Revisit of time-optimal control of a triple integrator system

The time-optimal control method is generally a bang-bang type control approach.

By switching the output state of the controller from one extreme to another, if switched under the precise conditions, time-optimization can be achieved.

The ideal time-optimal controller for the triple integrator plant introduced by Kalyon [26], uses a control law of

 which is the controller output which K is the gain of the triple integrator system. _g X , ₁

X2 are switching surface related functions described by



In view of state space, according to the switching law (8), there are two switching surfaces of the TOC, being x₁X₁and x₂X₂, letting the states of the system initially move toward the first switching surface. When the system reaches the first switching surface, it moves along the first switching surface toward the second switching surface.

15

After the system states reach the second switching surface, the representative point then moves along the switching curve which is the intersection of the two switching surfaces toward the desired system states.

Fig. 3- 1 Block diagram of time-optimal control of a third-order system

Fig. 3- 2 Trajectory of time-optimal control of a third-order system

Fig. 3-2 is the trajectory of the switching of the TOC in state space. The x, y, z axis are ~() ( ) ( )

1 t r t x t

e  _f  , ~ ( ) ( ) ( )

2 t r t x t

e  _f  and ~( ) ( ) ( )

3 t r t x t

e  _f   , where r_f 2,

TOC ∫ ∫ ∫

x x

u x

Kg

16

0

rf and r_f 0. The trajectory starts off at with the system error



~e₁ ~e₂ ~e₃







^{2 0 0}



, it then experiences two switching points before reaching the origin of state space.

17 3.2 Jerk-constrained time-optimal control

Consider a servomotor described by bu x a

x  (11) where x(t) is the displacement, u(t) is the control input, and a and b are plant parameters. The tracking error variables are defined as ~( ) ( ) ( )

1 t r t x t

e  _f  ,

) ( ) ( )

~(

2 t r t x t

e _f   and ~( ) () ( )

3 t r t x t

e  _f   in which r_f(t) is the position reference. The objective of the JCTOC design is to track the position reference, r_f(t) , under the following constraint on the maximum magnitude of jerk: e~(t) k

3 , where the constant k is the admissible maximum jerk.

Fig. 3- 3 Block diagram of JCTOC applied to a servo-motor

Fig. 3-3 shows the block diagram of the JCTOC applied to the servomotor, in which the TOC denotes the non-linear time-optimal controller [26]. Here, the control input to the plant u(t) generated by the JCTOC has an integral relationship to the output of the TOC, (t); that is,

TOC ∫ ∫ ∫

a

b

1

b a





rf ^x

x

 u

JCTOC

Plant

rf



18

Examining the error dynamics of the system gives ) (11) becomes the TOC of a triple-integrator system (13)-(15) with the following control input the plant. But unlike the TOC of a triple-integrator, the plant of the JCTOC only consists of two integrators. The extra integrator is contained in the controller, thus alleviating the bang-bang characteristics of the TOC. Thus, by rearranging the block diagram, the JCTOC method can be simplified to the form of Fig. 3-4. In this block diagram, the disturbance is neglected. The JCTOC with the addition of disturbance is discussed in chapter 3.3.

Fig. 3- 4 Block diagram of rearranged JCTOC

TOC k ^ ∫ ^~e3 ∫ ∫

JCTOC Plant

 ~e2 ~e₁

19

According to system response time-optimal control [27], E₁ and E₂ are functions related to the time-optimal switching planes, which are defined as

)

control approach become



The switching instants of the system are

3 time, the calculation of the switching instants are shown in Appendix A.

In order to avoid the singularity problem around the origin of the state space, an additional switching plane is introduced for tt_f . The switching plane, S 0, is defined as

~ 0

in which ~c and ₁ ~c are constants determined through pole assignment. Here the assigned ₂

20

poles of the S-plane dynamics are both _J , yielding c~ 2_J

2  and ~₁ ²

c J . The switching region of the S-plane is called the Ω region, which is defined as

 region, the control method moves the trajectory toward the S-plane. The borders of the Ω region are set to let the trajectory be switching on the S-plane while changing from the former control scheme, the calculation of the borders of the Ω region are shown in Appendix B. The phase trajectory behaves as it does after the second switching point but without the singularity problem near the origin due to the sliding surface properties.

0 0.1 0.2 0.3 0.4 0.5 0.6

Fig. 3- 5 Control method of the JCTOC

21

By combining the control laws above, the switching law of the JCTOC becomes



Fig. 3-5 shows the control method of the JCTOC, in which the top figure shows the errors of the system ~e , ₁ ~e and ₂ ~e . The bottom figure shows the surface switching variables ₃

1

~1 E

e  , ~e₂E₂ and .

22 3.3 JCTOC with IDOB compensation

The IDOB can help compensate the disturbance that is applied to the system, which may be system uncertainties, external or internal disturbance…etc., the IDOB is added to the system as shown in figure 3-6 [28], which u^* is the output of the nominal controller, d is the disturbance of the system and dˆ is the disturbance estimate of the IDOB. When the disturbance of the system is considered, with the IDOB added to the JCTOC, the switching characteristics can then be less affected by the disturbance and more correct.

Fig. 3- 6 Block diagram of the system control method

The JCTOC is compared with a JCTOC added with an IDOB in the following experiment. The JCTOCs both have a jerk limit of k1024and Ω region pole of

10

J 

 . The IDOB used in the experiment has an integral gain of K100. The reference is set at r_f2(rad).

Fig. 3-7 is the response of the JCTOC with and without the IDOB, it is obvious that the JCTOC without disturbance compensation does not have an ideal response which stops too early and moves slowly toward e0 due to the sliding property of the Ω region, while the JCTOC with disturbance compensation has an ideal convergence

JCTOC or PD

Plant

IDOB

dˆ d

r u ^x

x



 

 



u*

23

characteristic. Fig. 3-8 and Fig. 3-9 are figures of the system states and switching surface variables of the two methods. It can be seen that in Fig. 3-8, the switching functions do not response as smooth as in the simulation, while in Fig. 3-9 with the IDOB added, the response is better.

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1 2 3

time (s)

error (rad)

JCTOC+IDOB JCTOC

0 0.1 0.2 0.3 0.4 0.5 0.6

−1

−0.5 0 0.5 1

time (s)

nominal control effort (V)

JCTOC+IDOB JCTOC

Fig. 3- 7 Step response of JCTOC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−100

−50 0 50 100

time (s)

variables

e1 e2 e3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−10

−5 0 5

time (s)

variables

ρ e1−E1 e2−E2

Fig. 3- 8 Control variables of JCTOC without IDOB compensation

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−100

−50 0 50 100

time (s)

variables

e1 e2 e3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−10

−5 0 5

time (s)

variables

ρ e1−E1 e2−E2

Fig. 3- 9 Control variables of JCTOC with IDOB compensation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

time (s)

ρ

JCTOC (k=1024)

Fig. 3- 10 Mode switching characteristic of the JCTOC with IDOB compensation

25

The switching characteristic of the JCTOC is shown in Fig. 3-10. The ideal switching instants of the JCTOC of r_f2(rad) and k1024 can be calculated from (20) - (22), which are t_s10.0992(s), 298t_s2 0. (s) and t_f 0.397(s). In the experiment results of the JCTOC with IDOB, switching instants are t_s1 0.1001(s), 2995t_s2 0. (s) and 3978t_f 0. (s), which are very close to the ideal switching instants.

26 3.4 JCTOC and PD comparison

The JCTOC is compared with the PD controller in the following experiments. The PD controller’s gains are tuned through pole assignment, with the system dynamics of

0 )

(   

 a bK e bK e

e _d  _p , (26) where K and _p K are the proportional and differential gains of the PD controller, a and _d b are plant parameters. Letting the system have a damping ratio of 1, the gains of the PD controller will then become

b

K_p _pd / , (27) b

a

K_d (2_pd  )/ , (28) in which the system poles are set at _pd. When compared, the JCTOC and PD controller both are added with an IDOB for disturbance compensation, in which the integral gain of the IDOB is K100. Several experiments are conducted with different conditions for comparison:

Exp.3-1 Same settling time: JCTOC of k 1024 compared with PD controller withr_f 2(rad).

The JCTOC of k 1024and r_f2(rad) has a settling time of t0.33(s). Thus, if the PD controller is set at having the same settling time, the pole of the controller is then set at _pd 18.5. The response and control effort of the JCTOC and PD controller are shown in Fig. 3-11. While having the same settling time (2% of reference) of 0.33 (s), the JCTOC has a maximum nominal control effort of 0.564 V while the PD controller has a much greater maximum nominal control effort of 3.79 V.

Thus, with the results of this experiment, it can be clear that the JCTOC needs a much less maximum nominal control effort than the PD controller, which can lead to less possibility of plant input saturation.

27

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1 2 3

time (s)

error (rad)

JCTOC PD (ω_pd=18.5)

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1 2 3 4

time (s)

nominal control effort (V)

JCTOC PD (ω_pd=18.5)

Fig. 3- 11 Response of JCTOC and PD controller with same settling time (k 1024,

2

rf (rad))

28

Exp.3-2 Same maximum nominal control effort: JCTOC of k 1024compared with PD controller withr_f2(rad).

The JCTOC of k 1024and r_f2(rad) has a maximum nominal control effort of 0.564 (V). Thus, if the PD controller is set at having the same maximum nominal control effort, the pole of the controller is then set at _pd 7.13. The response and control effort of the JCTOC and PD controller are shown in Fig. 3-12. The JCTOC and PD controller have the same maximum nominal control effort of 0.564 V. The JCTOC has a settling time of 0.33 (s), while the PD controller has a settling time of 0.93 (s).

Thus, it can easily be seen that with the same maximum nominal control effort, the JCTOC has a much faster response than the PD controller.

0 0.2 0.4 0.6 0.8 1

−1 0 1 2 3

time (s)

error (rad)

JCTOC PD (ω_pd=7.13)

0 0.2 0.4 0.6 0.8 1

−1

−0.5 0 0.5 1

time (s)

nominal control effort (V)

JCTOC PD (ω_pd=7.13)

Fig. 3- 12 Response of JCTOC and PD controller with same maximum nominal control effort (k 1024, r_f2(rad))

29

Exp.3-3 Double reference: JCTOC of k 1024compared with PD controller of same settling time and PD controller of same maximum nominal control effort, r_f4(rad).

The comparison of the JCTOC and PD controller are continued with a double reference of r_f4(rad), the jerk limit of the JCTOC is still k 1024. The comparison objectives are the same as the previous two experiments: the same settling time and same maximum nominal control effort. The poles of the PD controllers with the same settling time and maximum nominal control effort are respectively _pd 16 and _pd 5.67.

The comparison of the JCTOC and PD controller with same settling time is shown in Fig. 3-13. Both control methods have a settling time of 0.43 (s), the JCTOC has a maximum nominal control effort of 0.711 V while the PD controller has already caused saturation of the plant, having a maximum nominal control effort of 5.5 V.

The comparison of the JCTOC and PD controller with maximum nominal control effort is shown in Fig. 3-14. With both controllers having the same maximum nominal control effort of 0.711 (V), the JCTOC has a settling time of 0.43 (s) while the PD controller has a much larger settling time of 1.4 (s).

0 0.1 0.2 0.3 0.4 0.5 0.6

−2 0 2 4 6

time (s)

error (rad)

JCTOC PD (ω_pd=16)

0 0.1 0.2 0.3 0.4 0.5 0.6

−2 0 2 4 6

time (s)

nominal control effort (V)

JCTOC PD (ω_pd=16)

Fig. 3- 13 Response of JCTOC and PD controller with same settling time (k 1024,

4

rf (rad))

30

0 0.2 0.4 0.6 0.8 1

−2 0 2 4 6

time (s)

error (rad)

JCTOC PD (ω_pd=5.67)

0 0.2 0.4 0.6 0.8 1

−1

−0.5 0 0.5 1

time (s)

nominal control effort (V)

JCTOC PD (ω_pd=5.67)

Fig. 3- 14 Response of JCTOC and PD controller with same maximum nominal control effort (k 1024, r_f4(rad))

31

Exp.3-4 Double admissible jerk of JCTOC: JCTOC of k2048compared with PD controller of same settling time and PD controller of same maximum nominal control effort, r_f2(rad).

If the jerk-constraint of the JCTOC is doubled and set to k2048, the settling time and maximum control effort will both be different. The JCTOC is compared with the PD controller with the same comparison objectives: the same settling time and same maximum nominal control effort, the poles of the PD controllers with the same settling time and maximum nominal control effort are respectively _pd 23.6 and

01 .

9

pd 

 . The velocity signals of the experiments are differentiated to examine the approximated acceleration, which have a direct relation with the torque and command input of the plant. The approximated acceleration signal can also be observed to examine the jerk of the plant.

The JCTOC and PD controller with the poles of _pd 23.6 have the same settling time of 0.256 (s), the response of the plant and control effort of the controllers are shown in Fig. 3-15, the approximated acceleration is shown in Fig. 3-16, which is acquired by differentiating the velocity signal. With the same settling time, the PD controller has caused the plant to saturate for a certain amount of time while the JCTOC only has a maximum nominal control effort of 0.9 V. By observing the approximated acceleration signal, it can be seen that it is very similar to the nominal control effort of the controller due to the plant dynamics. The change of acceleration of the PD controller can be very steep in the beginning, leading to large jerk of the plant, while the change of acceleration of the JCTOC is constant due to the jerk-constraint.

The JCTOC and PD controller with the poles of _pd 9.01 have the same maximum nominal control effort of 0.9V, the response of the plant and nominal control effort of the controllers are shown in Fig. 3-17, the approximated acceleration is shown in Fig. 3-18. With the same maximum nominal control effort, the JCTOC has a settling time of 0.256 (s) while the PD controller has a settling time of 0.71 (s), which is much larger.

32

Although the acceleration of the PD controller is smoother than the PD controller with same settling time, the initial nominal control effort is not continuous.

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1 2 3

time (s)

error (rad)

JCTOC PD (ω_pd=23.6)

0 0.1 0.2 0.3 0.4 0.5 0.6

−2 0 2 4 6

time (s)

nominal control effort (V)

JCTOC PD (ω_pd=23.6)

Fig. 3- 15 Response of JCTOC and PD controller with same settling time (k2048,

2

rf (rad))

0 0.1 0.2 0.3 0.4 0.5 0.6

−1000

−800

−600

−400

−200 0 200 400

time (s) acceleration (rad/s2 )

JCTOC PD (ω_pd=23.6)

Fig. 3- 16 Approximated acceleration signal of control methods

33

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−1 0 1 2 3

time (s)

error (rad)

JCTOC PD (ω_pd=9.01)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−1

−0.5 0 0.5 1

time (s)

nominal control effort (V)

JCTOC PD (ω_pd=9.01)

Fig. 3- 17 Response of JCTOC and PD controller with same maximum nominal control effort (k 2048, r_f2(rad))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−300

−200

−100 0 100 200 300

time (s) acceleration (rad/s2 )

JCTOC PD (ω_pd=9.01)

Fig. 3- 18 Approximated acceleration signal of control methods

34

Exp.3-5 Comparison of JCTOC with different constraints: k 512 , k 1024 and

2048

k , with r_f2(rad).

The following experiment compares the JCTOC of 3 different jerk-constraints,

512

k , k 1024 and k 2048. The 3 JCTOCs have a separate settling time and maximum control effort, the response of the plant and control effort of the controllers are shown in Fig. 3-19 and Fig. 3-20. The JCTOC with the jerk-constraint of k512 has a settling time of 0.418 (s) and maximum nominal control effort of 0.357 V. The JCTOC with the jerk-constraint of k1024 has a settling time of 0.33 (s) and maximum nominal control effort of 0.564 V. The JCTOC with the jerk-constraint of k2048 has a settling time of 0.256 (s) and maximum nominal control effort of 0.9 V. The JCTOC with a larger jerk-constraint has a shorter settling time, but a larger jerk would lead to a larger maximum control effort.

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.5 0 0.5 1 1.5 2 2.5

time (s)

error (rad)

JCTOC (k=512) JCTOC (k=1024) JCTOC (k=2048)

Fig. 3- 19 Response of JCTOCs with different jerk-constraints

35

0 0.1 0.2 0.3 0.4 0.5 0.6

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

time (s)

nominal control effort (V)

JCTOC (k=512) JCTOC (k=1024) JCTOC (k=2048)

Fig. 3- 20 Nominal control effort of JCTOCs with different jerk-constraints

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1

time (s)

ρ

JCTOC (k=512)

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1

time (s)

ρ

JCTOC (k=1024)

0 0.1 0.2 0.3 0.4 0.5 0.6

−1 0 1

time (s)

ρ

JCTOC (k=2048)

Fig. 3- 21 Switching instants of the JCTOCs with different constraints

36

The switching instants of the JCTOCs are shown in Fig. 3-21. The ideal and experimental switching instants of the different controller gains are shown in Table. 3-1.

Experiment results are all very similar to the ideal switching instants, which assure the correct switching of the TOC at different gains.

Table 3- 1 Indexes of JCTOCs with different jerk constraints Ideal

1

ts (s)

1

ts experiment

result (s)

Ideal

2

ts (s)

2

ts

experiment result (s)

Ideal tf (s)

tf

experiment result (s)

Max. u^* (V) JCTOC

(k 512) 0.125 0.1247 0.375 0.3768 0.5 0.5106 0.357

JCTOC

(k 1024) 0.0992 0.1001 0.298 0.2995 0.397 0.3978 0.564

JCTOC

(k2048) 0.0787 0.7919 0.236 0.2367 0.315 0.3186 0.9

37

Exp.3-6 Maximum jerk-constraint of JCTOC while avoiding plant saturation,

2

rf (rad).

While the previous experiments of the JCTOC have a small maximum nominal control effort, the following experiment raises the constraint of the JCTOC to let the maximum nominal control effort almost exceed the saturation limit of the plant. The maximum jerk-constraint can be found by a simple method, due to the maximum nominal control effort appearing at the first two switching instants of the JCTOC and the slope of the nominal control effort having a direct relation to the jerk-constraint k , the limit can be found by the function

5 .

15 kts

 ^, (29) where is a plant constant that can be experimentally found. Thus, by using this method, the maximum jerk limit can be found.

0 0.05 0.1 0.15 0.2

−1 0 1 2 3

time (s)

error (rad)

JCTOC (k=30000)

0 0.05 0.1 0.15 0.2

−6

−4

−2 0 2 4 6

time (s)

nominal control effort (V)

JCTOC (k=30000)

Fig. 3- 22 Response of JCTOC with maximum jerk-constraint

38

The system response and nominal control effort can be seen in Fig. 3-22. The JCTOC with jerk-constraint of k 30000 has a maximum nominal control effort of 5.4 V, which is very close to the saturation 5.5 V. The settling time of the JCTOC is 0.103 (s), which is very fast and unable to be accomplished with a normal PD controller. The nominal control effort is still continuous regardless of the high gain of the controller due to the integrator in the JCTOC.

39

CHAPTER 4

在文檔中 急動度限制下之定位伺服及其干擾補償方法 (頁 28-53)