Robust two-degrees of freedom linear quadratic gaussian position control for the front axle actuator of a steer-by-wire system

For the linear plant model with objective output equationan optimal control u_c,p is needed, which drives the state vector \(\underline{\mathbf{x}}_{\boldsymbol{p}}\) from any initial state to the zero state, such that the value of a performance index is minimized. For the performance index, a quadratic cost function J_r build by the weighted control input and objective output is used

The weighting factors \(Q\geq 0\) and \(R> 0\) penalize large values of the respective signals in the minimization of the cost function. A simple and physically meaningful choice is to select Q and R as [14]:

Following this rule, the variables that appear in the cost function J_r are scaled so that the maximum acceptable value of each term is 1. By application of this method, good starting values for the optimal controller design are obtained, which can be further refined by additional controller design and analysis iterations. The state-feedbackis the optimal control, which minimizes the quadratic cost function over an infinite time horizon \([0{,}\infty ]\). The controller gain matrix \(\underline{\mathbf{K}}_{\boldsymbol{p}}\) follows from [16]where \(\underline{\mathbf{S}}\) is the steady-state solution of the matrix Riccati equation

The optimal state-feedback control determines the dynamics of the disturbance response and the stability.

For steady-state disturbance rejection, a static disturbance feedforward is further added. The extended control law isand requires that the disturbance variables \(\underline{\mathbf{x}}_{\mathrm{dist}}\) can be measured or estimated. The gains \(\underline{\mathbf{K}}_{\boldsymbol{d}}\) of the static disturbance feedforward can be analytically calculated for the linear plant model with objective output equation by [17]

For optimal state-feedback controller design, it was supposed that the disturbance variables are available as measurements. However, this is not the case in practical implementation. Therefore, a state-estimator is designed, which provides estimates of the plant states and reconstructs the unknown disturbance variables.

For the design of the state-estimator, the linear plant model with measurement output equationis augmented by unknown input integrator disturbance models

Using the substitution \(\underline{\mathbf{u}}_{\boldsymbol{d}{,}\boldsymbol{p}}=\underline{\mathbf{y}}_{\mathrm{dist}}\) the extended linear plant model results [14, 16]respectively in compact matrix-/vector notation

Based on the augmented linear plant model, an optimal state-estimator (Kalman filter) is designed. For the optimal estimation problem, it is assumed that the augmented plant is disturbed by process noise \(\underline{\mathbf{w}}\) and measurement noise \(\underline{\mathbf{v}}\). In this context, the process noise is modeled additive to the inputs for penalizing the unknown disturbance model inputs and preparing for input Loop Transfer Recovery (LTR). The latter is a Kalman filter adjustment procedure that asymptotically recovers the excellent robustness properties for full-state feedback and thereby contributes to the robust design of the LQG control [14, 16, 19, 20].

For the augmented plant model subject to process and measurement noise an optimal state estimate \(\underline{\hat{\mathbf{x}} }_{\boldsymbol{a}}\left(\mathrm{t}\right)\) is needed, which minimizes the steady-state value of the sum of the squared estimation errors

The optimal solution is given by the state-estimatorthat minimizes the quadratic cost function over the infinite time horizon \([0{,}\infty ]\). The state-estimator feedback gain matrix \(\underline{\mathbf{L}}\) is computed from [16]with \({\underline{\mathbf{P}}}_{\boldsymbol{e}}^{\mathbf{*}}\) the steady-state solution of the Riccati equation

The weighting matrices \(\underline{\mathbf{W}}\) and \(\underline{\mathbf{V}}\) can be treated as design parameters for the determination of the estimator feedback gain \(\underline{\mathbf{L}}\). By assuming that the noise processes are uncorrelated, they become diagonal matrices whose elements are the variances of the respective noise vector components. Suitable starting values for the noise variances may be obtained by considering the environment of the estimator in the digital implementation. The control inputs will be fixed point values with a defined quantization that is given by the software scaling. The measurement outputs will also be subject to quantization, which is due to the limited resolution in the AD conversion of these signals. The resulting round-off errors can be modeled as stochastic white noise processes that are added to the respective signals. According to [14] the variance of quantization noise can be defined as:with q being the signal quantization. Noise that does not originate from signal quantization, such as for example sensor noise, can be measured by experiment and the determined variance value used as a direct design parameter. By applying this rule to the control input and measurement output signals, suitable estimator design parameters can be found. All that remains is finding appropriate values for the noise variance of the integrator disturbance model inputs. As these inputs are unknown, their noise variances must be selected as high as possible (i.e. high uncertainty) to make the estimator fully rely on the measurements for their reconstruction. Summarizing these guidelines, physically meaningful initial values for the weighting matrices \(\underline{\mathbf{W}}\) and \(\underline{\mathbf{V}}\) are obtained by selecting:with

The estimated state and disturbance variables are provided as input to the control law. By incorporating the integrator disturbance models into the state-estimator, the resulting LQG controller has integral action and is robust against plant uncertainty.

The LQG control defines the dynamics of the disturbance response and the stability of the closed-loop system. For an independent design of the dynamics of the command response, the model-based dynamic feedforward control is introduced (see Fig. 5). It consists of a virtual control loop, which is executed online to generate a control signal \(\tilde{u}_{c{,}p}\) for command following. The generated control signal is then provided to the real plant. For hiding the effect of the control signal on the system states from the LQG feedback controller, the virtual state vector \(\underline{\tilde{\mathbf{x}} }_{\boldsymbol{p}}\) is subtracted from the state estimates \(\underline{\hat{\mathbf{x}} }_{\boldsymbol{p}}\) (see Fig. 4).

Thereby, the LQG feedback controller accepts any intervention from dynamic feedforward control and remains inactive [15].

The virtual control loop is designed for the desired command response. For this purpose, the synthesis of an optimal state-feedback controller with static reference feedforward is suitable as the state vector of the plant model is directly accessible [18]. The control law is

The optimal feedback gain matrix \(\underline{\tilde{\mathbf{K}} }_{\boldsymbol{p}}\) is analogous to Sect. 3.1 designed by minimization of a quadratic cost function. The static reference feedforward is analytically computed by [16]

As the dynamic reference feedforward is a pure feedforward control, it has no impact on the stability of the control loop.

By combining the model-based dynamic feedforward control with the LQG feedback control, the 2DOF LQG position control according to Fig. 4 results. For the absence of model error, disturbances, or different initial conditions the command response of the virtual control loop propagates to the real system. The LQG feedback controller remains inactive in this case. However, if different initial conditions, external disturbances, or model error occur, the LQG feedback controller becomes active and determines the disturbance response. Hence, the 2DOF control structure allows for a separate design of the command and the disturbance response of the control system [15, 17].

For the modeling of plant uncertainties, parametric and dynamic uncertainties are considered. Parametric uncertainty is the deviation of a physical parameter from its nominal value and can be mathematically modeled as [19]:with

With this respect the following parametric uncertainties are considered for the linearized plant model (Table 1).

Furthermore, a dynamic multiplicative input uncertainty is assumed to account for simplifications in the actuator modeling, and neglected high-frequency dynamics of the plant. The dynamic uncertainty is modeled as [19]:with

The uncertainty weight is frequency-dependent and modeled by a lead-lag filterwith \(K_{l}=0.05\) and \(K_{u}=1.5\) for a 5% gain uncertainty at low frequencies and 150% gain uncertainty at high frequencies. The gain cross-over is selected in line with the actuator bandwidth to \(f_{c}=50\,\mathrm{Hz}\).

For robust performance analysis, weighting functions W_i(s) are defined in the frequency domain. These weighting functions specify upper bounds on the command response and the disturbance response of the uncertain control system [19]. With this regard, the performance requirements for the command response are modeled by a second-order lag filter with as shown in Fig. 10. Thus, the requirements are that the maximum gain must not exceed 0.83 dB (abs 1.1). Moreover, a maximum cut-off frequency of 30 Hz and a high-frequency gain roll off-of at least −40 dB/dec is required to ensure acceptable measurement noise attenuation and avoid the excitation of unmodeled high-frequency dynamics. The performance requirements for the disturbance response are modeled by a lead-lag filter with

Therefore, it is required that the minimum disturbance attenuation is −14 dB (abs 0.2) over the entire frequency range, which provides a strong rejection of external load torques. Furthermore, a steady-state disturbance attenuation of at least −80 dB (abs 0.0001) is required to ensure stationary accuracy of the position control.

The previously defined parametric uncertainties, dynamic uncertainty, and performance weighting functions are now added to the nominal control system. For this purpose, the nominal physical parameters are replaced by uncertain physical parameters in the state-space equations of the plant model. Furthermore, the dynamic uncertainty is inserted at the plant control input. Finally, the performance weighting functions are connected to the reference and the disturbance input. This gives the uncertain control loop with performance weighting as shown in Fig. 11.

To get a suitable format for μ-analysis, the uncertain control loop must be transformed into a Linear Fractional Representation (LFR). This is accomplished by introducing virtual inputs w and outputs z, which allow a separation of the unknown uncertainty variables \(\delta _{i}{,}\Updelta _{i}\).

The separated uncertainty variables are collected in a perturbation matrix \(\underline{\mathbf{\Delta }}\)(s). The remaining “known part” of the system can be transformed into a transfer function matrix \(\underline{\mathbf{N}}\left(\mathrm{s}\right)\) that is connected to \(\underline{\mathbf{\Delta }}\)(s) in a feedback configuration. Thereby, the \(\underline{\mathbf{N}}{,}\underline{\mathbf{\Delta }}\)—LFR of the uncertain control loop shown in Fig. 12 is obtained, which gives the foundation for robust stability and performance analysis [19].

Partitioning of \(\underline{\mathbf{N}}\) and merging it with \(\underline{\mathbf{\Delta }}\) gives the transfer function matrix \(\underline{\mathbf{F}}\) of the uncertain control loop:

For analyzing robust stability, only the elements \(\underline{\mathbf{N}}_{\mathbf{11}}\) are needed, which are connected to the perturbation matrix \(\underline{\mathbf{\Delta }}\) in a feedback configuration, that is (Fig. 13).

The structured singular value (\(\mu _{\underline{\mathbf{\Delta }}}\)) is introduced to formulate necessary and sufficient conditions for robust stability. The mathematical definition is given as:

Therefore, \(\mu _{\underline{\mathbf{\Delta }}}\) is defined as the reciprocal value of the smallest perturbation \(\underline{\mathbf{\Delta }}\) (measured by \(\overline{\sigma }\left(\underline{\mathbf{\Delta }}\right)\)), for which \(\det \left(\underline{\mathbf{I}}-\underline{\mathbf{N}}_{\mathbf{11}}\underline{\mathbf{\Delta }}\left(j\omega \right)\right)\) becomes 0 and thus instability occurs. Hence, the system is robustly stable if and only if \(\mu _{\underline{\mathbf{\Delta }}}< 1\) for all frequencies [19].

Figure 14 shows the computed \(\mu _{\underline{\mathbf{\Delta }}}\) upper bound over frequency, which provides a numerical approximation to the exact \(\mu _{\underline{\mathbf{\Delta }}}\) value. The peak value of the upper bound is 0.21. Therefore, the control system is robustly stable for the modeled uncertainty. It can tolerate up to 474% of the considered uncertainty before instability occurs.

Hence, the peak value of \(\mu _{\underline{\mathbf{\Delta }}}\) provides a quantitative measure for the robustness of the 2DOF LQG position control system.

For analyzing robust performance, the entire transfer function matrix \(\underline{\mathbf{N}}\) is needed, and a fictitious complex uncertainty \(\underline{\mathbf{\Delta }}_{\boldsymbol{F}}\) connecting the reference input and the objective output respectively the disturbance input and the objective output is introduced (Fig. 15).

Thus, the robust performance analysis can be transformed into a robust stability analysis problem [19]. The structured singular value \(\mu _{{\underline{\mathbf{\Delta }}_{\boldsymbol{P}}}}\) for the augmented perturbation matrixand the matrix \(\underline{\mathbf{N}}\) is introduced to formulate necessary and sufficient conditions for robust performance. The mathematical definition for \(\mu _{{\underline{\mathbf{\Delta }}_{\boldsymbol{P}}}}\)is given by:

Therefore, the designed controller achieves robust performance if \(\mu _{{\underline{\mathbf{\Delta }}_{\boldsymbol{P}}}}< 1\) for all frequencies [19].

Figure 16 illustrates the \(\mu _{{\underline{\mathbf{\Delta }}_{\boldsymbol{P}}}}\) upper bounds over frequency for the command response and the disturbance response control performance. The peak value of the upper bound of \(\mu _{{\underline{\mathbf{\Delta }}_{\boldsymbol{P}}}}\) for the command response is 0.72. Therefore, the control system has robust performance and fulfills the performance requirements for all of the modeled uncertainty. The uncertainty can be increased by up to 139% before requirements are violated. Similarly, by computing the upper bound of \(\mu _{{\underline{\mathbf{\Delta }}_{\boldsymbol{P}}}}\) for the disturbance response, a peak value of 0.5 is found, thereby proving robust performance. The control system can tolerate up to 200% of the modeled uncertainty before performance requirements are no longer met.

The controller design was fine-tuned iteratively in the vehicle together with a professional test driver compromising performance, robustness, NVH, and steering feel. Thereby an optimal setting was identified and used as the basis for all driving maneuvers presented in the following. Figure 18, 19 and 20 exemplarily show the control performance in the real vehicle against nonlinear simulation for step steering at 30 kph, sweep steering at 100 kph, and slalom at 40 kph. The step and sweep response driving maneuver are carried out to test the dynamic vehicle response and the command following of the position control.

The controller dynamically tracks the reference steering position with minor phase delay given by the control bandwidth. Disturbances originating from external rack forces, nonlinear friction or plant parameter uncertainty are strongly rejected and compensated in steady state. Moreover, the controller showed a stable behavior over the complete operating range of the system in all driving maneuvers. Thereby the robustness of the designed control in practical application is confirmed. Besides the performance evaluation, Fig. 18 to 19 reveal a good fit between the vehicle results and simulation, thereby providing verification to the simulation results. Figure 20 shows the results of a slalom driving maneuver at 40 kph.

The controller accurately follows the given reference profile and robustly rejects unknown disturbances. Again, an accurate fit between vehicle and simulation was achieved, which verifies the validity of the nonlinear simulation. Minor deviations during motion reversals are observable, which are due to the difficultly of modeling nonlinear friction.

Additional tests for parking conditions were performed using artificially generated reference step and sine sweep excitation signals. Hereby, the control performance was extensively tested for high load and high steering velocity conditions. In all of the tests performed, a good correspondence between vehicle measurement and simulation was achieved.