MESB374 System Modeling and Analysis PID Controller Design ME375 Handouts - Spring 2002 MESB374 System Modeling and Analysis PID Controller Design

PID Controller Structure of Controller ME375 Handouts - Spring 2002 PID Controller Structure of Controller Effects of Proportional, Integral and Derivative Actions Design of PID Controllers

(prediction of) future information ME375 Handouts - Spring 2002 PID Controller GC (s) H(s) + - Reference Input R(s) Error E(s) Output Y(s) Disturbance D(s) Plant G(s) Control Input U(s) Gf (s) Proportional plus Integral plus Derivative (PID) Control In time domain current information passed information (prediction of) future information In s-domain

Effect of P-I-D Actions Derivative Action (KCD s): Provides added damping to the closed-loop system; reduces overshoot and oscillation in step response; tends to slow down the closed-loop system response. Dominant during initial transient, due to the effect of differentiation. Integral Action (KCI /s): Eliminates steady-state error to step inputs; tends to destabilize the closed-loop system; has averaging effect. Dominant during steady-state by producing an accumulation of steady-state error to increase control effort. Proportional Action (KCP): Introduces immediate action due to error; improves system response time. Has similar control authority for both transient and steady-state.

Effect of PID Actions Ex: For a given process Reference input R(s) + Output Y(s) Disturbance D(s) Ex: For a given process Unit step reference response Unit step disturbance response

(A.) Let’s try Proportional control: GC(s)=KCP Effect of P Action Objective: Design a system that has zero steady state error for step inputs with %OS<10% and Ts (2%)<6 [sec] GC (s) + - Reference Input R(s) Error E(s) Output Y(s) Disturbance D(s) Plant Control Input U(s) (A.) Let’s try Proportional control: GC(s)=KCP CLTF:

(A.1) Design P Control using Root Locus: Effect of P Action Stability: 0<K CP <14.1 (A.1) Design P Control using Root Locus: CLCE: Real Axis -25 -20 -15 -10 -5 5 10 15 Img. Axis -7.5 -0.667 -10 -15 Not valid

Effect of P Action (A.2) Check for Steady State Error: Unit Step Reference Response Do NOT forget Stability: 0<K CP <14.1 Larger gain results in smaller steady-state error. Unit Step Disturbance Response Larger gain results in stronger attenuation of disturbance. Think about the change of overshoot when gain increases

(B.) Add integral action (PI control): Effect of PI Actions GC (s) + - Reference Input R(s) Error E(s) Output Y(s) Disturbance D(s) Plant Control Input U(s) (B.) Add integral action (PI control): CLTF:

(B.1) Design PI Control using Root Locus: Effect of PI Actions Stability: 0<K CP <11 (B.1) Design PI Control using Root Locus: Real Axis -25 -20 -15 -2 -1 1 5 Img. Axis -10 -5 10 15 CLCE: Choose zPI = -0.5: -7.3 Not valid

Effect of PI Actions (B.2) Check for Steady State Error: Unit Step Reference Response By using Integral action. steady state error is eliminated. Unit Step Disturbance Response By using Integral action, the effect of a constant disturbance can also be eliminated. Has transient performance been improved? Not much

(C.) Add derivative action (PID control): Effect of PID Actions GC (s) + - Reference Input R(s) Error E(s) Output Y(s) Disturbance D(s) Plant Control Input U(s) (C.) Add derivative action (PID control): CLTF:

(C.1) Design PI Control using Root Locus: Effect of PID Actions Stability: 0<K CD (C.1) Design PI Control using Root Locus: Real Axis -20 -10 -15 -2 -1 1 5 Img. Axis -5 10 15 CLCE: Choose z1 = -0.5, z 2=-2.1 -9.95 -2.1 -9.92 -2.6 -1.62

Effect of PID Actions (C.2) Check for Steady State Error: Unit Step Reference Response By using Integral action. steady state error is eliminated. Unit Step Disturbance Response By using Integral action, the effect of a constant disturbance can also be eliminated. Has transient performance been improved? Yes

PID Controller Design via Root Locus Pole-Zero structure of PID Controller: PID Controller adds one open-loop pole at origin and two open-loop zeros, z1 and z2. These two open-loop zeros could be either real or complex conjugate pair. Design Procedure Step1: Select the position of the two zeros such that the root locus will intersect with the desired performance region. Step 2: Pick the controller gain KC such that CL poles are in the performance region Step 3: Find the corresponding PID gain using the above formula.

Design of PID Controller Ex: ( Motion Control of Hydraulic Cylinders ) Recall the example of the flow control of a hydraulic cylinder that takes into account the capacitance effect of the pressure chamber. The plant transfer function is: A V C M B Output V(s) GC (s) + - Reference Velocity R(s) Error E(s) Plant Control Input QIN (s) Disturbance D (s) qIN

Design of PID Controller Output V(s) GC (s) + - Reference Velocity R(s) Error E(s) Plant Control Input QIN (s) Disturbance D (s) We would like to design a controller such that the closed loop system is better damped (smaller OS%) CLTF:

Design of PID Controller PID controller design Closed-loop Characteristic Equation Real Axis -30 -25 -20 -15 -10 -5 5 10 15 Img. Axis 20 Transient performance region Can a PID controller be designed to satisfy the transient design specifications (smaller overshoot and faster settling time) ?