3.1 Overview 3.2 Process and control-loop interactions

Transcription

1 3. Multivariable 3.1 Overview 3.2 and control-loop interactions Interaction analysis Closed-loop stability 3.3 Decoupling control Basic design principle Complete decoupling Partial decoupling Decoupling by linear combination of inputs Decoupling by nonlinear variable combinations 3.4 Decentralized control KEH Plantwide 3 1

2 3. Multivariable 3.1 Overview In the previous chapter, control problems with one (primary) controlled variable and (in principle) one manipulated variable were emphasized. Such control problems are referred to as single-input, single-output (SISO), or single-loop, control problems. In most practical control problems, many variables have to be controlled using many manipulated variables. Such control problems are multiple-input, multiple-output (MIMO) control problems. [Figure from SEMD] KEH Plantwide 3 2

3 3. Multivariable 3.1 Overview Important special cases of MIMO control are decentralized, or multiloop SISO, control decoupling control The most extreme case of MIMO control is centralized control where a single controller controls the full MIMO process. However, for processes with more than a few variables, centralized control on the regulatory control level is not practical. Multiloop SISO control [Marlin] Centralized MIMO control [Marlin] KEH Plantwide 3 3

4 3. Multivariable 3.2 and control-loop interactions MIMO control is difficult compared to SISO control because of process and control loop interactions. Interactions occur when there are inputs used as control variables that affect more than one controlled variable When significant process interactions are present, the most effective control configuration may be difficult to find Example A 2x2 system can be described by the transfer function model y 1 y = G 11 G 12 u 1 2 G 21 G 22 u 2 Interaction is present if at least 3 transfer functions are 0, e.g., G 11 and G 22 0 G 12 and/or G 21 0 [WÅJ] KEH Plantwide 3 4

5 3. Multivariable 3.2 and control-loop interactions Interaction analysis A 2x2 system is considered for simplicity. There are two possibilities for multiloop SISO control: to control y 1 by u 1 and y 2 by u 2 (the 1 1/2 2 pairing) y 1 by u 2 and y 2 by u 1 (the 1 2/2 1 pairing) Which one to choose, depends on the interaction properties. For an n n system, there are n! different basic multiloop SISO configurations. Systematic ways of selecting proper variable pairings are considered in the context of decentralized control. R 1 C 11 G 11 R 1 C 12 G 11 G 12 G 12 G 21 G 21 [SEMD] R 2 C 22 G 22 R 2 C 21 G 22 KEH Plantwide 2 5

6 3.2 and control-loop interactions Interaction analysis The hidden feedback control loop Consider, e.g., the 1 1/2 2 pairing with the controllers C 11 controlling y 1 by u 1 C 22 controlling y 2 by u 2 Due to interaction, there is a third hidden feedback control loop containing the transfer functions G 21 C 11, G 21, C 22 and G 12. Because of R 2 C this, there is for the 1 1 loop 22 G 22 (and similarly of the 2 2 loop) a direct effect of u 1 on y 1 through G 11 ; [SEMD] an indirect effect of u 1 on y 1 through the hidden feedback loop, which depends on the controller C 22. The hidden feedback loop causes two potential problems: it increases the danger of instability of the closed loop system it makes controller tuning more difficult, since both controllers affect both outputs R 1 C 11 G 11 G 12 KEH Plantwide 3 6

7 3.2 and control-loop interactions Interaction analysis Illustration of multiloop SISO control [Marlin] Setpoint change of CV 1 (CV i = controlled var. i, MV i = manipulated var. i) Both loops tuned for good SISO control Loop 2 tuned for tighter control Loop 1 tuned for tighter control Loops tuned for minimal IAE sum KEH Plantwide 3 7

8 3. Multivariable 3.2 and control-loop interactions Closed-loop stability Let us consider a 2 2 process with the transfer function matrix Y 1 Y 2 = G 11 G 12 G 21 G 22 U 1 U 2 The 1 1/2 2 pairing uses the control configuration U 1 = C 11 0 R 1 Y 1 U 2 0 C 22 R 2 Y 2 Combining these equations gives the closed-loop transfer function matrix where Y 1 Y 2 = 1 A B 21 B 22 B 11 B 12 KEH Plantwide 3 8 R 1 R 2 B 11 = G 11 C 11 + C 11 C 22 detg, B 12 = G 12 C 22 B 21 = G 21 C 11, B 22 = G 22 C 22 + C 11 C 22 detg A = 1 + G 11 C 11 + G 22 C 22 + C 11 C 22 detg detg = G 11 G 22 G 12 G 21

9 3.2 and control-loop interactions Closed-loop stability For the 1 2/2 1 pairing we have the control configuration U 1 U 2 = 0 C 12 C 21 0 R 1 Y 1 R 2 Y 2 and the closed-loop transfer function matrix where Y 1 Y 2 = 1 A B 21 B 22 B 11 B 12 R 1 R 2 B 11 = G 12 C 21 C 12 C 21 detg, B 12 = G 11 C 12 B 21 = G 22 C 21, B 22 = G 21 C 12 C 12 C 21 detg A = 1 + G 12 C 21 + G 21 C 12 C 12 C 21 detg KEH Plantwide 3 9

10 3.2 and control-loop interactions Closed-loop stability The characteristic equation The closed-loop stability is determined by the characteristic equation. For the 1 1/2 2 pairing, it is A = 1 + G 11 C 11 + G 22 C 22 + C 11 C 22 detg = 0 and for the 1 2/2 1 pairing, it is A = 1 + G 12 C 21 + G 21 C 12 C 12 C 21 detg = 0 For a (multiloop SISO) n n system Y = GG, U = C(R Y) where U, Y, and R are n-vectors, G and C are n n matrices (C having one nonzero element in each row and column if multiloop SISO control is used), the characteristic equation is A = det I + GG = 0 The closed-loop system is stable iff all roots of the characteristic equation are in the complex left-half plane (LHP) KEH Plantwide 3 10

11 3.2 and control-loop interactions Closed-loop stability Example [SEMD] Consider a 2 2 system with the transfer function matrix G = 2 10s s s s+1 From a steady-state point of view, the 1 1/2 2 pairing is suggested for multiloop SISO control. (Why?) Consider multiloop SISO control with two P-controllers and the 1 1/2 2 pairing, i.e., C = k k 22 The characteristic equation is A = 1 + 2k 11 10s+1 + 2k 22 10s+1 + k 11k s s s s+1 = 0 For s 1 and s 0.1, this can in the usual way be rewritten as KEH Plantwide 3 11

12 3.2.2 Closed-loop stability Example where a 0 s 4 + a 1 s 3 + a 2 s 2 + a 3 s + a 4 = 0 a 0 = 100 a 1 = 20k k a 2 = 42k k k 11 k a 3 = 24k k 22 37k 11 k a 4 = 2k k k 11 k Since the characteristic equation in this case is a pure polynomial in s, the stability can be analyzed by the Routh-Hurwitz method. The stability limits in terms of k 11 and k 22 are shown in the figure. k 11 k 22 KEH Plantwide 3 12

13 3.2.2 Closed-loop stability Example Consider now the 1 2/2 1 pairing with two P-controllers, i.e., C = 0 k 12 k 21 0 A similar analysis as for the 1 1/2 2 pairing gives the stability region showed in the figure to the right. In this case, the stability region with the 1 2/2 1 pairing is much larger, which indicates that it is a better pairing than the 1 1/2 2 pairing. Comments: If controllers with integral action are used, e.g., PI-controllers, stability requires k ii > 0. This would increase the advantage of the 1 2/2 1 paring. Time-delays or higher order dynamics would significantly restrict the maximum allowable values of k ii. k 12 k 21 KEH Plantwide 3 13

14 3. Multivariable 3.3 Decoupling control Consider a 2x2 system described by the model y 1 y = G 11 G 12 u 1 2 G 21 G 22 u 2 The block diagram is a representation of this model. If G 12 and G 21 are small compared to G 11 and G 22, decentralised control can be used. Then y 1 is controlled by u 1 and y 2 is controlled by u 2. If G 12 and G 21 are not small, there are considerable interactions in the system. These should be taken into account in the control system design. One way of doing this is through decoupling. KEH Plantwide 3 14

15 3. Multivariable 3.3 Decoupling control Basic design principle Consider a general MIMO system with the transfer function matrix G(s). The input-output relationship is then y s = G s u(s) where u and y are vectors of inputs and outputs, respectively. We are interested in finding a linear variable transformation yielding u s = D s m s y s = G s D s m s = H(s)m s such that the transfer function matrix H s = G s D s has nice properties for controller design using m as control signals. m D u H G y KEH Plantwide 3 15

16 3.3 Decoupling control Basic design principle In principle, H(s) can be chosen as desired. The decoupler D(s) is then calculated by D s = G 1 s H(s) However, it should be noted all elements of D(s) have to be stable and implementable. This often restricts the choice of H(s). In particular, the following should be observed: If G(s) contains one or more time delays, the inversion of G(s) yields negative time delays, which are impossible to implement. This can be avoided by including time delays in H s that cancel out the negative time delays. If G(s) contains a zero in the right-half complex plane, D(s) may become unstable. This can be avoided by including the same zero in H(s). Often, only steady-state decoupling is chosen, which is obtained by using the steady-state part D(0) of the decoupler. KEH Plantwide 3 16

17 3.3 Decoupling control Basic design principle The next step is to design a multiloop SISO controller, based on H(s), with the control signal m s = C(s) r s y(s) which is the input to D(s). C(s) is a diagonal controller transfer matrix. The block diagram illustrates the implementation for a 2x2 system. r 1 C 11 r 2 C 22 H KEH Plantwide 3 17

18 3. Multivariable 3.3 Decoupling control Complete decoupling If H s is chosen as a diagonal matrix of desired transfer functions, complete decoupling is obtained. This simplifies the controller design. For a 2x2 system, complete decoupling is also called two-way decoupling. The choice of H s such that D s becomes implementable, can be facilitated by solving D s = G 1 s H(s) analytically. For a 2x2 system with H s = H 11(s) 0 0 H 22 (s) this gives where D s = 1 det G(s) G 22 (s)h 11 (s) G 12 (s)h 22 (s) G 21 (s)h 11 (s) G 11 (s)h 22 (s) det G(s) = G 11 (s)g 22 (s) G 12 (s)g 21 (s) From this expression it can be seen which choices of H 11 (s) and H 22 (s) make the elements of D s implementable are sufficiently simple for a convenient controller design KEH Plantwide 3 18

19 3.3 Decoupling control Complete decoupling Example Two-way decoupling of a simple system Consider the system G s = 1 s+2 3 s+2 2s+7 (s+2)(s+1) s+11 (s+2)(s+1) Explicit solution of D s = G 1 s H(s) with a diagonal H s yields D s = 1 (s + 11)H 11 (s) (2s + 7)H 22 (s) 5 3(s + 1)H 11 (s) (s + 1)H 22 (s) H 11 (s) and H 22 (s) can obviously be selected in countless ways. For example, the choices H 11 s = 5, H s s = 5 s+1 yield (2s+7) 1 s+1 D s = 3(s+1) 1 s+11 where the diagonal elements are straight lines and the off-diagonal elements are PD controllers with derivative filtering. KEH Plantwide 3 19

20 3.3.2 Complete decoupling Example If strictly proper decoupler elements are desired, a possible choice is This yields H 11 s = D s = 5 (s+1)(s+11), H 22 s = 1 s+1 3 s+11 1 s+1 1 2s+1 5 (s+1)(2s+7) where all elements are simple first-order systems, which are easy to implement. Also H 11 s and H 22 s are simple, but note that they are never implemented, they are only used for controller design. KEH Plantwide 3 20

21 3.3 Decoupling control Complete decoupling Standard choices of decoupler elements In Example 3.3.1, the first choice of H s resulted in a decoupler, where the diagonal elements were static elements with the gain 1. This means that the blocks D 11 (s) and D 22 (s) are replaced by straight lines in a block diagram (see p. 3 17). More generally, for a 2x2 system the decoupler 1 G D s = 12 (s) G 11 (s) G 21 (s) G 22 (s) 1 corresponds to the choice H 11 s = G 1 22 (s) det G(s), H 22 s = G 1 11 (s) det G(s) This decoupling scheme is known as simplified decoupling (due to the simplicity of D). For this scheme to be feasible, it is required that G 11 and G 22 do not have larger time delays than G 12 and G 21, respectively contain RHP zeroes that are not cancelled out by G 12 and G 21, resp. Also H 11 (s) and H 22 (s) may be complicated for controller design. KEH Plantwide 3 21

22 3.3.2 Complete decoupling Standard choices of decoupler elements If the simplified decoupling scheme with diagonal decoupler elements = 1 is not implementable, there are other possibilities with one decoupler element = 1 in each column of D. For a 2x2 system they are D s = 1 1 G 21 (s) G 22 (s) G 11 (s) G 12 (s) H 11 s = G 1 22 (s) det G(s), H 22 s = G 1 12 (s) det G(s) D s = G 22(s) G 21 (s) 1 1 G 11 (s) G 12 (s) H 11 s = G 1 21 (s) det G(s), H 22 s = G 1 12 (s) det G(s) D s = G 22(s) G 21 (s) G 12 (s) G 11 (s) 1 1 H 11 s = G 1 21 (s) det G(s), H 22 s = G 1 11 (s) det G(s) KEH Plantwide 3 22

23 3.3.2 Complete decoupling Standard choices of decoupler elements Another interesting choice is which corresponds to D s = G 22(s) G 12 (s) G 21 (s) G 11 (s) H 11 s = H 22 (s) = det G(s) This decoupler is always implementable (if the system G(s) is implementable). m 1 m 2 D KEH Plantwide 3 23

24 3. Multivariable 3.3 Decoupling control Partial decoupling Even if the choice of a diagonal H s to make D s implementable and not too complicated is quite flexible, there are situations when it is difficult to find a suitable diagonal H s. Another issue is that H s or D s will contain det G(s). If the system is ill-conditioned, det G(s) will be close to zero. Errors in G(s), even if small, will then have a large relative effect on det G(s). Even the sign of det G(s) could change, giving the elements of H s or D s the wrong sign. Complete decoupling of an ill-conditioned system is thus very sensitive to modelling errors. In this situation, partial decoupling could be preferable. In partial decoupling, H s is chosen as a triangular matrix. The resulting D s matrix may be triangular (but doesn t have to be) contain more elements = 1 than in complete decoupling KEH Plantwide 3 24

25 3.3 Decoupling control Partial decoupling One-way decoupling Partial decoupling of a 2x2 system is called one-way decoupling, because only one interactive relationship is cancelled out by the decoupling. The following structures of a triangular H s are possible: H s = H 11(s) 0 H 21 (s) H 22 (s) which yields D s = 1 det G(s) The particular choice G 22 (s)h 11 (s) G 12 (s)h 21 (s) G 12 (s)h 22 (s) G 21 (s)h 11 (s) + G 11 (s)h 21 (s) G 11 (s)h 22 (s) H s = G 11(s) 0 G 21 (s) det G(s) gives D s = 1 G 12(s) 0 G 11 (s) which is always implementable. Because of H s, control loop 1 1 is probably easier to tune than control loop 2 2. KEH Plantwide 3 25

26 3.3.3 Partial decoupling One-way decoupling H s = H 11(s) H 12 (s) 0 H 22 (s) which yields D s = 1 det G(s) G 22 (s)h 11 (s) G 22 (s)h 12 (s) G 12 (s)h 22 (s) G 21 (s)h 11 (s) G 21 s H 12 s + G 11 (s)h 22 (s) The particular choice gives H s = det G(s) G 12(s) 0 G 22 (s) D s = G 22(s) 0 G 11 (s) 1 which is always implementable. Because of H s, control loop 2 2 is probably easier to tune than control loop 1 1. KEH Plantwide 3 26

27 3.3 Decoupling control Partial decoupling ler tuning Consider a decoupled system y s = H s m(s) with the triangular H matrix H s = H 11 (s) 0 0 H 21 (s) H 22 (s) 0 H 31 (s) H 32 (s) H 33 (s) loop 1 1 can be tuned independently of other control loops since only m 1 affects y 1. Thus, the controller of this loop should be tuned first. loop 2 2 is not affected by m i, i > 2. Thus, the controller of this loop should be the second to be tuned. The interaction from loop 1 1 can be taken into account when loop 2 2 is tuned. loop j j is not affected by m i, i > j. Thus, the controller of this loop should be tuned before loop i i, i > j. The interaction from the already tuned loops can be taken into account when this loop is tuned. And so on to the last control loop. KEH Plantwide 3 27

28 3. Multivariable 3.3 Decoupling control Decoupling by linear combination of inputs Decoupling is a way of combining the controller outputs m(s) to obtain the process inputs u(s) according to u s = D s m(s) If D(s) is invertible, it can also be seen as a way of defining new inputs m(s) as a (linear) combination of the process inputs according to m s = D 1 (s)u s If a steady-state decoupling is desired, this is a convenient way of deriving the decoupler. Consider the steady-state relationship of a 2x2 system, i.e., y 1 y = K 11 K 12 u 1 2 K 21 K 22 u 2 Defining the new inputs m 1 = K 11 u 1 + K 12 u 2, m 2 = K 21 u 1 + K 22 u 2 now yields y 1 y = 1 0 m m, D = K 1 2 KEH Plantwide 3 28

29 3.3 Decoupling control Linear combination of inputs Example Decoupling of a crude-oil distillation column The crude-oil distillation column in the figure can be approximately described by the steady-state model T 1 T 2 T 3 T 4 = k 1 k 2 k k 3 k 3 k 3 0 k 4 k 4 k 4 k 4 u 1 u 2 u 3 u 4 where T i, i = 1 4, are boiling temperatures of different oil fractions, u i are their flow rates, and k i are stead-state gains. All non-zero gains in a row are equal because it is assumed that the flow rates of a side stream and those above the side stream have an additive effect on the boiling temperature of the side stream. This is a reasonable assumption because the flow rates have an additive effect on the internal liquid flow in the column below the side stream. KEH Plantwide 3 29 u 2 u 3 u 4 u 1 T1 T2 T3 T4

30 3.3 Decoupling control Linear combination of inputs Example Decoupling of a crude-oil distillation column The crude-oil distillation column in the figure can be approximately described by the steady-state model T 1 T 2 T 3 T 4 = k 1 k 2 k k 3 k 3 k 3 0 k 4 k 4 k 4 k 4 u 1 u 2 u 3 u 4 where T i, i = 1 4, are boiling temperatures of different oil fractions, u i are their flow rates, and k i are stead-state gains. All non-zero gains in a row are equal because it is assumed that the flow rates of a side stream and those above the side stream have an additive effect on the boiling temperature of the side stream. This is a reasonable assumption because the flow rates have an additive effect on the internal liquid flow in the column below the side stream. KEH Plantwide 3 30 u 2 u 3 u 4 u 1 T1 T2 T3 T4

31 3.3.4 Decoupling by linear combination of inputs Example Because the gain matrix is triangular, the system is already partially decoupled. A complete decoupling can easily be obtained by defining new inputs as follows: m 1 = u 1, m 2 = u 1 + u 2, m 3 = u 1 + u 2 + u 3, m 4 = u 1 + u 2 + u 3 + u 4 This gives the model T 1 T 2 T 3 T 4 = k 1 0 k k k 4 m 1 m 2 m 3 m 4 The decoupling is obtained by combining the controller outputs to obtain the flow rates as u 1 = m 1, u 2 = m 2 m 1 u 3 = m 3 m 2, u 4 = m 4 m 3 This corresponds to the decoupler D = KEH Plantwide 3 31 u 2 u 3 u 4 u 1 m 1 m 2 T1 T2 m 3 T3 m 4 T4

32 3. Multivariable 3.3 Decoupling control Decoupling by nonlinear variable combinations If a nonlinear MIMO process model is known, it can be linearized at an operating point and decoupled by standard linear techniques at this operating point. Sometimes it is possible to achieve linearization and decoupling by a nonlinear combination of variables. The process inputs are then calculated as a nonlinear combination of the controller outputs The advantage of this approach is that it applies over the full operating range for which the model applies. This technique has several names: feedback linearization exact linearization global linearization KEH Plantwide 3 32

33 3.3 Decoupling control Nonlinear variable combinations Example Mixing of hot and cold liquid streams Consider the mixing of a hot and a cold liquid stream as illustrated by the figure. It is desired the control the mixing by manipulating the mass flow rates m A and m B to achieve a desired flow rate m C and a desired temperature T C after the mixing point. The temperature of stream A and B is T A and T B, respectively. How should the mixing be controlled? This is simple process, which can be modelled by a mass balance and an energy balance. Mass balance: m A + m B = m C Energy balance: m A T A + m B T B = m C T C These equations can be solved for m C and T C : m C = m A + m B T C = m AT A +m B T B m A +m B KEH Plantwide 3 33

34 3.3.5 Nonlinear variable combinations Example From the previous equations it is clear that it would be perfect to control m C by a variable u m = m A + m B T C by a variable u T = m AT A +m B T B m A +m B This gives the static model m C T = 1 0 C 0 1 which is linear and completely decoupled. The controller outputs are u m and u T, but it is m A and m B that have to be adjusted to control m C and T C. However, these can be calculated from the above variable definitions as m A = u m u T T B T A T B m B = u m u T T A T B T A u m ut The calculation of m A and m B requires values for the stream temperatures T A and T B. These can be obtained by measurement, estimation, or use of nominal ( guessed ) values. In the case of measurement or estimation, this control structure gives automatic disturbance elimination through feed forward. Note that m A and m B are here true process variables, u m and u T are controller outputs. This is contrary to the variable definitions in u = DD. KEH Plantwide 3 34