Adaptive Observer for PTWV
8.1 Complete PTWV Model
8.1.1 State Space Representation
In most situation, the motorcycle develop a combined scenarios and applied simultaneously longitudinal forces (braking or acceleration) and lateral forces (cornering). Since this dependency in motorcycle motions, obtaining a “good” estimation require that the coupled behavior must be taken into account. Thereafter, this section is devoted to the modeling of the state space sub-models of the longitudinal and lateral dynamics.
In this work, the first sub-system of the rectilinear motion under the influence of lateral dynamics is con-sidered, this analytic model is derived from the single-corner model (defined in section 2.9.4) to describe the rotations of the tires with respect to the front and rear braking systems. Thereby, the equations of the longitudinal motion will be modeled by the following quasi LPV sub-state space model:
8.1. Complete PTWV Model 141
ζ˙1(t) =A¯(ζ1)ζ1(t) +Bu¯ B(t) +D¯(ζ2)ζ2(t) (8.1) Whereas the state vectorζ1(t) = [vx,ωf,ωr,Fxf,Fxr]T and the input vector is uB = [Bf, Br+T]T. The estimation of this model required to know the longitudinal acceleration and the rotational speed of wheels, and to suppose that during the acceleration phase, the engine torque is applied only to rear wheel. We consider this sub-model to estimate the longitudinal speed and forces based on some of the lateral estimates under an acceptable convergence time. This quasi LPV sub-model (8.1) depends on longitudinal velocity vx, the longitudinal front and rear stiffness̺f,r, which are considered as external varying parameters, with:
̺i= max(R1
iωi,vx), i=f,r.
The lateral dynamics study quantifies the vehicle’s ability to support lateral accelerations and to develop lateral forces to follow a steering rider input. The good compromise between simplicity and accuracy for the modeling of lateral dynamics is the Sharp two-body model, defined in section2.9.1. Thereby, the quasi LPV sub-model of the lateral motion is described by:
ζ˙2(t) =A˘(ζ1)ζ2(t) +Bu˘ τ(t) +D˘(ζ2)ζ1(t) (8.2) Withζ2(t) = [φ,δ,vy,ψ,˙ φ,˙ δ,˙ Fyf,Fyr]T anduτ =τ(t).
The motorcycle interconnected dynamics is given by:
The design of the observer can be handled in the domain of polytopic models.
1. The longitudinal model has 8 sub-models comes from the fact that there is 3 nonlinearities on the model:
z1=vx, z2=̺f, z3=̺r (8.4)
z1min≤z1≤z1max z2min≤z2≤z2max z3min≤z3≤z3max The membership functions of the fuzzy sets are defined as:
X
where the variablesηi(ρ)are computed as follows:
2. The lateral model has 2 sub-models comes from the froward speed coupling. With regard to the motorcycle stability, this forward speed is considered bounded in the interval where the motorcycle is
142 Chapter 8. Interconnected Observers for PTWV stable. Consequently, the membership functions are given by:
X
The variables ηi and ϑj are called the weighing functions and they must satisfy the following convex sum
property:
For the quasi LPV interconnected model (8.1), applying TS representation would lead to an exact form well-suited to design the appropriate observer.
With this model, we propose a Interconnected Fuzzy Observer (IFO) design for nonlinear systems whose TS form has unmeasured premise variables. In the next section, we derive the synthesis steps of the observer for joint states and time-varying parameter estimation. The observer for this type of system should take into account the fact that the weighting functions would be depending on estimated premise variables, rather than exact ones.
8.2 Observer Design
Motivated by the need of observers to acquire certain states used in safety and control systems to prevent possible dangerous situation, this section investigates the design of an interconnected observers. The design is done in two stages, first an observer is associated with the longitudinal subsystem, and then a second observer based on the results of the first observer is proposed for the estimation of lateral dynamics. Thereby, the sub-observer of the lateral dynamics gives the unmeasured variable, then, the lateral velocity is connected to the longitudinal sub-observer to estimate the forward speed and the longitudinal forces. An overall scheme of the system/observer structure is given in figure8.1.
PTWV IFO
Figure 8.1: General diagram of the interconnected estimation of longitudinal and lateral dynamics
8.2. Observer Design 143
8.2.1 Preliminaries
The following nonrestrictive assumptions are considered:
Assumption 8.1. Assume that, for the design of each observer, the states of other subsystems are available.
Assumption 8.2. Suppose that the signalsuB anduτ are known, bounded and sufficiently persistent inputs for each observer respectively.
Assumption 8.3. The state vectorζ1 andζ2 of the two models are considered bounded.
Assumption 8.4. The pair(A˘(ζ1),C˘) and(A¯(ζ2),C¯) are observable or detectable.
The following lemmas are used in the proof of the observer convergence study.
Lemma 8.1. Consider Υ and Ξ matrices with appropriate dimensions. For every positive definite matrix Λ>0. the following property holds (Boyd et al.,1994).
ΥTΞ+ΞTΥ≤ΥTΛΥ+ΞTΛ−1Ξ (8.10)
Lemma 8.2. Given the following matrices Υ, Ξ andℵ, with appropriate dimensions, where Υ = ΥT and ℵ=ℵT (Boyd et al., 1994), the Schur’s lemma apply:
Definition 8.1. The state estimation error verifies the Input To State Practical Stability (ISpS) if there exists a KL function β : Rn×R −→ R,and a K function α : R −→ R such that for each input ∆(t)
Based on the connection between the two lateral and longitudinal subsystems in equation (8.9), the following observer is proposed:
Using equations (8.9) and (8.13), the state estimation error obeys the following differential equations:
( e˙ζ1 = Ppi=11 ηi(ζ1(t))(Φ¯ieζ1+D¯ieζ2) +∆ζ1(t)
Notice that if the state estimation errors converge to zero, the terms∆ζ
1(t)and∆ζ
2(t)converge also towards zero. In addition, since the weighting functions are bounded and the state vector ζ1(t) and ζ2(t) are also bounded (see assumption8.1), the term ∆ζ
i(t)are thus bounded.