1. Bibliographic Information

• Title: Multi-loop PID tuning strategy based on non-iterative linear matrix inequalities
• Authors: Diego José Trica. The author is affiliated with a university in Rio de Janeiro, Brazil.
• Journal/Conference: The paper does not explicitly state the journal or conference it was published in. Based on the formatting and content, it appears to be a formally published article in a control engineering or chemical engineering journal.
• Publication Year: The specific year is not mentioned in the provided text, but the references go up to 2023, suggesting the paper was written in or after 2023.
• Abstract: The paper addresses the challenge of tuning Proportional-Integral-Derivative (PID) controllers in multi-loop (decentralized) control systems, which are common in chemical plants. Traditional methods based on Linear Matrix Inequalities (LMIs) for Static Output Feedback (SOF) tuning often result in a bilinear (non-convex) problem. While iterative LMI (ILMI) approaches exist to solve this, they can be computationally expensive and sensitive to initial conditions. This work introduces a non-iterative LMI strategy. The key innovation is to express the Lyapunov matrix as a function of the controller gain matrices, making only the gains the decision variables. This creates quadratic matrix terms, which are handled using the congruency property and an $\boldsymbol{s}$-procedure with a slack variable, transforming the problem back into a convex LMI form. The method's effectiveness is demonstrated by solving an SOF problem that maximizes the system's stability (decay rate) while satisfying a disturbance rejection constraint ($\mathcal{H}_{\infty}$ norm).
• Original Source Link: The paper provides a relative link (/files/papers/68ef099f58c9cb7bcb2c7efe/paper.pdf) and a GitHub link for associated code: https://github.com/diegotrica/LM I_PIDloops.git.

2. Executive Summary

• Background & Motivation (Why):

  • Core Problem: In industrial settings like chemical plants, complex systems with multiple inputs and outputs (MIMO) are often controlled by a set of simpler single-input, single-output (SISO) PID controllers. This is called a multi-loop or decentralized control system. The main challenge is tuning these PID controllers simultaneously to ensure the entire system is stable and performs well, especially because the control loops interact with each other.
  • Existing Gaps: One powerful modern technique for controller design is using Linear Matrix Inequalities (LMIs), which frame the problem as a convex optimization task called Semi-Definite Programming (SDP). However, when applied to PID tuning (a form of Static Output Feedback or SOF), the problem naturally contains products of unknown variables (the controller gains and the Lyapunov stability matrix), resulting in Bilinear Matrix Inequalities (BMIs). BMIs are non-convex and computationally very hard to solve (NP-hard).
  • Limitations of Prior Work: To overcome the BMI issue, researchers developed iterative LMI (ILMI) methods. These methods linearize the problem and solve it in steps. However, they suffer from significant drawbacks: they can be very slow, their success often depends heavily on a good initial guess for the controller parameters, and they may converge to a suboptimal solution.
  • Paper's Innovation: This paper proposes a novel non-iterative LMI approach that completely avoids the iterative process. It reformulates the problem so that the Lyapunov matrix (a key variable in stability analysis) is no longer an independent decision variable but is instead mathematically defined as a function of the PID controller gains. This clever parametrization converts the difficult BMI problem into a convex LMI problem that can be solved directly and efficiently in a single step.

• Main Contributions / Findings (What):

  • A Novel Non-Iterative LMI Formulation: The primary contribution is a new method to solve the SOF problem for multi-loop PID tuning without iteration. This is achieved by:
    1. Parametrizing the Lyapunov Matrix: The Lyapunov matrix $\bar{P}$ is explicitly defined as a function of the PID gain matrices ($K_P, K_I, K_D$) and a set of new auxiliary matrix variables.
    2. Handling Quadratic Terms: This parametrization leads to quadratic terms in the stability conditions. These are managed using the matrix congruency property and an $\boldsymbol{s}$-procedure, a mathematical technique that relaxes the quadratic terms into linear ones using a slack variable.
  • Reduced Computational Cost: The proposed method is demonstrated to be significantly faster than state-of-the-art ILMI approaches. In the provided examples, it was up to 23 times faster, reducing tuning time from many minutes to just a few seconds or minutes.
  • Effective and Robust Tuning: The method successfully tuned controllers for two challenging chemical process models: the classic Wood and Berry distillation column and a complex Fluid Catalytic Cracker (FCC) unit. The resulting controllers showed performance comparable or superior to those from ILMI and other established tuning methods, particularly in ill-conditioned systems.

3. Prerequisite Knowledge & Related Work

• Foundational Concepts:

  • PID Controller: A PID (Proportional-Integral-Derivative) controller is a feedback control mechanism widely used in industrial control systems. It calculates an "error" value as the difference between a measured process variable and a desired setpoint. The controller attempts to minimize the error by adjusting a control input. It has three components:
    • Proportional ($K_P$): Responds to the current error.
    • Integral ($K_I$): Accumulates past errors to eliminate steady-state offset.
    • Derivative ($K_D$): Responds to the rate of change of the error to improve transient response.
  • Multi-loop (Decentralized) Control: For a system with multiple inputs and multiple outputs (MIMO), instead of designing a complex central controller that handles all variables at once, a simpler approach is to use one PID controller for each input-output pair (e.g., control temperature with a heater, control level with a valve). This is a decentralized architecture. The main difficulty is that adjusting one input can affect multiple outputs, causing loop interactions.
  • State-Space Representation: A mathematical model of a physical system as a set of input, output, and state variables related by first-order differential equations. For a Linear Time-Invariant (LTI) system, it is written as: $$\dot{x}(t) = Ax(t) + B_u u(t) + B_w w(t) \\ y(t) = Cx(t)$$
    • x(t): The state vector (internal variables of the system).
    • u(t): The control input vector.
    • w(t): The disturbance input vector.
    • y(t): The output vector (measured variables).
    • $A, B_u, B_w, C$: System matrices that define the system's dynamics.
  • Lyapunov Stability: A fundamental concept in control theory for assessing system stability. A system is stable if there exists a scalar function (the Lyapunov function, V(x)) that is positive definite (like energy) and its time derivative is negative definite (energy is always decreasing). For an LTI system $\dot{x} = A_{cl}x$, the Lyapunov stability conditions are: $$P > 0, \quad A_{cl}^T P + P A_{cl} < 0$$ Here, $P$ is a symmetric positive-definite matrix called the Lyapunov matrix, and these inequalities are Linear Matrix Inequalities (LMIs) in $P$.
  • Linear Matrix Inequality (LMI): A constraint of the form $F(x) = F_0 + \sum_{i=1}^m x_i F_i > 0$, where $x$ is a vector of decision variables and the $F_i$ matrices are known symmetric matrices. The set of all $x$ satisfying an LMI is a convex set, which makes optimization problems with LMI constraints computationally tractable.
  • Semi-Definite Programming (SDP): A subfield of convex optimization that deals with optimizing a linear objective function over the intersection of the cone of positive semi-definite matrices with an affine space. It is the standard method for solving problems involving LMIs.
  • Static Output Feedback (SOF): A control design where the control law is a static gain applied to the system's outputs: $u(t) = K y(t)$. Finding the optimal gain matrix $K$ is the SOF problem. PID control is a specific, structured form of SOF. The closed-loop system matrix becomes $A_{cl} = A - B_u K C$.
  • Bilinear Matrix Inequality (BMI): When the Lyapunov stability condition $A_{cl}^T P + P A_{cl} < 0$ is written for the SOF problem, it becomes $(A - B_u K C)^T P + P(A - B_u K C) < 0$. This inequality contains product terms like $P B_u K C$, where both $P$ and $K$ are unknown decision variables. Such an inequality is a BMI and is non-convex, making the problem NP-hard.

• Previous Works:

  • Traditional Multi-loop Tuning: Methods like detuning, sequential loop closing, and independent design rely on SISO tuning rules for each loop and often require analytical transfer functions, which are cumbersome for large systems.
  • Iterative Autotuning: Methods like relay feedback autotuning are iterative and can be time-consuming, with no guarantee of convergence.
  • Iterative LMI (ILMI) Approaches (e.g., Wang et al., 2008): These methods are the direct predecessors to the paper's contribution. They tackle the BMI problem by fixing one variable (e.g., $K$) and solving for the other ($P$), then fixing $P$ and solving for $K$, and iterating until convergence. The paper identifies their main flaws: high computational cost, sensitivity to the initial guess (a bad guess can lead to slow convergence or a poor local minimum), and potential failure in specific cases (e.g., systems with integrator dynamics).
  • Other Non-iterative Approaches: Some non-iterative LMI methods exist but are often limited to systems with specific structures or lead to very conservative (i.e., overly safe but low-performance) results. For example, the two-step procedure of Carvalho and Rodrigues (2019) is non-iterative but relies on user-defined weighting matrices, making the tuning process less direct.

• Differentiation: The proposed method is distinct from all previous works because it is both non-iterative and generally applicable. Unlike limited non-iterative methods, it does not require special system structures. Unlike ILMI methods, it avoids iteration entirely by mathematically eliminating the Lyapunov matrix as an independent variable, thereby directly solving a convex problem for the PID gains. This makes it faster and more reliable.

4. Methodology (Core Technology & Implementation)

The core of the paper is a novel mathematical reformulation of the PID tuning problem. The goal is to find the PID gain matrices $K_P, K_I, K_D$ that stabilize the system and meet performance objectives.

• Principles: The central idea is to avoid the bilinear terms that arise from having both the Lyapunov matrix $\bar{P}$ and the controller gain matrix $\bar{K}$ as decision variables. The authors achieve this by creating a mathematical relationship that defines $\bar{P}$ in terms of $\bar{K}$.

• Steps & Procedures:

  Step 1: Augmented State-Space Model for PID Control A standard LTI system cannot directly represent the integral and derivative actions of a PID controller. Therefore, the system state is augmented. The new state vector becomes $\bar{x} = [x, -\int e, -u_D]^T$, where $x$ are the original states, $\int e$ is the integral of the error, and $u_D$ is the filtered derivative action. This results in a larger LTI system described by augmented matrices $\bar{A}, \bar{B}_u, \bar{C}$ and the augmented gain matrix $\bar{K} = [K_P, K_I, K_D]$. The closed-loop dynamics are $\dot{\bar{x}} = (\bar{A} - \bar{B}_u \bar{K} \bar{C})\bar{x} + \dots$.

  Step 2: Parametrization of the Lyapunov Matrix $\bar{P}$ (Lemma 1) This is the key innovation. Instead of treating $\bar{P}$ as an unknown to be solved for, the paper proposes a specific structure for it that is a function of the controller gains.

  • Key Equality: The authors establish the following matrix equality: $$(B_u^T B_u)^{-1} \bar{B}_u^T \bar{P} = \bar{K} \bar{C}$$ This equation links $\bar{P}$ and $\bar{K}$. For this equality to hold, $\bar{P}$ must have a specific structure.
  • Structure of $\bar{P}$: The Lyapunov matrix is defined as: $$\bar{P} = \left[ \begin{array} { c c c } { P _ { P } } & { B _ { u } K _ { I } } & { S _ { D } } \\ { \star } & { K _ { I } ^ { T } K _ { I } } & { 0 } \\ { \star } & { \star } & { K _ { D } ^ { T } \phi _ { D } ^ { - 1 } K _ { D } } \end{array} \right] > 0$$ where $\star$ denotes symmetric elements. The sub-matrix $P_P$ is itself a complex function involving $K_P$, $K_D$, and new auxiliary matrix variables $X_P, Y_P, Z_P$. These new variables are introduced to ensure $P_P$ can represent any symmetric matrix, providing enough degrees of freedom for the optimization.

  Step 3: Transforming the Stability Condition using Congruency The standard Lyapunov stability inequality for the closed-loop system is $\bar{F} = (\bar{A} - \bar{B}_u \bar{K} \bar{C})^T \bar{P} + \bar{P}(\bar{A} - \bar{B}_u \bar{K} \bar{C}) \le 0$.

  • By substituting the key equality from Step 2, this inequality becomes quadratic in the controller gains $\bar{K}$.
  • The authors then show that this quadratic inequality can be factored using the congruency property. They find a matrix $\hat{M}$ (which depends on $K_I$ and $K_D$) such that the inequality can be written as: $$\bar{F} = \hat{M}^T \hat{F} \hat{M} \le 0$$
  • Due to the properties of matrix congruence, this is equivalent to checking if $\hat{F} \le 0$. This step cleverly removes the bilinear terms involving $K_I$ and $K_D$ from the main structure of the inequality.

  Step 4: Relaxation of Remaining Quadratic Terms using an $\boldsymbol{s}$-Procedure The new matrix inequality $\hat{F} \le 0$ is much simpler, but it still contains quadratic terms in $K_P$ and $K_D$.

  • For example, a term like $-C^T K_P^T (B_u^T B_u) K_P C$ appears. This term is quadratic in the decision variable $K_P$.
  • To handle this, the authors use an $\boldsymbol{s}$-procedure with a slack variable. The quadratic term is replaced by a linear term involving a new matrix variable, $\Sigma_P$. For example, $-K_P^T R K_P$ is replaced by $-W_P^{1/2} \Sigma_P W_P^{1/2}$, where $W_P$ depends on a user-defined upper bound for $K_P$.
  • To ensure this replacement is valid and not overly conservative, an additional LMI constraint is introduced to bound the slack variable $\Sigma_P$. The final LMI derived from this procedure is (for the $K_P$ term): $$\left[ \begin{array} { c c } { { \varepsilon _ { P } I - \Sigma _ { P } } } & { { \pi _ { P } ^ { T } } } \\ { { \pi _ { P } } } & { { \frac { 1 } { \tau } I } } \end{array} \right] \geq 0$$ where $\pi_P$ is a normalized term related to $K_P$, $\varepsilon_P$ is a scalar decision variable, and $\tau$ is a parameter of the $\boldsymbol{s}$-procedure (set to 1 in the paper). A similar LMI is derived for the quadratic term involving $K_D$.

  Step 5: Final SDP Problem Formulation With all non-convexities removed, the PID tuning is formulated as a single, convex optimization problem. The specific problem solved in the paper is a Generalized Eigenvalue Problem (GEVP) to maximize the stability decay rate ($\alpha$).

  • Minimize: $\alpha$
  • Decision Variables: $K_P, K_I, K_D$, the parametrization variables ($X_P, Y_P, Z_P$), and the slack variables from the $\boldsymbol{s}$-procedure ($\Sigma_P, \varepsilon_P, \Sigma_D, \varepsilon_D$).
  • Subject to (Constraints):

    1. $\hat{P} > 0$: The parametrized Lyapunov matrix must be positive definite.

    2. $\hat{F} \le 2\alpha \hat{P}$: The $\alpha$-stability LMI.

    3. $\hat{H} \le 0$: An LMI for the $\mathcal{H}_{\infty}$ norm bound (for disturbance rejection).

    4. The LMIs from the $\boldsymbol{s}$-procedure (Step 4).

    5. User-defined upper and lower bounds on the PID parameters ($K_P$, integral time $T_I$, and derivative time $T_D$).

       This sequence of steps transforms a non-convex, hard-to-solve BMI problem into a convex GEVP that can be solved efficiently with standard SDP solvers.

5. Experimental Setup

• Datasets (Case Studies):

  • Case A: Wood and Berry (1973) Distillation Column: This is a classic 2-input, 2-output (2x2) MIMO benchmark problem in process control. The system model is given by a transfer function matrix with time delays. To use LMI methods, which require a state-space model, the time delays were approximated using a semi-discretization technique, resulting in an LTI model with $n=21$ states. It is used to test performance on a well-known, interactive system.
  • Case B: Santander et al. (2022) Fluid Catalytic Cracker (FCC) Unit: This is a much more complex and realistic case study. The model is a nonlinear, large-scale representation of an FCC unit, a core process in oil refineries. The authors linearized the nonlinear model around an operating point to get an LTI state-space model with $n=28$ states and 5 input-output pairs. This system is known to be ill-conditioned, meaning it has both very fast and very slow dynamics, which makes it numerically challenging for control design.

• Evaluation Metrics:

  • Decay Rate ($\alpha$):
    1. Conceptual Definition: This metric quantifies the stability of the closed-loop system. It represents the guaranteed exponential decay rate of the system's energy (Lyapunov function). A more negative value of $\alpha$ means the system returns to its equilibrium state faster after a disturbance. The optimization problem aims to make $\alpha$ as negative as possible.
    2. Mathematical Formula: The objective is to $\min \alpha$ subject to the LMI: $$A_{cl}^T P + P A_{cl} \le 2\alpha P$$
    3. Symbol Explanation: $A_{cl}$ is the closed-loop system matrix, $P$ is the Lyapunov matrix.
  • Computational Time: The wall-clock time in seconds required for the SDP solver to find a solution. This directly measures the efficiency of the proposed method versus the iterative baseline.
  • $\mathcal{H}_{\infty}$ norm ($\|G_{load}\|_{\infty}$):
    1. Conceptual Definition: Measures the system's robustness to external disturbances. It is the maximum "amplification" (gain) from the disturbance input ($w$) to the controlled output ($y$) across all frequencies. A smaller $\mathcal{H}_{\infty}$ norm means better disturbance rejection.
    2. Mathematical Formula: $$\|G(s)\|_{\infty} = \sup_{\omega} \bar{\sigma}(G(i\omega))$$
    3. Symbol Explanation: G(s) is the transfer function from disturbance to output, $\sup$ is the supremum (least upper bound), $\omega$ is frequency, and $\bar{\sigma}$ is the maximum singular value of the matrix $G(i\omega)$.
  • Integral Time Absolute Error (ITAE):
    1. Conceptual Definition: A performance index that measures the quality of a system's transient response to a step change in setpoint or disturbance. It penalizes errors that persist for a long time more heavily than initial errors. Lower ITAE values indicate a better response (less overshoot, faster settling).
    2. Mathematical Formula: $$\text{ITAE} = \int_{0}^{\infty} t |e(t)| dt$$
    3. Symbol Explanation: $t$ is time, and e(t) is the error signal (setpoint - measured output).
  • Settling Time: The time it takes for the system's response to a step input to enter and remain within a specified error band (e.g., $\pm 2\%$) of its final value.

• Baselines:

  • ILMI approach (Wang et al., 2008): This is the main point of comparison, representing the standard iterative LMI technique for SOF-$\mathcal{H}_{\infty}$ design.
  • Other Methods (for Wood and Berry):
    • BHA16: A method by Boyd, Hast, and Åström (2016).
    • GAR21: A frequency-domain method by Garrido et al. (2021).
    • ITAE-SP: A standard single-loop tuning rule based on minimizing ITAE for setpoint tracking.
  • Original Tunings (for FCC Unit): The PI controller settings provided in the original Santander et al. (2022) paper, which were based on the Tyreus-Luyben rules.

6. Results & Analysis

• Core Results:

  Case A: Wood and Berry Distillation Column

  The paper compares the proposed non-iterative method (This Work - TW) against the ILMI approach for both PI and PID controllers, under different disturbance rejection requirements (specified by an upper bound $\gamma$ on the $\mathcal{H}_{\infty}$ norm).

  ---

  Manual transcription of Table 1 from the paper. Table 1: Comparison of PI tunings for the Wood and Berry (1973) distillation column model.

  $\gamma$	Approach	I/O	$K_P^a$	$T_I$ [min]	$\alpha^*$ (optimal)	Comp. time [s]	$\|G_{load}\|_{\infty}^b$	#
  \multirow{4}{*}{10$^3$}	\multirow{2}{*}{TW}	r-D	0.4984	16.67	\multirow{2}{*}{-0.0476}	\multirow{2}{*}{37.1}	\multirow{2}{*}{2.0595}	\multirow{2}{*}{1}
  		v-B	-0.2651	16.67
  	\multirow{2}{*}{ILMI}	r-D	0.4875	16.60	\multirow{2}{*}{-0.0671}	\multirow{2}{*}{183.5}	\multirow{2}{*}{1.9963}	\multirow{2}{*}{2}
  		v-B	-0.2600	16.66
  \multirow{4}{*}{10$^2$}	\multirow{2}{*}{TW}	r-D	0.4368	16.55	\multirow{2}{*}{-0.0476}	\multirow{2}{*}{58.2}	\multirow{2}{*}{1.9182}	\multirow{2}{*}{3}
  		v-B	-0.2535	16.68
  	\multirow{2}{*}{ILMI}	r-D	0.4544	16.68	\multirow{2}{*}{-0.0493}	\multirow{2}{*}{242.2}	\multirow{2}{*}{1.7105}	\multirow{2}{*}{4}
  		v-B	-0.2301	16.70
  \multirow{4}{*}{10}	\multirow{2}{*}{TW}	r-D	0.3225	10.03	\multirow{2}{*}{-0.0266}	\multirow{2}{*}{34.0}	\multirow{2}{*}{1.8432}	\multirow{2}{*}{5}
  		v-B	-0.1517	10.11
  	\multirow{2}{*}{ILMI}	r-D	0.4700	16.64	\multirow{2}{*}{-0.0401}	\multirow{2}{*}{190.6}	\multirow{2}{*}{1.7613}	\multirow{2}{*}{6}
  		v-B	0.2367	16.66

  a $K_P$ in units of [lb/min/wt. frac.]. b Computed by hinfnorm MATLAB command.

  ---

  Manual transcription of Table 2 from the paper. Table 2: Comparison of PID tunings for the Wood and Berry (1973) distillation column model.

  $\gamma$	Approach	I/O	$K_P^a$	$T_I$ [min]	$T_D$ [min]	$\alpha^*$ (optimal)	Comp. time [s]	$\|G_{load}\|_{\infty}^b$	#
  \multirow{4}{*}{10$^3$}	\multirow{2}{*}{TW}	r-D	0.5052	16.65	0.40	\multirow{2}{*}{-0.0476}	\multirow{2}{*}{14.0}	\multirow{2}{*}{0.9507}	\multirow{2}{*}{7}
  		v-B	-0.2669	16.66	3.09
  	\multirow{2}{*}{ILMI}	r-D	0.5182	16.78	2.95	\multirow{2}{*}{-0.0671}	\multirow{2}{*}{284.7}	\multirow{2}{*}{0.9510}	\multirow{2}{*}{8}
  		v-B	-0.2675	16.72	2.83
  \multirow{4}{*}{10$^2$}	\multirow{2}{*}{TW}	r-D	0.4709	16.39	0.11	\multirow{2}{*}{-0.0476}	\multirow{2}{*}{15.3}	\multirow{2}{*}{1.3889}	\multirow{2}{*}{9}
  		v-B	-0.2510	16.66	0.29
  	\multirow{2}{*}{ILMI}	r-D	0.4970	16.54	2.42	\multirow{2}{*}{-0.0513}	\multirow{2}{*}{399.9}	\multirow{2}{*}{0.9945}	\multirow{2}{*}{10}
  		v-B	-0.2525	16.52	2.73
  \multirow{4}{*}{10}	\multirow{2}{*}{TW}	r-D	0.3228	10.01	0.00	\multirow{2}{*}{-0.0266}	\multirow{2}{*}{11.9}	\multirow{2}{*}{1.8481}	\multirow{2}{*}{11}
  		v-B	-0.1520	10.03	0.00
  	\multirow{2}{*}{ILMI}	r-D	0.5228	16.51	3.41	\multirow{2}{*}{-0.0401}	\multirow{2}{*}{217.6}	\multirow{2}{*}{0.9918}	\multirow{2}{*}{12}
  		v-B	-0.2528	16.65	2.75

  a $K_P$ in units of [lb/min/wt. frac.]. b Computed by hinfnorm MATLAB command.

  • Analysis of Tables 1 & 2:

    • Computational Efficiency: The most striking result is the difference in computation time. The TW method is consistently and dramatically faster. For the PI case (Table 1), TW takes 34-58 seconds, while ILMI takes 183-242 seconds (about 4-5 times slower). For the PID case (Table 2), the difference is even more pronounced: TW takes 11-15 seconds, while ILMI takes 217-400 seconds (about 20 times slower).
    • Performance: For looser constraints ($\gamma=10^3, 10^2$), both methods find similar controller tunings and achieve comparable performance. However, for the stricter disturbance rejection requirement ($\gamma=10$), the TW method produces a more aggressive PI controller (smaller integral time) that performs better in terms of ITAE and settling time (as shown in Table 3). The ILMI approach seems to produce conservative tunings that do not change much.
    • Decay Rate ($\alpha$): The ILMI approach finds slightly more negative (better) decay rates, suggesting it might find a less conservative solution for stability. However, the authors argue this is because the fixed structure of the parametrized Lyapunov matrix in the TW method is more restrictive, which is a trade-off for non-iteration and speed.

  • Time-Domain Response Comparison:

    

    This figure, labeled as Fig. 1 in the paper, shows the closed-loop response of the Wood and Berry column to step changes in setpoints. The y-axis represents the deviation of the distillate and bottom compositions from their initial values.

    

    This figure, labeled as Fig. 2 in the paper, shows the corresponding manipulated variable movements (reflux and vapor flow rates) for the setpoint changes in Fig. 1.

  • Analysis of Figures 1 & 2: These plots compare the controller from this work (TW, result #5 from Table 1) against three other literature methods. The TW controller demonstrates a competitive performance, with a response that is well-damped and settles quickly, comparable to the specialized tuning methods BHA16 and GAR21. This validates that the non-iterative LMI approach can produce high-quality controllers in practice.

  Case B: Santander et al. (2022) FCC Unit

  This case tests the method on a larger, more challenging, ill-conditioned system.

  ---

  Manual transcription of Table 6 from the paper. Table 6: Comparison of PI tunings between this work (TW) and the Wang et al. (2008) ILMI approach for the Santander et al. (2022) FCC model.

  Approach	I/O	K	T_I [min]
  \multirow{5}{*}{TW$^a$}	Preg - Ffg	-1.9516	159.9
  	Tpre - Ff	7.8978	80.0
  	Trea - Frgc	0.6499	160.0
  	Treg - Fa	2.3618	640.0
  	Lrea - Fsc	1.2998	800.0
  \multirow{5}{*}{ILMI$^b$}	Preg - Ffg	-1.7660	160.7
  	Tpre - Ff	7.7670	80.0
  	Trea - Frgc	0.7450	160.2
  	Treg - Fa	2.3236	640.0
  	Lrea - Fsc	-1.5501	800.0

  a $\alpha^* \approx -O(10^{-6})$ after 355.6 s. b $\alpha^* = 1.0646$ after 2564 s.

  ---

  • Analysis of Table 6:

    • Stability and Numerical Robustness: The ill-conditioned nature of the FCC model poses a significant challenge. The proposed TW method successfully found a stabilizing controller with a (marginally) negative decay rate $\alpha^* \approx 0$. In contrast, the ILMI approach failed to find a stabilizing solution from a stability analysis perspective, yielding a positive $\alpha^* = 1.0646$. This highlights the numerical robustness of the TW formulation.
    • Computational Efficiency: The TW method was vastly superior, taking about 6 minutes (355.6 s) to compute all five controller tunings. The ILMI approach took about 43 minutes (2564 s), making it over 7 times slower.
    • Sensitivity: The authors report that the ILMI approach was highly sensitive to the initial guess for the Lyapunov matrix $\bar{P}$ and only converged when started from a null matrix. The TW method has no such dependency.

  • Nonlinear Simulation Results:

    

    This figure, labeled as Fig. 3 in the paper, shows the response of key FCC and fractionator process variables (temperatures, pressures, level) to a setpoint change in the reactor temperature. It compares the original controller, the TW controller, and the ILMI controller.

    

    This figure, labeled as Fig. 7 in the paper, shows the response of the same variables to a disturbance (a change in the feed properties). This tests the disturbance rejection capabilities of the controllers.

  • Analysis of Figures 3 and 7 (and others): The plots show the performance of the computed controllers when applied to the full nonlinear simulation model. Both the TW and ILMI controllers provide a slightly faster and better response for the FCC variables compared to the original tunings from Santander et al. (2022). This demonstrates that the tunings derived from a linearized model using the proposed LMI method are effective and can be successfully applied to the real-world nonlinear process. The performance of TW and ILMI are very similar in simulation, despite the ILMI's poor theoretical stability result ($\alpha > 0$).

7. Conclusion & Reflections

• Conclusion Summary: The paper successfully develops and validates a non-iterative LMI-based approach for tuning multi-loop PID controllers. By parametrizing the Lyapunov matrix as a function of the controller gains, the method transforms the traditionally non-convex BMI problem into a single, convex SDP problem. This core contribution directly addresses the major drawbacks of existing iterative LMI methods, namely high computational cost and sensitivity to initialization. The experimental results on two chemical process benchmarks confirm that the proposed method is significantly faster (5-20x), more numerically robust for ill-conditioned systems, and produces controllers with competitive or superior performance.

• Limitations & Future Work:

  • Restrictive Lyapunov Structure: The authors acknowledge that the specific structure imposed on the Lyapunov matrix to enable parametrization might be restrictive. This could lead to more conservative results (i.e., lower performance) in some cases compared to a method that could search over all possible Lyapunov matrices. This was hinted at in the Wood and Berry results where the ILMI method found a better decay rate $\alpha$.
  • Dependence on Gain Bounds: The performance of the resulting controller depends on the user-provided upper and lower bounds for the PID gains. Poorly chosen bounds can lead to a limited feasible search space and suboptimal tunings.
  • Future Work Suggestion: Instead of relying on gain bounds, the authors suggest incorporating pole placement constraints as LMIs. This would allow users to directly specify desired closed-loop dynamic characteristics (like settling time and damping), making the tuning process more intuitive.

• Personal Insights & Critique:

  • Novelty and Impact: The paper's contribution is elegant and practically significant. It provides a robust and efficient solution to a long-standing problem in advanced process control. The idea of parametrizing the Lyapunov matrix is a powerful technique for convexification that could potentially be applied to other control problems plagued by BMIs.
  • Strengths: The primary strength is the massive reduction in computational time without a significant sacrifice in performance. This makes advanced, model-based tuning methods more accessible for industrial applications where quick and reliable results are needed. The method's robustness on the ill-conditioned FCC model is another major plus.
  • Potential Weaknesses: The parametrization of the Lyapunov matrix, while clever, introduces a large number of auxiliary variables ($X_P, Y_P, Z_P$). As noted by the authors, for systems with many states ($n$) but few control loops ($p$), this can paradoxically increase the total number of decision variables compared to the original SOF problem. The scalability to systems with very high dimensionality ($n > 100$) might still be a practical concern.
  • Open Questions: While the method performed well, it would be interesting to see a theoretical analysis of the degree of conservatism introduced by the Lyapunov matrix parametrization. Is there a way to relax this structure slightly to find better solutions while still avoiding iteration? Furthermore, the paper focuses on LTI models obtained from linearization; its performance on systems with significant uncertainty or time-varying dynamics would be a valuable area for future investigation.