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.
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. Section 2 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.
Let z∈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:|z|≤M⟹|q(z)-z|≤Δ,|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]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.
If-then form:
Ri: IF x1(t) is Mi1, x2(t) is Mi2… and xn(t) is Min
thenẋ(t)=Aix(t)+Biu(t).
Input-output form:
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.
If-then form:
Ri: if x1(t) is Mi1, x2(t) is Mi2… and xn(t) is Min
thenu(t)=Lix(t),
oru(t)=Liqμ(x).
Input-output form:
u(t)∑i=1r=hiLix(t),
oru(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:ẋ(t)=∑i,j=1rhihj(Ai+BiLj)x(t)=∑i,j=1rhihjHijx(t),
orẋ(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
-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 Assumption 1 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 Assumption 1 holds. an arbitrary σ>0, and M is large enough compared to Δ such that
λmin(P)λmax(P)M>ΘxΔ(1+σ),
where
Θx:=2θ/λ,λ:=min{λ(Qij):i,j=1,2,…,r},θ:=max{‖PBiLj‖:i,j=1,2,…,r}.
Let
E1(μ):={x:xTPx≤λmin(P)M2μ2},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 candidate V(x)=xTPx for the closed-loop system (11) the derivative of V(x) along solutions of (11) is computed as
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
ΘxμΔ(1+σ)≤|x|≤Mμ.
This is also true in the case of x=0, where we set μ=0 as an extreme case and consider the output of the quantizer as zero.
When ΘxμΔ(1+σ)≤|x|≤Mμ holds, we have(xTPx)′≤-|x|λΘxμΔσ.
Claim 1.
Both ℰ1(μ) and ℰ2(μ) are invariant sets of the system (11).
Proof.
we only prove that ℰ1(μ) is an invariant set of the system (11). Assuming x(t0)∈ℰ1(μ), we denote
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
x(τ)TPx(τ)=λmin(P)M2μ2.
By the virtue of condition (13), we have
λmax(P)Θx2Δ2(1+σ)2μ2<λmin(P)M2μ2=x(τ)TPx(τ)≤λmax(P)|x(τ)|2.
Hence, we obtain
ΘxμΔ(1+σ)<[|x(t)|]∣t=τ.
Using (13), we have ℰ2(μ)∈ℰ1(μ) and
[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
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 arbitrary x(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 havexT(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 obtainxT(t0+T)Px(t0+T)≤xT(t0)Px(t0)-TΘx2μ2Δ2σ(1+σ)λ≤λmin(P)M2μ2-TΘx2μ2Δ2σ(1+σ)λ.
If we chooseT:=Tx=λmin(P)M2-λmax(P)Θx2Δ2(1+σ)2Θx2Δ2σ(1+σ)λ,
we have x(t0+Tx)∈ℰ2(μ).
Using Lemma 1 and assuming that the fuzzy control law (9) of the system (5) satisfiesu(t)={0,0≤t<t0,∑i=1rhiqμ(x)[Liqμ(x)],t≥t0,
We have the following theorem 1.
Theorem 1.
Assume that Assumption 1 holds. Assume that M is large enough compared to Δ such that
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. Let u=0. In this case, we rewrite the system (11) for
ẋ(t)=∑i,j=1rhihjqμ(x){Hijx(t)+BiLjμ[q(xμ)-xμ]}=∑i=1rhiAix(t).
Let
A:=argmaxAiji,j∈{1,2,…,r}‖Aij‖
Let μ0=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|x(t)μ(t)|≤λmin(P)λmax(P)M-2Δ.
By condition (1) in Section 2, it is implied
|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
|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 satisfyingx(t0)∈E1(μ(t0)),tj+1=tj+Tx,
and the sequence of positive real numbersμ0=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 lawu(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 equilibrium x=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 |x(t)|≤Mμ(tK-1)=MΩK-1μ(t0)=MΩK-1≤ϵ.
This implies ℰ1(ΩK-1)∈{x:|x|<ϵ}.
By the virtue of q(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 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 |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: 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, andh(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.
If-then rule: if x1(t) is F1, then ẋ=A1x(t)+B1u(t); IF x1(t) is F2, then ẋ=A2x(t)+B2u(t), where
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 asUρ={±ui,ui=ρiu0,i=1,2,…}∪{ρiu0}∪{0},
and the associated quantizer q(·) is defined as follows: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 asû=q(z)=q[∑i=12hi(x(t))L̂ix(t)].
Now define the quantization error bye(z)=q(z)-z=Φz.
Therefore, û(t) can be expressed asû=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 followsẋ(t)=∑i,j=12hihj[Ai+(1+Φ)BiL̂j]x(t).
In this paper, the system parameters are a=-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: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.
System state.
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.
Acknowledgments
The authors are very grateful to all the anonymous reviewers and the editors for their helpful comments and suggestions. This paper was supported by the National Natural Science Foundation of P. R. China under Grant 60874006, Doctoral Foundation of Henan University of Technology under Grant 2009BS048, by the Natural Science Foundation of Henan Province of China under Grant 102300410118, Foundation of Henan Educational Committee under Grant 2011A120003, and Foundation of Henan University of Technology under Grant 09XJC011.
LiberzonD.MorseA. S.Basic problems in stability and design of switched systems199919559702-s2.0-003331118110.1109/37.793443LiberzonD.2003Boston, Mass, USABirkhäuserBrockettR. W.LiberzonD.Quantized feedback stabilization of linear systems2000457127912892-s2.0-003422392510.1109/9.867021IshiiH.FrancisB. A.Stabilizing a linear system by switching control with dwell time20024712196219732-s2.0-003694794110.1109/TAC.2002.805689LiberzonD.Hybrid feedback stabilization of systems with quantized signals2003399154315542-s2.0-004309399910.1016/S0005-1098(03)00151-1IshiiH.FrancisB. A.Quadratic stabilization of sampled-data systems with quantization20033910179318002-s2.0-004251377410.1016/S0005-1098(03)00179-1LiuJ.EliaN.Quantized feedback stabilization of non-linear affine systems20047732392492-s2.0-154236531410.1080/00207170310001655336DelchampsD. F.Stabilizing a linear system with quantized state feedback19903589169242-s2.0-002547177410.1109/9.58500EliaN.MitterS. K.Stabilization of linear systems with limited information2001469138414002-s2.0-003543960810.1109/9.948466FuM.XieL.The sector bound approach to quantized feedback control20055011169817112-s2.0-2864443599410.1109/TAC.2005.858689TatikondaS.MitterS.Control under communication constraints2004497105610682-s2.0-384307426410.1109/TAC.2004.831187De PersisC.On feedback stabilization of nonlinear systems under quantizationProceedings of the 44th IEEE Conference on Decision and Control (CDC '05)December 2005769877032-s2.0-3374604694710.1109/CDC.2005.1583405GaoH.ChenT.A new approach to quantized feedback control systems20084425345422-s2.0-3784901019810.1016/j.automatica.2007.06.015CeragioliF.De PersisC.Discontinuous stabilization of nonlinear systems: quantized and switching controls2007567-84614732-s2.0-3424902926810.1016/j.sysconle.2007.01.001TakagiT.SugenoM.Fuzzy identification of systems and its applications to modeling and control19851511161322-s2.0-0021892282TanakaK.SugenoM.Stability analysis and design of fuzzy control systems19924521351562-s2.0-0026644930TanakaK.IkedaT.WangH. O.Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs1998622502652-s2.0-0032070670WangH. O.TanakaK.GriffinM. F.An approach to fuzzy control of nonlinear systems: stability and design issues19964114232-s2.0-0030082891TuanH. D.ApkarianP.NarikiyoT.YamamotoY.Parameterized linear matrix inequality techniques in fuzzy control system design2001923243322-s2.0-003530279610.1109/91.919253TsengC. S.ChenB. S.UangH. J.Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model2001933813922-s2.0-003536094210.1109/91.928735TeixeiraM. C. M.AssunçãoE.AvellarR. G.On relaxed LMI-based designs for fuzzy regulators and fuzzy observers20031156136232-s2.0-1074421978310.1109/TFUZZ.2003.817840WangR. J.Observer-based fuzzy control of fuzzy time-delay systems with parametric uncertainties200435126716832-s2.0-1064426621610.1080/00207720412351297910LinC.WangQ. G.LeeT. H.HeY.Design of observer-based H∞ control for fuzzy time-delay systems20081625345432-s2.0-4254913264610.1109/TFUZZ.2006.889934HouL.MichelA. N.YeH.Some qualitative properties of sampled-data control systems19974212172117252-s2.0-0031382032