Ind. Eng. Chem. Res. 2001, 40, 2069-2078
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Continuous-Time, Nonlinear Feedback Control of Stable 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 stable, single-input single-output processes, whether minimum phase or nonminimum phase. This study addresses the problem of nonlinear control of nonminimum-phase processes by exploiting the connections between model-predictive control and input-output linearization. Continuous-time, long-prediction-horizon, model-predictive control laws with the shortest control horizon are derived. These are approximate, input-output, linearizing control laws with one tunable parameter and, thus, are very easy to tune. Their application and performance are illustrated using numerical simulation of a chemical reactor that exhibits nonminimum-phase behavior. 1. Introduction Nonlinear processes with unstable zero dynamics are common in chemical engineering. Examples are chemical reactors whose temperatures are controlled by adjusting the flow rates of cold inlet streams, azeotropic distillation towers whose temperatures respond to adjustments in the feed flow rate, and reboilers whose liquid level is controlled, in response to increases in the feed flow rate, by adjusting the bottoms flow rate. During the past decade, the problem of nonlinear control of nonminimum-phase processes has received considerable attention, leading to several solutions. While these solutions seem different, they are essentially based on factoring the process model into minimum-phase and nonminimum-phase parts and designing a model-based controller that includes an inverse of the minimum-phase part, as in linear modelbased control. One of the solutions centers around finding a minimum-phase equivalent output which at steady state is equal to the original system output. An input-output, linearizing controller is then designed to control the minimum-phase equivalent output.1 In another approach, a control law is derived2 that leads to integralof-squared error (ISE) optimal, closed-loop responses to changes in the set point, but the process must be secondorder. An attempt was made to extend this idea to general nonlinear processes,1 but the ISE optimization is only solvable for r ) n - 1, where n and r are the system order and relative order (degree), respectively. Subsequently, a control method was developed for the class of nonminimum-phase, nonlinear processes, with all modes of the zero dynamics unstable.3 There is a class of processes for which these three methods lead to the same model-based controller. By applying these methods to linear processes, one can see easily that they are essentially extensions of linear internal-model control.4 Other approaches to analytical control of nonminimum-phase, nonlinear processes include output regulation5 and stable inversion.6-9 While the nonlinear output * To whom all correspondence should be addressed. Email:
[email protected]. Telephone: (215) 8951710. Fax: (215) 895-5837. † University of Pennsylvania. ‡ Drexel University.
regulation theory ensures internal stability for a class of nonlinear processes, nonlinear output regulation has the following disadvantages: (a) the controller is obtained by solving a set of nonlinear partial differential equations, and (b) the closed-loop system shows large transient error when the process is nonminimum-phase. In contrast to output regulation, the stable-inversion approach does not lead to transient errors. However, it requires off-line calculation of manipulated input profiles and desired state trajectories and the use of a stabilizing feedback. Another drawback is that it needs pre-actuation (the controller is noncausal) when the process is nonminimum-phase. In the framework of model-predictive control, it is well-known that large prediction horizons should be used for nonminimum-phase processes. For example, a p-inverse (long-prediction-horizon) control law for nonlinear, single-input single-output (SISO), nonminimumphase, discrete-time processes with arbitrary order and relative order was proposed.10 These authors have shown that as the prediction horizon goes to infinity the Jacobian of the closed-loop system approaches the Jacobian of the open-loop system. The main objective of this work is to derive nonlinear control laws for SISO, stable, nonminimum-phase, nonlinear, continuous-time processes. To this end, the connections between model-predictive control and inputoutput linearization shown in ref 11 are further exploited. This approach results in the derivation of continuous-time, long-prediction-horizon, model-predictive control laws with the shortest control horizon. The resulting control laws can be considered as approximate, input-output, linearizing control laws.12 The organization of the paper is as follows. The scope of the study and some mathematical preliminaries are given in section 2. Section 3 presents a method of nonlinear feedforward/state-feedback design. Methods for designing nonlinear feedback controllers with integral action are given in section 4. These methods are applied to linear processes in section 5. Finally, the application and performance of the derived control laws are illustrated by numerical simulation of a chemical reactor in section 6. 2. Scope and Mathematical Preliminaries Consider the class of SISO, nonlinear processes of the form
10.1021/ie000406k CCC: $20.00 © 2001 American Chemical Society Published on Web 04/07/2001
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dxj(t) ) f[xj(t),u(t)], xj(0) ) xj0 dt yj(t) ) h[xj(t)] + d
(1)
where xj ) [xj1, ..., xjn]T ∈ X is the vector of the process state variables, u ∈ U is the manipulated input, yj is the process output, d ∈ D is a constant unmeasured disturbance, f(.,.) is a smooth vector field on X × U, and h(.) is a smooth function on X. The nominal steady-state (equilibrium) pair of the process is denoted by (xjss, uss); that is, f[xjss,uss] ) 0. Here X ⊂ R n is a connected open set that includes xss and xj0, U ⊂ R is a connected open set that includes uss, and D ⊂ R is a connected set. The relative order (degree) of the process is represented by r, where r is the smallest integer for which dryj/dtr explicitly depends on u for every x ∈ X. The set point and the set of the values that it can take are denoted by ysp and Y, respectively, where Y ⊂ R 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 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 y(t) ) h[x(t)]
(2)
where x ) [x1, ..., xn]T ∈ X is the vector of model state variables and y is the model output. The following notation is used:
[ ]
h
r-1
[ [ [
] ] ] [
hp(x,u(0),u(1),...,u(p-r)) } f(x,u) +
[
]
]
∂hp-1(x,u(0),...,u(p-r-1)) × ∂x
]
∂hp-1(x,u(0),...,u(p-r-1)) (1) u + ... + ∂u
[
∂h
p-1
dpy ) p dt where p g r and u(l) ) dlu/dtl.
(0)
]
(x,u ,...,u(p-r-1)) ∂u(p-r-1)
yj +
()
()
r dyj r r dryj + ... + r ) ysp 1 dt r dt
(5)
where
()
a a! } b b!(a - b)!
It is straightforward to show that when the relative order, r, is finite, the feedforward/state feedback
h(xj) +
()
( ) ()
r r h1(xj) + ... + r-1hr-1(xj) + 1 r-1 r r r h (xj,u) ) ysp - d (6) r
induces the linear response of eq 4. When the process exhibits nonminimum-phase behavior, the inputoutput behavior of the closed-loop system under the feedforward/state feedback of eq 6 is governed by the linear response of eq 4 but the internal dynamics (unobservable modes) of the closed-loop system are unstable. The dynamic feedforward/state feedback
(7)
u(p-r) (3)
() ()
p h1(xj) + ... + 1 p p r r r-1hr-1(xj) + h (xj,u) + r-1 r p r+1hr+1(xj,u,u(1)) + ... + r+1 p p p h (xj,u,u(1),...,u(p-r)) (8) p
( ) ( )
∂hr(x,u) f(x,u) + ∂x ∂hr(x,u) (1) dr+1y u ) r+1 ∂u dt
[
where D is the differential operator (i.e., D } d/dt) and is a positive, constant, adjustable parameter that sets the speed of the response of the closed-loop process output. This response has the expanded form
Φp(xj,u,u(1),...,u(p-r)) } h(xj) +
∂hr-1(x) dry hr(x,u) } f(x,u) ) r ∂x dt
l
(4)
where
∂hr-2(x) dr-1y (x) } f(x,u) ) r-1 ∂x dt
hr+1(x,u(0),u(1)) }
(D + 1)ryj ) ysp
Φp(xj,u,u(1),...,u(p-r)) ) ysp - d
∂h(x) dy f(x,u) ) h1(x) } ∂x dt l
2.1. Output Linearization. For a process in the form of eq 1, a response of the closed-loop, process output is requested, having the linear form
()
with ∂Φp/∂u(p-r) * 0, ∀ x ∈ X, also induces a linear, closed-loop, process output response of the form
(D + 1)pyj ) ysp
(9)
Similarly, the dynamic feedforward/state feedback of eq 7 cannot ensure asymptotic stability of the closed-loop system when the process exhibits nonminimum-phase behavior. Consequently, the main objective of this study has been to design feedback control laws that ensure asymptotic stability of the closed-loop system, whether the process is minimum-phase or nonminimum-phase. 3. Nonlinear Feedforward/State-Feedback Design Assume that, for every x ∈ X, every d ∈ D, and every ysp ∈ Y, the algebraic equation
Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001 2071
φp(xj,u) ) ysp - d
(10)
φp(xj,u) } Φp(xj,u,0,...,0)
(11)
where
has a real root inside U for u and that, for every xj ∈ X and every u ∈ U
∂φp(xj,u) *0 ∂u
(12)
The corresponding feedforward/state feedback that satisfies eq 10 is denoted by
u z Ψp(xj,ysp-d)
(13)
Note that the preceding feedforward/state feedback was obtained by setting all of the time derivatives of u in eq 7 to zero. Theorem 1. For a process in the form of eq 1, the closed-loop system under the feedforward/state feedback of eq 13 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 parameter p is chosen to be sufficiently large. (c) The tunable parameter is chosen such that all eigenvalues of Jol } [(∂/∂x)f(x,u)](xjss,uss) lie inside the unit circle centered at (-1, 0i). Furthermore, as p f ∞, the state feedback places the n eigenvalues of the Jacobian of the closed-loop system evaluated at the nominal equilibrium pair at the n eigenvalues of the Jacobian of the open-loop process evaluated at the nominal equilibrium pair. The proof is given in the Appendix. Condition c states that should be chosen such that |λl + 1| < 1, with l ) 1, ..., n, where λ1, ..., λn are the eigenvalues of Jol. For an overdamped, stable process, should be chosen such that 0 < /τl < 2, with l ) 1, ..., n, where τ1, ..., τn are the open-loop time constants of the process. In other words, should be chosen to be less than 2τmin, where τmin is the smallest time constant of the process [i.e., τmin ) min (τ1, ..., τn)]. Note that the feedforward/state feedback of eq 13 does not induce the linear, closed-loop, output response of eq 9, because in the derivation of the feedforward/state feedback the time derivatives of u were set to zero. The nonlinearity of the resulting process output response is the price of ensuring closed-loop stability for nonminimum-phase processes. The extent of the deviation of the closed-loop, output response under the feedforward/state feedback of eq 13 from the linear, closed-loop, output response of eq 9 depends on the rate of change of the manipulated input profile corresponding to the linear, closed-loop, output response of eq 9. If inducing the linear, closed-loop, output response of eq 9 does not require very aggressive manipulated input changes, the loss of tracking performance (compared to eq 9) is small even when the process is far from its steady-state conditions. 4. Nonlinear Feedback Controller Design The feedforward/state feedback of eq 13 cannot be implemented, because the value of the constant unmea-
sured disturbance is unknown. Furthermore, the feedforward/state feedback lacks integral action, and thus it cannot induce an offset-free response of the process output when process-model mismatch exists. To resolve these problems, the feedforward/state feedback of eq 13 is implemented with a disturbance estimator, leading to the derivation of feedback control laws with integral action. An estimate of the disturbance (d) is dˆ ) yj - y˜ , where y˜ is a “disturbance-free” estimate of the controlled output yj. Here, two feedback control laws that induce an offset-free response are derived using two approaches to estimate the disturbance. The first approach involves (a) using measurements of state variables (when measurements of all state variables are available) and the manipulated input and (b) integrating
dη ) Acη + bcΦp(xj,u,u(1),...,u(p-r)) dt y˜ ) ccη
(14)
where η ∈ R p and eq 14 is a minimal-order, state-space realization of
(D + 1)py˜ ) Φp(xj,u,u(1),...,u(p-r))
[
] []
and Ac, bc, and cc are p × p, p × 1, and 1 × p constant matrices, respectively. A typical matrix triplet (Ac, bc, cc) is
0
1
0
0
· · · 0
0 · · · Ac ) 0
0 · · · 0
1 · · · 0
0 · ·
0
0
0
0
· · · 0 · ‚‚‚ · · · 0 · · · · · 1
-
0
·
β1 β2 β3 βp-1 1 · · · βp βp βp βp βp cc ) [1 0 ... 0]
0
0 · · · , bc ) 0 0
1 βp
where
βl ) l
()
p , l ) 1, ..., p l
In this case, the disturbance estimate is dˆ ) yj - ccη. The second approach is to integrate the model of eq 2 and estimate the disturbance using dˆ ) yj - h(x). These approaches differ in the following aspects: (i) The system of eq 14 is of order p, while that of eq 2 is of order n; that is, the system of eq 14 is, in general, of a lower order. (ii) The first approach requires measurements of all state variables, while these measurements are not needed in the second approach. 4.1. Nonlinear Control Laws. In this subsection, two control laws, one for nonlinear processes with full state measurements and one for nonlinear processes with incomplete state measurements, are presented. 4.1.1. Processes with Full State Measurements. Theorem 2. For a process in the form of eq 1 with the state measurements, xj, the closed-loop system under the
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Figure 2. Block diagram of the error-feedback control law of theorem 3. Figure 1. Block diagram of the mixed error- and state-feedback control law of theorem 2.
mixed error- and state-feedback control law
dη ) Acη + bcΦp[xj,u,u(1),...,u(p-r)] dt u ) Ψp(xj,e+ccη)
(15)
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 parameter p is chosen to be sufficiently large. (c) The tunable parameter is chosen such that all eigenvalues of Jol } [(∂/∂x)f(x,u)](xjss,uss) lie inside the unit circle centered at (-1, 0i). Furthermore, the mixed error- and state-feedback control law of eq 15 has integral action. The proof is given in the Appendix. A block diagram of the mixed error- and state-feedback control law is depicted in Figure 1. The (p - r) time derivatives of u needed in Φp[xj,u,u(1), ...,u(p-r)] of eq 15 are obtained by analytical differentiation of u ) Ψp(xj,e+ccη) 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:
dη ) Acη + bcφp[xj,u] dt u ) Ψp[xj,e+ccη]
(16)
which is much easier to implement. This approximate form of eq 15 is adequate in many practical cases. 4.1.2. Processes with Incomplete State Measurements. Theorem 3. For a process in the form of eq 1, the closed-loop system under the error-feedback control law
dx ) f[x,u] dt u ) Ψp[x,e+h(x)]
5. Application to Linear Processes Consider the class of time-invariant linear processes with a mathematical model of the form
x˘ (t) ) Ax(t) + bu(t), x(0) ) 0 y(t) ) cx(t)
where A, b, and c are n × n, n × 1, and 1 × n constant matrices, respectively. This class of processes is a special case of eq 1 for f(xj,u) ) Axj + bu and h(xj) ) cxj. It is assumed that the process is asymptotically stable. For this class of processes, the function φp(x,u) is linear in u
∑ (l )lcAlx + ∑l)r(l )lcAl-1bu l)0 p
φp(x,u) }
p
p
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 parameter p is chosen to be sufficiently large. (c) The tunable parameter is chosen such that all eigenvalues of Jol } [(∂/∂x)f(x,u)](xjss,uss) lie inside the unit circle centered at (-1, 0i). Furthermore, the error-feedback control law of eq 17 has integral action. The proof is in the Appendix. A block diagram of the error-feedback controller is depicted in Figure 2.
p
) ysp - d
(19)
and thus u can be solved for analytically:
lcAlx ∑ ( ) l l)0 p
ysp - d u ) Ψp(x,ysp-d) )
∑ ( l)r p
p
)
(20)
p l l-1 cA b l
Application of the control laws of eqs 15-17 to the models in the form of eq 18 leads to the following linear control laws respectively:
{∑ ( ) p
η˘ ) Acη + bc
l)0
p l l cA xj + l
u)
)
p
p l l cA xj + l
e + cc η -
lcAl-1bu ∑ ( ) l l)r p
p
∑ (l )lcAl-1b l)r p
p
p
lcAlxj ∑ ( ) l l)0 p
u)
(21)
p l l cA xj l
{∑( ) l)0
}
)∑ l-r
p l cAl-j-1bu(j) l j)0
lcAl-1b ∑ ( ) l l)r p
p
η˘ ) Acη + bc
∑ ( l)r p
∑ ( l)0 p
e + cc η (17)
(18)
} (22)
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x˘ ) Ax + bu
∑ ( l)0 p
e + cx u)
(23)
)
p l l cA x l
lcAl-1b ∑ ( ) l l)r p
p
As stated in theorems 2 and 3, in the limit as p f ∞, the preceding control laws place the n eigenvalues of the linear closed-loop system at the n eigenvalues of the linear open-loop process. This fact is illustrated by the example below. Example 1. Consider a linear process with
A)
[
-11 1
]
[]
-10 1 , b) , c ) [-1 2] 0 0
s-2 (s + 1)(s + 10)
It is nonminimum-phase [has a right-half-plane (RHP) zero at s ) 2] and asymptotically stable [has two lefthalf-plane (LHP) poles at s ) -1 and -10]. Its relative order is 1. With ) 0.1, eigenvalues of A are -0.1 and -1, which are inside the unit circle centered at (-1, 0i), and F(A+I) ) 0.9. The Jacobian of the closed-loop system under the feedforward/state feedback of eq 20 for processes in the form of eq 18 is
bc[A + I]p Jclp ) A p r r-1 p p p-1 cA b + ... + cA b r p
()
p
λ1(Jcl)
λ2(Jcl)
1 2 3 4 5 6 7 8 10 12 14 16 18 22 26
-10.00 -10.00 -10.00 -10.00 -63.09 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00
2.00 2.86 4.65 10.70 -10.00 -8.75 -4.93 -3.54 -2.39 -1.89 -1.62 -1.45 -1.33 -1.20 -1.12
for A and B are
dCA F ) (CAi - CA) - k1CA - k3CA2 dt V
which has the transfer function
G(s) )
Table 1. Closed-Loop Eigenvalues of Example 1 for Several Values of p
()
The location of the closed-loop eigenvalues, λ1(Jclp) and λ2(Jclp), for several values of p are given in Table 1; the closed-loop eigenvalues converge to the open-loop eigenvalues, s ) -1 and -10, as p f ∞. 6. Illustrative Example To illustrate the application and performance of the derived control laws and to compare them to the control law of Kravaris and Daoutidis,2 a chemical reactor with nonminimum-phase behavior is simulated. Consider a constant-volume, isothermal, continuousstirred-tank reactor (CSTR) in which the following liquid-phase van de Vusse reactions13 take place:
AfBfC 2A f D The rates of formation of A and B are
RA ) -k1CA - k3CA2 RB ) k1CA - k2CB The feed stream consists of pure A. The mole balances
dCB F ) (-CB) + k1CA - k2CB dt V where F is the inlet volumetric flow rate of A, V is the reactor volume, CA and CB are the concentrations of A and B in the reactor, and CAi is the concentration of A in the feed stream. Note that it is desired to maintain CB at the set point by manipulating u ) F/V. Let x1 ) CA, x2 ) CB, u ) F/V, and y ) CB. The mole balances are a model in the form of eq 2 with
f(x,u) )
[
-k1x1 - k3x12 + (CAi - x1)u k1x1 - k2x2 - x2u
]
h(x) ) x2 where CAi ) 10 mol/L. Using this model, the controllers in eqs 16 and 17 and Kravaris and Daoutidis2 are developed and simulated on the following process:
dxj1 ) -k10xj1 - k30xj21 + (CAi0 - xj1)u dt dxj2 ) k10xj1 - k20xj2 - xj2u dt yj ) xj2 with k10 ) 50 h-1, k20 ) 100 h-1, and k30 ) 10 L mol-1 h-1. Two different cases are considered: (i) Case A: the reaction rate constants in the three controllers are equal to those in the process (k10 ) k1, k20 ) k2, and k30 ) k3). (ii) Case B: the reaction rate constants in the three controllers are 20% higher than those in the process (k1 ) 1.2k10, k2 ) 1.2k20, and k3 ) 1.2k30). 6.1. Control Laws. The process steady state correh Ass ) 3, sponding to ysp ) 1.117 and CAi ) 10 mol/L (C C h Bss ) 1.117, and uss ) 34.29) is asymptotically stable (eigenvalues of the Jacobian evaluated at the steady state, λ1(Jol) ) -134.3 and λ2(Jol) ) -144.3, lie in the LHP).
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dx1 ) -k1x1 - k3x12 + (10 - x1)Ψ2(x,e+x2), dt x1(0) ) x10 (26)
The process zero dynamics are governed by
[
k1 0 dξ ) -k10ξ - k30ξ2 + (10 - ξ) ξ - k20 dt 1.117
]
dx2 ) k1x1 - k2x2 - x2Ψ2(x,e+x2), x2(0) ) x20 dt
whose Jacobian evaluated at ξss ) x1ss ) 3 has an eigenvalue in the RHP at λ ) 169.05. Thus, the process is nonminimum-phase. For this process, eq 10 with p ) 2, d ) 0, and xj ) x takes the form
u ) Ψ2(x,e+x2) In contrast, the controller of Kravaris and Daoutidis2 is
φ2(x,u) ) a(x1,x2) u2 + b(x1,x2) u + c(x1,x2) ) ysp
dx1 ) -k1x1 - k3x12 + (10 - x1)Ψ ˆ (x,e+x2), dt x1(0) ) x10 (27)
where
a(x1,x2) z 2x2
dx2 ) k1x1 - k2x2 - x2Ψ ˆ (x,e+x2), x2(0) ) x20 dt
b(x1,x2) z -2x2 + k12(10 - x1) + k2x22 - k12x1 + k22x2 c(x1,x2) z x2 + 2k1x1 - 2k2x2 - k122x1 2 2
u)Ψ ˆ (x,e+x2) where
2
2
2
k1k3x1 - k1k2x1 + k2 x2
Ψ ˆ (x,v) ) The quadratic equation has two solutions. Because only
u)
-b + xb2 - 4a(c - ysp) 2a
satisfies Ψp)2(xss)[3 1.117]T,ysp)1.117) ) uss, the state feedback is
u ) Ψp)2(xj,ysp) )
-b + xb2 - 4a(c - ysp) (24) 2a
In case A, selecting ) 1/144.3, F(Jol + I) ) 10/144.3 < 1 and the closed-loop eigenvalues approach the openloop eigenvalues as p f ∞. The closed-loop Jacobian under Ψp)2(x,ysp) has eigenvalues λ1(Jclp) ) -153.4 and λ2(Jclp) ) -144.3, which indicates that the closed-loop eigenvalues approach the open-loop eigenvalues [λ1(Jol) ) -134.3 and λ2(Jol) ) -144.3] for p ) 2 and ) 1/144.3. In case B, selecting ) 1/144.3, F(Jol + I) ) 28.8/ 144.3 < 1 and the closed-loop eigenvalues approach the open-loop eigenvalues as p f ∞. The closed-loop Jacobian under Ψp)2(x,ysp) has eigenvalues λ1(Jclp) ) - 111.2 and λ2(Jclp) ) -152.8, which indicates that the closedloop eigenvalues approach the open-loop eigenvalues [λ1(Jol) ) -161.1 and λ2(Jol) ) -173.1] for p ) 2 and ) 1/144.3. With the aforementioned settings, the mixed errorand state-feedback control law in eq 16 is
dη ) Acη + bcφ2(xj,u), η1(0) ) y(0), η2(0) ) y˘ (0) dt (25) u ) Ψ2(xj,e+ccη) and the error-feedback controller of eq 17 is
v - h/opt(x) - γ1Lfh/opt(x) γ1Lgh/opt(x)
h/opt(x) z x2 -
A(x1,x2) B(x1,x2)
A(x1,x2) z 2x2(k1x1x2 + k3x12x2 - (10 - x1)(k1x1 k2x2)) B(x1,x2) z -x2(k1x1 + k3x12 - (10 - x1)(-k1 2k3x1)) - (10 - x1)(-k1x1 - k1(10 - x1)) Lgh(x) z
∂h1(x) ∂h2(x) g1(x) + g (x) ∂x1 ∂x2 2
[f, g] z
f(x) )
[
∂g(x) ∂f(x) f(x) g(x) ∂x ∂x
]
[
10 - x1 -k1x1 - k3x12 , g(x) ) -x k1x1 - k2x2 2
]
h(x) ) x2 Thus, the system
dx ) f(x) + g(x) u dt y/ ) h/opt(x) is minimum-phase and its relative order r* ) 1. Furthermore, it has the following properties: (a) y/ss ) yss and (b) ISE-optimal control of y* implies ISE-optimal control of y. The controller induces the first-order,
Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001 2075
Figure 3. Servo responses of the measured process output under the three controllers (case A).
Figure 4. Servo responses of the measured process output under the three controllers (case B).
closed-loop response
γ1
dy* + y* ) ysp - d dt
implying y/ss ) yss ) ysp - d. As was shown above, the derivations of the controllers of eqs 15 and 17 are easier than that of the controller of Kravaris and Daoutidis.2 In terms of closed-loop performance, as we will show through simulations below, closed-loop responses under the controllers of eqs 15 and 17 and that of Kravaris and Daoutidis2 have comparable ISEs. In what follows, the servo and regulatory performances of the controllers of eqs 26, 25, and 27 with γ1 ) 0.01 h are compared for cases A and B. The three controllers will be referred to as controllers I-III, respectively. 6.2. Servo Performance. The process is initially at h B(0) ) 1. The set point the steady state C h A(0) ) 2.5 and C ysp ) 1.117 mol/L and CAi0 ) 10 mol/L. Figure 3 depicts the closed-loop responses of the process output for case A. All three are offset-free and show inverse response. The latter must be present in the closed-loop response to have stable control of a nonminimum-phase process. Recall that a standard input-output linearizing controller for a nonminimumphase process leads to closed-loop instability because the requested response is “inverse-free”. The process output response under controller II differs slightly from that under controller I. It is likely that this is due to controller II being an approximation of eq 15. Figure 4 shows the closed-loop responses of the process output for case B. Because the three controllers have integral action, their responses are offset-free in the presence of the process-model mismatch. These offset-free responses are despite mismatch between the states of the process and the model and unequal process and model outputs. 6.3. Regulatory Performance. The process is initially at the steady state corresponding to ysp ) 1.117 h A(0) ) 3 and C h B(0) mol/L and CAi0 ) 10 mol/L; that is, C ) 1.117. The set point is maintained constant at ysp ) 1.117 mol/L, but a step change in the unmeasured disturbance, CAi0, from 10 to 12 mol/L is introduced at t ) 0. Figure 5 shows the closed-loop responses of the
Figure 5. Regulatory responses of the measured process output under the three controllers (case A).
process output for case A. Again, integral action in the controllers leads to offset-free responses; that is, the integral action compensates for the unmeasured disturbance in the state equations. The regulatory response under controller II has a significantly lower ISE than those under controllers I and III. This is because controller II becomes aware of the disturbance sooner through the state measurements. Figure 6 shows the closed-loop responses of the process output for case B. Here, the integral action compensates for the unmeasured disturbance and the errors in the values of k1, k2, and k3. 7. Conclusions The derived nonlinear control laws are applicable to SISO, stable, nonminimum-phase, nonlinear, continuous-time systems with no input constraints. The control laws ensure asymptotic tracking of the measured output in the presence of constant model errors and disturbances as long as the nominal, closed-loop, asymptotic stability is preserved. There are no limitations on the order, relative order, and number of RHP zeros of processes to which the control laws are applicable. The control laws are continuous-time model-predictive controllers with a long prediction horizon and the shortest control horizon. They are also approximate, input-
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where δ ) ysp - d. Under the conditions of the theorem, it is proved that, as p f ∞, Jclp(xjss,uss) f Jol(xjss,uss). Because φp(x,u) ) δ and u ) Ψp(x,δ), the partial derivative of φp(x,u) ) δ with respect to x yields
∂φp(x,u) ∂φp(x,u) ∂Ψp(x,u) + ) [0‚‚‚0] ∂x ∂u ∂x
(A2)
Using the definition of φp(x,u) in eq 11:
Figure 6. Regulatory responses of the measured process output under the three controllers (case B).
output linearizing controllers. The derived feedforward/ state feedback is a continuous-time analogue of the p-inverse control law developed by Hernandez and Arkun10 for discrete-time processes. The derived feedback control laws do not induce the linear, closed-loop, output response of eq 9, because in the derivation of the feedforward/state feedback of eq 13, the time derivatives of u were set to zero. The nonlinearity of the resulting process output response is the price of ensuring closed-loop stability for nonminimum-phase processes. The extent of the deviation of this closed-loop, output response from the linear, closed-loop, output response of eq 9 depends on the rate of change of the manipulated input profile corresponding to the linear, closed-loop, output response of eq 9. If inducing the linear, closed-loop, output response of eq 9 does not require very aggressive manipulated input changes, the loss of tracking performance (compared to eq 9) is small even when the process is far from its steady-state conditions. The unconstrained control laws presented in this paper were derived using a direct synthesis approach. To derive controllers of the same type that can perform optimally in the presence of input constraints, we have used an optimization formulation. The resulting controllers minimize a performance index subject to input constraints. Because of the use of the two fundamentally different approaches to derive the control laws for constrained and unconstrained processes, the controllers for input-constrained processes were not presented here but will be included in a forthcoming publication.
()
1 ∂φp(x,u) ∂h(x) p ∂h (x) ) + + ... + 1 ∂x ∂x ∂x r ∂hr-1(x) p p r ∂h (x,u) + + r-1 r-1 r ∂x ∂x p ∂hr+1(x,u,0) p p p ∂h (x,u,0,...,0) r+1 + ... + r+1 p ∂x ∂x (A3)
( ) ( )
()
Using the definition of h1, ..., hp, at the nominal equilibrium pair (xjss, uss):
∂h1(x) ∂h(x) ∂f(x,u) ∂h(x) ) ) Jol ∂x ∂x ∂x ∂x ∂h2(x) ∂h1(x) ∂f(x,u) ∂h(x) 2 ) ) J ∂x ∂x ∂x ∂x ol · · · ∂hp(x,u,0,...,0) ∂hp-1(x) ∂f(x,u) ∂h(x) p ) ) Jol ∂x ∂x ∂x ∂x
(A4)
and therefore
∂φp(x,u) ∂h(x) ) [Jol + I]p ∂x ∂x
(A5)
Because is chosen such that all of the eigenvalues of Jol(xj ss,uss) are in the unit circle centered at (-1, 0), F[Jol(xjss,uss) + I] < 1. Consequently
lim [Jol(xjss,uss) + I]p ) 0 pf∞
and
lim pf∞
[
]
∂φp(x,u) ∂x
(xjss,uss)
)0
(A6)
Because the term
Acknowledgment J.M.K. and W.D.S. gratefully acknowledge partial support from the National Science Foundation through Grant CTS-963-2992. M.S. acknowledges financial support from the National Science Foundation through Grant CTS-970-3278.
[ pf∞
∂f(x,u) ∂f(x,u) ∂Ψp(x,δ) + ∂x ∂u ∂x
(A1)
(xjss,uss)
is nonzero (see eq 12), eqs A2 and A6 imply that
lim
Proof of Theorem 1. The Jacobian of the closed-loop system under the state feedback is
]
∂φp(x,u) ∂u
Appendix
Jclp(x,u) )
()
[
]
∂Ψp(x,δ) ∂x
(xjss,uss)
)0
(A7)
This together with eq A1 proves that
lim Jclp(xjss,uss) ) pf∞
[
∂f(x,u) ∂x
]
(xjss,uss)
) Jol(xjss,uss) (A8)
Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001 2077
η1ss + ess ) φp(xjss,uss)
Proof of Theorem 2. The closed-loop system under the control law of eq 15 is given by
xj˘ ) f[xj,Ψp(xj,δ-h(xj)+η1)]
(A9)
η˘ ) Acη + bcΦp(xj,u,u(1),...,u(p-r))
Therefore, ess ) 0. This concludes the proof of the second part of the theorem. Proof of Theorem 3. The closed-loop system under the control law of eq 17 is given by
Using eq 14,
xj˘ ) f(xj,Ψp[x,δ+h(x)-h(xj)]) p
(D + 1) [η1 - h(xj)] ) 0 or equivalently
θ˙ ) Acθ
(A10)
θl ) ηl -
d
h(xj)
dtl-1
with xjss ) xss, and its Jacobian at the equilibrium pair ([xjss, xss], δ) by
Jclp([xjss,xss],uss) )
[
where l-1
Jol(xjss,uss) - W -W
, l ) 1, ..., p
xj˘ ) f[xj,Ψp(xj,δ+ccθ)]
(A11)
θ˙ ) Acθ
W)
[
Jol(xjss,uss) + Q 0
E Ac
]
(A12)
]
Jol(xjss,uss) -W
,
(xjss,uss)
[
[
]
∂f(xj,u) ∂Ψp(xj,v) ∂u ∂xj
∂f(xj,u) ∂Ψp(xj,v) cc ∂u ∂v
(xjss,uss)
]
(xjss, uss)
(A15)
-I I
]
The matrix of eq A15 is lower block triangular, and thus its eigenvalues are those of the 1,1 and 2,2 blocks. Eigenvalues of the 1,1 block are in the LHP. The eigenvalues of the 2,2 block are those of the process under the feedforward/state feedback of eq 13. Theorem 1 proves that the eigenvalues of the 2,2 block approach the open-loop eigenvalues when conditions a-c hold. The proof of the last part of this theorem is almost identical to that of theorem 2. Consider the closed-loop system at steady state:
0 ) f{xjss,Ψp[xss,ess+h(xss)]} 0 ) f{xss,Ψp[xss,ess+h(xss)]} uss ) Ψp[xss,ess+h(xss)]
(A16)
According to the definitions of φp(.,.) and Ψp(.,.), uss ) Ψp(xss,ess+h(xss)) implies that φp(xss,uss) ) ess + h(xss). Because at steady state
dy dpy ) ... ) p ) 0 dt dt
0 ) f[xjss,Ψp(xjss,ess+ccηss)] (A13)
The first and last equations in eq A13 respectively imply that
η1ss ) Φp(xjss,uss,0,...,0) ) φp(xjss,uss)
]
are similar, the eigenvalues of Jclp([xjss,xss],uss) are the same as the eigenvalues of the matrix of eq A15. The similarity transformation is
0 ) Acηss + bcΦp(xjss,uss,0,...,0) uss ) Ψp(xjss,ess+η1ss)
(xjss,uss)
0 Jol(xss,uss) + Q
I 0
Because the Jacobian of eq A12 is upper block triangular, its eigenvalues are those of the 1,1 and 2,2 blocks. The eigenvalues of the 1,1 block are those of the process under the feedforward/state feedback of eq 13. Theorem 1 proves that the eigenvalues of the 1,1 block lie in the LHP when conditions a-c hold. The 2,2 block has p eigenvalues at -1/. Thus, the first part of the theorem is proven. To prove that the control law has integral action (i.e., ess ) 0 for every constant d), consider the closed-loop system at steady state:
and
]
∂f(xj,u) ∂Ψp(xj,v) ∂h(x) ∂u ∂v ∂x
[ Q)
E)
[
[
where
[
]
Because the closed-loop Jacobian above and the matrix
The Jacobian of eq A11 at the equilibrium pair ([xjss, θss], δ) has the form
∂f(xj,u) Jol(xjss,uss) ) ∂xj
Q+W Jol(xss,uss) + Q + W
where
In terms of xj and θ, the closed-loop system of eq A9 takes the form
Jclp([xjss,θss],uss) )
(A14)
x˘ ) f(x,Ψp[x,δ+h(x)-h(xj)])
φp(xss,uss) ) h(xss), and thus ess ) 0. Notation A, b, c ) matrices in the state-space description of a linear system CA ) concentration of A in CSTR, mol/L CAi ) inlet concentration of A, mol/L CB ) concentration of B in CSTR, mol/L
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Ind. Eng. Chem. Res., Vol. 40, No. 9, 2001
d ) additive, unmeasured, constant disturbance; see eq 1 dˆ ) estimate of d, dˆ ) yj - y˜ D) derivative operator, Dly } dly/dtl e ) error in measured output, ysp - yj f(xj,u) ) right-hand side in nonlinear state equations; see eq 1 F ) feed and product flow rate, L/h G(s) ) process transfer function h(x) ) output map h/opt(x) ) ISE-optimal output map Jol ) open-loop Jacobian Jclp ) closed-loop Jacobian k1, k2, k3 ) reaction rate constants n ) process order p ) order of requested response of the process output r ) relative order (degree) of the process r* ) relative order of the auxiliary system x˘ ) f(x) + g(x) u, y* ) h/opt(x) RA, RB ) rate of formation of species A and B, mol/(L‚h) t ) time, h u ) process input (manipulated variable) V ) reactor volume, L x ) vector of model states xj ) vector of process states y ) model output yj ) measured process output y* ) ISE-optimal output ysp ) process output set point y˜ ) estimate of disturbance-free process output Greek Symbols γ1 ) adjustable parameter in ISE-optimal output-based state feedback ) controller adjustable parameter λi ) ith eigenvalue η ) defined in eq 14 Φp(x,u) ) defined in eq 8 φp(x,u) ) defined in eq 11 Ψ ˆ (x,v) ) state feedback based on ISE-optimal output Ψp(x,v) ) state feedback for the current value of p F(A) ) spectral radius of matrix A, max {|λ1(A)|, |λ2(A)|, ...} τ ) open-loop time constant of the process ξ ) x1 ) CA Superscript - ) measured value Subscripts opt ) optimal
ss ) steady state sp ) set point 0 ) initial Math Symbols n (∂h/∂xi)fi Lfh(x) ) Lie derivative of h with respect to f, ∑i)1 [f, g] ) Lie bracket of f and g, (∂g/∂x)f - (∂f/∂x)g
()
a ) a!/[b!(a - b)!] b
Literature Cited (1) Wright, R. A.; Kravaris, C. Nonminimum Phase Compensation for Nonlinear Processes. AIChE J. 1992, 38, 26. (2) Kravaris, C.; Daoutidis, P. Nonlinear State Feedback Control of Second-Order Nonminimum Phase Nonlinear Systems. Comput. Chem. Eng. 1990, 14, 439. (3) 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. (4) Morari, M.; Zafiriou, E. Robust Process Control; PrenticeHall: Englewood Cliffs, NJ, 1989. (5) Isidori, A.; Byrnes, C. I. Output Regulation of Nonlinear Systems. IEEE Trans. Autom. Control 1990, 35, 131. (6) Chen, D.; Paden, B. Stable Inversion of Nonlinear Nonminimum-phase Systems. Int. J. Control 1996, 64, 81. (7) Devasia, S.; Chen, D.; Paden, B. Nonlinear Inversion-based Output Tracking. IEEE Trans. Autom. Control 1996, 41, 930. (8) Hunt, L. R.; Meyer, G. Stable Inversion for Nonlinear Systems. Automatica 1997, 33, 1549. (9) Devasia, S. Approximated Stable Inversion for Nonlinear Systems with Nonhyperbolic Internal Dynamics. IEEE Trans. Autom. Control 1999, 44, 1419. (10) Herna´ndez, E.; Arkun, Y. Study of the Control-relevant Properties of Back-propagation Neural Network Models of Nonlinear Dynamical Systems. Comput. Chem. Eng. 1992, 16, 227. (11) Soroush, M.; Muske, K. Analytical Model Predictive Control. In Nonlinear Model Predictive Control; Allgo¨wer, F., Zheng, A., Eds.; Kluwer: Dordrecht, The Netherlands, 1999. (12) Allgo¨wer, F.; Doyle, F. J., III. Approximate I/O-Linearization of Nonlinear Systems. In Nonlinear Model-Based Process Control; Berber, R., Kravaris, C., Eds.; Kluwer: Dordrecht, The Netherlands, 1998. (13) van de Vusse, J. G. Plug-Flow-Type Reactor Versus Tank Reactor. Chem. Eng. Sci. 1964, 19, 994.
Received for review April 17, 2000 Revised manuscript received February 13, 2001 Accepted February 15, 2001 IE000406K
