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arXiv:0901.3764v2 [math.OC] 3 Mar 2009

Electronic Journal of Differential Equations, Vol. 2009(2009), No. 37, pp. 1–32.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu

CONTROLLABILITY, OBSERVABILITY, REALIZABILITY, AND

STABILITY OF DYNAMIC LINEAR SYSTEMS

JOHN M. DAVIS, IAN A. GRAVAGNE, BILLY J. JACKSON, ROBERT J. MARKS II

Abstract. We develop a linear systems theory that coincides with the ex-

isting theories for continuous and discrete dynamical systems, but that also

extends to linear systems defined on nonuniform time scales. The approach

here is based on generalized Laplace transform methods (e.g. shifts and con-

volution) from the recent work [13]. We study controllability in terms of the

controllability Gramian and various rank conditions (including Kalman’s) in

both the time invariant and time varying settings and compare the results.

We explore observability in terms of both Gramian and rank conditions and

establish related realizability results. We conclude by applying this systems

theory to connect exponential and BIBO stability problems in this general

setting. Numerous examples are included to show the utility of these results.

1. Introduction

In this paper, our goal is to develop the foundation for a comprehensive linear

systems theory which not only coincides with the existing canonical systems theories

in the continuous and discrete cases, but also to extend those theories to dynamical

systems with nonuniform domains (e.g. the quantum time scale used in quantum

calculus [9]). We quickly see that the standard arguments on R and Z do not go

through when the graininess of the underlying time scale is not uniform, but we

overcome this obstacle by taking an approach rooted in recent generalized Laplace

transform methods [13, 19]. For those not familiar with the rapidly expanding area

of dynamic equations on time scales, excellent references are [6, 7].

We examine the foundational notions of controllability, observability, realiz-

ability, and stability commonly dealt with in linear systems and control theory

[3, 8, 22, 24]. Our focus here is how to generalize these concepts to the nonuni-

form domain setting while at the same time preserving and unifying the well-known

bodies of knowledge on these subjects in the continuous and discrete cases. This

generalized framework has already shown promising application to adaptive control

regimes [16, 17].

2000 Mathematics Subject Classification. 93B05, 93B07, 93B20, 93B55, 93D99.

Key words and phrases. Systems theory; time scale; controllability; observability; realizability;

Gramian; exponential stability; BIBO stability; generalized Laplace transform; convolution.

c ?2009 Texas State University - San Marcos.

Submitted January 23, 2009. Published March 3, 2009.

Supported by NSF Grants EHS#0410685 and CMMI#726996.

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2 J. M. DAVIS, I. A. GRAVAGNE, B. J. JACKSON, R. J. MARKS IIEJDE-2009/37

Throughout this work, we assume the following:

• T is a time scale that is unbounded above but with bounded graininess,

• A(t) ∈ Rn×n,B(t) ∈ Rn×m,C(t) ∈ Rp×n, and D(t) ∈ Rp×mare all rd-

continuous on T,

• all systems in question are regressive.

The third assumption implies that the matrix I + µ(t)A(t) is invertible, and so

on Z, the transition matrix will always be invertible. We are therefore justified in

talking about controllability rather than reachability which is common [8, 14, 25]

since the transition matrix in general need not be invertible for T = Z.

In the following sections, we begin with the time varying case, and then proceed

to treat the time invariant case. We will get stronger (necessary and sufficient)

results in the more restrictive time invariant setting relative to the time varying

case (sufficient conditions). Although some of the statements contained in this work

can be found elsewhere [4, 5, 15], in each of these cases proofs are either missing,

are restricted to time invariant systems, or are believed to be in error [15] when

T has nonuniform graininess. Moreover—and very importantly—the methods used

here are rooted in Laplace transform techniques (shifts and convolution), and thus

are fundamentally different than the approaches taken elsewhere in the literature.

This tack overcomes the subtle problems that arise in the arguments found in [15]

when the graininess is nonconstant.

2. Controllability

2.1. Time Varying Case. In linear systems theory, we say that a system is con-

trollable provided the solution of the relevant dynamical system (discrete, continu-

ous, or hybrid) can be driven to a specified final state in finite time. We make this

precise now.

Definition 2.1. Let A(t) ∈ Rn×n, B(t) ∈ Rn×m, C(t) ∈ Rp×n, and D(t) ∈ Rp×m

all be rd-continuous matrix functions defined on T, with p,m ≤ n. The regressive

linear system

x∆(t) = A(t)x(t) + B(t)u(t),x(t0) = x0,

y(t) = C(t)x(t) + D(t)u(t),

(2.1)

is controllable on [t0,tf] if given any initial state x0 there exists a rd-continuous

input u(t) such that the corresponding solution of the system satisfies x(tf) = xf.

Our first result establishes that a necessary and sufficient condition for con-

trollability of the linear system (2.1) is the invertibility of an associated Gramian

matrix.

Theorem 2.2 (Controllability Gramian Condition). The regressive linear system

x∆(t) = A(t)x(t) + B(t)u(t),x(t0) = x0,

y(t) = C(t)x(t) + D(t)u(t),

is controllable on [t0,tf] if and only if the n × n controllability Gramian matrix

given by

?tf

is invertible, where ΦZ(t,t0) is the transition matrix for the system X∆(t) =

Z(t)X(t), X(t0) = I.

GC(t0,tf) :=

t0

ΦA(t0,σ(t))B(t)BT(t)ΦT

A(t0,σ(t))∆t,

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EJDE-2009/37 CONTROLLABILITY, OBSERVABILITY, REALIZABILITY3

Proof. Suppose GC(t0,tf) is invertible. Then, given x0and xf, we can choose the

input signal u(t) as

u(t) = −BT(t)ΦT

and extend u(t) continuously for all other values of t. The corresponding solution

of the system at t = tf can be written as

A(t0,σ(t))G−1

C(t0,tf)(x0− ΦA(t0,tf)xf),t ∈ [t0,tf),

x(tf) = ΦA(tf,t0)x0+

?tf

t0

ΦA(tf,σ(t))B(t)u(t)∆t

= ΦA(tf,t0)x0

?tf

= ΦA(tf,t0)x0− ΦA(tf,t0)

?tf

= ΦA(tf,t0)x0− (ΦA(tf,t0)x0− xf)

= xf,

−

t0

ΦA(tf,σ(t))B(t)BT(t)ΦT

A(t0,σ(t))G−1

C(t0,tf)(x0− ΦA(t0,tf)xf)∆t

×

t0

ΦA(t0,σ(t))B(t)BT(t)ΦT

A(t0,σ(t))∆tG−1

C(t0,tf)(x0− ΦA(t0,tf)xf)

so that the state equation is controllable on [t0,tf].

For the converse, suppose that the state equation is controllable, but for the sake

of a contradiction, assume that the matrix GC(t0,tf) is not invertible. If GC(t0,tf)

is not invertible, then there exists a vector xa?= 0 such that

?tf

?tf

and hence

xT

aΦA(t0,σ(t))B(t) = 0,

However, the state equation is controllable on [t0,tf], and so choosing x0= xa+

ΦA(t0,tf)xf, there exists an input signal ua(t) such that

0 = xT

aGC(t0,tf)xa=

t0

xT

aΦA(t0,σ(t))B(t)BT(t)ΦT

A(t0,σ(t))xa∆t

=

t0

?xT

aΦA(t0,σ(t))B(t)?2∆t,(2.2)

t ∈ [t0,tf). (2.3)

xf= ΦA(tf,t0)x0+

?tf

t0

ΦA(tf,σ(t))B(t)ua(t)∆t,

which is equivalent to the equation

xa= −

?tf

t0

ΦA(t0,σ(t))B(t)ua(t)∆t.

Multiplying through by xT

tion. Thus, the matrix GC(t0,tf) is invertible.

aand using (2.2) and (2.3) yields xT

axa= 0, a contradic-

?

Since the controllability Gramian is symmetric and positive semidefinite, Theo-

rem 2.2 can be interpreted as saying (2.1) is controllable on [t0,tf] if and only if

the Gramian is positive definite. A system that is not controllable on [t0,tf] may

become so when either tf is increased or t0is decreased. Likewise, a system that

is controllable on [t0,tf] may become uncontrollable if t0is increased and/or tf is

decreased.

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4 J. M. DAVIS, I. A. GRAVAGNE, B. J. JACKSON, R. J. MARKS IIEJDE-2009/37

Although the preceding theorem is strong in theory, in practice it is quite lim-

ited since computing the controllability Gramian requires explicit knowledge of the

transition matrix, but the transition matrix for time varying problems is generally

not known and can be difficult to approximate in some cases. This observation

motivates the following definition and our next theorem.

Definition 2.3. If T is a time scale such that µ is sufficiently differentiable with the

indicated rd-continuous derivatives, define the sequence of n × m matrix functions

K0(t) := B(t),

Kj+1(t) := (I + µ(σ(t))A(σ(t)))−1K∆

j(t) −

?

(I + µ(σ(t))A(σ(t)))−1(µ∆(t)A(σ(t))

+ µ(t)A∆(t))(I + µ(t)A(t))−1+ A(t)(I + µ(t)A(t))−1?

j = 0,1,2,....

Kj(t),

A straightforward induction proof shows that for all t,s, we have

∂j

∆sj[ΦA(σ(t),σ(s))B(s)] = ΦA(σ(t),σ(s))Kj(s),

Evaluation at s = t yields a relationship between these matrices and those in

Definition 2.3:

∂j

∆sj[ΦA(σ(t),σ(s))B(s)]

This in turn leads to the following sufficient condition for controllability.

j = 0,1,2,...

Kj(t) =

???

s=t,j = 0,1,2,...

Theorem 2.4 (Controllability Rank Theorem). Suppose q ∈ Z+such that, for

t ∈ [t0,tf], B(t) is q-times rd-continuously differentiable and both of µ(t) and A(t)

are (q − 1)-times rd-continuously differentiable. Then the regressive linear system

x∆(t) = A(t)x(t) + B(t)u(t),x(t0) = x0,

y(t) = C(t)x(t) + D(t)u(t),

is controllable on [t0,tf] if for some tc∈ [t0,tf), we have

rank?K0(tc)

where

∂j

∆sj[ΦA(σ(t),σ(s))B(s)]

K1(tc)...Kq(tc)?= n,

Kj(t) =

???

s=t,j = 0,1,...,q.

Proof. Suppose there is some tc∈ [t0,tf) such that the rank condition holds. For

the sake of a contradiction, suppose that the state equation is not controllable on

[t0,tf]. Then the controllability Gramian GC(t0,tf) is not invertible and, as in the

proof of Theorem 2.2, there exists a nonzero n × 1 vector xasuch that

xT

aΦA(t0,σ(t))B(t) = 0,

If we choose the nonzero vector xbso that xb= ΦT

t ∈ [t0,tf).

A(t0,σ(tc))xa, then (2.4) yields

(2.4)

xT

bΦA(σ(tc),σ(t))B(t) = 0,t ∈ [t0,tf).

In particular, at t = tc, we have xT

to t,

bK0(tc) = 0. Differentiating (2.4) with respect

xT

bΦA(σ(tc),σ(t))K1(t) = 0,t ∈ [t0,tf),

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EJDE-2009/37CONTROLLABILITY, OBSERVABILITY, REALIZABILITY5

so that xT

bK1(tc) = 0. In general,

dj

∆tj

?xT

bΦT

A(σ(tc),σ(t))B(t)????

xT

b

?K0(tc)

t=tc= xT

bKj(tc) = 0,j = 0,1,...,q.

Thus,

K1(tc)...Kq(tc)?= 0,

which contradicts the linear independence of the rows guaranteed by the rank con-

dition. Hence, the equation is controllable on [t0,tf].

?

When T = R, the collection of matrices Kj(t) above is such that each member is

the jth derivative of the matrix ΦA(σ(t),σ(s))B(s) = ΦA(t,s)B(s). This coincides

with the literature in the continuous case (see, for example, [3, 8, 24]). However,

while still tractable, in general the collection Kj(t) is nontrivial to compute. The

mechanics are more involved even on Z, which is still a very “tame” time scale.

Therefore, the complications of extending the usual theory to the general time

scales case are evident even at this early juncture.

Furthermore, the preceding theorem shows that if the rank condition holds for

some q and some tc ∈ [t0,tf), then the linear state equation is controllable on

any interval [t0,tf] containing tc. This strong conclusion partly explains why the

condition is only a sufficient one.

2.2. Time Invariant Case. We now turn our attention to establishing results

concerning the controllability of regressive linear time invariant systems. The gen-

eralized Laplace transform presented in [13, 19] allows us to attack the problem in

ways that simply are not available in the time varying case.

First we recall a result from DaCunha in order to establish a preliminary technical

lemma.

Theorem 2.5. [12] For the system X∆(t) = AX(t), X(t0) = I, there exist scalar

functions {γ0(t,t0),...,γn−1(t,t0)} ⊂ C∞

representation

n−1

?

Lemma 2.6. Let A,B ∈ Rn×nand u := ux0(tf,σ(s)) ∈ Crd(T,Rn×1). Then

??tf

Proof. Let {γk(t,t0)}n−1

nential matrix as guaranteed by Theorem 2.5. This collection forms a linearly

independent set since it can be taken as the solution set of an n-th order sys-

tem of linear dynamic equations. Apply the Gram-Schmidt process to generate an

orthonormal collection {ˆ γk(t,t0)}n−1

?γ0(t,t0)

rd(T,R) such that the unique solution has

eA(t,t0) =

i=0

Aiγi(t,t0).

span

t0

eA(s,t0)Bux0(tf,σ(s))∆s

?

= span{B,AB,...,An−1B}.(2.5)

k=0be the collection of functions that decompose the expo-

k=0. The two collections are related by

γn−1(t,t0)?

γ1(t,t0)···

=?ˆ γ0(t,t0)ˆ γ1(t,t0)···ˆ γn−1(t,t0)?

p11

0

...

0

p12

p22

...

0

···

···

...

···

p1n

p2n

...

pnn

,