# A Real-Time Optimization Framework for the Iterative Controller Tuning Problem

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## Abstract

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## 1. Introduction

**Figure 1.**Qualitative schematic of a single-input-single-output system with the controller ${G}_{c}(\mathit{\rho})$. Elements such as disturbances and sensor dynamics, as well as any controller-specific requirements, are left out for simplicity. We use the notation $y(t,\mathit{\rho})$ to mark the (implicit) dependence of the control output on the tuning parameters

**ρ**(likewise for the input and the error).

**ρ**, and in every application, some sort of design phase precedes the actual implementation and acts to choose a set of

**ρ**that is expected to track the reference, ${y}_{ref}$, “well”, while meeting any additional specifications. The classic example for PID controllers is the Ziegler-Nichols tuning method [1], with methods such as model-based direct synthesis [2] and virtual reference feedback tuning [3] acting as more advanced alternatives. Though not as developed, both theoretical and heuristic approaches exist for the design of MPC [4] and general fixed-order controllers [5,6] as well.

- assumptions on the plant, such as linearity or time invariance, that are made at the design stage,
- modeling errors and simplifications,
- conservatism in the case of a robust design,
- time constraints and/or deadlines that give preference to a simpler design over an advanced one.

**Figure 2.**The basic idea of iterative controller tuning. Here, a step change in the setpoint represents the repetitive control task. We use ${\mathit{\rho}}_{*}$ to denote a sort of “anti-optimum” that might be achieved with a bad adaptation algorithm.

- the tracking of a temperature profile in a laboratory-scale stirred tank by an MPC controller,
- the tracking of a periodic setpoint for a laboratory-scale torsional system by a general fixed-order controller with a controller stability constraint,
- the setpoint tracking and disturbance rejection for a five-input, five-output multi-loop PI system with imperfect decoupling and a hard output constraint.

## 2. The RTO Formulation of the Iterative Controller Tuning Problem

#### 2.1. The Cost Function ${\varphi}_{p}$ → The Control Performance Metric

**Assumption 1**(Repeatability)

**.**Let $\mathit{\rho}\in {\mathbb{R}}^{{n}_{\rho}}$ denote the tuning parameters of a controller and ${J}_{k}$ the observed value of the user-defined performance metric at run k for a fixed control task that is identical from run to run. The closed-loop process is repeatable with respect to performance if:

#### 2.2. The Uncertain Inequality Constraints ${\mathbf{G}}_{p}$ → Safety and Economic Constraints

**ρ**with the additive errors ${\delta}_{min}$ and ${\delta}_{max}$.

#### 2.3. The Certain Inequality Constraints $\mathbf{G}$ → Controller Specifications and Stability Considerations

#### 2.4. The Box Constraints ${\mathbf{v}}^{L}\u2aaf\mathbf{v}\u2aaf{\mathbf{v}}^{U}$ → Controller Parameter Limits

#### 2.5. The ICT Problem in RTO Form: Summary

## 3. The SCFO Solver and Its Configuration

- the RTO scheme converges arbitrarily close to a Karush-Kuhn-Tucker (KKT) point that is, in the vast majority of practical cases, a local minimum,
- the constraints ${\mathbf{G}}_{p}(\mathbf{v})\u2aaf\mathbf{0}$ and $\mathbf{G}(\mathbf{v})\u2aaf\mathbf{0}$ are never violated,
- the objective value is consistently improved, with ${\varphi}_{p}({\mathbf{v}}_{k+1})<{\varphi}_{p}({\mathbf{v}}_{k})$ always,

**Figure 3.**The iterative tuning scheme, where the results obtained after each closed-loop experiment on the plant (denoted by the dashed lines) are sent to the RTO loop (denoted by the dotted box), which then appends these data to previous data and uses the full data set to prompt the SCFO solver, as well as to update any data-driven adaptive settings (we refer the reader to Table 1 for which settings are fixed and which are adaptive).

Solver Setting | Chosen As | Justification | Type |
---|---|---|---|

Initialization | ${n}_{\mathit{\rho}}+1$ closed-loop experiments | See Section 3.1 | – |

Optimization target | Scaled gradient descent | See Section 3.2 | Adaptive |

Noise statistics | Initial experiments at ${\mathit{\rho}}_{0}$ | See Section 3.3 | Fixed |

Constraint concavity | None assumed | No reason for assuming this property in ICT context | Fixed |

Constraint relaxations | None assumed | For simplicity (should be added if some constraints are soft) | Fixed |

Cost certainty | Cost function is uncertain | The performance metric is an unknown function of ρ | Fixed |

Structural assumptions | Locally quadratic structure | Recommended choice for general RTO problem [17] | Fixed |

Minimal-excitation radius | $0.01\left({\mathit{\rho}}_{1}^{U}-{\mathit{\rho}}_{1}^{L}\right)$ | Recommended choice for general RTO problem [17] | Fixed |

Lower and upper limits, ${\mathbf{v}}^{L}$ and ${\mathbf{v}}^{U}$ | Controller-dependent or set adaptively | See Section 3.4 | Fixed/Adaptive |

Lipschitz and quadratic bound constants | Initial data-driven guess followed by adaptive setting | See Section 3.5 | Fixed/Adaptive |

Scaling bounds | Problem-dependent; easily chosen | See [17] | Fixed |

Maximal allowable adaptation step, $\Delta {\mathbf{v}}_{max}$ | $0.1{\left({\mathit{\rho}}^{U}-{\mathit{\rho}}^{L}\right)}^{T}$ | Recommended choice for general RTO problem [17] | Fixed |

#### 3.1. Solver Initialization

#### 3.2. The Optimization Target

#### 3.3. The Noise Statistics

#### 3.4. Lower and Upper Input Limits

#### 3.5. Lipschitz and Quadratic Bound Constants

- If ${\widehat{\varphi}}_{p}({\mathbf{v}}_{k})-4{\sigma}_{\varphi}\ge \underset{i=0,...,k-1}{min}{\widehat{\varphi}}_{p}({\mathbf{v}}_{i})$, then set $\eta :=\eta +1$;
- otherwise, set $\eta :=\eta -0.5$, with $\eta <0\to \eta :=0$,

## 4. Case Studies

#### 4.1. Batch-to-Batch Temperature Tracking in a Stirred Tank

**Figure 4.**Schematic of the jacketed stirred tank and the cascade control system used to control the water temperature inside the tank. The reference (${F}_{j,ref}$) for the water flow to the jacket (${F}_{j}$) was fixed at 2 L/min.

- the output weight that controls the trade-off between controller aggressiveness and output tracking,
- the bias update filter gain, which acts to ensure offset-free tracking,
- the control and prediction horizons that dictate how far ahead the MPC attempts to look and control,

**Figure 5.**The parameter adaptation plot (left) and the measured performance metric (right) for the solution of Problem (18). Hollow circles on the left indicate batches that were carried out as part of the initialization (prior to applying the solver). Likewise, the dotted vertical line on the right shows the iteration past which the parameter adaptations were dictated by the SCFO solver.

**Figure 6.**The visual improvement in the temperature profile tracking from Batch 1 to Batch 20. The dotted (red) lines denote the setpoint, while the solid (black) lines denote the actual measured temperature.

**Figure 7.**The measured performance metric for the solution of Problem (17), together with the tracking obtained for the final batch.

#### 4.2. Periodic Setpoint Tracking in a Torsional System

**Figure 9.**The generalization of “run-to-run” tuning to a system with a periodic setpoint trajectory. Only the setpoint is given here.

**Figure 10.**A twenty-bin histogram representation of the observed scaled performance metric values for a hundred runs with the initial parameter set (Problem (19)).

**Figure 11.**Performance improvement over 100 runs of operation for three different trials (dashed lines) of Problem (19).

**Figure 12.**Difference in control input and output profiles between the first and final runs of Problem (19), with the dashed green line used to denote the input (motor voltage) values.

#### 4.3. PID Tuning for a Step Setpoint Change

**Figure 13.**Performance obtained by iterative tuning for both the noiseless (left) and noisy (right) cases of Study 1 of Problem (20), with the solid blue line used to denote the “true” performance of the closed-loop system and the green dashed line used to denote what is actually observed (and provided to the solver). In both cases, the SCFO solver brings the closed-loop performance metric value close to its global minimum of zero (marked by the black dashed line in the lower plots).

**Figure 14.**Performance obtained by iterative tuning for Study 2 of Problem (20).

**Figure 15.**Performance obtained by iterative tuning for Study 3 of Problem (20).

#### 4.4. Tuning a System of PI Controllers for Setpoint Tracking and Disturbance Rejection

**Figure 16.**Performance obtained by iterative tuning for the system of PI controllers in Problem (22)—the noiseless case is given on the left and the noisy case on the right. For the output profiles, we note that the initial profiles are given as dashed lines, with the final profiles given by solid lines of the same color.

^{th}run), the algorithm remains, on the whole, reliable, as it keeps the performance metric at low values for the majority of the runs despite significant noise corruption.

**Figure 17.**Performance obtained by iterative tuning for the system of PI controllers in Problem (22) (without an output constraint).

## 5. Concluding Remarks

- No solution has been proposed for how to treat the case where the repeatability assumption is not a good approximation of reality. Instead of hoping that the approximation suffices in practice, it would be beneficial to propose alternatives that would still allow one to use the RTO framework to deal with the problem. In particular, one could attempt to make the repeatability assumption on the input and output trajectories rather than making it directly on the performance metric. This could allow one to establish a closer link between the lack of repeatability and the input/output noise in the control system.
- Although the proposed configuration has been shown to be largely successful here, many of the elements involved still remain heuristic in nature. Either improving on these heuristics or finding ways to avoid them are desired.
- The method is currently limited to solving ICT problems where the control task remains the same, which may significantly limit its domain of applicability. It would be interesting to attempt to extend it to cases where the control tasks were similar, rather than identical, and then somehow penalize the method based on the degree of similarity (e.g., one could attempt to lump non-similarity into the noise element δ of the repeatability assumption).

## Acknowledgments

## Conflicts of Interest

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## Appendix

#### A.1. Description of the Initialization Scheme

- Initialize $\mathbf{P}\in {\mathbb{R}}^{{n}_{v}\times {n}_{v}}$ as a diagonal matrix with ${P}_{11}:=1$ and all other elements set to 0. Set $k:=1$. Define by $\Delta {\mathbf{v}}_{pert}\in {\mathbb{R}}_{++}^{{n}_{v}}$ the perturbation vector, and set $\Delta {\mathbf{v}}_{pert}:=\Delta {\mathbf{v}}_{max}$.
- Define ${\mathbf{v}}_{k}:={\mathbf{v}}_{0}+\mathbf{P}\Delta {\mathbf{v}}_{pert}$, and compute the following matrix:$$\Delta \mathbf{V}:=\left[\begin{array}{c}{({\mathbf{v}}_{0}-{\mathbf{v}}_{1})}^{T}\\ {({\mathbf{v}}_{1}-{\mathbf{v}}_{2})}^{T}\\ \vdots \\ {({\mathbf{v}}_{k-1}-{\mathbf{v}}_{k})}^{T}\end{array}\right]$$If the condition number of $\Delta \mathbf{V}$ is greater than 50, re-define ${\mathbf{v}}_{k}$ as ${\mathbf{v}}_{k}:={\mathbf{v}}_{k-1}+{\mathbf{R}}_{k}\Delta {\mathbf{v}}_{pert}$, where ${\mathbf{R}}_{k}$ is a diagonal matrix of zeros with the sole ${k}^{\mathrm{th}}$ diagonal element equal to 1.
- Obtain the corresponding ${\widehat{\varphi}}_{p}({\mathbf{v}}_{k}):={J}_{k}$ by running a closed-loop experiment with the controller parameters ${\mathit{\rho}}_{k}:={\mathbf{v}}_{k}$. Define:$$\Delta \Phi :=\left[\begin{array}{c}{\widehat{\varphi}}_{p}({\mathbf{v}}_{0})-{\widehat{\varphi}}_{p}({\mathbf{v}}_{1})\\ {\widehat{\varphi}}_{p}({\mathbf{v}}_{1})-{\widehat{\varphi}}_{p}({\mathbf{v}}_{2})\\ \vdots \\ {\widehat{\varphi}}_{p}({\mathbf{v}}_{k-1})-{\widehat{\varphi}}_{p}({\mathbf{v}}_{k})\end{array}\right]$$$$\nabla {\widehat{\varphi}}_{p}:={\left(\Delta \mathbf{V}\right)}^{\u2020}\Delta \Phi $$
- Re-define $\mathbf{P}$ as a diagonal matrix with the diagonal elements set as:$${P}_{ii}:=\left\{\begin{array}{cc}\hfill 1,& \nabla {\widehat{\varphi}}_{p,i}\le 0\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}i\le k\hfill \\ \hfill -1,& \nabla {\widehat{\varphi}}_{p,i}>0\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}i\le k\hfill \\ \hfill 1,& i=k+1\hfill \\ \hfill 0,& i>k+1\hfill \end{array}\right.$$
- Set $k:=k+1$. If $k>{n}_{v}$, terminate. Otherwise, return to Step 2.

- This scheme starts like the simple perturbation scheme, where only one parameter is perturbed at a time (only ${\rho}_{1}$ is perturbed for the first experiment), but adapts based on the results of the perturbation. For example, if we see that setting ${\rho}_{1,1}:={\rho}_{0,1}+\Delta {v}_{pert,1}$ improves performance, then we will maintain this perturbation while additionally perturbing ${\rho}_{2}$ in the following experiment. On the other hand, if we see that this perturbation leads to worse control performance, then we simply negate it for the following experiment, with this experiment being defined by the perturbations ${\rho}_{2,1}:={\rho}_{0,1}-\Delta {v}_{pert,1}$ and ${\rho}_{2,2}:={\rho}_{0,2}+\Delta {v}_{pert,2}$. The (partial) linear estimate (23) of the gradient acts as a guide in which directions to perturb.
- Due to the pseudo-inversion of $\Delta \mathbf{V}$, it follows that we also require an additional safeguard to ensure that the matrix remains well-conditioned, as not doing this could lead to a poor estimate of the gradient (assuming the inputs $\mathbf{v}$ to be well-scaled, which we do). Since the perturbation scheme alone does not ensure this, an override is introduced, where only a single input is perturbed once the condition number goes over a certain threshold (chosen here as 50). This essentially ensures that the conditioning does not get any worse as it forces $\Delta \mathbf{V}$ to be block diagonal.
- The choice of $\Delta {\mathbf{v}}_{pert}:=\Delta {\mathbf{v}}_{max}$ is only a recommendation, as the recommended definition for $\Delta {\mathbf{v}}_{max}$ as given in Table 1 (i.e., $0.1({\mathit{\rho}}^{U}-{\mathit{\rho}}^{L})$) tends to provide sufficient excitation without perturbing “too far”. However, if there is a fear that applying perturbations of this size will violate some of the problem constraints or destabilize the system, then $\Delta {\mathbf{v}}_{pert}$ should be reduced accordingly.

#### A.2. Data-Driven Estimations of the Performance Gradient and Hessian

- If $k<2{n}_{v}+1$, fit a linear model to all of the available data:$${\varphi}_{p}(\mathbf{v})\approx {a}_{0}+\sum _{i=1}^{{n}_{v}}{a}_{i}{v}_{i}$$$$\frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}:={a}_{i},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{H}_{k,ij}:=\left\{\begin{array}{cc}\hfill 0.5{\kappa}_{\varphi ,i},& i=j\hfill \\ \hfill 0,& i\ne j\hfill \end{array}\right.$$$$\begin{array}{c}{\displaystyle \frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}>{\kappa}_{\varphi ,i}\to \frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}:={\kappa}_{\varphi ,i}}\hfill \\ {\displaystyle \frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}<-{\kappa}_{\varphi ,i}\to \frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}:=-{\kappa}_{\varphi ,i}}\hfill \end{array}$$
- If $2{n}_{v}+1\le k<2{n}_{v}+1+{\sum}_{i=1}^{{n}_{v}-1}i$, fit a diagonal quadratic model to the data (quadratic without interaction terms):$${\varphi}_{p}(\mathbf{v})\approx {a}_{0}+\sum _{i=1}^{{n}_{v}}{a}_{i}{v}_{i}+\sum _{i=1}^{{n}_{v}}{a}_{ii}{v}_{i}^{2}$$$$\frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}:={a}_{i}+2{a}_{ii}{v}_{k,i},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{H}_{k,ij}:=\left\{\begin{array}{cc}\hfill 2{a}_{ii},& i=j\hfill \\ \hfill 0,& i\ne j\hfill \end{array}\right.$$$$\begin{array}{c}{H}_{k,ij}>0.5{\kappa}_{\varphi ,i}\to {H}_{k,ij}:=0.5{\kappa}_{\varphi ,i}\hfill \\ {H}_{k,ij}<-0.5{\kappa}_{\varphi ,i}\to {H}_{k,ij}:=-0.5{\kappa}_{\varphi ,i}\hfill \end{array}$$
- If $k\ge 2{n}_{v}+1+{\sum}_{i=1}^{{n}_{v}-1}i$, fit a full quadratic model to the data:$${\varphi}_{p}(\mathbf{v})\approx {a}_{0}+\sum _{i=1}^{{n}_{v}}{a}_{i}{v}_{i}+\sum _{i=1}^{{n}_{v}}\sum _{j=1}^{{n}_{v}}{a}_{ij}{v}_{i}{v}_{j}$$$$\frac{\partial {\widehat{\varphi}}_{p}}{\partial {v}_{i}}{|}_{{\mathbf{v}}_{k}}:={a}_{i}+\sum _{j=1}^{{n}_{v}}{a}_{ij}{v}_{k,j},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{H}_{k,ij}:=2{a}_{ij}$$

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Bunin, G.A.; François, G.; Bonvin, D. A Real-Time Optimization Framework for the Iterative Controller Tuning Problem. *Processes* **2013**, *1*, 203-237.
https://doi.org/10.3390/pr1020203

**AMA Style**

Bunin GA, François G, Bonvin D. A Real-Time Optimization Framework for the Iterative Controller Tuning Problem. *Processes*. 2013; 1(2):203-237.
https://doi.org/10.3390/pr1020203

**Chicago/Turabian Style**

Bunin, Gene A., Grégory François, and Dominique Bonvin. 2013. "A Real-Time Optimization Framework for the Iterative Controller Tuning Problem" *Processes* 1, no. 2: 203-237.
https://doi.org/10.3390/pr1020203