Idle speed control using Model-based sliding mode control

May 12th, 2017 by Mahdieh Mehrabi

Written by: Xiaoran Han – Project Engineer

  1. Introduction

Following the previous blog post, ‘Linearised Mean Value Engine Model-based Idle Control‘, which describes how to derive a linearised model of crank angle resolved engine for idle speed controller design, this article describes the design of a sliding mode controller based on the linearised model and shows simulation results. Sliding mode controllers are known to have the following features:

  1. Controller design with sliding mode and integral action

Consider a tracking control law for a linear delay system

F1

where F2 is states, F3 is inputs, F4 are the outputs and F5 is the delay.

We introduce additional states

F6

where signal F7 satisfies

F8

and F9 is idle speed request. Augment the states with the error states

F10

and partition the augmented states as

F11

Then the nominal system can be written in the form:

F12

The proposed controller seeks to induce a sliding motion on the surface:

F13

where F14 are design parameters. Partition the matrix S into:

F15

If a controller exists which induces an ideal sliding motion on the sliding surface then the ideal sliding motion is given by

F16     (2)

where F17.

Design objective 1: Choose sliding surface matrix elements F18 and F19 such that (2) is stable and the influence of reference signal F20 on F21 is minimised. A change of coordinates exists such that

F22

Then the nominal system can be written as:

F23

Where F24, F25, F26, F27F28F28.2 F28.2, F29, F30, F31 F31.1 and F31 is a nonsingular diagonal design matrix.

Design objective 2: Design controller F33 such that sliding surface F34 is achieved

F35

where F36 is a stable design matrix. The overall control law is then given by

F37

The nonlinear switching control is:

F38

where F39 is a small positive number for smoothing the control signal, and F40 is a symmetric positive definite matrix satisfying F41. In terms of the original coordinates the linear control is

F42

With control gains

F43

where F44, and F45F46 is manifold pressure, F47 is engine speed. Signals required for feedback control can be listed as follows:

T1

  1. Simulation results

The simulation results in Figure 1 show the idle control performance using the sliding mode controller for the crank angle resolved engine model:

Figure 1

Figure 1: Manifold pressure and engine speed for the crank angle resolved Dymola engine model, state space nonlinear mode and linearised model, where engine speed is controlled to 710 rpm

Figure 2

Figure 2: Sliding surface and control signal for throttle opening angle

Analysing Figure 1 and 2, the manifold pressure of the nonlinear model is in a similar range to that of the crank angle resolved engine model. The pressure of the linearised model remains at zero at the equilibrium operating point. The engine speed of all the three models is controlled at 74.5 rad/s (710 rpm). The sliding surface variable is not zero because switching control is smoothed by a filter as strictly switching at infinite or very fast frequencies is not practical even though smaller simulation time steps would allow the controller to respond with less oscillations in the outputs.. The throttle opening angle oscillates heavily in the first 10 seconds because a controller with higher control gains is used. A controller with delayed state takes over after 10 seconds and the controller oscillation is reduced significantly. Engine speed remains very close to nominal operating point without significant oscillations.

Figure 3

Figur3: Manifold pressure and engine speed for Dymola crank angle resolved engine model, state space nonlinear model and linearised model, where engine speed is 620 rpm

Figure 4

Figure 4: Sliding surface and control signal for throttle opening angle

Figure 3 and 4 show manifold pressure and throttle control input for engine speed controlled at 620 rpm. Figure 5 shows control performance for engine speed at 800 rpm, 900 rpm and 1000 rpm. Figure 6 shows control performance under load disturbance for engine speed at 710 rpm. It is shown that engine speed is maintained at the desired speed despite the presence of a load torque disturbance. Note that same controller has been applied for all the simulation results.

Figure 5

Figure 5: Manifold pressure and engine speed for Dymola crank angle resolved engine model, state space nonlinear mode and linearised model, where engine speed is 800 rpm, 900 rpm and 1000 rpm.

Figure 6

Figure 6: Control performance under load disturbance for engine speed at 710 rpm.

  1. Conclusion

A sliding mode controller has been designed for tracking control of engine idle speed. The control takes delayed information between induction and power output at shaft. Robust control performance is shown under a load torque disturbance. Engine speed is controlled between 600 rpm to 1000 rpm. The control requires fast switching which is difficult to implement in the simulation platform considered. A smoothed nonlinear control has been used to counteract this problem.

Reference

C. Edwards and S. Spurgeon, “Sliding mode control: theory and applications”, 1998.

 

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