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1. Reinschke, K. J.
Arbitrary Pole Assignability by Static Output Feedback under Structural
Contraints
Using a few elementary concepts from algebraic geometry such as multidimensional projective space, quadric hyper-
surface and its tangential variety, the known problem of arbitrary pole placement is transformed into a system of
well-structured (partly non-linear) algebraic equations. Necessary and sufficient solvability conditions are derived.
Finally, it is outlined how to calculate admissible output feedback matrices which ensure the desired pole assignment.
1. Problem formulation
The pole assignment problem by output feedback has been investigated for more than a quarter of a century, see [1-
8]. This contribution takes into account additional structural constraints of the feedback law and does not require
deeper knowledge of algebraic geometry. Consider a linear time-invariant plant model ẋ = Ax+Bu, y = Cx, where
x ∈ IRn
, u ∈ IRm
, y ∈ IRr
, and a controller u = Ky, where K has nDF freely adjustable real-valued entries whereas
the remaining m · r − nDF entries are structurally fixed at zero.
The problem of arbitrary pole assignment is to find admissible K such that the coefficients q1(K), . . . , qn(K) of
the closed-loop characteristic polynomial CLCP(s) = det(sIn − A − BKC) = sn
+
Pn
i=1 qi(K)sn−i
are equal to an
arbitrarily chosen vector qd
∈ IRn
. Because of
det(sIn − A − BKC) = det


Ir −C 0
0 sIn − A −B
K 0 Im

 ,
one obtains by Laplace expansion of this big determinant CLCP(s) =
PN
ν=0 pν(s)zν, where the z0, . . . , zN symbolize
the N + 1 non-vanishing m × m minors of the m × (r + n + m) matrix (K 0 Im), and the polynomials pν(s) are the
complementary (r + n) × (r + n) minors. The preceding polynomial equation can be re-written, with prespect to
the different powers of s, as a matrix equation
zN





1
q1
.
.
.
qn





=





0 0 · · · 0 1
p10 p11 · · · p1,N−1 p1,N
.
.
.
.
.
.
...
.
.
.
.
.
.
pn0 pn1 · · · pn,N−1 pn,N










z0
z1
.
.
.
zN





=: M z. (1)
The minors z0, . . . , zN may be interpreted as homogeneous coordinates in the N-dimensional projective space IRIPN
.
These coordinates are subject to N − nDF =: ℓ non-zero homogeneous polynomial equations of total degree 2:
fλ(z) = 0 (λ = 1, . . . , ℓ), in short, f(z) = 0. In terms of algebraic geometry, the variety V(f) = {z ∈ IRIPN
: f(z) = 0}
is called a quadric hypersurface or simply quadric. The tangential variety to V(f) at z = z̃ is denoted by Tz̃(V(f))),
i.e., Tz̃(V(f))) = {z ∈ IRIPN
: (∂f/∂z)|z=z̃ = f∗z̃ · z = 0}. It is not difficult to verify that f∗z · z = 2f(z) and
f∗z̃ · z = f∗z · z̃. After these preliminaries, the problem under consideration may be formulated like this: For the
desired qd
∈ IRn
find real solutions of the (non-linear) system of algebraic equations
zN


1
qd
0

 =
µ
M
f∗z
¶
z. (2)
E x a m p l e 1. Let n = 5, m = 2, r = 4, K =
¡k11 k12 k13 0
0 k22 k23 k24
¢
. Obviously, nDF = 6. We enumerate the
minors zν according to the selected columns of (K , I2) and obtain z0 = (1, 2) = k11k22, z1 = (1, 3) = k11k23,
z2 = (1, 4) = k11k24, z3 = (1, 6) = k11, z4 = (2, 3) = k12k23 − k13k22, z5 = (2, 4) = k12k24, z6 = (2, 5) = −k12,
z7 = (2, 6) = k12, z8 = (3, 4) = k13k24, z9 = (3, 5) = −k23, z10 = (3, 6) = k13, z11 = (4, 5) = −k24, z12 = (5, 6) = 1.
Hence, N = 12, and the ℓ = N −nDF = 12−6 = 6 homogeneous equations of degree 2 read f1(z) = z0z12 +z3z6 = 0,
f2(z) = z1z12 + z3z9 = 0, f3(z) = z2z12 + z3z11 = 0, f4(z) = z4z12 − z6z10 + z7z9 = 0, f5(z) = z5z12 + z7z11 = 0,
f6(z) = z8z12 + z10z11 = 0.
PAMM · Proc. Appl. Math. Mech. 2, 535–536 (2003) / DOI 10.1002/pamm.200310249

2. 2. Solvability conditions
The system of equations (2) to be solved consists of n + 1 + ℓ equations with N + 1 = nDF + 1 + ℓ unknowns
and n arbitrarily chosen real numbers qd
1, . . . , qd
n on the left-hand side. Its generic solvability implies two obviously
necessary conditions: nDF ≥ n, rank M = n + 1. As for sufficient solvability conditions, we confine ourselves here
to the case nDF > n: The first step is to look for a solution z̃ of (2) with z̃N = 0, i.e.,
z̃ ∈ S := {z ∈ IRIPN
: Mz = 0, f(z) = 0}. (3)
Provided the set S is not empty, rank M = n + 1, and rank
¡M
f∗z̃
¢
= n + 1 + k with 0 ≤ k ≤ ℓ, then the assignability
problem is solvable.
3. Determination of solutions
If nDF > n and the sufficient solvability conditions are satisfied, then the solutions of the assignability problem can
be obtained as follows: First, choose z̃ ∈ S defined by (3) and select k rows of f∗z̃ as well as n + 1 + k columns of the
augmented coefficient matrix
¡M
f∗z̃
¢
such that the resulting (n + 1 + k) × (n + 1 + k) matrix Mreg becomes regular.
Then, for any qd
∈ IRn
and any zN ∈ IR+, the system of linear equations Mreg zreg = zN (1 qd
1 · · · qd
n, 0 · · · 0)T
has
a unique solution zreg ∈ IRn+1+k
. After completing zreg with N − n − k zeros, we get ẑ ∈ IRIPN
, and z̄ = z̃ + ẑ
satisfies the equation
¡M
f∗z̃
¢
z̄ =
¡M
f∗z̃
¢
ẑ = Mreg zreg = z̄N (1 qd
1 · · · qd
n, 0 · · · 0)T
. In geometric terms, the point z̄ ∈ IRIPn
satisfies (1) and belongs to the tangential variety Tz̃(V(f))). For zN = z̄N sufficiently small, the point z̄ is very close
to the quadric V(f) since f(z̄) = f(z̃) + f∗z̃ẑ + f(ẑ) = 0 + 0 + f(ẑ) = O(z2
N ). Next, determine a point z0
∈ V(f) that
is “adjacent” to z̄ ∈ Tz̃(V(f))). To do this, we define z0
ν = z̄ν for nDF + 1 components and calculate the remaining ℓ
components taking into account the constraints f(z0
) = 0. As for the example, we obtain z0
12 = z̄12 = z12, z0
ν = z̄ν
for ν = 3, 6, 7, 9, 10, and 11, whereas z0
0z12 = −z0
3z0
6 = −z̄3z̄6, z0
1z12 = −z0
3z0
9 = −z̄3z̄9, etc.
Generally, there holds M z0
= zN (1 qd
1 · · · qd
n)T
+ O(z2
N ). Consequently, for zN sufficiently small, the matrix
equation (1) can be satiesfied with any desired numerical accuracy. In other words, the point z0
∈ V(f) corresponds
to an admissible feedback matrix K0
such that det(sIn −A−BK0
C) = sn
+
Pn
i=1(qd
i +δi)sn−i
with δi = O(zN ). For
the example system, we get K0
= z−1
12
¡z0
3 z0
7 z0
10 0
0 −z0
6 −z0
9 −z0
11
¢
. If K0
is not (yet) good enough, then a Newton-type iterative
method can be used to generate a sequence K0
, K1
, K2
, . . . of admissible K such that det(sIn − A − BK∞
C) =
sn
+
Pn
i=1 qd
i sn−i
. To see this, we re-write (1) as



q1
.
.
.
qn


 =



p10 p11 · · · p1N
.
.
.
.
.
.
...
.
.
.
pn0 pn1 · · · pnN


 z/zN =: P g(K).
If n non-vanishing entries of K are allowed to vary, then the Jacobian ∂g/∂K and the matrix product P∂g/∂K be-
come n×(N+1) and n×n matrices, respectively. The truncated Taylor expansion qd
= P(g(Kj
)+(∂g/∂K)|j
K(Kj+1
−
Kj
)) =: qj
+ P(∂g/∂K)|Kj (Kj+1
− Kj
) yields Kj+1
= Kj
+ (P(∂g/∂K)|Kj )−1
(qd
− qj
) for j = 0, 1, 2, . . . Finally,
it should be mentioned that the set of all admissible K solving the assignability problem forms a smooth manifold
of dimension nDF − n in the nDF -dimensional affine space. A procedure how to compute this smooth manifold
numerically can be found in [6, Sect. 33.4].
4. References
1 Wonham, W.M. : On pole assignment in multi-input controllable linear systems. IEEE Trans.AC-12(1967), 660-665.
2 Fallside, F.; Seraji, H.: Pole-shifting procedure for multivariable systems using output feedback. Proc. IEE 118(1971),
1648-1654.
3 Kimura, H.: Pole assignment by gain output feedback. IEEE Trans.AC-20(1975), 509-516.
4 Herrmann, R. ; Martin, C.F.: Applications of algebraic geometry to systems theory. IEEE Trans.AC-22(1977), 19-25.
5 Brockett, R.W.; Byrnes, C.I.: Multivariable Nyquist criteria, root loci and pole placement: a geometric viewpoint.
IEEE Trans.AC-26(1981), 271-284.
6 Reinschke, K. J.: Multivariable Control – A Graph-theoretic Approach. Springer, 1988.
7 Leventides, J.; Karcanias, N.: Global asymptotic linearization of the pole placement map. Automatica 31 (1995),
1303-1309.
8 Wang, X. A.: Grassmannian, central projection, and output feedback pole assignment. IEEE Trans.AC-41(1996), 786-796.
Prof. Dr. Kurt J. Reinschke, Institut für Regelungs- und Steuerungstheorie, TU Dresden, Mommsenstr. 13, 01062
Dresden, Germany, Email: kr@erss11.et.tu-dresden.de.
Section 15: Mathematical Methods of the Natural and Engineering Sciences 536