Predictive control of general nonlinear systems using approximation

Predictive control of general nonlinear systems usingapproximation

W.H. Chen

Abstract: The paper addresses a tracking problem for general nonlinear systems using modelpredictive control (MPC). After approximation of the tracking error in the receding horizon by itsTaylor-series expansion to any specified order, an analytic solution to the MPC is developed and aclosed-form nonlinear predictive controller is presented. Unlike other nonlinear-model predictivecontrol (NMPC), there is a built-in integral action in the developed scheme and the implementationissues are discussed. It is pointed out that the proposed NMPC derived using approximation canstablise the original nonlinear systems if certain conditions, which can be met by properly choosingpredictive times and the order for Taylor expansion, are satisfied. Simulation demonstrates theeffectiveness of the proposed NMPC.

1 Introduction

Approximation of nonlinear systems by series expansionprovides a powerful technique to tackle nonlinear systems.Series-expansion-linearisation theory is well established butis strictly confined to dynamic analysis, locally to a singletrajectory or an equilibrium operating point, of smoothnonlinear systems. The frozen-input technique extends locallinearisation theory from a single equilibrium point to afamily of equilibrium points. This lays the foundation forseveral widespread control methods for nonlinear systemssuch as gain scheduling. However, there are two mainshortcomings in these techniques: One is that the linearisa-tion is restricted to around equilibria, and the other is thatmainly the stability property is considered and little insightinto other aspects of dynamic behaviour is provided [1].It was pointed out that, while indicating stability, thelinearisation provides, in general, a somewhat poorindication of the time response of the original nonlinearsystem. Several approaches have been proposed to over-come these shortcomings such as local model networks [2],local control networks [3, 4] and velocity-based linearisa-tion [5]. In these approaches, the linearisation can be carriedout not only on equilibria but also on any operating point,and the transient dynamics can be improved.

To develop model predictive control (MPC) for nonlinearsystems, another approach to approximation of dynamicbehaviour of a nonlinear system was adapted in [6, 7] where,instead of linearisation of a nonlinear system aroundequilibria or offequilibria, the output of a nonlinear systemis approximated by high series expansion. Althoughapproximation of the output of a nonlinear system ratherthan the nonlinear system itself might render the stabilityanalysis of the nonlinear system more difficult, it can

provide direct insight into other dynamic properties liketransient response. A nonlinear MPC (NMPC) algorithmwas developed in [6, 7] using Taylor-series expansion.

As for other NMPC algorithms (for example, see [8–10]),a differential-optimisation problem has to been solved forthe NMPC algorithm developed in [6, 7], which imposestwo main obstacles in engineering implementation. One isthe optimality of the online optimisation and the other is thesampling-time restriction. In general, the on-line optimis-ation involved in NMPC is nonconvex, which implies thatthe optimisation procedure might be terminated at a localminimum. Consequently, poor performance even instabilitymight be resulted after the control sequence yielded by theon-line optimiser is implemented. The on-line nonlinearoptimisation also imposes a heavy computational burden,which requires extensive computing power and a longsampling time in engineering implementation. For a systemwith fast dynamics, this problem becomes even morechallenging due to the fast sampling requirement.

The best way to address these difficulties arising inpredictive control of nonlinear systems is to pursue ananalytic approach, where it is intended to develop a closed-form NMPC and online optimisation is not required. In theNMPC algorithm in [7], the optimality condition gives a setof highly nonlinear equations in terms of control, andvarious derivatives of control (depending on the order forTaylor-series expansion). It is very unlikely (if notimpossible) to find the analytic optimal solution and it isalso very difficult to find a numerical optimal solution byusing nonlinear optimization algorithms since it is, ingeneral, nonconvex. To obtain analytic solution, theconstraint on the order of the Taylor-series expansion isimposed, i.e. the control order is limited to be zero, or thenonlinear system is only expanded by Taylor-seriesexpansion to its relative degree [6]. In this case, the controleffort appears linearly in the optimality condition and thereare no derivatives of the control effort due to zero controlorder. Then the analytic solution for nonlinear MPC wasfound, and it is also found that the nonlinear system islinearised by the derived analytic MPC law, and therelationship between this analytic nonlinear control lawand feedback linearisation was established in [11]. A similaridea was employed in [12, 13] independently to develop

q IEE, 2004

IEE Proceedings online no. 20040042

doi: 10.1049/ip-cta:20040042

The author is with the Department of Aeronautical and AutomotiveEngineering, Loughborough University, Loughborough, Leics. LE113TU, UK

Paper received 3rd October 2003

IEE Proc.-Control Theory Appl., Vol. 151, No. 2, March 2004 137

analytic NMPC using Taylor expansion, and the relation-ships between the analytic control laws they developed andthe feedback-linearisation technique were also explored. Inall these studies, the control order is restricted to be zero andthere is no stability analysis for the developed algorithms.

The drawback of restricting the control order to zero isobvious. This implies that the control effort to be optimizedin the receding horizon is constant, which is a veryrestrictive requirement. In many cases, this implies thatthe nonlinear system can only be approximated by a Taylor-series expansion to its relative degree. With this limitation,it is understandable that poor performance might result.In fact [14] shows that a nonlinear system with high relativedegree under the NMPC law developed in this way isunstable, no matter how small a receding horizon is chosen.

The first attempt to develop an analytic solution of NMPCfor arbitrary control order was made in [6]. Since a set ofhighly nonlinear equations derived from the optimalitycondition has to be solved to find an analytic solution forMPC, two simplifications are used: one is that the controlweighting is not included in the performance index and theother is that the control order is chosen as the differencebetween the order for Taylor expansion and the relativedegree (the price for these two simplifications is that, unlikethe original NMPC algorithm [7], the analytic MPC law isnot applicable for nonlinear systems with unstable zerodynamics). Then an analytic NMC was derived on theassumption that the control variables appear in theoptimality condition linearly, which, unfortunately, is nottrue for arbitrarily chosen Taylor-expansion order. To findthe analytic solution for NMPC, a set of very complicatednonlinear coupling equations in terms of control effort andits derivatives, which is derived from the optimal condition,still has to be solved. The analytic solution of the nonlinearcoupling equations and then the closed-form NMPC fornonlinear systems approximated by Taylor-series expansionto arbitrarily chosen order was found in [14]. Further, thestability result for analytic NMPC was first established, andthe influence of the control order and predictive horizon onthe transient performance was investigated. This alsoimplies that the closed-form MPC developed based on theTaylor-series expansion can approach the optimal MPC fororiginal nonlinear systems with any specified accuracy. Thiswork was further extended to nonlinear systems with ill-defined relative degree [15]. All the existing analytic NMPCalgorithms are confined to affine systems.

This paper further develops a closed-form NMPC for‘general’ nonlinear systems using the above output-approximation technique. An analytic solution to NMPCfor the tracking problem of general nonlinear systems ispresented, where the tracking error in the receding horizonis approximated by its Taylor-series expansion to any order.Stability of the proposed NMPC is established. Unlike theNMPC schemes for affine systems, there is a built-inintegral action in the control structure proposed in this paperand the implementation issues are discussed.

2 Predictive control of general nonlinear systems

To simplify the notation and concentrate on the maincontribution of this paper, only single-input single-output(SISO) nonlinear systems are considered, and most of theresults developed in this paper are capable of beingextended to multivariable general nonlinear systems. AnSISO general nonlinear system can be described by

_xxðtÞ ¼ f ðxðtÞ; uðtÞÞyðtÞ ¼ hðxðtÞÞ

)ð1Þ

where x 2 Rn; u 2 R and y 2 R are state, input and output,

respectively.For the sake of simplicity, the following notation is

introduced in this paper:

Df xhðxÞ ¼ @hðxÞ

@xf ðx; uÞ ð2Þ

D kf x

hðxÞ ¼@D k�1

f xhðxÞ

@xf ðx; uÞ; for k > 1 ð3Þ

and

DuD kf x

hðxÞ ¼@Dk

f xhðxÞ

@u; ð4Þ

To develop our results, the following assumptions areimposed on the system (1):

Assumption 1: The general nonlinear system (1) issufficiently differentiable with respect to time to anyorder.

Assumption 2: f ð0; 0Þ ¼ 0:

Assumption 3: DuDkf x

hðxÞ ¼ 0; for k ¼ 1; . . . ; m� 1; andDuDm

f xhðxÞ 6¼ 0; for all x and u.

Assumption 1 implies that the nonlinear system (1) can beapproximated by its Taylor-series expansion to anyspecified accuracy. Assumption 2 means that the origin isan equilibrium of the nonlinear system when there is nocontrol. Assumption 3 is similar to the well defined relativedegree for affine nonlinear systems, although, as will beshown below, the derivative of the control is used incontroller design for general nonlinear systems.

Unlike most of the conventional NMPC algorithms fornonlinear systems [8–10], the tracking problem is con-sidered in this paper, following [7, 14], i.e. a predictivecontroller is designed such that the output y optimallyfollows an reference signal o in terms of a givenperformance index. The most widely used quadraticperformance index is adopted in this paper, given by

J ¼ 1

2

Z T2

T1

eðt þ tÞ2dt ð5Þ

where T1 and T2 are the lower and upper predictive times,respectively. e is the tracking error, defined as

eðt þ tÞ ¼ yðt þ tÞ � oðt þ tÞ ð6Þ

3 Tracking-error approximation

Suppose that the output of the nonlinear system in theprediction horizon is approximated by its Taylor-seriesexpansion up to order r m: One has

yðt þ tÞ ’ yðtÞ þ t_yyðtÞ þ . . .þ tr

r!y½t�ðtÞ T1 t T2 ð7Þ

The derivatives of the output required in the approximationare given by

_yyðtÞ ¼ @hðxÞ@x

_xx

¼ @hðxÞ@x

f ðx; uÞ

¼ Df xhðxÞ ð8Þ

IEE Proc.-Control Theory Appl., Vol. 151, No. 2, March 2004138

and

€yyðtÞ ¼ @

@x

@hðxÞ@x

f ðx; uÞ� �

_xx þ @

@u

@hðxÞ@x

fðx; uÞ� �

_uu

¼ D2f x

hðxÞ ð9Þ

The last equality follows from the notation in (2), (3) andassumption 3. Similarly, one has

y½k�ðtÞ ¼ Dkf x

hðxÞ; for k ¼ 3; . . . ; m ð10Þ

and

y½mþ1�ðtÞ ¼@Dm

f x

@xfðx; uÞ þ

@Dmf x

@u_uu

¼ Dmþ1f x

hðxÞ þ DuDmf x

hðxÞ_uu ð11Þ

Differentiation of y½mþ1� with respect to time t andsubstitution of the system’s dynamics gives

y½mþ2�ðtÞ ¼@Dmþ1

f xhðxÞ

@xf ðx;uÞþ

@Dmþ1f x

hðxÞ@u

_uu

þ@DuDm

f xhðxÞ

@xf ðx;uÞ_uuþ

@DuDmf x

hðxÞ@u

_uu2

þDuDmf x

hðxÞ€uu

¼Dmþ2f x

hðxÞþDuDmf x

hðxÞ€uuþ z1ðx;u; _uuÞ ð12Þ

where

z1ðx; u; _uuÞ ¼@Dmþ1

f xhðxÞ

@u_uu þ

@DuDmf x

hðxÞ@x

fðx; uÞ_uu

þ@DuDm

f xhðxÞ

@u_uu2 ð13Þ

Repeating the above procedure, the higher-order derivativescan be calculated until the rth-order derivative, which isgiven by

y½r�ðtÞ ¼ Drf x

hðxÞ þ DuDmf x

hðxÞu½r�m�

þ zr�m�1ðx; u; _uu; . . . ; u½r�m�1�Þ ð14Þ

where zr�m�1ðx; u; _uu; . . . ; u½r�m�1�Þ is a complicated non-linear function of x; u; _uu; . . . ; u½r�m�1�:

Invoking (8–14) into (7), the output in the recedinghorizon is approximated by its Taylor expansion to the orderr as

yðt þ tÞ ’ tY T1 t T2 ð15Þ

where

t ¼ 1 t . . .tr

r!

� �ð16Þ

and

YðtÞ ¼

y½0�ðtÞ y½1�ðtÞ � � � y½m�ðtÞ y½mþ1�ðtÞ y½mþ2�ðtÞ � � � y½r�ðtÞ� �T

¼

hðxÞDf x

hðxÞ

..

.

Dmf x

hðxÞ

Dmþ1f x

hðxÞ

Dmþ2f x

hðxÞ

..

.

Drf x

hðxÞ

26666666666666666664

37777777777777777775

þ

0

0

..

.

0

DuDmf x

hðxÞ_uuDuDm

f xhðxÞ€uu þ z1ðx; u; _uuÞ

..

.

DuDmf x

hðxÞu½r�m�

þzr�m�1ðx; u; _uu; . . . ; u½r�m�1�Þ

2666666666666666666664

3777777777777777777775

ð17Þ

In the same fashion, the command oðt þ tÞ in the recedinghorizon can also be approximated by its Taylor-seriesexpansion to order r as

oðt þ tÞ ¼ tv ð18Þwhere

vðtÞ¼ o½0�ðtÞ o½1�ðtÞ � � � o½m�ðtÞ o½mþ1�ðtÞ � � � o½r�ðtÞ� �T

ð19Þ

4 Nonlinear predictive controller

After the output of the nonlinear system (1) and thereference to be tracked by series expansion has beenapproximated, the tracking error then can be calculated by

eðt þ tÞ ¼ yðt þ tÞ � oðt þ tÞ� tðYðtÞ �vðtÞÞ ð20Þ

Thus the predictive control performance (5) can beapproximated by

J ’ 1

2

Z T2

T1

ðYðtÞ �vðtÞÞTtTtðYðtÞ � oðtÞÞdt

¼ 1

2ðYðtÞ �vðtÞÞT

Z T2

T1

tTtdtðYðtÞ � oðtÞÞ

¼ 1

2ðYðtÞ �vðtÞÞTT ðYðtÞ �vðtÞÞ ð21Þ

where

T ¼Z T2

T1

tTtdt

¼

T2 � T1

T22 � T2

1

2� � � Trþ1

2 � Trþ11

ðr þ 1Þ!T2

2 � T21

2

T32 � T3

1

3� � � Trþ2

2 � Trþ21

r!ðr þ 2Þ� � � � � � � � � � � �

Trþ12 � Trþ1

1

ðr þ 1Þ!Trþ2

2 � Trþ21

r!ðr þ 2Þ � � � T2rþ12 � T2rþ1

1

r!r!ð2r þ 1Þ

266666666664

377777777775

ð22ÞAt the time t, MPC attempts to find the optimal controlprofile in the receding horizon, uðt þ tÞ; 0 t T2; tominimise the tracking error. After the nonlinear system (1)is approximated by its Taylor-series expansion up to theorder r, the highest derivative of the control required isr � m: Hence all the control profile in the receding horizoncan be parameterised by


uðt þ tÞ ¼ uðtÞ þ t_uuðtÞ þ � � � þ tr�m

ðr � mÞ! u½r�m�ðtÞ ð23Þ

Therefore, instead of minimising the performance index J interms of the control profile uðt þ tÞ; 0 t T2 directly, theperformance index J can be minimised in terms of variablesuðtÞ; _uuðtÞ; . . . ; u½r�m�ðtÞ: However, for a general nonlinearsystem under consideration as in (1), the control u does notappear in a linear manner and it is difficult to give theexplicit solution for u. To avoid this problem, the control uis considered as a new state variable and the performance Jis minimised in terms of _uuðtÞ; . . . ; u½r�m�ðtÞ: Later in thispaper, we discuss the implementation issue for the proposedcontroller.

The necessary condition for the optimality is given by

@J

@u¼ 0 ð24Þ

where

u ¼ _uuðtÞ; . . . ; u½r�m�ðtÞh iT

ð25Þ

Theorem 1: Consider a general nonlinear system (1)satisfying assumptions 1– 3 and with the trackingperformance index (5). When the tracking error isapproximated by its Taylor-series expansion up to orderr m; the NMPC is given by

uðtÞ ¼Z

_uuð�Þd� ð26Þ

where

_uuð�Þ ¼ �ðDuDmf x

_hhðxÞÞ�1ðKMmð�Þ þ Dmþ1f x

hðxÞ � o½mþ1�ð�ÞÞð27Þ

Mmð�Þ ¼

hðxÞ � oð�ÞD1

f xhðxÞ � o½1�ð�Þ

� � �Dm

f xhðxÞ � o½m�ð�Þ

0BB@

1CCA ð28Þ

and K 2 Rmþ1 is the first row of the matrix T �1

rr T Tmr;

T mr ¼

Tmþ22 � T

mþ21

ðmþ 1Þ!ðmþ 2Þ � � � T2mþ22 � T

2mþ21

m!ðmþ 1Þ!ð2mþ 2Þ... ..

. ...

Trþ12 � Trþ1

1

ðr þ 1Þ! � � � Trþmþ12 � T

rþmþ11

m!r!ðr þ mþ 1Þ

2666664

3777775ð29Þ

T rr ¼T

2mþ32 � T

2mþ31

ðmþ 1Þ!ðmþ 1Þ!ð2mþ 3Þ � � � Trþmþ22 � T

rþmþ21

ðmþ 1Þ!r!ðr þ mþ 2Þ... ..

. ...

Trþmþ22 � T

rþmþ21

ðmþ 1Þ!r!ðr þ mþ 2Þ � � � T2rþ12 � T2rþ1

1

r!r!ð2r þ 1Þ

26666664

37777775;

ð30Þ

Proof: See Appendix (Section 9.1).

Remark 1: Theorem 1 presents an analytic solution to theNMPC problem for general nonlinear systems usingapproximation and the NMPC is given in a closed form.

The existing results for analytic NMPC are extended fromaffine nonlinear systems to general nonlinear systems. Thiscontroller overcomes the difficulties imposed by the on-linenonlinear optimisation involved in NMPC, such as attaininglocal-minimum and long-sampling-time requirements. TheNMPC developed in this paper is suboptimal for the originalNMPC problem. Since there is no restriction on the order forTaylor-series expansion, theoretically the analytic solutiondeveloped by approximation can approach the solution ofthe original NMPC in any specified accuracy.

4.1 Structure of predictive control

Since the closed-form NMPC for the general nonlinearsystem is derived, no on-line optimisation is required.Actually from the control law (26) and (27), one can see thatit is a nonlinear state-variable-feedback control law. Afterthe order for Taylor-series expansion, the order m and thepredictive times T1; T2 are chosen, the control gain K can becalculated off-line.

The block diagram of the proposed NMPC scheme isshown in Fig. 1. Unlike previous analytic nonlinearpredictive control schemes for affine systems, the derivativeof the control action is calculated using the control law (27)based on the state variable, the current control and thereference, and then the control action is obtained byintegrating it with respect to time.

4.2 Choice of the initial control

In implementation of the above control strategy for generalnonlinear systems, the initial control u(0) should bespecified. Suppose that, at the initial state, the system is atits equilibrium. It follows from assumption 3 that the initialcontrol u(0) should be chosen as zero.

5 Stability analysis

Substituting the control law (27) into (11) gives

y½ mþ1�ðtÞ ¼ Dmþ1f x

hðxÞ � KMmðtÞ þ Dmþ1f x

hðxÞ � o½ mþ1�ðtÞ� �

ð31Þwhich implies that

y½mþ1�ðtÞ � o½mþ1�ðtÞ þ KMmðtÞ ¼ 0 ð32Þ

Eqn (28) is used for calculating the control law whichexplicitly shows that _uuðtÞ depends on the state and thecurrent control u(t). Using the relationship in (8–10), (28)can also be written as

Mm ¼

yðtÞ � oðtÞ_yyðtÞ � o½1�ðtÞ

� � �y½m� � o½m�ðtÞ

0BB@

1CCA ¼

eðtÞ_eeðtÞ� � �

e½m�ðtÞ

0BB@

1CCA ð33Þ

Invoking (33) into the error dynamics (32) obtains

e½mþ1�ðtÞ þ kmþ1e½m�ðtÞ þ � � � þ k2 _eeðtÞ þ k1eðtÞ ¼ 0 ð34Þ

Fig. 1 Structure of predictive control for general nonlinearsystems


where

K ¼ ½k1 . . . kmþ1� ð35Þ

T rm and T rr in (29) and (30) can be written as

T mr ¼ diagTmþ22

ðmþ 1Þ! ;Tmþ32

ðmþ 2Þ! ; . . . ;Trþ1

2

r!

( )

�

1 � Tmþ21

T2

mþ 2� � �

1 � T2mþ21

T2

2mþ 2

..

. ... ..

.

1 � Trþ11

T2

r þ 1� � �

1 � Trþmþ11 z

T2

r þ mþ 1

266666666664

377777777775

� diag 1; T2; . . . ;Tm2

m!

� �ð36Þ

and

T rr ¼diagTmþ22

ðmþ 1Þ! ;Tmþ32

ðmþ 2Þ! ; . . . ;Trþ1

2

r!

( )

�

1 � T2mþ31

T2

2mþ 3� � �

1 � Trþmþ21

T2

r þ mþ 2

..

. ... ..

.

1 � Trþmþ21

T2

r þ mþ 2� � �

1 � T2rþ11

T2

2r þ 1

266666666664

377777777775

� diagTmþ12

ðmþ 1Þ! ;Tmþ22

ðmþ 2Þ! ; . . . ;Tr

2

r!

( )ð37Þ

Let

Mm ¼

1 � Tmþ21

T2

mþ 2� � �

1 � T2mþ21

T2

2mþ 2

..

. ... ..

.

1 � Trþ11

T2

r þ 1� � �

1 � Trþmþ11

T2

r þ mþ 1

26666666664

37777777775

ð38Þ

and

Mn ¼

1 � T2mþ31

T2

2mþ 3� � �

1 � Trþmþ21

T2

r þ mþ 2

..

. ... ..

.

1 � Trþmþ21

T2

r þ mþ 2� � �

1 � T2rþ11

T2

2r þ 1

26666666664

37777777775

ð39Þ

Then one has

T �1rr T rm ¼ diagfT

�ðmþ1Þ2 ðmþ 1Þ!; T

�ðmþ2Þ2 ðmþ 2Þ!; . . . ;

T�r2 r!gM�1

n Mm diag 1; T2; . . . ;Tm2

m!

� �ð40Þ

Let zm ¼ ½m1;m2; . . . ;mmþ1� denote the first row of the

matrix M�1n Mm: Since the gain K is determined by the first

row of the matrix T �1rr T rm; it follows from (40) and (35)

that

ki ¼ miT�m�2þi2

ðmþ 1Þ!ði � 1Þ! i ¼ 1; . . . ; mþ 1 ð41Þ

where 0! ¼ 1: Substituting the gains into the error dynamics(34) yields

e½mþ1�ðtÞ þ mmþ1T�12 ðmþ 1Þe½m�ðtÞ þ mmT�2

2 ðmþ 1Þme½m�ðtÞ

þ� � �þm2T�m2

ðmþ 1Þ!1!

_eeðtÞ þ m1T�m�12 ðmþ 1Þ!eðtÞ ¼ 0

ð42ÞOne can conclude that the stability of the error dynamics isdetermined by the polynomial

smþ1 þ mmþ1T�12 ðmþ 1Þsm þ mmT�2

2 ðmþ 1Þmsm�1 þ � � �

þ mmT�m2 ðmþ 1Þ!s þ m1T

�m�12 ðmþ 1Þ! ¼ 0

ð43Þi.e.

Tmþ12 smþ1 þmmþ1ðmþ1ÞTm

2 smþmmðmþ1ÞmTm�12 sm�1 þ��

þm2ðmþ1Þ!T2sþm1ðmþ1Þ!¼ 0

ð44ÞNow using the transform ~ss ¼ T2s; the above polynomialbecomes

~ssmþ1 þ mmþ1ðmþ 1Þ~ssm þ mmðmþ 1Þm~ssm�1 þ � � �

þ m2ðmþ 1Þ!~ss þ m1ðmþ 1Þ! ¼ 0ð45Þ

The error dynamics (34) are stable if and only if all roots ofthe polynomial in (45) have negative real parts since thetransform ~ss ¼ T2s does not change the sign of the roots.Therefore, the stability of the error dynamics only dependson the coefficients mi; i ¼ 1; . . . ;mþ 1; i.e. the first row ofthe matrix M�1

n Mm:However, to ensure the stability of the overall closed-loop

system, the stability of the error dynamics alone is notenough. All the driven zero dynamics under the referencesignal o should be stable. One can conclude that, under theassumption that the internal zero dynamics of the system (1)driven by o are defined for all t 0; bounded and uniformlyasymptotically stable, the general nonlinear system (1) isstable under the NMPC (27) if all the roots of thepolynomial (45) have negative real parts [16].

The foregoing stability analysis is summarised intheorem 2.

Theorem 2: Consider a general nonlinear system (1)satisfying assumptions 1–3 and suppose that its internalzero dynamics driven by the reference o are defined for allt 0; bounded and uniformly asymptotically stable. Thenonlinear system (1) under the NMPC (26), (27) isasymptotically stable if all the roots of the polynomial(45) have negative real parts.

A test of the stability of the error dynamics can beperformed as follows. After the predictive times T1 and T2

are chosen and the order for Taylor expansion of the generalnonlinear system is determined, calculate the matrix Mn andMm according to (38) and (39). Then the first row of thematrix M�1

n Mm can be determined and, after substitutingthese coefficients in the polynomial (45), the stability can betested.


The stability of the closed-loop system under thedeveloped nonlinear control depends on the predictivetimes T1 and T2; the order m that relates to the physicalcharacteristics of the nonlinear system, and the order ofTaylor-series expansion. Further, when T1 is chosen aszero, it follows from (38) and (39) that the coefficients inthe polynomial (45) do not depend on the predictive timeT2: This implies that, in this case, the stability onlydepends on m and the Taylor-expansion order t: Ingeneral, the stability can be achieved by using a large r,i.e. approximating a general nonlinear system with aTaylor-series expansion to a high order [14].

Remark 2: The relationship between the analytic NMPC andthe feedback linearisation for affine systems has beendiscussed in [12, 13, 7]. This paper shows that the generalsystem (1) is also linearised by the developed analyticNMPC. This provides a new method to linearise generalnonlinear systems using feedback, while minimising certainquadratic-tracking performance specification.

6 Illustrative example

The NMPC developed in this paper is illustrated by asecond-order general nonlinear system, which was takenfrom [6, 7]

_xx1ðtÞ ¼ x2ðtÞ þ tanhðuðtÞÞ_xx2ðtÞ ¼ �x1ðtÞ þ x2

2ðtÞ þ uðtÞð46Þ

with the output

_yyðtÞ ¼ x1ðtÞ ð47ÞTo design the NMPC developed in this paper, first the orderm needs to be calculated:

Df xhðxÞ ¼ @hðxÞ

@xf ðx; uÞ

¼ 1 0½ �x2ðtÞ þ tanhðuðtÞÞ

�x1ðtÞ þ x22ðtÞ þ uðtÞ

� �¼ x2ðtÞ þ tanhðuðtÞÞ ð48Þ

DuDf xhðxÞ ¼ 1

cosh2ðuÞ ð49Þ

which is not equal to zero for all x and u. Hence m ¼ 1:

It follows from theorem 1 that the NMPC law is given by

_uuðtÞ ¼ 1

cosh2ðuÞ

�� ð�x1ðtÞ þ x2

2ðtÞ þ uÞ þ o½2�ðtÞ

þ k2ð _ooðtÞ � _yyðtÞÞ þ k1ðoðtÞ � yðtÞÞ�

ð50Þ

where k1 and k2 are control gains determined by (41).To investigate the performance of the NMPC developed

using approximation, the first test is to fix the predictivetimes but approximate the nonlinear system with Taylor-series expansion to a different order. Control performancewith the Taylor expansion order r ¼ 2; 3; 4; 5 are shown inFigs. 2 and 3, where the predictive times are chosen asT1 ¼ 1ðsÞ and T2 ¼ 2ðsÞ:

The second test is to investigate the influence of thechoice of the predictive times. The order for Taylor-seriesexpansion is chosen as r ¼ 5 for all simulations and variouspredictive times are considered. The tracking performanceand the control profile are shown in Figs. 4 and 5. In thesimulation study, the length of the predictive horizon is keptas the same, i.e. T2 � T1 ¼ 1ðsÞ: It can be seen from Figs. 4and 5 that, with the same order for Taylor-series expansion,the rising time increases with the predictive times, while thecontrol effort decreases as the predictive times increase.

Fig. 2 Tracking performance under the nonlinear NMPC withseries expansion to different orders

Fig. 3 Time histories of control for series expansion to differentorders

Fig. 4 Tracking performance with fifth-order Taylor-seriesexpansion and different predictive series


7 Conclusions

For a general nonlinear system, an analytic solution toNMPC was developed in this paper by tracking errorapproximation. In this setting, the tracking performance isdefined by a receding-horizon performance index in termsof the tracking error and then the tracking error isapproximated by its Taylor-series expansion. Since thereis no restriction on the order for series expansion,theoretically the analytic solution presented in this papercan approach to the original nonlinear-predictive-controlproblem to any specified accuracy. The implementationissues, including the controller structure and the choice ofthe initial control action, are discussed. It is shown that theNMPC derived using approximation can stabilise theoriginal general nonlinear system if certain conditions aresatisfied. Since no online optimisation is required, thisprovides an easy way to implement suboptimal NMPC forgeneral nonlinear systems, in particular for nonlinearsystems with fast dynamics.

8 References

1 Leith, D.J., and Leithhead, W.E.: ‘Gain-scheduled and nonlinearsystems: dynamic analysis by velocity-based linearisation families’,Int. J. Control, 1998, 70, (2), pp. 289–317

2 Prasad, G., Swidenbank, E., and Hogg, B.W.: ‘A local model networksbased multivariable long-range predictive control strategy for thermalpower plants’, Automatica, 1998, 34, (10), pp. 1185–1204

3 Hunt, K.J., and Johansen, T.A.: ‘Design and analysis of gain-scheduledcontrol using local controller networks’, Int. J. Control, 1997, 66, (5),pp. 619–651

4 Murray-Smith, R., and Johansen, T.A. (Eds.): ‘Multiple modellingapproach to modelling and control’ (Taylor and Francis, London, 1997)

5 Leith, D.J., and Leithhead, W.E.: ‘Gain-scheduled controller design: ananalytic framework directly incorporating non-equilibrium plantdynamics’, Int. J. Control, 1998, 70, pp. 289–317

6 Siller-Alcala, I.: ‘Generalised predictive control of nonlinear systems’.PhD thesis, Faculty of Engineering, Glasgow University, 1998

7 Gawthrop, P.J., Demircioglu, H., and Siller-Alcala, I.: ‘Multivariablecontinuous-time generalised predictive control: a state-space approachto linear and nonlinear systems’, IEE Proc., Control Theory Appl.,1998, 145, (3), pp. 241–250

8 Chen, H., and Allgower, F.: ‘A quasi-infinite horizon nonlinear modelpredictive control scheme with guaranteed stability’, Automatica, 1998,34, (10), pp. 1205–1217

9 de Oliveira Kothare, S.L., and Morari, M.: ‘Contractive modelpredictive control for constrainted nonlinear systems’, IEEE Trans.Autom. Control, 2000, 45, pp. 1053–1071

10 Allgower, F., and Zheng, A. (Eds.): ‘Nonlinear model predictivecontrol. Progress in systems and control theory’ (Birkhauser Verlag,Boston, 1998)

11 Gawthrop, P.J., and Arsan, T.: ‘Exact linearisation is a special case ofnon-linear GPC (abstract only)’, in Allgower, F., and Zheng, A., (Eds.):‘Preprints of international symposium on Nonlinear Model Predictive

Control: Assessment and Future Directions’ (Ascona, Switzerland,1998) 7, p. 37

12 Soroush, M., and Soroush, H.M.: ‘Input–output linearising nonlinearmodel predictive control’, Int. J. Control, 1997, 68, (6), pp. 1449–1473

13 Lu, P.: ‘Optimal predictive control for continuous nonlinear systems’,Int. J. Control, 1995, 62, (3), pp. 633–649

14 Chen, W.-H., Ballance, D.J., and Gawthrop, P.J.: ‘Optimal control ofnonlinear systems: a predictive control approach’, Automatica, 2003,39, (4), pp. 633–641

15 Chen, W.-H.: ‘Analytic predictive controllers for nonlinear systemswith ill-defined relative degree’, IEE Proc., Control Theory Appl.,2001, 148, (1), pp. 9–16

16 Isidori, A.: ‘Nonlinear control systems: an introduction’ (Springer-Verlag, New York, 1995, 3rd edn.)

9 Appendices

9.1 Proof of theorem 1

It follows from (17) and (19) that

YðtÞ � oðtÞ ¼ MmðtÞHðuðtÞÞ

� �ð51Þ

Mm ¼

hðxÞ � oD1

f xhðxÞ � _oo

..

.

Dmf x

hðxÞ � o½m�

26664

37775 ð52Þ

and

HðuÞ

¼

Dmþ1f x

hðxÞ þ DuDf xhðxÞ_uu � o½ mþ1�

..

.

Dmþ1f x

hðxÞ þ DuDf xhðxÞu½r�m� þ zr�m�1ðx; u; uÞ � o½r�

26664

37775

ð53Þ

Differentiating HðuÞ in (53) with respect to u gives

@HðuÞ@u

¼

DuDm�1f x

hðxÞ 0 0 � � � 0

� DuDm�1f x

hðxÞ 0 � � � 0

..

. ... ..

. ... ..

.

� � � � � � DuDm�1f x

hðxÞ

26666664

37777775

ð54Þ

where ‘�’ denotes the nonzero element in the matrix@HðuÞ@u

:The necessary condition for the optimal control u is

given by

@J

@u¼ 0 ð55Þ

Partition the matrix T in (22) into the submatrices

T ¼T mm T mr

T Tmr T rr

� �ð56Þ

where

T mm 2 Rm�m;T mr 2 R

m�ðrþ1Þ;T rr 2 Rðrþ1Þ�ðrþ1Þ

Differentiating the performance index J in (21) with respectto control vector u and using the notations in (17), (52) and(54) yields

Fig. 5 Control profiles for fifth-order Taylor-series expansionand different predictive times


@J

@u¼ @ðY � oÞ

@u

� �T

T ðY � oÞ

¼0

@HðuÞ@u

" #T

T ðY � oÞ

¼ 0@HðuÞ@u

� �T� � T mm T mr

T Tmr T rr

� �Mm

HðuÞ

� �ð57Þ

It can be shown that the necessary condition (55) can bewritten as

@HðuÞ@u

��u¼u�

� �T

T TmrMm þ

@HðuÞ@u

��u¼u�

� �T

T rrHðu�Þ ¼ 0

ð58Þwhere u� denotes the solution satisfying the necessary

optimality condition. Since DuDm�1f x

hðxÞ is invertible

according to assumption 3, then @HðuÞ=@u in (54) is alsoinvertible. Note that the matrix T rr is positive definite. (58)implies

Hðu�Þ ¼ �T �1rr T T

mrMm ð59Þ

Following (53), the first equations in (59) can be written as

DuD mf x

hðxÞ_uuðtÞ� þ KMm þDmþ1f x

hðxÞ � oðtÞ½mþ1� ¼ 0 ð60Þ

where K denotes the first row of the matrix T �1rr T T

mr: Theoptimal _uuðtÞ� can be determined uniquely by

_uuðtÞ� ¼ �ðDuDmf x

hðxÞÞ�1ðKMmðtÞ þ Dmþ1f x

hðxÞ � o½mþ1�ðtÞÞð61Þ

since this is the only solution to (59) and thus optimalcondition (55).

In the moving time frame t þ t located at time t,differentiating uðt þ tÞ in (23) with respect to t gives

duðt þ tÞdt

¼ _uuðtÞ� þ t€uuðtÞ� � � � þ t½r�m�1�

r � m� 1!u½r�m�ðtÞ�

0 t T2 ð62Þ

In the implementation of NMPC,

_uuðtÞ ¼ duðt þ tÞdt

for t ¼ 0 ð63Þ

The control law (27) is obtained by combining (62), (63)and (61).

9.2 Optimal control vector u(t)

Although, in the implementation of the NMPC, only theoptimal _uu is required, all variables to be optimised in uðtÞcan be uniquely determined by (59). Following (53), (12)and (13), the second equations in (59) can be written as

DuDmf x

hðxÞ€uuðtÞ� þ z1ðx; uðtÞ; _uuðtÞ�Þ þ Dmþ2f x

hðxÞ

� o½ mþ2�ðtÞ ¼ �K2Mm ð64Þ

Hence the optimal €uuðtÞ� is determined by

€uuðtÞ� ¼ � ðDuDmf x

hðxÞÞ�1�K2Mm þ z1ðx; uðtÞ; _uuðtÞ�Þ

þ Dmþ2f x

hðxÞ � o½mþ2�ðtÞ

ð65Þwhere K2 denotes the second row in the matrix T �1

rr T Tmr:

After the optimal control _uuðtÞ� is obtained from (61), €uuðtÞ�can be calculated by substituting (61) into (65). Recursively,optimal u½3�ðtÞ�; . . . ; u½r�m�ðtÞ� can also be uniquely deter-mined from the other equations in (59).


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Predictive control of general nonlinear systems using approximation