1ME 233 Advanced Control II
Lecture 22
Indirect Adaptive Pole Placement,
Disturbance Rejection and Tracking
Control2

Adaptive Control
Adaptive Control Principle
Controller parameters are not constant, rather,
they are adjusted in an online fashion by aParameter Adaptation Algorithm (PAA)
When is adaptive control used?

• Plant parameters are unknown• Plant parameters are slowly time varying

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Self-Tuning Regulator (STR):
performance
specification

plantparameters

controller
designreferenceadjustable
controller

model
identification
u(t)

y(t)
plant

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Self-tuning Regulator Approach• Control Design Procedure:– Pole-placement, tracking control anddeterministic disturbance rejection for
ARMA models (Lecture 16).
• Model Identification:– Series-parallel with Recursive Least
Squares (RLS) identification with or without
forgetting factor.

5Certainty Equivalence Principle
Use pole-placement, tracking control and deterministic
disturbance rejection controller synthesis methodology.

1. Estimate plant parameters using RLS PAA.

2. Controller parameters are re-calculated at every
sample instance by assuming that the latest plantparameters estimates are the real parameters.

6Direct 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).

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Outline
1. Review lecture 16: Pole-placement, trackingcontrol and deterministic disturbancerejection for ARMA models.

2. Formulate the plant’s Parameter Adaptation
Algorithm (PAA).
3. Implement an indirect adaptive controller,
using the certainty equivalence principle.
4. For plants with minimum phase zeros, wewill simplify the indirect adaptive controller.

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Deterministic SISO ARMA models
SISO ARMA model

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

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10Deterministic SISO ARMA models
We factor the zero polynomial as:

whereis anti-Schurhas the zeros that we
do not want to cancel

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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

12Control 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
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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Control Law• Feedback and feedforward actions:reference output
u (k)

-1

T( 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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Feedback Controller
Diophantine equation: Obtain polynomials
that satisfy:

Plant poles
Close loop
Unstable plant zeros
poles minus
Disturbance annihilating polynomial
cancelled zeros

Feedback Controller

16

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Zero-phase error feedforward

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Outline
1. Review lecture 16: Pole-placement, tracking
control and deterministic disturbancerejection for ARMA models.

2. Formulate the plant’s Parameter Adaptation
Algorithm (PAA).
3. Implement an indirect adaptive controller,
using the certainty equivalence principle.
4. For plants with minimum phase zeros, wewill simplify the indirect adaptive controller.

19

Certainty Equivalence Principle
Use pole-placement, tracking control and deterministic
disturbance rejection controller synthesis methodology.

1. Estimate plant parameters using RLS PAA:
•Polynomial estimates:

2. Controller polynomials
Feedforward compensatorare computed using

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Parameter Adaptation Algorithm (PAA)
1. Use a series-parallel RLS algorithm to estimate plantparameters.
2. Pre-filter input u(k) and output y(k) using thedisturbance annihilating polynomial, to prevent
parameter biasing.
3. Use “parameter projection” to prevent unboundedcontrol input

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PAA: sequence pre-filtering
Plant dynamics:

Disturbance:

Filtered input and output sequences:

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PAA: sequence pre-filtering
Multiply plant dynamics by annihilating polynomial:

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PAA: series parallel RLS
Filtered plant dynamics

Can be written as

Where:

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PAA: parameter projection
Assume that we know:
1. Minimum magnitude of DC gain of

2. Sign and minimum value of leading coefficient of25

Series parallel RL with projection
PAA:

A-priori parameter estimate (prior to projection)

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Series parallel RL with projection
PAA: ProjectionReplacesmall.

byif it becomes too

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Outline
1. Review lecture 16: Pole-placement, tracking
control and deterministic disturbancerejection for ARMA models.

2. Formulate the plant’s Parameter Adaptation
Algorithm (PAA).
3. Implement an indirect adaptive controller,
using the certainty equivalence principle.
4. For plants with minimum phase zeros, wewill simplify the indirect adaptive controller.

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Indirect Adaptive Controller
After each PAA iteration:1) Update2) Factorize

polynomials:

polynomial:: has constant coefficient 1

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Indirect Adaptive Controller
3) Calculate controller polynomials:

by solving the Diophantine equation:

Plant parameter polynomial estimates are usedinstead of actual polynomials

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Indirect Adaptive Controller
4) Calculate feedforward filter:

Where:

Replacebyif it becomes too small.

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Indirect Adaptive Controller5) Calculate polynomial:

Notice thatandeach have constant coefficient 1Thus,has constant coefficient32

Indirect Adaptive Controller
6) Adaptive control law is given by:

d(k)
u (k)

-1

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

S( q )

u(k) +

q-d B(q-1)-1

A( q )y(k)33Outline
1. Review lecture 16: Pole-placement, tracking
control and deterministic disturbancerejection for ARMA models.

2. Formulate the plant’s Parameter Adaptation
Algorithm (PAA).
3. Implement an indirect adaptive controller,
using the certainty equivalence principle.
4. For plants with minimum phase zeros, wewill simplify the indirect adaptive controller.

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Indirect Adaptive Controller with Stable Zeros
After each PAA iteration:1) Update•(no need to factorize

polynomials:)35

Indirect Adaptive Controller with Stable Zeros2)

Calculate controller polynomials:

by solving the Diophantine equation:

Plant parameter polynomial estimates are usedinstead of actual polynomials

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Indirect Adaptive Controller with Stable Zeros•

Feedforward filter

is constant and known

Thus, there is no need to update it at every sample step.

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Indirect Adaptive Controller with Stable Zeros
3) Calculate polynomial:

Notice that bothandhave constant coefficient 1Thus,

This coefficient is always

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Indirect Adaptive Controller with Stable Zeros
4)

Adaptive control law is given by:

d(k)
u (k)

-1

B (q )

r(k) +-

u(k) +1

^-1

^

-1

R( q )

S( q )

q-d B(q-1)-1

A( q )

y(k)