" Closed Loop Control with Second Derivative Gain Saves the Day Peter Nachtwey, President, Delta Computer Systems Room II Thursday Sept. 29, 2016 2:00 pm
Closed Loop Control Position-Speed Pressure/Force Only Position-Force Pressure/Force Limit Position-Pressure
First order vs. Second order control Motors look like first order systems Hydraulic systems look like 2nd order systems Modeled as a Mass between two springs as a representative, effective, simple model. The linear model for a hydraulic system The linear model for a servo motor system K τ s + 1 s ω K 2 s +2 ζ ω +1 The same type of controller will not work well for both systems.
What are the typical problems? Three Typical issues Heavy masses on the end of long cylinders are difficult to control. Low natural frequency. Non-linear friction makes the problem worse resulting in stick/slip jerky motion even in open loop. Friction near frictionless actuators have very low damping factor.
Symptoms The stick/slip systems have jerky motion even if the control signal is a constant open loop signal. If the system exhibits jerky motion in open loop it will be difficult or impossible to tune in closed loop. The underdamped/low natural frequency systems are difficult/impossible to tune well due to mechanical and hydraulic design. Think about tuning a system that acts like a slinky.
Real World Problem Applications Metals Industry - Moving steel coils using long thin cylinders Wood Industry - Curve sawing where the saws or chippers move on oil/air bearing surfaces. General Any System with hose between the valves and cylinder
Experience shows PI works just as well as PID Most people have discovered empirically that PI control works just as well as PID control for servo control, but why is this so? Take a look at Transfer Functions and placement of poles and zeros And we know, the optimal placement for closed loop poles is the same for both PI and PID control.
Physics Hasn t Changed Control Theory Has! Simple PI and PID will only work well with servo motors. Simple P only, PI or PID has performance limitations due to the natural frequency and damping factor of the hydraulic/mechanical design. The errors decay as a function of e ζζζ 2 for both PI and PID control when optimally tuned There is nothing the controller or programmer can do. The limitation is from the natural frequency and damping factor
Insights from Control Theory A second derivative gain is required to place the closed loop poles to any desired location, in theory. The second derivative gain allows the control person to get around the limitations due to hydraulic and mechanical design. Now the errors will decay at e λt but the value for λ can be chosen by the programmer
What is D gain and DD gain? Derivative or D gain is applied to errors between the target and actual velocity. It provides an electric form of viscous damping. Second derivative gain (DD or D2 gain) is applied to the error between the target and actual accelerations. Using these gains require the motion controller to generated target velocities and accelerations.
PID with Second Derivative Gain Allows placement of all the modeled closed loop poles where ever desired There are practical limitations Feedback resolution At higher frequencies the models become more complex.
Mass and Two Springs
First vs Second Order Response 4 First Order vs Second Order Response 3 Velocity 2 1 0 0 0.05 0.1 0.15 0.2 First Order Velocity Second Order Velocity Open Loop Control Output Time
First and Second Order Controllers First Order Controllers have a PID and velocity and acceleration feed forward. Second order controllers have a PID with a second derivative and velocity, acceleration and jerk feed forwards.
Why 2 nd Order Control? It s costly to design hydraulic systems with natural frequencies high enough for higher production rates. Response is limited by -ξω n /2 without 2 nd order motion control One answer is to control the system with 2nd order motion controllers quicker accels and decels (under control) than what 1st order systems permit. Controlling lower damping factor & natural frequency, allows greater advantage over 1st order controllers Compensate for mechanical cost in the electronic controls
Three Challenges Challenge 1. Must have smooth motion profiles where the jerk changes smoothly for the jerk feed forward. Simple motion or target profile generators aren t good enough.
Higher Order PID Higher Order Target Generator ) ( 2* ) ( * ) ( * ) ( * ) ( 2 2 t e K t e Kd t e Kp dt t e Ki t u dt d dt d + + + = 5 5 4 4 3 3 2 2 0 0 0 ) ( t c t c t c t t v s t s a + + + + + = 4 5 3 4 2 3 0 0 5 4 3 ) ( t c t c t c t a v t v + + + + = 3 5 2 4 3 0 20 12 6 ) ( t c t c t c a t a + + + = 2 5 4 3 60 24 6 ) ( t c t c c t j + + = Second Order Motion Profile
Three Challenges Challenge 2. Using the double derivative gain is problematic. The derivative gain is difficult enough! quantizing error due to lack of resolution. Sample jitter Noise. These problem has been solved
Three Challenges
Three Challenges Challenge 3. How does one tune a second order? Use a 5th order motion profile or target generator. Use model based control. Auto tuning determines the jerk feed forward and second derivative gain.
Solutions Solutions to 2nd order controller implementation problems Use model based control. Use Auto tuning to determine the jerk feed forward and second derivative gain. Use a 5 th order motion profile or target generator.
Model Based Control Why Bother?
Model Based Control The PID and feed forwards use the positions, velocities, and accelerations generated by the model, not the feedback. The feedback continuously updates the model to keep the model from going astray. The advantage is that the PID sees a nearly perfect system virtually free of quantizing errors, sample jitter and noise.
Model Based Control The result is a smoother output which allows use of higher gains. However, one should ask, Where does the model come from?
System Identification and Auto Tuning The result is The information needed is in the plots/graphs Need time, control output and actual position or velocity Gain and time constant for a first order model Gain, damping factor and natural frequency second order. Choose the model for the best fit.
First Order Model G = 3.095512 α = 377.177281 1 α = 0.002651 ERR = 0.248201
Second Order Model G = 2.99991 ζ 0.10225 = C 5.935563 10 6 ω = 125.183235 = ERR = 0.009074
Estimated Velocity vs Measured Velocity 10 Actual vs Estimated Velocity 10 8 6 5 Velocity 4 0 Control 2 5 0 2 0 0.2 0.4 0.6 0.8 10 Time ActVel EstVel Control
Actual and Estimated Accelerations
Estimated State Feedback
Selecting the Closed Loop Gain. Feed-Forward Gains are calculated from the model only Only one parameter to choose the desired bandwidth. Closed Loop Gains are calculated from the model and the desired bandwidth.
Auto Tuning via Tuning Wizard
Step Response for Different Bandwidths
Summary Why Bother? Machines can be simpler and less costly to manufacture. Technology allows advances in machine motion control
" Closed Loop Control with Second Derivative Gain Saves the Day Peter Nachtwey, President, Delta Computer Systems Room II Thursday Sept. 29, 2016 2:00 pm
Practical Hydraulic Design Guide 3
Thank You for Your Time and Attention! Questions?