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

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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