1: SP16.1: To a transmission, draw the Bode diagram, find GM,PM, find the critical K value; The critical K value is determined by Rouse criterion, and the critical K value is determined by drawing the root locus
2: Simplify the structure diagram
3: Draw the root track and determine the range of K according to the requirements
4: Time domain analysis: first order, second order, high order,
① Given a step response diagram, let you write out the various performance parameters
(2) the reduced order
Matlab and the calculator
Matlab: Use MATLAB/calculator to solve auxiliary equations in Rous criterion
Root locus verification Bode diagram verification:
- ① Smooth curves can only be compared approximately (starting point and end point can be used to test!!)
- ② Can output amplitude margin, phase margin, the corresponding cut-off frequency, through frequency!! Do do
The calculator
- The transfer function solves the poles
- Solving polynomial equation
Transfer function and simplification of structure diagram
Simplify the structure diagram
Transfer function form and points to note :juejin.cn/post/684490…
- The order of the denominator is the order of the system
- The highest order coefficient of even_form is 1 (for second-order systems), and the constant term of Bode_form is 1 (for one-section system, Bode diagram analysis, and high-order system reduction)
Structural diagram simplification:
Taken the steps:
- Step1: find the feedback according to the above picture (press and hold a comparison point and compare the above picture), series, parallel and other basic links
- Step2: move the comparison point and the extraction point without it, and then proceed Step1 again
Supplement:
- Feedback: do addition and subtraction at the input point, output back; “Forward channel as numerator, 1± forward channel × feedback channel as denominator (synthesis point does not move it, if there is no other input can be removed)”
- ★★ feedback formula: numerator × denominator do numerator, denominator product -+ numerator product do denominator; The unit feedback just writes the numerator in the denominator
- ★★ lead point movement: more than what divide what
- ★★ Comparison point movement: more what fill what
- Adjacent extraction points can be exchanged or merged. Adjacent synthesis points can be interchanged and merged
- Adjacent outlets and complex stores are not recommended to change location
special case
Do the format:
Mason formula
Time domain analysis of the system
A first order system
- And the most important thing is: in standard Bode form, find the time constant T
Steady-state error solution
Step response
Features:
- The steady state error
=r(t)-c(t)=0
- τ can be used to determine the first-order system. Over A time constant τ, the response can reach 63% of the final value (A*1)
=r(t)-c(t)=0The slope response
- The steady-state error is equal to A
First order closed loop system
- K ‘and
Standard second order system
If it is not a standard second-order system, but the following second-order system, then you have to see the PO-γ diagram of the second-order analysis of the higher-order system!
- And the most important thing is: even form in standard form
and
and
Improvement of second-order system performance
By increasing Zeta, PO was reduced
Increase steady-state error
The steady state error is reduced by increasing the open loop gain K of the original system
High order system
Go to another note if you see two situations that require overshoot: the negative point of the third-order system becomes the standard second-order system; The zero-elimination of the second order system becomes the standard second order
General order reduction method :(if only the order reduction is required, follow the following method)
- ① Denominator factorization (quadratic can be removed on the first order, there is a complex root is not removed)
- (2) Bode_form:
;
- Comparing the relative distance of poles, non-dominant poles can be omitted if the distance is five times smaller (the corresponding term of non-dominant poles can be directly removed from Bodeform). A pair of zeros and poles close together cancel out
- If it cannot be eliminated, the influence of zero and pole of additional closed-loop should be considered in drawing, and some adjustments should be made in drawing: adding zero of closed-loop will advance the peak time of the system and increase the overshoot; The addition of closed loop poles will delay Tp and reduce the overshoot
;
★2 Case analysis of third-order system
Research Objects:
2.1 Role of real poles
Real Pole activity: Make reponse Sluggish. The closer you get to the virtual axis, the more resistance it is
2.2 Influence of poles
Define a parameter:
Conclusion u u u:
- When β>=10, the system can be reduced to the second order system, the real poles do not play the role of resistance, and the overshoot of response satisfies the relation diagram of PO and ζ of the second order system
- Beta <=0.1, reduced to a first-order system, no more overshoot
- 0.1
- 1
★3 Case analysis of second-order system
3.1 Function of the closed loop zero
Research Objects:
Make reponse more-oscillatory, the closer you are to the virtual axis, the more dynamic you are
3.2 Influence of zero point
Define a parameter:
Conclusion u u u:
- γ>=10, the real zero does not play the role of the [dynamic], and the overshoot of the response satisfies the relationship diagram of PO and ζ of the second-order system
- When γ<10, the action of the real zero is taken into account. PO is determined according to the values of ζ in the figure above
4 Sample Problem 7.1 Level-4 System Degradation example
For the following four-order system, the unit step response of the well is roughly drawn:
Matlab verify:
The notes
Type and steady-state error of the system
Steady-state error problem type and corresponding solution method
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Before calculating the steady-state error, judge whether the system is stable
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Know closed loop transfer function: final value theorem method
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* To multi-input structure block diagram to find the steady-state error: ① find out
; For the single input common case
; For the disturbed input
② Use the final value theorem
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To find the steady state error of the open loop transfer function: fill in the form, either unit feedback or unit feedback
; For the single input common case
; For the disturbed input
② Use the final value theorem
1. Final value theorem method:
The following question is debatable for the practice of having H
Find the steady-state error for the open loop transfer function
★ Research object: open loop transfer function G(s) with unit negative feedback H=1
Judge type: how many order 1/s can be put forward in the denominator of open loop function is the type of system
★ To solve the steady-state error:
Judge the order of the system:
Steady state error derivation:
Lawes criterion
[Note] : GH molecules have K, then D (S) note many with K!!
- Write the characteristic equation D(s)=0, that is, 1+GH(s)=0
- Make the highest degree coefficient positive, to ensure that there is no lack of terms, coefficients are all positive
- The Rous table is listed below
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- If the first item is not 0 and the second item is 0, then the item is equal to the first item
- Judgment rules: ① the coefficients are stable >0 ② the number of sign changes is the number of poles in the positive real part ③ The occurrence of 0 is replaced by E and approximated by limit
【 Test 】 The last element in the first column of the Rous table, in the case of no division, is equal to the constant term of the equation, when the test can check whether they have miscalculated the Rous table.
[Two special cases: the occurrence of 0 system is definitely unstable]
(1) The first column appears 0, with ε instead of continuing to calculate the Rous table, finally take the limit ε tends to infinitesimal integer (0.0001) (0 +), for the missing term of the characteristic equation, still can use the Rous criterion to determine how many right half plane poles, as follows
② All zero lines (if two lines are the same, the next line must be all zero lines)
Step1: construct the auxiliary equation F(s) by adding a line of coefficients, and the coefficients of the equation dF/ds after derivation are all 0 line coefficients
Step2: The system is not stable without judgment; See if there’s a sign change in the first column to see how many poles are in the right half plane
Step3: ** solve the auxiliary equation F(s), work out the poles, and say that the system has conjugate complex roots on the virtual axis, so the system is unstable (solve the auxiliary equation using matlab/ calculator)
Root locus
When you solve for K, you have to turn it into the first polynomial, and the constant term in the numerator is the open root trajectory gain = is the value that you read from the root trajectory that you draw; What MATLAB reads is the value of the molecular constant that you give the function
Rule1: The starting point is the open-loop pole and the ending point is the open-loop zero or infinity; There are n root trajectories; We have n minus m asymptotes
Rule2: Determines the root locus on the real axis
If there is an asymptote: Rule3, Rule4: determine the Angle and intersection of the asymptote and the real axis (technique: about the real axis symmetry, only one side) if there is no asymptote test write: Rule3 Rule4: none
N – m = 3: plus or minus
Draw an asymptote:
If there are complex roots: Rule5: used only for complex roots: (pole) exit Angle =180- the Angle between the vector with other poles pointing to it and the real axis + zero; Angle of incidence =180- zero + pole
If the root locus and the virtual axis have intersection point: Rule6: ① solve the root locus and the virtual axis intersection point: S =jω substitute 1+KGH=0(molecule =0) make the real part of the imaginary part is 0 solve the ω and K(directly see the MATLAB diagram to write the results, and finally have time to write s=jω after substituting the equation)
1, I squared is -1, 2, I to the third is -i, 3, I to the fourth position 1, 4, I to the fifth is I
{② sum of the roots of Rule10 (n-m≥2) ③ Rawls criterion let the first column containing K =0 ②③ get K, substitute back to the characteristic equation D(s)=0, let s=jω solve ω}
☆ To find the separation point: Rule7 Rule8: separation point and separation Angle of the root trajectory: (write the second method, directly see the results of MATLAB output, and finally have time to write k derivative of the equation of S)(separation point may be a pair of conjugate complex roots, useful!)
Angle of separation:
Separation Angle of the Angle included with the real axis:Combined with symmetry, the number of poles encountered is L, and the included Angle between L rays is
Or:
Rule9: Given the closed-loop pole S, calculate K
K = the product of all open loop poles from S/the product of all open loop zeros from SCopy the code
If there is no zero or pole, the denominator or numerator is 1
Rule10: n-m≥2, then the sum of the real parts of the closed loop poles = the sum of the real parts of the open loop poles; To find the Angle with the imaginary axis of the root locus
The separation point is the conjugate complex root:
The root locus closes the inner ring
Template:
Byrd figure
☆☆ Byrd drawing method:
The open-loop transfer function GH is used to draw the amplitude frequency characteristic curve of open-loop logarithm to summarize the problem:
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① Into bodeform: tail one polynomial
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② Write the starting point and turning point of all links. (Change slope)
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③ Write the convergent line of turning point, slope of change and phase frequency curve
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(4) Draw the radial phase curve of each link in turn, and mark the link beside the curve
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⑤ Synthesis curve: amplitude frequency: from
Draw the slope of– x 20 vIt changes slope at every turning pointNotice the slope for each part of the drawing; Phase frequency: from– 90 * vStart by drawing a straight line, changing the slope at every turning point;Notice the slope for each part of the drawing
Draw the slope of– x 20 vIt changes slope at every turning pointNotice the slope for each part of the drawing; Phase frequency: from– 90 * vStart by drawing a straight line, changing the slope at every turning point;Notice the slope for each part of the drawing
Pay attention to
- When you draw a composition, mark the slope as you draw it!
- Draw a typical link phase frequency and amplitude frequency together!
- 1. The typical link should not be drawn on K! Draw at the 0 db line! 2. Two same typical links, pay attention to draw double!! Don’t ignore it!
Do the template:
Write the transfer function for the Bode diagram
- Step1: column frame: mark all turning points and write them in G(s); The slope of the line to the left of the minimum turning point is determined by v and the height by K
- Step2: Find the parameters
Bode diagram to judge system stability
Find GM, first find the phase crossing frequency :(phase crossing frequency of -180° line), make a straight line, 0-dbline is the GM of the bode diagram (> 0 stable)
☆ Find PM, first find the amplitude across the frequency (zero db line), make a straight line, the phase value minus -180° is the PM of the Byrd diagram (>0 stable)
☆ Judge system stability only by PM, not GM! PM>0, the system is stable. GM is used to calculate the critical K value
I’m increasing the multiples of K: NK (N > 0),
Calculate the critical K value (K can be maximized): let GM=0 (that is, the corresponding frequency PM page =0)! So 20lgN=GM is N, and K’=NK is the critical K value
The system design
The relation between frequency domain properties and time domain properties
Two conclusions:
1.
2.
Examples:
Solution: Truth value, the following is estimated by using the Bode diagram
Topic analysis
The time domain analysis
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There are two special cases of high order systems,Find the maximum response
: ① Calculate β/γ+ζ and judge overshoot PO② by looking at the picture
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The dynamic performance index of standard second-order system is obtained
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Analyze the first order system under open loop and closed loop
And K, the time constant
: ① Calculate β/γ+ζ and judge overshoot PO② by looking at the picture
Comprehensive question type: Find the critical K value:
Root trajectory analysis system performance
- Critical stability of the system: the value of K can be solved by the first column 0 of Rous table; The root locus finds K on the imaginary axis
- Given the value of a closed-loop pole, determine the corresponding value of K: Rule9: K = the product of all open loop poles to S/the product of all open loop zeros to S; Given the value of one closed-loop pole, find the other poles. If a single real pole is given, long division is used, and the value of second-order ζ is given, and the comparison coefficient method is used. If the value of the second order pole is known, another single real pole is found with Rule10
- ζ=0.5 is required to determine the corresponding K value: ① Plotting method: sinθ=ζ, draw a ray from the original point to find the K value of the point where it intersects with the root locus (Rule9) ② calculation method, using the comparison coefficient method
- The value range of K when the system is stable: 0< K < K value on the virtual axis; The range of K of the attenuated oscillation of the system response is as follows: k value at the separation point < K
Bode diagram type
- Given a K draw a Bode diagram
- GM and PM
Changing the Byrd diagram:
- Calculate the critical K’ value from the graph (K can be maximized): let GM=0! So 20lgN=GM is N, and K’=NK is the critical K value
- K prime in terms of PM and GM.
- If you make 20lgN equal to a positive value, you raise the curve, and if you make 20lgN equal to a negative value, you lower the curve
- I give you the Byrd picture transfer function
System modeling
Forward pathway: R->C, pairwise not fully contained
Independent loop: the comparison point passes through the forward path to see if it can form a circle,