1、Chapter 7 Analysis and Design of Linear Discrete-Time System （Sampled-data System）7.1 Introduction7.2 The Sampling Process and Sampling Theorem7.3 Signal Recovery and Zero-Order Hold 7.4 Z-Transform and Inverse Z Transform 7.5 Mathematical Models of Discrete-Time Systems7.6 Performance Analysis of D

2、iscrete-Time Systems7.7 Digital Control Design for Discrete-Time Systems7.1 Discrete-Time Control Systems Discrete-systems: There is one or more impulse series or digital signals in the system. Sampled-Data System: a system that is continuous except for one or more sampling operations.kTkTtTttkTtt)(

3、)()()()( -2 T -T 0 T 2 T Unit Impulse sequence1、Unit Impulse Sequence (Unit Impulse Train)2、Sampling Signal kkTttete)()()(* 7.2 Signal Sampling and Shannon Theorem)()(*teLsE0)(kkTsekTekskjETE1j*hsT 22 Shannon Sampling Theorem The Necessary Condition for signal recoveryhsT 22 Ideal filterSampling swi

4、tchhs 2 hsT 22 hT 3 Frequency Spectrum Analysis of Sampled Signal)( 1)( 1)(Ttttk 7.3 Zero-Order Hold0)( 1)( 1)()(khTkTtkTtkTxtxTssxkkTshessekTxsx11)()()(0*)(1)()(*sGsesxsxhTshUsing L-TransformUsing L-Transform，we getwe get7.4 Z-Transform and Inverse Z Transform7.4.1. Z-transformDefinition： 0*skkTsek

5、TeESet Tsez zTsln1 0kkzkTezETsezsEzE| )()(*0*)()()()(nkezzkTesEtezETsRmk：)()()()()(*teZsEZsEZtezE The z-transform is only for discrete signal.E(z) is only mapping to a unique e*(t), but not a unique e(t).7.4.2 Methods of z-Transform 0*)()(kkTsekTesE1Tsze By the definition.Partial fraction expansion.

6、 )()()()(21*2*1zEbzEatebteaZ 1. linear property )()(zEznTteZn 7.4.3 Properties of z-Transform 2. Real shifting theorem 實位移定理實位移定理 Lag 延時定理延時定理 10)()()(nkknzkTezEznTteZ Lead 超前定理超前定理 aTtaezEeteZ )(3. Complex shifting theorem 復位移定理復位移定理 )(lim)(lim0zEnTezn 4. Initial value Theorem )()1(lim)(lim1zEznTez

7、n 5. Final value theorem 7.4.4 Inverse z-Transform Long Division（長除法）（長除法）Partial-Fraction expansionResidue（留數法）（留數法）zzE)(Expansion of 1)(Res)( nzzEnTe Inverse Z-transform can only provide discrete-time signal x*(t), instead of continuous signal x(t)。 Tips:1ZX zx nT ( )()7.5.1 Linear Time-Invariant

8、Difference Equations (1) Definition of difference Forward Backward 7.5 Mathematical Models of Discrete-Time Systems (2) The difference equation and its solving method Iteration Z-transformation7.5.2 Impulse-Transfer Function (1) Definition (2) Properties7.5.3 Impulse Transfer Function of Open-Loop S

9、ystems (1) Switch between factors(2) No switch between factors(3) With ZOH7.5.4 Impulse Transfer Function of Closed-Loop Systems (1) General Method (2) Masons formula7.6 Performance Analysis of Discrete-Time Systems Stability Dynamic Performance Steady-state Errorss-Domain to z-Domain Mapping Necess

10、ary and Sufficient Condition for Stability of Linear Discrete-Time Systems All poles of F F(z) lie in the unit circle of z planeRouth criterion in w domain (Generalized Routh Criterion) 7.6.1 StabilitysTze11zwzweve learned three methods to determine the stability of a discrete-time systems.7.6.2 Ana

11、lysis of discrete-time dynamic performance Obtain s s, ts by definition (1)General method0( )( )( )()nnG zzC zc nT z F F *0( )() ()nc tc nTtnT (2) Closed-loop poleskp Response()nkkkcnTC p ()1, 2,nkkkcnTc pkn1kp1kp1kp：0kp:0kp，1kp1kp7.6.3 Steady-state error of discrete systemsFinal value theorem( )( )

12、eG zz F F1()lim(1) ( )( )ezezR zz F F( )D z Stability(3) Static error constant( ),pvaG zv KKK()e ObtainGeneral method: obtain system response0011( )( )(1)lim( )vzGH zGHzzGHzK pKA最少拍設計中，最少拍設計中， F F(z)(z)和和F Fe(z) )選取時應遵循的原則：選取時應遵循的原則：1。D(z)零點的數目不能大于極點的數目；零點的數目不能大于極點的數目；2。 F Fe(z) )應把應把G(z)G(z)在單位圓上及單位圓外的極點作為自己在單位圓上及單位圓外的極點作為自己的零點；的零點；3 3。 F F(z)應把應把G(z)G(z)在單位圓上及單位圓外的零點作為自己在單位圓上及單位圓外的零點作為自己的零點；的零點；4。當。當G(z)含有含有z -1因子時，要求因子時，要求F F(z)也含有也含有z -1的因子；的因子；5。 因為因為F F(z)=1- F Fe(z) )，他們應該是關于，他們應該是關于z -1同樣階次的多同樣階次的多項式，而且項式，而且F Fe(z) )還應包含常數項還應包含常數項1 1。6。當最小拍系統還有無紋波要求時，閉環脈沖傳函。當最小拍系統還有無紋波要求時，閉環