Concepts of LPV Control with Applications in Automotive Engine Control

Abstract

In this paper, a linear parameter-varying sys- tem (A(θ) ,B (θ) ,C (θ)) with the parameter vector θ is considered, which is obtained by linearizing a nonlinear system around its instantaneous state and input vectors. The controller is designed with the H∞ method us- ing parameter-varying shaping weights W•(θ, s). Thus, a parameter-varying bandwidth ωc(θ) is obtained. Two applications in the area of automotive engine control are discussed. In one of them, a parameter-varying time delay T (θ) is involved. I. INTRODUCTION In this paper, a linear parameter-varying plant (A(θ) ,B (θ) ,C (θ)) with the parameter vector θ is con- sidered with continuously differentiable system matrices A(θ), B(θ) ,a ndC(θ). As described in Section II, such an LPV plant description is typically obtained by linearizing the model of a nonlinear plant about a nominal trajectory. The control problem which is considered in this paper is finding a linear parameter-varying controller with the system matrices F (θ), G(θ) ,a ndH(θ) of its state space model. In Section III, the control problem is formulated as an H∞ problem using the mixed sensitivity approach. The shaping weights We(θ, s), Wu(θ, s) ,a ndWy(θ, s) are allowed to be parameter-varying. The most appealing fea- ture of this approach is that it yields a parameter-varying bandwidth ωc(θ) of the robust control system. Choosing appropriate shaping weights is described in Section IV. For more details about the design methodology, the reader is referred to (1)-(6). In Section V, it is shown, how parameter-varying time- delays in the plant dynamics can be handled in the framework proposed in Sections III and IV. For more details, consult (5) and (7). In Section VI, two applications in the area of automo- tive engine control are discussed. In the first application (15), (5), (4), the design of an LPV feedback controller for the fuel injection is shown which is suitable over the whole operating envelope of the engine. In the second application (16), (23), the philosophy of designing an LPV feedback controller is carried over to the problem of designing an additional LPV feedfor- ward controller compensating the parameter-varying wall- wetting dynamics in the intake manifold of the port- injected gasoline the engine. II. STATEMENT OF THE CONTROL PROBLEM We consider the following nonlinear time-invariant dynamic system (plant) with the unconstrained input vector U (t) ∈ R m , the state vector X(t) ∈ R n , and the output vector Y (t) ∈ R p : ˙ X(t )= f (X(t) ,U (t)) Y (t )= g(X(t)) , where f and g are fairly smooth continuously differen- tiable functions. Let us assume that we have found a reasonable or even optimal open-loop control strategy Unom(t) for a rather large time interval t ∈ (0 ,T ) (perhaps T = ∞) which theoretically generates the nominal state and output trajectories Xnom(t) and Ynom(t), respectively. In order to ensure that the actual state and output trajectories X(t) and Y (t) stay close to the nominal ones at all times, we augment the open-loop control Unom(t) with a (correcting) feedback part u(t). Thus, the combined open-loop/closed-loop input vector becomes U (t )= Unom(t )+ u(t) . Assuming that the errors