# Semiglobal Stabilization via Output-Feedback for a Class of Nontriangular Nonlinear Systems with an Unknown Coefficient

Authors: Mengliang Liu; Yungang Liu; Fengzhong Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is devoted to the semiglobal stabilization via output-feedback for a class of uncertain nonlinear systems. Remark that the systems in question contain an unknown control coefficient which inherently depends on the system output and allow larger-than-two order growing unmeasurable states which is the obstruction of global stabilization via output-feedback. By introducing a recursive reduced-order observer and combining with saturated state estimate, a desired output-feedback controller is explicitly constructed for the systems. Under the appropriate choice of design parameters, the controller can make the closed-loop system semiglobally attractive and locally exponentially stable at the origin. A simulation example is provided to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In this paper, we consider the semiglobal stabilization via output-feedback for a class of nonlinear systems described by(1)x˙i=xi+1+ϕi(t,x),i=1,…,n-1,x˙n=g(t,x)u+ϕn(t,x),y=x1,where x=[x1,…,xn]T is the system state with the initial value x(t0)=x0;  u∈R and y∈R are the input and output of the system, respectively; ϕi:[t0,+∞)×Rn→R,  i=1,…,n, and g:[t0,+∞)×Rn→R are unknown but continuous functions, callednonlinearities andcontrol coefficient of the system, respectively. In what follows, suppose that only the output y is measurable.Rigorously speaking, the control objective of this paper is, for any given constantρ0>0 (may be arbitrarily large), to seek a dynamic output-feedback controller for system (1) as follows: (2)η˙=ψ(η,y),η∈Rn-1,u=φ(η,y),where ψ(·) and φ(·) are continuous functions satisfying ψ(0,0)=0 and φ(0,0)=0, such that the closed-loop system is (i)
semiglobally attractive; that is, by starting from the compact set Γρ0x×Γρ0η, all the trajectories of the closed-loop system converge to the origin, where Γρ0x={x∣x∈[-ρ0,ρ0]×⋯×[-ρ0,ρ0]⊂Rn} and Γρ0η={η∣η∈[-ρ0,ρ0]×⋯×[-ρ0,ρ0]⊂Rn-1};(ii)
locally exponentially stable; that is, the closed-loop system is locally exponentially stable at the origin (x,η)=(0,0).As in [1, 2], we introduce the dilation δɛ(x)=(x1,ɛx2,…,ɛn-1xn) with ɛ being any positive constant. Based on this, the following assumptions are made on the nonlinearities and the control coefficient of system (1).Assumption 1.
There exist known positive continuous functionsg0(y) and g1(y) such that, for any ɛ≥1, (3)g0(y)≤g(t,δɛ(x))≤g1(y).Assumption 2.
There exist a known constantσ∈[0,1) and known nonnegative continuous function ci(x), i=1,…,n, such that, for any ɛ≥1, (4)ϕi(t,δɛ(x))≤ɛi-1+σcixx1+⋯+xn,i=1,…,n.For nonlinear systems, semiglobal stabilization via output-feedback has attracted a great deal of attention (see, e.g., [3–13] and references therein), since it has a control objective meeting the needs of practical application and, compared to the global case, it requires rather weak restrictions on the system nonlinearities. We would like to stress that our problem is nontrivial and rather difficult to solve, which is mainly due to the generality of Assumptions 1 and 2. On the one hand, system (1) allows additional unknowns and nonlinearities in the control coefficient. Assumption 1 shows that the control coefficient of system (1) is unknown and inherently depends on the system output, essentially different from [7–10, 14–16] where the control coefficients are known/unknown constants or known functions of the system output. On the other hand, system (1) can be of nontriangular structure and has inherent nonlinearities. Assumption 2 indicates that, for any i=1,…,n, ϕi(·) inherently depends on all the states, rather than merely on x1,…,xi like [8, 10, 11]. Also, due to the presence of ci(·)’s, system (1) allows more serious nonlinearities than those in [10, 11, 17]. Particularly, in [10], the system nonlinearities ψi(·)’s are required to satisfy (5)ψit,η,u≤μψiy∑j=1i‍ηj+νψiy∑j=2i-1‍ηji-1/j-1,i=1,…,n,where μψi(·), i=1,…,n, and νψi(·), i=3,…,n, are known nonnegative smooth functions and νψ1(·)=νψ2(·)≡0. This is a special case of Assumption 2 with ci(x)≥μψi(y)+νψi(y)∑j=1i‍|xj|(i-1)/(j-1)-1, i=1,…,n. Therefore, one can see that the nonlinearities ϕi(·)’s of system (1) permit larger-than-two order growing unmeasurable states, which is the obstruction of global stabilization via output-feedback.In this paper, a semiglobal stabilization scheme via output-feedback is proposed for uncertain nontriangular nonlinear system (1) with serious nonlinearities and the unknown control coefficient depending on the system output. Specifically, a state-feedback controller is first constructed for a nominal system (where ϕi(·)≡0, i=1,…,n), which ensures that the closed-loop system (corresponding to the nominal system) is globally exponentially stable. Then, a recursive reduced-order observer is introduced to recover the unmeasurable states of system (1). Based on these and combining with saturated state estimate [3, 4], a semiglobal output-feedback stabilizer is explicitly constructed for system (1). By appropriately choosing design parameters, the controller can guarantee that the closed-loop system (corresponding to system (1)) is semiglobally attractive and locally exponentially stable at the origin.The remainder of this paper is organized as follows. Section2 presents semiglobal output-feedback control design for system (1). Section 3 provides the main results and the rigorous performance analysis of the closed-loop system. Section 4 gives a numerical example to illustrate effectiveness of the proposed method. Section 5 addresses some concluding remarks. This paper ends with an appendix that collects two proofs of important propositions.


## 2. Semiglobal Output-Feedback Control

The section is to design a semiglobal stabilizer via output-feedback for system (1) under Assumptions 1 and 2.To achieve this, we introduce the coordinate transformation(6)zi=xiLi-1,i=1,…,n,v=uLn,which changes system (1) into the following: (7)z˙i=Lzi+1+ϕi(t,δL(z))Li-1,i=1,…,n-1,z˙n=Lg(t,δL(z))v+ϕn(t,δL(z))Ln-1,y=z1,where z=[z1,…,zn]T and L≥1 is a design parameter to be determined later.For simplicity, we denoteg~(t,z)=g(t,δL(z)) and ϕ~i(t,z)=ϕi(t,δL(z))/Li-1, i=1,…,n. By Assumption 2, it can be verified that, for i=1,…,n,(8)ϕ~i(t,z)≤Lσcizz1+⋯+zn.Moreover, it is worth stressing that the semiglobal stabilization of system (1) is implied by that of system (7). In the sequel, we turn to the controller design of system (7).We first establish the following proposition, which gives a state-feedback controller for system (7) without considering the nonlinearities ϕ~i(·)’s.Proposition 3.
Under Assumption1, consider the following nominal system:(9)z˙i=Lzi+1,i=1,…,n-1,z˙n=Lg~(t,z)v~.There exist positive constants ki,  i=1,…,n, such that system (9) is globally exponentially stabilized by the state-feedback controller:(10)v~(z)=-kng0(z1)(zn+kn-1zn-1+⋯+k1z1).Proof.
Choose constantski,  i=1,…,n-1, such that polynomial p(s)=sn-1+kn-1sn-2+⋯+k2s+k1 is Hurwitz. Then, we rewrite system (9) as(11)z˙[n-1]=LA1-B1K1z[n-1]+LB1zn+K1zn-1,z˙n=Lg~(t,z)v~,where z[n-1]=[z1,…,zn-1]T, K1=[k1,…,kn-1],(12)A1=01⋯0⋮⋮⋱⋮00⋯100⋯0∈Rn-1×n-1,B1=0⋮01∈R(n-1)×1.
Noting thatA1-B1K1 is a Hurwitz matrix, there exists a positive definite matrix such that(13)A1-B1K1TP1+P1(A1-B1K1)≤-2In-1.Then, we define the Lyapunov function(14)Vz(z)=z[n-1]TP1z[n-1]+zn+K1zn-122,whose derivative along system (11) is as follows:(15)V˙zz≤-2Lzn-12+2Lzn-1TP1B1zn+K1zn-1+L(zn+K1z[n-1])g~(t,z)v~+(zn+K1zn-1)·K1LA1-B1K1zn-1+B1zn+K1zn-1≤-2Lzn-12+L2P1B1+K1A1-B1K1·zn-1zn+K1zn-1+Lkn-1zn+K1zn-12+Lg~t,zzn+K1zn-1v~.Using Young’s inequality, we have (16)2P1B1+K1A1-B1K1zn-1zn+K1zn-1≤zn-12+Czn+K1zn-12,where C=(2P1B1+K1(A1-B1K1))2/4. Substituting the above estimation into (15) yields (17)V˙zz≤-Lzn-12+L(kn-1+C)zn+K1zn-12+Lg~(t,z)(zn+K1z[n-1])v~.
Choosekn≥kn-1+C+1 and construct the state feedback controller (18)v~(z)=-kng0z1zn+K1zn-1.Then, by Assumption 1, we derive(19)V˙z(z)≤-Lz[n-1]2-Lzn+K1zn-12,which, together with (14), implies that the closed-loop system consisting of (9) and (10) is globally exponentially stable.
This completes the proof.Based on controller (10) of nominal system (9), we construct a desired output-feedback controller for system (7). Motivated by [8, 18, 19], we introduce the following recursive reduced-order observer to recover unmeasurable states zi, i=2,…,n, of system (7): (20)η˙2=-L(l2η2+l22y),z^2=η2+l2z^1,η˙3=-L(l3η3+l32z^2),z^3=η3+l3z^2,⋮⋮η˙n=-L(lnηn+ln2z^n-1),z^n=ηn+lnz^n-1,where z^1=z1 and li>1, i=2,…,n, are the gains to be determined later. For the later use, we denote z^=[z^1,…,z^n]T and η=[η2,…,ηn]T.Furthermore, using saturated state estimate, we construct the following output-feedback controller for system (7):(21)vz^=-kng0z^1satz^n+kn-1satz^n-1+⋯+k2satz^2+k1z^1,where ki, i=1,…,n, are the same as those in Proposition 3; the saturation function sat:R→[-M,M] is defined as (22)sat(a)=-M,a<-M,a,a≤M,M,a>M,(23)M≥max⁡z∈Ωzz∞,with Ωz={z∈Rn∣Vz(z)≤1+max⁡z∈Γρ0zVz(z)}, Γρ0z={z∣z∈[-ρ0,ρ0]×⋯×[-ρ0,ρ0]⊂Rn}, and z∞ standing for the ∞-norm of vector z.


## 3. Main Results

This section is devoted to the performance analysis of the closed-loop system consisting of (7), (20), and (21) and summarizes the main results of this paper.Defineei=zi-lizi-1-ηi, i=2,…,n, and denote e=[e2,…,en]T. Then, by (7) and (20), we deduce(24)e˙i=Lzi+1+ϕ~i·-Lliei-liϕ~i-1·-Lli2zi-1-z^i-1,i=2,…,n-1,e˙n=Lg~(t,z)v(z^)+ϕ~n(·)-Llnen-lnϕ~n-1(·)-Lln2(zn-1-z^n-1).Define the Lyapunov functionVe(e)=∑i=2n‍(ei2/2) and choose the same Lyapunov function Vz(z) used in Proposition 3. On the set ΓMz×Rn-1={(z,η)∣z∈[-M,M]×⋯×[-M,M]⊂Rn,η∈Rn-1}, there hold Propositions 4 and 5 (whose detailed proofs are given in Appendices A and B, resp.), which will play a key role in the later performance analysis.Proposition 4.
For system (7) on the set ΓMz×Rn-1, there exist constants c>0 and d1>0, such that(25)V˙zzΓMz×Rn-1≤-cL2z2+Lσd1z2+Lmin⁡α^nen2+∑i=2n-1‍α^ili+1,…,lnei2,α^n,where α^n≥1 is a constant independent of li’s and α^i(li+1,…,ln)≥α^n, i=2,…,n-1, are certain polynomial functions of their arguments.Proposition 5.
For system (24) on the set ΓMz×Rn-1 and constant c>0 given in Proposition 4, there holds(26)V˙eeΓMz×Rn-1≤cL4z2-Lln-α~n-2cen2-L∑i=2n-1‍(li-α~i(li+1,…,ln)-2c)ei2+∑i=2n‍eiϕ~i·-liϕ~i-1·,where α~n>0 is a constant independent of li’s and α~i(li+1,…,ln)>0,  i=2,…,n-1, are certain polynomial functions of their arguments.In view of (25) and (26), we recursively determine the observer gains li’s as follows:(27)ln(l)=lα^n+α~n+2c,lil=lα^ili+1,…,ln+α~ili+1,…,ln+2c,i=n-1,…,2,such that(28)V˙e(e)≤cL4z2-LlWe(e)+∑i=2n‍eiϕ~i·-liϕ~i-1·,where l is a positive constant to be determined later and We(e)=α^nen2+∑i=2n-1‍α^i(li+1,…,ln)ei2.Now, we are ready to address the main results of this paper, which are summarized in the following theorem.Theorem 6.
Consider system (1) under Assumptions 1 and 2. For any given constant ρ0>0 (may be arbitrarily large), there exist appropriate L and li, i=2,…,n, depending on ρ0, such that the closed-loop system, consisting of (1), (20), and (21), is locally exponentially stable, and, by starting from the given compact set Γρ0x×Γρ0η, all the trajectories of the closed-loop system converge to the origin.Proof.
Motivated by [5, 6, 13], for the closed-loop system consisting of (7), (20), and (21), we choose Lyapunov function (29)Vz,η(z,η)=Vz(z)+ln⁡(1+Ve(e))ln⁡(1+μ(l)),where μ(l)=(1/2)∑i=2n‍(2ρ0+li(l)ρ0)2≥max⁡Γρ0z×Γρ0ηVe(e). From the definitions of Vz(z) and Ve(e), we see that Vz,η(z,η) is positive definite with respect to (z,η). Moreover, by (25) and (28), we obtain that, on the set ΓMz×Rn-1, (30)V˙z,ηz,ηΓMz×Rn-1≤-cL2z2+Lσd1z2+Lmin⁡{We(e),α^n}+Lln⁡⁡1+μl1+Vee·c4z2-lWe(e)+1ln⁡⁡1+μl1+Vee·∑i=2n‍eiϕ~i·-liϕ~i-1·.From α^i(·)≥α^n≥1, i=2,…,n-1, it follows that (31)Wee1+Vee≥α^nWee/α^n1+Wee/α^n≥α^n2min⁡Weeα^n,1≥12min⁡Wee,α^n,which implies that, on the set ΓMz×Rn-1,(32)V˙z,ηz,ηΓMz×Rn-1≤-cL2z2+Lσd1z2+cL4ln⁡(1+μ(l))z2-Ll2ln⁡⁡1+μl-1·min⁡Wee,α^n+1ln⁡⁡1+μl·∑i=2n‍eiϕ~i·-liϕ~i-1·.
From the choice ofli(l)’s, we see that li(l)’s are polynomial functions of l. Then, we can find l>1 to be sufficiently large such that(33)l2ln⁡(1+μ(l))>2,ln⁡(1+μ(l))>1,which, together with (32), implies that, on the set ΓMz×Rn-1,(34)V˙z,ηz,ηΓMz×Rn-1≤-cL4z2+Lσd1z2-Lmin⁡Wee,α^n+∑i=2n‍eiϕ~i·-lilϕ~i-1·.
Moreover, define the compact setΩ={(z,η)∣Vz,η(z,η)≤1+max⁡z∈Γρ0zVz(z)}. Then, it is easy to verify the following relation:(35)Γρ0z×Γρ0η⊂Ω⊂ΓMz×Rn-1.We now estimate the last term in the right-hand side of (34) on the compact set Ω. By (8) and Young’s inequality and noting the boundedness of ci(z)’s and We(e) on Ω, there exist positive constants d2 and d3 depending on li’s, such that, on the set Ω, (36)∑i=2n‍eiϕ~i·-lilϕ~i-1·≤Lσ∑i=2n‍ciz-lilci-1zei∑j=1n‍zj≤Lσd2z2+Lσd3min⁡{We(e),α^n}.Substituting this into (34), we obtain that, on the set Ω, (37)V˙z,ηz,ηΩ≤-cL4-Lσd1-Lσd2z2-L-Lσd3min⁡Wee,α^n.
By0≤σ<1, we can choose L≥1 satisfying(38)L1-σ≥max⁡d1+d2+14c,d3+1.Then, we obtain that, on the set Ω,(39)V˙z,η(z,η)Ω≤-Lσz2-Lσmin⁡Wee,α^n.By the definitions of ηi’s and We(e), we derive V˙z,η(z,η)<0 on Ω∖{(0,0)}, which implies that, by starting from Γρ0z×Γρ0η, (z(t),η(t)) remains in the compact set Ω, and (40)limt→+∞(z(t),η(t))=(0,0).
ByWe(0)=0 and the continuity of We(e) on Ω, there exists a neighborhood Ω0⊆Ω of the origin, such that We(e)≤α^n for all (z,η)∈Ω0. Then, by (39), we obtain that, on the set Ω0,(41)V˙z,η(z,η)Ω0≤-Lσz2-LσWe(e).By the definition of Ve(e) and We(e), we deduce that, on the set Ω0, (42)We(e)≥μ1Ve(e)≥μ2ln⁡(1+Ve(e))ln⁡(1+μ(l))with some positive constants μ1 and μ2. Substituting this into (41) and noting the definitions of Vz(z), we have that, on the set Ω0, (43)V˙z,η(z,η)Ω0≤-Lσμ3Vz,ηz,η,with positive constant μ3. Therefore, the closed-loop system consisting of (7), (20), and (21) is locally exponentially stable.
By (6), (35) and noting L≥1, there holds (44)x∈Γρ0x,η∈Γρ0η⟹z∈Γρ0z,η∈Γρ0η⟹z,η∈Ω.Furthermore, by the invertibility of (6), we arrive at the fact that the closed-loop system, consisting of (1), (20), and (21), is locally exponentially stable, and, by starting from Γρ0x×Γρ0η, the trajectories (x(t),η(t)) converge to the origin.
This completes the proof.


## 4. A Simulation Example

In this section, we consider the following nonlinear system to illustrate the correctness and effectiveness of the theoretical results:(45)x˙1=x2+x12/3x31/3,x˙2=x3+x1ln⁡(1+x22),x˙3=g(t,x)u+x1,y=x1,with 2(x12+1)≤g(t,x)≤4(x12+1). It can be verified that system (45) satisfies Assumption 2 with σ=2/3, c1(x)=1, and c2(x)=2(1+x22).According to the design procedure of the state feedback controller, we choosek1=0.8 and k2=1.2 and then, by solving the matrix inequality (13), yield (46)P1=1.6875-0.5-0.50.75.Furthermore, by Proposition 3, we choose k3=5.Letρ0=3. Then, by the above P1, (14), and (23), we choose M=6. Moreover, from (33) and (38), we choose l=1.5 and L=2 and, in terms of (27), recursively determine l2=10 and l3=8. Thus, by the design procedure in Section 2, we construct the following observer: (47)η˙2=-2(10η2+100y),z^2=η2+10z^1,η˙3=-2(8η3+64z^2),z^3=η3+8z^2,and design the following output-feedback controller: (48)v(z^)=-52z^12+1satz^3+1.2satz^2+0.8z^1.Letg(t,x)=(3-sinx2)(x12+1), and let the initial values of the states be x(0)=[1,-0.5,-0.5]T and η(0)=[1,-1]T. Using MATLAB, Figures 1–3 are obtained to exhibit the trajectories of the closed-loop system states and the controller, from which the effectiveness of the design method is indeed demonstrated.Figure 1
Trajectory of statesx1, x2, and x3.Figure 2
Trajectories of statesη2 and η3.Figure 3
Trajectories of controlu.


## 5. Concluding Remarks

In this paper, the semiglobal stabilization via output-feedback has been investigated for a class of uncertain nontriangular nonlinear systems. Essentially different from the existing related works, the control coefficient of the system is unknown and inherently depends on the system output, and, hence, the scope of the nonlinear systems is considerably broadened. By introducing a recursive reduced-order observer and combining with saturated state estimate, a semiglobal output-feedback controller has been constructed for the uncertain system. Under the appropriate choice of design parameters, the controller can guarantee that the closed-loop system achieves semiglobal attractivity and locally exponential stability. Along this direction, another interesting research problem is how to design a semiglobal finite-time stabilizer via output-feedback for system (1).



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