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SimultaneousLQControlofa Set ofLTISystemsusing
ConstrainedGeneralizedSampled-DataHoldFunctions
Javad Lavaei and Amir G. Aghdam
Department of Electrical and Computer Engineering, Concordia
UniversityMontreal, QC Canada H3G 1M8
{j lavaei, aghdam}@ece.concordia.ca
Abstract
In this paper, sampled-data control of a set of continuous-time
LTI systems is considered. It is assumed that a predefined
guaranteed continuous-time quadratic cost function, which is, in
fact, the sum of the performance indices for all systems,is given.
The main objective here is to design a decentralized periodic
output feedback controller with a prespecified form,e.g.,
polynomial, piecewise constant, exponential, etc., which minimizes
the above mentioned guaranteed cost function. Thisproblem is first
formulated as a set of matrix inequalities, and then by using a
well-known technique, it is reformulated as aLMI problem. The set
of linear matrix inequalities obtained provides necessary and
sufficient conditions for the existence ofa decentralized optimal
simultaneous stabilizing controller with the prespecified form
(rather than a general form). Moreover,an algorithm is presented to
solve the resultant LMI problem. Finally, the efficiency of the
proposed method is demonstratedin two numerical examples.
Key words: Simultaneous stabilization, H2 optimal control,
Generalized sampled-data hold function, Decentralized, LMI
1 Introduction
There has been a considerable amount of interest inthe past
several years towards control of continuous-time systems by means
of periodic feedback, or so-calledgeneralized sampled-data hold
functions (GSHF). Theidea of using GSHF instead of a simple
zero-order hold(ZOH) in control systems was first introduced in
Cham-mas and Leondes (1978). Kabamba investigated sev-eral
applications and properties of GSHF in control sys-tems (Kabamba,
1987; Kabamba and Yang, 1991). Heshowed that many of the advantages
of state feedbackcontrollers, without the requirement of using
state esti-mation procedures, can be obtained by using a GSHF.
A
comprehensive frequency-domain analysis was presentedby Feuer
and Goodwin to examine robustness, sensitiv-ity, and intersample
effect of GSHF (Feuer and Good-win, 1994).
Simultaneous stabilization of a set of systems, on theother
hand, is of special interest in the control literature(Fonte,
Zasadzinski, and Bernier-Kazantsev, 2001), and
1 This work has been supported by the Natural Sciencesand
Engineering Research Council of Canada under
grantRGPIN-262127-03.
has applications in the following problems:
A system which is desired to be stabilized by a fixedcontroller
in different modes of operations, e.g., failuremode.
A nonlinear plant which is linearized at several
equi-libria.
A system which is desired to be stabilized in presenceof
uncertainties in its parameters.
Despite numerous efforts made to solve the simultane-ous
stabilization problem, it still remains an open prob-lem. In the
special case, when there are only two plants,the problem is
completely solved in Youla, Bongiorno,
and Lu (1974); Vidyasagar and Viswanadham (1982),and for the
case of three and four plants, some neces-sary and sufficient
relations in the form of polynomialare presented in Jia and
Ackermann (2001). However, nonecessary and sufficient condition has
been obtained forsimultaneous stabilization of more than four
plants, sofar. Moreover, it is proved in Blondel and Gevers
(1993)that if the number of plants is more than two, then
theproblem is rationally undecidable. It is also shown inToker and
Ozbay (1995) that the problem is NP-hard.These results clearly
demonstrate complexity level of theproblem. Since there does not
exist any LTI simultane-
Preprint submitted to Automatica 8 August 2006
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ous stabilizing controller in many cases, a
time-varyingcontroller is considered in Miller and Rossi (2001).
Itis shown that for any set of stabilizable and observableplants,
there exists a time-varying controller consistingof a sampler, a
ZOH, and a time-varying discrete-timecompensator which not only
stabilizes all of the plants,
but also acts as a near-optimal controller for each plant.This
result points to the usefulness of sampling in simul-taneous
stabilization problem. Nevertheless, fast sam-pling requirement and
large control gain are the draw-backs of this approach.
Stabilizing a set of plants simultaneously by means of aperiodic
controller is also investigated in the literature(Kabamba and Yang,
1991; Tarn and Yang, 1988). Amethod is proposed in Kabamba and Yang
(1991) to notonly minimize a guaranteed cost function
correspondingto all of the systems, but also accomplish desired
poleplacement. The drawback, however, is that the problemis
formulated as a two-boundary point differential equa-tion whose
analytical solution is cumbersome, in general.Some algorithms are
proposed to solve the resultant dif-ferential equation numerically,
in the particular case ofonly one LTI system, which is no longer a
simultane-ous stabilization problem (Hyslop, Schattler, and
Tarn,1992; Werner, 1996). Design of a high-performance
si-multaneous stabilizer in the form of a piecewise constantGSHF is
investigated in Cao and Lam (2001).
On the other hand, it is not realistic in many practicalproblems
to assume that all of the outputs of a systemare available to
construct any particular input of thesystem. In other words, it is
often desired to have some
form of decentralization. Problems of this kind appear,for
example, in electric power systems, communicationnetworks, large
space structures, robotic systems, eco-nomic systems and traffic
networks, to name only a few.
This paper deals with the problem of simultaneous sta-bilization
of a set of systems by means of a decentral-ized periodic
controller. It is assumed that a discrete-time decentralized
compensator is given for a set of de-tectable and stabilizable LTI
systems. This compensatoris employed to simplify the simultaneous
stabilizer de-sign problem. In certain cases, however, the
problemmay not be solvable without using a proper compensator
(e.g., in presence of unstable fixed modes (Davison andChang,
1990)). The objective here is to design a GSHFwhich satisfies the
following constraints:
i) The GSHF along with the discrete-time compen-sator
simultaneously stabilize the plants.
ii) It has the desired decentralized structure.iii) It has a
prespecified form such as polynomial, piece-
wise constant, etc.iv) It minimizes a predefined guaranteed cost
function,
which is the sum of the performance indices of allplants.
It is to be noted that condition (iii) given above is mo-tivated
by the following practical issues:
In many problems involving robustness, noise rejec-tion,
simplicity of implementation, elimination of fixedmodes, etc., it
is desired to design GSHFs with a spe-
cific form, e.g. piecewise constant, exponential, etc.(Wang,
1982; Kabamba, 1987). Design of a high performance simultaneously
stabi-
lizing piecewise constant GSHF with no compensatoris studied in
Cao and Lam (2001) for the centralizedcase, which can be simply
extended to the decentral-ized case. However, the present paper
solves the prob-lem in the general case by considering any
arbitraryform for the GSHF, such as exponential, and by in-cluding
a compensator in the control configuration aswell.
In the case of a sufficiently small sampling period, theoptimal
simultaneous stabilizer (whose exact solution,as pointed out
earlier, involves complicated computa-
tions), can be approximated by a polynomial (e.g., thetruncated
Taylor series).
This paper is organized as follows. The simultaneous
sta-bilizing periodic feedback problem is formulated in Sec-tion 2.
A necessary and sufficient condition for the exis-tence of a
stabilizing GSHF and compensator with thedesired structure is
obtained as a set of matrix inequal-ities in Section 3. The problem
is then converted to alinear matrix inequality (LMI) problem by
using a well-known technique, and an algorithm is presented to
solveit. The effectiveness of the proposed method is demon-strated
in two numerical examples in Section 4. Finally,
some concluding remarks are given in Section 5.
2 Problem formulation
Consider a set of continuous-time detectable and sta-bilizable
LTI systems S1, S2, ..., S with the followingstate-space
representations:
xi(t) = Aixi(t) + Biui(t) (1a)
yi(t) = Cixi(t) (1b)
where xi ni , ui
m and yi l, i :=
{1, 2,...,}, are the state, the input and the output of
Si,respectively. Assume that the discrete-time compensatorKic, i ,
with the following representation is given:
zi[ + 1] = Ezi[] + F yi[]
i[] = Gzi[] + Hyi[](2)
and assume also that zi[0] = 0. It is to be noted that
thediscrete argument corresponding to the samples of anysignal is
enclosed in brackets (e.g., yi[] = yi(h)). K
ic
can be either decentralized with block-diagonal trans-fer
function matrix or centralized. Suppose now that the
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system Si, i is desired to be controlled by the com-pensator Kic
and the hold controller K
ih represented by:
ui(t) = f(t)i[], h t < ( + 1)h, = 0, 1, 2,...
where h is the sampling period, and f(t) = f(t + h),
t 0. Note that f(t) is a sampled-data hold function,which is
desired to be described by the following set ofbasis functions:
f := {f1(t), f2(t),...,fk(t)}
where fi(t) mli , i = 1, 2,...,k. Thus, f(t) can be
written as a linear combination of the basis functions inf as
follows:
f(t) = f1(t)1 + f2(t)2 + + fk(t)k (3)
where some of the entries of the variable matrices i
lil, i = 1, 2,...,k, are set equal to zero and the otherentries
are free variables so that the structure of f(t)complies with the
desired control constraint, which isdetermined by a given
information flow matrix (Davisonand Chang, 1990). This is
illustrated later in Example 2.Furthermore, the set of basis
functions f is obtained ac-cording to the desirable form of GSHF
(e.g, exponential,polynomial, etc.). This will be demonstrated in
Exam-ples 1 and 2. Note that the motivation for considering
aspecial form for f(t) is discussed in the introduction.
For any i {1, 2,...,k}, put all of the indices of the ze-roed
entries ofi in the set Ei. Assume now the expected
value of xi(0)xi(0)T
, which is referred to as the covari-ance matrix of the initial
state xi(0), is known and de-noted by Xi0 for any i . The objective
is to obtain theconstrained matrices 1,...,k such that the
followingperformance index is minimized:
J = E
i=1
0
xi(t)
TQixi(t) + ui(t)TRiui(t)
dt
(4)
where Ri mm and Qi
nini are symmetric posi-tive definite and symmetric positive
semi-definite matri-ces, respectively, and E{} denotes the
expectation op-erator. Note that by minimizing the cost function
given
above, the stability of the system Si under the discrete-time
compensator Kic and the hold controller K
ih, for any
i , is achieved because the cost function becomes in-finity
otherwise. Note also that since (4) is a continuous-time
performance index, it takes the intersample rippleeffect into
account.
The equation (3) can be written as f(t) = g(t), where:
g(t) := [f1(t) f2(t) ... fk(t)] , :=
T1 T2 ...
Tk
T(5)
Define a new set E based on the sets E1, ..., Ek, suchthat any
of the entries of whose index belongs to E isequal to zero. On the
other hand, it is known that:
xi(t) = e(th)Aixi(h) +
t
h
e(t)AiBiui()d
for any h t ( + 1)h, 0. Let the followingmatrices be defined for
any i :
Mi(t) := etAi , Mi(t) :=
t0
e(t)AiBig()d
Therefore:
xi(t) = Mi(t h)xi[] + Mi(t h)i[k] (6)
for any h t ( + 1)h. It can be easily concludedfrom (1b), (2),
and (6) by substituting t = ( + 1)h,
that xi[ + 1] = Mi(h, )xi[] for any 0, where
xi[] =
xi[]T zi[]
T
T, and:
Mi(h, ) :=
Mi(h) + Mi(h)H Ci Mi(h)G
F Ci E
(7)
It is straightforward to show that:
xi[] =
Mi(h, )
xi[0], = 0, 1, 2,...
3 Optimal Structurally Constrained GSHF
It is desired now to find out when the structurally con-strained
GSHF f(t) exists such that the system Si is sta-ble under the
compensator Kic and the hold controllerKih, for any i .
Lemma 1 There exists a GSHF f(t) with the desiredform (given by
the equation (3)) such that the systemSi isstable under the
compensatorKic and the hold controllerKih for any i , if and only
if there exists an outputfeedback with the constant gain, with the
properties that:
1. Each entry of whose index belongs to the set E isequal to
zero.
2. It simultaneously stabilizes all of the systemsS1, S2, ...,
S, where the system Si, i , is repre-sented by:
xi[ + 1] =
Mi(h) 0
F Ci E
xi[] +
Mi(h)
0
ui[]
yi[] =
HCi G
xi[]
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(note that each of the two 0s in the above equationrepresents a
zero matrix with proper dimension).
Proof: The proof follows from the fact that the systemSi, i , is
stable under K
ic and K
ih if and only if all of
the eigenvalues of the matrix Mi(h, ) given in (7) are
located inside the unit circle in the complex plane.
Remark 1 Lemma 1 presents a necessary and sufficientcondition
for the existence of a structurally constrainedGSHF f(t) with a
desired form, which simultaneouslystabilizes all of the systems S1,
S2, ..., S along with agiven discrete-time compensator. The
condition obtainedis usually referred to as simultaneous
stabilization of aset of LTI systems via structured static output
feedback,which has been investigated intensively in the
literature.For instance, one can exploit the LMI algorithm
proposedin Cao and Lam (2001) to solve the simultaneous
stabi-lization problem given in Lemma 1 in order to obtain a
stabilizing matrix denoted by (which is later used asthe initial
point in the main algorithm), or conclude thenon-existence of such
GSHF, otherwise.
Define now the following matrices for any i :
Pi0 :=
h0
Mi(t)
TQiMi(t)
dt
Pi1 :=
h0
Mi(t)
TQiMi(t)
dt
Pi2 := h
0 Mi(t)
TQiMi(t) + g(t)TRig(t)
dt
qi0() := Pi0 + Pi1H Ci + (Pi1HCi)T
+ (HCi)TPi2(HCi)
qi1() := Pi1G + (HC)
TPi2G
Ni() :=
qi0() q
i1()
qi1()T GTTPi2G
Theorem 1 The optimal GSHFf(t) can be obtained byminimizingJ
given below:
J = trace
i=1
Ki Xi0 0
0 0 (8)
where K1, K2,...,K satisfy the following inequalities:
Ki + Ni() M
Ti (h, )Ki
KiMi(h, ) Ki
< 0, i = 1, 2,..., (9)
Proof: The proof is in line with that given in Lavaei andAghdam
(2006), and is omitted here.
Lemma 2 The matrixPi2 is positive definite if and onlyif there
does not exist a constant nonzero vector x suchthatg(t)x = 0 for
allt [0, h].
Proof: The proof is given in Lavaei and Aghdam(2006).
Lemma 2 presents a necessary and sufficient conditionfor the
positive definiteness of the matrix Pi2, which al-most always holds
in practice. It is assumed in the re-mainder of the paper that the
matrix Pi2 is positive defi-nite, as this assumption is required
for the developmentof the main result.
Theorem 2 The matrix inequality (9) is equivalent tothe
following matrix inequality:
i1 (i2)
T (i4)T
i2 i3 (
i5)
T
i4 i5 I
< 0 (10)
where
i1 := Ki +
Pi0 + P
i1HCi +
Pi1H Ci
TPi1G
(Pi1G)T 0
,
i2 := Ki
Mi(h) 0
F Ci E
,
i3 := Ki Ki Mi(h)(P
i2)1Mi(h)
T 0
0 0Ki,
i4 := (Pi2)
1
2
HCi G
,
i5 :=
(Pi2)
1
2 Mi(h)T 0
Ki
Proof: One can write the inequality (9) as follows:
i1 (
i2)
T
i2 i3
(i4)
T
(i5)T
(I)
i4
i5
< 0
The matrix inequality (10) yields by applying the
Schurcomplement formula to the above inequality.
It can be easily verified that in the absence of the blockentry
i3, the matrix given in the left side of (10) isin the form of LMI.
Moreover, this block entry can-not be converted to the LMI form due
to the negativequadratic term inside it. Thus, the technique
introducedin Cao, Sun, and Lam (1999) will now be used to rem-edy
this drawback. Consider the arbitrary positive def-inite matrices
1, 2, ...., with the same dimensionsas K1, K2,....,K, respectively.
Since P
i2 is assumed to
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be positive definite, one can write (Ki i)i(Ki i) 0, i ,
where:
i :=
Mi(h)(P
i2)1Mi(h)
T 0
0 0
Therefore:KiiKi i, i (11)
where i := iii Kiii iiKi, i .
Theorem 3 There exist positive definite matricesK1, K2,...,K
satisfying the matrix inequality (10)if and only if there exist
positive definite matricesK1, K2,...,K and 1, 2, ..., satisfying
the followingmatrix inequalities:
i1 (i2)
T (i4)T
i2 Ki + i (i5)
T
i4 i5 I
< 0, i (12)
Proof: If there exist positive definite matrices K1, K2, ...,K
and 1, 2, ..., satisfying (12), then accordingto (11), the matrices
K1, K2,...,K satisfy (10) as well.On the other hand, suppose that
there exist positivedefinite matrices K1, K2,...,K satisfying the
matrixinequality (10). Choosing i = Ki for i = 1, 2,...,, onecan
easily verify that i3 = Ki + i. Hence, the in-equality (10) is
equivalent to the inequality (12) in thiscase.
It is to be noted that the matrix inequality (12) is LMIfor the
variables Ki, i , and , if the matrices i,i , are set to be fixed.
The following algorithm isproposed based on Theorems 1, 2 and 3, to
compute thecoefficients 1, 2,..., in order to obtain the
desiredGSHF f(t).
Algorithm 1:
Step 1) Set = (where is defined in Remark 1)and solve the
(linear matrix) inequality (9) in orderto obtain K1, K2,...,K.Step
2) Set i = Ki for all i , where the matrices
Ki, i , are obtained in Step 1.Step 3) Minimize J given by (8)
for K1, K2,...,K and subject to The LMI constraint (12) Ki > 0
for all i The constraint that each entry of whose index
belongs to the set E must be zero.Step 4) If
i=1 Ki i < , where is a predeter-
mined error margin, go to Step 6.Step 5) Set i = Ki for i = 1,
2,...,, where the matri-ces Ki, i , are obtained by solving the
optimizationproblem in Step 3. Go to Step 3.
Step 6) The value obtained for is sufficiently closeto the
optimal value, and substituting the resultantmatrices Ki, i , into
(8) gives the minimum valueof J. Note that the coefficients 1,
2,...,k can beobtained from (5).
Remark 2 It can be easily verified that the value ofJ de-creases
each time that the optimization problem of Step 3is solved, which
indicates that the algorithm is monotonedecreasing. On the other
hand, since the inequality (11)will be converted to the equality if
i = Ki, Algorithm 1should ideally stop when
i=1 Ki i = 0 in order to
obtain the exact result. However, since it is desirable thatthe
algorithm be halted in a finite time, Step 4 is added. Itis to be
noted that determines (indirectly) the closenessof the performance
index obtained to its minimum value.
4 Numerical examples
Example 1: This example can be found in Howitt et al.(1993), and
represents the ship-steering system with twodistinct modes.
Consider two systems with the followingparameters:
A1 =
0.298 0.279 0
4.370 0.773 0
0 1 0
, B1 =
0.116
0.773
0
A2 =
0.428 0.339 0
2.939 1.011 0
0 1 0
, B2 =
0.150
1.011
0
and C1 = C2 = I. Assume that the initial state of eachof these
systems is a random variable whose covariancematrix is equal to the
identity matrix, and that h =0.1sec. Assume also that it is desired
to find a GSHFwhich minimizes the performance index J given by
(4)with Ri = Qi = I, i = 1, 2, while it has the
followingstructure:
f(t) =
+ sin(t) + et
(13)
where the symbol * represents constant values whichare to be
found. Note that no compensator is consideredin this example. The
following basis functions and coef-ficient matrices can therefore
be defined for (13):
f1(t) = sin(t), f2(t) = 1, f3(t) = et
1 =
0 0
, 2 =
, 3 =
0 0
It is to be noted that the * elements used above implythat these
entries of the vectors 1, 2 and 3 are theones that are not set
equal to zero. The optimal GSHF
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obtained from Algorithm 1 (by using the initial point given in
Lavaei and Aghdam (2006)) will be:
3.925 + 2.805 sin(t) 2.117 0.188 + 1.480 et
The corresponding performance index is 31.581.
Example 2: Consider a two-input two-output system Sconsisting of
two single-input single-output (SISO)agents with the following
state-space matrices:
A = diag([0.5 2.5 5.5])
B =
0 2 4
2 2 0
T, C =
2 0 2
0 2 2
It is desired to design a high-performance
decentralizedcontroller with the diagonal information flow
structure
for this system. It can be easily concluded from Davi-son and
Chang (1990) that = 0.5 is a decentralizedfixed mode (DFM) of the
system. Thus, there is no LTIcontroller to stabilize the system. As
a result, the avail-able methods to design a continuous-time LTI
controller(e.g., see Cao et al. (1999)) are incapable of handling
thisproblem. Choose now h = 1sec, and denote the discrete-time
equivalent model of the system S with Sd. If thealgorithm presented
in Cao and Lam (2001) is exploitedto design a discrete-time static
stabilizing controller forthe system Sd, it will fail. This
signifies that in orderto design a decentralized controller for the
system S,two different types of controllers can be used: a
dynamicdiscrete-time controller or a periodic controller. These
two possibilities are explained in the following:
i) Let a deadbeat dynamic stabilizing controller Kcfor the
system Sd be designed by using the methodgiven in Davison and Chang
(1990). Assume thatQ = R = I, and that the initial state of the
sys-tem S is a random variable with the identity covari-ance
matrix. In this case, the corresponding per-formance index will be
equal to 83439.49, which isinadmissibly large. To improve the
performance, ahold controller Kh is desired to be added to the
con-trol system. Assume that the hold function f(t) isdesirable to
have the following form:
diag([ + cos(850t) + cos(850t)])
Using Algorithm 1 with several iterations re-sults in the hold
function f(t) = diag([1.003 0.071cos(850t) , 0.958 +
0.754cos(850t)]), and thecorresponding performance index turns out
to be81517.97. This indicates an improvement of about2.36% by using
the hold controller Kh. However,this enhancement is not
noticeable.
ii) It is desired to find out whether there exists a
holdcontroller Kh to stabilize the system S by itself
(i.e., without any compensator Kc). Consider thefollowing basis
functions for the hold function f(t):
fi(t) = ue
t
i 1
2
ue
t
i
2
, i = 1, 2
where ue() denotes the unit-step function. It is tobe noted that
this GSHF is equivalent to a piecewiseconstant function with two
different levels. Apply-ing the result of Cao and Lam (2001) to
Lemma 1leads to the controller Kh with the hold function:
diag([1.4f1(t) 0.185f2(t) 0.5f1(t) + f2(t)])(14)
The resulting performance index is equal to2121.18. Hence,
Algorithm 1 can now be utilized toadjust the coefficients of this
hold function prop-erly. The optimal f(t) obtained will be equal
to:
diag([2.71f1(t)+1.08f2(t) 0.97f1(t)0.30f2(t)])(15)
and the performance index of the closed-loopsystem will be
301.73. This implies that a high-performance stabilizing controller
is designed forthe ill-controllable system S. The outputs of
thefirst and the second agents of S under the holdfunctions (14)
and (15) are illustrated in Fig-ures 1(a) and 1(b) for x(0) = [0.5
0.5 0.5]T. Inaddition, the inputs of the first and the secondagents
of S are depicted in Figures 2(a) and 2(b).The value of the cost
function J is plotted for thefirst 150 iterations in Figure 3.
0 5 1015
10
5
0
5
10
(a)0 5 10 15
3
2
1
0
1
2
(b)
Fig. 1. The outputs of the first agent and the second agentare
depicted in (a) and (b), respectively, under the GSHFs(14) (dotted
curves), and (15) (solid curves).
0 5 10 155
0
5
10
(a)0 5 10 15
1
0.5
0
0.5
1
1.5
(b)
Fig. 2. The inputs of the first agent and the second agentare
depicted in (a) and (b), respectively, under the GSHFs(14) (dotted
curves), and (15) (solid curves).
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0 50 100 150
500
1000
1500
2000
Fig. 3. The value ofJ for the first 150 iterations.
5 Conclusions
In this paper, a method is proposed to design a de-centralized
periodic output feedback with a prescribedform, e.g. polynomial,
piecewise constant, sinusoidal,etc., to simultaneously stabilize a
set of continuous-timeLTI systems and minimize a predefined
guaranteedcontinuous-time quadratic performance index, whichis, in
fact, the sum of the performance indices of all of
the systems. The design procedure is accomplished inthree
phases: First, the problem is formulated as a setof matrix
inequalities. Next, it is converted to a set oflinear matrix
inequalities, which represent necessaryand sufficient conditions
for the existence of such astructurally constrained controller with
the prespecifiedform. An algorithm is then presented to solve the
resul-tant LMI problem. Simulation results demonstrate
theeffectiveness of the proposed method.
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