Ind. Eng. Chem. Res. 1993,32, 2647-2652
2647
Proportional plus Integral MIMO Controller for Regulation and Tracking with Anti-Wind-Up Features Pablo F. Puleston' and Ricardo J. Mantz CIC Pcia. Bs.As. and Laboratorio de Electronica Industrial, Control e Znstrumentacion (LEICI), Facultad de Zngenierh, Universidad Nacional de La Plata, CC 91, 1900 La Plata, Argentina
A proportional plus integral matrix control structure for MIMO systems is proposed. Based on a standard optimal control structure with integral action, it permits a greater degree of independence of the design and tuning of the regulating and tracking features, without considerably increasing the controller complexity. Fast recovery from load disturbances is achieved, while large overshoots associated with set-point changes and reset wind-up problems can be reduced. A simple effective procedure for practical tuning is introduced.
I. Introduction
PI (PID) control is the most widespread controller type in industry. Among other reasons, this is due to its robustness in conventional process control and tuning simplicity for trained personnel. Traditionally, proportional plus integral strategies have been preferred for controlling complex industrial processes, for instance multiinput-multioutput coupled systems. Although the conventionalPI (PID) controller has wellknown remarkable properties, occasionallythe number of tuning parameters may be scarce. This is a major issue when confronting restrictive design requirements. These structural constraints are unveiled when tracking and regulation are simultaneouslydemanded. Very often, controllers tuned for optimal regulation tend to produce large overshoots for set-point changes, whereas trackingtuned controllers result in sluggish recovery from load disturbances. In this paper, we propose a multivariable P+I control structure (MPI), particularly suited to MIMO systems. It is based on a standard optimal control structure with integral action (SOCI) (Kwakernaak and Sivan, 1972; Wong and Seborg,19851,requiring knowledge of the system matrices (triplet S(AJ3,C)). The MPI controller keeps SOCI optimal regulation properties, while yielding extra tracking-tuning parameters. It thus provides far better tuning for the tracking mode, without losing regulation properties. Despite the inherent differences because of its matrix nature, the proposed controller presents analogies with a modified single loop PI (PID) controller (Hhgglud and Astrom, 1985;Gawthrop, 1986;Eitbelrg, 1987;Mantz and Tacconi, 1989,1990;Astram et al., 1992). In the present case, each single gain became a gain matrix. The new structure effectively reduces reset wind-up overshoota (Shinskey,1979;Astrom and Wittenmark, 1984; Hanus et al., 1987;Hanus and Kinnaert, 1989;Hanus and Peng, 1991) during set-point changes. 11. Approaches to Standard Optimal Control The background needed is presented in this section (Kwakernaak and Sivan, 1972;Bryson, 1985). Consider a time-invariant linear system, assumed to be controllable and observable, described by state-space equations:
The technique to generate a SOCI is as follows: (a) Shift the origin of the state-space, referring x, y, and u to their equilibrium points 80, yo, and UO; (b) Augment the shifted system by adding a set of integral variables, given by q = Ji(y - yo) dt. At this point the augmented system can be written:
x,' = A,x,'
+ B,u,'
y,' = c,x,'
where
A, =
[$I!]
Be =
(1)
y =cx
(2)
(4)
[4]
4,error integral =
x,', extended state =
[c
C, = O I I I
K(Y- yo)dt
[XI [ [t] [ =
x-x O]
= Y - Yo
y,', extended outputs =
]
u', shifted input = [u - uol Define the performance index as
J = Lm(yfTR1y' + ufTR2uf+ qTRl,q) dt = = Sbl)(ya/rRlj,' + ufTR2uf) dt
(5)
with
and Rz = weighting matrices. Whenever the augmented system is controllable, the optimal control law is given by
uf = -h, * '
(6)
u = - F ~ x- FIq + Hc1y0
(7)
where F = (R;'B,TP,)
X=Ax+Bu
(3)
= [Fp,FIl
(8)
HL1 = [C(-A + BFp)-'B]-l (9) and matrix P, satisfies the algebraic Riccati equation for
0888-588519312632-2647$04.O0/0 0 1993 American Chemical Society
2648 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
the augmented system: 0 = CTRleCe- P J 3 , R ~ 1 B ~ + PA e TP,
+ PeAe
(10)
The matrices Fp and FI are obtained from partition of F, the augmented-system state-feedback matrix. The Fp matrix characterizes the feedback of the original state x . On the other hand, FI is related in the same way to the integral state q. It is easy to see that the set-point weighting matrix HC-', forces the steady-state-transfer matrix of the closed-loop system to be identity. We have exposed the basic ideas to develop a standard optimal control structure with integral action, considering nonzero set points (SOCI). This structure has been optimized for fast recovery from load disturbances. Integral action has been included in order to eliminate the offset caused by either unmeasurable nonzero mean disturbances or modeling errors.
111. MIMO Modified PI Controller (MPI) 1II.A. Initial Considerations. Before discussion of the proposed structure, it will be helpful to make some considerations: (a) It can be seen in eq 9 that HC-ldepends not only on the system matrices but also on Fp. Then, when computing Fp and FI for optimal fast recovery from disturbances, the Hc-1 matrix is determined as well. This rigid relation between the feedback gain matrix and the closed-loop transfer function matrix could result in an unacceptable tracking behavior. For instance in many processes, particularly thermal and chemical ones, an underdamped disturbance rejection is sought (i-e.,quarter decay response (Shinskey, 1979)). This kind of tuning generally leads to a similar tracking dynamic behavior, i.e., equal ratios of overshoots to jumps. This is frequently undesirable for the plant. Indeed, equilibrium deviations due to set-point changes normally are muchlarger than those caused by disturbances. Hence, overshoots that could be tolerated in regulation are not suitable for tracking. (b) Even though the regulation problem could be optimized looking for an overdamped behavior, integrators overload might cause important overshoots associated with set-point changes (Wong and Seborg, 1985). (c) In the previous analysis we have considered a linear system; nevertheless real actuators are limited to a certain range. The combination between actuator saturation and the integral action may produce unexpected overshoots. This phenomenon is known as reset wind-up (Astrom and Wittenmark, 1984;Hanus et al., 1987;Hanus and Kinnaert, 1989; Hanus and Peng, 1991). The preceding concepts enumerate some possible causes of large overshoots due to set-point changes. Naturally, some of these problems could be counteracted by adding a new term (considering the tracking response) in the performance index J. However, this procedure would end up in a trade-off with the dynamic response to disturbances. 1II.B. Proposed Strategy. This paper develops a control strategy that can be interpreted as a MIMO version of a modified PI controller. The proposed controller (MPI) provides an easy solution to the inconveniences mentioned in section 1II.A. We considered the derivation of our control structure more interesting if done following optimal control theory. However, other approaches are also possible.
Figure 1. Proposed control structure (MPI).
The proposed strategy emerges from imposing the integrator final state ( q o ) as a function of the reference vector (yo): qo = f ( Y 0 )
(11) Thereupon, it is possible to redefine the extended state (x:) and outputs (ye')(eqs 3 and 4), referring q to a steadystate vector, which depends on the set point:
xl extended state =
[i:]= [x --xf & o ) ]
y i , extended outputs =
["9:] = [Yq -- Yo ] f(Y0)
The optimal control law that minimizes the performance index J (eq 5) is given by eq 6, leading to
u = - F x ~+ u0 = -FPx - FIq + q0
(12)
with *o = Fpxo + Fj(Y0) + uo
(13) To express the vector u as a function of x , q, and yo, we substitute eq 12 in eq 1:
where
Evaluating eq 14 for t
-
a,it
becomes
Applying eq 15, eq 12'is rewritten as
u = -FPx - FIq + HC1yo + F j ( y o )
(16) Proper selection of the function f(y0) yields different tracking behaviors without affecting the regulating response. This paper emphasizes a linear relation between the final integral state and the reference vector ( q o = Kyo). For this case eq 16 can be written as
+
u = -FPx - FIq Hyo
(17)
where
H = (H;' + FIK) Figure 1shows a block diagram of the MPI. As we can see, the proposed strategy reduces to the inclusion (in the SOCI structure) of a gain matrix K , = FIK in parallel with Hc-l. We should note the following:
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2649
Figure 2. MPI represented as a MIMO modified PI structure.
(a) These simple modifications do not affect the regulation optimal characteristics of the SOCI, because the K, matrix is placed out of the feedback loop. (b) Since H does not alter the loop transfer matrix of the SOCI, regulation robustness of the MPI is the same as regulation robustness of the SOCI (Lehtomaki et al., 1981). (c) The integral action eliminates the steady-state error for a nonzero set point. This means that the inclusion of K, does not modify the equality between yo and y when t+-. (d) However, as will be shown later, K, does modify the transient tracking behavior. This is essential to solve the problems mentioned in section 1II.A. The MPI control scheme, under certain hypothesis, can be interpreted as a MIMO PI structure, resembling different single input-single output (SISO) PI structures used in industrial processes. Therefore, for those readers acquainted with SISO PI controllers, this representation permits a better understanding of the proposed structure. We proceed as follows to obtain this MIMO PI representation. The structure shown in Figure 1is described by
For the purpose of the present interpretation (assuming that C and F p are of the same size and invertible) eq 21 can be expressed as U = H Y o -FpCIY -FI(Y - Y ~ ) / s
controller (Isermann, 1981;Astrom and Wittenmark, 1984; Gerry, 1987). (b) Canceling W,sets analogies to the controller known as IP (Isermann, 1981; Nandam and Sen, 1986). (c) W, selected as a diagonal matrix, with elements bounded between 0 and 1, resembles a SISO modified PI control structure (Hhgglud and Astrlim, 1985;Gawthrop, 1986; Eitbelrg, 1987; Mantz and Tacconi, 1989, 1990; Astrom et al., 19921, where each single gain has turned into a gain matrix. 1II.C. Closed-LoopZero-Pole Location. For Laplace domain analysis of the proposed strategy, the closed-loop transfer function matrix (T(s))is found. From T(s),we can infer some conclusions about how the features of the system are modified when H is used for tuning. Using eqs 18-21, it is possible to write M(s)Y = N(s)Yo (27) M(s)and N(s)are polynomials ins with matrix coefficients. Then if M(s) is invertible, T(s)takes the following form:
T(s)= M(s)-'N(s)
(28)
with
+
+ BFp) + BFIC]-'B N(s) = (sH + FI)
M(s)-'
C[s21 s(-A
(29) (30)
In these conditions, the elements of the M(s1-l matrix (mi(s)ij)are rational ins. All the elements have a common polynomial denominator, given by the determinant of M ( s ) (A(M)(s)). On the other hand, the numerator polynomials are given by the elements (am(s)ij)of the adjugate matrix of M(s):
It is important to emphasize that the elements of M(s)-l do not depend on the H matrix. With the system and its feedback matrices FPand FIdefined, the mi(s)ijzero poles are senseless to changes in the tuning of H. Instead, the elements of N(s) (n(s)ij)are first-order polynomials in s, whose zeros directly depend on the H elements (hij): N(s) =
(22)
f [s,hii + fiij)l
[ ? ~ ( ~ )= ij]
(32)
Finally, the system transfer function matrix can be written in explicit form using eq 2 8
Then U = Kp[WpYo- Y - KI(Y - Y~)/s]
(23)
Kp = FPC1
(24)
W,= (FpC-')-'H
(25)
where
(26) K, = (FPC-')-'FI Figure 2 shows the MIMO PI structure based on eq 23. The control matrices Kp and KIare determined by F p and FI,which have been tuned for optimal disturbance rejection. On the other hand, an adequate tuning of W, (equivalenttoadjusting K in Figure 1)enhanced the system tracking behavior without affecting its regulation properties. For some particular values of W,, resemblances with familiar single-loop control structures are found: (a) Tuning W, to be the identity (W,= I) leads to a MIMO control structure parallel to a classic SISO PI
with
+ am21n21 + ~ ~ ~=( (amlln12 s ) + am21n22+ 7 2 1 w = (am,,n,, + am22n21 + ~ ~ ~=( (am12n12 s ) + am22n22+
711(s)
= (am1,n,i
...)(8)
Each element t(s)ijcan be considered as the transfer function between the output Yi and the reference input Yoj. Now, clearly, we observe that the pole location of t(s)ijis independent of H, hence remaining at the position
2650 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
dictated by the optimal regulator design. On the other hand, tuning H (or W,) allows different locations for the zeros of the transfer functions. However, notice that there might be some zero-pole cancellations. In summary, a proper closed-loop zero location may increase or decrease (ideally cancel) the influence of the dominant poles on the tracking time response, without affectingoptimal disturbance rejection. Hence, MPI extra degrees of freedom can be used to improve the tracking behavior without compromising the regulating performance of the feedback loop. 1II.D. Practical Tuning Criterion. From the former concepts we can infer a tuning criterion for the proposed structure: (I)Design matrices Fp and FI for optimal disturbance rejection (like in the SOCI structure), without considering tracking. (11)Improve tracking by adjusting the matrix H (or W, in Figure 2). Adequate closed-loop zeros assignment can weaken undesirable pole contribution to tracking behavior. Although many other tuning criteria can be proposed, a very practical tuning method is suggested. In this method, step I1 is accomplished by a tuning technique based on a time domain analysis of the control action vector u. For this purpose, we consider the effects of the adjusting matrix W, over the control action. It can be easily observed that (a) shortly after starting, the control action is basically constituted by the proportional contribution (up = Kp(WPyo - y)) and (b) when the elements of y become greater than the elements of W,yo ( y i > (Wpyo)j), the elements of upchange their signs, restraining the output rise. According to this, we define W, as
w,= aw,
(34) where a is a scalar factor and W, is the set-point weighting matrix corresponding to the SOCI structure (W, = (FpC-l)-lH,). A simple design possibility emerges: large values of a lead to stronger immediate controller action, while low values of a soften the controller action after a reference change. Therefore, smaller tracking overshoots can be obtained by choosing 0 5 a < 1. Even though the analysis of the MPI in the presence of time-delay systems surpasses the scope of this paper, we would like to make a remark in regard to this. It is important to note that independence between tracking and regulation is one of the main features of the proposed structure. This feature depends only on the control scheme, and not at all on the plant. Then, this remarkable property is valuable to fulfill simultaneously tracking and regulationrequirements, even when confrontingtime-delay systems. Of course, like all other control structures, the MPI have limitation, but they are less restrictive than those of the SOCI. 1II.E. Reset Wind-Up. In this section, we make a brief discussion about the behavior of the MPI in the presence of nonlinear actuators. Due to their limited capability, real system actuators fall into saturation in demanding control situations (ui> usah). This fact, combined with the controller integral action, may generate significant output overshoots, even larger than expected in an ideal linear system. This phenomenon is known as reset wind-up (Shinskey, 1979; Astrom and Wittenmark, 1984; Hanus et al., 1987; Hanus and Kinnaert, 1989; Hanus and Peng, 1991). Generally in industrial processes, deviations (about the equilibrium state) due to set-point changes are larger than those caused by disturbances; thus for set-point changes a stronger control signal will be needed to reinstate the
I
I1
V
i'
P , C
Figure 3. Stirred tank.
system equilibrium. Therefore, the most critical situation as far as actuator saturation is concerned will be with the system acting as a tracker. As detailed in section III.B, the proposed structure permits a fast-recovery tuning, not imposing an underdamped tracking response. Thus, the adequate selection of matrix H (or W,)may reduce tracking overshoots when compared to the SOCI time response. This reduction is a direct consequence of a smaller tracking control action. We can summarize how the MPI decreases reset wind-up overshoots with the following idea: smaller control actions in tracking mode, saturate actuators for shorter periods, shirking integrator overload. IV. Numeric Example An example is given based on the stirred tank used by Kwakernaak and Sivan (1972). The tank (Figure 3) is fed with two incoming flows F1 and F2, which can be adjust independently. Both feeds contain dissolved material with constant concentrations c1 and c2, respectively. We define the volume of the fluid in the tank as V, the tank concentration as c, and the outgoing flow rate as F. The state model matrices of the system are
The control vector is u = [F1F2IT,the state vector is x = [V cIT, and the output vector is y = [F cIT. The steady-state output values are FO= 0.02 m3/s and co = 1.25 kmol/m3. As it was explained, in some processes underdamped regulating responses are desired (Ziegler and Nichols, 1942; Shinskey, 1979). These responses provide fast recovery from small disturbances, driving rapidly the controlled variables into the tolerance zone. Following the previous ideas, the SOCI and the MPI structures have been equally tuned for regulation, minimizing the index J (eq 5) with 50 0
60 0
This results in 0.39 -1.701 Fl - [lo3 -1.101 = L0.12 0.88 2 80 0.44 As said in section 111, identical optimal disturbance rejection is obtained with both SOCI and MPI structures. However, in the SOCI case a highly underdamped tracking behavior with large undesirable overshoots is achieved, although it has a suitable regulating response. Using the MPI controller, these overshoots can be reduced without altering the desired disturbance rejection (Fp and FIremain unmodified). The tracking-tuning criterion from eq 34 is applied to compute the MPI parameters. Fp
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2651 -1
0.005
I
.._ ,." _.--._
I
-0.015 I 0
I
,-
5
10
Time [Sa]
I
15
7.0
25
15
7.0
25
15
m
25
Time [Sec]
Figure 4. Tracking response when a flow reference step is applied (curve A, a = 1 (SOCI); curve B, a = 0.8; curve C, a = 0.6). (a, top) Flow output. (b,bottom) Concentration output.
0
5
10
Time [Sec]
0
5
10
Time [Sec]
[&I Figure 6. SOCI reset wind-up when a concentration reference step is applied (curve A, no saturation; curve B, F1,t = 0.85 and Faut = 0.45; curve C, F1,t = 0.7 and Fkt = 0.35). (a, top-d, bottom) (a) Concentration output. (b) Flow output. (c) Control action Fl. (d) Control action F2.
Time [&I
Finally, reset wind-up effects in both structures are compared for different actuator saturation values (F1,t andFht). Applying a concentration reference step, Figure 6a,b shows how saturation can spoil the SOCI tracking response, while the MPI (Figure 7a,b) remarkablyreduces reset wind-up overshoots. Control actions are depicted in Figures 6c,d and 7c,d. As we have described in section IILE, the MPI control actions are less demanding, so actuators saturate for shorter periods. Analogous results are found when a flow reference step is applied.
mhE
-0.01I 0
I 5
15
10
7.0
25
Figure 5. Tracking response when a concentration reference step is applied (curve A, a = 1(SOCI); curve B, a = 0.8; curve C, a = 0.6). (a, top) Concentration output. (b, bottom) Flow output.
In Figures 4 and 5, the tracking system response to steps in both references, flow and concentration, are displayed for a = 1(SOCI case), a = 0.8, and a = 0.6. It can be seen that, with this simple tuning method (decrementing parameter a), an important overshoot reduction is achieved. Further improvementswould be possible using this method to find a basic W,matrix and then iteratively adjusting its elements one by one.
V. Conclusions In this paper we have proposed a proportional plus integral matrix control structure. It may be derived by defining a functional relation between the reference vector and the steady-state integral vector. The proposed PI control strategy, in spite of intrinsic differences due to its
2652 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
-1 1.6 1.8
I
t
properties have been analyzed: the proposed structure does not modify optimal regulator features, yields extra tuning parameters to improve tracking behavior, and effectively reduces reset wind-up overshoot. We also presented a numerical example to illustrate the main features of this control scheme.
Literature Cited
Time [Sa]
3
P
-0.01
-0.015
0
5
15
10
20
25
15
20
25
15
20
25
Time [Sec]
0
5
'
10
Time [Sec]
11
0
5
10
Time [h]
Figure 7. MPI reset wind-up when a concentration reference step is applied (curve A, no saturation; curve B, F1,t = 0.85 and Fkt = 0.45; curve C, Flmt= 0.7 and Fkt = 0.35). (a, top-d, bottom) (a) Concentration output. (b) Flow output. (c) Control action F1. (d) Control action Fz.
multivariable nature, can be regarded as a generalized MIMO version of a modified PI controller recently published. Regulation, tracking, and anti-reset wind-up
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Received for review March 2, 1993 Revised manuscript received July 8, 1993 Accepted July 27, 1993. Abstract published in Advance ACS Abstracts, October 1, 1993. @
