LMI-Based Digital Redesign of Linear Time-Invariant

Systems with State-Derivative Feedback

Rodrigo Cardim, Marcelo C. M. Teixeira∗, Member IEEE, Flavio A. Faria and Edvaldo Assuncao

Abstract - A simple method for designing a digital state-derivative feedback gain and a feedforward gain such thatthe control law is equivalent to a known and adequate statefeedback and feedforward control law of a digital redesignedsystem is presented. It is assumed that the plant is a linearcontrollable, time-invariant, Single-Input (SI) or Multiple-Input(MI) system. This procedure allows the use of well-knowncontinuous-time state feedback design methods to directlydesign discrete-time state-derivative feedback control systems.The state-derivative feedback can be useful, for instance,in the vibration control of mechanical systems, where themain sensors are accelerometers. One example considering thedigital redesign with state-derivative feedback of a helicopterillustrates the proposed method.

Index Terms - Digital redesign, state-derivative feedback,control of mechanical systems, linear matrix inequalities.

I. INTRODUCTION

In the last years, the proportional and state-derivative

feedback have been very useful [1], for instance, to design

controllers for the following problems: derivative feedback

for multivariable linear systems using Linear Matrix Inequa-

lities (LMIs) [2], robust state-derivative pole placement LMI-

based designs for linear systems [3], [4], robust stabilization

of descriptor linear systems [5], [6], feedback control of

singular systems [7], nonlinear control with exact feedback

linearization [8], and H∞-control of continuous-time sys-

tems with state-delay [9].

There exist some practical problems where the state-

derivative signals are easier to obtain than the state signals.

For instance, for controlled vibration supression of mechani-

cal systems, where the main sensors are accelerometers and

it is possible to get the velocities with a good precision but

not the displacements [10], [11]. Defining the velocities and

displacements as the state variables, then one has available

for feedback the state-derivative signals.

In [11] a method to design a state-derivative feedback

gain and a feedforward gain, such that the control law

is equivalent to a known and suitable state feedback and

feedforward control law was proposed. This method extends

the results described in [10] to a more general class of

control systems, such as the noninteracting control problem

and also presents a theoretical analysis simpler and easier

to understand. It was assumed that the plant is a linear

controllable, time-invariant, single-input (SI) or multiple-

input (MI) system. This procedure allows the designers to

R. Cardim, M. C. M. Teixeira, F. A. Faria and E. Assuncao are withthe Department of Electrical Engineering, Faculdade de Engenharia de IlhaSolteira, UNESP-Sao Paulo State University, Ilha Solteira, Sao Paulo, Brazil.* Corresponding author: marcelo@dee.feis.unesp.br.

use the well-known state feedback design methods to directly

design state-derivative feedback control systems.

The designs presented in [1]-[11] considered continuous-

time state-derivative feedback. The authors did not find

papers with results about the redesign of discrete-time state-

derivative feedback.

In this paper a new method to design a state-derivative

feedback gain for digital control systems is proposed. This

method is based on the digital redesign theory proposed in

[12], and state-derivative feedback theory for continuous-

time systems proposed in [11], described above. The so-

called digital redesign problem ([12], [13]) is to design

a suitable analogue controller first and then convert the

obtained analogue controller to the equivalent digital con-

troller maintaining the properties of the original analogously

controlled system, by which the benefits of both continuous-

time controllers and the advanced digital technology can be

obtained [12]. In [12] a simple design methodology for the

digital redesign of static state feedback controllers by using

Linear Matrix Inequalities (LMI) was presented. The method

provides close matching of the states between the original

continuous-time system and those of the digitally redesigned

system with a guaranteed stability. It is very useful for the

solution of the proposed method in this paper. An example

considering the pole-placement for the control problem of a

helicopter illustrates the proposed design procedure.

II. DIGITAL REDESIGN WITH STATE FEEDBACK

This section describes the main results presented in [12].

These results will be used in the proof of the new method

proposed in this paper.

Consider a controllable linear time-invariant plant descri-

bed by{

xc(t) = Axc(t) + Buc(t), xc(0) = x0,yc(t) = Cxc(t),

(1)

where xc(t) ∈ Rn

is the state vector, uc(t) ∈ Rm

is

the control vector, yc(t) ∈ Rp

is the output vector, and

A ∈ Rn×n

, B ∈ Rn×m

and C ∈ Rp×n

are time-invariant

matrices. The control vector uc(t) is given by

uc(t) = −Kcxc(t) + Ecr, (2)

where Kc ∈ Rm×n

is the state feedback gain, Ec ∈ Rm×p

is

the feedforward gain, and r ∈ Rm

is the constant reference

vector. Note that the gain Kc can be specified using well-

known methods available in the literature, for instance, such

that the poles of the closed-loop of (1) and (2) are placed in

the wanted positions [14], [15], [16].

18th IEEE International Conference on Control ApplicationsPart of 2009 IEEE Multi-conference on Systems and ControlSaint Petersburg, Russia, July 8-10, 2009

978-1-4244-4602-5/09/$25.00 ©2009 IEEE 745

The closed-loop system of (1) with (2) becomes{

xc(t) = (A − BKc)xc(t) + BEcr,yc(t) = Cxc(t).

(3)

The discrete model of the closed-loop system (3), at t =kT + T , where T is the sampling period, is given by [12]

{

xc(kT + T ) = Gcxc(kT ) + HcEcr,yc(kT ) = Cxc(kT ),

(4)

where Gc = e(A−BKc)T and

Hc = (Gc − In)(A − BKc)−1B.

Considering the same analysis presented in [12], let the

state equation of the continuous-time system in (1) with a

digital control input be represented by{

xd(t) = Axd(t) + Bud(t), xd(0) = x0,yd(t) = Cxd(t),

(5)

where

ud(t) = ud(kT ) = −Kdxd(kT ) + Edr,

kT ≤ t < kT + T, (6)

where Kd ∈ Rm×n

is the digital feedback gain and Ed ∈R

m×pis the digital feedforward gain. Then, the closed-loop

system is given by

xd(t) = Axd(t)−BKdxd(kT )+BEdr, kT ≤ t < kT +T,(7)

and the discrete model of the closed-loop system (5) with

(6) is{

xd(kT + T ) = (G − HKd)xd(kT ) + HEdr,yd(kT ) = Cxd(kT ),

(8)

where G = eAT and H = (G − In)A−1B. If A is singular,

the matrix H can be computed by the following equation

[12]:

H =∞∑

i=1

1

i!(AT )i−1BT.

The problem proposed in [12] was the following:

Problem 1: [12] For the well designed stable analogue

control gains Kc and Ec in (2), determine a digital control

gains Kd and Ed for the control law (6) such that:

(i) The digitally controlled system in (7) is stable in the sense

of Lyapunov stability criterion;

(ii) The outputs of the digital control system in (8) match

those of the analogue system in (4) as closely as possible.

Theorem 1 solves the Problem 1 proposed in [12].

Theorem 1: [12] If there exist a symmetric positive de-

finite matrix Γ, a matrix F , and a scalar α > 0 such that

the following constrained minimisation problem is solved,

then the digital control law in (6) satisfies the given design

objectives in Problem 1.

min α

[

−αΓ ∗GcΓ − GΓ + HF −αI

]

< 0, (9)

[

−Γ ∗GΓ − HF −Γ

]

< 0, (10)

where F = KdΓ, ∗ denotes the transposed elements in the

symmetric positions and the digital state feedback gain Kd

and feedforward gain Ed are given by

Kd = FΓ−1, (11)

Ed = ((I − (G − HKd))−1H)+1(I − Gc)

−1HcEc, (12)

where (·)+1 denotes the pseudo-inverse of (·).Remark 1: The solution of the minimisation problem gi-

ven in Theorem 1, can be easily solved using softwares based

on convex programming, for instance the LMI optimisation

toolbox in MATLAB [17], considering that[

−αΓ ∗GcΓ − GΓ + HF −αI

]

< 0

is equivalent to

Z

[

−αΓ ∗GcΓ − GΓ + HF −αI

]

Z

=

[

−Γ ∗GcΓ − GΓ + HF −α2I

]

< 0, (13)

where the nonsingular matrix Z is given by

Z =

[

(√

α)−1I 00

√α I

]

. (14)

Therefore, (9) is equivalent to:

min µ

[

−Γ ∗GcΓ − GΓ + HF −µI

]

< 0, (15)

where µ = α2.

Note that (15) is an LMI and the problem in Theorem 1

described by (9) and (10), is referred to as a generalized

eigenvalue problem. The procedure above is convenient,

because the solution of LMI is easier to solve, than the

solution of a generalized eigenvalue problem [17].

Remark 2: Note that in Problem 1 the state feedback

control law (6) is such that the design specification (ii),

related to the outputs of the controlled systems, holds.

Remark 3: [12] If both the original (3) and redesigned

(8) systems are asymptotically stable, then the following

condition hold for the outputs of these systems:

limk→∞

(yc(kT ) − yd(kT )) = 0.

III. DIGITAL REDESIGN WITH STATE-DERIVATIVE

FEEDBACK

Considering the results presented in [12], the Problem 2

is proposed in this paper.

Problem 2: Consider that the conditions of Theorem 1

holds and let Kd and Ed be a solution of Problem 1. Then,

determine the state-derivative feedback gain Kdf and the

feedforward gain Edf such that, for the controlled system

(5), (6) and k = 0, 1, ..., the discrete-time state feedback

746

law (6) is equal to the discrete-time state-derivative feedback

described below:

ud(kT ) = −Kdxd(kT ) + Edr = −Kdf xd(kT ) + Edfr.(16)

Equation (16) shows that the system (5) presents the same

state vector xd(t), for t > 0, with the control signal

ud(kT ) = udf (kT ) = −Kdf xd(kT )+ Edfr (state-derivative

feedback) and ud(kT ) = −Kdxd(kT ) + Edr (state feed-

back).

To establish the proposed method, the following assump-

tions are considered:

(i) The matrix A in equation (1) has a determinant different

from zero;

(ii) The determinant of (A − BKc) is different from zero;

(iii) The matrix B has rank equal to m.

The Assumption (i) was also considered in [10] and

is important for the stability of the system (1), with the

proposed method and the control law ud(kT ) = udf(kT ) =−Kdf xd(kT ) + Edfr. The Assumption (ii) is necessary for

the system (1), with the control law uc(t) = −Kcxc(t), given

by

xc(t) = (A − BKc)xc(t), xc(0) = xc0, (17)

to be an asymptotically stable system, because otherwise the

matrix (A−BKc) would have one or more eigenvalues equal

to zero.

The following theorem shows the main result of this paper.

Theorem 2: Consider that the system (5), whose the con-

trol law is given in (6), has a wanted performance. Then, if

the Assumptions (i), (ii) and (iii) are satisfied, the control

signal for state-derivative feedback

ud(t) = ud(kT ) = udf(kT ) = −Kdf xd(kT ) + Edfr,

kT ≤ t < kT + T, (18)

Kdf = Kd(A − BKd)−1, (19)

Edf = (Im + KdfB)Ed, (20)

is such that, for the controlled system (5) and (18),

ud(kT ) = −Kdf xd(kT ) + Edf r = −Kdxd(kT ) + Edr.(21)

Proof: From (5), (18) and t = kT it follows that

xd(kT ) = Axd(kT ) − BKdf xd(kT ) + BEdf r,

(In + BKdf)xd(kT ) = Axd(kT ) + BEdfr. (22)

Now, from Edf given in (20) and (22) one has

(In + BKdf)xd(kT ) = Axd(kT ) + B(Im + KdfB)Edr,

= Axd(kT ) + (In + BKdf)BEdr. (23)

Note that from Kdf given in (19), then

(In + BKdf ) = [In + BKd(A − BKd)−1]

= [(A − BKd)(A − BKd)−1

+BKd(A − BKd)−1]

= A(A − BKd)−1 (24)

and so, from Assumptions (i) and (ii) this matrix is invertible.

Therefore, from (23) and (24) it follows that

xd(kT ) = (In + BKdf)−1Axd(kT ) + BEdr

= [A(A − BKd)−1]−1Axd(kT ) + BEdr

= (A − BKd)xd(kT ) + BEdr

= Axd(kT ) + B(−Kdxd(kT ) + Edr). (25)

Finally, from Assumption (iii), (5) and (25) observe that

(6) is satisfied. Hence (21) holds and the proof is concluded.

IV. PRACTICAL IMPLEMENTATION OF THE CONTROLLER

Assume that in (5) xd(t) is available but xd(t) is not

available, for t = kT , k = 0, 1, .... Note that from (5), for

ud(kT ) = udf(kT ) given in (18), then xd(kT ) depends on

udf(kT ) and udf(kT ) depends on xd(kT ). To avoid this

problem, an alternative implementation of (18) is presented

below.

Consider the following signals:

xA(kT ) = Axd(kT ) + Budf (kT − T ), (26)

xB(kT ) = Axd(kT ) + Budf (kT ), (27)

udf (kT ) = −Kdf xB(kT ) + Edfr. (28)

Then, the control law ud(kT ) = udf (kT ) given in (18) can

also be obtained as follows: from (26) note that

Axd(kT ) = xA(kT ) − Budf (kT − T ), (29)

and from (27) it follows that:

xB(kT ) = xA(kT ) − Budf(kT − T ) + Budf(kT ), (30)

= xA(kT ) + B(udf (kT )− udf(kT − T )). (31)

Now, from (28) and (30) one has:

xB(kT ) = xA(kT ) − Budf(kT − T )

−BKdf xB(kT ) + BEdfr,

xB(kT ) = (In + BKdf)−1(xA(kT )

−Budf(kT − T ) + BEdfr). (32)

Now, from (28) and (32)

udf (kT ) = −Kdf(In + BKdf )−1(xA(kT )

−Budf(kT − T ) + BEdf ) + Edfr, (33)

udf(kT ) = −Q1xA(kT ) + Q2udf(kT − T ) + Q3r, (34)

where Q1 = Kdf(In + BKdf )−1, Q2 = Q1B and Q3 =−Kdf(In + BKdf )−1BEdf + Edf .

For the implementation of the control law, note that from

(5), (26) and (27),

xA(kT ) = xd(kT ) for ud(kT ) = udf(kT − T ),(35)

xB(kT ) = xd(kT ) for ud(kT ) = udf(kT ). (36)

Thus, xA(kT ) can be approximately obtained as follows:

xA(kT ) ≈ xd(t), for t such that t < kT and t ≈ kT .

Considering that ud(kT ) = udf (kT ), note that for t defined

747

above, ud(t) = udf(kT −T ) and from (5) and (26) it follows

that xd(t) ≈ xA(kT ).

Remark 4: Note that in (26), if k = 0, xA(0) = Ax(0) +Budf (−T ). For this case it is assumed that udf(−T ) = 0.

V. EXAMPLES

A. Example 1

Consider the control problem of a Vertical Take Off and

Landing (VTOL), presented in [18]. The linearized dynamic

equation of the VTOL helicopter is shown in equation (37)

below:

xc(t) =

−0.0366 0.0271 0.0188 −0.45550.0482 −1.010 0.0024 −4.02080.1002 0.3681 −0.707 1.4200

0 0 1 0

xc(t)

+

0.4422 0.17613.5446 −7.5922−5.52 4.49

0 0

uc(t) = Axc(t) + Buc(t), (37)

yc(t) =

[

1 0 0 00 1 0 0

]

xc(t) = Cxc(t). (38)

The physical description of the dynamic equations is as

follows:

xc1(t) - horizontal velocity, [knots];

xc2(t) - vertical velocity, [knots];

xc3(t) - pitch rate, [degree/s];

xc4(t) - pitch angle, [degree];

uc1(t) - collective pitch control;

uc2(t) - longitudinal cyclic pitch control,

where xc(t) = [xc1(t) xc2(t) xc3(t) xc4(t)]T and uc(t) =

[uc1(t) uc2(t)]T .

Figure 1 shown the VTOL helicopter with some state

variables of the system.

x1(t)

x2(t)

x4(t)

Fig. 1. Illustration of the Vertical Take Off and Landing with some statevariables.

Suppose that, for the implementation of the control law,

only accelerometers are used as sensors. Then, xc1(t),xc2(t) and xc3(t) are available. From the signals xc1(t),xc2(t) and xc3(t) it is possible to directly get the veloci-

ties xc1(t), xc2(t) and xc3(t) with a good precision, but

not the angle xc4(t) [4], [11]. Thus, the vector xc(t) =[xc1(t) xc2(t) xc3(t) xc3(t)]

T is available and a discrete-

time state-derivative feedback can be implemented, following

the proposed method.

For this system the open-loop poles are equal to 0.2758±j0.2576, −0.2325 and −2.0727. Consider the pole place-

ment as design technique, and the following closed-loop

poles for the system:

−1, − 5, − 3 ± j15.

The gain Kc such that the controlled system (3), presents

these poles, can be easily obtained with the software MA-

TLAB (command place):

Kc =

[

34.6217 7.3049 1.2743 −25.777628.4481 4.2729 0.7815 −20.7768

]

. (39)

From Theorem 1, considering a sampling period T =0.01s, the gain matrix Kd is given by:

Kd =

[

34.7490 6.9373 1.0923 −26.002927.6946 3.9035 0.6216 −20.3129

]

. (40)

Note that the system (37) with (39), satisfies the Assump-

tions (i), (ii) and (iii). Then, from Theorem 2, the gain matrix

Kdf is the following:

Kdf = Kd(A − BKd)−1,

Kdf =

[

−0.3537 0.1295 0.2411 0.1689−0.9042 0.1031 0.0055 0.0013

]

. (41)

Therefore, considering that Ec = Ed = Edf = 0, for the con-

trolled system (5), (18) and (41), ud(kT ) = −Kdf xd(kT ) =−Kdxd(kT ).

For the implementation of the control law ud(kT ) =udf(kT ), as discussed in Section IV, was considered udf (kT )given in equation (34), where

Q1 = Kdf (In + BKdf )−1

=

[

69.4906 111.1519 318.7109 224.847743.5076 88.3402 249.7908 176.1937

]

, (42)

Q2 = Q1B = 1 × 103

[

−1.3346 0.5994−1.0465 0.4585

]

. (43)

Furthermore, a delay δ = T/100 s was used to estimate

xA(kT ): xA(kT ) ≈ xd(kT − δ), (see the end of Section IV

for details).

Figures 2, 3 and 4 show the simulation results, considering

the initial conditions x(0) = [1 − 0.5 0 0]T and sampling

period T = 0.01 s. Note that the controlled systems present

almost the same responses.

VI. CONCLUSION

A simple method for redesigning of linear time-invariant

systems with state-derivative feedback was proposed. This

procedure allows the design of discrete-time state-derivative

feedback, given an adequate continuous-time state feedback.

It uses the results presented in [12], that are based on LMI,

and can be useful in the digital control of mechanical systems

using accelerometers as sensors.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support

by FAPESP, CNPq and CAPES, from Brazil.

748

0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

3

vel

oci

ties

[kn

ots

]

t [s]

x1(t) (continuous)

x1(kT ) ([12])

x2(t) (continuous)

x2(kT ) ([12])

T = 0.01 s

Fig. 2. Transient responses of the system (37) with the control laws (2)(continuous) and (6) ([12]).

0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

3

vel

oci

ties

[kn

ots

]

t [s]

x1(kT ) ([12])

x1(kT ) (derivative feedback)

x2(kT ) ([12])

x2(kT ) (derivative feedback)

T = 0.01 s

Fig. 3. Transient responses of the system (37) with the control laws (6)([12]) and (18) (derivative-feedback).

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