1ME 233 Advanced Control II
Lecture 23
Direct Adaptive Pole Placement, andTracking Control

2Direct vs. Indirect Adaptive Control•

Both use pole-placement, tracking control and deterministic
disturbance rejection controller synthesis methodology.

•Indirect adaptive control:
1.
2.
–•Plant parameters are estimated using a RLS PAA.
Controller parameters are calculated using the certainty
equivalence principle.
Use with plants that have non-minimum phase zeros.
(Plant unstable zeros are not cancelled).

Direct adaptive control:
1.
–Controller parameters are updated directly using a RLS
PAA.
Use with plants that do not have non-minimum phasezeros. (Plant zeros are cancelled).

3

Direct Adaptive Control
1. Plants with minimum phase zeros andno disturbances:• Controller design (review)
1. Controller PAA
2. Adaptive Controller
2. Plants with minimum phase zeros andconstant disturbances:•

Read section: Direct adaptive control withintegral action for plants with stable zeros
in the ME233 class notes, part II.

4Deterministic SISO ARMA models
SISO ARMA model

Where all inputs and outputs are scalars:•control input•output

d is the known pure time delay

5Deterministic SISO ARMA models

Where polynomials:

are co-prime andisanti-Schur

6

Control Objectives
1. Pole Placement: The poles of the closed-loopsystem must be placed at specific locations in thecomplex plane.
•Closed-loop pole polynomial:

Where:•

cancelable plant zeros•

anti-Schur polynomial chosen by the designer

7

Control Objectives
2. Tracking: The output sequencereference sequence•must follow awhich is known

Reference model:

Where:•known reference input control input sequence•(from the previous slide)•zero polynomial, chosen by the designer

8Control Law• Feedback and feedforward actions:u (k)d(k) =0-1

B (q )

r(k) +-u(k) +1-1

R( q )

q-d B(q-1)

y(k)

-1

A( q )-1

S( q )

Feedforward is causal

9Feedback Controller
Diophantine equation: Obtain polynomials
that satisfy:

Closed-looppoles

Plant poles

Plant pure delays

10Diophantine equation

Solution:

11

Feedback Controller

where

12

Closed-loop dynamics
u (k)d(k) =0-1

B (q )

r(k) +-u(k) +1-1

R( q )-1

S( q )

q-d B(q-1)-1A( q )

y(k)

13

Direct Adaptive Control
1. Plants with minimum phase zeros andno disturbances:•Controller design

1. Controller PAA
2. Adaptive Controller

14Controller parameters
We want to identify the controller polynomials

directly, where

15Controller parameters
Start with the Diophantine equation

Multiply both sides by y(k)16Controller parameters

Insert plant dynamics

17

PAA – version 1

Filter by(normally a low-pass filter)18

PAA – version 1

Is linear in the controller parameters:

19

PAA – version 1
Plant dynamics:

20

PAA – version 1
Plant dynamics:RLS PAA:21

PAA – version 2

filtered output signal

22

PAA – version 2
Plant dynamics:
RLS PAA:23

PAA – version 1 Vs version 2•

is normally a high-pass filter•

is normally a low-pass filter
Bode Diagram
5Example0

Version 1 is preferable

A’c (z -1)

-5
60

30

0-30

filters high frequencynoise

1
A’c (z -1)

6 10 -1
0

A’c (z -1)
1
A’c (z -1)
Frequency (rad/sec)

100

24

PAA projection
PAA: Projection

Replacesmall.

byif it becomes too

Control law will divide by. Thus, the projection
algorithm prevents the control action from becoming toolarge.

25

PAA Gain matrix
Gain matrix:

are adjusted so that the maximum singular value of F(k) is
uniformly bounded, and26

Direct Adaptive Control
1. Plants with minimum phase zeros and nodisturbances:•Controller design

1. Controller PAA
2. Adaptive Controller

27

Fixed Controller
d(k)
u (k)

-1

B (q )

r(k) +-u(k) +1-1

R( q )-1

S( q )

Use this equation to solve for u(k)q-d B(q-1)-1

A( q )y(k)28Adaptive Controller
d(k)

u (k)

-1

B (q )

r(k) +-u(k) +1-1

R( q )-1

S( q )

Use this equation to solve for u(k)q-d B(q-1)-1

A( q )y(k)29Fixed Controller
d(k)
u (k)

-1

B (q )

r(k) +-u(k) +1-1

R( q )-1

S( q )

q-d B(q-1)-1

A( q )y(k)30

Adaptive Controller
d(k)

u (k)

-1

B (q )

r(k) +-1^ -1

R( q )-1^ q)
S(u(k) +q-d B(q-1)-1

A( q )y(k)31

Direct Adaptive Control
1. Plants with minimum phase zeros and nodisturbances:• Controller design
1. Controller PAA
2. Adaptive Controller

2. Plants with minimum phase zeros andconstant disturbances:•Read section: Direct adaptive control withintegral action for plants with stable zeros
in the ME233 class notes, part II.