robust design: pole placement - laas
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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
2
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
−1.5
−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
0.3
0.4
0.5
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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
−1.5
−1
−0.5
0
0.5
1
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
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real
Imag
Robust closed-loop root-locus for random admissibleellipsoidal uncertainty
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