现代控制系统分析与设计

1、Department of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory现代控制理论Modern Control Theory刘向杰 教授 Email: 华北电力大学控制科学与工程学院Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 Introductionn1.1 Motivation nWhat is a system?lA physical pr

2、ocess or a mathematical model of a physical process that relates one set of signals to another set of signals Two general categories of signals/systems: Continuous-time (CT) signals/systems Examples: Speed/car, current/circuit, temperature/room Described by differential eqs., e.g., dy/dt = ay(t) + b

3、u(t) Signals themselves could be discontinuous. But defined for each time instant. System input/ excitation/ cause output/ response/ result Examples: Air conditioner, cars, circuits, bank accountsDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 Introdu

4、ction Discrete-time (DT) signals/systems Examples: Money in a bank account, quarterly profit Sequence of numbers Input/output related by difference equations, e.g., yk+1 = ayk + buk, (on a daily or monthly base) DT and CT are quite similar, and will be treated in parallel The goal of System Theory?

5、Establish input/output relationship through models, Predict output from input, know how to produce desired output Alter input automatically (via controller) to produce desired output ControllerSysteminputoutputrequirement k y(k) Department of AutomationNorth China Electric Power UniversityNCEPUModer

6、n Control TheoryChapter 1 IntroductionUse physical laws to model/describe the behavior of the system: lWhat are the components? What properties do they have?dtdvCi,dtdiLv,RivCCLLRR Relationship among the variables by physical law: KCL: Current to a node = 0, iR= iC= iL= i. KVL: Voltage across a loop

7、 = 0. Example: A simple electric circuit Output R L C u(t) i(t) Input Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 Introduction)()(1:00tuvdiCdtdiLRiKVLtlAn integral-differential or differential equationlInput-output description or external descript

8、ionHow to analyze the input-output relationship?lFor example, find the output i(t) given u(t) and IC.We can use Laplace transform -Note: only effective for LTI systems Output R L C u(t) i(t) Input Department of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory Chapter 1 Intro

9、ductionLaplace Transform, A Quick ReviewnConverting linear constant coefficient differential equations into algebraic equationsnOther propertieslDifferentiation in the frequency domain: tf (t) (-1)F (s) lConvolution: h(t)f (t) H(s)F(s)lTime and frequency shifting: f (t-t0) u(t-t0) e-st0 F(s); l es0t

10、 f (t) F(s - s0)-0)()()(dtetfsFtfst)()()()(22112211sFasFatfatfa-ssFdtffssFtf/ )()(),0()()( Key Properties Linearity: Derivative theorem: Department of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory Chapter 1 IntroductionlTime and frequency scaling: f (at) 1/a F(s/a) for a

11、0 lInitial Value Theorem: f (0+) = lims sF(s)lFinal Value Theorem: f () = lims0 sF(s) if all the poles of sF(s) have strictly negative real parts)()(1:00tuvdiCdtdiLRiKVLt)( )()()(00susvCssiisi sLsiR- Output R L C u(t) i(t) Input Example (Continued)Department of AutomationNorth China Electric Power U

12、niversityNCEPUModern Control Theory Chapter 1 Introduction1)( 1)(2002-RCsLCscvLCsisuRCsLCscssiIs there any pattern with the equation?lIt has two components, one caused by input, and the other by IC How about the voltage across the capacitor? An algebraic equation vs integral-differential equation. S

13、olution:1)( 11)()( 20020RCsLCsvRCLCsLisuRCsLCssvCssisv-svLisusicsRLs00)( )(1 What is the systems transfer function?Department of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory Chapter 1 Introductiong(s) i(s) = g(s) u(s) u(s) lFrequency domain analysisHow to obtain the resp

14、onse in time domain? Assume that the ICs are zero, then)()(1siLti-) s (u 1RCsLCsCs) s ( i21RCsLCsCs) s (g 2 Suppose that L = C = 1, R = 2, v0 = i0 = 0, and u(t) = U(t) (unit step function). ThenDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory Chapter 1 Introduc

15、tion2221s11s2s11RCsLCs) s (u Cs) s ( is1) s (u tte) t ( i-00.050.10.150.20.250.30.350.4051015ti(t)Department of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory Chapter 1 IntroductionLimitation of Laplace transform: not effective for time varying/nonlinear systems such as Th

16、e state space description to be studied in this class will be able to handle more general systems How can we do it? Properties can be characterized without solving for the output )(),()(),()(),()(01tutybtytyatytyaty To get some general idea about state space description, we consider the same circuit

17、 system.Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 Introduction Output R L C u(t) i(t) Input State-Space Description State variables: Voltage across C and current through L uLvLiLRdtdiiCdtdv)/1 ()/1 ()/()/1 (-nA set of first-order differential eq

18、uationsuLivLRLCiv-/10/1/10Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 Introduction It describes the behaviors inside the system by using the state variables v(t) and i(t) How to describe the output?iviy10 The output equation Combined with the stat

19、e equation, we have the state-space description or internal description Output R L C u(t) i(t) Input v(t)-ivyuLivLRLCiv10,/10/1/10CxyBuAxxxABCx Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 Introduction A general form:qpnyyyyuuuuxxxxtuxhytuxfx212121

20、,),(),( Main features of the state-space approach It describes the behaviors inside the systemCharacterizes stability and performances without solving the differential equationsApplicable to more general systems, nonlinear, time-varying, uncertain, hybridMost recent advancements in control theory ar

21、e developed via a state-space approachDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 IntroductionTextbook:lChi-Tsong Chen, Linear System Theory and Design, 3rd Edition, Oxford University Press, 1999 (Required)Reference: l刘豹 现代控制理论。 机械工业出版社，2005。 1.2

22、Course OverviewDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 IntroductionnGoals: To achieve a thorough understanding about systems theory and multivariable system design nTentative Outline :lIntroduction lModeling: How to model a physical system lTh

23、e fundamentals of linear algebra lAnalysis: uQuantitative: How to derive response for a given input uQualitative: How to analyze controllability, observability, stability and robustness without knowing the exact solution?Department of AutomationNorth China Electric Power UniversityNCEPUModern Contro

24、l TheoryChapter 1 IntroductionlDesign: uHow to realize a system given its mathematical description uHow to design a control law so that desired output response is produceduHow to design an observer to estimate the state of the system lWill treat continuous-time and discrete-time systems in parallelD

25、epartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 1 IntroductionnPrerequisites:lLinear Feedback SystemslBackground on uLinear algebra: Matrices, vectors, determinant,u eigenvalue, solving a system of equationsuz-transformuOrdinary differential equationsuLa

26、place transform, and uModeling of electrical and mechanical systemsDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control Theory Chapter 2 Mathematical Descriptions of SystemsnBasic assumption: When an input signal is applied to the system, a unique output is obtainednQ. Ho

27、w do we classify systems?lNumber of inputs/outputs; with/without memory; causality; dimensionality; linearity; time invariancenThe number of inputs and outputslWhen p = q = 1, it is called a single-input single-output (SISO) systemlWhen p 1 and q 1, it is called a multi-input multi-output (MIMO) sys

28、temlMISO, SIMO defined similarly2.1 Classification of SystemsDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of Systems Memoryless vs. with Memory If y(t) depends on u(t) only, the system is said to be memoryless, otherwise,

29、it has memory An example of a memoryless system? + - u(t) R1 R2 + y(t) - A purely resistive circuit) t (uRRR) t (y212 MemorylesslAn example of a system with memory? + - u(t) R L i udtdiLRiuLiLRdtdior1-tttLRttLRdueLtieti00)(1)()(0Department of AutomationNorth China Electric Power UniversityNCEPUModer

30、n Control TheoryChapter 2 Mathematical Descriptions of Systems i(t) depends on i(t0) and u() for t0 t, not just u(t) A system with memory Causality: No output before an input is applied Input System Output A system is causal or non-anticipatory if y(t0) depends only on u(t) for t t0 and is independe

31、nt of u(t) for t t0 Is the circuit discussed last time causal? + - u(t) R L i t u(t) 1 t y(t) 1 Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemslAn example of a non-causal system?ly(t) = u(t + 2) t u(t) 1 t y(t) 1 C

32、an you truly build a physical system like this? All physical systems are causal!Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsnThe Concept of StatelThe state of a system at t0 is the information at t0 that, togeth

33、er with ut0,), uniquely determines the behavior of the system for t t0 lThe number of state variables = the number of ICs needed to solve the problemlFor an LRC circuit, the number of state variables = the number of C + the number of L (except for degenerated cases)lA natural way to choose state var

34、iables as what we have done earlier: vc and iL lIs this the unique way to choose state variables?Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemslAny invertible transformation of the above can serve as a state, e.g.

35、, )()()(2)()(1012)()(21tititvtitvtxtx Although the number of state variables = 2, there are infinite numbers of representations Order of dimension of a system: The number of state variables If the dimension is a finite number Finite dimensional (or lumped) system Otherwise, an infinite dimensional (

36、or distributed) system Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of Systems An example of an infinite dimensional system u(t) System y(t) = u(t-1) A delay line t u(t) t y(t) 1 Given u(t) for t 0, what information is nee

37、ded to know y(t) for t 0? We need an infinite amount of information An infinite dimensional systemDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsLinearitynDouble the efforts double the outcome?lSuppose we have the

38、following (state,input)-output pairs:010101),(),()(tttytttutx020202),(),()(tttytttutx0210201),()()()(tttututxtx What would be the output of021),()(tttyty2.2 Linear SystemsDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of Sys

39、temslIf this is true AdditivitylHow about0101),()(tttutx01),(ttty02211022011),()()()(tttututxtx If this is true Homogeneity Combined together to have:)()(2211tyty If this is true Superposition or linearity property A system with such a property: a Linear SystemDepartment of AutomationNorth China Ele

40、ctric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsnAlso, KVL and KCL are linear constraints. When put together, we have a linear system Are R, L, and C linear elements?dtdvCi,dtdiLv,RivCCLLRR Yes (differentiation is a linear operation) i v v = Ri i v Linea

41、rNonlinearDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsnThe additivity property implies that Response = zero-input response + zero-state response0101),()(todue)(tttutxty0101101),(0)(todue)(0)()(todue)(tttutxtytut

42、xtynA linear system can be described by)()()()()()()()()()(tutDtxtCtytutBtxtAtxState-Space DescriptionDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsnTime Invariance: The characteristics of a system do not change o

43、ver timelWhat are some of the LTI examples? Time-varying examples?lWhat happens for an LTI system if u(t) is delayed by T? t u(t) y(t) t u(t-T) If the same IC is also shifted by T, then2.3 Linear Time-Invariant (LTI) SystemsDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Con

44、trol TheoryChapter 2 Mathematical Descriptions of SystemslThis property can be stated as:00(0)( ),( ),0 xxy tttu tt0( )(),(),x Txy t TtTu t TtT-Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 3 Mathematical Descriptions of SystemsnA linear system can be

45、 described by)()()()()()(tDutCxtytButAxtxState-Space Description) s ( ) s ( ) s ( ) s ( ) s ( )0() s ( suDxCyuBxAxx-Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of Systems) s ( ) s ( )()0()() s ( ) s ( )()0()() s ( 1111uDu

46、BAICxAICyuBAIxAIx-ssss) s ( )() s ( 1uDBAICy-sDBAICG-1)() s (sIf the initial state is zero, thenDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsConsider the system shown in the figure. It consists of two blocks, wit

47、h masses and , connected by three springs with spring constants and as shown. To simplify the discussion, we assume that there is no friction between the blocks and the floor. The applied force must overcome the spring forces and the remainder is used to accelerate the block1m2m1k2kDepartment of Aut

48、omationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of Systemswe have 11212111)(ymyykyku -or 12212111)(uykykkym- For the second block, we have21222122)(uykykkym- Let us define24231211 yxyxyxyxDepartment of AutomationNorth China Electric Power Uni

49、versityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsThen we can readily obtain-2121432122122121214321 100001000)(0100000)(0010uummxxxxmkkmkmkmkkxxxxxy 0100000121yyDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical

50、Descriptions of SystemsConsider a cart with an inverted pendulum hinged on top of it as shown in the figure. For simplicity, the cart and the pendulum are assumed to move in only one plane, and the friction, the mass of the stick, and the gust of wind are disregarded. The problem is to maintain the

51、pendulum at the vertical position. Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsLet H and V be, respectively, the horizontal and vertical forces exerted by the cart on the pendulum as shown. The application of Ne

52、wtons law to the linear movements yields HudtydM-22cos)(sin)cos(222 -mlldtdmVmgsin)(cos)sin(222 mlmlymlydtdmH-Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsThe application of Newtons law to the rotational movement

53、 of the pendulum around the hinge yields 1cos sinVmg SinceWe obtain mlymy- uMylg Which imply mgy- uM ugmMMl-)( cossingl ymlmllm Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsLet us define 4321 xxyxyxu-MlMxxxxMlgmM

54、Mmgxxxx10100)(001000000001043214321xy 0001Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsConsider the network shown in the figure. We assign the Ci-capacitor voltages as xi, i=1,2 and the inductor current as x3. Th

55、en their currents and voltage are, respectively The voltage across the resistor is , Thus its current is 32211 ,xLxCxC1xu -Rxu)(1-Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsApplying Kirchhoffs current law at no

56、de A yield ; at node B it yields 31122111)(xxCxCxCRxu-Thus we have 113111RCuCxRCxx-232Cxx Applying Kirchhoffs voltage law to the right-hand-side loop yield Lxxx)(213-The output y is given by 213xxxLy-322xxCDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter

57、 2 Mathematical Descriptions of SystemsThey can be combined in matrix form as uxx-0010111001011211RCLLCCRCxy 011-Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of SystemsHome work1. Find a state equation to describe the netw

58、ork shown in the figure. Find also its transfer functionDepartment of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 2 Mathematical Descriptions of Systems2. Find a state equation to describe the network shown in the figure. Find also its transfer functionDepartment

59、 of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 3 Linear AlgebraConsider and n-dimensional real linear space, denoted by . Every vector in is an n-tuple of real numbers such as nnn21 xxxxTo save space, we write it as where the prime denotes the transpose n21 xxxx

60、Department of AutomationNorth China Electric Power UniversityNCEPUModern Control TheoryChapter 3 Linear AlgebraThe set of vectors in is said to be linearly dependent if there exist real number not all zero, such that m,xxx21n0 xxxm21m21If the only set of for which the above equation holds is , then