128
International of Control, Automation, vol. 5, no. 2, pp. 128-137, April 2007 JinJournal Zhu, Hong-sheng Xi, Hai-bo Ji,and andSystems, Bing Wang
A Robust Adaptive Controller for Markovian Jump Uncertain Nonlinear Systems with Wiener Noises of Unknown Covariance Jin Zhu, Hong-sheng Xi*, Hai-bo Ji, and Bing Wang Abstract: A robust adaptive controller design for a class of Markovian jump parametric -strictfeedback systems is given. The disturbances considered herein include both uncertain nonlinearities and Wiener noises of unknown covariance. And they satisfy some boundconditions. By using stochastic Lyapunov method in Markovian jump systems, a switching robust adaptive controller was obtained that guarantees global uniform ultimate boundedness of the closed-loop jump system. Keywords: Markovian jump nonlinear systems, robust adaptive control, Wiener noise, uncertain nonlinearity.
1. INTRODUCTION The passed decades have witnessed substantial research activities in the development of Markovian jump systems, and much effort is directed towards jump linear systems [1]. With many linear problems (Kalman filtering [2,3], LQG [4,5], Slide-mode control [6] etc.) solved, more attention is focused on the study of Markovian jump non-linear systems. Some results can be found in the works of Aliyu [7] and Zhu [8]. However, noise and disturbance are not considered in these works. On the other hand, many physical systems do be disturbed by noises. Thus Markovian jump nonlinear systems disturbed by Wiener noise (or Brown motion) have been the subject of numerous studies in recent years. For this class of jump systems, Mao [9] gives the sufficient condition to ensure existence and uniqueness of the solution; Yuan [10,11] introduce the notions of asymptotic stability and robust stability; Boukas [12] presents the notions of mean square stability. However, their work mainly focus on the notions and definitions of stochastic stability, not the practical controller design in jump systems. At the knowledge of the authors, the practical control design for Markovian jump nonlinear systems with stochastic __________ Manuscript received November 11, 2005; revised September 27, 2006; accepted January 31, 2007. Recommended by Editor Tae-Woong Yoon. This work was supported by the National Science Foundation of China under Grant 60674029 and the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant 20050358044. Jin Zhu, Hong-sheng Xi, Hai-bo Ji, and Bing Wang are with the Department of Automation, University of Science and Technology of China, Hefei, Anhui, 230027, China (e-mails:
[email protected], {xihs,Jihb}@ustc.edu.cn,
[email protected]). * Corresponding author.
noises has received little attention in literature. In this paper, we are interested in the practical switching robust adaptive controller design for Markovian jump nonlinear systems disturbed by Wiener noise. As parametric-strict-feedback systems could rep-resent many practical systems in real world, and many other nonlinear systems could be converted to this form via mathematical transformation [13]. We choose the Markovian jump parametric-strictfeedback systems as the research model. Considering that modeling errors, parametric uncertainty and time variations may exist, the systems structure uncertainty is thus taken into account. Here the covariance of Wiener noise is assumed to be unknown but bounded, and the structure uncertainty is assumed to satisfy some growth conditions [14]. With the control law and the parameter adaptive law designed, all signals of the closed-loop system are globally uniformly ultimately bounded. The rest of this paper is organized as follows: Section 2 briefly introduces some mathematical notions and the Markovian jump nonlinear system model. The robust adaptive controller for the system is then proposed in Section 3. In Section 4, an example is shown to illustrate the validity of the controller design. Finally, conclusions are drawn in Section 5.
2. PROBLEM AND PRELIMINARIES 2.1. Notation Throughout the paper, unless otherwise specified, we denote by (Ω, F,{Ft }t ≥0 , P), a complete probability
space with a filtration {Ft }t≥0 satisfying the usual
conditions (i.e., it is right continuous and F0 contains all p-null sets). Let | ⋅ |P stand for the p-th Euclidean
A Robust Adaptive Controller for Markovian Jump Uncertain Nonlinear Systems with Wiener Noises of…
norm for vectors. The superscript T will denote transpose and we refer to Tr (⋅) as the trace for matrix. In addition we use L2 ( P) to denote the space of Lebesgue square integrable vector. Let r (t ), t ≥ 0, be a right continuous Markov chain on the probability space taking values in finite state space S = {1,2,", N}, and we introduce Φ(t ) = [Φ1 (t ), Φ 2 (t )" , Φ N (t )]T regime r (t ), as: Φ j (t ) =
the indicator process for the
r (t ) = j r (t ) ≠ j, j ∈ S
1 0
t
Φ (t ) = Φ (t0 ) + Π ∫ Φ ( s )ds + M (t ) t0
impose a hypothesis [9]: (H) Both f and g satisfy the local Lipschitz condition and the linear growth condition. That is for each h = 1, 2," , there is an Lh > 0 such that
| f ( x, t, k ) − f ( y, t, k ) | ∨ | g ( x, t, k ) − g ( y, t, k ) | ≤ Lh | x − y | for all (t , k ) ∈ R+ × S and those x, y ∈ R n with | x | ∨ | y |≤ h. Moreover there is an ν > 0 such that: | f ( x, t , k ) | ∨ | g ( x, t, k ) |≤ ν (1+ | x |)
(1)
and Φ (t ) satisfies the following equation: (2)
with M (t ) = [ M1 (t ), M 2 (t )," , M N (t )]T , an N-dimensional Ft -martingale satisfying M (t ) ∈ L2 ( P) and Π = [π kj ] the chain generator, an N × N matrix. The entries π kj , k , j = 1, 2," , N are interpreted as transition rates such that
for all ( x, t , k ) ∈ R n × R+ × S . In general, the hypothesis (H) will guarantee a unique local solution to (4). Let C 2,1 ( R n × R+ × S ) denote the family of all functions F ( x, t , k ) on R n × R+ × S which are continuously twice differentiable in x and once in t. Furthermore, we will given the stochastic differrentiable equation of F ( x, t , k ) : Fix any ( x0 , t0 , k ) ∈ R n × R+ × S and suppose x(t) is the unique solution to (4). By the generalized Ito formula, we have
P(r (t + dt ) = j | r (t ) = k ) π kj dt + o(dt ) = 1 + π kj dt + o(dt )
if
k≠ j
if
k = j,
(3)
where dt>0. Here π kj > 0(k ≠ j ) is the transition rate from k to j. Notice that the total probability axiom imposes π kk negative and N
Consider a stochastic differential equation with Markovian switching of the form dx = f ( x, t , r (t ))dt + g ( x, t , r (t ))dω t≥0
with initial data
x(0) = x0 ∈ R n
(4) and
r (0) = k0 ∈ S , where f : R n × R+ × S → R n , g : R n ×S → R n×m , x = x(t ) is system state vector, ω (t ) = T
(ωt1 , ωt 2 ," , ωtm ) is an independent m-dimensional Wiener noise defined on the probability space, with covariance E{dω ⋅ dω T } = ϒ(t ) ϒ(t )T dt , where ϒ(t ) is an unknown boundedmatrix-valued function. Furthermore, we assume that the Wiener noise ω (t ) is independent of the Markov chain r(t). For the existence and uniqueness of the solution, we shall
t
t0
+∫
t
+∫
t
+∫
t
t0
t0
+∫
j =1
on
F ( x, t , r (t )) = F ( x0 , t0 , k ) + ∫
t0
∑ π kj = 0, ∀k ∈ S.
129
∂F ( x, s, r ( s )) ds ∂s
∂F ( x, s, r ( s )) f ( x, s, r ( s ))ds ∂x
1 ∂ 2 F ( x, s, r ( s )) Tr[ ϒT g T ( x, s, r ( s )) g ( x, s, r ( s )) ϒ]ds 2 ∂x 2 ∂F ( x, s, r ( s )) g ( x, s, r ( s ))dω ∂x
t N
t0
∑ [ F ( x, s, j ) − F ( x, s, k )]d Φ j (s). j =1
(5) According to (2), the differential equation of the indicator Φ (t ) is as following: d Φ (t ) = ΠΦ (t )dt + dM (t ).
(6)
Submit (6) into (5) and notice that N
∑ π kj F ( x, t , k ) = 0.
∀k ∈ S .
j =1
Therefore, the stochastic differentiable equation of F ( x, t , k ) is given by the following:
dF ( x, t , k ) =
∂F ( x, t , k ) ∂F ( x, t , k ) dt + f ( x, t , k )dt ∂t ∂x
130
Jin Zhu, Hong-sheng Xi, Hai-bo Ji, and Bing Wang
vector of unknown constant parameters; The Markov chain r(t) and m-dimensional Wiener noise ω are as defined in Section 2.1. ϕi ( x i, t , r (t )), gi ( x i, t , r (t ))
1 ∂ 2 F ( x, t , k ) + Tr[ ϒT g T ( x, t , k ) g ( x, t , k ) ϒ]dt 2 ∂x 2 N
+ ∑ π kj F ( x, t , j )dt + j =1
∂F ( x, t , k ) g ( x, t , k ) d ω ∂x
(7)
N
+ ∑ [ F ( x, t , j ) − F ( x, t , k )]dM j (t ). j =1
We take the expectation in (7), so that the infinitesimal generator produces [9] LF ( x, t , k ) =
∂F ( x, t , k ) ∂F ( x, t , k ) + f ( x, t , k ) ∂t ∂x
1 ∂ 2 F ( x, t , k ) + Tr[ ϒT g T ( x, t , k ) g ( x, t , k ) ϒ ] 2 ∂x 2
are vector-valued smooth functions. ∆i ( x i, t , r (t )) is an unknown function which could be due to modeling errors, parametric uncertainty, time variations in the systems, or a combination of these. And it maybe different with each regime r (t ). It is assumed that the control designer has, at least, partial knowledge of bounds for the function uncertainty ∆i ( x i, t , r (t )). In particular, we assume that ∆i ( x i, t , r (t ) = k ) ≤ ψ i∗ p i ( x i, r (t ) = k ),
(8)
∀ x i ∈ Ri , ∀t ∈ R+ , ∀k ∈ S ,
(12)
N
where pi( x i, r (t )) ∈ C1 ( Ri × S , R+ ) is a known smooth
j =1
function and ψ i∗ ≥ 0 is a constant parameter, which
+ ∑ π kj F ( x, t , j ). Lemma 1 (Martingale Representation) [15]: Let B(t) = [B1(t), B2 (t),", BN (t)] be N-dimensional standard
Wiener noise. Suppose M (t ) is an
FtN
-martingale
2
(w.r.t. P) and that M (t ) ∈ L ( P) for all t ≥ 0. Then there exists a stochastic process Ψ ∈ such that dM (t ) = Ψ ⋅ dB(t ).
L2 (Ft , P), (9)
Lemma 2 (Young’s inequality): For any two vectors x, y ∈ R n , the following holds T
x y≤
εp p
p
| x| +
1 qε q
q
| y| ,
(10)
where ε > 0 and the constants p > 1, q > 1 satisfy ( p − 1)(q − 1) = 1. 2.2. Problem description Consider the following Markovian jump uncertain nonlinear systems with Wiener noises:
dxi = xi +1dt + ϕi ( x i, t , r (t ))T θ ∗ dt + ∆i ( x i, t , r (t ))dt + gi ( x i, t , r (t ))T dω T ∗ dxn = u dt + ϕ n ( x, t , r (t )) θ dt + ∆ n ( x, t , r (t ))dt (11) + g n ( x, t , r (t ))T dω y = x , i = 1, 2," , n − 1, 1 where x = ( x1 , x2 ," , xn )T ∈ R n is the state vector, here x i ( x1 , x2 ," , xi )T , u ∈ R is the input, and y ∈ R is the output of the system. θ ∗ ∈ R P is a
is not necessarily known. Note that ψ i∗ is not unique, ∗
since any ψ i > ψ i∗ satisfies in-equality (12). To avoid confusion, we define ψ i∗ the smallest(non-negative) constant such that (12) is satisfied. In this paper, the equilibrium x=0 is assumed a common one for all the regimes, which means ϕi (0, t , k ) = 0, gi (0, t , k ) = 0,
∀k ∈ S . With ϕi ( x i, t , r (t )), gi ( x i, t , r (t )), ∆i ( x i, t , r (t )) satisfying hypothesis (H), Markovian jump system (11) has a unique solution.
3. CONTROL DESIGN Now we begin to design a robust adaptive controller for system (11), where the parameter θ ∗ and ψ i∗ need to be estimated. Denote the estimation
of θ ∗ with θ , and the estimation of ψ i∗ with ψ i . First we employ a coordinate transformation: zi = xi − α i −1 ( x i −1,θ ,ψ i , t , r (t ) = k ),
(13)
where α 0 = 0, ∀k ∈ S , and the new coordinate is Z = ( z1 , z2 ," , zn ). For simplicity, we just denote
α i −1 ( x i −1,θ ,ψ i , t , k ), ϕi ( x i, t , k ), gi ( x i, t , k ), ∆i ( x i, t , k ) by α i −1 (k ), ϕi (k ), gi (k ), and ∆i (k ).
According to (7), the (13) can be written as:
dzi = [ xi +1 + ϕiT (k )θ ∗ + ∆i (k )]dt + giT (k )dω − dα i −1 (k ) = {zi +1 + α i (k ) + ϕiT (k )θ ∗ + ∆i (k )}dt −
i −1 ∂α i −1 (k ) ∂α (k ) ∂α (k ) dt − i −1 θdt − ∑ i −1 ψ j dt ∂t ∂θ j =1 ∂ψ j
A Robust Adaptive Controller for Markovian Jump Uncertain Nonlinear Systems with Wiener Noises of…
∂α i −1 (k ) [ x j +1 + ϕ Tj (k )θ ∗ + ∆ j (k )]dt ∂ x j j =1
where γ > 0, σ i > 0 are constants. θ = θ ∗ − θ and χ i
1 i −1 ∂ 2α i −1 (k ) T g p (k ) ϒϒT g q (k )dt ∑ 2 p,q =1 ∂x p ∂xq
ψ iM max{ψ i∗ ,ψ i0 }, and ψ i0 are given positive
i −1
−∑ −
131
= ψ iM − ψ i are the parameter estimation errors, where
i −1
N
∂α i −1 (k ) T g j (k )]dω ∂x j j =1
− ∑ π kjα i −1 ( j )dt + [ giT (k ) − ∑ j =1
constants. We set out to choose the function α i −1 (k ) and adaptive functions to make LV non-positive. Along the solutions of (15), we have
N
+ ∑ [α i −1 (k ) − α i −1 ( j )]dM j (t ).
(14)
j =1
n
LV = ∑ zi3{zi +1 + α i (k ) + ϕiT (k )θ ∗ − i =1
N ∂α i −1 (k ) i −1 ∂α i −1 (k ) θ −∑ ψ j − ∑ π kjα i −1 ( j ) ∂θ j =1 ∂ψ j j =1
By Lemma 1, there exists a Ψ ∈ L2 (Ft , P) and an Ndimensional standard Wiener noise B(t ), such that
−
dM (t ) = ΨdB(t ), where E{ΨΨT } = φφ T ≤ Q < ∞ and Q is a positive bounded constant. Moreover, we define an N-dimensional vector
−∑
i −1
∂α i −1 (k ) [ x j +1 + ϕ Tj (k )θ ∗ ] + Λi (k ) ∂x j j =1
Γi (k ) [α i −1 (k ) − α i −1 (1),α i −1 (k ) − α i −1 (2), " ,α i −1 (k ) − α i −1 ( N )].
Therefore (14) is as following:
−
1 i −1 ∂ 2α i −1 (k ) T g p (k ) ϒϒT g q (k )} ∑ 2 p,q =1 ∂x p ∂xq
+
i −1 ∂α (k ) T 3 n 2 T zi [ gi (k ) − ∑ i −1 g j (k )]ϒϒT ∑ 2 i =1 ∂ x j j =1 i −1
∂α i −1 (k ) g j (k )] ∂x j j =1
[ gi (k )− ∑
dzi = [ zi +1 + αi (k ) + ϕiT (k )θ ∗ + Λi (k )]dt −
i −1 ∂αi −1 (k ) ∂α (k ) ∂α (k ) dt − i −1 θdt − ∑ i −1 ψ j dt ∂t ∂θ j =1 ∂ψ j
+
i −1
∂αi −1 (k ) [ x j +1 + ϕ Tj (k )θ ∗ ]dt ∂ x j j =1
−∑ −
i −1
γ
∂ αi −1 (k ) T 1 g p (k ) ϒϒT g q (k )dt ∑ 2 p,q =1 ∂x p ∂xq i −1
∂αi −1 (k ) T g j (k )]dω ∂x j j =1
−
+ Γi (k )ΨdB,
i −1
N ∂α i −1 (k ) x j +1 − ∑ π kjα i −1 ( j ) ∂x j j =1 j =1
where Λi (k ) is: i −1
+ λ zi3
∂α (k ) Λi (k ) ∆i (k ) − ∑ i −1 ∆ j (k ) ∂x j j =1
i −1
∑
p, q =1
[
∂ 2α i −1 (k ) 2 T ] g p (k ) g p (k ) ∂x p ∂xq
g qT (k ) g q (k ) + µ1 zi [ ρiT (k ) ρi (k )]2
and according to inequality (12), it is easily seen that there exist a series of continuous function p i ( x i, k ) ∈ C ( Ri × S , R+ ), such that
n
n
9 Q 2 + ι | ϒ |4 16 µ 2 i =1
+ ∑ µ 2 zi [Γi (k )ΓTi (k )]2 } + ∑ i =1
n n 1 1 − θ T [ θ − ∑ zi3τ i (k )] − ∑ [ χ iψ i − zi3 Λi (k )]
| Λi (k ) |≤ ψ i∗ p i ( x i, k ), ∀ x i ∈ Ri , ∀t ∈ R+ , ∀k ∈ S . (16)
γ
i =1
i =1
σi
(18)
Choose a Lyapunov function of the form 1 n 4 1 T n 1 2 ∑ zi + 2γ θ θ + ∑ 2σ χ i , 4 i =1 i i =1
∂α i −1 (k ) i −1 ∂α i −1 (k ) θ −∑ ψ j + τ iT (k )θ ∂θ ∂ ψ j j =1
−∑
(15)
V=
i =1 σ i
N ∂α (k ) 3 4 1 ≤ ∑ zi3{( δ i3 + 4 ) zi + α i (k ) − i −1 4 ∂t 4δ i −1 i =1
− ∑ π kjαi −1 ( j )dt + [ giT (k ) − ∑ j =1
3 N 2 ∑ zi Γi (k )φφ T ΓTi (k ) 2 i =1
n 1 1 − θ Tθ − ∑ χ iψ i
2
N
∂α i −1 (k ) ∂t
with (17)
i −1
∂α i −1 (k ) ϕ j (k ), ∂x j j =1
τ i (k ) = ϕi (k ) − ∑
132
Jin Zhu, Hong-sheng Xi, Hai-bo Ji, and Bing Wang
According to [14], we suggest the following adaptive laws:
i −1
∂α i −1 (k ) g j (k ). ∂x j j =1
ρi ( k ) = g i ( k ) − ∑
n
In (18), the following inequalities are used, which can be reduced from Young’s inequalities and norm inequalities with the help of changing the order of summations or exchanging the indices of the summations: n
∑ zi3 zi +1 ≤ i =1
3 n −1 43 4 1 n −1 1 4 ∑ δ zi + 4 ∑ δ 4 zi +1 4 i =1 i =1 i
θ = γ [∑ zi3τ i (k ) − l (θ − θ 0 )],
(20)
ψ i = σ i [ zi3ϖ i (k ) − mi (ψ i − ψ i0 )],
(21)
i =1
ϖ i (k ) = p i (k ) ⋅ tanh[
zi3 p i (k )
εi
(22)
].
Here l > 0, mi > 0, ε i > 0,θ 0 ∈ R p are given constants. Denote
n 3 4 1 = ∑ ( δ i3 + 4 ) zi4 , 4δ i −1 i =1 4
βi (k ) = ψ i ⋅ϖ i (k ).
(23)
where δ 0 = ∞, δ n = 0, and δ i > 0, i = 1, 2," , n − 1.
Substituting (20),(21),(23) into (18), and we suggest the virtual control as
1 n 3 i −1 ∂ 2α i −1 (k ) T zi ∑ g p (k ) ϒϒT g q (k ) ∑ 2 i =1 p,q =1 ∂x p ∂xq
α i (k ) = −ci zi − ( δ i3 +
n
≤∑
λ zi6
i =1
n
∂ 2α (k ) ∑ [ ∂x i −∂1x ]2 | g p (k ) |2 | gq (k ) |2 p q p, q =1 i −1
−λ zi3
1 +∑ ∑ |ϒϒT |2 λ 16 i =1 p, q =1 n
i =1
+
i −1
∑
(
p,q =1
∂ 2α i −1 (k ) 2 T ) g p (k ) g p (k ) gqT (k ) gq (k ) ∂x p ∂xq
∑[
(24) However, if adaptive law (20) is adopted, θ concerning with z1 , ⋅ ⋅ ⋅, zn exists in (24). Therefore it is impossible to get α i (k ) directly. For this reason, the following transitions are necessary: n
∑ zi3
∂α (k ) ≤ ∑ µ1 zi4 {[ gi (k ) − ∑ i −1 g j (k )]T ∂ x j i =1 j =1
i =1 n
i −1
n ∂α (k ) 9 [ gi (k ) − ∑ i −1 | ϒϒT |2 . g j (k )]}2 + ∑ 16 ∂ µ x j 1 j =1 i =1 n
3 ∑ zi2Γi (k )φφ T ΓTi (k ) 2 i =1
∂α i −1 (k ) θ ∂θ
= ∑ zi3 i =1 n
= ∑ zi3 i =1
∂α i −1 (k ) i 3 γ [ ∑ z jτ j (k ) + ∂θ j =1
3 ∑ zi2Γi (k )QΓTi (k ) 2 i =1
j =1 i =1 n
n
9 Q2 , ≤ ∑ µ 2 zi4 [Γi (k )ΓTi (k )]2 + ∑ 16 µ 2 i =1 i =1
∂α i −1 (k ) γ l (θ − θ 0 ) ∂θ
= γ ∑ zi3 [
∂α i −1 (k ) i 3 ∑ z jτ j (k ) ∂θ j =1
i =1
i −1
(19)
j =i +1
z 3jτ j (k ) − l (θ − θ 0 )]
∂α i −1 (k ) 3 γ z jτ j (k ) ∂θ
−∑ zi3 i =1 n
where λ > 0, µ1 > 0, µ 2 > 0 are design parameters,
∑
∂α i −1 (k ) ∑ z3jτ j (k ) ∂θ j =1
n j −1
n
n
i
+ ∑ ∑ zi3
(n − 1)n(2n − 1) 9n + . 96λ 16µ1
∂ 2αi −1 (k ) 2 T ] g p (k ) g p (k ) gqT (k ) gq (k ) x x ∂ ∂ p q p,q =1 i −1
N ∂α i −1 (k ) x j +1 + ∑ π kjα i −1 ( j ) − β i (k ). ∂x j j =1 j =1
i −1
ι=
∂α i −1 (k ) ∂t
i −1
∂α (k ) ϒϒT [ gi (k ) − ∑ i −1 g j (k )] ∂x j j =1
n
) zi +
+∑
i −1
≤
4δ i4−1
∂α i −1 (k ) i −1 ∂α i −1 (k ) ψ j (k ) − τ iT (k )θ θ +∑ ∂θ j =1 ∂ψ j
+
| ϒϒT |2 (n − 1)n(2n − 1) 96λ i −1 ∂α (k ) T 3 n 2 T zi [ gi (k ) − ∑ i −1 g j (k )] ∑ ∂x j 2 i =1 j =1
n
1
− µ1 zi [ ρiT (k ) ρi (k )]2 − µ 2 zi [Γi (k )ΓTi (k )]2
i −1
=∑ λ zi6
4
3 4
+( ∑ z 3j j =1
∂α j −1 (k ) ∂θ
)τ i (k ) − l (θ − θ 0 )].
(25)
A Robust Adaptive Controller for Markovian Jump Uncertain Nonlinear Systems with Wiener Noises of…
Substituting (25) into (24), and the virtual control design is 3 4
4
α i (k ) = −ci zi − ( δ i3 + +γ
1 4δ i4−1
) zi +
∂α i −1 (k ) ∂t
∂α j −1 (k ) ∂α i −1 (k ) z 3jτ j (k ) + γ ∑ z 3j τ i (k ) ∑ ∂θ ∂θ j =1 j =1
−γ l
i −1 ∂α i −1 (k ) ∂α (k ) (θ − θ 0 ) + ∑ i −1 ψ j ∂θ j =1 ∂ψ j
−τ iT (k )θ − µ1 zi [ ρiT (k ) ρi (k )]2 − µ 2 zi [Γi (k )ΓTi (k )]2 −λ zi3
i −1
∑
p, q =1
[
∂ 2α i −1 (k ) 2 T ] g p (k ) g p (k ) g qT (k ) g q (k ) ∂x p ∂xq
i −1
N ∂α i −1 (k ) x j +1 + ∑ π kjα i −1 ( j ) − β i (k ), ∂x j j =1 j =1
+∑
(26) where α 0 (k ) = 0, ci > 0, i = 1, 2," , n with the actual control : u (k ) = α n (k ),
1 1 lθ T (θ − θ 0 ) = − lθ Tθ − l (θ − θ 0 )T (θ − θ 0 ) 2 2 1 ∗ + l (θ − θ 0 )T (θ ∗ − θ 0 ) 2 1 T 1 ∗ ≤ − lθ θ + l (θ − θ 0 )T (θ ∗ − θ 0 ) 2 2 1 1 0 mi χ i (ψ i − ψ i ) = − mi χ i2 − mi (ψ i − ψ i0 )2 2 2 1 M + mi (ψ i − ψ i0 ) 2 1 1 ≤ − mi χ i2 + mi (ψ iM − ψ i0 )2 , 2 2 therefore n 1 1 LV ≤ −∑ ci zi4 − lθ Tθ + l (θ ∗ − θ 0 )T (θ ∗ − θ 0 ) 2 2 i =1
(27)
+
then the infinitesimal generator of V becomes : n
n
i =1
i =1
n
= −∑ ci zi4 + lθ T (θ − θ 0 ) + i =1 n
9n 2 Q + τ | ∆ |4 16µ 2
9n 2 Q + τ | ∆ |4 16µ2
1 n 9n 2 mi (ψ iM − ψ i0 )2 + Q + ι | ∆ |4 ∑ 2 i =1 16µ 2 ≤ −cV + κ , (30) where m = min(mi ),
κ=
n
+ ∑ [ zi3 Λi (k ) − (ψ i + χ i ) zi3ϖ i (k )] + ∑ mi χ i (ψ i − ψ i0 ) i =1 n
i =1 n
n
+ ∑ mi χ i (ψ i − ψ i0 ) + i =1
i =1
9n 2 Q + τ | ∆ |4 . 16µ 2 (28)
Consider (16), (22), we get zi3 Λi (k ) − ψ iM zi3ϖ i (k ) ≤ ψ i∗ | zi3 pi (k ) | −ψ iM zi3ϖ i (k ) ≤ ψ iM | zi3 pi (k ) | −ψ iM zi3 pi (k )tanh [
zi3 pi (k )
εi
and according to
η 1 0 ≤| η | −η tanh( ) ≤ 0.2785ε < ε , 2 ε such that
]
c = min(4ci , lγ , mσ i )
n
1 1 n M ε ψ + ∑ i i 2 ∑ mi (ψ iM − ψ i0 )2 + ι | ∆ |4 2 i =1 i =1 1 9n 2 + l (θ ∗ −θ 0 )T (θ ∗ −θ 0 )+ Q . 2 16µ 2
i =1
= −∑ ci zi4 + lθ T (θ − θ 0 ) + ∑ [ zi3 Λi (k ) − ψ iM zi3ϖ i (k )]
1 n 1 n ε iψ iM − m∑ χ i2 ∑ 2 i =1 2 i =1
+
LV ≤ −∑ ci zi4 − ∑ [ zi3 β i (k ) + zi3 χ iϖ i (k ) − zi3 Λi (k ) − χ i mi (ψ i − ψ i0 )] + lθ T (θ − θ 0 ) +
(29)
By using the following inequalities:
i −1
i
1 zi3 Λi (k ) − ψ iM zi3ϖ i (k ) ≤ ψ iM ε i . 2
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Theorem 1: Considering Markovian jump nonlinear system(11), if adaptive law (20), (21) and controller (27) is adopted, the equilibrium of the closed-loop system is globally uniformly ultimately bounded in the 4th-moment. Furthermore, for any given ε > 0, there is
lim E (| Z |44 ) < ε .
(31)
t →∞
Proof: According to the conclusion in [11], we have
κ
κ
EV ≤ e−ct [V ( x0 , t0 , r0 ) − ] + c c and there is V=
1 n 4 1 T n 1 2 1 n 4 ∑ zi + 2γ θ θ + ∑ 2σ χi ≥ 4 ∑ zi . 4 i =1 i i =1 i =1
134
Jin Zhu, Hong-sheng Xi, Hai-bo Ji, and Bing Wang
Take expectation in the above equation
ξ 2 ( x, t,1) = x1 , ξ 2 ( x, t, 2) = x1sinx2 ,
κ 4κ , (32) E (| Z |44 ) ≤ 4 EV ≤ 4e−ct [V ( x0 , t0 , r0 ) − ] + c c which
means
that Z = ( z1, z2 ,", zn ) is
globally
uniformly bounded in 4th-moment, thus x = ( x1 , x2 , " , xn ) is globally uniformly bounded in 4th-moment.
Moreover, ∃T > 0, if t ≥ T , there is 4e− ct [V ( x0 , κ 4κ 8κ t0 , r0 ) − ] ≤ , and E | Z |44 ≤ . So for any given c c c ε > 0, appropriate control design parameters ci , l , mi 8κ < ε. can be chosen to guarantee c Therefore when t ≥ T , E | Z |44 ≤
8κ < ε. c
4. EXMAPLE Consider a 2-order Markovian jump nonlinear system with the regime transition space S = {1, 2}, and the transition rate matrix is −2 2 Π= . 5 −5
The system is as follows: =
x2 dt + ξ1 ( x1, t, r (t ))θ ∗dt + ∆( x1, t, r (t ))dt,
= u dt + ξ2 ( x, t, r (t ))θ ∗dt + g2 ( x, t, r (t ))dω , = x1.
Here
ξ1 ( x1, t ,1) = x12 , ξ1 ( x1, t , 2) = x1 ,
where θ ∗ is an unknown parameter and ∆( x1 , t , r (t )) is an unknown bounded disturbance. For simulation purposes, we let θ ∗ = 2, ∆( x1 , t ,1) = 0.6 sin2t , ∆( x1 , t , 2) = cos 4t , and the noise covariance ϒ = 2. The control law and the adaptive law are taken as follows (here δ1 = 1 ) Case 1: The system regime is 1: 3 4 1 α 2 (1) = −(c2 + ) z2 − τ 2 (1)θ − x12γ [ z13τ1 (1) + z23τ 2 (1)] 4
α1 (1) = −(c1 + ) z1 − τ1 (1)θ − β1 (1),
+ γ l (θ − θ 0 ) x12 − β 2 (1)
(33)
Remark 1: From Theorem 1 above, it could be seen that all signals in the closed-loop system are globally uniformly ultimately bounded in the 4thmoment. According to the knowledge of stochastic stability [15], the closed-loop jump system (11) is thus stable in probability. Remark 2: According to (26), only the bound of noise ω and uncertainty ∆i are needed, and we do not care for the detailed information of them. Meanwhile the increase of ci , l, γ , m, σ i will reduce the effect of noise and structure uncertainty. As long as these design parameters are large enough, the system output y = x1 could be as close to the equilibrium as possible, disregarding the difference of system equations caused by regime r (t ) switching stochastically.
dx1 dx2 y
g 2 ( x, t,1) = sinx2 , g 2 ( x, t, 2) = 1 − 2cosx2 ,
+ γ l (θ − θ 0 ) x12 − β 2 (1) ∂α (1) − 1 x2 − ϖ1 (1)ψ 1 + π11α1 (1) + π12α1 (2), ∂x1 − µ1 z1 ( sinx2 )4 − µ 2 z2 [α1 (1) − α1 (2)]4 ,
θ = γ [ z13τ1 (1) + z23τ 2 (1) − l (θ − θ 0 )], ψ 1 = σ 1[ z13 w1 (1) − m1 (ψ1 − ψ10 )], ψ 2 = σ 2 [ z23 w2 (1) − m2 (ψ 2 − ψ 20 )], where z1 = x1 , z2 = x2 − α1 (1),
τ1 (1) = x12 , p1(1) = 1, ϖ1 (1) = p1(1)tanh(
z13 p1(1)
ε1
), β1 (1) = ψ 1ϖ1 (1),
∂α1 (1) 3z 2ψ 3 |=| c1 + + 2θ x1 + 1 1 [1 − ϖ12 (1)] |, ∂x1 4 ε1 ∂α (1) τ 2 (1) = x1 − 1 x12 , ∂x1 p 2(1) =|
ϖ 2 (1) = p 2(1)tanh(
z23 p 2(1)
ε2
),
β 2 (1) = ψ 2ϖ 2 (1) Case 2: The system regime is 2:
3 4 1 α 2 (2) = −(c2 + ) z2 − τ 2 (2)θ − x1γ [ z13τ1 (2) + z23τ 2 (2)] 4
α1 (2) = −(c1 + ) z1 − τ1 (2)θ − β1 (2),
+ γ l (θ − θ 0 ) x1 − β 2 (2) ∂α (2) − 1 x2 − ϖ1 (2)ψ 1 + π 21α1 (1) ∂x1
A Robust Adaptive Controller for Markovian Jump Uncertain Nonlinear Systems with Wiener Noises of…
135
+π 22α1 (2) − 4 µ1 z1 (1 − 2cosx2 )4 − µ 2 z2 [α1 (1) − α1 (2)]4 ,
θ = γ [ z13τ1 (2) + z23τ 2 (2) − l (θ − θ 0 )], ψ 1 = σ 1[ z13 w1 (2) − m1 (ψ1 − ψ 10 )], ψ 2 = σ 2 [ z23 w2 (2) − m2 (ψ 2 − ψ 20 )], where z1 = x1 , z2 = x2 − α1 (2),
τ1 (2) = x1 , p1(2) = 1,
ϖ1 (2) = p1(2)tanh(
z13 p1(2)
ε1
), Fig. 2. System control input u with time t.
β1 (2) = ψ1ϖ1 (2), ∂α1 (2) 3z 2ψ 3 |=| c1 + + θ + 1 1 [1 − ϖ12 (2)] |, ∂x1 4 ε1 ∂α (2) τ 2 (2) = x1sinx2 − 1 x1 , ∂x1 p 2(2) =|
ϖ 2 (2) = p 2(2)tanh(
z23 p 2(2)
ε2
),
β 2 (2) = ψ 2ϖ 2 (2). In computation we take design constants c1 = c2 = 5, θ 0 = 1, ψ 10 = ψ 20 = 1, σ 1 = σ 2 = γ = 1, m1 = m2 = l = 1, µ1 = 50, µ 2 = 25, ε1 = ε 2 = 0.4, the initial values as
choose
x1 = 8.7, x2 = −2, θ (0) = 0,
ψ 1 (0) = ψ 2 (0) = 0 and the time step is 0.01s. The simulation figures are as follows: In simulation, Fig. 1 shows the regime switching with time; Fig. 2 shows the corresponding input control u, and Fig. 3 shows the time response of the state variation (solid line for state x1 or output y, and dashed line for x2 ). As could be seen from the figures, the system regime may switch stochastically
Fig. 3. System states variation with time t. such that the control input will vary with the regime, thus they are not continuous to time t. However, at the time-point of regime switching, the system states will have an abrupt “jump”, but they are still continuous to time t, and this is an important quality of Markovian jump systems. The simulation results illustrate the global uniform ultimate boundedness of the closedloop system.
5. CONCLUSION
Fig. 1 Regime switching with time t.
The robust adaptive control of Markovian jump uncertain nonlinear parametric-strict-feedback systems disturbed by Wiener noises of unknown covariance was investigated. A robust adaptive control scheme was obtained by using a stochastic Lyapunov method and backstepping techniques, which guarantees that the closed-loop system is globally uniformly ultimately bounded. Backstepping design for non-jump stochastic system has been developed before by Deng [16]. In contrast to the design of Deng, the main difference is the introduction of the Markovian stochastic switching, thus the difficulties in controller design lie
136
Jin Zhu, Hong-sheng Xi, Hai-bo Ji, and Bing Wang
in follows: 1. The introduction of stochastic Markovian switching added the complexity of system analysis largely. Consider an N-regime Markovian jump systemsand it consists of N different subsystems. For each regime k, the actual controller u(k) would present a different expression. Therefore the regime-dependent controller u(k) represents the stochastic switching controller between the N different subsystem. In the infinitesimal generator for Markovian jump systems(11), the coupling of regime π kj is added, thus the switching controller u(k) concerns not only with the current regime k, but with all the regime j = 1, 2,", N . 2. Since the generalized Ito formula only deals with problems of stochastic systems perturbed by Wiener noise, it would be hard to handle the martingale process caused by Markovian switching. In this paper the martingale is converted to the correspondent Wiener noise B(t) so that infinitesimal generator could be applied. Thus in the switching controller u(k), the term µ 2 zi [Γi (k )ΓTi (k )]2 is included to reduced the effect of martingale. From above, it could been seen Deng’s controller would regarded as a specific example of our work in which system regime N ≡ 1. Thus this paper extends the controller design for stochastic nonlinear systems to a more general form. And the design accuracy could be ensured with appropriate parameters chosen. [1] [2]
[3]
[4]
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Jin Zhu received the B.S. and Ph.D. degrees in Control Science and Engineering from University of Science & Technology of China in 2001 and 2006 respectively. He is currently a Post-doc in Han-Yang University, Korea. His research interests include Markovian jump systems and nonlinear control.
Hai-bo Ji received the B.S. degree and Ph.D. degree in Mechanical Engineering from Zhe Jiang University and Beijing University in 1984 and 1990 respectively. He is currently a Professor in the Dept. of Automation, USTC. His research interests include nonlinear control and adaptive control.
Hong-sheng Xi received the M.S. degree in Applied Mathematics from University of Science & Technology of China (USTC) in 1977. He is currently a Professor in the Dept. of Automation, USTC. His research interests include stochastic control and network security.
Bing Wang received the Ph.D. degree in Control Science and Engineering from USTC in 2006. His research interest is nonlinear control.
