# Hybrid Feedback Stabilization of Fuzzy Nonlinear Systems

Authors: Yingqi Zhang; Caixia Liu; Xiaowu Mu
Journal: Journal of Control Science and Engineering
 (2011)

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## Abstract

This paper is concerned with the problem of stabilizing one family of fuzzy
nonlinear systems by means of fuzzy quantized feedback. The hybrid control strategy
originating in earlier work by Brockett and Liberzon (2000) and Liberzon (2003) relies on the possibility of making discrete online adjustments of quantizer parameters. We explore this method here for one class of fuzzy nonlinear systems with fuzzy quantizers affecting the state of the system. New results on the
stabilization of the family of fuzzy nonlinear systems are obtained by choosing
appropriately quantized strategies. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

In recent years, there has been increasing interest in stability analysis and controller design for hybrid and switched systems see, for example, [1, 2]. In the presence of quantization, the state space of the system is divided into a finite number of quantization regions, each corresponding to a fixed value of the quantizer. At the time of passage from one quantization region to another, the dynamics of the closed-loop system change abruptly. Therefore, systems with quantization can be naturally viewed as hybrid systems. Thus, considerable efforts have been devoted to the study of quantized control, for instance, see [3–7] and the references therein. Among these results, mainly two approaches for studying control problems with quantized feedback are chosen, which are called static quantization policies (e.g., [8–10]) and dynamic quantization policies (e.g., [5, 11]).Liberzon [5] gave the conditions of hybrid feedback stabilization of systems with quantized signal under the assumption of the systems being stabilized by a feedback law.  De Persis [12] extended Liberzon's [5] results to the problem of stabilizing a nonlinear system by means of quantized output feedback.  Gao and Chen [13] presented a new approach to quantized feedback control systems which provided stability and H∞ performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers. The most significant feature is the utilization of a quantization dependent Lyapunov function. Ceragioli and De Persis [14] discussed discontinuous stabilization of nonlinear systems with quantized and switching controls, that is, considering the classical problem of stabilizing nonlinear systems in the case of the control laws which take values in a discrete set.The well-known Takagi-Sugeno (T-S) fuzzy model (e.g., [15]) has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems. Thus, over the past ten years, the study of T-S systems has been attracting increasing attention, for instance, see [16–23]. However, so far, the study of fuzzy systems with quantized feedback was rare, for instance, [24]. In this paper, we concentrate on the problem of stabilizing fuzzy nonlinear systems via fuzzy quantized feedback. We extend the results (see, [5]) to a class of T-S fuzzy nonlinear systems with general types of quantizers affecting the state of the system. New results on the stabilization of fuzzy nonlinear systems are obtained by choosing appropriately quantized strategies and applying the Lyapunov function approach.The paper is organized as follows. Section2 gives the concept of quantizer and the description of fuzzy systems. New results on the stabilization of fuzzy nonlinear systems with fuzzy quantized feedback are presented in Section 3. In Section 4, an example is given to show the effectiveness of the proposed method. Conclusions are presented in Section 5.


## 2. Problem Statement

In this section, some notations and definition of quantizer are introduced, and the problem statement is given.As in [5], a quantizer with general form is defined as follows.Letz∈ℝl be the variable being quantized. A quantizer is defined as a piecewise constant function q:ℝl→D, where D is a finite subset of ℝl. This leads to a partition of ℝl into a finite number of quantization regions of the form {z∈ℝl:q(z)=i}, i∈D. These quantization regions are not assumed to have any particular shapes. We assume that there exist positive real numbers M and Δ such that the following conditions hold:(1)|z|≤M⟹|q(z)-z|≤Δ,(2)|z|>M⟹|q(z)|>M-Δ.Throughout this paper, we denote by|·| the standard Euclidean norm in the n-dimensional vector space ℝn and denote by ∥·∥ the corresponding induced matrix norm in ℝn×n. Condition (1) gives a bound on the quantization error when the quantizer does not saturate. Condition (2) provides a way to detect the possibility of saturation. We will refer to M and Δ as the range of q and the quantization error, respectively. We also assume that {x:q(x)=0} for x in some neighborhood of the origin which is needed to preserve the origin as an equilibrium.In the control strategy to be developed below, we will use quantized measurements of same the form as in [3, 4](3)qμ(z):=μq(zμ),
where μ>0 is an adjustable parameter, called the “zoom” variable, that is updated at discrete instants of time.To be convenient, we denoted that∑i,j=1r:=∑i=1r∑j=1r,  hi:=hi(x(t)), hiqμ(x):=hi(qμ(x(t))), and wi:=wi(x(t)).The T-S fuzzy system, suggested by  Takagi and Sugeno [15] can represent a general class of nonlinear systems. It is based on “fuzzy partition” of input space and it can be viewed as the expansion of piecewise linear partition. Considering a nonlinear dynamic multi-input-multi-output system modeled by the T-S fuzzy system, it can be represented by the following forms.(i)
If-then form:Ri: IF x1(t) is Mi1, x2(t) is Mi2… and xn(t) is Minthen(4)ẋ(t)=Aix(t)+Biu(t).(ii)
Input-output form:(5)ẋ(t)=∑i=1rwi[Aix(t)+Biu(t)]∑i=1rwi=∑i=1rhi(x(t))[Aix(t)+Biu(t)],wi=∏j=1nMij(xj(t)),∑i=1rwi>0,wi≥0,hi=wi∑i=1rwi,∑i=1rhi=1,hi≥0,
where x(t)=[x1(t),x2(t),…,xn(t)]T is the state, u(t)∈ℝm is the control input, Ri  (i=1,2,…,r) is the ith fuzzy rule, r is the number of rule, Mi1,Mi2,…,Min are fuzzy variable, and hi is fuzzy basis function.For the nonlinear plant represented by (4) or (5), we consider the fuzzy controller as follows.(iii)
If-then form:Ri: if x1(t) is Mi1, x2(t) is Mi2… and xn(t) is Minthen(6)u(t)=Lix(t),
or(7)u(t)=Liqμ(x).(iv)
Input-output form:(8)u(t)∑i=1r=hiLix(t),
or(9)u(t)=∑i=1rhiqμ(x)[Liqμ(x)].The system (5) with (8) or the system (5) with (9) can, respectively, be written in the form of the T-S fuzzy control system as follows:(10)ẋ(t)=∑i,j=1rhihj(Ai+BiLj)x(t)=∑i,j=1rhihjHijx(t),
or(11)ẋ(t)=∑i,j=1rhihjqμ(x){(Ai+BiLj)x(t)+BiLjμ[q(xμ)-xμ]}=∑i,j=1rhihjqμ(x){Hijx(t)+BiLjμ[q(xμ)-xμ]},
where Hij denotes Hij:=Ai+BiLj.


## 3. Fuzzy Hybrid Feedback Stabilization

In this section, in order to find some sufficient conditions which stabilize the fuzzy nonlinear systems (11) by choosing appropriately quantized strategies, we require the following assumption 1 and an important lemma is given as in Lemma 1 as follows.Assumption.
Assume that there exists a sequence of matrices{Li}i=1r and a common positive definite matrix P and a sequence of positive matrices {Qij}i,j=1r such that
(12)-Qij:=(Ai+BiLj)TP+P(Ai+BiLj)=HijTP+PHij.
Moreover, both Bi and Li for all i,j∈{1,2,…,r} are nonzero matrices, which cause no loss of generality because the case of interest is when Ai is not a stable matrix for all i∈{1,2,…,r}.Remark 1.
If Assumption1 holds, the system (5) with fuzzy control law (8) or the T-S fuzzy system (10) is asymptotically stable by using Lyapunov approach (e.g., see [16, 17]).Remark 2.
As in [5], it is necessary to suppose that systems are stabilizable. To be convenient, we suppose that Assumption 1 holds so that the system (5) is stabilizable.Lemma 1.
Assume that Assumption1 holds. an arbitrary σ>0, and M is large enough compared to Δ such that
(13)λmin⁡(P)λmax⁡(P)M>ΘxΔ(1+σ),
where
(14)Θx:=2θ/λ,λ:=min⁡{λ(Qij):i,j=1,2,…,r},θ:=max⁡{‖PBiLj‖:i,j=1,2,…,r}.
Let
(15)E1(μ):={x:xTPx≤λmin⁡(P)M2μ2},(16)E2(μ):={x:xTPx≤λmax⁡(P)Θx2Δ2(1+σ)2μ2}.
Then all solutions of (11) that start in the ellipsoid ℰ1(μ) enter the smaller ellipsoid ℰ2(μ) in finite time.Proof.
We consider the Lyapunov function candidateV(x)=xTPx for the closed-loop system (11) the derivative of V(x) along solutions of (11) is computed as
(17)V̇(x)=(xTPx)′=∑i,j=1rhihjqμ(x){[q(xμ)-xμ]xT(HijTP+PHij)x(t)+2xTPBiLjμ×[q(xμ)-xμ]}=∑i,j=1rhihjqμ(x){(xμ)-xTQijx(t)+2xTPBiLjμ×[q(xμ)-xμ]}≤∑i,j=1rhihjqμ(x){-λ|x|2+2θ|x|μΔ}≤∑i,j=1rhihjqμ(x){-λ|x|(|x|-ΘxμΔ)}=-λ|x|(|x|-ΘxμΔ).
According to (13), for any nonzero x, we can find a positive scalar μ such that
(18)ΘxμΔ(1+σ)≤|x|≤Mμ.
This is also true in the case ofx=0, where we set μ=0 as an extreme case and consider the output of the quantizer as zero.
WhenΘxμΔ(1+σ)≤|x|≤Mμ holds, we have(19)(xTPx)′≤-|x|λΘxμΔσ.Claim 1.
Bothℰ1(μ) and ℰ2(μ) are invariant sets of the system (11).Proof.
we only prove thatℰ1(μ) is an invariant set of the system (11). Assuming x(t0)∈ℰ1(μ), we denote
(20)τ:=sup⁡{t:x(t)∈E1(μ),∀t∈[t0,τ)},
where x(t) is a solution of the system (11) with the initial condition x(t0). If τ<+∞, then there exists a positive constant τ such that
(21)x(τ)TPx(τ)=λmin⁡(P)M2μ2.
By the virtue of condition (13), we have
(22)λmax⁡(P)Θx2Δ2(1+σ)2μ2<λmin⁡(P)M2μ2=x(τ)TPx(τ)≤λmax⁡(P)|x(τ)|2.
Hence, we obtain
(23)ΘxμΔ(1+σ)<[|x(t)|]∣t=τ.
Using (13), we have ℰ2(μ)∈ℰ1(μ) and
(24)[V̇(x(t))]∣t=τ≤-λ[|x(t)|(|x(t)|-μΔΘx)]∣t=τ<-λ[|x(t)|]∣t=τΘxμΔσ≤0.
By the continuity of V(x(t)), there exists a positive constant ϵ such that  for all t∈[0,ϵ] satisfying
(25)λmin⁡(P)|x(t+τ)|2≤x(t+τ)TPx(t+τ)=V(x(t+τ))≤V(x(τ))=λmin⁡(P)M2μ2.
Hence, we have V(x(t+τ))≤λmin(P)M2μ2 for all t∈[0,ϵ] that is to say that x(t+τ)∈ℰ1(μ) holds for all t∈[0,ϵ] this is a contradiction with the definition τ. Thus, τ=+∞. we complete the proof of Claim 1.Fixed an arbitraryx(t0)∈ℰ1(μ), and for all x, we can find a positive scalar μ satisfying (18).  Then, integrating (19) from t0 to t0+T, we have(26)xT(t0+T)Px(t0+T)-xT(t0)Px(t0)=∫t0t0+T[x(s)TPx(s)]′ds≤-∫t0t0+T[|x(s)|λΘxμΔσ]ds≤-∫t0t0+T[ΘxμΔ(1+σ)λΘxμΔσ]ds≤-TΘx2μ2Δ2σ(1+σ)λ.
Hence, we obtain(27)xT(t0+T)Px(t0+T)≤xT(t0)Px(t0)-TΘx2μ2Δ2σ(1+σ)λ≤λmin⁡(P)M2μ2-TΘx2μ2Δ2σ(1+σ)λ.
If we choose(28)T:=Tx=λmin⁡(P)M2-λmax⁡(P)Θx2Δ2(1+σ)2Θx2Δ2σ(1+σ)λ,
we have x(t0+Tx)∈ℰ2(μ).Using Lemma1 and assuming that the fuzzy control law (9) of the system (5) satisfies(29)u(t)={0,0≤t<t0,∑i=1rhiqμ(x)[Liqμ(x)],t≥t0,
We have the following theorem 1.Theorem 1.
Assume that Assumption1 holds. Assume that M is large enough compared to Δ such that
(30)λmin⁡(P)λmax⁡(P)M>2Δmax⁡{1,Θx}
holds, where Θx is the same as in Lemma 1. Then there exists a fuzzy quantized feedback control strategy such that the system (5) with fuzzy quantized control law (9) or the closed fuzzy nonlinear system (11) is globally asymptotically stable.Proof.
The “zooming-out” stage. Letu=0. In this case, we rewrite the system (11) for
(31)ẋ(t)=∑i,j=1rhihjqμ(x){Hijx(t)+BiLjμ[q(xμ)-xμ]}=∑i=1rhiAix(t).
Let
(32)A:=argmaxAiji,j∈{1,2,…,r}⁡‖Aij‖
Letμ0=μ(0)=1, and then increase μ in a piecewise constant fashion, fast enough to dominate the rate of erAt. Then, there is a time t≥0 such that(33)|x(t)μ(t)|≤λmin⁡(P)λmax⁡(P)M-2Δ.
By condition (1) in Section 2, it is implied
(34)|q(x(t)μ(t))|≤λmin⁡(P)λmax⁡(P)M-Δ.
We can pick a t0 such that (34) holds with t=t0. Again, applying conditions (1) and (2) of Section 2, we obtain
(35)|x(t0)μ(t0)|≤λmin⁡(P)λmax⁡(P)M.
Hence, we have x(t0)∈ℰ1(μ(t0)) given by (15).The “zooming-in” stage. Define the sequence of times{tj}j∈ℕ satisfying(36)x(t0)∈E1(μ(t0)),tj+1=tj+Tx,
and the sequence of positive real numbers(37)μ0=μ(t0)=1,μj=μ(tj)=Ωμ(tj-1)=Ωjμ0=Ωj.
where Ω denotes λmax(P)/λmin(P)(ΘxμΔ(1+σ)/M) and Tx is the same as in (28).Define also the control law(38)u(t)=∑j=1rhjqμj(x)Ljqμj(x(t)),t∈[tj,tj+1),j∈N.By (30) and Lemma 1, we have Ω<1 and ℰ2(μ(tj+1))=ℰ1(μ(tj)). Hence, μj=Ωjμ0=Ωj→0 as t→+∞, and the above analysis implies x(t)→0 as t→+∞.In order to prove the stability of the equilibriumx=0 of system (11), take an arbitrary ϵ>0 and notice that u(t)=0 as 0≤t≤t0 firstly, finding a positive integer K:=ln(ϵ/M)/Ω+1, t∈[(K-1)Tx,KTx), we have (39)|x(t)|≤Mμ(tK-1)=MΩK-1μ(t0)=MΩK-1≤ϵ.
This implies ℰ1(ΩK-1)∈{x:|x|<ϵ}.By the virtue ofq(x), there exists a positive constant ϵ0 such that q(x)=0 holds for all x∈{x:|x|<ϵ0}. With no loss of generality, we assume ϵ0≤ϵ. We define (40)δ:=min⁡{ϵ0e-r‖Aj‖Tx:j=1,2,…,K}=ϵ0e-r‖A‖KTx.
Then for all |x(0)|<δ and for all j=1,2,…,K, we have (41)|x(t)μ(tj)|≤|x(0)|er‖Aj‖Tx≤|x(0)|er‖A‖KTx≤ϵ0,
Hence there exists a positive constant δ:=ϵ0e-r∥A∥KTx, and the solutions of ẋ=∑i=1rhiAix with |x(0)|<δ stay in the intersection of this ϵ0 with the region {x:q(x)=q(x/Ω)=q(x/Ω2)=⋯=q(x/ΩK-1)}=0 for all t∈[0,KTx]. Therefore, these solutions satisfy |x(t)|≤ϵ for all t≥0.


## 4. Numerical Example

In this section, we consider the following nonlinear system:(42)ẋ1=ax1(t)+bx2(t),ẋ2=cx2(t)+dh(x1(t))x2(t)+u(t),
where a,b,c,d are constants, u(t) is control input, and(43)h(x1(t))={sinx1(t)x1(t),x1(t)≠0,1,x1(t)=0.
It follows that the nonlinear system can be represented by the following T-S fuzzy model.(i)
If-then rule: ifx1(t) is F1, then ẋ=A1x(t)+B1u(t); IF x1(t) is F2, then ẋ=A2x(t)+B2u(t), where(44)x(t)=[x1(t)x2(t)],A1=[ab0c+d],A2=[ab0c],B1=B2=[01].
Moreover, the F1 and F2 are fuzzy sets defined as F1(x(t))=h(x1(t)) and F2(x(t))=1-h(x1(t)).For the simplicity of simulation, the quantizer is chosen to be logarithmic, which satisfies general quantizer (9), see [5, 9, 10]. That is to say, we choose the quantization level to be described as(45)Uρ={±ui,ui=ρiu0,i=1,2,…}∪{ρiu0}∪{0},
and the associated quantizer q(·) is defined as follows:(46)q(z)={ui,1+ρ2ui<z≤1+ρ2ρui,z>0,0,z=0,-q(-z),z<0.
Thus, the corresponding fuzzy quantized controller can be chosen as(47)û=q(z)=q[∑i=12hi(x(t))L̂ix(t)].
Now define the quantization error by(48)e(z)=q(z)-z=Φz.
Therefore, û(t) can be expressed as(49)û=q(z)=(1+Φ)q[∑i=12hi(x(t))L̂ix(t)],
where Φ∈[-δ,δ]. Thus the above closed-loop system with quantized control law can be written as follows(50)ẋ(t)=∑i,j=12hihj[Ai+(1+Φ)BiL̂j]x(t).In this paper, the system parameters area=-10, b=2, c=0.2, d=0.1, and quantized parameters are δ=0.4. It can be easily seen that both matrices A1 and A2 are unstable and the corresponding feedback gain matrix and Lyapunov function matrix of the fuzzy system with quantized controller (49) in Lemma 1 are obtained, respectively:(51)L̂1=L̂2=[-0.9865-2.0583]T,P=[0.09900.12160.12161.0403].
Moreover, for the quantized control of system (42), we can obtain M>43.8764Δ from Theorem 1. Then the response of state and control law with quantized control law (49)  is  showed in Figures 1 and 2, respectively, where the initial condition is x0=[1.2,-0.85]T.Figure 1
System state.Figure 2
System control law.


## 5. Conclusions

In this paper, we extend the results (see, [5]) to a class of T-S fuzzy nonlinear systems and obtain the conditions of stabilizing a fuzzy nonlinear system via fuzzy quantized feedback. We present new results on the stabilization of fuzzy nonlinear systems by choosing appropriately quantized strategies and applying the Lyapunov function approach. An example has been given to illustrate the effectiveness of the proposed method.



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*Source: 579871-2011-02-06.xml*