Explicit solutions to optimal control problems for constrained continuous-time linear systems

Explicit solutions to optimal control problems forconstrained continuous-time linear systems

V. Sakizlis, J.D. Perkins and E.N. Pistikopoulos

Abstract: An algorithmic framework is presented for the derivation of the explicit optimal controlpolicy for continuous-time linear dynamic systems that involve constraints on the process inputsand outputs. The control actions are usually computed by regularly solving an on-line optimisationproblem in the discrete-time space based on a set of measurements that specify the current processstate. A way to derive the explicit optimal control law, thereby, eliminating the need for rigorouson-line computations has already been reported in the literature, but it is limited to discrete-timelinear dynamic systems. The currently presented approach derives the optimal state-feedbackcontrol law off-line for a continuous-time dynamic plant representation. The control law is provedto be nonlinear piecewise differentiable with respect to the system state and does not require therepetitive solution of on-line optimisation problems. Hence, the on-line implementation is reducedto a sequence of function evaluations. The key advantages of the proposed algorithm aredemonstrated via two illustrative examples.

1 Introduction

Linear model predictive control (MPC) almost exclusivelyemploys discrete-time dynamic models. The reason for thisis that even for linear models the presence of continuousdynamics combined with state constraints results incomplex infinite-dimensional nonlinear programs [1, 2]that can be computationally formidable. However, discrete-time representation leads to unavoidable inaccuracies if thesampling interval is not sufficiently small and increases thecomputational load if a detailed discretisation is chosen.This may not matter in many applications such as data-driven models and slowly-varying systems. Nevertheless,for safety critical applications with large sampling timessuch as biomedical systems [3], for fast-varying systems andfor systems represented by continuous-time first principlesmodels it can be a crucial issue.On-line implicit control schemes for continuous-time

MPC usually rely on numerical dynamic optimisationtechniques such as multiple shooting [4, 5], control vectorparameterisation [6–9], or modified variational approaches[10]. Additionally, significant developments towards enhan-cing the stability characteristics of these schemes have beenreported [11], based on infinite horizon predictive control.These techniques, despite being successfully used, requirea large number of computations that restrict theirapplicability.

An approach for moving off-line the expensive on-linecalculations involved inMPC has been reported [12, 13], basedon recently proposed parametric programming algorithms,developed at Imperial College [14]. This approach derivesoff-line the explicit mapping of the optimal control actions inthe space of the current states resulting in a closed-form controllaw. However, this method has been developed only fordiscrete-time linear dynamic systems. Nonlinear explicitcontrollers have been developed for continuous-time systemsusing different approaches. However, these approaches rely onthe assumptions that there are no state constraints [9, 15–17],that the sets of active constraints are fixed [18] or thatthe control inputs are finitely parameterised [19].

In this work we will propose a model-based controlframework to derive off-line the explicit control law forconstrained linear continuous-time dynamic systems. Here,the optimal control problem will be formulated as amultiparametric dynamic optimisation problem: the vectorof the current state will be considered as a set of parametersand the control inputs as optimisation variables. The solutionof this problem will be attempted by merging the parametricprogramming theory with the variational first-order optim-ality conditions of the open-loop optimal control problem.Our technique will provide a set of explicit functions for thecontrol variables in terms of the state variables, thus,featuring a control law for the plant. These findings willallow theoretical insight to the structure of the optimalcontrol law for continuous-time dynamic plants. However,their practical applicability will be limited to low-dimen-sional systems since complex nonlinear control expressionsare usually derived in high-dimensional cases.

2 Multiparametric dynamic optimisation

2.1 Problem formulation

Consider the following optimal control problem:

ff ¼ minxðtÞ;vðtÞ

1

2xðtfÞTP1xðtfÞ

þ 1

2

Z tf

to

½xðtÞTQ1xðtÞ þ vðtÞTR1vðtÞ�dt

q IEE, 2005

IEE Proceedings online no. 20059041

doi: 10.1049/ip-cta:20059041

E.N. Pistikopoulos is with the Centre for Process Systems Engineering,Department of Chemical Engineering, Imperial College London, London,SW7 2AZ, UKJ.D. Perkins is with the Office of the President and Vice Chancellor,University ofManchester,MainBuilding, Sackville,Manchester,M601QD,UKV. Sakizlis is with Parametric Optimisation Solutions Ltd., 90 Fetter Lane,London, EC4 1JP, UK

E-mail: [email protected]

Paper received 15th April 2004. Originally published online 8th June 2005

IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005 443

subject to

_xxðtÞ ¼ A1xðtÞ þ A2vðtÞ0 � c gðxÞ ¼ D1xðtfÞ þ b2

0 � gðx; vÞ ¼ C1xðtÞ þ C2 vðtÞ þ b1

xðtoÞ ¼ xo; to � t � tf ð1Þ

x 2 X � <nare the states, such as the concentration ofcomponents, mass, energy or momentum, v 2 U � <nv arethe control manipulating inputs, such as flows of streams, orpower supply. The constraints pertain to product specifica-tions, or safety and environmental restrictions and lower andupper bounds on x and v. Consider: g : <n � <nv 7!<q andc g : <n 7!<Qg : Matrices A1;2;C1;2;D1;2 and vector b1;2 aretime-invariant of appropriate dimensions. The pair ðA1; A2Þis assumed to be stabilisable and the pair

�Q

1=21 ; A1

�detectable. Matrices Q1 � 0; R1 P1 � 0 constitute thequadratic performance index. The terminal quadratic costP1 is calculated according to stability criteria, from thesolution of the steady-state Riccati equation or thecontinuous Lyapunov equation [11]. It is useful here, todefine the order of a path state constraint.

Definition 1: The constraint giðx; vÞ is said to be of orderll � 1 with respect to the dynamics, if:

@giðx; vÞ j

@vk¼ 0; j ¼ 1; 2; . . . ; ll� 1; k ¼ 1; nv and

@giðx; vÞll

@vk6¼ 0; for at least one k; k ¼ 1; nv

ð2ÞThe constraint giðx; vÞ is said to be of zeroth order withrespect to the dynamics, if:

@giðx; vÞ@vk

6¼ 0 for at least one k; k ¼ 1; nv

where the index j denotes time derivatives, e.g.g1i ¼ _ggi; g

2i ¼ €ggi; . . . and the index i denotes the numbering

of the constraints i ¼ 1; q:The purpose of this work is not the solution of problem

(1) for fixed values of the initial condition, but the derivationof the mapping of the optimal conditions in the space of theinitial state values. This mapping comprises a set ofexpressions for the optimal value of the performanceindex and the optimal profiles of the control inputs as afunction of the initial state: ffðxoÞ; vvðt; xoÞ: To achieve thisgoal, the initial state conditions are treated as parametersand problem (1) is recast as a multiparametric dynamicoptimisation problem (mp-DO). The following Sectiondiscusses the solution of the mp-DO problem that providesthe required parametric expressions.

2.2 Solution procedure: theoreticaldevelopments for mp-DO

2.2.1 Optimality conditions: The foundation forthe solution of the multiparametric dynamic optimisation(mp-DO) problem (1) is based on applying sensitivity analysis[20] to the stationary conditions of the dynamic optimisationproblem. TheKarush-Kuhn-Tucker conditions for the optimalcontrol problem (1) derive from the Euler-Lagrange equationsand were first presented by Bryson et al. [21] for a scalarstate constraint and a scalar control variable for a genericnonlinear dynamic system with a nonlinear objective andconstraints. Recently, Malanowski and Maurer [22]

andAugustin andMaurer [23] presented stationary conditionsfor the same problem, however, they traded the Hamiltonianpart of the transversality conditions for additional junctionconditions at the exit points. Following [21] when ll � 1the optimality conditions comprise:The ordinary differential equation (ODE) system

_xxðtÞ ¼ A1xðtÞ þ A2vðtÞ; t 2 ½to; tf � ð3Þ

xðtoÞ ¼ xo ð4Þ

The Boundary conditions for the adjoints

lðtfÞ ¼ P1 xðtfÞ þ@c gð:Þ@x

��t¼tf

� �T

n ð5Þ

Complementarity conditions

0 ¼ nj cgj ðxðtfÞÞ ð6Þ

nj � 0; j ¼ 1; Qg ð7Þ

The Adjoint differential system

miðtÞ � 0; gið:Þ mi ¼ 0; i ¼ 1; q ð8Þ

_llðtÞ ¼ �Q1 xðtÞ � AT1 lðtÞ �

Xqi¼1

@gllii ð:Þ@x

!T

miðtÞ ð9Þ

vðtÞ ¼ � R�11 AT

2 lðtÞ þXqi¼1

@gllii ð:Þ@v

!T

miðtÞ

8<:

9=;

for t 2 ½to; tf �

ð10Þ

Assume: tnktþnkxþ1 ¼ tf ; and define:

tkt entry point ) mjðt�ktÞ ¼ 0; mj tþkt

� �� 0;

k ¼ 1; 2; nkttkx entry point ) mj t

þkx

� �¼ 0; mj t

�kxð Þ � 0;

k ¼ 1; 2; nkxFor at least one j ¼ 1; q

Junction conditions (entry point)

0 ¼ gji ðxðtktÞ; vðtktÞÞ; j ¼ 0; lli � 1 ð11Þ

0 ¼ gllii x tþkt� �

; v tþkt� ��

; k ¼ 1; 2; . . . ; nkt;

i ¼ 1; qð12Þ

The Jump conditions (entry point - exit point). The jumpconditions (13)–(16) are also called Weierstrass - Erdmannconditions or transversality conditions in the optimal controlliterature [24, 25]

l tþkt� �

¼ l t�ktð Þ �Xqi¼1

Xj¼lli�1

j¼0

@g ji ð:Þ@x

��tkt

!T

jj;iðtktÞ !

ð13Þ

H tþkt0� �

¼ H t�kt0ð Þ; k ¼ 1; 2; . . . ; nkt ð14Þ

l tþkx� �

¼ l t�kxð Þ; ð15Þ

H tþkx0� �

¼ H t�kx0ð Þ; k ¼ 1; 2; . . . ; nkx ð16Þ

IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005444

HðtÞ ¼ _xxðtÞTlðtÞ þ 1

2ðxðtÞTQ1xðtÞ þ vðtÞTR1vðtÞÞ

þ gðxðtÞ; vðtÞÞTmðtÞð17Þ

tkðt; xÞ ¼ fminðtkðt;xÞ0 ; tfÞ _maxðtkðt;xÞ0 ; toÞg ð18Þ

For a zeroth-order constraint (11) and (12) are omitted and(13) and (14) are written as:

l tþkt� �

¼ l t�ktð Þ; H tþkt� �

¼ H t�ktð Þ;k ¼ 1; 2; . . . ; nkt; j ¼ 0

where l 2 <n is the vector of adjoint (co-state) time-varying variables associated with the dynamic ODE; m 2<q is the vector of Lagrange multipliers associated with thepath constraints; n 2 <Qg are the Lagrange multipliersassociated with the end-point constraints; wi 2 <lli ;i ¼ 1; q are the Lagrange multipliers linked with thejump conditions and H(t) is the scalar Hamiltonian functionof the system.

Remark 1: The time points tk where the jump conditionsapply are called corners or switching points. The timeintervals t 2 ½tk; tkþ1�; k ¼ 1; ðnkt þ nkxÞ between twoconsecutive corners are termed as constrained or boundaryarcs if at least one constraint is active or unconstrained arcsotherwise. nkt is the maximum number of entry points thatmay exist in the problem and nkx is the maximum number ofexit points.

Parametric programming replaces the optimality vari-ational conditions of the optimal control problem (1) with:(i) functions for the optimal profiles of the control inputs interms of the initial state conditions; and (ii) compactmultidimensional regions in the space of the initialconditions where these functions hold. The following twoSections address these points.

2.2.2 Derivation of the parametric controlprofile: Consider for simplicity initially that all theconstraints are first-order constraints. The extension tohigher-order and zeroth-order constraints readily followsbut is omitted for brevity. The steps followed for thederivation of the control parametric profiles are:Step 1: The dynamic system described by (3), (8)–(10) is ingeneral an index 2 linear differential algebraic equationsystem [26]. However, by taking twice the derivative of theconstraints (8) with respect to time we can generate an ODEof the following matrix form:

_xxðtÞ_llðtÞ

� �¼ A1 �A2R

�11 AT

2

�Q1 �AT1

" #xðtÞlðtÞ

� �

þXqi¼1

�A2R�11 AT

2CT1i mðtÞi

�AT1C

T1i mðtÞi

" # ð19Þ

0 ¼ miðtÞ ð _mmiðtÞ � K1i xðtÞ � K2ilðtÞ � K3imiðtÞÞ;i ¼ 1; q

ð20Þ

where:

K1i ¼ G�1i C1iA

21 þ C1iA2R

�11 AT

2Q1

� K2i ¼ G�1

i C1iA2R�11 AT

2AT1 � C1iA1A2R

�11 AT

2

� K3i ¼ G�1

i C1iA2R�11 AT

2AT1C

T1i � C1iA1A2R

�11 AT

2CT1i

� Gi ¼ C1iA2R

�11 AT

2CT1i

Where C1i is the ith row of the C1 matrix. Note that on anunconstrained arc (19)–(20) is simplified to:

_xxðtÞ_llðtÞ

� �¼ A1 �A2R

�11 AT

2

�Q1 �AT1

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

G

xðtÞlðtÞ

� �ð21Þ

Define transition matrices [24]:

Matkðtkþ1; tkÞ ¼ eGðtkþ1�tkÞ; ð22Þ

Matkðtkþ1; tkÞ ¼Matkðtkþ1; tkÞ11 Matkðtkþ1; tkÞ12Matkðtkþ1; tkÞ21 Matkðtkþ1; tkÞ22

� �ð23Þ

Similarly, over a constrained arc, ~qq constraints are activeand the pertaining multipliers ~mmi � 0: Accordingly, themultipliers of the �qq inactive constraints are zero �mm ¼ 0:Thus, (19)–(20) reduces to:

_xxðtÞ_llðtÞ_~mm~mmðtÞ

24

35 ¼

A1 �A2R�11 AT

2 �A2R�11 AT

2~CCT1

�Q1 �AT1 �AT

1~CCT1

~KK1~KK2

~KK3

24

35

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~GG

xðtÞlðtÞ~mmðtÞ

24

35

ð24Þ

Define transition matrices:

M �aatkðtkþ1; tkÞ ¼ e~GGðtkþ1�tkÞ; ð25Þ

M �aatkðtkþ1; tkÞ

¼M �aatkðtkþ1; tkÞ11 M �aatkðtkþ1; tkÞ12 M �aatkðtkþ1; tkÞ13Matkðtkþ1; tkÞ21 M �aatkðtkþ1; tkÞ22 M �aatkðtkþ1; tkÞ23M �aatkðtkþ1; tkÞ31 M �aatkðtkþ1; tkÞ32 M �aatkðtkþ1; tkÞ33

264

375

ð26Þ

where ~CC1 is a matrix comprising the active rows of C1;similarly for K:Step 2: The following vector j is defined as:j ¼ xTf ; lT

o ; mTðt1Þ; mTðt2Þ; . . . ; mTðtnktÞ; wTðt1Þ; ;�

wTðtnktÞ; nT �T : Parametric expressions for j are readilyderived by substituting the multi-point boundary valuesystem (19)–(20), into the initial (4), the boundary (5),the jump (13), (15) and the junction (11), (12) conditions.The result is a linear system:

I H12

PT1 H22

� �zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{H

xf

lo

� �þ

J11 J12 J1;2nkt 0

J21 J22 J2;2nkt DT1

� �zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{J

mðt1Þmðt2Þ...

mðtnktÞwðt1Þ...

wðtnktÞn

26666666666666664

37777777777777775

¼L11

L21

� �zfflfflffl}|fflfflffl{S

xo ð27Þ


n

wðt1Þwðt2Þ...

wðtnktÞmðt1Þmðt2Þ...

mðtnktÞ

26666666666666666664

37777777777777777775

�D1 0

0 H42

0 H52

264

375

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}M

xf

lo

� �8>>>>>><>>>>>>:

þ0 0 0 0

J41 J42 J4;2nkt 0

J51 J52 J5;2nkt 0

264

375

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}U

mðt1Þmðt2Þ...

mðtnktÞwðt1Þ...

wðtnktÞn

26666666666666664

37777777777777775

:

�0

L41

L51

264

375

|fflfflfflffl{zfflfflfflffl}F

xo þb2

b1

0

264

375

|fflffl{zfflffl}B

9>>>>>=>>>>>;

¼ 0

ð28Þ

mðtktÞ � 0; k ¼ 1; nkt; n � 0 ð29Þ

where matrix I denotes the identity matrix of dimensionn � n: The zeros denote null matrices of appropriatedimensions. Matrices H ; J ; L are explicit functions of:(i) the constraint matrices; and (ii) the transition matricesMatk,M �aatk: Thus, they are an explicit nonlinear function ofthe corner times as indicated in (22)–(23), (25)–(26). Theirfunctional form depends on the sequence of the switchingpoints. For a case of single first-order constraint with asingle constrained arc over the horizon the functions forH12;H22 for instance are:

H12 ¼ � Matkðtf ; t2Þ11ðM �aatkðt2; t1Þ11Matkðt1; toÞ12þ M �aatkðt2; t1Þ12Matkðt1; toÞ22Þ� Matkðtf ; t2Þ12ðM �aatkðt2; t1Þ21Matkðt1; toÞ12þ M �aatkðt2; t1Þ22Matkðt1; toÞ22Þ

H22 ¼ � Matkðtf ; t2Þ2;1ðM �aatkðt2; t1Þ1;1Matkðt1; toÞ1;2þ M �aatkðt2; t1Þ1;2Matkðt1; toÞ2;2Þ� Matkðtf ; t2Þ2;2ðM �aatkðt2; t1Þ2;1Matkðt1; toÞ1;2þ M �aatkðt2; t1Þ2;2Matkðt1; toÞ2;2Þ

Where t1 ¼ t1t; t2 ¼ t1x:Step 3: Next, theorem 1 is stated that enables the derivationof the control functions vðt; xoÞ:

Theorem 1: Let Q1;P1 be positive semi-definite matrix andR1; P1 be a positive definite matrix. Let also the strictcomplementarity slackness condition, i.e. ~nn; ~mm>0; and thelinear independence condition of the binding constraintshold. Then xf ;lo;mðtktÞjk¼1;nkt ;wðtktÞjk¼1;nkt and n are

affine functions of xo for a given finite set and values ofcorners tk ¼ ft1; t2; tnktþnkx

g ft1t; t1x; t2t; tnkxg:

Note that the form of the sequence ft1t; t1x; t2t; tnkxxg isproblem dependent, thus it could be for instance: ft1t; t2t;t1x; tnkxg:

Proof: From (27) we have:

xflo

� �¼ �H�1 J

mkt

wkt

n

24

35þH�1 S xo ð30Þ

where mkt ¼ ½mðt1ÞT ; ;mðtnktÞT�T and accordingly for w:

Then for the active constrains, (25) yields:

~MM xflo

� �þ ~UU

~mmkt

~wwkt

~nn

24

35� ~FF xo � ~BB ¼ 0 ð31Þ

where ~MM ; ~UU; ~FF ; ~BB are the matrices consisting of the rowsof M;U;F;B respectively, that correspond to active cons-traints. Substituting (30) into (31) we obtain:

0 ¼� ~MM �H�1 ~JJ

~mmkt

~wwkt

~nn

264

375þH�1 S xo

264

375

þ ~UU~mmkt

~wwkt

~nn

264

375� ~FF xo � ~BB

,~mmkt

~wwkt

~nn

264

375 ¼ ½ ~MM H�1 ~JJ þ ~UU��1 ½ ~MM H�1 Sþ ~FF � xo

þ ½ ~MM H�1 ~JJ þ ~UU��1 ~BB

ð32ÞNote that sub-matrix ~JJ pertains to the active constraints andalso that ½ ~MM H�1 ~JJ þ ~UU��1 exists because of the assump-tion that the active constraints are linearly independent.Once tk are given, matrices H; J; S;U;M;F;B becomeconstant. Thus, from (32) it follows that the multipliers ~mmkt;~wwkt; ~nn>0 corresponding to the active constraints are anaffine function of the initial states xo: Hence, once (32) issubstituted into (30) along with the condition �mmkt; �wwkt �nn ¼ 0;an affine expression of xTf ; lT

o

� Tin terms of xo is derived.

Theorem 1 implies that j is an affine function of xo but also anonlinear function of tk; since all the matrices in (27)–(29)are explicit nonlinear functions of tk: Hence it follows:j ¼ jðxo; ½t1; t2; tnktþnkx

�Þ ¼ jðxo; tkÞ

Step 4: Next, the goal is to derive expressions for tkðxoÞ andvvcðt; tkðxoÞ; xoÞ: This is performed as follows: (i) solve theODEs (21)–(26) symbolically and substitute into theirsolution the expressions (30), (32) to determine llðt; t k; xoÞ;xxðt; tk; xoÞ; mmðt; tk; xoÞ and (ii) obtain tkðxoÞ from (14), (16),(18). Derive the control profile vvcðt; tkðxoÞ; xoÞ via (10).The form of the control function is shown here as an example,for the case of a single path constraint with a singleconstrained arc and in the absence of endpoint constraints:

For to < t< t1t :

xðtÞlðtÞ

� �¼ Matkðt; toÞ

xo

loð½t1t; t1x; tf �; xoÞ

� �vðtÞ ¼ �R�1

1 AT2lðtÞ

ð33Þ


For t1t < t< t1x :

xðtÞlðtÞmðtÞ

264

375¼M �aatkðt; t1tÞ

Matkðt1t; toÞxo

loð:Þ

� ��

0

CT1wð:Þ

� �� m1tðt1t; ½t1t; t1x; tf �;xoÞ

264

375

vðtÞ¼�R�11 AT

2lðtÞþðC1A2ÞTmðtÞ� �

ð34Þ

A similar expression holds for t1x< t< tf :

2.2.3 Derivation of the region boundar-ies: The boundaries of a critical region, where eachcontrol expression holds, originate from the inequalities:

�ggðxxðt; tkðxoÞ; xoÞ; vvðt; tkðxoÞ; xoÞÞ< 0;

~mmmmðt; tkðxoÞ; xoÞ>0; ~nnðtkðxoÞ; xoÞ>0

Here, we are not interested in all t 2 ½to; tf � but only in thetime instant t ¼ tcr where the inequalities �gg< 0; ~mm>0 arecritically satisfied. Thus, after computing tcrðxoÞ andsubstituting it back into these inequalities we derive a setof inequalities merely in terms of xo: This undertaking isperformed in a compact way by solving parametrically thefollowing problems:

� Take first the constraints that are inactive throughout thecomplete time horizon. Derive from them the followingparametric expressions:

�GGiðxoÞ¼maxtf�ggiðxxðt; tkðxoÞ;xoÞ; vvðt;tkðxoÞ;xoÞÞjt2 ½to; tf �g

i¼1; . . .; �qq

ð35Þ

� Take the path constraints that have at least one constrainedarc ½ti;~kkt; ti;~kkx� and derive from them the parametricexpressions:

~GGiðxoÞ¼maxtf~ggiðxxðt;tkðxoÞ;xoÞ;vvðt;tkðxoÞ;xoÞÞjft2½to;tf �g^

ft 62 ½ti;~kkt;ti;~kkx�g; k¼1;2;...;ni;~kktg; i¼1;...; ~qq

ð36Þ

where, ni;~kkt is the total number of entry points associatedwith the ith active constraint.� Finally, take the multipliers of the constraints that have atleast one constrained arc:

�mmiðxoÞ ¼ mintf ~mm~mmiðt; tk; xoÞjt ¼ ti;kt ¼ ti;kx k ¼ 1; 2; . . . ; ni;ktg;

i ¼ 1; . . . ; ~qq

ð37Þ

Note that when the multipliers assume their minimum value,the corresponding constraint is critically satisfied, hence, thepath constraint reduces to a point constraint. This explainsthe presence of the constraint t ¼ ti;kt ¼ ti;kx (37) thatcaptures exactly this feature.This formulation involves univariant multiparametric

optimisation. Here, we assume that a unique globaloptimum of (35)–(37) can be found (e.g. via the methodof Dua et al. [27]). If multiple solutions occur then oneconstraint may provide more than one parametricexpression, e.g. in (35) we can obtain: �GGijðxoÞ; i ¼ 1; �qq;j ¼ 1; ms instead of �GGiðxoÞ; where ms is the number ofsolutions of program (35). Equations (35)–(37) yield thecritical region of the parametric solution defined as follows:

CRcc ¼ f �GGðxoÞ< 0; ~GGðxoÞ>0; �mmðxoÞ>0; ~nnðxoÞ>0g \ CRIG ð38Þ

From the parametric inequalities the redundant ones areremoved and a compact representation of the region CRcc isderived. The boundaries of the region in general arerepresented by parametric nonlinear expressions in termsof xo: Note that conditions (35)–(37) imply that each regiondiffers from another by its number of boundary arcs overdifferent time intervals and its set of active constraints.

2.2.4 Summary of the algorithm: The varia-tional mp-DO algorithm is summarised below:

Step 1: Define an initial region CRIG: Set index c ¼ 1:Step 2: Fix xoc at a feasible point within region CRIG andsolve the resulting deterministic dynamic optimisationproblem. Thus, obtain the active constraints ~ggc and cornerpoints tck:Step 3: Compute matricesH; ~JJ ; ~SS ; ~MM ; ~UU; ~FF and ~BB and obtainexpressions for xf ;lo; n;m

kt;wkt in terms of the initial statesxo and the corner points tk; k ¼ 1; nkt þ nkx via (30)–(32).Step 4: Next, solve the ODEs (21)–(26), symbolicallyand use (30)–(32) to determine llðt; tk; xoÞ; xxðt; tk; xoÞ;mmðt; tk; xoÞ: Then obtain tkðxoÞ from (14), (16), (18) andthe control profile vvcðt; tkðxoÞ; xoÞ via (10).Step 5: Construct the region boundaries from (38). Removethe redundant inequalities resulting in a compact region CRc

and the optimum control policy vvcðt; tkðxoÞ; xoÞ:Step 6: Define the rest of the region as CRrest ¼ CRIG �Sc

i¼1 CRi:Step 7: If there are no more regions to explore, go to nextstep, otherwise set CRIG ¼ CRrest and c ¼ cþ 1 and go tostep 2.Step 8: Collect all the solutions and unify the regions havingthe same solution to obtain a compact representation.

Figure 1 shows a schematic representation of the algorithm.The algorithm provides a piecewise time-dependentparametric control function of the following form:

CRIG, xoc

CR rest

CR rest = empty

CR rest = CR IG

define initial region and feasibleparametric point

solve DO

active set:corner points:

derive expressions

define rest of region

yes

no

STOP

construct region boundaries

gc~

t k

l(t, t k, xo), m(t, t k, xo), x(t, t k, xo)

CRc(tk(xo),xo)< 0

tk(xo), vc(t, tk(xo), xo)

Fig. 1 Summary of algorithm 1


vvðt; tkðxoÞ; xoÞ ¼ Acðt; tkðxoÞÞ xo þBcðt; tkðxoÞÞ if

0 � CR1cðtkðxoÞÞ xo þ CR2

cðtkðxoÞÞ;for c ¼ 1; . . .Nc

ð39Þ

Theorem 2: The feasible space of initial conditions whereproblem (1) has a solution is convex. The mapping of theoptimal value of the performance index ffðxoÞ of problem(1) in the space of the initial conditions is continuous,differentiable and convex.

Theorem 3: The expression (39) for the optimal value of thecontrol variables in terms of the initial conditions vvðt; xoÞ isnonlinear, continuous and piecewise differentiable.

The proofs of the two aforementioned theorems areindependent of the methodology that is used to derive theoptimal control functions and rely on the linearity andconvexity characteristics of problem (1) [20]. The details ofthe proof are omitted here for brevity.

Remark 2: The mapping tk 7! vv is in general an explicitexponential or trigonometric function with a uniquesolution. Note, however, that the mapping xo 7! tk is ingeneral an implicit nonlinear function. However, either viastochastic approaches or using multiparametric nonlinearglobal optimization [27] we can directly derive explicitexpressions for

tkt ¼ tktðxoÞ; kt ¼ 1; ; nkt; tkx ¼ tkxðxoÞ;kx ¼ 1; ; nkx ð40Þ

Equation (40) may yield: (i) multiple solutions; or (ii) aninfinite number of solutions in terms of tkt; tkx: Here,although, we assume non-Zeno behaviour [28] of thedynamics thus, ruling out the possibility of an infinitenumber of switching points within a finite time interval(case (ii)). Case (i) poses no limitations to the solutionprocedure as it follows from the continuity results oftheorems 2 and 3 that the transversality conditions (14),(16), (18) have as many feasible solutions as the number ofcorner points within the horizon.

3 Control law derivation

Equation (39) features the explicit open-loop optimalcontrol policy for system (1) as a function of the initialstate condition xo: When the implementation of this controlpolicy is performed in an MPC fashion it can be readilytranslated into an optimal control law for the linear dynamicplant. This is achieved by first treating the current processstate as initial state xðt Þ xo where t

is the time where thestate value becomes available from the plant. Then thecontrol action obtained from (39) is applied only throughoutthe interval t � t � t þ Dt and at the next time instantt þ Dt the current state values are updated (i.e. t ¼ t þ Dtand xðt ÞÞ and the control computation is repeated. Thiscontrol law is applied over a time interval equal to the plantsampling time: vðxðt ÞÞ ¼ fvvcðt; tkðx Þ; x Þjxðt Þ ¼ x ; 0 �t � Dt g: Note that the control law derivation is indepen-dent of the length of the sampling interval Dt : However, itsimplementation does depend on the size of the samplingtime. In the case where a continuous realisation of the statevariables is available the control law is given by the

expression: vðxðtÞÞ ¼nlimt0!0

vvðt0; tkðx Þ; x ÞjxðtÞ ¼ x o:

4 Comparison between the continuous- anddiscrete - time parametric controller

The parametric controller derived in this work (continuous-time parco) has the following benefits comparing to theparametric controller for discrete-time dynamic systems(discrete-time parco) [12, 13]:

. The constraints are satisfied over the complete time horizonirrespective of the length of the sampling interval, whereas thecontroller for the discrete-time dynamic system can guaranteeconstraint satisfaction only at discrete points in time.. The value of the cost function is more precise inasmuch asthe integral and not the sum of the input=output deviations,over the time domain is considered.. Here, the complexity of the control law derivation doesnot depend on any form of time discretisation and iscontingent solely upon: (i) the number of constraints; (ii) thesystem dynamics; and (iii) the number of control variables.In particular, the maximum number of regions that aregenerated at the worst case by a discrete-time parametriccontroller are computed from the following formula:

Nc ¼Xnv N

i¼0

q N þ Qg

i

� �¼Xnv N

i¼0

ðq N þ QgÞ!ðq N þ Qg � iÞ!i! ð41Þ

where N is an integer representing the number of discretetime elements within the problem time horizon.The corresponding maximum number of regions of thecontinuous-time parametric controller is (for higher than1st order systems):

Nc ¼Xnvi¼0

Pqj¼1

ðj nbjÞ þ Qg

i

0@

1A ð42Þ

where nbj is the maximum number of boundary constrainedarcs of constraint j. In most well-behaved and thoroughlytuned closed-loop systems it holds that: nbj ¼ 1: A measureof the controller complexity is the number of regions that itgenerates, thus, it can be deduced from (41) and (42) that forlong time horizons, the continuous-time control scheme issignificantly less complex. However, the derivation and theon-line implementation of the continuous-time parametriccontroller may involve the solution of nonlinear equationsand a number of nonlinear function evaluations respect-ively, whereas only linear computations are required in thediscrete-time case. Hence, there is clearly a trade-offbetween the complexity of the state partition and thecomplexity of the respective control functions when puttingthe two control schemes into perspective.

5 One-state SISO illustrative example

The example is concerned with deriving the control lawfor the integrating system in Scokaert and Mayne [29].Maple v. 5.1. [30] and MATLAB v. 5. [31] were used for thesymbolic and algebraic manipulations in both this and thefollowing illustrative example.

fðxoÞ ¼1

2xðtfÞ2 þ

1

2

Z tf

to

½xðtÞ2 þ vðtÞ2�dt

subject to

_xxðtÞ ¼ vðtÞ; yðtÞ ¼ xðtÞ � vðtÞ; to � t � tf

2 � yðtÞ � �1:2; to � t � tf

to ¼ 0; tf ¼ 2

xðtoÞ ¼ xo; �2 � xo � 2 ð43Þ


The initial states xo are treated as parameters, and (43)is recast as a multiparametric dynamic optimisationproblem.

5.1 Steps of algorithm 1

Iteration 1.1. Define an initial region �2 � xo � 2:2. Choose a point in the state space xo ¼ �1 and for thatpoint solve the dynamic optimisation problem. Identifythat there are no active constraints for that point so we donot have to divide the control horizon into constrained=unconstrained arcs.3,4. The optimal control, state and adjoint profile for theunconstrained case are shown in Table 1 under region CR01.5. The region boundaries are identified as follows:

� Solve the following two problems parametrically:

Gc1ðxoÞ ¼ maxto�t�tf

ð�xðtÞ þ vðtÞ þ 1:2Þ

subject to

xðtÞ ¼ e�txo; vðtÞ ¼ �e�txo; �2 � xo � 2 ð44Þ

Gc2ðxoÞ ¼ maxto�t�tf

ðxðtÞ � vðtÞ � 2Þ

subject to

xðtÞ ¼ e�txo; vðtÞ ¼ �e�txo; �2 � xo � 2 ð45Þ

Table 1: Control and state functions for the example in Section 5.1

CR01 - region bounds: �0:6 � x� � 1

For to � t � tf : vðtÞ ¼ �e�t x�

xðtÞ ¼ e�t x�

CR02 - regions bounds: �1:1186 � x� � �0:6

For to � t � t1x : For t1x � t � tfvðtÞ ¼ 1:2et þ etx vðtÞ ¼ �e�ðt�t1x Þ � ð1:2et1x þ et1x x � 1:2ÞxðtÞ ¼ 1:2et þ etx � 1:2 xðtÞ ¼ e�ðt�t1x Þ � ð1:2et1x þ et1x x � 1:2Þ

CR03 - region bounds: 1 � x � 1:86

For to � t � t1x : For t1x � t � tf :

vðtÞ ¼ �2et þ etx vðtÞ ¼ �e�ðt�t1x Þ � ð�2et1x þ et1x x þ 2ÞxðtÞ ¼ �2et þ etx þ 2 xðtÞ ¼ e�ðt�t1x Þ � ð�2et1x þ et1x x þ 2ÞCR04 - regions bounds: �2 � x � �1:1186

For to � t � tf : vðtÞ ¼ 1:2et þ etx

xðtÞ ¼ 1:2et þ etx � 1:2

CR05 - regions bounds: 1:86 � x � 2

For to � t � tf : vðtÞ ¼ �2et þ etx

xðtÞ ¼ �2et þ etx þ 2

Fig. 2 Control input, output and constraint profile for the example in Section 5.2


The solution is:

Gc1ðxoÞ ¼ �2xo � 1:2 � 0 if xo � 0

Gc1ðxoÞ ¼ �0:27xo � 1:2 � 0 if xo � 0

Gc2ðxoÞ ¼ 2xo � 2 � 0 if xo � 0

Gc2ðxoÞ ¼ 0:27xo � 2 � 0 if xo � 0

� Remove the redundant constraints from the followinginequalities:

Cc1ðxoÞ � 0

Cc2ðxoÞ � 0

� 2 � xo � 2

ð46Þ

� The non-redundant constraints comprise the uncons-trained region boundaries CR01 shown in Table 1:6. The rest of the region is here trivially defined as CRrest ¼CRrest

1 [ CRrest2 ¼ f�2 � xo � �0:6 [ 1:01 � xo � 2g:

Iteration 2.1. Select the critical region CRrest

1 ¼ f�2 � xo � �0:6g:2. Choose a point in the state space xo ¼ �1:0 and for thatpoint solve the dynamic optimisation problem. Identify thatthere is an active constraint xðtÞ � vðtÞ � �1:2; oneconstrained arc and one unconstrained arc.3. The ODE systems (21)–(26) take the form:For t1x � t � tf : _xxðtÞ ¼ �lðtÞ; _llðtÞ ¼ �xFor t1t � t � t1x : _xxðtÞ ¼ �lðtÞ � mðtÞ; _llðtÞ ¼ �xþ mðtÞ;mðtÞ ¼ �ðxðtÞ þ lðtÞ þ 1:2Þ where t1t ¼ to: Using (27)-(29)we obtain the following expressions for lo; xf ; mðt1tÞ :

lo ¼2:4ðet1x � 1Þ þ 2et1x xo � e�t1x xo

e�t1xð47Þ

xf ¼ e�ðtf�t1xÞ½1:2et1x�t1t � 1:2þ et1x�t1t xo� ð48Þ

mðt1tÞ ¼ �½xo þ lo þ 1:2� ð49Þ

4. The optimum time trajectories of the state and controlvariables for that region are given in Table 1 under regionCR02 characteristics. The optimum profiles of the adjointand Lagrange multiplier are derived similarly and areomitted for brevity.5. To determine the region boundaries the followinginequality is first posed:

0 � �mmðxoÞ ¼ minto�t�tf

fmðt; t1t; t1x; xoÞjt ¼ t1t ¼ t2xg ð50Þ

The expression for mðtÞ ¼ �½e�txo þ e�tlo þ 1:2� is sub-stituted into (50) to yield:

�mmðxoÞ ¼ � ½e�txo þ e�tlo þ 1:2�jt¼to¼t1t¼t2x� 0

) �½2:4eto � 1:2þ 2etoxo� � 0 ) xo � �0:6

Then from the inequality (36) posed on the constraint yðtÞ>�1:2 we obtain: xo � �1:1186: This implies that for xo ��1:1186 the solution comprises a single constrained arccovering the complete horizon and no unconstrained arc.Inequality (50) does not pose any additional constraints.Removing the redundant constraints from fCRrest

1 ; xo ��0:6; xo � �1:1186g results in a critical region CR02 thatis a subset of CRrest

1 as shown in Table 1. A point in the partof region CRrest

1 that has not been explored �2 � xo ��1:1186 is chosen and the procedure is repeated from step2. in this iteration stage.Iteration 3.For region CRrest

2 ¼ f1:01 � xo � 2g the same procedure isrepeated. That region is also divided to two subregions

similarly to region CRrest1 where the constraint 2 � yðtÞ is

now the active one. For the sake of brevity, Table 1 showsonly the parametric control profiles and the boundaries ofthat region.

5.2 Implementation of the control law

Given an x from the plant set xo ¼ x and identify fromTable 1 the region where the system resides. If the systemresides in the unconstrained region CR01 then theimplementation of the control policy is straight-forwardfrom the first two rows of Table 1. If, however, the systemresides in a constrained region we follow the steps:1. First generate the appropriate time expressions for theLagrange multipliers, adjoint, state and control variablesfrom Table 1 depending on what region the system stateresides.2. Substitute xx; ll; mm in the Hamiltonian expression (17) andthen use (14), (16) and the modified version of (18): t1ðt;xÞ ¼fminðt1ðt;xÞ0 ; tfÞ _maxðt1ðt;xÞ0 ; toÞg to obtain the corner pointsparametrically in terms of xo: In region CR02 we derive thefollowing functions: t1tðxoÞ ¼ to ¼ 0; t1x ¼ �18:8 ðx Þ3 �45:851 ðx Þ2 � 34:762 x � 9:6101 within an integralsquare error accuracy of 99.96%. These expressions werederived by solving the nonlinear equations (14), (16) and (18)for a large number of initial state realisations within regionCR02. Then nonlinear regression was used to generate thepolynomial expressions that relate xo with t1t; t1x:3. Substitute the corner point functions t1xðxoÞ; t1xðxoÞ backto vvðt; tk; xoÞ thus, deriving the control profile.The execution of the control law for a sampling time ofDt ¼ 0:5 s is shown in Fig. 2, where the state initialisesfrom the perturbed point xo ¼ �1:

6 Two-state SISO illustrative example

A two-state SISO numerical example is presented here.Consider the open-loop unstable SISO plant example fromKwakernaak and Sivan [32]:

SðsÞ ¼ 0:003396ðsþ 0:8575Þðs� 1Þðs� 0:6313Þ ð51Þ

corresponding to the following numerical values of theproblem matrices in formulation (1) where also a pathconstraint is added.

Q1 ¼0:01156 0:00986

0:00986 0:00841

� �;

R1 ¼ 10�4; g1 ¼ C1 xðtÞ � 2:4;

8t 2 ½to; tf �; to ¼ 0; tf ¼ 1:5 s

where

A1 ¼1:6313 �0:6313

1 0

� �;

A2 ¼1

0

� �; C1 ¼ 1:5 1½ �

The terminal cost P1 is evaluated from the solution of theRiccati equation. The initial states are treated as parametersand a mp-DO problem is formulated. Note that the pathconstraint is first-order. The aim here is to derive a controllaw for this continuous-time dynamic system over the statespace: CRIG ¼ f�10 � xo1 � 10;�15 � xo2 � 10g: Thedetails of the control law derivation are omitted for brevity.The application of the algorithm presented in Section 2results in the state-space partition shown in Fig. 3 where


a comparison with the partition of the discrete-time parcois also displayed. The boundaries of region CR01 arerepresented by the following initial state expressions:

xo1>�10; xo2 < 10 ð52Þ

If 88:2543xo1þ 74:3203xo2 � 0; 0:1329xo1þ 1:4309xo2 � 0or 88:2543xo1þ74:3203xo2� 0; 0:1329xo1þ1:4309xo2� 0;xo2� 10or 88:2543xo1þ74:3203xo2� 0; 0:1329xo1þ1:4309xo2� 0;xo2� 10 then:

�tt ¼ 0; �GG : 1:5xo1 þ xo2 � 2:4< 0 ð53Þ

If 88:2543xo1þ74:3203xo2 � 0; 0:1329xo1þ1:4309xo2 � 0;xo2 ��15 then:

�tt¼ 0:100 877 6354 ln 81:21:0912xo1þ0:935xo20:1329xo1þ1:4309xo2

� ��GG : 1:4011e�10:77�ttð1:0912xo1þ0:9352xo2Þ

�0:216e�0:8577�ttð0:1329xo1þ1:4309xo2Þ�2:4<0

ð54Þ

The control policy in the same region is given by thefunction:

vvðtÞ¼�10 000½0:001 240e�10:76tþ10:76toð1:0912xo1þ0:93xo2Þ�0:000 2098e�0:8570tþ0:857toð0:1329xo1þ1:4309xo2Þ�

The control law implementation for this example asdescribed in Section 3 involves: (i) setting x ¼xo; (ii)identifying the region where the system resides from theinequalities of the form of (52)–(54); and (iii) implementingthe control function pertaining to each region for a timeinterval 0� t�Dt : The discrete-time parco was designedby considering the control input as a piecewise constantfunction of time parameterised over ten-equidistant time-elements. Interior-point constraints were added to ensureconstraint satisfaction over the complete prediction horizon.It is clear from Fig. 3 that the discrete-time parco results in alarger number of critical polyhedral regions comparing tothe continuous-time parco that generates only three regionswith nonlinear boundaries. The execution of the derivedcontrol law is also shown in Fig. 3, where the system is

steered to the origin starting from a perturbed point ofx¼½7; �13�T :

7 Conclusions

An approach has been developed to derive: (i) a closed-formparametric controller for linear constrained continuous-timedynamic systems; and (ii) the state domain where thiscontroller provides a feasible control action. This method issuitable for low-dimensional systems that involve: (i) largesampling times; (ii) safety critical constraints; and (iii) highprecision requirements. The on-line implementation of thecontrol action was simply achieved by function evaluations,obviating the need for on-line optimisation. It was shownthat while the convexity of the feasible state space and thecontinuity of the controller are retained, the optimal controlfunctions are nonlinear piecewise differentiable and thestate-space partition is in general nonlinear. The longwithstanding issue of expressing the switching time points,when future changes into the system dynamic behaviouroccur, as a function of the plant states was addressed. Thisparticular feature gives an insight to the structure of thecontroller and the behaviour of the closed-loop system.

8 Acknowledgments

The financial support from the Department of EnergyTransport and the Regions (DETR-ETSU), Shell Chemicalsand Air Products and Chemicals Inc. (APCI) is gratefullyacknowledged.

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Explicit solutions to optimal control problems for constrained continuous-time linear systems