Submission to International Journal of Control, Automation, and Systems 1

Optimal LPV control with hard constraints

Andrew White, Guoming Zhu, and Jongeun Choi

Abstract: This paper considers the optimal control of polytopic, discrete-time linear parameter

varying (LPV) systems with a guaranteed ℓ2 to ℓ∞ gain. Additionally, to guarantee robust stability

of the closed-loop system under parameter variations, H∞ performance criterion is also consid-

ered as well. Controllers with a guaranteed ℓ2 to ℓ∞ gain and a guaranteed H∞ performance

(ℓ2 to ℓ2 gain) are a special family of mixed H2/H∞ controllers. Normally, H2 controllers are

obtained by considering a quadratic cost function that balances the output performance with the

control input needed to achieve that performance. However, to obtain an optimal controller with a

guaranteed ℓ2 to ℓ∞ gain (closely related to the physical performance constraint), the cost function

used in the H2 control synthesis minimizes the control input subject to maximal singular-value

performance constraints on the output. This problem can be efficiently solved by a convex opti-

mization with linear matrix inequality (LMI) constraints. The main contribution of this paper is

the characterization of the control synthesis LMIs used to obtain an LPV controller with a guar-

anteed ℓ2 to ℓ∞ gain and H∞ performance. A numerical example is presented to demonstrate the

effectiveness of the convex optimization.

Keywords: Linear parameter varying (LPV) systems, hard constraints, ℓ2 to ℓ∞ gain, linear matrix

inequality (LMI), LPV control.

1. INTRODUCTION

The design of multi-objective, mixed H2/H∞ controllers

has been a topic of interest for sometime [1–9]. The goal

of using both H2 and H∞ performance criteria is to de-

sign a controller which can meet multiple performance

objectives. In [1] and [2] mixed H2/H∞ control was

introduced by minimizing the H2 norm of a closed-loop

transfer function subject to an H∞ norm constraint of an-

other closed-loop transfer function. In [3], mixed H2/H∞

state-feedback and output-feedback controllers were de-

signed for continuous-time systems by using a convex op-

timization approach to solve the coupled nonlinear matrix

Riccati equations and in [4] a similar approach is used for

discrete-time systems. The state-feedback H2/H∞ design

with regional pole placement was addressed by [5] using

the linear matrix inequality (LMI) approach. In [6] and

[7], the LMI approach for multi-objective control synthe-

sis for output-feedback controllers is presented. In [10], an

extra instrumental variable was added to the LMI stability

conditions to build a parameter dependent Lyapunov func-

tion capable of proving the stability of uncertain linear-

time-invariant (LTI) systems. The new extended LMI con-

ditions in [10] were used in [8] to develop H2 and H∞

LMI conditions for linear state-feedback and output-feedback

Andrew White, Guoming Zhu, and Jongeun Choi are withthe Department of Mechanical Engineering, Michigan StateUniversity, East Lansing, MI 48824 (e-mail: {whitea23,zhug, jchoi}@egr.msu.edu)

Guoming Zhu, and Jongeun Choi are also with the Depart-ment of Electrical Engineering, Michigan State University, EastLansing, MI 48824

controllers for uncertain LTI systems. The extended LMI

conditions provided by [8] were utilized in [11] and [12]

to develop linear parameter varying (LPV) static output

feedback controllers that meet H2 [11] and H∞ [12] per-

formance bounds for linear time-varying (LTV) systems

with polytopic uncertainty. The results presented in [11]

and [12] were extended in [9] to cover the synthesis of

multi-objective H2/H∞ gain-scheduled output feedback

controllers. These LPV control techniques have been suc-

cessfully applied to engine system applications [13–16].

Gain scheduling controllers designed using the LPV method

have traditionally included H∞ performance constraints.

This is largely due to the fact that H∞ controllers can pro-

vide robust stability margins that H2 controllers cannot

provide [17]. However, since the H∞ norm is defined as

the root-mean-square gain, or ℓ2 to ℓ2 gain, from the ex-

ogenous input to the regulated output, controllers designed

with only H∞ performance constraints are not suitable for

applications when hard constraints on responses or actua-

tor signals must be met.

When hard constraints on responses or actuator signals

must be met, a controller with a guaranteed ℓ2 to ℓ∞ gain

is required, which is a special type of H2 controller [18].

The ℓ2-ℓ∞ filtering has been a recent topic of interest for

various nonlinear and stochastic systems [19, 20]. A con-

troller with a guaranteed ℓ2 to ℓ∞ gain provides hard bounds

on the regulated output while minimizing the control in-

put as much as possible. This problem was solved for LTI

systems in [18], where it is referred to as the output co-

variance constraint (OCC) problem. The OCC problem

2 Submission to International Journal of Control, Automation, and Systems

defined in [18] is to find a controller for a given system to

minimize the weighted control input cost subject to a set of

output constraints. The OCC problem has two interesting

interpretations: stochastic and deterministic. The stochas-

tic interpretation is obtained by first assuming that the H2

exogenous inputs are uncorrelated zero-mean white noises

with a given intensity. Then the OCC problem minimizes

the weighted control input covariance subject to the output

covariance constraints, such that the constraints are inter-

preted as constraints on the variance of the performance

variables. The deterministic interpretation is obtained by

assuming that the H2 exogenous inputs are unknown but

belong to a bounded ℓ2 energy set. Then the OCC prob-

lem minimizes the weighted control input while ensuring

that the maximum singular values, or ℓ∞ response, of the

regulated outputs are less than the corresponding output

constraints. In other words, the OCC problem is the prob-

lem of minimizing the weighted sum of worst-case peak

values on the control signal subject to the constraints on

the worst-case peak values of the performance variables.

This interpretation is important in applications where hard

constraints on responses or actuator signals cannot be ig-

nored, such as space telescope pointing [21] and machine

tool control. For both interpretations, a solution to the

OCC control problem results in a controller with a guar-

anteed ℓ2 to ℓ∞ gain.

The idea of an LPV controller design with a guaranteed

ℓ2 to ℓ∞ gain is not an entirely new concept and has been

previously studied before. In [22], the ℓ2 to ℓ2 and ℓ2 to

ℓ∞ gains of LPV sampled-data systems was investigated.

LMI conditions for the synthesis of output-feedback con-

trollers that provide the closed loop sampled-data system

with the desired gain were provided as well. However,

all of the LMI conditions provided in [22] are infinite di-

mensional, such that gridding of the parameter space is

required to obtain a finite dimensional set of LMI condi-

tions. In contrast, the LMI conditions presented in this pa-

per are finite dimensional such that parameter space grid-

ding is not required.

The main contributions of this paper are the guaranteed

ℓ2 to ℓ∞ gain controller synthesis LMIs for gain-scheduled

state-feedback and dynamic output-feedback control for

discrete-time polytopic LPV systems in Section 4.. When

these LMIs are satisfied, the optimal state-feedback or dy-

namic output-feedback LPV controller obtained guaran-

tees that for a finite disturbance energy, hard constraints

on the regulated output are met. The guaranteed ℓ2 to ℓ∞

gain is achieved by modifying H2 control synthesis LMIs

provided by [9] to minimize the weighted control input

cost while ensuring the output covariances meet the per-

formance constraints. A preliminary version of this paper

has been included as a chapter in [16].

The paper is organized as follows. The H∞ and guar-

anteed ℓ2 to ℓ∞ gain performances are provided in Sec-

tion 2. Then, in Section 3, the modeling of the uncer-

tainty domain where the time-varying barycentric coordi-

nates takes their values is presented. The controller syn-

thesis conditions for both state-feedback and dynamic out-

put feedback are presented in Section 4. In section 5, a nu-

merical example is presented for both state-feedback and

output-feedback control to illustrate the performance of

the control synthesis LMIs. Conclusions of this work are

given in Section 6.

2. PERFORMANCE OF DISCRETE-TIME

SYSTEMS WITH HARD CONSTRAINTS

Consider the closed-loop, asymptotically stable, discrete-

time LPV system H with the following finite-dimensional

state space realization:

H :=

{

x(k+ 1) = A (αk)x(k)+B(αk)w(k)

z(k) = C (αk)x(k)+D(αk)w(k)(1)

where x is the state, w is the exogenous input, and z is the

system output. The matrices A (α(k)), B(α(k)), C (α(k)),and D(α(k)) belong to the polytope

D=

{

(A ,B,C ,D)(α(k)) : (A ,B,C ,D)

=N

∑i=1

αi(k)(A ,B,C ,D)i,α(k) ∈ ΛN

}

,

(2)

where Ai, Bi, Ci, and Di are the vertices of the polytope

and α(k) ∈ RN is the vector of time-varying barycentric

coordinates lying in the unit simplex ΛN , given by

ΛN =

{

ζ ∈ RN :

N

∑i=1

ζi = 1,ζi ≥ 0, i = 1, · · · ,N

}

. (3)

2.1. H∞ Performance

The H∞ performance of system H, given by (1), is de-

fined as

‖H‖∞ = sup‖w(k)‖2 6=0

‖z(k)‖2

‖w(k)‖2

(4)

with w(k) ∈ ℓr2 and z(k) ∈ ℓp

2 . Based on the bounded real

lemma, an upper bound for the H∞ performance of system

H can be computed using an extended LMI characteriza-

tion, as shown in the following lemma given by [9, 23].

Lemma 1: Consider the system H given by (1). If

there exist a bounded matrix G(αk)∈Rn×n and a bounded

symmetric positive-definite matrix P(αk) ∈ Rn×n, for all

αk ∈ΛN , such that the LMI (5) is satisfied, then the system

H is exponentially stable and

‖H‖∞ ≤ infP(αk),G (αk),η

η . (6)

Submission to International Journal of Control, Automation, and Systems 3

P(αk+1) A (αk)G (αk) B(αk) 0

G (αk)T A (αk)

T G (αk)+G (αk)T −P(αk) 0 G (αk)

T C (αk)T

B(αk)T 0 ηI D(αk)

0 C (αk)G (αk) D(αk) ηI

> 0, (5)

2.2. Minimum Energy Control with Guaranteed ℓ2 to ℓ∞

Gain

To define the minimum energy control with guaranteed

ℓ2 to ℓ∞ gain problem for the system H, the following as-

sumptions are first made:

1. The system output z(k) is partitioned into

z(k) = [zp(k)T , zu(k)

T ]T ,

2. and the feed-through matrix D(αk) = 0

such that the system output z(k) is given by

z(k) :=

[zp(k)zu(k)

]

=

[Cp(αk)Cu(αk)

]

x(k) (7)

where the vector zp(k)∈Rc contains all the variables whose

dynamic responses are of interest and the vector zu(k) con-

tains the weighted control variables to be minimized. De-

fine the ℓ2 and ℓ∞ norms as

‖zp‖2∞ := sup

k≥0

zTp (k)zp(k), (8)

‖w‖22 :=

∞

∑ℓ=0

wT (ℓ)w(ℓ). (9)

Then the ℓ2 to ℓ∞ gain of (1) is less than or equal to the

square root of the maximum singular value of the matrix

constraint Zp if

supw∈ℓ2−{0}

‖zp‖∞

‖w‖2

≤√

σ(Z p) (10)

where σ(·) denotes the maximum singular value operator.

Corollary 2: Consider the asymptotically stable sys-

tem (1) with the performance output defined by (7). The

ℓ2 to ℓ∞ gain of (1) is less than or equal to the square root

of σ(Zp

), where Z p = Z

Tp > 0 is a given matrix constraint,

if there exists a bounded an continuous matrix function

P(αk) = P(αk)T > 0 such that

A (αk)P(αk)A (αk)T −P(αk+1)+B(αk)B(αk)

T < 0

(11)

Cp(αk)P(αk)Cp(αk)T −Zp < 0

(12)

The proof for Corollary 2, which follows from Theo-

rem 4 of [22], is provided in the following text.

Proof: Using the Schur complement formula, condi-

tion (11) is equivalent to[

P(αk)−1 0

0 I

]

−

[A (αk)

T

B(αk)T

]

P(αk+1)−1

[A (αk) B(αk)

]> 0.

(13)

Let w be any signal having finite energy, and x be the so-

lution of (1). Then, multiply (13) by [xT (k) wT (k)] and its

transpose from left and right, to obtain

xT (k)P(αk)−1x(k)+wT (k)w(k)

− xT (k+ 1)P(αk+1)−1x(k+ 1)< 0

Taking the summation from k = 0 to k = n− 1, we have

xT (n)P(αn)−1x(n)<

n−1

∑k=0

‖w(k)‖22.

Using the Schur complement formula again, this is equiv-

alent to[

∑n−1k=0 ‖w(k)‖2

2 xT (n)x(n) P(αn)

]

> 0 (14)

which implies[

1 00 Cp(αn)

][

∑n−1k=0 ‖w(k)‖2

2 xT (n)x(n) P(αn)

][1 0

0 C Tp (αn)

]

+

[

wT (n)0

]

[ w(n) 0 ]≥ 0

(15)

or equivalently[

∑n−1k=0 ‖w(k)‖2

2 +‖w(n)‖22 zT

p (n)

zp(n) Cp(αn)P(αn)Cp(αn)T

]

≥ 0.

(16)

Using the Schur complement formula again, we obtain

n

∑k=0

‖w(k)‖22−zT

p (n)[Cp(αn)P(αn)Cp(αn)

T]−1

zp(n)≥ 0.

(17)

After re-arranging and noting that (12) implies

Zp > Cp(αn)P(αn)Cp(αn)T

σ(Zp

)I > Cp(αn)P(αn)Cp(αn)

T(18)

we have that

zTp (n)

[Cp(αn)P(αn)Cp(αn)

T]−1

zp(n)≤n

∑k=0

‖w(k)‖22

zTp (n)

[σ(Zp

)]−1zp(n)≤

n

∑k=0

‖w(k)‖22

4 Submission to International Journal of Control, Automation, and Systems

such that

zTp (n)zp(n)≤ σ

(Z p

) n

∑k=0

‖w(k)‖22.

This implies that

‖zp‖∞ ≤√

σ(Zp

)‖w‖2 (19)

such that the ℓ2 to ℓ∞ gain is less than or equal to the square

root of σ(Zp

). �

Suppose that some a priori information about the con-

straints on the performance of zp are known such that an

output covariance bound Zp can be constructed. Then

it would be desirable to design a minimum energy con-

troller such that the closed-loop system H has the follow-

ing property:

‖zp‖2∞ ≤ σ

(Zp

)‖w‖2

2, (20)

where

Zp(αk) = Cp(αk)P(αk)Cp(αk)T ≤ Zp, (21)

and P(αk) is the solution to the time-varying Lyapunov

equation

P(αk+1) = A (αk)P(αk)A (αk)T +B(αk)B(αk)

T

(22)

This problem, which we call the minimum energy control

with guaranteed ℓ2 to ℓ∞ gain problem, is defined as fol-

lows: find a state feedback or full-order dynamic output

feedback controller to minimize the control energy

Zu(αk) = trace{Cu(αk)P(αk)Cu(αk)

T}, (23)

of the closed-loop system H, subject to the hard constraint

Z p.

Theorem 3: Consider the system H, given by (1) with

the performance output given by (7). Given the output

covariance Zp, if there exist parameter-dependent matrices

G (αk), P(αk) = P(αk)T > 0, and W (αk) = W (αk)

T >0, for all αk ∈ ΛN , such that

P(αk+1) A (αk)G (αk) B(αk)⋆ G (αk)+G (αk)

T −P(αk) 0

⋆ ⋆ I

> 0,

(24)[

W (αk) Cu(αk)G (αk)⋆ G (αk)+G (αk)

T −P(αk)

]

> 0,

(25)

Zp −Cp(αk)P(αk)Cp(αk)T > 0,(26)

where ⋆ represents entries that follow from symmetry, then

the closed-loop system (1) is exponentially stable with a

guaranteed ℓ2 to ℓ∞ performance given by

supw∈ℓ2

‖zp‖2∞

‖w‖22

≤ σ(Z p

), (27)

and a control energy bounded by

Zu = infP(αk),G (αk),W (αk)

supαk∈ΛN

trace{W (αk)} ,

≥ trace{Cu(αk)P(αk)Cu(αk)

T}≥ Zu(αk).

(28)

Proof: The LMIs (24) and (25) ensure the stability of

the closed-loop system H with the control energy bounded

by (28) [9]. However, the guaranteed ℓ2 to ℓ∞ gain perfor-

mance (27) is a result of (26). Since (24) implies that

P(αk+1)> A (αk)P(αk)A (αk)T +B(αk)B(αk)

T ,(29)

there exist matrices M(αk) = M(αk)T > 0 such that

P(αk+1)=A (αk)P(αk)A (αk)T +B(αk)B(αk)

T +M(αk).(30)

Consequently, P(αk)> P(αk) ∀k≥ 0, which shows that

Z p ≥ Cp(αk)P(αk)Cp(αk)T

≥ Cp(αk)P(αk)Cp(αk)T = Zp(k).

(31)

Thus, it follows that the guaranteed ℓ2 to ℓ∞ gain (27) is

satisfied. The Schur complement of the LMI (25) is

W (αk)−Cu(αk)G (αk)(G (αk)+G (αk)

T −P(αk))−1

×G (αk)TCu(αk)

T > 0.

(32)

In order for the LMI (25) to be feasible, it follows that

G (αk)+G (αk)T > P(αk)> 0, such that

W (αk)> Cu(αk)G (αk)(G (αk)+G (αk)

T −P(αk))−1

×G (αk)TCu(αk)

T ,

> Cu(αk)P(αk)Cu(αk)T ,

> Cu(αk)P(αk)Cu(αk)T ,

(33)

such that (28) holds. �

3. MODELING THE UNCERTAINTY DOMAIN

In this section, the modeling of the uncertainty domain,

which is covered in full detail in [9], is briefly presented.

For all k ∈ Z≥0, the rate of variation of the parameters

∆αi(k) = αi(k+ 1)−αi(k), i = 1, · · · ,N, (34)

is assumed to be limited by an a priori known bound b ∈R

such that

−b ≤ ∆αi(k)≤ b, i = 1, . . . ,N, (35)

Submission to International Journal of Control, Automation, and Systems 5

with b ∈ [0,1]. Since α(k) ∈ ΛN , it is clear from (34) that

N

∑i=1

∆αi(k) = 0. (36)

The uncertainty domain, where the vector (α(k),∆α(k))T ∈R

2N takes values, can be modeled by the compact set

Γb =

{

δ ∈ R2N : δ ∈ co{g1, . . . ,gM},

g j =

(f j

h j

)

, f j ∈ RN , h j ∈R

N ,

N

∑i=1

fj

i = 1 with fj

i ≥ 0, i = 1, . . . ,N,

N

∑i=1

hji = 0, j = 1, . . . ,M

}

(37)

defined as the convex combination of the vectors g j, for

j = 1, . . . ,M, given a priori. This definition of Γb ensures

that α(k) ∈ ΛN and that (36) holds for all k ≥ 0. For a

given bound b, the columns of Γb can be generated as

shown in [9].

4. CONTROLLER SYNTHESIS

This section considers the design of minimum energy

gain-scheduled controllers that provide guaranteed hard

constraints for the closed-loop system, while also satis-

fying some other H∞ performance criteria for robustness.

Thus, in this section we consider the following discrete-

time polytopic time-varying systems: the system Hh with

hard constraints that must be satisfied given by

Hh :=

xp(k+ 1) = A(αk)xp(k)+Bh(αk)wh(k)

+Bu(αk)u(k)

zp(k) =Cp(αk)xp(k)

zu(k) = Dhu(αk)u(k)

y(k) =Cy(αk)xp(k)+Dyh(αk)wh(k)

(38)

and the H∞ weighted system H∞ given by

H∞ :=

xp(k+ 1) = A(αk)xp(k)+B∞(αk)w∞(k)

+Bu(αk)u(k)

z∞(k) =C∞(αk)xp(k)+D∞(αk)w∞(k)

+D∞u(αk)u(k)

y(k) =Cy(αk)xp(k)+Dy∞(αk)w∞(k)(39)

where xp(k) ∈ Rn is the state, wh(k) ∈ R

rh and w∞ ∈ Rr∞

are the exogenous inputs, and u(k) ∈ Rm is the control

input. The outputs zu(k) ∈ Rph and z∞(k) ∈ R

p∞ are the

weighted system performance outputs for the mixed con-

trol synthesis, while the output zp(k) ∈ Rc contains all

variables whose dynamic responses have hard constraints

that must be met. The output vector y(k) ∈Rq is the mea-

surement to be used for control. The goal is to provide

a finite-dimensional set of LMIs for the synthesis of both

gain-scheduled state feedback controllers of the form

u(k) = K(αk)x(k), (40)

and gain-scheduled, strictly proper, output feedback con-

trollers of the form

xc(k+ 1) = Ac(αk)xc(k)+Bc(αk)y(k),

u(k) =Cc(αk)xc(k),(41)

such that the closed-loop systems given by

Hhcl :=

x(k+ 1) = A (αk)x(k)+Bh(αk)wh(k),

zp(k) = Cp(αk)x(k),

zu(k) = Cu(αk)x(k),

(42)

and

H∞cl :=

{

x(k+ 1) = A (αk)x(k)+B∞(αk)w∞(k),

z∞ = C∞(αk)x(k)+D∞(αk)w∞(k),(43)

are exponentially stable and satisfy hard constraints on de-

sired performance outputs for all possible trajectories of

the parameter αk ∈ ΛN , while minimizing the control en-

ergy for Hhcl and satisfying a robustness criteria defined as

an H∞ performance bound for H∞cl .

4.1. Gain-Scheduled State Feedback Control Synthesis

In this section, it is assumed that the state vector, xp(k),is available for feedback without corruption from the ex-

ogenous inputs wh(k) or w∞(k). This is a standard as-

sumption, and as in [8] can be enforced on the measure-

ment equation in (38) and (39) by assigning to the matri-

ces Cy(αk), Dyh(αk), and Dy∞(αk) the values Cy(αk) = I,

Dyh(αk) = 0, and Dy∞(αk) = 0. The feedback structure

provided by the gain-scheduled state-feedback controller

(40), produces the closed-loop systems Hhcl and H∞

cl in (42)

and (43) where x(k) = xp(k) and the closed-loop system

matrices are given by

A (αk) = A(αk)+Bu(αk)K(αk),

Bh(αk) = Bh(αk),

Cp(αk) =Cp(αk),

Cu(αk) = Dhu(αk)K(αk),

B∞(αk) = B∞(αk),

C∞(αk) =C∞(αk)+D∞u(αk)K(αk),

D∞(αk) = D∞(αk).

(44)

The parameter-dependent, full-state feedback controller

is solved for by performing a convex optimization over a

set of linear matrix inequalities. The LMIs in this sec-

tion are an extension of the work presented in [9]. To ob-

tain a finite-dimensional set of LMI conditions, an affine

6 Submission to International Journal of Control, Automation, and Systems

parameter-dependent structure is imposed on the Lyapunov

matrix P(αk) such that

P(αk) =N

∑i=1

αi(k)Pi, αk ∈ ΛN . (45)

With the uncertainty set Γb, each αi(k) and ∆αi(k) for i =1,2, . . . ,N are given by

αi(k) =M

∑j=1

fj

i γ j(k) and ∆αi(k) =M

∑j=1

hji γ j(k) (46)

such that

P(αk) = P(γ(k)) =M

∑j=1

γ j(k)Pj (47)

with Pj =∑Ni=1 f

ji Pi as shown in [9]. Using the same struc-

ture for αk, the system matrices in H∞ and Hh are also

converted to the new representation in terms of γ(k)∈ΛM ,

such that

A(αk) = A(γ(k)) =M

∑j=1

γ j(k)A j (48)

with A j = ∑Ni=1 f

ji Ai. All other matrices in H∞ and Hh are

converted the same way. Also, by combining (46) with the

fact that αk+1 = αk +∆αk,

P(αk+1) = P(γ(k)) =M

∑j=1

γ j(k)Pj (49)

with Pj = ∑Ni=1

(

fj

i + hji

)

Pi. Using these parameteriza-

tions, the finite-dimensional LMIs in the following theo-

rem can be solved to obtain a full-state feedback controller

(40) such that the closed-loop systems for Hh and H∞ have

a guaranteed ℓ2 to ℓ∞ and H∞ gain, respectively.

Theorem 4: Consider the system Hh, given by (38).

Assume that the vectors f j and h j of Γb are given. Given

Z p, if there exists, for i = 1,2, . . . ,N, matrices, Gi ∈ Rn×n

and Zi ∈ Rm×n, and symmetric positive-definite matrices

Ph,i ∈ Rn×n and Wi ∈ R

ph×ph such that

Ph, j ⋆ ⋆GT

j ATj + ZT

j BTu, j G j + GT

j − Ph, j ⋆

BTh, j 0 I

= Φ j > 0

(50)

for j = 1,2, . . . ,M, and

Ph, j + Ph,ℓ ⋆ ⋆Φ21, jℓ Φ22, jℓ ⋆

BTh, j + BT

h,ℓ 0 2I

= Φ jℓ > 0 (51)

with

Φ21, jℓ = GTj AT

ℓ + GTℓ AT

j + ZTj BT

u,ℓ+ ZTℓ BT

u, j

Φ22, jℓ = G j + GTj + Gℓ+ GT

ℓ − Ph, j − Ph,ℓ

for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, and

[Wj ⋆

ZTj DT

hu, j G j + GTj − Ph, j

]

= Ψ j > 0 (52)

for j = 1,2, . . . ,M and

[Wj +Wℓ ⋆

ZTj DT

hu,ℓ+ ZTℓ DT

hu, j Ψ22, jℓ

]

= Ψ jℓ > 0 (53)

with

Ψ22, jℓ = G j + GTj + Gℓ+ GT

ℓ − Ph, j − Ph,ℓ

for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M and

Zp −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N, (54)

with

Ph, j =N

∑i=1

(

fj

i + hji

)

Ph,i, Ph, j =N

∑i=1

fj

i Ph,i,

G j =N

∑i=1

fj

i Gi, Z j =N

∑i=1

fj

i Zi, and Wj =N

∑i=1

fj

i Wi.

then the parameter-dependent full state feedback gain

K(αk) = Z(αk)G(αk)−1 (55)

with

Z(αk) =N

∑i=1

αi(k)Zi and G(αk) =N

∑i=1

αi(k)Gi (56)

stabilizes the the system Hh with a guaranteed (weighted)

control energy bounded by Zu given by

Zu = minP∞,i,Pσ ,i,Gi,Zi ,Wi

maxi

trace{Wi}

≥ trace{Dhu(α)K(α)Ph(α)K(α)T Dhu(α)T }= Zu(αk)

(57)

while also ensuring that the hard constraint Zp is satisfied.

The equality for Zu(αk) is obtained by substituting the

value for Cu(αk) from (44) into the expression (23) and

by noting that the Lyapunov variable is represented here

by Ph(α). Additionally, consider the system H∞, given by

(39). If there exist, for i = 1,2, . . . ,N, symmetric positive-

definite matrices P∞,i ∈ Rn×n such that

P∞, j ⋆ ⋆ ⋆GT

j ATj + ZT

j BTu, j G j + GT

j − P∞, j ⋆ ⋆

BT∞, j 0 ηI ⋆

0 C∞, jG j + D∞u, jZ j D∞, j ηI

= Θ j > 0

(58)

Submission to International Journal of Control, Automation, and Systems 7

for j = 1,2, . . . ,M and

P∞, j + P∞,ℓ ⋆ ⋆ ⋆Θ21, jℓ Θ22, jℓ ⋆ ⋆

BT∞, j + BT

∞,ℓ 0 2ηI ⋆

0 Θ42, jℓ D∞, j + D∞,ℓ 2ηI

= Θ jℓ > 0

(59)

with

Θ21, jℓ = GTj AT

ℓ + GTℓ AT

j + ZTj BT

u,ℓ+ ZTℓ BT

u, j

Θ22, jℓ = G j + GTj + Gℓ+ GT

ℓ − P∞, j − P∞,ℓ

Θ42, jℓ = C∞, jGℓ+ C∞,ℓG j + D∞u, jZℓ+ D∞u,ℓZ j

for j = 1,2, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, where

P∞, j =N

∑i=1

(

fj

i + hji

)

P∞,i, P∞, j =N

∑i=1

fj

i P∞,i,

then the parameter-dependent full-state feedback gain K(αk)given by (55) also stabilizes the system H∞ with a guaran-

teed H∞ performance bounded by η .

Proof: The proof is organized as follows. The follow-

ing properties are a consequence of applying Theorems 8

and 9 of [9]:

• The system H∞ is stabilized with a guaranteed H∞ per-

formance bounded by η when the LMIs (58) and (59) are

satisfied.

• The system Hh is stabilized with a guaranteed (weighted)

control energy bounded by Zu (57) when the LMIs (50),

(51), (52), and (53) are satisfied.

However, the fact that the output constraint (20) is satisfied

for i = 1,2, . . . ,N follows from the LMI constraint (54)

Z p −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N.

Since the LMIs (50),(51),(52), and (53) are all required to

be positive-definite, from [9] it can be shown that

Ph(αk) =N

∑i=1

αi(k)Ph,i > P(αk), ∀k ≥ 0,

where P(αk) is the controllability Gramian satisfying (22).

Thus, it is also true that

Z p −Cp(αk)P(αk)Cp(αk)T ≥ 0

such that

Zp(αk) =Cp(αk)P(αk)Cp(αk)T ≤ Zp.

�

4.2. Output Feedback LMIs

The feedback structure provided by the gain-scheduled

dynamic output-feedback controller (41), produces the closed-

loop systems Hhcl and H∞

cl in (42) and (43) where x(k) =[xT

p (k) xTc (k)]

T with the closed-loop system matrices given

by

A (αk) =

[A(αk) Bu(αk)Cc(αk)

Bc(αk)Cy(αk) Ac(αk)

]

,

Bh(αk) =

[Bh(αk)

Bc(αk)Dyh(αk)

]

,

Cp(αk) =[

Cp(αk) 0],

Cu(αk) =[

0 Dhu(αk)K(αk)],

B∞(αk) =

[B∞(αk)

Bc(αk)Dy∞(αk)

]

,

C∞(αk) =[

C∞(αk) D∞u(αk)Cc(αk)],

D∞(αk) = D∞(αk).

(60)

As for the full-state feedback problem, the parameter-

dependent, strictly proper output feedback controller is

also solved for by performing a convex optimization over

a set of linear matrix inequalities. It is clear that when sub-

stituting the closed-loop matrices (60) into the matrix in-

equalities of the performance conditions in Lemma 1 and

Theorem 3 nonlinear matrix inequalities result, due to the

multiplication between the unknown controller matrices

and the slack variable G (αk). Also, to obtain a set of LMI

conditions to synthesize an LPV controller that will rely

on measurements of only the current time-varying param-

eter αk, it is necessary to enforce the slack variable G (αk)to be independent of the scheduling parameter αk, such

that G (αk) = G (see [23]). Then the slack variable G , its

inverse G −1, the Lyapunov matrix P(αk), and the con-

troller matrices K (αk) are partitioned as

G :=

[X Z1

U Z2

]

, G−1 :=

[Y T Z3

V T Z4

]

,

P(αk) :=

[P(αk) P2(αk)

P2(αk)T P3(αk)

]

,

K (αk) :=

[Ac(αk) Bc(αk)Cc(αk) 0

]

.

From the definition of G and G −1, it is clear that the fol-

lowing relationship must hold:

G G−1 =

[X Z1

U Z2

][Y T Z3

V T Z4

]

=

[XY T +Z1V T XZ3 +Z1Z4

UY T +Z2V T UZ3 +Z2Z4

]

=

[I 0

0 I

]

,

such that XY T + Z1V T = I and UY T + Z2V T = 0. Now,

the parameter-independent transformation matrix

T :=

[I Y T

0 V T

]

(61)

8 Submission to International Journal of Control, Automation, and Systems

used in [6–8,23] is introduced and the following nonlinear

parameter-dependent change of variables are defined:

[Q(αk) F(αk)L(αk) 0

]

:=

[Y

0

]

A(αk)[

X 0]

+

[V Y Bu(αk)0 I

]

K (αk)

[U 0

Cy(αk)X I

]

,

(62)

[P(αk) J(αk)J(αk)

T H(αk)

]

:= TTPT , (63)

S := Y X +VU. (64)

The nonlinear matrix inequalities that result from sub-

stituting the closed-loop matrices into the matrix inequal-

ities in Theorem 3 can be transformed into the LMIs (65),

(66), and (67) by using the congruence transformations

T1 = diag(T ,T , I) and T2 = diag(I,T )

on the first and second nonlinear matrix inequalities, re-

spectively. While the matrix inequalities in (65), (66), and

(67) are linear, they are still infinite dimensional as they

must be evaluated for all values of time-varying parameter

αk. To obtain a finite-dimensional set of LMIs, the param-

eter dependent structure imposed on the Lyapunov matrix

in (45) is imposed on the Lyapunov matrix here as well.

In the following, the same parameterizations used for the

state-feedback case are utilized for the gain-scheduled dy-

namic output-feedback control problem.

Theorem 5: Consider the system Hh, given by (38).

Assume that the vectors f j and h j of Γb are given. Given

Z p, if there exists matrices X ∈ Rn×n, Y ∈ R

n×n, and for

i = 1,2, . . . ,N, matrices, Jh,i ∈ Rn×n, Li ∈ R

m×n, and Fi ∈R

n×q, and symmetric positive-definite matrices Ph,i ∈Rn×n,

Hh,i ∈Rn×n, and Wi ∈ R

ph×ph such that

Ph, j Jh, j Ψ13, j A j Bh, j

⋆ Hh, j Q j Ψ24, j Ψ25, j

⋆ ⋆ Ψ33, j Ψ34, j 0

⋆ ⋆ ⋆ Ψ44, j 0

⋆ ⋆ ⋆ ⋆ I

= Ψ j > 0 (68)

with

Ψ13, j = A jX + Bu, jL j

Ψ24, j = YA j + FjCy, j

Ψ25, j = YBh, j + FjDyh, j

Ψ33, j = X +XT − Ph, j

Ψ34, j = I+ ST − Jh, j

Ψ44, j = Y +YT − Hh, j

for j = 1, . . . ,M, and

Ψ11, jℓ Ψ12, jℓ Ψ13, jℓ A j + Aℓ Bh, j + Bh,ℓ

⋆ Ψ22, jℓ Q j + Qℓ Ψ24, jℓ Ψ25, jℓ

⋆ ⋆ Ψ33, jℓ Ψ34, jℓ 0

⋆ ⋆ ⋆ Ψ44, jℓ 0

⋆ ⋆ ⋆ ⋆ 2I

= Ψ jℓ > 0

(69)

with

Ψ11, jℓ = Ph, j + Ph,ℓ

Ψ12, jℓ = J j + Jℓ

Ψ13, jℓ = A jX + AℓX + Bu, jLℓ+ Bu,ℓL j

Ψ22, jℓ = Hh, j + Hh,ℓ

Ψ24, jℓ = YA j +YAℓ+ FjCy,ℓ+ FℓCy, j

Ψ25, jℓ = YBh, j +YBh,ℓ+ FjDyh,ℓ+ FℓDyh, j

Ψ33, jℓ = 2X + 2XT − Ph, j − Ph,ℓ

Ψ34, jℓ = 2I+ 2ST − Jh, j − Jh,ℓ

Ψ44, jℓ = 2Y + 2YT − Hh, j − Hh,ℓ

for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, and

Wj Dhu, jL j 0

⋆ X +XT − Ph, j I + ST − Jh, j

⋆ ⋆ Y +Y T − Hh, j

= Φ j > 0

(70)

for j = 1, . . . ,M, and

Wj +Wℓ Φ12, jℓ 0

⋆ Φ22, jℓ Φ23, jℓ

⋆ ⋆ Φ33, jℓ

= Φ jℓ > 0 (71)

with

Φ12, jℓ = Dhu, jLℓ+ Dhu,ℓL j

Φ22, jℓ = 2X + 2XT − Ph, j − Ph,ℓ

Φ23, jℓ = 2I+ 2ST − Jh, j − Jh,ℓ

Φ33, jℓ = 2Y + 2YT − Hh, j − Hh,ℓ

for j = 1, . . . ,M− 1 and ℓ= j+ 1, . . . ,M, and

Zp −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N, (72)

where

Ph, j =N

∑i=1

( fj

i + hji )Ph,i,

Jh, j =N

∑i=1

( fj

i + hji )Jh,i,

Hh, j =N

∑i=1

( fj

i + hji )Hh,i,

(73)

Submission to International Journal of Control, Automation, and Systems 9

P(αk) J(αk) A(αk)X +Bu(αk)L(αk) A(αk) Bw(αk)

⋆ H(αk) Q(αk) YA(αk)+F(αk)Cy(αk) Y Bw(αk)⋆ ⋆ X +XT −P(αk) I + ST − J(αk) 0

⋆ ⋆ ⋆ Y +Y T −H(αk) 0

⋆ ⋆ ⋆ ⋆ I

> 0 (65)

W (αk) Dzu(αk)L(αk) 0

⋆ X +XT −P(αk) I+ ST − J(αk)⋆ ⋆ Y +YT −H(αk)

> 0 (66)

Zp −Cp(αk)P(αk)Cp(αk)T > 0 (67)

and

Ph, j =N

∑i=1

fj

i Ph,i, Jh, j =N

∑i=1

fj

i Jh,i,

Hh, j =N

∑i=1

fj

i Hh,i, Q j =N

∑i=1

fj

i Qi,

L j =N

∑i=1

fj

i Li, Fj =N

∑i=1

fj

i Fi,

A j =N

∑i=1

fj

i Ai, Bu, j =N

∑i=1

fj

i Bu,i,

Bw, j =N

∑i=1

fj

i Bw,i, Cz, j =N

∑i=1

fj

i Cz,i,

Dzu, j =N

∑i=1

fj

i Dzu,i, Dzw, j =N

∑i=1

fj

i Dzw,i,

Cy, j =N

∑i=1

fj

i Cy,i, Dyw, j =N

∑i=1

fj

i Dyw,i,

(74)

then the parameter-dependent, strictly proper output feed-

back controller

xc(k+ 1) = Ac(αk)xc(k)+Bc(αk)y(k),

u(k) =Cc(αk)xc(k),(75)

with matrices computed by

Ac(αk) =V−1

(

Q(αk)−YA(αk)X

−YBu(αk)L(αk)−F(αk)Cy(αk)X

)

U−1,

Bc(αk) =V−1F(αk),

Cc(αk) = L(αk)U−1,

(76)

stabilizes the system Hh with a guaranteed (weighted) con-

trol energy bounded by Zu while also ensuring that the

hard constraint Zp is satisfied. Additionally, consider the

system H∞, given by (39). If there exist, for i= 1,2, . . . ,N,

matrices J∞,i ∈ Rn×n and symmetric positive-definite ma-

trices P∞,i ∈Rn×n and H∞,i ∈ R

n×n such that

P∞, j J∞, j Θ13, j A j B∞, j 0

⋆ H∞, j Q j Θ24, j Θ25, j 0

⋆ ⋆ Θ33, j Θ34, j 0 Θ36, j

⋆ ⋆ ⋆ Θ44, j 0 CT∞, j

⋆ ⋆ ⋆ ⋆ ηI DT∞, j

⋆ ⋆ ⋆ ⋆ ⋆ ηI

= Θ j > 0

(77)

with

Θ13, j = A jX + Bu, jL j

Θ24, j = Y A j + FjCy, j

Θ25, j = Y B∞, j + FjDy∞, j

Θ33, j = X +XT − P∞, j

Θ34, j = I+ ST − J∞, j

Θ36, j = XTCT∞, j + LT

j DT∞u, j

Θ44, j = Y +Y T − H∞, j

for j = 1, . . . ,M, and

Θ11, jℓ Θ12, jℓ Θ13, jℓ A j + Aℓ Θ15, jℓ 0

⋆ Θ22, jℓ Q j + Qℓ Θ24, jℓ Θ25, jℓ 0

⋆ ⋆ Θ33, jℓ Θ34, jℓ 0 Θ36, jℓ

⋆ ⋆ ⋆ Θ44, jℓ 0 Θ46, jℓ

⋆ ⋆ ⋆ ⋆ 2ηI Θ56, jℓ

⋆ ⋆ ⋆ ⋆ ⋆ 2ηI

= Θ j > 0

(78)

10 Submission to International Journal of Control, Automation, and Systems

with

Θ11, jℓ = P∞, j + P∞,ℓ

Θ12, jℓ = J∞, j + J∞,ℓ

Θ13, jℓ = A jX + AℓX + Bu, jLℓ+ Bu,ℓL j

Θ15, jℓ = B∞, j + B∞,ℓ

Θ22, jℓ = H∞, j + H∞,ℓ

Θ24, jℓ = Y A j +YAℓ+ FjCy,ℓ+ FℓCy, j

Θ25, jℓ = Y B∞, j +YB∞,ℓ+ FjDy∞,ℓ+ FℓDy∞, j

Θ33, jℓ = 2X + 2XT − P∞, j − P∞,ℓ

Θ34, jℓ = 2I+ 2ST − J∞, j − J∞,ℓ

Θ36, jℓ = XTCT∞, j +XTCT

∞,ℓ+ LTj DT

∞u,ℓ+ LTℓ DT

∞u, j

Θ44, jℓ = 2Y + 2YT − H∞, j − H∞,ℓ

Θ46, jℓ = CT∞, j + CT

∞,ℓ

Θ56, jℓ = DT∞, j + DT

∞,ℓ

for j = 1, . . . ,M − 1 and ℓ= j+ 1, . . . ,M with

P∞, j =N

∑i=1

( fj

i + hji )P∞,i, P∞, j =

N

∑i=1

fj

i P∞,i,

J∞, j =N

∑i=1

( fj

i + hji )J∞,i, J∞, j =

N

∑i=1

fj

i J∞,i,

H∞, j =N

∑i=1

( fj

i + hji )H∞,i, H∞, j =

N

∑i=1

fj

i H∞,i.

(79)

then the parameter-dependent, strictly proper output feed-

back controller also stabilizes the system H∞ with a guar-

anteed H∞ performance bounded by η .

Proof: The proof, which is similar to the proof for The-

orem 4, is organized as follows:

• The system H∞ is stabilized with a guaranteed H∞ per-

formance bounded by η when the LMIs (77) and (78) are

satisfied.

• The system Hh is stabilized with a guaranteed (weighted)

control energy bounded by Zu when the LMIs (68), (69),

(70), and (71) are satisfied.

The output constraint (20) is satisfied for i = 1,2, . . . ,N as

a result of the LMI constraint (72)

Z p −Cp,iPh,iCTp,i ≥ 0, i = 1,2, . . . ,N.

Since the LMIs (68), (69), (70), and (71) are all required

to be positive definite, it can be shown that

[Ph(αk) Jh(αk)Jh(αk)

T Hh(αk)

]

=N

∑i=1

αi(k)

[Ph,i Jh,i

JTh,i Hh,i

]

> P(αk), ∀k ≥ 0,

where P(αk) is the controllability Gramian satisfying (22).

Thus, it is also true that

Zp −Cp(αk)P(αk)Cp(αk)T ≥ 0

such that

Zp(αk) = Cp(αk)P(αk)Cp(αk)T ≤ Zp.

�

5. NUMERICAL EXAMPLE

The approach proposed in this paper is demonstrated

with a numerical example. Consider the discrete-time LPV

system (originally used in [24], and later used in [25] and

[8])

xp(k+ 1) =

2+ δ1 0 1

1 0.5 0

0 1 −0.5

︸ ︷︷ ︸

A(δ1(k))

xp(k)

+

1+ δ2

0

0

︸ ︷︷ ︸

Bu(δ2(k))

u(k)+

0

1

0

︸ ︷︷ ︸

Bh

wp(k)

zp(k) =

1 0 0

0 1 0

0 0 1

︸ ︷︷ ︸

Cp

xp(k)

zu(k) = u(k)

(80)

where δi, i = 1,2 are the time-varying parameters. In this

section, two design examples with different performance

constraints are considered for both state-feedback and dy-

namic output-feedback control.

5.1. State Feedback Control

For the state-feedback controller design, the time-varying

parameters were assumed to have the following parameter

variation bounds:

δ1 ∈ [−1, 1], and δ2 ∈ [−0.5, 0.5]. (81)

The discrete-time LPV system (80) is converted to the

discrete-time polytopic LPV system (38) by solving A(δ1)and Bu(δ2) at the vertices of the parameter space polytope

of δ1 and δ2. The exogenous ℓ2 disturbance wp is a scalar

and the performance variable zp has three components.

In the following, we consider two different ℓ2 to ℓ∞ gain

designs. The designs differ in the grouping of the perfor-

mance variables inside of zp used to define the constraints

(20). The constraints for each design are given as follows:

Design 1: Zp ≤ 1.85× I3, (82)

Design 2: Zp,1 ≤ 1.85, Zp,2 ≤ 1.85× I2, (83)

Submission to International Journal of Control, Automation, and Systems 11

where for design 1, Zp denotes the (3 × 3) output co-

variance matrix corresponding to the all performance out-

puts in zp grouped together. In design 2, Zp,1 denotes the

(1× 1) output variance corresponding to the first perfor-

mance output of zp and Zp,2 denotes the (2× 2) output

covariance matrix corresponding to the second and third

performance outputs grouped together.

For each design, to enhance the robustness of the closed-

loop system with the controller K(αk) with respect to un-

certainty in the measurements of the time-varying param-

eters δ1 and δ2, the closed-loop H∞ norms of the trans-

fer functions of some appropriately defined extra inputs

and outputs that ‘pull out’ [8, 24] the uncertain parame-

ters are bounded. Specifically, the following H∞ system is

defined:

H∞ =

x(k+ 1) = A(δ1(k))x(k)+Bu(δ2(k))u(k)

+

1

0

0

w∞,1(k)+

1

0

0

w∞,2(k)

z∞,1(k) =[

1 0 0]

x(k)

z∞,2(k) = u(k)(84)

so that the robustness requirement is given by

‖Hz∞,iw∞,i(α)‖∞ < η = 100, i = 1,2, (85)

where η defines the robustness level. Note that the nota-

tion used here, specifically w∞,1(k) and w∞,2(k) with the

same input matrix, was selected to match what is found in

the literature [8, 24].

For each of the ℓ2-ℓ∞ designs (82)-(83), the LMIs in

Theorem 4 are programmed into MATLAB using the LMI

parser YALMIP [26] and solved with SeDuMi [27] to min-

imize the cost function (57). As shown in Fig. 1 and

Fig. 3A, each design is feasible and the achieved covari-

ance bound is tight with the design bound in at least one

dimension. The constraint in design 1 ensures that the

covariance bound ellipsoid of Zp remains inside of the

sphere displayed in Fig. 1A. Side views of the covariance

bound Zp are displayed in Fig. 1B, Fig. 1C, and Fig. 1D.

As displayed in Fig. 1C, the output covariance Zp is tight

with the bound in the zp,2-zp,3 plane.

For design 2, the constraints ensure that the variance of

the first output of zp will be below 1.85 and the covariance

bound of second and third outputs of zp will remain inside

of the circle in Fig. 3A. The dashed ellipses in Fig. 3A are

the obtained output covariances at each of the vertices for

i = 1, . . . ,4, and as shown they are tight with the bound.

To test the performance of each design, we simulate

each of the controllers with a positive impulse (I1), an

ℓ2 excitation, followed by a negative impulse (I2) as dis-

played in Fig. 4A. To see the effect of the time-varying

parameters, the parameters δ1 and δ2 are varied as dis-

played in Fig. 4B. The values used to compute the con-

troller at each time step k are the noisy measurements dis-

played with a gray dashed line. The response to the ℓ2

zp,1

zp,1zp,1

zp,2

z p,2

zp,2

z p,3

z p,3

z p,3

A B

C D 2

2

2

2

2

2

2

22

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0 0

-1

-1

-1

-1

-1

-1

-1

-2-2

-2-2

-2-2

-2

-2 -2

Fig 1: Design 1: The covariance bound Zp achieved com-

pared to the constraint (82).

disturbance wp(k) for design 1 is displayed in Fig. 2. The

response in Fig. 2 is plotted inside of the ℓ∞ norm con-

straint (the square root of the covariance bound) sphere

and the achieved ℓ∞ norm bound ellipsoid. In Fig. 3B, the

response of design 2 is plotted inside of the ℓ∞ norm con-

straint circle and the achieved ℓ∞ norm bound ellipse. The

path of the response, with respect to each of the impulses

(I1) and (I2), is also displayed in Fig. 3B. As shown in

Fig. 2 and Fig. 3B, the response for each design stays in-

side of the ℓ∞ bound.

5.2. Dynamic Output-Feedback Control

For the dynamic output-feedback controller design, the

measurement equation in [8, 24, 25] is given as

y(k) =[

0 1+ δ3(k) 0]

x(k), (86)

where δ3 is an additional time-varying parameter. As one

might guess, this is much more restrictive than the state

feedback case. The good news is that the system is still

observable (assuming δ3 6=−1). However, the bad news is

that in order to obtain a feasible controller with the LMIs

provided by Theorem 5, some modifications need to be

made to the problem. The modified discrete-time LPV is

12 Submission to International Journal of Control, Automation, and Systems

zp,1

zp,1zp,1

zp,2

z p,2

zp,2

z p,3

z p,3

z p,3

A B

C D 1.51.5

1.51.5

11

11

1

0

0

0

0

0

0

0

0 0

-1-1

-1-1 -1

-1.5-1.5

-1.5

-1.5

Fig 2: Design 1: The output response of zp,1, zp,2, and

zp,3 plotted against each other for design 1 simu-

lated with a positive (I1) and negative (I2) impulse

function and compared with the ℓ∞ norm bound.

zp,2zp,2

z p,3

z p,3 I1I2

A B2

2

1.5

1.5

1

1

0

0

0

0

-1

-1-1.5

-1.5-2-2

Fig 3: Design 2: A. The covariance bound Zp,2 achieved

compared to the second constraint in (83). B. The

output response of zp,2 plotted against zp,3 for de-

sign 2 simulated with a positive and negative im-

pulse function and compared with the ℓ∞ norm

bound.

δ1(k)

δ2(k)

k

wp(k)

A

B

I1

I2

1

1

0.5

0

0

0

0-0.5

-1

-1

100 200 300

-0.8 -0.6 -0.4 -0.2 0.80.60.40.2

True

Measured

Fig 4: The ℓ2 disturbance (A) and the parameter variation

(B) used to simulate each controller design.

given by

xp(k+ 1) =

2+ δ (k) 0 1

1 0.5 0

0 1 −0.5

︸ ︷︷ ︸

A(δ (k))

xp(k)

+

1+ δ (k)0

0

︸ ︷︷ ︸

Bu(δ (k))

u(k)+

0

0.10

︸ ︷︷ ︸

Bh

wp(k),

zp(k) =

1 0 0

0 1 0

0 0 1

︸ ︷︷ ︸

Cp

xp(k),

zu(k) = u(k),

y(k) =[

0 1+ δ (k) 0]

︸ ︷︷ ︸

Cy(δ (k))

x(k)+ 0.01v(k).

(87)

Notice that in the modified system, each of the time-varying

parameters δi, i = 1,2,3 have been set equal to each other

such that δ = δ1 = δ2 = δ3, which was originally done

by the authors in [25]. Also, as in [25], the time-varying

parameter δ is assumed to have the following parameter

variation bound:

δ ∈ [−0.2, 0.2]. (88)

As for the state-feedback design case, the discrete-time

LPV system (87) is converted to the discrete-time poly-

topic LPV system (38) by solving A(δ ), Bu(δ ), and Cy(δ )at the vertices of the parameter space polytope of δ . For

the dynamic output-feedback design, the exogenous ℓ2 dis-

turbance is given by the process disturbance wp(k) and the

measurement disturbance v(k), such that w(k)= [wp(k), v(k)]T .

The performance variable zp, again has three components.

As we did for the state-feedback control example, we

also consider two different ℓ2 to ℓ∞ gain designs for the

dynamic output-feedback control example. As before, the

designs differ in the grouping of the performance variables

inside of zp used to define the constraints (20). The con-

straints for each design are given as follows:

Design 1: Zp ≤ 5× I3, (89)

Design 2: Zp,1 ≤ 5, Zp,2 ≤ 5× I2, (90)

where for design 1, Zp denotes the (3 × 3) output co-

variance matrix corresponding to the all performance out-

puts in zp grouped together. In design 2, Zp,1 denotes the

(1× 1) output variance corresponding to the first perfor-

mance output of zp and Zp,2 denotes the (2× 2) output

covariance matrix corresponding to the second and third

performance outputs grouped together.

As in the state-feedback control example, H∞ perfor-

mance criteria is used to enhance the robustness of the

Submission to International Journal of Control, Automation, and Systems 13

zp,1

zp,1zp,1

zp,2

z p,2

zp,2

z p,3

z p,3

z p,3

A B

C D 5

5

5

5

5

5

5

55

0

0

0

0

0

0

0

0 0

-5-5

-5-5

-5-5

-5

-5 -5

Fig 5: Design 1: The covariance bound Zp achieved com-

pared to the constraint (82).

closed-loop system using the dynamic output-feedback con-

troller with respect to uncertainty in the measurements of

the time-varying parameter δ . The system H∞ used for

the dynamic output-feedback design is the same as given

in (84), with the following additions:

y(k) =Cy(δ (k))x(k)+w∞,3(k)

z∞,3(k) =[

0 1 0]

x(k)(91)

such that the robustness requirement is now given by

‖Hz∞,iw∞,i(α)‖∞ < η = 100, i = 1,2,3, (92)

where η defines the robustness level.

For each of the ℓ2 to ℓ∞ designs (89)-(90), the LMIs

in Theorem 5 are programmed into MATLAB and solved

with LMI Lab [28] to minimize the control energy Zu.

As shown in Fig. 5 and Fig. 7A, each design is feasible

and the achieved covariance bound is tight with the de-

sign bound in at least one dimension. The constraint in

design 1 ensures that the covariance bound ellipsoid of

Zp remains inside of the sphere displayed in Fig. 1A. Side

views of the covariance bound Zp are displayed in Fig. 5B,

Fig. 5C, and Fig. 5D. As displayed in Fig. 5C, the output

covariance Zp is tight with the bound in the zp,2-zp,3 plane.

For design 2, the constraints ensure that the variance of

the first output of zp will be below 5 and the covariance

bound of second and third outputs of zp will remain inside

of the circle in Fig. 7A. The dashed ellipses in Fig. 7A are

the obtained output covariances at each of the vertices for

i = 1,2, and as shown they are tight with the bound.

To test the performance of each design, we again sim-

ulate each of the controllers with a positive impulse (I1)

followed by a negative impulse (I2) as displayed in Fig. 8A.

To see the effect of the time-varying parameter, the param-

eter δ was varied as displayed in Fig. 8B. As before, the

zp,1

zp,1zp,1

zp,2

z p,2

zp,2

z p,3

z p,3

z p,3

A B

C D2

2

2

2

2

2

2

22

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0 0

-1

-1

-1

-1

-1

-1

-1

-2

-2

-2

-2

-2

-2

-2

-2 -2

Fig 6: Design 1: The output response of zp,1, zp,2, and

zp,3 plotted against each other for design 1 simu-

lated with a positive (I1) and negative (I2) impulse

function and compared with the ℓ∞ norm bound.

zp,2zp,2

z p,3

z p,3

I1

I2

A B5

5

2

2

1

1

0

0

0

0

-1

-1

-2

-2-5

-5

Fig 7: Design 2: A. The covariance bound Zp,2 achieved

compared to the second constraint in (83). B. The

output response of zp,2 plotted against zp,3 for de-

sign 2 simulated with a positive and negative im-

pulse function and compared with the ℓ∞ norm

bound.

k

δ(k)

k

wp(k)

A

B

I1

I2

1

0.5

0

0

0

0

-0.5

-1

0.2

0.1

-0.1

-0.2

100

100

200

200

300

300

True

Measured

Fig 8: The ℓ2 disturbance (A) and the parameter variation

(B) used to simulate each controller design.

14 Submission to International Journal of Control, Automation, and Systems

values used to compute the controller at each time step k

are the noisy measurements displayed with a gray dashed

line. The response to the ℓ2 disturbance wp(k) for design 1

is displayed in Fig. 6. The response in Fig. 6 is plotted

inside of the ℓ∞ norm constraint (the square root of the co-

variance bound) sphere and the achieved ℓ∞ norm bound

ellipsoid. In Fig. 7B, the response of design 2 is plotted

inside of the ℓ∞ norm constraint circle and the achieved ℓ∞

norm bound ellipse. The path of the response, with respect

to each of the impulses (I1) and (I2), is also displayed in

Fig. 7B. As shown in Fig. 6 and Fig. 7B, the response for

each design stays inside of the ℓ∞ bound. It is interesting

to note that the response to the positive impulse (I1) is

larger than the response to the negative impulse (I2). This

is caused by the different time-varying parameter δ (k) val-

ues at the time of each impulse, as is displayed in Fig. 8.

6. CONCLUSION

In this paper, discussion motivating the necessity of the

ℓ2 to ℓ∞ gain performance criteria was provided. Then, the

ℓ2 to ℓ∞ gain performance criteria was introduced to allow

for the specification of hard constraints when designing

gain-scheduling controllers. Controller synthesis LMIs

are provided for the synthesis of minimum energy state-

feedback and dynamic output-feedback controllers with

guaranteed ℓ2 to ℓ∞ gain and H∞ performance. To demon-

strate the effectiveness of the controller synthesis LMIs

provided in this paper, gain-scheduled state-feedback and

dynamic output-feedback controllers are designed for a

numerical example. One of the future work will be to

apply our approach to a practical application such as en-

gine control problems [13–15] with hard constraints on

responses or actuators.

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Andrew White received his Ph.D., M.S.

and B.S. degrees in Mechanical Engi-

neering from Michigan State University

in 2012, 2008 and 2006 respectively. Cur-

rently, he is working in the powertrain

virtual analysis department at Chrysler.

His research interests include adaptive

and robust control, and parameter esti-

mation, with applications to engine con-

trol, robotics, and non-destructive testing and evaluation. He

is a member of IEEE and ASME.

Dr. Guoming (George) Zhu is a profes-

sor of mechanical engineering and elec-

trical/computer engineering at Michigan

State University. Prior to joining the ME

and ECE departments, he was a techni-

cal fellow in advanced powertrain sys-

tems at the Visteon Corporation. He also

worked for Cummins Engine Co. as a

technical advisor. Dr. Zhu earned his

PhD (1992) in aerospace engineering at Purdue University.

His BS and MS degrees (1982 and 1984 respectively) were

from Beijing University of Aeronautics and Astronautics in

China. His current research interests include closed-loop com-

bustion control, adaptive control, closed-loop system identi-

fication, LPV control of automotive systems, hybrid power-

train control and optimization, and thermoelectric generator

management system. Dr. Zhu has over 30 years of experience

related to control theory and applications. He has authored or

co-authored more than 140 refereed technical papers and re-

ceived 40 US patents. He was an associate editor for ASME

Journal of Dynamic Systems, Measurement and Control and

a member of editorial board of International Journal of Pow-

ertrain. Dr. Zhu is a Fellow of SAE and ASME.

Jongeun Choi received his Ph.D. and

M.S. degrees in Mechanical Engineer-

ing from the University of California at

Berkeley in 2006 and 2002 respectively.

He also received a B.S. degree in Me-

chanical Design and Production Engineer-

ing from Yonsei University at Seoul, Re-

public of Korea in 1998. He is currently

an Associate Professor with the Depart-

ments of Mechanical Engineering and Electrical and Com-

puter Engineering at the Michigan State University. His cur-

rent research interests include systems and control, LPV and

robust control, system identification, and Bayesian methods,

with applications to mobile robotic sensors, environmental

adaptive sampling, engine control, human motor control sys-

tems, and biomedical problems. He is an Associate Editor for

the Journal of Dynamic Systems, Measurement and Control

(JDSMC) and a Co-Editor for a Special Issue on Stochas-

tic Models, Control, and Algorithms in Robotics in JDSMC.

His papers were finalists for the Best Student Paper Award

at the 24th American Control Conference (ACC) 2005 and

the Dynamic System and Control Conference (DSCC) 2011

and 2012. He was a recipient of an NSF CAREER Award in

2009. Dr. Choi is a member of ASME and IEEE.