Feedback Linearization of a Multi-Input SI-Engine System for Idle Speed Control

The idle speed problem is a classical example of an automotive control application. The set-up corresponds to a disturbance rejection problem where the main plant output (engine speed) has to be maintained at a (low) constant value despite the torque disturbances acting on the engine crank-shaft (servosteering pump, air-conditioning, etc.). The relevance (comfort, fuel consumption, etc.) and the technical challenges (nonlinear plant with large time delays) of this control problem have led to many different control strategies. PID [7], LQ [6], ‘H, [3, 111, Ci [2], fuzzy control [l], adaptive control [8], sliding mode [5] and neural networks [lo] are some of the frameworks used to tackle this problem. Feedback linearization was also investigated in some papers [5], but the engine’s induction to power stroke delay was neglected. Unfortunately, this effect, that depends also on the engine speed, is often the limiting factor for the controller design. For this reason, in the here presented work, this delay is approximated with first order low pass elements, which have an enginespeed dependent time constant. The resulting nonlinear plant with two inputs (airbypass valve and spark-advance) is not affine in the inputs. Nevertheless, by introducing additional static compensations the plant is shown to be exactly feedback linearizable. The linearized plant permits the application of well-known linear control design methods. In this paper the different bandwidths of the two input-channels are used in a setting similar to the one presented by [3, 111 to guarantee an optimal engine operation, both under transient and steady-state conditions. 1. Notation The following notation is used in this paper e : riz : P : Pa : T : R : v, : IL.2 : N : T, : Tl : K, : ; I Ki : J, : first input, air-bypass valve air-bypass valve mass flow rate intake manifold pressure atmospheric pressure intake manifold air temperature air gas constant intake manifold volume cylinder air mass flow rate engine speed net engine torque engine load torque delay parameter induction to power stroke delay second input, spark-advance regular load torque parameter effective engine rotational inertia. 2. Nonlinear SI-Engine Model The nonlinear engine model is based on a work of Powell and Cook [9]. A block diagram of it is presented in Fig. 1. Figure 1: Nonlinear engine model Assumptions: A constant intake manifold temperature, no intake manifold leaks and constant stoichiometric air/fuel-ratio is assumed in this work. Under these assumptions the model of the analyzed plant is described by the following set of equations. For the throttle-plate behavior ,riJ = fl(e,P> = J%(WP(P)