adaptive control theory: introduction

NATIONAL CHENG KUNG UNIVERSITY

Department of Mechanical Engineering

ADAPTIVE CONTROL

HOMEWORK 1

Instructor: Ming – Shaung Ju

Student: Nguyen Van Thanh

Student ID: P96007019

Department: Inst. of Manufacturing & Information Systems

Class: 1001- N164400 - Adaptive Control

April 20, 2023

http://class-qry.acad.ncku.edu.tw/qry/classmate.php?syear=0100&sem=1&co_no=N164400&class_code=

Contents

Problem 1.........................................................................................................................2

Problem 2.........................................................................................................................5

Problem 3.......................................................................................................................10

Problem 4.......................................................................................................................15

1Adaptive Control Class – HW1

Problem 1

Consider the system depicted in Figure 1.3 of the text book. If an input disturbance, –

d(t), is added at the input to the process.

(a) Prove that high loop-gain can be utilized to reject step disturbance.

(b) If a measurement noise, n(t), is added at the output of the process. Could the noise be

rejected by using high loop-gain?

Solution:

Figure 1.1. Block diagram of a robust high – gain system [1]

The system is given in figure 1.1 that is two – degree – of freedom system. A design

procedure for 2 DOF is,

Step 1: Design feedback controller, the feedback transfer function Gfb(s) is chosen for

disturbance rejection, and insensitivity of closed – loop transfer function to process

variations.

Step 2: Design feedforward controller, the feedforward transfer function Gff (s) is then

chosen to give desired response to command signals that means y(t) becomes closely to

uc(t) as t goes to infinite.


(a) Prove that high loop-gain can be utilized to reject step disturbance.

Now consider the system is given in figure 1.2 below,

Figure 1.2. A system with disturbance

+ If d(t) = 0, the transfer function from ym to y is

Y (s )Y m (s )

=GfbGp

1+G fbG p

(1.1)

+ If ym = 0; the transfer function from d(t) to y is

Y (s )D(s )

=G p

1+Gfb G p

(1.2)

Combine two equations (1.1) and (1.2) we get,

Y (s )=G fbG p

1+GfbG p

Y m (s )+G p

1+G fbG p

D(s) (1.3)

From equation (1.2), if we choose a high – gain loop (for example Gfb = Kp (Kp >>1)), so

Y= DGfb

(1.4)

If Gfb is large enough, disturbance D is rejected, and the most suitable for this case is d(t)

is a step function.


(b) If a measurement noise, n(t), is added at the output of the process. Could the noise

be rejected by using high loop-gain?

Consider a system with a noise is added at the output of the process (see figure 1.3).

Figure 1.3. A system with noise

The transfer function form n to y is given in equation below,

Y ( s )N ( s )

=−GfbG p

1+Gfb G p (1.5)

By observing equation (1.5), if Gfb is chosen as a high – gain loop, then the magnitude of

the transfer function it doesn’t change. So, the high – gain loop cannot reject the noise.


Problem 2

Consider systems with open-loop transfer functions

Where p = - a, 0 and a.

(a) Let a = 0.01, k = 1. Show that the unit step responses of these systems are quite

different but by introducing proportional feedback of u = uc – k*y the unit-step responses

of the closed-loop systems become quite similar.

(b) Compare the Bode diagrams of the open-loop system and the closed-loop system and

discuss the effect of high loop gain on closed-loop system dynamics

(c) Try larger values of ‘a’ such that the closed-loop response becomes different. What

happen if k is increased?

Solution:

(a) Let a = 0.01, k = 1. Show that the unit step responses of these systems are quite

different but by introducing proportional feedback of u = uc – k*y the unit-step

responses of the closed-loop systems become quite similar.

Firstly, consider the transfer function of the open – loop,

Go (s )= 1( s+1 ) (s+ p )

= 1

s2+ (1+ p ) s+ p (2.1)

Figure 2.1. A open – loop system


The step response of the open – loop is shown in figure 2.2 below,

Figure 2.2. Step response of the open – loop with varying parameter p

From figure 2.1, we see that the responses are significantly different. The system with p

= 0.01 is stable; the others are unstable.

Now consider the closed – loop systems are show as figure below,

Figure 2.3. A closed – loop system


The closed – loop systems are obtained by introducing the proportional feedback with

unit gain (k = 1), that is, u = uc – y, give the step responses show in figure 2.4 below.

Figure 2.4. Closed – loop step responses with varying parameter p, k = 1.

We can see that, from figure 2.4, the responses of the closed – loop systems are virtually

identical.

(b) Compare the Bode diagrams of the open-loop system and the closed-loop system and discuss the effect of high loop gain on closed-loop system dynamics

Bode diagrams for the open and closed loops are shown in Fig.2.5. Notice that the Bode

diagrams for the open-loop systems differ significantly at low frequencies but are

virtually identical for high frequencies. Intuitively, it thus appears that there is no

problem in designing a controller that will work well for all systems, provided that the

closed-loop bandwidth is chosen to be sufficiently high. This is also verified by the Bode


diagrams for the closed-loop systems shown in Fig.2.5 (b), which are practically

identical.

(a)

(b)


Figure 2.5. Bode diagrams, (a) Open – loop systems and (b) closed – loop systems

Now consider the case that k becomes larger. We have the transfer function of the closed

– loop systems are,

Gcl ( s)= k

s2+(1+ p ) s+ p+k (2.2)

The damping coefficient of the systems are given by equation,

¿ 1+ p2∗√ p+k

(2.3)

From this equation, if k becomes larger so becomes smaller, thus the system will

increasingly oscillate at the transient state.

Figure 2.6. Step responses of the closed – loop systems with k = 10

The larger k becomes the smaller steady – state value will be.


Problem 3

Consider systems with open-loop transfer functions

Where T = 0, 0.02 and 0.04.

(a) Show that the unit-step responses of the open-loop systems are quite similar but the

unit-step responses of the closed-loop systems (with a proportional feedback of u = uc-y)

are quite different.

(b) Use the Bode diagrams of the open-loop systems and the closed-loop systems to

explain the origin of these different responses.

(c) If T is slowly time-varying, what kind of controller can be used to solve this problem?

Solution

(a) Show that the unit-step responses of the open-loop systems are quite similar but the

unit-step responses of the closed-loop systems (with a proportional feedback of u = uc

- y) are quite different.

Figure 3.1. A open – loop system

Figure 3.2. A closed – loop system


(a)

(b)

Figure 3.3. Step responses: (a) Open – loop and (b) Closed – loop


From figure 3.3, we can see that the unit-step responses of the open-loop systems are

quite similar. The open – loop systems are stable for all values of T’s. But the unit-step

responses of the closed-loop systems are quite different, the systems are stable for T = 0,

0.02, but not stable for T = 0.04.

(b) Use the Bode diagrams of the open-loop systems and the closed-loop systems to

explain the origin of these different responses.

Figure 3.4. Bode diagrams for the open – loop system


Figure 3.5. Bode diagrams for the closed – loop system

The Bode diagrams for the open- and closed-loop systems are shown in Fig. 3.4. Notice

that the frequency responses of the open-loop systems are very close for low frequencies

but differ considerably in the phase at high frequencies (especially for T = 0.04). The

Bode diagrams of the closed-loop systems in Fig. 3.5, we can see that for T = 0.04, the

output signal leads the input signal (phase lead).

(c) If T is slowly time-varying, what kind of controller can be used to solve this

problem?


For every T < 0, the open – loop and closed loop systems are unstable. And clearly see

that if T’s > 0.04, the closed – loop systems are unstable. As we known, in the two

previous sections, the open – loop systems slowly change dynamic behavior if T slowly

time – varying , but the systems cannot reach to the commands and if T becomes large,

there has critical phase – lag in these systems. So, we need a controller can deal with this

case (T slowly time - varying), and maybe everyone all knows that is an “Adaptive

Controller”. We will mention it later.


Problem 4

Find an example of application of adaptive system in your research area or engineering

discipline. Use the definition of an adaptive system to examine the example. Identify the

parameter adjustment loop and the feedback control loop. Is there any learning

mechanism in the system?

Solution

Adaptive control is applied in many control system, for example, aircraft control system,

automobile, ship steering control system, etc. Tell you the true, I don’t have any

knowledge of adaptive control, that’s why I’m taking the adaptive control class, but I will

try hard to give an example about adaptive control in AC servo motor control system. I

want to track the positions and speeds of the systems.

Firstly, I design a feedback controller. The feedback controller basically combines a

position loop with a velocity loop. More specifically, the result of the position error

multiplied by Kp (proportional term) becomes a velocity correction command. The

integral term Ki now operates directly on the velocity error instead of the position error as

in the PID (proportional integral and derivative controller) case and finally, the Kd term in

the PID position loop is replaced by a Kv term in the PIV (proportional integral and

velocity (Kv)) velocity loop. Note, however, they have the same units, Nm/ (rad/sec). The

controller schematic is shown in figure 4.1.

In the figure 4.1,

Kt - Torque constant, V/ (rad/s);

J - Motor inertia, Kg-m2;

B – Damping coefficient, N.m/ (rad/s);

- Position of the shaft of motor, rad;

- Speed of the shaft of the motor, rad/s.


Figure 4.1. Block diagram of a Feedback controller

The transfer function of the system is given in equation (4.1).

G (s )=K i K p K t /J

s3+K t K v+B

Js2+

K i K t

Js+K i K p K t /J

(4.1)

The standard third-order form is given in equation (4.2).

G3 (s )=wn

3

(s+wn)(s2+2 wn s+wn

2) (4.2)

For equation (4.1) and equation (4.2), we roughly define three parameters: Kp, Ki, and Kv as given in equation (4.3).

K p=wn

1+2

K i=wn

2 (1+2 ) JK t

(4.3)

K v=wn (1+2 ) J−B

K t

After that, we need to refine the controller’s parameters to get a better behavior response of the systems.

For simulating, a motor is selected. The motor parameters used in the study are listed in

table 4.1 below.


Table 4.1. Servo motor specifications

Specifications of servo motor (ECMA series, Model C206)

Rated output power (kW) 0.4

Rated torque (N-m) 1.27

Maximum torque (N-m) 3.82

Rated speed (r/min) 3000

Maximum speed (r/min) 5000

Rated current (A) 2.60

Maximum current (A) 7.80

Power rating (kW/s) (without brake) 57.6

Rotor moment of inertia - J (× 10-4 kg.m2) (without brake)

0.277

Mechanical time constant (ms) (without brake) 0.53

Torque constant – Kt (N-m/A) 0.49

Voltage constant – Ke (mV/(r/min)) 17.4

Armature resistance - Ra (Ohm) 1.55

Armature inductance - La (mH) 6.71

Electrical time constant (ms) 4.30

Viscous friction coefficient – B (x 10-3 N.m.s) 0.277


The step response of the system with: n = 2*pi*fn = 2*pi*60 rad/s; = 1.0 is show in

figure 4.2 below.

Figure 4.2. Step response of the system

We can see that, the rough design is not satisfy. We need to refine the controller’s

parameters. But, for now we want to know the effective of an adaptive control. Assume

the feedback controller is okay. Now, we consider a case, the motor’s parameters (plant’s

parameters) is time – varying. So, we need to use an adaptive controller (estimation of

parameters loop). But, now it is difficult and takes a lot of time of us to do it. We can do

it after we have some knowledge about this field.


adaptive control theory: introduction

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