Design of a Class of Stabilizing Nonlinear State Feedback Controllers

Ind. Eng. Chem. Res. 1998, 37, 131-144

131

Design of a Class of Stabilizing Nonlinear State Feedback Controllers with Bounded Inputs Antonio A. Alonso*,† and Julio R. Banga‡ Department of Chemical Engineering, Facultad de Ciencias, Universidad de Vigo, Aptdo 874, 36200 Vigo, Spain, and Chemical Engineering Lab, Instituto Investigacions Marinas (CSIC), E/Eduardo Cabello 6, 36208 Vigo, Spain

This paper addresses the problem of designing stabilizing global linearization controllers subject to bounded inputs. It is well-known that saturation of the input variables usually leads to a serious performance degradation of even instability of processes if undesired attractors are present. Instability occurs in open-loop unstable plants once the controller hits the constraints. If the plant is open-loop stable, saturation may also lead to instability in the form of limit cycles. This paper provides conditions under which the existence of stable global linearization controllers for SISO and MIMO plants is ensured. These conditions will set up the basis to derive a tuning technique that, preserving stability, renders a quite acceptable performance in terms of closedloop response. Two problem categories, and solution strategies, will be considered: (1) If the set of constrained inputs Ub is given, a feasible region in the state space is calculated and a low-gain static controller computed to preserve stability. Performance is then improved by adding and saturating and external controller. (2) If the feasible set in the state space and the desired dynamic behavior are given, a feasible set, Ub, and thus a global linearization controller, is computed. Performance is improved through an external and possibly saturated controller. The methodology we propose represents a simple way of designing nonlinear state feedback controllers for systems affected by severe nonlinearities and undesired attractors such as chemical reactors and it is particulary attractive for open-loop unstable plants. It applies to SISO as well as to MIMO systems and is illustrated through two simulation examples that involve continuousstirred tank reactors. The first of them consists of the control of a non-isothermal reactor at the unstable state, while the second consists of the control of level and concentration. 1. Introduction During the past decade, developments in differentialgeometric nonlinear control theory (Isidori, 1988; Kravaris, 1988; Kravaris and Kantor, 1990a,b) have made it possible to design control structures that, by using a physically meaningful model of the plant, are able to globally stabilize systems exhibiting strong nonlinearities such as chemical reactors. The design techniques consist of transforming the original nonlinear system into a linear equivalent one by appropriate coordinate transformations. The desired response is then obtained by allocating the poles of the resulting linear system. However, when the plant is perturbed, the controller may reach the physical limits on the control inputs. If this happens, global linearization fails and performance degradation or instability may occur (Calvet and Arkun, 1988). It is a fact that any plant operates under input constraints. Sometimes, those are specified before designing the controller. In other occasions, and on the basis of cost considerations and on the region of the state space where the plant is expected to operate, the control designer may recommend certain minimum acceptable limits on the actuators. These constraints, if not * Author to whom correspondence should be addressed. Tel.: +34-86-812383. Fax: +34-86-812382. E-mail: aalvarez@ seinv.uvigo.es. † Universidad de Vigo. ‡ Instituto Investigacions Marinas (CSIC). Tel.: +34-86231930. Fax: +34-86-292762. E-mail: [email protected].

satisfied, can make the system drift away from the operating region to undesired attractive points. Even if stability is guaranteed, saturation of control variables could result in serious control performance degradation and undesired travels of the state variables. Physically, those travels may represent, for instance, temperatures at which secondary reactions activate, explosion phenomena, or phase changes in the reaction mixture. Under these circumstances, it seems reasonable to avoid saturation or if that happens, to have the plant back under control as soon as possible. In linear systems, that can be achieved by direct reparameterization of the controllers (Rotea and Marchetti, 1987) or by inclusion of saturation compensation schemes (antiwindup) (Campo and Morari, 1990). Special attention is paid to stability issues such as limit cycles, derived from the implementation of those nonlinear elements. In this context, common approaches, like stable saturation schemes, have been successfully employed to preserve stability of linear and open-loop stable systems (Campo and Morari, 1990; Gutman and Hagander, 1985). Different approaches have been considered to handle constraints in designing nonlinear controllers. Calvet and Arkun (1988) opted for the reconstruction of the external linear input by observing the current states as well as the original input variable. The observations were fed into an external feedback controller designed on an IMC framework. Although performance was considerably improved (the mismatch originated on the external linear loop was reduced), stability was not

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132 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998

guaranteed. Alternatively, the input constraints can be directly included in the control structure through the solution of an optimal control problem as that formulated in the field of model predictive control (see, for example, Li et al. (1990)). For instance, Lee and coworkers (Lee et al., 1991) employed a constrained optimization framework integrated into their generic model control algorithm to solve the constrained problem notwithstanding a high computational effort. In a recent paper, Kurtz and Henson (1997) employ global linearization concepts to derive linear time-varying constraints that are integrated into a model predictive scheme. Independently of the control structure, it is clear that if the plant is open-loop unstable, saturation should not occur at any moment unless the “size” of the actuators is such that the reachable region in the state space encloses all the equilibrium points, stable and unstable. This was shown in a very elegant manner by Alvarez et al. (1992). However, the same authors recognize the possibility of instability in the form of limit cycles. Besides, one can think of situations where extremely big (and probably too expensive) actuators are required to enclose all stationary points. Consequently, if there are attractors not reachable for the set of bounded inputs, the only alternative is to operate in a way such that the inputs are restricted to their physical limits. Since in closed loop, they depend on the operation region, that must be either specified or estimated as part of the control design problem. If the plant is open-loop stable, there exists a potential for the appearance of limit cycles under control saturation, and that must be also taken into account on the design of the nonlinear controller. This problem inspired a significant number of contributions in the recent control literature (see, for instance, Lin and Sontag (1991) and Zhang and Hirschorn (1994)). The approach taken requires the knowledge of a Lyapunov function for the nonlinear system in order to derive a globally stabilizing nonlinear state feedback controller for bounded inputs. However, despite its theoretical importance, the derivation of these control laws seems quite involved. The approach we adopt in this contribution is that of designing global linearization controllers so to be stable for arbitrary state space regions. What we mean by arbitrary state space regions depends on the particular problem, although two basic categories can be stated: (1) Given a set of constrained inputs, design a stable global linearization controller and thus estimate the reachable set in the state space (if possible maximal) compatible with it. (2) Given an operation region, estimate a set of suitable actuators (if possible minimal) or, equivalently, a set of constrained inputs and design a stable global linearization controller compatible with it. The so-called reachable set, in the first problem category, or the operation region, in the second, will be designated from now on as Xf. This set is assumed to contain a set X0, in the state space, of all acceptable perturbations that the plant can handle under control. In this way, and for both problem categories, either X0 is limited in a way such that any trajectory starting in X0 remains in Xf, or the controller is tuned so to accomplish that objective for a prescribed X0. In the first case, actions must be taken, usually on a plant design context, to bound the set of admissible distur-

bances that induce X0. In the second case, the set of perturbations is known from plant measures and X0 is computed accordingly. The results presented in this article allow the solution of both problem categories in a simple way. Once the set of bounded inputs is given (or estimated from the proposed theory) a low-gain nonlinear controller that satisfies the input constraints for a prescribed or estimated operation region is computed. Performance is then improved by adding and saturating an external controller. This controller operates on the external loop (i.e., on the equivalent linear system) and it is designed to be stable under saturation. In this way, the proposed methodology parallels and extends the results of Gutman and Hagander (1985) and Lin and Saberi (1993) to nonlinear systems. It must be pointed out that once the input set has been determined in the second problem category, the control tuning follows the same steps as for the first problem category. For this reason, the methodology will be illustrated by assuming that the bounded input set is given. The paper is as follows: In section 2, input-output global linearization is briefly described together with some useful results from linear system theory. The methodology for SISO (single input single output) systems is presented in section 3. There, we first develop the theoretical aspects, summarized in two fundamental results. One refers to the sufficient conditions for the existence of nonlinear controllers, feasible for the set of constrained inputs. The other (developed in the Appendix) allows the construction of an external controller, stable under saturation. The design methodology is then presented and illustrated on a simulation example that consists of the control of a nonisothermal CST reactor at the unstable steady state. In section 4, the methodology is extended to MIMO (multiple inputs multiple outputs) systems by employing the same line of arguments. Finally, in this section, the control of level and concentration on a isothermal CST reactor will serve to illustrate the design procedure. 2. Global Linearizing Controllers: A Brief Description Next, we summarize some concepts regarding global linearization control as well as some results from linear system theory that will be employed in the subsequent sections. For a more detailed description, the reader is referred to Isidori (1988), Kravaris and Kantor (1990a,b), and Kailath (1980). Linear Equivalent Systems. Consider the following dynamic SISO system:

x˘ ) f(x) + g(x)u

(1)

y ) h(x)

(2)

where x ∈ R η represents the state and u,y ∈ R are the input and output, respectively. f(x) and g(x) are smooth vector fields of appropriate dimensions and h(x) a scalar C∞ function that relates the state to the output. The vector fields are assumed to be Lipschitz continuous in a neighborhood of a certain equilibrium point x0. The necessary and sufficient condition for the existence of a global linearizing transform relies on the notion of relative degree. It is defined for the system (1) and (2) as the smallest integer number F for which LgLF-1 h(x) f * 0. Lfh and Lgh represent Lie derivatives of the output

Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 133

scalar function h(x) along the vector fields f(x) and g(x), respectively. If the system described by eqs 1 and 2 has relative degree, there exists a nonlinear transformation z ) Φ(x), defined as

( )

h(x) Lfh(x) Φ(x) ) ‚‚‚ F-1 Lf h(x)

and (7) into an equivalent system of the form

z˘ i1 ) zi2 ‚‚‚ z˘ Fi i-1 ) zFi i

(3)

so that the system (1)-(2) can be transformed into a partially linear system of the following form:

z˘ 1 ) z2 z˘ 2 ) z3 ‚‚‚ z˘ F-1 ) zF z˘ F ) v η˘ ) q(z,η) y ) z1

(10)

z˘ Fi i ) vi

η˘ ′ ) q′(z,η′) yi ) zi1

where the vector η′ stands for the states that constitute the zero dynamics. vi represents the new manipulated variable, related to the physical one by the following equation: m

(4)

vi ) LFf ihi +

ujLg LFf -1hi ∑ j)1

(11)

i

j

Thereinafter, we will refer to the complete state by the vector θT ) (zT l ηT). Equation η˘ ) q(z,η) represents the internal dynamic of the system (zero dynamics). If F ) n, the system does not have internal dynamic and transformation (3) is invertible. The transformed input v is related to u by the following equation:

For convenience, we group the states into the following vectors: A vector of transformed states ξ ) (z1T, ..., ziT, ..., zmT)T and the vector of complete states Θ ) (ξT,η′T)T. Linear Control Laws. Once the equivalent partially linear systems have been constructed (eqs 4 or 10), it is possible to induce a predefined dynamic behavior by acting on the new manipulated variables v with a proper control law. For an SISO system, this law takes the following form:

v ) LFf h(x) + LgLF-1 h(x)u f

v ) -KT(z - zsp)

(5)

Global linearization is extended to MIMO systems of the following form:

(12)

where K constitutes the vector of gains K ) (k0, ..., kF-1)T, and zsp the set point. With a proper translation of coordinates, the linear control law can be written as

m

x˘ ) f(x) +

gi(x)ui ∑ i)1

y1 ) h1(x) ‚‚‚ ym ) hm(x)

(6)

-S(KT,z) ) KTz

(7)

From a geometrical point of view, eq 13 can be interpreted as a surface that under closed loop contains all the z-trajectories. Combining eqs 5 and 13, the following nonlinear control law results:

if the system has a vector of relative degrees (F1, ..., Fm) at x ) x0 or, equivalently, if the following conditions hold: (i) LgjLkf hi (x) ) 0 for every 1 e j e m, 1 e i e m, and 0 e k < Fi - 1, for all x in a neighborhood of x0. (ii) The m × m matrix

(

Lg1LF1-1 h1(x) f

F1-1 ‚‚‚ LgmLf h1(x) Π(x) ) ‚‚‚ ‚‚‚ ‚‚‚ Fm-1 Lg1Lf hm(x) ‚‚‚ LgmLFm-1 hm(x) f

)

() ( )

kiLifh(x) ∑ i)0

F-1

-LFf h(x) u)

(14)

LgLF-1 h(x) f and the structure for the closed-loop system becomes

z˘ ) Acsz (8)

is nonsingular at x ) x0. These conditions allow us to define a set of nonlinear transformations:

hi zi1 L fhi zi ) ‚‚‚ ) Φi(x) ) ‚‚‚ zFi i LFf i-1hi

(13)

(9)

for i ) 1, ..., m, that transform the original system (6)

η˘ ) q(z,η)

(15)

where η refers to the internal (uncontrollable) dynamics and Acs is of the form

(

0 0 · Acs ) · · 0 -k0

1 0 · · · 0 -k1

0 1 · · · 0 -k2

... ... · · · ... ...

0 0 · · · 1 kF-1

)

(16)

The corresponding linear control law for a MIMO


system is defined in the same way:

v ) -Kξ

(17)

where v ) (v1, ..., vm)T represents now a vector of manipulated variables and K the corresponding gain m matrix, of dimension m × ∑i)1 Fi. The coordinate system has been translated so the set point ξsp ) 0. Thereinafter, we will refer to each element vi under control as

vi ) Si ) -(Kξ)i

(18)

where (‚)i represents the ith row. The closed-loop dynamics become of the form

in open loop. That means that in the presence of undesired attractors, the system will never return to the desired operating point. The results we give in sections 3 and 4 state sufficient conditions under which feasible stabilizing nonlinear controllers of the form (12) or (17) exists for arbitrary operation regions in the state space. 3. Control in the Presence of Constraints for SISO Systems Let us consider the system described by eqs 1 and 2. The control law is constrained by the set Ub in the following manner:

kiLifh(x) ∑ i)0

F-1

ξ ) Acmξ

-LFf h(x) -

η ) q(ξ,η)

(19)

where, as for the SISO case, the structure of the matrix Acm depends on K. Stability of the Closed-Loop System. For either the SISO or the MIMO linear equivalent systems described above, stability will be assured as long as the internal dynamics are stable and the roots of the polynomials p(s) ) |sI - C|, with C equal to Acs in the SISO case and Acm in the MIMO case, lie all in the negative part of the complex plane (Isidori, 1988). In fact, the state vectors z and ξ in systems (15) and (19) respectively become uniformly and exponentially stable. In the SISO case, the norm of z is bounded as follows:

|z(t)| e γe-λ(t-t0)|z0|

for all t g t0

(20)

for γ,λ positive constants. An equivalent bound can be found for ξ. If there is no internal dynamics, this bound can be employed to relate the initial condition region (X0) with the region of evolution of the states (Xf). To that purpose, let us consider the solution of eq 4:

z(t) ) eAcstz0

(21)

The following bound on the initial conditions will restrict the states z to the region |z(t)| e R ∀t:

|z0| e

R where δ ) maxt|eAcst| δ

(22)

Similar conclusions can be drawn for ξ. It must be pointed out that if an internal dynamics exists, that must be considered in deriving a bound for the region X0. Because, in general, the internal dynamics will be nonlinear, analytic bounds will not be easily found. For these cases, R would be better estimated through simulation. Effects of Control Saturation. What has been described so far is valid as long as the system does not reach the constraints imposed by the actuators. If it does, the nonlinear transformations will no longer yield a linear (or partially linear) equivalent system. In the presence of input constraints, the input vector is restricted to a region (Ub) bounded by a maximum and a minimum control action, umax and umin. The following relation holds for the elements of the set (Ub):

uj ) uj min + R(uj max - uj min)

0eRe1

(23)

When saturation occurs, the system behaves as it were

u)

umin < u < umax (24)

LgLF-1 h(x) f

u ) umin u e umin u ) umax u g umax

(25)

For a given set of parameters ki, a perturbation can drive the plant to a point in the state space where limits are met. Then linearization fails and performance degradation or instability will occur. If the plant is open-loop stable, instability could appear in the form of limit cycles. For open-loop unstable plants control is not possible and the system will evolve out of the operation region to undesired attractors. The objective of this section is to develop sufficient conditions under which stability is preserved. These results will be used to derive simple design procedures that allow the solution of both problem categories discussed in the Introduction. No optimallity in the control performance is claimed. However, as it will be illustrated on the examples, the proposed methodology results in controllers with quite acceptable performances for a set of bounded perturbations around the operation point. Before embarking into the formal aspects of the theory, let us consider the following, very simple case. Suppose a system of the form (1)-(2) with y ) x1:

(

-x + x2 f(x) ) -x1 1

)

g(x) )

() 0 1

This system is already linear and therefore eq 14 acts as a proportional controller that relocates the poles of the original plant. However, it will be useful in motivating the theory below. For this system (F ) 2), transformation (3) becomes

() (

z1 x1 z2 ) -x1 + x2

)

(26)

If the input u lies in the interval Ub ) [-1, +1], eq 24 results in the form

u ) x2 - k0z1 - k1z2

-1 < u < +1

u ) -1

u e -1

u ) +1

u g +1

(27)

(28)

Defining the functions Ψmin ) -x2 - 1, Ψmax ) -x2 + 1,


with y ) x1, the relative degree is F ) 2. Now, the control law becomes of the form v ) -(k0 + k1)x1 - k1x2. For this system, the constraints and the control law are presented in Figure 1b. As it can be seen, for this system to be stable, the region Xf where the control does not saturate gets smaller than that corresponding to the open-loop stable plant (Figure 1a). Since the original system is unstable, global stabilization cannot be achieved and the operation must be restricted to Xf. These observations are formalized next as sufficient conditions to ensure stabilization. 3.1. Definitions and Preliminary Results. Equations 24 and 25 bound a region in the x-space (feasible region Xf) of points accessible by the bounded control set Ub. The following functions are obtained by reordering inequalities in (24):

g(x,K,u*) ) KTΦ(x) + LFf h(x) + LgLF-1 h(x)u* f u* ) umin, umax H(x) ) LgLF-1 h(x)(umax - umin) f

(30)

The feasible region Xf is now defined as

Xf ) x ∈ Rn/

{

g(x,K,umax) sgn(LgLF-1 h(x)) g 0 f F-1 g(x,K,umin) sgn(LgLf h(x)) e 0

}

(31)

Let us define the set Zc as Figure 1. Cross section of the planes (at x1 ) 0) defined by the constraints Ψmin,Ψmax and the control surfaces S. δ represents the size of the feasible region Xf. (a) Open-loop stable plant. S1(K,z) corresponds to k0 ) k1 ) 1 and S2(K,z) to k1 > 1. (b) The same representation for an open-loop unstable plant and a stabilizing controller S(K,z).

and v ) -k0z1 - k1z2, eqs 27 and 28 can be rearranged as

Ψmin e v e Ψmax

(29)

where, using (26), v becomes v ) (k1 - k0)x1 - k1x2. The geometrical interpretation of inequalities in (29) is simple: designating Ψmin, Ψmax, and v by x3, it can be said that the control law (27) does not saturate as long as the surface x3 + (k0 - k1)x1 + k1x2 ) 0 does not intersect the planes x3 + x2 + 1 ) 0 and x3 + x2 - 1 ) 0. In fact, choosing k0 ) k1 ) 1, the plane defined by the control law is parallel to the planes defined by the constraints and saturation never occurs. Obviously, this implies u ) 0 and its dynamics corresponds to that of the unforced system. Any other choices of K will make the control law to intersect the planes defined by the constraints in x-space, thus defining a feasible set Xf. These situations are illustrated in Figure 1a. Since the original system is open-loop stable, global stability can be preserved by defining a v ) - KTz + w, with w selected so that u ∈ Ub. This w corresponds to the external controller subject to saturation referred to in the Introduction. In Figure 1b we represent the terms of inequalities (29) for an open-loop unstable plant of the form

(

x1 + x2 f(x) ) -x 1

)

g(x) )

() 0 1

Zc ) {θ/θTθ e c2}

(32)

which represents the set of points in the θ-space lying in the circle of radius c. We note that there exists a one-to-one map between the x-space and the θ-space (constituted by the linearized space plus the internal dynamics) (Isidori, 1988). Therefore, there is an equivalent in the x-space to the set Zc that will be represented as Zc(θ(x)). The set Zc is constructed in a way such that Zc(θ(x)) ⊂ Xf. The following functions will be employed in the theoretical development:

Ψ*(x) ) LFf h + u*LgLF-1 h f

(33)

ζ(x) ) -H(x)

(34)

without loss of generality, we assume ζ(x) > 0 in all of our developments. The same conclusions can be drawn for ζ(x) < 0 by inverting the sign of the inequalities. The symbol (*), represents either the minimum (Ψβ) or the maximum (ΨR) value of the input variable. It is pointed out that ζ(x) should never be zero in order for the transformation in (3) to be applied (see section 2); hence, it will have a definite sign (either positive or negative) in the state space region of interest. From definition (31) and eqs 33 and 34, we restate the feasibility conditions as follows:

KTΦ + ΨR e 0

(35)

KTΦ + ΨR + ζ g 0

(36)


where eqs 35 and 36 must hold everywhere in ZR, being R ) |θ|. This is equivalent to state the existence of numbers dR and dβ (with S defined as in eq 13):

dR ) min(S - ΨR)

(37)

dβ ) min(Ψβ - S)

(38)

θ∈ZR

θ∈ZR

tions, ΨR < 0 and Ψβ ) ΨR + ζ > 0 everywhere in ZR. Selecting a positive number

a ) inf (-ΨR,Ψβ) θ∈ZR

(45)

Inequalities (40) will be satisfied in the region ZR and, consequently, the control law S ) a cos ω will lead to

Both are nonnegative everywhere in ZR. The bounded control problem becomes equivalent to a feasibility problem for each gain vector KT. In general, this problem can be very complex and difficult to solve. However, it is possible to state sufficient conditions that guarantee the existence of nonnegative numbers dR and dβ in the region ZR. Such conditions are stated in the following proposition: Proposition 1. Let us define the set CR ) {θ/θTθ ) R2}. The following conditions are necessary in order to satisfy eqs 35 and 36 in CR:

which is equivalent to the existence of dR and dβ in (37) and (38) nonnegative. ∆ The proposition states conditions under which it is possible to design stable low-gain controllers compatible with Ub in ZR(θ(x)) ⊂ Xf (feasible region). Feasibility is achieved by choosing a gain candidate such that

(i) ζ(θ) > 0. (ii) There exist vectors θ ) (zTi lηT)T and θ′ ) (-zTi lηT)T; θ,θ′ ∈CR and a number g 0 such that ΨR (θ) e and ΨR(θ′) e -.

This can be shown by noting that |θ| g |z| and, therefore, -Smin e |KT|R e a. From (45) and the second part of Proposition 1, it follows that

If, moreover, ΨR(θ) < 0 and Ψβ(θ) > 0 (this implying |ζ| g |ΨR|) everywhere in ZR, then there exist some a > 0 such that dR and dβsas defined by (37) and (38)sare nonnegative in ZR. Proof. To prove the first of the two statements, let us define the following terms related to the gain vector KT:

ω ) angle(z,-K)

θ ) (zTlηT)T ∈ ZR

a ) |KT|R′ -KTΦ

The function S ) R′, with R′ e R, as

a>0

will be represented, for |Φ| )

S ) |KT|R′ cos ω ) a cos ω

(39)

Substituting (39) in eqs 35 and 36, we obtain the following inequalities that must be satisfied for all θ ∈ CR:

ΨR(θ) ζ ΨR(θ) + g cos ω g a a a

(40)

Inequalities (40) can be rewritten as

cos ω e

Ψβ(θ) g S g ΨR(θ)

0 < |K|
0. f Once the set Ub is determined, the existence of a stabilizing nonlinear controller is assured and its parameterization follows the same steps as in the first case. We note that, for this case, the set Xf is known and, therefore, the radius of the set ZR can be easily computed. To construct the surface S, we employ the results obtained in subsection 3.1. For a particular system, we start by checking the assumptions under which Proposition 1 relies, namely max(ΨR) < 0 and min(Ψβ) > 0 at different radius R and determining the region

ZR ) {θ/θTθ e R2} where those requirements are met. Then, according to Proposition 1, any control law with a gain |K| ) a/R (where a is computed as in eq 45) will automatically satisfy the constraints. In particular, we will choose a stable gain with the required dynamic behavior. If there is not internal dynamics, the region of initial conditions X0 will be chosen according to the linear dynamic behavior exhibited by the system and eq 22 can be used to that purpose. In the presence of an internalsand generally nonlinearsdynamics, simulation is the best way of estimating such a region. If, on the other hand, X0 is prescribed, the controller will be tuned so the states will remain in the set Xf. The performance of the equivalent linear system is then improved by adding an external controller and saturating it. Stability will be preserved, as stated in Theorem 1 (Appendix), even under saturation of the new input variable. Next, we summarize the different steps of the design algorithm: (1) Compute the region ZR(θ(x)) ⊂ Xf. We do this by first checking conditions (i) and (ii) of Proposition 1 and calculating r1 and r2 such that

max|θ|er1 ΨR < 0

min|θ|er2 Ψβ > 0

Figure 2. Block diagram for the GLC control structure subject to saturation in the external controller (SISO case). Observer block, calculates saturations on w from g(θ) and h(θ) in inequalities (47).

Then ZR is computed as

ZR ) {θ/θTθ e R2 and R ) inf(r1,r2)} (2) Select a stable gain vector such that |K| ) a/R (see 46). The direction of K will be defined according to the desired performance and the initial condition region ZR0 of interest. Different methods such as pole placement techniques can be employed to select the candidate gain KT. (3) Closed-loop performance is improved by adding an external controller of the form proposed in Theorem 1 (see the Appendix). Saturations on the external input v will be estimated from g(θ) and h(θ) in (47). It must be pointed out that for open-loop stable plants, this structure guarantees global stability if w is chosen in a way such that u ∈ Ub. A schematic of the complete control structure is shown in Figure 2. The region ZR can be extended to r > R, by solving the associated feasibility problem (eqs 37 and 38) for a candidate KT at Zr. Although there is not a general rule to select such a candidate, KT, it seems reasonable to choose the vector KT in the direction where the inferior constraint is maximum. Its norm will be modified until the control S fits into the region bounded by the constraints. This way of selecting the gain vector can be well understood once it becomes clear that S represents a scalar product of the gain vector and the transformed states, so its maximum must go in the opposite direction where constraints reach the maximum. An equivalent conclusion can be drawn for the minimum. If the resulting system is unstable, then the rule is not applicable and a new stable candidate must be selected. Once the new feasible region is determined, go to step 3. 3.3. Case Study: Control of a CSTR in the Presence of Constraints. The proposed tuning technique is illustrated with an example taken from Alvarez et al. (1991). It consists of a non-isothermal CST reactor where a first-order exothermic reaction takes place. The dynamic of the system is described by the following set of ordinary differential equations, derived from mass and energy balances:

x˘ 1 ) -F + θ(xi1 - x1) x˘ 2 ) βF + θ(xi2 - x2) - γ(x2 - Tc)

(50)

where x1 and x2 represent composition and temperature, respectively. The inlet concentration and temperature are xi1 ()1.0 mol L-1) and xi2 ()350 K). The kinetic expression, F, as a function of the composition of the


reactant and the reactor temperature is of the following form:

(

F ) x1 exp a -

b x2

)

(51)

The following dimensionless parameters, taken from the literature (Alvarez et al., 1991), are employed in the simulation example:

hwAw )1 dVCp

(52)

-∆H ) 200 dCp

(53)

q )1 V

(54)

γ) β)

θ)

The description of each parameter is summarized in the Nomenclature section. The control objective is to keep the concentration of reactant at a certain constant set point by acting on the jacket temperature (u ) Tc). In this case, the input variable is assumed to be constrained by a maximum and minimum value, Up ) 365 K and Ul ) 340 K, respectively, near the bifurcation + region (Ub ) 337, Ub ) 368). The desired steady state corresponds to the unstable stationary state x1 ) 0.5 mol L-1 and x2 ) 400 K associated to a nominal input u ) 350 K. The nonlinear control law (14) is derived on the basis of the following vector fields:

f(x) )

(

-F + θ(xi1 - x1) βF + θ(xi2 - x2) ) γx2 g(x) )

()

Figure 3. Maximum and minimum values of the functions ΨR and Ψβ for different radii. r is defined as the distance of the current state to the operation condition.

(a)

)

(b)

0 γ

The system has relative degree 2, which implies that the function

ζ ) (Ul - Up)LgLfh(x) is nonzero. In fact, for this particular example, LgLfh < 0 for all the state space. The lower and upper constraints, then become

ΨR ) L2f h + UpLgLfh

(55)

Ψβ ) L2f h + UlLgLfh

(56)

For this system, we follow the design steps proposed in subsection 3.2. First, the feasible region ZR is computed. To do that, we calculate at different radius max(ΨR) and min(Ψβ) (Figure 3). Since the maximum of the inferior constraint lies on the first quadrant of the state space, the largest surface S ) -KTz compatible with that constraint would require unstable gains. In fact, a gain candidate KT ) (-2.31,-1.1) would be feasible up to regions R ) 0.15. This, however, will lead to an unstable equivalent linear system. A feasible region with R < 0.09 is calculated by applying conditions of Proposition 1 (see Figure 3). Different stable gain candidates, feasible for R e 0.08, that satisfy |K| ) a/R (a is calculated from eq 45), are presented in Figure 4a,b. On the basis of the dynamic response, a gain candidate KT ) (0.1054,0.6581) that induces a linear

Figure 4. Values of gain candidates in region CR. (a) K ) (0.4, 1.8)T at R ) 0.06. (b) K ) (0.5,0.0)T at R ) 0.08.

system with real eigenvalues and |K| e 0.675 has been chosen. The region of initial conditions is calculated according to (22), thus ensuring that all the states will remain in the region ZR. The evolution of the linear trajectories starting in CR0 (R0 ) 0.06) is presented in Figure 5. The performance of the controller is illustrated in Figure 6 for an initial state in CR0. Once the equivalent linear system is configured, performance is improved through an external linear controller. This controller is designed, as proposed in Theorem 1, so to preserve stability under saturation of the new input w (eq A2) and to increase the speed of the response. The gain of the external controller is obtained from eq A1, for Λ ) 1 and a Lyapunov matrix P calculated from (A4) as the solution of the following equation:

ATcsP + PAcs ) Q with Q ) diag(-1,-1). Its value resulted in KTe ) (4.74,7.97). Saturations on the new input are estimated


Figure 5. Evolution of the trajectories in ZR (R ) 0.08) for the linear equivalent system with KT ) (0.1054,0.6581) from the initial condition region CR0 (R0 ) 0.06).

Figure 6. Regulatory response in temperature and concentration for a CSTR under feasible GLC control (KT ) (0.1054,0.6581)). Dashed lines represent input constraints. Initial conditions x ) (0.52,396.86)T.

Figure 7. Regulatory response in temperature and concentration for a CSTR under feasible GLC control (KT ) (0.1054,0.6581)) and external linear control subject to saturation, KTe ) (4.74,7.97). Dashed lines represent input constraints. Initial conditions xT ) (0.52,396.86).

on-line from (47). The responses obtained with the complete structure are presented in Figure 7 for the same initial conditions as in Figure 6.

Figure 8. Effect of a nonfeasible controller on level and concentration in the CSTR (KT ) (0.5,0)). Dashed lines represent constraints. Initial conditions xT ) (0.55,394.51).

We note that stability is guaranteed as long as the states remain in the feasible set ZR. This requires the initial conditions to lie in ZR0 ⊂ ZR which depends on the gain candidate KT chosen. However, the external controller can sometimes make the region of initial conditions to match the feasible region so the complete structure is compatible with the input constraints. This is illustrated on Figures 8 and 9. A gain candidate KT ) (0.5,0), feasible on ZR (R ) 0.08), induces an oscillatory dynamic behavior that results in constraint violation for initial conditions in ZR. Consequently, the reactor evolves out of the desired steady state (extinction) as it is shown in Figure 8. An external controller KTe ) (-0.0996,1.8) relocates the poles so to make the system stable and feasible for the same initial conditions (Figure 9). As a remark, we note that the proposed technique results especially attractive in those situations that involve bounded inputs and open-loop unstable plants or states constrained by physical limitations. For both cases the technique represents a reliable way of deriving stable nonlinear controllers. Although no optimallity is claimed, the performance of the resulting control


Figure 9. Stabilizing effect of the external controller (Ke ) (-0.0996,1.8)) with nonfeasible GLC. Dashed lines represent constraints. Initial conditions xT ) (0.55,394.51).

configurations proved to be quite acceptable with small computational effort. 4. Control in the Presence of Constraints for MIMO Systems Here, the ideas developed previously will be extended to a multivariable system with the same number of inputs as outputs. However, the formalization of the problem from the SISO case is not straightforward since now saturation of the input variables will lead to an infinite number of constraints in the state space domain. This can be shown by writing up the global linearization controller (GLC) transformations (11) in vectorial form:

v ) H(x) + Π(x)u

(57)

T Fm where H ) (-LF1 f h1, ..., Lf hm) , the matrix Π(x) is of the form of eq 8 and u,v are the vectors of manipulated variables in the original and the transformed space, respectively. Without loss of generality, we will consider each input variable ui normalized to the range 0 e ui e 1 and represent the desired steady-state operation point by the pair of vectors (x*,u*). The equivalent to the set (31) for a multivariable system of the form (6)-(7), under a feedback control law (17), can be written as

m

Xf ) {x ∈ Rn/-Kξ(x) ) H(x) + ∀0 < Rj < 1

Rjπj(x) ∑ j)1 j ) 1, ..., m} (58)

where each πj represents the columns of the matrix Π. Let us define a set D as m

D ) {v ∈ Rm/v ) H(x) +

Rjπj(x) ∑ j)1

is not possible to state, as in the SISO case, an optimization problem of the form (37) and (38). Such difficulty is overcome by constructing a convex set E ⊂ D bounded by a finite set of linear equations. On this set, feasibility conditions can be derived using the same line of arguments as those employed in the SISO case. Before going into the construction, we note that eq 59 describes a set of manifolds of dimension m - 1 with the following properties: (1) At each point x*, the manifolds G0j (R;x*) and are parallel. (2) At each point x*, Gβj (R;x*) intersects every Gβi (R;x*), for i * j. G1j (R;x*)

The first property is evident from the construction of (59). The second property follows from the definition of relative degree (i.e., Π(x) nonsingular implies linear independence of the column vectors πj(x)). Consequently, equations of the form (59) define a polyhedron with 2m faces and D becomes simply connected at each point x*. To illustrate the construction of the set E ⊂ D, let us consider the following example that involves a MIMO system consisting of two inputs u and two outputs y. Example. Construction of a Set E ⊂ D for a 2 × 2 System. We start defining a set Zr ) {Θ ∈ Z/ΘTΘ e r2} in the transformed space and a set of vectors:

χ0 ) H(Θ) χ2 ) H(Θ) + π2(Θ)

χ1 ) H(Θ) + π1(Θ) χ3 ) H(Θ) + π1(Θ) + π2(Θ) (60)

∀x ∈ Xf

∀0 < Rj < 1

j ) 1, ..., m}

The set D is bounded by a set of real parameter vector fields: m

Gβj (R;x) ) βπj +

Figure 10. Representation of vectors χi, the constraints Gi, and the bounding spheres (Example of section 4) in the input-output space. The linear convex set E is bounded by linear constraints (dashed lines) tangent to the spheres.

Riπi + H ∑ i*j

(59)

where β is an integer that can take the value 1 or 0, and Ri ∈ [0,1]. Since parameters Ri vary continuously, the number of constraints becomes infinity. Physically, this represents saturation in one of the input variables while the others remain in their bounds. Therefore, it

such that limrf0 χi ) χ* i and there exist positive numbers γi ) supΘ∈Zr|χi(Θ) - χ* i | for each i ) 0, ..., 3. Each vector χi, χ* i and the associated manifolds (59) are represented in Figure 10. The set E is then defined, as shown in the figure, as that bounded by straight lines, tangent to the spheres of radius γ ) supi(γi). Obviously, we assume in the construction that the resulting set E is nonempty. Next, these ideas are formalized for a general m × m MIMO system. Definition and Construction of a Convex Set E ⊂ D. We define a linear convex set E ⊂ D bounding all constraints (59) as follows:

E ) {v ∈ Rm

c* l e F(x*)v e c* p}

(61)


where the matrix F* and the vectors c*l, c* are p ∈ calculated at the desired operation point (x*,u*) from the vector H(x*) and the columns of the matrix Π(x*). The algorithm to obtain the set E is sketched next: (1) A polyhedron {χi} (with vertices located at χi) is constructed from the set of vectors H(Θ) and πi(Θ) for i ) 1, ..., m, as shown in the example above. (2) Define a set Zr ) {Θ/ΘTΘ e r2} and calculate for i ) 0, ..., m - 1: Rm

χ*i ) limrf0 χi(Θ)

γi ) supΘ∈Zr |χi(Θ) - χ*i | γ ) supi(γi)

through external controllers. The vector of new manipulated variables is decomposed as follows:

v ) -Kξ + w

(65)

External controllers, stable under saturation, can now be derived from Theorem 1 (see Appendix) and implemented on the external input vector w. However, the structure of the set E (61) does not provide a unique way of determining saturations on w, as it occurred in the SISO case. In this contribution, we decided to define saturations in the external inputs in a way such to avoid directionality changes on the vector w (Campo and Morari, 1989). This is summarized next. From the structure of the set E, the vector w ∈ E if and only if

(c* l )i + (F*Kξ)i e (F*w)i e (c* p)i + (F*Kξ)i

(66)

(3) For each face of the polyhedron {χi}, calculate a plane parallel to each face and tangent to the spheres centered on each vertex of radius γ. (4) F(x*) is obtained from the elements πi(Θ) and c*l, c*p from the independent terms of each plane. The different steps formalize the idea of constructing a linear hyper-polyhedron with faces tangent to the spheres (v - χ*i )T(v - χ*i ) ) γ2. These represent bounds on the evolution of the vector fields in the input-output space for states moving in the region Zr. We are now in a position to formalize the equivalent to Proposition 1 for a MIMO system. Proposition 2. Let us define the set CR ) {Θ/ΘTΘ ) R2} and a set E′ ) {v ∈ Rm/ΨRi e vi e Ψβi}. The following conditions are necessary for the control law (17) (or equivalently, eq 18) to belong to the set E′ ⊂ D for all Θ ∈CR: (i) Π(Θ) > 0. (ii) There exist ΘT ) (ξTi , ηT) and Θ′T ) (-ξTi , ηT). Θ,Θ′ ∈ CR and some numbers i g 0 such that ΨRi(Θ) e i and ΨRi(Θ′) e -i, for i ) 1, ..., m. If, moreover, ΨRi(Θ) < 0 and Ψβi(Θ) > 0 everywhere in ZR, then there exist some ai > 0 such that dRi and dβi, defined for each i ) 1, ... m as

This fact can be verified by noting that - |(F)* i ||w| e (F)*i w e |(F)* i ||w| and substituting it into inequalities (66). Equation 67 will be part of the control structure. If it is not satisfied (this implying saturation as it was defined) a new set of inputs is selected, being of the form

dRi ) min(Si - ΨRi)

(62)

wi ) φ(Ke)iξ for i ) 1, ..., m

dβi ) min(Ψβi - Si)

(63)

Θ∈ZR

Θ∈ZR

ZR.

are nonnegative in Proof. Condition (i) states the fact that the elements (row or columns) of Π(x) must be linearly independent. If they are not, GLC will not be applicable. The rest of the proposition follows immediately from the same line of arguments as those employed in the proof of Proposition 1 for each i ) 1, ..., m. ∆ Since for an MIMO system is, in general, difficult to find functions ΨRi(x) and Ψβi(x) bounding all constraints of the form Gβi (R;x), we will use the linear set E at the expenses of obtaining more conservative control laws. In this way, the candidates K will be chosen as those that, providing the desired dynamic behavior, satisfy the set of inequalities:

c*l e -F*Kξ e c* p

(64)

everywhere in ZR. Performance of the control structure is then improved, as described for the SISO case,

where (‚)i represents the ith row of a matrix or vector. Defining functions,

gi ) (c* l )i + (F*Kξ)i hi ) (c* p)i + (F*Kξ)i It can be easily shown that saturation will never occur as long as

(67)

|w| < Ω where Ω is defined as

(

Ω ) inf i

φ)

Ω κR

-gi

hi

,

|(F)* i | |(F)* i |

)

(68)

(69)

m

κ2 )

|(Ke)i|2 ∑ i)1

Thus ensuring that condition (67) is satisfied. The different steps of the proposed design are summarized next. Parameterization of the Control Law. As in the SISO case, the idea consists of deriving a low-gain controller stable and feasible in some set included in D. The response can then be improved by an external controller that can be subject to saturation. As in section 3, we now summarize the different steps that constitute the proposed tuning method for MIMO nonlinear model-based controllers: (1) Construct the convex set E ⊂ D following the steps proposed in the preceding section. We note that whenever possible, the employment of the set E′ (see Proposition 2) will give less conservative control laws. This would imply better performance and larger feasible regions in the state space ZR. However, it is usually very difficult to define proper functions ΨRi(Θ) and Ψβi(Θ).


(2) Check conditions of Proposition 2 on the set E (if possible on E′). If they hold, the existence of feasible and stable control laws in ZR is guaranteed. (3) Choose a stable candidate with the desired performance and compatibility with the constraints in (64). (4) Performance is improved by adding and saturating a stabilizing external controller of the form (A1). Saturations on w are defined from (67) and (69). The following example will illustrate the different design steps. 4.1. Case Study: Level and Concentration Control in a CSTR. In this example, the objective is to control level and concentration in an isothermal reactor, by acting on reactant flows (Matsuura and Kato, 1967). The reaction that takes place is A + B f P. The plant is described by the following set of ordinary differential equations:

(a)

(b)

x˘ 1 ) u1 + u2 - a1xx1 x2 u1 u2 + (CB2 - x2) - R2 x1 x1 (1+x2)2

x˘ 2 ) (CB1 - x2)

where R1 ) 0.2 and R2 ) 1.0, u1 and u2 represent the flow rates of concentrated and diluted streams, respectively. CB1 ()24.9 mol L-1) and CB2 ()0.1 mol L-1) are the inlet concentrations of each component. x1 is liquid level and x2 represents the concentration of B in the reactor. The control objective consists of maintaining the system at the unstable stationary state x1 ) 100 and x2 ) 2.79. Two attractors are observed at the points (100,0.633) and (100,7.07). Feasible manipulations lie in the region 0 e ui e 5 for i ) 1, 2. The GLC transformation for the multivariable case (11) is constructed from the following vector fields:

(

)

-R1xx1 x2 f(x) ) -R2 (1 + x2)2 1 1 x ) (C (C B1 2 B2 - x2) g(x) )

(

x1

x1

)

Figure 11. Regulatory control of the MIMO CST reactor under feasible GLC control. K ) diag (0.0913,0.0913). Initial conditions xT ) (101,1.058). (a) Evolution of level and concentration. (b) Movements of the manipulated variables.

(a)

(b)

To derive the control structure, we follow the steps proposed in the preceding section. The set E is calculated for the region ZR(R ) 2) and is defined by inequalities (64) with matrix F* and vectors c*l and c* p of the following form:

F* )

(

-0.2221 1 0.0269 1

)

c* l )

(

-0.9371 -0.1915

)

c* p )

(

0.1921 0.9405

)

The existence of a stable and feasible control law v ) -Kξ is guaranteed by Proposition 2. A feasible matrix gain candidate was selected so to provide the desired performance and satisfy feasibility conditions, being its value K ) diag(0.0913,0.0913). The evolution of level and concentration as well as the movements of the manipulated variables are presented in Figure 11a,b. The trajectories are radially distributed and pointing inward the state sphere CR which makes the initial condition region to be equal to ZR. Performance is improved through an external controller derived from eq A1 for P ) diag(1,1), Ke ) diag(5.48,5.48). Saturation

Figure 12. Regulatory control of the MIMO CST reactor under feasible GLC control. K ) diag (0.0913,0.0913). External controller subject to saturation Ke ) diag (5.48,5.48). Initial conditions xT ) (101,1.058). (a) Evolution of level and concentration. (b) Movements of the manipulated variables.

on w was defined as described in the section above and checked from condition (67). If that was violated, the current inputs were selected according to eq (69). The evolution of the system with the external controller is presented in Figure 12a,b.


Conclusions In this contribution, a general method for tuning global linearization control laws has been presented. The idea consists of designing low-gain feasible controllers that stabilize the system for arbitrary state space regions. This feasible region is given by the available equipment or by state constraints such as phase changes or activation of secondary reactions, for instance. The performance of the resulting low-gain controller is afterward improved by adding and saturating an external controller. The stability issues derived from these schemes have been analyzed and stabilization proved. The technique allows the designer to link process design and control. In this way, disturbances can be prevented by a robust control structure that takes the main advantages of linear as well as nonlinear control theory in dealing with saturations on the input variables. Two case studies of importance for the chemical industry have been selected to illustrate the methodology. One of them, the control of a nonisothermal CSTR at the unstable point, corresponds to a well-known example to test robust control for SISO systems with strong nonlinearities. The other example consists of the control of level and concentration on an isothermal reactor. In both cases, this methodology gives a new insight into the effects induced by saturation on the system dynamics as well as a reliable way to overcome them. Acknowledgment We thank the anonymous referees for their valuable comments. This research was partially funded by EC Project CT96-1192. Nomenclature a ) parameter of Arrenhius temperature relation (25) Acs ) linear system matrix (SISO) Acm ) linear system matrix (MIMO) Aw ) heat exchange area b ) parameter of Arrenhius temperature relation (10 000 K) Cp ) heat capacity of the reacting mixture d ) density of the reacting mixture D ) set of feasible transformed inputs hw ) heat transfer coefficent ∆H ) enthalpy of reaction K ) gain vector or gain matrix Tc ) cooling temperature q ) input and output flow u ) input variable (scalar or vector) Ul ) lower constraint on the input variable Up ) upper constraint on the input variable v ) input variable in the transformed space (scalar or vector) xT ) (x1,x2) ) state vector (concentration and temperature in the reactor, respectively) Xf ) feasible region y ) output signal (scalar or vector) z ) linear transformed state (SISO) Greek Letters η ) internal dynamics ξ ) linear transformed state for MIMO systems θ ) complete state for SISO systems

Θ ) complete state for MIMO systems Ψ ) constraints

Appendix In this section, we present a modified version of the result given by Gutman and Hagander (1985) regarding the stabilization of linear open-loop stable systems with bounded controls. In order to keep its generality, the result is proved for the MIMO case. The particularization to the SISO case is straightforward. First, let us define a saturation function sat(‚) as the following: Definition 1. Let v be an m × 1 vector and r a scalar bounded by numbers gi and hi such that gi e r e hi. Then

sat(v) ) (sat1(v1), ..., sati(vi), ..., satm(vm))T where

{

gi if r e gi sati(r) ) r if gi < r < hi hi if hi e r

}

Theorem 1. Assume that there exists a transformation (9) that partially linearizes the nonlinear system (6)-(7) that the internal dynamics is stable and that conditions of Proposition 2 hold, thus ensuring the existence of a stabilizing controller (17) such that -Kξ ∈ E′ for every Θ ∈ ZR. Then, an external control law of the form

w ) sat(-ΛBTcmPξ)

(A1)

for some P ) PT > 0 and Λ ) diag(Λ1, ..., Λi, ..., Λm) with Λi > 0 for i ) 1, ..., m stabilizes the equivalent linear system ξ˘ ) Acmξ + Bcmw. Proof. First, we note that under the conditions of the theorem, the linear equivalent system in ZR is of the form ξ˘ ) Acmξ and K is such that its dynamics is uniformly and exponentially stable. Redefining the vector v so that

v ) -Kξ + w

(A2)

and substituting in (10), the linear transformed system becomes of the following form:

ξ˘ ) Acmξ + Bcmw

(A3)

Since ξ˘ ) Acmξ is stable and satisfies constraints in ZR, there exists a Lyapunov function V ) ξTPξ such that

V˙ ) ξT(ATcmP + PAcm)ξ ) ξTQξ < 0

(A4)

Computing V˙ for the system (A3):

V˙ ) ξT(ATcmP + PAcm)ξ + 2ξTPBcmw According to eq A4, the first term of the right-hand side is negative definite. In order to ensure that V is a Lyapunov function for the system (A3), we define the external variable w as

w ) -λBTcmPξ

(A5)


where λ ) diag(λ1, ..., λi, ..., λm) and λi > 0 for every i ) 1, ..., m. Therefore

2ξTPBcmw ) -2ξTPBcmλBTcmPξ < 0 Stability of the constrained controller is proved by showing that (A1) and (A5) are in fact equivalent for an appropriate selection of λ. Such selection is done by noting that Π(Θ) is nonsingular in ZR (Proposition 2) and thus the set E′ is nonempty. Therefore, there exist functions

gi(Θ) ) ΨRi + (Kξ)i hi(Θ) ) Ψβi + (Kξ)i such that gi(Θ) e wi e hi(Θ) and Λi ) hi(Θ) - gi(Θ) > 0 everywhere in ZR. Then, for each i ) 1, ..., m we can choose numbers λi such that 0 < λi e Λi, which makes (A1) and (A5) equivalent and V˙ < 0 also for the constrained system and the theorem is proved. Literature Cited Alvarez, J.; Alvarez, J.; Suarez, R. Nonlinear bounded control for a class of continuous agitated reactors. Chem. Eng. Sci. 1991, 46 (12), 3235. Calvet, J. P.; Arkun, Y. Feedforward and feedback linearization of nonlinear systems and its implementation using Internal Model Control (IMC). Ind. Eng. Chem. Res. 1988, 27, 1822. Campo, P. J.; Morari, M. Robust control of processes subject to saturation nonlinearities. Comput. Chem. Eng. 1990, 14 (4/5), 343. Gutman, P.; Hagander, P. A new design of constrained controllers for linear systems. IEEE Trans. Autom. Control 1985, AC30 (1), 22.

Isidori, A. Nonlinear Control Systems; Springer Verlag: New York, 1988. Kailath, T. Linear Systems; Prentice-Hall Information & System Science Series; Prentice Hall: Englewood Cliffs, NJ, 1980. Kravaris, C. Input/output linearization: A nonlinear analog of placing poles at process zeros. AIChE J. 1988, 34 (11), 1803. Kravaris, C.; Kantor, J. Geometric methods for nonlinear process control: 1. Background. Ind. Eng. Chem. Res. 1990a, 29, 2295. Kravaris, C.; Kantor, J. Geometric methods for nonlinear process control: 2. Controller synthesis. Ind. Eng. Chem. Res. 1990b, 29, 2310. Kurtz, M. J.; Henson, M. A. Input-output linearizing control of constrained nonlinear processes. J. Process Control 1997, 7 (1), 3. Lee, P. I.; Zhou, W.; Cameron, I. T.; Newell, R. B.; Sullivan, G. R. Constrained generic model control of a surge tank. Comput. Chem. Eng. 1991, 15 (3), 191. Li, W. C.; Biegler, L. T.; Economou, C. G.; Morari, M. A constrained pseudo-Newton control strategy for nonlinear systems. Comput. Chem. Eng. 1990, 14 (4/5), 451. Lin, Y.; Sontag, E. D. A universal formula for stabilization with bounded controls. Syst. Control Lett. 1991, 21, 225. Lin, Z.; Saberi, A. Semi-global exponential stabilization of linear systems subject to “input saturation” via linear feedbacks. Syst. Control Lett. 1993, 21, 225. Matsuura, T.; Kato, M. Concentration stability of the isothermal reactor. Chem. Eng. Sci. 1967, 22, 171. Rotea, M. A.; Marchetti, J. L. Internal model control using the linear quadratic regulator. Ind. Eng. Chem. Res. 1987, 26, 577. Zhang, M.; Hirschorn, R. M. Feed-back stabilization of nonlinear systems by locally bounded controls. Syst. Control Lett. 1994, 21, 225.

Received for review September 9, 1996 Revised manuscript received October 3, 1997 Accepted October 6, 1997X IE9605591 X Abstract published in Advance ACS Abstracts, December 1, 1997.

Design of a Class of Stabilizing Nonlinear State Feedback Controllers

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