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Ali & Rashid | 85

B. Mathematical model for Autotuning PID Controller.

A PID controller consists of three terms: the

proportional (P) term, the integral (I) term, and the derivative

(D) term. In an ideal form, the output u(t) of a PID controller

is the sum of the three terms,

??(??) = ???? ??(??) + ???? ?0?? ??(??)???? + ???? ???? ????(??) (6)
??1 ????

where e(t) = r(t) - y(t) is the feedback error signal between

the reference signal r(t) and the output y(t), and ??D is the

derivative control gain. The Laplace transfer function of the

PID controller is

??(??) = ??(??) = ???? (1 + 1 + ???? ??) (7)
??(??) ??????

The P action (mode) adjusts the controller output

dependent on the error magnitude. An I action (mode) may

eliminate the steady-state offset, but the D action anticipates

the future trend (mode) (mode). These helpful functions are

adequate for a broad variety of process applications, and the

features' transparency leads to widespread user acceptance

[23].

To design the PID controller, can assume that two

Fig. 3: D5204 in Basrah Refinery. estimated frequency response points from the relay testing

Balance over Total Process System (PS total): data are the fundamental frequency Gp(j?1), ?1 = 2p/N?t
and Gp(j?2). If there is no disturbance in the system, then ?2
is selected as 3?1.Otherwise, ?2 is selected as 2?1.

- Total mass balance: Converting the ideal PID structure into another

???? = ??2? ?? + ??2? ?? - ???3 - ???4 (1) polynomial form:
????
??(??) = ??2??2+??1??+??0 (8)
M: the total mass, ??? ??: the mass flow of current i. ??
where Kc = c1, tI = c1/c0 and tD = c2/c1, the PID controller
With subscript V for vapor and L for liquid in multiphase

currents. design using the estimated frequency response is to find the

- Energy balance: parameters c2, c1 and c0.

???? - ???? + ??2? ?? * ?^?2?? + ??2? ?? * ?^?2?? - ???3 * ?^?3 - Automatically adjust PID gains in real time depending
???? ???? on the expected plant frequency response from a closed-loop
?????? experiment. Utilize the Frequency Response Estimator to do
???4 * ?^?4 = ???? (2) a real-time experiment-based estimate on a physical plant.
To acquire a frequency response estimate.
With ???? the heat flow and ???? the workflow of the system, ?^??? ? Injects sinusoidal test signals into the plant at the nominal
???? ????

the specific enthalpy of current ET the total energy of total

PS.

? Balance over Liquid Process System (PS liquid): operating point.

- Total mass balance: ? Collects response data from the plant output.

?????? = 1 * (??2? ?? - ???4) (3) ? Computes the estimated frequency response.
???? ????*??*??2

With ???? the liquid level in the tank, ???? density of phase ?? A recursive least squares (RLS) algorithm [24-26] to

(liquid or vapor), and r radius of the tank. compute the estimated frequency response. Assume that the
plant frequency response is ??(????) = ??????? . When a signal
? Balance over Vapor Process System (PS vapor): ??(??) = ????????(????) excites the plant, the steady-state plant is
??(??) = ????sin(???? + ??), which is equivalent to:
- Total mass balance: ??(??) = (??????????)????????(????) + (??????????)????????(????)

???? = 1 * ( ??*?? * ??2? ?? - ??*?? * ???3 - ?? * ??????) (4) At any given time, ????????(????) and ????????(????) are known,
???? ???? ?????? therefore, they can be used as regresses in an RLS algorithm
?????? ???? to estimate ?? cos(??) and ??sin(??) from the measured plant
output y(t) at run time.
With P the pressure in the vapor phase inside the tank, R the
When the excitation signal contains a superposition of
universal gas constant the operating temperature, ?????? the multiple signals, then:
molecular mass of vapor, and ???? the volume of vapor phase ??(??) = ??1 sin(??1??) + ??2 sin(??2??) + ?

inside the flash tank. In this case, the plant output becomes:
??(??) = (??1????????1)??1 sin(??1??) + (??1 sin ??1)??1 cos(??1??) +
Liquid or vapor flow through output flash tank valve, ?

required to calculate ???3 and ???4 , vapor and liquid flows (??2????????2)??2 sin(??2??) + (??2????????2)??2 cos(??2??) + ?

exiting the flash tank:

??? ?? = ?????? * ???? * v????? * ???? (5)
100

With ?????? the coefficient of valve for phase i, ???? the valve

opining percentage, and ????? the pressure drop through the

valve acting over phase ??.In valve sizing ???? is taken equal

to 50% for nominal or design flow.