The purpose of this paper is to construct a smooth state-feedback control law such that the solution process of system (1) is bounded in probability. For clarity, the case that x0(t0)≠0 is firstly considered. Then, the case where the initial x0(t0)=0 is dealt with later. The triangular structure of system (1) suggests that we should design the control inputs u0 and u1 in two separate stages.
Assumption 6.
For f0(t,x0), there exists a nonnegative smooth function γ0(t,x0), such that |f0(t,x0)|≤|x0|γ0(t,x0).
For each fi(t,x0,x-i), φi(x-i), there exist nonnegative smooth functions γi(t,x0,x-i) and ρi(x-i), such that |fi(t,x0,x-i)|≤(∑k=1i|xk|)γi(t,x0,x-i), |φi(x-i)|≤(∑k=1i|xk|)ρi(x-i).
3.1. Designingu0forx0-Subsystem
For x0-subsystem, the control u0 can be chosen as
(6)u0=-λ0x0,
where λ0=(k0+γ0)/λ01 and k0 is a positive design parameter.
Consider the Lyapunov function candidate V0=x02/2. From (6) and Assumptions 5 and 6, we have
(7)LV0=x0(d0u0+f0(t,x0))≤d0u0x0+x02γ0≤-k0x02=-2k0V0.
So, we obtain the first result of this paper.
Theorem 7.
The x0-subsystem, under the control law (6) with an appropriate choice of the parameters k0, λ01, λ02, is globally exponentially stable.
Proof.
Clearly, from (7), LV0≤0, which implies that |x0(t)|≤|x0(t0)|e-k0(t-t0). Therefore, x0 is globally exponentially convergent. Consequently, x0 can be zero only at t=t0, when x(t0)=0 or t=∞. It is concluded that x0 does not cross zero for all t∈(t0,∞) provided that x(t0)≠0.
Remark 8.
If x(t0)≠0, u0 exists and does not cross zero for all t∈(t0,∞) independent of the x-subsystem from (6).
3.2. Backstepping Design foru1
From the above analysis, the x0-state in (1) can be globally exponentially regulated to zero as t→∞, obviously. In this subsection, we consider the control law u1 for the x-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation:
(8)ηi=eαtxiu0n-i, i=1…,n, u=eαtu1.
Under the new x-coordinates, x-subsystems is transformed into
(9)dηi=diηi+1dt+f-idt+ϕiT ∑T(t)dω, i=1,…,n-1,dηn=dnudt+f-ndt+ϕnT∑T(t)dω,
where
(10)f-i=αηi+eαtfiu0n-i-(n-i)ηiu0∂u0∂x0(d0u0+f0),ϕi=eαtφiu0n-i.
In order to obtain the estimations for the nonlinear functions f-i and ϕi, the following Lemma can be derived by Assumption 6.
Lemma 9.
For i=1,2…n,there exist nonnegative smooth functions γ-i(·), ρ-i(·), such that
(11)|f-i|≤(∑k=1i|ηk|)γ-i(x0,x-i),(12)|ϕi|≤(∑k=1i|ηk|)ρ-i(x-i).
Proof.
We only prove (11). The proof of (12) is similar to that of (11). In view of (6), (8), (10) and Assumption 6, one obtains
(13)|f-i|=|αηi+eαtfiu0n-i-(n-i)ηiu0∂u0∂x0(d0x0+f0)|≤|αηi|+(∑k=1ieαt|xk|u0n-k|u0i-k|)γi +(n-i)(λ0λ02+γ0)|ηi|≤|α||ηi|+(∑k=1i|ηk|λ0i-k|x0i-k|)γi +(n-i)(λ0λ02+γ0)|ηi|≤(∑k=1i|ηk|)(|α|+|λ0i-k||x0i-k|γi+(n-i)(λ0λ02+γ0))≤(∑k=1i|ηk|)γ-i(x0,x-i),
where γ-i(x0,x-i)≥|α|+|λ0i-k|x0i-k|γi+(n-i)(λ0λ02+γ0).
To design a state-feedback controller, one introduces the coordinate transformation
(14)z1=η1,zi=ηi-αi(z-i-1), i=1,2…,n,
where α2,…,αn are smooth virtual control laws and will be designed later and α1=0. θ^ denotes the estimate of θ, where
(15)θ=supt≥0 {max{∥∑(t)∑T(t)∥2,∥∑(t)∑T(t)∥4/3, ∥∑(t)∑T(t)∥∥∑(t)∑T(t)∥2,∥∑(t)∑T(t)∥4/3}}.
Then using (9), (10), (14) and It o^ differentiation rule, one has
(16)dzi=d(ηi-αi)=(diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙)dt+GiT(z-i)∑T(t)dω -12∑k,m=1i-1∂2αi∂zk∂zmϕkT(z-k)∑T(t)∑(t)ϕm(z-m)dt,=(diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙)dt+GiTi=1,2…n,
where ηn+1=u, Fi(z-i,x0)=f-i+∑k=1i-1(∂αi/∂zk)(dkηk+1+f-k), and Gi(z-i,x0)=ϕi+∑k=1i-1(∂αi/∂zk)ϕk,where i=1,2…n. Using Lemmas 2, 4, and 9 and (14), we easily obtain the following lemma.
Lemma 10.
For 1≤i≤n,there exist nonnegative smooth functions γi1(z-i,x0), pi1(z-i), and p-i(z-i), such that
(17)|Fi|≤(∑k=1i|zk|)γi1(z-i,x0),|Gi|≤(∑k=1i|zk|)pi1(z-i),|Φi|≤(∑k=1i|zk|)p-i(z-i).
The proof of Lemma 10 is similar to that of Lemma 9, so we omitted it.
We now give the design process of the controller.
Step 1.
Consider the first Lyapunov function V1(z1,θ^)=(1/4)z14+(1/2)(θ^-θ)2. By (14), (15), and (16), we have
(18)LV1=z13(d1η2+F1)+32z12Tr(G1T∑T(t)∑(t)G1) +(θ^-θ)θ^˙.
Using Lemma 10 and Lemma 4, we have
(19)|z13F1|≤z14γ11(z1,x0)|32z12Tr(G1T∑T(t)∑(t)G1)| ≤z14p112(z1,x0)|∑T(t)∑(t)|≤z14p112(z1,x0)θ.
Substituting (19) into (18) and using (14), we have
(20)LV1≤d1z13(η2-α2)+d1z13α2+z14p112(z1,x0)θ +z14γ11(z1,x0)+(θ^-θ)θ^˙≤d1z13z2+d1z13α2+z14p112(z1,x0)θ +z14γ11(z1,x0)+(θ^-θ)θ^˙,
where α2=-z1β1=-z1((c1+γ11+p112θ^)/λ11). Substituting α2 into (20), we have
(21)LV1≤d1z13z2-c1z14+(θ^-θ)(θ^˙-τ1),
where τ1=z14p112.
Step i.
(2≤i≤n). Assume that at step i-1, there exists a smooth state-feedback virtual control αi=-zi-1βi-1(z-i-1,θ^)=-zi-1((ci-1+θ^1+(ψi-12+ψi-13)2+bi-1+ψi-11+ψi-14)/λi-11), such that
(22)LVi-1≤-∑j=1i-2(cj-εj-ej)zj4-ci-1zi-14+di-1zi-13zi +(θ^-θ-∑k=2i-1zk3∂αk∂θ^)(θ^˙-τi-1),
where Vi-1=∑j=1i-1(1/4)zj4+(1/2)(θ^-θ)2, τi-1=τ1+∑k=2i-1zk4(ψi-12+ψi-13), and εj=∑k=1j(εk1+εk2+εk3+εk4),where j=1,…,n.
Then, define the ith Lyapunov candidate function Vi(z-i,θ^)=Vi-1+(1/4)zi4. From (16) and (22), it follows that
(23)LVi≤-∑j=1i-2(cj-εj-ej)zj4-ci-1zi-14+di-1zi-13zi +zi3(∑k,m=1i-1diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙ +zi3(-12∑k,m=1i-1∂2αi∂zk∂zmϕkT(z-k)∑T(t)∑(t)ϕm(z-m)) +32zi2Tr(GiT(z-i)∑T(t)∑(t)Gi(z-i)) +(θ^-θ-∑k=2i-1zk3∂αk∂θ^)(θ^˙-τi-1).
Using Lemmas 9 and 4, there are always known nonnegative smooth functions ψi1(z-i), ψi2(z-i), ψi3(z-i), ψi4(z-i) and constant εi>0, εij>0, where i=1,…,n and j=1,2,3,4.
Consider
(24)zi3Fi≤|zi3|(∑k=1i-1|zk|)γi1(z-i,x0)≤γi1zi4+∑k=1i-1(εk1zk4+34(4εk1)-1/3γi14/3zi4)≤∑k=1i-1εk1zk4+ψi1zi4,
where ψi1≥γi1+∑k=1i-1(3/4)(4εk1)-1/3γi14/3. (25)-12zi3∑k,m=1i-1∂2αi∂zk∂zmϕkT∑T(t)∑(t)ϕm ≤12zi3∑k,m=1i-1|∂2αi∂zk∂zm|(∑j=1k|zj|)p-k(z-k) ×(∑j=1m|zj|)p-m(z-m)|∑T(t)∑(t)| ≤zi3(∑k=1i-1zk2)p--i(z-i)|∑T(t)∑(t)| ≤zi4ψi2(z-i)θ+∑k=1i-1εk2zk4,
where ψi2≥∑k=1i-1(3/4)(4εk2)-1/3(p--i(z-i))3/4.
(26)32zi2Tr(GiT∑T(t)∑(t)Gi) ≤32zi2pi22(z-i)(∑k=1i|zk|)2|∑T(t)∑(t)| ≤32zi2ipi22(z-i)(∑k=1izk2)|∑T(t)∑(t)| ≤32zi4ipi22(z-i)θ+∑k=1i-1εk3zk4+∑k=1i-114εk3(32ipi22)2zi4θ ≤∑k=1i-1εk3zk4+ψi3zi4θ,
where ψi3≥(3/2)ipi22+∑k=1i-1(1/4εk3)((3/2)ipi22)2. (27)di-1zi-13zi≤λi-12|zi-13zi|≤ei-1zi-14+14(43ei-1)-3zi4λi24≤ei-1zi-14+bizi4,
where bi≥(1/4)((4/3)ei-1)-3λi24, τi-1=z14p112+∑k=2i-1zk4(ψk2+ψk3), and τi=τi-1+(ψi2+ψi3)zi4.
(28)-zi3∂αi∂θ^τi ≤zi3|∂αi∂θ^|(τi-1+zi4(ψi2+ψi3)) ≤zi41+(zi3∂αi∂θ^)2(ψi2+ψi3) +zi3|∂αi∂θ^|(z14p112+∑k=2i-1zk4(ψk2+ψk3)) +34(4)-1/3(|∂α2∂θ^|z13p112)4/3εi4-1/3zi4 ≤εi4z14+34(4)-1/31+(∂αi∂θ^z13P112)24/3εi4-1/3zi4 +∑k=2i-1εk4zk4 +∑k=2i-134(4)-1/31+(∂αi∂θ^zk3(ψk2+ψk3))24/3εk4-1/3zi4 ≤∑k=1i-1εk4zk4+ψi4zi4,
where ψi4≥(3/4)(4εi4)-1/31+((∂αi/∂θ^ )z13P112)24/3+∑k=2i-1(3/4)(4εk4)-1/31+((∂αi/∂θ^ )zk3(ψk2+ψk3))24/3.
(29)αi+1(z-i,θ^)=-ziβi(z-i,θ^),βi(z-i,θ^)=ci+ψi1+ψi4+bi+1+(ψi2+ψi3)2θ^λi1,
where ci>0 is a design parameter to be chosen.
With the aid of (24)–(29) and (14), (23) can be simplified as
(30)LVi≤-∑j=1i-1(cj-εj-ej)zj4-cizi4 +dizi3zi+1+(θ^-θ-∑k=2i∂αk∂θ^zk3)(θ^˙-τi).
Finally, when i=n, zn+1=u is the actual control. By choosing the actual control law and the adaptive law,
(31)u(z-n,θ^)=-znβn(z-n,θ^),θ^˙=τn=z14p112+∑k=2nzk4(ψk2+ψk3), βn(z-n,θ^)=cn+bn+ψn1+ψn4+1+(ψn2+ψn3)2θ^λn1,u1=e-αtu,
where cn>0 is a design parameter to be chosen and ψni,i=1,…4 are smooth functions; we get
(32)LVn≤-∑j=1n(cj-εj-ej)zj4,
where Vn(z,θ^)=∑k=1n(1/4)zk4+(1/2)(θ^-θ)2, z=(z1,…zn). We have finished the controller design procedure for x0(t0)≠0 and the parameter identification. Without loss of generality, we can assume that t0≠0.
3.3. Switching Control and Main Result
In the preceding subsection, we have given controller design for x0≠0. Now, we discuss how to choose the control laws u0 and u1 when x0=0. We choose u0 as u0=-λ0x0+u0*, u0*>0. And choose the Lyapunov function V0=(1/2)x02. Its time derivative is given by LV0=-λ0x02+u0*, which leads to the bounds of x0. During the time period [0,ts), using u0=-λ0x0+u0*, new control law u can be obtained by the control procedure described above to the original x-subsystem in (1). Then, we can conclude that the x-state of (1) cannot be blown up during the time period [0,ts). Since at x(ts)≠0,we can switch the control inputs u0 and u to (6) and (31), respectively.
Now, we state the main results as follows.
Theorem 11.
Under Assumption 5, if the proposed adaptive controller (31) together with the above switching control strategy is used in (1), then for any initial contidion (x0,x,θ^)∈Rn, the closed-loop system has an almost surely unique solution on [0,∞), the solution process is bounded in probability, and P{limt→∞θ^(t) exists and is finite}=1.
Proof.
According to the above analysis, it suffices to prove in the case x0(0)≠0. Since we have already proven that x0 can be globally exponentially convergent to zero in probability in Section 3.1, we only need prove that x(t) is convergent to zero in probability also. In this case, we choose the Lyapunov function V=Vn, and ci>εi+ei; from (32) and Lemma 3, we know that the closed-loop system has an almost surely unique solution on [0,∞), and the solution process is bounded in probability.