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Robust Generalized

Proportional Integral control of

a DC-Motor

Hebertt Sira-Ramrez

Jesus Linares-Flores

October 15, 2010

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The DC motor model

Consider the popular model of a DC motor

Ldia

dt= (uV Ria Kw)

Jdwdt

= Kia Bw (t)

where ia is the armature current, w is the angu-

lar velocity of the motor shaft (t) is the load

torque applied to the shaft, The parameters L,

R and K represent respectively the inductance

in the armature circuit, the armature winding

resistance and the current gain of the motor.

V is the fixed input voltage amplitude, mod-

ulated by the control input u, taking values

on the segment [1, 1]. On the other hand

J and B represent the motor inertia and theviscous friction coefficient. The torque gain is

assumed to be identical to the current gain K.

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Flatness of the DC motor model

The unperturbed motor system ((t) = 0, t)

is represented by a linear controllable system

model. Hence, the system is flat with flat out-

put given by the angular velocity w. Indeed, all

variables can be written in terms of the angularvelocity and its time derivatives:

ia =1

K[Jw + Bw]

u =JL

KVw +

BL

KV+

JR

KV

w +

RB

K+ K

w

The perturbed version of the above differential

parametrization is simply given by

ia =J

Kw +

B

Kw +

(t)

K

u =JL

KVw + BL

KV +RJ

KV

w +RB

KV +K

V

w

+L

KV(t) +

R

KV(t)

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Sensor aided control problem formulation

We first present the GPI observer-based solu-

tion to the problem of robust angular velocity

reference trajectory tracking when the angular

velocity is gathered from the motor with the

aid of a tachometer. We formulate the prob-

lem as follows:

Given a desired angular reference trajectory

w(t) device, on the basis of the measured w

and the relevant system parameters, a feed-

back control law which, irrespectively of the

uniformly absolutely bounded values of the torque

input (t), forces the motor angular velocity, w

to asymptotically exponentially track the given

reference signal, w

(t), modulo a small as de-sired tracking error neighborhood around the

origin of the tracking error phase space.

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Robust angular velocity feedback control

A simplified non-phenomenological relation link-ing the angular velocity to the control input isreadily obtained from the above perturbed dif-ferential parametrization of the control input.We have,

w =

KV

JL

u + (t)

where (t) is, both, an exogenous and a statedependent unknown disturbance input given

by:

(t) =

B

J+

R

L

w+

RB

JL+

K2

JL

w+

1

J(t)+

R

JL(t)

We assume (t) is an unknown time signal,but otherwise known to be uniformly abso-

lutely bounded combined state dependent andexogenous disturbance input signal. We alsoassume that the time derivatives: (j)(t), j =1, 2, 3, 4 are also uniformly absolutely bounded.

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A GPI observer based feedback controller is

readily proposed to be

u = u(t) +JL

KV

s(t) k1(w2s w

(t))

k0(w w(t))

where s(t) and w2s are, respectively, a smoothedestimate of the unknown signal (t) a n d a

smoothed estimate of the angular accelerationw, given by the following GPI linear observer

w1 = w2 + 4(w w1)

w2 =KV

JLu + z1 + 3(w w1)

z1

= z2

+ 2

(w w1

)

z2 = z3 + 1(w w1)

z3 = 0(w w1)

where w1 is the redundant estimate of the mea-sured angular velocity w. The signal w2 isthe estimate of the angular acceleration w and

z1 represents an internal, self-updating, timepolynomial model (also called Taylor polyno-

mial) of just second degree, estimating theperturbation input (t).

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The estimation error vector (e1, e2) = (w

w1, w w2) satisfies

e1 = e2 4e1

e2

= (t) z1

3e

1z1 = z2 + 2e1

z2 = z3 + 1e1

z3 = 0e1

The estimation error e = e1 = w w1 obeys,upon elimination of the variables z1, z2 and

z3, the following perturbed linear differential

equation

e(5) + 4e(4) + + 0e =

(3)(t)

We have the following result:

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Theorem. The linear feedback control law

u = u(t) + JLKV

(t) k1(w2 w

(t))

k0(w w(t))

with (t) = z1 and w2 being, respectively, es-

timates of (t) and w which are generated by

the observer:

w1 = w2 + 4(w w1)

w2 =KV

JLu + z1 + 3(w w1)

z1 = z2 + 2(w w1)

z2 = z3 + 1(w w1)

z3 = 0(w w1)asymptotically exponentially drives the obser-

vation error variables e1 = ww1, e2 = ww2,to an arbitrarily small neighborhood of zero,

while z1 asymptotically exponentially converges

to a small as desired neighborhood of the un-

known signal, (t), provided the observer gains,{4,...,0}, and the controller gains, {k1, k0},are chosen so that the dominating characteris-

tic polynomials:

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pc(s) = s2 + k1s + k0po(s) = s

5 + 4s4 + + 1s + 0

are Hurwitz polynomials with their roots lo-

cated deep into the left half of the complex

plane. Moreover, the further the roots of these

polynomials are located in the left half of thecomplex plane the smaller the estimation er-

rors and the tracking errors ew = w w(t),

ew = w w(t)

The proof of this result stems from the fact

that the closed loop and injected dynamics aregiven by

e(5) + 4e(4) + + 0e =

(3)(t)

ew + k1ew + k0ew = (t) (t) + l(e, e, ...)

where l(e, e, ...) is a linear, rapidly fading, func-

tion of the estimation error e and its time deriva-tives, circumscribed to a small as desired disk

centered on the origin of the estimation error

phase space.

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Remark

The observer variables z1 = (t) and w2 = w(t)

being produced by a high gain observer nat-

urally exhibit the peaking phenomenon at

the initial stages of their existence. In order

to avoid the effects of these initially large sig-

nals in the controller, we propose, as custom-

arily made, a smoothing of its initial values

by means of a clutching function smoothly

interpolating between the values 0 and 1 dur-

ing a small time interval [0, ] whose value isa function of the large negative real part of

the eigenvalues, or characteristic polynomial

roots, defining the dominating estimation error

injected dynamics for e. We propose

s(t) = sf(t)(t)

sf(t) =

sin8(t2) t 1 for t >

with = w2, z1.

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Simulation Results

CONTINUOUS SYSTEM MOTORDC"this program simulates a dc motor robustly controlled via" a GPI observer based controller rejecting the

" additive disturbance containing the unknown load torque" and other system terms. A stabilization task is presented.

STATE ia w

STATE w1 w2 z1 z2 z3DER dia dwDER dw1 dw2 dz1 dz2 dz3

TIME t

" Plant equations:

dia=1/L*(v-R*ia-K*w)dw=1/J*(K*ia-B*w-tau)

" definition of load torque disturbance input

tau=0.5*exp(-sin(3*t)*sin(3*t))*cos(0.3*t)*cos(0.3*t)

"Linear controller with disturbance rejection

v=1/ga*(-z1s-k1*(w2s-wsd)-k0*(w-ws))

k0=wnc^2k1=2*zetac*wnc

wnc:1.5zetac:1

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"GPI observer

dw1=w2+g4*(w-w1)

dw2=ga*v+z1+g3*(w-w1)

dz1=z2+g2*(w-w1)dz2=z3+g1*(w-w1)

dz3=g0*(w-w1)

ga=K/(L*J)

"smoothing of observer signals

z1s=z1*sfw2s=w2*sf

sf=if t < eps then sin8 else 1

sin8=sin4*sin4sin4=sin2*sin2

sin2=sin1*sin1sin1=sin(pi*t/(2*eps))

pi:3.14159265

eps:0.5

" pole placement for GPI observer

g4=po+4*zetao*wno

g3=4*zetao*po*wno+2*wno^2+4*zetao^2*wno^2g2=4*zetao*wno^3+2*po*wno^2+4*zetao^2*po*wno^2

g1=wno^4+4*zetao*po*wno^3g0=po*wno^4

po=40

wno=40zetao=1

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" reference angular velocity signal

ws:100wsd:0

" Motor parameter values:

L:7e-3

R:2.33B:.00937K:.479

J:.01164

END

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0 2 4 6 8 10

0

2

4

0 2 4 6 8 100

50

100

0 2 4 6 8 100

50

ia(t)

v(t)

w(t), w(t)

Performance of direct GPI observer based

control for angular velocity tracking

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0 2 4 6 8 104

3

2

1

0

1x 10

5

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

(t)

(t)

Disturbance estimation and applied load

torque

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A two stage robust controller design

Consider again the DC motor dynamics

d

dtia = u

V

L

R

Lia

K

Lw

d

dtw =

K

Jia

B

Jw

1

J(t)

In contrast with the previous controller design

procedure, we now adopt a two stage con-

troller design approach. This is accomplished

by first designing an outer loop robust GPI

observer based angular velocity reference tra-

jectory tracking controller taking the armature

current, ia, as an effective control input. Once

the robust feedback control law is established

for the armature current, we design an inner

loop controller specifying a robust controller

for the armature voltage so that the armaturecurrent tracks, on-line, the generated time val-

ues of the designed outer loop armature cur-

rent inner loop controller.

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Outer loop robust controller design

The design of the outer loop controller designentitles the angular velocity dynamics alone

taking as an auxiliary control input the arma-ture current, ia,

We consider the torque perturbed angular ve-locity dynamics:

d

dtw =

K

Jia

B

Jw

1

J(t)

We specifically assume that the load torque(t) is unknown and that the viscous frictioncoefficient B is also unknown.

We, thus, propose the following perturbed sim-plified model of the mechanical part of the sys-

tem, lumping all the uncertainties into a sin-gle time-varying function of unknown, but uni-formly absolutely bounded, nature denoted by1(t).

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ddt

w = KJ

ia + 1(t)

where the unknown perturbation input 1(t) isgiven by the sum of uncertain quantities:

1(t) = B

Jw

1

J(t)

The outer loop robust armature current feed-back controller design may be formulated as

follows:

Given a smooth, desired, reference trajectory

for the motor shaft angular velocity, w(t), de-

vice an auxiliary feedback control law for thearmature current, denoted by ia,ref, which achievesthe reference trajectory tracking, even if in

an approximate manner, regardless of the per-

turbation inputs represented by the exogenous

torque input, (t) and regardless of the un-

known friction coefficient, B, i.e., regardlessof the lumped unknown but bounded uncer-

tain term, 1(t), in the simplified descriptionof the perturbed system.

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The robust, GPI observer based controller for

the angular velocity trajectory tracking is givenby

ia,ref = ia(t) +

J

K

z1s k1(w w

(t))

where z1s is the smoothed version of the GPI

observer variable, z1, generated by the observer:d

dtw =

K

Jia,ref + z1 + 3(w w)

z1 = z2 + 2(w w)

z2 = z3 + 1(w w)

z3 = 0(w w)

The nominal armature current, ia(t), cannot

be exactly pre-computed, due to the uncer-

tainty surrounding the viscous friction coeffi-

cient, B, and the torque input, (t). We, thus,

propose to compute such a pre-compensation

in the following faulty manner:

ia(t) =J

Kw(t)

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Let ew = w w denote the, redundant, angular

velocity estimation error. Subtracting from theangular velocity equation the angular velocity

estimation equation and eliminating the z vari-

ables, we obtain, after some algebraic manip-

ulations it is found that ew satisfies,

e(4)w + 3e

(3)w + 2ew + 1ew + 0ew =

(3)

(t)

while the tracking error, e, closed loop system

satisfies,

e + k1e = (t) (t)

Under the assumption of an absolutely uni-

formly bounded right hand side in the above

set of equations, by choosing the observer gains

and the controller gains sufficiently large, it

follows that the trajectories of the estimation

error and of its time derivatives, as well as thetracking error converge to a small neighbor-

hood of the origin of the corresponding phase

space.

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Inner loop robust controller design

The second stage for the controller design is

concerned with the electrical system,

d

dtia =

V

L u R

Lia K

L w

which is controlled by the armature circuit in-

put voltage, u. In this instance, we assume

that the armature resistance, R, is unknown

while we try to synthesize the control withoutthe aid of the angular velocity w. We thus pro-

pose the following simplified perturbed model:

d

dtia =

V

Lu + 2(t)

where

2(t) = R

Lia

K

Lw

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The inner loop robust armature input voltage

feedback controller design may be formulated

as follows:

Given the smooth, desired, reference trajectory

for the armature current, ia,ref(t), obtained

from the first design stage, device a feedback

control law for the armature input voltage, u,

which achieves the reference trajectory track-

ing, even if in an approximate manner, regard-

less of the perturbation inputs represented by

the endogenous inputs, RL ia and regardless of

the angular velocity influence through the term

KL w i.e., regardless of the lumped unknown

but bounded uncertain term, 2(t), in the sim-

plified description of the perturbed system.

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The robust feedback control law for the arma-

ture circuit input voltage is given by

u = u(t) +L

V

z1is p1(ia ia,ref)

where

u(t) =LJ

V Kw(t)

and where z1is is the smoothed version of the

observer variable z1i defined by the following

observer

ddt

a = uVL

+ z1i + 3(ia a)

z1i = z2i + 2(ia a)

z2i = z3i + 1(ia a)

z3i = 0(ia a)

(Note that we have used the same design pa-

rameters for the observers appearing in each

design stage)

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Since ia(t) = (J/K) w(t) and u(t) = (L/V)ia(t),

the closed loop system is given by,

d

dtia

d

dtia(t) =

d

dt(ia i

a(t)) = p1(ia ia,ref)

Note that if eia = ia ia,ref then

eia = p1eia + (d

dtia(t)

d

dtia,ref)

For large p1, the tracking error eia converges to

a vicinity of zero in spite of the bounded per-

turbation input represented by the time deriva-

tive of ia(t) ia,ref.

Let ei = ia a. The injected estimation error

dynamics evolves, after elimination of the z1i,

in accordance with,

e(4)i + 3e(3)i + 2e

(2)i + 1ei + 0ei =

(3)2

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CONTINUOUS SYSTEM MOTORDC4

" This program simulates a two stage robust gpi observer based

" controller for a dc motor in a sensor aided scheme.

" A two stage desgin controller is proposed. An outer loop" controller for angular velocity control through the current

" acting as a control, with due cancellation of

" unknown torque and viscous friction effects" thanks to a gpi observer. The GPI observer based inner

" loop controller specifies the input voltage to" regulate the current towards the current

" reference signal acting as a control in the previous stage." the main advantage is that B, R, and tau may be unknown

STATE ia w wh z1 z2 z3

STATE iah z1i z2i z3iDER dia dw dwh dz1 dz2 dz3

DER diah dz1i dz2i dz3iTIME t

"dc motor plant

dia=u*V/L-R/L*ia-K/L*w

dw=K/J*ia-B/J*w-tau/J

"time varying load torque


"outer loop PI controller

iar=ias+J/K*(-z1s-k1*(w-ws))ias=J/K*wsd

phi1=-B/J*w-tau/J

k1:4

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"inner loop PI controller

u=us+L/V*(-z1is-p1*(ia-iar))

us=L*J/(V*K)*wsddphi2=-R/L*ia-K/L*w

p1=20

" armature current and disturbance observer

diah=u*V/L+z1i+g3*(ia-iah)

dz1i=z2i+g2*(ia-iah)dz2i=z3i+g1*(ia-iah)

dz3i=g0*(ia-iah)

" angular velocity and disturbance observer

dwh=K/J*ia+z1+g3*(w-wh)

dz1=z2+g2*(w-wh)dz2=z3+g1*(w-wh)dz3=g0*(w-wh)

"observer design gains

g3=4*zo*wnog2=(2*wno^2+4*zo^2*wno^2)

g1=4*zo*wno^3g0=wno^4

zo=1

wno=40

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"angular velocity trajectory

ws=Fs "desired constant angular velocitywsd=Fsd

wsdd=Fsdd

"smoothing of observer variables

whs=wh*sf

z1s=z1*sf

z1is=z1i*sf


sin8=sin2*sin2*sin2*sin2sin2=sin1*sin1

sin1=sin(pi*t/(2*eps))

eps:0.5pi:3.14159265

"DC motor parameter values:

L:7e-3

R:2.33B:0.00937

K:0.479J:0.01164

V:100

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"rest to rest trajectory planning for angular velocity profile

t1=0 " initial and final instants of rest-to-restt2=4

delt=t2-t1

tdif = abs(t-t1)

z1f = 100 " initial and final values

z1in = 0

z1dif = z1f-z1in

tau1=(tdif/delt)

tau2=tau1*tau1

tau3=tau1*tau2tau4=tau1*tau3


tau7=tau1*tau6tau8=tau1*tau7tau9=tau1*tau8

r1=12870r2=91520

r3=288288r4=524160

r5=600600r6=443520

r7=205920r8=54912

r9=6435

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z1st1 = z1in

fr=r1-r2*tau1+r3*tau2-r4*tau3+r5*tau4-r6*tau5+r7*tau6-r8*tau7+r9*tau8z1stt = z1in+z1dif*tau8*fr

z1st2 = z1f

z1std1 = 0

frd2=14*r7*tau6-15*r8*tau7+16*r9*tau8frd=8*r1-9*r2*tau1+10*r3*tau2-11*r4*tau3+12*r5*tau4-13*r6*tau5+frd2

z1stdt= z1dif*tau7*frd/deltz1std2 = 0

z1stdd1 = 0

frdd2=-12*13*r6*tau5+13*14*r7*tau6-14*15*r8*tau7+15*16*r9*tau8frdd=7*8*r1-8*9*r2*tau1+9*10*r3*tau2-10*11*r4*tau3+11*12*r5*tau4+frdd

z1stddt = z1dif*tau6*frdd/(delt*delt)z1stdd2 = 0

z1st3d1 = 0frddd3=-13*14*15*r8*tau7+14*15*16*r9*tau8

frddd2=10*11*12*r5*tau4-11*12*13*r6*tau5+12*13*14*r7*tau6+frddd3

frddd=6*7*8*r1-7*8*9*r2*tau1+8*9*10*r3*tau2-9*10*11*r4*tau3+frddd2z1st3dt = z1dif*tau5*frddd/(delt*delt*delt)

z1st3d2 = 0

Fs=if t < t1 then z1st1 else if t < t2 then z1stt else z1st2

Fsd=if t < t1 then z1std1 else if t < t2 then z1stdt else z1std2Fsdd=if t < t1 then z1stdd1 else if t < t2 then z1stddt else z1stdd2

Fs3d=if t < t1 then z1st3d1 else if t < t2 then z1st3dt else z1st3d2

END

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Simulation Results

0 1 2 3 4 5 6 7 8 9 102

0

2

4

0 1 2 3 4 5 6 7 8 9 10100

0

100

200

ia,ref(t), ia(t)

w(t), w(t)

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0 1 2 3 4 5 6 7 8 9 10150

100

50

0

0 1 2 3 4 5 6 7 8 9 1010000

5000

0

5000

1(t), 1(t)

2(t), 2(t)

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Sensorless feedback control of the DC mo-

tor: A GPI observer approach

Sensorless control of DC motors refers to the possi-

bility of controlling the motor without measuring the

mechanical variables such as angular position or angu-

lar velocity. The control scheme is then to be carried

out solely on the basis of armature current measure-

ments and, possibly, the knowledge of the control input

voltage. The usual approach is to use an observer, of

the reduced order type, in order to estimate the angular

velocity from an indirect output equation representing

the angular velocity in terms of the input voltage, the

armature current and its time derivative.

However, most sensorless designs are not particularly ro-

bust with respect to the mechanical loads and specially

when these are not of constant nature, but rather time

varying. Here, we propose a GPI observer approach for

the efficient simultaneous asymptotic, though approxi-

mate, estimation of the angular velocity as well as of

the unknown, time-varying, load torque input.

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Problem formulation

Given a desired angular velocity profile, w(t),

for the motor shaft, device a linear feedback

control law, on the basis of the armature cur-

rent, ia, and knowledge of the system parame-

ters, which asymptotically exponentially drives

the angular velocity of the motor, w(t), to-wards the desired reference signal within an

uniformly absolutely bounded negligible error,

regardless of the unknown, but uniformly ab-

solutely bounded, load torque input, (t).

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We will be proposing, in the next paragraphs,a reduced order GPI observer which simultane-

ously estimates in a rather close manner, both,

the angular velocity, w, and the torque input,

(t), based only on knowledge of the armature

current, ia. In principle, such an observer isgiven by

dw

dt=

K

Jia

B

Jw + z1 + 3(w w)

z1 = z2 + 2(w w)

z2 = z3 + 1(w w)

z3 = 0(w w)

with the injection of w given by

w = L

K

dia

dt

+

V

Ku

R

Kia

where we have adopted a second degree time

polynomial for the internal model of the torque

input, (t).

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The angular velocity estimation error, e = w

w, is seen to formally satisfy the following per-turbed injected dynamics,

e(4)+(3+B

J)e(3)+2e+1e+0e =

1

J(3)(t)

Thus, the design coefficients may be chosen

by comparison with those of a desired, domi-

nating, fourth degree characteristic polynomialof the form pd(s) = (s

2 + 2onos + 2no)2. i.e.,

3 = 4ono B

J2 = 2

2no + 4

2o

2no

1 = 4o

3

no0 =

4no

As long as the roots of the characteristic poly-

nomial are chosen sufficiently far from the imag-

inary axis in the left portion of the complex

plane, the effect of the third order derivative

of (t) is diminished having the estimation er-ror and its time derivatives converge towards

a small as desired disk in the estimation error

phase space.

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Observer assessment Before presenting the

complete solution to the problem, we carryout a partial assessment of the proposed load

torque GPI observer by simulating the observer

as if ia were the control input and w were ac-

tually the measurable output. We propose a

rather complex torque input given by,

(t) = e sin2(3t) cos2(0.3t) [N m]

We set, for the observer gains,

o = 1, no = 40

The feedback controller for the current, ia, was

set to be:ia = i

a +

J

K

1

Jz1s k1(w w

(t))

k0

t0

(w w())d

with w(t) being a rest-to-rest velocity profile

starting at 0 and ending at 100 [r/s] in 2[s]. We have set k1 = 2cnc, k0 =

2nc, with

c = 1, nc = 0.3. The smoothing interval:

= 0.4[s].

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0 2 4 6 8 100

50

100

0 2 4 6 8 10

0

0.2

0.4

0 2 4 6 8 100

2

4

6

w(t), w(t), w(t)

(t), (t)

ia(t)

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The sensorless GPI controller

However, we must modify the form of the GPI

observer since the expression for w contains

time derivatives which are inconvenient to com-

pute in practise. Substituting the expression

for w and defining:

= w + 3L

Kia

j = zj + 3jL

Kia, j = 1, 2, 3

one obtains the following modified GPI ob-server:

ddt

=

KJ 2 L

K 3 R

K+ L

K

BJ

+ 3

3

ia

B

J+ 3

+ 1 + 3

V

Ku

d1

dt= 2 + 2

V

Ku

2

R

K+ 1

L

K 23

L

K

ia 2

d2

dt = 3 + 1V

Ku

1R

K + 0L

K 13L

K

ia 1

d3

dt= 0

V

Ku

0

R

K 03

L

K

ia 0

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The angular velocity estimate is thus given by

w = 3L

Kia

and the torque estimate is obtained as

(t) = J1 2L

K

iaIn order to complete the solution to the prob-lem we propose a customary two stage pro-cedure with the advantageous standpoint ofhaving now accurate angular velocity and loadtorque input estimations. For this, we firstview the armature current, ia, as an interme-diate auxiliary control input we set,

ia =J

K

(t)

J+

B

J(w w(t)) 1(w w

(t))

0

t0

(w w()) d

+ ia(t)

where ia(t) may be taken to be

ia(t) = JK(t) + B

K(t)

and 1, 0 are control design constants

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Under the assumption of fast convergence of

w and (t) to their actual values w and , theclosed loop behavior of the tracking error, ew =ww(t), caused by this auxiliary control input,is found to be described by the following lineardynamics,

ew

+ 1

+B

J

ew+

0t

0 ew(

)

d= 0

We may set 1 and 0 so that the closed loopcharacteristic polynomial coincides with

s2 + 2cncs + 2nc

and thus, have an asymptotic exponential con-

vergence of w to w.We regard this control law as part of an outercontrol loop specifying the desirable currentsignal. We let this specification of the arma-ture current to act as a current reference anddenote it, from now on, as ia,ref, i.e.,

ia,ref =

J

K (t)

J +

B

J(w w

(t)) 1(w w

(t))

0

t0

(w w()) d

+ ia(t)

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At the second controller design stage, the in-

ner control loop specifies now the input volt-age, u, to the armature circuit, so that the ac-

tual armature current, ia, asymptotically expo-

nentially tracks the desired current reference,

ia,ref, specified in the first stage. We set

u = u(t) + LV

RL

(ia ia(t)) +

KL

(w w(t))

1(ia ia,ref(t)) 0

t0

(ia ia,ref())d

where

u(t) = LV d

dti

a(t) +

R

Vi

a(t) +

K

V(t)

and

ia(t) =J

K(t) +

B

K(t)

The proposed controller design produces a closed

loop current behavior tracking error, eia = ia

ia,ref, ultimately governed by

d

dteia +

1 +

R

L

eia + 0

t0

eia()d = 0

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where the coefficients 1, 0, are set so that

the characteristic polynomial coincides with the

desired closed loop polynomial,

pdi(s) = s2 + 2cincis +

2nci

This is achieved by simply setting

1 = 2cinci R

L0 =

2nci

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Simulation results

CONTINUOUS SYSTEM MOTORDC3

" This program simulates a robust gpi observer based" controller for a dc motor in a sensorless scheme.

" A two stage desgin controller is proposed. An outer loop" controller for angular velocity control through the current

" acting as a control, with due cancellation of unknown torque

" thanks to a gpi observer. The inner loop controller specifies" the input voltage to regulate the current towards the current

" reference signal acting as a control in the previous stage.

STATE ia w iwmwo iiamir

STATE wh z1 z2 z3DER dia dw diwmwo diiamirDER dwh dz1 dz2 dz3

TIME t

"dc motor plant

dia=u*V/L-R/L*ia-K/L*wdw=K/J*ia-B/J*w-tau/J

"time varying load torque


"inner loop PI controller

u=us+L/V*(R/L*(ia-ias)+K/L*(wh-ws)-p1*(ia-iar)-p0*iiamir)diiamir=ia-iar

us=L*J/(V*K)*wsdd+(L*B/(V*K)+R*J/(V*K))*wsd+ussuss=(R*B/(V*K)+K/V)*ws

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p1=2*zic*wnicp0=wnic^2

zic:1wnic:8

"outer loop PI controller

iar=ias+J/K*(tauh/J+B/J*(wh-ws)-k1*(wh-ws)-k0*iwmwo)diwmwo=wh-ws

ias=J/K*wsd+B/K*ws

ws=Fs "desired constant angular velocity

wsd=Fsdwsdd=Fsdd

k1=2*zc*wc

k0=wc*wczc:1wc:4

"reduced order angular velocity and torque observer

dwh=1/J*(K*ia-B*wh)+z1+g3*(wm-wh)dz1=z2+g2*(wm-wh)

dz2=z3+g1*(wm-wh)dz3=g0*(wm-wh)

wm=-L/K*dia+V/K*u-R/K*ia

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"smoothing of observer variables

whs=wh*sf

z1s=z1*sftauh=-J*z1s


sin8=sin2*sin2*sin2*sin2

sin2=sin1*sin1sin1=sin(pi*t/(2*eps))

eps:0.2pi:3.14159265

"observer design gains

g3=4*zo*wno-B/Jg2=(2*wno^2+4*zo^2*wno^2)

g1=4*zo*wno^3g0=wno^4

zo=1

wno=40

"DC motor parameter values:

L:7e-3R:2.33

B:0.00937K:0.479

J:0.01164V:100

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"rest to rest trajectory planning for angular velocity profile

t1=0 " initial and final instants of rest-to-restt2=2

delt=t2-t1

tdif = abs(t-t1)

z1f = 100 " initial and final values

z1in = 0

z1dif = z1f-z1in

tau1=(tdif/delt)

tau2=tau1*tau1



tau7=tau1*tau6tau8=tau1*tau7tau9=tau1*tau8

r1=12870r2=91520

r3=288288r4=524160

r5=600600r6=443520

r7=205920r8=54912

r9=6435

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z1st1 = z1in

fr=r1-r2*tau1+r3*tau2-r4*tau3+r5*tau4-r6*tau5+r7*tau6-r8*tau7+r9*tau8z1stt = z1in+z1dif*tau8*fr

z1st2 = z1f

z1std1 = 0

frd2=14*r7*tau6-15*r8*tau7+16*r9*tau8frd=8*r1-9*r2*tau1+10*r3*tau2-11*r4*tau3+12*r5*tau4-13*r6*tau5+frd2

z1stdt= z1dif*tau7*frd/deltz1std2 = 0

z1stdd1 = 0

frdd2=-12*13*r6*tau5+13*14*r7*tau6-14*15*r8*tau7+15*16*r9*tau8frdd=7*8*r1-8*9*r2*tau1+9*10*r3*tau2-10*11*r4*tau3+11*12*r5*tau4+frdd

z1stddt = z1dif*tau6*frdd/(delt*delt)z1stdd2 = 0

z1st3d1 = 0frddd3=-13*14*15*r8*tau7+14*15*16*r9*tau8

frddd2=10*11*12*r5*tau4-11*12*13*r6*tau5+12*13*14*r7*tau6+frddd3

frddd=6*7*8*r1-7*8*9*r2*tau1+8*9*10*r3*tau2-9*10*11*r4*tau3+frddd2z1st3dt = z1dif*tau5*frddd/(delt*delt*delt)

z1st3d2 = 0

Fs=if t < t1 then z1st1 else if t < t2 then z1stt else z1st2

Fsd=if t < t1 then z1std1 else if t < t2 then z1stdt else z1std2Fsdd=if t < t1 then z1stdd1 else if t < t2 then z1stddt else z1stdd2

Fs3d=if t < t1 then z1st3d1 else if t < t2 then z1st3dt else z1st3d2

END

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0 5 10 15 200

2

4

6

8

[A]

0 2 4 6 8 10

0

50

100

[rad/s]

0 5 10 15 200.5

0

0.5

1

[Nm]

w(t), w(t)w(t)

ia(t), ia,ref(t)

(t), (t)

Performance of sensorless GPI observer based

control for angular velocity tracking

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0 5 10 15 200.1

0

0.1

0.2

0.3

0.4

0.5

0.6

u(t)

Applied input voltage in a sensorless GPI

observer based control for angular velocity

tracking

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dc-motor (2)