CH2计算机控制技术（英文）

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chapter 2,Discrete-time Systems,Analysis,Discrete-time systems,Theory of the z-transform,Signal sampling and reconstruction,Pulse transfer function of sampled-data systems,Stability , transient response and steady-state error,What is a discrete time system?,They are systems in which the inputs and outputs are described by discrete samples in time domain.,Discrete Time,System,u,k,y,k,k,y,k,k denotes the sampling instant at time t=kT,s,Inputs and outputs are not continuous in time but instead are sampled at t=kT,s,where T,s,is the sampling interval.,Sampling frequency = 1/T,s,Hz or 2,p,/T,s,rad/s.,Continuous time,Discrete samples,k,u,k,1,2,3,Discrete-Time Systems,A Discrete-Time System,transforms discrete-time inputs to discrete-time outputs.,The output at a particular time index depends on both the input at specific index values and output values at previous indices.,In contrast to a continuous-time system whose operation is described (or modeled) by a set of differential equation, a discrete-time system can be described by a set of,difference equations,(,差分方程,),.,How do you describe the input-output behavior of discrete time systems?,Do so with,difference equations,instead of differential equations,Examples of difference equations (DE),1,st,order DE :,2,nd,order DE :,3,rd,order DE :,Compare with ordinary differential equations (ODE),1,st,order ODE :,2,nd,order ODE :,Converting ODEs to difference equations,Approximate,by,Hence, 1,st,order ODE will lead to 1,st,order difference eqns,2,nd,order ODE will lead to 2,nd,order difference eqns,Thus, easy to see how continuous time systems can be converted into approximate discrete time models.,Transform Methods,In linear time-invariant (LTI) continuous-time systems, the,Laplace transform,can be used in system analysis and design.,In linear time-invariant discrete-time systems, the,z-transform,is utilized in the analysis of the system described by difference equations.,What is,z,-transform ?,Signal Sampling,x(t,),载波器,脉冲调制器,x*,（,t,）,x,(,t,),t,x,*,(,t,),t,x,(,t,),x,*,(,t,),S,Sampling Switch,The L-transform of,x,*(t):,=1 + e,-Ts,+ e,-2Ts,+ ,Example: Unit step signal,x,(t) =1(t).,Transcendental function !,X(z),is called the,z-transform,of discrete signal,x,*(t). Since,x*(t),is a sampled series from the signal,x(t),we may also say,X(z),is the z-transform of,x(t).,Hence the following notation:,Let,e,Ts,= z, then,In some cases,x(kT),is written simply as,x(k).,The Unilateral Z transform,(,单侧,z,变换）,In control systems analysis, we use the unilateral z transform.,Justified because in control systems, we only deal with signals that are causal.,2,.,k,1,0,Due to the infinite sum, convergence is an important issue. Ideally, the region of convergence (ROC) should be stated. ROC refers to the region on the complex plane on which the transform exists,Bilateral z Transforms,(,双侧,z,变换）,Given a sample sequence, x(-2), x(-1), x(0), x(1), x(2), ,we define the bilateral Z-transform as,Example : Unit Impulse,The discrete version of an unit impulse (with delay),d,(t-t,0,), is defined to be,By definition of the z-transform :,If k,0,= 0, D(z)=1!,Example,：,impulse series,Example : Unit Step,2,.,k,1,0,A step sequence,Region of convergence is |,z,| 1.,Pole at z = 1,2,.,k,1,0,a1,Region of convergence is |z| a,Pole at z = a,What does this tells us about the relationship between stability and poles?,Power series,2,.,k,1,0,a ramp sequence,How to show this ?,For unit-step signal,：,Multiply both sides with,(-Tz),，,and obtain the z-transform of unit ramp function:,Take derivative with respect to z:,Proof:,Example,：,Exponential function,(,指数函数,),x,(,t,)=,e,-,a,t,(a : constant parameter,This is a geometric series with a common ratio of (,e,-,aT,z,-1,),，,When,|,e,-aT,z,-1,|,1,，,this series is convergent and can be written in closed form as follows:,Example,：,Sinusoidal signal (,正弦信号,),x,(,t,)=,sin t,Properties of z-transform,Linearity:,If,X,(,z,)=,Z,x,(,t,),，,Then,Real Translation (Time Shift,实数位移定理,),实数位移,定理,若,X,(,z,)=,Z,x,(,t,),，,则,Proof,：,假定,k 1 lead to unstable systems!,Poles with magnitudes |z| 1 are stable.,Signal Sampling and Reconstruction,Signal Sampling,x,(,t,),x,*,(,t,),S,Sampling Switch,(,a,),t,x(t),(,b,),t,x*(t),Obviously, is a periodic function,，,hence can be expanded into Fourier series:,where is the sampling frequency.,Hence,Taking Laplace transform and using complex translation theorem:,Its spectrum,（频谱）,can be given by,-,-,X(j ),0,0,-,-,(,a),Spectrum of x(t).,(,b),Spectrum of,x*,（,t,）,（ ,2,）,Ideal low-pass filter,Spectrum preserved,Signal x(t) can be recovered,Aliasing (,混叠,):,s, 2,max,Spectrum overlap,Signal distorted,Can not be recovered.,Nyquist Sampling Theorem (,采样定理,),Nyquist Sampling Theorem,:,One can recover a signal from its samples if the sampling frequency (,s,= 2/T) is at least,twice,the highest frequency (,max,) in the signal,i.e.,Put in another way:,For a given sampling frequency,s, only when the highest frequency (,max,) of the signal is no larger than,half,of sampling frequency (,s,) can we recover the signal without any distortion,i.e.,Nyquist frequency,Ideal low-pass filter,-,Ideal Reconstruction of Signal,After filtering:,Impulse response:,Noncausal,！,Can not be implemented physically.,t/T,1,2,3,-1,-2,-3,Signal,Reconstruction: a,polynomial extrapolation approach.,Using a,Taylors series expansion,about,t = nT,We define,as the reconstructed version of,x(t).,Such a mechanism is called,data hold,and,x,h,(t),is the output of the data hold.,If only the first term of the Taylors series is used, the data hold is called a,zero-order hold,(,零阶保持器,),i.e.,If the first two terms of the Taylors series are used, it is the,first-order hold,(,一阶保持器,),i.e.,We approximate the derivatives by,backward difference,.,Zero-Order Hold,(,ZOH,零阶保持器,),ZOH is the most commonly used data hold,，,it maintains the sampled value for the whole sampling period,，,and output a staircase signal.,x,h,(,t,),x,*,(,t,),x,*,(t),t,ZOH,x,h,(,t,),t,Taking Laplace transform:,Hence the transfer function of ZOH is given by,Then,Sampling and Hold,x,h,(t),G,h,（s）,x*(t),x(t),Sampler,Data Hold,For ZOH:,Frequency response of ZOH,Amplitude:,Phase:,Frequency response of ZOH,T,High frequency components are attenuated,but can not be totally erased;,Phase delay related to T.,First-Order Hold (FOH,一阶保持器,),0,T 2T 3T .,Its frequency response:,The transfer function of FOH is given by,where,Frequency response of FOH,Conclusion: FOH is not better than ZOH.,FOH,ZOH,Pulse transfer function,(,脉冲传递函数,),1.,Open loop,Pulse transfer function,G(s),r*( t ),r ( t ),y*( t ),y ( t ),Pulse transfer function,(z,transfer function, discrete,transfer function,) is defined as the ratio of the z-transform of output y*(t),or Y(z), to that of input r*(t), or R(z),，,i.e,., H(z)=Y(z)/R(z).,Any continuous-time signal r(t) sampled by an ideal sampler with period T will produce a train of pulse signal as:,If,is input into G(s),If the input is ,Assuming that the continuous output c(t) is also sampled by an ideal sampler as that of input, then the output sample at t=nT,is,By the theorem of discrete convolution,：,G(z)=,Z,G(s),Generally G(z),can be written as :,Caution,：,G(z) is determined by the structure and parameters of the discrete system, and is independent of the reference input.,Example,: find the z transfer function for the system with the following s transfer function:,Solution:,Example,:,determine the pulse transfer function for the following open-loop sampled-data system :,r*( t ),r ( t ),y*( t ),y ( t ),Solution:,Pulse transfer function of cascaded systems,Case 1: No sampler between two cascaded subsystems,G,1,(s),G,2,(s),r*(t),r ( t ),y*(t),y(t),The block diagram can be reduced to,G,1,(s)G,2,(s),r*(t),r (t),y*(t),y(t),Then,Let,Case 2: There is a sampler between two cascaded subsystems, and samplers are synchronized.,y*(t),y(t),G,1,(s),G,2,(s),r*(t),r ( t ),y,1,*(t),Case 3: Open loop system preceded by a ZOH.,G,p,(s),r*(t),r ( t ),y*(t),y(t),3.,Pulse transfer function of closed-loop discrete systems,y*(t),G,1,(s),G,2,(s),H(s),r(t),e(t),e*(t),d(t),b(t),y(t),-,+,+,Figure: Linear,discrete system with disturbance,By assuming d(t)=0, the diagram can be reduced to:,Figure: Linear,discrete system,By the definition of pulse transfer function:,G,1,(s),G,2,(s),H(s),r(t),e*(t),y*(t),y(t),b(t),Define the error pulse transfer function G,e,(z),as:,Hence the closed-loop pulse transfer function G,B,(z) is given by,Now assume r(t)=0, and obtain the following diagram with disturbance as an equivalent input:,G,2,(s),G,1,(s),H(s),r(t)=0,e*(t),y*(t),-,y(t),d(t),+,+,Figure: Linear,discrete system with disturbance,as input.,Example: Consider the following sampled-data system:,G(s),H(s),r(t),b*(t),y*(t),y(t),Analysis of Discrete Systems,Transient response,Stability,Steady-state error,1.,Transient response,Closed-loop transfer function of a typical discrete system:,N(z),and,D(z),are monic polynomial of,z,.,The unit step response is given by,By partial fraction expansion:,where,(1) p,k,is real:,Case a: p,k,=1,y,k,(n) is a,constant sequence.,The output series:,Case b: 0p,k,1, expanding geometric sequence.,Case e: -1p,k,0,decaying geometric sequence with alternating signs.,Case d: p,k,=-1,alternating sequence.,Case f: p,k,-1,expanding geometric sequence with alternating signs.,Summary: transient response with,a single real pole p,k,Im,Re,Z,f,f,d,d,a,a,c,c,b,b,e,e,(2),p,k,is conjugate complex (in pairs),Then, c,k,and c,k+1,form a conjugate pair:,The magnitude of pole, |p,k,|, will determine whether the response is convergent or divergent.,The transient response:,Case a: |p,k,|1,exponentially expanding sinusoidal sequence,A larger means faster oscillation in the transient response.,Let,k,=,d,T,then,is the oscillating frequency of the response, and the period of oscillation,is given by,The impact of the argument,(3). Deadbeat system (,有限时间响应系统,),When all the closed-loop poles are at the origin, the transient response will settle down within limited periods. Such a system is called deadbeat system.,The unit impulse response:,The transient process will die out after n periods. This property is never found in a continuous-time system.,A very important qualitative property of a dynamic system is,stability,.,Internal stability,is concerned with the responses at all the internal variables.,External stability,is concerned with the input-output relation.,The most common definition of,appropriate response,is that for every,Bounded Input, we should have a,Bounded Output,.,i.e.,we call the system,BIBO stable,.,2. Stability Analysis,Linear Discrete System:,G（s）,r(t),y,*,(t),y (t),_,If all closed-loop poles of a system is,inside,the unit circle, the system is,stable,.,If at least one pole is,on or outside,the unit circle, the corresponding system is,not BIBO stable,.,1+,G,(z)=0,T,he stability boundary,of,discrete-time,systems (,in the,z-plane,),is different from that,of continuous systems (,in the s-plane,).,How does this happen?,Consider the following mapping (,from s to z):,z,= e,Ts,For any point in the s-plane: s =,+j,then after mapping, the point in z-plane is:,case 1,:,=0，the imaginary axis in s-plane is mapped into the unit circle in z-plane stability boundary.,case 2,:,0，the RHP of s-plane is mapped into the exterior of the unit circle in z-plane instability region.,s =,+j,Re,Re,Im,Im,Mapping the s-plane into z-plane,s-plane,z-plane,s =,+j,Ways to check stability,Direct calculation: for simple cases;,Bilinear transform + Routh test;,Jurys test,: similar to Hurwitz test in continuous-time case.,Other ways,:,root locus, Nyquist stability criterion, Lyapunov theorem, etc.,Solution,：,The open loop pulse transer function is,Example,: Check the stability of the following sampled-data system with T=1s.,r (t),y,*,(t),y (t),1+,G(z)=0,z,2,+4.952z+0.368=0,z,1,=-0.076 z,2,=-4.876,There is one pole outside the unit circle, hence the,system is unstable.,The closed-loop C.E. is given by,Define,(,1),The two complex variables,z,and,w,can be written as,z=x+jy,w=u+jv,(,2),(,3),Substitute (2) and (3) into (1):,Bilinear transform + Routh test,then,Bilinear transform,w-transform,Case 1: x,2,+y,2,=1 , the unit circle in z-plane,u=0 the imaginary axis in w-plane.,Case 2: x,2,+y,2,1, the interior of unit circle in z-plane,u1, the exterior of unit circle in z-plane,u0 the right half of w-plane.,z=x+jy,w=u+jv,For Discrete-time systems: poles are inside unit circle (,z,plane)?,Stability ?,For Continuous-time systems: poles are on the left half plane (,w,domain) ?,Bilinear transform,Routh test,Given a sampled-data system with,T,=1s .,Check its stability for the case when,K,=10, and find the critical gain K.,Example:,Solution,： ,The closed-loop CE is:,z,2,+2.31,z,+3=0,By manual calculation:,1,=-1.156,j,1.29,2,=-1.156-,j,1.29,Both poles are outside the unit circle,，,hence the system is unstable.,C,(,s,),R,(,s,),Open-loop pulse TF:,when,K,=10, the closed-loop TF:,CE: 1+G(z)=0,z,2,-(1.368-0.368,K,),z,+(0.368+0.264,K,) =0,Open loop pulse transfer function:,The critical value of,gain,K is :,K,c,=2.4,Routh array:,w,2,0.632,K,2.736-0.104,K,w,1,1.264-0.528,K,0,w,0,2.736-0.104,K,For stability, we need,After,w,-transform:,0.632,Kw,2,+(1.264-0.528K),w,+(2.736-0.104K) =0,0,K,0,f(-1)0,|f(0)|,1,3,.,Steady-state error in discrete-time systems,Consider a discrete-time system with unit feedback:,G（s）,r(t),y (t),e,*,(t),-,The error transfer function:,By final value theorem:,Based on the number of open-loop poles at,z=1, the open-loop system G(z) can be,categorize,d as Type 0, Type 1, Type 2,The error is given by,1),Unit step input,Define as position error constant.,Type 0:,Type 1+:,2),Unit ramp input,Define the velocity error constant K,v,as,Then,System Type 0,：,K,v,=0,Type 1,：,Where G,1,(z) has no pole at,z=1.,Type 2,+ :,3),Acceleration input,Define the acceleration error constant K,a,as,Type 0,1,:,K,a,= 0,Type 2,：,Type 3,+:,Steady-state error versus Di