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New LMI Conditions for Static Output Feedback

Synthesis with Multiple Performance ObjectivesHakan Köroğlu∗ and Paolo Falcone∗

∗ Dept. Signals and Systems, Chalmers University of Technology, Gothenburg,SwedenE-Mail : {hakanko,paolo.falcone}@chalmers.se

Abstract— The static output feedback synthesis problem isconsidered with H ∞ and generalized H 2 performance objec-tives. New sufficient LMI conditions are derived for guarantee-ing the required performance objectives. These conditions alsodepend on scalar parameters that need to be fixed beforehand.The output feedback gain matrix is computed from the involvedmatrix variables without the use of system matrices. Hencethe conditions can directly be used to solve the multi-objectivesynthesis problem also for parameter-dependent systems withconstant or parameter-dependent gain matrices. The method isillustrated in the adaptive cruise control problem.

I. INTRODUCTION

Static output feedback synthesis is a conceptually simply

and yet theoretically challenging problem. It is well-known

that the dynamic, fixed-order output feedback controller

design problem can also be formulated as a static output

feedback synthesis by system augmentation. Referring the

reader to [21] for a survey through the relatively old litera-

ture, we cite below only some selected recent works that are

relevant for our discussion.

As is the case for various problems, the synthesis of static

output feedback controllers can also be based on matrix

inequality conditions that ensure stability as well as some

performance objectives (H ∞,H 2, etc.). One is then facedwith optimization problems under bilinear matrix inequality

(BMI) constraints, which are non-convex and hence hard to

deal with. It is also possible to reformulate the constraints in

the form of linear matrix inequalities (LMI) accompanied by

a rank constraint or alternatively a coupling condition among

the matrix variables. Various approaches are developed in

the literature to deal with the non-convex constraints. One

typically identifies an iterative procedure in which a convex

optimization problem is solved repeatedly until a stopping

criterion is satisfied (see e.g. [12], [3], [15], [13], [10] and

the references therein). It would be fair to say that these

procedures can provide solutions only with limited efficiency.

It is hence natural to consider making a trade off be-tween optimality and tractability in static output feedback

synthesis problems with specified performance objectives.

This is usually done by representing the systems in suitable

bases and then choosing the matrix variables with particular

structures that facilitate a reformulation of BMI constraints

in the form of LMIs [5], [16], [18]. Exact LMI formulations

can be made only when the plant satisfies some restrictive

assumptions [18], [9]. It is hence necessary to seek for

ways to reduce the potential conservatism. To this end, it

is possible to use dilated versions of the matrix inequality

conditions for performance, which have proven useful in

multi-objective, structured and robust synthesis problems

(see e.g [1], [17] and the references therein). This approach

has been applied also to derive sufficient LMI conditions for

(single or multiple objective) static output feedback synthesis

problems for fixed or uncertain systems [2], [14], [7], [22],

[20], [6], [8].

The method developed in this paper also falls in the

category of suboptimal solutions to static output feedback

problems. Our approach is inspired by the solution of [4] tothe multi-objective static state-feedback synthesis problem.

By following the same way of dilation, we have been able

to modify the conditions in [5] in a way to reduce their

potential conservatism. One of the modified conditions is

actually quite similar to the recent work [8], in which the

considered system model is less general. The LMI conditions

depend (in the same way as in [4]) on scalar parameters that

need to be fixed beforehand (cf. [6]).

For the convenience of our presentation, we formulate

the problem as a multi-objective synthesis for linear time-

invariant systems in the following section. In Section III,

we derive two alternative sets of LMI conditions for H ∞

performance objectives. We then adapt one of these togeneralized H 2 performance in Section IV. These results

are combined in Section V to express our solution to the

multi-objective synthesis problem. Extensions to parameter-

dependent systems are also discussed briefly. The synthesis

is illustrated for the adaptive cruise control problem before

the concluding remarks.

I I . PROBLEM F ORMULATION

We consider a set of linear time-invariant plants indexed

with i = 1, . . . ,η as

Σi : ˙ xi(t ) zi(t )

yi(t )

= A Bi BC i Di E iC H i 0

xi(t )wi(t )ui(t )

, xi(0) = 0, (1)where xi(t ) ∈ Rk denotes the state vector, while ui(t ) ∈ Rnand y(t ) ∈ Rm represent the control input and the measuredoutput vectors respectively. These plants might refer to differ-

ent systems or a single system expressed for the assessment

of different performance objectives. Indeed, the realizations

differ only in the matrices that relate to the generalized

disturbance input wi(t ) ∈ Rqi and the performance output zi(t ) ∈R pi . In fact, we can also allow for different ( A, B,C )

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matrices provided that ui’s and yi’s have identical dimensions

for all i’s. We avoid this only for notational reasons.

In this paper, we consider the synthesis of a controller

based on static output feedback. The control input to each

plant will hence be formed as

ui(t ) = Kyi(t ), (2)

where K ∈Rn×

m

is the (common) gain matrix to be designed.We emphasize at this point that static (partial) disturbance

feed-forward synthesis can also be included in this formula-

tion thanks to the presence of H i. Indeed, one can then extend

C and H i matrices and thereby the measurement vector y in

a way to have the measurable disturbances as a part of the

extended measurement vector.

The closed-loop dynamics of the i’th plant are obtained

with the control input of (2) as ˙ xi zi

=

A + BKC Bi + BKH iC i + E iKC Di + E iKH i

xiwi

, xi(0) = 0, (3)

where the time dependencies of the signals are suppressed

for notational simplicity.

In this paper, we will consider the static output feedback

synthesis problem with H ∞ and generalized H 2 performance

objectives. Since we will work with time-domain formula-

tions, we recall the definitions of L 2 and L ∞ norms as

∥w∥2 ∫ ∞

0∥w(t )∥2dt

1/2, (4)

∥ z∥∞ supt ≥0

∥ z(t )∥, (5)

where ∥a∥ √

aT a. We can now state the core problem

considered in this paper precisely as follows:

Problem 1: Given a set of linear time-invariant plantsΣi, i = 1, . . . ,η as in (1), find a matrix K ∈Rn×m such thatthe closed-loop systems in (3) are all stable (i.e. A + BKC is Hurwitz) and satisfy the following performance objectives

for all wi(·) with 0 < ∥wi∥2 < ∞:∥ zi∥2 < σ i∥wi∥2, ∀i ∈I ∞ (6)∥ zi∥∞ < γ i∥wi∥2, ∀i ∈I 2 (7)

In these expressions, σ i’s represent the desired levels of H ∞performance for i ∈I ∞ and γ i’s represent the desired levelsof generalized H 2 performance for i ∈I 2.

III . SUFFICIENT C ONDITIONS FOR G UARANTEEDH

∞PERFORMANCE

In this subsection, we derive two alternative solvability

conditions for the generic H ∞ objective in (6). These condi-

tions are both sufficient and are inspired by the work of [4]

on multi-objective controller synthesis based on static state-

feedback. Moreover, they can be viewed as modified versions

of the conditions in [5], which are retrieved when particular

equality constraints are imposed on the design variables. We

describe the derivations of these conditions separately in the

following subsections.

A. First Set of Conditions

The basic idea behind the dilation of the state-feedback

LMIs as in [4] is to assume a feedback gain matrix as

K = NW −1, (8)

and construct a quadratic Lyapunov function with a matrix

that is different from W . Since we consider output feedback,

the dimensions of the matrices are identified in our case as N ∈Rn×m and W ∈ Rm×m. We represent the symmetric andpositive-definite matrix that will be used to construct the

Lyapunov function by Y i ∈ Sk +. Being indexed with i, theLyapunov matrix Y i can be chosen different for different i’s.

On the other hand, since we need to have a common output

feedback design for all the plants (see (2)), we avoid any

index in N and W , which means that they will be identical

when we consider the performance objective for different i’s.

In terms of the matrices introduced so far, we now define

a transformed state vector θ i and a new signal ζ i as

θ i Y −1

i xi, (9)

ζ i C θ i −W −1

yi. (10)

Multiplication of (10) from the left by N leads to

Kyi = NC θ i − N ζ i. (11)This allows us to express the closed-loop plant dynamics in

(3) as

˙ xi = ( AY i + BNC )θ i − BN ζ i + Biwi, zi = (C iY i + E i NC )θ i − E i N ζ i + Diwi. (12)

We have thus obtained a description of the closed-loop that

will help us avoid a bilinear dependence on the design

variables when matrix inequality conditions are derived in

the usual way. This is achieved by the introduction of a newinput signal ζ i, the relation of which to θ i and wi has to beused to end up with useful conditions. By multiplying (10)

from the left with W , we derive this relation as

oi (WC −CY i)θ i − H iwi −W ζ i = 0. (13)Let us now form a quadratic Lyapunov function as

V i( xi) = xT i Y

−1i xi, (14)

and recall that stability and H ∞ performance requirement

of (6) will be satisfied if the following condition is ensured

along the trajectories of (3):

d V

i( xi(t ))dt

+ ∥ zi(t )∥2

σ i−σ i∥wi(t )∥2

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By using these expressions, we put (16) into the form (18),

from which a condition is obtained as in (26).

Remark 3: As derived in [5], a dual solvability condition

to (20) can be obtained for the case in which E i = 0 and Bhas full row rank. This condition is also formed by an LMI

accompanied by an equality constraint as

He{ X i A + BMC } ∗ ∗( X i Bi + BMH i)T −σ i I ∗C i Di −σ i I

≺ 0 and BV − X i B = 0.(27)

We again note that the feasibility of these conditions imply

the feasibility of (26).

IV. SUFFICIENT C ONDITIONS FOR G UARANTEED

GENERALIZED H 2 PERFORMANCE

In this section, we adapt the derivation in Section III-A

to obtain a set of sufficient LMI conditions for generalized

H 2 performance. It is also possible to adapt the derivation

in Section III-B, which we avoid for reasons of space.

Throughout this section, we assume that the plant satisfies

Di = 0 and E i = 0, ∀i ∈I 2. (28)

This guarantees that the feed-through term of the closed-

loop system in (3) is zero, which is necessary for having a

bounded generalized H 2 norm.

Let us now recall the Lyapunov function in (14) and

note that the conditions that guarantee the generalized H 2performance requirement of (7) are expressed as follows:

d V i( xi(t ))

dt − γ i∥wi(t )∥2 0, ∀t ≥ 0. (30)

With x(0) = 0, we observe by integrating the first inequalitythat V i( xi(t )) 0. A gain matrix K ∈R

n

×m

that solves the problem can then be computed as in (8).

In multi-objective synthesis, we typically need to consider

minimization over one particular σ i or γ i and fix all others tofeasible values. As is emphasized in the theorem statement,

we also need to fix (φ ,ψ ) in order to obtain a genuineLMI optimization problem. In order to reduce the potential

conservatism in our synthesis, we need to perform a plane

search over (φ ,ψ ). In fact, we can also consider usingdifferent scalars for different i’s. Nevertheless, this is not

quite desirable as it will lead to increased computational

complexity when we have to search over three or more

scalars.

Problem 1 can also be considered for parameter-dependentsystems. A parameter-dependent system is described by a

state-space model in which all the system matrices depend

on an uncertain parameter vector δ , which can take its valuesfrom a known compact set R . The system matrices are then

described by continuous, matrix-valued maps, i.e. A(·) : R →R

k ×k , B(·) : R → Rk ×n, etc. When δ is time-invariant orarbitrarily time-varying, knowledge of R will be enough to

describe the uncertainty. On the other hand, when parameters

are assumed to vary slowly in time, it becomes convenient

to introduce an extended parameter vector as δ e (δ , δ̇ ). Inthis case, the uncertainty description can be captured by a

compact set U , which represents the set of admissible values

for δ e.We will be faced with two different synthesis problems

for parameter-dependent systems, depending on whether δ (t )is measurable during online operation or not. When δ (t ) isonline-measurable, we can consider synthesizing an output

feedback gain that is also parameter-dependent, i.e. K (·) : R → Rn×m. If this is not the case, we will have to considerthe synthesis of a constant matrix K ∈Rn×m with which theperformance objectives are achieved.

It is straightforward to extend Theorem 4 in a way to pro-

vide solutions to the multi-objective synthesis problems for

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10−1

100

101

−60

−50

−40

−30

−20

−10

0

Singular Values

Frequency (rad/s)

S i n g u l a r V a l u e s ( d B )

T1

sfb

T1

ofb

10−1

100

101

−40

−30

−20

−10

0

Singular Values

Frequency (rad/s)

S i n g u l a r V a l u e s ( d B )

T2

sfb

T2

ofb

Fig. 1. Singular value plots of T 1( jω ) and T 2( jω ).

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

Time [sec]

a 1

[ m / s 2 ]

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

e 1

[ m ]

SFB

OFB

a0

Fig. 2. Simulation results.

feedback synthesis. As we have noted in Remark 1, the H ∞synthesis condition in [5] is evidently more conservative than

(19). This is what we also observed in our numerical investi-

gations. We have also made a comparison with Example 5.1

of [6], in which the optimum σ is found as 4.17 (and thecorresponding output feedback led to an H ∞ norm of 2.49).Based on (19), we obtained the minimum σ value as 1.6716and the associated output feedback led to an H ∞ norm of

1.4433. A theoretical investigation is needed to determinewhether (19) always provides less conservative results or not.

VII. CONCLUDING R EMARKS

We have derived new sufficient LMI conditions for static

output feedback synthesis for guaranteed H ∞ and gener-

alized H 2 performance objectives. It is not necessary to

perform any state transformation to apply these conditions.

Hence they can be directly used to synthesize robust or

scheduled controllers for parameter-dependent systems. It is

also possible to use them for structured controller synthesis

as the feedback gain matrix is computed without the use of

system matrices. Seemingly the conditions need some im-

provement to be useful in reduced-order controller synthesis.

ACKNOWLEDGEMENT

This work is supported by SAFER Vehicleand Traffic SafetyCenter.

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