计算机控制系统 Ch5

1、125.1 IntroductionnConventional methods such as root-locus and frequency-response methods are useful for dealing with single-input-single-output systems.nConventional methods are simple, but applicable only to linear time-invariant systems having a single input and single output.nConventional method

2、s are based on the input-output relationship of the system, i.e. the transfer function or the pulse transfer function.35.1 IntroductionnConventional methods do not apply to nonlinear systems except in simple cases.nConventional methods do not apply to the design of optimal and adaptive control syste

3、ms, which are mostly time varying and/or nonlinear.nThe state-space methods for the analysis and synthesis of control systems are best suited for dealing with multiple-input-multiple-output systems.45.1 IntroductionnConcept of the state-space Method.The state-space method is based on the description

4、 of system equations in terms of n first-order difference equations or differential equations, which may be combined into a first-order vector-matrix difference equation or differential equation.n State: The state of a dynamic system is the smallest set of variables (called state variables) such tha

5、t the knowledge of these variables at t=t0, together with the knowledge of the input for tt0, completely determines the behavior of the system for any time tt0.55.1 Introductionn State Variables: variables making up the smallest set of variables that determine the state of the dynamic system. n Stat

6、e Vector: If n variables are needed to completely describe the behavior of a given system, then these n state variables can be considered the n components of a vector x.n State Space: The n-dimensional space whose coordinate axes consist of x1-axis, x2-axis, , xn-axis is called a state space. Any st

7、ate can be represented by a point in the state space.n State-space equations: The state space representation for a given system is not unique.65.1 IntroductionFor time-varying (linear or nonlinear) discrete-time systems, the state equation may be written asAnd the output equation asFor linear time-v

8、arying discrete-time systems, the state equation and output equation maybe simplified to(1) ( ), ( ), x kf x k u k k( ) ( ), ( ), y kg x k u kk(1)( ) ( )( ) ( )( )( ) ( )( ) ( )x kG k x kH k u ky kC k x kD k u k75.1 Introductionwhere( )vector (state vector)( )vector (output vector)( )vector (input v

9、ector)( )matrix (state matrix)( )matrix (input matrix)( )x kny kmu krG kn nH kn rC km n matrix (output matrix)( )matrix (direct transmission matrix)D km r 85.1 IntroductionFor linear time-invariant discrete-time systems, the state equation and output equation maybe simplified to( ) ( ), ( ), ( ) ( )

10、, ( ), x tf x t u t ty tg x t u t t(1)( )( )( )( )( )x kGx kHu ky kCx kDu k( )( ) ( )( ) ( )( )( ) ( )( ) ( )x tA t x tB t u ty tC t x tD t u t( )( )( )( )( )( )x tAx tBu ty tCx tDu tLinear time-varyingLinear time-invariant9Block diagram of system represented in state space10Outline of the chapternI

11、ntroductionnState-space representations of LTI discrete-time systemsnSolution of discrete-time state-space equationsnPulse-transfer function matrixnDiscretization of continuous time state space equations115.2 State space representationsnCanonical Forms for Discrete-Time State-Space EquationsConsider

12、 the discrete time system described by10110111101( )(1).()( )(1).()(55)It can be written in the form of the pulse transfer function as.( )(56)( )1.or.( )( )nnnnnnnny ka y ka y knb u kbu kb u knbb zb zY zU za za zb zb zbY zU z11(57).nnnnza za125.2 State space representationsnControllable Canonical Fo

13、rm.The state space representation of the discrete time system given by (5-5,6,7) can be given by the following controllable canonical form:112211121011 01(1)0100( )0(1)0010( )0( )(58)(1)0001( )0(1)( )1( )nnnnnnnnnnnx kx kx kx ku kxkxkx kaaaax ky kba b babb 121 00( )( )( )(59)( )nx kx kabb u kx k135.

14、2 State space representationsnObservable Canonical Form.The state space representation of the discrete time system given by (5-5,6,7) can be given by the following observable canonical form:1102211 011122 0211 01(1)( )000(1)( )100( )(5 12)(1)( )000(1)( )001(nnnnnnnnnnx kx kba bax kx kbabau kxkxkba b

15、ax kx kbabay120( )( )0 00 1( )(5 13)( )nx kx kkb u kx k 145.2 State space representationsnDiagonal Canonical Form.If the poles of the pulse transfer function given by (5-5,6,7) are all distinct, then the state space representation can be given by the following diagonal canonical form:1112221111212(1

16、)000( )1(1)000( )1( )(5 16)(1)000( )1(1)000( )1( )( )( )( )nnnnnnnnx kpx kx kpx ku kxkpxkx kpx kx kx ky kc ccx k 0( )(5 17)b u k155.2 State space representationsnJordan Canonical Form.If the poles of the pulse transfer function given by (5-5,6,7) involve a multiple pole of order m at z=p1, and all o

17、ther poles are distinct, then the state space representation can be given by the following Jordan canonical form:165.2 State space representationsobservable canonical formcontrollable canonical form175.2 State space representationsdiagonal canonical form185.2 State space representationsnNon-uniquene

18、ss of state-space representations.For a given pulse transfer function system, the state space representation is not unique. Consider the system(1)( )( )(520)( )( )( ) (5-21)Let us difine a new state vector ( ) by ( )( ) (5-22)x kGx kHu ky kCx kDu kx kx kPx k195.2 State space representations205.2 Sta

19、te space representations215-3 Solving discrete time state space equationsnSolution of LTI discrete time state equationnRecursive procedurenZ transform methodnComputing (zI - G)-1nSolution of the linear time-varying discrete time state equation225-3 Solving discrete time state space equationsnSolutio

20、n of LTI discrete time state equationnRecursive procedureConsider the system(1)( )( )(5-28)( )( )( ) (5-29)The solution of Eq. (5-28) for any integer k may be obtaineddirectly by recursion, as followsx kGx kHu ky kCx kDu k235-3 Solving discrete time state space equations2(1)(0)(0)(2)(1)(1)(0)(0)(1)x

21、GxHuxGxHuG xGHuHu1110110( )(0)(0)(1)(0)( )1,2,3, (530)( )(0)( )( ) (531)kkkkkjjkkkjjx kG xGHuHu kG xGHu jky kCG xCGHu jDu k 245-3 Solving discrete time state space equationsnState Transition Matrix It is possible to write the solution of the homogeneous state equation 255-3 Solving discrete time sta

22、te space equationsnThe unique matrix (k) is called the state transition matrix. And output equation as265-3 Solving discrete time state space equationsnSolution of LTI discrete time state equationnZ transform method275-3 Solving discrete time state space equations285-3 Solving discrete time state sp

23、ace equationsExample 5-2Consider the system(1)( )( )( )( )x kGx kHu ky kCx kwithObtain the state transition matrix and then obtain state x(k) and output y(k) when the input u(k)=1 for k=0,1.295-3 Solving discrete time state space equations305-3 Solving discrete time state space equations315-3 Solvin

24、g discrete time state space equations325-3 Solving discrete time state space equations335-3 Solving discrete time state space equationsnComputing (zI-G)-11()()|withadj zIGzIGzIG345-3 Solving discrete time state space equationsThe ais can be given by use of the trace ( The trace of a nxn matrix is th

25、e sum of its diagonal elements) :355-3 Solving discrete time state space equationsnSolution of the linear time-varying discrete time state equationConsider the system(1)( ) ( )( ) ( )(5-52)( )( ) ( )( ) ( ) (5-53)Its solution maybe found easily by recursion, as followsx kG k x kH k u ky kC k x kD k

26、u k(1)( ) ( )( ) ( )(2)(1) (1)(1) (1)(1) ( ) ( )(1)( ) ( )(1) (1)x hG h x hH h u hx hG hx hH hu hG hG h x hG hH h u hH hu h365-3 Solving discrete time state space equations37nSummary on (k,h)5-3 Solving discrete time state space equations385-4 Pulse-Transfer-Function MatrixnA single-input-single-out

27、put discrete time system may be modeled by a pulse transfer function.nMultiple-input-multiple-output discrete system pulse-transfer-function matrixnRelationship between state-space representation and representation by the pulse-transfer-function matrix?395-4 Pulse-Transfer-Function MatrixnPulse-Tran

28、sfer-Function MatrixThe state-space representation of an nth-order linear time-invariant discrete-time system with r inputs and m outputs can be given by where x(k) is an n-vector, u(k) an r-vector, y(k) an m-vector, G a nxn matrix, H an nxr matrix, C an mxn matrix, and D an mxr matrix.(1)( )( )(558

29、)( )( )( ) (5-59)x kGx kHu ky kCx kDu k405-4 Pulse-Transfer-Function MatrixTaking Z tranform of Equations (5-58) and (5-59), we obtain Assume zero initial state x(0), then we have( )(0)( )( )( )( )( )zX zzxGX zHU zY zCX zDU z111( )()( )and( ) () ( )( ) ( )where ( )() (5-60)( ) ispulse-transfer-funct

30、ion matrix. called the X zzIGHU zY zC zIGHD U zF z U zF zC zIGHDF z415-4 Pulse-Transfer-Function Matrix The pulse-transfer-function matrix F(z) characterizes the input-output dynamics of the given discrete time system. 1Since the inverse of matrix () can be written as()()|the pulse-transfer-function

31、 matrix F(z) can be given by the equation()( )|The poles of F(z) are the zeros zIGadj zIGzIGzIGCadj zIG HF zDzIG12121of | 0. Then the characteristic equation of the discrete time system is given by| 0or0nnnnnzIGzIGza za zaza425-4 Pulse-Transfer-Function MatrixnSimilarity TransformationFor the system

32、The pulse-transfer-function matrix isDefine a new state vector (1)( )( )( )( )( )x kGx kHu ky kCx kDu k1( )()F zC zIGHD( )( )x kPx k435-4 Pulse-Transfer-Function MatrixThen we have445-4 Pulse-Transfer-Function MatrixnThe pulse-transfer-function matrix is invariant under similarity transformation.nTh

33、e characteristic equation (zI-G)=0 is invariant under similarity transformation.nThe eigenvalues of G are invariant under similarity transformation.455-5 Discretization of continuous-time state-space equationsnReview of solution of continuous-time state equationsThe matrix exponential is defined by

34、It can be differentiated term by term as465-5 Discretization of continuous-time state-space equationsThe matrix exponential has the property If s=-t, then The inverse of eAt is e-At 。 eAt is nonsingular。Please note: 47nSolution of the continuous-time state equationRewrite it as 5-5 Discretization of

35、 continuous-time state-space equations485-5 Discretization of continuous-time state-space equationsnDiscretization of continuous-time state-space equations.Consider the continuous-time state equation and output equationThe discrete time representation of Equation (5-66) is(566)(567)xAxBuyCxDu(1)() (

36、)() ()(568)xkTG T x kTH T u kT495-5 Discretization of continuous-time state-space equationsAssumeWhere =T-t.Define We haveNote: G(T) and H(T) depend on the sampling period T. 0( )(573)( )() (574)ATTAG TeH TedB(1) )( ) ()( ) ()(575)x kTG T x kTH T u kT50For nonsingular A， 5-5 Discretization of contin

37、uous-time state-space equationsComments：1. The discrete-time state equation given by Equation (5-68) is called zero-order hold equivalent of the continuous-time state equation given by Equation (5-66).2. Equation (5-75) involves no approximation, provided the input vector is constant between any two consecutive sampling instants.3. As the sampling period T becomes very small, G(T) approaches the identity matrix.515-5 Discretization of continuous-time state-space equationsExample 5-4Consider the continuous-time system given by-0( )1( )( )The continuous-time state-space