Page 1

Controllability and Observability of Uncertain Systems: A Robust Measure

Somayeh Sojoudi, Javad Lavaei and Amir G. Aghdam

Abstract—This paper deals with the class of polynomially

uncertain continuous-time linear time-invariant (LTI) systems

whose uncertainties belong to a semi-algebraic set. The ob-

jective is to determine the minimum of the smallest singular

value of the controllability or observability Gramian over the

uncertainty region. This provides a quantitative measure for the

robust controllability or observability degree of the system. To

this end, it is shown that the problem can be recast as a sum-of-

squares (SOS) problem. In the special case when the uncertainty

region is polytopic, the corresponding SOS formulation can be

simplified significantly. One can apply the proposed method to

any large-scale interconnected system to identify those inputs

and outputs that are more effective in controlling the system.

This enables the designer to simplify the control structure

by ignoring those inputs and outputs whose contribution to

the overall control operation is relatively weak. A numerical

example is presented to demonstrate the efficacy of the results.

I. INTRODUCTION

There has been a growing interest in recent years in

robust control of systems with parametric uncertainty [1],

[2], [3], [4], [5]. The dynamic behavior of this type of

systems is typically governed by a set of differential equa-

tions whose coefficients belong to fairly-known uncertainty

regions. Although there are several methods to capture the

uncertain nature of a real-world system (e.g., by modeling it

as a structured or unstructured uncertainty [6]), it turns out

that the most realistic means of describing uncertainty is to

parameterize it and then specify its domain of variation.

Robust stability is an important requirement in the control

of a system with parametric uncertainty. This problem has

been extensively studied in the case of linear time-invariant

(LTI) control systems with specific types of uncertainty

regions. for instance, sum-of-squares (SOS) relaxations are

numerically efficient techniques introduced in [2] and [3]

for checking the robust stability of polynomially uncertain

systems. Moreover, a necessary and sufficient condition is

proposed in [1] for the robust stability verification of this

class of uncertain systems, by solving a hierarchy of semi-

definite programming (SDP) problems.

On the other hand, real-world systems are often composed

of multiple interacting components, and hence possess so-

phisticated structures. Such systems are typically modeled

as large-scale interconnected systems, for which classical

control analysis and design techniques are usually inefficient.

This work has been supported by the Natural Sciences and Engineering

Research Council of Canada (NSERC) under grant RGPIN-262127-07.

Somayeh Sojoudi and Javad Lavaei are with the Department of Control

and Dynamical Systems, California Institute of Technology, USA (emails:

sojoudi@cds.caltech.edu, lavaei@cds.caltech.edu).

Amir G. Aghdam iswith the

ComputerEngineering,Concordia

dam@ece.concordia.ca).

Department

University,

ofElectrical

(email:

and

agh-Canada

Several results are reported in the literature for structurally

constrained control of large-scale systems in the contexts

of decentralized and overlapping control, to address the

shortcomings of the traditional control techniques [7], [8],

[9], [10], [11].

The concepts of controllability and observability were

introduced in the literature, and it was shown that they play

a key role in various feedback control analysis and design

problems such as model reduction, optimal control, state

estimation, etc. [6]. Several techniques are provided in the

literature to verify the controllability and/or observability

of a system. However, in many applications it is important

to know how much controllable or observable a system

is. Gramian matrices were introduced to address this issue

by providing a quantifying measure for controllability and

observability [6]. While these notions were originally intro-

duced for fixed known systems, they have been investigated

thoroughly in the past two decades for the case of uncertain

systems [12], [13], [14].

This work aims to measure the minimum of the smallest

singular value for the controllability and observability Grami-

ans of parametric systems, over a given uncertainty region.

Given a polynomially uncertain LTI system with uncertain

parameters defined on a semi-algebraic set, it is asserted

that the controllability (observability) Gramian is a rational

matrix in the corresponding variables. Since it is desired to

attain the minimum singular value of this matrix over the

uncertainty region, the rational structure of the matrix does

not allow to use the efficient techniques such as SOS tools.

To bypass this obstacle, it is shown that said rational matrix

can be replaced by a polynomial approximation satisfying

an important relation. An SOS formula is then obtained to

find the underlying infimum. The special case of polytopic

uncertainty regions is also investigated, due to its importance

in practice. An alternative approach is proposed for this

special case, with a substantially reduced computational

burden.

This work can be used to determine if a system which is

controllable for the nominal parameters, is also controllable

with a sufficiently safe margin in a practical environment,

where the parameters are subject to variation around the

nominal values. Another application of the current work is

in large-scale systems, which will be discussed thoroughly.

II. PRELIMINARIES AND PROBLEM FORMULATION

Consider an uncertain LTI system S(α) with the following

state-space representation:

˙ x(t) = A(α)x(t) + B(α)u(t)

y(t) = C(α)x(t) + D(α)u(t)

(1)

Page 2

where

• x(t) ∈ ℜnis the state, and u(t) ∈ ℜmand y(t) ∈ ℜr

are the input and output of the system, respectively.

• α = [α1,α2,...,αk] represents the uncertain param-

eters of the system, which are assumed to be fixed,

but unknown. By assumption, this uncertainty vector

belongs to a given semi-algebraic set D defined below:

D = {α ∈ ℜk|f1(α) ≥ 0,...,fz(α) ≥ 0}

where f1(α),...,fz(α) are given scalar polynomials.

• A(α),B(α),C(α) and D(α) are matrix polynomials

in the variable α.

Assume that the matrix A(α) is robustly Hurwitz over the re-

gion D for all α ∈ D (this condition is required to define the

infinite-horizon Gramians, and can be systematically checked

using a variation of the SOS method proposed in [1]).

The controllability and observability of the system can be

measured by the following parametric Gramian matrices,

respectively:

∫∞

Wo(α) =

0

for all α ∈ D. The matrices Wc(α) and Wo(α) can alterna-

tively be obtained by solving the following continuous-time

Lyapunov equations:

(2)

Wc(α) =

0

eA(α)tB(α)B(α)TeA(α)Ttdt

(3a)

∫∞

eA(α)TtC(α)TC(α)eA(α)tdt

(3b)

A(α)Wc(α) + Wc(α)A(α)T= −B(α)B(α)T

A(α)TWo(α) + Wo(α)A(α) = −C(α)TC(α)

Hence, one can write:

(4a)

(4b)

[I ⊗ A(α) + A(α) ⊗ I]vec{Wc(α)} =

vec{−B(α)B(α)T}

vec{−C(α)TC(α)}

(5a)

[I ⊗ A(α)T+ A(α)T⊗ I]vec{Wo(α)} =

where ⊗ denotes the Kronecker product, and vec{·} is an

operator which takes a matrix and converts it to a vector by

stacking its columns on top of one another. It can be inferred

from these equations that although A(α),B(α) and C(α)

are polynomial matrices, the solutions Wc(α) and Wo(α)

are symmetric rational matrices in α (recall that the inverse

of a polynomial matrix is a rational matrix).

Definition 1: The system S(α) is said to be robustly

controllable (observable), if it is controllable (observable)

for every α ∈ D.

It is well-known that the system S(α) is robustly con-

trollable (respectively, observable) if and only if Wc(α)

(respectively, Wo(α)) is positive definite for all α ∈ D [6].

It follows from a celebrated result (Proposition 4.5 in [6])

that the input energy required for controlling the system is,

roughly speaking, proportional to the inverse of the matrix

Wc(α), and more specifically, to the inverse of its smallest

singular value. A similar result holds for the observability

matrix (in a dual manner). This motivates the derivation

(5b)

of the minimum singular value of the matrices Wc(α) and

Wo(α) over the region D, which is central to this paper.

Derivation of the minimum singular value will be addressed

in the sequel for the matrix Wc(α) (evidently, the results

developed for Wc(α) hold for the matrix Wo(α) as well).

Notation 1: σ{M} denotes the minimum singular value

of the matrix M.

Notation 2: Given a vector β = [β1,β2,··· ,βk], define

β2to be equal to β2= [β2

The following mild assumption on the region D is essential

for the main development of this work.

Assumption 1: The set D is compact, and there exist SOS

scalar polynomials w0(α),w1(α),..., wz(α) such that all

points α satisfying the inequality:

1,β2

2,··· ,β2

k].

w0(α) + w1(α)f1(α) + ··· + wz(α)fz(α) ≥ 0

form a compact set.

Notice that checking Assumption 1 amounts to solving an

SOS problem, and that if this assumption does not hold, the

results of this paper will become only sufficient, rather than

both necessary and sufficient.

(6)

III. MAIN RESULTS

As stated in the preceding section, the matrix Wc(α)

satisfying the equality (4a) is a rational function, which

impedes the use of SOS techniques. Since Wc(α) can be

obtained from (3a), substituting the exponential matrices in

the integral with their truncated Taylor series would result in

a polynomial approximation of the Gramian matrix. How-

ever, the polynomial obtained using this simple technique

would not necessarily satisfy an important property, namely

inequality (11), which will be introduced later and is essential

in developing the main results of this paper. This is due to

the fact that the error of this truncation is sign-indefinite

in general; i.e., it is neither positive definite nor negative

definite. This obstacle will be overcome in the sequel.

One can adopt an approach similar to the one given in

the proof of Lemma 1 in [1] for discrete-time systems to

conclude that Wc(α) can be expressed asH(α)

and h(α) are positive definite matrix and positive definite

scalar polynomials (of known degrees), respectively, over the

region D. It follows from the positiveness of h(α) as well as

the compactness of D that there exist reals µ1and µ2such

that:

0 < µ1< h(α) < µ2,

h(α), where H(α)

∀α ∈ D

(7)

Definition 2: A sequence of matrices {Mi}∞

converge to a matrix M from below if M1≤ M2≤ M3≤

··· ≤ M and limi→∞∥Mi− M∥ = 0.

Definition 3: Define Pi(α) to be:

(

1

is said to

Pi(α) := Wc(α) ×

1 −

(

1 −h(α)

µ2

)2i)

,i = 1,2,...

(8)

Theorem 1: The following statements are true:

i) Pi(α) is a matrix polynomial.

Page 3

ii) Given α ∈ D, the sequence {Pi(α)}∞

Wc(α) from below. In particular,

(

µ2

1

converges to

1 −

(

1 −µ1

)2i)

min

α∈Dσ{Wc(α)} ≤ min

α∈Dσ{Pi(α)}

(9)

and

min

α∈Dσ{Pi(α)} ≤ min

for i = 1,2,....

iii) Pi(α) satisfies the following matrix inequality:

α∈Dσ{Wc(α)}

(10)

A(α)Pi(α) + Pi(α)A(α)T+ B(α)B(α)T≥ 0 (11)

for all α ∈ D.

Proof of Part (i): The proof of this part is an immediate

consequence of the fact that the term:

(

1 −

(

1 −h(α)

µ2

)2i)

(12)

is divisible by h(α) (the denominator of Wc(α)).

Proof of Part (ii): It is straightforward to conclude from

Definition 3 and the inequality (7) that:

P1(α) ≤ P2(α) ≤ P3(α) ≤ ···

(13)

and:

Pi(α) ≤ Wc(α) ≤

(

1 −

(

1 −µ1

µ2

)2i)−1

Pi(α)

(14)

for all α ∈ D. The proof of part (ii) follows directly from

(13) and (14).

Proof of Part (iii): One can write:

A(α)Pi(α) + Pi(α)A(α)T+ B(α)B(α)T=

(

µ2

×(A(α)Wc(α) + Wc(α)A(α)T)+ B(α)B(α)T

= B(α)B(α)T

µ2

1 −

(

1 −h(α)

)2i)

(

1 −h(α)

)2i

≥ 0

(15)

This completes the proof.

Theorem 1 shows that Wc(α) can be approximated by a

matrix polynomial satisfying a matrix inequality. Moreover,

implicit bounds on the smallest singular value of the control-

lability matrix are provided. It is to be noted that as pointed

out in [1], one can use the interpolation technique developed

in [16] to calculate h(α) (unlike H(α), whose calculation is

involved). This leads to the possibility of obtaining µ1and

µ2, and subsequently using (9) in order to roughly determine

a proper range of values for the degrees of the polynomials

which can approximate Wc(α) satisfactorily.

Let an optimization problem be introduced in the sequel.

Optimization 1: Given the system S(α) and the uncer-

tainty region D, maximize µ ∈ ℜ subject to the constraint

that there exist a symmetric matrix polynomial P(α) and

?

SOS matrix polynomials S0(α),...,Sz(α),˜S0(α),...,˜Sz(α)

such that:

A(α)P(α) + P(α)A(α)T+ B(α)B(α)T=

z

∑

P(α) = µIn+˜S0(α) +

S0(α) +

i=1

Si(α)fi(α)

(16a)

z

∑

i=1

˜Si(α)fi(α)

(16b)

where Inis the n × n identity matrix. Denote the solution

of this optimization problem with µ∗.

Theorem 2: The quantity minα∈Dσ{Wc(α)} is equal to

µ∗.

Proof: Let P(α) be a matrix polynomial for which

there exista real

µ

and

S0(α),...,Sz(α),˜S0(α),...,˜Sz(α) satisfying the equalities

given in (16). One can write:

SOSmatrix polynomials

A(α)P(α) + P(α)A(α)T+ B(α)B(α)T≥ 0

P(α) ≥ µIn

for all α ∈ D. It follows from (4a) and (17a) that P(α) ≤

Wc(α), ∀α ∈ D. This, together with the inequality (17b),

yields:

µ ≤ min

As a result:

µ∗≤ min

On the other hand, notice that:

(17a)

(17b)

α∈Dσ{Wc(α)}

(18)

α∈Dσ{Wc(α)}

(19)

Pj(α) > (min

α∈Dσ{Pj(α)} − ε)In,

∀α ∈ D

(20)

for any j ∈ N, ϵ ∈ ℜ+(note that ℜ+represents the

set of positive real numbers). Therefore, it can be in-

ferred from Theorem 1, Assumption 1 and the results of

[17] (Theorem 2) that there exist SOS matrix polynomials

S0(α),...,Sz(α),˜S0(α), ...,˜Sz(α) such that:

A(α)Pj(α) + Pj(α)A(α)T+ B(α)B(α)T=

z

∑

S0(α) +

i=1

Si(α)fi(α)

(21)

and:

Pj(α) = (min

α∈Dσ{Pj(α)} − ε)In+˜S0(α) +

z

∑

i=1

˜Si(α)fi(α)

(22)

This implies that:

min

α∈Dσ{Pj(α)} ≤ µ∗

(23)

The proof is completed by using part (ii) of Theorem 1, while

taking the inequalities (19) and (23) into consideration.

Remark 1: It can be observed that Optimization 1 is an

SOS optimization problem, which can be efficiently handled

using proper software such as YALMIP or SOSTOOLS [18],

[19]. Nevertheless, some upper bounds on the degrees of

the polynomials involved in the corresponding optimization

problem must be assumed, from which a lower bound on

?

Page 4

the solution can be found. In other words, this optimization

problem is tantamount to a hierarchy of SDP problems,

whose solutions asymptotically converge to the quantity of

interest, i.e. minα∈Dσ{Wc(α)}, from below.

Remark 2: Despite the fact that a high-order rational ma-

trix Wc(α) cannot, in general, be approximated satisfactorily

by a low-order polynomial matrix P(α), it will be illustrated

later in an example that a relatively low-order polynomial

typically works well, as it suffices that these two functions

be equal only at a critical point corresponding to the solution

of the optimization problem (as opposed to everywhere in the

region D).

A. Special case: A polytopic region

Although Theorem 2 provides a numerically tractable

method for measuring the robust controllability of a system,

the proposed optimization problem can be simplified signifi-

cantly for specific types of uncertainty. For instance, assume

that D is a polytopic region given by:

P = {α|α1+ ··· + αk= 1, α1,...,αk≥ 0}

This type of uncertainty region is of particular interest, due

to its important applications.

Assumption 2: Assume that A(α) and B(α) are homoge-

neous matrix polynomials, and let their degrees be denoted

by ζ1and ζ2, respectively.

Note that Assumption 2 holds automatically for polytopic

systems, with ζ1= ζ2= 1.

Theorem 3: The quantity minα∈PσWc(α) is equal to the

maximum value of µ for which there exists a homogeneous

matrix polynomial˜P(α) satisfying the following inequalities

for all α ∈ ℜk:

˜P(α2) ≥ 0,

[

+AT(α2)

(24)

(25a)

A(α2)

(˜P(α2) + µ(ααT)ζ3In

(˜P(α2) + µ(ααT)ζ3In

+ B(α2)B(α2)T(ααT)max(0,ζ3+ζ1−2ζ2)≥ 0

where ζ3denotes the degree of the polynomial˜P(α).

Proof: The proof follows by refining the proofs of The-

orems 1 and 2 in such a way that the homogeneity of the

system matrices as well as the polytopic structure of the

uncertainty region are both taken into account. To this end,

the homogenization technique given in [3] is adopted. First

of all, observe that the equation (5a) yields:

)

)

]

(ααT)max(0,2ζ2−ζ1−ζ3)

(25b)

vec{Wc(α)} = −[I ⊗ A(α) + A(α) ⊗ I]−1

× vec{B(α)B(α)T}

Note that I ⊗ A(α) + A(α) ⊗ I and B(α)B(α)Tare

both homogeneous polynomials. Therefore, one can conclude

from the above equation that the solution Wc(α) is a

rational matrix of the form

(26)

H(α)

h(α), where H(α) and h(α)

are homogenous matrix and scalar polynomials, respectively.

Now, let Pi(α) be defined as:

Pi(α) := Wc(α) ×

)ζ4

(

k

∑

i=1

αi

)2iζ4

−

(

k

∑

i=1

αi

−h(α)

µ2

2i

(27)

in lieu of the one introduced in (8), where ζ4 denotes the

degree of h(α). Note that Pi(α) defined above is identical

to the one defined in (8) over the region P, while its subtle

difference is that it is homogenous. In other words, an

approximation of Wc(α) by a homogenous polynomial is

desired here. It is easy to show that Theorem 1 holds for the

polynomial Pi(α) defined in (27). On the other hand, from

the property of polytopic set P given in (24), the inequality

(11) in Theorem 1 can be rewritten as:

(A(α)Pi(α) + Pi(α)A(α)T)

(

k

∑

i=1

αi

)max(0,2ζ2−ζ1−ζi

3)

+ B(α)B(α)T

(

k

∑

i=1

αi

)max(0,ζ1+ζi

3−2ζ2)

≥ 0, ∀α ∈ P

(28)

where ζi

3denotes the degree of Pi(α). One can write:

˜Pi(α) : = Pi(α) − min

α∈P{Pi(α)}

α∈P{Pi(α)}In≥ 0,

(

k

∑

i=1

αi

)ζi

∀α ∈ P

3

In

= Pi(α) − min

(29)

Hence:

˜Pi(α) ≥ 0,

∀α ∈ P

(30)

Notice now the following facts:

• The region P is identical to the following:

D = {α|α2

after changing the variable α to α2.

• Assume that M(α) is a homogeneous matrix polyno-

mial of degree γ. One can write:

1+ ··· + α2

k= 1}

(31)

M(β) = ∥β∥γ× M

(

β

∥β∥

)

(32)

Since

nonnegative over D if and only if it is nonnegative over

the whole space ℜk.

Thus, the proof is completed by pursuing the argument given

in the proof of Theorem 2, and using the inequalities (28) and

(30) after replacing α with α2and removing the constraint

α ∈ P (note that the left side of the inequality (28) is a

homogeneous polynomial).

The following optimization problem is defined for the

system S.

β

∥β∥

∈ D, one can conclude that M(α) is

?

Page 5

Optimization 2: Maximize µ subject to the constraint that

there exist a homogeneous matrix polynomial P(α) and SOS

matrix polynomials S1(α) and S2(α) such that:

P(α2) = S1(α),

[

(P(α2) + µ(ααT)ζ3In

+ B(α2)B(α2)T(ααT)max(0,ζ3+ζ1−2ζ2)= S2(α) (33b)

(33a)

A(α2)(P(α2) + µ(ααT)ζ3In

)+

)AT(α2)

]

(ααT)max(0,2ζ2−ζ1−ζ3)

where ζ3denotes the degree of the polynomial˜P(α). Denote

the solution of this optimization problem with ˜ µ∗.

The following lemma is required in order to delve into the

properties of the defined quantity ˜ µ∗.

Lemma 1: Let M(α) be a homogeneous matrix polyno-

mial with the property that M(α2) is positive definite for

every α ∈ ℜk\{0}. There exists a natural number c so that

(ααT)cM(α2) is SOS.

Proof: Since M(α) is positive definite over the polytope

P, it follows from Theorem 3 in [17] that there exists a

natural number c such that (α1+ α2+ ··· + αk)cM(α)

has only positive definite matrix coefficients. This implies

that the matrix coefficients of (ααT)cM(α2) are all positive

definite, and in addition, its monomials are squared terms.

As a result, (ααT)cM(α2) is SOS.

Theorem 4: The quantity minα∈P{Wc(α)} is equal to

˜ µ∗.

Proof: The proof will be performed in two steps. First,

observe that if (33) holds for some appropriate matrices

P(α), S1(α) and S2(α), then (25) is satisfied for˜P(α) =

P(α).

Conversely, assume that a matrix polynomial˜P(α) sat-

isfies the inequalities given in (25). It is easy to show

that the non-strict inequalities in (25) can be replaced by

strict inequalities. Now, one can apply Lemma 1 to these

inequalities to conclude that there exists a natural number c

such that if the expressions in the left side of the inequalities

(25a) and (25b) are multiplied by (ααT)c, then they become

SOS matrix polynomials. It is enough to choose P(α) as

˜P(α)(ααT)cin order for the inequalities given in (33) to

hold (for some appropriate matrices S1(α) and S2(α)). ?

Remark 3: Optimization 2 is obtained from Theorem 3 by

replacing the nonnegativity constraints with SOS conditions.

Although one may argue that this can potentially introduce

some conservatism due to the known gap between the set

of SOS polynomials and the set of nonnegative polynomials,

Theorem 4 shows that this is not the case. In other words, the

proposed replacement does not make the resultant conditions

conservative at all.

Note that Optimization 2 can be handled using proper

software as noted in Remark 1, provided some a priori

upper bounds on the degree of the relevant polynomials being

sought are set. The question arises as to whether choosing

higher degree polynomials would result in a tighter (less

conservative) solution. This question is addressed in the next

corollary.

?

Corrolary 1: The solution of Optimization 2 is a mono-

tone nondecreasing function with respect to ζ3 (the degree

of the polynomial˜P(α)).

Proof: The proof is a direct upshot of the fact that if˜P(α)

satisfies the constraints of Optimization 2 for some µ, then

˜P(α)(α1+···+αk) also satisfies them for the same µ, but

for some other suitable matrices S1(α) and S2(α).

?

B. Application to large-scale systems

In large-scale interconnected systems, typically the input

and output vectors u(t) and y(t) have several entries. Due

to the practical limitations, in this type of systems it is

desired to simplify the control structure and employ as few

communication links as possible (a communication link in

an interconnected system is referred to a data transmission

channel between a pair of local controllers). In other words,

it would be very useful to extract two subvectors ˜ u(t) and

˜ y(t) from the vectors u(t) and y(t) such that the system is

controllable from the reduced input ˜ u(t) and observable from

the reduced output ˜ y(t). Moreover, the controllability and

observability of the new configuration must be sufficiently

close to those of the original system. To find the proper

subvectors ˜ u(t) and ˜ y(t), one can consider all (possible)

desirable subsets of the inputs and outputs, and calculate

the minimum singular values of the Gramians for each

combination to assess the corresponding controllability and

observability degrees. One can then choose the optimal

subset of the input and output vectors, accordingly. This idea

is further clarified in the next section.

IV. NUMERICAL EXAMPLE

Consider an LTI uncertain fourth-order system with the

state-space matrices:

where α1and α2are the uncertain parameters of the system,

which belong to the polytope P = {(α1,α2)|α1+ α2 =

1, α1,α2≥ 0}.

Regard this system as an interconnected system with four

inputs. It is desired to determine which inputs contribute

weakly to the control of the system, and hence can be

A(α) =

−0.65

−0.05

−0.7

−1.5

−0.75

−1.35

0.8

−0.5

−1.2

0.35

0.25

−0.3

0.45

−1.7

0.55

−0.2

−0.6

−0.95

1.6

−0.1

1.4

−1.75

2.2

1.7

−0.3

−1.05

1.2

1.45

1.7

−0.7

1

0.25

−11.5

−0.4

0.2

−0.35

2.3

−1.1

−2.5

0.45

1.55

−1.8

−0.8

−0.3

0.75

0.7

−1.7

0.75

−1.85

−3.05

−2.85

0

−0.65

−3.25

0.15

−0.85

−1.3

−2.5

−1

0.85

0.65

−0.75

1.15

α1

α1

+

α2,

α2

B(α) =

+

(34)