a linear matrix inequality solution to the input

A LINEAR MATRIX INEQUALITY SOLUTION TO THE INPUT COVARIANCECONSTRAINT CONTROL PROBLEM

Andrew White, Guoming Zhu, Jongeun Choi

Department of Mechanical Engineering

Michigan State UniversityEast Lansing, Michigan 48824

Email: {whitea23, zhug, jchoi}@egr.msu.edu

ABSTRACT

In this paper, the input covariance constraint (ICC) con-

trol problem is solved by a convex optimization with linear ma-

trix inequality (LMI) constraints. The ICC control problem is an

optimal control problem that is concerned with finding the best

output performance possible subject to multiple constraints on

the input covariance matrices. The contribution of this paper is

the characterization of the control synthesis LMIs used to solve

the ICC control problem. To demonstrate the effectiveness of the

proposed approach a numerical example is solved with the con-

trol synthesis LMIs. Both discrete and continuous-time problems

are considered.

INTRODUCTION

In this paper, we consider the input covariance constraint

(ICC) control problem. The ICC control problem is an optimal

control problem in which the output performance is minimized

subject to multiple constraints on the control input covariance

matrices Ui of the form Ui ≤U i, where U i is given. The ICC con-

trol problem has two interesting interpretations: stochastic and

deterministic. For the stochastic interpretation the exogenous in-

puts are assumed to be uncorrelated zero-mean white noises with

a given intensity. With the exogenous input defined in this way,

the ICC control problem minimizes the weighted performance

output covariance subject to the control input covariance con-

straints, such that the constraints can be interpreted as constraints

on the variance of the control actuation. For the deterministic in-

terpretation the exogenous inputs are assumed to be unknown

disturbances that belong to a bounded ℓ2 energy set. Then the

ICC control problem minimizes the maximum singular value of

the performance outputs while ensuring that the maximum sin-

gular value of the control inputs are less than the corresponding

control input constraints. In other words, the ICC control prob-

lem is the problem of minimizing the weighted sum of worst-case

peak values on the performance outputs subject to the constraints

on the worst-case peak values of the control input. This inter-

pretation is important in applications where hard constraints on

the actuator signals are present, such as space telescope pointing

control [1], system identification, and machine tool control.

The ICC control problem is closely related to the output co-

variance constraint (OCC) control problem which was originally

studied in [2]. The OCC control problem is an optimal control

problem that minimizes the control input subject to output co-

variance constraints. The OCC control problem is solved by a

linear quadratic Gaussian (LQG) controller with a special choice

of output weights, which can be obtained by using the iterative

OCC algorithm detailed in [2]. While the ICC control problem

can also technically be solved by an LQG controller with a spe-

cial choice of input weights, developing an iterative algorithm to

directly obtain such an input weighting matrix has been an ex-

tremely difficult problem to solve. However, after reconsidering

the OCC control problem as a convex optimization with linear

matrix inequality (LMI) constraints in [3], it became clear that

the more difficult ICC control problem could also be solved as a

convex optimization with LMI constraints.

This paper is organized as follows. First, the continuous-

time ICC control problem is introduced and then Theorems 1

and 2 provide LMIs that can be solved to obtain state-feedback

and dynamic output feedback controllers, respectively, that min-

imize an upper bound on the ICC cost. In the next section, the

discrete-time ICC control problem is presented and Theorems 3

and 4, which find state-feedback and dynamic output feedback

controllers that minimize the upper bound of the ICC cost, are

1 Copyright © 2013 by ASME

Proceedings of the ASME 2013 Dynamic Systems and Control Conference DSCC2013

October 21-23, 2013, Palo Alto, California, USA

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given. To demonstrate the effectiveness of the proposed ap-

proach, a numerical example is solved with the LMIs introduced

in this paper. Concluding remarks are given in the final section.

CONTINUOUS TIME SYSTEMS

Consider the following continuous-time system:

xp(t) = Apxp(t)+Bpu(t)+Dpwp(t)

yp(t) =Cpxp(t)

z(t) = Mpxp(t)+ v(t)

(1)

where xp, u, wp, and v represent the state, control, process noise,

and measurement noise. The vector yp contains all variables

whose dynamic responses are of interest and the vector z is a

vector of noisy measurements.

Suppose that we apply to the plant (1) a full state feedback

stabilizing control law of the form

u(t) = Gxp(t), (2)

or a strictly proper output feedback stabilizing control law given

by

xc(t) = Acxc(t)+Fz(t),

u(t) = Gxc(t).(3)

Then the resulting closed-loop system is

x(t) = Ax(t)+Dw(t)

y(t) =

[

yp(t)u(t)

]

=

[

Cy

Cu

]

x(t) =Cx(t)(4)

where for the state feedback case we have x = xp and w = wp,

while for the output feedback case we have x =[

xTp xT

c

]Tand

w =[

wTp vT

]T.

Considering the closed-loop system (4), let Wp and V de-

note positive definite symmetric matrices with dimensions equal

to the process noise wp and measurement vector z, respectively.

Then define W =Wp, if the state feedback controller (2) is used

or W = block diag [Wp,V ] if the strictly proper, output feedback

controller (3) is used. Let P denote the closed-loop controlla-

bility Gramian from the (weighted) disturbance input W−1/2w.

Since A is stable, P satisfies

0 = AP+ PAT +DWDT . (5)

The control input u(t) in (4) is partitioned into

u = [uT1 ,u

T2 , . . . ,u

Tm]

T , (6)

such that each ui for i = 1,2, . . . ,m is given by

ui =Cu,ixp = ΦiCuxp ∈ Rmi , (7)

where Φi is an appropriately selected projection matrix for each

input ui. In this paper, we are interested in finding controllers of

the form (2) or (3) that minimize the (weighted) output perfor-

mance trace(

QCpPCTp

)

with Q > 0, and satisfy the constraints

Ui = ΦiCuPCTu Φ

Ti ≤U i, i = 1,2, . . . ,m, (8)

where U i > 0 (i = 1,2, . . . ,m) are given and P solves (5). This

problem, which is called the input covariance constraint (ICC)

problem, is defined as follows.

The ICC Problem. Find a static state feedback or full-order

dynamic output feedback controller for the system (1) to mini-

mize the ICC cost

JICC = trace(

QCpPCTp

)

, Q > 0, (9)

subject to (5) and (8). �

In this paper, we consider a convex optimization solution to

the ICC problem using LMIs.

State Feedback

With the state feedback controller (2) the closed-loop system

matrices in (4) are given by

A = Ap +BpG, D = Dp, Cy =Cp, Cu = G. (10)

Theorem 1. There exists a controller in the form (2), given by

G = LP−1, (11)

that minimizes JICC (9) and satisfies the input constraints (8) if

there exists a matrix L ∈ Rm×n and a symmetric positive definite

matrix P ∈ Rn×n that minimize the upper bound of the ICC cost

JICC = minP,L

trace(

QCpPCTp

)

> trace(

QCpPCTp

)

= JICC, (12)

subject to the LMIs

[

PATp +ApP+LT BT

p +BpL DpW1/2p

⋆ −I

]

< 0, (13)

QCpPCTp > 0, (14)

[

U i ΦiL

⋆ P

]

> 0, (15)

for i = 1,2, . . . ,m.



Proof. According to [4], the Lyapunov equation (5) can be writ-

ten as the following inequality:

0 > AP+PAT +DWDT , (16)

where P = PT > 0. Notice that (16) is the Schur complement of

the following LMI:

[

AP+PAT DW 1/2

W 1/2DT −I

]

< 0. (17)

Since P = PT > 0, to ensure that (17) is satisfied the closed-loop

state matrix A must be Hurwitz. The LMI (13) is constructed

from the LMI (17) by first substituting the closed-loop matrices

(10) into the LMI (17), then by using the change of variables

L = GP, and finally by recalling that for state feedback W =Wp.

Notice that since (16) is less than zero, there exist a matrix

M = MT > 0 such that

0 = AP+PAT +DWDT +M. (18)

Consequently, P > P. From (14) it follows that

QCpPCTp > QCpPCT

p . (19)

Therefore,

JICC = trace(

QCpPCTp

)

> trace(

QCpPCTp

)

= JICC. (20)

Likewise, it follows from (15) that

U i > ΦiGPGTΦ

Ti > ΦiGPGT

ΦTi =Ui, (21)

for i = 1,2, . . . ,m, since

ΦiGPGTΦ

Ti = ΦiGPP−1PGT

ΦTi = ΦiLP−1LT

ΦTi . 2

Dynamic Output Feedback

The extension of the state feedback case to the full-order

dynamic output feedback case is straightforward. In fact, the

state feedback LMIs in Theorem 1 are applied to solve the

full-order dynamic output feedback OCC problem. It is known

that the performance of a full information state feedback con-

troller cannot be improved upon by the use of dynamic compen-

sation [4]. However, the full state information without corruption

from measurement noise is not usually available. Thus, under the

assumption that the pair (Mp,Ap) is detectable, a dynamic output

feedback controller can be designed by using the state estimator

xc(t) = Apxc(t)+Bpu(t)+F (z(t)−Mpxc(t)) (22)

where F = PMTp V−1 and P is the unique positive definite matrix

P that satisfies the Riccati equation [2]

0 = ApP+ PATp − PMT

p V−1MpP+DpWpDTp . (23)

Then, as shown in (3), the estimated states are used to compute

the control input such that the state estimator becomes

xc(t) = (Ap +BpG−FMp)xc(t)+Fz(t). (24)

Thus, the only remaining question is how to obtain the state feed-

back gain G, which is covered in the next theorem.

Theorem 2. If the pair (Mp,Ap) is detectable and there exists

a matrix L ∈ Rm×n and a symmetric positive definite matrix P ∈

Rn×n that minimize

JICC = minP,L

trace[

QCp

(

P+P)

CTp

]

, (25)

subject to the LMIs

[

PATp +ApP+LT BT

p +BpL FV 1/2

⋆ −I

]

< 0, (26)

QCp

(

P+P)

CTp > 0, (27)

[

U i ΦiL

⋆ P

]

> 0, (28)

for i = 1,2, . . . ,m where P is the unique positive solution to the

Riccati equation (23), then there exists a controller that mini-

mizes the ICC cost JICC while satisfying the input constraints (8)

in the form (3), given by

xc(t) = (Ap +BpG−FMp)xc(t)+Fz(t),

u(t) = Gxc(t),(29)

with G = LP−1 and F = PMTp V−1.

Proof. A proof of this theorem can be obtained by combining

Theorem 1 of this paper with Lemma 4.2 and Theorem 4.1 of [4].

One of the main results of [4] demonstrates that the ICC prob-

lem (and other H2 like problems) with dynamic output feed-

back reduces to an equivalent problem with state feedback. Thus

the output feedback problem can be solved by first designing a

standard Kalman filter with (23) and then using the state feed-

back synthesis LMIs in Theorem 1 after replacing Dp, Wp, and

QCpPCTp with F , V , and QCp

(

P+P)

CTp , respectively. 2



DISCRETE TIME SYSTEMSConsider the following discrete-time system:

xp(k+ 1) = Apxp(k)+Bpu(k)+Dpwp(k),

yp(k) =Cpxp(k),

z(k) = Mpxp(k)+ v(k).

(30)

Suppose that we apply to the plant (30) a full state feedback sta-

bilizing control

u(k) = Gx(k), (31)

or a strictly proper stabilizing control

xc(k+ 1) = Acxc(k)+Fz(k),

u(k) = Gxc(k).(32)

Then the closed-loop system has the following form:

x(k+ 1) = Ax(k)+Dw(k),

y(k) =

[

yp(k)u(k)

]

=

[

Cy

Cu

]

x(k) =Cx(k),(33)

where the definitions of matrices A, D, and C, and vectors x, w,

and y are the same as in the continuous-time case.

As in the continuous-time case, let Wp > 0 and V > 0 denote

symmetric matrices with dimensions equal to wp and z, respec-

tively. Also, define W = Wp if state feedback (31) is used or

W = block diag[Wp,V ] if dynamic output feedback (32) is used.

Then, let P denote the closed-loop controllability Gramian from

the input W−1/2w. Since A is stable, P is given by

P = APAT +DWDT . (34)

As in the continuous-time case, we seek a solution to the follow-

ing optimal control problem.

The Discrete-Time ICC Problem. Find a state feedback

stabilizing controller (31) or a strictly proper output feedback

stabilizing controller (32) for the system (30) to minimize the

ICC cost

JICC = traceQCpPCTp , Q > 0, (35)

subject to

Ui = ΦiCuPCTu Φ

Ti ≤U i, i = 1,2, . . . ,m, (36)

where P is given by (34) and the matrices Φi for i = 1,2, . . . ,mare, as in the continuous-time case, appropriately selected pro-

jection matrices for each ui corresponding to the constraint U i.

State FeedbackWith the state feedback controller (31), the closed-loop ma-

trices are the same as in the continuous-time case (10). To for-

mulate the LMIs for the discrete-time case, we use the H2 state

feedback LMIs given by Theorem 5 of [5] as a starting point.

Theorem 3. There exists a controller in the form (31), given by

G = LX−1 (37)

that minimizes JICC (9) and satisfies the input constraints (36) if

there exist matrices L ∈ Rm×n and X ∈ R

n×n and a symmetric

positive definite matrix P ∈ Rn×n that minimize the upper bound

of the ICC cost

JICC = minP,L,X

trace(

QCpPCTp

)

> trace(

QCpPCTp

)

= JICC, (38)

subject to the LMIs

P ApX +BpL DpW1/2p

⋆ X +XT −P 0

⋆ ⋆ I

> 0, (39)

QCpPCTp > 0, (40)

[

U i ΦiL

⋆ X +XT −P

]

> 0, (41)

for i = 1,2, . . . ,m.

Proof. The fact that the LMI (39) is equivalent to the correspond-

ing LMI in Theorem 5 of [5] comes from noticing that DpW1/2p

is the weighted disturbance input matrix. Since (39) implies that

P > APAT +DWDT (42)

there exists a matrix M = MT > 0 such that

P = APAT +DWDT +M. (43)

Consequently, P > P. Thus, from (40) it can be shown that

QCpPCTp > QCpPCT

p . (44)

Therefore,

JICC = trace(

QCpPCTp

)

> trace(

QCpPCTp

)

= JICC. (45)

Similarly, it follows from (41) that

U i > ΦiGPGTΦ

Ti > ΦiGPGT

ΦTi =Ui, (46)

for i = 1,2, . . . ,m. 2



Dynamic Output Feedback

As in the continuous-time case, the discrete-time state feed-

back results are extended to the discrete-time OCC problem with

output feedback with the use of a state estimator. Under the as-

sumption that the pair (Mp,Ap) is detectable, then the state esti-

mator is given by

xc(k+ 1) = Apxc(k)+Bpu(k)+F (z(k)−Mpxc(k)) , (47)

where F = ApPMTp

(

V +MpPMTp

)−1and P is the unique positive

definite matrix P that satisfies the Riccati equation [2]

P = ApPATp −ApPMT

p

(

V +MpPMTp

)−1MpPAT

p +DpWpDTp .(48)

Then, as in the continuous-time case, the estimated states are

used to compute the control input such that the state estimator

becomes

xc(k+ 1) = (Ap +BpG−FMp)xc(k)+Fz(k), (49)

and G is given by the following theorem.

Theorem 4. If the pair (Mp,Ap) is detectable and there exists

matrices L ∈ Rm×n and X ∈R

n×n and a symmetric positive defi-

nite matrix P ∈ Rn×n that minimize

JICC = minP,L,X

trace[

QCp

(

P+P)

CTp

]

, (50)

subject to the LMIs

P ApX +BpL F(

V +MpPMTp

)1/2

⋆ X +XT −P 0

⋆ ⋆ I

> 0, (51)

QCp

(

P+P)

CTp > 0, (52)

[

U i ΦiL

⋆ X +XT −P

]

> 0, (53)

for i = 1,2, . . . ,m where P is the unique positive solution to the

Riccati equation (48), then there exists a controller that mini-

mizes the ICC cost JICC while satisfying the input constraints

(36) in the form (32), given by

xc(t) = (Ap +BpG−FMp)xc(t)+Fz(t)

u(t) = Gxc(t)(54)

with G = LX−1 and F = ApPMTp

(

V +MpPMTp

)−1.

Proof. Note that, as in the continuous-time case, the full-order

dynamic output feedback OCC problem can be solved by first

designing a standard Kalman filter and then using the state feed-

back LMIs in Theorem 3 by replacing Dp, Wp, and QCpPCTp with

F , V +MpPMTp , and QCp

(

P+P)

CTp , respectively. 2

Numerical Example

To show the effectiveness of the LMIs presented in this pa-

per, we first demonstrate that the ICC problem considered in this

paper can be solved through the use of an iterative approach.

Then we show that similar and sometimes better results can be

obtained directly by solving the LMIs provided in Theorems 1-4.

For this demonstration, we use the example given in [2], which

considers the continuous-time OCC problem for the plant (1)

with the following system matrices:

Ap =

0 1 0

−1 −0.1 1

0 0 −10

, Bp =

0

0

1

, Dp =

0

0

1

,

Cp =

1 0.5 0

0 0 0.51 1 0

,

Mp =[

1 1 0]

.

(55)

The process and measurement noises wp and v are weighted by

Wp = 1 and V = 0.01. (56)

As mentioned in the introduction, the OCC problem is an op-

timal control problem that minimizes the control input subject

to specified output covariance constraints. For this example, the

output covariance constraint is taken to be

Y1 ≤ 0.035, Y2 ≤ σ× I2, (57)

where Y1 denotes the (1× 1) output variance corresponding to

the first performance variable and Y2 denotes the (2× 2) output

covariance matrix of the second and third performance outputs

grouped together. To show how an iterative approach can be

used to solve the ICC problem, we start with a σ value of 0.05

and then reduce it gradually down to 0.005. When this is done,

the input energy required to meet the demand increases as shown

in Fig. 1. Notice also in Fig. 1 that at a certain point, the control

energy required for additional performance increases exponen-

tially. Then at a certain point, increasing the control energy will

no longer provide better control. This is especially true for the

output feedback control problem.

This indicates that for this specific example in order to solve

the ICC problem, it is possible to use an iterative algorithm by it-

erating level of performance required for the OCC problem until



U

σ0 0.01 0.02 0.03 0.04 0.05

10−2

10−1

100

101

State Feedback ControlOutput Feedback Control

Figure 1. The input energy, U , plotted against the covariance costraint σ

for both the state feedback control and output feedback control problems.

the desired input covariance constraint is met. To steer the it-

erative algorithm towards the desired input covariance, a simple

bisection algorithm like Algorithm 1 could be used. In the iter-

Algorithm 1 Iterative Bisection ICC Algorithm

Input: Plant matrices Ap, Bp, Dp, Cp, Mp, weighting matrices

Wp and V , a desired input covariance constraint U , and an

upper and lower bound for the output performance level σ

denoted by σU and σL, respectively.

Output: A state feedback (2) or output feedback controller (3)

with an input covariance given by U ≤U .

1: Set σ = σU .

2: while σU −σL ≥ ε do

3: Solve the OCC problem to obtain a controller that satis-

fies the performance constraint σ and compute the input

covariance U .

4: if U ≤U then

5: Set σU = σ

6: else

7: Set σL = σ

8: end if

9: σ = σL +σU−σL

2

10: end while

ative algorithm, any method that is capable of solving the OCC

problem, such as the OCC algorithm [2] or the LMI method [3],

can be used. Algorithm 1 is used to solve the following two ICC

problems:

Problem 1: U ≤ 0.25, (58)

Problem 2: U ≤ 1.00, (59)

Table 1. STATE FEEDBACK DESIGN: PROBLEM 1 COMPARISON

U Algorithm 1 SeDuMi LMI Lab

Iterations: 23

U 0.250000 0.250000 0.250000 0.249933

JICC 0.033735 0.033652 0.033688

GT

−0.7648

−5.6956

−0.5543

−0.7083

−5.5831

−0.7014

−0.6992

−5.5739

−0.7085

Table 2. OUTPUT FEEDBACK DESIGN: PROBLEM 1 COMPARISON


Iterations: 22

U 0.250000 0.250000 0.250000 0.249449

JICC 0.054937 0.054935 0.055090

GT

−1.0757

−6.6391

−0.6431

−1.0687

−6.7709

−0.8755

−1.3283

−6.7550

−1.0910

with the system matrices given by (55) for both state feedback

and output feedback control. For the dynamic output feedback

controller, the controller input matrix F is precomputed accord-

ing to (22) and (23). In this case we have

F = [0.4412 0.7633 0.4796]T . (60)

Since F is precomputed and independent of the control synthesis

for the output matrix, G, of the dynamic output feedback con-

troller, it is same for both problems 1 and 2. With the precision

ε set at 1× 10−8, Algorithm 1 typically needs about 22-23 itera-

tions before it finds a solutions of acceptable accuracy, as shown

in Tables 1-4.

The benefit of the LMI solution method for the ICC problem

detailed in this paper over ad-hoc methods like Algorithm 1 is

that a solution can be obtained directly without any iterations. To

show this, we solve the same two problems ((58) and (59)) using

two different LMI solvers: SeDuMi [6] and LMI Lab [7]. To use

SeDuMi as the LMI solver, the LMIs are first programmed into

MATLAB using YALMIP [8]. SeDuMi and YALMIP are both

free software that can be installed in MATLAB as toolboxes. The

LMI Lab is included in the Robust Controls Toolbox. The results

obtained using each of the solvers are compared to the results

obtained when using Algorithm 1.

For problem 1 (58), we can see in Table 1 for the state feed-

back control problem that LMI method found a solution with



Table 3. STATE FEEDBACK DESIGN: PROBLEM 2 COMPARISON


Iterations: 23

U 1.000000 0.999996 1.000000 0.998416

JICC 0.017615 0.015359 0.015441

GT

−1.3217

−10.7071

−4.2923

−6.2728

−17.2734

−2.8007

−6.1818

−17.2283

−2.7912

Table 4. OUTPUT FEEDBACK DESIGN: PROBLEM 2 COMPARISON


Iterations: 22

U 1.000000 0.999999 1.000000 0.999441

JICC 0.043040 0.043003 0.043167

GT

−8.5133

−20.1235

−1.8452

−9.9072

−22.9509

−4.1005

−12.7348

−25.9277

−7.6070

a lower JICC cost and therefore somewhat slightly better per-

formance than was found with the iterative algorithm. Also as

shown in Table 2, for the output feedback control problem, the

solution found using SeDuMi was just ever so slightly better than

the solution found with the iterative algorithm. We also note that

as expected, the output performance obtained is better with state

feedback control than with output feedback control.

For problem 2 (59), we can see in Table 3 for the state feed-

back control problem that again the LMI method found a solution

with a lower JICC cost than was found with the iterative algo-

rithm. Also as shown in Table 4, for the output feedback con-

trol problem, the solution found using SeDuMi was again ever

so slightly better than the solution found with the iterative algo-

rithm.

CONCLUSION

In this paper the input covariance constraint (ICC) control

problem is solved using a convex optimization with linear matrix

inequality (LMI) constraints for both continuous and discrete-

time linear time invariant systems. The theorems provided in

this paper provide a set of control synthesis techniques based on

LMI optimization to obtain a controller, state feedback or dy-

namic output feedback, that solves the ICC problem. That is a

controller that obtains the best possible performance subject to

multiple constraints on the input covariance matrices.

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a linear matrix inequality solution to the input

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