具有输入非线性的离散时间系统预测控制的稳定性分析

第 29卷 第 6期 2003年 11月 自 动 化 学 报 ACTA AUTOM ATICA SINICA Vo1．29．No．6 Nov．．2003 Stability Analysis on Predictive Control of Discrete-Time Systems with Input Nonlinearity DING Bao——Cang XI Yu——Geng LI Shao——Yuan (Institute of Automation，Shanghai Jiaotong University，Shanghai 200030) (E—mail：dbc309011@mail．sjtu．edu．cn) Abstract For systems with input nonlinearities，a two—step control scheme is adopted．For linear part the control law with Riccati iteration matrices satisfying certain conditions is used to get the Lyapunov function． The stability conditions are investigated， considering the reversion errors coming from solving nonlinear algebraic equation and desaturation computation，which give tuning guidelines for the real systems．Simulation studies validate the results of theoretical analysis． Key words Input nonlinearity，two step control，predictive control，Riccati iteration，stability 1 Introduction In process industries，many system s have input nonlinearities such as saturation，dead time．relay cycle，etc． M oreover，chemical processes represented by Hammerstein model in the form of“static nonlinear+ dynamic linear”，such as pH neutralization，high purity distillation，etc．，can also be taken as input nonlinear systems．Generally，two—step con— trol can be applied to this kind of system sE -3]，in which a desired intermediate variable is firstlv obtained by applying linear model and then the real control action is obtained by sol— ving nonlinear algebraic equation group(NAEG)，desaturation，etc．The advantage of two— step control is that the controller design is still within the scope of linear system s，which is much simpler than nonlinear control incorporating the nonlinearities into system equation or objective function． For an input nonlinear control system designed by two—step scheme，if the real control input recurs desired intermediate variable exactly through static nonlinearities，the stabili— ty of the system could be guaranteed by properly designing the linear system．However，it is difficult to meet this perfect condition in real applications． For the Hammerstein sys— tem ．solving NAEG will inevitably have error，and for the input saturated system ，the re— stricted input is often largely different from the desired one．In these cases，the stability a— nalysis of two—step controller becomes very difficult． It is the aim of this paper to study the stability property of two step model predictive control systems with input nonlinearities(TSM PC)，including Hammerstein nonlinearity， input saturation，etc．Applying Lyapunov s stability theory，we obtain some stability con— clusions of this kind of systems． Section 2 describes the main idea of TSM PC． Section 3 gives the stability conditions．Section 4 illustrates the stability tuning of TSM PC and Sec— tion 5 gives a simulation example． 2 The description of TSM PC Consider the following discrete—time system with input nonlinearity， zk+1一 Az + Bx ， Y 一 Cz ， x 一 (“ ) (1) where zERn，xER ，yERp，uE R are state，intermediate variable，output and input，re— spectively． represents the relationship between input and the intermediate variable satis一 1)Supported by National Natural Science Foundation of P．R．China(69934020) Received March 4，2002；in revised form December 19，2002 收稿 日期 2002—03—04；收修改稿 日期 2002—12—09 维普资讯 http://www.cqvip.com 828 ACTA AUT0M ATICA SINICA VolI 29 fying (0)一 O． (A ，B)is assumed stabilizable． W e also assume that includes Hammer— stein static nonlinearity，and input saturation constraints sat ，i一1，⋯ ， ． For the above nonlinear system ，the structure of general two—step controller is shown in Fig．1． In the following we describe the concrete realization of TSM PC． - _。 ‘] ⋯·⋯ i二 _1一 i s ’li ‘ i ：i妒 !⋯⋯⋯⋯⋯ ⋯⋯ ： ： General input nonlinear system - ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ Fig．1 The structure of controller and system cascade form In the first step，we only consider the linear system z +1= Az 4-Bx ，Y 一 and de— fine the obj ective function as N -- 1 -，(N， )一∑ Ed+ Qz蚪 4- R 州]+z NQNz抖N (2) where Q—Q ≥0 and R>0 are weighting matrices of the state and the intermediate varia— ble；QN≥0 is the terminal state weighting matrix and Q≠QN is the usual case in predictive contro1．The following Riccati iteration is adopted： P 一 Q 4- A P +1A — A P 1 B(R+ B P +l B) B Pj+lA， J< N ，PN— QN (3) to obtain the linear controllaw ： x 一一 (R 4- B P1B)一 B P1Az (4) Note that x in (4)may be unable to be implemented by real control action，so we denote it as }一 Kz 一一 (R+B PlB) B PlAz (5) In the second step，we solve }一f(u )一0 to obtain 一 ( })，then obtain u by desatu— ration，u 一sat{ }，which is formalized as ll 一g( })as shown in Fig．1．Note that for one several as well as ll may be obtained．However，adding extra conditions(such as choosing u to be the closest to u 一1，choosing u with smallest amplitude，etc．)we can ob— tain a most suitable u ． W hen u has been implemented，the corresponding x is denoted as x 一，(sat{ll })一 ，(sat{g( })})一 ( ·g)( })一 h( })．Then，the control law of TSM PC represented by the intermediate variable becom es x 一 h( })一 h(一 (R+B Pl B) B PlAz ) (6) W ith (6)we obtain the closed—loop system z计1一 Az 4- Bx ： (A—B(R+B P1 B) B P1A)z +B(Jl( })一 }) (7) The closed—loop structure of TSMPC is shown in Fig．2． Fig．2 The closed—loop structure of TSM PC W hen h一 1一 r1，1，⋯ 1] ，the nonlinear item in (7) disappears． However， generally h一 1 can not hold。since in the actuaI solution of TSM PC，h may incorporate： i)the solution error of NAEG； ii)the desaturation action that makes z ≠z ． M oreover。for a real system h may also in— corporate： iii)the modeling error of the Hammerstein nonlinearity； iv)the uncertainty in the actuator and the execution error[ ． Hence，in real applications ≠1(i．e．，z}≠ )will be always true．It is just this rea— 维普资讯 http://www.cqvip.com No．6 DING Bao—Cang et a1．：Stability Analysis on Predictive Control of Discrete-Time⋯ 829 son that brings the complexity of the stability analysis for TSM PC ． In the next chapter we adopt Lyapunov’s theory to analyze the stability of TSM PC ． 3 Stability analysis of TSM PC In the following，we choose R— I for convenience ． Theorem 1．For system represented by (1)，TSM PC is adopted． Then lowing two conditions the closed—loop system is exponentially stable： i)The control parameters fQN，Q， ，N )satisfy Q> Po—P】 ii)The nonlinearity h satisfies — as V2h(s)一s]+ (Jl(s)一s) B P1B(JIl(s)一s)≤ 0 Proof．Define Lyapunov function as V(z )一zip】z ．Then under the fol (8) V( 抖1)～ V (z )一 z2[A—B(aI+B P1 B) B P1A] P1 EA—B(aI+B P1 B)一 B P1A]z 一z 2'P1 z + 2az~A PlB(aI+B PlB)一 ( ( })一 })+ ( ( })一 }) B P1B( ( })一 )一 z [一Q+P。一P1一A P1B(aI+B P1B) (aI+B P1B)一 B P1A]z + 2az[A P1 B(aI+ B P1 B)一 (Jl( })一 })+ (Jl( })一 }) B P B(Jl( })一 x1)一 z2(一Q+ P。一P1)z 一 ( ) }一 2,t( }) ( ( })一 })+ ( ( })～ }) B P1B( ( })一 })一 z (一Q+ P。一 P1)z 一 ( }) (2h(x})一 )+ (Jl( })～ }) B P1B(Jl( })一 ) Apparently the system will be stable under the conditions i)～ii)．Moreover，since Q> P0 一 P1，V(z +1)一V(z )≤一 一(Q—Po+P1)ll z lld0，V z ≠0 and“exponentially”holds． 口 Theorem 1 em bodies the characteristic of two—step control，where condition i) is a stability requirement on the linear controllaw while condition ii)on the nonlinearity h． From the proof of Theorem 1 we can easily know that condition i) is a sufficient stability condition of the linear control law and can be satisfied by properly choosing the control pa— rameters．M oreover，if there is not reversion error，then h(s)一 5 and (8)becomes一 s ≤ 0 which will be always true，and by Theorem 1 we obtain the stability condition for line— ar control law． W ith condition i)satisfied，we can further investigate the stability requirement on h under condition ii)．To do this we assume (s)『I≥b lI s If， (s)一sI『≤ f b～1『．『I s ll (9) where 6>o and b1>0 are scalars；ll ll stands for 2一norm of vector ；ll Jl(s)0≥b1 ll sll mainly stands for the requirement on desaturation level while (s)一s 0≤l 6—1⋯s 0 em— bodies the restriction on the total reversion error． Now we can obtain the following re— quirement on the reversion error． Corollary 1． For system represented by (1)，assume that the control law (4)satisfies Q> Po—P1．Then the control law (6)will exponentially stabilize the system if b，b1 in(9) satisfy the following requirement： 一 Eb}一 (6～1) ]+ (6—1) (B P B)≤ 0 Proo f．Applying (9)we make following deductions： 一 as V2h(s)一s]+ (Jl(s)一s) B P1B(Jl(s)一s)一 一 Ah(s) Jl(s)+ (Jl(s)一 s) ( J+ B P1 B)(Jl(s)一 s)≤ (10) 一 }sTs+ (Jl(s)一 s) ( J+B P1 B)(Jl(s)一 s)≤ 一 ab{s s+ (6— 1) ⋯ ( J+B P1B)s s一 一 sTs+ (6— 1) s s+ (6— 1) (B P】B)s S一 一 Eb}一 (6—1) IS S+ (6— 1) ⋯ (B P1B)s S Hence，if condition (10) holds， (8) can be deduced from (9)． Therefore，the corollary 维普资讯 http://www.cqvip.com 830 ACTA AUT0M ATICA SINICA Vo1．29 holds． 口 In real applications，f is often chosen as totally decoupled form．This simplifies not only the NAEG solving but also f identifying．In the case f is totally decoupled，we can aSSU m e b州s ≤ h (s )s ≤ b s ， i一 1，⋯ ， (11) where b ≥6 >0 are scalars．Apparently(11)has clearer meaning than(9)． Since h (s )has the same sign with s ，l h (s )一s l— l l h (s )l—l s l l≤max{l b州一1 l， l b 一1 l}·l s 1．Let b 一min{b⋯ ，b2．1，⋯ ，b ，1}and l 6—1 l—max{l b⋯一1 l，⋯ ，l b ，l一1 l， l b1，2—1 l，⋯ ，lb ．2—1 l}；then(9)can be deduced from (11)，so Corollary 1 will still hold for(1 1)．M oreover，we can obtain the following conclusion． Corollary 2．For system represented by(1)，assume that f has totally decoupled form and the controllaw (4)satisfies Q> Po—P1．Then the controllaw (6)will exponentially stabilize the system if the following condition holds： 一 (2b1— 1)+ (b一 1) (B P1B)≤ 0 (12) Proof．By using(11)，it is easy to conclude that s (s )一s ]≥s 6¨ s —s ]，i一1， ⋯ ， ． Since — As r2Jz(s)一s]+ (h(s)一s) B Pl B(h(s)一s)= 一 As s一 2As rJz(s)一 s]+ (h(s)一 s) B P1 B(h(s)一s)≤ 一 As s 2A∑(6¨ 一1)s +(Jz(s)一s) B P1B(Jz(s)一s)≤ i一 1 一 As s一 2A(b1— 1)sTs+ (b一 1)。o- (B P1B)sTs一 一 (2b1— 1)s s+ (b一 1) o-⋯ (B P1B)s s if condition (1 2)is satisfied，then (1 1)satisfies(8)．Therefore，the corollary holds． 口 Remark 1．If，一1，that is，if there is only input saturation nonlinearity，then b2—1， (6— 1)。一 (b1— 1)。，and both conditions(10)and (12)will have the form — (2b1— 1)+ (b1— 1) (B P1 B)≤ 0． 4 Stability tuning of TSM PC 4．1 Evaluatinl~the bounds of h for a real system According to (10)and (12)，we can give the tuning guideline for TSM PC，that is，if the controlled system is not stable yet，we can tune QN，Q， and N to restabilize it．To do this we should first determine b】and l 6—1 1．In the following we illustrate how to evalu— ate b】and l 6—1 l for a single input system．Denote 训 A)The solution error of z 一 -厂( )is ≤ ht⋯。 “⋯。 ≤ ≤ ht⋯ (13) ≥ ht⋯ restricted to b( )。≤ -厂·9(x )．27 ≤ (z )。 where — b> O and b> O are constants； B)The design of TSMPC satisfies z 。 ≤ z ≤ z and z 一-厂( )always has real—val ued solution； C)Let z ． 一-厂(“s,min)and z ． 一-厂(“ ． )；then zL ≤z$,min<0 and O Po—P1．If it is，go to step5，else tune {QN，Q，N }to satisfy it and go to step2．If(19)and Q> Po— P1 can not be satisfied by repeated tunings，(19) needs only to be satisfied as much as possible． Step5．Check if the system designing has been satisfactory．If it is，stop tuning，else change QN，Q，N and go to step2． Take single input system with only symmetric in— put saturation constraint as an example． Fig．3 is the curve of h( )．Now (19) becomes B P1 B／A≤ (2bl一 1)／(b1— 1) for which the necessary condition is bl>1／2．Since 一maxI }I，if b1≤1／2，we can increase to decrease }and consequently decrease to make b1> 1／2． Regarding the above parameter tuning guideline， we can give the following conclusion to show that TSM PC is tunable whenever A is stable． Theorem 2．For system (1)with stable A ，assume saturation constraint．Then，by tuning {QN，Q， ，N }， Corollary 2)can be satisfied． h( ) ／ h= b z ， ^一 ￡ ⋯ 一 ⋯ ： ／ Fig．3 The sketch map shows the saturation s restriction on b that 3(b，b1)> O if there isn’t input all the conditions in Corollary 1(or Proof．Take Corollary 2 as an example．Corollary 1 will be analogous．At first，if sat— uration constraint is not considered，then determining (b，b1) is independent of control parameters．W hen there is saturation constraint，consider the following two cases： Case A：As — 。，b1> O．5．Always take QN—Q+A QNA — A QNB ( I+ B QNB ) B QNA．This is equivalent to infinite horizon control ，so Po— P1一O．Further take Q> 维普资讯 http://www.cqvip.com 832 ACTA AUTOMATICA SINICA VoL 29 0，then Q> P0一 P1．Moreover，since A is stable，P1 will be bounded for any ≤ ∞． Hence，there exists sufficient large 1 such that whenever ≥ 1≥ 。， (261——1)≥ (6——1) o-⋯ (B Pl B)and all the conditions in Corollary 2 are satisfied． Case B：As — 。，bl≤ 0．5．Since (b，b1)> 0 if there isn’t saturation constraint，b1 ≤ 0．5 is only due to the heavy constraints on control actions by saturation．According to e— quation(5)and the reason stated in Case A we can conclude：for any arbitrary large ll ll， there is a sufficiently large 2，such that whenever ≥ 2≥ 0，h doesn’t violate saturation constraint，that is，take ≥ 2 then b1> 0．5． In a word，if — n doesn’t satisfy the conditions in Corollary 2，then choosing ≥ max{ l， 2)and suitable{QN，Q，N )may satisfy it．Therefore，the theorem holds． 口 The concrete design procedure is referred to the following simulation example． 5 Simulation example The system adopted is open—loop unstable，where ． 2 0．34 0．12 ] 2] A — l。．123。．223 ：：； l，B—l 3 。l and c一[ 0 。0：]． 1 0．3 0．3 0．4 1 J l 0．3 1．2 J It is easy to know that the linear system is controllable and observable．The Hammerstein nonlinearity is described by ， 、 f0．04480。一0．2512 。+1．28020， l l≤ 2 z1一 l 1 一1 0．99220， 1 0 I>2 and (0)一 The saturation constraints are一1≤u1≤1 and一1≤u2≤ 1．sat {0)一sign{ )· min{1，l 1)，i一1，2． Adopt TSMPC proposed in this paper．The initial state is 0一EO．55，一0．6，0．55， 一 0．61．Tune the system by applying the guideline Step1～Step5 given above．In the sim— ulation，the linear part always satisfies condition i)in Theorem 1．And since u2 has never saturated in the simulation，in the following we concentrate on the computation of u1． Case a．Hammerstein nonlinearity is not considered in computation of u1，that is，gl— sat1．The parameters are chosen as N一 4， 一0．005，Q— C C，QN一0．1I4+ Q+A QNA — A QNB (aI2+ B QNB) B QNA where I2(I4)is second—order(fourth—order)identity matrix．This choice of Q is equivalent to quasi—infinite horizon control j． Case b．Consider Hammerstein nonlinearity，that is，the formula used to calculate u1 is ， ， 、 f satl{一0．0488(z}) +0．2674(z})。+0．7075x~)， l z}l≤2 Ul—g (z 1 f {0．9963z}}， I z}I>2 Other parameters are chosen the same as Case a． Case c．Same as Case b except that 一 20． The simulation results are shown in Fig．4，where (a)，(b)，(c)correspond to the a— bove three cases，and 1，2 to system output and input respectively．Solid and dotted lines mean two output(input)variables． M oreover，to illustrate whether all the conditions in Theorem 1 are satisfied，Fig．4(a3)，(b3) and (c3) protract c 一 一 ( ) (2x 一 )+ (x 一 }) B P1B(x 一 })，named as stability condition testing curves．That is，when ck ≤ 0，the condition ii)in Theorem 1 is satisfied at time是． For Case a，according to the stability condition testing curve we know that the condi— tion ii)in Theorem 1 is not satisfied．The simulation results Fig．4(a1)and (a2)also show that the system is unstable．For Case b，h1 does not satisfy the condition ii)in Theorem 1． The svstem is stable but is not of good quality．And for Case c，the conditions in Theorem 1 are alwavs satisfied．The system is stable and the control quality is better than in (b)． The simulation results above illustrate the conclusions in Theorem 1． 维普资讯 http://www.cqvip.com No．6 DING Bao—Cang et a1．：Stability Analysis on Predictive Control of Discrete-Time⋯ 833 O — O 一 1 testin g curve O — O 一 1 testin g curve Fig．4 Simulation results，where the abscissa is the sample time (a1)～ ( 3) = 0．005，not considering Hammerstein nonlinearity；(b1)～ (b3) 一 0·005’considering Ham— merstein nonlinearity；(。1)～ (。3) 一 20，considering Hammerstein nonlinearity 6 Conclusion This paper studies a two—step predictive controller for systems with input nonlineari— ties，including Hammerstein nonlinearity， saturation constraint and static uncertainty． Soree stability conclusions are obtained．M ost of the existing stable predictive controllers dealing with real constraints add extra artificial constraints and discuss the feasibility prob— lem after having guaranteed the stabilityE ]． Deducing the stability property of TSM PC for systems with constraints and／or input nonlinearities，on the other hand’starts from the feasible linear controllaw ，and its main problem iS to investigate the domain Ot attrac— tion of this kind of systems。The fact that the latter does not add artificial constraints is a main characteristic of the studies in this paper． Deducing the domain of attraction based-on (8)needs further investigation． M oreo— ver。TSM PC with state observer(i．e．，output feedback TSM PC)，robustness of TSM PC for the linear Dart with uncertainties，etc．，all deserve further investigation．The results in this paper can serve as a basis for these future studies． Refefences 1 Fruzzetti K P，Palazoglu A ，M cdonald K A．Nonlinear model predictive control using Hammerstein models·Journal ofProcess Control，1997，7(1)：31～41 维普资讯 http://www.cqvip.com 834 ACTA AUT0M ATICA SINICA Vo1．29 Zhu Q M，Warwich K，Douce J L．Adaptive genera1 predictive controller for nonlinear systems．Proceedings of the Institute of Electrical Engineering，Part D，1991，138(1)：33～40 Zhu X F，Seborg D E．Nonlinear predictive control based on Hammerstein models．Control Theory and Application， 1994。11(6)：564～575 De Nicolao G，M agni L。Scattolini R．On the robustness of receding—horizon control with terminal constraints．n EE Transactions on Automatic Control，1996，41(3)：451～ 453 De Dona J A，Goodwin G C．Elucidation of the state-space regions wher