2758
Ind. Eng. Chem. Res. 2006, 45, 2758-2768
Model-Based Controller Design for Unstable, Non-Minimum-Phase, Nonlinear Processes Chanin Panjapornpon and Masoud Soroush* Department of Chemical and Biological Engineering, Drexel UniVersity, Philadelphia, PennsylVania 19104
Warren D. Seider Department of Chemical and Biomolecular Engineering, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104-6393
This paper presents a nonlinear control method that is applicable to stable and unstable processes, whether non-minimum- or minimum-phase. The closed-loop stability is ensured by forcing all process state variables to follow their corresponding reference trajectories. These trajectories are linear and lead the process state variables to their steady-state values corresponding to desired output set-points. This state-tracking approach results in a nonlinear state feedback that tries to induce linear responses to the state variables. The controller system includes the nonlinear state feedback and a reduced-order, nonlinear state observer to reconstruct unmeasured state variables. The application and performance of the proposed control system are illustrated for a bioreactor and two chemical reactors with multiple steady states. The controller is used to operate these reactors at steady states that are unstable and non-minimum-phase. The simulation results show that the proposed control method can achieve closed-loop stability successfully. 1. Introduction Non-minimum-phase behavior of a process limits the achievable control quality and complicates the controller design for the process. The behavior can be due to the presence of an unstable mode in the zero dynamics of the process (finite righthalf-plane zero in the linear case), a time delay (infinite righthalf-plane zero in the linear case), or both. In a process, the source of a time delay, such as a transportation lag or a measurement delay, can be identified more easily than that of an unstable mode in the zero dynamics. Consequently, in the process design stage, it is easier to identify and develop a design without time delays than one without unstable modes in the zero dynamics. To achieve greater profitability, process designers have been creating designs in regions involving complex nonlinearity where process controllers continue to face stiff challenges. Steady-state multiplicity, limit cycles, chaos, and parametric sensitivity are manifestations of the nonlinearity. In addition to the nonlinearity, there are many process designs that have unstable and/or non-minimum-phase steady states. In many cases, operation is more profitable at an unstable steady state, or at a stable steady state in the close proximity of an unstable steady state, often involving non-minimum-phase behavior (inverse response). Examples of process designs that can show such behavior are those of chemical reactors,1-3 fermentation reactors,4 fluidized catalytic crackers, reactor-separator-recycle plants, azeotropic distillation columns,5 and reboilers. Processes with such designs are known to be more challenging to control than processes with stable and minimum-phase steady states. During the past 20 years, many advances have been made in nonlinear model-based control, mainly within the frameworks * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: 215-895-1710. Fax: 215-8955837.
of model-predictive control, differential-geometric control, and Lyapunov-based control. In model-predictive control, the controller action is the solution to a constrained optimization problem that is solved numerically on-line.6-8 In contrast, differential-geometric control is a direct synthesis approach in which the controller is derived by requesting a desired closedloop response in the absence of input constraints.9,10 In other words, model-predictive control involves numerical model inversion, while differential-geometric control involves analytical model inversion. In model-predictive control, non-minimumphase behavior can be handled simply by increasing prediction horizons, but in differential-geometric control, special treatment is needed. In Lyapunov-based control, closed-loop stability plays a central basic role in the controller design.11-13 The asymptotic decay of a norm of the state variables is ensured by the use of a proper Lyapunov function in the controller design. Input-output (IO) linearization has been the most widely used differential-geometric control method, because it is easy to implement (does not require solving a set of partial differential equations). However, IO linearization cannot be applied directly to processes with a non-minimum-phase (NMP) steady state, because it includes an inverse of the process model, causing closed-loop instability in the case of processes with a nonminimum-phase steady state. IO linearizing control was initially developed for unconstrained, minimum-phase (MP) processes. During the past 15 years, several successful attempts have been made to develop differential-geometric controllers applicable to NMP nonlinear processes.1-3,10,14-25 Isidori,10 Isidori and Byrnes,14 and Isidori and Astolfi15 studied total linearization, which is applicable to a limited class of minimum- and non-minimum-phase processes and requires solving a nonlinear partial differential equation (PDE). Kravaris and Daoutidis1 presented a nonlinear statefeedback controller for second-order, non-minimum-phase, nonlinear systems. Niemiec and Kravaris2 proposed a systematic
10.1021/ie050724p CCC: $33.50 © 2006 American Chemical Society Published on Web 03/16/2006
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2759
procedure for the construction of statically equivalent outputs with prescribed transmission zeros. They then designed a nonlinear state-feedback controller on the basis of the synthetic outputs. Kanter et al.3 developed nonlinear control laws for input-constrained, multiple-input, multiple-output, stable processes, whether their delay-free part is minimum- or nonminimum-phase. They addressed the nonlinear control of the processes by exploiting the connections between model-predictive control and input-output linearization. Kravaris et al.16 presented a systematic method of arbitrarily assigning the zero dynamics of a nonlinear system by constructing the requisite synthetic output maps. The minimum-phase, synthetic output maps constructed can be made statically equivalent to the original output maps, and therefore, they can be directly used for non-minimum-phase compensation purposes. The method requires solving a system of first-order, nonlinear, singular PDEs. Mickle et al.17 developed a tracking controller for unstable, non-minimum-phase, nonlinear processes by using trajectory linearization. Tomlin and Sastry18 derived tracking control laws for non-minimum-phase, nonlinear systems with both fast and slow, possibly unstable, zero dynamics. van der Schaft19 developed a nonlinear state feedback H∞ optimal controller. Devasia et al.20 and Devasia21 introduced an inversion procedure for nonlinear systems that constructs a bounded input trajectory in the pre-image of a desired output trajectory. The pre-image trajectory is noncausal (rather than unstable) in the non-minimum-phase case. Hunt and Meyer22 showed that under appropriate assumptions the bounded solution of the partial differential equation of Isidori and Byrnes14 for each trajectory of an exosystem must be given by an integral representation formula of Devasia et al.20 Chen and Paden23 studied the stable inversion of non-minimum-phase nonlinear systems. They derived a stable, but noncausal, inverse that can be incorporated into a stabilizing controller for output tracking. Doyle et al.24 presented a control synthesis scheme for nonlinear, single-input, single-output systems which have completely unstable zero dynamics. The approach is similar to linear approaches for nonminimum-phase systems and involves the derivation of an input-output linearizing controller for a suitably defined, nonlinear, minimum-phase approximation to the original system. The linearizing controller achieves an approximately linear input-output response and internal stability. McLain et al.25 proposed a controller design strategy for nonlinear systems with more manipulated inputs than controlled outputs. The controller can be used for processes with an NMP steady state. While controllers in refs 20-25 are applicable to multiinput, multi-output (MIMO), NMP processes, either sets of partial differential equations must be solved,2,15,16 or the controllers are applicable to very limited classes of processes.1,10,14,15,20-24 This work uses the same continuous-time, model-predictivecontrol framework employed by Kanter et al.3 and Soroush and Soroush.26 However, it is conceptually different. First, the controllers in refs 3 and 26 are not applicable to processes operating at an unstable, non-minimum-phase steady state, unlike the controller introduced herein. Second, the controllers in refs 3 and 26 were derived by requesting desired responses for the controlled outputs, while the controller introduced herein was obtained by requesting desired responses for the state variables. The nonlinear state feedback in ref 26 was derived by minimizing a function norm of the deviations of the controlled outputs from linear reference trajectories with orders equal to the (output) relatiVe orders. The resulting state feedback can be used to operate processes at minimum-phase steady states
and is input-output linearizing in the absence of constraints. In ref 3, a nonlinear state feedback that can be used to operate processes at a non-minimum- or minimum-phase, stable steady state was developed. It was derived by minimizing a function norm of the deviations of controlled outputs from linear reference trajectories with orders higher than the (output) relatiVe orders. The resulting state feedback is, approximately, input-state linearizing in the absence of constraints. The controllers presented herein are applicable to processes with a non-minimum-phase unstable steady state, but the differentialgeometric control laws in ref 3 are not. Because of the optimization (numerical) and shortest-prediction-horizon forms of the controller system proposed herein, an analytical proof of the asymptotic stability of the closed-loop system remains an open problem. This paper presents a nonlinear control method that is applicable to stable and unstable processes, whether nonminimum- or minimum-phase. The closed-loop stability is ensured by forcing all of the process state variables to follow their corresponding reference trajectories. The proposed control system includes a nonlinear state feedback and a reduced-order, nonlinear state observer. The nonlinear state feedback is derived by minimizing a function norm of the deviations of the state variables from their linear reference trajectories, which have orders higher than the state-Variable relatiVe orders. The resulting state feedback approximately induces linear responses to the state variables in the absence of constraints. Integral action is added to ensure offset-free output responses in the presence of constant disturbances and process-model mismatch. A state observer estimates unmeasured state variables. This paper is organized as follows. The scope of the study and some mathematical preliminaries are given in Section 2. Section 3 presents the nonlinear feedback control method. Finally, in Section 4, the application and performance of the control method are illustrated by numerical simulation of a bioreactor and two chemical reactors with multiple steady states.
2. Scope and Mathematical Preliminaries Consider general, square, multi-input, multi-output processes having a mathematical model in the form:
dx ) f(x, u) x(0) ) x0 dt y ) h(x)
(1)
where x ) [x1 ‚‚‚ xn]T ∈ Rn is the vector of state variables, u ) [u1 ‚‚‚ um]T ∈ Rm is the vector of manipulated inputs, y ) [y1 ‚‚‚ ym]T ∈ Rm is the vector of controlled outputs, f(x, u) ) [f1(x, u) ‚‚‚ fn(x, u)]T and h(x) ) [h1(x) ‚‚‚ hm(x)]T are smooth vector functions. The relative order (degree) of a state variable, xi, is denoted by ri, where ri is the smallest integer for which
( )
ri ∂ d xi *0 ∂u dtri
It is assumed that the relative orders, r1, ‚‚‚, rn, are finite and that the process is controllable and observable locally around a desired steady state.
2760
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006
Let H0(x) ) x, and define the following notation:
Hi1(x) )
If (5) has a real root for u at each time instant and ∂φp(x, u)/∂u * 0 at the root, a state feedback of the form exists:
dxi dt
u ) ψP(x, ysp)
l Hiri-1(x) ) Hiri(x, u) ) ri+1
Hi
(x, u , u ) ) (0)
(1)
dri-1xi dtri-1 drixi dtri dri+1xi dtri+1
l Hij(x, u(0), u(1), ‚‚‚, u(j-ri)) )
djxi dtj
(2)
where j g ri, i ) 1, ‚‚‚, n, and
u(l) )
Theorem 1.3 The closed-loop system under the state feedback of (6) is asymptotically stable, if the following conditions hold: (a) The nominal equilibrium pair of the process corresponding to ysp is hyperbolically stable. (b) The tunable parameters P1, ‚‚‚, Pm are chosen to be sufficiently large. (c) The tunable parameters, τ1, ‚‚‚, τm, are chosen such that for every l ) 1, ‚‚‚, m all eigenvalues of [I + τlJol(xss, uss)] lie inside the unit circle. Furthermore, as P1, ‚‚‚, Pm f ∞, the eigenvalues of the Jacobian of the closed-loop system [Jcl(xss, uss)] approach the eigenvalues of the Jacobian of the open-loop process [Jol(xss, uss)]. 2.2. Reduced-Order State Observer Design. This subsection presents a brief review of the nonlinear, reduced-order state observer described in ref 27. The state observer will be used in this paper to reconstruct unmeasured state variables of the process under consideration. Consider a nonlinear process in the form of (1) with additional output measurements Y1, ‚‚‚, Yq; that is,
dlu dtl
dx ) f(x, u) x(0) ) x0 dt y ) h(x)
For a given output set-point, ysp, the corresponding desired steady-state pair (xss, uss) satisfies
0 ) f(xss, uss)
Y ) K(x) The nonredundancy of the measured outputs ensures the existence of a locally invertible state transformation of the form
ysp ) h(xss) These relations are used to describe the dependence of a nominal (desired) steady state, xssN, on the set-point, say xssN ) F(ysp). 2.1. Approximate Input-Output Linearization.3 To place the new control method in perspective, it is instructive to review briefly the control method presented in ref 3. For a process in the form of (1), in ref 3, closed-loop process output responses of the linear form:
[] [ ] Qx η y ) T(x) ) h(x) Y K(x)
where η ) [η1, ‚‚‚, ηn-m-q]T, and Q is a constant (n - m - q) × n matrix which, for the sake of simplicity, is chosen such that (a) each row of Q has only one nonzero term equal to one and (b) locally,
{ [ ]}
(τ1D + 1)P1y1 ) ysp1
Qx ∂ h(x) rank ∂u K(x)
l (τmD + 1)Pmym ) yspm
(3)
are requested. Here, D ) d/dt; P1 g R1, ‚‚‚, Pm g Rm (R1, ‚‚‚, Rm are the relative orders of the process outputs, y1, ‚‚‚, ym, respectively); and τ1, ‚‚‚, τm are positive constants that set the speed of the responses of the process outputs, y1, ‚‚‚, ym, in a closed-loop, respectively. Taking the output time derivatives in (3) leads to the derivation of a dynamic state feedback in the compact form:
ΦP(x, u(0), u(1), ‚‚‚, u(P-R)) ) ysp
(4)
where P ) max(P1, ‚‚‚, Pm) and R ) min(R1, ‚‚‚, Rm). Setting all the time derivatives of u in (4) to zero results in the static state feedback:
φP(x, u) ) ΦP(x, u(0), 0, ‚‚‚, 0) ) ysp
(6)
(5)
)n
Thus, the state transformation [η y Y]T ) T(x) is locally invertible. The system of (1), in terms of the new state variables, η1, ‚‚‚, ηn-m-q, y, and Y, takes the form
η˘ ) Fη(η, y, Y, u) y˘ ) Fy(η, y, Y, u) Y˙ ) FY(η, y, Y, u) where
Fη(η, y, Y, u) ) Qf[T-1(η, y, Y), u] ∂h(x) | f[T-1(η, y, Y), u] ∂x x)T-1(η,y,Y) ∂K(x) | f[T-1(η, y, Y), u] FY(η, y, Y, u) ) ∂x x)T-1(η,y,Y) Fy(η, y, Y, u) )
(7)
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2761
A closed-loop, reduced-order observer is then designed in the form:
z˘ ) Fη(z + L1y + L2Y, y, Y, u) - L1Fy(z + L1y + L2Y, y, Y, u) - L2FY(z + L1y + L2Y, y, Y, u) xˆ ) T-1(z + L1y + L2Y, y, Y)
(8)
where the constant [(n - m - q) × m] and [(n - m - q) × q] matrixes, L1 and L2, are the observer gains. The observer gains should be set such that the observer error dynamics are asymptotically stable, which requires all eigenvalues of the [(n - m - q) × (n - m - q)] matrix
∂Fη(η, y, Y, u) ∂Fy(η, y, Y, u) ∂FY(η, y, Y, u) - L1 - L2 ∂η ∂η ∂η evaluated at the nominal steady-state pair, to be in the left half of the complex plane. 3. Nonlinear Control Method A state feedback that induces approximately linear responses to the state variables is first derived. A reduced-order state observer is then designed to reconstruct unmeasured state variables from the output measurements. To add integral action to the state feedback-state observer system, a dynamic subsystem is finally added. 3.1. State Feedback Design. For a process in the form of (1) with input constraints:
uli e u e uhi, i ) 1, ‚‚‚, m a linear response is requested of the following form for each of the state variables:
(1D + 1)p1x1 ) xssN1 l (nD + 1) xn ) xssNn pm
(9)
where p1 g r1, ‚‚‚, pm g rm, and 1, ‚‚‚, n are positive constants that set the speed of the state responses in closed-loop. The state responses in (9) can be achieved only when m g n. However, since in many processes m < n (there are more state variables than manipulated inputs), the state responses in (9) can rarely be achieved. To relax the requirement of linear responses for all state variables, state responses are requested that are as close as possible to the linear ones described by (9). To derive a state feedback that achieves the relaxed state responses, the following constrained optimization problem is solved at each time instant: n
min
∑
u i)1
[
]
(iD + 1) xi - xssNi
θi
pi
ipi
variables: the higher the value of θi, the smaller the deviation of the xi response from the desired linear response for xi. For a process in the form of (1), the optimization problem in (10) can be recast as
2
(10)
subject to
ui(l) ) 0, l g1, i ) 1, ‚‚‚, m uli e u e uhi, i ) 1, ‚‚‚, m where θ1, ‚‚‚, θn are adjustable, positive, scalar weights whose values are set according to the relative importance of the state
n
min u
∑ i)1
[
xi +
θi
ri-1
∑ l)1
()
il
pi Hil(x) + l
()
pi
∑
il
l)ri
]
pi Hil(x, u, 0, ‚‚‚, 0) - xssNi l
ipi
2
(11)
subject to
uli e u e uhi, i ) 1, ‚‚‚, m where
()
pi! pi ) l (pi - l)!l!
In the case that the minimum of the minimization problem of (10) is zero at every time instant, the linear, closed-loop state responses of (9) are achieved. Let the static state feedback
u ) Ψ(x, xssN)
(12)
denote the feasible solution to the m-dimensional optimization problem of (11). Like any constrained optimization problem, the constrained optimization problem of (11) may not have a feasible solution. Furthermore, the performance index may be nonconvex, and therefore, more than one minimizer may exist. The optimization problem is solved at discrete time instants (as explained in detail in the Controller System Implementation section of this article). It is easy to solve, because it is of dimension m and static. The optimization dimension is very low, compared to the dimension of the optimization problems solved in typical model-predictive controllers. The order of the linear state responses, p1, ‚‚‚, pm, should be chosen such that p1 gr1, ‚‚‚, pm grm. Note that the closed-loop eigenvalues do not move monotonically to the left in the complex plane, as the parameters p1, ‚‚‚, pm increase. Choosing p1 . r1, ‚‚‚, pm . rm places all of the closed-loop eigenvalues in the left half of the complex plane and, therefore, ensures the asymptotic stability of the closedloop system. Example: A Linear, Unstable, Non-Minimum-Phase Process. Consider the linear process:
dx1 ) x2 dt dx2 ) 10x1 + 9x2 + u dt y ) 2x1 - x2 This process is unstable (has poles at -1 and 10) and nonminimum-phase (has a transmission zero at 2). The relative orders of the states, x1 and x2, are the following: r1 ) 2 and r2 ) 1. For 1 ) 0.8, 2 ) 0.01, θ1 ) 1, and θ2 ) 1, the closedloop eigenvalues of this process under the state feedback in (11) are given in Table 1. The state feedback of (11) is capable of operating the process at any steady state when p1 and p2 are chosen such that p1 g 2 and p2 g 1.
2762
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006
Table 1. Closed-Loop Eigenvalues of the Linear Example for Several p1 and p2 Values p1
p2
λ1
λ2
1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
0.000 0.000 0.000 0.000 0.000 -1.090 -1.020 -0.970 -0.930 -0.900 -0.150 -0.153 -0.153 -0.153 -0.153 -0.017 -0.017 -0.017 -0.017 -0.017
-100.00 -47.89 -30.59 -21.99 -16.86 -1.44 -1.52 -1.60 -1.67 -1.73 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00
Figure 1. Parametrized controller system.
3.2. Reduced-Order State Observer. In general, measurements of all state variables are not available. In such cases, estimates of the unmeasured state variables can be obtained from the output measurements. Here, we use the “closed-loop”, reduced-order, nonlinear state observer of (8) to reconstruct the unmeasured state variables. The state observer is applicable to both stable and unstable processes. 3.3. Integral Action. To ensure offset-free response of the closed-loop system in the presence of constant/asymptotically constant, output disturbances and model-process mismatch, a control system should have integral action. Integral action is added to the state feedback using the approach described next. This approach is similar to the one used in internal-model control and approximate input-output linearization.4 Estimates of disturbance-free process outputs are calculated first using the closed-loop process model:
w˘ ) f(w,Ψ(w, xssN)) ξ ) h(w)
subsystem of (13) and (14) leads to the following controller system that has integral action:
z˘ ) Fη(ηˆ , y, Y, u) - L1Fy(ηˆ , y, Y, u) - L2FY(ηˆ , y, Y, u) ηˆ ) z + L1y + L2Y xˆ ) T-1(ηˆ , y, Y) w˘ ) f(w, Ψ(w, V)) V ) F(ysp - y + h(w)) u ) Ψ(xˆ , V)
(13)
where ξ is the vector of the estimates of the disturbance-free controlled outputs. The difference between these estimates and the measured controlled outputs, y, is then added to the output set-point to calculate xssN according to
xssN ) F(ysp + (h(w) - y))
Figure 2. Schematic of the nonisothermal, continuous stirred-tank reactor.
(14)
An interesting property of this approach to adding integral action is that the addition of the dynamic system of (13) to the state feedback of (12) [calculation of xssN according to (14)] adds no additional conditions for closed-loop, asymptotic stability. In other words, the asymptotic stability of the closed-loop system under the state feedback of (12) subjected to constant/asymptotically constant, output disturbances implies that the asymptotic stability of the closed-loop system under (12), (13), and (14) is subjected to the same disturbances. This property can be shown using Lyapunov’s first method. Other methods of rejecting unmeasured disturbances include the one presented in ref 28. 3.4. Controller System. Combining the state feedback of (12), the reduced-order observer of (8), and the dynamic
(15)
The control system parameters, 1, ‚‚‚, n, set the speed of the closed-loop state responses; the smaller the value of i, the faster the xi response. When the process is to operate at a minimum-phase steady state, choosing p1 ) r1, ‚‚‚, pn ) rn is adequate to ensure the asymptotic stability of the closed-loop system. When the process is to operate at a non-minimum-phase steady state, one may have to choose p1 > r1, ‚‚‚, pn > rn. A block diagram of the controller system is shown in Figure 1. Controller System Implementation. To implement the controller system, one needs to time-discretize the system of (15) or let the numerical method that is used to integrate the differential equations in (15) perform the time-discretization. The state feedback of (12) requires solving the m-dimensional static optimization problem of (11) at each time instant of the discretized system or at the end of each numerical integration time interval. 4. Illustrative Examples 4.1. Single-Input, Single-Output Chemical Reactor. Consider the constant-volume, nonisothermal, continuous stirredtank reactor (CSTR) shown in Figure 2, in which the reaction
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2763 Table 2. Values of the Parameters of the Nonisothermal Reactor Model parameter
value
unit
s Ea γ Q CAi Ti R V
5.0 × 108 8100 3.9 -2.519 × 10-2 12 300 8.314 0.1
1/s kJ/kmol (m3 K)/kmol K/s kmol/m3 K kJ/(kmol K) m3
A f B takes place in the liquid phase. The reactor model has the form:
( ) ( )
A11 )
V 12s 2
VRxˆ 2
{
22γsxˆ 1
A21 )
} ( )
C Ai - ζ C Ai - ζ -Ea dζ s exp ζQ ) -1 - γ dt Ti - ysp Rysp Ti - ysp whose Jacobian evaluated at the desired steady state has an eigenvalue in the right half of the complex plane at 36.27. Thus, the middle steady state is unstable and non-minimum-phase. Controller System. For this process, the controller system of (15) with θ1 ) 12, θ2 ) 22, p1 ) 2, and p2 ) 2 takes the form:
Ea {-R(CAi - 2xˆ 1)xˆ 22 + Eaxˆ 1(xˆ 2 - Ti)} Rxˆ 2 12(CAi - xˆ 1) V2
( )} ( ){ ( )
A20 ) x2 + 22 Q + γxˆ 1s exp exp -
Rxˆ 2
where F is the volumetric flow rate of the reactor feed and product streams, V is the reactor volume, and CAi is the concentration of A in the feed stream. The reactor parameter values are given in Table 2. This reactor has multiple steady states. The reactor temperature, T, is the controlled output; it is desired to operate the reactor at its middle (unstable) steady state by manipulating the feed flow rate, F. The operating range of the feed flow rate is 0.2 e F e 1.5 m3/h. Here, x ) [CA T]T, u ) [F], and y ) [T]. The steady-state pair corresponding to ysp ) 302 is (x1ss ) 6.319, x2ss ) 302, uss ) 0.45), which is unstable (eigenvalues of the process Jacobian evaluated at the steady-state pair are -4.5 and 0.309). The relative orders of the state variables are both unity (r1 ) r2 ) 1). The zero dynamics of this process are governed by
( )
exp -
2
Ea dT F ) γs exp C + (Ti - T) + Q dt RT A V
+
A12 ) -
dCA Ea F ) -s exp C + (CAi - CA) dt RT A V
{
1(2CAi - 2xˆ 1)
-Ea Rxˆ 2
+
}
2Ea Ea QEa exp + xˆ 1sγEa - xˆ 22Rs Rxˆ 2 Rxˆ 2
2(2Ti - 2xˆ 2 - 2Q) + V Ea 22 γs exp {(Ti - xˆ 2)xˆ 1Ea + (CAi - 2xˆ 1)Rxˆ 22} 2 Rxˆ 2 VRxˆ
( )
2
A22 ) -
22(Ti - xˆ 2) V2
xˆ 1 ) z + Ly xˆ 2 ) y
{
( ) { ( )
}
Ea u (z + Ly) + (CAi - z - Ly) Ry V Ea u L γs exp (z + Ly) + (Ti - y) + Q Ry V
z˘ ) -s exp -
( ) ( )
w˘ 1 ) -s exp w˘ 2 ) γs exp -
( ( )
V1 ) - exp
Ea Ψ(w, V) w + (CAi - w1) Rw2 1 V
Ea Ψ(w, V) w + (Ti - w2) +Q Rw2 1 V
Ea Q - s(CAiγ + Ti - V2) + RV2
x{ ( ) exp
}
}
Ea Q - s(CAiγ + Ti - V2) RV2
2
( )
+ 4 exp
)
Ea sγQCAi /(2γs) RV2
V2 ) ysp - xˆ 2 + w2 u ) Ψ(xˆ , V) ) arg{min[(-V1 + A10 + A11u + A12u ) + 2 2
u
(-V2 + A20 + A21u + A22u2)2]} subject to
0.2 e u e 1.5 where
A10 ) xˆ 1 +
{
1xˆ 1s Rxˆ 22
( )
-(1QEa + 2Rxˆ 22) exp -
Ea + Rxˆ 2
( )}
1s(-Eaγxˆ 1 + Rxˆ 22) exp -
2Ea Rxˆ 2
The following controller parameter values are used: 1 ) 468 s, 2 ) 468 s, and L ) 20. The closed-loop eigenvalues of this process under the state feedback of (11), for several p1 and p2 values, are given in Table 3. MATLAB optimization toolbox is used to solve the constrained optimization problem of the controller system. The number of the optimizing variables of this optimization problem is equal to the number of manipulated inputs (m ) 2). Therefore, the computational cost of this controller is much less than that of a typical model-predictive controller. Controller Performance. Servo and regulatory responses of the controller are shown in Figures 3 and 4. Figure 3 depicts the evolution of the process state variables for two initial conditions, [x1(0), x2(0)] ) [9.162, 290] in the minimum-phase
2764
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006
Figure 3. Closed-loop responses of the state variables of the chemical reactor.
Figure 4. Manipulated input and unmeasured disturbance profiles corresponding to Figure 3.
Table 3. Closed-Loop Eigenvalues of the CSTR Example for Several p1 and p2 Values
4.2. Single-Input, Single-Output Bioreactor. Consider the constant-volume, continuous bioreactor described in ref 29. The process dynamics are described by
p1
p2
λ1
λ2
1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3
7.1628 -3.1375 - 0.5928i -4.264 16.118 -3.4071 + 0.7204i -4.3881 0.0003 -3.4752 + 0.3415i -4.3489
-3.3332 -3.1375 + 0.5928i -1.4298 -3.024 -3.4071 - 0.7204i -1.5140 -4.9778 -3.4752 - 0.3415i -1.5317
region and [x1(0), x2(0)] ) [2.923, 320] in the non-minimumphase region, with CAi ) 12 kmol/m3. The set-point, ysp ) 302, corresponds to (x1ss ) 6.319 and x2ss ) 302). In addition, when the process is at the desired set-point, the regulatory performance of the controller is studied. A step change from 12 to 10 kmol/ m3 in CAi is made at t ) 2 h (unmeasured disturbance). The simulation results show that the controller successfully operates the reactor at the desired steady state, which is non-minimumphase and open-loop unstable. The controller is capable of operating the process at the desired steady state, regardless of the initial conditions of the process. The integral action of the controller ensures offset-free response in the presence of the unmeasured disturbance. Although the disturbance enters in one of the state equations, the control system is capable of ensuring closed-loop stability and an offset-free response in the presence of this disturbance.
x˘ 1 ) -x1u + x1(1 - x2) exp x˘ 2 ) -x2u + x1(1 - x2) exp
()
() x2 γ1
x 2 1 + β1 γ1 1 + β 1 - x 2
y ) x1 where x1 is the dimensionless cell-mass concentration in the bioreactor, x2 is the dimensionless substrate concentration, and u is the dilution rate. The model parameters are β1 ) 0.02 and γ1 ) 0.48. It is desired to maintain the cell-mass concentration, x1, at a set-point by manipulating the dilution rate, u, within the range, 0.4 e u e 1.0. The process steady-state pair corresponding to ysp ) 0.1448 (x1ss ) 0.1448, x2ss ) 0.8455, uss ) 0.9) is unstable (eigenvalues of the open-loop Jacobian evaluated at the steady state are 0.061 + 1.731i and 0.061 - 1.731i). The relative orders of the state variables are the following: r1 ) 1 and r2 ) 1. The zero dynamics of the process are governed by
()
dζ ζ ) -yspζ(1 - ζ) exp + dt γ1 ysp(1 - ζ) exp
()
ζ 1 + β1 γ1 1 + β 1 - ζ
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2765 Table 4. Closed-Loop Eigenvalues of the Bioreactor Example for Several p1 and p2 Values p1
p2
λ1
λ2
1 1 1 2 2
0 1 2 1 2
-10 -2.91 -2.91 -2.91 -2.91
3.4557 -0.85 -0.83 -0.83 -0.83
whose Jacobian evaluated at the steady state has an eigenvalue in the right half of the complex plane at 1.374. Thus, the desired steady state is unstable and non-minimum-phase. Controller System. For this process, the controller system of (15), with θ1 ) 1, θ2 ) 2, p1 ) 1, and p2 ) 1, takes the form:
u ) Ψ(xˆ , V) ) arg{min[(-V1 + A10 + A11u)2 + u
(-V2 + A20 + A21u)2]} subject to
0.4 e u e 1.0 where
A10(xˆ ) ) xˆ 1 + 1xˆ 1(1 - xˆ 2) exp
() xˆ 2 γ1
A11(xˆ ) ) -xˆ 11 A20(xˆ ) ) xˆ 2 + 2xˆ 1(1 - xˆ 2) exp
()
xˆ 2 1 + β1 γ1 1 + β1 - xˆ 2
A21(xˆ ) ) -xˆ 22 xˆ 1 ) y Figure 5. Closed-loop responses of the bioreactor state variables.
xˆ 2 ) z + Ly
{
z˘ ) -(z + Ly)u + x1{1 - z - Ly} exp
(
}
)
1 + β1 z + Ly γ1 1 + β1 - z - Ly
{
L -x1u + x1{1 - z - Ly} exp w˘ 1 ) -w1q + w1(1 - w2) exp w˘ 2 ) -w2q + w1(1 - w2) exp
()
()
(
)}
z + Ly γ1
w2 γ1
w2 1 + β1 γ1 1 + β 1 - w 2
ν1 ) ysp - xˆ 1 + w1 ν2 ) F2(V1) q ) Ψ(w, V) The following controller parameter values are used: 1 ) 0.001, 2 ) 0.15, and L ) 0.01. The closed-loop eigenvalues of this process under the state feedback of (11), for several p1 and p2 values, are given in Table 4. Controller Performance. Figures 5 and 6 show the servo and regulatory responses of the controller. For two initial conditions, [x1(0), x2(0)] ) [0.1, 0.75] and [0.25, 0.2], the process state profiles are depicted in Figure 5. The set-point, ysp ) 0.1448, corresponds to the steady state (x1ss ) 0.1448 and x2ss ) 0.8453). Figure 5 shows that the controller successfully operates the bioreactor at the desired steady state, irrespective of the initial conditions of the process. To evaluate
the regulatory performance of the controller, a step change from 0.02 to 0.022 in β1 of the process (unmeasured disturbance) is made at t ) 8 h when the process is at the desired steady state. As Figure 5 shows, the controller rejects the unmeasured disturbance asymptotically. Also in this example, the disturbance enters in one of the state equations and the control system is capable of ensuring closed-loop stability and an offset-free response in the presence of this disturbance. 4.3. Multi-Input, Multi-Output Chemical Reactor. Consider the constant-volume, nonisothermal, continuous chemical reactor shown in Figure 7, in which the series reactions, A f B f C, take place in the liquid phase. The process dynamics are represented by the following model:
( )
E1 dCA F ) (CAi - CA) - s1 exp C 2 dt V RT A E1 E2 dCB F ) - CB + s1 exp C 2 - s2 exp C dt V RT A RT B (-∆H1) E1 dT F s1 exp C 2+ ) (Ti - T) + dt V Fcp RT A E2 (- ∆H2) US s2 exp C + (T - T) Fcp RT B FcpV j
( ) ( ) ( ) ( )
dTj Fj US ) (Tji - Tj) (T - T) dt V FjcpjVj j y 1 ) CB y2 ) T
2766
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006
Figure 6. Manipulated input and unmeasured disturbance profiles corresponding to Figure 5.
Figure 8. Closed-loop responses of the first two state variables of the chemical reactor. Table 5. Parameter Values of the Nonisothermal Jacketed Reactor
Figure 7. Schematic of the multivariable, nonisothermal chemical reactor.
The reactor parameter values are given in Table 5. It is desired to maintain CB and T at set-point values by manipulating F and Fj within the operating ranges, 10 e F e 150 L/h and 10 e Fj e 150 L/h. In this process, only the state variable, CA, is not measured. Here, x ) [CA CB T Tj]T, u ) [F Fj]T, y ) [CB T]T, and Y ) [Tj]. The steady state corresponding to the set-points, y1sp ) 5.233 and y2sp ) 443.92, is (x1ss ) 0.701, x2ss ) 5.233, x3ss ) 443.92, x4ss ) 403.24), which is unstable (eigenvalues of the Jacobian of the process evaluated at the steady state are -491.65, -90.71, 86.29, and -15.36). The relative orders of the state variables are r1 ) 1, r2 ) 1, r3 ) 1, and r4 ) 1. The zero dynamics of this process are governed by
[ ( ) ( ) ]
E1 dζ1 1 ) s1 exp ζ 2 - s2 dt y1sp Ry2sp 1 E2 E1 exp y1sp (CAi - ζ1) - s1 exp ζ2 Ry2sp Ry2sp 1
(
)
whose Jacobian evaluated at the steady state has an eigenvalue
parameter
value
unit
F Fj CAi s1 s2 -E1/R -E2/R Ti Tji -∆H1 -∆H2 F Fj cp cpj U S V Vj
80.95 100 12 2.5 × 1010 1.5 × 1010 8000 9100 320 298.15 20 80 1 1.1 2.25 3 3825 0.225 5 5
L/h L/h mol/L L/(mol h) 1/h K K K K kJ/mol kJ/mol kg/L kg/L kJ/(kg K) kJ/(kg K) kJ/(m2 K s) m2 L L
at 589.59. Thus, the desired steady state is unstable and nonminimum-phase. For this process, the controller system (15), with θ1 ) 12, θ2 ) 22, θ3 ) 32, θ4 ) 4, p1 ) 2, p2 ) 2, p3 ) 2, and p4 ) 1, is implemented, with 10 e u1 e 150 and 10 e u2 e 150. The following controller parameter values are used: 1 ) 0.17, 2 ) 0.17, 3 ) 0.17, 4 ) 0.17, L11 ) -0.03, L12 ) -0.03, and L21 ) 0. Controller Performance. It is desired to maintain the controlled outputs at y1sp ) 5.233 and y2sp ) 443.92, which correspond to the steady state (x1ss ) 0.701, x2ss ) 5.233, x3ss ) 443.92, x4ss ) 403.24). Figures 8, 9, and 10 show the servo and regulatory responses of the controller. The process state
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2767
Figure 9. Closed-loop responses of the last two state variables of the chemical reactor.
variable profiles for two initial conditions, [x1(0), x2(0), x3(0), x4(0)] ) [1.814, 9.446, 387.69, 380.67] and [0.519, 3.5, 455, 410], are shown in Figures 8 and 9; the controller is capable of operating the process at the unstable, non-minimum-phase steady state. After the process reaches the desired steady state, a step change from 12 to 11.8 kmol/m3 in CAi (unmeasured disturbance) is made at t ) 8 h. From Figures 8 and 9, it can be seen that the controller provides offset-free responses in the presence of the unmeasured disturbance. The corresponding manipulated input profiles are shown in Figure 10. This disturbance also enters in one of the state equations. Despite this, the control system is capable of ensuring closed-loop stability and an offsetfree response in the presence of the disturbance. 5. Conclusions A control law is presented that is applicable to general multivariable processes, whether minimum- or non-minimumphase. The state feedback of the control law is obtained by requesting state responses closest to a set of desired linear state responses. Thus, the controller forces process state variables to be as close as possible to their corresponding desired linear responses. The integral action ensures offset-free, closed-loop output response in the presence of constant output disturbances and process-model mismatch. A closed-loop, reduced-order observer is used to estimate the unmeasured state variables. Compared to a general, multivariable model-predictive controller, this control system has less tuning parameters to achieve closed-loop, asymptotic stability. The control system does not have the limitation of the control method presented by Kanter
Figure 10. Manipulated input profiles corresponding to Figures 8 and 9.
et al.3 However, because of the optimization (numerical) and shortest-prediction-horizon forms of the proposed controller system, an analytical proof of the asymptotic stability of the closed-loop system is still an open problem. Acknowledgment This study was supported by the National Science Foundation Grants CTS-0101133 and CTS-0101237. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Nomenclature A, B, C ) chemical species cp ) heat capacity of feed and product, kJ/(kg K) cpj ) heat capacity of jacket fluid, kJ/(kg K) CAi ) inlet concentration of the reactant, kmol/m3 CA ) outlet concentration of the reactant, kmol/m3 CB ) outlet concentration of B, mol/L D ) differential operator, D ) d/dt. Ea, E1, E2 ) activation energy, kJ/mol F ) reactor feed flow rate, L/h Fj ) jacket coolant flow rate, L/h Jol ) open-loop Jacobian Jcl ) closed-loop Jacobian L1, L2 ) observer gains m ) number of manipulated inputs and controlled outputs n ) number of state variables ri ) relative order of state variable xi R ) universal gas constant, kJ/(kmol K)
2768
Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006
s, s1, s2 ) pre-exponential factor, 1/s, L/(mol h), 1/h S ) heat transfer surface area, m2 t ) time, s T ) reactor outlet temperature, K Ti ) reactor inlet temperature, K Tj ) jacket temperature, K Tji ) jacket inlet temperature, K u ) vector of manipulated inputs U ) overall heat-transfer coefficient, kJ/(m2 K s) V ) reactor volume, m3 Vj ) jacket volume, m3 w ) vector of the controller state variables x ) vector of state variables y ) vector of controlled outputs ysp ) vector of set-points -∆H1, -∆H2 ) heat of the reactions, kJ/mol Greek 1, ‚‚‚, n ) adjustable parameters of the controller γ ) reactor model parameter, K m3/kmol ζ ) vector of the state variables of the zero dynamics F ) density of the mixture in the reactor, kg/L Fj ) density of the jacket fluid, kg/L Superscript ˆ ) estimate Subscripts A, B ) chemical species ss ) steady state sp ) set-point Literature Cited (1) Kravaris, C.;Daoutidis, P. Nonlinear state feedback control of secondorder nonminimum phase nonlinear systems. Comput. Chem. Eng. 1990, 14 (4/5), 439. (2) Niemiec, M.; Kravaris, C. Nonlinear model-state feedback control for non-minimum-phase processes. Automatica 2003, 39, 1295. (3) Kanter, J. M.; Soroush, M.; Seider, W. D. Nonlinear controller design for input-constrained, multivariable processes. Ind. Eng. Chem. Res. 2002, 41, 3735. (4) Dochain, D. Adaptive control algorithms for nonminimum phase nonlinear bioreactors. Comput. Chem. Eng. 1992, 16 (5), 449. (5) Kumar, A.; Daoutidis, P. Modeling, analysis and control of ethylene glycol reactive distillation column. AIChE J. 1999, 45 (1), 51. (6) Mayne, D. Q.; Rawlings, J. B.; Rao, C. V.; Scokaert, P. O. M. Constrained model predictive control: Stability and Optimality. Automatica 2000, 36 (6), 789. (7) Allgower, F.; Zheng, A. Nonlinear Model PredictiVe Control; Progress in Systems and Control Theory series; Birkhauser Verlag: Basel 2000; Vol. 26. (8) de Oliveira, S. L.; Kothare, M. V.; Morari, M. Contractive model predictive control for constrained nonlinear systems. IEEE Trans.Autom. Control 2000, 45, 1053.
(9) Kravaris, C.; Daoutidis, P.; Wright, R. A. Output feedback control of nonminimum-phase nonlinear processes. Chem. Eng. Sci. 1994, 49 (13), 2107. (10) Isidori, A. Nonlinear control systems; Springer-Verlag: New York, 1995. (11) El-Farra, N. H.; Christofides, P. D. Integrating Robustness, Optimality and Constraints in Control of Nonlinear Processes. Chem. Eng. Sci. 2001, 56, 1841. (12) El-Farra, N. H.; Christofides, P. D. Bounded Robust Control of Constrained Nonlinear Processes. Chem. Eng. Sci. 2003, 58, 3025. (13) Dubljevic, S.; Kazantzis, N. A new Lyapunov design approach for nonlinear systems based on Zubov’s method. Automatica 2002, 38, 1999. (14) Isidori, A.; Byrnes, C. I. Output regulation of nonlinear systems. IEEE Trans. Autom. Control 1990, 35, 131. (15) Isidori, A.; Astolfi, A. Disturbance attenuation and H∞ control via measurement feedback in nonlinear systems. IEEE Trans. Autom. Control 1992, 37, 1283. (16) Kravaris, C.; Niemiec, M.; Kazantzis, N. Singular PDEs and the assignment of zero dynamics in nonlinear systems. Syst. Control Lett. 2004, 51, 67. (17) Mickle, M. C.; Huang, R.; Zhu, J. J. Unstable, nonminimum phase, nonlinear tracking by trajectory linearization control. Proceedings of the IEEE International Conference on Control Applications, Taipei, Taiwan, 2004; p 812. (18) Tomlin, C. T.; Sastry, S. S. Bounded tracking for nonminimum phase nonlinear systems with fast zero dynamics. Int. J. Control 1997, 68, 819. (19) van der Schaft, A. J. L2-Gain analysis of nonlinear systems and nonlinear state feedback H∞ control. IEEE Trans. Autom. Control 1992, 37, 770. (20) Devasia, S.; Chen, D.; Paden, B. Nonlinear inversion-based output tracking. IEEE Trans. Autom. Control 1996, 41, 930. (21) Devasia, S. Approximated stable inversion for nonlinear systems with nonhyperbolic internal dynamics. IEEE Trans. Autom. Control 1999, 44, 1419. (22) Hunt, L. R.; Meyer, G. Stable inversion for nonlinear systems. Automatica 1997, 33, 1549. (23) Chen, D. G.; Paden, B. Stable inversion of nonlinear non-minimum phase systems. Int. J. Control 1996, 64, 81. (24) Doyle, F. J., III; Allgower, 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. (25) McLain, R. B.; Kurtz, M. J.; Henson, M. A.; Doyle, F. J., III Habituating control for nonsquare nonlinear processes. Ind. Eng. Chem. Res. 1996, 35, 4067. (26) Soroush, M.; Soroush, H. M. Input-output linearizing nonlinear model predictive control. Int. J. Control 1997, 68 (6), 1449. (27) Soroush, M. Nonlinear state-observer design with application to reactors. Chem. Eng. Sci. 1997, 52 (3), 387. (28) El-Farra, N.; Christofides, P. D. Robust Inverse Optimal Control Laws for Nonlinear Systems. Int. J. Robust Nonlinear Control 2003, 13, 1371. (29) Brengel, D. D.; Seider, W. D. Multistep nonlinear predictive controller. Ind. Eng. Chem. Res. 1989, 28 (12), 1812.
ReceiVed for reView June 16, 2005 ReVised manuscript receiVed December 14, 2005 Accepted February 3, 2006 IE050724P
