ECE 388 Automatic Control

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1 Lead Compensator and PID Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage: Lead Compensator: Usage Goal Reshape frequency response curve to give additional phase lead in order to increase the phase margin Starting Point Task Plant transfer function G(s) Lead compensator transfer function C(s) = K α Desired phase margin Φ m Steady-state error e Determine the parameters K α, α and T 1 + T s 1 + α T s

2 Lead Compensator: Transfer Function Time-constant Representation Explanation C(s) = K α Attenuation factor < α < 1 Gain K α Pole at s = 1 α T Zero at s = 1 T Remarks 1 + T s 1 + α T s Phase increase (lead) up to a maximum value of sin(ϕ α ) = 1 α 1+α Frequency of maximum phase lead at ω α = 1 α T Magnitude at ω α is C(jω α ) = 1 α > 1 Lead Compensator: Bode Plot 4 Bode Diagram C(s) = 1 1+s 1+.1 s Magnitude (db) Phase (deg) Frequency (rad/s)

3 Lead Compensator: Procedure 1 Determine the gain K α to achieve the static error specification 2 Draw a Bode plot of K α G(jω) and determine the phase margin Φ m 3 Determine the required lead angle ϕ α = Φ m Φ m + 1 α = 1 sin(ϕ α) 1 + sin(ϕ α ) 4 Choose the gain crossover frequency ω g such that K α G(jω g ) db = 2 log( 1 α ) Choose ω α = ω g 5 Evaluate ω α = 1 α T T = 1 ω α α 6 Verify if the design fulfills the specified requirements. Go back to step 3. if the requirements are not fulfilled Lead Compensator: Example Computation Gap 1

4 Lead Compensator: Example Computation Gap 2 Lead Compensator: Example 4 Bode Diagram Magnitude (db) 2 2 Phase (deg) Frequency (rad/s)

5 PID Controller: Characteristics Gap 3 Ordinary Differential Equation (ODE) u = K p (e + 1 T I e + T D ė) Transfer Function (TF) U(s) = K p (E(s) + 1 T I s E(s) + T D s E(s) = K p (1 + 1 T I s + T D s) E(s) PID Controller: Parameters Proportional Action: K p e Depends on instantaneous value of error Can control any stable plant but usually with low performance Integral Action: K p e T I Realizes memory due to dependency on accumulated error Enforces steady state error of lim t e(t) = Derivative Action: K p T D ė Captures trend of the error due to dependency on rate of change of e Susceptible to amplification of high-frequency disturbances/noise PID-controller design requires appropriate assignment of the three parameters K p, T I and T D

6 PID Controller: Parameters Illustration Gap 4 PID Controller: Special Cases P-Controller C(s) = K p PI-Controller PD-Controller C(s) = K p (1 + 1 T I s ) C(s) = K p (1 + T D s) Design Task Determine the most suitable controller type and the controller parameters K p, T I and T D in order to fulfill given performance specifications

7 Ziegler-Nichols: Oscillation Method Assumption Stable, non-oscillatory plant: G(s) = K excluding first-order/second-order lag Note: plant is not modeled! e sτ (1 + st 1 ) (1 + st n ) Practical Experiment Start with K p = and increase K p gradually until y oscillates Critical gain K crit Note oscillation period T crit Control Loop with P-control Gap 5 Ziegler-Nichols: Oscillation Method PID-controller Parameters Controller K p T I T D P-.5K crit PI-.45K crit.85t crit PID-.6K crit.5t crit.12t crit Example (temperature control) Results Stable closed loop Addresses both reference tracking and disturbance rejection outdoors (θ O ) boiler (θ B ) pump temperature (θ) furnace (u F ) radiator (θ R )

8 Ziegler-Nichols: Oscillation Method Uncontrolled Plant Step Response Oscillation Experiment critical gain K crit = temperature change [ C] 1 5 temperature change [ C] input step response disturbance step response time [min] T crit time [min] Ziegler-Nichols: Example Computation Gap 6

9 Ziegler-Nichols: Oscillation Method Reference Step Response Disturbance Step Response P control PI control PID control.3.25 P control PI control PID control Nonzero static position error with P-control Larger overshoot for PI and PID control due to plant delay Similar dynamics for reference tracking and disturbance rejection Ziegler-Nichols: Reaction Curve Method Assumption Stable, non-oscillatory plant: G(s) = K K excluding G(s) = 1 + st 1 Note: plant is not modeled! Practical Experiment Approach desired set-point Apply small step input Record plant output: process reaction curve e sτ (1 + st 1 ) (1 + st n ) Step Response in Open Loop Gap 7

10 Ziegler-Nichols: Reaction Curve Method Characteristic Plant Parameters Gap 8 PID-controller Parameters Controller K p T I T D P- 1/K T /τ PI-.9/K T /τ 3.33τ PID- 1.2/K T /τ 2τ.5τ Ziegler-Nichols: Oscillation Method Reference Step Response Disturbance Step Response P control PI control PID control P control PI control PID control Similar behavior to Ziegler-Nichols Oscillation Method

ECE 388 Automatic Control

ECE 388 Automatic Control Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage: