A Tuning Strategy for Unconstrained SISO Model Predictive Control
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Ind. Eng. Chem. Res. 1997, 36, 729-746 729
A Tuning Strategy for Unconstrained SISO Model Predictive
Control
Rahul Shridhar and Douglas J. Cooper*
Chemical Engineering Department, University of Connecticut, U-222, Storrs, Connecticut 06269-3222
This paper presents an easy-to-use and reliable tuning strategy for unconstrained SISO dynamic
matrix control (DMC) and lays a foundation for extension to multivariable systems. The tuning
strategy achieves set point tracking with minimal overshoot and modest manipulated input
move sizes and is applicable to a broad class of open loop stable processes. The derivation of an
analytical expression for the move suppression coefficient, λ, and its demonstration in a DMC
tuning strategy is one of the significant contributions of this work. The compact form for the
analytical expression for λ is achieved by employing a first order plus dead time (FOPDT) model
approximation of the process dynamics. With tuning parameters computed, DMC is then
implemented in the classical fashion using a dynamic matrix formulated from step response
coefficients of the actual process. Just as the FOPDT model approximation has proved a valuable
tool in tuning rules such as Cohen-Coon, ITAE, and IAE for PID implementations, the tuning
strategy presented here is significant because it offers an analogous approach for DMC.
Introduction The derivation of an analytical expression that com-
putes λ and its demonstration in a DMC tuning strategy
This paper details and demonstrates a tuning strategy is one of the significant contributions of this work.
for single-input single-output (SISO), unconstrained Table 1 illustrates the step-by-step procedure to com-
dynamic matrix control (DMC) (Cutler and Ramaker, pute the DMC tuning parameters, including the ana-
1980, 1983) that is applicable to a wide range of open- lytical expression that computes λ, based on a user-
loop stable processes. The DMC control law is given specified control horizon and a given sample time.
by
Derivation of the analytical expression for λ considers
a first order plus dead time (FOPDT) approximation of
j ) (ATA + λI)-1ATej
∆u (1) the process dynamics. It must be emphasized that the
where A is the dynamic matrix, ej is the vector of FOPDT approximation is employed only in the deriva-
predicted errors over the next P sampling instants tion of the analytical expression for λ. The examples
(prediction horizon), λ is the move suppression coef- presented later in this work all use the traditional DMC
ficient, and ∆u j is the manipulated input profile com- step response matrix of the actual process upon imple-
puted for the next M sampling instants, also called the mentation.
control horizon. The ATA matrix, to be inverted in the The primary benefit of a FOPDT model approximation
evaluation of the DMC control law, is referred to in this is that it permits derivation of a compact analytical
work as the system matrix. expression for computing λ. Although a FOPDT model
Implementation of DMC with a control horizon greater approximation does not capture all the features of some
than one manipulated input move necessitates the higher-order processes, it often reasonably describes the
inclusion of a move suppression coefficient, λ (Maurath process gain, overall time constant, and effective dead
et al., 1988a). This coefficient serves a dual purpose of time of such processes (Cohen and Coon, 1953). In the
conditioning the system matrix before inversion and past, tuning strategies based on a FOPDT model such
suppressing otherwise aggressive control action. It is as Cohen-Coon, IAE, and ITAE have proved useful for
often used as the primary adjustable parameter to fine- PID implementations. The tuning strategy presented
tune DMC to desirable performance. Though the dual here is significant because it offers an analogous ap-
effect of λ has been discussed by others (e.g., Ogunnaike proach for DMC.
(1986)), past researchers have focused most attention The theoretical details in this paper are organized as
on the latter effect in the selection of λ. follows: (i) the DMC transfer function form is derived
This paper designs an analytical expression that and from it the existence of a gain-scaled move sup-
computes an appropriate λ by recognizing and exploiting pression coefficient is established; (ii) an approximation
the strong correlation between the condition number of of the scaled system matrix, ÃTÃ, is obtained from the
the system matrix, ATA, and resultant manipulated DMC control law; (iii) formulas for the maximum and
input move sizes. The move suppression coefficient, λ, minimum eigenvalues of the gain-scaled (ATA + λI)
which effectively modifies the condition number of the matrix are developed and an expression for its condition
system matrix, is computed such that the condition number is obtained; (iv) based on this expression for the
number is always bounded by a fixed low value. By condition number, an analytical expression for comput-
holding the condition number at a low value, desirable ing the move suppression coefficient is derived and an
DMC closed-loop performance is achieved while the overall DMC tuning strategy is formulated; (v) a straight-
manipulated input move sizes are prevented from forward extension of this tuning strategy to multivari-
becoming excessive. able systems is proposed; and (vi) issues important to
the implementation of the tuning strategy are discussed
* Author to whom correspondence should be addressed. with guidelines for selection of the sample time and the
Phone: (860) 486-4092. Fax: (860) 486-2959. Email: cooper@ control horizon.
eng2.uconn.edu. Through demonstrations of several simulation ex-
S0888-5885(96)00428-9 CCC: $14.00 © 1997 American Chemical Society730 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
Table 1. DMC Tuning Strategy
1. Approximate the process dynamics with a first order plus dead time (FOPDT) model:
-θ s
y(s) Kpe p
)
u(s) τps + 1
2. It is desirable but not necessary to select a value for the sampling interval, T. If possible, select T as the largest value that satisfies
T e 0.1τp and T e 0.5θp
3. Calculate the discrete dead time (rounded to the next integer):
k ) θp/T + 1
4. Calculate the prediction horizon and the model horizon as the process settling time in samples (rounded to the next integer):
P ) N ) 5τp/T + k
5. Select the control horizon, M (integer, usually from 1 to 6) and calculate the move suppression coefficient:
f)
{0
(
M 3.5τp
500 T
+2-
(M - 1)
2 )
M)1
M>1
λ ) fK2p
6. Implement DMC using the traditional step response matrix of the actual process and the following parameters computed
in steps 1-5:
sample time, T
model horizon (process settling time in samples), N
prediction horizon (optimization horizon), P
control horizon (number of moves), M
move suppression coefficient, λ
amples that encompass a range of process characteris-
tics, the tuning strategy is validated for set point
tracking performance using different choices of the
control horizon and sampling rates. The tuning strategy
is further validated for disturbance rejection. An ex-
ample application that validates an extension to the
tuning strategy applied to multivariable systems is also
presented.
Background
Model predictive control (MPC) has established itself
over the past decade as an industrially important form
of advanced control. Since the seminal publication of
Model Predictive Heuristic Control (later Model Algo-
rithmic Control) (Richalet et al., 1976, 1978) and
Dynamic Matrix Control (Cutler and Ramaker, 1980,
1983), MPC has gained widespread acceptance in aca-
demia and in industry. Several excellent technical
reviews of MPC recount the significant contributions in
the past decade and detail the role of MPC from an
academic perspective (Garcı́a et al., 1989; Morari and
Figure 1. Moving horizon concept of model predictive control.
Lee, 1991; Ricker, 1991; Muske and Rawlings, 1993) and
from an industrial perspective (Richalet, 1993; Clarke, Dynamic matrix control is arguably the most popular
1994; Froisy, 1994; Camacho and Bordons, 1995; Qin MPC algorithm currently used in the chemical process
and Badgwell, 1996). industry. Qin and Badgwell (1996) reported about 600
MPC refers to a family of controllers that employs a successful applications of DMC. It is not surprising why
distinctly identifiable model of the process to predict its DMC, one of the earliest formulations of MPC, repre-
future behavior over an extended prediction horizon. A sents the industry’s standard today. A large part of
performance objective to be minimized is defined over DMC’s appeal is drawn from an intuitive use of a finite
the prediction horizon, usually as a sum of quadratic step response (or convolution) model of the process, a
set point tracking error and control effort terms. This quadratic performance objective over a finite prediction
cost function is minimized by evaluating a profile of horizon, and optimal manipulated input moves com-
manipulated input moves to be implemented at succes- puted as the solution to a least squares problem.
sive sampling instants over the control horizon. Closed- Because of its popularity, this work focuses on an overall
loop optimal feedback is achieved by implementing only tuning strategy for DMC.
the first manipulated input move and repeating the Another form of MPC that has rapidly gained ac-
complete sequence of steps at the subsequent sample ceptance in the control community is Generalized
time. This “moving horizon” concept of MPC, where the Predictive Control (GPC) (Clarke et al., 1987a,b). It
controller looks a finite time into the future, is il- differs from DMC in that it employs a controlled
lustrated in Figure 1. autoregressive and integrated moving average (CA-Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 731
RIMA) model of the process which allows a rigorous phisticated analysis tools and an advanced knowledge
mathematical treatment of the predictive control para- of control concepts for their implementation. Hence,
digm. The GPC performance objective is very similar there still exists a need for easy-to-use tuning strategies
to that of DMC but is minimized via recursion on the for DMC.
diophantine identity (Clarke et al., 1987a,b; Lalonde and Tuning of unconstrained SISO DMC is challenging
Cooper, 1989). Nevertheless, GPC reduces to the DMC because of the number of adjustable parameters that
algorithm when the weighting polynomial that modifies affect closed-loop performance. These include the
the predicted output trajectory is assumed to be unity following: a finite prediction horizon, P; a control
(McIntosh et al., 1991). Therefore, without any loss of horizon, M; a move suppression coefficient, λ; a model
generality, the tuning strategy proposed in this paper horizon, N; and a sample time, T.
is directly applicable to GPC. The first problem that needs to be addressed is the
Unconstrained SISO DMC. DMC does not always selection of an appropriate set of tuning parameters
compete with, but sometimes complements, classical from among those available for DMC. Practical limita-
three-term PID (proportional, integral, derivative) con- tions often restrict the availability of sample time, T,
trollers. That is, it is often implemented in advanced as a tuning parameter (Franklin and Powell, 1980;
industrial control applications embedded in a hierarchy A° ström and Wittenmark, 1984). The model horizon is
of control functions above a set of traditional PID loops also not an appropriate tuning parameter since trunca-
(Prett and Garcı́a, 1988; Qin and Badgwell, 1996). tion of the model horizon, N, misrepresents the effect
The unconstrained SISO DMC formulation considered of past moves in the predicted output and leads to
in this work (eq 1) does not unleash the full power of unpredictable closed-loop performance (Lundström et
MPC. This restricted form of DMC does not allow al., 1995).
multivariable control while satisfying multiple process Sensitivity Study and Final Selection of DMC
and performance objectives. However, the analysis Tuning Parameters. Based on the above discussion,
presented here provides a foundation upon which more candidate parameters for developing a systematic tun-
advanced tuning strategies may be developed, and this ing strategy for DMC include the prediction horizon, P,
is illustrated later in the work with an extension to a the control horizon, M, and the move suppression
tuning strategy for multivariable DMC. coefficient, λ. Though this simplifies the task of sensi-
In any event, unconstrained SISO DMC does offer tivity analysis, the appropriate choice of these param-
some useful capabilities. For example, past comparison eters is strongly dependent on the sample time and the
studies between unconstrained DMC and traditional PI nature of the process.
control (e.g., Farrell and Polli, 1990) show that DMC Over the past decade, detailed studies of DMC pa-
provides superior performance when disturbance tuning rameters have provided a wealth of information about
differs significantly from servo tuning. DMC has also their qualitative effects on closed-loop performance
demonstrated superior performance in the case of plant- (Marchetti et al., 1983; Garcı́a and Morshedi, 1986;
model mismatch, except for process gain mismatch. Downs et al., 1988; Maurath et al., 1988a). In this
Additionally, incorporation of process knowledge in section, a brief sensitivity study investigates the extent
the controller architecture provides DMC with anticipa- to which various parameters affect DMC performance.
tory capabilities and facilitates control of processes with This study is targeted toward selection of appropriate
nonminimum phase behavior and large dead times. The tuning parameters for developing a DMC tuning strat-
form of the performance objective provides a convenient egy.
way to balance set point tracking with control effort, A base case process is employed to illustrate the effect
leading to an intuitive tradeoff between performance of adjustable parameters on DMC response for a step
and robustness. Also, the DMC form considered in this change in set point (Figures 2-4). The base case
work allows the introduction of feed-forward control in process has the transfer function
a natural way to compensate for measurable distur-
bances.
process 1
Challenges in SISO DMC Tuning. Tuning of
unconstrained and constrained DMC for SISO and e-50s
multivariable systems has been addressed by an array Gp(s) ) (2)
of researchers. In the past, systematic trial-and-error (150s + 1)(25s + 1)
tuning procedures have been proposed (e.g., Cutler,
1983; Ricker, 1991). Marchetti et al. (1983) presented Figure 2 illustrates the importance of λ in tuning
a detailed sensitivity analysis of adjustable parameters DMC with a control horizon greater than one manipu-
and their effects on DMC performance. The method of lated input move. The sample time is selected for this
principal component selection was presented by Maurath base case such that the T/τp ratio is 0.1. This sample
et al. (1985, 1988b) as a method to compute an ap- time to overall time constant ratio lies within the
propriate prediction horizon and a move suppression recommended range for digital controllers (Seborg, 1986;
coefficient (Callaghan and Lee, 1988). To simplify DMC Ogunnaike, 1994). A FOPDT model approximation of
tuning, Maurath et al. (1988a) also proposed the “M ) the process described by eq 2 yields a process gain, Kp
1” controller configuration of DMC. ) 1, an overall time constant, τp ) 157, and effective
Other tuning strategies for DMC have concentrated dead time, θp ) 70. The prediction horizon and model
on specific aspects such as tuning for stability (Garcı́a horizon are fixed at 54 and represent the complete
and Morari, 1982; McIntosh, 1988; Clarke and Scat- settling time of process 1 in samples.
tolini, 1991; Rawlings and Muske, 1993), robustness Graph a of Figure 2 illustrates the response to a step
(Ohshima et al., 1991; Lee and Yu, 1994), and perfor- change in set point when the control horizon is 1 (M )
mance (McIntosh et al., 1992; Hinde and Cooper, 1993, 1) and no move suppression is used (λ ) 0). Note that
1995). Although some of the above methods provide a for M ) 1, the set point step response is sluggish and a
complete design of DMC, they also require fairly so- λ > 0 will only further slow the process response.732 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
Figure 3. Effect of prediction horizon, P, and sample time, T, on
DMC performance for process 1 (M ) 4, λ ) 0.14).
prediction horizon (P ) 16 with T/τp ) 0.1) or a large
sample time (T/τp ) 0.15 with P ) 9) does not improve
closed-loop performance significantly. However, it has
Figure 2. Importance of the move suppression coefficient, λ, in been shown by past researchers (Garcı́a and Morari,
tuning DMC with the control horizon, M, greater than one 1982; Clarke and Scattolini, 1991; Rawlings and Muske,
manipulated input move (T ) 16, T/τp ) 0.1, P ) 54). 1993; Scokaert and Clarke, 1994) that a larger predic-
tion horizon improves nominal stability of the closed
Graphs b-d show the impact of λ on performance for loop. For this reason, the prediction horizon should be
a control horizon of M ) 4. Graph b demonstrates that, selected such that it includes the steady state effect of
with M > 1, the lack of move suppression results in all past computed manipulated input moves. Therefore,
dramatically aggressive control effort and a significantly the prediction horizon cannot be used as the primary
underdamped measured output response. Graph c DMC tuning parameter.
shows that an intermediate response can be achieved
Figure 4 illustrates the effect of control horizon, M,
by an appropriate choice of λ. Graph d shows that a
larger move suppression coefficient (λ ) 4.0) results in for fixed P ) 54, T/τp ) 0.15, and λ ) 0.14. Graphs a-c
a slower response. Further increasing λ can lead to an show that increasing the control horizon from 2 to 6 does
undesirable sluggish response for most applications. not alter the performance significantly. Actually, the
Consequently, this study shows that the choice of λ is only noticeable effect is a slight increase in overshoot
critical to the performance achieved by DMC. for a larger control horizon. This is due to the additional
Figure 3 demonstrates the interdependence of predic- degree of freedom from a larger control horizon. This
tion horizon, P, and sample time, T. A matrix of closed- allows more aggressive initial moves that are later
loop response results is generated for different choices compensated for by the extra moves available.
of P and T while maintaining the control horizon, M, Rawlings and Muske (1993) have shown that a
and move suppression coefficient, λ, constant. When the necessary condition to ensure nominal stability of
prediction horizon is altered, the model horizon, N, is infinite horizon MPC is that the control horizon should
always kept long enough to ensure that the DMC step be greater than or equal to the number of unstable
response model correctly predicts the steady state. By modes of the system. Nevertheless, it is clear from this
eliminating the truncation errors that result when the study that control horizon, due to its negligible effect
model horizon does not account for the process steady on closed-loop performance, is not well suited as the
state, Figure 3 isolates the effect of prediction horizon, primary DMC tuning parameter.
P, and sample time, T, on DMC performance. A conclusion from the above discussion is that the
The trend in the Figure 3 graphs h f e f b shows choice of prediction horizon, P, cannot be made inde-
that by reducing prediction horizon while maintaining pendent of the sample time, T. For stability reasons, P
a constant sample time (T/τp ) 0.1), the output response must be selected such that it includes the steady state
becomes increasingly underdamped. Also, the trend in effect of all past manipulated input moves; i.e., it should
graphs f f e f d illustrates that by decreasing the equal the open-loop settling time of the process in
sample time for a constant prediction horizon (P ) 9), samples. Stability considerations also restrict the choice
the output response becomes increasingly underdamped. of control horizon, M. Besides, the control horizon does
Thus, Figure 3 shows that the choice of P and T are not have a significant impact on closed-loop performance
interrelated. in the presence of move suppression. Therefore, this
Graphs f, h, and i in Figure 3 show that a large brief study supports the opinions of other researchersInd. Eng. Chem. Res., Vol. 36, No. 3, 1997 733
response coefficient of the process, and N is the process
settling time in samples.
The current and future manipulated input moves are
yet undetermined and are not used in the computation
of the predicted output profile. Therefore, eq 3 reduces
to
N-1
ŷ(n + j) ) y0 + ∑ (ai∆u(n + j - i)) + d(n + j)
i)j+1
(4)
where the term d(n + j) lumps together possible
unmeasured disturbances and inaccuracies due to plant-
model mismatch. From eq 4, the predicted output at
the current instant (j ) 0) is
N-1
ŷ(n) ) y0 + ∑
i)1
(ai∆u(n - i)) + d(n) (5)
Since the future values of d(n + j) are not available,
an estimate is used. In the absence of any additional
knowledge of d(n + j) over future sampling instants, the
predicted disturbance is assumed to be equal to that
estimated at the current time instant. Therefore
N-1
d(n + j) ) d(n) ) y(n) - y0 - ∑
i)1
(ai∆u(n - i)) (6)
where y(n) is the measurement of the current output.
A more accurate estimate of the d(n + j) is possible,
provided the load disturbance is measured and a reliable
Figure 4. Effect of the control horizon, M, on DMC performance
for process 1 (T ) 16, T/τp ) 0.1, P ) 54, λ ) 0.14).
load disturbance to measured output model is available.
Using the unit step response coefficients from this
model, an equation similar to eq 5 above can be used to
that the one candidate best suited as the final DMC
predict the future disturbance (Asbjornsen, 1984; Muske
tuning parameter is the move suppression coefficient,
and Rawlings, 1993). A DMC configuration that uses
λ.
a process model augmented with a disturbance to output
model is known as feed-forward DMC.
Analytical Framework Now, if the predicted output is to follow the set point
In this section, the form of the DMC transfer function in the wake of a set point change or unmeasured
is derived from the DMC control law. The difficulty in disturbance, then the current and future manipulated
using this form for development of an overall DMC input moves in eq 3 have to be determined such that
tuning strategy is highlighted. It is then shown how
some useful information relevant to the selection of ysp(n + j) - ŷ(n + j) ) 0 j ) 1, 2, ..., P (7)
move suppression coefficient, λ, can be extracted from
the form of the DMC transfer function. Substituting eq 3 in eq 7 gives
DMC Control Law. The cornerstone of the DMC
N-1
∑ a ∆u(n + j - i) - d(n) )
algorithm is a discrete convolution or step response
ysp(n + j) - y0 -
model of the process that predicts the output (ŷ(n + j)) i)j+1
i
j sampling instants ahead of the current time instant,
}}
n:
predicted error based on past moves, e(n + j)
j
j ∑a ∆u(n + j - i)
i j ) 1, 2, ..., P (8)
ŷ(n + j) ) y0 + ∑
i)1
ai∆u(n + j - i) + i)1
}
effect of current and future
effect of current and
moves to be determined
future moves
N-1
∑ ai∆u(n + j - i)
i)j+1
(3) The terms on the left in eq 8 represent the error between
the predicted output and the set point that will exist
over the next P sampling instants provided no further
}
effect of past moves manipulated input moves are made by the controller.
The term on the right represents the effect of the yet
undetermined current and future manipulated input
In eq 3, y0 is the initial condition of the measured moves.
output, ∆ui ) ui - ui-1 is the change in the manipulated Equation 8 is a system of linear equations which can
input at the ith sampling instant, ai is the ith unit step be represented as a matrix equation of the form[]
734 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
e(n + 1) the first element of ∆u
j in eq 14 to be implemented at
e(n + 2) the current instant, can be written as
e(n + 3) P
·
· ) ∆u(n) ) ∑
i)1
cie(n + i) (15)
e(n + M)
·
[ ]
·
· The expression for the predicted error from eq 8 can be
modified to eliminate y0 using eq 5:
e(n + P)
·
P×1
N-1
a1
a2
0
a1
0
0
· · · 0
0
e(n + j) ) ysp(n + j) - y0 - ∑ ai∆u(n + j - i) -
i)j+1
a3 a2 a1 ·
· 0 d(n)
× N-1
·
0
· · ·
∑
· · ·
a·M aM-1
· aM-2
· a1 ) ysp(n + j) - ŷ(n) - (ai+j - aj)∆u(n - j)
j)1
· · · · (16)
[ ]
· · · ·
a·
P a·P-1 a·
P-2 ... aP-M+1
·
P×M Substituting eq 16 in eq 15 gives
∆u(n) P
∆u(n + 1) ∆u(n) ) ∑ ci{ysp(n + i) - ŷ(n) -
∆u(n + 2) (9) i)1
N-1
∆u(n + M - 1) M×1 ∑
j)1
(ai+j - aj) ∆u(n - j)} (17)
or in a compact matrix notation as Let ysp(n) be a weighted average of the desired set
point profile over P future sampling instants:
ej ) A∆u
j (10)
P
where ej is the vector of predicted errors over the next
P sampling instants, A is the dynamic matrix, and ∆u j ∑
i)1
ciysp(n + i)
is the vector of manipulated input moves to be deter- ysp(n) ) (18)
mined. P
An exact solution to eq 10 is not possible since the ∑ ci
number of equations exceeds the degrees of freedom (P i)1
> M). Hence, the control objective is alternatively posed
as a least-squares optimization problem with a qua- Using eq 18 in eq 17, the latter becomes
dratic performance objective (cost function) of the form P
j ]T[ej - A∆u
Min J ) [ej - A∆u j] (11)
∆u(n) ) ∑
i)1
ci{ysp(n) - ŷ(n)} -
∆u
j
P N-1
In the unconstrained case, this minimization problem ∑
i)1
ci{ ∑ (ai+j - aj)∆u(n - j)}
j)1
(19)
has a closed-form solution which represents the DMC
control law
Replacing ysp(n) - y(n) ) e(n), the error at the current
T T sample time, eq 19 becomes
j ) (A A) A ej
∆u -1
(12)
P N-1 P
Implementation of DMC with the control law in eq
12 results in excessive control action, especially when
∆u(n) + ∑
i)1
ci{ ∑ (ai+j - aj)∆u(n - j)} ) ∑cie(n)
j)1 i)1
(20)
the control horizon is greater than 1. Hence, a qua-
dratic penalty on the size of manipulated input moves Transformation of eq 20 to the z-domain gives the DMC
is introduced into the DMC performance objective. The transfer function form:
( ( ))
modified performance objective has the form
u(z) 1
Min J ) [ej - A∆u T
j ] [ej - A∆u
j ] + [∆u
j ] λ[∆u
j ] (13) T D(z) ) )
∆u e(z) P
∑
j
ciai+j
N-1
where λ is the move suppression coefficient. For the 1
+ ∑
i)1
modified performance objective the closed form solution (1 - z )
-1
- aj z-j
P P
∑ ∑
takes the form (Marchetti et al., 1983; Ogunnaike, j)1
1986): ci ci
i)1 i)1
j ) (ATA + λI)-1ATej
∆u (14) (21)
DMC Transfer Function. The DMC transfer func- Although eq 21 provides insight into the form of the
tion is developed by building upon prior work by Gupta DMC transfer function, theoretical analysis of the
(1987, 1993). Let ci denote the ith first row element of closed-loop system is very complicated (even after the
the pseudoinverse matrix, (ATA + λI)-1AT. Then, ∆u(n), assumption of a FOPDT model approximation of theInd. Eng. Chem. Res., Vol. 36, No. 3, 1997 735
process). This is primarily because it is difficult to G(z) ) y(z)/u(z) ) KpG̃(z) (26)
represent the elements, ci, of the pseudoinverse matrix
analytically in terms of the step response coefficients.
Consequently, eq 21 does not serve as a convenient Using eq 25 and eq 26, the open-loop transfer function
starting point for devising the analytical expression for is of the form
computing λ. However, as shown in the next section,
some useful information regarding the form of the move D(z)G(z) ) y(z)/e(z) ) (D̃(z)/Kp)KpG̃(z) ) D̃(z)G̃(z)
suppression coefficient rule can be extracted from the (27)
DMC transfer function.
Gain Scaling of the Move Suppression Coef-
ficient. Previous work has proposed that the choice of and the closed-loop transfer function is of the form
the move suppression coefficient can be made indepen-
dent of the process gain (Lalonde and Cooper, 1989; y(z) D(z)G(z) D̃(z)G̃(z)
McIntosh et al., 1991). Gain-scaling is a term coined ) ) (28)
ysp(z) 1 + D(z)G(z) 1 + D̃(z)G̃(z)
to represent a transformation where a mathematical
expression is stripped of the effects of process gain for
analysis independent of gain. For example, gain-scaling By casting the move suppression, λ, as a scaled
of the move suppression coefficient can be done by
expressing it as a product of a scaled move suppression coefficient, f, times the square of the process gain, K2p,
the SISO DMC response looks similar for all first-order
coefficient, f, and the square of the process gain, K2p.
Note that this approach is restricted to DMC applied systems when the time dimension is scaled appropri-
to SISO systems and cannot be directly extended to ately. In other words, eq 28 shows that, by gain-scaling
multivariable systems. λ as in eq 22, the closed-loop performance is rendered
Consider a form of the move suppression coefficient independent of the process gain. As a result, derivation
given by of an analytical expression for λ yields a scaled coef-
ficient, f, that is a function of parameters other than
λ ) fK2p (22) the process gain.
The step response coefficient of any linear system can Derivation of an Analytical Expression for λ
be written as
The analysis of the gain-scaled system matrix, ÃTÃ,
ai ) Kpãi (23) involves the development of an approximate form of the
gain-scaled system matrix. Such an approximation is
where ãi represents the part of the unit step response made possible with the use of a FOPDT model ap-
coefficient that is independent of the process gain, Kp. proximation of the process. It must be emphasized that
Using eq 22 and eq 23, it is possible to separate the gain-
the use of this FOPDT approximation is employed only
related effects from the first row elements, ci, of the
in the derivation of the analytical expression for λ. The
pseudoinverse matrix:
examples presented later in this work all use the
traditional DMC step response matrix of the actual
ci ) ith first row element of {(ATA + λI)-1AT}
process upon implementation.
) ith first row element of {(K2pÃTÃ + fK2pI)-1KpÃT} An Approximate Form of the ÃTÃ Matrix. A
) 1/Kp × ith first row element of {(ÃTÃ + fI)-1ÃT} FOPDT model with zero-order hold is represented by a
discrete transfer function as
) c̃i/Kp (24)
Here, Ã is the gain-scaled dynamic matrix, ÃTÃ is the Kp(1 - e-(T/τp))z-k
H0Gp(z) ) (29)
gain-scaled system matrix, and c̃i is the ith first row (1 - e-(T/τp)z-1)
element of the gain-scaled pseudo-inverse matrix.
Substituting eq 23 and eq 24 in eq 21 shows that the
gain dependence of the SISO DMC transfer function is where Kp is the process gain, τp is the overall process
separable from the dependence on remaining process time constant, T is the discrete sample time, and k is
and controller parameters: the effective discrete dead time given by
( ( ))
D(z) ) u(z)/e(z) ) k ) θp/T + 1 (30)
1
) D̃(z)/Kp
P In eq 30, θp is the effective dead time in the process.
1 N-1
∑ c̃iãi+j From eq 29, the step response coefficients of a FOPDT
+ ∑
i)1 process are given by
Kp(1 - z )
-1
- ãj z-j
P P
∑ ∑
{
j)1
c̃i c̃i 0 0ejek-1
i)1 i)1 ãi ) (31)
(1 - e-(T/τp)(j-k+1)) kej
(25)
Similarly, the gain dependence of a linear process Using a FOPDT model approximation of the process
transfer function is separable from the remaining and the gain-scaled step response coefficients from eq
process parameters: 31, the dynamic matrix from eq 9 has the form736 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
e
e e
e e e
e e e e
e e e e
For this form of the dynamic matrix, the gain-scaled
system matrix, ÃTÃ, in the DMC control law has the
form
e e e e e
Figure 5. Comparison of the true and approximate condition
numbers of the ÃTÃ matrix for large values of the prediction
e e e e e horizon, P.
The validity of the approximation in eq 36 is explored
e e e e e
later in this section.
The other terms of the ÃTÃ matrix can be ap-
proximated using a similar procedure. The final ap-
[ ]
proximate form of the matrix that results is
An approximate form of the gain-scaled system matrix
can be obtained by approximating individual terms of ÃTÃ =
the matrix in eq 33 for large values of the prediction 3T 3T 3 3T
horizon, P, and small sample times, T (small T/τp). Let P-k- +2 P-k- + P-k-
2 τp 2 τp 2 2 τp
+1 · · ·
R̃ij (i,j ) 1, 2, ..., M) be the term in the ith row and jth
column of the gain-scaled system matrix. The ap- 3T 3 3T 3T 1
P-k- + P-k- +1 P-k-
2 τp 2 2 τp 2 τp 2
+ · · ·
proximation of one such term, R̃11, is shown in eq 34.
Recognizing that the summation terms in R̃11 are in 3T 3T 1 3T
P-k- +1 P-k- + P-k-
geometric progression results in the exact expression 2 τp 2 τp 2 2 τp · · ·
· · · ·
P-k+1 M×M
∑
· · · ·
(1 - e-i(T/τp))2 (37)
· · · ·
R̃11 )
[ ]
i)1
P-k+1 Let â ) R̃11 = P - k - (3/2)(T/τp) + 2, then
) ∑
i)1
(1 - 2e -i(T/τp)
+e -2i(T/τp)
)
1
â â- â-1
2
· · ·
-(T/τp) -(P-k+1)(T/τp)
2e (1 - e ) 1 3
) (P - k + 1) - + â- â-1 â-
(1 - e-(T/τp)) ÃTÃ ) 2 2
· · ·
(38)
3
e-(2T/τp)(1 - e-2(P-k+1)(T/τp)) â-1 â- â-2
2
· · ·
-(2T/τp)
(34)
(1 - e ) · · · ·
· · · · M×M
· · · ·
With a FOPDT model approximation available, the
Note that the approximate ÃTÃ matrix has a Hankel
prediction horizon, P, can be computed as the open-loop
matrix form with the added feature that the elements
process settling time in samples as P ) (5τp/T) + k. This
of every row successively decrease by 0.5 from left to
supports the findings of past researchers (Garcı́a and
right. The approximate matrix of eq 38 is singular when
Morshedi, 1986; Maurath et al., 1988a; Lundström et
M g 3. This supports the observation made by prior
al., 1995) that P should be large enough to include the
researchers (Marchetti et al., 1983) that the ATA matrix
steady state effect of all past input moves.
(hence, the ÃTÃ matrix) becomes increasingly singular
For large values of prediction horizon, the term in eq
for large values of the prediction horizon, P, and control
34 simplifies to
horizon, M.
Figure 5 provides evidence that the approximate gain-
2e-(T/τp) e-(2T/τp) scaled system matrix is a good approximation of the true
R̃11 = (P - k + 1) - -(T/τp)
+ (35)
(1 - e ) (1 - e-(2T/τp)) system matrix, especially for the analysis of the ÃTÃ +
fI matrix in the DMC control law. Specifically, Figure
Notice that the approximation in eq 35 becomes increas- 5 shows the condition number for the exact and ap-
ingly accurate as P increases and is exactly true for an proximate ÃTÃ + fI matrix as a function of the scaled
infinite horizon implementation of DMC. A first-order move suppression coefficient, f, for different choices of
Taylor series expansion, e-(T/τp) ) 1 - T/τp, that is valid the control horizon. The prediction horizon employed
for small sample times (T/τp e 0.1), can be applied to in Figure 5 is (5τp)/T + k, which is the open-loop settling
eq 35 to yield time of the process based on a FOPDT model ap-
3T
R̃11 = P - k - +2 (36)
2 τpInd. Eng. Chem. Res., Vol. 36, No. 3, 1997 737
proximation of the process. This value of prediction eq 45 by solving the system of equations that result from
horizon is recommended in the proposed tuning strategy the top left partitioned block using eq 38 and eq 42.
based on closed-loop stability considerations (Garcı́a and Thus, a and b are given by
Morari, 1982; Clarke and Scattolini, 1991; Rawlings and
Muske, 1993; Scokaert and Clarke, 1994). From Figure
5 it can be seen that, for large values of prediction
horizon, the condition number of the approximate
a)M â- { (M - 1)
2 }
system matrix in eq 38 closely follows the true condition Mx(M - 1)(M + 1)
number. b)- (46)
QR Factorization of the Approximate ÃTÃ + fI 2x12
Matrix. Since the approximate form of the system
matrix (eq 38) was shown above to be a reasonable Now, the ÃTÃ + fI matrix, to be inverted in DMC
[ ]
approximation of the true system matrix (eq 33), the control law, has the form
following treatment of the ÃTÃ + fI matrix is based on
this approximate form. ÃTÃ + fI )
Consider two linearly independent M-vectors: 1
â+f â- â-1
2
· · ·
T
h
h 1 ) (1 1 1 ‚‚‚ 1)1×M 1 3
â- â-1+f â-
2 2
· · ·
(47)
T 3
h
h 2 ) (0 1 2 ‚‚‚ M - 1)1×M (39) â-1 â- â-2+f
2
· · ·
The approximate ÃTÃ matrix (eq 38) can be written in ·
·
·
·
·
·
·
· M×M
terms of these vectors as · · · ·
Adopting the factored form in eq 45 and eq 46, eq 47 is
h 1T + h
ÃTÃ ) νj‚h h 1‚νjT (40) written as
where
[ ]
ÃTÃ + fI ) [ej 1 ej 2 ... 0]M×M ×
â 1
νj ) hh - h (41) a+f b
·
0
2 1 2 2
h ·
b f
·
Hence, h h 2 form a basis for the approximate ÃTÃ
h 1 and h [ej 1 ej 2 ... 0]TM×M (48)
· · · · · · · · ·
matrix.
0 fI
·
A Gram-Schmidt orthonormalization of h1 and h2 · M×M
·
yields the orthonormal basis for ÃTÃ:
Equation 48 can now be used to determine explicit
T 1 analytical expressions for the eigenvalues of ÃTÃ + fI.
ej 1 ) (1 1 1 ‚‚‚ 1)1×M From eq 47 it is clear that the approximate form of ÃTÃ
xM + fI has eigenvalues µ ) f with multiplicity (M - 2).
Therefore
x
12
ej 2T ) (0 1 2 ‚‚‚ M - 1)1×M -
M(M - 1)(M + 1) µmin ) f (49)
(M - 1)
(1 1 1 ‚‚‚ 1)1×M (42) The remaining two eigenvalues are found from the
2
top left partitioned block as the µ that satisfies
Therefore, h
h 1 and h
h 2 can be QR factored as
h1 h
[h h 2]M×2 ) [e1 ej 2]M×2R2×2 (43) |ab + f - µ b
f-µ
)0 | (50)
where R is upper triangular and invertible. Using eq A solution to eq 50 (using eq 46) yields the larger of the
43, eq 40 can be transformed to two eigenvalues as
h 1T + h
ÃTÃ ) νj‚h h 1‚νjT ) µmax )
[ej 1 ej 2]M×2 [
a
b
b
0 ]
2×2
T
[ej 1 ej 2]2×M (44) Mâ+2f -
M(M-1)
2
+
x M2â2-M2(M-1)â+
2
(M-1)M2(2M-1)
6
where a and b are simple linear functions of â. Equa- (51)
tion 44 can also be written as
Alternatively, the minimum and maximum eigenval-
T ues of ÃTÃ + fI can be determined by reasoning. Note
à à ) [ej 1 ej 2 ... 0]M×M ×
[ ]
that the ÃTÃ matrix (eq 38) is nearly singular for M )
2 and is perfectly singular for M > 2. Therefore, the
a b
·
· 0 minimum absolute eigenvalue of ÃTÃ for M g 2 is close
to or exactly zero. When a constant quantity, f, is added
b 0
·
[ej 1 ej 2 ... 0]TM×M (45) to the leading diagonal of such a matrix, all its eigen-
· · · · · · · · · values shift by that quantity (Hoerl and Kennard, 1970;
0 0 Ogunnaike, 1986). Hence, the minimum absolute eigen-
·
· M×M
· value of the resultant ÃTÃ + fI matrix, µmin, is equal to
Analytical expressions for a and b can be obtained from f (eq 49).738 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
Analytical expressions for the maximum eigenvalue (M - 1)(2M - 1)
can be derived for the square matrix, ÃTÃ + fI, with â2 - (M - 1)â +
6
=
( )
successively increasing dimensions (M × M). By rec- 2 2
(M - 1) (M - 1)
ognizing that the various coefficients in the analytical â2 - (M - 1)â + ) â- (55)
expressions follow a series pattern that is a function of 4 2
the matrix dimension (M × M), a general formula for
the maximum eigenvalue can be obtained as a function With this simplification, the expression for the gain
of M. This general analytical expression is identical to scaled coefficient becomes
( )
the analytical expression obtained in eq 51.
M (M - 1)
Formulation of the Analytical Expression for λ. f) â- (56)
Past researchers (e.g., Ogunnaike, 1986) have indicated c 2
that the move suppression coefficient, λ, serves a dual
purpose in the DMC control law. Its primary role in Past researchers (Maurath et al., 1988a,b; Callaghan
and Lee, 1988; Farrell and Polli, 1990) have indicated
DMC is to suppress otherwise aggressive controller
typical condition numbers for a moderately ill-condi-
action when M > 1. Additionally, λ serves to improve
tioned DMC system matrix. Condition numbers re-
the conditioning of the system matrix by rendering it
ported range from about 100 for single-input single-
more positive definite. output systems to about 2000 for multivariable systems.
One premise underlying this work is that both these Apart from the variety of systems considered, the
effects are interrelated. When λ is increased, the approximate relation, c ∝ M(P - k)/f, indicates that
manipulated input move sizes decrease, as does the some of the differences are due to the different predic-
condition number. Clearly, if the effect of λ on the tion and control horizons used by the individual design-
condition number of the system matrix can be analyti- ers. In this work, a condition number of 500 is selected
cally expressed, then this condition number can be to represent the upper allowable limit of ill-conditioning
maintained within specified bounds by an appropriate in the system matrix (corresponding to modest control
choice of λ. An upper bound on the condition number effort).
would, in turn, prevent the manipulated input move The choice of the condition number, and hence the
sizes from becoming excessive. upper allowable limit of ill-conditioning in the system
The condition number is defined for a square matrix matrix, lies with the individual designer. Depending
on the criterion for good closed-loop performance for a
as
given application, the condition number can be conve-
niently fixed. The choice of a condition number of 500
|µmax| was motivated by the rule of thumb that the manipu-
c) (52)
|µmin| lated input move sizes for a change in set point should
not exceed 2-3 times the final change in manipulated
input (Maurath et al., 1985; Callaghan and Lee, 1988).
where µmax and µmin represent the maximum and
However, if a faster or slower closed-loop response is
minimum eigenvalues of the matrix. From eqs 49, 51,
more desirable, a larger or smaller condition number
and 52, the condition number for the ÃTÃ + fI matrix respectively, can be used instead.
is Substituting the expression for â in eq 56, with P )
(
(5τp)/T + k, the analytical expression for the move
1 M(M - 1) suppression coefficient, λ, is given by
c) Mâ + 2f -
( )
2f 2
+
) M 7 τp (M - 1)
M
x 2
â - (M - 1)â +
(M - 1)(2M - 1)
6
(53)
f)
500 2 T
+2-
2
λ ) fK2p (57)
An interesting approximate relation from eq 53 is that
the condition number of the DMC gain-scaled system The analytical expression for λ in eq 57, applied in
matrix, c ∝ M(P - k)/f. conjunction with the recommended values for the other
Equation 53 is rearranged to give an expression for tuning parameters, gives the tuning strategy for DMC
the scaled move suppression coefficient as with M > 1.
Note that eq 57, which computes the move suppres-
f)
M
(
2c
â-
(M - 1)
2
+
sion coefficient, does not contain a dead time term. The
reason is not intuitively obvious but is an interesting
)
one. The elements, and the condition number, of the
x (M - 1)(2M - 1) (ATA + λI) matrix depend on the number of non-zero
â2 - (M - 1)â + (54) terms in the dynamic matrix, A. This is clearer from
6
eq 33, where the first term of the (ATA + λI) matrix is
a series summation performed over the P - k + 1 non-
where â ) P - k - (3/2)(T/τp) + 2. Equation 54
zero terms in the dynamic matrix A. (Equation 37 also
expresses the gain-scaled coefficient as a function of the conveys the dependence of condition number of (ATA +
condition number of ÃTÃ + fI, the control horizon, and λI) on P - k, rather than P alone). Additionally, the
FOPDT model parameters other than the process gain. choice of prediction horizon as P ) (5τp/T) + k causes
Evaluation of f in eq 54 can be simplified by recogniz- the elements of (ATA + λI) to depend on P - k ) (5τp/
ing the contribution of each term to the value of f. The T). Hence, the condition number of (ATA + λI) and the
last term within the square root in eq 54 can be modified move suppression from eq 57, are dependent on (5τp/T)
to ease evaluation: and are independent of dead time. Of course, this isInd. Eng. Chem. Res., Vol. 36, No. 3, 1997 739
[ ]
true only if the prediction horizon is chosen as P ) (5τp/ controlled variable weights, ΓTΓ, has the form
T) + k (settling time of the process).
γ21IP×P 0 0
· ·
Extension of Tuning Strategy to Multivariable · ·
Systems · · ·
·
· · · ·
·
· · · ·
T
The selection of tuning parameters for multiple-input γ22IP×P
· ·
Γ Γ) 0 · · 0 (61)
multiple-output (MIMO) DMC is significantly more · ·
challenging for the practitioner. The analytical expres- · · · · · · · · · · ·
sion that computes the move suppression coefficient (eq
· ·
0 · 0 ·
·
· P‚R×P‚R
57), developed in the previous section for SISO DMC, · · ·
provides the foundation upon which a similar analytical
expression can be developed for MIMO DMC. This is Hence, the controlled variable weights are γ2i (i ) 1, 2,
possible in a straightforward fashion even though the ..., R).
performance objective for MIMO DMC is defined over Similar to SISO DMC, adjustable parameters that can
several manipulated inputs and measured outputs and be used to alter closed-loop performance for MIMO DMC
results in a more complex MIMO DMC control law. include the prediction horizon, P, the control horizon,
For a system with S manipulated inputs and R M, the model horizon, N, the sample time, T, and the
measured outputs, the MIMO DMC performance objec- move suppression coefficients, λ2i . In addition, MIMO
tive (cost function) has the form (Garcı́a and Morshedi, DMC has yet another set of adjustable parameters in
1986) the form of the controlled variable weights, γ2i .
Just as with the SISO case, practical limitations often
j ]TΓTΓ[ej - A∆u
Min J ) [ej - A∆u j ]TΛTΛ[∆u
j ] + [∆u j] restrict the availability of sample time, T, as a tuning
∆u
j parameter for MIMO DMC (Franklin and Powell, 1980;
(58)
A° ström and Wittenmark, 1984). The model horizon, N,
is also not an appropriate tuning parameter since
In eq 58, ej is the vector of predicted errors for the R truncation of the model horizon can lead to unpredict-
measured outputs over the next P sampling instants, able closed-loop performance (Lundström et al., 1995).
A is the multivariable dynamic matrix, and ∆u j is the As demonstrated earlier for SISO DMC, the control
vector of M moves to be determined for each of the S horizon, M, does not have a significant effect on closed-
manipulated inputs. ΛTΛ is a square diagonal matrix loop performance, especially in the presence of move
of move suppression coefficients with dimensions (M‚S suppression.
× M‚S). Similarly, ΓTΓ is a square diagonal matrix that
The controlled variable weights, γ2i , serve a dual
contains the controlled variable weights (equal concern
purpose in MIMO DMC. These weights can be ap-
factors) with dimensions (P‚R × P‚R).
propriately chosen by the user to scale measurements
A closed-form solution to the MIMO DMC perfor-
of the R measured outputs to comparable units. Also,
mance objective (eq 58) results in the MIMO DMC
it is possible to achieve tighter control of a particular
control law (Garcı́a and Morshedi, 1986):
measured output by selectively increasing the relative
weight of the corresponding least-square residual. Hence,
j ) (ATΓTΓA + ΛTΛ)-1ATΓTΓej
∆u (59) controlled variable weights are usually specified by the
The MIMO DMC control law (eq 59) is similar to the user for a certain application and cannot be employed
SISO DMC control law (eq 14), except for the form of as the primary tuning parameters for MIMO DMC.
the move suppression coefficients, ΛTΛ, and the intro- Analogous to SISO DMC, the move suppression
duction of controlled variable weights, ΓTΓ. coefficients can be conveniently used to fine tune MIMO
The move suppression coefficients in multivariable DMC for desired closed-loop performance. Since the
DMC follow a notation slightly different from SISO dual benefit of the move suppression coefficients, λ2i , is
DMC. Λ is a square diagonal matrix with dimensions again to improve the conditioning of the MIMO DMC
(M‚S × M‚S). This matrix can be divided into S2 square system matrix (ATΓTΓA) and move size suppression for
blocks, each with dimensions (M × M). The leading the S inputs, a strategy analogous to SISO DMC can
diagonal elements of the first (M × M) matrix block be used to extend eq 57 to compute the move suppres-
along the diagonal of Λ are λ1, the leading diagonal sion coefficients for MIMO DMC.
elements of the next (M × M) matrix block along the Building upon the analogy, an approximation of the
diagonal of Λ are λ2, and so on. All off-diagonal MIMO DMC system matrix, (ATΓTΓA), has the form of
elements of the matrix Λ are zero. Hence, the matrix a mosaic Hankel matrix (not shown here) comprised of
[ ]
of move suppression coefficients, ΛTΛ, has the form S2 Hankel matrix blocks. The S2 Hankel matrix blocks,
each with dimensions (M × M), have a form identical
to that obtained earlier (eq 47) from a similar ap-
λ21IM×M 0 0 proximation of the scaled SISO DMC system matrix,
· ·
· ·
· · ÃTÃ.
The impact of a change in the ith manipulated input
· · · · · · · · · · ·
T
λ22IM×M
· ·
Λ Λ) 0 · · 0 (60) on all measured outputs is reflected in the ith diagonal
· · Hankel matrix block. Hence, it is possible to select the
ith move suppression coefficient, λ2i , such that the
· · · · · · · · · · ·
· ·
0 · 0 ·
·
· M‚S×M‚S condition number of the ith diagonal matrix block is
always bounded by a fixed low value. By holding the
· · ·
Thus, in the MIMO DMC control law (eq 59), the move condition number of the ith diagonal matrix block at a
suppression coefficients that are added to the leading low value, desirable closed-loop performance is achieved
diagonal of the multivariable system matrix, (ATΓTΓA), while preventing the ith manipulated input move size
are λ2i (i ) 1, 2, ..., S). Similarly, the matrix of from becoming excessive.740 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
With this understanding, an analytical expression An additional criterion for sample time selection is
that computes the move suppression coefficients for that the sampling rate should be high enough to sample
MIMO DMC can be obtained as two or three times per effective dead time of the process,
T e 0.5θp (Seborg et al., 1986). This criterion is not a
[ ( )]
stringent requirement for DMC. However, if sample
M R
3 τpij (M - 1)
λ2i ) ∑
500 j)1
γ2j Kp2ij P - kij -
2 T
+2-
2
time can be chosen, the prudent approach is to pick the
largest T that satisfies both criteria mentioned above.
(i ) 1, 2, ..., S) (62) Once T is fixed, the discrete dead time, k, is calculated
from the effective dead time of the process, θp in step 3.
Using eq 62, the S move suppression coefficients, one Step 4 computes a model horizon, N, and a prediction
for each input, can be computed for a given sampling horizon, P, from τp, θp, and T. Note that both N and P
time, T, control horizon, M, and controlled variable cannot be selected independent of the sample time, T.
weights, γi. For DMC, it is imperative that N be equal to the open-
In eq 63, it is not possible to substitute P ) (5τp)/T + loop process settling time in samples to avoid truncation
k as was done for the SISO case since the MIMO DMC error in the model prediction (Lundström et al., 1995).
architecture requires a single value of the prediction For computing P, a rigorous treatment by past research-
horizon to be selected for all manipulated input and ers (Garcı́a and Morari, 1982; Clarke and Scattolini,
measured output pairs. Nevertheless, the analytical 1991; Rawlings and Muske, 1993; Scokaert and Clarke,
expression for MIMO DMC is similar in form to that 1994) has shown that a larger prediction horizon
obtained for SISO DMC (eq 57). In fact, for a single- improves nominal stability of the closed loop. For
input single-output process, the analytical expression practical applications this translates to using a reason-
for MIMO DMC (eq 62) reduces to the analytical ably large but finite P. Bearing in mind this important
expression for SISO DMC (eq 57) when P ) (5τp)/T + k. requirement for stability, P is also set equal to the open-
loop process settling time in samples. With a FOPDT
model approximation available, prediction horizon and
Implementation of the DMC Tuning Strategy model horizon can both be computed as P ) N ) (5τp/
T) + k.
The proposed DMC tuning strategy, which includes
the analytical expression for the move suppression In an industrial setting, there can be barriers to
coefficient, λ, is presented in Table 1. This tuning selecting different P and T values for different SISO
strategy can be applied to unconstrained DMC in closed DMC controllers. Also, the MIMO DMC architecture
loop with a broad class of SISO processes that are open- requires that a single P and a single T be selected for
loop stable, including those with challenging control all manipulated input and measured output pairs. If
characteristics such as high process order, large dead the user has the freedom to select a sample time in such
time, and noniminimum phase behavior. a scenario, a conservative choice would be the smallest
value of T that satisfies eq 63 for all individual ma-
Step 1 (Table 1) involves the identification of a first
nipulated input and measured output pairs. Based on
order plus dead time (FOPDT) model approximation of
this sample time (or the fixed sample time), a single
the process. An accurate identification of the FOPDT
prediction horizon can be computed as P ) max {(5τpij)/T
model parameters is essential to the success of this
+ kij (i ) 1, 2, ..., S; j ) 1, 2, ..., R)} for a system with S
tuning strategy. Since a model is only as good as the
data it fits, it is necessary that the input-output data manipulated inputs and R measured outputs.
used for model fitting be rich in dynamic information Step 5 requires the specification of a control horizon,
of the process. Typically, this is achieved by perturbing M, and the computation of an appropriate move sup-
the process to obtain a signal-to-noise ratio greater than pression coefficient, λ. Recommended values of M range
10:1 (Seborg et al., 1986). Additionally, the data must from 1 through 6. However, selecting M > 1 provides
be collected over a reasonable period of time to capture advance knowledge of the impending manipulated input
the complete process dynamics. An FOPDT fit thus moves by the controller and can be very useful to the
obtained often reasonably describes the process gain, practitioner (Ogunnaike, 1986). Also, Rawlings and
Kp, overall time constant, τp, and effective dead time, Muske (1993) have shown that the stability of infinite
θp, of higher-order processes. horizon MPC can be guaranteed if M is greater than or
Step 2 involves the selection of an appropriate sample equal to the number of unstable modes in the process.
time, T. In most practical applications the user does The strategy for computing λ depends on the choice
not have the freedom to choose sample time. The tuning of M. As demonstrated earlier, with M ) 1 the need
strategy does require that T be known. In cases where for a move suppression is obviated and λ is set equal to
the designer can select the sample time, certain rules zero. However, if M > 1, a scaled move suppression
of thumb that guide in its selection are available. Rules coefficient, f, is first computed from an analytical
that select T for a desired closed-loop bandwidth, ωB, expression (eq 57). The product of the scaled move
have been proposed in the past, e.g., T e 2π/10ωB suppression coefficient, f, and the square of the process
(Middleton, 1991). Since the choice of T is related to gain, K2p, determines an appropriate value for λ.
the overall time constant, τp, and the effective dead time Step 6 summarizes the DMC tuning parameters. For
of the process, θp (Seborg et al., 1986), the estimated certain applications, more specific or stringent perfor-
FOPDT model parameters provide a convenient way to mance criteria regarding the manipulated input move
select T. A good rule of thumb is to select T such that sizes or the nature of the measured output response may
the sampling rate is not less than 10 times per time apply. For such cases, it may be necessary to fine tune
constant. Hence, for DMC, the recommended sample DMC for desired performance by altering P and λ from
time is the starting values given by the tuning strategy. The
recommended approach is to increase λ for smaller move
T e 0.1τp (63) sizes and slower output response.You can also read
