Invariant Sets in Saturated and Robust Vehicle Suspension Control

Abstract

This manuscript introduces a new robust control design of car active suspension systems using ellipsoidal techniques. The impact of road irregularities is regarded as an external disturbance. The suggested controller meets the following objectives: optimal passenger comfort, actuator control force limit satisfaction, effective disturbance rejection, and robustness against changes in passengers’ load. The passenger load variations cause system uncertainty that is modeled as norm-bounded. A new sufficient condition is established based on the invariant ellipsoid method and the linear matrix inequalities optimization to guarantee robust stability and performance for the system. Time-domain model of road roughness based on trigonometric functions is studied. The performance of the proposed controller is tested using a quarter-car model with an active suspension system. Comparative simulation with other techniques, e.g., H_∞ and regional pole placement, is given. Actuator dynamics and system nonlinearities are modeled and included in the design at the end.

Appendix

Proof of Theorem 2

The original minimization problem in terms of the ellipsoid ${\mathcal{E}}_{z} = z^{\prime } P^{ - 1} z, z = \left( {C + DK} \right)x$ can be written as

$$\hbox{min } tr\left[ {\left( {C + DK} \right)P\left( {C + DK} \right)} \right]$$

(A1)

Subject to (9).

The matrix Eq. (9) is nonlinear in the variables K, and P. To linearize it, the following new matrix is introduced

$$Y = KP$$

(A2)

Hence, (9) reduces to

$$\left( {PA + BY + *} \right) + \alpha P + \frac{1}{\alpha }EE' < 0$$

(A3)

The objective function (A1) to be minimized is rewritten as

$$\hbox{min } J\left( {P,Y} \right) = tr\left[ {CPC^{\prime } + \left( {CPK^{\prime } D^{\prime } + *} \right) + DYP^{ - 1} Y^{\prime } D^{\prime } } \right]$$

(A4)

Equation (A4) is nonlinear in Y. To reduce it to minimizing a linear function, another matrix is introduced as follows.

$$Z \ge YP^{ - 1} Y^{\prime } \Leftrightarrow \left[ {\begin{array}{*{20}c} Z & * \\ {Y^{\prime } } & P \\ \end{array} } \right] \ge 0$$

(A5)

Minimizing $J \left( {P, Y} \right)$ is equivalent to minimizing (13) subject to (15).

To include system uncertainty, the matrices $A, B$ are replaced by $A + \Delta A$, $B + \Delta B$, respectively, in (A3) to get (12). Applying the Schur complement $\varPsi \Delta \left( t \right)\varPhi + *\left\langle {\varepsilon \varPsi \varPsi^{\prime } + \varepsilon^{ - 1} \varPhi^{\prime } \varPhi ,\varepsilon } \right\rangle 0$ [21] to eliminate the uncertainty matrices Δ(t), Theorem 2 is obtained.