Ind. Eng. Chem. Res. 1997, 36, 2225-2230
2225
PROCESS DESIGN AND CONTROL Relay-Based On-Line Monitoring Procedures for 2 × 2 and 3 × 3 Multiloop Control Systems Jun Ju and Min-Sen Chiu* Department of Chemical Engineering, National University of Singapore, Singapore 119260, Singapore
A monitoring procedure is proposed to identify on-line the maximum closed-loop log modulus, Lc,max, for multiloop control systems. Lc,max is considered because it measures the robustness of the control systems and thereby can indicate the appropriateness of controller design with respect to the current operating conditions. The proposed method identifies Lc,max via relay feedback experiments without requiring a priori information of process conditions. This procedure is applied to both linear systems and nonlinear systems. The results show that the proposed method is efficient and accurate in on-line identification of Lc,max. Introduction Recently, Chiang and Yu (1993) developed an on-line monitoring procedure to calculate the maximum closedloop log modulus, Lc,max (Luyben, 1990). Lc,max provides a good measure of the robustness in the frequency domain for control systems and thereby can indicate the appropriateness of the controller’s parameters. In their work, Chiang and Yu used two to three relay experiments to obtain Lc,max for single-input-single-output (SISO) systems. In this paper, we extend their work to multivariable control systems. More specifically, the online monitoring procedures to calculate Lc,max for both 2 × 2 and 3 × 3 multiloop control systems are developed and treated in detail. Having said that, however, it is noted that the analytical framework presented in this paper can apply straightforwardly to n × n (n > 3) multiloop control systems. Relay Feedback Systems and Lc,max The A° stro¨m-Ha¨gglund (1988) relay feedback system as shown in Figure 1 is a simple means to identify the critical process information, the ultimate gain, and the ultimate frequency. In a relay feedback experiment (Figure 1) where N denotes the relay, the feedback system will exhibit a limit cycle as shown in Figure 1 if the following equation holds:
G(jω) ) -
1 N
(1)
πa 1 )N 4h
(2)
π π 1 ) - (a2 - 2)1/2 - j N 4h 4h
(3)
and
-
respectively. The height of the relay is denoted by h and the amplitude of the oscillation by a. According to Luyben (1990), the maximum closed-loop log modulus of a multivariable control system, Lc,max, is defined as follows:
(
Lc,max ) 20 log max
The describing functions of an ideal relay and a relay with hysteresis width () are given by
-
Figure 1. A° stro¨m and Ha¨gglund relay feedback experiments.
* To whom all correspondence should be addressed. Phone: (65)772-2223. Fax: (65)779-1936. E-mail:
[email protected]. S0888-5885(96)00773-7 CCC: $14.00
ω
|
|)
W 1+W
(4)
and
W ) -1 + det(I + GK)
(5)
where G and K denote the process and controller transfer function matrix, respectively. It can be shown from the stability analysis that, the closer W approaches the (-1,0), the closer the feedback control system is to closed-loop instability. Therefore, a large Lc,max would indicate that the control system can tolerate a small amount of uncertainties. If this is indeed the case, attention should be focused on the redesign of the controller. © 1997 American Chemical Society
2226 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
in the proposed monitoring scheme, culminating in the on-line procedures to identify Lc,max for 2 × 2 and 3 × 3 multiloop control systems as described in the previous section. The derivations of the equations of importance, i.e., eqs 6, 11, 14, 17, and 19, are given in Appendices A and B, respectively. Monitoring of 2 × 2 Multiloop Control Systems. In step 1, one relay (N1) is first placed in loop 1 as shown in Figure 2. When the relay feedback system exhibits sustained oscillation, the describing function analysis shows that
(
det(I + GK) ) -
)
1 - 1 (1 + g22k2) N1
(6)
and recall that Figure 2. Performance monitoring for a n × n multiloop control system: m stands for monitoring mode and c stands for control mode.
On-Line Monitoring Procedure for n × n Multiloop Control Systems The framework for on-line identification of Lc,max for a n × n multiloop control system is illustrated in Figure 2, where the process is denoted by G ) [gij]i,j)1-n and the multiloop controller denoted by K ) diag[ki]i)1-n, where diag stands for the diagonal matrix. For the purpose of monitoring, a series of relays are placed between the process and controller in each loop. In the monitoring mode, the controller output goes directly into the relay and the input of the process is the output of the relay. In the control mode, there is no relay and the controller output goes into process directly. Before the main results of this paper are presented below, it is noted that the condition for which the sustained oscillation exists in the proposed monitoring scheme is discussed by Ju (1996). However, it is omitted here for the sake of brevity. In order to streamline the presentation of the main results, the proposed monitoring procedures of identifying Lc,max for n × n multiloop control systems are given first. The theoretical analysis of the proposed monitoring schemes for 2 × 2 and 3 × 3 control systems, which lays the foundation of the proposed monitoring procedures, is detailed in the next section. For a n × n multiloop control system, there are n steps required to identify Lc,max on-line as described in the following. Step 1. Place a relay between the controller and the process while keeping all other controllers in the control mode. Repeat until n controllers are tested. The analysis in the following section will show that three tests with different hysteresis are required for each relay in step 1 and the subsequent steps. Step 2. Select a pair of controllers and place a relay between each controller in the pair and the process. Repeat until all possible pairs of controllers are tested. Step n - 1. Select n - 1 controllers and place a relay between each of the n - 1 controllers and the process. Repeat until all possible combinations of n - 1 controllers are tested. Step n. For all n controllers, place a relay between each controller and the process. Main Results This section summarizes the results of the describing function analysis of various relay experiments required
-
1 π π )(a 2 - 2)1/2 - j N1 4h1 1 4h1
(7)
where h1 is the relay height and a1 is the amplitude of the oscillation in loop 1. It is clear that the amplitude of the oscillation is a function of both GK and for a fixed relay height, which prevents one from obtaining an explicit relation between N1 and . Hence, an assumption on a1 is needed to approximate the computation of N1 in eq 7. Chiang and Yu (1993) made a linear assumption between a1 and for SISO systems. However, it is difficult to use the same assumption in multivariable control systems due to the interactive nature of such control systems. After numerous simulation tests, a quadratic relation between a1 and is recommended, i.e.,
a1 ) R22 + R1 + R0 ) f1()
(8)
With such simplification on a1, eq 7 shows that N1 depends only on for a given relay height (h1); i.e.,
-
1 π π )[f ())2 - 2]1/2 - j ) F1() N1 4h1 1 4h1
(9)
From eq 8, it is obvious that at least three relay experiments are required to obtain the coefficients R0, R1, and R2. For the sake of the simplicity and effectiveness of the proposed method, three relay experiments, i.e., one ideal relay and two relays with different hysteresis, are conducted in step 1 to achieve this goal. From our simulation experience, eq 8 gives satisfactory results. Likewise, this assumed relation between the amplitude of the oscillation and the hysteresis width applies to the other relay experiments in the proposed monitoring procedures. Based on the same analysis, the following relation can be derived when the relay (N2) is placed in loop 2:
(
det(I + GK) ) -
)
1 - 1 (1 + g11k1) N2
(10)
In step 2, two relays (N3 and N4) are placed in loop 1 and loop 2, respectively. The following relation can be derived from the describing function analysis:
(
det(I + GK) ) -
)(
) )
1 1 -1 + g22k2 N3 N4 1 - 1 (1 + g11k1) (11) N4
(
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2227
Finally, solving eqs 6, 10, and 11 gives W ) -1 -
(
(
)( ) (
)( )(
)( ) ) ( )(
)(
(
)
(12)
Substituting eq 12 into eq 4 yields
( |
Lc,max ) 20 log max
F2()
1 + F2()
|)
(13)
Therefore, instead of calculating Lc,max in the frequency domain (which requires the information of both G and K), we are able to utilize eq 13 to compute Lc,max online in the domain, which requires no a priori information of the process conditions. Monitoring of 3 × 3 Multiloop Control Systems. In step 1, one relay (N1) is first placed in loop 1. When the monitoring system exhibits sustained oscillation, the describing function analysis shows that
(
)|
1 + g22k2 g23k3 1 -1 det(I + GK) ) g32k2 1 + g33k3 N1
|
( (
)| )|
1 + g11k1 g13k3 1 -1 g31k1 1 + g33k3 N2
det(I + GK) ) -
1 + g11k1 g12k2 1 -1 g21k1 1 + g22k2 N3
| |
)( ) ) ( )(
(16)
(17)
)
(
(
)(
)(
)(
)(
)(
)(
)
)(
)
where
( ( (
)( )( )(
)( )( )(
1 1 1 -1 -1 -1 N2 N3 N10
A1 )
1 1 1 -1 -1 -1 N1 N3 N11 1 1 1 -1 -1 -1 N1 N2 N12
A3 )
){ ( ){ ( ){ (
)( )( )(
)} )} )}
|)
(20)
1+
1 1 1 -1 -1 N1 N11 β1
1+
1 1 1 -1 -1 N2 N12 β2
1+
1 1 1 -1 -1 N3 N10 β3
Finally, we have
( |
Lc,max ) 20 log max
F3()
1 + F3()
Examples In this section, the proposed on-line monitoring procedures for multiloop control systems are tested on literature examples including a nonlinear distillation column. Also, the robustness of the monitoring scheme is tested when the control system is corrupted with noise. Linear System. The distillation column model studied by Ogunnaike and Ray (1979) is used to test the proposed monitoring procedure, where
[
G(s) )
β1 )
(
)( ) (
)( )(
)( ) ) ( )(
0.66e-2.6s
1 1 1 1 -1 -1 -1 -1 N1 N2 N4 N5 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 N1 N2 N2 N4 N1 N5
)(
)
Likewise, the following results hold when the same analysis applies to the other two tests when relays N6 and N7 are placed in loop 2 and loop 3 and N8 and N9 in loop 1 and loop 3:
det(I + GK) ) -β2(1 + g11k1) ) -β3(1 + g22k2)
(18)
β2 )
(
)( ) (
)( )(
)( ) ) ( )(
1 1 1 1 -1 -1 -1 -1 N2 N3 N6 N7 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 N2 N3 N3 N6 N2 N7
)(
)
-0.61e-3.5s -0.0049e-s
6.7s + 1
8.64s + 1
1.11e
-6.5s
-3s
-2.36e
3.25s + 1
[
5s + 1
-34.68e-9.2s 46.2e-9.4s 8.15s + 1
10.9s + 1
9.06s + 1 -0.01e-1.2s 7.09s + 1 0.87(11.61s + 1)e-s (3.89s + 1)(18.8s + 1)
K(s) )
(
0.702 1 +
where
(
)( )(
W ) -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 N1 N2 N3 N10 N11 N12 1 1 1 -1 -1 - 1 - A1 - A2 - A3 N1 N2 N3 ) F3() (19)
(15)
where
(
)( ) (
In step 3, three relays (N10, N11, and N12) are placed simultaneously in loops 1, 2, and 3. Combining with the results from step 1 and step 2, the following relation can be derived:
(14)
In step 2, two relays (N4 and N5) are placed in loop 1 and loop 2, respectively. The following equation can be derived from the analysis of the resulting monitoring system:
det(I + GK) ) -β1(1 + g33k3)
)(
A2 )
Similarly, we can derive the following two equations when the relay is placed in loop 2 and loop 3, respectively:
det(I + GK) ) -
(
1 1 1 1 -1 -1 -1 -1 N1 N3 N8 N9 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 N1 N3 N3 N8 N1 N9
1 1 1 1 111N1 N2 N3 N4 1 1 1 1 1 1- 11- 11N2 N1 N4 N2 N3 1-
1 1N1 ) F2()
β3 )
1 35.26s
)
0
0
(
0
-0.137 1 +
0
0
1 38.7s
)
0
(
1.223 1 +
1 14.211s
)
]
]
Since this system is a 3 × 3 multiloop control system, the proposed monitoring procedure for 3 × 3 systems
2228 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
Figure 3. Closed-loop log modulus in hysteresis width domain. Table 1. Steady-State Operating Conditions for a Binary Distillation Column and Tuning Parameters of a 2 × 2 Multiloop Controller parameters
value
feed tray column pressure, atm distillate flow rate, kmol/min reflux ratio distillate comp, mole fraction relative volatility feed flow rate, kmol/min bottom flow rate, kmol/min feed comp, mole fraction bottom comp, mole fraction proportional gain of PI controller k1 integral time of PI controller k1 proportional gain of PI controller k2 integral time of PI controller k2
10 1.0 55.2 2.56 0.98 2.0 100 44.8 0.50 0.02 168.8 20.7 105.7 16.46
should be used. Lc,max obtained from the proposed procedure is 0.474 dB (Figure 3). It differs from the actual value of 0.457 dB, obtained directly from the frequency domain (eq 4), by 4%. Nonlinear System. The nonlinear example is a 20tray binary distillation column given in Luyben (1990). The PI controllers at both ends of the column will change the bottom flow rate and distillate flow rate to control the overhead composition (xd) and bottom composition (xb) at setpoint values of xdset ) 0.98 and xbset ) 0.02, respectively. This control scheme has been previously investigated by Finco et al. (1989). Table 1 gives the steady-state operating conditions and controllers’ parameters. In the dynamic simulation, the following assumptions are made: (1) 100% tray efficiency; (2) equal molar overflow; (3) saturated liquid feed; (4) total condenser and partial reboiler. The proposed identification procedure is applied to the composition loops to monitor the robustness of the closed-loop system. The Lc,max found is 14.899 dB. Assume that the PI controllers remain the same and the product specifications are changed to xdset ) 0.97 and xbset ) 0.03; the Lc,max becomes 27.299 dB based on the same monitoring procedure. Therefore, it can be concluded that the closed-loop system operated at the new setpoints is lack of robustness. Figure 4 verifies this observation by comparing the load responses for a 10% step change in the feed composition, which means a composition change from 0.5 to 0.55, for two different operating conditions. The controller indeed gives a more oscillatory load response for new operating conditions
Figure 4. Load responses for 10% step change in feed composition for the binary distillation column under two different operating conditions: (a) xdset ) 0.98, xbset ) 0.02 and (b) xdset ) 0.97, xbset ) 0.03. Solid line: xd. Dash dotted line: xb.
(xdset ) 0.97, xbset ) 0.03), which is exactly expectable from the proposed monitoring procedure. System with Noise. It is known that any practical monitoring procedure should be insensitive to process noise. The binary distillation column example is used again to test the robustness of the proposed monitoring method. The level of noise is set to be 10% of the amplitude of the limit cycle from the ideal relay feedback experiment for each loop. The Lc,max is bounded approximately between 16.02 and 13.86 dB, which differs from Lc,max ) 14.899 dB obtained in the absence of noise by approximately 7%. Therefore, the proposed method can handle the process noise to some extent. Conclusions A relay-based monitoring procedure to identify Lc,max for multiloop control systems is presented. The proposed method is a frequency domain approach and requires no a priori information of process conditions. The method is tested against a linear system, a nonlinear distillation column, and a system with noise. Simulation results show that the proposed method is very efficient and accurate in the identification of Lc,max.
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2229
Appendix A. Derivation for On-Line Identification of Lc,max for 2 × 2 Multiloop Control Systems
In step 2, two relays (N4 and N5) are placed simultaneously in loop 1 and loop 2. The following equation can be derived as below:
In step 1, one relay (N1) is placed in loop 1 as shown in Figure 2; the following relations can be derived:
0 ) (g11N1k1 + 1)(-y1) + g12k2(-y2)
(A1)
M2[y1 y2 y3]t ) 0 where
[ ])
(
N 0 det I + GK 1 0 1
)0
(A3)
Based on the property of determinant, eq A3 can be further reduced to
|
|
|
|
1 -1 0 g11k1 + 1 g12k2 + N1 ) 0 (A4) g21k1 g22k2 + 1 g21k1 g22k2 + 1
Hence, eq 6 can be readily obtained from eq A4. In step 2, two relays (N3 and N4) are placed in loop 1 and loop 2, respectively. After some algebra, the condition for stable limit cycle is
|
g11k1 +
1 g k N3 12 2 g22k2 +
g21k1
|
|
|
)(
)
Appendix B. Derivation for On-Line Identification of Lc,max for 3 × 3 Multiloop Control Systems
M1[y1 y2 y3]t ) 0
(B1)
(
1 + g11k1N1 g12k2 g13k3 M1 ) g21k1N1 1 + g22k2 g23k3 g31k1N1 g32k2 1 + g33k3
)
The condition for sustained oscillation requires det(M1) ) 0. After some algebraic manipulations, this condition becomes
|
1 -1 0 0 N1 det(I + GK) + g k ) 0 (B2) 1 + g22k2 g23k3 21 1 g31k1 g32k2 1 + g33k3 Thus, eq 14 is readily derived from eq B2.
)|
| (
1 + g11k1 g13k3 1 1 -1 + g31k1 1 + g33k3 N5 N4 1 -1 0 1 + g22k2 g23k3 + N5 )0 g32k2 1 + g33k3 g32k2 1 + g33k3 (B4)
det(I + GK) +
){|
(
|
|
|
|}
M3[y1 y2 y3]t ) 0 where
(B5)
(
1 + g11k1N10 g12k2N11 g13k3N12 M3 ) g21k1N10 1 + g22k2N11 g23k3N12 g31k1N10 g32k2N11 1 + g33k3N12
|
|
)
The condition det(M3) ) 0 can be further simplified to
In step 1, one relay (N1) is first placed in loop 1 as shown in Figure 2 and the next relationship can be obtained from Figure 2:
|
|
)
or
This completes the proof of eq 11.
where
|
Substituting eqs 14 and 15 into eq B4 gives eq 17. In step 3, three relays (N10, N11, and N12) are placed simultaneously in loops 1, 2, and 3. Again, the following expression can be derived:
g11k1 + 1 g12k2 1 + 0 -1 N4 1 1 -1 + g22k2 ) 0 (A6) N3 N4
(
|(
)
1 + g11k1 g12k2 g13k3 1 -1 0 det(I + GK) + 0 + N5 g32k2 1 + g33k3 g31k1 1 + g22k2 g23k3 1 - 1 N5 )0 N4 g32k2 1 + g33k3
(A5)
or
det(I + GK) +
Again, the condition det(M2) ) 0 reduces to
1
1 )0 N4
(
1 + g11k1N4 g12k2N5 g13k3 M2 ) g21k1N4 1 + g22k2N5 g23k3 g31k1N4 g32k2N5 1 + g33k3
0 ) (g21N1k1 + 1)(-y1) + (g22k2 + 1)(-y2) (A2) The condition for sustained oscillation is as follows:
(B3)
1 + g11k1 g12k2 g13k3 1 +g k 22 2 g21k1 g23k3 + N11 1 +g k 33 3 g31k1 g32k2 N12 1 -1 0 0 N10 1 +g k )0 22 2 g23k3 g21k1 N11 1 +g k 33 3 g31k1 g32k2 N12
|
|
or
(
)[
]
1 + g11k1 g12k2 1 -1 + g21k1 1 + g22k2 N12 1 + g11k1 g13k3 1 -1 1 +g k + 33 3 g31k1 N11 N12 1 + g22k2 g23k3 1 N 11 -1 ) 0 (B6) 1 +g k N10 33 3 g32k2 N12
det(I + GK) +
( (
)[ )
[
]
]
2230 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
Substituting eqs 14-16 into eq B6 yields
{
}
1 1 1 -1 -1 -1 N12 N11 N10 det(I + GK) 1 + 1 1 1 -1 -1 -1 N3 N2 N1
( (
)( )(
) )
1 1 -1 - 1 (1 + g11k1) + N11 N12
(
)(
)
Finco, M. V.; Luyben, W. L.; Polleck, R. E. Control of Distillation Columns with Low Relative Volatilities. Ind. Eng. Chem. Res. 1989, 28, 75. Ju, J. On-Line Performance Monitoring Procedure for Intelligent Control. M.Eng. Thesis, National University of Singapore, 1996. Luyben, W. L. Process Modelling, Simulation, and Control for Chemical Engineers; McGraw-Hill: New York, 1990.
1 1 -1 - 1 (1 + g22k2) + N10 N12
)(
Chiang, R. C.; Yu, C. C. Monitoring Procedure for Intelligent Control: On-Line Identification of Maximum Closed-Loop Log Modulus. Ind. Eng. Chem. Res. 1993, 32, 90.
Ogunnaike, B. A.; Ray, W. H. Multivariable Controller Design for Linear Systems Having Multiple Time Delays. AIChE J. 1979, 25, 1043.
1 1 1 -1 -1 + g33k3 ) 0 N10 N11 N12 (B7)
Finally, substituting eqs 17 and 18 into eq B7 and solving for det(I + GK) gives eq 19. This completes the proof.
Received for review December 6, 1996 Revised manuscript received March 4, 1997 Accepted March 4, 1997X IE960773P
Literature Cited A° stro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of PID Controllers; Instrument Society of America: Research Triangle Park, NC, 1988.
X Abstract published in Advance ACS Abstracts, April 15, 1997.
