Tutorial on control theory

Tutorial on control theory
Feng QIU (KEK) Dec. 17, KEK Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Main content Transfer function & Frequency response
Benefits of feedback system (Why FB?) Stability Criteria (FB is stable or not?) Analytical study of LLRF system Detune=0 Detune≠0 Discrete system Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Question We know that LLRF system is an feedback control system, but
Why we need feedback? Why FB improves system. Why we does not implement infinite gains in the feedback? Is the control (LLRF) system stable or it will be oscillated? What will influence the stability? To understand all of that, we need to learn Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Control Theory Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

LTI system (Assuming) Assume our system is LTI system Linear
Time-invariant If we have a system as follows: Is this system a linear system? Is it a time-invariant system? Why? Linear Time-invariant Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

SISO vs. MIMO SISO (Single Input Single Output) and MIMO (Multi-input Multi-output) We will study the SISO system at first and then expand to MIMO system. SISO MIMO Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

What is a control system?
A control system is a device, or set of devices, that manages, commands, directs or regulates the behavior of other devices or systems. Industrial control system are used in industrial production for controlling equipment or machines. (Definition from Wikipedia) A control system mainly include a plant, a sensor and a controller. Requirements: maintain some characteristics or behavior of the “plant”. Systems or devices that need to be controlled A device that regulate or control the behavior of the “Plant” Measuring tool that measured the response of the “Plant” Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Main components (Examples)
Examples: Digital low-level RF system. The field-programmable-gate-array (FPGA) plays roles of both controller and sensor (core component, will be introduced later). Requirements: making the electric field inside the cavity stable (most significant goal). Plant: Cavities, power source, RF Gun, antenna,… Controller: Electrical control phase shifter or attenuator, FPGA Target: Stabilize the field inside the cavity Cavity Sensor: Phase detector, amplitude discriminator, or FPGA FPGA Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Close Loop vs. Open loop Open loop control systems vs. closed loop control systems. In open loop control systems output is generated based on feedforward (FF). In close loop control systems current output is taken into consideration and corrections are made based on feedback. A closed loop system is also called a feedback (FB) control system (from wiki). To analyze how does a FB control system works, let’s start from “Transfer function” Close loop Open loop Open loop Closed loop Advantages Generally stable Easier construction Good disturbances rejection Disadvantages Poor disturbances rejection Unstable risk Difficult construction

Transfer function Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Transfer function A Transfer Function is the ratio of the output of a system to the input of a LTI system. The X(s) and Y(s) are the Laplace-transform of the input/output signal, respectively. Key point: The transfer function H(s) includes information of a system (usually can be seen as a representation of a given system). i.e. if we know the transfer function H(s) of a specified system (assume initial states = 0), we can calculate the output Y(s) by input X(s). Laplace-transform Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Inverse Laplace transformation
Formula of the Laplace-Transform Impulse response, (time domain representation of a system) Time domain Laplace-transform Complex freq. domain Transfer function Inverse Laplace transformation Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Laplace-Transform Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

How to obtain a TF (Example: RC circuits)
TF is related with differential equation. We have the differential equation: Current on C Current on R Laplace Transform Transfer function Transfer function of RC circuit Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Examples: RC circuit If we know the TF (assume initial state = 0), in principle, we know the system output uc(t) according to the given input u(t) (unit step). U(s) 1 u(t)= unit step time Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Examples: RC circuit Go back to the differential equation:
Solution of the differential equation 1 u(t)= unit step Input time 1 Usually, we also call the parameter τ time constant… Output τ time Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Examples: RC circuit If the time constant (τ) is 1ms. τ = 1ms
u(t)= unit step 1 uc(t) 1 time time τ = 1ms Time constant 0.632 ~ 0.632 Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Frequency Response Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Frequency response Frequency response It is a measure of magnitude and phase of the output as a function of frequency. From TF to transfer function: H(s)→H(jω). Freq. domain Fourier-transform Time domain Laplace-transform Complex freq. domain Transfer function

Amplitude vs. frequency
Frequency response If we know H(jω), we also know H(j2π∙f). And then A( f ) & P( f ). Amplitude vs. frequency Phase vs. frequency To go a little bit further. We are able to easily plot A( f ) & P( f ). if we know their expressions. Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Frequency response (Bode plot)
Bode diagram: plots of the amplitude-frequency and phase-frequency response of the system H(s). Something like that… dB frequency f frequency f degree We called these two plots “Bode plots” Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Frequency response Go back to the RC circuit.
Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Bode diagram Bode diagram: plots of the amplitude-frequency and phase-frequency response of the system H(s). Here τ =1 ms Amplitude vs. frequency Phase vs. frequency

Bode diagram How about some special case,
ω =1/τ = 1000 [rad/s]. f=160 Hz bandwidth The half power point is that frequency at which the output power (not voltage) has dropped to half of its mid-band value.

Frequency response If we know the frequency response of a system or its bode diagram, then for any sinusoidal signal… Still 50 Hz (steady state) 50 Hz E.g. a 50 Hz sinewave Steady state output Still 50 Hz, but,… Magnitude gain Phase shift Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Frequency response We can easily simulate with matlab/Simulink, the input is a 160 Hz sinusoidal signal. Special case, ω =1/τ = 1000 [rad/s]. f = ω/2π=160 Hz Input Output

Frequency response How about f = 600 Hz ?

Example of cavity Cavity transfer function More popular
Resonance radian frequency Quality factor or Q value Cavity RLC is rather difficult to measure, but we can easily measure Q and ω0

Example of cavity (cont’d)
Cavity transfer function H(s)→ Frequency response H(jω) Resonance radian frequency

Bode diagram ( Plots of the amplitude-frequency and phase-frequency response of the system) Bode Plot Magnitude vs. frequency Phase vs. frequency On-resonance

Bode Plot (Example1) 𝜔 0 =1.3𝑒9∙2𝜋 [𝑟𝑎𝑑/𝑠] 𝑄=1.3× 10 6 -3 dB Bandwidth 𝑓 2 𝑓 1 Bode Plot (Example2) 𝜔 0 =1.3𝑒9∙2𝜋 [𝑟𝑎𝑑/𝑠] 𝑄=7000 The half power point of an electronic amplifier stage is that frequency at which the output power (not voltage) has dropped to half of its mid-band value. That is a level of -3 dB. 3-dB Bandwidth (BW): Example1: BW1=100 [Hz], Example2: BW2=186 [kHz] Bandwidth Half BW: 𝑓 2 − 𝑓 1

Resonance frequency of a cavity (1.3 GHz)
Bode plot (SC vs. NC) Ex. 1 Bode Plot (SC) 𝜔 0 =1.3∙2𝜋 [𝑟𝑎𝑑/𝑠] 𝑄 1 =1.3× 10 6 𝐵𝑊 1 =100 [𝐻𝑧] Ex. 2 Bode Plot (NC) 𝜔 0 =1.3∙2𝜋 [𝑟𝑎𝑑/𝑠] 𝑄 2 =7000 𝐵𝑊 2 =186 [𝑘𝐻𝑧] Seems that BW is related to the Q value. 𝐵𝑊 1 =100 [𝐻𝑧] 𝐵𝑊 2 =186 [𝑘𝐻𝑧] 𝑄 1 =1.3× 10 6 𝑄 2 =7000 Resonance frequency of a cavity (1.3 GHz)

TF of a FB system Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Transfer function of a FB system
Let us go back to the basic control system, now let’s review it in the viewpoint of the transfer functions. Basic control system TF representation LLRF system K(s): Controller. generally, a proportional & integral (PI) controller P(s): Plant you want to control F(s): Detector (sensor), to measured the response of the plant Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Block diagram transformations
How to calculate the TF of the whole system if we know the TF of each subsystem? Serial Parallel Feedback Minus

Example I How to calibrate the transfer function from X(s) to Y(s)?
Serial Forward gain=P(s)K(s) Feedback

Mason’s rule To go a little bit further. In some cases, to calculate the TF is not so easy, for example (too many loops): Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Mason’s rule Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Mason’s rule “±” is important

Mason’s rule Forward path is only P(s) in this case.

Mason’s rule If we remove M1, some loops disappeared, the only residual loop is L1. Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Practice Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

We have learned We would discuss why we need FB in the next stage?
Feedback system We would discuss why we need FB in the next stage?

Benefits of FB control Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

With FB vs. w/o FB FB w/o FB
Now we are going to find out the answer of “why we need feedback”. FB w/o FB Let’s use the knowledge of TF Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Disturbance Rejection
In the real case, disturbances exists in the system (not only LLRF system, but almost al of the control system), so the actual system is something like: How to evaluate the influence of disturbances? Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Obviously, the existence of the disturbances (or perturbations) will influence the system, but is it same for FB and FF? We can analyze it what we have learned. With FB w/o FB Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

How about give some meaning for each component (H(s)). FB w/o FB z

Further more, if GP =1, then w/o FB FB z Considering the H(jω) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

The best way is to compare their frequency response or bode plot? Bode diagram: plots of the amplitude-frequency and phase-frequency response of the system H(s). FB w/o FB Let’s find out the 0 Hz in the plot w/o FB How about 500 Hz disturbances? If there is 5 Hz disturbances 500 Hz

Frequency Response (review)
If we know the frequency response or bode diagram, then… Still 50 Hz (steady state) 50 Hz E.g. a 50 Hz sinewave Steady state output Still 50 Hz, but,… Magnitude gain Phase shift Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

FB vs. FF (disturbance rejection)
If the disturbances is, for example, a sine wave like d(t)=sin(2π∙500t) d(t)=sin(2π∙500t) d(t)=sin(2π∙500t) d(t)=sin(2π∙500t) w/o FB FF K0=10 Feedback K0=100 Feedback K0=1 ~1% (-40 dB) Feedback

Power convertor ripples in RF system
Low level RF (sensor & controller) That d(t)=sin(2π∙300t) will be suppressed by LLRF feedback. Plant: P(s) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Sensitivity of model parameter variation
Assume there are no disturbances, but the model parameter GP varied due to some reason. FB W/O FB z Not constant, but varies due to some reason time Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Sensitivity of model parameter variation
In the FB control, the effects of the system parameter variation becomes smaller. time z w/o FB K0=20 w/o FB K0=40 K0=60 Feedback Feedback Feedback

Time response The time response can be faster (bandwidth can be larger) in the FB. w/o FB K0=60 K0=40 K0=20 z Feedback w/o FB Feedback Feedback Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Stabilize an unstable system
In some cases, the system itself is unstable (this definition will be discussed later, now just accepted), we can use FB to make the system stable. w/o FB To infinite z w/o FB With FB Feedback Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

We have learnt We would discuss it in the next topic?
Feedback system Benefits of FB Parameter variation We would discuss it in the next topic?

Stability Criteria Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Stability Criteria Stable is the most important thesis in a feedback system, if the system is not stable, there is no meaning for any efforts. The Stability Criteria for a feedback system includes Root locus Solve the characteristic equation Open loop bode plot & Nyquist Criterion Routh–Hurwitz stability criterion All of them are important, but… Stability criteria Open loop-based Close loop-based Nyquist Criterion Bode plot Characteristic equation Routh–Hurwitz Root locus Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Stability Criteria (poles position)
Definition: A stable system is a dynamic system with a bounded response to a bounded input. GO TO infinite (dangerous) Unstable x(t) y(t) bounded input Stable time time We can not try all of the bounded input signal anyway Characteristic equation Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Stability Criteria (poles position)
zeros Complex number A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.

Example The system H (s) should be stable because all of the poles is in the have negative real part. The system G(s) should be unstable because some poles have positive real part. positive real part Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Example (pole-zero map)
Imag H (s): stable Real G (s): unstable poles Unstable: right half plane poles (RHP poles) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Example (pole-zero map)
No RHP poles Go to infinite (unstable) G(s) Step H(s) Has RHP poles Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Stability Criteria (bode plot)
It is easy to solve the linear equation, but can not answer questions like “if I increase the gain in K(s), what would happen? The FB system is still stable or not? If not, why it becomes unstable? ”. Further more, in some system, the characteristic equation is not like a polynomial, thus it is difficult find out the poles directly (such as system with time delay). Time delay Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Still remember the bode plot? It is a very popular method to judge the system stability and also analyze the system performance by its open loop TF. Feedback frequency Minus dB f IMPORTANT: open loop TF f degree

Suitable for bode plot: 1) Minimum phase, 2) SISO system. dB H(s): Unstable G(s): Stable frequency f minimum phase frequency f degree -180 deg The closed loop is stable if the open loop gain is less than 1 (0 dB) if the phase of the open loop is -180 degree (or +180, -540, -720 ….). Non-minimum phase

Gain Margin and Phase Margin
dB G(s): Stable frequency f Gain margin degree frequency f Phase margin -180 deg Gain Margin: Phase Margin Larger margin→ Better robustness Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Stability Criteria (Nyquist diagram)
Benefits of Nyquist diagram More information Non-minimum phase okay MIMO also okay Minimum phase Bode Nyquist Nyquist diagram Clockwise:-1 Anti-clockwise:+1 Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Nyquist diagram Clockwise:-1 Anticlockwise:+1 End (ω=inf)
Start (ω=0, H(0)=0.9) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Summary Parameter variation Characteristic equation Bode plot
Feedback system Benefits of FB Parameter variation Characteristic equation RHP poles Stability criteria Bode plot Gain margin & Phase margin Nyquist diagram Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

TF Model of LLRF system (Δω=0)

LLRF system Plant: Cavities, power source, RF Gun, antenna,…
Controller: Electrical control phase shifter or attenuator, FPGA Target: Stabilize the field inside the cavity Cavity Sensor: Phase detector, amplitude discriminator, or FPGA FPGA Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Cavity (detuning=0) Cavity is like a parallel resonance circuit.
First of all, we consider the simplest case: no cross component (detuning=0) If detuning=0 baseband RF frequency Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

PI controller PI control is very popular in the FB control system (& LLRF FB control system) PI Controller Usually performed in the FPGA (or DSP) Transfer function Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Sensor Detector can be also seen as a low pass filter but with higher bandwidth than cavity.. low pass filter Usually performed in the FPGA Cavity half bandwidth detector -3dB bandwidth Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Analytical Study (components)
PI control is very popular in the FB control system (& LLRF FB control system) Cavity and detector are a low-pass filters with different bandwidth. Sys. Components Transfer function PI Controller Frequency response Cavity Half bandwidth Detector Frequency response

Loop delay Usually, system need a time to have a response (NOT immediately). Time delay exists in the majority control system (also in LLRF system) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Laplace transform Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

TF of overall LLRF Now, the overall model can be constructed.
What we can do with this LLRF system model Overall TF (closed-loop & open loop) Bode diagram (open-loop) Gain-margin and critical gain Loop gain r r f y Loop Delay Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Bode plots (review) dB G(s): Stable frequency f frequency f degree
Gain margin frequency f degree Phase margin -180 deg Gain Margin: Phase Margin Larger margin→ Better robustness Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Analytical Study (Bode plot)
Bode diagram judge the stability criterial based on OPEN LOOP. Is the system stable? What is gain-margin & phase margin? What is the Q value of the cavity (f0=1.3 GHz)? Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Nyquist diagram Benefits of Nyquist diagram More information
Non-minimum phase okay MIMO okay Minimum phase Nyquist diagram Clockwise:-1 Anti-clockwise:+1 Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Analytical Study (Nyquist)
Clockwise:-1 Anticlockwise:+1 Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Gain Margin and Phase Margin
Gain margin & phase margin? Unit circle GM PM Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Analytical Study(Loop Delay effect)
How does time delay influence the bode plot and the system? Time delay dB frequency Gain margin Gain margin frequency degree Phase margin -180 deg

Analytical Study(Loop Delay effect)
How does time delay influence the bode plot and the system? 0 dB Which system is stable? How does the loop delay influence the bode plot?

Analytical Study (Loop Gain effect)
How does loop gain influence the bode plot and the system? dB 20 dB G(s): Stable frequency f frequency f degree Phase margin -180 deg Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Example (Loop gain effect)
How does gain influence the bode plot and the system? Which system is stable? How does the loop gain influence the bode plot?

Stability Criteria (critical gain)
Bode Plot (Open loop) Magnitude vs. frequency ( 𝐿(𝑗𝜔) ) Phase vs. frequency (∡𝐿(𝑗𝜔)) Let us consider some very special case: Find out the frequency where phase is -180 degree lower plot), and then find out the frequency where the magnitude becomes 0 dB upper plot), if these two frequency are same, then the gain is CRITICAL GAIN. What is the gain margin (phase margin) under critical gain? If the gain is larger than critical gain, what would happen?

Analytical Study (I gain effect)
How does integral gain influence the bode plot and the system? Effect of Integral gain What is the effect of the I gain?

Feedback system Benefits of FB Parameter variation Characteristic equation RHP poles Stability criteria Bode plot Gain margin & Phase margin Nyquist diagram Overall TF of LLRF Gain margin & Phase margin loop gain &loop delay Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Cavity equation & TF Model of LLRF system (Δω≠0)

Cavity Differential Equation
In our previous discussion, we assume that there is no detuning in the cavity, thus the cavity (in baseband) can simplified to be a simple low-pass filter. If detuning=0 baseband RF frequency Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Cavity equation However, in the actual case, there is detuning especially in the presence of the large field. If detuning exist baseband RF frequency Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Similar with RC circuit
Cavity equation In the presence of the detuning, the cavity equation is some thing like: Δω≠0 Δω=0 detuning Based-band equation If Δω=const. Similar with RC circuit Laplace TF Step response Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

State Equation We can also use the state space model (another way to represent the system, we will not talk about it in this lecture). Cavity equation State vector State equation Standard state space model Measurement equation Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

State equation Actually, the TF is related with state equation.
If Δω=const. Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Cavity Model (TF) Then the TF model is a matrix (a 2×2 matrix, Δω=const.), and there are cross components. Cross Cross Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

SISO vs. MIMO (review) SISO (Single Input Single Output) and MIMO (Multi-input Multi-output) SISO MIMO Cross Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

TF of total LLRF (with detune)
The whole FB system is also MIMO if one of the component is MIMO. Unfortunately, LLRF system is also an MIMO system. SISO MIMO MIMO Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

TF of LLRF Another issue could generate the cross component is loop phase due to some errors (E.g. calibration errors). MIMO system

TF of LLRF We modified the LLRF system with loop phase, then we have the total TF of the LLRF system with constant detune. The stability of the LLRF system is mainly decided by the three main factors. Loop gain Loop Delay Loop Phase Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Bode plot (NOT work) Bode plot does not work in the MIMO system (the bode plot is not only one, but actually four). dB G(s): Stable frequency f Gain margin frequency f degree Phase margin -180 deg Gain Margin: Phase Margin

Analysis of the stability by Nyquist plot

Nyquist plot (MIMO case)
Fortunately, the Nyquist diagram still works for MIMO system. Only a little bit modification, let’s considering the determinant of the TF matrix (unique). Nyquist diagram for this determinant Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Loop Phase How does loop influence the system stability? We assume detuning is zero. If the loop phase error is up to 90 degree, the system is stable or not? Loop gain Detune=0 means P12=P21=0 Loop Delay Loop Phase Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Nyquist plot (loop phase)
Plot the Nyquist diagram. (0+0i) not (-1+0i)

Nyquist plot (loop phase)
Which case is not stable? Clockwise:-1 Anticlockwise:+1 Zoom (0+0i) not (-1+0i)

Nyquist plot (detune) If detuning is not zero, cavity is MIMO, and then FB system is also MIMO. Detuning is NOT zero, cavity is then MIMO Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Nyquist plot (detune) Is the system stable? Clockwise:-1
Anticlockwise:+1 Zoom (0+0i) not (-1+0i)

Bode vs Nyquist (Loop Gain)
If detune and loop gain is zero, we can also use bode plot to judge the stability. (-1+0i) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Bode vs Nyquist (Loop Delay)
If detune and loop gain is zero, we can also use bode plot to judge the stability. (-1+0i) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Cavity dynamic models

Cavity dynamic model Simulink model of the cavity equation (dynamic, so the Δω can be also dynamic)

LLRF dynamic model Construction is very simple.

LLRF dynamic model We can check what we have learned.
GO TO inf (unstable) Loop phase = 90 degree Detune = 600 Hz Still Stable

LLRF dynamic model We can check what we have learned. Gain = 1500
GO TO inf (unstable) Gain = 1500 Delay = 15µs GO TO inf (unstable)

Feedback system Benefits of FB Parameter variation Characteristic equation RHP poles Stability criteria Bode plot Gain margin & Phase margin Nyquist diagram Overall TF of LLRF Overall TF of LLRF Loop gain Bode plot & Nyquist loop gain &loop delay Loop delay Bode plot & Nyquist Loop Phase Nyquist diagram Detune Nyquist diagram

Discrete systems

Digital LLRF Since the digital LLRF system becomes more and more popular in accelerators. Its better to have some basic knowledge of the discrete systems. Both the sensor and controller are performed by FPGA (digital devices) Controller: Electrical control phase shifter or attenuator, FPGA DAC Cavity FPGA ADC Sensor: Phase detector, amplitude discriminator, or FPGA

Continuous to Discrete
It is easy to transform analog signal to digital one by ADC and DAC. digital digital analog ADC DAC DSP 7 6 f(kTs) 5 Amplitude 4 3 2 f(t) 1 kTs Time Ts 2Ts 3Ts Ts (k-1)Ts (k+1)Ts Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Discrete system DAC ADC h(t) h[k] u(k) y(k) u(k) y(k)

Z-transform Z-transform is very similar with Laplace transform
Impulse response Continuous time Discrete time Laplace-transform z-transform S-domain Z-domain Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

H(z) and H(s)

H(z) and H(s) Similar characteristics. y(t) x(t)
Almost exactly same with H(s) x(k) y(k) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Block diagram transformations
The transformation rules for block diagram is also exactly same. Serial Parallel Feedback Minus

Mappings Z transform is not one-to-one mapping.
For stable system H(z), poles should be inside the unit circle. Unit circle Unstable Unit circle Stable Unstable Stable Unstable

Discrete cavity equation
Mapping the poles of H(s) to H(z) Mapping Poles Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Discrete cavity equation
Mapping the poles of H(s) to H(z). Discrete-time cavity equation. It can be performed by FPGA (cavity simulator) Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Feedback system Benefits of FB Parameter variation Characteristic equation RHP poles Stability criteria Bode plot Gain margin & Phase margin Nyquist diagram Overall TF of LLRF Overall TF of LLRF Loop gain Bode plot & Nyquist loop gain &loop delay Loop delay Bode plot & Nyquist Loop Phase Nyquist diagram Discrete-form ( FPGA implementation) Detune Nyquist diagram Feng QIU, 10th International Accelerator School for Linear Colliders, Japan, 2016

Reference S. J. Mason, Feedback theory: Further properties of signal flow graphs, in Proceedings of the Institute of Radio Engineers (IRE, New York, 1956), pp. 920–926. C. Schmidt, Ph.D. thesis, Technische Universität Hamburg-Harburg, 2010 T. Schilcher, Ph. D. Thesis of DESY, 1998

Tutorial on control theory

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