• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Probabilistic Robust Linear Parameter-varying Control of a Small Helicopter Using Iterative Scenario Approach

    2015-08-11 11:57:35ZhouFangHuaTianandPingLi
    IEEE/CAA Journal of Automatica Sinica 2015年1期

    Zhou Fang,Hua Tian,and Ping Li

    Probabilistic Robust Linear Parameter-varying Control of a Small Helicopter Using Iterative Scenario Approach

    Zhou Fang,Hua Tian,and Ping Li

    —In this paper,we present an iterative scenario approach(ISA)to design robust controllers for complex linear parameter-varying(LPV)systems with uncertainties.The robust controller synthesis problem is transformed to a scenario design problem,with the scenarios generated by identically extracting random samples on both uncertainty parameters and scheduling parameters.An iterative scheme based on the maximum volume ellipsoid cutting-plane method is used to solve the problem. Heuristic logic based on relevance ratio ranking is used to prune the redundant constraints,and thus,to improve the numerical stability of the algorithm.And further,a batching technique is presented to remarkably enhance the computational efficiency. The proposed method is applied to design an output-feedback controller for a small helicopter.Multiple uncertain physical parameters are considered,and simulation studies show that the closed-loop performance is quite good in both aspects of modeltracking and dynamic decoupling.For robust LPV control problems,the proposed method is more computationally efficient than the popular stochastic ellipsoid methods.

    Index Terms—Probabilistic robust control(PRC),linear parameter-varying(LPV)control,scenario approach(SA),iterative algorithm,small helicopter.

    I.INTRODUCTION

    G AIN scheduling based on linear parameter-varying (LPV)systems has been a popular method for nonlinear controlsystem design in the pastdecade[1].Using linearization techniques,the inherently nonlinear system is described as a set of linear models,which are driven by several so-called scheduling parameters that can be measured online or a-priori known.Then based on these LPV models,the controller is synthesized in a linear-like fashion.

    For LPV systems with uncertainties,controller synthesis to achieve optimal robustness is a challenging problem,as both scheduling parameters and uncertainties can enter the model in arbitrary ways.Several robust techniques were applied to uncertain LPV problems,e.g.,μ-synthesis[2?3]and Lyapunovbased methods[4?6],allof which involved solving some linearmatrix inequality(LMI)based convex programs.The parameter dependence of the LMIs is not easy to handle,one natural way was to grid the parameter set[4],which,although is a direct way,cannot guarantee the feasibility of solution on the entire parameter set.One way bypassing gridding was to restrict the dependences on either uncertainties or scheduling parameters to some specific classes,e.g.,affine structure[5], and linear fractional transformations[2?3,6],making problems more tractable while bringing about some conservativeness.

    Probabilistic robust control(PRC)alleviates the drawbacks of the methods mentioned above by shifting the usual deterministic sense ofrobustness to a probabilistic one.A controller that is robust for most of uncertainty samples is satisfactory, which means a risk of instability is acceptable if it is very small.This evolutionary concept makes many robust design problems easier to compute.Moreover,PRC methods can handle very general uncertainty structures compared to the existing deterministic approaches,and the conservativeness of results is controllable.

    SA,developed by Calafiore and Campi[11?12],gives a one-shot solution to a convex optimization problem with constraints generated through uncertainty randomization.The number of samples that ought to be made in the randomization procedure is a first-order polynomial to the number of design variables[13].So when facing high-dimensional design problems,such as robust semidefinite programming,the computational load of SA is very heavy,and even the computing resources would be used up.This is why SA is more valuable in theory than in practice.

    The main contribution ofthis paperis three-fold.Firstly,wepropose a batch iterative algorithm for SA,which demonstrates a good solution performance as the SE methods for LPV controller design,while being more efficient.Secondly,in the probabilistic robust LPV control context,using randomization technique,we provide a unified and concise way to cope with the parameter dependency of the LMI constraints,thus remarkably reducing the number of constraints needed for optimization computation,compared to the commonly-used gridding approach[1].Thirdly,the proposed method is applied to design a robust LPV controller for a small helicopter.To the bestof our knowledge,small helicopters would be one of the most difficult aerial vehicles to control.The state of art has explored various control synthesis methods,focusing on the nonlinearand robustdesign issues(see,forexample,Wang et al.[14],Liu et al.[15],Cai etal.[16],and Simpl′?cio et al.[17]). Both problems of robustness and nonlinearity are dealt with simultaneously in this paper,and such a challenging task is a true touchstone to the proposed controllersynthesis algorithm.

    The paper is organized as follows.In Section II,the scenario design problem is introduced,where two iterative algorithms,including the batch one,are proposed.Section III gives a sufficient condition for the existence of a solution for a robust output-feedback LPV control problem,where the controller formulation is presented,and computational issues associated with iterative scenario approach(ISA)are discussed.In Section IV,using the batch algorithm developed in previous sections,a robust output-feedback controller is designed for a small helicopter with uncertainties in multiple physical parameters,and performance is verifi ed through both comparisons and nonlinear simulations.Conclusion is made in Section V.

    II.ISA

    A.Standard SA

    A generalrobustconvex program(RCP)takes the following form:

    whereΔ?Rnδis the uncertainty set and?(θ,δ)is convex inθ∈ Θ ? Rnθ.Despite the convexity assumption on both objective and constraints,RCP is in general an NP-hard problem due to the possible infinity of constraints.

    SA looses the constraint satisfaction from “all”to“almost”by applying the so-called randomization technique on the uncertainty parameters.Assum n e that N independe o nt identically distributed samples ofδ,named“scenarios”,are extracted according to some probability Pr(δ∈Δ),then a scenario convex program(SCP)is constructed as

    Solution to problem(2)guarantees the robustness in a probabilistic sense,which is stated by the following theorem.

    Theorem 1[13].Fix two real numbersε∈ (0,1)(level parameter)andβ∈(0,1)(confidence level).If then with probability of no smaller than 1?β,one of the following two things happens:1)SCPNis unfeasible,and then(1)is unfeasible;2)SCPNis feasible and then its optimal solution?θNisε-level robustly feasible,which means Pr{δ∈Δ:?(θ,δ)>0}≤ε.Although a lower bound of the constraint number has already been given,the computational load of a one-shot solution to SCPNis still too heavy.For example,if we setε=0.05,β=10?5,nθ=157,as seen later in Section IV,then 6 742 constraints are generated according to(3).This solving scale exceeds the capability of most current convex optimization softwares.

    B.ISA

    Inspired by Wada's work[18],we present here an iterative scheme based on the maximum volume ellipsoid(MVE)cutting plane method,which is known to be particularly efficient for large scale convex optimization problems.

    Firstly,we introduce the MVE within a polytope.It is assumed that the parameterθ's space is an nθ-dimensional Euclidean space.Then,an ellipsoid in the parameter space is defined as

    where Q=QT∈Rnθ×nθ+defines the ellipsoid's shape, θcis the ellipsoid center.The volume of the ellipsoid is given by vol(E(Q,θc))=det(Q).Given a convex polytope, P={θ∈Rn:aaaTiθ≤bi,i=1,···,L}the MVE within a polytope P is defined by

    It is easily seen that the MVE can be found by solving the following optimization problem:

    To facilitate the plane-cutting procedure,we limitthe search space to a hypercube that is known a priori.

    Assumption 1.Letθ1be the initial guess of parameter θ,and define rrrm=[r1r2···rnθ]T,then the hypercube Θ1? Rnθ,which is assumed to contain the optimaθ?of problem(11),is given as

    where ri> 0,and eeei∈Rnis the unit vector with its i th element being 1.

    Then a basic iterative algorithm to solve problem(2)is presented as follows.

    Algorithm 1.

    Inputs:θ1;rrrm;{δ(k)}Nk=1;τ.

    Initialize:l=1,A1=[IIIn?IIIn]T, B1=[θ1+rrrm?θ1+rrrm]T,{?θj}=NULL,j=0 where j counts the feasible solutions.The initial polytope,which is a hypercube,is defined as

    1)Perform oracle on the N scenarios.

    If?(θl,δ(k))≤ 0 for all k∈{1,2,···,N},then a feasible solution is obtained,and we set

    else set

    2)Update.Add a new cutting plane to Pl:

    find the MVE El+1in the new polytope:

    The algorithm presented above consists of two iteration loops.The inner is for feasibility verification,while the outeris for update.Notice that the number of linear inequalities that comprise Plincreases by one for each outer loop iteration. Consequently,the growing computational burden would cause numerical instability,especially for some high dimensional cases.Therefore a mechanism to remove the redundant ones from the constraint set is quite necessary.In this paper, a heuristic constraint-pruning logic,suggested by[19],is adopted.We fix the number of linear inequalities in Plto be an integer between 3nθand 5nθtypically,and then rank the constraints according to the relevance ratio,defined by

    where F=nθQl+1defines the shape of an ellipsoid centered atθc,which is supposed to cover the polyhedron Pl+1.And then we substitute the least relevant inequality(the inequality with the largest relevance ratio)with the new one.Thus(5) of the update procedure in Algorithm 1 is modified as

    where aaanis the n th row vector of Al+1,and bnis the n th element of Bl+1.In this paper,we limit the total number of inequalities to be 3nθ.

    The convergence of the algorithm is guaranteed by the characteristic of the MVE cutting plane method.A theoretical upper bound for the number of outer iterations of the MVE-based SE method,suggested in[18],is also applicable to Algorithm 1,despite its conservativeness.For details about cutting plane methods,readers can refer to[19].If Assumption 1 is notsatisfied,there would be otherwise two specialcases. First,if the primal SCPNproblem(2)is unfeasible,the algorithm will not output any feasible results after proceeding for a large quantity of cycles,even if the number of iterations reaches its upper bound.Second,ifΘ1does not contain the optimaθ?,which is possibly encountered in practice as the initial hypercube may fail to satisfy Assumption 1,the algorithm just finds a sub-optimal solution withinΘ1.

    Given an SCPNproblem,Algorithm 1 proceeds in a deterministic framework,although the underlying interpretation is in a probabilistic sense.Different from SE methods,the scenarios used for test are prepared in advance.However,for Algorithm 1,it is still cumbersome to test all the N scenarios in every inner iteration,because the quantity N depends on the dimension of the optimization parameter(see(3)),usually being a large number,larger than that used in SE methods. Consequently,Algorithm 1 has little advantage over the SE methods in respect of algorithm's total execution time.

    The basic idea to amend Algorithm 1 is to reduce the numberofscenarios consumed in each innerloop.One natural way is to randomly batch the N scenarios for verification;the underlying idea is thatsuccessive demonstrations of feasibility for the randomly-sampled batches will steadily enhance our confidence on the solution's feasibility for allthe N scenarios. To illustrate the rationale of this idea,we presentthe following theorem.

    Theorem 2.Suppose we have to inspect a set of N scenarios.A k-length subset randomly sampled from the N scenarios,called a k-sample batch,is defined to be“good”if and only if the solution is verified to be feasible for all the k scenarios.Then,if m successive k-sample batches have been verified to be good,ourconfidence thatthe solution is feasible for all the N scenarios,denoted by

    Proof.For a given solution,letλbe the chance of a scenario to be good,then in a k-sample batch,the number of good scenarios,denoted by“x”,obeys the binomial distribution, i.e.,

    where l(x|λ)is a likelihood function.If x = k,(9)is simplified as l(k|λ)=λk.

    Then for m successive good k-sample batches that are i.i.d, the likelihood function can be written as

    In the Bayesian sense,the posteriorprobability ofλis given by

    where g(λ)isλ's prior distribution.For a given N-length set of scenarios,λis a discrete stochastic variable that can be assigned(N+1)values:0,1/N,2/N,···,1.Once a good k-sample batch has been obtained,it is definitely known that λ≥k/N.And further,from a Bayesian point of view,if we have no preference aboutλ's evaluation in interval[k/N,1], it is most reasonable to assumeλto be uniformly distributed in this interval.So g(λ)is given as

    Inserting(10)and(12)into(11),the confidence thatλ equals to 1 is then obtained as

    Theorem 2 indicates the number of times we have to repeat for sampling inspections if k andηare determined in advance. It then inspires Algorithm 2,for which we just give the iteration part below,as the rest of Algorithm 2 is similar to Algorithm 1.

    Algorithm 2.

    Iterations:

    1)Randomly pick K samples from the set of N scenarios, and perform oracle on this subset.

    If?(θl,δ(k))≤0 for all k∈{1,2,···,K}

    If Do Batch Check/=1,set

    else set f cnt=f cnt+1, θl=θl?1,and go to step 3);

    else set f=?(θl,θ(k)),ggg=?θ?(θl,δ(k)),f cnt=0.

    2)Update.Add a new cutting plane to Placcording to(7), find the MVE El+1in polytope:

    3)Stopping rule.

    If f cnt>M,the algorithm stops,

    else set l=l+1,and go to step 1).

    In the algorithm,N,K and M respectively stand for the total amount of scenarios generated by the SCP problem,the number of scenarios comprising one batch and the number of successive good batches.

    Remark 1.For Algorithm 2,at the beginning,everything goes the same way as in Algorithm 1 except the number of scenarios used for feasibility verification,until the objective function cccTθ's descending rate reaches a prefixed threshold τ.Then the algorithm stops improving cccTθ,and tries to find a solution that is able to pass successive verifications on M randomly-picked K-sample batches.

    Remark 2.Only the last one of the output set{?θj}is the solution we want,and we recommend a complete feasibility verification ofthis solution on allthe N scenarios.Fortunately, if we have set a high confidence,e.g.,η=0.95,the last one is almost always the“right”one.

    Remark 3.The above batch algorithm is designed forrobust optimization problems,and a slight modification to the algorithm will fit it for robust feasibility problems,while merely removing the parts aboutobjective function improvementaway from the iteration process.

    III.ROBUST OUTPUT-FEEDBACK CONTROL OF LPV SYSTEM USING ISA

    A.Preliminary on Robust LPV Control

    Given two compact sets P?RnρandΔ?Rnδ,consider an uncertain LPV systemΣP,Δ,i.e.,

    In this paper,we are concerned with the induced L2-norm performance of closed-loop LPV system(13)and(14),which is defined as

    where FPdenotes the setofallpiecewise continuous functions mapping time t into P with a finite number of discontinuities in any interval[20].Following the loop-shifting techniques[21], it is possible to transform any general LPV plant as in(13) to a simplified form PSP,Δ,written as(15),through normpreserving transformations on zzz/ddd and invertible transformations on yyy/uuu:

    B.Induced L2-norm Control of LPV System with Uncertainty

    For the L2-norm robust control of an LPV system in the presence of model uncertainty,we give a sufficient condition as follows.

    Theorem 3.Given the uncertain LPV system PSP,Δ,and a controller KP,the closed-loop system is guaranteed to have an L2-norm performance levelγ>0 if there existtwo symmetric matrices X>0 and Y>0,such thatforallρ∈P andδ∈Δ, the following LMIs hold:

    The theorem can be easily proved by directly applying Wu's theorem(see[20],Theorem 3.3.1)on parameter dependent output-feedback controller synthesis,so long as we treat uncertain parameterδthe same way as scheduling parameter ρ,and use the single quadratic Lyapunov function(SQLF) as a degraded form of the parameter-dependent Lyapunov function(PDLF).Once such X and Y are found,we can then construct the robust output-feedback LPV controller based on the nominal state-space matrices.Defi ne[20]

    then the n-order strictly proper controller KPachieving γ-performance is presented as

    The RCP problem associated with Theorem 3 can be expressed as

    whereθdenotes the vector form of the tuple{X,Y,γ}, and Fi(θ,ρ,δ)< 0 represents the LMI constraints arising from Theorem 3.Note that while bothρandδenter the RCP problem in the same manner,they act differently on the system;ρenters the control law as a scheduling parameter, whileδis treated as parameter perturbation whose adverse impactshould be restrained by the controllaw.Therefore,(18) is indeed a robust optimal LPV control problem,taking both demands of gain scheduling and robustness into account.

    C.Computational Issues for Applying ISA

    To make the ISA algorithms applicable to problem(18), firstly it is necessary to define an equivalent scalar function as well as the associated sub-gradient.Let

    where?Fiand ?Fi's dependence onθ,ρandδis omitted for brevity.Then a scalar feasibility indicator function[22]is adopted.Define

    where?F+denotes the projection of matrix?F onto the cone of positive semidefinite matrices,and the subscript“fro”denotes the Frobenius norm that we adopt in this paper.It is easy to verify that?(θ,ρ,δ)>0 if and only if?F(θ,ρ,δ)≤0 does not hold,and it is zero,otherwise.Define

    Then,the sub-gradient of?(θ,δ)is given as

    Then we cope with the parameter dependency problem. Note that?(θ,ρ,δ)≤0(?(θ,ρ,δ)=0 in our case)must hold for allρ∈P.To handle the LMI's dependence onρ, a so-called gridding technique is commonly used[1,23].After L gridding points are selected,i.e.,and scenarios ofδare collected,problem(18)is further approximated as

    which has totally L×N constraints.

    In this paper,we use a different way to deal with the scheduling parameter,that we cast it as a role of the scenario, generating random samples rather than making grids upon it.This means the scheduling parameters act like uncertainty parameters during the computation,which can be easily understood by observing thatρandδenter problem(18)in thesame manner.So problem(18)is finally translated into the following scenario design problem thatcan be solved by ISA:

    The benefit of doing so is obvious that randomly sampling scheduling parameter does not increase the sample size of scenarios,while greatly reducing the number of LMI constraints for each scenario,compared to the gridding approach. And if the solution of(19)exists,it is supposed to be better generalized to allρ∈ P than that of using the gridding approach,as it has experienced more instances ofρ.

    IV.APPLICATION TO THE CONTROL OF A SMALL HELICOPTER

    A.Model Setup

    In this section,the ISA proposed in the above section is applied to design an LPV longitudinal controller of a small helicopter for its cruise flight.We employ a Hirobo 90-class model scale helicopter,shown in Fig.1,as the testbed of our design method.A small helicopter is accurately modeled by the fuselage's 6-DOF rigid body model augmented with the first-order main rotor and stabilizer bar's flapping dynamics, whose complete modeldescription includes 13 states:3 linear velocity components in body-axis(u,v,w),3 angular rates (p,q,r),3 Euler angles(φ,θ,?),2 flapping angles of main rotor's tip-path-plane(TPP)(a1,b1)and 2 flapping angles of Bell-Hiller bar's TPP(c,d).A minimum complexity nonlinear model[24]is used both for linear modelextraction and simulation verification.Earlier work on modeling a small helicopter with stabilizer bar[25]showed thatthe main rotor and stabilizer bar's flapping dynamics could be lumped and modeled just by main rotor's 2-order TPP flapping dynamics,with the stabilizer bar's effect counted into main rotor's equivalent flapping time constant.So for longitudinalmotion,the state vector is defined to be[u a1w q θ]T,where u is the body axis forward speed,a1is the longitudinal flapping angle,w is the vertical speed,q is the pitch rate andθis the pitch angle.

    Fig.1. The equipped 90-class model helicopter in flight.

    In order to describe the flight dynamics,many physical parameters need to be evaluated,some of which are related to aerodynamics,and rotor flapping are hard to be determined, as listed in Table I.

    The listed items are the uncertain parameters we care about for controller design,whose nominalvalues are determined by citing the parameter data of an X-Cell 60-class helicopter[25], and we assume they are uniformly distributed around the nominal values.

    The model dynamically varies depending on the advanced ratioμ,defined byμ=u/?R,where?and R are main rotor's revolution,speed and radius,respectively.It is reasonable to select u to be the scheduling parameter,i.e.,ρ=u,as?is usually kept constant by an independent engine controller during the flight.The flight envelope of interest covers the speed regime of 0~25 m/s,so the scheduling parameterρis assumed to be uniformly distributed in this interval.

    Following the trim and linearization procedures,the helicopters'longitudinal LPV model can be extracted from its nonlinear simulation model,expressed as(20)in the next page,whereδlonandδcolare longitudinal cyclic input,and collective input,respectively;θeandφeare trimmed angles; X,D,Z,M are the stability derivatives and K is the control derivative;and y denotes the output vector.

    B.Controller Design

    Similarto the synthesis procedures for LTIsystems,weighting functions are defined here to characterize the overall closed-loop performance.The open-loop interconnection of the corresponding weighted system is shown in Fig.2,where

    Fig.2. Weighted open loop interconnection of helicopter plant.

    From the interconnection shown in Fig.2,it is easy to construct a 12-order augmented open-loop LPV model of the form(13)for the longitudinal dynamics.The corresponding SCP has 157 design variables.To get an initial guess of the design variables,standard SA is implemented on 1 000 scenarios using YALMIP[26]under Matlab environment,which nearly uses up the computational capacity of a personal computerequipped with 2 GB memory.The calculated optimal performance level is 2.09,and the initial guess ofγis set to be 6.0,allowing some gap above the calculated value. The initialhypercube constraints,centered on the initialguess mentioned above,are chosen to be large enough to guarantee the confidence that the optima are included in the hypercube, whose volume is about 2.62×10365.

    We setτ=0.07,ε=0.05,andβ=10?5to execute the proposed two ISA algorithms.According to(3),6 742 scenarios are created.For Algorithm 2,we selectη=0.95,and K=400,and then using(8),the repeattimes ofsampling inspection,M,is set to be 51.For comparison,we also implementin the same settings,the SE algorithm based on the MVE cutting-plane method,which is proposed by Wada etal.[18]and regarded as the most sophisticated framework among current SE methods.All the algorithms are implemented on a personal computer with 2.26 GHz CPU and 2 GB memory. Table IIshows the process information ofthe three algorithms.

    TABLE I UNCERTAIN PARAMETERS OF A SMALL HELICOPTER AND THEIR DISTRIBUTIONS

    In Table II,the execution time(the 4th column)of the algorithms involves both the time spent on the iterations and the time to generate enough scenarios for the algorithm;the latter is particularly listed in the brackets to facilitate the comparison.As observed from the table,Algorithm 1 is the least efficient in the respect of execution time;the scenarios consumed by Wada's algorithm are nearly 3 times that of Algorithms 1 and 2,thus making itrather sluggish;the batch algorithm(Algorithm 2)spentjust 45%of the time taken by Wada's algorithm.

    C.Simulation Results

    We apply Monte Carlo approach to verify the robust feasibility of the solutions obtained from different algorithms. Firstly,we estimate the empirical probability of robust feasibility for both nominal solution and the solutions by the three algorithms mentioned above,by checking whether the LMIs (16 a)~(16 c)(or?(δ,ρ,θ)≤0)hold for each test scenario. Secondly,to verify the actualclosed-loop robustness,we adopt the closed-loop LMI formula[20]that conditions theγ-level L2-norm performance,given by

    and W is the matrix associated with the Lyapunov function of the closed-loop system,defined here by

    Ten thousand scenarios are generated for the verifications, and the results are shown in Table III.Forcomparison purpose, we also putinto the table the results obtained from other three algorithms:LPV control(LPV-C)on nominal plant,SA on 1 000 scenarios(whose solution is used as the initial guess of the iterative algorithms),and Wada's algorithm.

    From Table III,it seems that the robustness improvement is encouraging after performing the first kind of verification,as the empirical estimates of probabilistic robust feasibility for both Algorithms 1 and 2 are much better than their theoretical bounds.However,the results of closed-loop LMI verification are a bit disappointing.Recall that in order to make the controller implementable,we constructitbased on the nominal plant,see(17)where we use A(ρ)rather than A(ρ,δ).As discussed in Section III,bothρandδactin the same way from a pure mathematical point of view,and in the general SQLF settings,A(·)should depend on bothρandδ.So neglecting its dependence on the uncertaintyδresults in a loss ofrobustness. But from a practical point of view,things are still acceptable. Similar results were reported in[23],but the performance degradation of ours is remarkably smaller.

    Finally,the closed-loop performance of robust LPV controller constructed using the solution of Algorithm 2 is verified through simulations on helicopter's nonlinear model.To construct the controller as formulated in(17),nominal LPV matrices A(ρ)and B(ρ)are approximated by polynomialmatrices driven byρ.We exert step-like reference signal to either forward or verticalchannels respectively,while keeping the other at zero state.Ten uncertain nonlinear models are used for the verification,and the results are shown in Figs.3 and 4,where the dashed lines representthe desired responses (outputofthe reference model)and the solid lines representthe actualresponses.Note thatthe nominal LPV controller cannot pass the closed-loop feasibility verification for any scenario candidate,as shown in Table III,so there is no need to verify it on the nonlinear model again.

    TABLE II PROCESS INFORMATION OF DIFFERENT ALGORITHMS

    TABLE III FEASIBILITY VERIFICATION RESULTS OF DIFFERENT ALGORITHMS

    Fig.3. Forward and vertical speed responses to step-like excitation on the forward speed channel.

    Fig.4. Forward and vertical speed responses to step-like excitation on the vertical speed channel.

    From Figs.3 and 4 seen that in the presence of model perturbations,the closed-loop system tracks reference model outputs very well.This means that governed by the LPV controller,the system consistently presents a second-order closed-loop characteristic specified by the reference modelin a wide speed range(0~23 m/s),which is quite desirable for the high-layer motion planning design.Itis further observed from Fig.3 that the variation of vertical speed is kept very small during the high-speed forward flight,which means that the longitudinal phugoid mode is eliminated,demonstrating that the decoupling performance of the controller is satisfactory.

    D.Discussion

    For a complex system like a small helicopter,neither trim nor linearization procedures are simple,as seen from Table Iit takes more than 1 s to create a scenario.In this situation,those SE algorithms are too time-consuming.The proposed batch algorithm(Algorithm 2)adopts both advantages of Algorithm 1 and SE algorithms,i.e.,it consumes the same amount of scenarios in all as the former,and queries as many as the latter for feasibility verifications.

    Numerical stability is a problem of optimization algorithms using cutting-plane techniques.For the controller design of a smallhelicopter's longitudinalmotion,a 157-dimensionaloptimization problem,numericalstability would gradually deteriorate,i.e.,the condition numbers of the matrices generated from the algorithm become much larger,as the cutting-plane-based algorithm proceeds.The algorithm might stop unexpectedly with fatal errors,if we cancel the constraint pruning codes. The aforementioned heuristic logic for removing redundant constraints can effectively improve the numerical stability as no fatal error occurs once the logic works.However,the curse of dimension is still a problem that we cannot thoroughly circumvent.For SCP problems oflarger-scale,e.g.,largerthan 157-dimensions,a more elaborate constraintpruning technique would be necessary.

    V.CONCLUSIONS

    In this paper,we have presented iterative algorithms for the SA.For the robustcontrolof a complex LPV system which is time-costly to create through linearization,the proposed batch algorithm has been shown to be more efficientthan the popular SE methods.In the robust LPV control context,we have also proposed to cope with parameter dependency using the randomization technique identically on both uncertainty andscheduling parameters,and to prune the redundant constraints using a heuristic logic based on relevance ratio ranking. All these techniques together comprise a complete method, which has been applied to design robust LPV controller for small helicopter's cruise flight.Multiple uncertain physical parameters have been considered.Simulation studies have been provided to demonstrate its effectiveness.

    REFERENCES

    [1]Rugh WJ,Shamma JS.Research on gain scheduling.Automatica,2000, 36(10):1401?1425

    [2]Helmersson A.μ-synthesis and LFT gain scheduling with realuncertainties.International Journal of Robust and Nonlinear Control,1988,8(7): 631?642

    [3]Helmerson A.Application of real-μgain scheduling.In:Proceedings of the 35th IEEE Conference on Decision and Control,Kobe,Japan:IEEE, 1996.1666?1671

    [4]Apkarian P,Adams R J.Advanced gain-scheduling techniques for uncertain systems.IEEE Transactions on Control System Technology, 1998,6(1):21?32

    [5]Aouani N,Salhi S,Garcia G,Ksouri M.Robust control analysis and synthesis for LPV systems under affine uncertainty structure.In:Proceedings of the 6th International Multi-conference on Systems,Signals and Devices,Djerba,Tunisia:IEEE,2009.187?191

    [6]Veenman J,Scherer C W.On robustLPV controllersynthesis:a dynamic integralquadratic constraintbased approach.In:Proceedings ofthe 49th IEEE Conference on Decision and Control,Atlanta,USA:IEEE,2010. 591?596

    [7]Calafiore G,Dabbene F,Tempo R.A survey of randomized algorithms for control synthesis and performance verification.Journal of Complexity,2007,23(3):301?316

    [8]Kanev S,De Schutter B,Verhaegen M.An ellipsoid algorithm for probabilistic robust controller design.System&Control Letters,2003, 49(5):365?375

    [9]Calafiore G,Dabbene F.A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs.Automatica,2007,43(12): 2022?2033

    [10]Calafiore G,Dabbene F,Tempo R.Research on probabilistic methods for control system design.Automatica,2011,47(7):1279?1293

    [11]Calafiore G,CampiMC.The scenario approach to robustcontroldesign. IEEE Transactions on Automatic Control,2006,51(5):742?753

    [12]Calafiore G,Fagiano L.Stochastic model predictive control of LPV systems via scenario optimization,Automatica,2013,49(6):1861?1866

    [13]Campi M C,Garatti S,Prandini M.The scenario approach for systems and control design.Annual Reviewsin Control,2009,33(2):149?157

    [14]Wang X,Lu G,Zhong Y.Robust H∞attitude controlofa laboratory helicopter.Roboticsand Autonomous Systems,2013,61(12):1247?1257

    [15]Liu C,Chen W,Andrews J.Tracking control of small-scale helicopters using explicit nonlinear MPC augmented with disturbance observers. Control Engineering Practice,2012,20(3):258?268

    [16]Cai G,Chen B M,Dong X,Lee T H.Design and implementation of a robust and nonlinear flight control system for an unmanned helicopter. Mechatronics,2011,21(5):803?820

    [17]Simpl′?cio P,Pavel M D,van Kampen E,Chu Q P.An acceleration measurements-based approach for helicopter nonlinear flight control using incremental nonlinear dynamic inversion.Control Engineering Practice,2013,21(8):1065?1077

    [18]Wada T,Fujisaki Y.Sequential randomization algorithms:a probabilistic cutting plane technique based on maximum volume ellipsoid center.In:Proceedings of the 2010 IEEE International Symposium on Computer-Aided ControlSystem Design,Yokohama,Japan:IEEE,2010. 1533?1538

    [19]Localization methods[Online],available:http://www.stanford.edu/ class/ee392o,September 1,2011

    [20]Wu F.Controlof Linear Parameter Varying Systems[Ph.D.dissertation], University of California at Berkeley,USA,1995

    [21]Safonov M,Limbeer D,Chiang R.Simplifying the H∞theory via loop shifting,matrix pencil,and descriptor concepts.International Journal of Control,1989,50(6):2467?2488

    [22]Calafiore G,Polyak B T.Stochastic algorithms for exact and approximate feasibility of robust LMIs.IEEE Transactions on Automatic Control,2001,46(11):1755?1759

    [23]Lu B,Wu F.Probabilistic robust linear parameter-varying control of an F-16 aircraft.AIAA Journal of Guidance,Control,and Dynamics,2006, 29(6):1454?1459

    [24]Heffley R K,Mnich M A.Minimum-complexity Helicopter Simulation Math Model.Technical Report 19880020435,NASA,USA,1988

    [25]Gavrilets V.Autonomous Aerobatic Maneuvering of Miniature Helicopters[Ph.D.dissertation],Massachusetts Institute of Technology, 2003.

    [26]L¨ofberg J.YALMIP:A toolbox for modeling and optimization in MATLAB.In:Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design,Taipei,China:IEEE,2004. 284?289

    Zhou Fang Associate professor in the School of Aeronautics and Astronautics,Zhejiang University, China.He received his Ph.D.degree from the Department of Control Science and Engineering, Zhejiang University in 2008.His research interests include navigation,guidance and control of UAVs and advanced learning controlmethods.Corresponding author of this paper.

    Hua Tian Ph.D.candidate in the Department of Control Science and Engineering,Zhejiang University,China.Her research interests include modeling and control of UAVs.

    Ping Li Professor of both the Schoolof Aeronautics and Astronautics and the Departmentof ControlScience and Engineering,Zhejiang University,China. His research interests include process control,UAV projects and intelligenttransportation systems.

    RC has

    lots of attentions in control community since late 1990s.Various algorithms and their applications are documented,which are classified into three main methodologies by Calafiore etal.[7]:the approach based on the statistical learning theory,the sequential approximation methods based on gradientiterations orstochastic ellipsoid(SE)iterations,and the scenario approach(SA).SE algorithms[8?10]proceed in an iterative scheme,online generating scenarios for oracle,and the sample size per iteration is independent of both the number of uncertainty parameters and the number of the design variables.So SE approaches seem to perform quite efficiently,and have been the mostpopular PRC methods currently.However,for an SE algorithm,the total amount of scenarios consumed by all the iterations is very large, even much larger than that of SA.So for controller synthesis problems of LPV systems,whose linearization procedure may need a great deal of calculation,SE methods might be too time-consuming.

    Manuscriptreceived October 10,2013;accepted April11,2014.This work was supported by National Natural Science Foundation of China(61004066). Recommended by Associate Editor Bin Xian

    :Zhou Fang,Hua Tian,Ping Li.Probabilistic robust linear parameter-varying control of a small helicopter using iterative scenario approach.IEEE/CAA Journal of Automatica Sinica,2015,2(1):85?93

    Zhou Fang is with the School of Aeronautics and Astronautics,Zhejiang University,Hangzhou 310027,China(e-mail:zfang@zju.edu.cn).

    Hua Tian is with the Departmentof Control Science and Engineering,Zhejiang University,Hangzhou 310027,China(e-mail:htian@iipc.zju.edu.cn).

    Ping Li is with the School of Aeronautics and Astronautics,Zhejiang University,Hangzhou 310027,China and Department of Control Science and Engineering,Zhejiang University,Hangzhou 310027,China(e-mail: pli@iipc.zju.edu.cn).

    国产蜜桃级精品一区二区三区| a在线观看视频网站| 亚洲黑人精品在线| 给我免费播放毛片高清在线观看| 精品午夜福利视频在线观看一区| 国产激情欧美一区二区| 精品少妇一区二区三区视频日本电影| av天堂在线播放| 国产一卡二卡三卡精品| 精品国产一区二区三区四区第35| 国产一卡二卡三卡精品| 欧美在线一区亚洲| 夜夜夜夜夜久久久久| 99久久综合精品五月天人人| 又黄又粗又硬又大视频| 自线自在国产av| 美女午夜性视频免费| a级毛片a级免费在线| 欧美zozozo另类| 18禁黄网站禁片午夜丰满| 亚洲国产精品久久男人天堂| 9191精品国产免费久久| 少妇 在线观看| 搡老岳熟女国产| 亚洲熟妇熟女久久| 国产精品久久久久久人妻精品电影| 欧美日韩一级在线毛片| 国产精品久久电影中文字幕| 人人澡人人妻人| 欧美绝顶高潮抽搐喷水| 国产爱豆传媒在线观看 | 久久香蕉精品热| 国产av在哪里看| 亚洲精品色激情综合| 国产亚洲精品久久久久5区| 一级作爱视频免费观看| 国产成人av激情在线播放| 成人永久免费在线观看视频| 老汉色∧v一级毛片| 天天躁夜夜躁狠狠躁躁| 宅男免费午夜| 18禁裸乳无遮挡免费网站照片 | 成人一区二区视频在线观看| 一边摸一边做爽爽视频免费| 在线永久观看黄色视频| 夜夜爽天天搞| 久久久国产成人精品二区| 国产又黄又爽又无遮挡在线| 在线十欧美十亚洲十日本专区| 日韩精品青青久久久久久| 深夜精品福利| 中国美女看黄片| 欧美黄色淫秽网站| 国产亚洲精品一区二区www| 国产亚洲精品久久久久久毛片| 97人妻精品一区二区三区麻豆 | 久久久国产成人精品二区| 国产精品电影一区二区三区| 亚洲电影在线观看av| 一级a爱片免费观看的视频| 久久婷婷人人爽人人干人人爱| 亚洲狠狠婷婷综合久久图片| 三级毛片av免费| 亚洲欧洲精品一区二区精品久久久| 91九色精品人成在线观看| 亚洲中文字幕一区二区三区有码在线看 | 精品久久久久久,| 日本 av在线| 他把我摸到了高潮在线观看| 国产伦人伦偷精品视频| xxxwww97欧美| 欧美一级a爱片免费观看看 | 国产精品久久久久久亚洲av鲁大| 欧美黄色片欧美黄色片| 黄片大片在线免费观看| 亚洲男人的天堂狠狠| 国产精品久久电影中文字幕| 激情在线观看视频在线高清| 中文字幕人成人乱码亚洲影| 免费观看人在逋| 日韩有码中文字幕| 夜夜躁狠狠躁天天躁| 国产精品国产高清国产av| 人人妻人人澡欧美一区二区| 久久久精品欧美日韩精品| 久久午夜亚洲精品久久| 91老司机精品| 一边摸一边做爽爽视频免费| 精品乱码久久久久久99久播| av在线天堂中文字幕| 非洲黑人性xxxx精品又粗又长| 欧美久久黑人一区二区| 天天一区二区日本电影三级| 好男人电影高清在线观看| 男女床上黄色一级片免费看| 久9热在线精品视频| 怎么达到女性高潮| 久久国产亚洲av麻豆专区| 精品国产亚洲在线| 婷婷六月久久综合丁香| 男女之事视频高清在线观看| 精品午夜福利视频在线观看一区| 国产黄片美女视频| 午夜a级毛片| 男女之事视频高清在线观看| 精品午夜福利视频在线观看一区| 亚洲av五月六月丁香网| 国产野战对白在线观看| 一级片免费观看大全| 天堂影院成人在线观看| 人人澡人人妻人| 国产精品亚洲一级av第二区| 欧美激情极品国产一区二区三区| 成年人黄色毛片网站| 两个人看的免费小视频| 成人特级黄色片久久久久久久| 国产三级在线视频| 国内毛片毛片毛片毛片毛片| 一级作爱视频免费观看| 青草久久国产| 精品国产超薄肉色丝袜足j| 一级a爱片免费观看的视频| 欧美av亚洲av综合av国产av| 久久久水蜜桃国产精品网| 视频在线观看一区二区三区| 亚洲国产精品久久男人天堂| 国产精品98久久久久久宅男小说| 久久久久久免费高清国产稀缺| 啦啦啦免费观看视频1| 亚洲人成77777在线视频| 精品熟女少妇八av免费久了| 日日夜夜操网爽| 日韩欧美 国产精品| 日本精品一区二区三区蜜桃| 真人一进一出gif抽搐免费| 黑人欧美特级aaaaaa片| 夜夜看夜夜爽夜夜摸| 一区福利在线观看| 久久久久久大精品| 精品一区二区三区四区五区乱码| 韩国av一区二区三区四区| 日韩欧美国产一区二区入口| 99国产综合亚洲精品| 精品无人区乱码1区二区| 亚洲国产欧美网| 欧美+亚洲+日韩+国产| 十八禁网站免费在线| 欧美成狂野欧美在线观看| 中文字幕最新亚洲高清| 999久久久精品免费观看国产| 少妇被粗大的猛进出69影院| 嫁个100分男人电影在线观看| 亚洲精品国产一区二区精华液| 欧美乱色亚洲激情| 丰满的人妻完整版| 最近在线观看免费完整版| 人人澡人人妻人| 不卡一级毛片| 黄色视频不卡| 午夜老司机福利片| 国产免费男女视频| 亚洲成人精品中文字幕电影| 亚洲国产欧美一区二区综合| 色尼玛亚洲综合影院| 国产精品免费视频内射| 国产精品二区激情视频| 中文字幕精品亚洲无线码一区 | 亚洲专区中文字幕在线| 国产黄a三级三级三级人| 日本一区二区免费在线视频| 亚洲av片天天在线观看| 91成人精品电影| 国产精品av久久久久免费| 亚洲一卡2卡3卡4卡5卡精品中文| 久久久久久久久中文| 国产精品久久久久久亚洲av鲁大| а√天堂www在线а√下载| 精品欧美一区二区三区在线| 国产精品一区二区精品视频观看| 国产精品久久久人人做人人爽| 久久久精品欧美日韩精品| 非洲黑人性xxxx精品又粗又长| 人人妻人人看人人澡| 国产高清视频在线播放一区| or卡值多少钱| 成年女人毛片免费观看观看9| 无限看片的www在线观看| 啦啦啦免费观看视频1| 国产一区二区在线av高清观看| 一级作爱视频免费观看| 亚洲专区字幕在线| 少妇被粗大的猛进出69影院| 18禁黄网站禁片午夜丰满| 好看av亚洲va欧美ⅴa在| 亚洲成人精品中文字幕电影| 老鸭窝网址在线观看| 亚洲精品国产一区二区精华液| 国产精品免费视频内射| 中文字幕人妻丝袜一区二区| 日韩一卡2卡3卡4卡2021年| 成人永久免费在线观看视频| 亚洲成人精品中文字幕电影| a在线观看视频网站| 精品国内亚洲2022精品成人| 波多野结衣高清无吗| 丝袜美腿诱惑在线| 精品久久久久久久久久久久久 | 免费高清视频大片| 免费看日本二区| 天天一区二区日本电影三级| 99热只有精品国产| 国产熟女午夜一区二区三区| 看免费av毛片| 亚洲全国av大片| 国产精品日韩av在线免费观看| 无人区码免费观看不卡| 激情在线观看视频在线高清| 中文字幕人成人乱码亚洲影| 99riav亚洲国产免费| 国内精品久久久久精免费| 欧美一区二区精品小视频在线| 91成年电影在线观看| 欧美色欧美亚洲另类二区| 超碰成人久久| 日韩精品免费视频一区二区三区| 国产亚洲av嫩草精品影院| 国产精品野战在线观看| 自线自在国产av| 一夜夜www| 丰满人妻熟妇乱又伦精品不卡| 国产一级毛片七仙女欲春2 | 一区二区三区国产精品乱码| 国产三级在线视频| 免费人成视频x8x8入口观看| 免费观看精品视频网站| 亚洲美女黄片视频| 叶爱在线成人免费视频播放| www.精华液| 中文亚洲av片在线观看爽| 国产av不卡久久| 亚洲成a人片在线一区二区| 欧美不卡视频在线免费观看 | 搞女人的毛片| 波多野结衣高清无吗| 久久精品影院6| 久久久精品国产亚洲av高清涩受| 国产激情欧美一区二区| 日本 av在线| 久久九九热精品免费| 热re99久久国产66热| 日韩成人在线观看一区二区三区| 国产精品 国内视频| 我的亚洲天堂| 欧美av亚洲av综合av国产av| 久久九九热精品免费| 久久久久久久久中文| 午夜福利18| 亚洲av熟女| 精品日产1卡2卡| 欧美绝顶高潮抽搐喷水| 亚洲自拍偷在线| 一级片免费观看大全| 久久 成人 亚洲| 两个人免费观看高清视频| 美女 人体艺术 gogo| 一级a爱视频在线免费观看| 国产精品自产拍在线观看55亚洲| 国产激情欧美一区二区| 99热只有精品国产| www.熟女人妻精品国产| 精品久久久久久,| 一级作爱视频免费观看| 久久香蕉精品热| 亚洲九九香蕉| 男女那种视频在线观看| 亚洲真实伦在线观看| 18美女黄网站色大片免费观看| 久久午夜综合久久蜜桃| 一边摸一边做爽爽视频免费| 久久久久久大精品| 中文字幕最新亚洲高清| 亚洲 国产 在线| 亚洲欧美一区二区三区黑人| 久久人妻av系列| av中文乱码字幕在线| 精品久久久久久久末码| 男男h啪啪无遮挡| 久久亚洲精品不卡| 精品一区二区三区四区五区乱码| 午夜激情av网站| 久久香蕉国产精品| 国产精品久久电影中文字幕| 午夜福利在线在线| 在线观看午夜福利视频| 香蕉久久夜色| 中文字幕人妻丝袜一区二区| 欧美丝袜亚洲另类 | av超薄肉色丝袜交足视频| 欧美大码av| 最好的美女福利视频网| 午夜精品在线福利| 免费在线观看视频国产中文字幕亚洲| 老汉色av国产亚洲站长工具| 国内久久婷婷六月综合欲色啪| 欧美亚洲日本最大视频资源| 18禁美女被吸乳视频| 久久人妻av系列| 精品人妻1区二区| 色播亚洲综合网| 亚洲在线自拍视频| 国产一区在线观看成人免费| 亚洲五月婷婷丁香| 免费在线观看视频国产中文字幕亚洲| 男女床上黄色一级片免费看| 熟女少妇亚洲综合色aaa.| 好男人电影高清在线观看| 国产成人精品久久二区二区91| 国产久久久一区二区三区| 国产午夜福利久久久久久| 欧美日韩乱码在线| 黑人欧美特级aaaaaa片| 无遮挡黄片免费观看| 男人的好看免费观看在线视频 | 国产精品一区二区精品视频观看| 欧美激情极品国产一区二区三区| 成熟少妇高潮喷水视频| 国产爱豆传媒在线观看 | 女警被强在线播放| 亚洲午夜精品一区,二区,三区| 岛国视频午夜一区免费看| 欧美乱妇无乱码| 99国产精品一区二区三区| 欧美黑人欧美精品刺激| 亚洲一区高清亚洲精品| 999久久久精品免费观看国产| 午夜亚洲福利在线播放| 成人精品一区二区免费| 午夜免费观看网址| 99久久综合精品五月天人人| 精品人妻1区二区| 午夜免费激情av| 亚洲国产欧美网| 欧美激情 高清一区二区三区| xxxwww97欧美| 久久婷婷成人综合色麻豆| 啪啪无遮挡十八禁网站| 亚洲中文日韩欧美视频| 日韩欧美三级三区| 国产精品久久电影中文字幕| 亚洲精品在线观看二区| 大香蕉久久成人网| 不卡av一区二区三区| 亚洲精品粉嫩美女一区| 国产片内射在线| 午夜视频精品福利| 麻豆av在线久日| 黑人操中国人逼视频| 黄色女人牲交| 亚洲成a人片在线一区二区| 一边摸一边做爽爽视频免费| 欧美日韩乱码在线| 午夜日韩欧美国产| 亚洲成av片中文字幕在线观看| 免费在线观看亚洲国产| 一边摸一边做爽爽视频免费| 亚洲aⅴ乱码一区二区在线播放 | 手机成人av网站| 90打野战视频偷拍视频| 国产成人精品久久二区二区免费| 啦啦啦 在线观看视频| 日本一本二区三区精品| 18禁国产床啪视频网站| 91大片在线观看| 国内揄拍国产精品人妻在线 | 精品国产亚洲在线| 成人手机av| 一边摸一边做爽爽视频免费| 桃红色精品国产亚洲av| 亚洲人成网站在线播放欧美日韩| 嫁个100分男人电影在线观看| 欧美日韩精品网址| 香蕉丝袜av| 日韩大尺度精品在线看网址| 男人舔女人下体高潮全视频| 又大又爽又粗| 欧美精品啪啪一区二区三区| 日本免费a在线| 国产激情偷乱视频一区二区| 国产精品二区激情视频| 久久国产精品人妻蜜桃| 久久99热这里只有精品18| 老司机午夜十八禁免费视频| 亚洲av成人不卡在线观看播放网| 国产成人一区二区三区免费视频网站| 亚洲熟女毛片儿| 一个人观看的视频www高清免费观看 | 欧美另类亚洲清纯唯美| 久久久久亚洲av毛片大全| 听说在线观看完整版免费高清| 日韩欧美国产在线观看| 日韩欧美一区二区三区在线观看| 久久草成人影院| 国产精品久久久久久亚洲av鲁大| 久久午夜亚洲精品久久| 一边摸一边做爽爽视频免费| 欧美zozozo另类| 成人永久免费在线观看视频| 国产亚洲欧美在线一区二区| 精品日产1卡2卡| 欧美+亚洲+日韩+国产| 嫁个100分男人电影在线观看| 亚洲午夜精品一区,二区,三区| 制服丝袜大香蕉在线| 国产99久久九九免费精品| 国内揄拍国产精品人妻在线 | 成人18禁高潮啪啪吃奶动态图| ponron亚洲| 久久九九热精品免费| 精品福利观看| 欧美成人性av电影在线观看| 国产精品亚洲美女久久久| 国产精品九九99| 高潮久久久久久久久久久不卡| 久久精品aⅴ一区二区三区四区| 午夜精品在线福利| 侵犯人妻中文字幕一二三四区| 51午夜福利影视在线观看| 国产激情偷乱视频一区二区| 亚洲成av片中文字幕在线观看| 国产又黄又爽又无遮挡在线| 久久婷婷人人爽人人干人人爱| 成年女人毛片免费观看观看9| 亚洲国产欧美日韩在线播放| 午夜亚洲福利在线播放| 9191精品国产免费久久| 亚洲中文日韩欧美视频| 国产精品二区激情视频| 久久国产精品人妻蜜桃| 精品国产美女av久久久久小说| 色综合欧美亚洲国产小说| 最好的美女福利视频网| 国产国语露脸激情在线看| 国产午夜福利久久久久久| 亚洲精华国产精华精| 精品福利观看| 亚洲国产精品成人综合色| 熟妇人妻久久中文字幕3abv| 一二三四社区在线视频社区8| svipshipincom国产片| 精品久久久久久,| 99久久精品国产亚洲精品| 男人舔奶头视频| 日本一区二区免费在线视频| 精品国产亚洲在线| 亚洲五月婷婷丁香| 国产精品乱码一区二三区的特点| 欧美性猛交╳xxx乱大交人| 亚洲中文日韩欧美视频| 精品国产美女av久久久久小说| 国产色视频综合| 老司机在亚洲福利影院| 亚洲无线在线观看| 在线国产一区二区在线| 亚洲中文av在线| 中文字幕最新亚洲高清| 亚洲男人天堂网一区| www.精华液| av欧美777| 成人三级做爰电影| 美女午夜性视频免费| 午夜福利成人在线免费观看| 欧美在线黄色| 大香蕉久久成人网| 国产99白浆流出| 国产亚洲av嫩草精品影院| 夜夜看夜夜爽夜夜摸| 成年人黄色毛片网站| 欧美成人午夜精品| 久久九九热精品免费| 欧美日韩黄片免| 黄片大片在线免费观看| 身体一侧抽搐| 麻豆国产av国片精品| 亚洲中文字幕一区二区三区有码在线看 | 一级毛片精品| 天堂动漫精品| 欧美一级a爱片免费观看看 | 日韩欧美三级三区| 午夜免费鲁丝| 国产亚洲欧美在线一区二区| 久久 成人 亚洲| 欧美性猛交黑人性爽| 啦啦啦韩国在线观看视频| 男人舔女人下体高潮全视频| 欧美丝袜亚洲另类 | 国产主播在线观看一区二区| 正在播放国产对白刺激| 99热只有精品国产| 国产99久久九九免费精品| 久久久久久国产a免费观看| 色综合婷婷激情| 丝袜美腿诱惑在线| 欧美激情久久久久久爽电影| 一级片免费观看大全| 欧美成人午夜精品| tocl精华| 每晚都被弄得嗷嗷叫到高潮| 免费在线观看亚洲国产| 丁香欧美五月| 精品久久久久久成人av| 91麻豆精品激情在线观看国产| 制服人妻中文乱码| 十八禁网站免费在线| 人妻丰满熟妇av一区二区三区| 色综合站精品国产| 亚洲中文日韩欧美视频| 欧美激情高清一区二区三区| 又黄又爽又免费观看的视频| 一边摸一边抽搐一进一小说| 在线十欧美十亚洲十日本专区| a级毛片在线看网站| 欧美三级亚洲精品| 最近在线观看免费完整版| 神马国产精品三级电影在线观看 | 18禁裸乳无遮挡免费网站照片 | 啦啦啦免费观看视频1| 国产亚洲欧美精品永久| 在线视频色国产色| 亚洲男人的天堂狠狠| 啦啦啦 在线观看视频| 色婷婷久久久亚洲欧美| 欧美成狂野欧美在线观看| www.自偷自拍.com| 免费在线观看成人毛片| 少妇被粗大的猛进出69影院| 色播在线永久视频| 日韩高清综合在线| 熟女电影av网| 亚洲一卡2卡3卡4卡5卡精品中文| 在线观看一区二区三区| 可以在线观看毛片的网站| 亚洲国产精品成人综合色| 91在线观看av| 国产亚洲av嫩草精品影院| 成人三级黄色视频| 精品午夜福利视频在线观看一区| 变态另类丝袜制服| 色播在线永久视频| 黄片大片在线免费观看| 国产午夜福利久久久久久| 成年女人毛片免费观看观看9| 欧美又色又爽又黄视频| 精品欧美一区二区三区在线| 在线天堂中文资源库| 男男h啪啪无遮挡| 一边摸一边抽搐一进一小说| 婷婷精品国产亚洲av| 日韩大码丰满熟妇| 国产精品影院久久| 精品久久蜜臀av无| 免费高清视频大片| 男男h啪啪无遮挡| 国产成人精品无人区| 天天一区二区日本电影三级| 在线观看免费午夜福利视频| 嫩草影视91久久| 欧美大码av| 男女之事视频高清在线观看| 激情在线观看视频在线高清| 国产成人一区二区三区免费视频网站| 亚洲av五月六月丁香网| 日韩三级视频一区二区三区| 搡老熟女国产l中国老女人| 欧美乱妇无乱码| 久久久久久免费高清国产稀缺| 色综合欧美亚洲国产小说| 色在线成人网| svipshipincom国产片| 国产精品日韩av在线免费观看| 人人妻人人看人人澡| 久久久久九九精品影院| 亚洲一区二区三区色噜噜| 91麻豆av在线| 午夜福利欧美成人| 午夜日韩欧美国产| 国产一区二区激情短视频| 国产精品99久久99久久久不卡| 一个人免费在线观看的高清视频| 亚洲国产精品合色在线| 听说在线观看完整版免费高清| 女同久久另类99精品国产91| 一边摸一边抽搐一进一小说| 午夜福利高清视频| 午夜福利视频1000在线观看| 嫩草影视91久久| 十分钟在线观看高清视频www| АⅤ资源中文在线天堂| 人人澡人人妻人| 亚洲黑人精品在线| 日本熟妇午夜| 人人妻人人澡人人看| 精品国产美女av久久久久小说| 久久久久久久久免费视频了| 国产精品久久久久久人妻精品电影| 日韩中文字幕欧美一区二区| 91麻豆av在线| 少妇被粗大的猛进出69影院| 搞女人的毛片| av视频在线观看入口| 精品少妇一区二区三区视频日本电影| 国产色视频综合| 日韩欧美一区视频在线观看| 欧美成人免费av一区二区三区| 国产精品爽爽va在线观看网站 | 午夜免费观看网址|