Model-Based Controller Design

1. Model-Based Controller Design Introduction Direct synthesis method Internal model control (IMC) IMC derived PID tuning rules Simulink example

2. Model-Based Control PID controller tuning Restrict controller to PID form Seek �best� tuning parameters Can be perform with FOPTD model if available Model-based controller design Controller is not restricted to PID form Requires a process model that is used to determine the controller form as well as the tuning parameters Not restricted to FOPTD models Makes full use of available model Generates PID controllers for some model types

3. Direct Synthesis Method Closed-loop transfer function for setpoint changes Simplification of CLTF

4. Control Objective Rearrange CLTF Desired setpoint response Gd is the desired CLTF Gd The controller Gc depends explicitly on the inverse of the process model G The equation for Gc is known as the control law

5. Desired Closed-Loop Transfer Function The desired CLTF Gd is specified such that: The resulting Gc has a single tuning parameter with an easily understood effect on closed-loop stability and performance Gc is implementable � does not require prediction and has the appropriate properness Properness If n >= m, the controller is proper ? no derivative control If n = m-1, the controller is improper ? derivative control If n = m-2, the controller is improper ? requires second derivative of measured output (not implementable) Seek controllers that are proper or improper with n = m-1

6. Selecting the Desired CLTF Common choice tc > 0 is the desired closed-loop time constant Gd is stable for all tc > 0 Gd has a steady-state gain of unity ensuring offset-free performance due to integral action in Gc Closed-loop speed of response is determined by tc; typical choice is tc = 0.5t Other choices of Gd may be required to ensure that Gc is implementable

7. Simple Examples First-order system This is a PI controller! Second-order system This is a PID controller!

8. Systems with Time Delays Model representation: Desired CLTF FOPTD model

9. Non-Minimum Phase Systems Process Model Zeros: N(s) = 0 Systems with right-half plane zeros can exhibit inverse response Such systems are said to be non-minimum phase Direct synthesis controller Zeros of model become poles of controller Controller is unstable if model is non-minimum phase

10. Internal Model Control Applicable to both minimum-phase and non-minimum phase systems Do not invert non-invertible elements: time delays and right-half plane zeros IMC approach Factor model into invertible and non-invertible parts Design IMC controller using the IMC control structure Convert IMC controller into standard feedback controller Implement standard feedback controller as usual

11. IMC Structure

12. IMC Design Factor the process model contains any time delays and right-half plane zeros, has unity gain and is an all-pass element Construct the IMC controller f is the IMC filter, tc is the desired closed-loop time constant and r is chosen to G*c proper Resulting closed-loop relation

13. First-Order System This is a PI controller Same result as direct synthesis method Two methods always yield same result when G+ = 1

14. Non-Minimum Phase Examples Right-half plane zero Time delay

15. PID Tuning Rules

16. Example: IMC Design

17. Example: Simulink Implementation

18. Example: Setpoint Tracking

19. Example: Disturbance Rejection