# Control of MIMO processes. 1. Introduction. Control of MIMO processes. Control of Multiple-Input, Multiple Output (MIMO) Processes

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1 Control of MIMO processes Control of Multiple-Input, Multiple Output (MIMO) Processes Statistical Process Control Feedforward and ratio control Cascade control Split range and selective control Control of MIMO processes Structure of discussion: Introduction Interactions Case study: Interaction between flow and temperature on the Instrutek VVS-400 Heating and Ventilation rig Multiloop control strategy Control of MIMO processes Pairing of controlled and manipulated variables Strategies for dealing with undesirable closed loop interactions Virtual laboratories Questions and answers 2. Introduction Two control approaches Process plants have many variables that need to be controlled. The engineers must provide: the necessary sensors adequate manipulated variables decide how the controlled variables and manipulated variables are paired (they will also be linked through the controller design). Most of the techniques learned for single loop systems still apply. Reference: Marlin, T.E. (2000). Process control, Chapter

2 2. Interactions Interactions (continued) In practical control problems, there typically are a number of process variables which must be controlled and a number which can be manipulated. Example: as-liquid separator Reference: Seborg, D.E. et al. (2004). Process dynamics and control, Chapter Interactions. Single-Input, Single-Output (SISO) Process Example: Distillation column Multi-Input, Multi-Output (MIMO) Process 2x2 nxn 7 Controlled Variables: x D, xb, P, hd, and hb Manipulated Variables: D, B, R, Q,and Q D B Note: Possible multiloop control strategies 5! 20 8

3 Interactions. Thus, we are interested in characterizing process interactions and selecting an appropriate multiloop control configuration. If process interactions are significant, even the best multiloop control system may not provide satisfactory control. In these situations, there are incentives for considering multivariable control strategies. 3. Case study: Interaction between flow and temperature on the VVS- 400 heating and ventilation rig Multiloop control: Each manipulated variable depends on only a single controlled variable, i.e., a set of conventional feedback controllers. Multivariable Control: Each manipulated variable can depend on two or more of the controlled variables. How can we determine how much interaction exists? One method is to use empirical modelling. 9 0 Investigation into process interactions The inputs to both processes were held constant and allowed to settle. Then one of the process inputs underwent a step change. The output of the other process was observed to see if this change had any effect on it. Investigation into process interactions The left hand plot shows when the temperature process input (i.e. heater setting) is held constant (so that the temperature measured is 0.45 or 45 C) and the flow process input (i.e. fan speed) undergoes a step change (from 5% to 75% of full range), the output temperature measured reduces considerably (to 0.3 or 3 C). The right hand plot shows that when the flow process input (i.e. fan speed) is held constant and the temperature process input (i.e. heater setting) undergoes a step change, the output flow measured does not change. This relationship in block diagram form is: Heating and Ventilation System Temperature Process r + y + Interaction 2 Flow Process r2 y2 2

4 Investigation into process interactions This section attempts to estimate the interaction transfer function 2. A large step was applied to the flow process (at r 2 ) and the output of the temperature process (y ) was observed over a range of temperature inputs (at r ). The alternative tangent and point method was used to approximate the process as a first order lag plus time delay (FOLPD) model. The figure outlines this method of identification. Result of one of the tests 3 The left-hand plot shows a test carried out with a constant heater setting of 70%. The flow process receives a step input at 00seconds. The output flow change is large (20-75%). This results in a change in the temperature output. The right-hand plot displays a zoomed plot of the temperature output with the alternative tangent and point method applied. This test was carried out at three different constant temperature inputs (30, 50, and 70 per cent), as the process is nonlinear. 4 Transfer functions developed: Low heater setting (30% of max.) Medium heater setting (50% of max). High heater setting (70% of max). All test results e (s) + 63s 8s 0.8e (s) + 85s 0.24e (s) + 70s An inverse relationship exists between the flow process input and measured temperature output, as expected The effect of the interaction is more noticeable at higher heater settings (again, as expected). 7s 6s 5 4. Multiloop control strategy Typical industrial approach Consists of using n standard feedback controllers (e.g., PID), one for each controlled variable. Control system design. Select controlled and manipulated variables. 2. Select pairing of controlled and manipulated variables. 3. Specify types of feedback controllers. Example: 2 x 2 system Two possible controller pairings: U with Y, U 2 with Y 2 or U with Y 2, U 2 with Y Note: For n x n system, n! possible pairing configurations. 6

5 Block diagram 2x2 process Control loop interactions Process interactions may induce undesirable interactions between two or more control loops. Example: 2 x 2 system: Control loop interactions are due to the presence of a third (hidden) feedback loop. 7 Problems arising from control loop interactions i. Closed-loop system may become destabilized. ii. Controller tuning becomes more difficult Control of MIMO processes Control of MIMO processes 9 20

6 Control of MIMO processes Control of MIMO processes From block diagram algebra 2 Example 23 24

7 Control of MIMO processes Control of MIMO processes Control of MIMO processes 27 28

8 Control of MIMO processes Control of MIMO processes Control of MIMO processes 6. Pairing of controlled and manipulated variables We have seen that interaction is important. It affects the performance of the feedback control systems. Do we have a quantitative measurement of interaction? The answer is yes, we have several. We will concentrate on the Relative ain Array (RA). 3 32

9 Relative gain array This method, developed by Bristol (966), is used to determine the best pairing of controlled and manipulated variables for MIMO processes. The method is based solely on steady state information of the process. Relative gain array Take m and m 2 to be the inputs (or manipulated variables) and y and y 2 to be the outputs (or controlled variables) of the temperature and flow processes respectively. m Process y The VVS-400 heating and ventilation process (for example) has two inputs and two outputs (for flow and temperature). 33 m2 (constant) Assuming m 2 is constant, a step change bin input of magnitude m is introduced. y is the corresponding output change. So then the open loop gain between m and y is y m m cons tan t 2 y2 34 Relative gain array Now consider the loop gain between y and m when m 2 can vary (in a feedback loop controlling the other output, y 2 ): m y Relative gain array Now, the relative gain between y and m, λ, is defined by λ ( y / m ) m2const. ( y / m) y const. 2 Direct Effect Direct + Indirect Effect r2 + - C2 m2 y2 (Constant) The controller C2 attempts to hold y 2 constant; m 2 must change for this to happen, provoking a change in y (y ); this change is a result of the direct effect (from m ) and the indirect effect (from m 2 ). The open loop gain is then y m 35 y2cons tan t Similarly, the remaining relative gains may be defined as follows: Relative gain (m 2 y ): Relative gain (m y 2 ): Relative gain (m 2 y 2 ): λ λ λ 2 2 ( y / m ) 2 mconst. ( y / m2 ) y const. ( y / m ) 2 m2const. ( y2 / m) y const. ( y / m ) 2 2 mconst. ( y2 / m2 ) y const. 2 36

10 Relative gain array The relative gains are arranged into a relative gain array (RA): It can be shown that λ λ λ 2 λ + λ λ + λ λ + λ λ + λ 2 Then, for a 2x2 process Examples: λ : λ λ λ λ λ λ 2 2 λ λ λ λ i.e. direct effect direct + indirect effect, Pairings: (y, m ); (y 2, m 2 ). 37 λ 0: 0 λ 0 Relative gain array i.e. no direct effect. Pairings: (y, m 2 ); (y 2, m ). 0 < λ < : i.e. the control loops will interact.2 λ λ λ λ Pairings: (y, m ); (y 2, m 2 ). Pairings: (y, m 2 ); (y 2, m ). Most severe interaction 0.2 Closing the second loop reduces the gain between y.2 and m ; as λ increases, the degree of interaction becomes more severe λ Relative gain array Opening loop two gives a negative gain between y and m ; closing loop two gives a positive gain between y and m, i.e. the control loops interact by trying to fight each other. In general, y should be paired with m when λ 0.5; otherwise y should be paired with m 2. Provides two pieces of useful information Measure of process interactions Recommendation about the best pairing of controlled and manipulated variables It requires knowledge of process steady state gains only. 39 Relative ain Array (RA) For a 2 x 2 system: Steady-state process model, y + k.m k2. m2 2 k2.m k. m2 y + Where k, k 2, k 2, and k are the steady state gains i.e. y k m m2cons tan t Putting y 2 0: k 2 0 k 2.m + k. m2 i.e. m2. m k k2k 2 y k.m. m k2k 2 Therefore, i.e. y k. m k k k Therefore, y m y2const. k k 2k kk 2 i.e. λ k2k k k 2 λ 40

11 Relative ain Array (RA) Overall recommendation: Pair controlled and manipulated variables so that the corresponding relative gains are positive and as close to one as possible. Example : K K K K2 K Λ Recommended pairing is Y and U, Y 2 and U 2. Example 2: K. 5 2 Λ Recommended pairing is Y with U and Y 2 with U RA can be misleading s s + Steady State RA (0.3) (2) Note that the off-diagonal terms possess dynamics that are 0 times faster than the diagonal terms. As a result, adjustments in u to correct y result in changes in y 2 long before y can be corrected. Then the other control loop makes adjustments in u 2 to correct y 2, but y changes long before y 2. Thus adjustments in u cause changes in y from the coupling long before the direct effect s s + 7. Strategies for dealing with undesirable closed loop interactions. "Detune" one or more feedback controllers. 2. Select different manipulated or controlled variables e.g. nonlinear functions of original variables 3. Use a decoupling control scheme. 4. Use some other type of multivariable control scheme. r r2 Decoupling Basic Idea: Use additional controllers to compensate for process interactions and thus reduce control loop interactions Ideally, decoupling control allows setpoint changes to affect only the desired controlled variables. Typically, decoupling controllers are designed using a simple process model (e.g., a steady-state model or transfer function model). + e + e2 C2 - m C + + m D2 D2 + m m y y2 D2( s) 2 D

12 Decoupling Consider the 2x2 process shown in the figure (i.e. y is coupled with m and y 2 is coupled with m 2 ). Assume both outputs are initially at their desired values, and a disturbance causes the controller of loop 2 to vary the value of m 2. This will then cause an unwanted disturbance in loop and hence cause y to vary from its desired value. iven that; y.m +. 2 m2 then to keep y constant (i.e. Y (s) 0), m must be adjusted by (. 2 / ). m2 So by introducing a decoupler of transfer function ( 2 / ), the interacting effect of loop 2 on loop is eliminated. The same argument can be applied to the effect of loop on loop 2, and hence yield a decoupler with transfer function ( ). The decouplers D 2 and D 2 may not always be physically realizable especially when dealing with models with different time delays. For example, sometimes it may occur that the ideal decoupler has a time advance term (i.e. e +2s ), which is obviously impossible to Decoupling implement. A less ambitious approach to full decoupling but still very effective is static decoupling. This is where the decouplers are designed based on the steady state process interactions only. The design equations for the decouplers can be adjusted by setting s 0, i.e. the process transfer functions are simply replaced by their corresponding steady state gains, so that D K / and D K /. Since static decouplers are 2 2 K merely constants they are always physically realizable and easily implemented. 2 / r r2 + e m C e2 C2 - m D2 D2 2 2 K + m m y y2 2 D2 2 D2 Example 47 48

13 Tutorial example

14 Without decoupling. Servo and regulator responses step input on r Without decoupling. Servo and regulator responses step input on r2 Best decoupling (one) block diagram 55 56

15 Static decoupling (one) block diagram

16 Tutorial example

17 8. Virtual laboratories Other virtual laboratories ECOSSE Control HyperCourse Virtual Control laboratory The following virtual laboratories under Multiple Loop Control Systems are particularly relevant: - The blending process: deals mainly with (a) the matching up of manipulated variables with controlled variables using RA analysis, (b) viewing control loop interaction - Continuous Still (distillation column example). Control Systems Design, Australia See Real-Life Case Studies; of interest are - Distillation column control - Four coupled tank apparatus - Shape control (Rolling-mill shape control) National University of Singapore There is an interesting on-line experiment dealing with the PID control of a MIMO coupled-tank apparatus; details at

18 9. Questions and Answers Question ives the best performance Answer No, we cannot ensure that this simple approach gives the best control performance. Is the only possible approach No, another possibility is the centralized controller approach (see lecture notes). We often select multiloop control design because it ives the best performance Is the only possible approach Retains the well known PID controller Is simple to implement by the engineer and to use by the operator. Retains the well known PID controller Yes, the familiar and reliable PID algorithm can be used for each of the loops. Is simple to implement by the engineer and to use by the operator Yes, implementing the PID controller is easy, because the algorithm is programmed in all commercial control systems. Also, operators have experience using single-loop control. Therefore, the multiloop approach is typically used unless improvement is possible with other approaches Question Answer one controlled variable (CV) influences more than one CV No, we are investigating the following question, "When we adjust one manipulated variable, does this affect any other part of the feedback process?" one valve influences more than one CV Process interaction occurs when one controlled variable (CV) influences more than one CV one valve influences more than one CV one valve influences all CV s always exists, there is no need to check. Yes, this is the definition. If no interaction exists, we can design individual feedback loops independently, because all loops will be independent. This is not the case in most realistic processes; therefore, we have to understand interaction and take it into account when we design control systems. one valve influences all CV s No, however, this is close. always exists, there is no need to check No, interaction does not always exist. 7 72

19 Question Answer How process gains change due to interaction No, the process gains, K ij, do not change because of interaction. However, the behaviour of a control system with several control loops can be strongly affected by interaction. All types of interaction No, the relative gain does not distinguish between one-way interaction and no interaction. The relative gain provides an indication of How process gains change due to interaction All types of interaction Two way interaction Deviation of multiloop dynamic behaviour from single loop dynamic behaviour. Two way interaction Yes, the relative gain gives us an indication of the "strength" of two-way interaction. Deviation of multiloop dynamic behaviour from single loop dynamic behaviour No, the relative gain does not provide much information about the dynamic control performance of control systems For the 2x2 system shown, what is true if no interaction exists? All ij (s) 0 2 (s) 2 (s) 0 (s) (s) 0 None of the above Question and Answer Answer: All ij (s) 0. Yes, if all the gains are zero, no interaction exists. However, no feedback control is possible either! 2 (s) 2 (s) 0. Yes, in this case, the two loops shown in the diagram would be independent. (s) (s) 0. Yes, interaction would not be present in this situation. However, the control design in the figure would have poor integrity; the pairing of manipulated and controlled variables should be switched. The continuous stirred tank reactor (CSTR) in the figure has the steady-state gain matrix shown. Which of the multi-loop control designs will function correctly, i.e. could yield zerooffset, stable control? Question Control T with v C and control Ca with v A Control T with v A and control Ca with v C Either of the above will work None of the above will work None of the above No

20 First, calculate the RA: λ k2k k k 2 λ Answer λ (0.0047)(0) (0.0097)( 0.339) Therefore, answer is control T with v C and control Ca with v A. 77

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