Objective function and constraints of the optimal control problem in

## Abstract

This chapter introduces the closed-form analytical design of proportional-integral (PI) controller parameters for the optimal control subjected to operational constraints. The main idea of the design is not only to minimize the control performance index but also to cope with the constraints in the process variable, controller output, and its rate of change. The proposed optimization-based approach is examined to regulatory and servo control of integrating processes with three typical operation constraints. To derive an analytical design formula, the constrained optimal control problem in the time domain was transformed to an unconstrained optimization in a new parameter space associated with closed-loop dynamics. By taking the advantage of the proposed analytical approach, the optimal PI parameters can be found quickly based on the graphical analysis without complex numerical optimization. The resulting optimal PI controller guarantees the globally optimal closed-loop response and handles the operational constraints precisely.

### Keywords

- constrained optimal control
- industrial PI controller
- analytical design
- constraint handling
- integrating process
- optimal servo and regulatory control

## 1. Introduction

Many units used in the chemical process industry, such as heating boilers, batch chemical reactors, liquid storage tanks, or liquid level systems, are integrating processes in which the dynamic response is very slow with a large dominant time constant. In modern control, the integrating process also appears in many applications including space telescope control systems, lightweight robotic arms, and pilot crane control systems. Constraints are inherent in any industrial control systems, either implicitly or explicitly. They are generally associated with both the process variable and controller output. Indeed, typical operational constraints usually include the actuator magnitude and its rate saturation, process/output variable, and internal state variables. The objective of constrained optimal control is to minimize the control cost subjected to constraints on state variables and/or output variables. The importance of taking constraints into account during the design stage of the controller is no more questioned. In fact, a well-designed optimal control would fail in a real-life situation if the constraints are not taken into account while designing the controller. However, optimal control of a process with multiple constraints is still challenging even for a process with simple dynamics. In a popular approach using Pontryagin’s principle or the Hamilton-Jacobi-Bellman equation for a classical optimal control framework [1, 2], the optimal controller parameters are obtained via numerical solution of the nonlinear constrained optimization. However, the existing numerical methods neither guarantee a global optimal solution nor provide useful insights and physical interpretations of the complex relationships existing between the process parameters and control performance. To address this issue, the analytical solutions of optimal proportional-integral (PI) controller under constraints were previously proposed using the optimization-based approach for integrating systems [3–7] and extended to first-order systems [8–11]. This chapter introduces the optimization-based approach for the analytical design of optimal PI controller parameters for integrating processes without violating the operational constraints under a unified framework.

## 2. Formulation of constrained optimal PI control problem

Figure 1 presents the schematic diagram of an integrating process considered in this chapter. It is a type-C PI controller, also called I-P controller, which is a modified type of PID controller where the set point is removed from the proportional term in order to avoid the initial quick on the manipulated variable for a step change in the set point.

The major resulting transfer functions of this closed-loop system are expressed as

where

The closed-loop damping ratio of the above system becomes

The goal of a constrained optimal problem is to minimize the weighted sum of the process variable error, e(t), and the rate of change in the manipulated variable, u′(t), for a given step change,

Consequently, the constrained optimal control problem is formulated as

subject to

Through some mathematical operations, the above optimal control problem formulated in the time domain can be transformed to the form with the two new design parameters

Regulatory control | Servo control | |
---|---|---|

Objective function | ||

Constraints | ||

Parameters | ||

Functions | where | |

## 3. PI controller design

### 3.1. Optimal regulatory control

Applying the Lagrangian multiplier [12], it converts the constrained optimization problem in Table 1 to an equivalent unconstrained problem. In regulatory control, the constrained problem can be converted as

where

The necessary conditions for an optimal solution are then

The simultaneous solutions of Eqs. (7a)–(7d) for possible combinations of

Figure 2 presents seven possible cases for the location of global optima: the global optimum can be found inside the feasible region (case A), or on the boundary of one constraint (cases B, C, and E), or on the intersection point of two constraints (cases D, F, and G).

The global optima of the seven cases can be evaluated by inspecting their geometrical characteristics in

**Case A**

**Case B**

**Case C**

**Case D**

**Case E**

**Case F**

**Case G**

After the global optimum is determined in

One of main advantages of the optimization-based graphical approach is that the conditions for the seven possible cases can be directly evaluated based on a meticulous analysis of the graphical shape of the constraints and contours in

Using similar reasoning, the conditions to discriminate each of seven cases associated with the global optimum can be established according to the relative locations between the extreme point and its projections to the constraints. Table 2 lists the results for the conditions and characteristics of the global optima.

Case | Constraint specification | Condition | Global optimum | Location of global optimum | Calculation of global optimum |
---|---|---|---|---|---|

A | Mild Mild Mild | In the interior of the feasible region | |||

B | Mild Tight Mild | On | |||

C | Tight Mild Mild | On | |||

D | Tight Tight Mild | On the vertex by | |||

E | Mild Mild Tight | On | |||

F | Tight Mild Tight | On the vertex by | |||

G | Mild Tight Tight | On the vertex by |

### 3.2. Optimal servo control

The constrained optimization problem in Table 1 can be converted into an equivalent unconstrained problem by applying the Lagrangian multiplier [12] as follows:

where

Applying the same way used in the regulatory control case, the seven optimal cases can be found by solving the necessary conditions of the above unconstrained problem for the corresponding combination of slack variable and Lagrange multiplier. Figure 4 illustrates the seven possible locations of the global optimum.

After obtaining the global optimum for a particular case, the optimal parameters of the PI controller can be calculated using Eq. (15), i.e.,

Case | Constraint specification | Condition | Global optimum | Location of global optimum | Calculation of global optimum |
---|---|---|---|---|---|

A | Mild Mild Mild | where | In the interior of the feasible region | ||

B | Mild Tight Mild | On | |||

C | Tight Mild Mild | On | |||

D | Tight Tight Mild | On the vertex by | |||

E | Mild Mild Tight | On | |||

F | Tight Mild Tight | On the vertex by | |||

G | Mild Tight Tight | On the vertex by |

## 4. Design and evaluation of feasible constraints

The optimal solutions in Tables 2 and 3 are only true if the constraint set is such that a solution exists. Indeed, depending on the constraint set, the optimal solution may not have a feasible solution. Therefore, before applying the constrained optimal control formulation, either any given constraint set should first be screened quickly to determine its basic feasibility or a constraint set should be designed to be feasible.

### 4.1. Optimal regulatory control

**Conditions for a feasible**

As indicated in Figure 5, the feasible region reduces in size as

Let

Once

To sum up, if

Similarly, for a given

To sum up, if

**Feasible** **for a given feasible set**

Because

A vertex

which yields

Overall, for a given feasible

Figure 6 illustrates a procedure applied to design a feasible constraint set

### 4.2. Optimal servo control

It is clear from their approaching values of

## 5. Closed-loop performance

### 5.1. Optimal regulatory control

Consider the following integrating process as

Table 4 presents the examples of the seven possible aforementioned cases, as based on various constraint specifications. Simulations are carried out for weighting factors

Example | Case | Constraint specification | PI parameter | |||
---|---|---|---|---|---|---|

1 | A | 0.70 | 2.70 | 2.70 | 1.41 | 1.41 |

2 | B | 0.70 | 2.70 | 1.11 | 1.10 | 1.10 |

3 | C | 0.36 | 2.70 | 2.70 | 1.93 | 1.51 |

4 | D | 0.285 | 2.70 | 2.10 | 2.10 | 0.69 |

5 | E | 0.70 | 1.105 | 2.70 | 1.96 | 2.95 |

6 | F | 0.30 | 1.20 | 2.70 | 2.18 | 0.98 |

7 | G | 0.70 | 1.20 | 1.37 | 1.37 | 1.56 |

Figure 8 presents the resulting process variable,

### 5.2. Optimal servo control

Consider the following integrating process

Table 5 lists the examples of the seven possible cases based on various constraint specifications. Simulations are carried out for weighting factors,

Example | Case | Constraint specification | PI parameter | |||
---|---|---|---|---|---|---|

1 | A | 1.2 | 0.5 | 1.5 | 1.41 | 1.414 |

2 | B | 1.2 | 0.5 | 0.5 | 0.79 | 1.58 |

3 | C | 1.03 | 0.5 | 1.5 | 1.52 | 1.46 |

4 | D | 1.03 | 0.5 | 1.03 | 1.51 | 1.47 |

5 | E | 1.2 | 0.3 | 1.5 | 1.53 | 2.42 |

6 | F | 1.03 | 0.41 | 1.5 | 1.38 | 1.61 |

7 | G | 1.2 | 0.4 | 1.17 | 2.15 | 1.83 |

## 6. Conclusions

A novel analytical design approach is introduced for optimal regulatory and servo PI control subjected to operational constraints and examined to integrating processes. Owing to incisive parameterization, a complex constrained optimal control problem can be reformulated and converted to a simple algebraic form in the new design parameter

## Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015R1D1A3A01015621) and by the Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A6A1031189).

## Conflict of interest

The authors confirm there are no conflicts of interest.