Gimbal Control Using Matlab

Basic Gimbal Control Using Matlab Steve Rogers 28 June 2012

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Gimbal Control Design • • • • •

Gimbal model Feedforward (mathworks) Integral control using root locus (mathworks) LQR (mathworks) PIDtune & sisotool verification for PI & PID

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Denavit Hartenburg Gimbal Model

Θ1 & θ2 are the pan & tilt angles respectively. Θ2 = cos-1((z-d1)/d3) Θ1 = tan-1(y/x)

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differential equations defining a permanent magnet DC motor

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Open-Loop DC Motor Step Rate Response Step Response 0.25 System: untitled1 Settling time (seconds): 1.07 System: untitled1 Rise time (seconds): 0.598

0.2

Amplitude

0.15

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (seconds)

5

Motor Step Rate Response With Disturbance Setpoint tracking and disturbance rejection 1.2 cl_ff 1

Disturbance

0.6

To: w

Amplitude

0.8

0.4

disturbance T = -0.1Nm d

0.2

0

-0.2

0

5

10

15

Time (seconds)

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Root Locus Integral design for Disturbance Rejection Root Locus 15

Setpoint tracking and disturbance rejection 1.6 feedforward feedback w/ rlocus

1.4 System: untitled1 Gain: 4.99 Pole: -1.66 - 1.33i Damping: 0.782 Overshoot (%): 1.95 Frequency (rad/s): 2.13

0

1.2 1

-5

To: w

5

Amplitude

Imaginary Axis (seconds-1)

10

0.8 0.6 0.4

-10

0.2 -15 -15

0 -10

-5 Real Axis (seconds -1)

0

5

-0.2

0

5

10

15

Time (seconds)

Integrator design

K = 5; C = tf(K,[1 0]);

% compensator K/s 7

LQR Design for Disturbance Rejection Setpoint tracking and disturbance rejection 1.6

• LQR Control Design To further improve performance, a linear quadratic regulator (LQR) for the feedback structure is designed.

• In addition to the integral of error, the LQR scheme also uses the state vector x=(i,w) to synthesize the driving voltage Va. • The resulting voltage is of the form Va = K1 * w + K2 * w/s + K3 * i where i is the armature current.

• For better disturbance rejection, use a cost function that penalizes large integral error.

feedforward feedback (rlocus)

1.4

feedback (LQR) 1.2

To: w

Amplitude

1 0.8 0.6 0.4 0.2 0 -0.2

0

5

10

15

Time (seconds)

8

PID Design for Disturbance Rejection Using pidtune Root Locus Editor for Open Loop 1(OL1)

Open-Loop Bode Editor for Open Loop 1(OL1) 20

80 0.85

0.74

0.62 0.48 0.320.16

• pidtune tunes the parameters of the PID controller C to balance performance (response time) and robustness (stability margins).

• Phase Margin shows stability.

• Rise time is much faster now.

• Overshoot is probably excessive @ ~20%.

0

60 0.93

-20 40 0.98

-40

20

140 120 100 0

80

60

40

-60 G.M.: inf Freq: NaN Stable loop -80 0

20

System: Closed Loop r to y I/O: r to y Step Response Peak amplitude: 0.596 Overshoot (%): 19.3 At time (seconds): 0.284

0.7

0.6

-20 -45

0.98

0.5

-40

-60 0.93

-135 0.85

-80 -150

Amplitude

-90

0.74

P.M.: 120 deg Freq: 6.4 rad/s

0.62 0.48 0.320.16 -50

Real Axis

0

-2

10

0.3

0.2

-180 -100

System: Closed Loop r to y I/O: r to y Rise time (seconds): 0.129

0.4

0

2

10 10 Frequency (rad/s)

4

10

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (seconds)

9

1

PID Design for Disturbance Rejection Using pidtune Setpoint tracking and disturbance rejection

• pidtune tunes the parameters of the PI & PID controllers C to balance performance (response time) and robustness (stability margins). • Comparison with other controllers shows less magnitude impact and quicker recovery from disturbance.

1.6 feedforward feedback (rlocus)

1.4

feedback (LQR) pidtune (PI) (PID)

1.2

To: w

Amplitude

1 0.8 0.6 0.4 0.2 0 -0.2

0

5

10

15

Time (seconds)

10

Bode Plots Bode Diagram From: w ref

From: Td

50

To: w

feedforward feedback (rlocus)

-50

feedback (LQR) pidtune (PI) (PID)

-100

-150 90 0

To: w

Magnitude (dB) ; Phase (deg)

0

-90 -180 -270 -2 10

0

10

2

10

-2

10

0

10

2

10

Frequency (rad/s)

11