1ME 233 Advanced Control II
Lecture 14
Deterministic Input/Output Approach toSISO Discrete Time Systems
Pole Placement, Disturbance Rejection
and Tracking Control

2

SISO ARMA models• SISO State space model

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

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SISO transfer function

relative degree

4

SISO transfer function
U(z)
Y(z)

control input
outputrelative degree

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ARMA Models
Define:• the back-step operator

such that• the polynomials• relative degree (pure time delay)

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Back-step operator
Relationship to Z-transformSimilarly,

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SISO ARMA models

Auto-Regressive

Moving Average

8SISO ARMA models with persistent disturbances
SISO ARMA model with disturbance

Where all inputs and outputs are scalars:•control input•persistent (deterministic) but unknown disturbance•output

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Deterministic SISO ARMA models

Where polynomials:

are co-prime andd is the known pure time delay10Polynomials in q-1• Monic polynomialconstant coefficient is 1

leading coefficient is 1
monic

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Polynomials in q-1• Co-prime polynomials have no common roots

The polynomials

are co-prime if and only ifwhenever p satisfies A(p) = 0

12Polynomials in q-1• Anti-Schur polynomials have all of their roots outside theunit circle

For example, if the polynomial

is anti-Schur, then |q-1| > 1 whenever A(q-1) = 0

13Polynomials in q-1

is anti-Schur if and only if

for all sequencesthat satisfy

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Factorization of the zero polynomial B(q-1)
Assume the m order zero polynomialhas• mu zeros that we do not want to cancel.
• its remaining ms zeros inside the unit circle; these are
the zeros we will cancel

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Factorization of the zero polynomial B(q-1)
Assume the m order zero polynomialhas• mu zeros that we do not want to cancel.
• its remaining ms zeros inside the unit circle; these are
the zeros we will cancel

is anti-Schurhas the zeros (in q) that we
do not want to cancel

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ExampleRoot Locus
1*

Imaginary Axis

0.5

*0*

-0.5

*
-1-1

-0.5

00.5
Real Axis

11.5

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ExampleExampleRoot Locus
1*

Imaginary Axis

0.5

*0*-0.5*
we could have chosen to cancel these 2-1-1-0.5

00.5
Real Axis

11.5

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Deterministic SISO ARMA models
The zero polynomial:

Without loss of generality, we will assume that

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

Where:•

cancelable plant zeros•

anti-Schur polynomial of the form

chosen by
the designer

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Control Objectives
2. Tracking: The output sequencea reference sequencemust follow

which is known

In general, yd(k) can be generated by a reference model ofthe form

anti-Schur polynomial

The design of Am(q -1) and Bm(q -1) is not a part of thiscontrol design technique and these polynomials do notenter into the analysis

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Control Objectives
2. Tracking: The output sequencea reference sequence•must follow

which is known

Reference model:

Where:•

reference output sequence, which is known inadvance (i.e. yd(k+L) is available at instance k forsome L>d).

•anti-Schur polynomial•

polynomialchosen bythe designer

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Control Objectives
3. Disturbance rejection: The closed-loop system
must reject a class of persistent disturbances•Disturbance model:

Where•

is a known annihilating polynomialwith zeros on the unit circle•

are co-prime

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Deterministic disturbance examplesa) Constant disturbance:

Then,b) Sinusoidal disturbance of known frequency:

Then,

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Deterministic disturbance examplesc) Periodic disturbance of known periodThen,

In all of these three examples, the polynomial
has its roots on the unit circle.

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Control Law• Feedback and feedforward actions:reference outputu (k)-1T( q ,q)r(k) +-d(k)1
R( q-1)u(k) +

q-d B(q-1)

y(k)

-1

A( q )-1

S( q )

Feedforward action(a-causal)

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Closed-Loop TF from r(k) to y(k)y(k))

Closed-loop characteristic polynomial

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Closed-Loop TF from r(k) to y(k)If we lethave the special structure

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Closed-Loop TF from r(k) to y(k)y(k)

Closed-loop characteristic polynomial

Given Ac(q-1), we would like to find polynomialsR(q-1) and S(q-1) so that

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The Diophantine (Bezout) equation• Given the co-prime polynomials–is order n and has constant term 1–

is order m• and a polynomial of order nc with constant term 1

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The Diophantine (Bezout) equation
We wish to find polynomials

same order asthat satisfy the Diophantine equation:

given

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The Diophantine (Bezout) equation
Expanding in terms oforderordercoefficients:

order

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The Diophantine (Bezout) equation
Expanding in terms ofcoefficients:

We obtain:

given on next slide

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The Diophantine (Bezout) equation
Where the matrix
is given by:34

columnscolumns

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If ns = n - 1, then these rows of zeros will not be here

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The Diophantine (Bezout) equation
Theorem: D is nonsingular iff the polynomials

andare co-prime.

The solution to the Diophantine equation is:

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Example:Letorder nc = 2

order n = 2

order m = 2
order m = 2
Solve fororder ns

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Example:4 equations and 4 unknowns

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Example:Equating coefficients of powers of q-1

Solution:

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Return to the Control Problem…
• Feedback and feedforward actions:
d(k)
u (k)

-1

T( q ,q)r(k) +-1
R( q-1)u(k) +

q-d B(q-1)

y(k)

-1

A( q )-1

S( q )

Feedforward (a-causal)

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Feedback Controller
Diophantine equation: Obtain polynomials
that satisfy:

Closed-loop
poles

Plant poles

plant zeros

We will factor
out thepolynomial next

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Controller Diophantine equation
Factor out

polynomial

Therefore, we want to find R’(q-1) and S(q-1) such that

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Feedback Controller
Diophantine equation: Obtain polynomials
which satisfy:

Plant poles
Closed-loop
Unstable plant zeros
poles without
Disturbance annihilating polynomial
cancelled zeros

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Use previous solution of the Diophantine equation45Diophantine equation

Solution:
These are thecontroller
parameters!

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Feedback Controller

whereIf the degree of the disturbance annihilator
polynomial, nd is large (e.g. N is large),then nr and ns are also large
Then, the solution of the Diophantine equation
may be ill conditioned.

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ExamplePlant:

Zeros:

Disturbance:
Select closed-loop poles:

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Diophantine equation

Solution:

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ExampleControl:

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Feedback Control Law
Feedback control action:

Proof – block diagram algebra

The closed-loop dynamics is from r(k) and d(k) to y(k)51Proof – block diagram algebra
The closed-loop dynamics from d(k) to y(k) ( r(k) = 0 )Substitute:1

pole-zerocancellation

Diophantine equation52

Proof – block diagram algebra
The closed-loop dynamics from d(k) to y(k) ( r(k) = 0 )1pole-zerocancellation

Diophantine equation

53

Proof – block diagram algebra

The closed-loop dynamics from r(k) and d(k) to y(k)54Proof – block diagram algebra

Substitute:1

pole-zerocancellation

Diophantine equation

55

Proof – block diagram algebra1pole-zerocancellation

Diophantine equation

56

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ExampleControl:

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ExampleClosed-loop dynamics:

Unit step response
Step Response
454035
Amplitude30
252015105005101520Time (sec)

2530

35

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Feedback Control Law
The feedback control action:

Results in the following closed-loop input/output dynamics:

well-damped zeros

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Feedforward Control
Feedforward control objective is to make y(k)follow yd(k) as closely as possible.

Goal:

or

how well the objective met depends onwhether the plant has unstable zeros or not61

Feedforward Control Synthesis

closed loop dynamics
from r(k) to y(k)62Feedforward control principle: plant inversion≈1yd (k)pure step delays
unstable zeros
cannot be invertedr(k)q-d Bu(q-1)y(k)

A’c (q )-1

approximate inverse
we need to know yd(k+d)

closed-loop pole
polynomial(anti-Schur)63

Perfect Tracking Feedforward Control
Perfect tracking can be achieved if all plant zeros arecancelable, e.g.
in this caseyd (k)r(k)q-d Bu(q-1)
A’c (q )
-1

y(k)

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Tracking with unstable zeros• When the plant has unstable zeros we need to find
an approximate inverse

yd (k)r(k)q-d Bu(q-1)
A’c (q )-1y(k)

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A-causal Bounded-Input Bounded-Output (BIBO)
realization of a purely unstable operator

Leti.e. all zeros ofoutside the unit circle

areThen we can interpretin two ways:•

is causal but unstable•

is a-causal but BIBO

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A-causal Bounded-Input Bounded-Output (BIBO)
realization of a purely unstable operator

Example:unstable causal operator
Using an infinite series expansion,

infinite dimensional a-causal operator

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A-causal BIBO realization of a purely unstable operator
Thus,

Can be realized either as:(unstable)or(a-causal BIBO)

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A-causal BIBO approximation of a purely unstable operator
We will now describe two methods of approximating apurely unstable operator:
1) Truncated a-casual series expansion:2) Zero-phase error feedforward operator:(developed by Prof. Tomizuka)

Not q-1

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Example: realizingLet,1) Truncated a-casual series expansion:2) Zero-phase error feedforward operator:

Not q-1

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Zero-phase error tracking
One of the most popular feedforward techniques forsystems with unstable zeros.

Define the zero-phase operator

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Zero-phase error transfer function
A-causal zero-phase transfer function:

Properties:• It has zero-phase, i.e.

• It has unity dc gain, i.e.

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Example: realizingLet,•

Zero-phase feedforward:
Zero-phase transfer function:
Bode Diagram
Magnitude (dB)0-10-20-30-40-50
1Phase (deg)

•0.5
0-0.5
-1 -2
10-1

010
10
Frequency (rad/sec)

110

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Sinusoidal zero-phase error tracking
If yd(k) is a sinusoidal, there will be no phase shift
between yd(k) and y(k)

1

0.5

0-0.5

-1
05

10

15

2025

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Zero-phase error feedforward

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Zero-phase error feedforward

Closed-loop dynamics: