PARETO OPTIMAL SENSITIVITY BOUNDS

Transcription

1 ARETO OTIMA SENSITIVITY OUNDS FOR A STRONGY INTERACTING -OO RAZING TEMERATURE CONTRO SYSTEM Eduard Eitelberg ORT raude College, Karmiel, Israel & University of KwaZulu-Natal, Durban, South Africa Eitelberg@braude.ac.il Alan de Freitas Universidade Federal de Minas Gerais, elo Horizonte, Minas Gerais, razil alandefreitas@gmail.com Abstract: In the industry, feedback loops are designed, or tuned, individually whereby a loop design/tuning depends on the characteristics and closure of the other loops. In this paper, the nature of areto optimal sensitivity bounds is investigated for a -loop brazing temperature control system. It is found that there are a few attractive small ranges of sensitivity pairings on the areto optimality bound some of which sacrifice performance in one loop for significant improvement in the other loop. Keywords: ristol gain, relative gain, multi-loop feedback. Introduction For many good reasons, industrial plants are controlled by multiple singleinput-single-output (SISO) feedback loops. The design, or tuning, of these SISO loops is carried out sequentially both in practice and in such theoretical methods as the Quantitative Feedback Theory (QFT) of the late Isaac Horowitz (993). In case of plant interaction, an individual loop design/tuning depends on the closure of the other loops the loop designs interact too. Close to 50 years ago, Edgar H. ristol of Foxboro Company asked: How is the measured transfer function between a given manipulated variable [u j ] and a given controlled variable [y i ] affected by the complete control of all other controlled variables in a general square multivariable system? [ristol, 966] He observed that industry has often found it desirable to control the multivariable process as if it were made up of isolated single variable processes. This is still the practically well motivated case and will be followed presently. The essential components of interest of a multivariable regulation structure are shown in Figure.. Figure.: Multi-loop regulation: multivariable regulation with diagonal controller G. The plant is square. File: aretorazinginteraction.doc Dr. Ed. Eitelberg

2 ARETO OTIMA SENSITIVITY OUNDS There are n feedback loops around the plant (s) through the n scalar regulators G i (s). In the context of loop design, in isolation from tracking or regulating requirements, the sensor and actuator transfer functions may be allocated to or to G, whichever is more convenient. The scalar regulators are the diagonal elements of G(s), with all off-diagonal elements of G being zero. It is convenient to define the n loop transfer functions through the diagonal elements of the plant as G, i, n (.) i i ii, For any i, this is the loop transfer function when all other loops are open. Such is the case in sequential loop closure when loop number i is the first one to be tuned. Generally however, when any number (or all) of the other loops are closed, then the loop i (s) through the regulator G i (s) is different from i. ristol suggested that this loop modification, or interaction, should be measured by the ratio of two gains representing first the process gain in an isolated loop and, second, the apparent process gain in that same loop when all other control loops are closed. [ristol, 966] Nevertheless, he did not strictly follow his eminently reasonable suggestion. Instead, he assumed that all the other loops are ideal in the sense that all the corresponding regulated variables y j (j i) are independent of u i. This ideal situation requires infinite loop gains with infinite bandwidth. In the QFT notation, the above apparent process gain is denoted by Qi (s). Following the suggestion of McAvoy (983), Eitelberg (006) uses the expression ristol gain i (s) to denote this ideal ratio i (s)/ Qi (s) and defines relative gain i (s) as the real ratio i (s)/ i (s) with the as designed loop transfer functions i (s) and i (s). Note that the ristol gains are properties of the plant and do not depend on the controllers, while relative gains generally do [see for example Eitelberg (006)]. Hence, there is a significant difference between the plant and loop interactions. Relative gains in systems. After output-input, or control authority, allocation, the two ristol gains are identical: The relative gains, however, are not necessarily equal: (.) ; ; with with Q Q (.) (s) leads automatically to (s) (s) independently of the design of (s) or (s) in eq. (.). ( jω) << is of little interest in feedback design as it should lead to swapping of the output-input allocation and the corresponding s jω >> jω changes ( ). Design problems are encountered when ( ) or when ( ) phase angle within the design bandwidth. ( jω) 0, in any relevant frequency range, indicates a strict load sharing system configuration see Eitelberg (999a). Generally, in systems i i when when Qj Qj j j > < and and j j > < (.3) That means the relative gains of a simply regulated system are independent of the other loop design particulars, except between and around the gain File: aretorazinginteraction.doc Dr. Ed. Eitelberg

3 ARETO OTIMA SENSITIVITY OUNDS cross-over frequencies of the other loop in the form of j and Qj. This is precisely the frequency range where not only the plant but also the loop designs interact. As a practical consideration, this loop design interaction is most difficult to deal with when both loop bandwidth ranges are overlapping. If however, one of the loops is, or can be made, significantly slower than the other then the following observations are apposite when the design is carried out in a single pass sequence. et and be the slow and fast loop respectively. The slow loop design (within its bandwidth) is independent of the design as long as bandwidth is sufficiently high. However the slow loop tuning does depend on whether the fast loop is closed or open the difference is characterised by the ristol gain. In case of high gain (jω) within the slow loop bandwidth, the slow loop should (generally but not necessarily always) be tuned first. This is a practical recommendation discovered in the context of load sharing control (Eitelberg, 999a). The fast loop design is independent of the slow loop or its closure near the Nyquist point, because and beyond the slow loop gain cross-over frequency. In the same higher frequency range, uncertainty comes only from and it is independent of any uncertainty in, and. It does not follow, that stability of the closed fast loop is necessarily independent of the slow loop closure. This question does not arise when the slow loop is closed before the tuning of the fast loop as recommended above and remains operational (out of limits) during all operating conditions and disturbances. Tuning the fast loop first, may lead to higher bandwidth and improved small signal regulation. However such regulated process is often inoperable during start-up, shutdown and under other large signal conditions, when the (conditionally stable) fast loop saturates, temporarily or permanently. Some more insight is gained from eq. (.) directly. Without restriction of generality, the controller G in loop number is being tuned. The loop transfer function is G + Q (.4) + The poles of this loop number are those of the controller G, the diagonal plant element, and some or all of the zeros of the first loop characteristic equation when the second loop is open. If, for example, is stable and an unconditionally stable closed loop system is desirable then + should have no right half-plane zeros. Furthermore, if the plant must remain stable when is (temporarily) saturated then + must have no right half-plane zeros. The zeros of this loop number are those of the controller G, the diagonal plant element, and some or all of the zeros of the ideal first loop characteristic equation when the second loop is open. If, for example, one wishes to have no additional bandwidth limitations beyond those of then + Q should have no right half-plane zeros. Within the first loop bandwidth, those zeros are (approximately) equal to zeros of. 3 I control for sensitivity minimisation. To paraphrase ristol s wording, after all those years, industry is still finding it desirable and satisfactory to control its plants with I or ID controllers. The ID controller adds to the loop transfer function some phase lead while the simpler I controller cannot do so. Not all loop designs benefit sufficiently from the additional phase lead and most industrial control loops use I control only. A I controller is characterised by the proportional gain k and the integral time T i in the following structure ( ω /T i ) ci File: aretorazinginteraction.doc 3 Dr. Ed. Eitelberg

4 G ARETO OTIMA SENSITIVITY OUNDS ω I + Tis s (3.) ( ) ( ) + ci s G s k k The sensitivity function for a closed-loop feedback system around a plant (s) is defined as S, + G (3.) The task of feedback design is to minimise the sensitivity in some (usually low) frequency range with a given upper bound M (usually somewhere around 3 to 6 d, near the gain and phase cross-over frequencies): ( jω) M, ω S (3.3) 4 Two-loop sensitivity minimization for a brazing furnace. A certain industrial brazing furnace temperature control structure was shown to be inoperable in Eitelberg (999b). The reason advanced at the time was the large steady state ristol gain of 6. A solution to the problem was designed and successfully implemented then without understanding or using the dynamic ristol or relative gains. It is appropriate to revisit this problem presently with the new understanding. The non-linear plant could be approximated by 8. T s 5.9 d Td s e e + T s + T s (4.) 7.6 Td s 5.9 Td s e e + T s + T s All Tij are about 0 minutes and both loop delays Tdj are assumed here equal to 0.5 min. All loops have bandwidth limitations, usually due to a right half-plane zero at or dead-time T d. The gain cross-over frequency is bound by β ω gc <, ωgc < (4.) T d These limitations are due to plant and not often modified by controller design. In specific situations, they can be modified (see the above possibility of designing the faster loop first), however there are good engineering reasons to advise against such designs. Here both loop bandwidths are limited by ω < 0.5 rad/min rad/min. gc If not only unconditional stability, but also unconditional stability margins are required, the low frequency loop magnitude slope should be approximately 30 d/dec. Accordingly, a well-designed sensitivity at, say two decades below gain crossover, should be about 60 d. A simple I control structure does not permit achievement of the best theoretical performance. Hence one should not expect 60 d sensitivity at 0.0 rad/min, perhaps more realistically at around 0.0 rad/min. The inverse of the ristol gain is here ( + Ts )( + Ts) ( + T s)( + T s) (4.3) This example indicates why highly interacting plants ( 6 at low frequency) are likely to have non-minimum phase-lag problems too. T and T only need to be a few per cent greater than T and T to push one or both zeros of transmission zeros of into the right half-plane. All parameters of this plant vary much more than a few per cent when furnace loading and other operating conditions change. For the purposes of the present investigation, it is assumed that T T File: aretorazinginteraction.doc 4 Dr. Ed. Eitelberg

5 ARETO OTIMA SENSITIVITY OUNDS T 0 min and T is varied. When also T 0 min then 6. Otherwise is frequency-dependent: ( + Ts ) ( + 0s) ( 0.938T ) ( + 0s) s Its interaction related zero is at (4.4) 0.06 β i (4.5) 0.938T 0 It is in the right half-plane for T >.33. If the ristol number, then there is no loop design interaction and the two sensitivities are completely independent they are minimised independently. If, however,, then loop designs interact. Accordingly, the achievable minimum value of, say, S depends on the chosen design of. It stands to reason that there are many loop designs that jω + jω jω + jω lead to areto optimal sensitivity pairs ( ) ( ) and ( ) ( ) S S meaning that one cannot reduce S without increasing S and vice versa. Here, a qualitative analysis is developed for what can be expected under some realistic conditions. There are two distinct mechanisms whereby interaction will reduce the achievable performance of about 60 d at ωs 0. 0rad/min one is due to the interaction magnitude and the other is due to the interaction induced right halfplane zeros β i. If ω S << ωgc of Q and does not introduce any right half-plane zeros β i, then S ( jωs ) will be increased by ( j S ) d ω. Such is the case with T 0. et ω S 0. 0 then the low-frequency sensitivity increases at most by a factor of 6 4 d because the loop gain is reduced by this amount. esser change is obtained in a loop if the other loop s bandwidth is near or below the test frequency at ω 0.0. S If however, introduces a right half-plane zero β i < ω gc of Q then β i becomes the new bandwidth limitation for. If ω S << β i then ωs S ( jωs ) d > 30log0. To this one needs to add up to 4 d from the ristol gain β i magnitude. For T.4 we get from eq. (4.5) β i and for T 3 we get β i The corresponding loop bandwidths are limited to ω gci < 0. 4 and to ω gci < 0.0 respectively. The approximate lower bounds for sensitivity at ω S are 30log 0 48 d and 30log 0 9 d respectively. Due to the limited complexity of the used I controllers, more realistic numbers would be something like 4 d and 3 d respectively. To the first number one should add up to 4 d depending on the other loop bandwidth. The test frequency ω S 0. 0 is so close to the gain cross-over frequency in the second case (T 3), that the sensitivity should not be expected to change much from the 3 d above. This qualitative analysis is confirmed in Figure 4. that displays the areto lower bounds of S ( j0.0 ) d and S ( j0.0 ) d for the indicated three choices of T. The best performance of 60 d in either loop is achieved when the other loop is open (the corresponding S 0 d) this is independent of the T -value. When the other loop is gradually closed and its performance improves from 0 d to about 0 d, the excellent performance worsens from 60 d to about 40 d. It is not wholly the above predicted 4 d increase but close. This is clearly visible for T 0 min and.4 min. The case of T.4 differs from that of T 0 by the introduction of β i Hence, when the other loop s bandwidth is sufficiently high, the first loop s performance degrades to about 7 d. This confirms the above calculated expectation of 4 d + 4 d 8 d. The case of T 3 is dominated by the File: aretorazinginteraction.doc 5 Dr. Ed. Eitelberg

6 ARETO OTIMA SENSITIVITY OUNDS introduction of the interaction zero at β i Therefore, neither loop performs better than about 4 d, except when both loops have similar bandwidths and performances up to about d. S ( ) d and ( ) d In the case of T 0 min, the best performance is around j S j simultaneously. It is assumed that noone wants one of the two loops to be open! 0 S d -0 T3-0 T T S d Figure 4.: areto optimality bounds for T 0 min, T.4 min, and T 3 min. File: aretorazinginteraction.doc 6 Dr. Ed. Eitelberg

7 ARETO OTIMA SENSITIVITY OUNDS 3.5 k/ti 3.5 T0.5 T T k/ti Figure 4.: areto optimal low-frequency controller gains (k/ti) for T 0 min, T.4 min, and T 3 min. In the case of T.4 min however, there are two obvious choices of best S j0.0 40, S ( j0.0) 7d and the performance. The one is around ( ) d other is around S ( j0.0) 7d, S ( j0.0) 39d solution around S ( j0.0) 46d, S ( j0.0) 6d open loop number.. One could argue about the, but this means an almost If one wants any significant performance at ω 0. 0, in the case of T 3 S j0.0 j min, the best performance is around S ( ) 48d, ( j0.0) 3d around S ( j0.0) 4d, ( ) d S or S. If comparatively moderate performance is acceptable and it is required in both loops simultaneously then it will be about d. Figures 4. and 4. were generated as follows. First, million random combinations of k, ω ci, k, and ω ci were generated in a hyper-cube [0, 5] 4. This choice of parameter sub-space is reasonable because any k-values close to and above 5 lead to unstable closed-loop system and any Ti-values below /5 lead to instability. Every single combination of the four controller parameters defines the feedback system fully. Second, every one of the million systems from above was checked for stability and only those combinations of k, ω ci, k, and ω ci that resulted in a stable system, were admitted. The rest were rejected. Third, for every one of the above admitted combinations of k, ω ci, k, and ω ci, the pair of frequency responses S ( jω), S( jω) was calculated. Only those combinations of k, ω ci, k, and ω ci for which both S( jω) and S( jω) satisfied the sensitivity bound in eq. (3.3) with M 6 d, were admitted. The rest were rejected. File: aretorazinginteraction.doc 7 Dr. Ed. Eitelberg

8 ARETO OTIMA SENSITIVITY OUNDS Fourth, for the remaining admitted combinations of k, ω ci, k, and ω ci the S j.0, S 0.0, were recorded. Then a simple 0 j low frequency sensitivity pairs, ( ) ( ) areto dominance relation (A dominates if A is better than for all objectives) as implemented in [7] was applied to the low-frequency sensitivities to obtain the fronts. 3 Conclusion. There are two distinct mechanisms whereby interaction will reduce the achievable performance of two-loop feedback systems. The one is due to the interaction magnitude and the other is due to the interaction induced right halfplane zeros β. i In the shown example of brazing temperature control, a lack of interaction induced right half-plane zeros permits achievement of similar performance in both loops. If however, the interaction introduces a significant right half-plane zero, then it becomes attractive to sacrifice performance in one loop for the sake of performance in the other loop. References.. ristol, Edgar, H. (966): On a New Measure of Interaction for Multivariable rocess Control. IEEE Trans. Autom. Control, AC (966), pp Eitelberg, Ed. (999a): oad Sharing Control. NOY ress, Durban. 3. Eitelberg, Ed. (999b): oad sharing in a multivariable temperature control system. Control Engineering ractice, Vol. 7, No., pp Eitelberg, Ed. (006): On multi-loop interaction and relative and ristol gains. ASME Journal of Dynamic Systems, Measurement and Control. Dec. 006, Vol. 8, pp Horowitz, Isaac (993): Quantitative Feedback Design Theory (QFT). oulder, QFT ublications. 6. McAvoy, J. Thomas (983): Interaction Analysis: rinciples and Applications. Instrument Society of America, Research Triangle ark File: aretorazinginteraction.doc 8 Dr. Ed. Eitelberg