MIMO Systems: Decentralized control

Advanced Modeling and Control

Outline

  • Multivariable process plants
  • Methods of controlling process plant
  • Decentralized Control vs. Centralized Control Systems
  • Issues in decentralized control system
  • Decentralized PID Control Design Methods
  • Decoupling controllers

Real life processes

🤔 how many measurements are there?

  • Real Process often have more than one input and one outputs.
  • Real process has multi-input and multi-output (MIMO)
  • Engineers attempt to select a set of controlled variables from a set of measurement.

Plantwide control hierarchy

  • Several layers: regulatory, supervisory, optimization, scheduling.

  • Regulatory: multi-loop PID controls levels, temperature, flow, pressure. Ensures stability.

  • Supervisory: sends setpoints to the regulatory layer. May be centralized.

  • Optimization: real-time monitoring and optimization provide optimal targets for supervision.

  • Trade-off: plantwide optimization acts slower than local control.

  • Scheduling: inventory and production planning, often offline.

Use fast regulatory PID loops for stability, and let slower supervisory and optimization layers set plantwide targets and schedules.

Plantwide control design flowchart

  • Define control objectives: explicit outcomes and implicit plantwide goals.

  • Select controlled and manipulated variables that support the explicit objectives and fit the plantwide philosophy.

    • directly related to Explicit Control Objectives
    • indirectly related to Implicit Control Objectives

  • Choose the control architecture: centralized or decentralized.

  • For decentralized (multi-loop PID) systems, decide controller pairings first to manage interactions, then design the controllers.

Plantwide control design tasks

  • Plantwide control design is iterative

    • Choose initial controller pairings.

      note: different pairings may require redesign of individual controllers.

    • Design and tune controllers.

    • Simulate and assess plantwide performance.

      Use a full-plant simulation to evaluate the complete design.

    • If targets are not met, revise pairings and retune.

    • Repeat until objectives are satisfied.

Validate controller pairings with full-plant simulation; if KPIs fall short, revise pairings and retune until objectives are met.

Operating objectives

  • Purpose: Define, prioritize, and measure what “good operation” means for the unit or plant.

  • Priority order (use as override order)

    1. Safety and compliance
    2. Equipment protection and mechanical limits
    3. Product quality and on-spec rate
    4. Throughput and stability
    5. Cost and profit

  • Typical objectives

    • Maintain smooth operations.
    • Protect equipment.
    • Maximize yield of the desired product.
    • Ensure safe operation.
    • Minimize operating cost.
    • Meet production specifications.
    • Maximize profit subject to the constraints above.

Examples of control objectives

Objective Indicator variable (unit) Typical limit or target Primary MV(s) Control notes
Avoid flooding/weeping in column Tray dP, % flood [Constraint] dP < limit, flood < 80% Reflux, reboiler duty, feed rate Add dP override selectors on reflux and duty.
Protect pump (no cavitation) NPSH margin (m) [Constraint] Margin > required Suction drum level, recycle valve Use minimum flow recycle and tight level control.
Keep reactor safe T_{reactor} (°C) [CV] < T_{max}, HH trip at T_{trip} Coolant flow, jacket duty Cascade temperature to coolant valve, HH trip independent of PID.
Minimize utilities Steam per ton, kWh per ton [Economic] ↓ vs baseline Reboiler duty setpoint, compressor load Supervisor trims setpoints, bounded by quality and safety.
Achieve throughput Feed rate (t/h) [Throughput] ≥ plan Feed valve, upstream schedule Throughput held unless any constraint controller intervenes.

Examples of control objectives

Objective Indicator variable (unit) Typical limit or target Primary MV(s) Control notes
Avoid flooding/weeping in column Tray dP, % flood [Constraint] dP < limit, flood < 80% Reflux, reboiler duty, feed rate Add dP override selectors on reflux and duty.
Protect pump (no cavitation) NPSH margin (m) [Constraint] Margin > required Suction drum level, recycle valve Use minimum flow recycle and tight level control.
Keep reactor safe T_{reactor} (°C) [CV] < T_{max}, HH trip at T_{trip} Coolant flow, jacket duty Cascade temperature to coolant valve, HH trip independent of PID.
Minimize utilities Steam per ton, kWh per ton [Economic] ↓ vs baseline Reboiler duty setpoint, compressor load Supervisor trims setpoints, bounded by quality and safety.
Achieve throughput Feed rate (t/h) [Throughput] ≥ plan Feed valve, upstream schedule Throughput held unless any constraint controller intervenes.

Mapping objectives to CVs and MVs

  • Pick CVs and MVs that directly support each objective.

  • Put constraint controllers in place first (safety, equipment limits, environmental).

  • Add quality and inventory loops next, then throughput, then cost optimization.

  • Use override selectors so higher-priority loops take control when limits approach.

  • Provide simulation or digital twin checks for KPI impact before moving setpoints.

  • Document alarms and trips (setpoint, alarm, trip) for each safety-critical KPI.

Control strategy must achieve the entire set of objectives

2x2 MIMO process system

  • Inputs: c_1, c_2. Outputs: y_1, y_2.
  • Plant ransfer function matrix
    \mathbf{G}=\begin{bmatrix} G'_{11} & G'_{12} \\ G'_{21} & G'_{22} \end{bmatrix}
  • G'_{12} is the transfer from c_2 to y_1.
  • Interactions: a change in c_1 or c_2 affects both outputs.
  • Cross terms G'_{12} and G'_{21} create loop coupling.
  • Interactions can limit decentralized control performance.

Distillation column: 2x2 MIMO example

  • Two inputs: reflux flow L, steam/boilup S.
  • Two outputs: distillate composition y_D, bottoms composition x_B.
  • Interactions: changes in L or S affect both y_D and x_B.
  • Level loop: weakly coupled to compositions.
  • Design focus: composition controllers; choose pairings/decoupling to handle interactions.

Controller pairings

  • One major task in decentralized control system design → to select controller pairings

  • Controller pairings are chosen based on 3 main factors:

    1. Process Interactions
    2. Dynamic responses
    3. Sensitivity to disturbances

  • Improper controller pairings can lead to severe process interaction → causes poor control performance

  • Factors can be conflicting with each other

    • e.g., pairings that lead to minimum process interactions may exhibit slow dynamic responses
    • it is desirable to have fast dynamic responses.

multi‐loop controllers (decentralized)

  • Two SISO loops, two controllers.
  • Independence is only valid when interactions are negligible.
  • Cross terms G'_{21} and G'_{12} couple the loops and affect tuning.
  • Direct pairings: c_1 \rightarrow y_1, c_2 \rightarrow y_2.
  • Indirect pairings: c_1 \rightarrow y_2, c_2 \rightarrow y_1.
  • Pairing choice changes interaction severity and overall performance.

Multi‐loop controllers – distillation column

  • Reflux flow L controls top composition y_D.
  • Steam/boilup S controls bottom composition x_B.
  • Bottoms flow B maintains level.
  • Top and bottom composition loops are strongly coupled, level loop coupling is weak.
  • Select composition pairings to manage interactions. Compare L\!\to\!y_D, S\!\to\!x_B with the swapped pairing using gains, dynamics, and disturbance paths.

🤔

Can we control the top composition using the steam flow, while the bottom composition using the reflux flow? Why ?

Coupling effect of loop 2 on y_1

  • Change y_{1,\mathrm{sp}} → controller G_{C1} moves c_1.
  • c_1 impacts y_1 via G'_{11} and y_2 via G'_{21}.
  • y_2 deviates → G_{C2} adjusts c_2.
  • c_2 impacts y_2 via G'_{22} and y_1 via G'_{12}.
  • Loop 1 reacts again to the cross effect on y_1.
  • The loops iterate until y_1 and y_2 settle, if no further setpoint or disturbance changes occur.

Multi‐loop controllers – distillation column

  • Reflux L holds the distillate composition y_D; changing L also disturbs the bottoms composition x_B.

  • The bottoms loop reacts: adjust steam/boilup S to bring x_B back to its setpoint.

  • That change in S disturbs y_D; the reflux loop trims L until y_D returns to setpoint.

Relative Gain Array - steady-state interaction

  • Relative Gain Array (RGA) is a steady-state interaction measure for MIMO systems.

\Lambda=K\circ (K^{-1})^{\mathsf T}

  • K is the steady-state gain matrix of linearized plant. Its entries are the open-loop gains

  • K^{-1} is the inverse of K

  • (\cdot)^{\mathsf T} is transpose,

  • \circ is the Hadamard (element-wise) product.

  • The RGA element is thus \lambda_{ij}=k_{ij}\,(K^{-1})_{ji}.

  • For a 2×2 system

\Lambda =\begin{bmatrix}\lambda_{11}&\lambda_{12}\\\lambda_{21}&\lambda_{22}\end{bmatrix}

\lambda_{11}=\lambda_{22}=1-\frac{k_{12}k_{21}}{k_{11}k_{22}} \lambda_{12}=\lambda_{21}=\frac{k_{12}k_{21}}{k_{11}k_{22}}

  • Properties
  • Row and column sums: \sum_j \lambda_{ij}=1 \qquad \sum_i \lambda_{ij}=1
  • For 2×2, the diagonal terms are equal: \lambda_{11}=\lambda_{22}; \quad \lambda_{11}+\lambda_{12} = 1; \quad \lambda_{21}+\lambda_{22} = 1

Relative Gain Array - steady-state interaction

  • For 2×2, only one element requires evaluation: \lambda_{11} = \frac{ \overbrace{ \left(\dfrac{\Delta y_{1}}{\Delta c_{1}}\right)_{c_{2}} } ^{ {\color{#9f1853} \text{main effect (}2^{\text{nd}}\text{ loop open)} } } }{ \underbrace{ \left(\dfrac{\Delta y_{1}}{\Delta c_{1}}\right)_{y_{2}} }_{ {\color{#9f1853} \text{main + coupling (}2^{\text{nd}}\text{ loop closed)} } } }
  • RGA for higher-order MIMO (3×3, 4×4)
    • Definition stays the same: \Lambda = K \circ (K^{-1})^{\mathsf T}
    • Key properties for n\times n:
    • Row sums and column sums equal 1.
    • Diagonals need not be equal and \Lambda need not be symmetric.
    • Scale invariant, but operating-point dependent.
  • RGA does not change if you rescale variables or change units
  • RGA needs to be recomputed when the operating point shifts

RGA: coupling effect (interpretation of \lambda_{11})

  • \lambda_{11}=1 — no coupling.

  • 0<\lambda_{11}<1 — coupling in the same direction as the main effect; severity grows as \lambda_{11}\!\downarrow.
    At \lambda_{11}=0.5: very severe.

  • \lambda_{11}>1 — coupling in the opposite direction; severity grows as \lambda_{11}\!\uparrow.

  • \lambda_{11}<0 — sign reversal / very strong coupling; risk of closed-loop instability → avoid that pairing.

  • Choosing pairings for n\times n:

    • Select one element per row and column (a permutation) with positive \lambda_{ij} near 1.
    • Practical rule: minimize cost matrix C_{ij}=|\lambda_{ij}-1| with an assignment method (for example Hungarian), or use a greedy pick (per row choose the available column with the smallest C_{ij} and \lambda_{ij}>0).
  • Negative or near-zero \lambda_{ij} indicates problematic pairings.
  • Many large off-diagonals in a row or column imply strong interaction → consider decoupling or multivariable control.

RGA calculation

  • Given a 2\times2 transfer-function matrix (FOPTD elements):

    G(s)= \begin{bmatrix} \dfrac{k_{11}e^{-\theta_{11}s}}{\tau_{11}s+1} & \dfrac{k_{12}e^{-\theta_{12}s}}{\tau_{12}s+1} \\[6pt] \dfrac{k_{21}e^{-\theta_{21}s}}{\tau_{21}s+1} & \dfrac{k_{22}e^{-\theta_{22}s}}{\tau_{22}s+1} \end{bmatrix}

  • Steady-state gain matrix:

    K=\begin{bmatrix}k_{11}&k_{12}\\k_{21}&k_{22}\end{bmatrix}

  • Relative Gain Array:

    \Lambda=K\circ (K^{-1})^{\mathsf T} =\begin{bmatrix}\lambda_{11}&\lambda_{12}\\\lambda_{21}&\lambda_{22}\end{bmatrix}

  • Diagonal element:

    \lambda_{11} =\frac{k_{11}}{\underbrace{k_{11}}_{\text{main}}-\underbrace{\dfrac{k_{12}k_{21}}{k_{22}}}_{\text{coupling}}} =\frac{1}{1-\dfrac{k_{12}k_{21}}{k_{11}k_{22}}}

  • MATLAB:

    Lambda = K .* (inv(K)).';
% numerically safer but computationally expensive
% Lambda = K .* (pinv(K)).';

Activity 4 (i, ii, iii): Steady-state RGA

RGA analysis

  • Good for: assessing the steady-state coupling effect of a configuration.

  • Applicable when input/output relationships have the same general dynamic behaviors.

  • Pairing rules

    • Prefer \lambda_{ij} closest to 1 (weakest interaction).
    • Avoid \lambda_{ij}\le 0 (sign reversal / integrity risk—one loop failure can destabilize the other).
    • Ensure a one-to-one mapping (one element per row and column).

  • Caution: can be misleading if transfer-function dynamics are significantly different.

  • Bottom line: RGA is a steady-state measure of process interaction.

Impact of dynamic behavior

  • Process Transfer Functions

G_{11}(s) = \frac{1.0}{100s+1}, \quad G_{12}(s) = \frac{0.3}{10s+1}

G_{21}(s) = \frac{-0.4}{10s+1}, \quad G_{22}(s) = \frac{2.0}{100s+1}

  • Steady-State Gain Matrix

K = \begin{bmatrix} 1.0 & 0.3 \\ -0.4 & 2.0 \end{bmatrix}

  • Relative Gain Array (RGA)

\lambda_{11} = \frac{1}{1 - \frac{K_{12}K_{21}}{K_{11}K_{22}}} = \frac{1}{1 - \frac{(0.3)(-0.4)}{(1.0)(2.0)}} = 0.94

\Lambda = \begin{bmatrix} 0.94 & 0.06 \\ 0.06 & 0.94 \end{bmatrix}

Impact of dynamic behavior

Direct vs reverse pairing

  • RGA suggests direct pairings: u_1 \to y_1, u_2 \to y_2.
  • But time-domain simulations show reverse pairings outperform direct pairings.
  • Dynamic responses of two transfer functions differ (time constants 100 s vs 10 s).
  • Steady-state RGA alone may be misleading; dynamic RGA is needed.

Dynamic RGA

  • When transfer function dynamics differ significantly, steady-state RGA can be misleading.

  • Dynamic RGA evaluates interaction strength as a function of frequency ω.

  • Helps identify correct pairings depending on operating frequency range.

  • Mathematical Formulation

\lambda_{11}(\omega) = \frac{1}{ 1 - \frac{|G_{12}(i\omega)||G_{21}(i\omega)|}{|G_{11}(i\omega)||G_{22}(i\omega)|} }

  • For a first-order process:

|G(i\omega)| = \frac{K_p}{\sqrt{\tau_p^2 \omega^2 + 1}}

  • For the example system:

\lambda_{11}(\omega) = \frac{1}{1 + \frac{16.7(100^2\omega^2 + 1)}{100^2\omega^2 + 1}}

Pairing Considerations

  • Choose pairing between manipulated and controlled variables that results in the least process interactions.

    • Use steady-state RGA for comparable dynamics.
    • Use dynamic RGA for dissimilar dynamics.

  • Choose pairing that leads to quick response of controlled variable to manipulated variable (fast dynamic criterion).

    • Example:

      \frac{Y_1}{U_1} = \frac{2 e^{-s}}{10s+1}, \quad \frac{Y_1}{U_2} = \frac{2 e^{-3s}}{12s+1}

      Prefer U_1 \sim Y_1 because U_1 responds faster. U_2 has longer deadtime, which reduces achievable control performance.

  • Choose pairing that is least sensitive to disturbance.

  • Key Insights

    • Direct pairings can suffer at higher frequency if dynamics are mismatched.
    • Reverse pairings may be more effective at higher frequency, even if steady-state RGA suggests otherwise.
    • Rule of thumb:
      • Low frequency operation → direct pairings
      • High frequency operation → reverse pairings

Sensitivity to disturbances

  • Each configuration has a different sensitivity to a disturbance.

  • Thick and thin line represent the results of different configurations.

  • Notice that, the configuration with thick line is less sensitive to disturbance than the one with thin line.

  • The one that is less sensitive to disturbance is a more efficient configuration (or pairings).

  • Less sensitive to disturbance is also good because it could lead to lower control action required.

Example ‐ configuration selection for a C3 splitter

Configuration RGA (λ₁₁) Comment
(L, B) 0.94 Least interactions
(L, V) 25.3 Severe interactions – coupling effect opposite to main effect
(L/D, V/B) 1.70 Mild interactions
(D, V) 0.06 Mild interactions – coupling effect in same direction as main effect; main effect weaker than coupling effect

(L,V) configuration applied to the C3 splitter

  • Reflux flow L is used to control top composition
  • Steam flow S is used to control bottom composition
  • Steam flow directly affects vapor flow V in the column → therefore V is used to control the bottom composition
  • This arrangement is known as the (L,V) configuration

Reflux ratio applied to the overhead of the C3 splitter

  • Ratio controller is used where the wild stream is the distillate flow, while the reflux stream is the manipulated flow
  • Thus, reflux ratio (L/D) is used as manipulated variable by the top composition controller
  • For the bottom composition controller, the boilup ratio (V/B) can be used as manipulated variable
    • Bottom flow B is wild stream
    • Steam flow S is the manipulated stream (S is directly related to vapor flow
  • Thus, this is called the (L/D, V/B) configuration

Other configurations

  • (L,B) configuration
    • Reflux flow L is used to control top composition
    • Bottom flow B is used to control bottom composition
  • (D,V) configuration
    • Distillate flow D is used to control top composition
    • Steam flow (related directly to V) is used to control bottom composition

Configuration selection of C3 splitter

  • L, L/D and V are the least sensitive to feed composition disturbances

  • L and V have the most immediate effect on the product compositions

  • L/D and V/B give an intermediate effect

  • D and B yield the slowest response

  • Each configuration involves conflicting factors

    • Example: (L,V) is the least sensitive to disturbance and has fast dynamic response
    • But it exhibits the most severe process interaction (λ11 = 25.3)

  • Therefore, simulation is required to evaluate configuration performance

Control performance

Configuration IAE for Overhead IAE for Bottoms
(L, B) 0.067 1.49
(L, V) 0.250 13.3
(L/D, V/B) 0.095 2.00
(D, V) 0.098 1.91
  • (L,B) configuration gives the smallest IAE for the overhead controller
  • Indicates best top controller performance
  • Also corresponds to the configuration with least interactions

Analysis of configuration selection example

  • (L,V) is the worst configuration
    • It is the least susceptible to disturbances and the fastest acting configuration
    • But it is the most coupled (λ11 = 25.3)
  • (D,V) has an RGA of 0.06 but shows decent control performance
    • D has slow dynamic, V has fast dynamic
    • If both had fast or both had slow dynamics, this configuration might show poor performance
  • (L,B) is the best configuration
    • Provides good decoupling
    • Overhead product is most important

Multi‐loop (decentralized) pid controller design

There are six main categories of methods to design multi-loop PID control systems:

  • Detuning
  • Sequential loop closing
  • Independent tuning
  • Simultaneous tuning
  • Optimization
  • Relay auto-tuning

Key Idea

Each method offers a different balance of: - Simplicity vs. performance
- Handling of interactions
- Practical implementation effort

Detuning method

  • Step 1: An individual controller is tuned according to an existing single-input and single-output tuning formula, e.g., classical Ziegler-Nichols and Skogestad IMC
    • Example: consider 2 multi-loop PID controllers
      G_{c1} = K_{c1} \left( 1 + \frac{1}{\tau_{I1} s} + \tau_{D1} s \right), \quad G_{c2} = K_{c2} \left( 1 + \frac{1}{\tau_{I2} s} + \tau_{D2} s \right)
  • Step 2: Detune each controller by a factor F
    • Detuned controllers
      G'_{c1} = \frac{K_{c1}}{F} \left( 1 + \frac{1}{\tau_{I1} s} + \tau_{D1} s \right), \quad G'_{c2} = \frac{K_{c2}}{F} \left( 1 + \frac{1}{\tau_{I2} s} + \tau_{D2} s \right)
  • Step 3: Evaluate the closed-loop responses – if not satisfied then readjust F

Example 2x2 MIMO – Wood and Berry (WB) column

  • The Wood and Berry column is represented by the transfer function matrix

G = \begin{bmatrix} \frac{12.8 e^{-s}}{16.7s + 1} & \frac{-18.9 e^{-3s}}{21s + 1} \\ \frac{6.6 e^{-7s}}{10.9s + 1} & \frac{-19.4 e^{-3s}}{14.4s + 1} \end{bmatrix}

  • RGA analysis

\Lambda = \begin{bmatrix} 2.0094 & -1.0094 \\ -1.0094 & 2.0094 \end{bmatrix}

  • Recommended pairings: U_1 \rightarrow Y_1 and U_2 \rightarrow Y_2 (direct pairings)
  • Direct pairings ensure RGA elements are positive
  • Always avoid negative RGA pairings

WB column - detuning method

  • Apply Ziegler-Nichols tuning (Matlab Control System Designer) to design two PID controllers

  • Design PID 1 based on
    g_{11} = \frac{12.8 e^{-s}}{16.7s+1}
    G_{c1} = 1.2895 \left( 1 + \frac{1}{2s} + 0.4602s \right), \quad GM = 5.02 \, dB, \, PM = 34.9^\circ

  • Design PID 2 based on
    g_{22} = \frac{-19.4 e^{-3s}}{14.4s+1}
    G_{c2} = -0.2548 \left( 1 + \frac{1}{5.6s} + 1.4s \right), \quad GM = 4.88 \, dB, \, PM = 40.1^\circ

WB column - detuning method

  • If F = 2
    G'_{c1} = 0.6448 \left( 1 + \frac{1}{2s} + 0.4602s \right), \quad G'_{c2} = -0.1274 \left( 1 + \frac{1}{5.6s} + 1.4s \right)

  • If F = 5
    G'_{c1} = 0.2579 \left( 1 + \frac{1}{2s} + 0.4602s \right), \quad G'_{c2} = -0.0510 \left( 1 + \frac{1}{5.6s} + 1.4s \right)

Sequential loop (SQL) closing

  • Steps
    1. Choose the fast loop first, before the slower one
    2. Design the controller for the faster loop using its transfer function
    3. Close the fast loop (activate controller)
    4. Derive a linearized model for the slower loop with the fast loop already closed
    5. Design the second controller using the linearized model
    6. Close the second (slower) loop
  • For additional loops, repeat steps 4 to 6

The order of loop closing is critical as it strongly influences overall control performance

WB column – SQL method

  • Step 1: Identify which loop is faster by comparing open-loop step responses of g_{11} and g_{22}

  • Use Matlab commands:

  s = tf('s');
  g11 = 12.8*exp(-s)/(16.7*s + 1);
  g22 = -19.4*exp(-3*s)/(14.4*s + 1);
  step(g11, g22);
  • From the step responses:
    • g_{22} has a shorter settling time (59.3 s)
    • g_{11} has a longer settling time (66.3 s)
    • Therefore, loop 2 is faster than loop 1

Sequential loop closing example

  • use skogestad imc (simc) tuning in matlab control system designer

  • design pid 2 using g_{22}

    g_{c2} = -0.1423 \left( 1 + \frac{1}{16.5s} + 1.3636s \right)

    gm = 9.71 \, db, \quad pm = 74.8^\circ

  • apply a 1-unit step change in r1 and plot t against y1

  • from this step response, obtain the fopdt parameters (to be used in the next step)

Sequential loop closing example

  • from the step response of y_1:
    • delay: \theta = 1
    • time constant: \tau_p = \dfrac{23 - 2}{4} = 5.25
    • gain: k_p = 6.53

  • linearized model for loop 1:

    g'_{11} = \frac{6.53 \, e^{-s}}{5.25s + 1}

Sequential loop closing example

  • design pid 1 using the linearized model g'_{11}

    g'_{11} = \frac{6.53 e^{-s}}{5.25s + 1}

    g_{c1} = 0.4405 \left( 1 + \frac{1}{5.7s} + 0.456s \right)

    gm = 9.92 \, db, \quad pm = 74.6^\circ

  • the plots show comparison between sql tuning and the previous detuning method

  • sql demonstrates substantial improvement in closed-loop performance compared to detuning

Independent tuning method

  • Independent tuning method relies on the Effective Open-Loop Transfer Function (EOTF)

  • For a 2×2 MIMO transfer function matrix:

    G = \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix}

  • Reduce to decentralized form:

    G = \begin{bmatrix} g_{e11} & 0 \\ 0 & g_{e22} \end{bmatrix}

  • EOTFs:

g_{e11} = g_{11} - \frac{g_{12} g_{21}}{g_{22}}, \quad g_{e22} = g_{22} - \frac{g_{12} g_{21}}{g_{11}}

  • Controllers G_{c1} and G_{c2} are designed independently based on g_{e11} and g_{e22}

  • Existing SISO tuning formulas can be used

  • Next, we illustrate application of independent tuning for the Wood and Berry column example

Multi‐loop pid controllers for WB column – independent tuning method

  • Recall WB column transfer function:

G = \begin{bmatrix} \dfrac{12.8 e^{-s}}{16.7s+1} & \dfrac{-18.9 e^{-3s}}{21s+1} \\ \dfrac{6.6 e^{-7s}}{10.9s+1} & \dfrac{-19.4 e^{-3s}}{14.4s+1} \end{bmatrix}

  • Effective Open-Loop Transfer Functions (EOTFs):

    g_{e11} = \dfrac{12.8 e^{-s}}{16.7s+1} - \dfrac{\left( \dfrac{-18.9 e^{-3s}}{21s+1} \right) \left( \dfrac{6.6 e^{-7s}}{10.9s+1} \right)} {\dfrac{-19.4 e^{-3s}}{14.4s+1}}

    g_{e11} = \dfrac{12.8 e^{-s}}{16.7s+1} - \dfrac{6.43 (14.4s+1) e^{-7s}}{(21s+1)(10.9s+1)}

Multi‐loop pid controllers for WB column – independent tuning method

  • Approximation is required to simplify the EOTF

  • Equalize delays of main and coupling transfer functions

  • Assume general formula:

    g = \dfrac{k_p e^{-(\theta_1+\theta_2)s}}{(\tau_1 s+1)(\tau_2 s+1)} \;\; \approx \;\; \dfrac{k_p e^{-\theta_1 s}}{(\tau_1 s+1)(\tau_2 s+1)(\theta_2 s+1)}

  • Applying to coupling transfer function:

    g_{e11} \approx \dfrac{6.43 (14.4s+1) e^{-s}}{(21s+1)(10.9s+1)(6s+1)}

  • Overall EOTF for loop 1:

    g_{e11} = \left[ \dfrac{12.8}{16.7s+1} - \dfrac{6.43(14.4s+1)}{(21s+1)(10.9s+1)(6s+1)} \right] e^{-s}

  • This simplified transfer functions can be used in Matlab Control System Designer since they contain only a single delay term

Multi‐loop pid controllers for WB column – independent tuning method

  • The second EOTF can be derived in the same manner

    Overall EOTF for loop 2:

    g_{e22} = \left[ \dfrac{-19.4}{14.4s+1} + \dfrac{9.745(16.7s+1)}{(21s+1)(10.9s+1)(6s+1)} \right] e^{-3s}

  • Controller Design using Skogestad IMC Tuning

    • For g_{e11}:

    G_{c1} = 0.775 \left( 1 + \dfrac{1}{8.5s} + 0.471s \right)

    GM = 8.96 \, dB, \quad PM = 75.2^\circ

    • For g_{e22}:

    G_{c2} = -0.16 \left( 1 + \dfrac{1}{9.3s} + 1.258s \right)

    GM = 9.27 \, dB, \quad PM = 78.7^\circ

Multi‐loop pid controllers for WB column – independent tuning method

Summary

  • Decentralized or multi-loop PID control is widely applied in process industries

  • Multi-loop control is often implemented at the regulatory layer, crucial for achieving stability

  • Proper controller pairing is essential in decentralized design to handle process interactions

  • Relative Gain Array (RGA) is used to address the configuration issue

  • Use steady-state RGA when transfer function dynamics are comparable

  • Use dynamic RGA when transfer function dynamics differ significantly

Correct pairing and RGA analysis are critical for robust and stable multi-loop control