Page 1

388

Asian Journal of Control, Vol. 6, No. 3, pp. 388-397, September 2004

Manuscript received September 13, 2002; revised February

11, 2003; accepted November 10, 2003.

The authors are with Institute of Automation, Shanghai

Jiaotong University, Shanghai, 200030, P.R. China.

B.C. Ding is also with Institute of Automation, Hebei Uni-

versity of Technology, Tianjin, 300130, P.R. China.

ON THE STABILITY OF OUTPUT FEEDBACK PREDICTIVE

CONTROL FOR SYSTEMS WITH INPUT NONLINEARITY

Bao-Cang Ding, Yu-Geng Xi, and Shao-Yuan Li

ABSTRACT

For input saturated Hammerstein systems, a two-step output feedback

predictive control (TSOFPC) scheme is adopted. A receding horizon state

observer is chosen, the gain matrix of which has a form similar to the linear

control law. Through application of Lyapunov’s stability theory, the closed-

loop stability for this kind of system is analyzed. The intermediate variable

may or may not be available in real applications, and these two cases are

considered separately in this paper. Furthermore, the domain of attraction for

this kind of system is discussed, and we prove that it can be tuned to be arbi-

trarily large if the system matrix is semi-stable. The stability results are

validated by means of an example simulation.

KeyWords: Two-step control, predictive control, state observer, stability,

domain of attraction.

I. INTRODUCTION

Model predictive control (MPC) has been widely

adopted for the control of constrained multivariable sys-

tems. Most of the available MPC software is designed for

linear models, because the computational burden induced

by a nonlinear model is generally much greater than that

induced by a linear model, and because a linear model is

easily identifiable. However, linear MPC can not meet the

increasingly stringent performance requirements, so

nonlinear MPC techniques are highly desirable. On the

other hand, incorporating constraints into the optimization

exponentially increases the computational burden, which

is the main shortcoming of the currently most available

synthesis procedures that add artificial constraints to

guarantee stability [1].

Compared with a pure linear model, the Hammerstein

model can represent with greater accuracy some processes,

such as pH neutralization and high purity distillation

column [2]. Moreover, some Volterra series models can

be approximated by the Hammerstein model [3]. The

Hammerstein model is also one of the most easily identi-

fiable nonlinear models. Another advantage of the Ham-

merstein model, which is most closely related to this pa-

per, is that the nonlinearity can be removed to simplify

controller design [4,5]. If the Hammerstein nonlinearity

needs to be identified on-line, then it can be removed by

solving the nonlinear algebraic equation (group) (NAEG)

[5]; i.e., the linear part of the model is applied first to ob-

tain the ‘desired intermediate variable’, and then the

real control action is obtained by solving the NAEG. If

input saturation also exists, we can deal with it by us-

ing the ‘removal’ technique (i.e., through desaturation or

anti-windup), in which case the nonlinear removal con-

troller is called a ‘two-step’ controller [6,7] (see Fig. 1).

As the unconstrained linear control law can be de-

signed off-line, the computational effort is largely re-

duced with the two-step controller.

Regarding MPCs for Hammerstein systems, apart

from the nonlinear removal method, there are also

methods that incorporate nonlinearity into the optimiza-

tion. The techniques in [8,9], which consider only an

unconstrained SISO system with a control horizon equal

to 1, are restrictive and not easy to generalize. In [10], the

authors transformed the inversion of the Hammerstein

nonlinearity into a polytopic description, thus convert-

ing the nominal control problem into a robust linear

control problem. A large number of LMIs are involved,

and the final control action is obtained by iteration and

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B.C. Ding et al.: On the Stability of Output Feedback Predictive Control for Systems with Input Nonlinearity

389

Fig. 1. The schematic structure of the two-step controller, where f is the Hammerstein nonlinearity, u the input, y the output, v the

intermediate variable, and vL the desired intermediate variable.

by changing LMIs in each iteration. In order to improve

the feasibility of this approach, the local control law and

end-point weighting matrix are both optimized on-line.

Not considering the on-line identification of Hammer-

stein nonlinearity, the input uncertainty or incorporation

of the state observer imposes a heavy computational

burden on the technique ascribed in [10].

The authors in [10] provided an algorithm with

guaranteed stability; more precisely, the stability of the

controller in [10] is equivalent to the initial feasibility of

the optimization problem. Inevitably, by applying poly-

topic description, one can reduce the domain of attrac-

tion for the original system, while nonlinear removal

can not reduce it. In two-step control, feasibility is eas-

ily guaranteed, while stability needs further investiga-

tion. As shown in Fig. 1, if the ‘desired intermediate

variable’ is implemented by means of a real control in-

put without error, then there will be no nonlinearity in

the closed-loop system, and the studied problem will be

simplified. Generally, this perfect case can not be guar-

anteed, so in the stability analysis of the two-step con-

troller, considering the nonlinearity reserved in the

closed-loop system (which is formed by combining the

middle four blocks shown in Fig. 1) is necessary. This

reserved nonlinearity, unlike the sector nonlinearity in

[11,12], is not fixed a priori. It can be tuned by defining

the accuracy of the NAEG solution and restricting the

saturation level.

In [6,7,13], we studied the stability of the two-step

predictive controller for input saturated Hammerstien

systems. The authors in [6] considered the input/output

model, while in [7] and [13], the authors considered the

state-space one, the linear parts of which are nominal

and uncertain, respectively. However, in [7,13], the au-

thors only dealt with the case where the system states

are measurable, so their results can not be directly gen-

eralized or extended to the case where the state observer

exists because the existence of observation error com-

plicates the closed-loop system and spans the system

dimension. This paper continues the research reported in

[7]. The MPC based on the state observer is described in

[14] and in the survey in [1], but none of these results

can be directly applied to TSOFPC. Moreover, different

from [6,7,13], this paper will consider the domain of

attraction separate from the stability conditions.

This paper is organized as follows. Section II de-

scribes the main idea behind TSOFPC. Sections III and

IV draw conclusions with respect to the stability of

TSOFPC. Section V gives an example.

II. TWO-STEP PREDICTIVE

CONTROLLER FOR SYSTEMS WITH

INPUT NONLINEARITY

Consider the following discrete-time system with

input nonlinearity:

(1)( )( ) ,x kAx kBv k

+=+

( )y k( ), ( )v k( ( )) ,u k

=φ

Cx k

=

(1)

where x∈ℜn, v∈ℜm, y∈ℜp and u∈ℜm are the state, in-

termediate variable, output, and input, respectively; φ

satisfying φ (0) = 0 represents the relationship between

the input and intermediate variable; (A, B, C) is com-

pletely controllable and observable. In this paper, we

assume that φ includes the Hammerstein nonlinearity f,

its uncertainty δf, and the input saturation constraint sati,

i = 1, …, m.

The first step in TSOFPC applies the linear system

x(k + 1) = Ax(k) + Bv(k), y(k) = Cx(k). The state estima-

tion is x ? , and the prediction model is

(1| )( | )i k

( | ), ( | )i kx k k

?

( ) x k

?

(2)

x k

?

ikAx k

?

Bv k

+ +=+++=

.

Define the objective function as

Two-step controller

u

φ

y

v

s

u

L

v

Hammerstein model

Generalized system

Controller

based-on

linear model

Linear part of

the model

Nonlinear

removal

via solving

NAEG

ˆ u

Desatu

-ration

f + δf

Page 3

390

Asian Journal of Control, Vol. 6, No. 3, September 2004

(| )(| )

T

N

Jx k

?

N kP x kN k

=++

?

1

0[ (| )j k( | )j k

N

∑ ?

T

j

x kQ x k

?

−

=

+++

( | ) j k(| )],j k

T

v kR v k

+++

(3)

where Q = QT ≥ 0, R > 0 are weighting matrices of the

state and intermediate variable; PN ≥ 0 is the weighting

matrix of the terminal state. The Riccati iteration

1

T

jj

PQA P A

+

=+

1

111

(),,

TTT

jjj

A P B RB PB B PAjN

−

+++

−+<

(4)

is adopted to obtain the predictive control law

*1

11

( )()( ).

TT

v kRB PBB PA x k

−

= −+

?

(5)

Noting that v*(k) in (5) may be impossible to implement

via a real control input, we formalize it as

1

11

( )( )()( ).

LTT

v kK x k

?

RB PBB PA x k

−

= −+

?

?

(6)

The second step in TSOFPC solves the NAEG

vL(k) – f(û(k)) = 0 to obtain û(k) = ϕ (vL(k)), and then the

real control input u(k) is obtained via desaturation, i.e.,

u(k) = sat{û(k)}. We formalize the second step as

u(k)? g (vL(k)). When u(k) is implemented, the resultant

v(k) is formalized as

{

=

}()

( )v k(( ))

δ

()( )

φ

L

h v k

+

ff sat u k

=+

?

=

δ

( )({ (( ))})k( )(( )).k

LL

ff sat g v g v

⋅

(7)

This is the control law of TSOFPC in terms of the in-

termediate variable.

Remark 1. We may off-line determine the f0 that is

more likely to be the actual f and the ϕ0 that best satis-

fies f0ϕ0 =1

= [1, 1, …1]T. When we solve NAEG, û(k) =

ϕ0 (vL(k)) serves as the initial guess. Various methods

[17] can be used, depending on f. In most cases, NAEG

does not need to be solved accurately and the Newton

recursive method or its modified form can be used. Note

that for one vL(k), several û(k) as well as u(k) may be

obtained. However, by adding extra conditions (such as

by choosing u(k) to be the closest to u(k − 1), by choos-

ing u(k) with the smallest amplitude, etc.), we can obtain

the most suitable u(k). For the decentralized f, the reader

may refer to [5].

Now, we turn our attention to designing the state

observer. First, we assume that v is not measurable, and

?

that δf ≠ 0; then, v is not available, and the observer can

be designed based on vL as follows:

(1)() ( )

?

( )( ).

L

x k

?

ALC x kB v k L y k

+=−++

(8)

When v is measurable or δf = 0, i.e., when v is available,

another observer can be applied although (8) is also ap-

plicable. The case in which v is available will be dis-

cussed later in section IV.

L in (8) is the observer gain matrix, defined as [15]

1

101

()

TT

LAPC

?

RCPC

?

−

=+

, (9)

where

1P? can be iterated from

01

T

jj

P

?

QAP A

?

+

=+

?

1

10110

(),

TTT

jjj

AP C RCP C

?

CP A

?

jN

−

+++

−+<

. (10)

In [15], Q0 and R0 are taken as the covariance of some

noise items. In this paper, since noise will not be con-

sidered, R0, Q0,

0

N

rameters.

By (1), denoting e = x − x ? , we can easily obtain the

closed-loop system with the observer (8):

P?

and N0 are taken as tunable pa-

(

(

)

)

(1)( )( )[ (

−

( ))( )],

(1)( )[ (( ))( )].

LL

LL

x kABK x kBKe kB h v k v k

e kALC e kB h v kv k

+

+

=

=

+

−

−

+

+−

⎧

⎨

⎩

(11)

The closed-loop system for (8) and (11) is shown in

Fig. 2.

?

, the nonlinear item in (11) will disap-

pear, and the studied problem will become linear. How-

ever, because of desaturation, the error encountered in

solving NAEG, the modeling error of Hammerstein

nonlinearity, and the execution error of the actuator,

generally, h = 1

can not hold. Moreover, the estimation

error in the state observer has to be considered. All these

factors make the stability analysis of TSOFPC difficult.

When h = 1

?

Fig. 2. The closed-loop system for TSOFPC, where v is

not available.

(

Bv k

)

(1)( )

( )( )

L

x k

?

ALC x k

+

Ly k

+=

+

−

?

x ?

y

v

L

v

h

K

x(k + 1) = Ax(k) + Bv(k)

x(k) = Cx(k)

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B.C. Ding et al.: On the Stability of Output Feedback Predictive Control for Systems with Input Nonlinearity

391

III. THE STABILITY OF THE TWO-STEP

PREDICTIVE CONTROLLER

Lemma 1. [16] Assume that X and Y are matrices, while

s and t are vectors, all with proper dimensions. Then

21, 0.

TTTTT

s XYts XX s

γ

t Y Yt

≤+⋅ ∀ >

γγ

(12)

Define vx(k)? Kx(k), and ve(k)? Ke(k), so that vx

= vL(k) + ve(k). In the following, we take R = λI.

(k)

Theorem 1. For systems represented by (1), TSOFPC

(6)-(8) is adopted. Assume that there exist positive sca-

lars γ1 and γ2 such that the system design satisfies the

following conditions:

(1)

01

QPP

>−

;

(2)

200

(1 1)(QPLR L

+−+−

γ

− < −+

?

λγ

−λ

0

)

T

?

11

1

h v k

1

( ))

111

(1 1)()

TTT

PA P BI B P BB P A

−−

+

λ

;

(3)

(( ))(

LTL

h v k

11

[ (h v

(

1

( ))k

)

2

γ

( )] [(1k) ()

LLTT

vIB P B

+−++

γλ

1

][ (( ))k( )]k0

TLL

B P B h v

?

λ

v

++−≤

.

Then, the equilibrium {x = 0, e = 0} of the closed-loop

system is exponentially stable.

Proof. Choose a quadratic function as V(k) = x(k)T P1 x(k)

+ λ e(k)T

P e k

?

Applying (11), we obtain after lengthy

deduction the following:

1( ).

01

( 1) ( )( ) (x k) ( )P x k

T

V kV kQP

+−=−+−

(( ))k( ( )) [ (k

+

( ))k( )]k

LTLLLT

h vh vh v v

−−

λ

11

()[ ( ( ))k( )] k

TTLL

IB P BB P B h v

?

λ

v

⋅++−

λ

1

2 ( ) (v k )[ (( )) k( )] k

eTTLL

IB P B h v v

−+−

λ

1

( ) (v k )( )

eTTe

I B P B v k

++

λ

11

( ) [( e k

λ

)()] ( )P e k

?

TT

A LC P A

?

LC

+−−−

()

1

2( )

e k

λ

[ ( ( ))( )].

T

TLL

ALCPB h v k

?

v k

+−−

By applying Lemma 1 twice, we can obtain

(1)( ) ( ) (

V k V kx k

+−≤

−λ

+−

()

21

1

B PB h v k

++γλ

()

1

1 1/( ) (

v k

++γ

()(

2

[ 1 1/

A

⋅+−

γ

( ) (

x kQP

=−+

01

) ( )

P x k

T

QP

−+−

( ( ))(( ))

LTL

h v k h v k

()

−

11

[ (

h v k

( ))( )] [ 1

v k

?

()

LLTT

I B PB

++γλ

][ (

λ

( ))( )]

TLL

v k

1

−

) ( )( )

e k

λ

eTTeT

IB PB v k

++

?

)()

11

] ( )

P e k

T

LC

−

P A

?

LC

−

01

) ( )

P x k

(( ))(( ))

TLTL

h v kh v k

−λ

()

11

[ (

h v k

( ))( )] [ 1

v k

()

LLTT

IB PB

+−++γλ

211

(1) ][ (( ))k( )] (1 1/k

+

)

TLL

B P B h v

?

λ

v

++−+

γγ

12

( ) (v k)( ) (1 1

+

)( )

eTTeT

IB P B v k e k

×++

λγλ

0001

() ( )e k( )e k( )

TT

QP

?

LR LP e k

?

−+−−

λ

.

With all the conditions in Theorem 1 satisfied,

V(k + 1) − V(k) < 0 for ∀ [x(k)T, e(k)T] ≠ 0. Hence, V(k) is

the Lyapunov function that proves the exponential sta-

bility. ■

Conditions (i)-(ii) in Theorem 1 are the require-

ments imposed on R, Q, PN, N, R0, Q0,

while (iii) is the requirement imposed on h. Generally,

decreasing the NAEG solution error, then γ1 and γ2 will

improve satisfaction of (iii). When h = 1

stability conditions of the linear system. On the other

hand, if (i)-(ii) are satisfied but h ≠ 1

tain more sensible stability conditions under (iii). To

accomplish this, we assume that

0

N

P?

, and N0,

?

, (i)-(ii) are the

?

, then we can ob-

1

|| ( )|| h s

|| ||, s|| ( ) h s|||1| || ||,

⋅

|| ||s

≤ ∆

bsb

∀

s

≥−≤−

,

(13)

where b > 0 and b1 > 0 are scalars; || s || is the 2-norm of

vector s. The method for determining b were b1 were

given in [7].

Corollary 1. For systems represented by (1), TSOFPC

(6)-(8) is adopted, where h satisfies (13). Assume the

following:

(A1) ∀[x(0), e(0)]∈ Ω ⊂ ℜ2n, || vL(k) || ≤ ∆ for all k ≥ 0;

(A2) there exist positive scalars γ1 and γ2 such that the

system design satisfies (i)-(ii) in Theorem 1;

(A3)

11

[(1)( 1) ] (bb

−−+−+λγ

+ +≤

?

γλ

.

222

max11

1)((1)

T

bB P B

−+σγ

21

(1))0

T

B P B

Then, the equilibrium (0, 0) of the closed-loop system is

exponentially stable with a domain of attraction Ω.

Proof. Apply condition (iii) in Theorem 1 and (13) to

obtain the following result:

11

( )h s

λ

( ) [ ( )h s

+

] [(1)()

TTT

h ssIB P B

−−++

γλ

21

(1)][ ( )]

T

B P B h s

?

λ

s

+ +−

γ

2

1

2

max11

( 1)((1)()

TT

b s s

λ

bIB P B

≤ −+−++

σγλ

21

(1))

TT

B P B s s

?

λ

+ +

γ

2

1

2

1

(1 ) (1)

TT

b s s

λ

bs s

= −++−γ λ

2

max1121

( 1)((1)(1))

TTT

b B P B B P B s s

?

λ

+−+++

σγγ

2

1

2

1

[ (1)(1) ]

T

bb s s

= −−+−

λγ

2

max1121

( 1)((1) (1)).

TTT

bB P BB P B s s

?

λ

+−+++

σγγ

Page 5

392

Asian Journal of Control, Vol. 6, No. 3, September 2004

Hence, if condition (iii) in Corollary 1 is satisfied, then

condition (iii) in Theorem 1 will also be satisfied. This

proves the stability. ■

Remark 2. If the nonlinearity is decentralized, then

similar as [12], we can assume that

22

,1,2

( ),1,, ,m,

iiiiiiii

b sh s s b sis

≤≤=∀≤ ∆

?

(14)

where bi,2 ≥ bi,1 > 0 are scalars. Apparently, (14) is

clearer in meaning than (13). Let b1 = min{b1,1, b2,1, …,

bm,1} and | b – 1| = max{ | b1,1 − 1|, …, | bm,1 −1|, | b1,2

−1|, …, |bm,2 −1|}; then, (13) can be deduced from (14),

so Corollary 1 is also suitable for h satisfying (14).

Remark 3. We can substitute (iii) into Corollary 1 by

means of the following more conservative condition:

( )()( )()

22

2

1

γ

12 max1

[1111( )]

T

bbbB PB

−−+−−+−

?

λγγσ

2

1 max1

(1)(1)()0.

T

bB P B

+ +−≤σ

The above inequality will be useful in the following.

Remark 4. For the case where there is no observer, the

authors in [13] studied the robust stability property of

the two-step MPC (A replaced by A + δA). By compar-

ing the results presented in [13] and [7] with those pre-

sented in the present paper, one can easily obtain the

robust stability condition for TSOFPC.

Corollary 1 introduces the domain of attraction. For

a real system, it is important to obtain the domain of

attraction, because by doing so, we can decide whether

the initial state lies in the domain of attraction. More-

over, the control parameters can be tuned to obtain the

desired domain of attraction if this is possible. The fol-

lowing theorem gives the ellipsoidal domain of attrac-

tion for this kind of system.

Theorem 2. For systems represented by (1), TSOFPC

(6)-(8) is adopted where h satisfies (13). Assume that

there exist positive scalars ∆, γ1, and γ2 such that condi-

tions (i)-(iii) in Corollary 1 are satisfied. Then, the do-

main of attraction for the closed-loop system is not

smaller than

2

11

{( , )x e| },c

nTT

c S x P xe P e

λ

=∈+≤

?

R

(15)

where

2

( / )

= ∆

=

cd

,

11 2

−

1 2

−

1 2

−

1111

||()[,]||;

TT

dIB PB B PA PP

?

−

+−λλ

(16)

i.e., for all [x(0), e(0)]∈Sc, the equilibrium (0, 0) of the

closed-loop system is exponentially stable.

Proof. Having satisfied conditions (i)-(iii) in Corollary 1,

we need only verify that ∀[ x(0), e(0)]∈Sc, ||vL(k)|| ≤ ∆

for all k ≥ 0. We adopt two transformations,

=

?

λ

; then, ∀[x(0), e(0)]∈Sc, || [ (0)T

1 2

1

P x

x

x

=

and

1 21 2

1

P ee

(0)T

e

] || ≤ c and

[]

1

11

|| (0)||||( )(0) (0) ||

LTT

vIB P BB P A xe

−

=+−λ

1 2

1

11

1

||[()

TT

IB P B B P AP−

−

≤+

λ

11 2

−

1 2

−

111

()]||

TT

IB P BB P AP

?

−

−+

λλ

||[ (0)x(0) ]||.

TT

e

≤ ∆ (17)

Thus, all the conditions in Corollary 1 are satisfied at

time k = 0 if [x(0), e(0)]∈Sc. According to the proof of

Theorem 1, [x(1), e(1)]∈Sc if [x(0), e(0)]∈Sc; therefore,

|| vL(1) || ≤ ∆ for all [x(0), e(0)]∈Sc, which shows that all

the conditions in Corollary 1 are satisfied at time k = 1.

By analogy, we can conclude that || vL(k) || ≤ ∆ for all

k ≥ 0 and all [x(0), e(0)]∈Sc. Thus, Sc is a domain of

attraction. ■

By applying Theorem 2, we can tune the control

parameters so as to satisfy conditions (i)-(iii) in Corol-

lary 1 and obtain the desired domain of attraction. The

following algorithm may serve as a guideline.

Algorithm 1. Parameter tuning guideline for achieving

the desired domain of attraction Ω

Step 1. Define the accuracy of the NAEG solu-

tion. Choose the initial ∆. Determine b1

and b.

Step 2. Choose {R0, Q0,

convergent observer.

Step 3. Choose {λ, Q, PN, N} (mainly Q, PN, N)

satisfying (i).

Step 4. Choose {γ1, γ2, λ, Q, PN, N} (mainly γ1, γ2,

λ) satisfying (ii)-(iii). If they can not both

be satisfied, then go to one of Step 1-Step

3, according to the actual situation.

Step 5. Check if (i)-(iii) are all satisfied. If they

are not, go to Step 3. Otherwise, decrease

γ1 and γ2, maintaining satisfaction of (ii),

and increase ∆ (b1 is decreased accord-

ingly), maintaining satisfaction of (iii).

Step 6. Calculate c using (16). If Sc ⊇ Ω, then

STOP, else turn to Step 1.

Of course, this does not mean that any desired do-

main of attraction can be obtained for any system. But if

A is semi-stable (that is, A has all its eigenvalues inside

or on the unit circle), we have the following conclusion.

0

N

P?

, N0}, rendering a

Page 6

B.C. Ding et al.: On the Stability of Output Feedback Predictive Control for Systems with Input Nonlinearity

393

Theorem 3. For systems represented by (1), TSOFPC

(6)-(8) is adopted where h satisfies (13). Assume that A

is semi-stable, and that there exist ∆ and γ1 such that

b1

ration constraint. Then, for the bounded set Ω ⊂ ℜ2n, the

closed-loop system can be tuned to possess a domain of

attraction not smaller than Ω.

2 – (1 + γ1)(b − 1)2 > 0 in the absence of an input satu-

Proof. In the absence of input saturation, b1 and b are

determined independently of the controller parameters.

Denote the parameters {γ1, ∆} that make b1

− 1)2 > 0 as {γ1

still choose γ1 = γ1

cases may occur:

2 – (1 + γ1)(b

0, ∆0}. When there is input saturation,

0 and ∆ = ∆0; then, the following two

Case 1: b1

the parameters in the following way:

(A) Choose PN = Q + AT PN A − AT PN B(λI + BT

PNB)−1BTPN A; then, P0 – P1 = 0. Furthermore,

choose Q > 0; then, Q > P0 – P1 and condition (i) in

Corollary 1 will be satisfied for all λ and N. Choose

an arbitrary N.

2 – (1 + γ1)(b − 1)2 > 0 as λ = λ0. Decide

(B) Choose R0 = ε I, Q0 = ε I > 0,

?

0

N

P?

= Q0 +

0

T

N

AP A −

?

00

=

0

1

0

()

TTT

NNN

AP CRCP C

?

?

CP A

?

−

+

, and an arbitrary

N0; then,

01

0

?

P

?

P

−

. On the other hand, if we

−

?

denote

0000

1

()

IITITIT

NNNN

P

?

I AP AAP CICP C

?

−

= + +

0

IT

N

CP A

?

, then there exist γ2 > 0 and ξ > 0 such that

?

γξ . Since

()

0

22

1 1/

+

?

1/

I

N

IPI

−≥γ

?

00

I

NN

P

?

P

?

=ε

and

0

1

N

PP

=

, ()

2021

1 1/

+

1/

QP

?

I

−≥γγεξ holds for

all ε. Choose a sufficiently small ε such that

( )()(

11

111bb

−+−−+γγ

At this point, ()

2

1 1/

+γ

= ()

200

1 1/(

Q LR L

+−−

γ

)(

−

)

22

2

2max

−

?

1

1

+

γ

()0

?

T

bB PB

−>

−

?

σ

?

.

00

P

01

()

T

Q

+

PLR L

≤ −

εξ .

P

21

) 1/

T

I

(C) Multiplying both sides of

+

λ

11

T

PQA PA

=+−

1

111

()

TTT

A PBIB PBB PA

−

by

1

λ−, we obtain

11

11111

()

TTTT

P

Since A is semi-stable, by [12] we know that

1P → 0 as λ → ∞. Hence, there exists λ1 ≥ λ0 such

that for all λ ≥ λ1:

+

λγλ

Q A P AA P B I B P BB P A

−−

=+−+

λ

.

(a)

11

11

+

11

(1 1)()

TTT

A P BIB P BB P A

−−

+=

1

1111

(1 1

+

condition (ii) in Corollary 1 is satisfied;

)()

TTT

A P B IB P BB P AI

−

<

γεξ , i.e.,

(b)

222

112max

[(1)( 1)(1)(1)bbb

−−+−−+−

γγσ

() (

1

)

()

2

11 max1

(

i.e., the inequality in Remark 3 is satisfied, and

in turn, condition (iii) in Corollary 1 is satisfied.

(D) Further, according to [12], there exists λ2 ≥ λ1 such

that for all λ ≥ λ2,

||

P

)]10

TT

B PBbB PB

+−+≤

?

γ σ

,

1 2

1

+

1 2

1

||2

AP−

?

≤

. Now, let

Ω ⊆

[

)

2

11

,,( , )

x e

sup

∞

∈

R

()

TT

c x P xe P e

∈∈Ω

=

λλ

; then,

cS

=

2

1

x

λ

1

{( , )

x e

|}.

nTT

x P xe P ec

+≤

?

Define

=?

λ

two

transformations:

1 2

1

P x

such that for all

=

and

1 2

1

P ee

; then,

λ

and

there exists

32

λ≥

∈

3

≥

all [ (0), (0)]

x

c

eS

,

[]

1

11

||(0)||||()(0)(0) ||

LTT

vIB PB

+

B PA xe

−

=

=

−

≤

+−

λ

λ

+

λ

1 1 2

−

1

)

11

||[(

λ

),

TT

IB PBB PAP

−

11 2

−

111

(][ (0) , (0) ] ||

xe

TTT T T

IB PB

+

B PAP

−

?

11 2

−

111

||[(),

TT

IB PBB PAP

−

1 2

−

1

11

1

()]||||[ (0) , (0) ]||x

TTTT

I B P BB P APe

−

−+

?

λ

11 2

11

B PAP

|| 2(),

TT

IB PBB P

?

−

≤+

11 2

−

111

()||.

TT

IB PBc

−

−+≤ ∆

Hence, for all [x(0), e(0)] ∈ Ω, the conditions in Corol-

lary 1 can all be satisfied at time k = 0, and by the

proof of Theorem 2, they are also satisfied for all k > 0.

Through the above decision procedure, the designed

controller will have the desired domain of attraction.

Case 2: b1

apparently is due to the fact that the control action is

restricted too much by the saturation constraint. For the

same reason as in (C) in Case 1 and by (5), we know

that for any bounded Ω, there exists λ4 ≥ λ0 such that for

all λ ≥ λ4 and all [x(k), e(k)]∈Ω, ˆ( )

the saturation constraint. This process is equivalent to

decreasing ∆ and re-determining b1 and b such that b1

(1 + γ1)(b − 1)2 > 0.

In a word, if the domain of attraction is not satis-

factory with λ = λ0, then it can be satisfied by choosing

λ ≥ max{λ3, λ4} and suitable {Q, PN, N, R0, Q0,

2 – (1 + γ1)(b − 1)2 ≤ 0 as λ = λ0, which

u k does not violate

2 –

0

N

P?

,

N0}. ■

In the proof of Theorem 3, we have emphasized the

effect of tuning λ. If A is semi-stable, then by prop-

erly fixing other parameters, we can tune λ to obtain the

arbitrarily large bounded domain of attraction. This is

very important because many industrial processes can be

represented by semi-stable models.

Page 7

394

Asian Journal of Control, Vol. 6, No. 3, September 2004

IV. THE ALTERNATIVES TO THE CASE IN

WHICH v IS AVAILABLE

When v is available, the following state observer

can be applied to obtain, at each time k, the state estima-

tion

( )

x k

?

:

()

(1)( )( )( ).x k

?

ALC x kBv kLy k

+=−++

?

(18)

As stated in section II, this case occurs when v is meas-

urable or δf = 0. Note that if δf = 0, then v(k) = f(u(k))

can be obtained at each time k. All other design details

are the same as those given in section II. In the follow-

ing, we will show that alternative results can be obtained

if (8) is replaced with (18).

Similar to (11), the closed-loop system with the

observer (18) is

(

(

1)

1)

(

(

) ( )

) ( ).

( ) [ (( ))k( )],k

LL

x k

e k

A BK x k

A LC e k

−

BKe kB h vv

⎧

⎪⎨

⎪ ⎩

+

+

=

=

+−+−

(19)

This time, we choose the Lyapunov function as

( )( )( ) ( )V kx kP x k e k P e k

=+

?

section III, we can obtain the following conclusions. For

the sake of brevity, we will not give the proofs, but will

illustrate them with an example in the next section.

?

11

( )

TT

. By analogy with

Theorem 4. For systems represented by (1), TSOFPC

(6), (7) and (18) are adopted. Assume that there exists a

positive scalar γ such that the design of the controller

satisfies the following conditions:

(i)

01

QPP

>−

?

;

(ii)

0010

T

QPP

?

LR L

−+−−

1

111

(1 1/ )

+

()

TTT

A P B RB P BB P A

−

< −+

γ

;

(iii)

( ( ))k(( ))k (1)[ ( h v( ))k

LTLL

h vRh v

−++γ

1

( )] (k)[ (( ))k( )]k0.

LTTLL

vRB P B h vv

−+−≤

Then, the equilibrium {x = 0, e = 0} of the closed- loop

system is exponentially stable.

Corollary 2. For systems represented by (1), TSOFPC

(6), (7) and (18) are adopted, where h satisfies (13). As-

sume the following:

(A1)

[ (0), (0)]

xe

∀ ∈Ω ⊂ R

0k ≥

;

(A2) there exists a positive scalar γ such that the system

design satisfies conditions (i)-(ii) in Theorem 4;

(iii)

1

[(1 )( 1) ]bb

−−+−

λγ

+ +−

γσ

Then, the equilibrium (0, 0) of the closed-loop system is

2

n

, ||( )||k

L

v

≤ ∆ for all

22

2

max1

(1)(1) ()0

T

bB P B

≤

.

exponentially stable with a domain of attraction Ω.

Theorem 5. For systems represented by (1), TSOFPC

(6), (7) and (18) is adopted, where h satisfies (13). As-

sume that there exist positive scalars ∆ and γ such that

the designed system satisfies conditions (i)-(iii) in Cor-

ollary 2. Then, the domain of attraction for the

closed-loop system is not smaller than

?

2

1

{( , )|x ex P x

=∈

R

1

},

nTT

c Se P e

?

c

?

+≤

?

(20)

where

? 2

d( / )

= ∆

c

?

,

?

d

1 2

−

1 1 2

−

111

1

||()[,]||;

TT

IB P B B P A PP

−

=+−?

λ

(21)

i.e., for all [

the closed-loop system is exponentially stable.

Theorem 6. For systems represented by (1), TSOFPC

(6), (7) and (18) is adopted, where h satisfies (13). As-

sume that A is semi-stable, and that there exist ∆ and γ

such that b1

saturation. Then, for the bounded set Ω ⊂ ℜ2n, the

closed-loop system can be tuned to possess a domain of

attraction not smaller than Ω.

]

?

S(0), (0)e

c

x

∈

?, the equilibrium (0,0) of

2 – (1 + γ)(b − 1)2 > 0 in the absence of input

V. NUMERICAL EXAMPLE

Take

1 0

1 1

⎣

A

⎡⎤

⎥

⎦

=⎢

,

1

0

B

⎡ ⎤

⎣ ⎦

=⎢ ⎥

and C = [0 1]. The

saturation constraint is | u | ≤ 1. Consider the following

two cases.

Case A: TSOFPC (6)-(8)

=+ϑϑϑϑ

( )1/3 { }sin(3 )fsign

ϑ .

0.1 ( )f

−

( )

ϑ ϑ

0.1 ( )f.f

≤≤ϑ ϑδϑ ϑ

At every time k, δf is randomly selected, and NAEG

ˆ ( )

vf u

=

tightest b1 and b2, i.e., b1 = 0.9 and b2 = 11. Also choose

the largest possible ∆, i.e., ∆ = f(1) /b1 = 1.1634.

L

is solved accurately. Initially, choose the

Choose R0 = 1, R = 45.9,

0

1

0

0

1

Q

⎡

⎣

⎤

⎥

⎦

=⎢

,

1

0

0

1

Q

⎡

⎣

⎤

⎥

⎦

=⎢

,

0

2.95

2.37

2.37

4.61

N

P

?

⎡

⎣

⎤

⎥

⎦

=⎢

,

34.9

8.99

8.99

4.89

N

P

⎡

⎣

⎤

⎥

⎦

=⎢

,

N0 = N = 5, γ1 = 1.1, and γ2 = 10. It is easy to verify that

these choices satisfy conditions (i)-(iii) in Corollary 1.

Further, change b1 to 0.86; then, conditions (i)-(iii) in

Corollary 1 are still satisfied. Correspondingly, ∆ =

1.2176. By Theorem 2, we obtain c = 125.6. The four

Page 8

B.C. Ding et al.: On the Stability of Output Feedback Predictive Control for Systems with Input Nonlinearity

395

sets of initial states and their estimation errors are

[ (0)x (0) ][2.5851,5.5, 0, 0],[ 2.5851, 5.5, 0, 0],

−

−−−

TT

e

=−

[1.3434,1,0.7, 2.3],[ 1.3434, 1,0.7,2.3]

−

.

[ (0), (0)]x

the four marked lines shown in Fig. 3, which validate the

stability conclusions in Theorem 1, Corollary 1, and Theo-

rem 2.

c

eS

∈

. The simulation results are indicated by

Next, we will test what happens when the true system

becomes x(k + 1) = [A + δA(k)] x(k) + Bu(k) + D(k) w(k),

y(k) = Cx(k), where D(k) = [1 1]Te−0.1k, w(k) is white noise

satisfying w(k)∈[−2 2], and δA(k) = Co {−E1, −E2, E2, E1}

0.1 0

0 0.1

⎥

⎣⎦

(ς4 − ς1)E1 + (ς3 − ς2)E2 with ς1 + ς2 + ς3 + ς4 = 1. All the

design details are same as those employed when δA(k) =

0 and w(k) = 0. At every time k, δA(k) and w(k) is se-

lected randomly. The simulation results are indicated by

the dashed line shown in Figure 3, which shows the ro-

bustness of Theorem 1, Corollary 1, and Theorem 2.

Here, for the sake of clarity, only one trajectory is

drawn.

with

1

E

⎡⎤

=⎢

and

2

0.1 0

0

⎣

0.1

E

⎡⎤

⎥

⎦

=⎢

−

, i.e., δA(k) =

Case B: TSOFPC (6), (7) and (18)

f (ϑ) = 4/3ϑ + 4/9ϑsign{ϑ}sin(40ϑ). δf = 0. A sim-

ple solution û = 3/4vL is applied to NAEG. f and the re-

sultant h are shown in Fig. 4. Choose b1 = 2/3 and b2 =

4/3 as shown in Fig. 4(b). Choose ∆ = f (1)/b1 = 2.4968.

-9

Fig. 3. State trajectories of the closed-loop system, where v is

not available.

Choose R0 = R = 36.8,

0

36.8

0

30.3 8.2

8.2

⎣

0

36.8

Q

⎡

⎣

⎤

⎥

⎦

=⎢

,

1

0

0

1

Q

⎡

⎣

⎤

⎥

⎦

=⎢

,

0

108.5

87

⎣

87

170

N

P

?

⎡⎤

⎥

⎦

=⎢

,

4.7

N

P

⎡⎤

⎥

⎦

=⎢

,

N0 = N = 5, and γ = 1.1. It is easy to verify that these

choices satisfy conditions (i)-(iii) in Corollary 2. By

Theorem 5, we obtain ĉ = 402.5. The four sets of initial

states and their estimation errors are

[x(0)T e(0)T] = [4.9723, −7.5, 0, 0], [−4.9723, 7.5, 0, 0],

[2.2023, 3, 0.7, 0.3], [−2.2023, −3, −0.7, −0.3].

[x(0), e(0)]∈ Ŝ ĉ. The simulation results are shown in Fig.

5, which validate the stability conclusions in Theorem 4,

Corollary 2, and Theorem 5.

Fig. 4(a) Curve of f

Fig. 4(b) Curve of h

1x

2x

-3

-2

-1

0

1

2

3

-6

-3

0

3

6

9

c S

v

s

u

-2.5 -1.25

0

1.25

2.5

-5

-2.5

0

2.5

5

L

v

v

-2.5

-1.25

0

1.25

2.5

-5

0

2.5

5

-2.5

4 3

L

vv

=

2 3

L

vv

=

Page 9

396

Asian Journal of Control, Vol. 6, No. 3, September 2004

Fig. 5. State trajectories of the closed-loop system, where v

available.

VI. CONCLUSIONS

This paper has considered the stability property of a

two-step output feedback predictive controller for sys-

tems with input nonlinearities, including Hammerstein

nonlinearity, an input saturation constraint and input

uncertainty. A few stability results have been obtained,

and the domain of attraction for this kind of system has

been analyzed. Compared with the method in [10], the

two-step control scheme is simpler, easier to implement,

and always feasible. The results of performance com-

parisons (including optimality), however, are ambiguous.

But for simple Hammerstein nonlinearity, the method in

[10] may result in better response and optimality.

The maximum domain of attraction for the two-step

predictive controller is much larger than the ellipsoidal

region given in this paper, which can be obtained by

using the maximal output admissible set technique given

in [18] and elsewhere. Also, the bounds of the nonlinear

item h can be tuned to obtain a larger domain of at-

traction. The incorporation of noise into the state and/or

output equation theoretically complicates the stability

analysis and the domain of attraction, which will be our

subjects of future investigation.

ACKNOWLEDGEMENT

This work was supported by the National Nature

Science Foundation of China (69934020). The authors

would thank the reviewers for their pertinent comments

and suggestions.

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Baocang Ding was born in 1972. He

received his Master’s degree from Uni-

versity of Petroleum in China in 2000,

Ph.D. degree from Shanghai Jiao Tong

University in 2003. Now, he is an Asso-

ciate Professor of the Institute of Auto-

mation, Hebei University of Technology.

His research interests include chemical process control

and predictive control.

Yugeng Xi was born in 1947. He re-

ceived his Ph.D. degree from Technical

University of Munich in Germany in

1984. Now he is a Professor of the Insti-

tute of Automation, Shanghai Jiao Tong

University. His research interest includes

theory and application of predictive con-

trol, generalization of predictive control

principles to planning, scheduling, decision problems,

multi-robot coordinative system and robot intelligent tech-

nology, new optimization algorithms.

Shaoyuan Li was born in 1965. He re-

ceived his B.S. and M.S. degrees from

Hebei University of Technology in 1987

and 1992 respectively. And he received

his Ph.D. degree from the Department of

Computer and System Science of

Nankai University in 1997. Now he is a

Professor of the Institute of Automation, Shanghai Jiao

Tong University. His research interest includes predictive

control, fuzzy systems, nonlinear system control and so on.

?