AERO 422: Active Controls for Aerospace Vehicles Proportional, ntegral & Derivative Control Design Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University
roportional Control
Proportional Control r + e u y C P y C(s) = K, constant gain u(t) = Ke(t) Use Routh s criterion to determine range for K Verify if the system is stabilizable with C(s) = K AERO 422, nstructor: Raktim Bhattacharya 3 / 3
Second order system r + e u y C P y A P (s) = (τ s+)(τ 2 s+), C(s) = K Characteristic equation: zeros of + P C τ τ 2 s 2 + (τ + τ 2 )s + AK + = Open loop poles at /τ, /τ 2 Closed loop poles at p = τ + τ 2 + τ 2 2 τ τ 2 + τ 2 2 4 AK τ τ 2 2 τ τ 2 p 2 = τ + τ 2 τ 2 2 τ τ 2 + τ 2 2 4 AK τ τ 2 2 τ τ 2 AERO 422, nstructor: Raktim Bhattacharya 4 / 3
Routh Stability Criterion r + e u y C P y P (s) = 5 A (τ s+)(τ 2 s+), C(s) = K Root Locus Range for stabilization K > s 2 τ τ 2 AK s τ + τ 2 s AK maginary Axis (seconds ) 5 5 5 25 2 5 5 5 Real Axis (seconds ) Poles move towards each other till K = K = (τ τ 2 ) 2 4τ τ 2 A K > K, poles are purely imaginary K > K, ω n, ζ AERO 422, nstructor: Raktim Bhattacharya 5 / 3
Proportional Control Steady State Error r + e u y C P y System Transfer Function G yr (s) := P (s) = Corresponding ODE Steady state A (τ s + )(τ 2 s + ), C(s) = K p P C + P C = AK p τ τ 2 s 2 + (τ + τ 2 )s + AK p + τ τ 2 ÿ + (τ + τ 2 )ẏ + (AK p + )y = AK p r ÿ = ẏ = = y(t) = AK p + AK p r(t) AERO 422, nstructor: Raktim Bhattacharya 6 / 3
Proportional Control Summary Root Locus 5 9 5 Step Response 8 7 Amplitude maginary Axis (seconds ) 5 5 Mp 6 5 4 3 5 2 2 4 6 8 2 4 6 8 2 Time 5 25 2 5 5 5 Real Axis (seconds ) 2 3 4 5 6 7 8 9 ζ (a) Step Response (b) Root Locus (c) M p vs ζ K > K, ω n, ζ t r = 8 ω n, t s = 46 σ Effect of Proportional Control Reduces rise time Good! ncreases overshoot Bad! Large gain small steady state error Amplifies noise and disturbances Bad! Look at frequency characteristics of all the transfer functions (later) AERO 422, nstructor: Raktim Bhattacharya 7 / 3
ntegral Control
ntegral Control r + e u y C P y C(s) = K /s t u(t) = K e(t) Use Routh s criterion to determine range for K Verify if the system is stabilizable with C(s) = K /s AERO 422, nstructor: Raktim Bhattacharya 9 / 3
Second Order System r + e u y C P y P (s) = A (τ s+)(τ 2 s+), C(s) = K /s Characteristic equation: τ τ 2 s 3 + (τ + τ 2 )s 2 + s + AK Root Locus 2 maginary Axis (seconds ) 5 5 5 5 2 2 5 5 5 5 2 Real Axis (seconds ) s 3 τ τ 2 s 2 τ + τ 2 AK s τ +τ 2 AK τ τ 2 τ +τ 2 s AK < K < τ + τ 2 Aτ τ 2 AERO 422, nstructor: Raktim Bhattacharya / 3
ntegral Control Steady State Error r + e u y C P y System P (s) = Transfer Function G yr (s) := Corresponding ODE A (τ s + )(τ 2 s + ), C(s) = K /s P C + P C = AK τ τ 2 s 3 + (τ + τ 2 )s 2 + AK τ τ 2 y + (τ + τ 2 )ÿ + AK y = AK r Steady state y = ÿ = = y(t) = r(t) AERO 422, nstructor: Raktim Bhattacharya / 3
ntegral Control Summary r + e u y C P y maginary Axis (seconds ) 2 5 5 5 5 Root Locus 2 2 5 5 5 5 2 Real Axis (seconds ) K = ζ = M p K = ω n = t r K > τ +τ 2 Aτ τ 2 unstable Zero steady state error Needs anti windup mechanism for saturated actuators discussed later Good disturbance rejection property read book AERO 422, nstructor: Raktim Bhattacharya 2 / 3
Control
Proportional-ntegral Control r + e u y C P y System P (s) = ( A (τ s + )(τ 2 s + ), C(s) = K + ) T s T is called the integral, or reset time /T is reset rate, related to speed of response u(t) is a mixture of two signals t ) u(t) = K (e(t) + T e(t)dτ t u(t) even when e(t) =, because of integral action AERO 422, nstructor: Raktim Bhattacharya 4 / 3
Proportional-ntegral Control Steady State Error r + e u y C P y System P (s) = ( A (τ s + )(τ 2 s + ), C(s) = K + ) T s Transfer Function G yr (s) := P C + P C = AK(T s + ) T τ τ 2 s 3 + T (τ + τ 2 )s 2 + T ( + AK)s + AK Corresponding ODE ( ) y + ( )ÿ + ( )ẏ + AKy = AKr + ( )ṙ Steady state y = ÿ = ẏ = ṙ = = y(t) = r(t) AERO 422, nstructor: Raktim Bhattacharya 5 / 3
Proportional-ntegral Control ndependent control over two of the three terms r + e u y C P y System P (s) = ( A (τ s + )(τ 2 s + ), C(s) = K + ) T s Transfer Function G yr (s) := P C + P C = AK(T s + ) T τ τ 2 s 3 + T (τ + τ 2 )s 2 + T ( + AK)s + AK Characteristic Equation T τ τ 2 s 3 + T (τ + τ 2 )s 2 + T ( + AK)s + AK = AERO 422, nstructor: Raktim Bhattacharya 6 / 3
Proportional-ntegral Control Two tuning variables r + e u y C P y Routh s Table Constraints s 3 T τ τ 2 T (AK + ) s 2 T (τ + τ 2 ) AK s T + AKT AKτ + AKτ 2 τ +τ 2 s AK K >, T > AK τ τ 2 + AK τ + τ 2 AERO 422, nstructor: Raktim Bhattacharya 7 / 3
Derivative Control
Derivative Control r + e u y C P y C(s) = K D s u(t) = K D ė(t) Use Routh s criterion to determine range for K D Verify if the system is stabilizable with C(s) = K D s Almost never used by itself usually augmented by proportional control Derivative control is not causal depends on future values ė e(t + t) e(t) t Knows the slope known from future values of e(t) e(t) = t, ė = e(t) = t 2, ė = 2t e(t) = sin(t), ė = cos(t) = sin(t + π/2) AERO 422, nstructor: Raktim Bhattacharya 9 / 3
Approximate Derivative Control Approximation over Frequency range C(s) = K D s u(t) = K D ė(t) Not implementable in applications Approximate as C(s) = K Ds, α >> pole at far left s/α + K D s, for small s What is the effect of this approximation? Look at a step response AERO 422, nstructor: Raktim Bhattacharya 2 / 3
Approximate Derivative Control Effect of the Approximation 5 Step Response Amplitude 5 2 4 6 8 2 4 6 8 2 Time e(t) = r(t) y(t) = y(t) ė(t) = ẏ(t) AERO 422, nstructor: Raktim Bhattacharya 2 / 3
Approximate Derivative Control Effect of the Approximation (contd) 6 4 True alp= alp= Error Derivative 2 2 4 6 8 2 4 6 8 2 4 6 8 2 Time AERO 422, nstructor: Raktim Bhattacharya 22 / 3
Proportional-Derivative Control r + e u y C P y Controller Structure ( ) s C(s) = K + T D s/α + s = K P + K D s/α + Tune K P and K D to get desired response Use Routh s table to determine range for stable values Derivative control increases damping reduces overshoot AERO 422, nstructor: Raktim Bhattacharya 23 / 3
Control
Control Basic dea Controller Structure C(s) = K P + K s + K s D s/α + Tune K P, K and K D to get desired response Use Routh s table to determine range for stable values Proportional term increases ω n and decreases ζ mproves rise time Needs large gain to reduce steady-state error ntegral term increases ω n and decreases ζ Zero steady-state May make the system unstable Derivative control increases damping reduces overshoot and oscillations AERO 422, nstructor: Raktim Bhattacharya 25 / 3
Control Example System P (s) = s 2 + 5s + Study behavior of this system with various control strategies Proportional (P) Proportional ntegral () Proportional Derivative () Proportional ntegral and Derivative () AERO 422, nstructor: Raktim Bhattacharya 26 / 3
Control Example: P Amplitude Amplitude Amplitude Amplitude 2 Open Loop K= 5 5 2 25 Time (seconds) 2 Open Loop K=2 5 5 2 25 3 Time (seconds) 2 Open Loop K= 5 5 2 25 Time (seconds) 2 Open Loop K=5 5 5 2 25 Time (seconds) AERO 422, nstructor: Raktim Bhattacharya 27 / 3
Control Example: Step Response 4 2 P Amplitude 8 6 4 2 5 5 2 25 3 35 4 Time (seconds) AERO 422, nstructor: Raktim Bhattacharya 28 / 3
Control Example: (contd) Amplitude Amplitude Amplitude Amplitude 5 5 Open Loop Kp= Kp=, Ki = 5 5 5 2 25 3 35 4 45 5 Time (seconds) 5 5 Open Loop Kp=2 Kp=2, Ki = 2 3 4 5 6 7 Time (seconds) 2 Open Loop Kp= Kp=, Ki = 5 5 5 2 25 Time (seconds) 2 Open Loop Kp=5 Kp=5, Ki = 25 2 4 6 8 2 Time (seconds) AERO 422, nstructor: Raktim Bhattacharya 29 / 3
Control Example: Step Response 5 Open loop P D Amplitude 5 2 4 6 8 2 4 6 8 2 Time (seconds) AERO 422, nstructor: Raktim Bhattacharya 3 / 3
Control Example: Step Response 5 Open loop P Amplitude 5 2 4 6 8 2 4 6 8 2 Time (seconds) AERO 422, nstructor: Raktim Bhattacharya 3 / 3