State and output feedback control for Takagi-Sugeno

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING

Int. J. Adapt. Control Signal Process. 2014; 00:1–27

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs

State and output feedback control for Takagi-Sugeno systems

with saturated actuators

Souad Bezzaoucha1∗ Benoıt Marx2,3, Didier Maquin 2,3 and Jose Ragot2,3

1 Bordeaux Institute of Technology, INP Enseirb-Matmeca, IMS-lab, 351 cours de la liberation, 33405 Talence, France

2 Lorraine University, CRAN, UMR 7039, 2 avenue de la foret de Haye, Vandoeuvre-les-Nancy Cedex, 54516, France.

3 CNRS, CRAN, UMR 7039, France

SUMMARY

In the present paper, a Takagi-Sugeno (T-S) model is used to simultaneously represent the behaviour

of a nonlinear system and its saturated actuators. With the T-S formalism and the Lyapunov approach,

stabilization conditions are expressed as Linear Matrix Inequalities (LMIs) for different controller designs.

Static Parallel Distributed Compensation (PDC) state feedback, static and dynamic PDC output feedback

controllers for nonlinear saturated systems are proposed. The descriptor approach is used to obtain relaxed

conditions to compute the controller gains. The nonlinear cart-pendulum system is used to illustrate the

proposed control laws. Copyright c© 2014 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: Nonlinear system, T-S model, actuator saturation, parallel distributed compensation

design, descriptor approach, static and dynamic output feedback

∗Correspondence to: Corresponding author’s address e-mail: [email protected], benoit.marx@univ-

lorraine.fr, [email protected], [email protected]

Copyright c© 2014 John Wiley & Sons, Ltd.

Prepared using acsauth.cls [Version: 2010/03/27 v2.00]

2 S. B. ET AL.

1. INTRODUCTION

Actuator saturation or control input saturation is probably one of the most usual nonlinearity

encountered in control engineering due to the physical impossibility of applying unbounded control

signals and/or safety constraints. Actuator saturation may cause decreasing performances or even

instability of the system under control, since the needed control input energy may not be provided to

the system, for instance, when the control requires large gains resulting in control law magnitudes

exceeding the range of the actuators. Motivated by these issues, some efforts have focused on

developing saturated controllers for regulation problems [21], [12], [31], [19], [17] and [6].

There are two main design strategies to deal with actuator saturation. The first strategy is a two-step

approach in which a nominal linear controller is first constructed by ignoring actuator saturation

and using standard linear design tools; thereafter, a so called anti-windup compensator is designed

to handle the saturation constraints [30], [22], [14]. A typical anti-windup scheme consists in

augmenting a nominal pre-designed linear controller with a compensator based on the discrepancy

between unsaturated and saturated control signals provided to the plant. [24]. A generalization of

the anti-windup strategy based on a modified sector condition is used to obtain stability conditions

from a quadratic Lyapunov function as proposed in [13]. However, for saturated Linear Parameter

Varying (LPV) systems, only few works are available on the subject, like [3], [23], [28], [33] and

[11] where the modified sector condition from [13] was applied to the LPV case under saturated

inputs and states.

On the contrary, the second strategy consists in considering the saturation from the beginning of

the controller design, then the controller gains are adjusted accordingly to the saturation levels.

Among these strategies, one can cite the invariant sets framework, which has been significantly

developed in control engineering over the last decades [1], [7]. This framework ensures that every

state trajectory initialized inside a so-called invariant set remains inside this set. An interesting

approach of the invariant set framework, applied in [9] and [27], consists in the so-called polytopic

rewriting of the saturation constraint and is used to determine the largest invariant set by maximizing

Copyright c© 2014 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2014)

Prepared using acsauth.cls DOI: 10.1002/acs

STATE AND OUTPUT FEEDBACK CONTROL FOR T-S SYSTEMS WITH SATURATED ACTUATORS 3

an estimate of the basin of attraction of the closed-loop system [18], [10]. Unlike in [20] where the

occurrence of saturation is allowed, in most papers the overall goal of the previous method could be

roughly understood as preventing the controller from reaching the saturation constraints, implying

a decrease of the control input energy possibly provided to the system. Contrarily, in the present

contribution, the saturation is explicitly considered and its occurrence is admitted. The Takagi-

Sugeno (or polytopic) modeling is here used to represent the saturation constraints and integrate

them in the controller design in order to compute the controller gains according to the saturation

levels and to ensure the closed-loop system stability.

The aim of this paper is to present a new approach for saturated control of nonlinear systems, where

the Sector Nonlinearity Transformation (SNT) is used to represent both the saturated actuators and

the nonlinearities of the system itself under a T-S form.

One should note that even if the expressions of the T-S saturation are similar to the polytopic ones

used in [9] and [27], the development, control strategy and objectives are completely different.

Indeed, in the proposed approach the invariant sets are not considered and the objective concerns

both the global stabilization of nonlinear systems represented with the T-S models and the synthesis

of a state feedback controller by parallel distributed compensation (PDC) with control gains

explicitly depending on the saturation level.

After that, using only measured plant output signals, output feedback controllers are considered

[16], [29]. These controllers may be static or dynamic. Static output feedback control is the simplest

approach since no further dynamics are introduced. However, a dynamic compensator introducing

extra dynamics may be required to increase the number of degrees of freedom in the design and

improve the closed-loop transient response. For this part, the descriptor approach is envisaged [25],

[15]. This approach is well known to avoid the coupling terms between the feedback gains and the

Lyapunov matrices and thus facilitates the LMIs resolution. As a consequence, the number of LMI

decreases and relaxed conditions are obtained [15], [2].

It is important to highlight that the proposed approach ensures the stability of nonlinear system that

may be destabilized by the control saturation when the saturation is not taken into account in the



4 S. B. ET AL.

control synthesis. Nevertheless, if the submodels are unstable (matrices Ai of the T-S model not

stable), the proposed state and static output approaches are not suitable, but the dynamic output

feedback controller (section 5) can be used to stabilize the closed-loop system. The proposed

dynamic control presents the advantage of stabilizing the closed-loop system with an additional

degree of freedom in the design procedure which is the controller order (only the dimensions of the

LMI variables are changed). This controller order may be set up according to a tradeoff between

reducing the controller order and the obtained closed-loop performance (assessed by the radius of

the ball in which the state trajectory converges) for example.

The rest of this paper is organized as follows. The T-S structure for modeling is first introduced in

section 2, also with some preliminary results, mathematical notations and a brief description of the

actuator saturation. The section 3 is devoted to the representation of the nonlinear saturation by a

T-S structure. In the section 4, the PDC saturated state feedback control is detailed. The static and

dynamic PDC output feedback controllers are studied in the section 5. To highlight the interest of

the paper, the nonlinear model of a cart-pendulum is used and stabilizing state and output feedback

controllers are designed to counteract the effect of the actuator saturations.

2. PRELIMINARIES

The T-S representation of a nonlinear system consists in a time-varying interpolation of a set of

linear submodels. Each submodel contributes to the global behavior of the nonlinear system through

a weighting function hi(ξ (t)) [26]. The T-S structure is given byx(t) =

n

∑i=1

hi(ξ (t))(Aix(t)+Biu(t))

y(t) =n

∑i=1

hi(ξ (t))(Cix(t)+Diu(t))(1)

where x(t) ∈ Rnx is the system state, u(t) ∈ Rnu the control input and y(t) ∈ Rm the system output.

ξ (t) ∈ Rq is the premise variable assumed to be measurable or known signal (including at least the

input u and the output y of the system, but also the state x when it is accessible like in the state

feedback case). The weighting functions hi(ξ (t)), also noted hi(t), satisfy the so-called convex sum




property n

∑i=1

hi(ξ (t)) = 1

0≤ hi(ξ (t))≤ 1, i = 1, . . . ,n

(2)

In the rest of the paper, the following lemmas are used.

Lemma 1 ([8])

For any matrices X , Y and G = GT > 0 of appropriate dimensions, the following inequality holds

XTY +Y T X ≤ XT GX +Y T G−1Y (3)

Lemma 2 (congruence lemma, [8])

Consider two matrices X and Y , if X is positive (resp. negative) definite and if Y is a full column

rank matrix, then the matrix Y T XY is positive (resp. negative) definite.

The following notations are used throughout the paper:

• A block diagonal matrix with the square matrices A1, . . . ,An on its diagonal is denoted

diag(A1, . . . ,An).

• For any square matrix M, S(M) means S(M) = M+MT .

• The smallest and largest eigenvalues of the matrix M are respectively denoted λmin(M) and

λmax(M).

• The saturation function of the signal u(t) is denoted sat(u(t)) and defined componentwise by (4)

where u jmax and u j

min are the upper and lower saturation level of the jth component of u(t), denoted

u j(t)

sat(u j(t)) =

u j(t) if u j

min ≤ u(t)≤ u jmax

u jmax if u j(t)> u j

max

u jmin if u j(t)< u j

min

(4)

3. T-S MODELLING OF THE INPUT SATURATION

The main idea of this work is to model the nonlinear actuator saturation using the T-S representation

and then propose PDC state and output feedbacks, as well as dynamic output feedback control law



6 S. B. ET AL.

ensuring the stability of the closed-loop system. For that, it is proposed to re-write each saturated

control input component (4) under the T-S form (1) introduced in [4].

Let us consider a piecewise decomposition into three parts of each saturated actuator sat(u j(t))

given by:

sat(u j(t)) =3

∑i=1

µj

i (u j(t)) (λj

i u j(t)+ γj

i ) (5)

with λj

1 = 0 λj

2 = 1 λj

3 = 0

γj

1 = u jmin γ

j2 = 0 γ

j3 = u j

max

(6)

and the activation functionsµ

j1(u j(t)) =

1−sign(u j(t)−u jmin)

2

µj

2(u j(t)) =sign(u j(t)−u j

min)−sign(u j(t)−u jmax)

2

µj

3(u j(t)) =1+sign(u j(t)−u j

max)2

(7)

where the sign function is defined as:

sign(x) :=

−1 if x < 0

0 if x = 0

1 if x > 0

(8)

Note that in equation (5), sat(u j(t)) depends on the activation functions µj

i (u j(t)).

In order to lighten the notations, µj

i (u j(t)) will be denoted µj

i (t).

However, based on the convex sum property (2) of the activation functions (7), each control input

vector component u jsat can be written in order to have the same activation functions for all the input

vector as:

sat(u j(t)) =3

∑i=1

µj

i (t)(λj

i u j(t)+ γj

i )nu

∏k=1,k 6= j

3

∑j=1

µkj (t) (9)

Then, for nu inputs, 3nu submodels are obtained in the following compact form:

sat(u(t)) =3nu

∑i=1

µi(t)(Λiu(t)+Γi) (10)

It is important to note that sat(u(t)) (10) is directly expressed in term of the control variable u(t).

The global weighting functions µi(t), the matrices Λi ∈ Rnu×nu and vectors Γi ∈ Rnu×1 are defined




as follows

µi(t) =nu

∏j=1

µj

σj

i(t)

Λi = diag(λ 1σ1

i, . . . ,λ nu

σnui)

Γi =

[γ1

σ1i, . . . ,γnu

σnui

]T

(11)

where the indexes σj

i (i = 1, . . . ,3nu and j = 1, . . . ,nu), equal to 1,2 or 3, indicate which partition of

the j th input (µ j1 ,µ

j2 or µ

j3) is involved in the i th submodel (see [4] for more details).

As a consequence, it is now possible to describe a nonlinear system with bounded inputs. For that

purpose, let us now consider a T-S system with actuator saturation:x(t) =

n

∑j=1

h j(ξ (t))(A jx(t)+B jsat(u(t)))

y(t) =n

∑j=1

h j(ξ (t))(C jx(t)+D jsat(u(t)))(12)

According to the T-S writing of the saturation (10), the system (12) can be written asx(t) =

3nu

∑i=1

n

∑j=1

µi(t)h j(ξ (t))(A jx(t)+B j(Λiu(t)+Γi))

y(t) =3nu

∑i=1

n

∑j=1

µi(t)h j(ξ (t))(C jx(t)+D j(Λiu(t)+Γi))

(13)

Remark 1

In the previous section, a T-S representation of the saturation (10) has been proposed. It is important

to highlight that this representation is directly expressed in terms of the control variable u(t) and that

the number of sub-models (3nu) depends on the number of inputs nu. This statement may introduce

some conservatism. In fact, as the reader will notice in the following manuscript, the number of

LMIs to solve to find the control gains depends on the number of sub-models used to describe

the saturation constraint. As a first contribution, the saturation is expressed with a three parts

piecewise decomposition and decreasing the number of LMI conditions by finding more efficient

T-S representations of the control input saturation may be an interesting point for future works.

4. SATURATED STATE FEEDBACK CONTROL LAW

In this section, the system state is supposed to be known and the objective is to design a state

feedback control ensuring the stability of the closed-loop system, even in the presence of control



8 S. B. ET AL.

input saturation. The control law is defined by a PDC state feedback:

u(t) =−n

∑j=1

h j(ξ (t))K jx(t) (14)

The controller design is performed by the T-S modeling of the saturation (10) and by solving an

optimization problem under LMI constraints. In order to highlight the interest of considering the

saturation when computing the controller, the controller design is envisaged without (section 4.1)

and with (section 4.2) taking into account the control input bounds. A comparison of the obtained

results is made in section 4.3.

4.1. Nominal control law (without saturation)

In the nominal case, the controller gains K j are synthesized without taking into account the

saturation limits. From (1) and (14), the closed-loop system is:

x(t) =n

∑i=1

n

∑j=1

hi(ξ (t))h j(ξ (t))(Ai−BiK j)x(t) (15)

In order to analyse the time evolution of the system state, using a quadratic Lyapunov function

V (x(t)) = xT (t)P−1x(t) (P = PT > 0), it easily follows that the stability condition V (x(t)) < 0 is

ensured if there exists P and R j such that the following LMI conditions hold [26]

S(AiP−BiR j)< 0 i = 1, . . . ,n; j = 1, . . . ,n (16)

The gains of the controller (14) are then given by

K j = R jP−1, j = 1, . . . ,n (17)

4.2. Controller with saturation constraint

In this section, the objective is to design a time-varying state feedback controller (14) to guarantee

the stability of the bounded input system (12). Replacing (14) into (13) allows to express the

dynamics of the system state as:

x(t) =n

∑i=1

n

∑j=1

3nu

∑k=1

hi(ξ (t))h j(ξ (t))µk(t)((Ai−BiΛkK j)x(t)+BiΓk) (18)




The controller synthesis consists in designing the gains K j ensuring the stability of the system (18)

and the convergence of the state to an origin-centered ball as proposed in Theorem 1.

Theorem 1

There exists a time-varying state feedback controller (14) for a saturated input system (12) ensuring

that the system state converges toward an origin-centered ball of radius bounded by β , if there

exist matrices P = PT > 0, R j, Σk = ΣTk > 0 solutions of the following optimization problem (for

i = 1, . . . ,n, j = 1, . . . ,n and k = 1, . . . ,3nu)

minP,R j ,Σk

β (19)

s.t. Qi jk I

I −β I

< 0 (20)

ΓTk BT

i ΣkBiΓk < β (21)

with

Qi jk =

S(AiP−BiΛkR j) I

I −Σk

(22)

The controller gains are given by

K j = R jP−1, j = 1, . . . ,n (23)

Proof

Let us define the following Lyapunov function

V (x(t)) = xT (t)P−1x(t) (24)

where P = PT > 0. According to equations (18) and (24), the time derivative of V (x(t)) is given by

V (x(t)) =n

∑i=1

n

∑j=1

3nu

∑k=1

hi(t)h j(t)µk(t)(S(xT (t)P−1BiΓk + xT (t)P−1(Ai−BiΛkK j)x(t))

)(25)



10 S. B. ET AL.

Using Lemma 1, with Σk = ΣTk > 0, the time derivative of the Lyapunov function (25) is bounded as

follows

V (x(t))≤n

∑i=1

n

∑j=1

3nu

∑k=1

hi(t)h j(t)µk(t)(

ΓTk BT

i ΣkBiΓk + xT (t)Qi jkx(t))

(26)

with:

Qi jk = S(P−1(Ai−BiΛkK j))+P−1Σ−1k P−1 (27)

Let us define

ε = mini=1:n, j=1:n,k=1:3nu

λmin(−Qi jk) (28)

δ = maxi=1:n,k=1:3nu

ΓTk BT

i ΣkBiΓk (29)

Since Σk > 0 and from inequality (26), V (x(t))<−ε ‖ x(t) ‖22 +δ . It follows that V (x(t))< 0 for

Qi jk < 0 and ‖ x(t) ‖22>

δ

ε(30)

which means that, according to Lyapunov stability theory [32], x(t) is uniformly bounded and

converges to an origin-centered ball of radius√

δ

ε.

Let us now analyse the condition Qi jk < 0.

Applying Lemma 2, this inequality becomes

S((Ai−BiΛkK j)P)+Σ−1k < 0 (31)

Defining R j according to (23) and with a Schur complement, the inequalities (31) are equivalent to

the (1,1) block of (20), i.e. Qi jk < 0.

As the weighting functions hi(t), h j(t), µk(t) satisfy (2) and Σk > 0, if Qi jk < 0 is satisfied for

i= 1, . . . ,n, j = 1, . . . ,n, k = 1, . . . ,3nu , and ‖ x ‖22>

δ

ε, then V (x(t))< 0, implying that x(t) converges

to an origin-centered ball of radius√

δ

ε.

In order to improve the convergence to zero, the objective is now to minimize the radius√

δ

ε. Firstly

δ is bounded by β from (29) and the LMIs (21). From (20), with a Schur complement, it obviously

follows that

(1/β ) I <−Qi jki, j=1,...,n,k=1,...,3nu (32)




implying that all the eigenvalues of (−Qi jk) are larger that 1/β . As a consequence 1/β < ε holds,

and finally the radius of the ball is bounded by β .

4.3. Numerical example

Let us consider a nonlinear model of the cart-pendulum system illustrated in figure 1. The pendulum

Figure 1. Cart-pendulum system

rotates in a vertical plan around an axis located on a cart. The cart can move along a horizontal rail,

lying in the rotation plane. The characteristic variables of the system are z(t) the cart position,

z(t) the cart velocity, θ(t) the angle between the upward direction and the pendulum and θ(t)

the pendulum angular velocity. A control force F(t) parallel to the rail is applied to the cart. The

pendulum mass and cart mass are denoted m = 1kg and M = 5kg respectively. l = 0.1m is the

pendulum length and I = 510−3kgm2 the moment of inertia of the pendulum with respect to its

rotation axis on the cart. The cart is subject to a viscous friction force f z(t), proportional to the

cart velocity (with f = 100Nm−1s), to a static friction force ksz(t), proportional to the cart position

(with ks = 0.001Nm−1) and to a friction torque kθ(t) proportional to the angular velocity (with

k = 0.045Nrad−1s).

Although the system may seem to be very academic, its popularity derives in part from the fact that

it may be unstable with saturated control and non null initial state. Additionally, the dynamics of the

system are nonlinear.



12 S. B. ET AL.

For the sake of clarity, small angles are considered and the nonlinear system may be simplified into:(m+M)z(t)+ ksz(t)+ f z(t)−mlθ(t)+mlθ 2(t)θ(t) = F(t)

−mlz(t)+(ml2 + I)θ(t)+ kθ(t)+mglθ(t) = 0(33)

In order to ensure the stability of this nonlinear system and its state convergence to the origine-

centred ball, first, the system equations (33) are written under a T-S form by applying the SNT

allowing to exactly represent the nonlinear system without any loss of information. Due to space

limitations, only the main steps to obtain the T-S model are detailed, the interested reader can refer

to [26] and in particular section 2.2.1, examples 2 and 3 for more details.

First, the state variables and the control input are defined by:

x(t) = ( z(t) z(t) θ(t) θ(t) )T , u(t) = F(t) (34)

From (33) and (34), using factorization of x and u the following quasi-LPV form is deduced:

x(t) = A(x(t))x(t)+Bu(t) (35)

with :

A(x(t)) =

0 1 0 0

a1 a2 a3(t) a4

0 0 0 1

a5 a6 a7(t) a8

, B =

0

b1

0

b2

(36)

b1 =1

(m+M)(ml)2

ml2+I

b2 =b1ml

ml2+I a1 =−ksb1 a2 =− f b1 a3(t) =−b1ml(g+ x24(t))

a4 =− b1mlkml2+I a5 =

a1mlml2+I a6 =

a2mlml2+I a7(t) =

a3(t)mlml2+I −g a8 =

a4ml−kml2+I

(37)

Analyzing (37), the premise variable is chosen as ξ (t) = x24(t), which is bounded (due to the angle

and speed limitation). The SNT transformation is applied to obtain:

ξ (t) = h1(t)ξ1 +h2(t)ξ2 (38)ξ1 = maxξ (t), ξ2 = minξ (t)

h1(t) =ξ (t)−ξ2

ξ1−ξ2, h2(t) = 1−h1(t)

(39)




h1(t) and h2(t) represent the weighting functions of the T-S model defined by:

x(t) =2

∑i=1

hi(t)(Aix(t)+Bu(t)) (40)

From the definition of ξ1,ξ2,h1 and h2 in (39), it obviously follows that the convex sum properties

0≤ hi(t)≤ 1 (for i = 1,2) and h1(t)+h2(t) = 1 are satisfied.

The matrices Ai (i = 1,2) are obtained by setting a3(t) respectively to (−b1ml(g + ξ1)) and

(−b1ml(g+ξ2)) in A(x(t)) (36).

A state feedback (14) is considered both in the nominal (without saturation) and the saturated cases

with respective gains Ki,N and Ki,T S. From relations (16) and (17), the calculated control gains for

the nominal case are equal to:

K1,N = ( 11.53 −79.84 14.34 6.48 )

K2,N = ( 9.95 −82.17 11.78 5.51 )

(41)

The gains Ki,T S, computed by solving the LMIs of the theorem 1 with umin = 0 and umax = 3, are

given by:

K1,T S = ( 0.43 2.14 1.17 0.05 )

K2,T S = ( 0.37 −9.19 −0.10 0.53 )

(42)

A fourth control strategy is performed in order to compare a conventional anti-windup controller

with the proposed one. For the conventional anti-windup, the main idea is to synthesize a

nominal feedback control and to add a compensator to handle the saturated input. The anti-windup

compensator is taken as a large gain matrix Ξ = αI with α = 5.

For the initial condition x0 = ( 0 0 π/12 0 )T , figure 2 shows the time evolution of the states

where:

• xN is the state trajectory obtained by applying the unsaturated nominal control. It is ruled by:

xN(t) = ∑2i=1 hi(t)(Ai−BKi,N)xN(t).

• xN,sat is the state trajectory obtained by applying the saturated nominal control. It is ruled by:

xN,sat(t) = ∑2i=1 hi(t)AixN,sat(t)+Bsat(−∑

2i=1 hi(t)Ki,NxN,sat(t)).

• xsat,T S is the state trajectory obtained by applying the controller designed in the theorem 1. It

is ruled by: xsat,T S(t) = ∑2i=1 hi(t)Aixsat,T S(t)+Bsat(−∑

2i=1 hi(t)Ki,T Sxsat,T S(t)).



14 S. B. ET AL.

• xAW is the state obtained when applying the nominal control with an anti-windup module. The

control input is adjusted using the difference usat(t)−u(t), but the controller gain is computed

without taking into account the saturation on the input control.

It can be seen on figure 2 that the saturation of the nominal control prevent the first state variable of

0 0.5 1 1.5 2 2.5 3

0

0.01

0.02

0.03

0.04

time (s)

Ca

rt p

ositio

n

x

1n

x1nsat

x1satTS

x1AW

0 0.5 1 1.5 2 2.5 3−0.2

−0.1

0

0.1

0.2

time (s)

An

gle

x

3n

x3nsat

x3satTS

x3AW

Figure 2. System states with state feedback control

xN,sat from converging close to the origin, whereas the proposed approach allows xsat,T S to do so.

It can also be seen that the transient state response obtained with the proposed controller is better

damped than the one with the anti-windup (AW) controller.

The four nominal and saturated control signals are displayed on figure 3




0 0.5 1 1.5 2 2.5 3−8

−6

−4

−2

0

2

4

6

8

10

12

time (s)

Sa

tura

ted

sta

te f

ee

db

ack

co

ntr

ol

un

unsat

uTS

uAW

Figure 3. Saturated state feedback control

5. OUTPUT FEEDBACK CONTROLLER: A DESCRIPTOR APPROACH

The state feedback controller, proposed in the previous section, allows to efficiently compensate the

input saturation, but it needs the whole state vector to be accessible. In this section, this limitation is

overcome by envisaging output feedback control: if the state vector is not entirely available, static

or dynamic output feedback controllers can be designed using only measured signal.

Static output feedback control is the simplest approach since no further dynamics are needed.

However, a dynamic compensator introducing extra dynamics may be required to increase the

number of freedom degrees in the design and improve the closed-loop transient response. It may

be noted that dynamic output controller encompasses the class of observer-based controllers.

In the following section, both static and dynamic output feedback controllers are proposed in a

unified framework, thanks to the descriptor approach. Sufficient LMI constraints are derived from

the Lyapunov stability theory. Compared to [5], in the present paper, additional degrees of freedom

are introduced in the Lyapunov function to obtain relaxed controller design conditions.

5.1. Static feedback controller

A PDC static output feedback control is envisaged:

u(t) =n

∑j=1

h j(ξ (t))Ksjy(t) (43)



16 S. B. ET AL.

The proposed approach relies on a descriptor formulation which is well known to avoid the coupling

terms between the feedback gains and the Lyapunov matrices. As a consequence, the number of LMI

decreases and relaxed conditions are obtained [15].

The control law (43) and the system (12) are written as a descriptor system

Esxa(t) =3nu

∑i=1

n

∑j=1

µi(t)h j(t)(A si jxa(t)+Bs

i j) (44)

where the augmented state vector and system matrices are defined by xa(t) =

( xT (t) uT (t) yT (t) ) and

Es = diag(Inx ,0nu+m), A si j =

A j B jΛi 0

0 −Inu Ksj

C j D jΛi −Im

, Bsi j =

B jΓi

0

D jΓi

The objective is to compute the controller gains Ksj of (43), according to the saturation limits, in

order to guarantee the stability of the closed-loop system (44) .

Theorem 2

There exists a static feedback controller (43) for a system with bounded inputs (12) such that the

system state converges toward an origin-centered ball of radius bounded by βs, if there exist matrices

Ps1 ∈ Rnx×nx ,Ps

1 = (Ps1)

T > 0,Ps2 ∈ Rnu×nu ,Ps

2 > 0,Ps31 ∈ Rm×nx ,Ps

32 ∈ Rm×nu ,Ps33 ∈ Rm×m,Rs

j ∈

Rnu×m,Σ1si j ∈ Rnx×nx ,Σ1s

i j = (Σ1si j )

T > 0, Σ3si j ∈ Rm×m,Σ3s

i j = (Σ3si j )

T > 0, solutions of the following

optimization problem (for i = 1, . . . ,3nu and j = 1, . . . ,n)

minPs

1 , Ps2 , Ps

31, Ps32, Ps

33, Rsj , Σ1s

i j , Σ3si j

βs (45)




under the LMI constraints (46) and (47)

Qs1i j Qs12

i j CTj Ps

33−PsT31 Ps

1 PsT31 Inx 0 0

∗ Qs2i j Rs

j +ΛiDTj Ps

33−PsT32 0 PsT

32 0 Inu 0

∗ ∗ −Ps33− (Ps

33)T 0 PsT

33 0 0 Im

∗ ∗ ∗ −Σ1si j 0 0 0 0

∗ ∗ ∗ ∗ −Σ3si j 0 0 0

∗ ∗ ∗ ∗ ∗ −βsInx 0 0

∗ ∗ ∗ ∗ ∗ ∗ −βsInu 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −βsIm

< 0 (46)

ΓTi DT

j Σ3si j D jΓi +Γ

Ti BT

j Σ1si j B jΓi < βs (47)

with

Qs1j = S(Ps

1A j +CTj Ps

31), Qs12i j Ps

1B jΛi +CTj Ps

32 +PsT31 D jΛ

Ti and Qs2

i j = S(−Ps2 +PsT

32 D jΛi) (48)

The controller gains are given by Ksj = ((Ps

2)T )−1Rs

j, j = 1, . . . ,n.

Proof

Let us define the Lyapunov function

V (xa(t)) = xTa (t)(E

s)T Psxa(t) (49)

with Ps defined by

Ps =

Ps

1 0 0

0 Ps2 0

Ps31 Ps

32 Ps33

(50)

where Ps1 = (Ps

1)T > 0 and Ps

2 > 0. It follows from (50) and the structure of E, that (Es)T Ps =

(Ps)T Es ≥ 0 and that the Lyapunov function is in fact quadratic in the state vector: V (xa(t)) =

xT (t)Ps1x(t).

Using the state equation (44), the time derivative of the Lyapunov function (49) is given by

V (xa(t)) =3nu

∑i=1

n

∑j=1

µi(t)h j(t)S(xTa (t)(P

s)T Bsi j + xT

a (t)(As

i j)T Psxa(t)) (51)



18 S. B. ET AL.

Using Lemma 1, V (xa(t)) is bounded as follows

V (xa(t))≤3nu

∑i=1

n

∑j=1

µi(t)h j(t)(ΓTi BT

j Σ1si j B jΓi +Γ

Ti DT

j Σ3si j D jΓi + xT

a (t)Qsi jxa(t)) (52)

with

Qsi j = (A s

i j)T Ps +(Ps)T A s

i j +diag(Ps1(Σ

1si j )−1Ps

1 ,0,0)

+( Ps31 Ps

32 Ps33

)T (Σ3si j )−1( Ps

31 Ps32 Ps

33)

(53)

Let us define

εs = min

i=1:3nu , j=1:nλmin(−Qs

i j) (54)

δs = max

i=1:3nu , j=1:n(ΓT

i (BTj Σ

1si j B j +DT

j Σ3si j D j)Γi) (55)

Since Σ1si j and Σ3s

i j are positive definite, from (52) with the convex sum property (2), V (xa(t)) <

−εs||xa||22 +δ s. It follows that V (xa(t))< 0 for

Qsi j < 0 and ‖ xa ‖2

2>δ s

εs (56)

which means that xa(t) is uniformly bounded and converges to the origin-centered ball of radius√δ s/εs according to Lyapunov stability theory [32]. (46) is proved with some Schur’s complements

applied to Qsi j < 0 and the variable change Rs

j = (Ps2)

T Ksj . The objective is now to minimize the

radius√

δ s/εs. Firstly, using definition (55), δ s is bounded by βs when considering LMIs (47).

Secondly, it can be shown that 1/εs < βs. From (54), it follows that

−Qsi j > (1/βs) I, i = 1, . . . ,3nu , j = 1, . . . ,n (57)

meaning that all eigenvalues of (−Qsi j), including εs, are bigger then 1/βs. Thus 1/εs < βs and the

radius√

δ s/εs is bounded by βs.

5.2. Dynamic output feedback controller

The objective is now to design a stabilizing dynamic output feedback control even in the presence

of control input saturation. As previously, the solution is obtained by representing the saturation as

a T-S system and by solving an optimization problem under LMI constraints. Let us consider the




following nthc order dynamic output feedback controller defined by

xc(t) =

n

∑j=1

h j(t)(Acjxc(t)+Bc

jy(t))

u(t) =n

∑j=1

h j(t)(Ccjxc(t)+Dc

jy(t))(58)

designed to guarantee the stability of the saturated system (13). The matrices Acj ∈ Rnc×nc , Bc

j ∈

Rnc×m, Ccj ∈ Rnu×nc and Dc

j ∈ Rnu×m are the controller gains, determined to ensure the stability

of the closed-loop system (13) with (58). The controller order nc can be adapted according to the

control objectives and system dynamics. The closed-loop system defined from (13) and (58) is

written under the descriptor form:

Ed xa(t) =3nu

∑i=1

n

∑j=1

µi(t)h j(t)(A di j xa(t)+Bd

i j) (59)

with xa(t) = ( xT (t) xTc (t) uT (t) yT (t) ) and

A di j =

A j 0 B jΛi 0

0 Acj 0 Bc

j

0 Ccj −Inu Dc

j

C j 0 D jΛi −Im

, Bd

i j =

B jΓi

0

0

D jΓi

, Ed =

Inx+nc 0

0 0nu+m

(60)

Theorem 3

There exists a dynamic feedback controller (58) for the saturated input system (12) such that the

system state converges toward an origin-centered ball of radius bounded by βd , if there exist matrices

Pd11 ∈ Rnx×nx ,Pd

11 = (Pd11)

T > 0,Pd22 ∈ Rnc×nc ,Pd

22 = (Pd22)

T > 0, Pd33 ∈ Rnu×nu > 0, Pd

41 ∈ Rm×nx ,

Pd42 ∈ Rm×nc , Pd

43 ∈ Rm×nu , Pd44 ∈ Rm×m, Ac

j ∈ Rnc×nc , Bcj ∈ Rnc×m, Cc

j ∈ Rnu×nc , Dcj ∈ Rnu×m,

Σ1di j = (Σ1d

i j )T > 0, and Σ2d

i j = (Σ2di j )

T > 0, solutions of the following optimization problem (for

i = 1, . . . ,3nu and j = 1, . . . ,n)

minPd

11,Pd22,P

d33,P

d41,P

d42,P

d43,P

d44,A

cj ,B

cj ,C

cj ,D

cj ,Σ

1di j ,Σ

2di j

βd (61)



20 S. B. ET AL.

under the LMI constraints (62) and (63).

Q1i j CT

j Pd42 Q13

i j CTj Pd

44− (Pd41)

T Pd11 (Pd

41)T I 0 0 0

∗ Acj +(Ac

j)T (Cc

j)T +(Pd

42)T D jΛi Bc

j− (Pd42)

T 0 (Pd42)

T 0 I 0 0

∗ ∗ Q3i j Dc

j− (Pd43)

T +ΛiDTj Pd

44 0 (Pd43)

T 0 0 I 0

∗ ∗ ∗ −Pd44− (Pd

44)T 0 (Pd

44)T 0 0 0 I

∗ ∗ ∗ ∗ −Σ1di j 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ −Σ2di j 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −βdI 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −βdI 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −βdI 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −βdI

< 0

(62)

ΓTi BT

j Σ1di j B jΓi +Γ

Ti DT

j Σ2di j D jΓi < βd (63)

with:

Q1i j = AT

j Pd11 +Pd

11A j +CTj Pd

41 +(Pd41)

TC j

Q13i j = Pd

11B jΛi +CTj Pd

43 +(Pd41)

T D jΛi

Q3i j =−Pd

33− (Pd33)

T +(Pd43)

T D jΛi +ΛiDTj Pd

43

The gains of the controller (58) are given by Acj = (Pd

22)−1Ac

j Bcj = (Pd

22)−1Bc

j

Ccj = ((Pd

33)−1)TCc

j Dcj = ((Pd

33)−1)T Dc

j

(64)

Proof

Let us define the Lyapunov function V (xa(t)) = xTa (t)(E

d)T Pdxa(t) with Ed and Pd respectively

defined in (60) and (66) that satisfy

(Ed)T Pd = (Pd)T Ed ≥ 0 (65)

The matrix Pd is partitioned according to A di j defined in (60) and, for satisfying (65), it is chosen as

Pd =

Pd11 0 0 0

0 Pd22 0 0

0 0 Pd33 0

Pd41 Pd

42 Pd43 Pd

44

(66)




with Pd11 = (Pd

11)T > 0, Pd

22 = (Pd22)

T > 0 and P33 > 0. One can note that from (60) and (66), the

Lyapunov function is quadratic in the system and controller states since V (xa(t)) = xT (t)Pd11x(t)+

xTc (t)P

d22xc(t).

Applying the same developments as for the static controller with the variable changesAc

j = Pd22Ac

j Bcj = Pd

22Bcj

Ccj = (Pd

33)TCc

j Dcj = (Pd

33)T Dc

j

(67)

and defining εd and δ d by

εd = mini=1:3nu , j=1:n

λmin(−Qdi j)

δ d = maxi=1:3nu , j=1:n

ΓTi (B

Tj Σ

1di j B j +DT

i Σ2di j Di)Γi

(68)

with Qdi j defined in the same way as Qs

i j was, i.e.

Qdi j = S((A d

i j )T Pd)+diag(Pd

1 (Σ1di j )−1Pd

1 +(Pd41)

T (Σ2di j )−1Pd

41,

(Pd42)

T (Σ2di j )−1Pd

42,(Pd43)

T (Σ2di j )−1Pd

43,(Pd44)

T (Σ2di j )−1Pd

44)

(69)

the stabilizing conditions are linearized and given by (62). As the weighting functions satisfy (2)

and Σ1di j ,Σ

2di j > 0, if (62) holds and ‖ xa ‖2

2>δ d

εd , then V (xa(t))< 0, implying that xa(t) converges to

an origin-centered ball of radius√

δ d

εd . Similarly to the proof of theorem 2, the radius of the ball is

bounded by β d due to (62) and is minimized in (61).

Remark 2

Compared to the state feedback control, the dynamic output feedback control introduces additional

degrees of freedom in the controller design. Thus the stability of the unsaturated open-loop

subsystems (namely the matrices Ai, for i = 1, . . . ,n) is no longer required for the LMI constraints

to be feasible.

5.3. Numerical example

Let us consider the same cart-pendulum system (40) in section 4.3, both static and dynamic output

feedback controllers are designed and their performances are compared. The horizontal and angular



22 S. B. ET AL.

velocity can be measured, then the output matrices are defined by

C =

1 0 0 0

0 0 0 1

, D =

0

0

(70)

From (70), the second measured output is x4(t), then the premise variable ξ (t)= x24(t) is measurable,

as assumed in the beginning of section 2. Solving the optimization problem given in theorem 2, the

static controller gains are given by:

Ks1 = (−0.002 −1.126), Ks

2 = (−0.001 −0.1 4) (71)

Applying theorem 3 with nc = 2, the following dynamic controller gains are obtained:

Ac1 =

−0.324 0.083

0.083 −0.392

,Bc1 =

−0.003 0.002

0.002 −0.823

Ac

2 =

−0.332 0.783

0.783 −0.314

,Bc2 =

−0.003 0.002

0.002 −0.749

Cc

1 =

(0.019 −0.026

), Dc

1 =

(−0.002 −0.0021

)Cc

2 =

(0.097 0.0127

), Dc

2 =

(−0.002 −0.005

)(72)

For the initial condition x0 = ( 0 0 π/12 0 )T , figure 4 shows the system states in the

following four cases. The system states xN and xN,sat are obtained from the nominal and saturated

nominal controls presented in the previous example. The improvement from the proposed approach

is also clear, since for the first state the oscillation amplitudes are much smaller than previously

especially on the first state component. xsat,T Ss denotes the state trajectory for the proposed T-

S approach with static controller and finally xsat,T Sd stands for the proposed T-S approach with

dynamic controller.

From the depicted figures, it is clear that with the proposed T-S approach the convergence to an

origin-centered ball is ensured. It is also clear that the obtained results with the dynamic controller

are slightly better than the ones obtained with the static one, since the first state variable converges




0 0.5 1 1.5 2 2.5 3

0

0.02

0.04

time (s)

Ca

rt

po

sitio

n

x

1n

x1nsat

x1satTSs

x1satTSd

0 0.5 1 1.5 2 2.5 3−0.5

0

0.5

1

time (s)

An

gle

x

3n

x3nsat

x3satTSs

x1satTSd

Figure 4. System states with output feedback control

closer to the origin and the oscillation damping of the second state variable is better.

The saturated control signals for each case are represented in figure 5

0 0.5 1 1.5 2 2.5 3−8

−6

−4

−2

0

2

4

6

8

10

12

time (s)

Sa

tura

ted

ou

tpu

t fe

ed

ba

ck c

on

tro

l

un

unsat

uTSs

uTSd

Figure 5. Saturated output feedback control

To compare the two controllers, in figure 6 are depicted the phase diagrams of the states x2 and

x1 for both static and dynamic output controllers (the initial value of (x1,x2) is (0,0)). One can



24 S. B. ET AL.

see that, applying the dynamic controller, the state converges to an origin-centered ball of radius

β d = 0.6×10−5, which is smaller than β s = 4.5×10−3 obtained with a static controller.

−1.5 −1 −0.5 0 0.5 1 1.5x 10

−3

−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

xsatTSs

xsatTSd

Figure 6. Phase diagram

6. CONCLUSION AND FUTURE WORKS

Thanks to a polytopic representation of the control input saturation, a nonlinear system with

bounded inputs can be modeled as a Takagi-Sugeno system. It is important to note that, using the

proposed representation, the actuator model is expressed in terms of the control variables and of

the saturation limits. As a consequence, these limits are taken into account when computing the

controller gains. Moreover, the proposed T-S approach allows to extend the use of linear tools,

namely the LMI formalism, to a nonlinear control problem.

A state feedback controller and both static and dynamic output feedback controllers were developed.

The dynamic controller order is a degree of freedom fixed by the user. Relaxed LMI constraints are

obtained with the use of the descriptor approach. The potentially destabilizing effect of the control

saturation is compensated by the proposed controller ensuring that the closed-loop system state

converges to an origin-centered ball.

In order to show the effectiveness of the proposed approach, all the results have been applied to the




nonlinear model of a pendulum cart with saturated actuator.

Future works may concern the extension of the present study to the T-S models with unmeasurable

premise variables (e.g. activating functions depending on the unmeasured state variables) by the

mean of state observer based controller. Another possible extension would be the use of more

sophisticated Lyapunov functions (e.g. nonquadratic candidate functions) in order to obtain relaxed

LMI constraints in the controller design procedure.

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State and output feedback control for Takagi-Sugeno