ME 4SS3 - L12 - R01 PDF

Smart Systems ME 4SS3 Dr. S. Andrew Gadsden Department of Mechanical Engineering, McMaster University L12.1 Introduction to Feedback L12.2 PID Controllers L12.3 Design of PID Controllers L12.4 Sliding Mode Control (SMC) ...

Smart Systems ME 4SS3 Dr. S. Andrew Gadsden Department of Mechanical Engineering, McMaster University L12.1 Introduction to Feedback L12.2 PID Controllers L12.3 Design of PID Controllers L12.4 Sliding Mode Control (SMC)  Monday Wednesday Thursday Deliverables (Virtual on A2L or MS Teams) (HH 305) (HH 305) - - L01: Introduction 09/04 L02: Introduction to 09/05 - to Course Project L03: Introduction to 09/09 L04: System 09/11 L05: Project - Laser 09/12 - Smart Systems Modeling I Cutting (JHE A104A) L06: System Modeling 09/16 L07: System Modeling 09/18 L08: Project - Assembly 09/19 - Practice II Check and Testing L09: System Modeling 09/23 L10: System Modeling 09/25 L11: Project - Modeling 09/26 Assignment 1 and Matlab I and Matlab II Examples (09/29) No Class 09/30 L12: Control Theory 10/02 L13: Project - Matlab 10/03 - (Truth and Simulation Reconciliation) L14: Signal 10/07 L15: Controllers 10/09 L16: Project - 10/10 - Conditioning and Matlab TA Consultations Where are we? No Class 10/14 No Class 10/16 No Class 10/17 Assignment 2 (Thanksgiving Monday) (Break) (Break) (10/20) L17: Kalman Filter I 10/21 L18: Kalman Filter II 10/23 L19: Project - Arduino 10/24 - Tutorial L20: Kalman Filter and 10/28 L21: Introduction to 10/30 L22: Project - TA 10/31 Assignment 3 L12 - Smart Systems Matlab Artificial Intelligence Consultations (11/03) L23: Machine Learning 11/04 L24: Resume 11/06 L25: HR Industry 11/07 Assignment 4 Techniques Workshop (Online) Panel (In-Person) (11/10) L26: Machine Learning 11/11 L27: Review of Smart 11/13 L28: Project - TA 11/14 Resume Asgmt. Applications Systems Material Consultations (11/17) Virtual Office Hours on 11/18 L29: Midterm Review 11/20 L30: Midterm 11/21 Midterm MS Teams and Help (11/21) Virtual Office Hours on 11/25 L31: Project - TA 11/27 L32: Project - 11/28 Project Demo MS Teams Consultations Demonstration Day (11/28) * Check A2L for the latest schedule Virtual Office Hours on MS Teams 12/02 Virtual Office Hours on MS Teams 12/04 Project - Report Due (No Class) 12/05 Project Report (12/05) 2 12.0 L10 Quiz Smart Systems 1. What are the following terms: 𝑥, 𝑧, 𝑢, 𝐴, 𝐵, 𝐶, 𝑘 2. What are the dimensions for the above terms? Parameter Definition Dimensions 𝑥 State vector 𝑛×1 𝑧 Measurement vector 𝑚×1 L12 - Smart Systems 𝑢 Input vector 𝑟×1 𝐴 System matrix 𝑛×𝑛 𝐵 Input gain matrix 𝑛×𝑟 𝐶 Measurement matrix 𝑚×𝑛 𝑘 Time step 1×1 3 12.1 Control Theory Introduction to Feedback Open-loop feedback: L12 - Smart Systems 4 12.1 Control Theory Introduction to Feedback Closed-loop feedback: L12 - Smart Systems 5 12.1 Control Theory Introduction to Feedback Digital closed-loop feedback: L12 - Smart Systems 6 12.2 Control Theory PID Controllers L12 - Smart Systems Spot Launch by Boston Dynamics (2019) Link: https://www.youtube.com/watch?v=wlkCQXHEgjA 7 12.2 Control Theory PID Controllers Continuous-time PID controller: 𝑡 𝑢 𝑡 = 𝐾𝑝 𝑒 𝑡 + 𝐾𝑑 𝑒ሶ 𝑡 + 𝐾𝑖 න 𝑒 𝑡 𝑑 𝑡 0 How do you select the gain values? How do you ‘tune’ them? What is the discrete-time error signal?  Usually some desired state minus the actual state L12 - Smart Systems 8 12.2 Control Theory PID Controllers Discrete-time PID controller: 𝑢𝑘 = 𝐾𝑝 𝑒𝑘 + 𝐾𝑑 (𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑒𝑘 ) + 𝐾𝑖 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑜𝑓 𝑒𝑘 What are the derivative and integral of the error term?  Derivative? Slope of the error curve (rise over run)… 𝑒 L12 - Smart Systems 𝑒𝑘+1 − 𝑒𝑘 𝑡𝑘 𝑇 𝑡𝑘+1 Error 𝒅 𝒆 𝒆𝒌+𝟏 − 𝒆𝒌 𝒆𝒌+𝟏 − 𝒆𝒌 ≅ = Time 𝒅𝒕 𝒕𝒌+𝟏 − 𝒕𝒌 𝑻 9 12.2 Control Theory PID Controllers Discrete-time PID controller: 𝑢𝑘 = 𝐾𝑝 𝑒𝑘 + 𝐾𝑑 (𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑒𝑘 ) + 𝐾𝑖 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑜𝑓 𝑒𝑘 What are the derivative and integral of the error term?  Integral? Area under the error curve…  We should add up this integral error throughout the entire 𝑒 න𝑒 ≅ 1 + 2 simulation (still an approximation) L12 - Smart Systems 1 = 0.5 𝑒𝑘+1 − 𝑒𝑘 𝑇 𝑒𝑘+1 − 𝑒𝑘 (1) (2) = 𝑒𝑘 𝑇 Error න 𝑒 ≅ 𝑒𝑘 𝑇 + 0.5 𝑒𝑘+1 − 𝑒𝑘 𝑇 (2) 𝑒𝑘 10 න 𝒆 ≅ 𝟎. 𝟓 𝒆𝒌+𝟏 + 𝒆𝒌 𝑻 Time 𝑡𝑘 𝑇 𝑡𝑘+1 12.2 Control Theory PID Controllers Discrete-time PID controller: 𝑒𝑘 − 𝑒𝑘−1 𝑢𝑘 = 𝐾𝑝 𝑒𝑘 + 𝐾𝑑 + 𝐾𝑖 𝑒𝑖𝑛𝑡 𝑇 Where 𝑒𝑖𝑛𝑡 = σ𝑡𝑘=1 0.5 𝑒𝑘 + 𝑒𝑘−1 𝑇 and is summed throughout time What is the effect of 𝑇 on the control signal? L12 - Smart Systems 11  12.2 Control Theory PID Controllers What about the error signal?  Most would be a tracking error  Difference between the desired state value and the actual state value  For example: 𝑒𝑘 = 𝑥𝑑,𝑘 − 𝑥𝑘 What does it look like in MATLAB (inside for loop)? L12 - Smart Systems error(k) = xd(k) - x(1,k); % Defines error used to calculate PID control signal (e.g., first state in this case) error_int = error_int + (error(k) + error(k-1))*0.5*T; % Adds the integral error to the summation term (error_int) u(k) = Kp*error(k) + Kd*(error(k) - error(k-1))/T + Ki*error_int; % Calculates PID control signal 12 12.2 Control Theory PID Controllers In general, but not always:  Increasing 𝐾𝑝 makes the response faster, and reduces the steady-state error (a bit)  Increasing 𝐾𝑑 reduces the overshoot  Increasing 𝐾𝑖 reduces the steady-state error Since there is interaction between the three gains, tuning them manually can be very challenging L12 - Smart Systems Engineering design is a skill utilized by your knowledge base! 13 12.2 Control Theory PID Controllers This has encouraged the development of various tuning rules and methods:  Two of the best known methods for tuning a PID controller were developed by Zeigler and Nichols (Z-N)  Although it was developed for continuous time PID control, it is often used for discrete time PID control tuning Other software tools and methods exist L12 - Smart Systems  MATLAB has a great toolbox for tuning PID gains 14 12.2 Control Theory PID Controllers – Z-N Method #1 Z-N Method #1: 1. Perform an open-loop step test and plot the response 2. From the plot, obtain R (maximum slope) and L (the ‘lag’) 3. The PID parameters are then:  𝐾𝑝 = 1.2/(𝑅𝐿) L12 - Smart Systems  𝐾𝑖 = 1/(2𝐿)  𝐾𝑑 = 0.5𝐿 15 12.2 Control Theory PID Controllers – Z-N Method #1 – Example Z-N Method #1: 9 Solve for PID gains using Z-N Method #1 for 𝐺 𝑠 = 𝑠 2 +0.6𝑠+9 1. Plot the open-loop step response in MATLAB  G = tf(9,[1 0.6 9]); Step Response  step(G); 1.8 1.6 L12 - Smart Systems Find maximum slope (R) 1.4 2. 1.2 and lag (L) Amplitude 1  xlim([0 1]); 0.8 0.6  Lag is about L = 0.175 0.4 1  Slope is about 𝑅 = 0.2 = 2.5 0.4 0 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (seconds) 12.2 Control Theory PID Controllers – Z-N Method #1 – Example Z-N Method #1: 3. Initial PID parameters are then calculated as follows: 1.2 1.2 𝐾𝑝 = = = 2.743 𝑅𝐿 2.5 0.175 1 1 𝐾𝑖 = = = 2.857 2𝐿 2 0.175 𝐾𝑑 = 0.5𝐿 = 0.5 0.175 = 0.0875 L12 - Smart Systems These are starting points…  You then tune further (manually) to obtain desired responses and characteristics:  Rise time, steady-state error (e.g., for step input), settling time, percent overshoot  What is a ‘very fast’ sampling rate? Fast? Slow? 17 12.2 Control Theory PID Controllers – Z-N Method #2 Z-N Method #2: 1. Use proportional control only (i.e., 𝑢 = 𝐾𝑝 and 𝐾𝑑 = 𝐾𝑖 = 0). Perform a step test and observe the plant output. Increase 𝐾𝑝 until continuous oscillations result (i.e., their amplitude remains constant). 2. Record the 𝐾𝑝 gain as 𝐾𝑢 and the oscillation period as 𝑃𝑢. The PID parameters are then: L12 - Smart Systems 3. 𝐾𝑝 = 0.6𝐾𝑢 𝐾𝑖 = 2/𝑃𝑢 𝐾𝑑 = 𝑃𝑢 /8 These are starting points…  You then tune further (manually) to obtain desired responses and characteristics… 18 12.3 Control Theory Design of PID Controllers – System Types The type number of a system refers to the number of 1/𝑠 factors in the open-loop transfer function For example, the system types for the following transfer functions are defined as: 5 is type 1 (second order) 𝑠(𝑠+10) 5 is type 0 (first order) L12 - Smart Systems 𝑠+10 A type 0 system has finite steady-state error (𝑒𝑠𝑠 ) to a step input and infinite 𝑒𝑠𝑠 to a ramp input A type 1 system has zero 𝑒𝑠𝑠 to a step input and finite 𝑒𝑠𝑠 to a ramp input Normally, we want the open-loop transfer function to be type 1 (i.e., lower steady state error) 19 12.3 Control Theory Design of PID Controllers – Bode Plots In general, continuous controller design in the frequency domain is much simpler than in the time domain or using root locus methods Note that:  Any time domain signal can be converted into a summation of sinusoids of different amplitudes and phases  The steady-forced output of a transfer function to be sinusoidal input 𝐴 sin 𝜔𝑡 equals: L12 - Smart Systems 𝐴𝑀 sin 𝜔𝑡 + 𝜙  where 𝑀 is the magnitude (or amplitude ratio) and 𝜙 is the phase angle. Note that both 𝑀 and 𝜙 are a function of the input frequency 𝜔 20 12.3 Control Theory Design of PID Controllers – Bode Plots A Bode plot is a plot of 𝑀 in dB (where 𝑥 in 𝑑𝐵 = 20 log 𝑥) and 𝜙 in degrees versus log 𝜔 For the example shown in the figure, L12 - Smart Systems 𝐺𝑀 = +14 𝑑𝐵 and 𝑃𝑀 = +42° If 𝐺𝑀 < 0 or 𝑃𝑀 < 0 the closed-loop system will be unstable Check appendix slides for additional details… 21 12.3 Control Theory Design of PID Controllers L12 - Smart Systems Ball Balancing PID System by Nathan (2014) Link: https://www.youtube.com/watch?v=7Jw8m4pbTYI 22 12.4 Control Theory Sliding Mode Control For nonlinear systems, a well known control strategy is sliding mode control (SMC) It is a type of variable structure control: utilizes a discontinuous switching plane along some desired trajectory L12 - Smart Systems 23 12.4 Control Theory Sliding Mode Control Utilizes a discontinuous switching plane along some desired trajectory (referred to as a sliding surface) Objective is to keep the state values along this surface by minimizing the state errors If state value is off, a switching gain would be used to push the state towards the sliding surface L12 - Smart Systems Once on the surface, the states slide along the surface in what is called a sliding mode 24 12.4 Control Theory Sliding Mode Control The sliding surface is defined based on the trajectory tracking error, for example: 𝑆 = 𝑒ሷ + 2𝜉𝜆𝑒ሶ + 𝜆2 𝑒 The typical control signal is as follows: 𝑢 = 𝑢𝑒𝑞 + 𝑢𝑠𝑤 Where 𝑑𝑆/𝑑𝑡 = 0 (to stay on surface) we have: L12 - Smart Systems 𝑥ഺ𝑑 − 𝑓መ 𝒙 − 2𝜆𝜉 𝑒ሷ − 𝜆2 𝑒ሶ 𝑢𝑒𝑞 = 𝑔ො 𝒙 𝑆 𝑢𝑠𝑤 = −𝐾𝑆𝑀𝐶 𝑠𝑎𝑡 𝜑 25 L12 Summary Key Takeaways Open-loop and closed-loop (feedback) control systems PID is the most popular, linear, control method  Zeigler and Nichols (Z-N) methods can be used to obtain initial gain values  Increasing 𝐾𝑝 makes the response faster, and reduces the steady-state error (a bit) L12 - Smart Systems  Increasing 𝐾𝑑 reduces the overshoot  Increasing 𝐾𝑖 reduces the steady-state error Other types of control systems and strategies include gain- scheduled, self-tuning, model-referenced, LQR, and SMC 26  Monday Wednesday Thursday Deliverables (Virtual on A2L or MS Teams) (HH 305) (HH 305) - - L01: Introduction 09/04 L02: Introduction to 09/05 - to Course Project L03: Introduction to 09/09 L04: System 09/11 L05: Project - Laser 09/12 - Smart Systems Modeling I Cutting (JHE A104A) L06: System Modeling 09/16 L07: System Modeling 09/18 L08: Project - Assembly 09/19 - Practice II Check and Testing L09: System Modeling 09/23 L10: System Modeling 09/25 L11: Project - Modeling 09/26 Assignment 1 and Matlab I and Matlab II Examples (09/29) No Class 09/30 L12: Control Theory 10/02 L13: Project - Matlab 10/03 - (Truth and Simulation Reconciliation) L14: Signal 10/07 L15: Controllers 10/09 L16: Project - 10/10 - Conditioning and Matlab TA Consultations What’s next? No Class 10/14 No Class 10/16 No Class 10/17 Assignment 2 (Thanksgiving Monday) (Break) (Break) (10/20) L17: Kalman Filter I 10/21 L18: Kalman Filter II 10/23 L19: Project - Arduino 10/24 - Tutorial L20: Kalman Filter and 10/28 L21: Introduction to 10/30 L22: Project - TA 10/31 Assignment 3 L12 - Smart Systems Matlab Artificial Intelligence Consultations (11/03) L23: Machine Learning 11/04 L24: Resume 11/06 L25: HR Industry 11/07 Assignment 4 Techniques Workshop (Online) Panel (In-Person) (11/10) L26: Machine Learning 11/11 L27: Review of Smart 11/13 L28: Project - TA 11/14 Resume Asgmt. Applications Systems Material Consultations (11/17) Virtual Office Hours on 11/18 L29: Midterm Review 11/20 L30: Midterm 11/21 Midterm MS Teams and Help (11/21) Virtual Office Hours on 11/25 L31: Project - TA 11/27 L32: Project - 11/28 Project Demo MS Teams Consultations Demonstration Day (11/28) * Check A2L for the latest schedule Virtual Office Hours on MS Teams 12/02 Virtual Office Hours on MS Teams 12/04 Project - Report Due (No Class) 12/05 Project Report (12/05) 27 Additional Resources Smart Systems Slides (PDF) and relevant code will be found on the course page within Avenue to Learn Relevant textbooks and resources are referenced in the syllabus or within the slides Please contact me if you have any difficulties throughout the course L12 - Smart Systems 28 Appendix Slides Smart Systems Extra resources for assignments or project, as you see fit You will generally not be tested on this material L12 - Smart Systems 29 12.3 Control Theory Design of PID Controllers – Bode Plots A second order transfer function is often used to approximate higher order systems The standard form of this transfer function is as follows: 𝑌 𝑠 𝜔𝑛2 = 2 𝑅 𝑠 𝑠 + 2𝜉𝜔𝑛 𝑠 + 𝜔𝑛2 where 𝜔𝑛 is the undamped natural frequency and 𝜉 is the damping ratio L12 - Smart Systems If the closed-loop system can be approximated as second order with 0.6 ≤ 𝜉 ≤ 0.9 the following relationships can be obtained and used: 2,920 𝑃. 𝑂. ≈ − 40 𝜙𝑚 840 𝑡𝑠 ≈ − 8.7 /𝜔𝑐 𝜙𝑚 𝜙𝑚 𝑡𝑟 ≈ 70 + 0.5 /𝜔𝑐 30 12.3 Control Theory Design of PID Controllers – Bode Plots Bode plots may be used to design P, PD, and PID controllers In this case, the open-loop transfer function is 𝐺𝐶 𝐺 where 𝐺𝐶 is the controller transfer function and 𝐺 is the plant transfer function In general, when designing using Bode plots we use the controller to alter the shape of the open-loop Bode plot such that the design specifications are met (assuming that they can be met) L12 - Smart Systems The P controller has the following simple transfer function: 𝐺𝑐 𝑠 = 𝐾𝑝 Since a P controller only consists of a gain factor, it may only alter the Bode plot by shifting the dB curve either upwards or downwards  This limits its ability to meet several specifications simultaneously 31 12.3 Control Theory Design of PID Controllers – Prop. Con. Example For the Bode plot given, assuming the closed-loop transfer function can be approximated as second order, determine the value of 𝐾𝑝 which will provide the faster response (i.e., the largest 𝜔𝑐 ) and P.O. less than or equal to 20%. 2,920 2,920 2,920 𝑃. 𝑂. ≈ − 40 such that 𝜙𝑚 ≈ = = 49° 𝜙𝑚 𝑃.𝑂.+40 20+40 So, we require 𝜙 = −180 + 49 = −131° at the new L12 - Smart Systems 𝜔𝑐. The current gain at this frequency is -7 dB. To make it the new 𝜔𝑐 , this must be increased to 0 dB so the P controller must shift the curve by +7 dB. Therefore, the proportional gain is found as: 𝐾𝑝 = 10 7/20 = 2.2 32 12.3 Control Theory Design of PID Controllers – Prop. Con. Example Step responses with 𝐾𝑝 = 1 and 𝐾𝑝 = 2.2 are shown in Figure 4.3.5. Note how the higher gain achieves a faster response at the cost of more overshoot. L12 - Smart Systems 33 12.3 Control Theory Design of PID Controllers – 𝑲𝑷+𝑰 A PI controller has the following transfer function: 𝐾𝑖 𝐾𝑖 𝜏1 𝑠 + 1 𝐺𝑐 𝑠 = 𝐾𝑝 + = 𝑠 𝑠 where 𝜏1 = 𝐾𝑝 /𝐾𝑖 , 𝐾𝑝 is the proportional gain, and 𝐾𝑖 is the integral gain. This controller is normally used primarily to increase the system L12 - Smart Systems type from 0 to 1, reducing or eliminating steady-state errors. 34 12.3 Control Theory Design of PID Controllers – 𝑲𝑷+𝑰 A step-by-step design procedure for a PI controller is as follows: 1. Select 𝐾𝑝 such that the desired 𝜙𝑚 + 5° is met. Find the new 𝜔𝑐 which results from shifting the dB curve by 20 log 𝐾𝑝. 2. Select 𝐾𝑖 such that: 10 10𝐾𝑖 = = 𝜔𝑐,𝑛𝑒𝑤 𝜏1 𝐾𝑝 Note that this choice stops the PI controller from significantly altering our 𝜙𝑚 L12 - Smart Systems from step 1. 3. The PI controller will change the 𝑀∗ value at 1 rad/s by 20 log 𝐾𝑖. 4. Manually fine tune based on the time response. 35 12.3 Control Theory Design of PID Controllers – 𝑲𝑷+𝑫 The transfer function for a PD controller is defined by: 𝐺𝑐 𝑠 = 𝐾𝑝 + 𝐾𝑑 𝑠 = 𝐾𝑝 𝜏1 𝑠 + 1 where 𝜏1 = 𝐾𝑑 /𝐾𝑝 , 𝐾𝑝 is the proportional gain, and 𝐾𝑑 is the derivative gain. This controller is normally used primarily to increase 𝜙𝑚 and 𝜔𝑐 resulting in a faster response with reasonable overshoot. L12 - Smart Systems 36 12.3 Control Theory Design of PID Controllers – 𝑲𝑷+𝑫 A step-by-step design procedure for a PD controller is as follows: 1. Select 𝐾𝑝 such that 𝑒𝑠𝑠 and/or 𝜔𝑐 specifications are met. 2. Shift the dB curve by 20 log 𝐾𝑝 and find the new 𝜙𝑚 and 𝜔𝑐 values. 3. Phase shift needed from the PD controller is 𝜙𝑃𝐷 = 𝜙𝑚,𝑑𝑒𝑠𝑖𝑟𝑒𝑑 + 5° − 𝜙𝑚,𝑛𝑒𝑤 If 𝜙𝑃𝐷 > 70° the specifications cannot be met by a PD controller, otherwise: log 𝜏1 𝜔 = 𝜙𝑃𝐷 − 45° /45° Note that 𝜏1 = 10log 𝜏1𝜔 /𝜔𝑐,𝑛𝑒𝑤. 4. Determine 𝐾𝑑 using 𝐾𝑑 = 𝜏1 𝐾𝑝 L12 - Smart Systems 5. If 𝜙𝑃𝐷 > 45°, graphically implement the PD controller using the straight-line approximation and determine the actual 𝜔𝑐 and 𝜙𝑚 values. 6. Manually fine tune based on the time response. 37 12.3 Control Theory Design of PID Controllers – 𝑲𝑷𝑰𝑫 The transfer function for a PID controller is defined by: 𝐾𝑖 1 𝐺𝑐 𝑠 = 𝐾𝑝 + + 𝐾𝑑 𝑠 = 𝐾𝑑 𝑠 2 + 𝐾𝑝 𝑠 + 𝐾𝑖 𝑠 𝑠 This controller is typically used to increase the system type from 0 to 1 (reducing or eliminating steady-state errors), and to increase 𝜙𝑚 and 𝜔𝑐 resulting in a faster response with reasonable overshoot. L12 - Smart Systems 38 12.3 Control Theory Design of PID Controllers – 𝑲𝑷𝑰𝑫 A step-by-step design procedure for a PID controller is as follows: 1. Solve for 𝜏1 using: 10/𝜏1 = 𝜔𝑐,𝑑𝑒𝑠𝑖𝑟𝑒𝑑 2. Phase shift needed from the PID controller is: 𝜙𝑃𝐼𝐷 = 𝜙𝑚,𝑑𝑒𝑠𝑖𝑟𝑒𝑑 + 5° − 𝜙𝑚, 𝑎𝑡 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝜔𝑐  If 𝜙𝑃𝐼𝐷 > 70° the specifications cannot be met by a PID controller, otherwise:  log 𝜏2 𝜔 = 𝜙𝑃𝐼𝐷 − 45° /45° and 𝜏2 = 10log 𝜏2 𝜔 /𝜔𝑐,𝑑𝑒𝑠𝑖𝑟𝑒𝑑 3. Gain shift needed at 𝜔𝑐,𝑑𝑒𝑠𝑖𝑟𝑒𝑑 is: 𝑔𝑎𝑖𝑛 𝑠ℎ𝑖𝑓𝑡 = 0 − 𝑑𝐵 𝑎𝑡 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝜔𝑐 L12 - Smart Systems  If 𝜙𝑃𝐼𝐷 > 45° then: log 𝐾𝑖 𝜏1 = 𝑔𝑎𝑖𝑛 𝑠ℎ𝑖𝑓𝑡 − 20 log 𝜏2 𝜔 /20  Else: log 𝐾𝑖 𝜏1 = 𝑔𝑎𝑖𝑛 𝑠ℎ𝑖𝑓𝑡 /20 and 𝐾𝑖 = 10log 𝐾𝑖 𝜏1 /𝜏1 4. Determine 𝐾𝑝 and 𝐾𝑑 using: 𝐾𝑑 = 𝐾𝑖 𝜏1 𝜏2 and 𝐾𝑝 = 𝐾𝑖 𝜏1 + 𝜏2 5. Note that the PID controller will change the 𝑀 ∗ value at 1 rad/s by 20 log 𝐾𝑖. 6. Manually fine tune based on the time response. 39