Robust Pole Assignment Using Interval Polynomials

Robust Pole Assignment Using Interval Polynomials

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ROBUST POLE ASSIGNMENT USING INTERV AL POLYNOMIALS Y. ·'· /),/}(I/III/I·III

c.

Soh*, R.

J.

11/ FI,.,'Ir;ml 11//11

Evans* and I. R. Petersen** (,'olll/JIIII'I' FlIg;III',T;lIg. ( 'II ;""n;l\ of .\" ''' '(1/111" .

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Abstract . TIlis paper pre se nt s the app lication of new interval pol~nomial r es ults to ~e desi gn of pole ass i gnment controllers. The new theo rv allows co ntroll ers to make us e of freedom in the placement of closed-loop poles to ac hi e ve increased robu s tness against plant parameter uncertainties. Kevwords.

Po l e as s ignment, Robust control, Interval polynomials.

INTERVAL POLYNOMIAL THEORY

INTRODUCTION

Suppose we have a set of real and complex roots {-PI' -P2' ... -Pk' -0 1 ± jw 1 , .. . -am ± jw } and m we write each pair of complex roots -oi ± jW in i second order form, i.e .

Pole-placement is now becoming an accepted technique for controller design (Astrom and Wi t tenmark 1984, Goodwin and Sin 1984) . There remains however some difficulty with the robustness of controllers designed using this method. The problem arises when selecting the desired closed-loop poles. It is well known that in order to design a robust controller the choice of closed-loop poles depends critically on the plant transfer function (Horowitz 1963). An arbi trary choice of stable closed-loop poles can lead to a very poor controller design for certain plants. Some assistance in the choice of desired closed-loop poles is given in Astrom and Wi t tenmark (1984) and As trom (1980) using plant knowledge and properties of the known dis turbances . However the robustness issues are still hidden. making good pole-placement design difficul t.

Then the corresponding polynomial in power form is f(s} +

+PIP2'

'Pk{o;+w~}

. .. (o;+w;)

where n = k + 2m. However if the real roots are intervals, i.e.

In this paper we present a new approach to pole assignment. Firstly we represent the desired pole locations by intervals. This provides considerable flexibility because now there are many controllers which will place the closed-loop poles within these desired intervals thus an optimization procedure over this set of acceptable controllers is possible. For example it is possible to find the controller with minimum power requirements which still achieves satisfactory control (i.e. places the closed-loop poles within the desired intervals). Secondly, we represent the plant polynomials as interval polynomials then search for a controller which will place the closed-loop poles wi thin their desired intervals for any choice of plant coefficients wi thin the plant interval polynomials.

+

Pi'

i = {1, 2,

(2)

... k}

what is the corresponding set of polynomials? If we define f:"

:::

[PI' P2"' " f:"

:::

Pk'

[p~, P2'"'' Pk'

0

1 " " , am' w1 ""

0

wm)

1 " " , am' w1 , .. ·wm)'

+

:::

then the set of root locations is described by the region

In order to proceed with the above problems it is necessary to establish relationships between the interval roots and the corresponding coefficients of the polynomial in power form. Several new results concerning this representation problem have recently been proven (Soh, Evans and Petersen 1985). Some of these results will be summarized in Section 2 of this paper since they are required for the pole placement design algori thms which follow in Section 3 and 4 .

R

where

:::+ ~:::

implies

v; ~ vtVi

This defines a degenerate hypercube in

211

mn

Y. C. Soh. R. J. [\'
212

An

th

n

order

polynomial

+

+ a .

...

in

power

form

S

is

n

such that

mapping from ~

where

there exist

PI' P2 . .. Pn

Hence we can define a for all

and

The

coeff icients

are then required to lie in the

a.

1

region !J. HR)

S

The problem is thus to characterise the region

S

in ~n. For the special case of only real roots, i.e. rn=<), the following theorem states that S is a closed convex polytope . Theorem 1 : + + + If Pn ~ Pn > Pn-1 ~ Pn-1 ~ P2 > PI ~ PI ' then the region S is characterized by the following linear inequalities:

In the following theorem, we will show that this region S is a closed convex polytope. We now introduce some notation which will be used to characterize this polytope . Let the full (n+m)x1

rank

vector 1

o

/3 1

1

/3 2

/3 1

(n+m)xn

b

matrix

and the

B

be defined by

o

n

I (_p~)i

(_l)k

a

~

. n-l

0,

a

B =

o

/3 1

i=O

b

=

(6)

(3) n

(_1)k+1

I (_p~)i a

o

.

n-l

o

i=O for

o

k = I , 2 , .. . , n

Or in more compact notation P a

~

o

/3 m

We will also use the matrix (4)



where

{~

S



(~

where

P

P

and the vector

~

defined previously . Theorem 2 :

R(~)

~m+n : !:(~'!D-1~,~ ~ E + !:(~,~-1~,!? - !?) € R(~)}

denotes the range space of the matrix

B. Proof :

(Soh, Evans and Petersen 1985) .

One can always find a matrix

N such that

R(~)

O} .

constraint Proof:

(Soh, Evans and Petersen 1985)

Let the desired set of fixed roots be defined to be the roots of the polynomial

m

+ /3 1 s

_1

+ . .. + /3m-1 s +

/3

m

R(~

the

can be replaced by the

equality constraint

We have shown that S is a polytope when the interval roots are real and non-overlapping . We now consider the case in which there is a set of fixed roots (which may be complex) in addition to the real and non-overlapping interval roots .

s

a - b €

Hence

N a

Furthermore, the above result can be extended to non-monic polynomials [Soh 1987]. Hence, as a generalisation, we can denote the set of

(5)

Furthermore, let

Nb

S

n+1

{~ € ~

for some matrix

9.

:

9.

~ ~

(7)

g)

and vector

~.

POLE ASSIGNMENT PROBLEM

define the desired real interval roots as above. Hence, the desired region in the coefficient space is

The results are presented for a single-input single-output discrete time system. However, the continuous time resul ts can be developed in a similar fashion .

Robust Pole .-\ssiglllllellt lIsillg Illten'al l'olnlOlllials

Consider a plant polynomial in the backward shift operator

q

-I

as

cost

(8)

We

will

assume

that

the

plant

is

causal

and

where both L(q -I) and P(q -I) are of order m-I, the 2m-1 closed-loop system polynomial is given by

Thus se t t i ng D(q-I) as the (2m_l)th order polynomial for the closed-loop poles we can, under the assumption that A(q-I) and B(q-I) are copr ime, solve for the unique polynomials

L(q-I)

and p(q-I) (Goodwin and Sin 1984). Now it is straight-forward to rewrite the above equation as M x d where M is a 2m x 2m matrix of plant parameters,

~

is a

controller parameters and

d

2m x is a

vector of 2m x I vector

of the coefficients of the polynomial representing the desired pole locations, i.e. a

o

o

a

b

o

o b

o

I!

o

a

b

m

o

a

min 11~112 x € S

(15)

The

I!I

- norm approach in (14) leads to a linear

programming solution because of the linear constraint (13) . Thus the above procedure allows the designer to specify the desired closed-loop polynomial as interval regions in the s- or z-plane and then find the "best" controller which will ensure that the resul ting closed-loop poles lie wi thin these desired regions. One added advantage of the above approach is that there is no need for the coprime assumption. ROBUST POLE PLACEMENT The approach presented in section 3 allows flexibility in the pl acement of closed-loop poles but it does not directly account for plant parameter variations. In this section, we pre sent a design procedure that will ensure closed-loop stability despite hard errors in the plant parameters. Let a

A(q-I)

Mx

equat ion o

and

[~' ,~']'

B(q-I) respectively. Define as the plant vector. Then the d

can be replaced by

I!

o

o

m

o

m

o

o

I!

o

b

o

o I!

a

d

o

X

o

Xv

where

o b

many

(14)

m

o

m

are

x

of

o

b

There

min II~III x € S x

~

o

Sx'

be vectors that represent polynomial coefficients

o

d

a

over

and

controllable and both B(q-l) and A(q-I) are of order m. Now for a controller of the form

(9)

functional

considerations in the choice of the cost however some typical examples are

o

( 11 )

m

o The case we consider here is when the closed-loop poles are specified as intervals as described in section 2 . Thus from the results in section 2, we may let the desired closed-loop vector d be characterized by the polytope

S

(12) for some matrix Since ~

d

which

g

!~,

map

the

and vector

Suppose that the set of all possible plant vectors is defined by the polytope

where

v ..

-1

i=l,

2.

...

k

are vertices of

the set of controller vectors poles

into

the

S

(17)

x

Thus any

desired set is given by

time-invariant controller

will take any and every plant vector S

(13)

x

To select one of these controllers



we need a

maps it into any closed-loop vector The set

the

robust

convex polytope. Then the set of all pole-placement controllers is given by

51..

closed-loop

(16)

v

Sx

~

v € ~ €

in S v

Sd'

as defined in (17) is not explicit.

Y. C. Soh, R. J. £vans and I. Petersoll

214

In

the

following derivations,

we

show

S

that

and let

d € D be given .

can be reformulated as an explicit set. For

a

fixed

controller ~,

corresponding

k ~

vector

(and

{d € ffi2m : d = X v : v € S } -v

D

Lemma 4.1: Proof:

Let

Xv

d

(18)

i.e.

d

M. x

which are contained in

1

-

DC S

Hence

be given.

convex combination of

-

therefore

S . V

v € Sv'

S

v

d € S . v

v

Thus from Lemma 4.2. we conclude that the set of all robust controller is defined by

i.e .

S

= '\L 'X.v . 1-1

{~€

x

ffi2m : 9.

~i

x

s: s.:

i=1.2 . ... k} (20)

i=l

i=l Hence to find the "best" robust controller, we can solve the linear programming problem

k

Therefore

~ L\~i

d =

min 11~111

i=l s.t.

k

L\ ~ ~i i .e .

€ conv {~~i:

d

Conversely. suppose is given.

. . . k}

i=l,2,

~ € conv{~ ~i'

1=1,2 . ... k}

Then we can write

k

L'Xi~ ~i

s:

(21)

S.

for

i=l,2,

... k

reformulated as an explicit set of non-linear ine qualities [Soh 1987]. The number of constraints in the non-linear programming approach is greatly reduced . It is clear that the set

1.\~OV1



9. ~i ~

The number of constraints in (21) may be very large . However wi th the modern linear prog ramming techniques. it is computationally feasible. Moreover for the case of interval plant coefficients. the set Sx in (12) can be

i=l

d

the points

However.

Then we can write

k

v

is a

vvv

D

d €

~ 0

1. \

i=l

is a c onvex set.

conv {~~i' i = 1 . 2 . . .. k )

where

L\ ~i ~:

d

we define the corresponding set

of all possible closed-loop poles as

D

From above we can write

x

Sx

as defined may be

empty . To combat this difficulty we have devised an alternative algorithm which finds the controller which allows the largest plant va riations while ensuring that the closed-loop poles lie within their intervals [Soh 1987] .

1=1 k

~ L\ ~i i=l

NUMERICAL EXAMPLES

Xv

where

v € S

v

Consider the plant

i.e . d € D

Hence the result

(1-Z-l)(l-3 Z- 1 )

vvv For each plant defined by the vector be the corresponding matrix .

Xv . - -1

M.

-1

~i'

let

M.

-1

b

and Hence

D

conv {~i~'

for all Proof: ~i x €

D

Q

Sd

i=1.2 . . . . k Suppose that

D V i.

9.~i x

k}

i=1.2.

Now suppose that the region locations is defined by (12). Lemma 4 . 2:

of

for

~

So for

(19)

desired

pole



bo o

[0. 0 . 9]

for all possible values of

Using the results in previous sections.

el'

Po

and

PI

must

satisfy

the

and D

C

S . v

i=1.2, ... k

i=l,2 . . . . k

Then

since 1.0

3.0

3.0

-4 . 0

2.0-b

-6.9

4 .9

2 . 0+0.9b

o

3 .0

o

o

-3 . 0

o

-b

o

b

-2 . 0

o

-b

0

o 0

-2 . 0 2.0

eo'

following

eo = 1

if and only if

Conversely, suppose 9. ~i ~

1 . 04]

inequalities :

we must have

s: So

€ [0 . 96.

o

We are interested to design a controller such that the closed-loop poles are PI = O. P2 € [-1.0. 0]

V.1

X -

where

Then

1.0

o 0.81

o o

Robust Pole .\ssignme nt using Interyal Polynomials

From which we obtain

eo

and the admissible region for shown in satisfying

and

is as

Figure 1. Using any controller the above condi tion. e.g . eo 1. Po = 11.0. PI = -11.1 will give a

stable closed-loop system wi th the desired pole locations.
13 . 0

11.0

(-6.BI8, 10 .221)

"

9.0 -8.0

-9.0 fijl;ure i:

-7.0

Ad=olsslble rel;lon f o r

tl

a "d

-6 .0

1'0

REFERENCES Astrom. K. (1 980). "Robustness of a Design Method Based on Assignment of Poles and Zeroes", IEEE Tra ns . Auto . Contr . , Vol AC-25 , No . 3 , pp 588-59 1. Astrom, K. and B. Wittenmark (1984) . Computer Controlled Systems. Prentice-Hall . Adaptive Goodwin. G. C. and K.S. Sin (1984) . Fi 1 tering Pr ediction a nd Control, Prentice-Hall. Horowi tz, I. (1963). SYnthesis of Feedback Control Systems , Wiley. Soh , Y.C . (1987) . Robust Estimation and Control PhD Thesis, University of Newcastle . In preparation . Soh, Y. C., Evans R.J . , Petersen I . R. (1985) "Characterization of a Family of Polynomials with Interval Roots", Technical Report EE8543. University of Newcastle. Newcastle , N.S . W. Australia .

215