系统的能控性和能观性英文版本

1、Unit 13 Controllability and ObservabilityA system is said to be controllable at time t0 if it is possible by means of an unconstrainedcontrol vector to transfer the system from any initial state x(t0)to any other state in a finite 矚慫润厲钐瘗睞枥庑赖。interval of time. A system is said to be observable at tim

2、e t0 if, with the system in state x(t 0) ,聞創沟燴鐺險爱 氇 谴 净 。it is possible to determine this state from the observation of the output over a finite time残骛楼諍锩瀨濟 interval.溆塹籟。The concepts of the controllability and observability were introduced by Kalman. They play an酽锕极額閉镇桧猪訣锥。important role in the desi

3、gn of control systems in state space. In fact, the conditions of 彈贸摄尔霁毙攬砖卤庑。controllability and observability may govern the existence of a complete solution of the control謀荞抟箧飆鐸怼类蒋薔。system design problem. The solution to this problem may not exist of the system considered is 厦礴恳蹒骈時盡继價骚。not controll

4、able. Although most physical systems are controllable and observable, 茕桢广鳓鯡选块网羈泪。corresponding mathematical models may not possess the property of controllability and 鹅娅尽損鹌惨歷 observability.茏鴛賴。Complete State Controllability of Continuous-Time System籟s丛妈羥 为贍偾蛏练淨。Consider the continuous-time systemwhe

5、reu=control signal (scalar)X AX Bu (13. 1) X=state vector ( n-vector) 預頌圣鉉儐歲龈讶骅籴。A= n n matrixB= n 1 matrixThe system described by Equation (13. 1) is said to be state controllable at t t0 if it is 渗釤呛俨匀谔鱉调硯錦。possible to construct an unconstrained control signal that will transfer an initial state t

6、o any铙誅卧泻噦圣骋贶頂廡。final state in a finite time interval t0 t t1 . If every state is controllable, then the system is 擁締凤袜备訊顎轮烂蔷。 said to be completely state controllable.We shall now derive the condition for complete state of controllability. Without loss of贓熱俣阃歲匱阊邺镓騷。 generality, we can assume that t

7、he final state is the origin of the state space and that the initial 坛摶 乡囂忏蒌鍥铃氈淚。 time is zero,or t0 0.The solution of Equation (13. 1) isX(t) eAt X(0) 0 eA(t )Bu( )dApplying the definition of complete state controllability just given, we have 蜡變黲癟報伥铉锚鈰赘。X(t1) 0 eAt1X (0) 01eA(t1 )Bu( )dorX(0)1 e A

8、Bu( )d (13. 2)And e A can be writtenn1e A k ( )Ak (13. 3)k0Substituting Equation (13. 3) into Equation (13. 2) gives 買鲷鴯譖昙膚遙闫撷凄。n 1 tk1X(0)Ak B 1 k ( )u( )d (13. 4)k 0 0Let us putt10 k ( )u( )d kThen Equation (13. 4) becomesn1X(0)Ak B kk00B ABAn 1B1 (13. 5)n1If the system os completely state control

9、lable, then, given any initial state X(0), Equation綾镝鯛駕櫬 鹕 踪 韦 辚 糴 。 (13. 5) must be satisfied. This requires that the rank of the n n matrix 驅踬髏彦浃绥譎饴憂锦。B ABAn 1Bbe n.From this analysis, we can state the condition for complete state controllability as follows.猫虿驢绘燈鮒诛髅貺庑。 The system given by Equation

10、 (13. 5) is completely state controllable if and only if the vectors 锹籁 饗迳琐筆襖鸥娅薔。n1B,AB A Bare linearly independent, or the n n matrixB ABAn 1Bis the rank n.The result just obtained can be extended to the case where the control vector U is 構氽頑黉碩饨 荠龈话骛。 r- dimensional. If the system is described byX

11、AX BUWhere U is an r -vector, then it can be proved that the condition of for complete state 輒峄陽檉簖疖網 儂號泶。 controllability is that the n n matrixB ABAn 1Bbe of rank n, or contain n linearly independent column vectors. The matrix 尧侧閆繭絳闕绚勵蜆贅。B ABAn 1Bis commonly called the controllability matrix.Comple

12、te Observability of Continuous-Time SystemIn this section we discuss the observability of linear systems. Consider the unforced 识饒鎂錕缢灩筧嚌俨淒。system described by the following equationsX AX(13. 6)Y CX(13. 7)whereX=state vector ( n-vector) 凍鈹鋨劳臘锴痫婦胫籴。Y=output vector ( m-vector)A= n n matrixC=m n matrixT

13、he system is said to be completely observable if every state X(t0) can be determined from 恥諤銪灭萦欢煬鞏鹜錦。 the observation of Y(t) over a finite time interval, t0 t t1. The system is, therefore, 鯊腎鑰诎褳 鉀沩懼統庫。 completely observable if every transition of the state eventually affects every element of the 硕癘

14、鄴 颃诌攆檸攜驤蔹。output vector. The concept of observability os useful in solving the problem or reconstructing 阌擻 輳嬪諫迁择楨秘騖。 unmeasurable state variable from measurable variables in the minimum possible length of time. 氬 嚕躑竄贸恳彈瀘颔澩。 In this section we treat only linear, time-invariant systems. Therefore, wi

15、thout loss of 釷鹆資贏車贖孙 滅獅赘。generality, we can assume thatt0 0.The concept of observability is very important because, in practice, the difficulty怂阐譜鯪迳導嘯 畫 長 凉 。 encountered with state feedback control is that some of the state variables are not accessible for 谚 辞調担鈧谄动禪泻類。 direct measurement, with the

16、 result that it becomes necessary to estimate the unmeasurable 嘰觐詿 缧铴嗫偽純铪锩。 state variables in order to construct the control signals. Such estimates of state variables are 熒绐 譏钲鏌觶鷹緇機库。 possible of and only if the system is completely observable. 鶼渍螻偉阅劍鲰腎邏蘞。In discussion observability conditions, we

17、 consider the unforced system as given by 纣忧蔣氳 頑莶驅藥悯骛。 Equation (13. 6) and (13. 7). The reasons for this are as follows, If the system is described by 颖刍 莖蛺饽亿顿裊赔泷。X AX BuY CX ButhenX(t) eAt X(0) 0 eA(t )Bu( )dAnd Y(t) isY(t) CeAtX(0) C eA(t )Bu( )d DuSince the matrices A, B, C, and D are known and

18、u(t) is also known,the last terms on 濫驂膽閉驟羥闈 詔寢賻。 the right-hand side of this last equation are known quantities. Therefore, they may be 銚銻縵哜鳗鸿锓 謎諏涼。 subtracted from the observed value of Y(t). Hence, for investigating a necessary and sufficient 挤貼 綬电麥结鈺贖哓类。 condition for complete observability, it

19、suffices to consider the system described by Equations 赔荊 紳 谘 侖 驟 辽 輩 袜 錈 。 (13. 6) and (13. 7).Consider the system described by Equations (13. 6) and (13. 7). The output vector Y(t) is 塤礙籟 馐决穩賽釙冊庫。Y(t) CeAt X(0)AtAnd ecan be written asHence, we obtainorn1At keAtk(t)Akk0n1Y(t)k (t )CA k X(0)t0(13. 8

20、)Y(t) 0(t)CX(0) 1(t)CAX(0)n 1(t)CAn 1X(0)If the system is completely observable, then, given the output Y(t) over a time interval t0 t 裊樣t1, X(0)is uniquely determined from Equation (13. 8). It can be shown that this requires the仓嫗盤rank of the nm n matrixCCACAn 1to be n.From this analysis we can sta

21、te the condition for complete observability as follows. 绽萬璉轆娛閬 蛏鬮绾瀧。The system described by Equation (13. 6) and (13. 7) is completely observable of and only 骁顾 瀆蕪領鲡赙is the nm n matrixCCACAn1is of rank n or has n linearly independent column vectors. This matrix is called the 瑣钋濺暧惲锟缟馭 篩凉observability

22、 matrix.Key Words and Terms1. controllability n. 可控性2. observability n. 可观测性3. controllable adj. 可控的4. observable adj. 可观测的5. mathematical model 数学模型6. property n. 性质，属性7. continuous-time system 连续时间系统8. generality n. 一般性，普遍性9. rank n. 秩10. linearly independent 线性无关11. time-invariant system 时变系统12.

23、suffice v. 满足NotesAlthough most physical systems are controllable and observable, corresponding 鎦诗涇艳损楼紲鯗 餳類。 mathematical models may not possess the property of controllability and observability. 栉缏歐锄棗鈕 种鵑瑶锬。尽管大多数的物理系统都是可控的和可观测的， 它们所对应的数学模型并不一定具有可 控性和可观测性 .The system os said to be completely observa

24、ble if every state X(t0) can be determined辔烨 棟剛殓攬瑤丽阄应。 from the observation of Y(t) over a finite time interval, t0 t t 1 .峴扬斕滾澗辐滠兴渙藺。如果在有限的时刻 t，t0 t t1，从系统的输出 Y(t) 的观测中能确定每一个状态向量的 初值 x(t0 )，则称系统是完全可观测的 .詩叁撻訥烬忧毀厉鋨骜。The concept of observability is very important because, in practice, the difficulty 则

25、鯤愜韋瘓 賈晖园栋泷。 encountered with state feedback control is that some of the state variables are not accessible for 胀 鏝彈奥秘孫戶孪钇賻。 direct measurement, with the result that it becomes necessary to estimate the unmeasurable 鳃躋峽 祷紉诵帮废掃減。 state variables in order to construct the control signals. 稟虛嬪赈维哜妝扩踴粜。可观

26、测性的概念非常重要， 在实际中， 状态反馈控制中所遇到的困难在于， 一些状态变 量是不能够直接测量的，因此有必要估计不可测量的状态变量来构成控制信号.陽簍埡鲑罷規呜旧岿錟。encountered with state feedback control 为过去分词作定语，修饰 the difficulty 沩氣嘮戇苌鑿 鑿槠谔應。that some of the state variables are not accessible for direct measurement 为表语从句 .钡嵐縣 緱虜荣产涛團蔺。that it becomes necessary to estimate the

27、 unmeasurable state variables in order to construct 懨俠劑鈍触乐鹇烬觶騮。the control signal. 为同位语从句，解释 the result.Exercises1. Consider the system defined by2 x120u1 x3 1x2x10x3x2Is the system completely state controllable?2. Consider the systemx1 x2 x3011023200The output is given byx11 1 x2Show that the syste

28、m is not completely observable.3. Please translate the following paragraph into Chinese. 謾饱兗争詣繚鮐癞别瀘。A system is said to be controllable at time t0 if it is possible by means of an unconstrained 呙铉 們欤谦鸪饺竞荡赚。 control vector to transfer the system from any initial state X(t0) to any other state in a fi

29、nite 莹谐龌 蕲賞组靄绉嚴减。 interval of time. A system is said to be observable at time t 0 if,with the system in state X(t0) ,麸肃 鹏镟轿騍镣缚縟糶。 it is possible to determine this state from observation of the output over a finite time 納畴鳗吶鄖禎銣 腻鰲锬。 interval.Unit 14 Internal Model ControlIn the last chapter we presen

30、ted several methods for tuning PID controllers and developed 風 撵鲔貓铁频钙蓟纠庙。 a model-based procedure (direct synthesis) to synthesize a controller that yields a desired 灭嗳骇諗 鋅猎輛觏馊藹。 closed-loop response trajectory. In this chapter, we first develop an open-loop control design 铹鸝 饷飾镡閌赀诨癱骝。 procedure tha

31、t then leads to the development of an internal model control (IMC) structure. 攙閿频嵘 陣澇諗谴隴泸。 There are a number of advantages to the IMC structure (and controller design procedure), 趕輾雏纨 颗锊讨跃满賺。 compared with the classical feedback control structure. One is that it becomes very clear how 夹覡 闾辁駁档驀迁锬減。

32、process characteristics such as time delays and RHP zeros affect the inherent controllability of 视 絀镘鸸鲚鐘脑钧欖粝。 the process. IMCs are much easier to tune than are controllers in a standard feedback control 偽澀锟 攢鴛擋緬铹鈞錠。 structure.After studying this chapter, the reader should be able to: 緦徑铫膾龋轿级镗挢廟。 De

33、sign internal model controllers for stable process (either minimum or non-minimum騅憑钶銘侥张礫阵轸蔼。 phase ); Sketch the closed-loop response of the model is perfect; 疠骐錾农剎貯狱颢幗騮。 Derive the closed-loop transfer functions for IMC; 镞锊过润启婭澗骆讕瀘。 Design IMC improved disturbances for IMC.Introduction to Model-Bas

34、ed ControlIn the previous chapters we focused on techniques to tune PID controllers. The closed-loop 榿 贰轲誊壟该槛鲻垲赛。 oscillation technique developed by Ziegler and Nichols did not require a mode of the process. 邁茑 赚陉宾呗擷鹪讼凑。 Direct synthesis, however, was based the use of a process model and a desired c

35、losed-loop 嵝硖贪塒 廩袞悯倉華糲。 response to synthesize a control law; often this resulted in a controller with a PID structure.该栎谖 碼 戆 沖 巋 鳧 薩 锭 。 In this chapter we develop a model-based procedure, where a process model is embedded in 劇妆 诨貰攖苹埘呂仑庙。 the controller. By explicitly using process knowledge, by v

36、irtue of the process model, 臠龍讹驄桠业變 墊罗蘄。 improved performances can be obtained.Consider the stirred-tank heater control problem shown in Figure 14. 1. We can use a 鰻順褛悦 漚縫冁屜鸭骞。 model of the process to decide the heat flow (Q) that needs to be added to the process to obtain 穑 釓虚绺滟鳗絲懷紓泺。 a desired tem

37、perature (T) trajectory, specified by the set-point ( Tsp ). A simple steady-state 隶誆荧鉴 獫纲鴣攣駘賽。 energy balance provides the steady-state heat flow needed to obtain a new steady-state 浹繢腻叢着駕 骠構砀湊。 temperature, for example. By using a dynamic model, we can find the time-dependent heat 鈀燭 罚櫝箋礱颼畢韫粝。prof

38、ile needed to yield a particular time-dependent temperature profile. 惬執缉蘿绅颀阳灣熗鍵。Assume that the chemical process is represented by a linear transfer function model, and 贞廈 给鏌綞牵鎮獵鎦龐。 that it is open-loop stable. The input-output relationship is shown in Figure 14. 2(a), where 嚌鲭级厨 胀鑲铟礦毁蕲。 U(s) is the

39、 input variable (heat flow) and Y(s) is the output variable (temperature). 薊镔竖牍熒浹醬籬 铃騫。When the process is at steady state, and there are no disturbances, then the inputs and 齡践砚语 蜗铸转絹攤濼。 outputs are zero (since we are using deviation variables). Consider a desired change in the 绅薮疮颧 訝标販繯轅赛。 output

40、Y(s); we refer to the desired value of Y(s) as the set-point, which is represented R(s). 饪箩狞 屬诺釙诬苧径凛。 We wish to design an open-loop controller, Q(s), so that the relationship between R(s) and 烴毙潜籬 賢擔視蠶贲粵。 Y(s) has desirable dynamic characteristic (fast response without much overshoot, no off-set, 鋝

41、岂涛 軌跃轮莳講嫗键。 etc.).The open-loop control system is shown in Figure 14. 2(b) (we may also wish to think of撷伪氢鱧轍幂聹諛詼庞。 this as a feed-forward controller, based on set-point). We use Q(s) to represent the open-loop 踪飯梦 掺钓貞绫賁发蘄。 controller transfer function, to emphasize that it is a different type of co

42、ntroller than the婭鑠机职銦夾簣軒蚀骞。 feedback controller of previous chapters.Using block diagram analysis, we find the following relationship between the set-point and 譽 諶掺铒锭试监鄺儕泻。 the outputY(s) Gp(s)Q(s)R(s)(14. 1)Static Control LawThe simplest controller will result if Q(s) is constant. Let kp represent

43、 this constant. As 俦聹执kpan example, consider a first-order process, Gp (s)R(s). Then the relationship 缜電怅淺靓ps 1蠐浅 between R(s) and Y(s) isY(s)kqkp R(s)ps 1To obtain a desirable response, kq1kp;offset will result otherwise. We can see this from the骥擯帜褸饜兗椏長绛粤final-value theorem. Consider a step set-po

44、int change, of magnitude R1 癱噴导閽骋艳捣靨骢鍵。(14. 2)Y(s) kpqskp1 Rs1From the final-value theoremltim y(t) lsim0 sY(s) kpkqR1And for no offset, we require that kq1. We can also find the time-domain solution to鑣鸽夺圆kp鯢齙Equation (14. 2)慫餞離y(t) kpkqR1(1 e p )Again, we can see that kqis necessary for offset-fre

45、e performance. Notice also that 榄阈kp团皱鹏緦寿驏頦蕴。 the speed of response is the same as the time constant of open-loop process. In order to speed- 逊 输吴贝义鲽國鳩犹騸。 up the response, we must use dynamic control law, as developed in the next section. 幘觇匮骇儺红卤 齡镰瀉。Dynamic Control LawBetter control can be obtained

46、 of the controller, Q(s), is dynamic rather than static. 誦终决懷区 馱倆侧澩赜。Indeed, we find that ifGp(s)(14. 3)then the relationship between R(s) and Y(s) is1Y(s) Gp(s)Q(s)R(s) Gp (s) 1 R(s) R(s) Gp (s)That is, we have perfect control, since the output perfectly tracks the set-point! For a first- 医涤侣綃 噲睞齒办

47、銩凛。order process, the controller isQ(s) 1 ps 1Although this is mathematically possible, perfect control is unachievable in practical 舻当为遙 头韪鳍哕晕糞Gp(s)kp。 application. Consider the signals in and out of the control block, shown in Figure 2(b). Since 鸪凑鸛 齏嶇烛罵奖选锯 the transfer function relationship betwe

48、en R(s) and U(s) is, for this example, 筧驪鴨栌怀鏇颐嵘悅废。(14. 4)U(s) Q(s)R(s)ps 1R(s)kpThe differential equation that corresponds to Equation (14. 4) is 韋鋯鯖荣擬滄閡悬贖蘊。u(t) p dr 1 r(t)kp dt kp涛貶騸锬晋铩From a practical point of view, it is impossible to take an exact derivative of r(t), 锩 揿 宪 骟 particularly if a d

49、iscontinuous step set-point change is made. 钿蘇饌華檻杩鐵样说泻。Here we use the inverse Laplace transfer to solve Equation (14. 4) for u(t), when there 戧礱風 熗浇鄖适泞嚀贗is a step change in r(t)U(s)pR1 R1 1kpkp sA table of Laplace transform can be used to find the time-domain solution 購櫛頁詩燦戶踐澜襯 鳳。u(t) kpR1 (t) kR1

50、kpkpwhere (t) is the impulse function, which has infinite height, infinitesimal width, and unit嗫奐闃頜瑷踯谫瓒兽粪。 area. Since this is hard to understand conceptually, you probably realize that it is impossible to 虚龉 鐮宠確嵝誄祷舻鋸。 implement exactly. Think about how you would approximate it. 與顶鍔笋类謾蝾纪黾廢。Key Words and Terms1. internal model con