# Analytic Model Predictive Control of Uncertain Nonlinear Systems: A Fuzzy Adaptive Approach

Authors: Xiuyan Peng; Shuli Jia; Xingmei Wang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

A fuzzy adaptive analytic model predictive control method is proposed in this paper for a class of uncertain nonlinear systems. Specifically, invoking the standard results from the Moore-Penrose inverse of matrix, the unmatched problem which exists commonly in input and output dimensions of systems is firstly solved. Then, recurring to analytic model predictive control law, combined with fuzzy adaptive approach, the fuzzy adaptive predictive controller synthesis for the underlying systems is developed. To further reduce the impact of fuzzy approximation error on the system and improve the robustness of the system, the robust compensation term is introduced. It is shown that by applying the fuzzy adaptive analytic model predictive controller the rudder roll stabilization system is ultimately uniformly bounded stabilized in theH-infinity sense. Finally, simulation results demonstrate the effectiveness of the proposed method.


---

## Body

## 1. Introduction

Currently, the study of nonlinear systems mainly includes feedback linearization method, backstepping, forwarding, and passivity method. Backstepping and passivity method apply for nonlinear systems with lower triangular structural features; forwarding method is suitable for systems with upper triangular structural features. These methods for nonlinear systems have a strict requirement in the system form. Feedback linearization method can turn the nonlinear system into a linear system, which is an effective solution to a nonlinear system control. However, analytic model predictive not only eliminates the restrictions of the nonlinear system form but also combines the advantages of predictive control and feedback linearization method; it can be seen as an optimized feedback linearization method applying predictive control thought. Compared with feedback linearization, it improves the control precision of the system. It is a control algorithm based on model optimization, with easy modeling, control performance, robustness, and simple logical structure and other characteristics, and has been widely used in recent years.Literatures [1–3] propose the analytic model predictive control algorithm. In those papers, the predictive control in the literature requires that the controlled object is known, and the system uncertainties impose greater impact on predictive control. Thus, analytic model predictive control usually requires the controlled object to be precisely known. However, due to uncertainties of various disturbances, modelling errors, and so forth, in the actual project, we are often unable to obtain the accurate controlled plant model. Therefore, how to design a good control effect analytic model predictive control law in the case of uncertainties in system is crucial.When the controlled object is known, many approaches are given by literatures [4–11] to solve this problem. But we hope to improve analytic model predictive control algorithm to complete control of the system, because the method can avoid the calculation of online optimization predictive control, thus saving the amount of computation and reducing the complexity of solving the problem. Considering that the fuzzy system has the ability to approximate the unknown nonlinear system function and uncertainties, applying the fuzzy system to control nonlinear uncertain systems has become a hot spot theory and engineering research and made a lot of research results (see [12–18]).For solving the analytic model predictive control law which cannot be determined precisely because of system uncertainty, a fuzzy adaptive analytic model predictive control law is given in this paper based on adaptive fuzzy concept. Using fuzzy system to approximate the uncertainties in the controller, weights of fuzzy systems are based on system feedback error online adjustment, to make fuzzy system approach the unknown functions of controller. Secondly, considering the impacts of the fuzzy modeling errors on the system, the dynamic performance of the system is therefore reduced. A robust compensation term based onH-infinity method is introduced to eliminate this influence. According to Lyapunov stability theory, the closed-loop system ultimately bounded stable is proved.Moreover, in those papers, the solved problem is that the dimensions of output are equal to the dimensions of input. But there were many situations in practical engineering in which the dimensions on input and output of system were not equal. Therefore, Moore-Penrose inverse of matrix is proposed in this paper; it can make the design process of the controller simple.Rudder roll stabilization system as an emerging antirolling method attracts extensive attention domestically and abroad. The ships’ motion environment and their movement characteristics show that it is difficult to get an accurate mathematical model. Model parameters uncertainty, unmodeled dynamics, and other issues increase the difficulty of determining rudder roll stabilization system control algorithm. The proposed method will be applied on rudder roll stabilization system; the simulation results show good effectiveness. The proposed method has important theoretical and practical value.


## 2. Problem for Mulation

A class of nonlinear system is given by(1)x˙=fx+gxu,y=hx,where x∈Rn, u∈Rn1, y∈Rm, fx∈Rn, gx∈Rn×n1, h(x)=h1x⋯hmxT, and n1≤m≤n.If there exist uncertainties inf(x) and g(x), system (1) is a class of uncertain nonlinear systems.The performance index adopted for system (1) is given by(2)J=12∫0T1y^t+τ-ydt+τTy^t+τ-ydt+τdτ,where y^(t+τ) is predictive value of output in the t,t+T1,τ∈0,T1,T1 is the predictive period, and yd(t+τ) is reference signal in the t,t+T1.The following assumptions are imposed on nonlinear system (1) (see [2]).(1)
Each of the system outputsyt has the same relative degree ρ and the zero dynamics are stable.(2)
All states are available.(3)
The outputyt and the reference signal ydt are sufficiently many times continuously differentiable with respect to t.If f(x) and g(x) are known, the control order of u is selected l. Similarly, the higher derivatives of the output y^(t), seeking ρ+l order derivative, are denoted by Y^t:(3)Y^t=y^ty^(1)tMy^(ρ)ty^(ρ+1)tMy(ρ+l)t=hxLfhxMLfρhxLfρ+1hxMLfρ+lhx+0ρ×1Lu-^,where ρ is the relative degree of system (1), l is control order, and u-^=[u^(t),u^˙(t),…,u^[l-ρ](t)]T. Consider(4)Lu-^=LgLfρ-1hxu^tLgLfρ-1hxu^˙t+p11u^t,xt⋮LgLfρ-1hxu^lt+pr1u^t,xt+Lprru^t,u^˙t,L,u^r-1t,xt,where r=l-ρ:(5)p11u^t,xt=LgLfρhxu^t+∂LgLfρ-1hxu^t∂xfx+gxu^t.Within the moving time frame, the output y^(t+τ) and yd(t+τ) at the time τ is approximately predicted by(6)y^t+τ=ΓτY^t,(7)ydt+τ=ΓτYdt,where(8)Γ(τ)=1,τ,…,τρ+lρ+l!,(9)Ydt=ydt,y˙dt,…,ydρ+ltT.Take (7) and (8) into the performance function (2):(10)J=12Y^t-YdtTY^t-Ydt∫0T1ΓτTΓτdτ.So the analytic model predictive control law of system (1) can be given by  [1] (11)Gxu+Fx+KMρ-Ydρ=0,where(12)Fx=Lfρh1x⋮Lfρhmxm×1,Mρ=yx-ydxy˙x-y˙dx⋮Lfρ-1hx-ydρ-1xmρ×1,Gx=Lg1Lfρ-1h1x⋯Lgn1Lfρ-1h1xLg1Lfρ-1h2x⋯Lgn1Lfρ-1h2x⋮⋱⋮Lg1Lfρ-1hmx⋯Lgn1Lfρ-1hmxm×n1.Because of n1≤m, if n1=m, the control law can be improved as follows:(13)u=-Gx-1Fx+KMρ-Ydρ.But if n1<m, Gx-1 does not exist, so (13) does not describe the control law of system (1).In order to deal with this problem, we give a new definition.Definition 1.
WhenG∈Rm×n, G+∈Rn×m; G+ is said to be the Moore-Penrose inverse of G, if G+ can satisfy the Penrose conditions (see [19]):(i)
GG+G=G;(ii)
G+GG+=G+;(iii)
GG+H=GG+;(iv)
G+GH=G+G,
where·H means conjugate transpose of matrix.
So the control law can be improved as follows whenn1<m:(14)u=-Gx+Fx+KMρ-Ydρ,where Gx+∈Rn1×m is defined as Moore-Penrose inverse of Gx and formula (14) is the minimum norm least-squares solution of formula (11) (see [19], for minimal norm least-squares solution).Remark 2.
If the Moore-Penrose inverse ofGx is used for formula (14), it can solve the problems of nonlinear system that the dimension of input and output cannot be necessarily equal. This is an expansion [1] in this paper, and it is more favorable for engineering applications.Remark 3.
When the Moore-Penrose inverse is used for formula (14), minimal norm least-squares solution for formula (11) can be obtained; it means that formula (14) is the minimum norm solution of (11), when -Gxu=Fx+KMρ-Ydρ is compatibility linear equation; if -Gxu=Fx+KMρ-Ydρ is incompatible linear equation, formula (14) is the least-squares solution of formula (11).K
∈
R
m
×
m
ρ are elements of the front m row of the matrix Γll-1ΓρlT:(15)Γll=Γ(ρ+1,ρ+1)…Γ(ρ+1,ρ+1)⋮⋱⋮Γ(ρ+l+1,ρ+1)⋯Γ(ρ+l+1,ρ+l+1)(l+1)×(l+1),Γρl=Γ(1,ρ+1)…Γ(1,ρ+l+1)⋮⋱⋮Γ(ρ,ρ+1)⋯Γ(ρ,ρ+l+1)ρ×(l+1),(16)Γ(i,j)=T1i+j-1(i-1)!(j-1)!(i+j-1),i,j=1,…,ρ+l+1.By the analytic model predictive control law (14) and its calculation, in order to get the control law of system (1), the f(x) and g(x) are known; otherwise F(x) and G(x) of predictive control law equation (14) cannot be accurately determined. When f(x) and g(x) are unknown, the controller will be unable to accurately calculate, so control performance will be affected. To solve this problem, we propose a fuzzy adaptive analytical model predictive controller.


## 3. Controller Design

### 3.1. Fuzzy Adaptive Analytical Model Predictive Controller Design

Fuzzy controller uses IF-THEN rules of fuzzy logic system, which uses a single point of obfuscation, product inference, and center of gravity defuzzification.i-rule is as follows.ri: if x1 is A1i,   x2 is A2i, and … and  xn is Ani, then y is θi, i=1,…,L, where L is the number of rules for fuzzy systems.Define the output of fuzzy system:(17)y=θTξx,where θ=[θ1,…,θr] is the weight coefficient that can be adjusted and ξx=ξ1x⋯ξLx is fuzzy basis function:(18)ξix=Πj=1nμajixj∑i=1rΠj=1nμajixj,where μajixj is the membership function of the fuzzy system.If the controlled object exists uncertainty or unknown function,F(x) and G(x) are unknown; then the predictive control law equation (14) cannot be accurately determined, so we propose using F^x∣θF and G^x∣θG of fuzzy system to approach F(x) and G(x), respectively:(19)F^x∣θF=θFTξFx,(20)G^x∣θG=θGTξGx,where ξFx∈RL×1 and ξGx∈RL×n1.Assumption 4.
There exist optimal parameter vectorsθF∗ and θG∗ of θF and θG, which can make the output of fuzzy system approach F(x) and G(x) arbitrary precision:(21)θF∗=argminθF∈MθF⁡sup⁡x∈Mx⁡Fx-F^x∣θF,θG∗=argminθG∈MθG⁡sup⁡x∈Mx⁡Gx-G^x∣θG.Define the approximation error of fuzzy system:(22)ω=F^x∣θF-F^x∣θF∗+G^x∣θG-G^x∣θG∗u.The equivalent fuzzy predictive controller of (14) is given by(23)up=-G^+x∣θgF^x∣θf+KMρ-Ydρ,where G^+x∣θg is Moore-Penrose inverse of G^x∣θg.In order to overcome the impact of the approximation errorω on system, it is necessary to introduce a robust compensation term ua to improve the performance of the whole system.So the new control law is given by(24)uc=-G^+x∣θgF^x∣θf+KMρ-Ydρ+ua,(25)ua=λaBTPe⌢,where the matrices P, B, and e⌢ and parameter λa are given in Section 3.2.The adaptive law ofθF, θG is given by(26)θ˙F=λFξFxe⌢TPB,θ˙G=λGξGxuce⌢TPB,where λF,λG are adaptive parameters of fuzzy system.


### 3.2. Stability Analysis and Proof

Theorem 5.
If uncertain nonlinear system (1) chooses control law (24) and the adaptive law of formula (26), it can be satisfied:(i)
closed-loop system is uniformly ultimately bounded;(ii)
for a given inhibition levelγ>0, output tracking error can achieve H∞ performance:(27)12∫0Te⌢TQe⌢dt≤12e⌢T0Pe⌢+12tr⁡θ~FT0λF-1θ~F0+12tr⁡θ~GT0λG-1θ~G0+12γ2∫0Tω2.Proof.
y
^
ρ of formula (4) is given by(28)y^ρ=Lfρhx+LgLfρ-1hxu^t=Fx+Gxu^t=Fx-F^x∣θF+Gx-G^x∣θGu^t+F^x∣θF+G^x∣θGu^t.Take (24) into (28):(29)y^ρ=Fx-F^x∣θF+Gx-G^x∣θGuct+F^x∣θF+G^x∣θGuct.Take (24) into (29):(30)y^ρ=Fx-F^x∣θF+Gx-G^x∣θGuct-KMρ+Ydρ-ua.It can be also written as(31)KMρ+y^ρ-Ydρ=Fx-F^x∣θF+Gx-G^x∣θGuct-ua,where Mρ=yx-ydxy˙x-y˙dx⋮Lfρ-1hx-ydρ-1x.
So error equation is obtained that theith error equation is given by(32)eiρ+ki,ρ-1eiρ-1+⋯+ki,0ei=Fix-F^ix∣θF+Gix-G^ix∣θGuc-uai,where ki,j is the ith row elements of the matrix Ki=1,…,m and the ki,j is determined by the predictive period T1, the relative degree ρ, and control order l.
So(33)he=eρt+ki,ρ-1eρ-1t+⋯+ki,0et.If we choose ki,j reasonable, it can make formula (33) as Hurwitz Polynomial.
Formula (33) can be also written as(34)e⌢˙=Ae⌢+Bθ~FTξFx+θ~GTξGxuc+ω-ua,where(35)e⌢=e⌢1⋯e⌢mT∈Rmρ,e⌢i=ei⋯eiρ-1T,A=diagAi,…,Am∈Rmρ×mρ,B=diagBi,…,Bm∈Rmρ×m,Ai=010⋯0001⋯0⋮⋮⋮⋱⋮-ki,0-ki,1-ki,2⋯-ki,ρ-1,Bi=00⋯1Ti=1,…,m,θ~F=θF∗-θF,θ~G=θG∗-θG.Lyapunov function is constructed as follows:(36)V=V1+V2+V3,V1=12e⌢TPe⌢,V2=12tr⁡θ~FTλF-1θ~F,V3=12tr⁡θ~GTλG-1θ~G,where tr⁡ represents trace of matrix.
P
=
diag
(
P
1
,
…
,
P
m
)
∈
R
m
ρ
×
m
ρ, Pi is positive symmetric matrix, and it satisfies the positive definite solution of Riccati equation: (37)PiAi+AiTPi+Qi-2λi-1γ2PiBiBiTPi=0,Qi=QiT>0,where Qi is design parameter.
V is derivative along the trajectory of the system:(38)V˙=V˙1+V˙2+V˙3,where M=B(θ~FTξF(x)+θ~GTξG(x)uc+ω-ua) and(39)V˙1=12e⌢˙TPe⌢+12e⌢TPe⌢˙=12e⌢TATPe⌢+MTPe⌢+12e⌢TPAe⌢+12e⌢TPM=12e⌢TATP+PAe⌢+e⌢TPM=-12e⌢TQe⌢+12e⌢T2λ-1γ2PBBTPe⌢+e⌢TPBθ~FTξFx+θ~GTξGxu+ω-ua,V˙2=λF-1tr⁡(θ~FTθ~˙F)=-λF-1tr⁡θ~FTθ˙F,V˙3=λG-1tr⁡(θ~GTθ~˙G)=-λG-1tr⁡θ~GTθ˙G.So(40)V˙=-12e⌢TQe⌢+e⌢TPBθ~FTξF(x)+θ~GTξG(x)uc+ω-ua-λF-1tr⁡(θ~FTθ˙F)-λG-1tr⁡(θ~GTθ˙G)=-12e⌢TQe⌢+e⌢TPBθ~FTξFx-λF-1tr⁡θ~FTθ˙F+e⌢TPBθ~GTξGxuc-λG-1tr⁡θ~GTθ˙G+12e⌢T2λ-1γ2PBBTPe⌢+e⌢TPBω-ua.
Because ofθ˙F=λFξF(x)e⌢TPB,θ˙G=λGξG(x)uce⌢TPB, ua=λaBTPe⌢, and we choose λa=1/λ.
Now it can obtain(41)V˙=-12e⌢TQe⌢+e⌢TPBω-12γ2e⌢TPBBTPe⌢+e⌢TPBω=-12e⌢TQe⌢-121γe⌢TPB-γω2+12γ2ω2≤-12e⌢TQe⌢+12γ2ω2≤-12λmin⁡(Q)e⌢2+12γ2ω-2,where λmin⁡(Q) is the smallest eigenvalues of matrix.
It can be known from (41), if(42)e≥γωλmin⁡Q,V˙≤0.So the closed-loop system is uniformly ultimately bounded.
The integration of (41) from t=0 to t=T is given by(43)12∫0Te⌢TQe⌢dt≤V0-VT+12γ2∫0Tω2.Because of VT≥0,(44)12∫0Te⌢TQe⌢dt≤V0+12γ2∫0Tω2=12e⌢T(0)Pe⌢+12tr⁡θ~FT0λF-1θ~F0+12tr⁡θ~GT0λG-1θ~G0+12γ2∫0Tω2.So output tracking error can achieve H∞ performance (27).


## 3.1. Fuzzy Adaptive Analytical Model Predictive Controller Design

Fuzzy controller uses IF-THEN rules of fuzzy logic system, which uses a single point of obfuscation, product inference, and center of gravity defuzzification.i-rule is as follows.ri: if x1 is A1i,   x2 is A2i, and … and  xn is Ani, then y is θi, i=1,…,L, where L is the number of rules for fuzzy systems.Define the output of fuzzy system:(17)y=θTξx,where θ=[θ1,…,θr] is the weight coefficient that can be adjusted and ξx=ξ1x⋯ξLx is fuzzy basis function:(18)ξix=Πj=1nμajixj∑i=1rΠj=1nμajixj,where μajixj is the membership function of the fuzzy system.If the controlled object exists uncertainty or unknown function,F(x) and G(x) are unknown; then the predictive control law equation (14) cannot be accurately determined, so we propose using F^x∣θF and G^x∣θG of fuzzy system to approach F(x) and G(x), respectively:(19)F^x∣θF=θFTξFx,(20)G^x∣θG=θGTξGx,where ξFx∈RL×1 and ξGx∈RL×n1.Assumption 4.
There exist optimal parameter vectorsθF∗ and θG∗ of θF and θG, which can make the output of fuzzy system approach F(x) and G(x) arbitrary precision:(21)θF∗=argminθF∈MθF⁡sup⁡x∈Mx⁡Fx-F^x∣θF,θG∗=argminθG∈MθG⁡sup⁡x∈Mx⁡Gx-G^x∣θG.Define the approximation error of fuzzy system:(22)ω=F^x∣θF-F^x∣θF∗+G^x∣θG-G^x∣θG∗u.The equivalent fuzzy predictive controller of (14) is given by(23)up=-G^+x∣θgF^x∣θf+KMρ-Ydρ,where G^+x∣θg is Moore-Penrose inverse of G^x∣θg.In order to overcome the impact of the approximation errorω on system, it is necessary to introduce a robust compensation term ua to improve the performance of the whole system.So the new control law is given by(24)uc=-G^+x∣θgF^x∣θf+KMρ-Ydρ+ua,(25)ua=λaBTPe⌢,where the matrices P, B, and e⌢ and parameter λa are given in Section 3.2.The adaptive law ofθF, θG is given by(26)θ˙F=λFξFxe⌢TPB,θ˙G=λGξGxuce⌢TPB,where λF,λG are adaptive parameters of fuzzy system.


## 3.2. Stability Analysis and Proof

Theorem 5.
If uncertain nonlinear system (1) chooses control law (24) and the adaptive law of formula (26), it can be satisfied:(i)
closed-loop system is uniformly ultimately bounded;(ii)
for a given inhibition levelγ>0, output tracking error can achieve H∞ performance:(27)12∫0Te⌢TQe⌢dt≤12e⌢T0Pe⌢+12tr⁡θ~FT0λF-1θ~F0+12tr⁡θ~GT0λG-1θ~G0+12γ2∫0Tω2.Proof.
y
^
ρ of formula (4) is given by(28)y^ρ=Lfρhx+LgLfρ-1hxu^t=Fx+Gxu^t=Fx-F^x∣θF+Gx-G^x∣θGu^t+F^x∣θF+G^x∣θGu^t.Take (24) into (28):(29)y^ρ=Fx-F^x∣θF+Gx-G^x∣θGuct+F^x∣θF+G^x∣θGuct.Take (24) into (29):(30)y^ρ=Fx-F^x∣θF+Gx-G^x∣θGuct-KMρ+Ydρ-ua.It can be also written as(31)KMρ+y^ρ-Ydρ=Fx-F^x∣θF+Gx-G^x∣θGuct-ua,where Mρ=yx-ydxy˙x-y˙dx⋮Lfρ-1hx-ydρ-1x.
So error equation is obtained that theith error equation is given by(32)eiρ+ki,ρ-1eiρ-1+⋯+ki,0ei=Fix-F^ix∣θF+Gix-G^ix∣θGuc-uai,where ki,j is the ith row elements of the matrix Ki=1,…,m and the ki,j is determined by the predictive period T1, the relative degree ρ, and control order l.
So(33)he=eρt+ki,ρ-1eρ-1t+⋯+ki,0et.If we choose ki,j reasonable, it can make formula (33) as Hurwitz Polynomial.
Formula (33) can be also written as(34)e⌢˙=Ae⌢+Bθ~FTξFx+θ~GTξGxuc+ω-ua,where(35)e⌢=e⌢1⋯e⌢mT∈Rmρ,e⌢i=ei⋯eiρ-1T,A=diagAi,…,Am∈Rmρ×mρ,B=diagBi,…,Bm∈Rmρ×m,Ai=010⋯0001⋯0⋮⋮⋮⋱⋮-ki,0-ki,1-ki,2⋯-ki,ρ-1,Bi=00⋯1Ti=1,…,m,θ~F=θF∗-θF,θ~G=θG∗-θG.Lyapunov function is constructed as follows:(36)V=V1+V2+V3,V1=12e⌢TPe⌢,V2=12tr⁡θ~FTλF-1θ~F,V3=12tr⁡θ~GTλG-1θ~G,where tr⁡ represents trace of matrix.
P
=
diag
(
P
1
,
…
,
P
m
)
∈
R
m
ρ
×
m
ρ, Pi is positive symmetric matrix, and it satisfies the positive definite solution of Riccati equation: (37)PiAi+AiTPi+Qi-2λi-1γ2PiBiBiTPi=0,Qi=QiT>0,where Qi is design parameter.
V is derivative along the trajectory of the system:(38)V˙=V˙1+V˙2+V˙3,where M=B(θ~FTξF(x)+θ~GTξG(x)uc+ω-ua) and(39)V˙1=12e⌢˙TPe⌢+12e⌢TPe⌢˙=12e⌢TATPe⌢+MTPe⌢+12e⌢TPAe⌢+12e⌢TPM=12e⌢TATP+PAe⌢+e⌢TPM=-12e⌢TQe⌢+12e⌢T2λ-1γ2PBBTPe⌢+e⌢TPBθ~FTξFx+θ~GTξGxu+ω-ua,V˙2=λF-1tr⁡(θ~FTθ~˙F)=-λF-1tr⁡θ~FTθ˙F,V˙3=λG-1tr⁡(θ~GTθ~˙G)=-λG-1tr⁡θ~GTθ˙G.So(40)V˙=-12e⌢TQe⌢+e⌢TPBθ~FTξF(x)+θ~GTξG(x)uc+ω-ua-λF-1tr⁡(θ~FTθ˙F)-λG-1tr⁡(θ~GTθ˙G)=-12e⌢TQe⌢+e⌢TPBθ~FTξFx-λF-1tr⁡θ~FTθ˙F+e⌢TPBθ~GTξGxuc-λG-1tr⁡θ~GTθ˙G+12e⌢T2λ-1γ2PBBTPe⌢+e⌢TPBω-ua.
Because ofθ˙F=λFξF(x)e⌢TPB,θ˙G=λGξG(x)uce⌢TPB, ua=λaBTPe⌢, and we choose λa=1/λ.
Now it can obtain(41)V˙=-12e⌢TQe⌢+e⌢TPBω-12γ2e⌢TPBBTPe⌢+e⌢TPBω=-12e⌢TQe⌢-121γe⌢TPB-γω2+12γ2ω2≤-12e⌢TQe⌢+12γ2ω2≤-12λmin⁡(Q)e⌢2+12γ2ω-2,where λmin⁡(Q) is the smallest eigenvalues of matrix.
It can be known from (41), if(42)e≥γωλmin⁡Q,V˙≤0.So the closed-loop system is uniformly ultimately bounded.
The integration of (41) from t=0 to t=T is given by(43)12∫0Te⌢TQe⌢dt≤V0-VT+12γ2∫0Tω2.Because of VT≥0,(44)12∫0Te⌢TQe⌢dt≤V0+12γ2∫0Tω2=12e⌢T(0)Pe⌢+12tr⁡θ~FT0λF-1θ~F0+12tr⁡θ~GT0λG-1θ~G0+12γ2∫0Tω2.So output tracking error can achieve H∞ performance (27).


## 4. Simulation Analysis

In order to verify the control performance of the controller and engineering applications, apply it to rudder roll stabilization system. This paper is a transport ship (see [20]). The controlled object model is given by(45)x˙=fx+Δfx+gx+Δgxu+d,y=hx,where Δfx and Δgx are unknown functions that are caused by changes of the speed and metacentric height; generally, these variations cannot be accurately determined. d is wave disturbance and its calculation is given by literature (see [20]).If there are no unknown functions, it can be known:(46)x˙=fx+gxu+d,y=hx,where(47)x=x1,x2,x3,x4,x5T=v,p,r,φ,ψT,y=y1,y2T,fx=f1x,f2x,f3x,f4x,f5xT,gx=b1,b2,b3,0,0T,f1x=a11x1+a12x3+a13x1x1+a14x1x3,f2x=a21x2+a22x4+a23x43,f3x=a31x1+a32x3+a33x1x3,f4x=x2;f5x=x3,h1x=x4;h2x=x5,a11=-0.0833,a12=-1.6355,a13=-0.0215,a14=-0.6048,a2=-0.0763,a22=-0.3588,a23=0.7363;a31=-0.0028,a32=-0.2706,a33=-0.3091;b1=-0.2121,b2=0.0182,b3=0.0166,where p,r,φ,ψ are roll velocity, yaw velocity, yaw, and roll angle, respectively.The initial speed is10m/s; assume that aij have perturbation of 15% and bi have perturbation of 20%. In the simulation, simulate the variations Δfx and Δgx using a set of random numbers within the given range.It can be calculated by the known conditions:(48)Lfh1x=0010f=x1,Lfh2x=0001f=x2,Lf2h1x=1000f=a22x1+a24x3+a29x33,Lf2h2x=0100f=a32x2+a39x22,LgLfh1x=1000g=b1,LgLfh1x=0100g=b2.From (48), we know that ρ1=ρ2=2, so ρ=4; if we choose yd=020/57.3, T1=60s, and l=1, the K is calculated:(49)K=0.050.320.901.33-0.03-0.22-0.55-0.61.We use the following fuzzy linguistic variables:Aj1 (negative large), Aj2 (negative middle), Aj3 (negative small), Aj4 (zero), Aj5 (positive small), Aj6 (positive middle), and Aj7 (positive large).The membership functions are given by(50)μajixj=exp⁡-xj-μi2,where μi=-0.5,-0.3,-0.1,0,0.1,0.3,0.5, i=1,…,7, j=1,2. This paper uses the seven fuzzy rules to approximate each variable of Fx and Gx. The rules are as follows:rL: if x1 is A1i, x2 is A2i, and … and xn is Ani, then y is FL, i=1,…,7.Remark 6.
Because of single input and multiple outputs for system (45) in this paper, we choose ξFx=ξGx∈RL×1. The inverses of Gx and G^x∣θG are used Moore-Penrose inverse.Initial weights values of fuzzy systems are given by(51)θF0=0,θg0=0,Q=3I8×8,λF=10,λG=2,λ=1,γ=5.So λa=1.The following simulation conditions and analysis of the simulation results to verify the effectiveness of the proposed method are given.Condition 1.
In system with the wave disturbance, wave periodTw=8s, significant wave height h1/3=4m, and relative damping coefficient ζ=0.3, encounter angle is 135∘, a reference yaw angle is 20∘, and wave disturbance calculation is given by [14].(i)
If there is no uncertain term in system (45), the control law is only analytic model predictive control for system simulation; the simulation results are shown in Figure 1.(ii)
If there are uncertain terms in system (45), the control law is only analytic model predictive control for system simulation; the simulation results are shown in Figure 2.(iii)
If there are uncertain terms in system (45), the control law is fuzzy adaptive analytic model predictive control for system simulation; the simulation results are shown in Figure 3.Figure 1
Analytic model predictive control on the accurate model.Figure 2
Analytic model predictive control on the uncertain model.Figure 3
Fuzzy adaptive analytic model predictive control on the uncertain model.Figures1(a), 2(a), and 3(a) are output of roll angle; the dotted line is the ship rolling motion without adding the controller, and the solid line is the rolling motion using controller. Figures 1(b), 2(b), and 3(b) are the output of yaw angle.In order to compare the effect of roll stabilization under different conditions, the roll reduction rate (RRR) is given by literature [12]: (52)RRR%=AP-RCSAP×100%,where RCS and AP are the standard deviation of roll angle with and without the roll damping system. Assume that CP is standard deviation of yaw angle.The roll reduction rate and standard deviation of yaw angle in each case can be seen in Table1.Table 1
Roll reduction rate and standard deviation of yaw whenh1/3=4 m.
Parameters
AP
RCS
RRR
CP
i
4.8626
1.8355
62.25%
3.1008
ii
4.8682
2.3629
51.46%
3.2504
iii
4.8682
1.9163
58.58%
3.1723It is apparent that it achieves better tracking performance on fuzzy adaptive analytic model predictive control law and the roll reduction rate achieves a satisfactory result when there are uncertain terms in system (45).Condition 2.
In a system with the wave disturbance, wave periodTw=8.5s, significant wave height h1/3=5.5m, relative damping coefficient ζ=0.3, encounter angle is 135∘, and a reference yaw angle is 20°.(i)
If there is no uncertain term in system (45), the control law is only analytic model predictive control for system simulation; the simulation results are shown in Figure 4.(ii)
If there are uncertain terms in system (45), the control law is only analytic model predictive control for system simulation; the simulation results are shown in Figure 5.(iii)
If there are uncertain terms in system (45), the control law is fuzzy adaptive analytic model predictive control for system simulation; the simulation results are shown in Figure 6.The roll reduction rate and standard deviation of yaw angle in each case can be seen in Table 2. As can be seen from Figure 5 when system (45) is uncertain, in dealing with wave disturbance strengthened, if we use only analytic model predictive control law, it will make the roll reduction rate significantly lower and part of roll angle will be increased. Figure 6 and Table 2 show that the proposed method in this paper has strong robustness.Table 2
Roll reduction rate and standard deviation of yaw whenh1/3=5.5 m.
Parameters
AP
RCS
RRR
CP
i
7.4459
2.9333
60.61%
3.1643
ii
7.4459
3.6529
50.94%
3.3310
iii
7.4459
3.1594
57.57%
3.2635Figure 4
Analytic model predictive control on the accurate model.Figure 5
Analytic model predictive control on the uncertain model.Figure 6
Fuzzy adaptive analytic model predictive control on the uncertain model.The simulation results under Condition2 are as shown in Figures 4, 5, and 6 and Table 2.


## 5. Conclusions

Robustness and control effect are important factors to measure the control quality of a nonlinear system. In the presence of uncertainty, the analytic model predictive control performance will be reduced significantly, and system cannot meet the robustness requirements. Using fuzzy system to approximate the uncertainties in the controller, weights of fuzzy systems are based on system feedback error online adjustment, to make fuzzy system approach the unknown functions of controller. Utilizing Taylor equation, the fuzzy adaptive predictive control law is achieved, and since online optimization is not required, the huge calculation burden of predictive control can be avoided. A robust compensation term is introduced to eliminate this influence which is the impact of the fuzzy modeling errors on the system.Finally, we apply it to the rudder roll stabilization system control, which is the ship motion model with four freedom degrees. Simulation results show that the proposed algorithm has better control effect and robustness.



---

*Source: 760490-2015-10-01.xml*