GRADUATE COURSE ONPOLYNOMIAL METHODS FOR

ROBUST CONTROLPART III.1

ROBUST DESIGN:

POLE PLACEMENT

Didier HENRION

www.laas.fr/∼henrion

henrion@laas.fr

Dancer seated on a pink divanHenri de Toulouse-Lautrec (1864-1901)

October-November 2001

Summary

In the first part of the course we studied robuststability analysis

We saw that checking robust stability can beeasy (polynomial-time algorithms) or moredifficult (NP-hard problem), depending namelyon the uncertainty description

We focused on polytopic uncertainty:• Interval scalar polynomialsKharitonov’s theorem (ct only)• Polytope of scalar polynomials(affine polynomial families)Edge theorem• Interval matrix polynomials(multiaffine polynomial families)Mapping theorem• Polytopes of matrix polynomials(polynomic polynomial families)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5

−0.6

−0.4

−0.2

0

0.2

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1.2

Real

Imag

origin w=0

w=1

−5 −4 −3 −2 −1 0 1 2 3 4−1.5

−1

−0.5

0

0.5

1

1.5

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2.5

3

3.5

w=0

w=1.5

Real

Imag

−4 −2 0 2 4 6 8−3.5

−3

−2.5

−2

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−1

−0.5

0

0.5

1

1.5

convex hull p(j,Q)

Real

Imag

convex hull p(j,Q2)

convex hull p(j,Q3)

convex hull p(j,Q4)

Robust design

Only in very special cases robust analysistechniques can be extended to design(for example the 32 plant theorem)

In this third part of the course we will describerobust design algorithms based either on• polynomial pole placement equations, or• rational Youla-Kucera parametrization ofstabilizing controllers

We will see that robust design is a difficultproblem when looking for low-order controllers

1 2 3 4 5 6 7 8 9 10

−0.5

0

0.5

1

Power of 1/z

Coe

ffici

ent

Coefficients of 1/(1 + 0.5z−1) = r0 + r1z−1 + r2z−2 + · · ·

Nominal Pole placement

We consider the SISO feedback system

ba

x

y

+

−

Closed-loop transfer function

bx

ax+ by

In the absence of hidden modes (a and b coprime

polynomials), pole placement amounts to findingpolynomials x and y solving the Diophantineequation (from Diophantus of Alexandria 200-284)

ax+ by = c

where c is a given closed-loop characteristicpolynomial capturing the desired system poles

Polynomial parametrization of controllers

Let x′,y′ be the least degree solution w.r.t y′ of theDiophantine design equation

ax+ by = c

All controllers achieving pole placement are given by

y

x=y′ + at

x′ − btwhere t is a free polynomial parameter

ExampleWater tank with inflow-to-level transfer function

b

a=

1

s+ 1

To counteract level disturbances, a PI controller is soughtthat places closed-loop poles at −6 and −10, i.e. wehave to solve the Diophantine equation

(s+ 1)x+ y = c = (s+ 6)(s+ 10)

Controllersy

x=

45 + (s+ 1)t

s+ 15− tPI controller for t = 15

y0

x0= 15 +

60

s

Pole placement

ExampleConsider the rotary hydraulic test rig with identifiedsampled transfer functionb

a=

z−3(−0.0036 + 0.1718z−1 + 0.3029z−2 − 0.0438z−3 − 0.0775z−4)

1− 2.8805z−1 + 3.7827z−2 − 2.8269z−3 + 1.1785z−4 − 0.2116z−5

in the backward shift operator z−1

The desired closed-loop characteristic polynomial is

c = (1− 0.3z−1)(1− 0.4z−1) = 1− 0.7z−1 + 0.12z−2

Solving the Diophantine equation ax+ by = c yields theunstable controllery′

x′=

2.9023− 6.7682z−1 + 7.4670z−2 − 4.3287z−3 + 1.0169z−4

1 + 2.1805z−1 + 2.6182z−2 + · · · − 0.3725z−6

With the choicet = 1.2222−0.1952z−1−0.1310z−2+0.5663z−3+0.8805z−4+0.5677z−5

as a free polynomial parameter we obtain a stablecontroller of higher order

y′′

x′′=

1.6801− 3.0525z−1 + 2.4125z−2 + · · ·+ 0.1201z−10

1 + 2.1085z−1 + 2.6182z−2 + · · · − 0.0440z−12

Pole placement: numerical aspects

The polynomial Diophantine equation

ax+ by = c

is linear in unknowns x and y, and denotinga(s) = a0 + a1s+ · · ·+ adas

da

x(s) = x0 + x1s+ · · ·+ xdbxdb

etc..

we can identify powers of the indeterminate s to builda linear system of equations

a0 b0

a1. . . b1

. . .... a0

... b0ada a1 bdb b1

. . . ... . . . ...ada bdb

x0x1...xdxy0y1...ydy

=

c0c1...cdc

The above matrix is called Sylvester matrix, it has aspecial Toeplitz banded structure that can be exploitedwhen solving the equation

James J Sylvester Otto Toeplitz(1814 London - 1897 London) (1881 Breslau - 1940 Jerusalem)

Pole placement for MIMO systems

Pole placement can be performed similarly for

a plant left MFD

A−1(s)B(s)

with a controller right MFD

Y (s)X−1(s)

The Diophantine equation to be solved is now

over polynomial matrices

A(s)X(s) +B(s)Y (s) = C(s)

and right hand-side matrix C(s) captures

information on invariant polynomials and

eigenstructure

For example C(s) may contain H2 or H∞optimal dynamics (obtained with spectral

factorization)

Robust pole placement

Now assume that the plant transfer function

b(q)

a(q)

contains some uncertain parameter q

The problem of robust pole placement will thenconsists in finding a controller

y

xsuch that the uncertain closed-loop charact.polynomial

a(q)x+ b(q)y = c(q)

is robustly stable

How can we find x, y to ensure robust stabilityof c(q) for all admissible uncertainty q ?

Coefficients of c are linear in x and y, but arestability conditions also linear in c ?

Stability criteria

Given a continuous-time polynomial

p(s) = p0 + p1s+ · · ·+ pn−1sn−1 + pns

n

with pn > 0 we define its n× n Hurwitz matrix

H(p) =

pn−1 pn−3 0 0pn pn−2

... ...0 pn−1

. . . 0 00 pn p0 0... ... p1 00 0 p2 p0

Hurwitz stability criterion: Polynomial p(s) isstable iff all principal minors of H(p) are > 0

ExampleFor n = 4 the stability conditions are

p3 > 0p2p3 − p1p4 > 0p1p2p3 − p0p2

3 − p21p4 > 0

p0p1p2p3 − p20p

23 − p0p2

1p4 > 0

Highly non-linear inequalities

Similar criteria for discrete-time polynomials(Jury matrix) or any other stability region(sector, parabola etc..)

Non-convexity of stability domain

Main problem: the stability domain in the spaceof polynomial coefficients pi is non-convex ingeneral

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

UNSTABLE

STABLE

q1

q 2

Discrete-time stability domain in (q1, q2) plane for poly-

nomial p(z, q) = (−0.825 + 0.225q1 + 0.1q2) + (0.895 +

0.025q1 + 0.09q2)z + (−2.475 + 0.675q1 + 0.3q2)z2 + z3

How can we overcome the non-convexity of thestability conditions in the coefficient space ?

Handling non-convexity

Basically, we can pursue two approaches:

• we can approximate the non-convex stabilitydomain with a convex domain (segment, polytope,

sphere, ellipsoid, LMI)

• we can address the non-convexity with thehelp of non-convex optimization (global or local

optimization)

0

10

20

30

40

0

10

20

30

400

500

1000

1500

2000

2500

3000

Approximation of the stability domain

From the tools of robust stability analysis, wecan build around a nominally stable polynomial• stability segments (eigenvalue criterion)

• stability boxes (Kharitonov’s theorem)

• stability polytopes (Edge theorem)

There exists other results, such as Rantzer’s growthcondition: a polynomial g(s) is a convex direction iff

d

dωarg g(jω) ≤

∣∣∣∣sin 2 arg g(jω)

2ω

∣∣∣∣ , ω > 0

It means that given any stable f(s) such that f(s)+g(s)is stable then the whole segment [f(s), g(s)] is stable

Exampleg(s) = 24 + 14s− 13s2 − 2s3 + s4 is a growth direction

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.1

0

0.1

0.2

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d(arg)/dw

|sin(2arg)/2w|

w

Stability polytopes

Largest hyper-rectangle around a nominallystable polynomial

p(s) + rn∑i=0

[−εi, εi]si

obtained with the eigenvalue criterion appliedon the 4 Kharitonov polynomials

In general, there is no systematic way toobtain more general stability polytopes, namelybecause of computational complexity(no analytic formula for the volume of a polytope)

Well-known candidates:

• ct LHP: outer approximation(necessary stab cond)positive cone pi > 0

• dt unit disk: inner approximation(sufficient stab cond)diamond |p0|+ |p1|+ · · ·+ |pn−1| < 1

Stability polytopes

Necessary stab cond in dt: convex hull ofstability domain is a polytope whose n+ 1vertices are polynomials with roots +1 or -1

ExampleWhen n = 2: triangle with vertices

(z + 1)(z + 1) = 1 + 2z + z2

(z + 1)(z − 1) = −1 + z2

(z − 1)(z − 1) = 1− 2z + z2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

p0

p 1

In this simple case it is the exactstability domain

Stability polytopes

ExampleThird degree dt polynomial: two hyperplanes and anon-convex hyperbolic paraboloid with a saddle pointat p(z) = p0 + p1z + p2z2 + z3 = z(1 + z2)

(z + 1)(z + 1)(z + 1) = 1 + 3z + 3z2 + z3

(z + 1)(z + 1)(z − 1) = −1− z + z2 + z3

(z + 1)(z − 1)(z − 1) = 1− z − z2 + z3

(z − 1)(z − 1)(z − 1) = −1 + 3z − 3z2 + z3

−1

−0.5

0

0.5

1

−1

0

1

2

3

−3

−2

−1

0

1

2

3

p0

p1

p 2

The convex hull is made of four hyperplanes

Stability hyper-spheres

Largest hyper-sphere around a nominallystable polynomial

p(s) +n∑i=0

qisi, ‖q‖ ≤ r

has radius

rmax = min{|p0|, |pn|, infω>0

√(Re p(jω))2

1 + w4 + w8..+

(Im p(jω))2

w2 + w6 + ..}

Example(2 + q0) + (1.4 + q1)s+ (1.5 + q2)s2 + (1 + q3)s3, ‖q‖ ≤ r

1.15 1.152 1.154 1.156 1.158 1.16 1.162 1.164 1.166 1.168 1.171

1.2

1.4

1.6

1.8

2

2.2x 10

−3

ω

f(ω

)

rmax = min{2, 1, infω>0 f(w)} = 1.08 · 10−3

Stability ellipsoids

A weighted and rotated hyper-sphere isan ellipsoid

Using recently developed algorithms based onLMI optimization one can obtain ellipsoidalinner approximation of stability domains

E = {p : (p− p)TP (p− p) ≤ 1}where

p coef vector of polynomial p(s)p center of ellipsoidP positive definite matrix

Stability ellipsoids

Example

Discrete-time second degree polynomial

p(z) = p0 + p1z + z2

We solve the LMI problem and we obtain

P =

[1.5625 0

0 1.2501

]p =

[0.2000

0

]which describes an ellipse E inscribed in the

exact triangular stability domain S

−1.5 −1 −0.5 0 0.5 1 1.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

p0

p 1

Stability ellipsoids

ExampleDiscrete-time third degree polynomial

p(z) = p0 + p1z + p2z2 + z3

We solve the LMI problem and we obtain

P =

2.3378 0 0.53970 2.1368 0

0.5397 0 1.7552

x =

00.1235

0

which describes a convex ellipse E inscribed in the exactstability domain S delimited by the non-convexhyperbolic paraboloid

−1

0

1−1

01

23

−3

−2

−1

0

1

2

3

A

D

p1

Q

C

B

p0

p 2

Very simple scalar sufficient stability condition2.4166p2

0 + 2.2088p21 + 1.8143p2

2 − 0.5458p1 + 1.1158p0p2 ≤ 1

Volume of stability ellipsoid

In the discrete-time case, the well-known sufficientstability condition defines a diamondR = {p : |p0|+ |p1|+ · · ·+ |pn−1| < 1}

For different values of degree n, we compared volumesof exact stability domain S, ellipsoid E and diamond D

n = 2 n = 3 n = 4 n = 5Stability domain S 4.0000 5.3333 7.1111 7.5852

Ellipsoid E 2.2479 1.4677 0.7770 0.3171Diamond D 2.0000 1.3333 0.6667 0.2667

E is “larger” than D, yet very small wrt S

LMI stability regions

In the fourth and last part of this course, we will proposebetter LMI inner approximations of the stability domain

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−2

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−1

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1.5

2

p0

p 1

Robust pole placement

Once we have a convex approximation of the

stability region, we can perform design with

• linear programming (polytopes)

• quadratic programming (spheres, ellipsoids)

• semidefinite programming (LMIs)

Complexity of design algorithm increases

Conservatism of control law decreases

Application in robust design

First exampleMIMO plant with right MFD

B(s)A−1(s) =[b 00 0

] [s+ 1 0

0 s+ 1

]−1

with uncertainty in parameterb ∈ [0.5, 1.5]

We seek a proper first order controller

X−1(s)Y (s) =[s+ x1 x2

0 1

]−1

=[y1s+ y2 y3s+ y4

0 y5

]assigning robustly the closed-loop polynomial matrix

C(s) =[s2 + αs+ β δ(s)

0 s+ γ

]whose coefficients live in the polytope −14 1 0

16 −2 0−2 1 00 0 −10 0 1

[ αβγ

]>

−19656−42−14

These specifications amounts to assigning the poles within the disk

|s+ 8| < 6

−15 −10 −5 0

−6

−4

−2

0

2

4

6

Real

Imag

Application in robust design

First example (end)Equating powers of indeterminate s in the polynomialmatrix Diophantine equation

X(s)A(s) + Y (s)B(s) = C(s)

we obtain the design inequalities−13 −7 0.514 8 −1−1 −1 0.5−13 −21 1.514 24 −3−1 −3 1.5

[ x1y1y2

]>

−182

40−2−182

40−2

characterizing all parameters x1, y1 and y2 of admissiblerobust controllers

−5

0

5

10

15

20

−10

0

10

20

300

50

100

150

200

250

300

350

400

y1

x1

y2

Corresponding polytope with 9 vertices

Application in robust design

Second exampleWe consider two mixing tanks arranged in cascade withrecycle stream

P

Fa

Fa Fb

Fa+FbTa

Tb

The controller must be designed to maintain temp Tbof the second tank at desired set point by manipulatingpower P delivered by heater located in first tank - onlyavailable measurement is temp Tb

Nominal plant identification is carried out with least-squares method, resulting in 2nd order dt model

b(z)

a(z)=

a1 + a2z

a3 + a4z + z2

with nominal plant vector

a =[

0.0038 0.0028 0.2087 −1.1871]

Positive definite matrix Q characterizing the uncertaintyellipsoid {a : (a− a)′Q(a− a) ≤ 1} is readily available asa by-product of the identification technique

Application in robust design

Second example (end)We perform robust pole placement in the innerellipsoidal approximation

{p : (p− p)′P (p− p) ≤ 1}of the 3rd degree non-convex stability region, with

P =

2.3378 0 0.53970 2.1368 0

0.5397 0 1.7552

p =

00.1235

0

We solve a convex quadratic optimization problem toobtain a first-order controller

y(z)

x(z)=

0.3377 + 166.0z

0.6212 + z

robustly stabilizing the plant

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−0.8

−0.6

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Real

Imag

Robust closed-loop root-locus for random admissibleellipsoidal uncertainty