Control Quality Loss in Analytical Control of Input-Constrained

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Ind. Eng. Chem. Res. 2006, 45, 8528-8538

Control Quality Loss in Analytical Control of Input-Constrained Processes Masoud Soroush* and Felix S. Rantow Department of Chemical and Biological Engineering, Drexel UniVersity, Philadelphia, PennsylVania 19104

Yiannis Dimitratos DuPont Engineering, Process Dynamics and Control, 1007 Market Street, Wilmington, Delaware 19898

Since the introduction of model predictive control (MPC), control practitioners have been faced with the following question: for what class of processes MPC should be implemented? MPC provides a control sequence that is optimal in the presence or absence of constraints. However, it requires (i) numerically solving a constrained optimization problem repeatedly on-line and (ii) an adequately reliable model. Alternatively, one can implement analytical control, such as proportional-integral-derivative (PID) control and differential geometric control, which does not require the on-line optimization but generally cannot provide the optimal performance. This paper presents an answer to the following question: for what class of processes can analytical control provide control quality that is close to the optimal control quality that MPC can provide? Here, an analytical controller is defined as the one whose implementation does not require solving a constrained optimization problem numerically. A measure that quantifies the degradation in the closed-loop performance of a given process when the process is controlled using analytical control instead of MPC has been defined. The measure is used to characterize the class of input-constrained processes for which MPC provides significantly higher control quality; processes with directionality and active input constraints benefit more from MPC. It is shown that structural information on the characteristic (decoupling) matrix of a process is often adequate for the characterization. Four input-constrained process examples are considered. On the basis of structural information on the characteristic matrices of the four processes, the processes that can be controlled satisfactorily using analytical control are specified. Simulated closed-loop responses are then presented, to show the validity of the characterization. 1. Introduction Constraints (limits on process variables) are abundant in the process industries. It is well understood that constraints have a degrading effect on control quality. To minimize the degrading effect, many strategies, such as anti-windup schemes,1 optimal directionality compensators,2,3 and optimal control methodologies (e.g., model predictive control (MPC)4), have been proposed, studied, and implemented. For a given process design, after disturbances have been identified and controlled outputs and manipulated inputs have been selected, an effective controller is chosen. Of course, the controller should ensure asymptotic closed-loop stability, whether the process is stable or unstable. The controller can be traditional (such as proportional-integral-derivative (PID) control), modelbased, analytical (such as differential geometric control and unconstrained linear quadratic control), and/or numerical (such as MPC). A question that process control engineers often face is for what class of processes one should use MPC that generally requires numerical solution of a constrained optimization problem repeatedly on-line. The alternative is to use analytical control, which does not require on-line optimization. In other words, control of which class of processes can benefit significantly from MPC? In current industrial practice, this question has been addressed in several ways, which generally is dependent on (i) the sector of the industry and the type of manufacturing operations (e.g., petrochemical, chemical, or food), (ii) the acceptance of the technology in operations (typically rooted in the history of * To whom correspondence should be addressed. Tel.: (215) 8951710. Fax: (215) 895-5837. E-mail address: [email protected].

introduction of the technology, experience with previous implementations, employee process control skills, and success in maintaining performance over time), and (iii) how strongly the financial benefits from a migration to MPC manifest themselves. It has often been an obvious value proposition or a leap of faith in the benefits of the technology that would have led to the development and commissioning of an MPC application and not a detailed analysis. Few companies have an in-house engineering capability to conduct an operability analysis, which allows for performance benchmarking and assessment of the financial benefits of the MPC migration. This analysis involves the development of a dynamic model of the process in question and the comparative assessment of an MPC versus an analytical implementation. In the best of the cases, a high-fidelity nonlinear simulator of the entire process is interfaced to the commercial model predictive controller of preference to conduct the evaluation. The lack of a simple analysis tool to determine whether an input-constrained process can be controlled adequately well using analytical control motivated the study that is presented in ref 5. Soroush and Dimitratos5 introduced a measure to quantify the degradation in closed-loop performance when analytical control is implemented instead of MPC for processes with active input constraints. The measure quantifies the deviation of (a) the response of the process under analytical control (unconstrained MPC) with saturation (clipping) from (b) the response of the same process under constrained MPC. Other studies related to this topic include those presented in refs 6-8. Soroush and Muske6 studied several special cases of MPC that take analytical forms. For example, the shortest prediction horizon MPC may have an analytical closed-form

10.1021/ie0601347 CCC: $33.50 © 2006 American Chemical Society Published on Web 09/12/2006

Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8529

solution. Furthermore, a general MPC can be tuned to obtain (i) input-output linearizing control laws that inherently include optimal windup and directionality compensators, (ii) model state feedback control and modified internal model control laws that inherently include optimal directionality compensators, and (iii) proportional-integral (PI) and PID controllers that inherently include optimal windup and directionality compensators. The MPC tuning includes the selection of the process model needed for the model-based controller design. Seron et al.7 characterized regions of the state-space wherein, for a general single-input single-output (SISO) linear system with no state constraints, the measure is zero; they identified the region in which constrained finite-horizon linear MPC and unconstrained finitehorizon linear MPC with clipping (saturation) provide identical solutions. Marjanovic et al.8 also showed that, for a general SISO system with input constraints and certain conditions imposed, the measure is zero;5 that is, for the systems, saturated infinite horizon linear quadratic regulator (SIHLQR) control provides the same control sequence as the constrained infinite horizon linear quadratic regulator (CIHLQR) control. They also showed that a saturated linear quadratic regulator (SIHLQR) is equivalent to the CIHLQR in the case of first-order SISO systems, subject to both state and control constraints. The work presented in this paper is built on the ideas originally introduced in ref 5. It generalizes the ideas and provides them with theoretical foundations. The connections between the closed-loop control quality degradation in a process and the directionality of the process are established. The measure presented in ref 5 is used to characterize the class of inputconstrained processes for which MPC provides significantly higher control quality; processes with directionality and active input constraints benefit more from MPC. It is shown that structural information on the characteristic (decoupling) matrix of a process is often adequate for the characterization. Four input-constrained process examples are considered. On the basis of structural information on the characteristic matrices of the four processes, the processes that can be controlled satisfactorily using analytical control are specified. Simulated closed-loop responses are then presented to show the validity of the characterization. Section 2 describes the scope of the study and some preliminaries. Cases in which constrained MPC admits an analytical solution are presented in Section 3. The index of the control quality loss is presented in Section 4. The application of the index is illustrated by four examples in Section 5. The main body of the paper ends with concluding remarks in Section 6. 2. Scope and Preliminaries Consider nonlinear processes with a model in the form

x(k + 1) ) Φ(x(k), u(k)), x(0) ) x0

(1)

y(k) ) h(x(k))

For a process in the form of eq 1, the relative order (degree) of an output yi, with respect to the vector of manipulated inputs, is denoted by ri, where ri is the smallest integer for which yi(k + ri) is dependent explicitly on the present value of a manipulated input. The relative order ri is the apparent (total) deadtime in yi. The deadtime in yi is (ri - 1) sampling periods, and the remaining one sampling period represents the time delay caused by time discretization of the process using the zeroorder hold. The present and future values of each controlled output yi are given by

yi(k) ) h0i (x(k)) z hi(x(k)) yi(k + l) ) hli(x(k)) z hl-1 i (Φ(x(k), u(k))) (for l ) 1, ..., ri - 1) yi(k + ri) ) hri i(x(k), u(k)) z hri i-1(Φ(x(k), u(k))) yi(k + ri + 1) ) hri i+1(x(k), u(k), u(k + 1)) z hri i(Φ(x(k),u(k)), u(k + 1)) yi(k + ri + l) ) hri i+l(x(k), u(k), ..., u(k + l)) z hri i+l-1(Φ(x(k), u(k)), u(k + 1), ..., u(k + l)) The m × m matrices

[

Qlj(x(k), u(k), ..., u(k + l)) z ∂hr11+l(x(k), u(k), ..., u(k + l))

∂u(k + j)

(for j ) 1, ..., m)

[ ]

∂ r1 h (x, u) ∂u 1 l Q00(x, u) ) ∂ rm h (x, u) ∂u m

(4)

The decoupling (characteristic) matrix has been used to check the degree of dynamic decoupling in processes and specify the class of processes that do not exhibit directionality.2 It has also been called instantaneous process gain,3 because the strength of a process response over a very short horizon is strongly dependent on the decoupling matrix of the process. For processes with a diagonal and independent-of-manipulated-inputs decoupling matrix, completely decentralized control is suitable. The matrix has also been used as basis for input-output pairing. Remark 1: For linear time-inVariant processes:

Φ(x,u) ) Ax + Bu h(x) ) Cx

(2)

where x, u, and y denote the vectors of state variables, manipulated inputs, and controlled outputs, respectively (x ) [x1‚‚‚xn]T, u ) [u1‚‚‚um]T, and y ) [y1‚‚‚ym]T), all in deviation form (that is, (xss, uss) ) (0,0) is the nominal steady-state pair and h(0) ) 0); Φ(x, u) and h(x) are smooth vector functions; and u1l, ..., uml and u1h, ..., umh are constant scalar quantities.

(3)

The matrix Q00 is simply the characteristic (decoupling) matrix of the process:

with the input constraints

ujl e uj(k) e ujh

]

∂u(k + j) (for j ) 0, ..., l; l ) 0, ..., P) l ∂hrmm+l(x(k), u(k), ..., u(k + l))

}

(5)

where A, B, and C are n × n, n × m, and m × n constant matrices, respectiVely,

[

c1Ar1+l-j-1B l Qlj z rm+l-j-1 B cm A

]

(for j ) 0, ..., l; l ) 0, ..., P)

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Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006

where ci is the ith row of C. Thus, for the linear systems,

Q(l+1)(j+1) ) Qlj

subject to

(for j ) 0, ..., l - 1; l ) 0, ..., P - 1)

Furthermore, the characteristic (decoupling) matrix is giVen as

where

[ ]

xˆ *(k + 1) ) Φ[xˆ *(k), w ˆ (k)], x*(0) ) xˆ 0 yˆ *(k) ) h[xˆ *(k)] xˆ (k + 1) ) Φ[xˆ (k), u(k)], xˆ (0) ) xˆ 0 yˆ (k) ) h[xˆ (k)]

c1Ar1-1B Q00 ) l cmArm-1B

}

The present Value of the manipulated inputs, u(k), affects each controlled output yi only after ri sampling periods and the gain of the effect is Q00:

[ ][ ]

y1(k + r1) c1Ar1 ) l x(k) + Q00u(k) l ym(k + rm) cmArm

Definition 1: A matrix every row of which does not have more than one nonzero entry is called a row-sparse matrix. Recall 1: Let L, K, and G be n × m, n × l, and diagonal m × m matrices independent of u, respectiVely. When the matrix L is row-sparse, the solution(s) to the constrained minimization problem (quadratic program),

min{||Lu - K||2 + ||Gu||2} u

(6)

subject to

ujl e uj e ujh is (are) giVen by

(for j ) 1, ..., m)

{

ujl, Vj < ujl

uj ) satj(Vj) z Vj, ujl e Vj e ujh (for j ) 1, ..., m) ujh, Vj > ujh where V represents the solution(s) to the unconstrained minimization problem: V

According to this definition, a SISO process that is not affine in its manipulated input may show/have the directionality. Using the developments in Appendix A and Recall 1, it is straightforward to show that a process in the form of eq 1 does not exhibit the directionality over N sampling periods beyond the apparent output deadtimes, if the block lower-triangular (N + 1) × (N + 1) matrix,

[

Q00 0 0 Q10 Q11 0 Q20 Q21 Q22 l l l Qz Q(N-1)0 Q(N-1)1 Q(N-1)2 QN0 QN1 QN2

‚‚‚ 0 0 ‚‚‚ 0 0 0 l l ‚ 0 l ‚‚ Q ‚‚‚ (N-1)(N-1) 0 QNN ‚‚‚ ‚‚‚

]

(8)

called the extended characteristic matrix, is independent of u and row-sparse. If (i) the matrices Q00, ‚‚‚, QNN can be made diagonal by row or column rearrangements and (ii) the matrices Qij ) 0, i, j ) 0, ..., N (i * j), then Q is row-sparse. Remark 2: Following Remark 1, a linear process in the form of eq 5 does not exhibit directionality oVer time horizons N beyond the apparent deadtimes, if (a) the characteristic matrix Q00 can be made diagonal by row or column rearrangements and (b) matrices Q10 ) ‚‚‚ ) QN0 ) 0. Remark 3: The class of processes whose characteristic (decoupling) matrix, Q00, is independent of u and can be made diagonal by row or column rearrangements does not exhibit the directionality oVer the apparent deadtime horizons, r1, ..., rm (when N ) 0 in Definition 2).

In this section, processes for which constrained MPC is identical to unconstrained MPC with saturation (clipping) are characterized. Let the time-varying static state feedback,

When (L′L + G′G) is nonsingular,

V ) [L′L + G′G]-1L′K 2.1. Directionality. This section presents a brief review of the notion of directionality in input-constrained processes.2,3 Directionality has been defined as a controller performance degradation that is associated with actuator saturation9 and as a systems property.2,3 As stated in refs 2 and 3, a process exhibits directionality when there is a command signal wˆ for which the output response of the process to sat(wˆ ) is not “closest” (in the controlled output space) to the output response of the process to wˆ . A precise statement of this definition is given below. Definition 2:2,3 A process in the form of eq 1 does not exhibit directionality over a time horizon N beyond the apparent output deadtimes, if and only if for every sequence wˆ (k) ∈ Rm, k ) 0, 1, ..., N, and for every initial condition xˆ 0 ∈ Rn, u(k) ) sat{wˆ (k)} minimizes N

∑ ∑[yˆ i(k + ri + l) - yˆ i*(k + ri + l)]2 i)1 l)0

}

3. Analytical Model Predictive Control

min{||LV - K||2 + ||GV||2}

m

(for j ) 1, ..., m)

ujl e uj(k) e ujh

(7)

u*(k) ) Ψ*(x(k), k, ysp(k))

(9)

represent the solution to the general MPC law,

{∑∑ m

min

P

u(k),‚‚‚,u(k+M-1) i)1 l)0

Wyil[yi(k + ri + l) - yspi(k + ri + l)]2 + m M-1

Wu [ui(k + l)]2 ∑ ∑ i)1 l)0 il

}

(10)

subject to

ujl e uj(k + l) e ujh

(for j ) 1, ..., m)

u(k + l) ) u(k + l - 1)

∀l

(for l ) M, ...)

for processes in the form of eq 1 with the input constraints of eq 2, where P g 0 is the prediction horizon beyond the apparent


deadtimes, and M g 1 is the control horizon (1 e M e P + 1). Also, let the time-invariant static state feedback,

w(k) ) Ψ(x(k), ysp(k))

u(k),‚‚‚,u(k+M-1)

{

m

P

∑ ∑Wy [yi(k + ri + l) - ysp (k + ri + l)]2 + i)1 l)0 il

i

m M-1

∑ ∑ Wu [ui(k + l)]2 i)1 l)0 il

}

(12)

subject to

(for l ) M, ...)

Using the results in Appendix B, one can show that the constrained MPC of eq 10 is identical to unconstrained MPC of eq 12 with saturation; that is,

u*j (k) ) satj(wj(k)) where

(for j ) 1, ..., m)

]

w(k) ) [Q ˜ (x(k))′W′yWyQ ˜ (x(k)) + W′uWu]-1Q ˜ (x l w(k + M - 1) ˜ (x(k))] (13) (k))′W′yWy[Ysp(k) - R

[

when the m(P + 1) × mM matrix, Q ˜ z Q00 Q10 Q20 l

]

Remark 5: Constrained MPC, with a prediction horizon of P e N and a control horizon of M ) P + 1, of a process that does not exhibit directionality oVer N sampling periods beyond the apparent output deadtimes, admits an analytical solution. With these MPC settings, Q ˜ ) Q; if Q is row-sparse, so is Q ˜. ˜ is row-sparse. Thus, WyQ Remark 6: For linear processes, the matrix Q ˜ is constant and is giVen by

[

Q ˜ z

u(k + l) ) u(k + l - 1)

[

Q00 0 Q10 Q11 Q21 + Q22 Q ˜ ) Q20 l l QP0 QP1 + ‚‚‚ + QPP

(11)

denote the solution to the same general MPC law for the process of eq 1 without the input constraints of eq 2; that is, the numerical solution to the general, unconstrained, moving horizon, optimization problem is

min

[

and when M ) 2, the matrix

0 Q11 Q21 l

0 0 Q22 l

Q(M-2)0 Q(M-2)1 Q(M-2)2 Q(M-1)0 Q(M-1)1 Q(M-1)2 QM0

QM1

QM2

l

l

l

QP0

QP1

QP2

‚‚‚ ‚‚‚ 0 ‚ ‚‚ ‚‚‚ ‚‚‚

0 0 ‚‚‚ 0

0 0 l l

0 0 l

Q(M-2)(M-2) 0 ‚‚‚ Q(M-1)(M-2) Q(M-1)(M-1) QM(M-1) + ‚‚‚ ‚‚‚ QM(M-2) QMM l l l l QP(M-1) + QP(M-2) ‚‚‚ ‚‚‚ + QPP

]

(14)

is independent of u and row-sparse. In the following sample cases, the matrix Q ˜ is row-sparse: (a) P ) 0 and Q00 is row-sparse; (b) m ) 1 and M ) 1, irrespective of the value of P; (c) M ) 1, and Qij, i ) 0, ‚‚‚, P, j ) 0, ‚‚‚, i, are row-sparse; and (d) Qij ) 0, i, j ) 0, ..., N (i * j) and Q00, ..., QNN are rowsparse. Remark 4: When M ) 1, the matrix

[

Q00 Q10 + Q11 Q ˜ ) l QP0 + ‚‚‚ + QPP

]

Q00 Q10 Q20 l

0 Q00 Q10 l

0 0 Q00 l

Q(M-2)0 Q(M-3)0 Q(M-4)0 Q(M-1)0 Q(M-2)0 Q(M-3)0 QM0 Q(M-1)0 Q(M-2)0 l l l QP0

‚‚‚ ‚‚‚ 0 ‚ ‚‚ ‚‚‚ ‚‚‚ ‚‚‚ l

0 0 ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ l

0 0 l 0 Q00 Q10 Q20 l

Q(P-1)0 Q(P-2)0 ‚‚‚ ‚‚‚ Q(P-M+2)0

0 0 l l 0 Q00 Q10 + Q00 l Q(P-M+1)0 + ‚‚‚ + Q00

and the model predictiVe controller of eq 10 takes the form

]

min{||Wy[Q ˜ U(k) + Dx(k) - Ysp(k)]||2 + ||WuU(k)||2} U(k)

subject to

ujl e uj(k), ‚‚‚, uj(k + M - 1) e ujh

(for j ) 1, ..., m)

where D is a constant matrix. In this case, clipping the unconstrained MPC sequence giVen by eq 12 yields exactly the solution to the preceding constrained optimization problem; that is,

u*j(k) ) satj(wj(k)) where

[

(for j ) 1, ..., m)

]

w(k) ) l w(k + M - 1) [Q ˜ ′W′yWyQ ˜ + W′uWu]-1Q ˜ ′W′yWy[Ysp(k) - Dx(k)]

if Q ˜ is row-sparse. A special case of interest is that QP0 ) ‚‚‚ ) Q10 ) 0, and Q00 is row-sparse, for which MPC has an analytical solution. Note that only structural information is need to determine whether a matrix is row-sparse. 4. Control Quality Loss in Analytical Control For linear processes, the general unconstrained MPC law of eq 12 is a linear state feedback that can be written in an analytical form. In the case of nonlinear processes, one may be able to find an analytical nonlinear state feedback equivalent

8532


to the general unconstrained MPC law of eq 12. Thus, the general unconstrained MPC law of eq 12, in many cases, can be viewed as an analytical time-invariant state feedback. A question that one may ask is when the analytical state feedback of eq 11, together with clipping (saturation), is adequate for constrained processes of the form shown in eq 1. In other words, for what class of processes can this analytical control scheme provide a satisfactory control quality (which is not much less than the quality provided by the numerical state feedback of eq 9)? To address this question, let us define the following index of control quality loss. Definition 3: The index of control quality loss in analytical control over a time horizon N, is defined as

IN ) m

1 m(N + 1)

max max x0∈X

k

N

∑ ∑ i)1 l)0

[

]

yi(k + ri + l) - y*i (k + ri + l) y*iul

2

x*(k + 1) ) Φ(x*(k), u*(k)), x*(0) ) x0 u*(k) ) Ψ*[x*(k), k, ysp(k)] y*(k) ) h[x*(k)] x(k + 1) ) Φ(x(k), sat{w(k)}), x(0) ) x0 w(k) ) Ψ[x(k), ysp(k)] y(k) ) h[x(k)]

} }

The definition of the directionality over N sampling periods beyond the apparent deadtimes r1, ..., rm, can be used to identify the class of processes that have a control-quality loss index of zero. For processes that do not exhibit directionality over N sampling periods beyond the apparent output deadtimes, IN ) 0. Remark 7: The index of control quality loss in analytical control oVer a time horizon of N ) 0 beyond the apparent deadtimes, r1, ..., rm, takes the form

1 m

m

max max x0∈X

k

∑ i)1

[

5.1. Multivariable Linear Example 1. Consider the twoinput two-output, time-invariant, linear process,

x1(k + 1) ) 0.95x1(k) + u1(k) x2(k + 1) ) 0.95x2(k) + x1(k) x3(k + 1) ) 0.95x3(k) + u2(k) x4(k + 1) ) 0.95x4(k) + x3(k) y1(k) ) 0.4x1(k) - 9x3(k) + 0.45x4(k) y2(k) ) -0.01x1(k) + 0.0005x2(k) + 0.4x3(k)

]

yi(k + ri + l) - y*i (k + ri + l) y*iul

2

(for k ) 0, ..., ∞) (16)

The definition of the directionality oVer the short horizons of r1, ..., rm, can be used to identify the class of processes that haVe a control-quality loss index of 0. The purpose of introducing these indices is not to calculate their values through many numerical simulations, but rather to identify the cases for which these indices are zero, especially if the identification of these cases requires little structural information on the process under consideration. Remark 8: For processes that do not exhibit the directionality oVer the time horizons r1, ..., rm, the index of control quality loss is I0 ) 0 when a model predictiVe controller with a prediction horizon of N ) 0 is implemented. Processes whose nonsingular characteristic matrices are independent of u and can be made diagonal by row or column rearrangements do not exhibit directionality oVer the time horizons r1, ..., rm. For processes that exhibit the directionality oVer the time horizons r1, ..., rm, I0 * 0, proVided that there is an actiVe input constraint.

}

(17)

with x1(0) ) 0, x2(0) ) 0, x3(0) ) 0, x4(0) ) 0, |u1(k)| e 2, and |u2(k)| e 2. The steady-state gain matrix (Kp) and the characteristic matrix (Q00) of the process are given as

[ ]

8 0 0 8 0.4 -9 Q00 ) -0.01 0.4 Kp )

(for k ) 0, ..., ∞) (15)

where X is the set of all admissible stabilizing initial conditions for the constrained model predictive controller of eq 10, y*iul is the smallest number that satisfies |y* i (k)| e y* iul for all k,

I0 )

5. Examples

[

]

The steady-state gain matrix is diagonal, implying that the process is “statically” and completely decoupled. However, the process is dynamically strongly coupled, because its characteristic matrix is not row-sparse; this process exhibits the directionality. Thus, for this process, I0 * 0, implying that if analytical control is used and an input constraint becomes active, the loss of control quality will be significant. The following two control schemes are implemented: 2

P

[yi(k + 1 + l) - ysp (k + 1 + l)]2 ∑ ∑ u(k) i)1 l)0

min

i

(18)

subject to

ujl e uj(k) e ujh u(k + l) ) u(k + l - 1)

(for j ) 1, 2)

∀k

(for l ) M, ..., P + 1)

and 2

P

[yi(k + 1 + l) - ysp (k + 1 + l)]2 ∑ ∑ u(k) i)1 l)0

min

i

(19)

subject to

u(k + l) ) u(k + l - 1)

(for l ) M, ..., P + 1)

with saturation. Here, ysp1 ) 2, ysp2 ) 2, u1l ) u2l ) -2, and u1h ) u2h ) +2. Controller 18 is a constrained linear model predictive controller with a prediction horizon of P beyond the apparent deadtimes (r1 ) 1, r2 ) 1) and a control horizon of M. Controller 19, which has an analytical solution, is the unconstrained counterpart of controller 18. Figures 1a, 1b, and 1c depict the closed-loop process, input, and output responses under controller 18 (solid line) and controller 19 with saturation (dashed line) for different values of P and M. They confirm that the loss of control quality in analytical control is indeed significant. For the given fixed initial conditions, I5 ) 1.892 × 101, 9.934 × 10-1, and 5.178 × 102 in Figures 1a, 1b, and 1c, respectively. Figure 1a shows that,


Figure 1. Controlled outputs and manipulated inputs of Example 1: (a) P ) 1, M ) 1; (b) P ) 5, M ) 1; and (c) P ) 5, M ) 3. In all cases, the solid line represents the response obtained under controller 18, and the dashed line represents the response obtained under controller 19.

under analytical controller 19 with P ) M ) 1, and unconstrained constrained MPC with saturation, the closed-loop system is unstable; the loss of control quality is significant (I5 ) 1.892 × 101). In Figures 1b and 1c (I5 ) 9.934 × 10-1 and 5.178 × 102, respectively), although the closed-loop system is stable under the unconstrained constrained MPC with saturation, the loss of control quality is easily observable.

teristic matrix (Q00) of the process are

5.2. Multivariable Linear Example 2. Consider the twoinput two-output, time-invariant, linear process,

implying that the process is statically highly coupled but is dynamically weakly coupled (the characteristic matrix is diagonal); the process does not exhibit the directionality over N ) 0. Thus, for this process, I0 ) 0, implying that if analytical control is used, the loss of control quality will not be significant when compared to MPC with P ) 0 and M ) 1. Figures 2a, 2b, and 2c compare the closed-loop responses of the input-constrained process under the two control schemes of controllers 18 and 19 with different values of P and M. Solid circles and solid lines represents responses under controller 18, and empty circles and dashed lines represent responses under controller 19. For the given fixed initial conditions, I5 ) 2.30 × 10-3, 1.34 × 10-9, and 3.10 × 10-3 in Figures 2a, 2b, and 2c, respectively. Here, ysp1 ) 2, ysp2 ) 4, u1l ) u2l ) -1 and

x1(k + 1) ) 0.5x1(k) + u1(k) x2(k + 1) ) -0.5x2(k) - x1(k) x3(k + 1) ) 0.5x3(k) + u2(k) x4(k + 1) ) 0.5x4(k) + x3(k) y1(k) ) x1(k) + x4(k) y2(k) ) x2(k) + 2x3(k)

}

(20)

with x1(0) ) 0, x2(0) ) 0, x3(0) ) 0, x4(0) ) 0, |u1(k)| e 1, and |u2(k)| e 1. The steady-state gain matrix (Kp) and the charac-

Kp )

[

2 4 -4/3 4

Q00 )

[ ]

]

1 0 0 2

8534


Figure 2. Controlled outputs and manipulated inputs of Example 2: (a) P ) 1, M ) 1; (b) P ) 5, M ) 1; and (c) P ) 5, M ) 3. In all cases, the solid circles and solid lines represent the response obtained under controller 18, and the empty circles and dashed lines represent the response obtained under controller 19.

u1h ) u2h ) +1. Although P and M were set to values greater than zero and one, respectively, as can be observed in the three parts of Figure 2, the two control systems showed very similar performances. This further confirms the central role that the characteristic matrix has in the constrained optimization, even when one implements MPC with a prediction horizon beyond the apparent process deadtime of greater than zero, with P > 0. 5.3. Multivariable Linear Example 3. Consider the twoinput two-output, time-invariant, linear process,

x1(k + 1) ) 0.99x1(k) + 0.40u1(k) - 3.00u2(k) x2(k + 1) ) 0.99x2(k) - 0.01u1(k) + 0.40u2(k) y1(k) ) x1(k) y2(k) ) x2(k)

}

(21)

with x1(0) ) 0, x2(0) ) 0, |u1(k)| e 2, and |u2(k)| e 2. For this linear example,

Q00 )

[

0.40 -3.00 -0.01 0.40

]

which is not row-sparse; thus, the process cannot be regulated effectively by analytical control when one of the input constraints becomes active. Figures 3a, 3b, and 3c compare the closed-loop responses of the input-constrained process under the two control schemes of controllers 18 and 19 with different values of P and M (the solid line represents data obtained under controller 18, and the dashed line represents data obtained under controller 19). For the given fixed initial conditions, I5 ) 8.258 × 100, 6.227 × 100, and 8.617 × 100 in Figures 3a, 3b, and 3c, respectively. Here, ysp1 ) 8, ysp2 ) 3, u1l ) u2l ) -2, and u1h ) u2h ) +2. The results confirm that the loss of control quality in analytical control is indeed significant. Under constrained MPC (controller 18), the y1 response has much smaller rise and response times than those under controller 19, whereas the y2 response has equal response times under the two controllers. Furthermore, under controller 18, the y2 response has no overshoot, whereas under controller 19, it shows some overshoot. 5.4. Single-Input Single-Output (SISO) Nonlinear Chemical Reactor. Consider an isothermal continuous stirred-tank


Figure 3. Controlled outputs and manipulated inputs of Example 3: (a) P ) 1, M ) 1; (b) P ) 5, M ) 1; and (c) P ) 5, M ) 3. In all cases, the solid line represents the response obtained under controller 18, and the dashed line represents the response obtained under controller 19.

reactor in which a nonelementary chemical reaction A f B occurs. The rate of production of B is given by

manipulating the reactant concentration, CA. Here, the characteristic matrix is

RB[CA] ) 0.036CA3 - 0.78CA2 + 4.4CA - 1.0

Q00 ) 0.108u2 - 1.56u + 4.4

where CA denotes the concentration of the reactant. The reactor dynamics are described by

dCB ) 0.036CA3 - 0.78CA2 + 4.4CA - 1.0 - 0.625CB dt (22) with CB(0) ) 0 and 0 e CA e 10, where CB denotes the concentration of the product. Exact time discretization (with a sampling period of 0.1 h) of the reactor model leads to

CB(k + 1) ) 0.9394CB(k) + 0.0969RB[CA(k)]

(23)

The control objective is to maintain the concentration of the product, CB, at a desired level (CB ) 8.0 kmol/m3) by

Because Q00 is dependent on u, the process has the directionality. For this process, the loss of control quality will be significant, if analytical control is used and an input constraint becomes active. For this process, we use the two model predictive controllers: 10

min{RB[u(k)] - 1.0320[e(k) + η(k)] - 0.4066CB(k)}2 u(k) (24) subject to

η(k + 1) ) 0.9η(k) + 0.0394CB(k) + 0.0969RB[u(k)] 0 e u(k) e 10 and

8536


min{RB[w(k)] - 1.0320[e(k) + η(k)] - 0.4066CB(k)}2 w(k)

η(k + 1) ) 0.9η(k) + 0.0394CB(k) + 0.09694RB[u(k)] u(k) ) sat(w(k))

(25)

with η(0) ) CB(0) and e ) 8 - CB. Numerical simulations are performed for the following three cases: (1) Case C1: the process of eq 23 without the input constraints, under controller 24 or 25. (2) Case C2: the process of eq 23 with the input constraints, under controller 24. (3) Case C3: the process of eq 23 with the input constraints, under controller 25. In the absence of the input constraints, controllers 24 and 25 induce the first-order linear input-output response,

CB(k + 1) - 0.9CB(k) ) 0.8

(26)

Controllers 24 and 25 are constrained and unconstrained nonlinear model algorithmic controllers (model predictive controllers with P ) 0, M ) 1, and a linear reference trajectory), respectively. Controller 25 is implemented with saturation (clipping). As shown in Figure 4, the closed-loop responses are considerably different for the three cases. The response in case C1 has, of course, the lowest possible integral of squared error [ISE] (ISE ) 3.47 × 102). The advantage of constrained MPC (case C2 with an ISE ) 3.90 × 102) over unconstrained MPC with saturation (case C3 with an ISE ) 1.45 × 103) is quite obvious in the CB plot. For the given fixed initial condition, I5 ) 2.381 × 10-1).

Figure 4. Controlled output and manipulated input of the chemical reactor example.

Appendix A

min

u(k), ‚‚‚, u(k + N)

6. Conclusions A measure that allows one to quantify the degradation in closed-loop performance when one implements analytical control instead of model predictive control (MPC) for processes with active input constraints was introduced. The measure quantifies the deviation of (a) the response of the process under analytical control (unconstrained model predictive control with saturation) from (b) the response of the same process under constrained MPC. The purpose of introducing the measure was not to calculate its value through many numerical simulations, but rather to identify the cases for which the measure is zero, especially if the identification of these cases requires little structural information on the process under consideration. The connections between the closed-loop control quality degradation in a process and the directionality of the process were established. The measure was used to characterize the class of input-constrained processes for which MPC provides significantly higher control quality; processes with directionality and active input constraints benefit more from MPC. It was shown that structural information on the characteristic (decoupling) matrix of a process, the row sparseness of the matrix, is often adequate for the characterization.

[| ]|

The minimization problem of eq 7 can be recast in the vector form,

subject to

[ [ [

]

yˆ 1(k + r1) - yˆ 1*(k + r1) l yˆ m(k + rm) - yˆ m*(k + rm) yˆ 1(k + r1 + 1) - yˆ 1*(k + r1 + 1) l yˆ m(k + rm + 1) - yˆ m*(k + rm + 1) l yˆ 1(k + r1 + N) - yˆ 1*(k + r1 + N) l yˆ m(k + rm + N) - yˆ m*(k + rm + N)


(for j ) 1, ..., m)

2

] ]

∀l

Substituting for the future values of the controlled outputs from the model:

min

u(k),‚‚‚,u(k+N)

||H(xˆ (k), u(k), ‚‚‚, u(k + N)) - Y*(k)||2 (A.1)

subject to

ujl e uj(k), ‚‚‚, uj(k + N) e ujh where

(for j ) 1, ..., m)

[ ] [ ]

H(xˆ (k), u(k), ‚‚‚, u(k + N)) z


subject to

hr11(xˆ (k),

u(k)) l hrmm(xˆ (k), u(k)) -------hr11+1(xˆ (k), u(k), u(k + 1)) l hrmm+1(xˆ (k), u(k), u(k + 1)) -------l -------hr11+N(xˆ (k), u(k), ‚‚‚, u(k + N)) l hrmm+N(xˆ (k), u(k), ‚‚‚, u(k + N))

hr11(xˆ *(k), w(k)) l hrmm(xˆ *(k), w(k)) -------hr11+1(xˆ *(k), w(k), w(k + 1)) l Y*(k) ) hrmm+1(xˆ *(k), w(k), w(k + 1)) -------l -------hr11+N(xˆ *(k), w(k), ‚‚‚, w(k + N)) l hrmm+N(xˆ *(k), w(k), ‚‚‚, w(k + N))

When the extended characteristic matrix, Q, which is defined by eq 8, is independent of u, the minimization of eq A.1 takes the form

min

u(k), ‚‚‚, u(k + N)

subject to

| [ ]

u(k) Q(xˆ ) l + R(xˆ ) - Y*(k) u(k + N)

ujl e uj(k), ‚‚‚, uj(k + N) e ujh where

2

|


u(k + l) ) u(k + l - 1)

[

Wu10 0 0 l 0 Wu z 0 0

0 0 ‚ ‚‚ 0 0 Wym0 l l 0 0 0 0 0 0

0 0 ‚ ‚‚ 0 0 Wum0 l l 0 0 0 0 0 0

[ ]

u(k) + R(xˆ ) ) H(xˆ (k), u(k), ‚‚‚, u(k + N)) Q(xˆ ) l u(k + N) Appendix B

‚‚‚ ‚‚‚ ‚‚‚ ‚ ‚‚ ‚‚‚ ‚‚‚ ‚‚‚

0 0 0 0 0 0 l l 0 Wy1p 0 0 0 0

0 0 0 0 0 0 l l 0 0 ‚ ‚‚ 0 0 Wymp

0 0 0 0 0 0 l l 0 Wu1(m-1) 0 0 0 0

0 0 0 0 0 0 l l 0 0 ‚ ‚‚ 0 0 Wum(m-1)

min

u(k),‚‚‚,u(k+M-1)

[ [

[

y1(k + r1) - ysp1(k + r1)

l ym(k + rm) - yspm(k + rm)

]

y1(k + r1 + 1) - ysp1(k + r1 + 1)

Wy

l ym(k + rm + 1) - yspm(k + rm + 1) l y1(k + r1 + P) - ysp1(k + r1 + P) l ym(k + rm + P) - yspm(k + rm + P)

2

] | [ ]| ] + Wu

u(k) l u(k+M-1)

2

(B.1)

]

U(k)

subject to

ujl e uj(k), ‚‚‚, uj(k + M - 1) e ujh

(for j ) 1, ..., m)

[ ] [

u(k) l u(k + M - 1)

]

hr11(x(k), u(k)) l hrmm(x(k), u(k)) -------hr11+1(x(k), u(k), u(k + 1)) l hrmm+1(x(k), u(k), u(k + 1)) -------l H ˜ (x(k),U(k)) z r1+M-1 -------(x(k), u(k), ‚‚‚, u(k + M - 1)) h1 l rm+M-1 (x(k), u(k), ‚‚‚, u(k + M - 1)) hm -------l -------r1+P h1 (x(k), u(k), ‚‚‚, u(k + M - 1)) l hrmm+P(x(k), u(k), ‚‚‚, u(k + M - 1))

{| [ ] | } [ ]

The general model predictive controller described in eq 10 can be recast in the vector form,

]

min{||WyH ˜ (x(k),U(k)) - WyYsp(k)||2 + Wu||U(k)||2} (B.2)

U(k) )

(for j ) 1, ..., m)

‚‚‚ ‚‚‚ ‚‚‚ ‚ ‚‚ ‚‚‚ ‚‚‚ ‚‚‚

Substituting for the future values of the controlled outputs from the model, the minimization problem of eq B.1 takes the form

where

(A.2)

[

(for l ) M, ...)

where the diagonal matrices are

Wy10 0 0 l 0 Wy z 0 0

∀l

(for j ) 1, ..., m)

ysp1(k + r1) l yspm(k + rm) --------ysp1(k + r1 + 1) l Ysp(k) z yspm(k + rm + 1) -------l -------ysp1(k + r1 + P) l yspm(k + rm + P)

When the matrix Q ˜ , given by eq 14, is independent of u, the minimization of eq B.2 takes the form

8538


min||Wy[Q ˜ (x)U(k) + R ˜ (x) - Ysp(k)]||2 + ||WuU(k)||2 (B.3) U(k)

subject to

ujl e uj(k), ‚‚‚, uj(k + N) e ujh

(for j ) 1, ..., m)

where

Q ˜ (x)U(k) + R ˜ (x) ) H ˜ (x(k),U(k)) Literature Cited (1) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, C. N. A Unified Framework for the Study of Anti-Windup Designs. Automatica 1994, 30 (12), 1869. (2) Soroush, M.; Mehranbod, N. Optimal Compensation for Directionality in Processes with a Saturating Actuator. Comput. Chem. Eng. 2002, 26 (11), 1633. (3) Soroush, M.; Valluri, S. Optimal Directionality Compensation in Processes with Input Saturation Nonlinearities. Int. J. Control 1999, 72 (17), 1555. (4) Allgower, F., Zheng, A., Eds. Nonlinear Model PredictiVe Control; Progress in Systems and Control Theory Series, Vol. 26; BirkhauserVerlag: Birkhauser Verlag: Basel, Switzerland, 2000.

(5) Soroush, M.; Dimitratos, Y. Control System Selection: A Measure of Control Quality Loss in Analytical Control. Presented at Dynamics and Control of Process Systems (DYCOPS 7) Symposium, Boston, MA, July 5-7, 2004. (6) Soroush, M.; Muske, K. Analytical Model Predictive Control. In Nonlinear Model PredictiVe Control; Allgower, F., Zheng, A., Eds.; Progress in Systems and Control Theory Series, Vol. 26; Birkhauser-Verlag: Birkhauser Verlag: Basel, Switzerland, 2000; pp 163-179. (7) Seron, M.; De Dona´, J.; Goodwin, G. Global Analytical Model Predictive Control with Input Constraints. In Proceedings of the 39th Conference on Decision and Control, Sydney, Australia, 2000; Vol. 1, p 154. (8) Marjanovic, O.; Lennox, B.; Goulding, P.; Sandoz, D. Minimising Conservatism in Infinite Horizon LQR Control. Syst. Control Lett. 2002, 46 (4), 271. (9) Campo, P. J.; Morari, M. Robust Control of Processes Subject to Saturation Nonlinearities. Comput. Chem. Eng. 1990, 14, 343. (10) Grantz, J.; Valluri, S.; Soroush, M. Discrete-Time Nonlinear Control of Processes with Actuator Saturation Nonlinearities. AIChE J. 1998, 44 (7), 1701.

ReceiVed for reView February 1, 2006 ReVised manuscript receiVed June 12, 2006 Accepted July 13, 2006 IE0601347

Control Quality Loss in Analytical Control of Input-Constrained

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