Discusión

M3 is more advanced than the M1 and M2 methods. M3 is a design method based on the internal model control (IMC) design that was introduced by Manfred Morari in 1986 [169]. The equivalent representations of the classical feedback structure and the IMC structure are shown in Figure 4-8. The IMC structure offers more benefits than the classical design in terms of the controller design procedure, in which the IMC is more direct and natural than the classical control and the requirement of solving the roots of the characteristic polynomial 1 + 𝐺𝑝𝐺𝑐 does not apply where only a simple examination of the poles of 𝑞𝑐

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Figure 4-8 (a) Classical feedback structure and (b) IMC structure

4.4.3.1 IMC Design Procedure

To obtain a near-optimal design of IMC, the IMC design procedure involves two main steps:

Step 1

Factorisation of the process plant model as

𝐺_𝑝(𝑠) = 𝐺_𝑝−(𝑠)𝐺_𝑝+(𝑠) (4-18)

where 𝐺_𝑝−(𝑠) is an invertible minimum phase and the noninvertible part contains all nonminimum phase elements e.g., delays and the zeros of the right half plane (RHP). However, the process model 𝐺𝑝 must be obtained, which is

modelled as (FOPTD) dynamics as

𝐺_𝑝(𝑠) =_𝜏𝑠+1𝐾 𝑒−𝜃𝑠 _(4-19)

or can also be expressed as

𝑑 + + 𝑟 𝑒 𝑢 𝑦 - + 𝐺𝑐 𝐺𝑝 - + 𝑢 𝑑 + + 𝑦 𝐺𝑝 𝑟 𝑒 + 𝑞𝑐 𝑞_𝑐 𝐺𝑐 = classical controller 𝐺𝑝 = process plant 𝑟 = reference input 𝑒 = error 𝑢 = input 𝑑 = disturbance 𝑦 = output 𝑞𝑐 𝑞 𝐺 (𝑠) (a) (b)

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𝐺𝑝(𝑠) =𝐾(−𝜃𝑠+1)_𝜏𝑠+1 𝜃, 𝜏 > 0 (4-20)

under the assumption that the inputs are steps, where 𝐾 is the gain, 𝜏 is the time constant and 𝜃 is the time delay of the FOPTD model of the plant transfer function. Thus,

𝐺𝑝+= −𝜃𝑠 + 1 𝐺𝑝−= _𝜏𝑠+1𝐾

The IMC controller is defined as

𝑞̃ (𝑠) = 𝐺𝑐 𝑝−(𝑠)−1, (4-21)

which is stable and causal. Thus,

𝑞̃ (𝑠) =𝑐 𝜏𝑠+1_𝐾 (4-22)

Step 2

Augment 𝑞̃ with an IMC filter, 𝑓(𝑠). The final IMC controller is defined as 𝑐

𝑞_𝑐(𝑠) = 𝐺_𝑝−(𝑠)−1_{𝑓(𝑠) (4-23)}

where

𝑓(𝑠) =_(λ𝑠+1)1 _𝑟 (4-24)

where the parameter λ is the adjustable parameter for the trade-off between the performance and the robustness of the inner loop and 𝑟 is the order of the filter, which should be sufficiently large to create a (semi-) proper IMC controller 𝑞_𝑐.A smaller λ provides a faster response but at the expense of more active control inputs; with a larger λ, the control is less aggressive but the closed-loop response is slower. In terms of robustness, the higher is the value of λ, the higher is the robustness of the control system.

To select the filter adjustable parameter λ, several approaches are available in the literature, such as reported by Saxena in [169], Tham in [172],

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Rivera in [170] and [171], Vilanova in [173], Lee et. al in [168] and Jeng et. al in [174]. As 𝑟 is 1 in this study, the selected 𝜆 is set as twice as fast as the open-loop response used by Tham in [172]. However, this selection complies with the rule

𝜆

𝜃> 1.7, as proposed by Rivera in [170] and 𝜆

𝜃> 0.2𝜏, as proposed by Shahrokhi

and Zomorrodi in [175].

The conventional feedback controller 𝐺𝑐 is related to the IMC controller

𝑞_𝑐, which is denoted as 𝐺𝑐(𝑠) =_1−𝐺𝑞(𝑠) 𝑝(𝑠)𝑞(𝑠) = 𝜏𝑠+1 𝐾 1 (λ𝑠+1)𝑟 (4-25)

This equivalent controller is not in the form of the PI controller. Therefore, the Maclaurin series expansion formula is applied to obtain a PI controller, which approximates 𝐺𝑐 given in (4-25) as

𝐺𝑐(𝑠) =_𝑠 1[𝑓(0) + 𝑓′(0)𝑠 + 𝑓 ′′₍₀₎ 2 𝑠 2_{+ ⋯ . . ] = 𝐾} 𝑐(1 +_𝑇1 𝑖𝑠+ 𝑇𝑑𝑠) (4-26)

where 𝑓(𝑠) = 𝑠𝐺_𝑐(𝑠). The PID parameters are obtained as 𝐾𝑐 = 𝑓′(0); 𝑇𝑖 = 𝑓

′₍₀₎

𝑓(0); 𝑇𝑑 = 𝑓′′₍₀₎

2𝑓′₍₀₎ (4-27)

Note that a PI controller or a P controller can be approximated using only the first two terms or the second term, respectively, in (4-26). Thus, the gain for the PI controller can be extracted as

𝐾𝑃 =_{𝐾(𝜃+λ)}𝜏 (4-28)

𝑇𝑖 = 𝜏 (4-29)

4.4.3.2 IMC-PI Cascade Controller Design

Cascade control is extensively used in industrial sectors to reduce the effects of possible disturbances that affect the secondary (inner) loop. In cascade

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design, the disturbance is effectively compensated prior to affecting the process variable of the primary (outer) loop. Cascade control may improve the dynamic performance of a closed-loop system [32]. This section explains how to obtain the parameters of the PI controllers of the proposed model (IVC and SLIVC) from a well-designed and well-tuned IMC cascade control system. Figure 4-9 shows the configuration of a typical cascade control system, where 𝐺_𝑝1and 𝐺_𝑝2 are the primary process and the secondary process, respectively. The primary process variable 𝑦1 (with setpoint 𝑟1) is used by the primary controller 𝐺𝑐1 to establish

the set-point (𝑟₂) for the secondary controller 𝐺_𝑐2. The secondary process variable 𝑦₂ is subsequently fed to the secondary controller, which fine-tunes the manipulated variable u. To obtain the PI parameter for the cascade design, the controller tuning began with the secondary controller (inner loop) and continued with the primary controller (outer loop) [168, 174].

Figure 4-9 Configuration of a typical cascade system

4.4.3.2.1 Design of the Secondary Controller

According to the method presented in the previous section, the secondary process model 𝐺_𝑝2 must be obtained; it is modelled as (FOPTD) dynamics as

𝐺_𝑝2(𝑠) = 𝐾2

𝜏2𝑠+1𝑒

−𝜃2𝑠 (4-30)

The IMC controller 𝑞𝑐2 is designed as

𝑞𝑐2(𝑠) = 𝐺𝑝2−(𝑠)−1𝑓2(𝑠) (4-31) 𝑦2 + + 𝑑1 + + 𝑟1 𝑒1 𝑟₂ 𝑦1 - + 𝐺𝑐1 𝐺𝑝2 𝑢2 𝑒2 - 𝐺𝑐2 𝑑2 + 𝐺𝑝1

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where 𝐺𝑝2− is the invertible part of the model 𝐺𝑝2 and 𝑓2 is the IMC filter that

addresses the robustness issue of the inner loop. In this case, the selected IMC filter is

𝑓₂(𝑠) =_(λ 1

2𝑠+1)𝑟2 (4-32)

where the parameter λ2 is the adjustable parameter of the secondary filter.

The conventional feedback controller 𝐺_𝑐2 is related to the IMC controller 𝑞𝑐2 as 𝐺_𝑐2(𝑠) = 𝑞2(𝑠) 1−𝐺𝑝2(𝑠)𝑞2(𝑠) = 𝜏2𝑠+1 𝐾2 1 (λ2𝑠+1)𝑟 (4-33)

Using Equations (4-28) and (4-29), the PI controller gains for the secondary controller can be obtained when compared with Equation (4-33).

4.4.3.2.2 Design of the Primary Controller

With the designed secondary controller, the primary controller is designed based on the apparent process 𝐺∗

𝑝1, as determined by the primary

controller as

𝐺∗

𝑝1(𝑠) =

𝐺𝑐2(𝑠)𝐺𝑝2(𝑠)

1+𝐺𝑐2(𝑠)𝐺𝑝2(𝑠)𝐺𝑝1(𝑠) (4-34)

With the model 𝐺∗

𝑝1, the IMC design is applied to design the primary

controller 𝐺_𝑐1. The IMC controller 𝑞_𝑐1 is designed as

𝑞_𝑐1(𝑠) = 𝐺∗

𝑝1−(𝑠)−1𝑓1(𝑠) (4-35)

where 𝑓₁ is the IMC filter that addresses the robustness issue of the outer loop. To maintain the simplicity of the control system, the typical IMC filter design is adopted for 𝑓1,

𝑓1(𝑠) =_(λ 1

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where λ1 is the adjustable parameter to create a trade-off between the

performance and the robustness of the outer loop.

The apparent process for the primary controller design is modelled using FOPTD dynamics as

𝐺𝑝1(𝑠) =_𝜏𝐾1

1𝑠+1𝑒

−𝜃1𝑠 (4-37)

In this primary FOPTD model, the conventional feedback controller 𝐺_𝑐1 is related to the IMC controller 𝑞𝑐1 as

𝐺_𝑐1(𝑠) = 𝑞1(𝑠) 1−𝐺∗_{𝑝1(𝑠)𝑞1(𝑠)} = 𝜏1𝑠+1 𝐾∗₁ 1 (λ1𝑠+1)𝑟 (4-38)

Using Equations (4-28) and (4-29), the PI controller gains for the primary controller can be obtained when compared with Equation (4-38).