Study notes of automatic control principle
Self control principle learning notes column
Chapter one —— Dynamic model of feedback control system
Chapter two —— Stability analysis of control system
The third chapter —— Continuous time system performance analysis
Chapter four —— Calibration and synthesis of automatic control system
The fifth chapter —— Linear discrete system

Linear discrete system

1. Discrete systems

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One or more signals in the system are pulse trains or digital systems

Types of discrete systems ：

• sampling system ： Time is discrete , Numerical continuity , The continuous signal is sampled somewhere

• digital system ： Time is discrete , Numerical quantification

A/D The process

• sampling ： Discrete in time ,

• quantitative ： Numerically discrete

  Ideal sampling process ：

  $\tau \ll T$, It is considered that the sampling is completed instantaneously

  The word is long enough , Think $e^{\ast }(kT)=e(kT)$

  

D/A The process

ZOH Zero order holder , Keep the discrete signal every beat , Get a continuous signal

2. Signal sampling and holding

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2.1 Signal sampling

1. Ideal sampling sequence

• Because it's just $nT$ Always valid , All direct use $e(nT)$

• $e(nT)$ Represents the value at the time of sampling ,$\delta (t-nT)$ It means the time is nT Effective at , It is only defined at the sampling point

$$\begin{array}{rl}& {\delta }_T(t)=\sum _{n=-\infty }^{\infty }\delta (t-nT)\\ & e^{\ast }(t)=e(t)\cdot {\delta }_T(t)=e(t)\cdot \sum _{n=0}^{\infty }\delta (t-nT)=\sum _{n=0}^{\infty }e(nT)\cdot \delta (t-nT)\end{array}$$

2.Laplace Transformation
$$\begin{array}{rl}L:E^{\ast }(s)& =L[e^{\ast }(t)]\\ & =L[\sum _{n=0}^{\infty }e(nT)\cdot \delta (t-nT)]=\sum _{n=0}^{\infty }e(nT)\cdot e^{-nTs}\end{array}$$

$$\begin{array}{rl}example：e(t)& =1(t),\text{ seek }{E}^{\ast }(s)\\ E^{\ast }(s)& =\sum _{n=0}^{\infty }1\cdot e^{-nTs}\\ & =1+e^{-Ts}+e^{-2Ts}+\cdots =\frac{1}{1-e^{-Ts}}=\frac{e^{Ts}}{e^{Ts}-1}\end{array}$$

give $E^{\ast }(s)$ And $e(t)$ The value relationship at the sampling point , Generally, it can be written in a closed form , Used to find $e^{\ast }(t)$ Of z Transform or time domain response of the system

3. Fourier transform ${\delta }_T(t)={\sum }_{n=-\infty }^{\infty }\delta (t-nT)$

• $E^{\ast }(s)=\frac{1}{T}{\sum }_{-n=\infty }^{\infty }E(s+jn{w}_s$),$w_s$ sampling frequency

• Generally, it cannot be written in a closed form , be used for $e^{\ast }(t)$ Spectrum analysis

$$\begin{array}{rl}{\delta }_T(t)=& \sum _{n=\infty }^{\infty }{c}_n{\mathrm{e}}^{-jn{\omega }_{s}t}dt\\ & {\omega }_s=2\pi /T\\ & c_n=\frac{1}{T}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}{\delta }_T(t)\cdot {\mathrm{e}}^{-jn{\omega }_{s}t}dt=\frac{1}{T}{\int }_{0^{-}}^{0^{+}}\delta (t)\cdot 1\cdot dt=\frac{1}{T}\\ {\delta }_T(t)=& \frac{1}{T}\sum _{n=\infty }^{\infty }{\mathrm{e}}^{-jn{\omega }_{s}t}dt\\ e^{\ast }(t)=& e(t)\cdot {\delta }_T(t)=\frac{1}{T}e(t)\sum _{n=0}^{\infty }{\mathrm{e}}^{-jn{\omega }_{s}t}=\frac{1}{T}\sum _{n=\infty }^{\infty }e(t)\cdot {\mathrm{e}}^{-jn{\omega }_{s}t}\\ L\left[e^{\ast }(t)\right]& =L\left[\frac{1}{T}\sum _{n=-\infty }^{\infty }e(t)\cdot {\mathrm{e}}^{-jn{\omega }_{s}t}\right]=\frac{1}{T}\sum _{n=-\infty }^{\infty }E(s+jn{w}_s)\end{array}$$

2.2 Sampling theorem （Shannon）

The signal is completely reproduced Necessary condition

$w_s=\frac{2\pi }{T}>2w_h$,$w_s$ Is the sampling frequency ,$w_h$ Is the maximum width of the spectrum

spectrum - The signal is decomposed by frequency

Frequency characteristics for the system , Spectrum for signals

2.3 Zero order holder ZOH

Adding a zero order holder can be seen as adding a $\frac{Ts}{2}$ Pure delay link

3. Z Transformation

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3.1 Definition

For sampled signals Laplace Transformation , Make $z^{-1}=e^{-Ts}$d
$$E(z)=Z\left[e^{\ast }(t)\right]=E^{\ast }(s){}_{z=e^{Ts}}=\sum _{n=0}^{\infty }e(nT)\cdot z^{-n}$$

$$\{\begin{array}{l}{e}^{\ast }(t)={\sum }_{n=0}^{\infty }e(nT)\cdot \delta (t-nT)\\ E^{\ast }(s)={\sum }_{n=0}^{\infty }e(nT)\cdot {\mathrm{e}}^{-nTs}\\ E(z)={\sum }_{n=0}^{\infty }e(nT)\cdot {\mathrm{z}}^{-\mathrm{n}}\end{array}$$

• E(z) Only corresponding to unique $e^{\ast }(t)$, It doesn't correspond to the only $e(t)$

• z Transformation is only for discrete signals

3.2 Z Change the way

1. Summation of series

Definition ：
$$E(z)=\sum _{n=0}^{\infty }e(nT)\cdot z^{-n}$$
Use the method of definition to find $E(z)$, Using the method of series

2. Partial fraction method

3. Residue method

$$E(z)=\sum _{i=1}^l{\left[\mathrm{Res}E(s)\frac{z}{z-e^{Ts}}\right]}_{s=s_i}$$

3.3 Z Basic theorem of transformation

Reset shift theorem ：
$$Z[e(t)\cdot e^{\mp at}]=E(z\cdot e^{\pm aT})$$
The initial value theorem ：
$$\underset{n\to 0}{\lim }e(nT)=\underset{z\to \infty }{\lim }E(z)$$
The final value theorem ：
$$\underset{n\to \infty }{\lim }e(nT)=\underset{z\to 1}{\lim }(z-1)\cdot E(z)$$

4.Z Reverse transformation

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4.1 Long division

Will be closed form z Field expression , Expand into series . Then the expression in time domain is obtained by transformation

4.2 Partial fraction expansion

Note that after expansion, there is a continuous time-domain signal , It is necessary to sample again to get the sampled discrete signal .

4.3 Residue method

$$e(nT)=\sum Res[E(z)\cdot z^{n-1}]$$

5. The difference equation

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Forward difference ：
$$\begin{array}{l}\Delta f(k)=f(k+1)-f(k)\\ {\Delta }^{2}f(k)=\Delta f(k+1)-\Delta f(k)=f(k+2)-2f(k+1)+f(k)\\ {\Delta }^{3}f(k)={\Delta }^{2}f(k+1)-{\Delta }^{2}f(k)=f(k+3)-3f(k+2)+3f(k+1)-f(k)\\ {\Delta }^{n}f(k)=f(k+n)-nf(k+n-1)+\dots +(-1{)}^{n-1}nf(k+1)+(-1{)}^{n}f(k)\end{array}$$
Backward difference ：
$$\begin{array}{l}\nabla f(k)=f(k)-f(k-1)\\ {\nabla }^{2}f(k)=\nabla f(k)-\nabla f(k-1)=f(k)-2f(k-1)+f(k-2)\\ {\nabla }^{3}f(k)={\nabla }^{2}f(k)-{\nabla }^{2}f(k-1)=f(k)-3f(k-1)+3f(k-2)-f(k-3)\\ {\nabla }^{n}f(k)={\nabla }^{n-1}f(k)-{\nabla }^{n-1}f(k-1)\end{array}$$

5.1 Iterative method

Will be continuous domain name $\dot{e}(t)$ In discrete fields $e(k+1)-e(k)$ Instead of

5.2 z Transformation method

6. Pulse transfer function

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Definition ： Output of discrete system under zero initial condition z Transform pair input z Transformation ratio .

• Unit impulse response function z Transformation

• It is only applicable to linear time invariant discrete systems

$$G(z)=\frac{C(z)}{R(z)}$$

$$\begin{array}{rl}\text{Convolution formula ：}& c(k)=\sum _{i=0}^{\infty }g(k-i)\cdot r(i)\\ & C(z)\sum _{k=0}^{\infty }\left[c(k)\cdot z^{-k}\right]=\sum _{k=0}^{\infty }\left[\left(\sum _{i=0}^{\infty }g(k-i)\cdot r(i)\right)\cdot z^{-k}\right]\\ & \stackrel{m=k-i}{=}\sum _{m=0}^{\infty }\left[\left(\sum _{i=0}^{\infty }g(m)\cdot r(i)\right)\cdot z^{-(m+i)}\right]=\sum _{m=0}^{\infty }g(m)z^{-m}\sum _{i=0}^{\infty }r(i)z^{-i}\\ & =G(z)\cdot R(z)\\ G(z)=& \sum _{k=0}^{\infty }g(k)z^{-k}=\frac{C(z)}{R(z)}\end{array}$$

6.1 Serial links

Including sampling switch , O, respectively, z Transform and multiply . No sampling switch , Multiply before z Transformation

6.2 ZOH

$$G(z)=Z\left(\frac{1-{\mathrm{e}}^{-sT}}{s}{G}_{\mathrm{p}}(s)\right)=\left(1-z^{-1}\right)Z\left(\frac{G_{\mathrm{p}}(s)}{s}\right)$$

Find the closed-loop transfer function

Generally, when the sampling switch is placed at the deviation , The closed-loop transfer function can be obtained in the following way .
$$\Phi (z)=\frac{C(z)}{R(z)}=\frac{G(z)}{G(z)H(z)}$$

plus： It works Mason The condition of the formula

7. Stability analysis of discrete systems

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7.1 Based on bilinear transformation ROUTH The criterion

Make $z=\frac{w+1}{w-1}$ Equivalent transformation of characteristic polynomial of closed-loop system , And then use it w Domain Routh Criterion to judge stability .P352

7.2JURY The criterion

Directly according to the coefficients of the closed-loop characteristic equation of the discrete system , Determine whether its root is located in the unit circle on the plane , So as to judge whether it is stable .

List Jury step ：

• The first row of coefficients is the coefficients of the characteristic equation
• Even row coefficients are the reverse order of odd row coefficients
• The calculation formula of other odd rows is as follows ：

8. Digital correction of discrete systems

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For design The controller also needs to satisfy the constraints ：

• $C(z)$ It is stable. ： The pole of the controller is in the unit circle or the pole is 1
• The controller is causal , Each item constituting the controller is the current or past data

The design method of the controller ：

• First design continuous controller , Then discretize the controller
• Analytical design method , The design of minimum beat system is the most typical
• Direct design method , It's based on z The root locus method of the plane is based on w Frequency domain method of plane

8.1 Typical continuous domain discretization method

The basic idea ： The frequency characteristics of digital controller and analog controller are equivalent , take s The controller of the domain is transformed into z Domain , First design the model controller , And then discretize it , The response of discrete system is not consistent with that of continuous system .

• First order difference method ： Forward difference Obtained controller The stability of the controller itself cannot be guaranteed , and Backward difference Discrete controller It can maintain its stability
• Impulse response method ： There is no serial discretization method
• Step response invariant method
• Pole zero matching
• bilinear transformation

8.2 discretization PID correction

• Integral separates numbers PID： First use PD…… still Will consider whether to add points link
• Incomplete differential number PID： Differential action is easy to cause high-frequency disturbance , In the typical PID Add a low pass filter （ First order inertial link ） To suppress high-frequency disturbances
• Differential antecedent number PID： In order to avoid the impact on the control system caused by the rise and fall of the given value , Only the output feedback is differentiated , Don't give differential
• Variable speed integral PID： Good continuity of integral separation , It's steady
• Bang-Bang PID： Choose according to the deviation , When The deviation is greater than the threshold , use Bang-Bang control , When the deviation is less than the threshold , use PID
• With dead band PID： in order to Prevent frequent action of mechanism , You can add a link with dead zone

8.3 Smith Pure lag compensation PID

The controlled object contains Pure hysteresis Often Cause overshooting and sustained oscillation , It will reduce the stability of the system , In serious cases, it may lose stability . Design PID The controller is not designed separately .

8.4Dahlin Algorithm ：

Expected closed-loop transfer function ： Equal to the pure lag time of the object

constraint ： Overshoot

Target ： First order with pure delay 、 Second order inertia link ( Linear systems )

Eliminate ringing : leave z=-1 The farther , The weaker the ringing phenomenon , You need to keep the static gain constant .

9. Minimum beat system

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Minimum beat system ： Under the action of typical input , Can be in The response process ends in a finite beat and there is no steady-state error at the sampling point .

• The impulse response of a system ends in the smallest finite beat , It means that the stability of the system is infinite .
• The designed controller must be stable or contain integral

The closed-loop transfer function of time invariant discrete systems is required to include a delay factor ： The delay factor is set according to the requirement that the controller cannot contain a lead factor , One of the sources of this delay factor is that the controlled object contains a pure lag link , One of the delay factors is adopted by the object ZOH Introduced after discretization P376

Ripple and no ripple ：

* Ripple ：* The steady-state error at the sampling point is 0

* ripple-free ：* The steady-state error between the sampling point and the sampling point is 0

• Ripple system is a minimum beat controller designed for a typical input, which makes the output response transient process achieve error free after finite beats at the sampling time point

• All ripples in a ripple system are mainly caused by the controller output not being synchronized with the output to achieve stability within a finite beat

• The output response of the ripple free system is synchronized with the controller output to achieve stability

Design steps of ripple and ripple free system ：

1. According to the typical input signal z The number of transformation integrals is determined m

   Typical signal	m
   1(t)	1
   t	2
   $t^2/2$	3

2. Determine the number of delayed beats according to the controlled object h（ Include ZOH Introduced 1 Beat lag and the inherent pure lag of the source object -> see $\frac{1}{1-z^{-1}}$ The number of times ）; Number of unstable poles p, Number of zeros q（q Is the unit outside the circle 、 Up zero — For ripple ,r For all zeros — For ripple free ）

3. The above coefficient can be determined by m+p Solve the equation P377

$$\Phi (z)=z^{-h}\prod _{i=1}^{q(r)}\left(1-{\beta }_i{z}^{-1}\right)\left({\phi }_0+{\phi }_1{z}^{-1}+\cdots +{\phi }_{m+p-1}{z}^{-(m+p-1)}\right)$$

$$1-\Phi (z)={\left(1-z^{-1}\right)}^m\prod _{i=1}^p\left(1-{\alpha }_i{z}^{-1}\right)\left(1+f_1{z}^{-1}+\cdots +f_{h+q(r)-1}{z}^{-(h+q(r)-1)}\right)$$

Solve the required equation ：
$$\begin{array}{l}{\Phi (z)|}_{z=1}=1,{{\Phi }^{\prime }(z)|}_{z=1}=0,{{\Phi }^{\prime \prime }(z)|}_{z=1}=0,{{\Phi }^{\prime \prime \prime }(z)|}_{z=1}=0,\cdots ,{{\Phi }^{(m-1)}(z)|}_{z=1}=0\\ {\Phi (z)|}_{z={\alpha }_1}=1,{\Phi (z)|}_{z={\alpha }_2}=1,{\Phi (z)|}_{z={\alpha }_3}=1,{\Phi (z)|}_{z={\alpha }_4}=1,\cdots ,{\Phi (z)|}_{z={\alpha }_p}=1\end{array}$$
Control system adjustment time estimation ：
$$t_s\le (h+p+q(r)+m-1)$$