Multi-Input Multi-output (MIMO) Processes CBE495 LECTURE III
Single-input single-output (SISO) processes
One CV and one MV: No need of pairing
CONTROL OF MULTI INPUT MULTI OUTPUT PROCESSES
Professor Dae Ryook Yang
Spring 2011 Dept. of Chemical and Biological Engineering Korea University
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Multi-input Multi-output (MIMO) processes
Several CVs and several MVs SIMO and MISO The numbers of CVs and MVs are not necessary same. One MV affects all or some of CVs. (process interaction) Pairing: Which MV will control which CV? Control loop interaction: One control loop affects the other control loops. Multiloop control: Multiple SISO controllers are applied. Multivariable control: All MVs will be manipulated to all or some CVs.
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MIMO Process Examples
Inline blending system
Component flows affect both product flow and composition.
Control loop interaction
2x2 control problem
Two-input and two-output process Transfer function (superposition principle for linear process)
Process interactions
Distillation column
Steam and reflux flows affect both top and bottom product compositions.
Multiloop control schemes for 2x2 process
Gas-liquid separator
Gas and liquid product flows affect both tank level and pressure.
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1-1/2-2 pairing
1-2/2-1 pairing
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Open-loop transfer function for Gp22
u1(s) G11 G12 G21 u2(s) G22 + + y2(s) + + y1(s)
Examples
When y1-u1 loop is closed. (automatic mode)
r1(s)
Gc1
u1(s)
G11 G12
+ +
y1(s)
Control loop interactions
G21 u2(s) G22 + + y2(s)
The controller Gc2 should be designed based on the TF which is altered by the other control loop.
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2x2 Multiloop Control
Closed-loop TF
Examples
Examine the poles from
1-1/2-2 pairing 1-2/2-1 pairing
Closed-loop stability depends on both Gc1 and Gc2. If either one or both of Gp12 and Gp21 are zero, the interaction term is vanished. The stability depends on two individual feedback control loops. For example, if Gp21 is zero, Gp12 U2 is a disturbance on y1.
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Pairing of CVs and MVs
Bristols relative gain array (RGA)
Relative gain is a measure of process interaction
Properties of RGA
It is normalized since the sum of the elements in each row or column is one. The relative gains are dimensionless and thus not affected by choice of units or scaling of variables. The value of RG is a measure of steady-state interaction. implies that closed-loop gain is same as open-loop gain. implies that the i-th output is not affected by the j-th input in open-loop mode or closed-loop gain becomes infinity. The value of represents the degree of alteration of openloop gain when other loops are closed. The negative RG implies the closed-loop gain will be different in sign compared to open-loop gain. This pairing is potentially unstable and should be avoided.
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RGA
The ratio between open-loop and closed-loop gains. The open-loop gain: is the gain between yi and uj while all loops are open. The closed-loop gain: is the gain between yi and uj while all other loops are closed. Choose so that it is close to unity or at least not negative for the multiloop pair.
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Proof of
Calculating RGA
2x2 system
All loops are open
Second loop is closed (Y2 is controlled perfectly controlled by U2)
Proof of
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nxn system
Except the i-th controller, all other control loops are closed.
Implications of RGA elements
1. : Indicating that the open-loop gain is identical to the closed-loop gain when Yi and Uj are paired. This loop can be tuned independently. (Ideal case) : Indicating that Yi will not be affected by Uj at all in open-loop mode. This loop should not be paired. : Indicating that the closed-loop gain will become larger than open-loop gain when the other loops are closed. This implies that the loops are interacting and the interaction from other closed loops is smaller if RG is close to one and is larger if RG is close to zero. Avoid the pairing if . : Indicating that the closed-loop gain will become smaller than open-loop gain when the other loops are closed. This implies that a high controller gain should be used for this pair. If some other controllers are open, this loop may become unstable. Avoid the pairing if RG is very high. : Indicating that the closed-loop gain has opposite sign of open-loop gain. This loop should not be paired.
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2. 3.
4.
5.
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Loop pairing rule
Pair input and output variables that have positive RGA elements that are closest to 1.0.
Example
?
Niederlinski index:
?
Niederlinski Index
For open-loop stable and 1-1/2-2//n-n configuration, the multiloop system will be unstable if the Niederlinski index is negative.
The 1-1/2-2/3-3 pairing may be stable, but not sure. When the first loop is open, the subsystem is unstable.
For 2x2 system, it is sufficient and necessary condition.
Therefore, if the is negative, then 1-1/2-2 pairing is unstable.
Such a system that is stable when all loops are closed, but that goes unstable if one of them become open is said to have a low degree of integrity.
Always pair on positive RGA elements that are closest to 1.0 in value, and thereafter use Niederlinskis condition to check the resulting configuration for structural instability
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For nxn systems (n>2), it is only sufficient condition.
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Example: Blending Process
Example: Pure integrator
Some gains become infinity. Replace 1/s as I and get the gains. Calculate RGA while I goes to infinity.
The RGA is dependent on the product composition. If x is greater than 0.5, use 1-1/2-2 pairing, else choose 1-2/2-1 pairing. If x is close to one, FB is very small and FB will not affect the product flow very much, but FB will change the composition significantly. This strategy implies that the larger flow of feeds is selected to control product flow and the smaller flow of feeds is selected to control composition.
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1-1/2-2 pairing is recommended. If I cannot be cancelled, use other approaches suggested by McAvoy.
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Pairing of non-square systems
Under-defined system: more outputs than inputs
Choose same number of outputs as inputs based on the importance of the output variables.
Multiloop Controller Tuning
The multiloop controllers have some performance limitation caused by the interaction.
For highly interacting systems, the performance cannot be improved very much by controller tuning.
Over-defined system: more inputs than outputs
Among possible combinations of inputs with same number of inputs as outputs, choose best subsystem based on the RGA analysis so that the subsystem has least interaction.
Comments on RGA
RGA is only based on the steady-state information. If there are some constraints on inputs, the best RGA pairing may perform poorly. Even though the RGA analysis indicates large interaction, some processes have less interaction dynamically when the time constants are quite different. If there are significant time delays, lags, or even inverse response, the best RGA pairing may perform poorly. Dynamic RGA or some other modification can be used.
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Practical tuning method
With the other loops on manual control, tune each control loop independently for satisfaction Then fine tune the controllers while all loops are on automatic. Detuning method for 2x2 (McAvoy)
Or, use optimization method to find tuning parameters so that the performance criteria such as ITAE is minimized.
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Change of Variables
By using transformations to create combinations of the original inputs and /or outputs of a process, it is possible to obtain an equivalent system with less interaction.
Use of Singular Value Decomposition (SVD)
Find A, B, and K so that the interaction for at least improved . Example:
is eliminated or
The controller should be designed based on new transformed inputs and outputs.
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Decoupling Control
The design objective
The reduction of control loop interactions by adding additional controllers called decouplers to a conventional multiloop control configuration.
Decoupler Design
The effect of u1 on y2 through Gp21 can be cancelled by using a decoupler D21 going through Gp22.
In the same manner,
Theoretical benefits
Control loop interaction are eliminated and the stability of the closed-loop system is determined by the stability characteristics of the individual feedback control loops A set-point change for one CV has no effect on the other CVs Ideal decoupler
Similar to a FF controller May be unstable or physically unrealizable Often implemented as a lead-lag module or a static decoupler
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In practice
Reduction of control loop interactions (not a perfect elimination of interactions due to the imperfect process model and the physical realizability of the decouplers)
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General Case Design
Example
2x2 case
D21(s) has a pole and zero which are very close each other. Thus, D21(s) =-0.5 is a quite reasonable approximation. D12(s) has time lead instead of time delay, which is physically unrealizable.
Time lead is 1 which is relatively small compared to time constants. Thus, neglect time lead and use a lead-lag type.
3x3 cases
Due to the modeling error of the process, the perfect decoupling would not be possible anyway. In many cases, the steady-state decouplers will be beneficial to reduce the control loop interactions.
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Alternative Decoupling Control System
The original configurations
The decoupler uses the controller output signal which may be different from actual input to process due to saturation. It may cause the wind up.
Experimental application to distillation column
Outperforms the conventional multiloop PI control
New approach
Use same input to process It is more sensitive to modeling error
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Other types of decoupling
Partial (one-way) Decoupling
Set some of decouplers zero. This is very attractive if one CV is more important than the others, or one interaction is much weaker than the others or absent. Less sensitive to modeling errors The partial decoupling can provide better control than the complete decoupling in some situations.
Sensitivity of the decouplers
For the imperfect model (static case)
Nonlinear decouplers
If the process is nonlinear or time-varying, the linear decoupler would be worse than conventional multiloop PID schemes. Then, the nonlinear decoupler can be considered.
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If the RG is large, the decoupled process gain becomes very small and large controller gain should be used. (It may cause trouble if there is model error.)
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If there is no modeling error (eij=1)
Regardless of , (no interaction)
Analysis in vector-matrix form (steady state)
If there is no interaction, (
No effect by modeling error ( )
) )
If there are large interaction (
Still large interaction even with decoupler
The RG becomes unity only when eij are ones. If the determinant of K is small
Thus, the high RGA processes may have strong sensitivity to modeling errors.
Small modeling errors will be magnified into very large error in y. Small change in controller output v will also result in large error in y. If the determinant is zero, then some outputs are dependent each other and independent control is impossible. (degeneracy)
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Less obvious ill-conditioned case
Example 1
Most reliable indicator of the interaction
Determinant of K, RGA, or condition number cannot be an reliable indicator of the ill conditioning in a matrix. Singular value: eigenvalue of the matrix KTK The condition number based on singular values is the most reliable indicator of the matrix condition.
No unusual indicator But effect of u1 is much greater than u2.
Example 2
Not ill-conditioned from eigenvalues Easy to decouple
Conclusion
Feasibility of decoupling is directly related to the conditioning of the process gain matrix. Decoupling is only feasible to the degree that the process is well conditioned; it is virtually impossible to achieve decoupling in a poorly conditioned process.
Example 3
Not ill-conditioned from eigenvalues But effect of u1 is much greater than u2.
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