v}}_d=a_{xd}-a_{dis}$

[00072]$x=v_d,$

[00073]$u={\left[a_{xd}{a}_{dis}\right]}^T,$

[00074]$y={\dot{v}}_d,$

[00075]$A=0,$

[00076]$B=\left[1-1\right],C=0,$

[00077]$D=\left[1-1\right]$

[0064] The lateral vehicle state is characterized by the sideslip angle β.sub.d and the yaw rate ω.sub.d. In order to allow individualization of the lateral vehicle dynamics that can be selected by the driver, the lateral single-track model is virtually controlled (VC) by a virtual rear-axle steering angle δ.sub.rd,VC. The idea behind the virtually controlled single-track model is to offer the driver of the electric vehicle improved transverse agility and yaw damping. The virtual rear-axle steering angle controller is realized as a linearly quadratic (LQ) state-space controller with gain K.sub.x together with a forward filter K.sub.w and disturbance variable compensation K.sub.s. The input to the feedforward filter is a quasi-static yaw rate ω.sub.stat, which is restricted by the road holding conditions. The driver’s front steering angle δ.sub.fd is regarded as a disturbance variable and is compensated in such a way that there is a vanishing stationary virtual steering angle. This avoids the driver having to react adaptively to a changing steady-state yaw gain caused by a non-zero virtual rear-axle steering angle. The open-loop side reference model is given in state space form by

[00078]$x=\left[\begin{array}{c}{\beta }_d\\ {\omega }_d\end{array}\right],u={\delta }_{rd,VC},s={\delta }_{fd},$

[00079]$A_{ol}=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right],$

[00080]$B_u=\left[\begin{array}{c}{b}_{u1}\\ b_{u2}\end{array}\right],$

[00081]$B_s=\left[\begin{array}{c}{b}_{s1}\\ b_{s2}\end{array}\right],$

[00082]$y=\left[\begin{array}{c}{\dot{\beta }}_d\\ {\dot{\omega }}_d\\ a_y\end{array}\right],$

[00083]$C_{ol}=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\\ c_{31}& c_{32}\end{array}\right],$

[00084]$D_u=\left[\begin{array}{c}{b}_{u1}\\ b_{u2}\\ b_{u3}\end{array}\right],$

[00085]$D_s=\left[\begin{array}{c}{b}_{s1}\\ b_{s2}\\ d_{s3}\end{array}\right]$

[0065] The matrix elements of the uncontrolled lateral vehicle model result from the vehicle parameters of mass m, yaw moment of inertia J.sub.z, lateral tire stiffness at the front and rear C.sub.f, C.sub.r and distance from the center of gravity to the front or rear axle I.sub.f, I.sub.r

[00086]$\begin{array}{l}{a}_{11}=-\frac{C_f+C_r}{m{v}_d},a_{12}=-1-\frac{C_f{l}_f-C_r{l}_r}{m{v}_d^2},\\ b_{u1}=\frac{C_r}{m{v}_d},b_{s1}=\frac{C_f}{m{v}_d},\end{array}$

[00087]$a_{21}=-\frac{C_f{l}_f-C_r{l}_r}{J_z},a_{22}=-\frac{C_f{l}_f^2+C_r{l}_r^2}{J_z{v}_d},b_{u2}=-\frac{C_r{l}_r}{J_z},b_{s2}=\frac{C_f{l}_f}{J_z},$

[00088]$c_{31}=-\frac{C_f+C_r}{m},c_{32}=-\frac{C_f{l}_f-C_r{l}_r}{m{v}_d},d_{u3}=\frac{C_r}{m},d_{s3}=\frac{C_f}{m}$

[0066] The control law for the virtual rear steering angle is given as follows, with all controller gains being specified by the longitudinal reference speed,

[00089]$u=-K_{x}x-K_{s}s+K_w{\omega }_{stat}$

[0067] The closed-loop control matrices, taking into account the front steering angle and the static yaw rate as new inputs, have the form

[00090]$A=A_{ol}-B_u{K}_x,$

[00091]$B=\left[B_s-B_u{K}_s{B}_u{K}_w\right],$

[00092]$C=C_{ol}-D_u{K}_x,$

[00093]$D=\left[D_s-D_u{K}_s{D}_u{K}_w\right]$

[0068] The static yaw rate is derived from a steady-state single-track model and is restricted by the tire-road friction, which is characterized by the coefficient .Math.. The remaining parameters are the gravitational acceleration constant g and the wheelbase / = /.sub.r+/.sub.r,

[00094]${\omega }_{stat}=\text{sign}\left({\delta }_{fd}\right)\cdot \min \left(\left|{\omega }_{drv}\right|,\left|{\omega }_{\mu }\right|\right),$

[00095]${\omega }_{drv}={\delta }_{fd}\frac{v_d}{l+EG{v}_d^2},$

[00096]${\omega }_{\mu }=\frac{\mu g}{v_d},$

[00097]$EG=\frac{m}{l}\left(\frac{l_r}{C_f}-\frac{l_f}{C_r}\right)$

[0069] The closed-loop single-track model, which includes both longitudinal and transverse motion, is processed in the “batch” form described further above, in order to provide the reference data for the entire forecast horizon.

[0070] The next step in the control chain is a non-linear feed-forward control with so-called reference global forces as the output. The reference global forces, i.e. the longitudinal force F.sub.xd, the transverse force F.sub.yd and the yaw moment M.sub.zd acting on the center of gravity, are derived from the following non-linear vehicle dynamics model in state space form. Of all the possible disturbing forces, only the wind resistance is taken into account by the aerodynamic drag coefficient parameter k.sub.x. The vehicle dynamics are described in a body-fixed coordinate system with its origin at the center of gravity. The non-linear system dynamics in input-affine form and the non-linear output equations are given by

[00098]$\left[\begin{array}{c}{\dot{v}}_x\\ {\dot{v}}_y\\ \dot{\omega }\end{array}\right]=\left[\begin{array}{c}{v}_y\omega -\frac{k_x}{m}{v}_x^2\\ -v_x\omega \\ 0\end{array}\right]+\left[\begin{array}{ccc}\frac{1}{m}& 0& 0\\ 0& \frac{1}{m}& 0\\ 0& 0& \frac{1}{J_z}\end{array}\right]\left[\begin{array}{c}{F}_{xd}\\ F_{yd}\\ M_{zd}\end{array}\right],$

[00099]$\left[\begin{array}{c}{v}_d\\ {\beta }_d\\ {\omega }_d\end{array}\right]=\left[\begin{array}{c}\sqrt{v_x^2+v_y^2}\\ a\text{rctan}\frac{v_y}{v_x}\\ \omega \end{array}\right]$

[0071] The flatness property of the reference outputs v.sub.d, β.sub.d and ω.sub.d is exploited to calculate the global reference forces by a flatness-based inversion, i.e. the global forces are functions of the reference outputs and their first derivatives as inputs. IFor example, for the longitudinal reference force, there is the inversion

[00100]$F_{xd}=m\left({\dot{v}}_d\cos {\beta }_d-v_d\left({\dot{\beta }}_d+{\omega }_d\right)\sin {\beta }_d\right)+k_x{v}_d^2\left(1-si{n}^2{\beta }_{dd}\right)$

[0072] Particular attention must be paid to the disturbances. The reference speed v.sub.d already includes longitudinal disturbances and using the previous equation may cause driver confusion. A desired zero acceleration should lead to a zero longitudinal reference force F.sub.xd. Therefore, all disturbance-related terms must be eliminated. For this purpose, the estimated disturbance a.sub.dis is described in the next section and a curve disturbance term is inserted into the equation. Altogether, the following model describes the desired global forces processed at each sampling time for an N-step prediction.

[00101]$\left[\begin{array}{c}{F}_{xd}\\ F_{yd}\\ M_{zd}\end{array}\right]=\left[\begin{array}{c}m\left({\dot{v}}_d\cos {\beta }_d-v_d\left({\dot{\beta }}_d+{\omega }_d\right)\sin {\beta }_d\right)+m\left(a_{dis}+{\beta }_d{a}_{yd}\right)\\ m\left({\dot{v}}_d\sin {\beta }_d+v{}_d\left({\dot{\beta }}_d+{\omega }_d\right)\cos {\beta }_d\right)\\ J_z{\dot{\omega }}_d\end{array}\right]$

[0073] There are two tasks of the vehicle observer: determining the forecast horizon of the driver input and estimating a longitudinal disturbance a.sub.dis acting on the vehicle.

[0074] There are two main approaches to predicting the driver horizon, presuming that no map or environmental sensor information is available. The first approach is simply to keep the current acceleration and steering value over the forecast horizon. The second approach is based on curve fitting. A few last sample values are stored and a curve is fitted to the data by interpolation, for example with a polynomial. The polynomial is then used for extrapolation over the entire forecast horizon. For the sake of simplicity, the first approach has been chosen below.

[0075] A Luenberger disturbance observer with two states, estimated vehicle speed v.sub.e and longitudinal disturbance a.sub.dis, and the desired driver acceleration a.sub.xd as the input is designed for the disturbance estimation. The output is the estimated vehicle speed v.sub.e, which is compared with the actual vehicle speed v in order to reduce the estimation error. The Luenberger observer has the following time-continuous, linear, time-invariant state space form

[00102]$\left[\begin{array}{c}{\dot{v}}_e\\ {\dot{a}}_{dis}\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{v}_e\\ a_{dis}\end{array}\right]+\left[\begin{array}{c}1\\ 0\end{array}\right]a_{xd}+\left[\begin{array}{c}{L}_1\\ L_2\end{array}\right]\left(v-v_e\right),$

[00103]$v_e=\left[10\right]\left[\begin{array}{c}{v}_e\\ a_{dis}\end{array}\right]$

[0076] The observer gains L.sub.1 and L.sub.2 are designed by the linearly quadratic (LQ) method and then the continuous disturbance observer is discretized by the Tustin method. The estimated longitudinal disturbance provided by the vehicle observer is kept constant over the forecast horizon.

[0077] The task of the chassis manager is to provide optimized longitudinal and transverse forces at each corner of the vehicle. FIG. 4 shows the geometrical and kinematic variables and also the forces that are taken into account in the motion control concept. The vehicle motion is described in a horizontal, body-fixed coordinate system (index “b”). Two types of auxiliary coordinate systems are used below. The first type are systems that are fixed in each chassis corner (index “c”), but have the same orientation as the body-fixed system. The second type are systems that are fixed in each wheel corner and turned by the wheel steering angle (index “w”). At each wheel, the chassis corner system and the wheel corner system have the same origin. For the motion control concept of the embodiment described, it was decided that the chassis manager would produce the optimized corner forces specified in the chassis corner system. The use of wheel corner forces would be an alternative, with the equations of motion being more complex.

[0078] The first task of the chassis manager is to ascertain the desired reference chassis corner forces, which result from the global reference forces produced by the vehicle manager. For this purpose, a static rule assignment approach on the basis of a pseudo-inverse with a weighting matrix W.sub.u is chosen. The relationship between the global forces and the chassis corner forces is given using the distribution matrix G and the parameters of the track width at the front and rear axles, b.sub.f and b.sub.r, by

[00104]${\left[F_{xd}{F}_{yd}{M}_{zd}\right]}^T=G{u}^c,$

[00105]$\begin{array}{l}{u}^c=\\ {\left[F_{xd,fl}{F}_{yd,fl}{F}_{xd,fr}{F}_{yd,fr}{F}_{xd,rl}{F}_{yd,rl}{F}_{xd,rr}{F}_{yd,rr}\right]}^T,\end{array}$

[00106]$G=\left[\begin{array}{cccccccc}1& 0& 1& 0& 1& 0& 1& 0\\ 0& 1& 0& 1& 0& 1& 0& 1\\ -b_f& l_f& b_f& l_f& -b_r& -l_r& b_r& -l_r\end{array}\right]$

[0079] The above relationship between global and chassis corner forces is an underdetermined linear system that cannot be uniquely solved for u.sup.c. There are numerous methods to exploit the remaining degrees of freedom. We have chosen a weighted Moore-Penrose pseudo-inverse, which results from minimizing the weighted 2-norm u.sup.TW.sub.u.sup.TW.sub.uu, where W.sub.u is a positive diagonal matrix.

[0080] The transformation between chassis-corner and wheel-corner coordinates is possible with the rotation matrix T.sup.cw,

[00107]$u^c=T^{cw}{u}^w,u^w=T^{wc}{u}^c={\left(T^{wc}\right)}^T{u}^c,v_w^w=T^{wc}{v}_w^c$

[00108]$\begin{array}{l}{T}^{cw}=\\ \left[\begin{array}{cccccccc}\cos {\delta }_f& -\sin {\delta }_f& 0& 0& 0& 0& 0& 0\\ \sin {\delta }_f& \cos {\delta }_f& 0& 0& 0& 0& 0& 0\\ 0& 0& \cos {\delta }_f& -\sin {\delta }_f& 0& 0& 0& 0\\ 0& 0& \sin {\delta }_f& \cos {\delta }_f& 0& 0& 0& 0\\ 0& 0& 0& 0& \cos {\delta }_r& -\sin {\delta }_r& 0& 0\\ 0& 0& 0& 0& \sin {\delta }_r& \cos {\delta }_r& 0& 0\\ 0& 0& 0& 0& 0& 0& \cos {\delta }_r& -\sin {\delta }_r\\ 0& 0& 0& 0& 0& 0& \sin {\delta }_r& \cos {\delta }_r\end{array}\right]\end{array}$

[0081] As already mentioned above, the reference corner forces are required in the chassis corner systems. However, it is more favorable for the static control assignment to weight the corner forces in the wheel coordinate systems and then to perform a transformation back into the chassis systems. The reason for this is that, to achieve the aim of energy efficiency, the primary purpose of the weighting matrix W.sub.u is to favor the electric motors on the front axle before the friction brakes on both axles are used. The longitudinal wheel corner forces correspond directly to the torques at the wheels, so that the rear longitudinal wheel corner forces are weighted lower than the other forces. The overall static control assignment process is then specified by

[00109]$u_d=T^{cw}{W}_u^{-1}{B}_g^T{\left(B_g{W}_u^{-1}{B}_g^T\right)}^{-1}{\left[F_{xd}{F}_{yd}{M}_{zd}\right]}^T,$

[00110]$B_g=G{T}^{cw},$

[00111]$W_u=\text{diag}\left({10}^{-6},{10}^{-6},{10}^{-6},{10}^{-6},1,{10}^{-6},1,{10}^{-6}\right).$

[0082] The second task of the chassis manager is to use a model-based predictive control (MPC) for the tracking of the reference states and inputs and to keep all states and control actions within their limits. In a first step, the non-linear vehicle model defined further above in the vehicle manager section is linearized along the reference state and input trajectory. The reference input u.sub.d is represented by the reference longitudinal and transverse forces in chassis corner coordinates, as explained above, and the reference state x.sub.d is calculated from the data provided by the vehicle manager,

[00112]$x_d=\left[\begin{array}{c}{v}_{xd}\\ v_{yd}\\ {\omega }_d\end{array}\right]=\left[\begin{array}{c}{v}_d\cos {\beta }_d\\ v_d\sin {\beta }_d\\ {\omega }_d\end{array}\right]$

[0083] The result of the linearization are the following matrices and vectors

[00113]$A=\left[\begin{array}{ccc}-\frac{2k_x{v}_{xd}}{m}& {\omega }_d& v_{yd}\\ -{\omega }_d& 0& -v_{xd}\\ 0& 0& 0\end{array}\right],$

[00114]$B=\left[\begin{array}{ccc}\frac{1}{m}& 0& 0\\ 0& \frac{1}{m}& 0\\ 0& 0& \frac{1}{J_z}\end{array}\right]G,$

[00115]$C=\left[\begin{array}{ccc}\frac{v_{xd}}{\sqrt{v_{xd}^2+v_{yd}^2}}& \frac{v_{yd}}{\sqrt{v_{xd}^2+v_{yd}^2}}& 0\\ -\frac{v_{yd}}{v_{xd}^2+v_{yd}^2}& \frac{1}{v_{xd}^2+v_{yd}^2}& 0\\ 0& 0& 1\end{array}\right],$

[00116]$D=0,$

[00117]$h_x=\left[\begin{array}{c}{v}_{yd}{\omega }_d-\frac{k_x}{m}{v}_{xd}^2\\ -v_{xd}{\omega }_d\\ 0\end{array}\right]+B{u}_d-A{x}_d-B{u}_d=\left[\begin{array}{c}-v_{yd}{\omega }_d+\frac{k_x}{m}{v}_{xd}^2\\ v_{xd}{\omega }_d\\ 0\end{array}\right],$

[00118]$h_y=\left[\begin{array}{c}\sqrt{v_{xd}^2+v_{yd}^2}\\ \text{aran}\frac{v_{yd}}{v_{xd}}\\ {\omega }_d\end{array}\right]-C{x}_d$

[0084] The state x, the output y and the control action u of the chassis manager MPC consists of

[00119]$x={\left[v_x{v}_y\omega \right]}^T,$

[00120]$y={\left[v\beta \omega \right]}^T$

[00121]$u=\left[F_{xd,fl,MPC}{F}_{yd,fl,MPC}\right)$

[00122]${\left(F_{xd,rr,MPC}{F}_{yd,rr,MPC}\right]}^T$

[0085] Constraints which are described below can be defined for the state and the control action. The equality constraints of the MPC are the above system equations, the driving lock and a balance of the longitudinal corner forces of the chassis with the global force provided by the vehicle manager, i.e.

[00123]$w_k=F_{xd},$

[00124]$C_{wk}=0,$

[00125]$D_{wk}=\left[10101010\right],$

[00126]$V_w=1$

[0086] The inequality constraints are given, as u.sub.min, the absolute force limits F.sub.max at each wheel specified by the chassis observer and, as u.sub.max, zero at the rear axle and the maximum propulsion force of the electric motor at the front axle. The rate limits are chosen to be symmetrical with +/- 100 Nm at each sampling time. A polytope constraint approximates the friction circle at each wheel with octagons. The vertices of the octagon are calculated by

[00127]$\left[\begin{array}{c}{P}_{x,j}\\ P_{y,j}\end{array}\right]=F_{max}\left[\begin{array}{c}\sin \left(2\pi \frac{j}{8}\right)\\ \cos \left(2\pi \frac{j}{8}\right)\end{array}\right],$

[00128]$j=1,.Math.,8$

[0087] The edges of the octagon can be described by linear functions and p.sub.m and p.sub.b can be derived from the corner points, i.e. the limit for wheel i is described by

[00129]$F_{xd,i,MPC}=p_{mj,i}{F}_{yd,i,MPC}+p_{bj,i},$

[00130]$j=1,.Math.,8,$

[00131]$i=1,.Math.4$

[0088] The control actions are restricted by the edges, but in principle can be controlled to any desired value within the polytope. This leads to the following inequality constraint, which is relaxed to ensure the viability of the optimization,

[00132]$H_{xj,i}{F}_{xd,i,MPC}+H_{yj,i}{F}_{yd,i,MPC}\le H_{j,i}+V_{Hj,i}\epsilon ,$

[00133]$j=1,.Math.,8,$

[00134]$i=1,.Math.,4$

[00135]$H_{xj,i}=\text{sign}\left(p_{bj,i}\right),$

[00136]$H_{yj,i}=-\text{sign}\left(p_{bj,i}\right)p_{mj,i},$

[00137]$H{l}_{j,i}=\text{sign}\left(p_{bj,i}\right)p_{bj,i},$

[00138]$V_{Hj,i}=1$

[0089] The second type of polytope constraints is a stability envelope that restricts the yaw rate ω and the rear slip angle a.sub.r in a stable driving range.

[0090] With the tire-road friction .Math., the gravitational constant g and the rear lateral tire stiffness C.sub.r, the corners of the envelope are defined by

[00139]${\omega }_{max}=\frac{\mu g}{v_d},$

[00140]${\alpha }_{r,max}=3\frac{F_{max,rl}+f_{max,rr}}{C_r}$

[0091] The rear slip angle is a simple non-linear function of the output, but a more complicated function of the state. In order to derive a linear constraint, the non-linear function is approximated by linearization along the reference states. The result is an affine linear function of the state, with the details of the time-dependent coefficients being omitted here,

[00141]${\alpha }_r=\beta -\frac{l_r}{v}\omega =arctan\left(\frac{v_y}{v_x}\right)-\frac{l_r}{\sqrt{v_x^2+v_y^2}}\omega \approx c_1{v}_x+c_2{v}_y+c_3\omega +h_{\alpha },$

[0092] Applying the same method as for the friction circle, the envelope constraint is obtained as follows

[00142]$H_{1l}{v}_x+H_{2l}{v}_y+H_{31}\omega \le H_{0l}+V_{Hl}\epsilon ,l=1,.Math.,4,$

[00143]$H_{1l}=-\text{sign}\left(p_{bl}\right)p_{ml}{c}_1,$

[00144]$H_{2l}=-\text{sign}\left(p_{bl}\right)p_{ml}{c}_2,$

[00145]$H_{3l}=\text{sign}\left(p_{bl}\right)\left(1-p_{ml}{c}_3\right),$

[00146]$H_{0l}=\text{sign}\left(p_{bl}\right)\left(p_{ml}{h}_{\alpha }+p_{bl}\right),$

[00147]$V_{Hl}=1$

[0093] Collecting all the constraints in the overall matrices H.sub.xk and H.sub.uk and the vector b.sub.H introduced further above leads to the following results, with the trailing and leading zeros respectively corresponding to the non-zero block in the other matrix,

[00148]$H_{xk+1}=\left[\begin{array}{ccc}{H}_{11}& H_{21}& H_{31}\\ .Math.& .Math.& .Math.\\ H_{14}& H_{24}& H_{34}\\ 0& 0& 0\\ .Math.& .Math.& .Math.\\ 0& 0& 0\end{array}\right],$