Robust Control of a Class of Time-Delay Nonlinear Processes

Ind. Eng. Chem. Res. 2006, 45, 8963-8972

8963

PROCESS DESIGN AND CONTROL Robust Control of a Class of Time-Delay Nonlinear Processes Yi-Shyong Chou* and Kuen-Tsuen Jih Department of Chemical Engineering, National Taiwan UniVersity of Science & Technology, Taipei 106, Taiwan

In this paper, the problem of designing a tracking controller for uncertain nonlinear state-delay systems that can suppress the effects of both unknown uncertainties and disturbances is investigated. The controller is designed by using the sliding-mode control concept and the polynomial approximation method. One of the features in this paper is that model uncertainties and state-delay terms are expressed as the Legendre polynomials expansion. The expansion coefficients of the Legendre polynomials can furthermore be modified by the update law derived from the Lyapunov stability theorem. The presented composite nonlinear controller, which consists of a sliding-mode controller with a coefficient update law and a sliding-mode observer, can achieve offsetfree performance. The major advantage of the proposed control system design is to track the set point without producing a vigorous control action and requiring the exact knowledge of model uncertainties. The control scheme is illustrated by an example of a chemical reactor with delayed states. Simulation results indicate that the proposed method can work for processes with time delays, despite unknown modeling uncertainties. 1. Introduction For the past decade, the analysis and control synthesis of timedelay processes has been one of the most active research areas in process system engineering. This is because time delays are frequently encountered in chemical processes. The existence of time delays degrades the control performance and sometimes makes the closed-loop stabilization difficult. Although a number of significant time-delay compensation techniques have been proposed to improve the control performance of linear timedelay processes, only a few time-delay compensation strategies have been developed for nonlinear state-delay processes involving unmodeled dynamics or unknown uncertainties.1 There are many chemical processes that are governed by nonlinear statedelay equations. Examples are recycled reactors, recycled storage tanks, cold rolling mills, and so forth.2 Shyu and Yan3 and Luo and de la Sen4 employed the technique of variable structure control to deal with the stabilizing of uncertain processes with delayed states. Velasco et al.5 proposed a linear approximation of the nonlinear model around an equilibrium point and a causal static-state feedback law to deal with the disturbance decoupling problem of nonlinear time-delay processes. Hu et al.6 introduced the linear matrix inequality technique and the sliding-mode control method to solve the control design problem of uncertain time-delay processes with mismatching uncertainties. Oguchi et al.7 presented a coordinate transformation involving the delayed states to deal with nonlinear processes with multiple time delays in the state. Lehman et al.8 developed the vibrational control technique for nonlinear time-lag processes with arbitrarily large but bounded delay. Niu et al.9 proposed a neural-network-based sliding-mode control scheme to solve the control problem of state-delay processes with mismatched parameter uncertainties, unknown nonlinearities, and external disturbances. Mounier and Rudolph10 * To whom all correspondence should be addressed. Fax: +8862-27376644. E-mail: [email protected].

introduced the concept of flatness to derive the trajectory tracking control law for nonlinear processes with time delays. In all of the above methods, the admissible nonlinear uncertainties, disturbances, and time delays are usually assumed to be known a priori or to be satisfied with certain conditions. However, a time delay in the state is often not measured and the bounds of uncertainties and disturbances are difficult to estimate. Control of nonlinear systems with unknown uncertainties has been a challenging problem. Slotine and Coetsee11 proposed an adaptive sliding control to reduce the effect of uncertainties on the process and further improve performance by on-line parameter estimation algorithms. In their study, there is no need to know the bounds of unknown parameters. Huang and Kuo12 presented a sliding control scheme for nonlinear processes with unknown bound time-varying uncertainties. Those uncertainties were represented in finite-term Fourier functions. Vecchio et al.13 developed a repeatable control design method for a class of feedback linearizable processes with an unknown nonlinear function. They used the Fourier series to approximate the nonlinear function of the process. In the previously mentioned papers, all uncertain elements that appeared in the process were lumped and expanded in finite-term Fourier functions. This requires us to choose sufficiently large terms of Fourier functions to approximate uncertainties. The aim of this paper is to design a sliding controller for nonlinear state-delay processes with time-varying uncertainties, despite unknown magnitudes on time delays and model uncertainties. The key idea of the paper is to use orthogonal polynomials to represent model uncertainties and delayed states. Because of the fact that the convergence of orthogonal polynomials is fast, one only needs to calculate a few terms of coefficients. This allows us to improve the transient performance. Once the unknown uncertainties and state-delay terms have been approximated and parametrized, a feedback-control law and an update law for adjusting the parameters of orthogonal

10.1021/ie0512901 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/14/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006

polynomials can be designed by using the Lyapunov stability theory. The sliding-mode control structure is the backbone of the proposed control strategy development. The presented composite nonlinear controller, which consists of a sliding-mode controller with a coefficient update law and a sliding-mode observer, can achieve offset-free performance. Furthermore, we discuss the stability of the closed-loop process. Finally, the effectiveness of the proposed method is demonstrated by illustrated simulations. The scope of this paper covers a class of time-delay nonlinear processes. It is not a general approach to all nonlinear problems. The class of problems is frequently encountered in process control of recycled reactors, recycled storage tanks, cold rolling mills, and so forth.

The Legendre polynomials have the following important recurrence property

(n + 1)Pn+1(t) ) (2n + 1)P1(t)Pn(t) - nPn-1(t) (10) The first several Legendre polynomials are given by

P0(t) ) 1 b+a 2t b-a b-a

(12)

3 2t b+a2 1 2b-a b-a 2

(13)

P1(t) ) P2(t) )

(11)

[

]

3. Problem Formulation 2. Function Approximation with Orthogonal Polynomials Let us suppose that a function f(t) defined on (a, b) is to be represented as a series of orthogonal polynomials {Φn(t)}, n ) 0, 1, 2, ‚‚‚, that is, ∞

f(t) )

∑ cnΦn(t), a < t < b n)0

(1)

The values cn are the coefficients. Define

z(t) ) [Φ0(t) Φ1(t) ‚‚‚ Φn(t)]T

(2)

W ) [c0 c1 ‚‚‚ cn]T

(3)

Consider a class of dynamic processes that may be modeled by

x(n) ) ft(X(t), X(t - τ)) + gt(X(t), X(t - τ))u(t) + D(t) (14) where X ) [x x(1) ‚‚‚ x(n-1)]T is the process state vector, ft(X(t), X(t - τ)) and gt(X(t), X(t - τ)) are nonlinear functions, τ > 0 is a delay time in the process state, u(t) is the control input, and D(t) is the disturbance. Generally, a large class of chemical processes can be represented with this type of model structure. Examples are recycled reactors, recycled storage tanks, cold rolling mills, and so on.2 Most of the recycling processes inherit delays in their state equations. Equation 14 also can be expressed in companion form,16 i.e.,

∞

E(t) )

∑ clΦl(t) l)n+1

x˘ 1 ) x2

(4)

x˘ 2 ) x3 l

Then eq 1 can be rewritten as

f(t) ) WTz(t) + E(t)

(5)

In applications, f(t) is approximated by a linear combination n

fn(t) )

∑akΦk(t)

(6)

k)0

and the mean-square error is defined as the form of

En )

∫abw(t)[f(t) - fn(t)]2 dt

(7)

where w(t) is the weight function. If the mean-square error approaches zero as n becomes infinite, the sequence {fn(t)} converges in the mean to f(t).14 In other words, as long as n is large enough, f(t) can be approximated as

f(t) ≈ W z(t) T

(8)

+∞ Note that f(t) must be in L2, that is, means ∫-∞ (f(x)2) dx < ∞. It is well-known that many orthogonal polynomials are frequently used to approximate the solution of time-varying systems because of their computation convergence advantages. In this paper, the Legendre polynomials expansion is demonstrated to deal with unknown uncertainties and delayed states. The Legendre polynomials are defined in a finite interval [a, b] by15

Pn(t) )

dn(t - a)n(t - b)n 1 n!(b - a)n dtn

(9)

x˘ n ) ft(X(t), X(t - τ)) + gt(X(t), X(t - τ))u(t) + D(t) (15) The control objective is to design a control law u to make x(t) track xd(t) in the presence of unknown modeling errors in ft(X(t), X(x - τ)) and gt(X(t), X(t - τ)) and unknown disturbances, D(t). The sliding-mode design approach consists of two components.16 The first one involves the design of a switching function so that the sliding motion satisfies design specifications. The second one is concerned with the selection of a control law which will make the switching function attractive to the system state. Thus, the first step is to define a time-varying surface s(t) in the state-space as s(X,t) ) 0 that is a differential operator acting on some error function

s)

(dtd + γ)

n-1

e(t)

(16)

e(t) ) x(t) - xd(t)

(17)

where xd(t) is the desired trajectory and γ is a strictly positive constant, determining the performance of the system on the sliding surface. The control goal is to force the process state x(t) to follow the specified trajectory xd(t). The problem is equal to design a control law u(t) to ensure

lim e(t) ) 0 tf∞

(18)

This condition can be achieved by making the system trajectory converge to the sliding surface. Thus, the problem of tracking

Ind. Eng. Chem. Res., Vol. 45, No. 26, 2006 8965

{ () () {

trajectory can be reduced to keep s at zero.

(dtd + γ)

n-1

e(t) ) 0

4. Controller Design

s)

(dtd + γ)

n-1

e(t)

(20)

The derivative of the sliding surface is calculated as follows,

s˘ ) x

(n)

- xd

(n)

+ Ω(e)

(21)

Ω(e) )

∑

(n - 1)!γn-k

k)1(n

e

(k)

(22)

- k)!(k - 1)!

(23)

gt(X(t), X(t - τ)) ) g(X(t))∆g(X(t), X(t - τ))

(24)

This assumption decomposes ft(X(t), X(t - τ)) and gt(X(t), X(t - τ)), respectively, into two parts, where f(X(t)) and g(X(t)) are referred to as the known functions whereas ∆f(X(t), X(t τ)) and ∆g(X(t), X(t - τ)) can be viewed as the time-varying uncertainties that are unknown functions of time. Here, ∆g(X(t), X(t - τ)) is, in general, referred to as the matched uncertainty, which is assumed to be bounded as below:

0 < ∆gmin e ∆g(X(t), X(t - τ)) e ∆gmax

[

(∆H - ∆H ˆ ) - ηsat

s˘ ) (f + ∆F) + g∆gu - xd(n) + Ω(e)

∆H ) ∆F∆g-1 ∆g-1

Assume that ∆H and can be approximated as

u)

s 1 -∆H ˆ - ∆gˆ -1(f - xd(n) + Ω(e)) - ηsat g φ

[

( )] (27)

The thickness of the boundary layer φ is defined to be a positive real scalar. Outside the boundary layer, |s| g φ, the sliding condition will be used to specify the s dynamics. Inside the boundary layer, |s| < φ, the control law will be modified to impose a smoothing process to the s dynamics. The tuning gain η is a strictly positive constant, and ∆H ˆ and ∆gˆ -1 are the estimates of the unknown functions ∆F∆g-1 and ∆g-1, respectively. The sat is the saturation function

(31)

are unknown bounded functions and

∆H ≈ WHTzF(t)

(32)

∆g-1 ≈ WgTzF(t)

(33)

zF(t) ) [P0(t) P1(t) ‚‚‚ Pn(t)]T

(34)

WH ) [wH0 wH1 ‚‚‚ wHn]T

(35)

Wg ) [wg0 wg1 ‚‚‚ wgn]T

(36)

where

Here, zF is the Legendre polynomials vector. WH and Wg are considered as the constant parameter vectors. Generally the number n should be properly selected so that the approximate errors are tolerable. ∆H ˆ and ∆gˆ -1 are the estimates of the unknown functions ∆H and ∆g-1, respectively, and can be represented as

∆H ˆ ≈W ˆ HTzF(t)

(37)

ˆ gTzF(t) ∆gˆ -1 ≈ W

(38)

W ˆ H ) [wˆ H0 wˆ H1 ‚‚‚ wˆ Hn]T

(39)

W ˆ g ) [wˆ g0 wˆ g1 ‚‚‚ wˆ gn]T

(40)

where

(26)

where ∆F ) ∆f + D. To make the value of s˘ equal to zero, we consider the following control input of the form

(φs )] (30)

where

(25)

∆f(X(t), X(t - τ)) can be viewed as the mismatched uncertainty. With the aid of eqs 23 and 24, by substituting eq 14 into eq 21, one obtains

(29)

s˘ ) ∆g (f - xd(n) + Ω(e))(∆g-1 - ∆gˆ -1) +

Here, we assume that the functions ft(X(t), X(t - τ)) and gt(X(t), X(t - τ)) can be represented as17

ft(X(t), X(t - τ)) ) f(X(t)) + ∆f(X(t), X(t - τ))

s +1 if > 0 s φ sgn ) s φ -1 if < 0 φ Substituting eq 27 into eq 26 gives

where n-1

(28)

()

(19)

The first step in controller design is to select a feedback control law u(t). The goal is to reach the sliding surface and remain on it. The second step is to estimate unknown functions in ft(X(t), X(t - τ)) and gt(X(t), X(t - τ)) by employing orthogonal polynomials. Classical orthogonal polynomials serve as an excellent tool for modeling processes in an approximative way. Here we select the Legendre polynomials in order to improve the accuracy of approximation of unknown functions. Recall eq 16,

s s if | | < 1 s φ φ sat ) s s φ if | | g 1 sgn φ φ

The update laws for adjusting the parameters W ˆ H and W ˆ g will be derived. Substituting from eqs 32 to 40 into eq 30 yields

[

s˘ ) ∆g (f - xd(n) + Ω(e))(W ˜ gTzF(t)) + W ˜ HTzF(t) - ηsat

(φs )] (41)

where

ˆH W ˜ H ) WH - W

(42)

ˆg W ˜ g ) Wg - W

(43)

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Consider the Lyapunov function candidate as12

1 1 ˜ HTQHW V(s, W ˜ H, W ˜ g) ) s2 + ∆g[W ˜H+W ˜ gTQgW ˜ g] 2 2

(44)

where QH and Qg are symmetric positive definite matrices. The derivative of V is

˜˙ H + W ˜ gTQgW ˜˙ g] V˙ ) ss˘ + ∆g[W ˜ HTQHW

(45)

Since WH and Wg are the constant vectors, W ˜˙ H ) -W ˆ˙ H and W ˜˙ g ˙ ) -W ˆ g, the above equation can be further expressed as

ˆ˙ H) + W ˜ gTQg(- W ˆ˙ g)] V˙ ) ss˘ + ∆g[W ˜ HTQH(-W

(46)

Substituting eq 41 into 46 and considering within the boundary layer, we obtain

[

˜ Tg(zFs(f - xd(n) + Ω(e)) - QgW ˆ˙ g) + V˙ ) ∆g W

]

2

s ˆ˙ H) - η (47) W ˜ HT(zFs - QHW φ Choose the update laws as

W ˆ˙ g ) Qg-1zFs(f - xd(n) + Ω(e))

(48)

W ˆ˙ H ) QH-1zFs

(49)

Then the derivative of V becomes

s2 φ

V˙ ) -∆gη

Figure 1. (a) Process responses and (b) controller responses.

Choose η ) η1/∆gmin, η1 > 0. Then

s2 s2 V˙ ) -∆gη e -η1 e 0 φ φ

(50)

This implies that s is bounded. The parameter vectors W ˜ H and W ˜ g are also bounded. To use Barbalat’s lemma,18 let us check the uniform continuity of V˙ . The derivative of V˙ is

2η ss˘ φ

V¨ ) -∆g

(51)

This shows that V¨ is bounded, since each term in s˘ (eq 41) is bounded, and s, W ˜ H, and W ˜ g were shown above to be bounded. Hence, V˙ is uniformly continuous. Application of Barbalat’s lemma then indicates that s f 0 as t f ∞. With the aid of the update laws eqs 48 and 49, the unknown functions, ∆H and ∆g, can be estimated and the control law eq 27 can be implemented. Notice that ∆H and ∆g can be approximated by a sum of n terms of orthogonal polynomials; they must be bounded in real applications. 5. Sliding Observer The previously described design method provides a dynamic feedback that effectively deals with unknown uncertainties and asymptotically tracks the system if all the states are available. If they cannot be measured, then a state estimator is necessary. Sliding observer is a high-performance state estimator well-suited for nonlinear uncertain systems with partial

state feedback. The basic sliding observer structure includes switching terms added to the design model. Here, we adopt the basic sliding observer as a state estimator that is added to the previous control scheme. A brief review of sliding observer has been presented in the literature.19 Here, we consider the state-space representation of nonlinear systems of the form

x˘ 1 ) f1(x1, x2,‚‚‚, xn, u) x˘ 2 ) f2(x1, x2,‚‚‚, xn, u) l x˘ n ) fn(x1, x2,‚‚‚, xn, u)

(52)

where [x1 x2 ‚‚‚ xn]T is the state-space vector and x1 is assumed to be the only measurable state. [f1 f2 ‚‚‚ fn]T is a nonlinear unknown function. Now, based on the sliding approach, the observer dynamics can be written as

xˆ˘ 1 ) ˆf 1(xˆ 1, xˆ 2,‚‚‚, xˆ n, u) - k1sat

() () x˜ 1 φ

xˆ˘ 2 ) ˆf 2(xˆ 1, xˆ 2, ‚‚‚, xˆ n, u) - k2sat

x˜ 1 φ

l xˆ˘ n ) ˆf n(xˆ 1, xˆ 2, ‚‚‚, xˆ n, u) - knsat

() x˜ 1 φ

(53)


6. Numerical Example In this section, simulations of the control in an uncertain nonlinear time-delay system are performed by using the proposed sliding control method. Consider the first-order, irreversible, exothermic reaction A f B, carried out in a wellmixed, continuously stirred tank reactor.8 Suppose that fresh feed of pure A is mixed with a recycled stream of unreacted A with recycle flow rate Fr(1 - λ). Notice that no recycled stream occurs in the limit of λ to 1. If there is a delay, τr, in the recycle stream, then the material and energy balances become8

dCA(t*) Frλ Fr(1 - λ) Fr ) CAf + CA(t* - τr) - CA(t*) dt* Vr Vr Vr E C (t*) (55) k0 exp RTr A

( )

dTr(t*) Frλ Fr(1 - λ) Fr ) T + Tr(t* - τr) - Tr(t*) + dt* Vr rf Vr Vr UAr (-∆H) E k exp C (t*) (T (t*) - Tj(t*)) FrCpr 0 RTr A FrCprVr r (56)

( )

UAr dTj(t*) Fj (T (t*) - Tj(t*)) ) (Tjf - Tj(t*)) + dt* Vj FjCpjVj r

(57)

Typically, the above equations are reduced to dimensionless form using the notation Figure 2. (a) Process responses and (b) controller responses.

ψ)

where {k1, k2, ‚‚‚, kn} are positive numbers, x˜ 1 ) xˆ 1 - x1 is the estimation error, and ˆf(xˆ , u) is the estimated value of f(xˆ , u). By using eqs 52 and 53, the resulting error dynamics can be written as

x˜˘ 1 ) δf1 - k1sat x˜˘ 2 ) δf2 - k2sat

() () x˜ 1 φ x˜ 1 φ

() x˜ 1 φ

βr ) βc )

UAr τ FrCprVr 0

UAr (-∆H)CAf Tjf - Trf τ0, B ) ψ, x3f ) ψ, FjCpjVj FrCprTrf Trf x1 ) 1 -

x2 )

l x˜˘ n ) δfn - knsat

Vr t* E , t ) , Da ) k0τ0 exp(-ψ), , τ0 ) RTrf Frλ τ0

CA CAf

Tr - Trf Tj - Trf Fj τr ψ, x3 ) ψ, u ) τ0, τs ) Trf Trf Vj τ0

Equations 55-57 in dimensionless variables become

(54)

where δf ) ˆf(xˆ , u) - f(x, u). The more sophisticated estimation techniques can be applied to improve state estimation; however, this is not the main concern in this paper. Here, we adopt the basic sliding observation method to estimate the process states without time delays. Certainly there exists a discrepancy between the actual state and the estimated state. This will degrade the performance of tracking. The merit of this paper is that we decompose ft(X(t), X(t - τ)) and gt(X(t), X(t - τ)), respectively, into two parts, where f(X(t)) and g(X(t)) are, in general, referred to as the known parts, whereas ∆f(X(t), X(t - τ)) and ∆g(X(t), X(t τ)) are viewed as the unknown parts, which can be compensated by the proposed controller.

1 1 x˘ 1(t) ) - x1(t) + - 1 x1(t - τs) + λ λ

(

)

Da(1 - x1(t)) exp

{ } x2(t)

x2(t) 1+ ψ

(58)

1 1 x˘ 2(t) ) - x2(t) + - 1 x2(t - τs) + λ λ x2(t) - βr(x2(t) - x3(t)) (59) BDa(1 - x1(t)) exp x2(t) 1+ ψ

(

)

{ }

x˘ 3(t) ) βc(x2(t) - x3(t)) + u(x3f - x3(t))

(60)

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Following the proposed method, the sliding surface can be expressed as

s ) e˘ (t) + γe(t) ) (x˘ 2(t) - x˘ d(t)) + γ(x2(t) - xd(t))

(65)

and then the derivative of s can be computed as

s˘ ) f + ∆f + g∆gu + D - x¨d(t) where

[

f ) -x2(t) + BDa(1 - x1(t)) exp

]( (

(

x3(t)) λ + BDa

1 - x1(t)

)

(66)

{ } { } ) { }) x2(t)

x2(t) 1+ ψ x2(t)

- βr(x2(t) -

- 1 - βr x2(t) 1+ ψ x2(t) -x1(t) + Da(1 - x1(t)) exp × x2(t) 1+ ψ x2(t) BDa exp - βrβc(x2(t) - x3(t)) - λx˘ d(t) (67) x2(t) 1+ ψ x2(t) 1+ ψ

2

( { }) ∆f + D ) ∆F

exp

( ( { } ) { }

) (∆x2(t, t - τs) + d) λ + BDa

exp Figure 3. (a) Process responses and (b) controller responses.

The main control objective for the reactor would be the regulation of the product concentration around a desired setpoint value. The close relationship between the concentration of component A and the temperature enables us to choose the regulation of the reactor outlet temperature about a corresponding set-point value as our control objective. The coolant flow rate is considered here as the manipulated variable. 6.1. Controller Design. To further conform to the design forms, the above equations are rearranged as

1 1 x˘ 1(t) ) - x1(t) + - 1 x1(t) + λ λ x2(t) + ∆x1(t, t - τs) (61) Da(1 - x1(t)) exp x2(t) 1+ ψ

(

)

{ }

1 1 x˘ 2(t) ) - x2(t) + - 1 x2(t) + BDa(1 - x1(t)) exp λ λ x2(t) - βr(x2(t) - x3(t)) + ∆x2(t, t - τs) + d(t) (62) x2(t) 1+ ψ

{ }

(

)

x˘ 3(t) ) βc(x2(t) - x3(t)) + u(x3f - x3(t))

(63)

y ) x2(t)

(64)

where d(t) is added to the system to represent the process disturbance.

x2(t)

x2(t) 1+ ψ

1 - x1(t)

)

x2(t) 1+ ψ

2

×

- 1 - βr +

∆x1(t, t - τs)BDa exp

x2(t)

+ x2(t) 1+ ψ ∆x˘ 2(t, t - τs) + d˙ (68)

g ) βr(x3f - x3(t))

(69)

∆g ) 1

(70)

The unknown function ∆F is assumed as a bounded continuous function of time and is approximated by the Legendre polynomials. The control law is

u)

1 s -W ˆ HTzF - W ˆ gTzF(f - x¨d) - ηsat g φ

[

( )]

(71)

W ˆ˙ g ) Qg-1zFs(f - x¨d)

(72)

W ˆ˙ H ) QH-1zFs

(73)

The parameters for all simulations are selected to be λ ) 0.5, B ) 8, Da ) 0.135, ψ ) 20, βc ) 1.5, βr ) 1.5, x3f ) -4, uin ) 0.97, x1in ) 0.65, x2in ) 3, and x3in ) 0.26, where the subscript {in} denotes the initial state. The controller parameters are selected to be γ ) 1.5, η ) 1.2, and Qg ) QH ) 0.1 × I, where I denotes the identity matrix with a proper dimension. The thickness of boundary layer φ is selected as 0.2 during the course of the simulation.




6.1.1. Case 1. In this case, all the states are available. Figure 1a shows that the proposed controller, the nonlinear controller based on the extension of the Lie derivative developed by Oguchi et al.7 (the new input V ) -4z1 - z2 is designed), and the nonlinear model predictive control (MPC)20 are compared for the process having a time delay τs ) 2 in the state and a set-point change from x2 ) 3 to x2 ) 4. The nonlinear MPC is designed with these parameters: the sampling interval is 0.01, the prediction horizon is 30, and the control horizon is 10. The controller moves are set within the range -2 e u e 1.2. The nonlinear constrained optimization problem is solved by using the MATLAB Optimization Toolbox routine FMINCON. The nonlinear controller based on the extension of the Lie derivative produces a large overshoot. The simulation results show that nonlinear MPC exhibits a response with the offset to the set point within the specified control constraints. By contrast, our proposed controller provides a rapid, smooth, and conservative response. Such a response is much more suitable for chemical process control applications. Figure 1b shows the input moves produced by the corresponding controllers. This result indicates that the superior performance of the proposed controller is not attributable to more aggressive control action but rather to a more judicious use of the input. Figure 2a shows that the proposed controller, the nonlinear controller based on the extension of the Lie derivative, and the nonlinear MPC are compared for the process having a time delay τs ) 2 in the state, a set-point change from x2 ) 3 to x2 ) 4, and an oscillatory disturbance d ) 0.1 sin(t). As before, the performance of the proposed controller shows less deviation from the set point.

Figure 2b shows the input moves produced by the corresponding controllers. Because of the additional oscillatory disturbance, the controller moves of nonlinear MPC display the chattering behavior. Figure 3a shows the simulation results of the process having a set-point change from x2 ) 3 to x2 ) 4 and a time delay τs ) 2 in the state for t e 10 and a τs ) 5 in the state for t > 10, with the proposed controller, the nonlinear controller based on the extension of the Lie derivative, and the nonlinear MPC. Figure 3b shows the input moves of the corresponding controllers. Notice that the proposed controller initially needs some time to update the estimation of model uncertainties. This results in a sluggish response. Once the work of updating parameters is finished, the control system possesses satisfactory robustness. It is evident that the performance retains a nearly similar response under a time-delay change τs ) 5 for t > 10, with respect to that obtained with a time delay τs ) 2 for t e 10. For the sake of demonstrating the choice of polynomial terms, Figure 4a compares the proposed controller with different polynomial terms for the process having a time delay τs ) 2 in the state and a set-point change from x2 ) 3 to x2 ) 4. The controller with three-term polynomials yields a larger deviation from the set point, while the controller with five-term polynomials provides more effective tracking ability. Figure 4b shows the input moves produced by the corresponding controllers. Figure 5a shows the simulation results of the process having a time delay τs ) 2 in the state, a set-point change from x2 ) 3 to x2 ) 4, and an oscillatory disturbance d ) 0.1 sin(t). The controller with five-term polynomials improves the performance of tracking. Figure 5b shows the input moves produced by the

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and a time delay τs ) 2 in the state for t e 10 and a τs ) 5 in the state for t > 10, with different polynomial terms. Figure 6b shows the input moves of the corresponding controllers. Observing the performance of the proposed design method, a set-point tracking and disturbance attenuation can be achieved for bounded uncertainties arising from delayed states and disturbances. 6.1.2. Case 2. In this case, only the reactor temperature x2 is available. The observer for this reactor system can be written of the form

( )

1 1 xˆ˘ 1(t) ) - xˆ 1(t) + - 1 xˆ 1(t) + λ λ xˆ 2(t) (x2 - xˆ 2) + k1sat (74) Da(1 - xˆ 1(t)) exp φ xˆ 2(t) 1+ ψ

{ }

( )

(

)

1 1 xˆ˘ 2(t) ) - xˆ 2(t) + - 1 xˆ 2(t) + BDa(1 - xˆ 1(t)) exp λ λ xˆ 2(t) (x2 - xˆ 2) - βr(xˆ 2(t) - xˆ 3(t)) + k2sat (75) φ xˆ 2(t) 1+ ψ

{ }

(

xˆ˘ 3(t) ) βc(xˆ 2(t) - xˆ 3(t)) + u(x3f - xˆ 3(t)) + k3sat


corresponding controllers. Figure 6a shows the simulation results of the process having a set-point change from x2 ) 3 to x2 ) 4

)

(

)

(x2 - xˆ 2) φ (76)

Since this state x2(t) is measurable, the computation of xˆ 1(t) and xˆ 3(t) is performed. The choice of {k1, k2, k3} ) {0.1, 1.6, 0.1} is done by trial and error.21 The initial values of estimate variables are selected as {0.65, 3, 0.26} during the course of the simulation. The thickness of boundary layer φ is designed as 0.05 to avoid the occurrence of chattering phenomena. Figure 7a shows the dynamic behavior of the process having a time delay τs ) 2 in the state in which the reference value of the dimensionless temperature x2 is changed from x2 ) 3 to x2 )

Figure 7. (a) Process responses, (b) controller responses, (c) estimation result of x1, and (d) estimation result of x3.




4. Figure 7b shows the input moves of the controller. The estimated states x1 and x3 of this system are depicted in parts c and d of Figure 7, respectively. By comparing the estimated value and the real value, one can see that they present a very close behavior. Figure 8a presents the simulation results for tracking problem with the process having a time delay τs ) 2 in the state, a set-point change from x2 ) 3 to x2 ) 4, and an oscillatory disturbance d ) 0.1 sin(t). In this case, the closed-

loop performance remains satisfactory despite the extremely noisy nature of the time-varying disturbance. Figure 8b shows the input moves of the controller. Parts c and d of Figure 8 present the estimated value and the real value of x1 and x3, respectively. Figure 9a shows the simulation results of the process having a set-point change from x2 ) 3 to x2 ) 4 and a time delay τs ) 2 in the state for t e 10 and a τs ) 5 in the state for t > 10, with the proposed controller. Figure 9b shows

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the input moves of the controller. Parts c and d of Figure 9 present the estimated value and the real value of x1 and x3, respectively. As a whole, the simulation results show that satisfactory control performances can be obtained by applying this proposed strategy. 7. Conclusions A control strategy to track a class of nonlinear state-delay systems with time-varying uncertainties is presented in this paper. Model uncertainties are represented as the expansion of the Legendre polynomials and are estimated by updating the expansion coefficients. The proposed algorithm takes advantage of the fast convergence characteristics of the orthogonal polynomials. The closed-loop performance of the proposed controller is illustrated by numerical simulation results. It is shown that the proposed method is able to track the desired trajectory despite time-varying modeling uncertainties. The sliding-mode control structure is the backbone of the proposed control strategy development. By using polynomials to approximate unknowns and adjusting polynomial coefficients by the update law derived from the Lyapunov stability theorem, it provides more robust capability than the traditional slidingmode control can offer. It can be viewed as a form of adaptive control. It is well-known that adaptive control is suitable for control problems with time-varying uncertainty. The presented technique is a general approach. It is applicable to those problems with any combination of uncertain delayedstate, unknown disturbance, and uncertainty. It can be used to solve problems with or without time delay in the states. Therefore, its applicability in chemical engineering is quite common; even our paper stresses its effectiveness with uncertain delayed state. Literature Cited (1) Richard, J. P. Time-delay Systems: An Overview of Some Recent Advances and Open Problems. Automatica 2003, 39, 1667-1694. (2) Malek-Zavarei, M. Time-delay Systems Analysis, Optimization and Applications; Elsevier Science: New York, 1987. (3) Shyu, K. K.; Yan, J. J. Robust Stability of Uncertain Time-delay and its Stabilizing by Variable Structure Control. Int. J. Control 1993, 57, 237-246. (4) Luo, N.; de la Sen, M. State Feedback Sliding Mode Control of a Class of Uncertain Time Delay Systems. IEE Proc. Control Theory Appl. 1993, 140, 261-274.

(5) Velasco, M.; Alvarez, J. A.; Castro, R. Approximate Disturbance Decoupling for a Class of Nonlinear Time Delay Systems. Proc. Am. Control Conf. 1993, 1046-1050. (6) Hu, J.; Chu, J.; Su, H. SMVSC for a Class of Time-delay uncertain Systems with Mismatching Uncertainties. IEE Proc. Control Theory Appl. 2000, 147, 687-693. (7) Oguchi, T.; Watanabe, A.; Nakamizo, T. Input-Output linearization of Retarded Nonlinear Systems by Using an Extension of Lie Derivative. Int. J. Control 2002, 75, 582-590. (8) Lehman, B.; Bentsman, J.; Lunel, S. V.; Verriest, E. I. Vibrational Control of Nonlinear Time Lag Systems with Bounded Delay: Averaging Theory, Stabilizability, and Transient Behavior. IEEE Trans. Autom. Control 1994, 39, 898-912. (9) Niu, Y.; Lam, J.; Wang, X.; Ho, D. W. C. Sliding-mode Control for Nonlinear State-delayed Systems Using Neural-network Approximation. IEE Proc. Control Theory Appl. 2003, 150, 233-239. (10) Mounier, H.; Rudolph, J. Flatness-based Control of Nonlinear Delay Systems; a Chemical Reactor Example. Int. J. Control 1998, 71, 871-890. (11) Slotine, J. J. E.; Coetsee, J. A. Adaptive Sliding Controller Synthesis for Nonlinear Systems. Int. J. Control 1986, 43, 1631-1651. (12) Huang, A. C.; Kuo, Y. S. Sliding Control of Nonlinear Systems Containing Time-Varying Uncertainties with Unknown Bounds. Int. J. Control 2001, 74, 252-264. (13) Vecchio, D. D.; Marino, R.; Tomei, P. Adaptive State Feedback Control by Orthogonal Approximation Functions. Int. J. Adapt. Control Signal Process. 2002, 16, 635-652. (14) Johnson, D. E.; Johnson, J. B. Mathematical Methods in Engineering and Physics; Ronald Press Co.: New York, 1965. (15) Sansone, G.; Diamond, A. H.; Hille, E. Orthogonal Functions; Robert E. Krieger Publishing Co.: Huntington, NY, 1977. (16) Slotine, J. J. E. Sliding Controller design for nonlinear systems. Int. J. Control 1984, 40, 421-434. (17) Li, M.; Wang, F.; Gao, F. PID-based Sliding Mode Controller for Nonlinear Processes. Ind. Eng. Chem. Res. 2001, 40, 2660-2667. (18) Slotine, J. J. E.; Li, W. Applied Nonlinear Control; Prentice Hall International Inc.: Englewood Cliffs, NJ, 1991. (19) Wang, G. B.; Peng, S. S.; Huang, H. P. A Sliding Observer for Nonlinear Process Control. Chem. Eng. Sci. 1997, 52, 787-800. (20) Henson, M. A. Nonlinear Model Predictive Control: Current Status and Future Directions. Comput. Chem. Eng. 1998, 23, 187-202. (21) Unsal, C.; Kachroo, P. Sliding Mode Measurement Feedback Control for Antilock Braking Systems. IEEE Trans. Control Syst. Technol. 1999, 7, 271-281.

ReceiVed for reView November 21, 2005 ReVised manuscript receiVed September 20, 2006 Accepted September 29, 2006 IE0512901

Robust Control of a Class of Time-Delay Nonlinear Processes

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