linear motor dynamics代做、代写Coulomb friction、代写Python/Java，c++编程设计

日期：2019-07-15 10:35

Advanced Mechatronic Control HOMEWORK SET 4 SUMMER 2019

OUT: 06/19/19 DUE: 07/10/19

PROBLEM #1 (100 points)

Let us consider the linear motor dynamics with the effect of Coulomb friction, cogging forces, and

external disturbance, i.e.,

where Asc is the magnitude of the Coulomb friction with Sy ky k ( ) sat , 1000, and

1 3 and Acog cog A are the unknown magnitudes of cogging forces, P ? 0.06 m is the known distance

between two adjacent magnets, and d t( ) represents the lumped external disturbance forces which is

assumed to be bounded by () 2 M dt d . Due to the large variations of the inertia load that the linear

motor can carry and the friction and cogging force characteristics, the exact values of , , Me B , Asc

1 3 and Acog cog A may not be known. However, through the specifications of the linear motor, the

variation ranges of these parameters are known and given by

A. First consider the situation of constant external disturbances, i.e., assuming 0 dt d t () . Design

an adaptive controller that achieves closed-loop stability and asymptotic output tracking (i.e.

() () () y m e t yt y t converges to zero asymptotically) for the linear motor described (H1) and (H2).

To receive full credit, you need to provide the detailed proof step by step.

B. Let the reference output be the output of a reference model given by

(H4)

in which the reference command input c u is a square wave type reference command with a half

period of 0.6 sec representing a back-forth movement of travel distance of 0.2m. Assuming a

sampling rate of 2kHz and using the Euler’s approximation algorithm, discretize the continuous

adaptive control law in part A. The initial values for the controller and estimator parameters

should be chosen according to the initial estimates of (0) 0.055 Me ,

B(0) 0.225 , (0) 0.125 Asc , 1 (0) 0.03 Acog and 3 (0) 0.03 Acog  . Simulate the adaptive control law with the

continuous plant (H1) for the following two sets of actual values for the linear motor:

Case 1: 1 30 0.025, 0.1, 0.1, 0.01, 0.05, 1 M BA A A d e sc cog cog

Case 2: 1 30 0.085, 0.35, 0.15, 0.05, 0.05, 1 M BA A A d e sc cog cog

The total simulation time is 4 seconds. Obtain the following time plots:

(i) A plot showing the reference command input c u , the reference output my , and the actual

output y .

(ii) A plot showing the output tracking error m eyy .

(iii) A plot showing the control input u.

(iv) A plot showing the parameter estimates and their actual values.

C. To see how well your adaptive controller can handle time-varying external disturbance, obtain the

simulation results of your adaptive controller for the following two sets of actual values for the

linear motor:

Case 3: round 10 sin(20 )

1 3 0.025, 0.1, 0.1, 0.01, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog cog

Case 4: round 10 sin(20 )

1 3 0.085, 0.35, 0.15, 0.05, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog

Consider the same linear motor system as in Problem 1.

D. Design a deterministic robust control (DRC) law that achieves a guaranteed transient and steadystate

tracking performance. To receive full credit, you need to provide the detailed proof step by

step.

E. Obtain the simulation results of your DRC controller for the same situations as in part C of Problem

1. Compare the results with those of AC controller in Problem 1.

PROBLEM #3 (100 points)

Consider the same linear motor system as in Problem 1.

F. Synthesize an adaptive robust controller (ARC) that achieves a guaranteed transient and steadystate

tracking performance in general, and asymptotic output tracking (i.e., () () () y m e t y t y t

converges to zero asymptotically) for constant external disturbances of 0 dt d t (). To receive

full credit, you need to provide the detailed proof as well.

G. Let the reference output be the output of a reference model given by

(H4)

in which the reference command input c u is a square wave type reference command with a half

period of 0.6 sec representing a back-forth movement of travel distance of 0.2m. Assuming a

sampling rate of 2kHz and using the Euler’s approximation algorithm, discretize the continuous

adaptive control law in part A. The initial values for the controller and estimator parameters

should be chosen according to the initial estimates of (0) 0.055 Me,

d (0) 0 . Simulate the ARC control law with

with the continuous plant (H1) for the following sets of actual values for the linear motor:

Case 1: 1 30 0.025, 0.1, 0.1, 0.01, 0.05, 1 M BA A A d e sc cog cog

Case 2: 1 30 0.085, 0.35, 0.15, 0.05, 0.05, 1 M BA A A d e sc cog cog

Case 3: round 10 sin(20 )

1 3 0.025, 0.1, 0.1, 0.01, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog cog

Case 4: round 10 sin(20 )

1 3 0.085, 0.35, 0.15, 0.05, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog

The total simulation time is 4 seconds. Obtain the following time plots:

(v) A plot showing the reference command input c u , the reference output my , and the actual

output y .

(vi) A plot showing the output tracking error m eyy.

(vii) A plot showing the control input u.

(viii) A plot showing the parameter estimates and their actual values.

PROBLEM #4 (100 points)

H. Optimize the performance of the ARC in B using the automated gain tuning procedure outlined in

the lecture notes. To receive full credit, you need to provide the detailed procedure and

justification as well. Obtain the simulation results for Cases 1-4 and compare them with those in

B.

PROBLEM #5 (100 points)

Consider the same linear motor system as in Problem 1.

I. Synthesize an indirect adaptive robust controller (IARC) that achieves a guaranteed transient

and steady-state tracking performance in general, and asymptotic output tracking (i.e., converges to zero asymptotically) for constant external disturbances of

0 dt d t (). To receive full credit, you need to provide the detailed designs and proof as well.

J. Let the reference output be the output of a reference model given by

(H4)

in which the reference command input c u is a square wave type reference command with a half

period of 0.6 sec representing a back-forth movement of travel distance of 0.2m. Assuming a

sampling rate of 2kHz and using the Euler’s approximation algorithm, discretize the continuous

adaptive control law in part A. The initial values for the controller and estimator parameters

should be chosen according to the initial estimates of (0) 0.055 Me ,

B(0) 0.225 (0) 0.125 Asc , 1(0) 0.03 Acog , 3 (0) 0.03 Acog and 0

d (0) 0 . Simulate the ARC control law with

with the continuous plant (H1) for the following sets of actual values for the linear motor:

Case 1: 1 30 0.025, 0.1, 0.1, 0.01, 0.05, 1 M BA A A d e sc cog cog

Case 2: 1 30 0.085, 0.35, 0.15, 0.05, 0.05, 1 M BA A A d e sc cog cog

Case 3: round 10 sin(20 )

1 3 0.025, 0.1, 0.1, 0.01, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog cog

Case 4: round 10 sin(20 )

1 3 0.085, 0.35, 0.15, 0.05, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog

The total simulation time is 4 seconds. Obtain the following time plots:

(ix) A plot showing the reference command input c u , the reference output my , and the actual

output y .

(x) A plot showing the output tracking error m eyy .

(xi) A plot showing the control input u.

(xii) A plot showing the parameter estimates and their actual values.

To receive full credit, you need to provide the details of all the controller gains used in the

simulation and how they are chosen as well.

Please attach your simulation program so that they can be run to check the correctness of

your simulation results as well.

PROBLEM #6 (100 points)

Consider the same linear motor system as in Problem 1.

K. Synthesize an integrated direct/indirect adaptive robust controller (DIARC) that achieves a

guaranteed transient and steady-state tracking performance in general, and asymptotic output

tracking (i.e., () () () y m e t yt y t converges to zero asymptotically) for constant external

disturbances of 0 dt d t () . To receive full credit, you need to provide the detailed designs and

proof as well.

L. Let the reference output be the output of a reference model given by

in which the reference command input c u is a square wave type reference command with a half

period of 0.6 sec representing a back-forth movement of travel distance of 0.2m. Assuming a

sampling rate of 2kHz and using the Euler’s approximation algorithm, discretize the continuous

adaptive control law in part A. The initial values for the controller and estimator parameters

should be chosen according to the initial estimates of (0) 0.055 Me ,

B(0) 0.225 ,

(0) 0.125 Asc , 1 (0) 0.03 Acog , 3 (0) 0.03 Acog and 0

d (0) 0 . Simulate the ARC control law with

with the continuous plant (H1) for the following sets of actual values for the linear motor:

Case 1: 1 30 0.025, 0.1, 0.1, 0.01, 0.05, 1 M BA A A d e sc cog cog

Case 2: 1 30 0.085, 0.35, 0.15, 0.05, 0.05, 1 M BA A A d e sc cog cog

Case 3: round 10 sin(20 )

1 3 0.025, 0.1, 0.1, 0.01, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog cog

Case 4: round 10 sin(20 )

1 3 0.085, 0.35, 0.15, 0.05, 0.05, ( ) 1 ( 1) t t M B A A A dt e sc cog

The total simulation time is 4 seconds. Obtain the following time plots:

(xiii) A plot showing the reference command input c u , the reference output my , and the actual

output y .

(xiv) A plot showing the output tracking error m eyy .

(xv) A plot showing the control input u.

(xvi) A plot showing the parameter estimates and their actual values.

To receive full credit, you need to provide the details of all the controller gains used in the

simulation and how they are chosen as well.

Please attach your simulation program so that they can be run to check the correctness of

your simulation results as well.

Note: When RLSE is used to estimate the physical parameter vector as in IARC or DIARC designs,

in continuous time domain, the resulting parameter estimation algorithm is of the following form:

(S4)

As mentioned in Remark 2 on page 26 of the course notes on parameter estimation, the above

adaptation rate updating law may be sensitive to the discretization effect due to the quadratic term

of involved, especially when simple discretization method such as Euler is used. So in practice

we will obtain ? using discretized version of the following equivalent differential equation of (S4):

(S4):

Specifically, when Euler discretization method is used, the discretized version of (S5) in

implementation becomes

(S6)

where kt kT is the k-th sampling instance and T is the sampling period. Using the matrix

inversion lemma with 2

(S7)

Note that this way of calculating the adaptation rate matrix 1 ( ) ktused in continuous time-domain

algorithm (S4) is equivalent to the discrete parameter estimation algorithm based on the sampled

values at each sampling time as follows. At each sampling time kt kT , the linear regression model

of T

f ef f u d in continuous time domain leads to the following linear regression model in

discrete-time domain when 0 f d :

() ()Tf k ef k ut t(S8)

Thus, the RLS parameter estimation algorithm in discrete-time domain based on this discrete-time

domain model would beis the equivalent forgetting factor in discrete-time domain. Comparing (S9)

to (S7) with 2 1 for LS estimation, it is easy to see that

1 1 () () Pt t T k k (S10)

which is what we would expect when we discretize the continuous parameter adaptation law using Euler method at the sampling instances.