Intro to Process Control--part 2

Process dynamics and time constants

Resistance, capacitance and inertia are perhaps the most important effects in industrial processes involving heat transfer, mass transfer and fluid flow operations. The essential characteristics of first and second order systems are summarized below, and they may be used to identify the time constant and responses of many processes as well as mechanical and electrical systems. In particular, it should be noted that most process measuring instruments will exhibit a certain amount of dynamic lag, and this must be recognized in any control system application since it will be a factor in the response and in the control loop tuning.

First order process dynamic characteristics

The general version of the process model for a first order lag system is a linear first order differential equation:

Where: T = the process response time constant

Kp = the process steady-state gain (output change/input change)

t = time

c(t) = process output response

m(t) = process input response.

The output of a first order process follows the step-change input with a classical exponential rise.

====

63.2% 86% 98% Time t, Time t, Input m

Output c

delta_c = KP delta_m

Input, Output, T 2T 4T

+++++ First order response

====

Important points to note: T is the time constant of the system and is the time taken to reach 63.2% of the final value after a step change has been applied to the system. After four time constants the output response has reached 98% of the final value that it will settle at.

P Final steady-state change in output is the steady-state gain

Change in input K =

The initial rate of rise of the output will be KP/T.

Application to the tank example: If we apply some typical tank dimensions to the response curve in +++++7 we can predict the time that the tank-level example will need to stabilize after a small step change around a target level H. For example, suppose the tank has a cross-sectional area of 2 m^2 and operates at H = 2 m when the outflow rate is 5 m^3 /h. The resistance constant R will be H/Q2 = 2 m/5 m^3 /h = 0.4 h/m^2 and the time constant will be AR = 0.8 h. The gain for a change in Q1 will also be R. Hence, if we make a small corrective change at Q1 of say 0.1 m3 /h the resulting change in level will be: RQ1 = 1 × 0.4 = 0.4 m, and the time to reach 98% of that change will be 3.2 h.

Resistance process

Now that we have seen how a first order process behaves, we can summarize the possible variations that may be found by considering the equivalent of resistance, capacitance and inertia type processes.

If a process has very little capacitance or energy storage the output response to a change in input will be instantaneous and proportional to the gain of the stage. For example, if a linear control valve is used to change the input flow in the tank example the output flow will rise immediately to a higher value with a negligible lag.

Capacitance type processes

Most processes include some form of capacitance or storage capability, either for materials (gas, liquid, or solids) or for energy (thermal, chemical, etc.). Those parts of the process with the ability to store mass or energy are termed 'capacities'. They are characterized by storing energy in the form of potential energy; For example, electrical charge, fluid hydrostatic head, pressure energy and thermal energy. The capacitance of a liquid- or gas-storage tank is expressed in area units. The gas capacitance of a tank is constant and is analogous to electrical capacitance. The liquid capacitance equals the cross-sectional area of the tank at the liquid surface; if this is constant then the capacitance is also constant at any head.

Consider now what happens if we have a steady-state condition, where flow into the tank matches the flow out via an orifice or valve with flow resistance R. If we change the inflow slightly by ?V the outflow will rise as the pressure rises until we have a new steady-state condition. For a small change we can take R to be a constant value. The pressure and outflow responses will follow the first order lag curve we have seen and will be given by the equation delta_p = R delta_V (1 – e -t/RC ) and the time constant will be RC.

It should be clear that this dynamic response follows the same laws as those for the liquid tank example and for the electrical circuit shown.

A purely capacitive process element can be illustrated by a tank with only an inflow connection such as shown. In such a process, the rate at which the level rises is inversely proportional to the capacitance and the tank will eventually flood. For an initially empty tank with constant inflow, the level c is the product of the inflow rate m and the time period of charging t divided by the capacitance of the tank C.

====

P

C, Capacitance

v = weight of gas in vessel, lb.

V = volume of vessel, ft 3

R = Gas constant of a specific gas, ft/deg

rho = pressure, lb-ft^2

n = polytropic exponent is between 1.0 and 1.2 for uninsulated tanks

Liquid capacitance is defined by C= dv/dh ft^2

Gas capacitance is defined by

C= dv/dp ft^2 = V/nRT

====

+++++ Capacitance of a liquid or gas storage tank expressed in area units m, Flow Physical diagram c, Head mc

Block diagram 1 Cs C dc dt =m=(Cs)c =m ?c = 1 Cs m

Where C= capacitance d t =time m= input variable (flow) c = output variable (head) dt = differential operator s =

+++++ Liquid capacitance calculation; the capacitance element

====

Inertia type processes

Inertia effects are typically due to the motion of matter involving the storage or dissipation of kinetic energy. They are most commonly associated with mechanical systems involving moving components, but are also important in some flow systems in which fluids must be accelerated or decelerated. The most common example of a first order lag caused by kinetic energy build-up is when a rotating mass is required to change speed or when a motor vehicle is accelerated by an increase in engine power up to a higher speed, until the wind and rolling resistances match the increased power input. For example consider a vehicle of mass M moving at V = 60 km/h, where the driving force F of the engine matches the wind drag and rolling resistance forces. If B is the coefficient of resistance, the steady state is F = VB, and for a small change of force ∆F the final speed change will be ∆V = ∆ F/B. The speed change response will be given by...

This equation is directly comparable to the versions for the tank and the electrical RC circuit. In this case, the time constant is given by M/B. Obviously, the higher the mass of the vehicle the longer it will take to change speed for the same change in driving force. If the resistance to speed is high, the speed change will be small and the time constant will be shorter.

Second order response

Second order processes result in a more complicated response curve. This is due to the exchange of energy between inertia effects and interactions between first order resistance and capacitance elements. They are described by the following second order differential equation:

Where:

T = the time constant of the second order process

xi = the damping ratio of the system

Kp = the system gain

t = time

c(t) = process output response

m(t) = process input response.

The solutions to the equation for a step change in m(t) with all initial conditions zero can be any one of a family of curves as shown. There are three broad classes of response in the solution, depending on the value of the damping ratio:

1. xi < 1.0, the system is underdamped and overshoots the steady-state value. If xi < 0.707, the system will oscillate about the final steady-state value.

2. xi > 1.0, the system is overdamped and won’t oscillate or overshoot the final steady-state value.

3. xi = 1.0, the system is critically damped. In this state it yields the fastest response without overshoot or oscillation. The natural frequency of oscillation will be omega_n = 1/T and is defined in terms of the 'perfect' or 'frictionless' situation where zeta = 0.0. As the damping factor increases, the oscillation frequency decreases or stretches out until the critical damping point is reached.

+++++Step response of a second order system.

Time, t

Output, c

xi_c =K_p delta m

Output

xi <1 Underdamped

xi >1 Critically damped

xi =1

Overdamped:

For practical application in control systems the most common form of second order system is found wherever two first order lag stages are in series, in which the output of the first stage is the input to the second. As we shall see: where the lags are modeled using transfer functions, the time constants of the two first order lags are combined to calculate the equivalent time constant and damping factor for their overall response as a second order system.

Important note: When a simple feedback control loop is applied to a first order system or to a second order system, the overall transfer function of the combined process and control system will usually be equivalent to a second order system. Hence, the response curves shown will be seen in typical closed loop control system responses.

Multiple time constant processes

In multiple time constant processes, say where two tanks are connected in series, the process will have two or more two time lags operating in series. As the number of time constants increases, the response curves of the system become progressively more retarded and the overall response gradually changes into an S-shaped reaction curve.

High order response

Any process that consists of a large number of process stages connected in series can be represented by a set of series-connected first order lags or transfer functions. When combined for the overall process, they represent a high order response, but very often one or two of the first order lags will be dominant or can be combined. Hence, many processes can be reduced to approximate first or second order lags, but they will also exhibit a dead time or transport lag as well.

Dead time or transport delay

For a pure dead-time process, whatever happens at the input is repeated at the output theta_d time units later, where theta_d is the dead time. This would be seen, for example, in a long pipeline if the liquid blend was changed at the input or the liquid temperature was changed at the input and the effects were not seen at the output until the travel time in the pipe has expired.

+++++Response curves of processes with several time constants

In practice, the mathematical analysis of uncontrolled processes containing time delays is relatively simple, but a time delay, or a set of time delays, within a feedback loop tends to lend itself to very complex mathematics.

In general, the presence of time delays in control systems reduces the effectiveness of the controller. In well-designed systems the time delays (dead times) should be kept to the minimum.

Using transfer functions

In practice, differential equations are difficult to manipulate for the purposes of control system analysis. The problem is simplified by the use of transfer functions.

Transfer functions allow the modeling blocks to be treated as simple functions that operate on the input variable to produce the output variable. They operate only on changes from a steady-state condition, so they will show us the time response profile for step changes or disturbances around the steady-state working point of the process. Transfer functions are based on the differential equations for the time response being converted by Laplace transforms into algebraic equations which can operate directly on the input variable. Without going into the mathematics of transforms, it’s sufficient to note that the transient operator (symbol S) replaces the differential operator such that d(variable)/dt = S. A transfer function is abbreviated as G(s) and it represents the ratio of the Laplace transform of a process output C(s) to that of an input M(s). From this, the simple relationship C(s) = G(s)M(s) is obtained.

Output C(s)=M(s) ×G(s) Control input M(s) Process transfer function G(s) Output C(s)

+++++12 Transfer function in a block diagram When applied to the first order system, we have already described the transfer function representing the action of a first order system on a changing input signal, where T is the time constant.

Control input M(s)

Output C(s) KP Ts +1

+++++ Transfer function for a first order process

As we have already seen, many processes involve the series combination of two or more first order lags. These are represented in the transfer function blocks as seen.

If the two blocks are combined by multiplying the functions together, they can be seen to form a second order system as shown here and as described.

Two first order lags in series

Control input M(s) Output C(s) K1 T1s + 1 T2s +1 K2

+++++ Two lags in series combine to produce a second order system.

Block diagram modeling of the control system proceeds in the same manner as for the process, and is shown by adding the feedback controller as one or more transfer function blocks. The most useful rule for constructing the transfer function of a feedback control loop is shown.

---- M(s) Process transfer function Gp(s)

Output C(s) Feedback transfer function H(s)

Controller transfer function Gc(s) + R(s)

- Combined transfer function: Gc(s)Gp(s) R(s) 1+G(s)Gp(s)H(s) = C(s)

+++++15 Block diagram and transfer function for a typical feedback control system

-----

The feedback transfer function H(s) (typically the sensor response) and the controller transfer function Gc(s) are combined in the model to give an overall transfer function that can be used to calculate the overall behavior of the controlled process. This allows the complete control system working with its process to be represented in an equation known as the closed loop transfer function. The denominator of the right hand side of this equation is known as the open loop transfer function. Notice that if this denominator becomes equal to zero, the output of the process approaches infinity and the whole process is seen to be unstable. Hence, control engineering studies place great emphasis on detecting and avoiding the condition where the open loop transfer function becomes negative and the control system becomes unstable.

Types or modes of operation of process control systems

There are five basic forms of control available in process control. These are:

1. On-off

2. Modulating

3. Open loop

4. Feed-forward

5. Closed loop.

The next five sections examine each of these in turn.

On-off control

The most basic control concept is on-off control, as found in a modern iron in our households. This is a very crude form of control, which nevertheless should be considered as a cheap and effective means of control if a fairly large fluctuation of the PV is acceptable. The wear and tear of the controlling element (solenoid valve etc.) needs special consideration. As the bandwidth of fluctuation of a PV increases, the frequency of switching (and thus wear and tear) of the controlling element decreases.

Modulating control

If the output of a controller can move through a range of values, we have modulating control. It’s understood that modulating control takes place within a defined operating range (with an upper and lower limit) only. Modulating control can be used in both open and closed loop control systems.

Open loop control

We have open loop control if the control action (Controller Output Signal OP) is not a function of the PV or load changes. The open loop control does not self-correct when these PVs drift.

Feed-forward control

Feed-forward control is a form of control based on anticipating the correct manipulated variables necessary to deliver the required output variable. It’s seen as a form of open loop control as the PV is not used directly in the control action. In some applications, the feed-forward control signal is added to a feedback control signal to drive the MV closer to its final value. In other more advanced control applications, a computer-based model of the process is used to compute the required MV and this is applied directly to the process.

+++++ A model based feedforward control system -- Set point (r ) Load (q)

Feedforward model

Manipulated variable (m)

Process a1 a3 a2

Controlled variable (c )

For example, a typical application of this type of control is to incorporate this with feedback - or closed loop control. Then the imperfect feedforward control can correct up to 90% of the upsets, leaving the feedback system to correct the 10% deviation left by the feedforward component.

Closed loop or feedback control

We have a closed loop control system if the PV, the objective of control, is used to determine the control action. The principle is given.

The idea of closed loop control is to measure the PV; compare this with the SP which is the desired or target value; and determine a control action which results in a change of the OP value of an automatic controller. In most cases, the ERROR (ERR) term is used to calculate the OP value.

ERR PV SP =- If ERR = SP - PV has to be used, the controller has to be set for REVERSE control action.

+++++17 The feedback control loop. Process. Process variable (PV) Setpoint (SP) Feedback controller ERR=PV-SP Output (OP) Manipulated variable (m) Load (q) a1 a2 a3

Closed loop controller and process gain calculations

In designing and setting up practical process control loops, one of the most important tasks is to establish the true factors making up the loop gain and then to calculate the gain.

Typically, the constituent parts of the entire loop will consist of a minimum of four functional items:

1. Process gain: (K_P) PV/ MV =? ?

2. Controller gain: (K_C) MV/ E =? ?

3. The measuring transducer or sensor gain, K_S and

4. The valve gain K_V.

The total loop gain is the product of these four operational blocks.

For simple loop tuning, two basic methods have been in use for many years. The Zeigler and Nichols method is called the 'ultimate cycle method' and requires that the controller should first be set up with proportional-only control. The loop gain is adjusted to find the ultimate gain, Ku. This is the gain at which the MV begins to sustain a permanent cycle. For a proportional-only controller the gain is then reduced to 0.5 Ku.

Therefore for this tuning the loop gain must be considered in terms of the sum of the four gains given above, and the tuning condition is given by the following equation:

Normally, only the controller gain can be changed, but it remains very important that the other gain components be recognized and calculated. In particular, the valve gain and process gain may change substantially with the working point of the process, and this is the cause of many of the tuning problems encountered on process plants.

Other gain settings are used in the Zeigler and Nichols method for PI and PID controllers to ensure stability when integral and derivative actions are added to the controller. See the next section for the meaning of these terms.

The alternative tuning method is known as the 1/4 damping method. This suggests that the controller gain should be adjusted to obtain an under-damped overshoot response having a quarter amplitude of the initial step change in setpoint. Subsequent oscillations then decay with 1/4 of the amplitude of the previous overshoot. This method does not change the gain settings, as integral and derivative terms () are added into the controller. Cautionary note: Rule-of-thumb guidelines for loop tuning should be treated with reservation since each application has its own special characteristics. There is no substitute for obtaining a reasonably complete knowledge of the type of disturbances that are likely to affect the controlled process, and it’s essential to agree with the process engineers on the nature of the controlled response that will best suit the process. In some cases, the above tuning methods will lead to loop tuning that is too sensitive for the conditions, resulting in high degree of instability.

Proportional, integral and derivative control modes

Most closed loop controllers are capable of controlling with three control modes, which can be used separately or together:

1. Proportional control (P)

2. Integral or reset control (I)

3. Derivative or rate control (D).

The purpose of each of these control modes is as follows: Proportional control

This is the main and principal method of control. It calculates a control action proportional to the ERROR. Proportional control cannot eliminate the ERROR completely.

Integral control (reset)

This is the means to eliminate the remaining ERROR or OFFSET value, left from the proportional action, completely. This may result in reduced stability in the control action.

Derivative control (rate)

This is sometimes added to introduce dynamic stability to the control LOOP. Note: The terms 'reset' for integral and 'rate' for derivative control actions are seldom used nowadays.

Derivative control has no functionality of its own.

The only combinations of the P, I and D modes are as follows:

• P For use as a basic controller

• PI Where the offset caused by the P mode is removed

• PID To remove instability problems that can occur in PI mode

• PD Used in cascade control; a special application

• I Used in the primary controller of cascaded systems.

An introduction to cascade control

Controllers are said to be 'in cascade' when the output of the first or primary controller is used to manipulate the SD of another or secondary controller. When two or more controllers are cascaded, each will have its own measurement input or PV, but only the primary controller can have an independent SP and only the secondary, or the most down stream, controller has an output to the process.

Cascade control is of great value where high performance is needed in the face of random disturbances, or where the secondary part of a process contains a significant time lag or has nonlinearity.

The principal advantages of cascade control are the following:

• Disturbances occurring in the secondary loop are corrected by the secondary controller before they can affect the primary, or main, variable.

• The secondary controller can significantly reduce phase lag in the secondary loop, thereby improving the speed or response of the primary loop.

• Gain variations due to nonlinearity in the process or actuator in the secondary loop are corrected within that loop.

• The secondary loop enables exact manipulation of the flow of mass or energy by the primary controller.

+++++ an example of cascade control where the primary controller TC is used to measure the output temperature T2, and compare this with the SP value of the TC; and the secondary controller, FC, is used to keep the fuel flow constant against variables like pressure changes. PV = T2 (output temp) SP OP TC PV SP FC OP Flow control Mode = Cascade (operational) SPFC ? OPTC Manual or starting value F

+++++An example of cascade control

The primary controller's output is used to manipulate the SP of the secondary controller, thereby changing the fuel feed rate to compensate for temperature variations of T2 only. Variations and inconsistencies in the fuel flow rate are corrected solely by the secondary controller - the FC controller. The secondary controller is tuned with a high gain to provide a proportional (linear) response to the range, thereby removing any nonlinear gain elements from the action of the primary controller.

NEXT: Process measurement and transducers

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