(64)
according to Definition 3.1, where m 1 and m 2 are positive constants, and lim k→∞ α( k) = 0.
Since lim k→∞ α( k) = 0, there exists a constant k 2 such that α( k) ≤ 1/ m 1, ∀ k > k 2. Therefore, it can be seen from (64) that
| e( k)| − | β( k − 1)| + λcβΔ s( k − n, n − 1) + α( k) m 2 ≥ (1 − α( k) m 1)| e( k)| ≥ 0, ∀ k > k 2 (65) From (50), it is clear that
| e( k)| − | β( k − 1)| + λcβΔ s( k − n, n − 1) + α( k) m 2
≤ ac( k)| e( k)| + λ Δ Z( k − n) + λcβΔ s( k − n, n − 1) + α( k) m 2
(66)
which implies that lim k→∞ e( k) = 0 according to (60)-(62), and (65), which further yields
lim k→∞ e( k) = 0 because of e( k) ∼ e( k). This completes the proof.
Remark 6.1. The underlying reason that the asymptotic tracking performance is achieved lies in that
the uncertain nonlinear term ν( k − n − τ) in the closed-loop tracking error dynamics (44) will converge
to zero because lim k→∞ Δ Z( k) = 0 as shown in (61).
7. Further discussion on output-feedback systems
In this section, we will make some discussions on the application of control design technique
developed before to nonlinear system in lower triangular form. The research interest of
lower triangular form systems lies in the fact that a large class of nonlinear systems can
be transformed into strict-feedback form or output-feedback form, where the unknown
parameters appear linearly in the system equations, via a global parameter-independent
diffeomorphism.
In a seminal work Kanellakopoulos et al. (1991), it is proved
that a class of continuous nonlinear systems can be transformed to lower triangular
parameter-strict-feedback form via parameter-independent diffeomorphisms. A similar result
is obtained for a class of discrete-time systems Yeh & Kokotovic (1995), in which the geometric
conditions for the systems transformable to the form are given and then the discrete-time
backstepping design is proposed. More general strict-feedback system with unknown control
gains was first studied for continuous-time systems Ye & Jiang (1998), in which it is indicated
that a class of nonlinear triangular systems T 1 S proposed in Seto et al. (1994) is transformable
to this form. The discrete-time counterpart system was then studied in Ge et al. (2008), in
which discrete Nussbaum gain was exploited to solve the unknown control direction problem.
In addition to strict-feedback form systems, output-feedback systems as another kind of
lower-triangular form systems have also received much research attention. The discrete-time
output-feedback form systems have been studied in Zhao & Kanellakopoulos (2002), in
which a set of parameter estimation algorithm using orthogonal projection is proposed and
it guarantees the convergence of estimated parameters to their true values in finite steps. In
Yang et al. (2009), adaptive control solving the unknown control direction problem has been
developed for the discrete-time output-feedback form systems.
As mentioned in Section 1, NARMA model is one of the most popular representations of
nonlinear discrete-time systemsLeontaritis & Billings (1985). In the following, we are going to
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Discrete Time Systems
show that the discrete-time output-feedback forms systems are transformable to the NARMA
systems in the form of (1) so that the control design in this chapter is also applicable to the
systems in the output-feedback form as below:
⎧
⎨ xi( k + 1) = θTφ
i
i( x 1 ( k)) + gixi+1( k) + υi( x 1( k)), i = 1, 2, . . . , n − 1
⎩ xn( k + 1) = θTnφn( x 1( k)) + gnu( k) + υn( x 1( k))
(67)
y( k) = x 1( k)
where xi( k) ∈ R, i = 1, 2, . . . , n are the system states, n ≥ 1 is system order; u( k) ∈ R, y( k) ∈ R is the system input and output, respectively; θi are the vectors of unknown constant
parameters; gi ∈ R are unknown control gains and gi = 0; φi(·), are known nonlinear vector
functions; and υi(·) are nonlinear uncertainties.
It is noted that the nonlinearities φi(·) as well as υi(·) depend only on the output y( k) = x 1( k), which is the only measured state. This justifies the name of “output-feedback” form.
According to Ge et al. (2009), for system (67) there exist prediction functions Fn− i(·) such that
y( k + n − i) = Fn− i( y( k), u( k − i)), i = 1, 2, . . . , n − 1, where y( k)=[ y( k), y( k − 1), . . . , y( k − n + 1)] T
(68)
u( k − i)=[ u( k − i), u( k − i − 1), . . . , u( k − n + 1)] T
(69)
By moving the i th equation ( n − i) step ahead, we can rewrite system (67) as follows
⎧
⎪
⎪ x
φ
⎪ 1( k + n) = θT 1 1( y( k + n − 1)) + g 1 x 2( k + n − 1) + υ 1( y( k + n − 1))
⎨ x 2( k + n − 1) = θTφ
2
2( y( k + n − 2)) + g 2 x 3 ( k + n − 2) + υ 2( y( k + n − 2))
⎪
.
(70)
⎪
⎪
.
⎩
.
xn( k + 1) = θTnφn( y( k)) + gnu( k) + υn( y( k))
Then, we submit the second equation to the first and obtain
x 1( k + n) = θTφ
φ
1
1 ( y( k + n − 1)) + g 1 θT
2
2 ( y( k + n − 2))
+ g 1 g 2 x 3( k + n − 2) + υ 1( y( k + n − 1)) + g 1 υ 2( Fn−2( y( k), u( k − 2)) (71) Continuing the iterative substitution, we could finally obtain
n
y( k + n) = ∑ θT φ
f i i( y( k + n − i)) + gu( k) + ν( z( k))
(72)
i=1
where
i−1
θ f = θ
= θ ∏ g
1
1, θ fi
i
j, i = 2, 3, . . . , n
j=1
i−1
n
g f = 1, g = ∏ g
∏ g
1
fi
j, i = 2, 3, . . . , n, g =
j
(73)
j=1
j=1
and
n
ν( z( k)) = ∑ gf νii( z( k)), z( k) = [ yT( k), uT( k − 1)] T
(74)
i=1
Discrete-Time Adaptive Predictive Control with Asymptotic Output Tracking
221
with
νi( z( k)) = υi( y( k + n − i)) = υi( Fn− i( y( k), u( k − i))), i = 1, 2, . . . , n − 1, νn( z( k)) = υn( y( k))
(75)
with z( k) defined in the same manner as in (1). Now, it is obvious that the transformed
output-feedback form system (72) is a special case of the general NARMA model (1).
8. Study on periodic varying parameters
In this section we shall study the case where the parameters θi and gj, i = 1, 2, . . . , n, j =
1, 2, . . . , m in (1) are periodically time-varying. The l th element of θi( k) is periodic with known
period Ni, l and the period of gi( k) is Ngi, i.e. θi, l( k) = θi, l( k − Ni, l) and gj( k) = gj( k − Ngj) for known positive constants Ni, l and Ngj, l = 1, 2, . . . , pi.
To deal with periodic varying parameters, periodic adaptive control (PAC) has been developed
in literature, which updates parameters every N steps, where N is a common period such
that every period Ni, l and Ngj can divide N with an integer quotient, respectively. However,
the use of the common period will make the periodic adaptation inefficient. If possible, the
periodic adaptation should be conducted according to individual periods. Therefore, we will
employ the lifting approach proposed in Xu & Huang (2009).
Firstly, we define the augmented parametric vector and corresponding vector-valued
nonlinearity function. As there are Ni, j different values of the j th element of θi at different
steps, denote an augmented vector combining them together by
¯ θi, l = [ θi, j,1, θi, j,2, . . . , θi, j, N ] T
(76)
i, l
with constant elements. We can construct an augmented vector including all pi periodic
parameters
Θ i = [ ¯ θT
] T = [ θ
] T
i,1, ¯
θTi,2, . . . , ¯ θTi, p
, . . . , θ
(77)
i
i,1,1, . . . , θi,1, Ni,1
i, pi,1, . . . , θi, pi, Ni, pi
with all elements being constant. Accordingly, we can define an augmented vector
Φ i( y( k + n − 1)) = [ ¯ φi,1( y( k + n − 1)), . . . , ¯ φi, p ( y( k + n − 1))] T
(78)
i
where ¯
φi, l( y( k + n − 1)) = [0, . . . , 0, φi( y( k + n − i)), 0, . . . , 0] T ∈ RNi, l and the element φi( k) appears in the q th position of ¯
φi, l( y( k + n − 1)) only when k = sNi, l + q, for i = 1, 2, . . . , Ni, l. It can be seen that n functions φi( k), rotate according to their own periodicity, Ni, l, respectively.
As a result, for each time instance k, we have
θT(
Φ
i
k) φi( y( k + n − i)) = Θ Ti i( y( k + n − 1))
(79)
which converts periodic parameters into an augmented time invariant vector.
Analogously, we convert gi( k) into an augmented vector ¯ gi = [ gi,1, gi,2, . . . , gi, N ] and gj
meanwhile define a vector
ϕj( k) = [0, . . . , 0, 1, 0, . . . , 0] T ∈ RNgj
(80)
where the element 1 appears in the q th position of ϕj( k) only when k = sNgj + q. Hence
for each time instance k, we have gj( k) = ¯ gj ϕj( k), i.e., gi( k) is converted into an augmented time-invariant vector.
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Discrete Time Systems
Then, system (1) with periodic time-varying parameters θi( k) and gj( k) can be transformed
into
n
m
y( k + n) = ∑ Θ TΦ
i
i( y( k + n − i)) + ∑ ¯ gj ϕj( k) u( k − m + j) + ν( z( k − τ)) (81)
i=1
j=1
such that the method developed in Sections 4 and 5 is applicable to (81) for control design.
9. Conclusion
In this chapter, we have studied asymptotic tracking adaptive control of a general class of
NARMA systems with both parametric and nonparametric model uncertainties. The effects
of nonlinear nonparametric uncertainty, as well as of the unknown time delay, have been
compensated for by using information of previous inputs and outputs. As the NARMA
model involves future outputs, which bring difficulties into the control design, a future output
prediction method has been proposed in Section 4, which makes sure that the prediction error
grows with smaller order than the outputs.
Combining the uncertainty compensation technique, the prediction method and adaptive
control approach, a predictive adaptive control has been developed in Section 5 which
guarantees stability and leads to asymptotic tracking performance. The techniques developed
in this chapter provide a general control design framework for high order nonlinear
discrete-time systems in NARMA form. In Sections 7 and 8, we have shown that the proposed
control design method is also applicable to output-feedback systems and extendable to
systems with periodic varying parameters.
10. Acknowledgments
This work is partially supported by National Nature Science Foundation (NSFC) under Grants
61004059 and 60904086. We would like to thank Ms. Lihua Rong for her careful proofreading.
11. Appendix A: Proof of Proposition 3.1
Only proofs of properties (ii) and (viii) are given below. Proofs of other properties are easy
and are thus omitted here.
(ii) From Definition 3.1, we can see that o[ x( k)] ≤ α( k) max k ≤ k+ τ x( k ) , ∀ k > k 0, τ ≥ 0, where lim k→∞ α( k) = 0. It implies that there exist constants k 1 and ¯ α 1 such that α( k) ≤ ¯ α 1 < 1,
∀ k > k 1. Then, we have
x( k + τ) + o[ x( k)] ≤ x( k + τ) + o[ x( k)] ≤ (1 + ¯ α 1) max x( k ) , ∀ k > k 1
k ≤ k+ τ
which leads to x( k + τ) + o[ x( k)] = O[ x( k + τ)]. On the other hand, we have max
x( k ) ≤
max
x( k ) + o[ x( k)] + o[ x( k)]
k 1<