# Adaptive Output Feedback Stabilization of Nonholonomic Systems with Nonlinear Parameterization

Authors: Yanling Shang; Ye Yuan; Fushun Yuan
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates the problem of adaptive output feedback stabilization for a class of nonholonomic systems with nonlinear parameterization and strong nonlinear drifts. A parameter separation technique is introduced to transform nonlinearly parameterized system into a linear-like parameterized system. Then, by using the integrator backstepping approach based on observer and parameter estimator, a constructive design procedure for output feedback adaptive control is given. And a switching strategy is developed to eliminate the phenomenon of uncontrollability. It is shown that, under some conditions, the proposed controller can guarantee that all the system states globally converge to the origin, while other signals remain bounded. An illustrative example is also provided to demonstrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Control of nonholonomic systems has received a great deal of attention over the last few years. It has been shown in [1] that mechanical systems with nonholonomic constraints such as mobile robots and wheeled vehicle can be either locally or globally converted to the so-called chained form under a coordinate transformation and a control mapping. As a result, the chained form has been used as a canonical form in analysis and control design for nonholonomic systems. Due to the nonsatisfaction of Brockett’s necessary condition [2], nonholonomic systems cannot be asymptotically stabilized by stationary continuous state-feedback, although it is controllable. To overcome this difficulty, a number of approaches have been proposed; see the review paper [3] and references therein. The proposed solutions for stabilization nonholonomic systems include discontinuous feedback, time-varying feedback, and hybrid control laws. Using there valid approaches, the robustness issue for the asymptotic and exponential stability properties has been extensively studied [4–20].However, it should be noticed that most of these papers were concerned with the systems with linear parameterization. Since nonlinear parameterization is exceptionally difficult to estimate, there are very few reports in literature for adaptive control of nonlinearly parameterized nonholonomic systems. Exceptions include [21–23] (and some references therein), where the authors consider that the whole state vector is measurable and adaptive control for chained form systems with nonlinear parameterization using state feedback.In this paper, the case of only partial state vector being measurable and adaptive control for chained form systems with nonlinear parameterization by output feedback is considered. The contributions of this paper are listed as follows: (i) a new observer design method is proposed, and, based on the observer, the unmeasurable states of the system involved are reconstructed; (ii) using parameter separation technique [24], the parameters nonlinearities are solved and the resulting adaptive regulator is of minimum dimension (1D) independent of the parameter dimension; (iii) output feedback adaptive control based switching strategy is adopted to handle the technical problem of uncontrollability at x0(t0)=0, which prevents the finite escape of systems and guarantees that all the states converge to the origin and other signals are bounded.The rest of this paper is organized as follows. In Section2 preliminary knowledge and the problem formulation are given. Section 3 presents the state-scaling technique and the backstepping design procedure, while Section 4 provides the switching control strategy and the main result. Section 5 gives a simulation example to illustrate the theoretical finding of this paper. Finally, concluding remarks are proposed in Section 6.


## 2. Problem Formulation

In this paper, we present an output feedback adaptive control design procedure for a class of uncertain chained form systems with nonlinear parameterization:(1)x˙0=u0+f1(x0,θ)x˙i=u0xi+1+fi(x0,x,θ)1≤i≤n-1x˙n=u1+fn(x0,x,θ)y=(x0,x1)T,
where (x0,x)T=(x0,x1,…,xn)T∈Rn+1, u=(u0,u1)T∈R2, and y∈R2 are the system state, control input, and system measurable output, fi, i=0,1,…,n are continuous functions of their arguments, and θ∈Rp is an unknown constant vector. The function fi is called the nonlinear drifts of the system (1).The objective of this paper is to design an output feedback adaptive stabilization control in the form(2)x^˙=ϑ(x^,y),μ˙=ν(x0,x,μ)u0=u0(y,μ),u1=u1(y,μ)
such that all signals of the closed loop system are bounded. Furthermore, the asymptotic regulation of the states is achieved; that is, lim⁡t→∞(|x0(t)|+|x(t)|)=0.A full characterization of the class of system (1) is given by the following assumption, which will be the base of the coming control design and performance analysis.Assumption 1.
Forf0, there exists a smooth nonnegative function φ0 such that
(3)|f0(x0,θ)|≤|x0|φ0(x0,θ).For1≤i≤n, there are smooth nonnegative function φi such that
(4)|fi(x0,x,θ)|≤|x1|φi(y,θ).Remark 2.
Assumptions1 implies that f0(0,θ)≡0 and fi(0,0,θ)≡0, that is, the origin is the equilibrium point of system (1).Lemma 3 (see [24]).
For any real-valued continuous functionf(x,y), where x∈Rm,y∈Rn, there are smooth scalar functions a(x)≥0,b(y)≥0,c(x)≥1 and d(y)≥1, such that
(5)|f(x,y)|≤a(x)+b(y)|f(x,y)|≤c(x)d(y).By Assumption1 and Lemma 3, we easily obtain the following lemma.Lemma 4.
For1≤i≤n, there are smooth nonnegative function φ¯i and unknown constant Θi≥1 such that
(6)|f0(x0,θ)|≤|x0|φ¯0(x0)Θ0|fi(x0,x,θ)|≤|x1|φ¯i(y)Θi.


## 3. Adaptive Output Feedback Control Design

In this section, we focus on designing the control inputu1 via output feedback provided that x0(t0)≠0. The case where the initial x0(t0)=0 will be treated in Section 4. The special structure of the system (1) suggests that we should design the control inputs u0 and u1 in two separate stages.LetΘ~0=Θ0-Θ^0, where Θ^0 is an estimate of Θ0. Assumption 1 leads us to choose the control law u0 as
(7)u0(x0,Θ^0)=x0g0(x0,Θ^0)(8)g0(x0,Θ^0)=-[φ¯0Θ^0+2k02+2(m0φ¯0+φ¯0Θ^0)2],
where k0 and m0 are positive design parameters to be specified later.Remark 5.
Note the controlu0 is an uncertain version of Sontag formula [25]. It is used to stabilize x0 with uncertainties. Because the particular choice of (8), g0(x0,Θ^0)≠0 is guaranteed irregardless of the values of x0. Hence, 1/g0 is well defined.Consider the Lyapunov function candidate(9)V0=12x02+12Θ~02.From (7) and (8) and Assumption 1, we have
(10)V˙0=x0[u0+f0(x0,θ)]-Θ~0Θ^˙0≤g0x02+x02φ¯0Θ-Θ~0Θ^˙0.Choose the adaptation lawΘ^0 as
(11)Θ^˙0=x02φ¯0(x0)
renders
(12)V˙0≤-k0x02.Therefore,x0 and Θ~0 (or, equivalently, Θ^0) are bounded; without loss of generality, we assume |Θ~0|≤M, so we can choose design parameter m0 in (8) as m0>M. Furthermore, we can conclude that x0→0 as t→∞ by using LaSalle’s Invariant Theorem [26].Under the control law (7), the solution of x0-subsystem can be expressed as
(13)x0(t)=x0(t0)exp⁡(∫t0t‍u0+f0x0ds)
which implies
(14)|x0(t)|≥|x0(t0)|exp⁡(-∫t0t‍|u0+f0x0|ds)≥|x0(t0)|exp⁡(-∫t0t‍λ(s)ds),
where λ(s)=2φ¯0Θ^0+2k0+2(m0φ¯0+φ¯0Θ^0)2+k0φ¯0.Consequently,x0 does not cross zero for all t∈(t0,∞) provided that x0(t0)≠0.From the above analysis, we can see thex0-state in (1) can be globally regulated to zero via u0 in (7) as t→∞. It is troublesome in controlling thex-subsystem via the control input u1 in the limit (i.e., u0=0), thex-subsystem is uncontrollable. This problem can be avoided by utilizing the following discontinuous state scaling transformation [4]:
(15)zi=xix0n-i1≤i≤n.
Under the newz-coordinates, with choice of u0 as in (7), thex-system is transformed into
(16)z˙1=g0z2-(n-1)g0z1+f1x0n-1z˙i=g0zi+1-(n-i)g0z1+fix0n-2z˙n=u1+fn.
It should be noted that the measurement of state z1 can be obtained if the to-be-designed control u0 is only dependent on output y. If x0(t)≠0 for every t≥0, the discontinuous state transformation (15) is applicable.We design the following observer for the system (16):
(17)z^˙1=g0z^2-(n-1)g0z^1+k1u0x0(z1-z^1)z^˙i=g0z^i+1-g0z^i+kiu0x0(z1-z^1)z^˙n=u1+knu0x0(z1-z^1),
where k1,…,kn are design parameters to be determined later.The estimation errorz~=z-z^ satisfies the dynamical equations:
(18)z~˙1=g0z~2-(n-1)g0z~1-k1g0z~1+f1x0n-1z~˙i=g0z~i+1-(n-i)g0z~i-kig0z~1+fix0n-iz~˙n=-kng0z~1+fn+ϕnd.
The differential equations (18) can be rewritten into the compact form
(19)z~˙=g0Az~+Φ,
where
(20)A=(-k1-n+1100⋯0-k2-n+210⋯0⋮0-n+31⋱0⋮⋮⋱⋱⋱0-kn-1⋮⋮⋱-11-kn0⋯⋯00)Φ=(f1x0n-1f2x0n-2⋮fn-1x0fn).
About matrix A defined by (20), there exists the following lemma.Lemma 6.
The eigenvalues of the matrix A defined by (20) can be arbitrarily assigned by a proper selection of the design parameters k1,…,kn.Proof.
The proof can be found in [16] and thus omitted here.Lemma 7.
By Assumption1 and Lemma 4, we easily obtain that, for every 1≤i≤n, there exists smooth nonnegative function γi such that
(21)|fi(x0,x,θ)x0n-i|≤|z1|γi(x0,z1,Θ^0)Θi.
In view of (18), (19), and (16), the overall system to be controlled can be expressed as
(22)z~˙=g0Az~+Φz˙1=g0z^2+g0z~2-(n-1)g0z1+f1x0n-1z^˙i=g0z^i+1-(n-i)g0z^i+kig0z~1z^˙n=u1+kng0z~1.We now turn to the constructive design procedure of the control.Step 1.
This step can be regarded as the initial assignation of the entire design procedure. At this step, we introduce a Lyapunov function for the estimation errorz~. Define Θ=max⁡{Θi,Θi2} and Θ~=Θ-Θ^, where Θ^ is the estimate of Θ. Consider the Lyapunov function
(23)U0=z~TPz~+Θ~22,
where P is the positive definite solution of the Riccati equation
(24)PA+ATP=In.
Then, taking time-derivation ofU0 along the solution of (19), we have
(25)U˙0=g0∥z~∥2+2z~TPΦ-Θ~Θ^˙.
By Lemma 7 and the expression of Φ in (20), we have
(26)∥Φ∥2≤z12Φ¯(x0,z1,Θ^0)Θ,
where Φ¯=∑i=1n‍γi2(x0,z1,Θ^0) is a nonnegative smooth function. Then, for the terms on the right-hand side of (25), by completing the square, we have
(27)2z~TPΦ=ɛ02∥z~∥2+ɛ0-2ΦTPPΦ-ɛ02∥z~-ɛ0-2PΦ∥2≤ɛ02∥z~∥2+ɛ0-2∥P∥2z12Φ¯2Θ,
where ɛ0 is a positive design parameter to be specified later.
Substituting (27) into (25) leads to
(28)U˙0=(g0+ɛ02)∥z~∥2+Ψ0(Θ^+Θ~)-Θ^˙Θ~,
where Ψ0=ɛ0-2∥P∥2z12Φ¯2.Step 2.
TakeV1=U0+z12/2 as the Lyapunov function of this step. Then, by (28) we have
(29)V˙1≤(g0+ɛ02)∥z~∥2+Ψ0(Θ^+Θ~)-Θ^˙Θ~+g0z1z^2+g0z1z~2+z12γ1(Θ~+Θ^);
with the choice of the virtual controller
(30)z^2*=-1g0z1[n+g02ɛ1-2+(ɛ0-2∥P∥2Φ¯2+γ1)Θ^2+1]=z1β1(x0,z1,Θ^)
we have
(31)V˙1≤(g0+ɛ02+ɛ12)∥z~∥2-nz12+g0z1(z^2-z^2*)+(Ψ1-Θ^˙)(Θ~+η1),
where Ψ1=z12(ɛ0-2∥P∥2Φ¯2+γ1)≥0 and η1=0.Step i (2 ≤I≤ n).
Suppose at stepi-1 we have designed a set of smooth virtual controllers z^1*,…,z^i* defined by
(32)z1*=0,z^2*=ξ1β1(x0,ξ1,Θ^0,Θ^),⋮z^i*=ξi-1βi-1(x0,ξ1,…,ξi-1,Θ^0,Θ^);ξ1=z1-z1*,ξ2=z^2-z^2*,⋮ξi=z^i-z^i*
with β1(x0,ξ1,Θ^)>0,…,βi-1(x0,ξ1,…,ξi-1,Θ^)>0 being smooth, such that
(33)V˙i-1=(g0+∑k=0i-1‍ɛk2)∥z~∥2-(n-i+2)(ξ12+⋯+ξi-12)+g0ξi-1(z^i-z^i*)+(Ψi-1-Θ^˙)(Θ~+ηi-1),
where
(34)0≤Ψi-1(x0,ξ1,…,ξi-1,Θ^0,Θ^)≤(ξ12+⋯+ξi-12)αi-1(x0,ξ1,…,ξi-1,Θ^0,Θ^)
for C∞αi-1≥0.We intend to establish a similar property for(z~,z1,…,z^i)-subsystem. Consider the Lyapunov function
(35)Vi(x0,ξ1,…,ξi,Θ^0,Θ^)=Vi-1(x0,ξ1,…,ξi-1,Θ^0,Θ^)+12ξi2.
Clearly
(36)V˙i=(g0+∑k=0i-1‍ɛk2)∥z~∥2-(n-i+2)(ξ12+⋯+ξi-12)+g0ξi-1(z^i-z^i*)+(Ψi-Θ^˙)(Θ~+ηi)+ξi{∑j=2i-1‍g0z^i+1-(n-i)g0z^i+kig0z~1-∂zi*∂Θ^0x02φ¯0ui-∂z^i*∂Θ^Θ^˙-∂zi*∂x0(u0+f0)-∂z^i*∂z1[g0z2-(n-1)g0z1+f1x0n-1]u-∑j=2i-1‍∂z^i*∂z^j[g0z^j+1-(n-j)g0z^j+kjg0z~1]}.Sincez^i* is a smooth function and satisfies
(37)z^i*(x0,0,…,0,Θ^0,Θ^)=0
therefore there exist some continuous functions ψi,1≤i≤n-1 such that
(38)z^i*(x0,ξ1,…,ξi-1,Θ^0,Θ^)=∑j=1i-1‍ξkψj(x0,ξ1,…,ξi-1,Θ^0,Θ^).After lengthy but simple calculations based on the completion of squares, there is a smooth nonnegative functionσi such that
(39)|ξi||∑j=2i-1‍(n-i)g0z^i-kig0z~1+∂zi*∂Θ^0x02φ¯0+∂z^i*∂Θ^Θ^˙+∂zi*∂x0u0+∂z^i*∂z1[g0z2-(n-1)g0z1]+∑j=2i-1‍∂z^i*∂z^j[g0z^j+1-(n-j)g0z^j+kjg0z~1]|≤ɛi2∥z~∥2+14∑j=1i-1‍ξj2+ξi2σi(x0,ξ1,…,ξi,Θ^0,Θ^),
where ɛi is a positive design parameter to be specified later.Using (38) and Lemma 4, we have
(40)|∂zi*∂x0f0|≤∑j=1i‍|ξj|ρi(x0,ξ1,…,ξi,Θ^0,Θ^)Θ0,
where ρi≥0 is a smooth function.By Lemma7, there is a smooth function ωi≥0 such that
(41)|∂z^i*∂z1||f1x0n-1|≤|z1|ωi(x0,z1,…,z^i,Θ^0,Θ^)Θ1.Based on the completion of squares, it is deduced that there is a smooth functionϖi≥0 satisfying
(42)|ξi||∂zi*∂x0f0+∂z^i*∂z1f1x0n-1|≤[∑j=1i-1ξj22(1+Θ^2)(1+ηi-12)+ξi2ϖ]Θ≤[∑j=1i-1ξj22(1+Θ^2)(1+ηi-12)+ξi2ϖi]Θ~+14∑j=1i-1‍ξj2+ξi2ϖiΘ^2+1
since
(43)|g0ξi-1(z^i-z^i*)|≤14ξi-12+ξi2γ~i(x0,ξ1,…,ξi,Θ^0,Θ^)
for a smooth function γ~i≥0.Substituting (39), (42), and (43) into (36) gives
(44)V˙i≤(g0+∑k=0i‍ɛk2)∥z~∥2-(n-i+54)(ξ12+⋯+ξi-12)+(Ψi-1-Θ^˙)(Θ~+ηi-1)-g0ξiz^i+1+ξi2[ϖiΘ^2+1+σi+γ~i]+[∑j=1i-1ξj22(1+Θ^2)(1+ηi-12)+ξi2ϖi]Θ~+ξi∂z^i*∂Θ^Θ^˙.
Define
(45)Ψi=Ψi-1+∑j=1i-1ξj22(1+Θ^2)(1+ηi-12)+ξi2ϖiηi=ηi-1+ξi∂z^i*∂Θ^.
Using (34), it is not difficult to verify that
(46)0≤Ψi(x0,ξ1,…,ξi,Θ^0,Θ^)≤(ξ12+⋯+ξi2)αi(x0,ξ1,…,ξi,Θ^0,Θ^)
for a C∞αi≥0.Moreover, (44) can be rewritten as follows:
(47)V˙i≤(g0+∑k=0i‍ɛk2)∥z~∥2-(n-i+54)(ξ12+⋯+ξi-12)+(Ψi-Θ^˙)(Θ~+ηi)-g0ξiz^i+1+ξi2[ϖiΘ^2+1+σi+γ~i]-[∑j=1i-1ξj22(1+Θ^2)(1+ηi-12)+ξi2ϖi]ηi-1-ξi∂z^i*∂Θ^Ψi.
By (46), we have
(48)|[∑j=1i-1ξj22(1+Θ^2)(1+ηi-12)+ξi2ϖi]ηi-1+ξi∂z^i*∂Θ^Ψi|≤18∑j=1i-1‍ξj2+ξi2ϖiηi-12+1+|ξi∂z^i*∂Θ^|(ξ12+⋯+ξi2)αi≤14∑j=1i-1‍ξj2+ξi2ϖiηi-12+1+ξi2∑j=1i‍β~i,j.
The last inequality from the following relation
(49)|ξi||ξj2∂z^i*∂Θ^αi|≤18ξj2+ξi2β~i,j(x0,ξ1,…,ξi,Θ^0,Θ^),
where β~i,j≥0,1≤j≤i-1 are smooth functions and β~i,i≥|ξi(∂zi*/∂Θ^)αi|, is a smooth function.Putting (47) and (48) together, we arrive at
(50)V˙i≤(g0+∑k=0i‍ɛk2)∥z~∥2-(n-i+1)(ξ12+⋯+ξi-12)+(Ψi-Θ^˙)(Θ~+ηi)+g0ξiz^i+1+ξi2[ϖiΘ^2+1+σi+γ~i+γ^i],
where γ^i=ϖiηi-12+1+∑j=1iβ~i,j(x0,ξ1,…,ξi,Θ^0,Θ^).Now, it easy to see that the smooth virtual controller(51)z^i+1*=-1g0ξi[n-i+1+κi(x0,ξ1,…,ξi,Θ^0,Θ^)]≔ξiβi(x0,ξ1,…,ξi,Θ^0,Θ^)κi=ϖiΘ^2+1+σi+γ~i+γ^i
renders
(52)V˙i≤(g0+∑k=0i‍ɛk2)∥z~∥2-(n-i+1)(ξ12+⋯+ξi2)+g0ξi(z^i+1-z^i+1*)+(Ψi-Θ^˙)(Θ~+ηi).Asi=n, the last step, we can construct explicitly a change of coordinates (ξ1,…,ξn), a positive-definite and proper Lyapunov function Vn=Vn-1+ξn2/2, and a smooth controller u1*=-ξnβn such that
(53)V˙n≤(g0+∑k=0n‍ɛk2)∥z~∥2-(ξ12+⋯+ξn2)+ξn(u1-u1*)+(Ψn-Θ^˙)(Θ~+ηn).Therefore, by choosing the smooth actual controlu1 and update law for Θ^ as
(54)u1=-ξnβn(x0,ξ1,…,ξi,Θ^0,Θ^)(55)Θ^˙=Ψn(x0,ξ1,…,ξi,Θ^0,Θ^)
we have
(56)V˙n≤(g0+∑k=0n‍ɛk2)∥z~∥2-(ξ12+⋯+ξn2).In Section4, we will show that an appropriate choice of the design parameters k0 and ɛi guarantees the asymptotic convergence of the state (x0,x) of the original system (1) in closed loop with our control laws (7), (8), (54), and (55) as long as the initial condition x0(t0) is nonzero. It will also be shown in the next section how to handle the singular case where x0(t0) is zero.


## 4. Switching Controller and Main Result

In the preceding section, we have given controller design forx0(t0)≠0. Without loss of generality, we can assume that t0=0. Now, we discuss how to select the control laws u0 and u1 when x0(0)=0. Choose u0 as
(57)u0=x0g0+u0*,2u0*>0,
where g0 is given by (7).Choosing the same Lyapunov function (9), its time derivative is given by
(58)V˙0≤-k0x02+u0*x0
which leads to the boundedness of x0 and consequently the bound of Θ^0 as well. Consequently, x0 does not escape and x0(ts)≠0 for any given finite ts>0. Thus state-scaling for the control design can be carried out.During the time period[0,ts), using u0 defined in (57), new control law u1=u1*(x0,x,μ) and new update law μ˙=ν*(x0,x,μ) can be obtained by following the control procedure described above to the originalx-system in (1). Then we can conclude that thex-state of (1) cannot blow up during the time period [0,ts). Since x(ts)≠0 at ts, we can switch the control input u0 and u1 to (7) and (54), respectively.We are now ready to state the main theorem of our paper.Theorem 8.
Under Assumption1, if the above switching control strategy is applied to (1) with an appropriate choice of the design parameters ki and ɛi′s, uncertain system (1) is globally regulated at the origin. Furthermore, the estimated parameters are bounded.Proof.
According to the above analysis, it suffices to prove the statement in the case wherex0(0)≠0.
Since we have already proven thatlim⁡t→∞x0(t)=0 and Θ^0 is bounded in Section 3, we just need to show that lim⁡t→∞x(t)=0 and Θ^ is bounded. In this case, as a result of (7), (11), (54), and (55), from (56), it holds that
(59)V˙n≤(g0+∑k=0n‍ɛk2)∥z~∥2-(ξ12+⋯+ξn2).
Choose the design parameters k0 and ɛi′s so that
(60)k0≥∑k=0n‍ɛk2.
Therefore, we have
(61)V˙n≤-(ξ12+⋯+ξn2)
which means ξ and Θ^ are bounded. Furthermore, we can conclude that ξ→0 as t→∞ by using LaSalle’s Invariant Theorem [26]. This implies that z→0 as t→∞. From the state scaling transformation (15), we conclude that x→0 as t→∞. This completes the proof of Theorem 8.Remark 9.
As seen from (54) and (7), the control law u1 may exhibit extremely large value when x0(t0)≠0 is sufficiently small. This is unacceptable from a practical point of view. It is therefore recommended to apply (57) in order to enlarge the initial value of x0 before we appeal to the asymptotically converging controllers (7) and (54).


## 5. Example

To verify our proposed controller, we consider the following low-dimensional system with parametric uncertainty:(62)x˙0=u0x˙1=u0x2+ln⁡⁡(1+(θx1)2)x˙2=u1,
where θ is unknown constant parameter. Define Θ=1+θ2; obviously, this system satisfies Lemma 4; that is, |ln⁡(1+(θx1)2)|≤|x1|Θ. The purpose is to design u0 and u1 based on only y=(x0,x1)T such that (x0,x1,x2)T→0 as t→∞.Ifx(0)=0, controls u0 and u1 are set as in Section 4 in interval [0,ts), such that x(ts)≠0; then we can adopt the controls developed below. Therefore, without loss of generality, we assume that x(0)≠0.For simplicity, in the first subsystem, we can choose the control lawu0=g0x0 and introduce the introduce the state scaling transformation:
(63)z1=x1x0,z2=x2.
In new z-coordinates, the (x1,x2)-subsystem of (62) is rewritten as
(64)z˙1=g0(z2-z1)+1x0ln⁡⁡(1+(θx1)2)z˙2=u1.
Design observer to reconstruct z1 and z2 of system (62) as follows:
(65)z^˙1=g0(z^2-z^1)+g0k1(z1-z^1),k1=-4z^˙2=u1+k2(z1-z^1),k2=-2.
Then, the estimation error z~=(z1-z^1,z2-z^2)T satisfies the equation
(66)z~˙=g0(-k1-11k20)z~+1x0(ln⁡(1+(θx1)2)0).
In this case, we have
(67)A=(31-20).Solving the matrix equationATP+PA=I2, we obtain
(68)P=(0.5000.5000.5000.100).
The eigenvalues of P are 0.1910,1.3090, and thus ∥P∥=1.3090.Using the method given in Section3, we can design the adaptive controller as follows:
(69)z^2*=-1g0z1[3+g02ɛ1-2+1.604ɛ0-2+Θ^2+1]=z1β1u1=-ξ2[9+β1+β14+β12Θ^2+1]Θ^˙=z12(1.604ɛ0-2+1)2+ξ22β12.The unknown parameterθ is assumed to be 1. In simulation, the design parameters are chosen as g0=-1 and ɛ0=ɛ1=ɛ2=0.25. The simulation results for initial conditions (x0(0),x1(0),x2(0))=(2,1,-1) and (z^1(0),z^2(0),Θ^(0))=(1,-1,1) are shown in Figures 1–4. From the figures, it is clear to see all the closed-loop system states are regulated to zero and the parameter estimates are bounded as proven in Theorem 8.Figure 1
System states.Figure 2
Observer states.Figure 3
Parameter estimate.Figure 4
Control inputs.


## 6. Conclusions

In this paper, a constructive adaptive output feedback control strategy is presented for a class of nonlinearly parameterized nonholonomic systems with strong nonlinear drifts. To deal with the nonlinear parameterization problem, a parameter separation technique is introduced to transform the nonlinear parameterized nonholonomic system into a linear-like parameterized nonholonomic system. We estimate onlyΘ, the bound of the unknown parameter rather than the parameter vector θ. This, in turn, results in a minimum dimension (1D) parameter update law. The controller design is developed by using the integrator backstepping approach based on observer and parameter estimator. With a switching strategy eliminating the phenomenon of uncontrollability, the system states asymptotically converge to the origin, while other signals remaining bounded is guaranteed. Simulation results have shown the effectiveness and feasibility of the proposed control strategy.



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*Source: 623605-2014-08-12.xml*