CG lab 3: Digital Control Systems PDF

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Summary

Control and GuidanceB. in Aerospace Systems EngineeringPolytechnic University of CataloniaDIGITALCONTROL SYSTEMSJohann SchwarzeMay 2020Contents 1 Satellite attitude control 1 Plant stability 1.1 Open loop 1.1 Closed loop 1 Sampling time 1.2 Open loop 1.2 Closed loop 1 Root locus 1 FOH discretization...

Description

Control and Guidance B.Sc. in Aerospace Systems Engineering Polytechnic University of Catalonia

DIGITAL CONTROL SYSTEMS

Johann Schwarze

May 2020

Contents 1 Satellite attitude control 1.1 Plant stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2

1.1.1 Open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Closed loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Sampling time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4

1.2.1 1.2.2

Open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closed loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 5

1.3 Root locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 FOH discretization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 8

1.5 Adding a zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Dead-beat controller feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 12

1.8 Dead-beat controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2 Additional considerations on the sampling period 2.1 Open loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Closed loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 16 18

ii

1.

Satellite attitude control

Space vehicles need attitude control systems in order to orientate their antennas and solar panels in the correct direction so they can work properly. The movement of the spacecraft around the y-axis (pitch angle) can be described by the following expression: d2 θ(t) = MD + FC (t)d (1.1) dt2 where θ(t) is the spacecraft orientation given by the pitch angle, MD is the momentum of the perturbations and FC (t) is the applied thrust.

Figure 1.1: Space vehicle axis system and attitude If no perturbations are considered and initial conditions are settled to zero in the Laplace domain: d2 θ(t) = FC (t)d dt2 ˜ θ(s) 1 Gp (s) = = 2 ˜ s FC (s) 1

(1.2) (1.3)

The design requirements of our attitude control system are the following:

(1.4) (1.5)

ωn ≥ 0.3 rad/s ζ ≥ 0.5

1.1

Plant stability

1.1.1

Open loop

The transfer function Gp (s) presents a double pole exactly at s = 0 in open loop, so the system will be marginally unstable, as we can appreciate in the impulse response and the root locus and the gain and phase margins in its bode diagram.

Root Locus

Impulse Response 45

1.5

40

Imaginary Axis (seconds -1 )

1 35

25 20 15 10

0.5

0

-0.5

-1 5 0 5

10

15

20

25

30

35

-1.5 -0.15

40

-0.1

-0.05

0

0.05

0.1

Real Axis (seconds -1 )

Time (seconds)

((a)) Open loop impulse response

((b)) Open loop root locus Bode Diagram

10

Magnitude (dB)

0

0 -10 -20 -30

-40 -179

Phase (deg)

Amplitude

30

-179.5 -180 -180.5 -181 100

10 1

Frequency (rad/s)

((c)) Open loop bode diagram

2

0.15

1.1.2

Closed loop

Giving a unitary feedback to the previous system, the following closed loop transfer function is obtained: Gc (s) =

s2

1 +1

(1.6)

which presents a pair of complex conjugated poles s = ±i, being i the imaginary unit. Notice that these poles present real part equal 0, so the system will also be marginally unstable but this time it will show oscillations, as it can be observed again in the impulse response and the root locus and the gain and phase margins in its bode diagram.

Root Locus

Impulse Response 5

0.8

4

0.6

3

Imaginary Axis (seconds -1 )

1

0.2 0 -0.2 -0.4

2 1 0 -1 -2

-0.6

-3

-0.8

-4

-1 0

10

20

30

40

50

60

70

80

90

-5 -0.5

100

-0.4

Time (seconds)

-0.3

-0.2

-0.1

Bode Diagram

Magnitude (dB)

0.1

0.2

0.3

((e)) Closed loop root locus

150 100 50 0 -50 0 -45 -90 -135 -180 10-1

0

Real Axis (seconds -1 )

((d)) Closed loop impulse response

Phase (deg)

Amplitude

0.4

10 0

Frequency (rad/s)

((f)) Closed loop bode diagram

3

10 1

0.4

0.5

1.2

Sampling time

1.2.1

Open loop

Different values of sampling time Ts will be tested in order to study the dependence of the system on this parameter. For those values, the following discrete transfer functions are obtained in the open loop case: 0.5z+0.5 z 2 −2z +1

• Ts = 1 −→ G(z) = • Ts = 0.1 −→ G(z) =

0.005z+0.005 z 2 −2z+1

• Ts = 0.01 −→ G(z) =

5×10−5 z +5×10−5 z 2 −2z +1

Notice that all three transfer functions keep the same poles and zeros, being the proportional gain the only affected parameter, which decreases as sampling time decreases too. The obtained step responses for each sampling time value are shown in the plot below.

6

Step Response

×107

5

1.0276

T = 1s T = 0.1s T = 0.01s

1.0274

Amplitude

4

Amplitude

Step Response

×107 1.0278

T = 1s T = 0.1s T = 0.01s

3

1.0272 1.027

2 1.0268 1 1.0266 0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000 10000

4531.5

Time (seconds)

4532

4532.5

4533

4533.5

Time (seconds)

((h)) Zoom

Figure 1.2: Open loop step response for different sampling times Notice that as the sampling time decreases, the system presents a more realistic behaviour and more similar to the continuous one.

4

1.2.2

Closed loop

In order to study the dependence on the sampling time Ts in a closed loop system, a unitary feedback will be added to the previous system. For the aforementioned sampling time values, the following discrete transfer functions are obtained in the closed loop case: 0.5z+0.5 • Ts = 1 −→ G(z) = z 2 −1.5 z +1.5 0.005z+0.005 • Ts = 0.1 −→ G(z) = z 2 −1.995z+1.005

• Ts = 0.01 −→ G(z) =

5×10−5 z +5×10−5 z 2 −2z+1

Notice that, in contrast with the open loop case, the poles do change with the sampling time this time. On the other hand, the proportional gain for each Ts value follows the same tendency as in the previous case. The obtained step responses for each sampling time value are shown in the plot below.

2.5

Step Response

×1026

Step Response

×1025 2

2

1.5 1 0.5

1

Amplitude

Amplitude

1.5

0.5 0

0 -0.5 -1

-0.5

-1.5 -2

-1

-2.5 -1.5 0

50

100

150

200

250

300

265

Time (seconds)

270

275

280

285

Time (seconds)

((a)) Ts = 1 s

((b)) Zoom

5

290

295

1.5

Step Response

×1027

Step Response

×1025 1

1

Amplitude

Amplitude

0.5

0

0

-0.5

-1 -1 -1.5 0

500

1000

1500

2000

2500

2298

2300

2302

2304

Time (seconds)

2306

2308

2310

2312

2314

2316

Time (seconds)

((c)) Ts = 0.1 s

((d)) Zoom

Step Response

Step Response

2.5 2.09 2 2.085

Amplitude

Amplitude

1.5

1

0.5

2.08

2.075

0 2.07 -0.5 0

10

20

30

40

50

60

70

80

90

100

34.2

34.3

Time (seconds)

34.4

34.5

34.6

34.7

34.8

34.9

Time (seconds)

((e)) Ts = 0.01 s

((f)) Zoom

Figure 1.3: Closed loop step response for different sampling times Notice that for Ts = 1 s no realistic behaviour is obtained.

1.3

Root locus

The root locus for different sampling time Ts values are shown in the following plots.

6

Root Locus

Root Locus 2.5

2.5 2

2

1.5

1.5 1

Imaginary Axis

Imaginary Axis

1 0.5 0 -0.5

0.5 0 -0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5 -7

-2.5 -7

-6

-5

-4

-3

-2

-1

0

1

2

System: Gz1 Gain: 16 Pole: -3 Damping: -0.33 Overshoot (%): 300 Frequency (rad/s): 3.33

-6

-5

-4

-3

-2

-1

0

1

2

1

2

1

2

Real Axis

Real Axis

((a)) Ts = 1 s Root Locus

Root Locus

2.5

2.5

2

2

1.5

1.5 1

Imaginary Axis

Imaginary Axis

1

System: Gz2 Gain: 1.6e+03 Pole: -3 Damping: -0.33 Overshoot (%): 300 Frequency (rad/s): 33.3

0.5 0 -0.5

0.5 0 -0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5 -7

-2.5 -7

-6

-5

-4

-3

-2

-1

0

1

2

-6

-5

-4

Real Axis

-3

-2

-1

0

Real Axis

((c)) Ts = 0.1 s Root Locus

Root Locus 2.5

2

2

1.5

1.5

1

1

Imaginary Axis

Imaginary Axis

2.5

0.5 0 -0.5

0.5 0 -0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5 -7

-2.5 -7

-6

-5

-4

-3

-2

-1

0

1

2

Real Axis

System: Gz3 Gain: 1.6e+05 Pole: -3 - 3.73e-08i Damping: -0.33 Overshoot (%): 300 Frequency (rad/s): 333

-6

-5

-4

-3

-2

Real Axis

((e)) Ts = 0.01 s

Figure 1.4: Root locus for different sampling time values

7

-1

0

Apparently, the root locus is the same for all three cases. Nevertheless, the gain and frequency values increase as sampling time decreases, whereas the root locus shape, damping and overshoot values stay the same.

1.4

FOH discretization method

Discretizing the initial open loop continuous transfer function using the first order hold (FOH) method instead of the zero order hold (ZOH) method and using a sampling time Ts = 0.1 s, the following discrete transfer function is obtained. G(z) =

0.001667z 2 + 0.006667z + 0.001667 z 2 − 2z + 1

(1.7)

The plot below shows a comparison of the different step responses for the continuous transfer function and both aforementioned discretization methods.

Step Response

Step Response

250 53

Continuous ZOH FOH

200

Continuous ZOH FOH

52

Amplitude

Amplitude

51 150

100

50 49 48

50

47 46

0 0

2

4

6

8

10

12

14

16

18

20

9.7

Time (seconds)

9.8

9.9

10

10.1

10.2

Time (seconds)

Figure 1.5: Step response for different discretization methods. As it can be observed, the FOH discretization method guarantees a good global approximation as does the ZOH method. In order to study again the effect of different sampling times on the FOH discretized step response, we will try again with the previous section values, which are shown in the following figure.

8

Step Response

Step Response

250 88

Ts = 1s Ts = 0.1s

86

Ts = 0.01s

200

Ts = 1s Ts = 0.1s Ts = 0.01s

Amplitude

Amplitude

84 150

100

82 80 78

50

76 74

0 0

2

4

6

8

10

12

14

16

18

20

12.2

12.4

12.6

Time (seconds)

12.8

13

13.2

Time (seconds)

Figure 1.6: FOH step response for different sampling time values. As in the ZOH method case, as the sampling time decreases, the step response presents a better approximation of the continuous case.

1.5

Adding a zero

From now on, we will focus on the ZOH discretization method with a sampling time Ts = 0.1 s. The chosen system presents a double pole at z = 1 and a zero at z = −1, as it can be observed in the p-z diagram and root locus below.

Root Locus

Pole-Zero Map 1

2.5

0.8

2

0.6

1.5 1

Imaginary Axis

Imaginary Axis

0.4 0.2 0 -0.2

0.5 0 -0.5

-0.4

-1

-0.6

-1.5

-0.8 -1 -1

-2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-2.5 -7

1

Real Axis

-6

-5

-4

-3

-2

-1

0

1

2

Real Axis

((a)) p-z diagram.

((b)) Root locus.

As we can appreciate, the key to stabilize the plant seems to be adding a zero between z = −1 and z = 1. In order to fulfill the design requirements, a zero at z = −0.15 is added, so the values for the natural frequency and damping factor are respectively: 9

(1.8)

ωn = 36.7 rad/s

(1.9)

ζ = 0.517 The root locus with the added zero is shown in the figure below. 1

Root Locus Editor for Open Loop 1(OL1)

0.8 0.6 0.4

Imag Axis

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.5

-1

-0.5

0

0.5

1

Real Axis

Figure 1.7: Root locus with zero at z = −0.15. However, a requirement for the system to be causal is to have more poles than zeros, so anyway adding only a zero would not be a feasible solution.

1.6

Compensation

Using a differentiator implies to add a zero at z = 0, so the same situation as in the previous section is achieved; no zeros can be added without adding poles if we want to keep the system causal and feasible. In order to properly stabilize the plant, a lead compensator can be used, with proportional gain = 3, a real zero at z = 0.9 and a real pole at z = 0.3. The root locus with the lead compensator is shown in the figure below.

10

Root Locus Editor for Open Loop 1(OL1)

1 0.8 0.6 0.4

Imag Axis

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.5

-1

-0.5

0

0.5

1

Real Axis

Figure 1.8: Root locus with lead compensator. Notice that the pink points stay within the unitary circumference whereas they also stay in the white area, which ensures stability and satisfying the design requirements respectively. The lead controller transfer functions stands for: C(z) = 3 with w =

z−1 Ts

1 + 1w 1 + 0.14w

(1.10)

and Ts = 0.1 s.

The response of the system to a discrete step with θref = 12° is shown in the following plot: Step Response 16 14

Amplitude

12 10 8 6 4 2 0 0

0.5

1

1.5

2

Time (seconds)

11

2.5

3

3.5

The response presents the following overshoot and peak time:

M = 15.24% tp = 0.9 s

1.7

(1.11) (1.12)

Dead-beat controller feasibility

The global closed loop transfer function of the system can be defined as: T (z) =

C(z )G(z) 1 + C(z )G(z)

(1.13)

From this point, the controller transfer function C(z) can be isolated: C(z) =

T (z) G(z)(1 − T (z))

(1.14)

If the global closed loop transfer function T (z) operates as a delay of a sampling period, the transfer function will stand for: T (z) =

1 z

(1.15)

As the global closed loop transfer function is known and the discrete plant transfer functions for different sampling time values were computed in section 1.2.2, now the controller transfer functions for different sampling times can be expressed as: 3

2

z −2z +z • Ts = 1 −→ C(z) = 0.5z 3 −3.886 ×10−6 z 2 −0.5z

• Ts = 0.1 −→ C(z) =

z 3 −2z 2 +z 0.005z 3 −2.602×10−1...