Linear Control of Nonlinear Processes: The Regions of Steady-state

7552

Ind. Eng. Chem. Res. 2006, 45, 7552-7565

Linear Control of Nonlinear Processes: The Regions of Steady-state Attainability Osvaldo J. Rojas,† Jie Bao,*,† and Peter L. Lee‡ School of Chemical Engineering and Industrial Chemistry, The UniVersity of New South Wales, Sydney NSW 2052, Australia, and Chancellery, LeVel 2, 160 Currie Street, The UniVersity of South Australia, Adelaide SA 5000, Australia

Despite the fact that most processes are intrinsically nonlinear, linear controllers are still widely favored in process-control applications. In this paper, we provide an analytical framework to assess the viability of applying linear control to stable nonlinear processes based solely on the process nonlinear steady-state information. Steady-state information is usually available from the process flowsheet; hence, the analysis described in this paper can be used as a tool for operability studies during process design. We define two operating spaces of interest: the steady-state region of attraction under linear control and the steady-state achieVable output space Via feedback. We show that these operating spaces provide an extension to a recently proposed steady-state operability analysis (Vinson, D. R.; Georgakis, C. A. J. Process Control 2000, 10, 185-194). Using the same steady-state information, we obtain additional insights into the range in which a linear controller can achieve offset-free control. 1. Introduction Most processes are inherently nonlinear. However, the majority of process-control solutions rely on linear controllers (such as proportional-integral-derivative (PID) controllers), because they are simpler to design, implement, operate, and maintain. The use of linear controllers is also justified for regulating control under disturbances with limited amplitude. In this case, linear control can often achieve acceptable closedloop performance when the process variables do not deviate from the operating point significantly. However, when the controlled system operates over a wider range, such as during start-ups, shut-downs, and set-point changes, linear control may produce significantly deteriorated control performance and can even lead to closed-loop instability.2 In principle, many nonlinear processes can be controlled locally using a linear controller (provided their local linear approximations are controllable); thus, the real issue is the range in which linear control is guaranteed to operate satisfactorily. In this paper, we address the key issue of whether a given nonlinear process is operable using linear control. Specifically, we want to investigate whether a certain operating point is attainable, i.e., offset-free controllable via linear output feedback. We call this condition the attainability of a given operating point using linear feedback control. Clearly, this issue can be studied locally using a linear approximation of the process dynamics. However, we depart from this approach in two ways: first, we assume that the only information available about the process is its steady-state nonlinear map; and second, we seek to determine the regions of steady-state attainability of the process using linear feedback control. It has long been recognized that the integration of process design and control is of great importance to designing modern process plants that need to operate efficiently.3-5 This is especially so in the context of current trends in process design with increased energy integration and product recycling. Operability studies in the early stages of process design should be a * Corresponding 2-9385-5966. Tel.: † The University ‡ The University

author. E-mail: [email protected]. Fax: +61+61-2-9385-6755. of New South Wales. of South Australia.

standard procedure to assist in screening out process configurations that are difficult to control.6 While information on the process dynamics is usually limited during process design, steady-state models based on first principles are often available. These steady-state models are usually well-understood and have been extensively tested and validated. The use of steady-state models to assess the static operability of a process has recently been suggested by Vinson and Georgakis,1 who have named their approach the steady-state operability analysis. This approach consists of using the steadystate input-output nonlinear map of the process to map the set containing the desired values of the controlled variables (the desired output space or DOS) into a corresponding set containing the required values of the manipulated variables (the desired input space or DIS). A measure of the process steady-state operability is then given by the hypervolume of the intersection between the DIS and the set that contains the available values of the manipulated variables (called the available input space or AIS). This is quantified by an operability index defined by7

OI }

µ[AIS ∩ DIS] µ[DIS]

(1)

where µ[‚] is the measure of the hypervolume of the corresponding regions (in a one-dimensional case, µ[‚] measures length; in a two-dimensional case, it measures area; in a threedimensional case, it measures volume; and so on). A process with an operability index of 1 is a process where every steadystate output in the desired output space can be reached using an input action not exceeding the available input space. In essence, the steady-state operability analysis is akin to a steadystate feasibility analysis in open loop. As such, it can be considered a necessary step during process design. However, it does not guarantee that the closed loop will have acceptable operability properties, as Vinson and Georgakis themselves conceded in their paper. The analytical framework described in this paper presents an extension to the steady-state operability analysis of Vinson and Georgakis in that we provide a characterization of the steady-state operating points that are attainable in closed loop using linear control. Similar to the steady-state operability analysis, our approach relies only on the steady-state information

10.1021/ie051226j CCC: $33.50 © 2006 American Chemical Society Published on Web 09/23/2006

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7553

of the process. In particular, we introduce two operating spaces of interest, namely, 1. The steady-state region of attraction under linear feedback control, which defines the largest set of steady-state initial operating conditions in the input space from which the closed loop with linear control is guaranteed to converge to the operating point of interest. 2. The steady-state achieVable output space Via linear feedback, which defines the set of steady-state operating points in the output space to which the closed loop with linear control is guaranteed to converge starting from the operating point of interest. Because of the nonlinear nature of the closed loop, the operating spaces defined above are not equivalent. Indeed, we will show that the steady-state region of attraction is related to the solution of a regulation problem for the nonlinear closed loop, while the achieVable output space Via linear feedback is related to the solution of a servo problem for the nonlinear closed loop. Using singular perturbation theory,8 we analyze the stability of the closed loop and we provide sufficient conditions on the process steady-state nonlinear map and the linear controller gain that guarantee that the closed loop is offset-free controllable. These conditions are then used to characterize the two operating spaces defined above. The proposed analysis provides the means to assess the viability of applying a linear controller with integral action to a stable nonlinear process. Using the same steady-state information assumed by Vinson and Georgakis,1 we obtain additional insights into the range in which a linear controller can achieve offset-free control. Recent work by Ekawati and Bahri9 has used the geometric interpretation of the operating spaces considered in the steadystate operability analysis to simplify the dynamic operability framework10 for regulatory cases. This approach differs from ours in that it is aimed at quantifying both the flexibility and the dynamic operability of the process, using nonlinear dynamic models and based on the solution of nested nonlinear optimization problems. The paper is organized as follows: In Section 2, we introduce the framework adopted in our study, and we provide the conditions that guarantee that a given operating point is offsetfree controllable with linear control based on the process nonlinear steady-state map. This leads to the definition of the steady-state region of attraction under linear control. In Section 3, we extend the analysis to include the case in which a step change in the reference is allowed, and we provide conditions that guarantee the stability of the new operating point. This, in turn, leads to the definition of the steady-state achieVable output space Via linear feedback. In Section 4, we discuss the insights and implications that can be derived from the analysis. Finally, in Section 5, we present a case study for a Van de Vusse reaction type process to substantiate the applicability of the results. 2. Steady-State Region of Attraction for Linear Feedback Control Consider a nonlinear process P defined by the following nonlinear state-space model,

x˘ ) f(x, u) y ) g(x, u)

(2)

where x ∈ Rn are the process states and u, y ∈ Rm are the process inputs and outputs, respectively. In addition, f(‚): Rn × Rm f Rn and g(‚): Rn × Rm f Rm are smooth vector-valued functions. Assume that the model in eq 2 defines a steady-state nonlinear

Figure 1. Internal model control configuration for stable processes.

map h(‚): Rm f Rm such that

yss ) h(uss)

(3)

where h(‚) is a smooth analytic vector-valued function. Consider an operating point (u/ss, y/ss) that satisfies the nonlinear relation in eq 3. We then have the following definition. Definition 2.1 (Feasible Operating Point). Consider the following nonempty and connected bounded subsets of Rm:

DOS ⊂ Rm

(4)

AIS ⊂ Rm

The operating point (u/ss, y/ss), such that y/ss ) h(u/ss), is said to be feasible if

y/ss ∈ DOS

(5)

u/ss ∈ AIS

We will investigate conditions such that a feasible operating point (u/ss, y/ss) can be attained using linear output feedback control. To develop our analysis, we will refer to the internal model control (IMC) configuration shown in Figure 1. We allow P, P0, and Q to be nonlinear dynamic systems. This is in agreement with the approach proposed by Economou et al.11 We restrict our attention to the case when the process P is openloop stable, in the sense of Definition A.1 (see Appendix A), and there is no model-plant mismatch, i.e., P0 ) P. In this case, from direct inspection of Figure 1 (and some additional mild assumptionsssee Appendix B), we have that

y(t) ) PQr(t)

∀t g 0

(6)

Thus, to obtain offset-free control for a constant reference r(t) ) rss, a sufficient condition is that the steady-state map of the IMC controller Q be a right inverse of the process steadystate nonlinear map h(‚) in eq 3. Finding the right inverse of a multivariable steady-state map is, in general, not a simple task. In many cases, an analytical inverse does not exist or, even if such solution exists, it may be impractical to derive. An alternative to finding an inverse analytically is to implement the inverse implicitly using a highgain feedback loop. Figure 2 shows how the IMC controller Q can be implemented as a feedback loop that contains a full dynamic nonlinear model P0 of the process and a linear controller C with infinite gain at steady state. Without loss of generality, we allow the controller C to be defined as follows,

C(s) ) C h (s)

K s

(7)

where K > 0 is an m × m constant matrix, C h (s) is a stable matrix transfer function, and s is the Laplace variable.

7554

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006

Figure 2. Implementation of the IMC controller Q as a high-gain feedback loop.

Remark 2.1. To keep the discussion general, we have considered the case in which a full dynamic model P0 of the process is available for the implementation of the implicit inverse in Figure 2 and the IMC scheme in Figure 1. We also assume that there is no process-model mismatch, i.e., P0 ) P, in which case the process P is effectively controlled in closed loop by the linear controller C(s) in eq 7. However, in Section 4, we will provide some comments on the more general case when the model P0 used for the implementation of the IMC controller Q in Figure 2 is allowed to differ from the model used in the IMC scheme in Figure 1. We will investigate the asymptotic stability of the IMC controller Q in Figure 2 when an exogenous constant reference signal r(t) ) r/ss is applied. We assume that r/ss is such that the operating point (u/ss, y/ss) with y/ss ) r/ss is feasible, in the sense of Definition 2.1. Observe that, if the IMC controller Q can be shown to be asymptotically stable, then u(t) in Figure 2 will asymptotically converge to a certain steady-state value uss. However, since we have assumed P0 ) P, we have that uss ) u/ss. Thus, at steady state, the IMC controller Q implements an exact right inverse of the process steady-state nonlinear map h(‚). Theorem 2.2 (Steady-State Attainability via Linear Feedback Control). Consider the closed-loop system shown in Figure 2. Assume that the technical conditions in Assumption C.1 (see the Appendix) hold true. Consider a constant reference r(t) ) r/ss such that the corresponding operating point (u/ss, y/ss) with y/ss ) r/ss is feasible, in the sense of Definition 2.1. Let the integral action gain K of the controller be

K ) K ˆ

(8)

where K ˆT ) K ˆ > 0 and > 0 is a detuning coefficient. Assume that there exists a nonempty region Λξ ⊂ Rm such that

[ξ - ξ/ss]T[φ(ξ) - y/ss] > 0 ∀ξ ∈ Λξ

(9)

where yss ) φ(ξss) in steady state. Then, there exists a (possibly small) 0 > 0 such that, for all 0 < e 0, the process equilibrium point (u/ss, y/ss) is asymptotically stable if the closed-loop trajectory is such that ξ(t) ∈ (Λξ ∪ {ξ/ss}) for all t g 0. Proof. See Appendixes C.1 and C.2. The importance of Theorem 2.2 is that it provides a sufficient condition for the asymptotic stability of a feasible equilibrium point (u/ss, y/ss) using linear output feedback control. Observe that the key condition in eq 9 depends only on the process steady-state nonlinear map yss ) h(uss) and the linear controller C h (s) steady-state gain matrix K h , since yss ) h(K h ξss) } φ(ξss). The actual dynamics of the stable controller C h (s) and the stable nonlinear model P0 are not relevant, since the result in Theorem 2.2 is valid for an arbitrarily small controller gain K ) K ˆ where 0 < e 0. Note that this requirement is not restrictive since

we are interested in the attainability of the equilibrium (u/ss, y/ss) using linear feedback control and not in the details of the closedloop transient response. We shall stress that the IMC scheme and the IMC controller Q in Figure 2 have been adopted primarily for analysis purposes. This does not necessarily imply that such a scheme and controller should be used for the synthesis of a controller for the nonlinear process P, the reason being that a full dynamic model P0 of P may not be readily available. Clearly, the result in Theorem 2.2 goes beyond the study of a linearized model around the feasible operating point (u/ss, y/ss). We shall use the result presented in Theorem 2.2 to introduce a new operating space, namely, the steady-state region of attraction under linear control. Definition 2.3 (Steady-State Region of Attraction under Linear Feedback Control). Consider a stable nonlinear process P and a feasible equilibrium point (u/ss, y/ss). Consider the following ellipsoidal region in Rm centered at u/ss,

Π(K ˆ , γ, u/ss) } {u ∈ Rm|(u - u/ss)TK h -TK ˆ -1K h -1(u - u/ss) e γ} (10) where K h is the steady-state matrix gain of the linear controller C h (s) and γ > 0 is a scalar parameter. Then, the steady-state region of attraction under linear feedback control for the feasible operating point (u/ss, y/ss) is given by

Ωu(u/ss) } max µ[Π(K ˆ , γ, u/ss)] K ˆ ,γ

(11)

subject to: Π(K ˆ , γ, u/ss) ⊂ Λu Π(K ˆ , γ, u/ss) ⊂ AIS where Λu is the region that results from mapping the region Λξ, where eq 9 in Theorem 2.2 holds, into the input u-space using the linear steady-state relation uss ) K h ξss. In addition, µ[‚] is a function that measures the hypervolume of the ellipsoid Π(K ˆ , γ, u/ss), similar to that used in eq 1. Remark 2.2. Observe that the steady-state region of attraction under linear feedback control Ωu(u/ss) is the largest ellipsoid completely inscribed in both the available input space (AIS) and the region Λu. The actual size of Ωu(u/ss) will depend on the size of the AIS and on the chosen operating point (u/ss, y/ss). This is because the region Λu in which the condition eq 9 holds changes for different operating points (u/ss, y/ss). The key property that distinguishes the steady-state region of attraction under linear feedback control Ωu(u/ss) is that it contains the set of steady-state initial operating conditions in the input space from which the closed loop with linear control is guaranteed to asymptotically converge to the operating point of interest (u/ss, y/ss). This is described in the following corollary. Corollary 2.4. The steady-state region of attraction under linear feedback control Ωu(u/ss) in Definition 2.3 is positively invariant with respect to u(t), for all 0 < e 0. This implies that, if a closed-loop trajectory originates from a steady-state initial condition (uss, yss) such that uss ∈ Ωu(u/ss), then u(t) will never leave the region Ωu(u/ss) for all t g 0. In addition, every closed-loop trajectory starting from a steady-state initial condition (uss, yss) such that uss ∈ Ωu(u/ss) will converge to the feasible operating point (u/ss, y/ss). Proof. See Appendix C.3.


Figure 3. Steady-state nonlinear map h(‚) of the process in Example 2.6.

Figure 4. Steady-state region of attraction under linear feedback control for the system in Example 2.6: (a) region Λu where eq 9 in Theorem 2.2 is / satisfied and points (M) of input multiplicity and (b) the steady-state region of attraction Ωu(uss ) and the resulting closed-loop input trajectories.

Corollary 2.4 shows that the steady-state region of attraction under linear feedback control Ωu(u/ss) provides an indication of the range (in terms of admissible initial steady-state conditions) in which linear feedback control is guaranteed to achieve offsetfree control of the feasible operating point (u/ss, y/ss) of interest. The next Lemma establishes that the steady-state gain matrix K h can be selected such that Ωu(u/ss) is guaranteed to be nonempty. Lemma 2.5. Consider the closed-loop system shown in Figure 2. Assume that the Jacobian of the process steady-state nonlinear map evaluated at the equilibrium point (u/ss, y/ss)

J}

|

∂h ∂u u ) uss/

(12)

is not singular, i.e., det(J) * 0. Let the steady-state gain matrix K h of the linear controller C h (s) be

K h )J

-1

(13)

Then the steady-state region of attraction under linear feedback control Ωu(u/ss) is not empty. Proof. See Appendix C.4. Definition 2.3 requires the solution of an optimization problem to determine the steady-state region of attraction under linear feedback control Ωu(u/ss). This is a standard problem in convex optimization theory,12 the details of which go beyond

the purpose of this paper. In the one- and two-dimensional cases, K ˆ and γ can be determined by direct inspection, as we shall demonstrate in the following numerical example. Example 2.6. Consider a two-input two-output stable nonlinear process P whose input-output steady-state nonlinear map is given by the surfaces shown in Figure 3. Consider the following equilibrium point

u/ss )

[]

[ ]

2 / 17.63 ,y ) 4 ss 24.69

(14)

and assume that this operating point is feasible as per Definition 2.1. Let the available input space (AIS) be the square region defined by the input range shown in Figure 3, i.e.,

AIS } {(u1, u2)| - 10 e u1 e 10 and -10 e u2 e 10} (15) On the basis of Lemma 2.5, the steady-state gain matrix K h is selected to be the inverse Jacobian matrix of the process steadystate nonlinear map evaluated at u ) u/ss. The shaded region in Figure 4a is the region Λu in the input space in which eq 9 of Theorem 2.2 is satisfied. The filled circle inside Λu is the operating point specified in eq 14. The dashed lines in Figure 4a indicate the level curves of the output map for y1 ) 17.63, while the dotted lines indicate the level curves of the output map for y2 ) 24.69. Observe that the points of intersection between the level curves (shown with an arrow and

7556


the letter M) are points of input multiplicity. From the expression for eq 9, we conclude that every point M of input multiplicity has to lie on the border of region Λu, since for those points [ξ - ξ/ss]T[y - y/ss ] ) 0. This is confirmed by the points shown in Figure 4a. The ellipse shown in Figure 4b is the steady-state region of attraction under linear feedback control Ωu(u/ss). It is the largest ellipse completely inscribed in Λu and in the AIS. The dashed lines in Figure 4b are the closed-loop trajectories obtained when the process P has steady-state initial conditions designated by the circles. Observe that every closed-loop trajectory with steady-state initial conditions inside Ωu(u/ss) converges to the equilibrium point (u/ss, y/ss) of interest using linear output feedback control. On the other hand, closed-loop trajectories with steady-state initial conditions outside the region Λu are seen to diverge and eventually move out of the input region shown in Figure 4b. Finally, observe the closed-loop trajectory with steady-state initial condition in the lower-right corner of Figure 4b. This initial condition is seen to be inside the region Λu where eq 9 is satisfied. However, the resulting closed-loop trajectory is not guaranteed to evolve inside Λu for all t g 0, since the steady-state initial condition is not inside the positively invariant set Ωu(u/ss)ssee Corollary 2.4. Indeed, in this case, the associated closed-loop trajectory does not converge to the equilibrium point (u/ss, y/ss). This numerical example has shown that, by applying the results of Theorem 2.2 and the definition of the steady-state region of attraction Ωu(u/ss), we can gain useful insights into the closed-loop properties of a stable nonlinear process subject to linear output feedback control. 3. Steady-State Achievable Output Space via Linear Feedback In the previous section, we have presented conditions that guarantee that a feasible operating point is attainable using linear feedback control. In general, for every different operating point (u/ss, y/ss), there will be a different steady-state region of h and attraction Ωu(u/ss) defined by different controller gains K K ˆ . In this section, we address a closely related issue: given that a feasible operating point (u/ss, y/ss) is attainable using linear feedback control, what is the set of operating points the closed-loop can switch to starting from (u/ss, y/ss)? This question is equivalent to solving a servo problem for the closed loop. In particular, in the analysis described in this section, we shall fix the controller designed for the operating point (u/ss, y/ss) and we shall derive conditions that guarantee that we can attain a new feasible operating point (u′ss, y′ss) using the same controller. We shall refer again to the IMC configuration in Figure 1 with the same conditions described in Section 2. Since y(t) ) PQr(t), we study the conditions such that the IMC controller Q in Figure 2 is asymptotically stable when the reference signal r(t) switches from r(t) ) y/ss to a new value r(t) ) y′ss. The main result is described in the following Theorem. Theorem 3.1 (Steady-State Attainability via Linear Feedback Control for Step Changes in the Reference). Consider the closed-loop system shown in Figure 2. Assume that the feasible operating point (u/ss, y/ss) is attainable using linear feedback control for all 0 < e 0, based on the assumptions and conditions outlined in Theorem 2.2. Consider a step change in the reference signal r(t) such that

r(t) )

{

y/ss, t < 0 y′ss t g 0

(16)

and assume that the new operating point (u′ss, y′ss) is also feasible. Suppose there exists a nonempty region Θu ⊂ AIS ⊂ Rm such that

∂h K hK ˆ >0 ∂u

∀u ∈ Θu

(17)

Then the new operating point (u′ss, y′ss) is asymptotically stable if the closed-loop trajectory is such that u(t) ∈ Θu for all t g 0. Proof. See Appendix C.5. We shall use the result described in Theorem 3.1 to introduce a new operating space, namely, the steady-state achieVable output space Via linear feedback control. Definition 3.2 (Steady-State Achievable Output Space via Linear Feedback Control). Consider a stable nonlinear process P and a feasible equilibrium point (u/ss, y/ss). The steady-state achievable output space via linear feedback control for the feasible operating point (u/ss, y/ss) is given by

Ωy(y/ss) } {y ∈ Θy | |y - y/ss| e min |y - w|} w∈Θ°y

(18)

where Θy is the region in the output space that results from mapping the region Θu where eq 17 holds, using the nonlinear steady-state relation yss ) h(uss). In addition, Θ°y is the contour of the region Θy. A geometric interpretation of the definition of the steadystate achievable output space via linear feedback control Ωy(y/ss) is as follows: a steady-state output value yss ∈ Θy is in Ωy(y/ss) if the largest circle centered at yss that is completely inscribed inside the region Θy contains the point y/ss. This is graphically shown in Figure 13 for an arbitrary region Θy. The key property that distinguishes the steady-state achievable output space via linear feedback control Ωy(y/ss) is that it contains the set of steady-state operating points in the output space to which the closed loop with linear control is guaranteed to converge starting from the operating point of interest (u/ss, y/ss). This is described in the following Corollary. Corollary 3.3. Consider a stable nonlinear process P and assume that P is initially at rest at a certain feasible operating point (u/ss, y/ss). Given a fixed linear controller C(s), the closed loop in Figure 2 can stably switch from (u/ss, y/ss) to a new feasible operating point (u′ss, y′ss) if y′ss ∈ Ωy(y/ss). Proof. See Appendix C.6. We will illustrate the application of the achievable output space via linear feedback control Ωy(y/ss) with the following example. Example 3.4. Consider the two-input, two-output stable nonlinear process P described in Example 2.6 with the same available input space (AIS) and the feasible operating point h and K ˆ are those selected in (u/ss, y/ss) in eq 14. Assume that K Example 2.6. We first determine the region Θu ⊂ AIS in which the condition (∂h/∂u)K hK ˆ > 0 is satisfied. We then map the region Θu into the output space using the nonlinear steady-state relation yss ) h(uss). The result is the irregular region Θy shown in Figure 5a. To compute the achievable output space via linear feedback control Ωy(y/ss), we approximate the contour Θ°y of the region Θy with the straight lines shown in Figure 5a. This simplification reduces the calculation of Ωy(y/ss) to simple point-to-point and point-to-line distance computations. The resulting achievable output space via linear feedback control Ωy(y/ss) is given by the


/ Figure 5. Achievable output space via linear feedback control Ωy(yss ) for Example 3.4: (a) region Θy resulting from mapping the region Θu where eq 17 / holds and (b) Ωy(yss ) and resulting closed-loop output trajectories.

region shown in Figure 5a around the steady-state output y/ss ) [17.63 24.69]T. A magnified image of Ωy(y/ss) is shown in Figure 5b. The dashed lines in Figure 5b show the simulated closed-loop trajectories of the process output when the reference signal r(t) changes from y/ss to the value y′ss marked with squares. Observe that, since y′ss ∈ Ωy(y/ss), the closed-loop trajectories are guaranteed to converge to the new operating point. Unlike the steady-state region of attraction for linear feedback control Ωu(u/ss) studied in Section 2, the steady-state achievable output space Ωy(y/ss) is not positively invariant for y(t). This is clearly shown in Figure 5b for the closed-loop output trajectory converging to y′ss ) [17.32 25.04]T, which is seen to exit and then re-enter the region Ωy(y/ss). However, in this case, the output trajectory y(t) still evolves inside the region Θy where eq 17 holds. 4. Discussion In this section, we explore some of the insights and implications that can be derived from the analysis presented in Sections 2 and 3. 4.1. Extensions to the Steady-State Operability Framework. The operating spaces defined in Sections 2 and 3 extend the steady-state operability analysis proposed by Vinson and Georgakis.1 In particular, the steady-state attainability regions Ωu(u/ss) and Ωy(y/ss) provide additional insights into the range in which a linear controller in closed loop with a nonlinear and stable process P can achieve offset-free control. The analysis proposed in this paper does not require any further information on the nonlinear process P, other than the information already assumed by the steady-state operability analysis of Vinson and Georgakis, namely, knowledge of the process steady-state nonlinear map yss ) h(uss). The steady-state operability analysis1 is essentially equivalent to an open-loop steady-state feasibility analysis, in the sense of Definition 2.1. As such, one may say that it is implicit in the execution of an acceptable process design, i.e., if the design is not feasible at steady state, then clearly it is not a satisfactory design. From this point of view, feasibility is a necessary condition for process design, and in the present paper, it has been assumed from the outset. By way of contrast, the analysis proposed in this article provides a characterization of the steadystate operating points that are attainable in closed loop using linear output feedback control.

We do not specify a method to find the set of all feasible operating points; therefore, in principle, the method outlined by Vinson and Georgakis1 could be employed. However, care needs to be taken when selecting a procedure to map the operating spaces of interest from the input space to the output space and vice versa. Subramanian and Georgakis,13 for example, recommend to map the contour of the available input space into the output space using the nonlinear relation yss ) h(uss). However, this approach can produce misleading results when the process has input multiplicities, the reason being that the mapped contour can intersect itself, thus generating ambiguity about the points that are contained and those that are not contained in the mapped region (note that the same can be said when mapping the desired output space into the input space and the process has output multiplicities). This problem was reported by Subramanian and Georgakis13 for a vinyl acetate reactor example. When the process has input or output multiplicities, it is necessary to map every point contained in the operating spaces of interest. In practice, this procedure is carried out for a grid of points inside the specified region. To illustrate, consider the nonlinear process in Example 2.3 with the available input space (AIS) given by the region in eq 15. Observing the process nonlinear map shown in Figure 3, we can see that the process exhibits input multiplicities (see also the input multiplicity points shown in Figure 4a). We map the contour of the AIS starting from the lower left corner (u1 ) u2 ) -10) and progress counterclockwise. The result is shown in Figure 6, where the numbered points denote the image in the output space of the vertexes of the AIS (vertex 1 is the starting point). Because of input multiplicities, the resulting mapped contour intersects itself several times and an estimation of the achievable output space (AOS) is not possible. The actual AOS for this example is the shaded area shown in Figure 6. This region was obtained by mapping a sufficiently dense grid of points inside the AIS. The difference between these two approaches is clear. 4.2 Linear Control Versus Nonlinear Control. Since the process model P0 used for the implementation of the IMC controller Q in Figure 2 is the same as that used in the IMC scheme in Figure 1, we have that P is effectively controlled in closed loop by the linear controller C(s) in eq 7. However, one may choose to use a different model of the process for the implementation of the IMC controller Q in Figure 2. In particular, one can use a model (say P′0) for which Ωy(y/ss) is

7558


Figure 6. Mapping in the output space of the contour of the available input space (AIS) for the system in Example 2.6. The shaded area is the actual achievable output space (AOS) for this example.

larger. Since for the IMC scheme in Figure 1 we have that

y(t) ) PQr(t)

∀t

(19)

we conclude that a larger Ωy(y/ss) for the closed loop in Figure 2 translates into a larger achievable output space for the IMC closed loop in eq 19. Recall that we are still assuming P0 ) P in Figure 1. In this case, the IMC scheme implements a full nonlinear controller since P′0 * P0. However, we see that offset-free control may be lost since the steady-state nonlinear map of the model P′0 inverted (at steady state) in the IMC controller Q could be different from that of P0 and of the real process P. This, in turn, suggests that it may be possible to find a model P′0 that has the same steady-state characteristics of the real process P in a certain region of interest (in particular, around the operating point (u/ss, y/ss)) but differs from it in other regions, so that Ωy(y/ss) for the closed loop in Figure 2 is larger. With reference to the IMC scheme in Figure 1, one may think of this possibility as being equivalent to using a linear controller in the region where the steady-state map of P′0 is equal to that of P0 and using a nonlinear controller in the region in which the steady-state maps differ. 4.3. When is Nonlinearity a Problem? The analysis that we have described in this paper contributes to the more general question of determining when a linear controller is adequate for a given nonlinear process. This problem has often been related to the issue of quantifying the degree of nonlinearity of the (open-loop) process for which a variety of nonlinearity measures have been proposed.14-18 As stated by Eker and Nikolaou,19 “the premise of these approaches is that if a nonlinear open-loop system is far from a linear one, then linear control will, most probably, be inadequate for the closed loop”. However, it has recently been recognized that the degree of nonlinearity of the open-loop process need not be always related to poor performance of a closed loop with a linear controller.2,19 The results presented in the current paper seem to confirm these studies. In particular, we have shown that, if linear control is assessed with reference to offset-free operability, then a highly nonlinear process may not be necessarily difficult to control, since closed-loop stability is guaranteed by eq 9 in Theorem 2.2. Indeed, Example 2.6 shows that a highly nonlinear process can satisfy eq 9 in a large region Λu. From the size of Λu and the shape of the surfaces shown in Figure 3, we can safely assume that a single linearized model of the process would have serious difficulties in describing the process P over the entire region Λu. Despite this, linear control can still guarantee offsetfree control if the steady-state initial condition is inside the

Figure 7. Steady-state nonlinear input-output relation for the Van de Vusse SISO reaction process.

steady-state region of attraction Ωu(u/ss). We emphasize that the analysis described in Sections 2 and 3 is valid for an arbitrarily small controller gain K, since we require 0 < e * for a (possibly) small value * > 0. However, this requirement is not restrictive since we are interested in the attainability of the equilibrium (u/ss, y/ss) using linear output feedback control and not in the details of the closed-loop transient response. 4.4. Connections to Passivity Theory. Observe that eq 9 in Theorem 2.2 is equivalent to a passivity type condition for the static-nonlinearity yss ) φ(ξss).20,21 Passivity theory is especially useful for control design and synthesis of large and complex nonlinear systems.22 Indeed, this connection suggests that, by using the full capabilities of passivity theory, it may be possible to extend the approach described in this paper to study the dynamic operability properties of nonlinear processes. 5. Case Study: A Van de Vusse Reaction Process In this section, we shall illustrate the approach described in this paper when applied to a Van de Vusse reaction process. We first analyze a single-input, single-output model to clarify the concepts. We will then apply the approach to a multivariable model of the process. The dynamics of a Van de Vusse reaction process can be described by the following set of nonlinear differential equations,23,24

x˘ 1 ) -k1x1 - k3x12 + (x1f - x1)u x˘ 2 ) k1x1 - k2x2 - x2u

(20)

where x1 is the concentration of the reactant and x2 is the concentration of the intermediate species. The input u is the dilution rate. The parameters

k1 ) 50 h-1 k2 ) 100 h-1

(21)

k3 ) 10 L(gmol h)-1 are the reaction rate constants, and x1f is the inlet concentration of the reactant x1. It can be verified that the process is openloop stable.23 5.1. SISO Case. We consider the case when x1f ) 10 gmol L-1, and we assume that the output of the process is y ) x2. Figure 7 shows the steady-state nonlinear map of the process.


Figure 8. Input and output closed-loop trajectories of the SISO Van de Vusse reaction process for steady-state initial conditoin (uss, yss) ) (130, 1.2013) (dashed line) and for steady-state initial condition (uss, yss) ) (20, 0.9065) (continuous line).

Note that the process exhibits input multiplicity, i.e., two different input values generate the same output value. We consider an available input space (AIS) given by

AIS } {u|0 e u e 300}

(22)

and a desired output space (DOS) that is equal to the region mapped by the nonlinear input-output relation shown in Figure 7, i.e.,

DOS } {y|0 e y e 1.266}

(23)

With this choice of AIS and DOS, we see that every point on the curve in Figure 7 is a feasible operating point, based on Definition 2.1. In addition, it is clear that the operability index1 is

OI }

µ[AIS ∩ DIS] )1 µ[DIS]

(24)

The value of the operability index confirms that every operating point on the steady-state curve is achievable in open loop. However, the steady-state operability analysis1 does not provide information on the closed-loop properties of a given operating point of the process. By way of contrast, the result in Theorem 2.2 gives a characterization of the region of attraction when using linear feedback control for a given operating point. We note that, in the SISO case, eq 9 can be tested graphically in a straightforward manner: for a given operating point on the steady-state curve of the process, it suffices to draw a new set of reference axes centered at the operating point of interest. Equation 9 is then satisfied for every point on the curve yss ) h(uss) that lies in the first or third quadrant of the new set of coordinates. To illustrate, we select the operating point

(u/ss, y/ss) ) (40, 1.1648)

(25)

as shown by the square in Figure 7 and we draw the auxiliary set of axes (dashed lines in Figure 7). We observe that eq 9 is satisfied for every pair (uss, yss) such that the input uss is inside the following region:

Λu ) {u|0 e u e 148.74}

attraction under linear feedback control Ωu(u/ss) is equal to the region Λu defined in eq 26. This is because every closed-loop trajectory that originates in its interior will always be contained in the same region. To confirm this, we select K h ) 1 and we simulate the closed loop for two different steady-state initial conditions inside Ωu(u/ss). The results are shown in Figure 8, where the dashed line is the response obtained when the steadystate initial condition is (uss, yss) ) (130, 1.2013) and the continuous line is the response obtained when the steady-state initial condition is (uss, yss) ) (20, 0.9065). As predicted, all the closed-loop trajectories converge to the desired operating point. Figure 9 shows the closed-loop response obtained when the initial steady-state condition is outside Ωu(u/ss). In particular, we choose the steady-state initial condition (uss, yss) ) (148.75, 1.1648). Observe that this steady-state initial condition is only slightly outside Ωu(u/ss). We see that the resulting closed-loop input and output trajectories do not converge to the desired operating point. Indeed, in the single-input, single-output case, eq 9 is not only a sufficient condition for closed-loop asymptotic stability but it can also be shown to be a necessary condition. This is substantiated by the result in Figure 9. 5.2. MIMO Case. We now illustrate the application of the analysis presented in this paper to a multivariable version of the Van de Vusse reaction process. With reference to the system’s dynamics in eq 20, we consider the following inputs and outputs:

u1 ) u u2 ) x1f y1 ) x1 y 2 ) x2 Figure 10 shows the steady-state nonlinear map of the multivariable Van de Vusse reaction process. The range of inputs u1 and u2 used to generate the surfaces in Figure 10 defines the available input space (AIS) considered, namely,

AIS } {(u1, u2)|0 e u1 e 300 and 0 e u2 e 20} (27)

(26)

Notice that the steady-state nonlinear map of the process exhibits input multiplicity in this region. Because of the scalar nature of the closed loop, we have that the steady-state region of

In addition, we consider a desired output space (DOS) such that the value of the operability index OI in eq 1 is equal to 1, i.e., we assume that every point on the surfaces shown in Figure 10 is feasible in open loop. We shall study the steady-state

7560


Figure 9. Input and output closed-loop trajectories of the SISO Van de Vusse reaction process for a steady-state initial condition outside the steady-state / region of attraction Ωu(uss ).

Figure 10. Steady-state nonlinear map of the multivariable Van de Vusse reaction process.

Figure 11. Multivariable Van de Vusse reaction process. (a) Region Λu where eq 9 holds; the square indicates the feasible operating point of interest. (b) / The steady-state region of attraction for linear control Ωu(uss ) and resulting closed-loop input trajectories.

region of attraction under linear feedback control for the following operating point:

u/ss )

[ ]

[

78 4.5065 , y/ss ) 10 1.2659

]

(28)

On the basis of Lemma 2.5, we take the steady-state gain matrix K h of the linear controller C h (s) to be the inverse of the

Jacobian matrix of the process steady-state map evaluated at the equilibrium point u/ss. Figure 11a shows the region Λu in the input space where eq 9 in Theorem 2.2 is satisfied. We observe that the region Λu covers a large portion of the AIS, indicating that the operating point of interest is attainable via linear feedback control with a large region of attraction. The dashed lines in Figure 11a are lines of constant outputs y1 )


Figure 12. Multivariable Van de Vusse reaction process: (a) region Θy resulting from mapping the region Θu where eq 17 holds and (b) the achievable / output space via linear feedback control Ωy(yss ) and resulting closed-loop output trajectories.

4.5065 and y2 ) 1.2659. These lines intersect each other only once (at the operating point of interest), thus confirming that, in the multivariable version of the Van de Vusse reaction process, there are no input multiplicity issues. The resulting steady-state region of attraction under linear feedback control Ωu(u/ss) is shown in Figure 11b. Every closedloop trajectory with steady-state initial conditions in Ωu(u/ss) is guaranteed to evolve inside Ωu(u/ss) and to converge to the operating point of interest. This is confirmed by the closedloop input trajectories shown as dashed lines in Figure 11b, where the circles indicate the steady-state initial condition. We also simulate a closed-loop trajectory with a steady-state initial conditions outside Ωu(u/ss). This closed-loop trajectory is also seen to converge to the operating point in eq 28. This result confirms the sufficient but not necessary nature of the results presented in Theorem 2.2. We next seek to determine the steady-state achievable output space via linear feedback Ωy(y/ss) for this case study. Figure 12a shows the region Θy in the output space where eq 17 holds. To compute the achievable output space via linear feedback control Ωy(y/ss), we approximate the contour Θ°y of the region Θy with the straight lines shown in Figure 12a. The resulting Ωy(y/ss) is then given by the region shown in Figure 12a. A magnified image of Ωy(y/ss) is shown in Figure 12b. The dashed lines in Figure 12b show the simulated closed-loop trajectories of the process output when the reference signal r(t) changes from y/ss to the value y′ss marked with squares. We observe that, since y′ss ∈ Ωy(y/ss), the closed-loop trajectories converge to the new operating point as predicted by Corollary 3.3 and Theorem 3.1. Figure 12b also shows a closed-loop output trajectory obtained for a reference value y′ss outside Ωy(y/ss). This trajectory is also seen to converge to the new operating point. Again, this result confirms that the conditions presented in Section 3 are only sufficient but not necessary to ensure asymptotic stability.

shown that these operating spaces can provide insights into the properties of the nonlinear process in closed loop with a linear controller, using the same steady-state information assumed by the steady-state operability approach.1

6. Conclusion

Appendix B: Nominal Stability of Nonlinear IMC

In this paper, we have developed an extension to the steadystate operability approach introduced by Vinson and Georgakis.1 On the basis of the steady-state information of the nonlinear process, we have characterized the steady-state region of attraction under linear feedback control and the steady-state achieVable output space Via linear feedback control. We have

The IMC scheme in Figure 1 has been adopted primarily for analysis purposes. Theorem B.4 below provides sufficient conditions that guarantee the asymptotic stability of the IMC closed loop in the nominal case, when P0 ) P. In particular, this result requires a stability condition for P that is slightly stronger than that considered in Definition A.1, namely, P is

Acknowledgment The support of the Australian Research Council (Grant DP0558755) is gratefully acknowledged. The authors wish to acknowledge the anonymous reviewers of this article for their helpful comments and suggestions. Appendix A: Definition of Asymptotic Stability on a Region In the nonlinear case, the concept of stability can have several definitions.21,25 We adopt the following. Definition A.1 (Asymptotic Stability on a Region). Consider a nonlinear process P defined by the following nonlinear statespace model:

x˘ ) f(x, u) y ) g(x, u)

(29)

where x ∈ Rn are the process states and u, y ∈ Rm are the process inputs and outputs, respectively. In addition, f(‚): Rn × Rm f Rn and g(‚): Rn × Rm f Rm are smooth vector-valued functions. The process P is said to be asymptotically stable on the region X0 ⊂ Rn if every steady-state operating point (uss, xss, yss) that satisfies

0 ) f(xss, uss) yss ) g(xss, uss)

(30)

with xss ∈ X0 is asymptotically stable for every initial condition x0 in X0.

7562


required to be input-to-state stable.26 However, we emphasize that, in order to prove the results in Theorem 2.2 and Theorem 3.1, P is only required to be stable in the sense of Definition A.1. We first present the following definitions: Definition B.1 (Class K Function).21 A continuous function υ(u): [0, a) f [0, ∞) is said to belong to class K if it is strictly increasing and υ(0) ) 0. Definition B.2 (ClassKL Function).21 A continuous function β(x, t): [0, a) × [0, ∞) f [0, ∞) is said to belong to class KL if, for each fixed t, the mapping β(x, t) belongs to class K with respect to x and, for each fixed x, the mapping β(x, t) is decreasing with respect to t and β(x, t) f 0 as t f ∞. Definition B.3 (Input-to-State Stability).21 Consider the system in eq 29 with steady-state equilibrium (uss, xss, yss). The system is said to be input-to-state stable if there exists a class KL function β(x, t) and a class K function υ(u) such that, for any initial state x0 ∈ X0 ⊂ Rn and any bounded input u(t), the process state vector x(t) satisfies the following:

|x(t) - xss| e β(|x0 - xss|, t - t0) + υ( sup |u(τ) - uss|) t0eτet (31) On the basis of these definitions, we have the following result. Theorem B.4 (Nonlinear IMC Nominal Stability). Consider the IMC scheme in Figure 1 where P, P0, and Q are nonlinear dynamic systems. Assume the following: (b1) The process P is input-to-state stable on X0 ⊂ Rn as per Definition B.3. (b2) xQ ∈ Rq is the state vector of the IMC controller Q. Moreover, Q is asymptotically stable on XQ ⊂ Rq as per Definition A.1. (b3) For each steady-state operating point (ess, xQss, uss) of Q such that xQss ∈ XQ, there exists a corresponding steady-state operating point (uss, xss, yss) of P such that xss ∈X0. (b4) There is no uncertainty in the process model, i.e., P0 ) P. (b5) The process P and the model P0 have the same initial conditions x0 ∈ X0. Then, the IMC closed loop in Figure 1 is asymptotically stable on the region X } X0 × XQ (where × stands for the Cartesian product). Proof. From conditions (b4) and (b5), we have that y0(t) ) y(t) for all t g 0. Thus, the signal y(t) - y0(t) fed back to the IMC controller in Figure 1 is identically zero. We then have that e(t) ) r(t) for all t g 0; hence, the IMC closed loop in Figure 1 is given by

y(t) ) PQr(t) ∀t g 0

where the existence of the process steady-state operating point (uss, xss, yss) is guaranteed by condition (b3). The above inequality implies that, for any bounded input u(t), the process state vector x(t) will also be bounded. From eq 33, we see that the input u(t) converges asymptotically to the steady-state value uss. This fact, in conjunction with inequality 34 and the properties of the class K function υ(‚) and class KL function β(‚,‚), implies that the process state vector x(t) converges asymptotically to xss. We conclude that the IMC closed loop in eq 32 is asymptotically stable on the region X } X0 × XQ. This completes the Proof. Appendix C. Proofs of Theorems and Corollaries in Sections 2 and 3 C.1. Preliminaries: Assumption C.1. Consider the following assumptions: (c1) The dynamics of the linear controller C h in Figure 2 are described by the state space equations:

C h:

{

z˘ ) Az + Bξ u ) Cz + Dξ

(35)

where z ∈ Rnz and ξ ∈ Rm. (c2) A in eq 35 is Hurwitz. (c3) The process P and the model P0 are stable in the sense of Definition A.1. In addition, there is no model-plant mismatch, i.e., P0 ) P. (c4) The algebraic equation f(x, u) ) 0 has a unique solution

xj ) hstate(u)

(36)

such that hstate(‚) is C2. (c5) The steady-state relation between ξ and y is given by

h ξ), K h ξ) } φ(ξ) y ) g(xj, K h ξ) ) g(hstate(K

(37)

where K h is the steady-state gain matrix of C h , i.e., K h ) -CA-1B + D, and xj is given in eq 36. C.2. Proof of Theorem 2.2. On the basis of the process nonlinear model in eq 2 and assumption (c1), we have that the dynamics of the closed loop in Figure 2 are described by the following set of differential equations:

ξ˙ ) K ˆ [r/ss - g(x, Cz + Dξ)] z˘ ) Az + Bξ

(38)

x˘ ) f(x, Cz + Dξ)

(32) We introduce the slow time scale variable τ given by

where PQ denotes the series interconnection of the nonlinear dynamic systems P and Q. With r(t) ) rss constant, condition (b2) implies that

lim xQ(t) ) xQss tf∞

(33)

lim u(t) ) uss tf∞

for every initial state in XQ ⊂ Rq. From condition (b1), we have that there exists a class KL function β(x, t) and a class K function υ(u) such that, for any initial state x0 ∈ X0 ⊂ Rn and any bounded input u(t), the process state vector x(t) satisfies

|x(t) - xss| e β(|x0 - xss|, t - t0) + υ( sup |u(τ) - uss|) t0eτet (34)

τ ) t

(39)

and rewrite the closed-loop dynamics in eq 38 in terms of τ as follows:

dξ )K ˆ [r/ss - g(x, Cz + Dξ)] dτ

dz ) Az + Bξ dτ

(40)

dx ) f(x, Cz + Dξ) dτ

The above set of equations are in standard singular perturbation form.8 We then use singular perturbation analysis to study the asymptotic stability of the feasible operating point of interest


(u/ss, y/ss) where y/ss ) r/ss. We first note that, from assumptions (c4) and (c5), the operating point (u/ss, y/ss) defines a unique operating point (ξ/ss, y/ss) where y/ss ) φ(ξ/ss). Using singular perturbation analysis, we let f 0 and we restrict the analysis to the slow time scale dynamics of the closed loop. This is equivalent to a time-scale separation approach: by detuning the controller (when f 0), we can separate the “slow” dynamics of the integrator from the “fast” dynamics (in relative terms) of the controller and the process. Thus, when f 0, the closedloop dynamics in eq 40 become

dξ )K ˆ [r/ss - g(x, Cz + Dξ)] dτ 0 ) Az + Bξ

(41)

0 ) f(x, Cz + Dξ) for which we require assumptions (c2) and (c3). Next, using assumptions (c4) and (c5), we can substitute the above set of equations with the following single equation:

|

∂h (u - u/ss) + h.o.t. ∂u u)u/ss

(46)

Then, there exists a nonempty neighborhood Uδ of u/ss such that

y ≈ y/ss +

|

∂h (u - u/ss) ∂u u)u/ss

(47)

Thus, for every ξ such that u ) K h ξ is inside the set Uδ (in steady state), we have that

(

(ξ - ξ/ss)T(y - y/ss) ≈ (ξ - ξ/ss)T y/ss +

|

∂h K h (ξ - ξ/ss) ∂u u)uss/

)

y/ss (48) ss

(42)

To study the stability of the above dynamics, we consider the Lyapunov function candidate:

1 ˆ -1(ξ - ξ/ss) (ξ - ξ/ss)TK 2

}

dV 1 dξT -1 dξ ) K ˆ (ξ - ξ/ss) + (ξ - ξ/ss)TK ˆ -1 dτ 2 dτ dτ

(ξ -

ξ/ss)T(y

-

y/ss)

≈ |ξ -

ξ/ss|2

>0

(49)

(43)

We conclude that there exists a nonempty neighborhood Λu ) Uδ of u/ss in which eq 9 of Theorem 2.2 is satisfied. Hence, the steady-state region of attraction under linear feedback control Ωu(u/ss) is nonempty since Ωu(u/ss) ⊂ Λu. C.5. Proof of Theorem 3.1. From the Proof of Theorem 2.2, we have that the slow time scale dynamics of the closed loop in Figure 2 are given by

(44)

dξ )K ˆ [y′ss - φ(ξ)] dτ

We then have that

{

y ) h(u) ) y/ss +

If K h is selected such that K h ) ((∂h/∂u)|u)u/ )-1, we obtain

dξ )K ˆ [r/ss - φ(ξ)] dτ

V(ξ) )

C.4. Proof of Lemma 2.5. Consider a multivariable Taylor series of the process steady-state nonlinear map yss ) h(uss) around the operating point of interest (u/ss, y/ss):

) -(ξ - ξ/ss)T(φ(ξ) - r/ss) where we have used the fact that K ˆT ) K ˆ . We conclude that a sufficient condition that guarantees the asymptotic stability of the equilibrium point (ξ/ss, y/ss) in the slow time scale domain is that (ξ - ξ/ss)T(y - y/ss) > 0 for every closed-loop trajectory ξ(τ). The existence of 0 > 0 such that the singular perturbation analysis can be applied for all 0 < e 0 is guaranteed by Theorem 3.18 on p 90 of ref 22. C.3. Proof of Corollary 2.4. The steady-state region of attraction under linear feedback control Ωu(u/ss) is given by the largest ellipsoid Π(K ˆ , γ, u/ss) in the u-space that is completely contained in the AIS and in the region Λussee eq 11. Using the linear relation u ) K h ξ, we can express the ellipsoid Π(K ˆ, γ, u/ss) in terms of ξ in the ξ-space as follows:

ˆ -1(ξ - ξ/ss) e γ} Π(K ˆ , γ, ξ/ss) } {ξ ∈ Rm|(ξ - ξ/ss)TK (45) We see that the above ellipsoid is defined based on the Lyapunov function V(ξ) in eq 43. For given K ˆ )K ˆ T > 0 and γ > 0, V(ξ) e γ/2 defines the same ellipsoid given by Π(K ˆ , γ, ξ/ss) in eq 45. From Definition 2.3, we have that K ˆ and γ are such that Π(K ˆ , γ, ξ/ss) is the largest ellipsoid completely contained in the region Λξ where eq 9 of Theorem 2.2 holds. Thus V(ξ) e γ/2 (therefore, also Ωu(u/ss)) is positively invariant, since (dV/dτ) < 0 for every closed-loop trajectory ξ(τ) originating in V(ξ) e γ/2. In addition, the result in Theorem 2.2 guarantees that the closed-loop trajectory ξ(τ) will asymptotically converge to the feasible operating point (ξ/ss, φ(ξ/ss)).

(50)

where the reference signal is r(t) ) y′ss constant. Differentiating the error signal e˜ ) r - y, we obtain

∂φ dξ de˜ )dτ ∂ξ dτ

(51)

since y ) φ(ξ). Substituting eq 50, we have

de˜ ∂φ ∂φ )K ˆ [y′ss - φ(ξ)] ) K ˆ e˜ dτ ∂ξ ∂ξ

(52)

Next, we show that the equilibrium point e˜ ) 0 is asymptotically stable. Consider the Lyapunov function candidate:

1 V(e˜ ) ) e˜ T e˜ 2

(53)

∂φ dV ) -e˜ T K ˆ e˜ dτ ∂ξ

(54)

Hence,

We conclude that (dV/dτ) < 0 if and only if

∂φ K ˆ >0 ∂ξ

(55)

∂h K hK ˆ >0 ∂u

(56)

or equivalently

If there exists a nonempty region Θu ⊂ Rm in the input space

7564


13). For a given y′ss ∈ Θy, the largest ball inscribed in Θe˜ (y′ss) has radius minw∈Θ°y |y′ss - w| where Θ°f is the contour of the region Θy and w are points on the contour. Thus, we conclude that the closed loop will converge to the new operating point (u′ss, y′ss) if the initial condition for the error signal e˜ (0) ) y′ss y/ss is contained in the ball with radius minw∈Θ°y |y′ss - w|, i.e., if

|y′ss - y/ss| e min |y′ss - w| w∈Θ° y

(60)

Figure 13 shows an example in which a set-point value y′ss satisfies the above condition. The region of all set-point values that satisfies eq 60 defines the steady-state achievable output space via linear feedback control Ωy(y/ss), that is,

Ωy(y/ss) } {y ∈ Θy| ||y - y/ss|| e min |y - w|} w∈Θ°y

Figure 13. Geometric interpretation of eq 60 that characterizes the set points contained in the steady-state achievable output space via linear / feedback Ωy(yss ).

such that the above condition is satisfied for every closed-loop trajectory, then the equilibrium e˜ ) 0 is asymptotically stable. C.6. Proof of Corollary 3.3. Consider the closed loop in Figure 2 and suppose that the reference signal r(t) is such that

r(t) )

{

y/ss, t < 0 y′ss, t g 0

(57)

where y′ss ∈ Ωy(y/ss). We will show that the resulting closedloop trajectory of the error signal e˜ (t) is always contained in a positively invariant set Ce˜ such that eq 17 in Theorem 3.1 is satisfied for all t g 0. Consider the region Θu defined in Theorem 3.1 in which eq 17 holds. Let Θy be the region in the output space that results from mapping the region Θu using the nonlinear steady-state relation yss ) h(uss). Next, for a given set-point value y′ss, consider the region Θe˜ (y′ss) given by

Θe˜ (y′ss) } {e˜ |e˜ ) y′ss - y such that y ∈ Θy}

(58)

This region is, by definition, a mirrored and shifted version of region Θy with the origin e˜ ) 0 coinciding with y′ss in Θy. This is shown graphically in Figure 13 for a two-input, two-output process. Observe that selecting a different set-point value y′ss will cause the region Θe˜ (y′ss) to shift with respect to the origin in the e˜ -space. However, the shape of Θe˜ (y′ss) will remain unchanged. We see that Θe˜ (y′ss) is the region in the e˜ -space in which eq 17 is satisfied for a given set-point value y′ss. Consider the ball

1 V(e˜ ) ) e˜ Te˜ e γ 2

(59)

centered at e˜ ) 0 with γ > 0. The proof of Theorem 3.1 shows that, if γ is chosen such that the ball V(e˜ ) e γ is completely contained in Θe˜ (y′ss), then it defines a positively invariant set. We conclude that every closed-loop trajectory of the error signal e˜ (t) that originates in V(e˜ ) e γ will always be contained in the ball V(e˜ ) e γ and it will asymptotically converge to the origin e˜ ) 0. We now seek to translate the argument described above in terms of conditions on the set of admissible set-point values y′ss. We observe that a ball centered at the origin in the e˜ -space is equivalent to a ball centered at y′ss in the y-space (see Figure

(61)

Literature Cited (1) Vinson, D. R.; Georgakis, C. A new measure of process output controllability. J. Process Control 2000, 10, 185-194. (2) Nikolau, M.; Misra, P. Linear control of nonlinear processes: Recent developments and future directions. Comput. Chem. Eng. 2003, 27, 10431059. (3) Morari, M. Effect of design on the controllability of chemical plants. In Interactions between process design and process control; Perkins, J. D., Ed.; Pergamon: Elmsford, NY, 1992; pp 3-16. (4) Perkins, J. D.; Walsh, S. P. K. Optimization as a tool for design/ control integration. Comput. Chem. Eng. 1996, 20, 315-323. (5) Perkins, J. The integration of design and controlsThe key to future processing systems? Presented at 6th World Congress of Chemical Engineering, Melbourne, Australia, 2001. (6) Seider, W. D.; Seader, J. D.; Lewin, D. R. Product and Process Design Principles: Synthesis, Analysis, and EValuation; John Wiley & Sons: New York, 2003. (7) Georgakis, C.; Uztu¨rk, D.; Subramanian, S.; Vinson, D. R. On the operability of continuous processes. Control Eng. Pract. 2003, 11, 859869. (8) Kokotovic´, P. V.; Khalil, H. K.; O’Reilly, J. Singular perturbation methods in control: analysis and design; Academic Press: New York, 1986. (9) Ekawati, E.; Bahri, P. A. The integration of the output controllability index within the dynamic operability framework in process system design. J. Process Control 2003, 13, 717-727. (10) Bahri, P. A.; Bandoni, J. A.; Romagnoli, J. A. Integrated flexibility and controllability analysis in design of chemical processes. AIChE J. 1997, 43, 997-1015. (11) Economou, C. G.; Morari, M.; Paisson, B. O. Internal model control. 5. Extension to nonlinear systems. Ind. Eng. Chem. Process Des. DeV. 1986, 25, 403-411. (12) Boyd, S.; Vandenberghe, L. ConVex Optimization; Cambridge University Press: New York, 2004. (13) Subramanian, S.; Georgakis, C. Steady-state operability characteristics of idealized reactors. Chem. Eng. Sci. 2001, 56, 5111-5130. (14) Desoer, C. A.; Wang, Y. T. Foundations of feedback theory for nonlinear dynamical-systems. IEEE Trans. Circuits Syst. 1980, 27, 104123. (15) Guay, M.; McLellan, P. J.; Bacon, D. W. Measurement of nonlinearity in chemical process control systems: The steady-state map. Can. J. Chem. Eng. 1995, 73, 868-882. (16) Stack, A. J.; Doyle, F. J. The optimal control structure: an approach to measuring control-law nonlinearity. Comput. Chem. Eng. 1997, 21, 10091019. (17) Helbig, A.; Marquardt, W.; Allgo¨wer, F. Nonlinearity measures: Definition, computation and applications. J. Process Control 2000, 10, 113123. (18) Schweickhardt, T.; Allgo¨wer, F. Quantitative nonlinearity assessments An introduction to nonlinearity measures. In The Integration of Process Design and Control, Vol. 17 of Computer-Aided Chemical Engineering; Seferlis, P., Georgiadis, M. C., Eds.; Elsevier B. V.: Amsterdam, The Netherlands, 2004; Chapter A3, pp 76-95. (19) Eker, S. A.; Nikolaou, M. Linear control of nonlinear systems: Interplay between nonlinearity and feedback. AIChE J. 2002, 48, 19571980.

Ind. Eng. Chem. Res., Vol. 45, No. 22, 2006 7565 (20) Desoer, C. A.; Vidyasagar, M. Feedback systems: input-output properties; Academic Press: New York, 1975. (21) Khalil, H. K. Nonlinear Systems, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 2002; p 07458. (22) Sepulchre, R.; Jancovic´, M.; Kokotovic´, P. ConstructiVe nonlinear control; Springer-Verlag: London, 1997. (23) Sistu, P. B.; Bequette, B. W. Model predictive control of processes with input multiplicities. Chem. Eng. Sci. 1995, 50, 921-936. (24) Kravaris, C.; Daoutidis, P. Nonlinear state feedback control of second-order nonminimum-phase nonlinear systems. Comput. Chem. Eng. 1990, 14, 439-449.

(25) Vidyasagar, M. Nonlinear Systems Analysis, Classics in Applied Mathematics, 2nd ed; SIAM: Philadelphia, PA, 2002. (26) Sontag, E. D.; Wang, Y. On characterizations of the input-to-state stability property. Syst. Control Lett. 1995, 24, 351-359.

ReceiVed for reView November 5, 2005 ReVised manuscript receiVed August 9, 2006 Accepted August 10, 2006 IE051226J

Linear Control of Nonlinear Processes: The Regions of Steady-state

Recommend Documents