Ind. Eng. Chem. Res. 2002, 41, 3735-3744
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Nonlinear Controller Design for Input-Constrained, Multivariable Processes Joshua M. Kanter,† Masoud Soroush,*,‡ and Warren D. Seider† Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6393 and Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19104
Nonlinear control laws are presented for input-constrained, multiple-input, multiple-output processes, whether their delay-free part is minimum or nonminimum phase. This study addresses the nonlinear control of input-constrained processes by exploiting the connections between modelpredictive control and input-output linearization. The continuous-time control laws involve the solution of constrained optimization problems online. They minimize the error between controlled outputs and their reference trajectories, subject to the input constraints. Their application and performance are illustrated using numerical simulation of two chemical reactors that exhibit nonminimum-phase behavior. 1. Introduction Over the past 20 years, significant advances have been made in nonlinear model-based control. These advances have been mainly in the frameworks of modelpredictive control (MPC) and differential geometric control. In model-predictive control, input constraints are explicitly accounted for, and the controller action is the solution of a constrained optimization problem. Differential geometric control, in contrast, is a direct synthesis approach in which a controller is derived by requesting a desired unconstrained response. While in model-predictive control nonminimum-phase behavior is handled simply by using larger prediction horizons, in differential geometric control special treatment is necessary. In the latter, advances first made for unconstrained, minimum-phase (MP) processes have been extended to unconstrained nonminimum-phase (NMP) processes.3-16 Those based on factorization of the process model3,4 and equivalent outputs5,6 are limited to single-input single-output, NMP processes. Others are applicable to multiinput multioutput (MIMO) processes, whether MP or NMP.7,8-15 However, either sets of partial differential equations must be solved8-10 or the process is subject to restrictions.11-15 Furthermore, input constraints and deadtimes are not considered during controller design for these MIMO processes. The main objective of this work is to derive nonlinear control laws for input-constrained, MIMO, stable, nonlinear processes with a nonminimum-phase or minimumphase delay-free part. Continuous-time, differential geometric control laws are derived that are approximately input-output linearizing in the absence of input constraints. The connections between these optimization-based controllers and those derived using a direct synthesis approach16 are established. This paper is organized as follows. The scope of the study and some mathematical preliminaries are given * Corresponding author. Telephone (215) 895-1710. Fax: (215) 895-5837. E-mail:
[email protected]. † University of Pennsylvania. ‡ Drexel University.
in section 2. Section 3 introduces a method of nonlinear feedforward/state feedback design for input-constrained processes. Methods of designing nonlinear feedback controllers with integral action in the presence of input constraints are given in section 4. Finally, the application and performance of the control laws are illustrated by numerical simulation of two chemical reactors in section 5. 2. Scope and Mathematical Preliminaries Consider the class of MIMO, nonlinear processes of the form
dxj(t) ) f[xj(t),u(t)], xj(0) ) xj0 dt yji(t) ) hi[xj(t - θi)] + di uli e ui(t) e uhi, i ) 1, ..., m
}
(1)
where xj ) [xj1...xjn]T ∈ X is the vector of the process state variables, u ) [u1...um]T ∈ U is the vector of manipulated inputs, yj ) [yj1...yjm]T is the vector of process outputs, θ1, ..., θm are the measurement delays, d ) [d1...dm]T ∈ D is the vector of constant unmeasured disturbances, f (‚,‚) is a smooth vector field on X × U, and h1(‚), ..., hm(‚) are smooth functions on X. Here X ⊂ Rn is a connected open set that includes xjss and xj0; U ) {u|uli e ui e uhi, i ) 1, ..., m} ⊂ Rm that includes uss; ul1, ..., ulm, uh1, ..., uhm are real scalars; D ⊂ Rm is a connected set; and (xjss, uss) denotes the nominal steady-state (equilibrium) pair of the process; that is, f[xjss,uss] ) 0. The system
dxj(t) ) f[xj(t),u(t)], xj(0) ) xj0 dt / yji (t) ) hi[xj(t)] + di uli e ui(t) e uhi, i ) 1, ..., m
}
(2)
is referred to as the delay-free part of the process. The relative orders (degrees) of the controlled outputs yj1, ..., yjm with respect to u are denoted by r1, ..., rm, respectively, where ri is the smallest integer for which driyj/i /dtri
10.1021/ie0105066 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/27/2001
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explicitly depends on u for every x ∈ X and every u ∈ U. The set point and the set of acceptable set point values are denoted by ysp and Y, respectively, where Y ⊂ Rm is a connected set. The following assumptions are made: (A1) For every ysp ∈ Y and every d ∈ D, there exists an equilibrium pair (xjss, uss) ∈ X × U that satisfies ysp - d ) h(xjss) and f[xjss, uss] ) 0. (A2) The nominal steady-state (equilibrium) pair of the process, (xjss, uss), is hyperbolically stable; that is, all eigenvalues of the Jacobian of the open-loop process evaluated at (xjss, uss) have negative real parts. (A3) For a process in the form of (1), a model in the following form is available:
dx(t) ) f[x(t),u(t)], x(0) ) x0 dt yi(t) ) hi[x(t - θi)] uli e ui(t) e uhi, i ) 1, ‚‚‚, m
}
[ [ [
hri i(x,u)
] ] ]
∂hri i(x,u) hri i+1(x,u(0),u(1)) } f(x,u) + ∂x ∂hri i(x,u) (1) dri+1y/i u ) r +1 ∂u dt i
]
l
[
[
]
∂hpi i-1(x,u(0),...,u(pi-ri-1)) f(x,u) + ∂x
]
∂hpi i-1(x,u(0),...,u(pi-ri-1)) ∂u
[
u(1) + ... +
]
∂hpi i-1(x,u(0),...,u(pi-ri-1))
)
∂u dpiy/i pi dt
(pi-ri-1)
where 1, ..., m are positive, adjustable, constant scalars
U ) [u(1)...u(max[p1-r1,...,pm-rm])]T,p ) [p1...pm] Assume that for every x ∈ X, every d ∈ D, and every ysp ∈ Y, the algebraic equation
φp(xj, u) ) ysp - d
(6)
φp(xj,u) } Φp(xj,u, 0)
(7)
has a real root for u, and that for every xj ∈ X and every u ∈ R m, ∂φp(xj,u)/∂u is nonsingular. The corresponding feedforward/state feedback that satisfies eq 6 is denoted by
dη ) Acη + BcΦp[xj,u,U] dt u ) Ψp(xj,e + Ccη)
driy/i ∂hri i-1(x) } f(x,u) ) r ∂x dt i
hpi i(x,u(0),u(1),...,u(pi-ri)) }
()
(8)
Theorem 1.16 For a process in the form of eq 1 without input constraints and with state measurements xj and θ1 ) ... ) θm ) 0, the closed-loop system under the mixed error- and state-feedback control law
dri-1y/i ∂hri i-2(x) f(x,u) ) r -1 ∂x dt i
[
)
()
u ) Ψp(xj,ysp - d )
[ ]
hri i-1(x) }
(
)
where
dy/i ∂hi(x) } f(x,u) ) ∂x dt l
(
(3)
where x ) [x1...xn]T ∈ X is the vector of model state variables, and y ) [y1, ..., ym]T is the vector of model outputs. (A4) The relative orders, r1, ..., rm, are finite. The following notation is used:
h1i (x)
()
[Φp(xj,u,U)]i } hi(xj) + pi ih1i (xj) + ... + 1 pi ri i-1hri i-1(xj) + pi ri ihri i(xj,u) + ri - 1 ri pi ri i+1hri i+1(xj, u, u(1)) + ... + ri + 1 pi pi ihpi i(xj,u,u(1),...,u(pi-ri)) (5) pi
u(pi-ri)
(4)
where pi g ri and u(l) ) dlu/dtl. 2.1. Unconstrained Feedback Controller Design via Direct Synthesis: Review. Consider the following notation:
}
(9)
is asymptotically stable, if the following conditions hold: (a) The nominal equilibrium pair of the process, (xjss,uss), corresponding to ysp and d, is hyperbolically stable. (b) The tunable parameters p1, ..., pm are chosen to be sufficiently large. (c) The tunable parameters 1, ..., m are chosen such that, for every l ) 1, ..., m, all eigenvalues of l Jol } l[∂/∂x f(x,u)](xj ss,uss) lie inside the unit circle centered at (-1, 0j). Furthermore, the mixed error- and state-feedback control law of eq 9 has integral action. Here Ac ) blockdiag(Ac1, ..., Acm), Bc ) block-diag(Bc1, ..., Bcm), and Cc ) block-diag(Cc1, ..., Ccm) are (p1 + ... + pm) × (p1 + ... + pm), (p1 + ... + pm) × m, and m × (p1 + ... + pm) block diagonal, constant matrices, respectively. A typical matrix triplet (Aci, Bci, Cci) is
[
0 0 l 0 A ci ) 0 -
1 0 l 0 0 1 βpi i
-
0 1 l 0 0 βi1 βpi i
-
0 0 ·
·
·
βpi i
-
‚‚‚
-
‚‚‚ ‚‚‚ ·
0 0 βi2
‚‚‚
0 0 l 0 1
‚‚‚
βi3 βpi i
·
·
βpi i-1 βpi i
]
[]
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3737
uli e ui(t) e uhi
0 0 l Bci ) 0 0 1 βpi i
u(1)(t) ) 0 l u(pi-ri)(t) ) 0,i ) 1, ..., m
Cci ) [1 0 ... 0],βil ) li pi , l ) 1, ..., pi l
()
Remark 1. An approximate form of eq 9 can be obtained by setting the time derivatives of u to zero, leading to the following mixed error- and state-feedback control law:
dη ) Acη + Bcφp[xj,u] dt u ) Ψp[xj,e + Ccη]
}
(10)
which is much easier to implement. This approximate form is adequate in many practical cases. Theorem 2.16 For a process in the form of eq 1 without input constraints and with incomplete state measurements, the closed-loop system under the errorfeedback control law
dx ) f[x,u] dt
[ [
h1[x(t - θ1)] u ) Ψp x, e + l hm[x(t - θm)]
]}
(11)
3. Feedforward/State Feedback Design for Input-Constrained Processes
m
∑
u(t) i)1
subject to the input constraints
i
i
hi
i
i
and w1, ..., wm are adjustable positive scalar weights whose values are set according to the relative importance of the controlled outputs: the higher the value of wi, the smaller the mismatch between yd/ i and yj/i . 3.1. Output Prediction Equation. For a process in the form of eq 1, the future value of the ith delay-free process output in eq 2 over the time interval [t, t + Thi] is predicted using a truncated Taylor series:
yjˆ /i (τ) ) yj/i (t) +
dpiyj/i (t) [τ - t]pi dyj/i (t) [τ - t] + ... + , dt pi! dtpi i ) 1, ..., m (13)
yj/i (t) ) hi(xj(t)) + di dyj/i (t) ) h1i (xj(t)) dt l dpiyj/i (t) pi dt
) hpi i(xj(t),u(0)(t),u(1)(t),...,u(pi-ri)(t))
3.2. Reference Trajectory. The ith component of the reference trajectory, yd/ i, describes the path that the ith delay-free process output, yj/i , is forced to follow in the absence of input constraints starting at time t. The ith reference trajectory is trackable when the following conditions are satisfied:17
yd/ i(t) ) yj/i (t) ) hi(xj(t)) + di
When input constraints are ignored during controller design, as in eqs 9 and 11, serious deterioration in control quality and even closed-loop instability can occur. To derive controllers that in the absence of input constraints are identical with those in eqs 9 and 11 and that in the presence of input constraints perform optimally, we formulate a moving-horizon, constrained, optimization problem in the form
wi||yd/ i(τ) - yjˆ /i (τ)||2qi,[t,t+Th ]
∫tt+T |yi(τ)|q dτ]1/q ,qi g 1
||yi(τ)||qi,[t,t+Th ] } [
where according to eqs 2 and 4
is asymptotically stable, if the following conditions hold: (a) The nominal equilibrium pair of the process, (xjss, uss), corresponding to ysp and d, is hyperbolically stable. (b) The tunable parameters p1, ..., pm are chosen to be sufficiently large. (c) The tunable parameters 1, ..., m are chosen such that, for every l ) 1, ..., m, all eigenvalues of l Jol } l[∂/∂x f(x,u)](xj ss,uss) lie inside the unit circle centered at (-1, 0j). Furthermore, the error-feedback control law of eq 11 has integral action.
min
where t represents the present time, yd/ i is the ith component of an output reference trajectory, yjˆ /i is the predicted value of the ith delay-free, controlled output, ||yi(τ)||qi,[t,t+Thi] denotes the qi-function norm of the scalar function yi(τ) over the finite time interval [t, t + Thi] with Thi > 0:
(12)
dyd/ i(t) dt
)
dyj/i (t) ) h1i (xj(t)) dt
l dri-1yjd/ i(t) dtri-1
)
dri-1yj/i (t) dtri-1
) hri i-1(xj(t))
Furthermore, every component of the reference trajectory, yd/ i, should lead its corresponding delay-free controlled output, yj/i , to its set point, yspi, asymptotically. A class of reference trajectories that has these properties is described by
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Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002
ysp )
[
(1D + 1)p1yd/ 1(τ) l (mD + 1)pmyd/ m(τ)
]
the feedforward/state feedback of eq 15 takes the simpler form: m
min u
subject to the “initial” conditions:
yd/ i(t)
dtri dri+1yd/ i(t) dtri+1
dt
pi-1
)
) 1, ..., m
) hpi i-1(xj(t),u(t),u(1)(t), ..., u(pi-1-ri)(t))
[τ - t]l
hli(xj(t)) ∑ l)1
l!
[τ - t]l
∑ hli(xj(t),u(t),..., u(l-r )(t)) l)r i
+
l!
i
ri-1
hi(xj(t)) -
+
{[
yspi - di -
li pi hli(xj(t)) ∑ l)1 ( )
]}
l
()
3.3. Feedforward/State Feedback. For a process in the form of eq 1, by using the series forms of the output prediction equations and reference trajectory components, the optimization problem in eq 12 is
u
∑ i)1
pi
∑
l)ri
li
{[
In this section, two control laws, one for inputconstrained nonlinear processes with full state measurements and no deadtimes and one for input-constrained nonlinear processes with incomplete state measurements and deadtimes, are presented. 4.1. Processes with Full State Measurements and No Deadtimes. Theorem 3. For a process in the form of eq 1 with state measurements xj and θ1 ) ... ) θm ) 0, the mixed error- and state-feedback control law
dη ) Acη + BcΦp(xj,u,U) dt u ) F[xj,e + Ccη]
[τ - t]pi il pi hli(xj(t),u(0)(t),...,u(l-ri)(t)) /pi i + pi! l higher order terms (14)
m
()
4. Feedback Control Laws
hri i+1(xj(t),u(t),u(1)(t)),i
pi-1
min
]}
2
pi hli(xj,u,0,...,0) /pi i l
uli e ui e uhi,i ) 1, ..., m
) hri i(xj(t),u(t))
yd/ i(τ) ) di + hi(xj(t)) +
∑
li
subject to the input constraints:
ri-1
l)ri
l
pi
∑
A series solution for the ith reference trajectory, yd/ i, has the following form:
pi-1
li pi hli(xj) ∑ l)1 ( )
l)ri
l dpi-1yd/ i(t)
{[
) hi(xj(t)) + di l
driyd/ i(t)
∑ i)1
ri-1
yspi - di - hi(xj) -
(l)
l [τ - t]pi
] } || 2
pi hli(xj,u,0,...,0)
li pi hli(xj) ∑ () l)1
pi
/i
pi!
||
2
(15) qi,[t,t+Thi]
dη ) Acη + Bcφp(xj,u) dt u ) F[xj,e + Ccη]
u ) F[x,ysp - d]
(16)
Remark 2. When the weights, w1, ..., wm, are chosen such that
wi||[τ - t]pi||2qi,[t,t+Thi] (pi!)2
) 1,i ) 1, ..., m
[
dx(t) ) f[x(t),u(t)] dt
u(t) ) F x(t), e(t) +
which is a feedforward/state feedback denoted by
(17)
}
(19)
which is much easier to implement. This approximate form of eq 18 is adequate in many practical cases. 4.2. Processes with Incomplete State Measurements and Deadtimes. Theorem 4. For a process in the form of eq 1 with incomplete state measurements, the error-feedback control law
subject to the input constraints:
uli e ui e uhi,i ) 1, ..., m
(18)
(a) at each time instant is the solution to the constrained optimization problem in eq 12 (b) in the absence of the input constraints, is identical with the mixed error- and state-feedback control law of eq 9 The proof is given in the Appendix. Remark 3. The time derivatives of u needed in Φp[xj,u,U] of eq 18 are obtained by numerical differentiation of the vector signal u with respect to time. Alternatively, the time derivatives of u are set at zero, leading to the following mixed error- and state-feedback control law:
ri-1
wi yspi - di - hi(xj) -
}
[
h1[x(t - θ1)] l hm[x(t - θm)]
]}
(20)
(a) at each time instant is the solution to the constrained optimization problem in eq 12. (b) in the absence of input constraints is the errorfeedback control law of eq 11. The proof is in the Appendix. Remark 4. Conditions for asymptotic stability of the closed-loop system under the controllers of eq 18 and
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3739
(1) the error-feedback control law of eq 20 with p1 ) p2 ) 2, 1 ) 1/8, 2 ) 1/10, and w1 and w2 set according to eq 17
dx1 u1 x (0) ) x10 ) -kAx12 + (CAi - x1) dt V1 1 dx2 u1 ) - x2 + kAx12 - kBx2 x2(0) ) x20 dt V1
(21)
dx3 u2 u1 ) -kAx32 + (CAi - x3) + (x1 - x3) x3(0) ) x30 dt V2 V2 u2 dx4 u1 ) (x2 - x4) - x4 - kBx4 + kAx32 x4(0) ) x40 dt V2 V2
( [
Figure 1. Schematic of the isothermal reactors in series.
u ) F x,e +
eq 20 in the absence of the input constraints are the same as those for the controllers of eq 9 and eq 11, respectively. In the presence of the input constraints, these conditions are necessary but not sufficient for the asymptotic stability of the closed-loop system. 5. Illustrative Examples
[
(
h h 21(x,u) )
RB ) kACA2 - kBCB
h1(x(t - θ1)) ) x2(t - θ1)
]
h2(x(t - θ2)) ) x4(t - θ2) where x(t) ) [CA1(t) CB1(t) CA2(t) CB2(t)]T, u(t) ) [F1(t) F2(t)]T, and y(t) ) [CB1(t - θ1) CB2(t - θ2)]T with θ1 ) θ2 ) 300/3600. In addition, CAi ) 7 mol/L, V1 ) V2 ) 0.01 L, ul1 ) ul2 ) 0 L h-1, uh1 ) uh2 ) 1.5 × 10-2 L h-1, kA ) 6 L mol-1 h-1, and kB ) 1 h-1. The following two nonlinear controllers are compared:
)[
]
u1 u1 k x 2 - kBx2 - x2 V1 A 1 V1
h2(x) ) x4
RA ) -kACA2
[
]
(CAi - x1) u1 V1
kB +
The rates of formation of A and B per unit volume are
u1 -kAx12 + (CAi - x1) V1 u1 - x2 + kAx12 - kBx2 V1 f(x,u) ) u2 u1 -kAx32 + (CAi - x3) + (x1 - x3) V2 V2 u2 u1 (x - x4) - x4 - kBx4 + kAx32 V2 2 V2
u1 x + kAx12 - kBx2 V1 2
h h 21(x,u) ) 2kAx1 -kAx12 +
AfBfC
Each reactor has a fresh feed stream consisting of pure A. The reactor dynamics are represented by the following model:
])
h1(x) ) x2 h h 11(x,u) ) -
5.1. A Chemical Reactor with Time Delay. Consider two constant-volume, isothermal, continuousstirred-tank reactors in series, as shown in Figure 1, in which the following series reactions take place in the liquid phase:
x2(t - θ1) x4(t - θ2)
u1 u2 (x2 - x4) - x4 + kAx32 - kBx4 V2 V2
(
)
u1 u1 - x2 + kAx12 - kBx2 + V2 V1 u2 u1 2kAx3 (x1 -x3) + (CAi - x3) - kAx32 V2 V2 u2 u1 u2 u1 + + kB (x2 - x4) - x4 + kAx32 - kBx4 V2 V2 V2 V2
h h 22(x,u) )
(
(
)(
)
)
(2) the nonlinear model-predictive controller 2
min u j
Pj
∑ ∑ aji[ysp (k + i) - yjˆ /j (k + i)]2 + j)1 i)1 j
2 Mj-1
∑ ∑ j)1 i)0
bji[uj(k + i) - uj(k + i - 1)]2 (22)
subject to
yjˆ /j (k + i) ) hij(xjˆ (k),u1(k),u2(k),...,u1(k + i - 1),u2(k + i - 1)) + dˆ j(k + i), i ) 1, ..., Pj, j ) 1, 2 xjˆ (k + 1) ) Φ(xjˆ (k),u(k)) uj(k + i) ) uj(k + i - 1),i ) Mj, Mj + 1, ..., j ) 1, 2
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dˆ j(k + i) ) dˆ j(k), i ) 1, ..., Pj, j ) 1, 2 dˆ j(k) ) yjj(k) - hj(xjˆ (k - θj)), j ) 1, 2 θj ) vj∆t, vj ∈ {1, 2, ‚‚‚}, j ) 1, 2 ulj e uj(k + i) e uhj, j ) 1, 2, i ) 0, 1, ..., Mj - 1 Here, Φ(xjˆ (k),u(k)) z xjˆ (k) + ∆t / (f(xjˆ (k)) + g1(xjˆ (k)) × u1(k) + g2(xjˆ (k)) u2(k)) and
hj(x(k + 1)) ) hj(Φ(x(k),u(k))) z h1j (x(k),u(k)) hj(x(k + 2)) ) hj(Φ(Φ(x(k),u(k)),u(k + 1))) z h2j (x(k),u(k),u(k + 1)) l where j ) 1, 2 is the output and manipulated variable counter, k denotes the current sampling instant, and u j )˘ [u1(k)...u1(k + M1 - 1) u2(k)...u2(k + M2 - 1)]. The following parameter values are used: prediction horizons: P1 ) P2 ) 2; control horizons: M1 ) M2 ) 1; sampling interval: ∆t ) 1/12(v1 ) v2 ) 1); a11 ) 0, a12 ) 4 × 106, a21 ) 0, a22 ) 2 × 102, b10 ) 1 × 1010, and b20 ) 1 × 106. Servo Performance. The process is initially at the steady state (C h A1(0) ) 3/4, C h B1(0) ) 2.1916, C h A2(0) ) 3/4, C h B2(0) ) 2.1916). The set points are adjusted to ysp ) [2 3]T. Figure 2 depicts the closed-loop responses of the process outputs and manipulated inputs. The output responses under the controller of eq 22 are more oscillatory and slower. This is because the controller is shortsighted. Even though the prediction horizons of both outputs are 2, the sample interval is small and therefore the time horizon for prediction of 2∆t is small. On the other hand, the controller in eq 21 does not suffer from shortsightedness when the same prediction horizons are used. A less oscillatory response results. The controller in eq 21 is easier to tune than that in eq 22. It is difficult to select better values for the tuning parameters a11, a12, a21, and a22 in eq 22 since the controller is somewhat over-parametrized, and their effects on speed of the response are not well understood and monotonic. However, the controller of eq 21 has only two tunable parameters, 1 and 2, which are easier to tune and whose effects on the response are predictable (the larger their values, the slower the response). Regulatory Performance. The process is initially at the steady state corresponding to ysp ) [2, 3]T and h A1(0) ) 0.6988, C h B1(0) ) 2, CAip ) 7 mol/L; that is, C h B2(0) ) 3. The set points are mainC h A2(0) ) 1.0133, C tained at ysp ) [2, 3]T, but a step change in the unmeasured disturbance, CAip, from 7 to 8.4 mol/L, is introduced at t ) 0. Figure 3 shows the responses of the process outputs and manipulated inputs. Both controllers in eq 21 and eq 22 return the outputs to their set points in the face of constant unmeasurable disturbances. However, the responses under the controller in eq 22 are again slower and more oscillatory due to the controller’s shortsightedness. 5.2. Chemical Reactor with All States Measurable and No Delays. Consider a constant-volume, nonisothermal, continuous-stirred-tank reactor, as shown in Figure 4, in which the following series reactions take place in the liquid phase:
Figure 2. Measured process outputs and manipulated inputs of the CSTRs in series under the controllers in eqs 21 and 22 (servo response).
AfBfC The rates of formation of A and B per unit volume are
( )
RA ) -A1 exp -
( )
RB ) A1 exp -
E1 C 2 RT A
( )
E1 E2 C 2 - A2 exp C RT A RT B
The reactor has a fresh feed stream consisting of pure A. The reactor dynamics are represented by the following model:
[
f(x,u) ) E1 u1 (C - x1) - A1 exp x2 V Ai Rx3 1 u1 E1 E2 - x2 + A1 exp x12 - A2 exp x V Rx3 Rx3 2 (-∆H1) E1 u1 A1 exp x 2+ (Ti - x3) + V Fcp Rx3 1 E2 (-∆H2) US A2 exp x + (x - x3) Fcp Rx3 2 FcpV 4 u2 US (T - x4) (x - x3) V ji FjVjcpj 4
( ) ( ) ( ) ( ) ( )
h1(x) ) x2 h2(x) ) x3
]
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3741
where x(t) ) [CA(t) CB(t) T(t) Tj(t)]T, u(t) ) [F(t) Fj(t)]T, and y(t) ) [CB(t) T(t)]T. Parameter values are in Table 1. ul1 ) ul2 ) 0, uh1 ) 200 L h-1 and uh2 ) 100 L h-1. The following two nonlinear controllers are compared: (1) the mixed-error and state feedback control law in eq 19 with p1 ) p2 ) 2, 1 ) 2 ) 1/100, and w1 and w2 set according to eq 17
dη ) Acη + Bcφp(xj,u) dt
(23)
u ) F(xj,e + Ccη) h1(xj) ) xj2 h h 11(xj,u) ) -
( )
u1 E1 xj + k1 exp xj 2 V 2 Rxj3 1
( )
k2 exp -
( ( ) )( ( ) ) ( ( )
E2 xj Rxj3 2
E1 u1 xj1 (C - xj1) Rxj3 V Ai E1 u1 E2 k1 exp - k2 exp xj12 + Rxj3 V Rxj3 E1 E2 u1 xj12 - k2 exp xj + xj2 + k1 exp V Rxj3 Rxj3 2 -∆H1 E1 u1 k1 exp xj 2 + (Ti - xj3) + V FCp Rxj3 1 E2 -∆H2 US (xj - xj3) k2 exp xj + FCp Rxj3 2 FCpV 4 E1 E2 E1 E2 k1exp xj12 -k2 exp xj2 2 Rxj3 Rxj3 Rxj 2 Rxj
h h 21(xj,u) ) 2k1 exp -
(
( ) ( ( )
3
( ))( ( )) ( ) ) ( ) )
Figure 3. Measured process outputs and manipulated inputs of the CSTRs in series under the controllers in eqs 21 and 22 (regulatory response).
3
h2(xj) ) xj3 h h 12(xj,u) )
h h 22(xj,u)
(
( )
u1 -∆H1 E1 (Ti - xj3) + k1 exp xj 2 + V FCp Rxj3 1 E2 -∆H2 US k2 exp xj + (xj - xj3) FCp Rxj3 2 FCpV 4
(
( ) ( ))
-∆H1 E1 ) 2 k1 exp xj [f(xj,u)]1 + FCp Rxj3 1
( ))
(
E2 u1 -∆H1 -∆H2 + k2 exp [f(xj,u)]2 + k × FCp Rxj3 V FCp 1
( )
exp -
( )
E1 E1 2 -∆H2 E2 E2 xj + k exp xj Rxj3 Rxj 2 1 FCp 2 Rxj3 Rxj 2 2 3 3
)
( )
US US [f(xj,u)]3 + [f(xj,u)]4 FCpV FCpV where k1 ) A1 and k2 ) A2. (2) the unconstrained controller of eq 10 with clipping
dη ) Acη + Bcφp(xj,u) dt u ) sat(Ψp(xj,e + Ccη))
(24)
Figure 4. Schematic of the nonisothermal reactor. Table 1. Parameters for the Nonisothermal Reactor C Ai V Vj E1/R E2/R F Fj cp cpj Ti Tji U S A1 A2 -∆H1 -∆H2
5 mol /L 10 L 5L 8023 K 9758 K 1 kg L-1 1.1 kg L-1 2.25 kJ kg-1 K-1 3 kJ kg-1 K-1 400 K 298.15 K 3825 kJ m-2 K-1 h-1 0.225 m2 9 × 109 L mol-1 h-1 1.287 × 1012 h-1 39 kJ mol-1 10 kJ mol-1
Servo Performance. The process is initially at the h B(0) ) 0.5, T h (0) ) 390, steady state (C h A(0) ) 0.9771, C T h j(0) ) 383.80). The set points are adjusted to ysp ) [0.9 407]T and CAip ) 5 mol/L. Figure 5 depicts the responses of the process outputs and manipulated inputs. The performance of the controller in eq 24 is
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Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002
Figure 5. Measured process outputs and manipulated inputs of the nonisothermal CSTR under the controllers in eqs 23 and 24 (servo response).
comparable to that in eq 23. In this case, clipping the unconstrained controller output is a good alternative to solving for an optimal manipulated input. However, this is not true in general, as shown for a regulatory response next. Regulatory Performance. The process is initially at the steady state corresponding to ysp ) [0.9 407]T and h A(0) ) 1.5672, C h B(0) ) 0.9, CAip ) 5 mol/L; that is, (C T h (0) ) 407, T h j(0) ) 377.51). The set point is maintained at ysp ) [0.9 407]T, but a step change in the unmeasured disturbance, CAip, from 5 to 6 mol/L, is introduced at t ) 0. Figure 6 depicts the responses of the process outputs and inputs, and illustrates the advantage of using the controller in eq 23. Clipping an unconstrained controller output leads to instability. 6. Conclusions Two nonlinear control laws for MIMO, stable, nonlinear, input-constrained processes were presented. The processes can be minimum phase or nonminimum phase. The control laws were derived by solving a moving-horizon, constrained optimization problem. In the absence of input constraints, the controllers are identical with those derived using a direct synthesis approach.16 Compared to a general model-predictive control law, the two control laws have far fewer tunable parameters, one for each controlled output, and are easier to tune (the larger the parameter, the slower the response of the controlled output). The two reactor simulations show that the control laws perform better than conventional model-predictive control and the control law derived in ref 16 used with clipping. The use of the latter can cause closed-loop
Figure 6. Measured process outputs and manipulated inputs of the nonisothermal CSTR under the controllers in eqs 23 and 24 (regulatory response).
instability, because input constraints are not accounted for in the controller design in ref 16. Acknowledgment J.M.K. and W.D.S. gratefully acknowledge partial support from the National Science Foundation through Grant CTS-0101237. M.S. acknowledges financial support from the National Science Foundation through Grant CTS-0101133. Notation a, b ) weighting coefficients in MPC A, B, C ) chemical species A, B, C ) matrices in state-space description of a linear system A1, A2 ) preexponential factors cp ) heat capacity of feed and product, kJ kg-1 K-1 cpj ) heat capacity of jacket fluid, kJ kg-1 K-1 Cij ) concentration of species i in tank j, mol L-1 CAi ) inlet concentration, mol L-1 d ) additive, unmeasured, constant disturbance vector; see eq 1 D ) differential operator, D ) d/dt e ) error in measured output vector, ysp - yj E1, E2 ) activation energy of reactions 1 and 2, kJ mol-1 F ) solution to optimization in eq 15 F1, F2 ) feed and product flow rate in reactors 1 and 2, L h-1 Fj ) jacket fluid flow rate, L h-1 hi(x) ) ith output map -∆H1, -∆H2 ) heats of reaction, kJ mol-1 Jol ) open-loop Jacobian
Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3743
k ) reaction rate constant; current sampling instant m ) number of inputs and outputs Mj ) jth manipulated variable horizon in MPC n ) process order p ) vector of the orders of the requested responses for the process outputs Pj ) jth prediction horizon in MPC r ) vector of relative orders of process outputs R ) universal gas constant, kJ mol-1 K-1; rate of reaction, mol L-1 h-1 S ) heat-transfer surface area, m2 t ) time, h ∆t ) sampling interval in MPC, h T ) product temperature, K Tj ) jacket temperature, K Ti ) feed temperature, K Tji ) jacket inlet temperature, K u ) process input vector uli ) lower bound on ith process input uhi ) upper bound on ith process input U ) overall heat transfer coefficient, kJ m-2 K-1 h-1 U ) vector of derivatives of process inputs V1, V2 ) reactor volume of tanks 1 and 2, L Vj ) jacket volume, L wi ) ith weighting coefficient in eq 12 x ) vector of model states xj ) vector of process states y ) model output vector yj ) measured process output vector ysp ) process output set point vector yj/d ) reference trajectory for delay-free process output vector y* ) delay-free model output vector yj* ) delay-free process output vector
{
zhi zi > zhi [sat(z)]i ) zi zli e zi e zhi zli zi < zli int(U) ) interior of U Appendix Proof of Theorem 3. (a) Substituting the expressions for Ac, Bc, and Cc into eq 18
η˘ i1 ) ηi2 l η˘ pi i-1
) ηpi i i
η˘ pi i ) -
βpi-1 βi1 i 1 i 1 η η - ... - i ηpi i + i [Φp(xj,u,U)]i i 1 i 2 βp i βp i βp βpi
u ) F[xj,e + Ccη] Thus
βpi iη˘ p1i ) -ηi1 - βi1ηi2 - ... - βpi i-1ηpi i +
[
]
dyi dpiyi pi p + ... + i yi + i i dt dtpi 1
()
Using the definition of βi1 and
ηi1 ) yi
Greek Symbols
ηi2 )
i ) ith adjustable parameter of controller η ) states in dynamic controllers in eqs 9, 10, 18 Φp(x,u) ) defined in eq 5 φp(x,u) ) defined in eq 7 Ψp(x,v) ) state feedback for current value of p F ) density of feed and product, kg L-1 Fj ) density of jacket fluid, kg L-1 θ1, θ2, ... ) time delays, h
dηi1 dt
l ηpi i
)
η˘ pi i )
dpi-1ηi1 dtpi-1 dpiηi1 dtpi
Superscripts
leads to
- ) measured value ∧ ) estimated value
dpi(yi - ηi1) dpi-1(yi - ηi1) pi-1 p pi i + + ... + i i dtpi dtpi-1 pi - 1
(
Subscripts
)
(yi - ηi1) ) 0
A, B ) chemical species p ) process ss ) steady state sp ) set point 0 ) initial value [‚]i ) ith row of vector
Since
ηi1(0) ) hi(xj(0)) ) yi(0) dyi dηi1 (0) ) h1i (xj(0)) ) (0) dt dt
Math Symbols
l a! a ) b!(a - b)! b
dpi-1ηi1 (0) pi-1
()
||xi(t)||qi,[t,t+Th ] ) [ i
∫tt+T |xi(t)|q dt]1/q ,qi g 1 hi
i
dt
i
the solution is
) hpi i-1(xj(0)) )
dpi-1yi dtpi-1
(0)
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Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002
dx ) f(x,u) dt
ηi1 ) yi ) hi(xj),i ) 1, ..., m; t g 0 Therefore
[
0 ) ei + hi(x(t - θi)) - hi(xj(t)) -
e + Ccη ) e + h(xj) ) ysp - h(xj) - d + h(xj) ) ysp - d
ri-1
∑ l)1
which implies
Thus, u is the solution to eq 15, which is equivalent to the optimization in eq 12. (b) In the absence of input constraints, under the assumption that the global minimum of the performance index is always reached, the control action is the solution of
[[
dη ) Acη + BcΦp(xj,u,U) dt
ei + [Cc]iη - hi(xj(t)) ri-1
∑ l)1
pi
∑
]]
il pi hli(xj(t)) il pi h h li(xj(t),u(t)) /ipi l)ri l l
()
()
()
∑
()
]
i ) 1, ..., m
u ) F[xj,e + Ccη] ) F[xj,ysp - d]
0)
pi
il pi hli(xj(t)) il pi h h li /ipi l)ri l l
i ) 1, ..., m Denoting the solution to the m algebraic equations as u ) Ψp(xj,e + Ccη) gives the control law in eq 9. Proof of Theorem 4. (a) Since x(0) ) xj(0), x(t) ) xj(t) ∀ t g 0. Therefore, the controller in eq 20 is
u(t) ) F[xj(t),e(t) + h(xj(t - θ))] dxj ) f[xj(t),u(t)] dt However, e + h(xj(t - θ)) ) ysp - yj + h(xj(t - θ)) ) ysp h(xj(t - θ)) - d + h(xj(t - θ)) ) ysp - d, implying the controller is
u ) F[xj,ysp - d] dxj ) f(xj,u) dt which is the solution to the optimization in eq 12. (b) In the absence of input constraints, under the assumption that the global minimum of the performance index is always reached, the control action is the solution of
Denoting the solution to the m algebraic equations as u ) Ψp(xj,e + h(x(t - θ))) gives the control law in eq 11. Literature Cited (1) De Nicolao, G.; Magni, L.; Scattolini, R. Stabilizing Receding-Horizon Control of Nonlinear Time-Varying Systems. IEEE Trans. Autom. Control 1998, 43, 1030. (2) Chen, H.; Allgower, F. A Quasi-infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability. Automatica 1998, 34, 1205. (3) Kravaris, C.; Daoutidis, P. Nonlinear State Feedback Control of Second-Order Nonminimum Phase Nonlinear Systems. Comput. Chem. Eng. 1990, 14, 439. (4) Doyle, F. J., III; Allgo¨wer, F.; Morari, M. A Normal Form Approach to Approximate Input-Output Linearization for Maximum Phase Nonlinear SISO Systems. IEEE Trans. Autom. Control 1996, 41, 305. (5) Wright, R. A.; Kravaris, C. Nonminimum Phase Compensation for Nonlinear Processes. AIChE J. 1992, 38, 26. (6) Kravaris, C.; Niemiec, M.; Berber, R.; Brosilow, C. B. Nonlinear Model-based Control of Nonminimum-phase Processes. In Nonlinear Model Based Process Control; Berber, R., Kravaris, C., Eds.; Kluwer: Dordrecht, 1998. (7) Niemiec, M.; Kravaris, C. Controller Synthesis for Multivariable Nonlinear Nonminimum-phase Processes. Proc. ACC 1998, 2076. (8) Isidori, A.; Byrnes, C. I. Output Regulation of Nonlinear Systems. IEEE Trans. Autom. Control 1990, 35, 131. (9) Isidori, A.; Astolfi, A. Disturbance Attenuation and H∞ Control via Measurement Feedback in Nonlinear Systems. IEEE Trans. Autom. Control 1992, 37, 1283. (10) van der Schaft, A. J. L2 Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H∞ Control. IEEE Trans. Autom. Control 1992, 37, 770. (11) Chen, D.; Paden, B. Stable Inversion of Nonlinear Nonminimum-phase Systems. Int. J. Control 1996, 64, 81. (12) Hunt, L. R.; Meyer, G. Stable Inversion for Nonlinear Systems. Automatica 1997, 33, 1549. (13) Devasia, S.; Chen, D.; Paden, B. Nonlinear Inversion-based Output Tracking. IEEE Trans. Autom. Control 1996, 41, 930. (14) Devasia, S. Approximated Stable Inversion for Nonlinear Systems with Nonhyperbolic Internal Dynamics. IEEE Trans. Autom. Control 1999, 44, 1419. (15) Isidori, A. Nonlinear Control Systems; Springer-Verlag: New York, 1995. (16) Kanter, J.; Soroush, M.; Seider, W. D. Continuous-time, Nonlinear Feedback Control of Multivariable Stable Processes. Submitted for publication in J. Process Control. (17) Hirschorn, R. M. Invertibility of Nonlinear Systems. SIAM J. Control Optim. 1979, 17, 289.
Received for review June 13, 2001 Revised manuscript received August 20, 2001 IE0105066
