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自动控制AC_l21_dt_systems Automatic Control – C. Novara Digital control Introduction to digital control Discrete-time signals Discrete-time systems Automatic Control – C. Novara L1 2 Introduction to digital control � At present, most control systems use digital computers to ...

自动控制AC_l21_dt_systems
Automatic Control – C. Novara Digital control Introduction to digital control Discrete-time signals Discrete-time systems Automatic Control – C. Novara L1 2 Introduction to digital control � At present, most control systems use digital computers to implement the controllers. � Digital computers allow one to calculate the control signal in software rather than in hardware. This gives relevant advantages: � Flexibility in making modifications to the control law after the hardware design is fixed. � Hardware and software design can proceed almost independently, saving a large amount of time. � Logic and nonlinear operations can be easily included in the controller. � Rapid prototyping. Automatic Control – C. Novara L1 3 Introduction to digital control � Digital control system design is typically based on three main steps: 1. Design of a continuous-time controller CCT(s). 2. Discretization of the controller: CCT(s) � C(z). 3. Verification of the design using discrete analysis and simulation. � The discrete-time controller is implemented on the real system to be controlled as follows: Automatic Control – C. Novara L1 4 Introduction to digital control � Analog-to-digital converter (A/D): This device samples a physical variable, and covert it into a binary number (truncation). The sampling is performed using a given sampling period T. � Digital-to-analog converter (D/A): 1. This device changes the binary number to an analog signal. 2. A zero-order hold (ZOH) hold maintains the same signal value throughout the sample period. Automatic Control – C. Novara L1 5 Introduction to digital control 0 1 2 3 4 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t e e(t) e(k) 0 1 2 3 4 5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 t u u(k) u(t) Automatic Control – C. Novara L1 6 Discrete-time signals: signal types � Analog signals: continuous in time and amplitude. � Example: voltage, current, temperature, mass position,… � Digital signals: discrete both in time and amplitude. � Example: attendance of this class, digitizes analog signals,… � Discrete-time signals: discrete in time, continuous in amplitude. � Example: hourly change of temperature. � Theory for digital signals is complicated. � Discrete-time signals: � Most convenient to develop theory. � Good enough approximation in most practical applications. � In practice we mostly process digital signals on processors. � Need to take into account finite precision effects. Automatic Control – C. Novara L1 7 Discrete-time signals: sampling � Sampling is a continuous-time to discrete-time conversion: T is the sampling period (s) fs = 1/T is the sampling frequency (Hz) ωs = 2pifs = 2pi/T is the sampling frequency (rad/s) ( ) ( ) K0,1,2,kkTxkx == 0 1 5 6 72 43 x t Automatic Control – C. Novara L1 8 Discrete-time signals: sampling � Sampling is not reversible. � Given a sampled signal there exist infinite continuous signals which fit the samples: � Sampling yields an information loss. � Under certain conditions an analog signal can be sampled without loss so that it can be reconstructed perfectly. 0 -1 20 40 60 80 100 -0.5 0 0.5 1 t x Automatic Control – C. Novara L1 9 Discrete-time signals: sampling Sampling Theorem (Nyquist-Shannon) � Let x(t) be a continuous-time bandlimited signal with where X(jω) is the continuous-time Fourier Transform of x(t). � Let x(k)=x(kT) be the discrete-time signal obtained from x(t) using a sampling frequency ωs=2π/T. � Then, x(t) is uniquely determined by its samples x(k) if ωs/2=π/T is called the Nyquist frequency. Bfor )j(X ≥ω=ω 0 B s 2≥ω Automatic Control – C. Novara L1 10 Discrete-time signals: sampling � Continuous-time Fourier transform: � Continuous-time Fourier inverse transform: ( ) ( )∫ ∞ ∞− ω− =ω dtetxjX tj ( ) ( )∫ ∞ ∞− ω ωω pi = dejXtx tj 2 1 � x(t) is written as a superposition of harmonic signals: ejωt = cos(ωt) + j sin(ωt). � X(jω) is the weight of the harmonic component with frequency ω. Nepster 高亮 Automatic Control – C. Novara L1 11 Discrete-time signals: sampling � continuous-time signal � Fourier transform B = 17 rad/s, ωs ≥ 2B � T ≤ 2π/ωs = π/B = 0.185 s Automatic Control – C. Novara L1 12 Discrete-time signals: sampling B=17 rad/s, ωs ≥ 2B � T ≤ 0.185 s full information maintained little loss of information relevant loss of information Automatic Control – C. Novara L1 13 Discrete-time signals: sampling � Let x(t)=sin(Bt). � If ωs < 2B, then this sinusoidal signal is not correctly sampled. � This effect is called aliasing and may occur for any type of signal if the sampling rate is not properly chosen. � Aliasing is very dangerous in real digital control systems since high- frequency noises may cause large low-frequency oscillations. t (s) x(k) x(t) Automatic Control – C. Novara L1 14 Discrete-time signals: sequences � A discrete-time signal is represented by a sequence of real numbers. 0 20 40 60 80 100 -10 0 10 t (s) 0 10 20 30 40 50 -10 0 10 k (samples) Automatic Control – C. Novara L1 15 Discrete-time signals: sequences � Unit impulse sequence: � Unit step sequence: � Exponential sequence: � Delayed (shifted) sequence: )kk(x)k(y o−=    = ≠ =δ 01 00 k k )k(    ≥ < =ε 01 00 k k )k( kA)k(y α= -10 -5 0 5 100 0.5 1 1.5 -10 -5 0 5 100 0.5 1 1.5 -10 -5 0 5 100 0.5 1 k (samples) Automatic Control – C. Novara L1 16 Discrete-time systems � A discrete-time system is an operator F mapping an input sequence u=[u0 u1 …] T (and an initial condition) into an output sequence y=[y0 y1 …] T: Fu y � A discrete-time system is in general described by a difference equation: ( ) ( ) ( ) ( ) ( 1), ( 2), , ( ), ( 1), ( 2), ( 1) ( ), ( ) ( ) ( ), ( ) y k f y k y k u k u k u k x k f x k u k y k h x k u k = − − − − + = = K K y=F(u) input-output state-space Nepster 高亮 离散时间系统 Automatic Control – C. Novara L1 17 Discrete-time systems: examples Traveling mass subject to friction M u(t) β y(t) � This simple model can be used for the design of cruise control systems. ( ) )t(u)t(ytyM speedmass:)t(y +β−=& Automatic Control – C. Novara L1 18 � Forward Euler discretization method. The time derivative is approximated by the difference quotient: where T is the sampling period. � Using this discretization, we obtain the following difference equation: Discrete-time systems: examples ( ) ( ) ( ) T kTyT)k(y ky −+ ≅ 1 & )k(u M T )k(y M T )k(y 111 −+−      β −= Nepster 高亮 欧拉 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 Automatic Control – C. Novara L1 19 Discrete-time systems: examples Classical example: Pendulum Variables: - y(t) : angular position - u(t) : external torque Parameters: - m : pendulum mass - L : pendulum length - g : gravity constant - J=mL2 : moment of inertia - β : friction - k=gmL mg u(t) y(t) L Automatic Control – C. Novara L1 20 Discrete-time systems: examples � Pendulum differential equation: � Forward Euler discretization method: where y(k)=y(kT) and T is the sampling period. ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 12211 T kykyky T kyky )k(y, T kyky )k(y ++−+ = −+ ≅ −+ ≅ && &&& � Discretized pendulum difference equation: [ ] )t(u)t(y)t(ysink)t(yJ +β−−= &&& ( ) ( ) ( ) [ ] J T b, J kT a, J T a, J T a )k(bu)k(ysinakyakyaky 22 321 321 12 2221 == β −=− β = −+−−−−−−= Automatic Control – C. Novara L1 21 Discrete-time systems: examples Matlab program for simulation of pendulum time evolution % Parameters m=1; l=0.8; J=m*l^2; K=9.81*m*l; beta=0.6; T=0.01; a1=2-T*beta/J; a2=-1+T*beta/J; a3=-T^2*K/J; b=T^2/J; % Initial conditions y=[pi/4;pi/4]; % Time evolution by iteration of the difference equation for k=3:4000 u(k)=sin(0.01*k); y(k)=a1*y(k-1)+a2*y(k-2)+a3*sin(y(k-2))+b*u(k-2); end Automatic Control – C. Novara L1 22 Discrete-time systems: examples � Moving Average: � Maximum: � Ideal Delay System: [ ] 4321 /)k(u)k(u)k(u)k(u)k(y −+−+−+= [ ])k(u),k(u),k(umax)k(y 21 −−= )kk(u)k(y 0−= Automatic Control – C. Novara L1 23 Discrete-time systems: memoryless systems � A system is memoryless if the output y(k) at each time k depends only on the input u(k) at the same time k. � Example of memoryless systems � Square: � Sign: � Counter Example � Ideal Delay System: 2)k(u)k(y = [ ])k(usign)k(y = )kk(u)k(y o−= Automatic Control – C. Novara L1 24 Discrete-time systems: causal systems � A system is causal if its output is a function of only the current and previous samples. Example � Backward Difference: Counter example, non-causal system � Forward Difference: )k(u)k(u)k(y ++= 1 )k(u)k(u)k(y 1−−= Automatic Control – C. Novara L1 25 Discrete-time systems: linear systems � Linear Time Invariant (LTI) discrete-time system: 1 2 0 1( ) ( 1) ( 2) ( ) ( 1) ( 1) ( ) ( ) ( ) ( ) ( ) y k a y k a y k b u k b u k x k Ax k Bu k y k Cx k Du k = − − − − + + + − + + = + = + K K � A discrete-time system is linear if for any input sequences u=[u0 u1 …] T and v=[v0 v1 …] T, and for any real numbers a and b (superposition principle). ( ) )v(bF)u(aFbvauF +=+ input-output state-space
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