Nominal and robust controller synthesis for an active suspension system

The file Demo_007.m is found in IQClab’s folder demos; it performs a nominal control design with the functions hinfsyn (for MATLAB) and fHinfsyn (from IQClab). In addition, the demo demonstrates the capability to robustify controllers using the functions musyn (from MATLAB) and again fHinfsyn (from IQClab).

The example is a MATLAB demo: The control design for an active suspension system as depicted in the following figure.

Active suspension of quarter-car model


  • x_b is the position of the body
  • x_w is the position of the wheel
  • r is the position of the road
  • m_b and m_t are the masses of the body and tire
  • k_s and k_t are the spring constants of the body and tire
  • b_s and f_s are the damping and active damping coefficients

With the notation (x_1,x_2,x_3,x_4)=(x_b,\dot{x}_b,x_w,\dot{x}_w), the linearized state equations are given by:

\left(\!\begin{array}{c}\dot{x}_1\\ \dot{x}_2\\  \dot{x}_3\\  \dot{x}_4\end{array}\!\right)\!=\! \left(\!\begin{array}{cccc}0&1&0&0\\-\tfrac{k_s}{m_b}&-\tfrac{b_s}{m_b}& \tfrac{k_s}{m_b} & \tfrac{b_s}{m_b}\\0&0&0&1\\ \tfrac{k_s}{m_t}&\tfrac{b_s}{m_w}&-\tfrac{k_s+k_t}{m_w} & -\tfrac{b_s}{m_w}  \end{array}\!\right)\!\left(\!\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\!\right)\!+\!\left(\!\begin{array}{cc}0&0\\ \tfrac{10^3}{m_b}&0\\0&0\\ -\tfrac{10^3}{m_w}&\tfrac{k_t}{m_w}\end{array}\!\right) \! \left( \! \begin{array}{c}f_s\\r\end{array}\!\right) \left(\!\begin{array}{c}x_1\\x_1-x_3\\ \dot{x}_2\end{array}\!\right)\!=\!\left(\!\begin{array}{cccc}1&0&0&0\\1&0&-1&0\\ -\tfrac{k_s}{m_b}&-\tfrac{b_s}{m_b}& \tfrac{k_s}{m_b} & \tfrac{b_s}{m_b}\end{array}\!\right)\!\left(\!\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\!\right)\!+ \!\left(\!\begin{array}{cc}0&0\\0&0\\ \tfrac{10^3}{m_b}&0\end{array}\!\right) \! \left( \! \begin{array}{c}f_s\\r\end{array}\!\right)

Clearly, road disturbances influence the motion of the car and the suspension. Moreover, passenger comfort is associated with small body accelerations. The allowable suspension travel is constrained by limits on the actuator displacement. The following plot depicts the open-loop gain from road disturbance and actuator force to body acceleration and suspension displacement.

Gain (frequency response) from road disturbance r and actuator force f_s to body acceleration a_b and suspension travel s_d.

Because of the zeros on the imaginary-axis, feedback control cannot improve the response from road disturbance r to body acceleration a_b at the so-called tire-hop frequency, and from r to suspension deflection s_d at the so-called rattle-space frequency. In addition, because x_w=x_b-s_d and x_w roughly follows r at low frequencies (less than 5 rad/s), there is an inherent trade-off between passenger comfort and suspension deflection; any reduction of body travel at low frequency will result in an increase of suspension deflection.

In this example, we consider an uncertain actuator model. This hydraulic actuator used for the active suspension control is connected between the body mass m_b and the wheel assembly mass m_w. The nominal actuator dynamics are represented by the first-order transfer function \tfrac{1}{(1+60/s)} with a maximum displacement of 0.05m.

Since the nominal model only approximates the physical actuator dynamics, we can use a family of actuator models to account for modelling errors and variability in the actuator model. This family consists of a nominal model with a frequency-dependent amount of uncertainty. At low frequency, below 3 rad/s, the model can vary up to 40% from its nominal value. Around 3 rad/s, the percentage variation starts to increase. The uncertainty crosses 100% at 15 rad/s and reaches 2000% at approximately 1000 rad/s. The weighting function W_\mathrm{unc} is used to modulate the amount of uncertainty with frequency.

As a result, we obtain Act, which is an uncertain state-space model of the actuator. The following figure depicts the Bode plots of 20 sample values of Act and compare with the nominal value.

Bode diagram of 20 samples of the uncertain actuator model Act

The main control objectives are formulated in terms of passenger comfort and road handling, which relate to body acceleration a_b and suspension travel s_d. Other factors that influence the control design include the characteristics of the road disturbance, the quality of the sensor measurements for feedback, and the limits on the available control force. To use the H_\infty-synthesis algorithms, we have to write done the weighted open-loop generalized plant interconnection as shown in the following figure.

Disturbance rejection formulation

As can be seen, the feedback controller uses measurements y_1, y_2 of the suspension travel s_d and body acceleration a_b to compute the control signal u driving the hydraulic actuator. On the other hand, there are three external sources of disturbance:

  • The road disturbance r, modeled as a normalized signal d_1 shaped by a weighting function W_\mathrm{road}. To model broadband road deflections of magnitude seven centimeters, we use the constant weight W_\mathrm{road}=0.07.
  • Sensor noise on both measurements, modelled as normalized signals d_2 and d_3 shaped by weighting functions W_{d_2} and W_{d_3}. We use W_{d_2}=0.01 and W_{d_3}=0.5 to model broadband sensor noise of intensity 0.01 and 0.5, respectively. In a more realistic design, these weights would be frequency dependent to model the noise spectrum of the displacement and acceleration sensors.

The control objectives can be interpreted as a disturbance rejection problem: Minimize the impact of the disturbances d_1, d_2, d_3 on a weighted combination of control effort u, suspension travel s_d, and body acceleration a_b. When using the H_\infty-norm to measure “impact”, this amounts to designing a controller that minimizes the H_\infty-norm from disturbance inputs d_1, d_2, d_3 to error signals e_1, e_2, e_3.

Let us also create the weighting functions. We use a high-pass filter for W_\mathrm{act} to penalize control at high-frequencies. In addition, we specify the closed-loop performance requirements for the gain from road disturbance r to suspension deflection s_d (handling) and body acceleration a_b (comfort). Because of the actuator uncertainty and control limitations, we only aim for attenuating disturbances below 10 rad/s.

The corresponding performance weights W_{s_d}, W_{a_b} are the inverses of these comfort and handling requirements. To investigate the trade-off between passenger comfort and road handling, we construct three sets of weights (\beta W_{s_d},1-\beta  W_{a_b}) corresponding to three different trade-offs: comfort (\beta=0.01), balanced (\beta=0.5), and handling (\beta=0.99).

We can now construct the uncertainty model of the generalized plant from above and perform an H_\infty-synthesis using hinfsyn (from MATLAB) and fHinfsyn (from IQClab). The three controllers achieve the following closed-loop H_\infty-norms.

Comfort classhinfsynfHinfsyn
Computed H_\infty-norms

Given the controllers K_1 and K_2 we can now construct the corresponding closed-loop models and compare the gains from road disturbance to x_b, s_d, a_b for the passive and active suspensions. This is shown in the following figure. As can be seen, all three controllers reduce suspension deflection and body acceleration below the rattle-space frequency (23 rad/s). Moreover, both algorithms show very similar performance (which is to be expected).

Frequency responses of the closed-loop system for the controllers obtained by hinfsyn (left) and fHinfysyn (right).

To further evaluate the three designs, let us also perform time-domain simulations using a road disturbance signal r(t) representing a road bump of height 5 cm. The simulation results are shown in the following two figures for the controllers obtained by both algorithms. Here we observe that the body acceleration is the smallest for the controller emphasizing passenger comfort and largest for the controller emphasizing suspension deflection. The “balanced” design achieves a good compromise between body acceleration and suspension deflection. As also can be seen, the results between the different design algorithms are again very similar.

Time-domain response for a road disturbance (hinfsyn)
Time-domain response for a road disturbance (fHinfsyn)

So far, we have designed H_\infty-controllers that meet the performance objectives for the nominal actuator model. As discussed earlier, this model is only an approximation of the true actuator and it is important to make sure that the controller performance is maintained in the face of model errors and uncertainty.

In fact, by performing a \mu-analysis one can show that the nominal controllers are not capable of guaranteeing robust performance for all possible actuator models in the uncertainty set.

To overcome this problem, we can exploit the \mu-synthesis tools to design a controller that achieves robust performance for the entire family of actuator models. Such robust controller can be synthesized by means of the function musyn using the uncertain model.

On the other hand, it is also possible to exploit fHinfsyn to include the uncertainty channel and impose a weighted small-gain argument to robustify the design.

Both options are presented next for the “balanced” scenario (\beta=0.5). The following two figures show the time responses for the robust controllers obtained by hinfsyn and fHinfsyn respectively. As can be seen, both controllers show good time responses for all samples taken from the uncertainty model of the actuator. In fact, the time responses for the controller obtained by fHinfsyn seem to be a little better.

Computing the worst-case gains for both closed-loop system, using wcgain we obtain 0.87 and 1.23 for musyn and fHinfsyn respectively, which implies that the \mu-synthesis algorithm yields slightly better robustness margins.

Time-domain responses of the closed-loop system and a batch of uncertain actuator models (musyn)
Time-domain responses of the closed-loop system and a batch of uncertain actuator models (fHinfsyn)

Previous page