A class of optimal state-delay control problems

Nonlinear Analysis: Real World Applications 14 (2013) 1536–1550

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

A class of optimal state-delay control problemsQinqin Chai a,b,∗, Ryan Loxton b, Kok Lay Teo b, Chunhua Yang a

a School of Information Science & Engineering, Central South University, Changsha, Chinab Department of Mathematics & Statistics, Curtin University, Perth, Australia

a r t i c l e i n f o

Article history:Received 23 July 2012Accepted 16 October 2012

Keywords:Time-delayOptimal controlNonlinear optimizationParameter identificationDelayed feedback control

a b s t r a c t

We consider a general nonlinear time-delay system with state-delays as control variables.The problem of determining optimal values for the state-delays tominimize overall systemcost is a non-standard optimal control problem – called an optimal state-delay controlproblem – that cannot be solved using existing optimal control techniques. We show thatthis optimal control problem can be formulated as a nonlinear programming problemin which the cost function is an implicit function of the decision variables. We thendevelop an efficient numerical method for determining the cost function’s gradient. Thismethod, which involves integrating an auxiliary impulsive system backwards in time, canbe combined with any standard gradient-based optimization method to solve the optimalstate-delay control problem effectively. We conclude the paper by discussing applicationsof our approach to parameter identification and delayed feedback control.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Time-delay systems arise in many real-world applications—e.g. chromatography [1], evaporation and purificationprocesses [2,3], aerospace models [4], and human immune response [5]. Over the past two decades, various optimal controlmethods have been developed for time-delay systems.Well-known tools include the necessary conditions for optimality [6]and numerical methods based on the control parameterization technique [7,8]. These existing optimal control methods arerestricted to time-delay systems in which the delays are fixed and known. In this paper, we consider a new class of optimalcontrol problems in which the delays are instead control variables to be chosen optimally. Such problems are called optimalstate-delay control problems.

As an example of an optimal state-delay control problem, consider a systemof delay-differential equationswith unknowndelays. This system is a dynamic model for some phenomenon under consideration. The problem is to choose values for theunknown delays (and possibly other model parameters) so that the model best fits a given set of experimental data. Thisso-called parameter identification problem can be formulated as an optimal state-delay control problem in which the delaysandmodel parameters are decision variables, and the cost function measures the least-squares error between the predictedoutput and the observed system output.

Parameter identification for time-delay systems is an active area of research (see [9] and the references cited therein).However, most of the available parameter identification tools, such as particle swarm optimization [10], frequency domainanalysis [11], and input–output representation [12], are restricted to linear systems or systems with a single delay.By formulating the parameter identification problem as an optimal state-delay control problem, and then applying thecomputational method to be developed in this paper, nonlinear systems with any number of delays can be considered.This computational approach is applicable to a broader class of systems than our recent parameter identification method

∗ Corresponding author at: School of Information Science & Engineering, Central South University, Changsha, China. Tel.: +86 731 88836876.E-mail addresses: [email protected] (Q. Chai), [email protected] (R. Loxton), [email protected] (K.L. Teo), [email protected] (C. Yang).

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Q. Chai et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1536–1550 1537

described in [13], which has two limitations: (i) it is only applicable to systems in which each nonlinear term containsa single delay and no unknown parameters; and (ii) it involves integrating a large number of auxiliary delay-differentialsystems (one auxiliary system for each unknown delay). Our work in this paper overcomes these limitations.

Another important application of optimal state-delay control problems is delayed feedback control. In delayed feedbackcontrol, the system’s input function is chosen to be a linear function of the delayed state, as opposed to traditional feedbackcontrol in which the input is a function of the current (undelayed) state. Voluntarily introducing delays via delayed feedbackcontrol can be beneficial for certain types of systems; see [14–16]. The problem of choosing optimal values for the delays in adelayed feedback controller can be formulated as an optimal state-delay control problem. Note that the control and stabilityof time-delay systems has been considered inmany recent papers—see, for example, [17–19]. However, these papersmainlyfocus on traditional state feedback rather than delayed state feedback, and as yet there are nomethods available for choosingoptimal values for the delays in a delayed feedback controller (the delays are usually chosen heuristically [20,21]). Byapplying the computational method to be developed in this paper, optimal delay values can be obtained that minimizesystem cost.

Our goal in this paper is to develop a unified computational approach for solving optimal state-delay control problems.This class of optimal control problems has yet to be considered in the literature. In fact, the latestmethods for solving optimalcontrol problems with time-delays (see [6–8] and the references cited therein) are based on the assumption that the time-delays are fixed values, not optimization variables. In this paper, we tackle the more difficult case in which the time-delaysneed to be chosen optimally to minimize a cost function. A key aspect of our work is the derivation of an auxiliary impulsivesystem, which is actually the analogue of the costate or adjoint system in classical optimal control. We derive formulae forthe cost function’s gradient in terms of the solution of this impulsive system. On this basis, the optimal state-delay controlproblem can be solved by combining numerical integration and nonlinear programming techniques. Aswewill demonstrate,this approach has proven very effective for the two specific applications mentioned above—parameter identification anddelayed feedback control.

The remainder of the paper is organized as follows. We first formulate the optimal state-delay control problem inSection 2, before introducing the auxiliary impulsive systemand deriving gradient formulae in Section 3. Section 4 is devotedto parameter identification problems, and Section 5 is devoted to delayed feedback control. We make some concludingremarks in Section 6.

2. Problem formulation

Consider the following nonlinear time-delay system:

x(t) = f (x(t), x(t − τ1), . . . , x(t − τm), ζ), t ∈ [0, T ], (1)

x(t) = φ(t, ζ), t ≤ 0, (2)

where T > 0 is a given terminal time; x(t) = [x1(t), . . . , xn(t)]⊤ ∈ Rn is the state vector; τi, i = 1, . . . ,m are state-delays;ζ = [ζ1, . . . , ζr ]

⊤∈ Rr is a vector of system parameters; and f : R(m+1)n

× Rr→ Rn and φ : R × Rr

→ Rn are givenfunctions.

System (1)–(2) is controlled via the state-delays and system parameters—these must be chosen optimally so that thesystem behaves in the best possible manner. We impose the following bound constraints on the state-delays and systemparameters:

ai ≤ τi ≤ bi, i = 1, . . . ,m, (3)

and

cj ≤ ζj ≤ dj, j = 1, . . . , r, (4)

where ai and bi are given constants such that 0 ≤ ai < bi, and cj and dj are given constants such that cj < dj.Any vector τ = [τ1, . . . , τm]

⊤∈ Rm satisfying (3) is called an admissible state-delay vector. Let T denote the set of all

such admissible state-delay vectors.Any vector ζ = [ζ1, . . . , ζr ]

⊤∈ Rr satisfying (4) is called an admissible parameter vector. Let Z denote the set of all such

admissible parameter vectors.Any combined pair (τ, ζ) ∈ T × Z is called an admissible control pair for system (1)–(2).We assume that the following conditions are satisfied.

Assumption 1. The given function f is continuously differentiable, and φ is twice continuously differentiable.

Assumption 2. There exists a real number L1 > 0 such that for all ξi ∈ Rn, i = 0, . . . ,m, and ω ∈ Rr ,f (ξ0, ξ1, . . . , ξm,ω) ≤ L1(1 + |ξ0| + |ξ1| + · · · + |ξm| + |ω|),

where | · | denotes the Euclidean norm.

1538 Q. Chai et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1536–1550

Assumptions 1 and 2 ensure that system (1)–(2) admits a unique solution corresponding to each admissible control pair(τ, ζ) ∈ T × Z [22]. We denote this solution by x(·|τ, ζ).

Our aim is to find an admissible control pair that minimizes the following cost function:

J(τ, ζ) = Φ(x(t1|τ, ζ), . . . , x(tp|τ, ζ), ζ), (5)

where Φ : Rpn× Rr

→ R is a given function and tk, k = 1, . . . , p are given time points satisfying

0 < t1 < · · · < tp ≤ T .

Unlike the standard Mayer cost function commonly used in optimal control (which depends solely on the final state), thecost function (5) depends on the state at a set of intermediate time points tk, k = 1, . . . , p. These time points are calledcharacteristic times in the optimal control literature [3,23,24]. As we will see, cost functions with characteristic times arisein parameter identification problems, where the aim is to minimize the discrepancy between the predicted output and theobserved system output over the time horizon.

Our optimal state-delay control problem is defined formally below.

Problem (P). Choose (τ, ζ) ∈ T × Z to minimize the cost function (5).

3. Gradient computation

Although the optimal control of time-delay systems has been the subject of numerous theoretical and practical inves-tigations [3,6,8,9,25], most research has focused on the simple case when the delays are fixed and known. The delays inProblem (P), however, are actually control variables to be determined optimally. Hence, Problem (P) differs considerablyfrom most time-delay optimal control problems in the literature.

The aim of this paper is to develop a computational method for solving Problem (P). Our approach is based on thefollowing key observation: Problem (P) can be viewed as a nonlinear optimization problem in which the decision vectorsτ and ζ influence the cost function J implicitly through the governing dynamic system (1)–(2). Thus, if the gradient of J canbe computed for each admissible control pair, then Problem (P) can be solved using existing gradient-based optimizationmethods, such as sequential quadratic programming (see [26,27]). However, since J is not an explicit function of τ and ζ,deriving its gradient is not straightforward. The purpose of this section is to develop a numerical algorithm for computingthe gradient of J .

3.1. Gradient with respect to state-delays

Define

ψ(t|τ, ζ) =

∂φ(t, ζ)

∂t, if t < 0,

f (x(t|τ, ζ), x(t − τ1|τ, ζ), . . . , x(t − τm|τ, ζ), ζ), if t ∈ [0, T ].

Furthermore, define

∂ f (t|τ, ζ)∂x

=∂f (x(t|τ, ζ), x(t − τ1|τ, ζ), . . . , x(t − τm|τ, ζ), ζ)

∂x,

∂ f (t|τ, ζ)∂ xi


∂ xi,

where ∂

∂ xi denotes differentiation with respect to the ith delayed state vector.Consider the following auxiliary impulsive system:

λ(t) = −

∂ f (t|τ, ζ)

∂x

⊤

λ(t) −

ml=1

∂ f (t + τl|τ, ζ)

∂ xl

⊤

λ(t + τl), t ∈ [0, tp], (6)

λ(t−k ) = λ(t+k ) +

∂Φ(x(t1|τ, ζ), . . . , x(tp|τ, ζ), ζ)

∂x(tk)

⊤

, k = 1, . . . , p, (7)

λ(t) = 0, t ≥ tp. (8)

Let λ(·|τ, ζ) denote the solution of system (6)–(8) corresponding to the admissible control pair (τ, ζ) ∈ T × Z.The following result gives formulae for the partial derivatives of J with respect to the state-delays.

Theorem 1. For each (τ, ζ) ∈ T × Z,

∂ J(τ, ζ)∂τi

= −

tp

0λ⊤(t|τ, ζ)

∂ f (t|τ, ζ)∂ xi

ψ(t − τi|τ, ζ)dt, i = 1, . . . ,m. (9)


Proof. Let v : [0, ∞) → Rn be an arbitrary function satisfying the following conditions:

(i) v is continuous on the intervals (tk−1, tk), k = 1, . . . , p + 1, where t0 = 0 and tp+1 = ∞;(ii) v is differentiable almost everywhere;(iii) v has finite left and right limits at t = tk, k = 1, . . . , p, and a finite right limit at t = 0.

Note that any discontinuity of v must lie in the set {t0, t1, . . . , tp}.We may express the cost function J as follows:

J(τ, ζ) = Φ(x(t1), . . . , x(tp), ζ)

= Φ(x(t1), . . . , x(tp), ζ) +

tp

0

v⊤(t)f (x(t), x(t − τ1), . . . , x(t − τm), ζ) − v⊤(t)x(t)

dt

= Φ(x(t1), . . . , x(tp), ζ) +

pk=1

tk

tk−1

v⊤(t)f (x(t), x(t − τ1), . . . , x(t − τm), ζ)dt −

pk=1

tk

tk−1

v⊤(t)x(t)dt,

where for simplicity we have omitted the τ and ζ arguments in x(·|τ, ζ). This notation will not cause confusion becauseτ and ζ are assumed to be fixed throughout this proof (where appropriate, we will also omit the τ and ζ arguments from∂ f (t|τ,ζ)

∂x , ∂ f (t|τ,ζ)∂ xi , and ψ(t|τ, ζ)).

Applying integration by parts to the last integral gives

J(τ, ζ) = Φ(x(t1), . . . , x(tp), ζ) +

pk=1

tk

tk−1

v⊤(t)f (x(t), x(t − τ1), . . . , x(t − τm), ζ)dt

−

pk=1

v⊤(t−k )x(tk) − v⊤(t+k−1)x(tk−1)

+

pk=1

tk

tk−1

v⊤(t)x(t)dt. (10)

Consider the third term on the right-hand side of (10):p

k=1

v⊤(t−k )x(tk) − v⊤(t+k−1)x(tk−1)

=

pk=1

v⊤(t−k )x(tk) −

pk=1

v⊤(t+k−1)x(tk−1)

=

pk=1

v⊤(t−k )x(tk) −

p−1k=0

v⊤(t+k )x(tk)

= v⊤(t−p )x(tp) +

p−1k=1

v⊤(t−k ) − v⊤(t+k )

x(tk) − v⊤(t+0 )x(t0). (11)

Substituting (11) into (10) yields

J(τ, ζ) = Φ(x(t1), . . . , x(tp), ζ) +

pk=1

tk

tk−1


+

pk=1

tk

tk−1

v⊤(t)x(t)dt − v⊤(t−p )x(tp) −

p−1k=1

v⊤(t−k ) − v⊤(t+k )

x(tk) + v⊤(0+)φ(0, ζ). (12)

Define the state variation with respect to τi as follows:

Λi(t) =∂x(t)∂τi

, t ∈ (−∞, T ].

If t < τl, then x(t − τl) = φ(t − τl, ζ), and thus

∂

∂τi{x(t − τl)} =

∂

∂τi{φ(t − τl, ζ)} = −δli

∂φ(t − τl, ζ)

∂t, (13)

where δli denotes the Kronecker delta function. On the other hand, if t ≥ τl, then

∂

∂τi{x(t − τl)} = Λi(t − τl) − δlix(t − τl). (14)

Combining (13) and (14) gives

∂

∂τi{x(t − τl)} = Λi(t − τl)χ[τl,∞)(t) − δliψ(t − τl), (15)


where χ[τl,∞) : R → R is the indicator function defined by

χ[τl,∞)(t) =

1, if t ≥ τl,0, otherwise.

Now, in view of (15), we can differentiate (12) with respect to τi to obtain

∂ J(τ, ζ)∂τi

=

pk=1

∂Φ(x(t1), . . . , x(tp), ζ)∂x(tk)

Λi(tk) +

pk=1

tk

tk−1

v⊤(t)∂ f (t)∂x

Λi(t)dt

+

ml=1

pk=1

tk

tk−1

v⊤(t)∂ f (t)∂ xl

Λi(t − τl)χ[τl,∞)(t)dt

−

pk=1

tk

tk−1

v⊤(t)∂ f (t)∂ xi

ψ(t − τi)dt − v⊤(t−p )Λi(tp)

−

p−1k=1

v⊤(t−k ) − v⊤(t+k )

Λi(tk) +

pk=1

tk

tk−1

v⊤(t)Λi(t)dt.

Thus,

∂ J(τ, ζ)∂τi

=

p−1k=1

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tk)− v⊤(t−k ) + v⊤(t+k )

Λi(tk) − v⊤(t−p )Λi(tp)

+∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tp)Λi(tp) +

tp

0

v⊤(t) + v⊤(t)

∂ f (t)∂x

Λi(t)dt

+

ml=1

tp

0v⊤(t)

∂ f (t)∂ xl

Λi(t − τl)χ[τl,∞)(t)dt −

tp

0v⊤(t)

∂ f (t)∂ xi

ψ(t − τi)dt. (16)

We now perform a change of variable in the second last integral term in (16): tp

0v⊤(t)

∂ f (t)∂ xl

Λi(t − τl)χ[τl,∞)(t)dt =

tp−τl

−τl

v⊤(t + τl)∂ f (t + τl)

∂ xlΛi(t)χ[0,∞)(t)dt

=

tp−τl

0v⊤(t + τl)

∂ f (t + τl)

∂ xlΛi(t)χ[0,∞)(t)dt. (17)

Substituting (17) into (16) gives,

∂ J(τ, ζ)∂τi

=

p−1k=1

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tk)− v⊤(t−k ) + v⊤(t+k )

Λi(tk)

+

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tp)− v⊤(t−p )

Λi(tp)

+

tp

0v⊤(t)Λi(t)dt +

tp

0v⊤(t)

∂ f (t)∂x

Λi(t)dt +

ml=1

tp−τl

0v⊤(t + τl)

∂ f (t + τl)

∂ xlΛi(t)χ[0,∞)(t)dt

−

tp

0v⊤(t)

∂ f (t)∂ xi

ψ(t − τi)dt. (18)

Recall that v is an arbitrary function satisfying conditions (i)–(iii) at the start of this proof. Thus, we can choose v = λ(·|τ, ζ).Substituting v = λ(·|τ, ζ) into (18) and using (7)–(8) give

∂ J(τ, ζ)∂τi

=

tp

0λ

⊤(t)Λi(t)dt +

tp

0λ⊤(t)

∂ f (t)∂x

Λi(t)dt +

ml=1

tp

0λ⊤(t + τl)

∂ f (t + τl)

∂ xlΛi(t)dt

−

tp

0λ⊤(t)

∂ f (t)∂ xi

ψ(t − τi)dt.

Hence, from (6),


∂ J(τ, ζ)∂τi

= −

tp

0λ⊤(t)

∂ f (t)∂ xi

ψ(t − τi)dt.

This completes the proof. �

3.2. Gradient with respect to system parameters

We now turn our attention to the gradient of J with respect to ζj, j = 1, . . . , r . As before, let λ(·|τ, ζ) be the solution ofthe impulsive dynamic system (6)–(8). Furthermore, for each j = 1, . . . , r , define

∂ f (t|τ, ζ)∂ζj


∂ζj.

Then we have the following result.

Theorem 2. For each (τ, ζ) ∈ T × Z,

∂ J(τ, ζ)∂ζj

=∂Φ(x(t1|τ, ζ), . . . , x(tp|τ, ζ), ζ)

∂ζj+

tp

0λ⊤(t|τ, ζ)

∂ f (t|τ, ζ)∂ζj

dt

+λ⊤(0+|τ, ζ)

∂φ(0, ζ)∂ζj

+

ml=1

0

−τl

λ⊤(t + τl|τ, ζ)∂ f (t + τl|τ, ζ)

∂ xl∂φ(t, ζ)

∂ζjdt, j = 1, . . . , r. (19)

Proof. Let v(·) be as defined in the proof of Theorem 1. Recall from Eq. (12) that

J(τ, ζ) = Φ(x(t1), . . . , x(tp), ζ) +

pk=1

tk

tk−1


+

pk=1

tk

tk−1

v⊤(t)x(t)dt − v⊤(t−p )x(tp) −

p−1k=1

v⊤(t−k ) − v⊤(t+k )

x(tk) + v⊤(0+)φ(0, ζ), (20)

where, as in the proof of Theorem 1, we omit the τ and ζ arguments for clarity.Differentiating (20) with respect to ζj gives

∂ J(τ, ζ)∂ζj

=∂Φ(x(t1), . . . , x(tp), ζ)

∂ζj+

pk=1

∂Φ(x(t1), . . . , x(tp), ζ)∂x(tk)

∂x(tk)∂ζj

+

pk=1

tk

tk−1

v⊤(t)∂ f (t)∂x

∂x(t)∂ζj

dt

+

pk=1

tk

tk−1

v⊤(t)∂ f (t)∂ζj

dt +

ml=1

pk=1

tk

tk−1

v⊤(t)∂ f (t)∂ xl

∂x(t − τl)

∂ζjdt − v⊤(t−p )

∂x(tp)∂ζj

−

p−1k=1

v⊤(t−k ) − v⊤(t+k )

∂x(tk)∂ζj

+ v⊤(0+)∂φ(0, ζ)

∂ζj+

pk=1

tk

tk−1

v⊤(t)∂x(t)∂ζj

dt.

Thus,

∂ J(τ, ζ)∂ζj

=∂Φ(x(t1), . . . , x(tp), ζ)

∂ζj+

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tp)− v⊤(t−p )

∂x(tp)∂ζj

+

p−1k=1

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tk)− v⊤(t−k ) + v⊤(t+k )

∂x(tk)∂ζj

+

tp

0

v⊤(t) + v⊤(t)

∂ f (t)∂x

∂x(t)∂ζj

dt

+

ml=1

tp

0v⊤(t)

∂ f (t)∂ xl

∂x(t − τl)

∂ζjdt +

tp

0v⊤(t)

∂ f (t)∂ζj

dt + v⊤(0+)∂φ(0, ζ)

∂ζj. (21)

We perform a change of variable in the second last integral term in (21) to obtain tp

0v⊤(t)

∂ f (t)∂ xl

∂x(t − τl)

∂ζjdt =

tp−τl

−τl

v⊤(t + τl)∂ f (t + τl)

∂ xl∂x(t)∂ζj

dt. (22)


Recall that x(t) = φ(t, ζ) for all t ≤ 0. Hence, from (22), tp

0v⊤(t)

∂ f (t)∂ xl

∂x(t − τl)

∂ζjdt =

0

−τl

v⊤(t + τl)∂ f (t + τl)

∂ xl∂φ(t, ζ)

∂ζjdt

+

tp−τl

0v⊤(t + τl)

∂ f (t + τl)

∂ xl∂x(t)∂ζj

dt. (23)

Substituting Eq. (23) into (21) gives,

∂ J(τ, ζ)∂ζj

=∂Φ(x(t1), . . . , x(tp), ζ)

∂ζj+

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tp)− v⊤(t−p )

∂x(tp)∂ζj

+

p−1k=1

∂Φ(x(t1), . . . , x(tp), ζ)

∂x(tk)− v⊤(t−k ) + v⊤(t+k )

∂x(tk)∂ζj

+

tp

0

v⊤(t) + v⊤(t)

∂ f (t)∂x

∂x(t)∂ζj

dt

+

ml=1

0

−τl

v⊤(t + τl)∂ f (t + τl)

∂ xl∂φ(t, ζ)

∂ζjdt +

ml=1

tp−τl

0v⊤(t + τl)

∂ f (t + τl)

∂ xl∂x(t)∂ζj

dt

+

tp

0v⊤(t)

∂ f (t)∂ζj

dt + v⊤(0+)∂φ(0, ζ)

∂ζj.

Choosing v = λ(·|τ, ζ) and substituting (6)–(8) into the above equation complete the proof of Eq. (19). �

3.3. Solving Problem (P)

On the basis of Theorems 1 and 2, we now present the following algorithm for computing the cost function (5) and itsgradient at a given admissible control pair (τ, ζ) ∈ T × Z.

Step 1. Solve the state system (1)–(2) from t = 0 to t = T to obtain x(·|τ, ζ).Step 2. Using x(·|τ, ζ), solve the impulsive system (6)–(8) from t = T to t = 0 to obtain λ(·|τ, ζ).Step 3. Using x(tk|τ, ζ), k = 1, . . . , p, compute J(τ, ζ) via Eq. (5).Step 4. Using x(·|τ, ζ) and λ(·|τ, ζ), compute ∂ J(τ,ζ)

∂τi, i = 1, . . . ,m and ∂ J(τ,ζ)

∂ζj, j = 1, . . . , r via Eqs. (9) and (19).

This algorithm can be integrated with a standard gradient-based optimization method (e.g. sequential quadraticprogramming) to solve Problem (P) as a nonlinear programming problem. The state system (1)–(2) evolves forwards in time(starting from an initial condition), while the auxiliary system (6)–(8) evolves backwards in time (starting from a terminalcondition). Thus, since the state and auxiliary systems evolve in opposite directions, and the auxiliary system depends onthe solution of the state system, these two systems cannot be solved simultaneously. Instead, the state system is solved firstin Step 1, and then the solution of the state system is used to solve the auxiliary system in Step 2. In practice, numericalintegration methods are used to solve the state and auxiliary systems. If, when solving the auxiliary system in Step 2, thevalue of the state vector is required at a point that does not coincide with one of the numerical integration knot points inStep 1, then an appropriate interpolationmethodmust be used (e.g. Hermite or Lagrange interpolation). The integrals in thegradient formulae (9) and (19) can be evaluated using standard numerical quadrature rules.

4. Application to parameter identification problems

4.1. Problem formulation

Consider the dynamic model (1)–(2). Suppose that τi, i = 1, . . . ,m and ζj, j = 1, . . . , r are unknown parameters thatneed to be identified. Furthermore, suppose that {(tk, yk)}

pk=1 is a given set of experimental data, where yk

∈ Rq is the systemoutput observed at sample time t = tk. Here, the output y(t) ∈ Rq is assumed to be a given function of the state and modelparameters:

y(t) = g(x(t|τ, ζ), ζ), t ∈ [0, T ], (24)

where g : Rn× Rr

→ Rq.


The aim is to choose appropriate values for the unknown parameters τi, i = 1, . . . ,m and ζj, j = 1, . . . , r so that thepredicted system output – obtained by solving (1)–(2) and (24) – best fits the experimental data. This leads to the followingparameter identification problem:

min(τ,ζ)∈T ×Z

pk=1

g(x(tk|τ, ζ), ζ) − yk2. (25)

This problem is clearly a special case of Problem (P). Hence, it can be solved using the computational approach outlined inthe previous section.

Remark 1. A similar (but less general) parameter identification problem was recently considered in Ref. [13]. Our newapproach has two important advantages over the method in [13].

(i) The method in [13] is only applicable to systems that can be expressed as the sum of nonlinear terms where eachnonlinear term contains a single delay and no unknown parameters. Our new approach is applicable to more generalsystems in which each nonlinear term can involve any number of delays and system parameters.

(ii) The method in [13] involves solving mn + n differential equations, where m is the number of unknown delays and n isthe number of state variables. Using the algorithm in Section 3.3, only 2n differential equations need to be solved. Thus,our approach is more efficient for systems with multiple delays. This makes it ideal for online applications in whichefficiency is paramount.

4.2. Example: continuously-stirred tank reactor

Ref. [13] gives the following dynamic equations for a continuously-stirred tank reactor with recycle stream:

x1(t) = −2x1(t) + 0.1(1 − x1(t)) exp

20x2(t)20 + 20x2(t)

+ x1(t − τ), (26)

x2(t) = −2.5x2(t) + 0.8(1 − x1(t)) exp

20x2(t)20 + 20x2(t)

+ x2(t − τ), (27)

and

x1(t) = 1, x2(t) = 1, t ≤ 0,

where x1 is the concentration of the solution; x2 is the temperature of the reactor; and τ is the unknown state delay to bedetermined. We assume that the terminal time is T = 10 and the system output is the reactor’s temperature. We generatethe observed data by sampling the output trajectory of the system with τ = 2 at the observation times t = tk = k/2,k = 1, . . . , 20. Thus, the state-delay identification problem for this system is: choose τ to minimize

J(τ ) =

20k=1

x2(tk|τ) − xk2

2,

where xk2 = x2(tk|2), k = 1, . . . , 20 are the observed output values at t = tk corresponding to τ = 2.The auxiliary impulsive system for this problem is

λ1(t) = 2λ1(t) + 0.1λ1(t) exp

20x2(t)20 + 20x2(t)

+ 0.8λ2(t) exp

20x2(t)

20 + 20x2(t)

− λ1(t + τ)

λ2(t) = −40λ1(t)(1 − x1(t))(20 + 20x2(t))2

exp

20x2(t)20 + 20x2(t)

+ 2.5λ2(t)

−320λ2(t)(1 − x1(t))

(20 + 20x2(t))2exp

20x2(t)

20 + 20x2(t)

− λ2(t + τ)

with jump conditions

λ1(t−k ) = λ1(t+k ), λ2(t−k ) = λ2(t+k ) + 2(x2(tk|τ) − xk2), k = 1, . . . , 20,

and boundary conditions

λ1(t) = 0, λ2(t) = 0, t ≥ 10.

According to Theorem 1, the derivative of J is

∂ J(τ )

∂τ= −

10

τ

{x1(t − τ)λ1(t) + x2(t − τ)λ2(t)} dt,

where x1 and x2 are given by the right-hand sides of (26) and (27).


Table 1State-delay estimate at the ith iteration for the example in Section 4.2.

Method i = 0 i = 4 i = 8τ 0 τ i J(τ i) τ i J(τ i)

Ref. [13] 1 1.8883 2.5747×10−3 2.0000 3.0413×10−14

Our method 1 1.8165 7.4069×10−3 2.0000 1.9334×10−13

Ref. [13] 3 2.1632 4.5562×10−3 2.0000 9.3648×10−11

Our method 3 1.9326 9.0169×10−4 2.0000 8.1544×10−13

To solve the parameter identification, we use a Matlab program that integrates the SQP optimization method with thegradient computation algorithm described in Section 3.3. Starting with the initial guesses τ = 1 and τ = 3, we recover theglobal optimal solution τ = 2 in both cases. The convergence history of our method and the method in [13] (taken fromSection 5 of [13]) are shown in Table 1. In this table, τ i and J(τ i) refer, respectively, to the value of τ and the value of thecost function at the ith iteration of the SQP optimization process (i = 0 refers to the initial guess). We see from Table 1 thatour method perfectly replicates the results in [13] (both methods recover the global solution for this example).

4.3. Example: zinc sulphate purification

Wenowdemonstrate the applicability of our approach to another realistic parameter identification problem. Specifically,we consider the industrial purification process described in [3,8]. In this process, zinc powder is added to a zinc sulphateelectrolyte to encourage deposition of harmful cobalt and cadmium ions. This is a key step in the production of zinc.

The concentrations of cobalt and cadmium ions in the electrolyte evolve according to the following differential equations:

x1(t) =1V

Qx01 − Qx1(t − τ) − α1u(t)x1(t − τ) + β1x2(t − τ)

, (28)

x2(t) =1V

Qx02 − Qx2(t − τ) − α2v(t)x2(t − τ) + β2x1(t − τ)

, (29)

and

x1(t) = 3.3 × 10−4, x2(t) = 4.0 × 10−3, t ≤ 0, (30)

where x1 is the concentration of cobalt ions; x2 is the concentration of cadmium ions; and u and v are control variables thatcorrespond to the amount of zinc powder added to the reaction tank. Furthermore, V is the volume of the reaction tank(V = 400); Q is the flux of solution (Q = 200); α1 and α2 are unknown model parameters; β1 and β2 are given modelparameters (β1 = 16.67, β2 = 710.7); and x01 and x02 are, respectively, the concentrations of cobalt and cadmium ions at theinlet of the reaction tank (x01 = 6 × 10−4, x02 = 9 × 10−3).

Ref. [8] considers system (28)–(30) with a given time-delay of τ = 2. Here, we suppose that τ is an unknown modelparameter that needs to be identified. We assume that the terminal time is T = 8. Furthermore, we set the input variablesu and v as equal to the optimal control functions obtained in [8]:

u(t) =

8l=1

σ l1χ[γl−1,γl)(t), t ∈ [0, 8], (31)

v(t) =

8l=1

σ l2χ[γl−1,γl)(t), t ∈ [0, 8], (32)

where the switching times γl and the control values σ l1 and σ l

2, l = 1, . . . , 8 are listed in Table 2.The system output is the concentration of cadmium ions:

y(t) = x2(t). (33)

Given system (28)–(30) and (33), and control input functions (31)–(32), our goal is to identify the model parameters α1and α2 and the state-delay τ . To generate the observed data for this parameter identification problem, we consider system(28)–(30) with the following data:

τ = τ = 2, α1 = α1 = 7.828 × 10−4, α2 = α2 = 2.823 × 10−4.

The corresponding output trajectory y(·|τ , α1, α2) = x2(·|τ , α1, α2) acts as our reference trajectory. Furthermore, we definethe sample times to be tk = k/2, k = 1, . . . , 16. Thus, the observed output is

yk = x2(tk|τ , α1, α2), k = 1, . . . , 16.


Table 2Control values and switching times for control functions (31) and (32).

l 1 2 3 4 5 6 7 8

γl 1 2 3 4 5 6 7 8σ l1 ×10−5 1.08 1.57 1.24 1.56 1.59 1.43 1.25 1.25

σ l2 ×10−5 5.20 4.70 4.97 4.60 4.53 4.64 4.74 4.62

Table 3Numerical convergence of the cost values for the example in Section 4.3.

Run Initial guess Cost value at ith iterationτ 0 α0

1 α02 i = 0 i = 10 i = 20 i = 50

1 0.0 0.0 0.0 9.264×10−5 1.514 × 10−6 3.690 × 10−9 2.525 × 10−11

2 0.5 0.5 0.5 7.360×1054 1.905 × 10−5 2.150 × 10−7 3.202 × 10−13

3 1.0 0.0 1.0 1.537×1020 1.330 × 10−7 9.813 × 10−10 1.290 × 10−10

4 1.0 1.0 1.0 3.392×1033 2.126 3.900 × 10−3 2.535 × 10−11

5 3.0 1.0 1.0 8.085×1013 4.841 × 10−6 7.072 × 10−9 8.882 × 10−11

Our parameter identification problem is now defined as follows: choose τ , α1, and α2 to minimize

J(τ , α1, α2) =

16k=1

y(tk|τ , α1, α2) − yk

2=

16k=1

x2(tk|τ , α1, α2) − x2(tk|τ , α1, α2)

2subject to the dynamic system (28)–(30).

Note that this problem cannot be solved using the identification method in [13], which is only applicable when eachnonlinear term in the system dynamics contains a single delay and no unknown parameters (the third term on the right-hand side of both (28) and (29) violates this requirement).

The auxiliary impulsive system corresponding to this problem is

λ1(t) =1V

[Qλ1(t + τ) + α1u(t)λ1(t + τ) − β2λ2(t + τ)]

λ2(t) =1V

[−β1λ1(t + τ) + Qλ2(t + τ) + α2v(t)λ2(t + τ)]


λ1(t−k ) = λ1(t+k ), λ2(t−k ) = λ2(t+k ) + 2(x2(tk|τ , α1, α2) − yk), k = 1, . . . , 16,


λ1(t) = 0, λ2(t) = 0, t ≥ 8.

Using Theorems 1 and 2, we obtain the following gradient formulae:

∂ J(τ , α1, α2)

∂τ= −

1V

8

τ

{[−Qλ1(t) − α1u(t)λ1(t) + β2λ2(t)] x1(t − τ)

+ [β1λ1(t) − Qλ2(t) − α2v(t)λ2(t)] x2(t − τ)} dt,

∂ J(τ , α1, α2)

∂α1= −

1V

8

0u(t)x1(t − τ)λ1(t)dt,

∂ J(τ , α1, α2)

∂α2= −

1V

8

0v(t)x2(t − τ)λ2(t)dt,

where x1 and x2 are given by the right-hand sides of (28) and (29).We solved the parameter identification problem using the same Matlab program that was used to solve the example in

Section 4.2. Computational results for different initial guesses are shown in Table 3. The convergence of the output trajectoryfor the initial guess τ = 3, α1 = 1, and α2 = 1 (run 5) is displayed in Fig. 1. This figure shows the output trajectory at twointermediate iterations of the optimization process, as well as the final (converged) trajectory. In Table 3 and Fig. 1, τ i, αi

1,and αi

2 are the values of τ , α1, and α2 at the ith iteration of the SQP optimization process (i = 0 refers to the initial guess).We see from Table 3 and Fig. 1 that the system trajectory converges quickly to the observed data, even when the initialtrajectory is far from the reference trajectory.


10×10–3

9

8

7

6

5

40 1 2 3 4 5 6 7 8

Fig. 1. Numerical convergence of the output trajectory for run 5 in Section 4.3.

5. Application to delayed feedback control

5.1. Problem formulation

Consider the following continuous-time control system:

x(t) = f (x(t), u(t)), t ∈ [0, T ], (34)x(t) = φ(t), t ≤ 0, (35)

where x(t) ∈ Rn is the state and u(t) ∈ Rr is the control input. Undelayed systems such as (34)–(35) are usually mucheasier to control than time-delay systems. Nevertheless, it has been shown that introducing delays to an undelayed systemcan be beneficial, especially for chaotic systems [14,16,20].

Delayed feedback control is one way of deliberately introducing delays to an undelayed system. In delayed feedbackcontrol, the control function u(t) is defined as follows:

u(t) = K0x(t) + K1x(t − τ1) + · · · + Kdx(t − τd), (36)

where Ki ∈ Rr×n, i = 0, . . . , d are feedback gain matrices and τi, i = 1, . . . , d are time-delays. Substituting (36) into(34)–(35) yields the following closed-loop system:

x(t) = f (x(t), x(t − τ1), . . . , x(t − τd), ξ) , t ∈ [0, T ], (37)

x(t) = φ(t), t ≤ 0, (38)

where ξ ∈ Rrn(d+1) is a vector containing the elements of the feedback gain matrices and

f (x(t), x(t − τ1), . . . , x(t − τd), ξ) = f (x(t),K0x(t) + K1x(t − τ1) + · · · + Kdx(t − τd)) .

The aim here is to choose the delays and feedback gain matrices in (36) to stabilize the closed-loop system (37)–(38). Thus,we consider the following optimization problem:

minτ,ξ

|x(T ) − x∗|2,

where x(·) is the solution of (37)–(38) and x∗ is a desired equilibrium point. This problem can be solved effectively using thecomputational approach outlined in Section 3.

5.2. Example 1: inverted pendulum

We consider the problem of controlling the position of a single-link rotational joint in robotics (a type of inverted pen-dulum system). The dynamics of the rotational joint are described as follows:

y(t) −gLy(t) = u(t), t ∈ [0, T ], (39)

with initial conditions

y(t) = 1, y(t) = 0, t ≤ 0, (40)

where y denotes the angular displacement of the inverted pendulum, g is the acceleration due to gravity (g = 9.8 ms−2), Lis the length of the pendulum (L = 0.4 m), and u is the external torque force.


In the absence of velocity measurements, the inverted pendulum system is difficult to stabilize using position feedbackcontrol [20]. Thus, it is necessary to instead consider the following delayed feedback controller:

u(t) = k1y(t − τ1) + k2y(t − τ2), (41)

where τ1 and τ2 are position delays, and k1 and k2 are parameters. We use the same values for k1 and k2 as given in [20]:

k1 = −63.73, k2 = 36.76. (42)

The second-order system (39)–(40), with u defined by (41), can be easily transformed into the following systemof first-orderdifferential equations:

x1(t) = x2(t), t ∈ [0, T ], (43)

x2(t) = k1x1(t − τ1) + k2x1(t − τ2) +gLx1(t), t ∈ [0, T ], (44)


x1(t) = 1, x2(t) = 0, t ≤ 0. (45)

Exponential stability conditions for system (43)–(44) were established in [20]. Here, we apply the computational methoddescribed in Section 3 to determine optimal values for the position delays so that the system becomes stable at the origin.Our optimal control problem can be stated as follows: given system (43)–(44) with initial conditions (45) and parametervalues (42), choose the position delays τ1 and τ2 to minimize the objective function

J = x1(T )2 + x2(T )2, (46)

where the terminal time T is chosen to be 20 s. The auxiliary impulsive system for this problem is

λ1(t) = −gLλ2(t) − k1λ2(t + τ1) − k2λ2(t + τ2), t ∈ [0, 20],

λ2(t) = −λ1(t), t ∈ [0, 20],


λ1(20−) = λ1(20+) + 2x1(20), λ2(20−) = λ2(20+) + 2x2(20),


λ1(t) = 0, λ2(t) = 0, t ≥ 20.

Using Theorems 1 and 2, the gradient formulae for this problem are

∂ J∂τ1

= −

20

τ1

k1λ2(t)x1(t − τ1)dt = −

20

τ1

k1λ2(t)x2(t − τ1)dt,

∂ J∂τ2

= −

20

τ2

k2λ2(t)x1(t − τ2)dt = −

20

τ2

k2λ2(t)x2(t − τ2)dt.

As in Section 4, we solved this problem using a Matlab program that implements the computational approach describedin Section 3.3. The optimal time-delays are τ1 = 0.1134 and τ2 = 0.2458. To compare, Ref. [20] reports optimal time-delaysof τ1 = 0.143 and τ2 = 0.286. Fig. 2 shows the angular displacement under our optimal feedback controller compared withthe optimal feedback controller in [20]. Note that our controller stabilizes the system more quickly with less oscillationsthan the controller in [20].

5.3. Example 2: Chen chaotic system

We now consider the problem of stabilizing the so-called disturbed Chen chaotic system, which is defined as follows:

x(t) =

−θ1 θ1 0

θ2 − θ1 θ2 00 0 −θ3

x(t) +

0−x1x3x1x2

+ ω(t) + u(t), t ∈ [0, T ], (47)


x(0) = [2, −3, 1]⊤, t ≤ 0, (48)

where ω(t) is a bounded exogenous disturbance and θ1, θ2, and θ3 are model parameters. Here, we assume that thedisturbance and model parameters are as given in [28]:

ω(t) = [0.2x1(t), −0.2x2(t), −0.2x3(t)]⊤, θ1 = 1, θ2 = 2, θ3 = 3. (49)


Fig. 2. Optimal angular displacement for the closed-loop inverted pendulum system.

Our aim is to stabilize the chaotic system (47)–(48) at the origin. Thus, the objective function is

J = |x(T )|2, (50)where the terminal time is T = 0.5. We design a delayed feedback controller in the following form:

u(t) = [k1x1(t − τ), k2x2(t − τ), k3x3(t − τ)]⊤, (51)where k1, k2, k3 are feedback gains and τ is the state-delay. Our optimal control problem can be stated as follows: given thesystem (47)–(48), with disturbance and parameters values defined by (49), and the feedback control (51), choose the state-delay and the feedback gains to minimize the objective function (50). The auxiliary impulsive system for this problem is

λ1(t) = (θ1 − 0.2)λ1(t) − (θ2 − θ1 − x3(t))λ2(t) − x2(t)λ3(t) − k1λ1(t + τ), t ∈ [0, 0.5],λ2(t) = −θ1λ1(t) − (θ2 − 0.2)λ2(t) − x1(t)λ3(t) − k2λ2(t + τ), t ∈ [0, 0.5],λ3(t) = x1(t)λ2(t) + (θ3 + 0.2)λ3(t) − k3λ3(t + τ), t ∈ [0, 0.5],

with jump conditionsλ1(0.5−) = λ1(0.5+) + 2x1(0.5),λ2(0.5−) = λ2(0.5+) + 2x2(0.5),λ3(0.5−) = λ3(0.5+) + 2x3(0.5),

and boundary conditionsλ1(t) = 0, λ2(t) = 0, λ3(t) = 0, t ≥ 0.5.

By Theorems 1 and 2, the gradient formulae for this problem are∂ J∂τ

= −

0.5

τ

{k1λ1(t)x1(t − τ) + k2λ2(t)x2(t − τ) + k3λ3(t)x3(t − τ)} dt,

∂ J∂k1

=

0.5

0λ1(t)x1(t − τ)dt,

∂ J∂k2

=

0.5

0λ2(t)x2(t − τ)dt,

∂ J∂k3

=

0.5

0λ3(t)x3(t − τ)dt,

where x1, x2, and x3 are given by the right hand side of (47).We solved this problem using the same Matlab program that was used to solve the examples in Sections 4 and 5.2. The

optimal delayed feedback control is

u(t) = [−47.08x1(t − 0.0076), −46.31x2(t − 0.0076), −46.40x3(t − 0.0076)]⊤. (52)Using the MISER optimal control software [29], we also computed the optimal undelayed feedback control:

u(t) = [−45.47x1(t), −61.84x2(t), −20.64x3(t)]⊤. (53)The optimal state variables under controls (52) and (53) are shown in Fig. 3. Note that for this system, delayed feedbackcontrol stabilizes the system quicker than the traditional feedback control.


a

b

c

Fig. 3. Optimal states of the Chen chaotic system in Section 5.3.

6. Conclusion

In this paper, we have considered a novel optimal control problem in which the delays in a nonlinear time-delay systemare control variables to be determined optimally. Such problems, which are called optimal state-delay control problems,arise in parameter identification and delayed feedback control. Our main contribution is a new computational methodfor determining the gradient of the cost function in an optimal state-delay control problem. This method requires lessnumerical integration than the existing method in [13], and is therefore faster in many situations. Furthermore, unlikethe method in [13], our new method is applicable to systems with nonlinear terms containing more than one state-delay.We have restricted our attention in this paper to systems with time-invariant (constant) time-delays. Our future work willinvolve combining the techniques in this paper with the control parameterization method [30,31] to solve optimal state-delay control problems with time-varying delays. Such problems arise in the control of crushing processes [9] and mixingtanks with recycle loops [32]. Another interesting topic for further researchwould be to extend the parameter identificationapproach proposed in Section 4 to adaptively identify the delays as further information about the system become available.

Acknowledgments

This work was supported by the Australian Research Council (under grant DP110100083), the China National ScienceFund forDistinguishedYoung Scholars (under grant 61025015), and theNational Natural Science Foundation of China (undergrant 61174133).

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A class of optimal state-delay control problems