Page 303 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics
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306 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics
19.3.3 Suspension Performance Analysis
The convergence of x 1 has been guaranteed based on Theorem 19.2.In
the following, we will address the other two suspension performance re-
quirements (19.20)and (19.21).
First, the boundedness of the state variables x 3 ,x 4 of system (19.19)
is studied. Substituting (19.26)into(19.19), one can obtain the following
dynamics
˙ x = Ax + ω (19.35)
where
0 1
x 3
x = ,A = (19.36)
k t b f
x 4 − −
m us m us
0
ω = (19.37)
k t b f m s
z r + ˙ z r + ω 1
m us m us m us
˜ T
where ω 1 = k ss 1 + x 2 −
denotes the effect of the residual error, which
is bounded because s 1 ,x 2 and
are all bounded. Therefore, ω is bounded,
˜
i.e., ω ≤ holds for a positive constant > 0.
Since the matrix A defined in (19.36) is stable, there exist positive ma-
T
trices P,Q so that the Lyapunov equation A P + AP =−Q holds. We
T
select a Lyapunov function as V = x Px,then
1
2
˙ V ≤−[λ min (Q) − λ max (P)] x + ηλ max (P) 2 (19.38)
η
Then for appropriately designed parameters fulfilling η> λ max (P)/
λ min (Q), it follows from (19.38)that
˙ V ≤−αV + β (19.39)
2
where α =[λ min (Q) − λ max (P)/η]/λ min (P) and β = ηλ max (P) are all pos-
itive constants. This implies that the state variables x 3 ,x 4 are all bounded
by
|x i | ≤ (V(0) + β α) λ min (P),i = 3,4 (19.40)
So that the upper bound of the tire load can be calculated as
|F t + F b | ≤ k t (V(0) + β α) λ min (P) + k t |z r | + b f |˙ z r | (19.41)