The chemical processes possess non-linear dynamic characteristics and the design
of controller for a non-linear chemical process involves linearizing the process
model around its steady state operating point and applying the linear control
theory. The decoupling and linearization control (Akkari
et al., 2009) provides satisfactory response when the process model
is available. However if there is a difference between the real process and
the process model, application of this model may give unsatisfactory results.
But the degree of the mismatch is generally not high in many chemical processes.
In such cases, it is sufficient to add external controllers which compensates
for the mismatch. In this work the state feedback law (Thosar
et al., 2008) is applied to the nonlinear process. That is linearized
as per the above process. The state feedback law is developed as a part of linearization
of the nonlinear process. This yields us a control structure called Global Linearizing
Control which responds like linear system. This type of control is tried in
this work for a MIMO nonlinear system with equal number of inputs and outputs.
For a linear non interacting MIMO system, Kalman and Unknown Input Observer (UIO) is designed from control point of view. The system states of a dynamic system is estimated using the unknown input observers which may receive input excitation of any kind. Recently there have been many researchers aiming to simultaneously estimate the system state and the unknown input. The estimation of parameters is important in many engineering applications.
APPLICATION TO CHEMICAL PROCESS CONTROL
Application of global linearizing control for the Level and Temperature Control Process is studied here.
The non-linear system is defined as:
where, x is the state vector of dimension n, u is an input vector of dimension m, y is an output vector of dimension of p, f(x) is a smooth function, h(x) is a (p,1) vector with a row element hj(x) also a smooth function and g(x) is an (n, m) matrix with elements of each column being gj(x).
The model is chemical process with a liquid level and temperature as variables
to be controlled (Kravaris and Chung, 1987) is shown
in Fig. 1.
The mathematical equations formed for the above system is:
where , x1 and x2 are the liquid level and the temperature
in the tank, respectively. u1 and u2 are the feed flow
rate to the tank and the heat flow rate from the heater, respectively.
||Schematic diagram for the level and temperature control process,
u1: Feed flow rate of the liquid, u2: Heater input,
Y1: Level sensor output, Y2: Temperature sensor output,
H: Height of the liquid
The feed flow rate and heat flow rate are constrained as 0=u1=22
cm3 sec-1 and 0=u2 = 2700 J sec-1.
k is the constant coefficient, 1.8, S is the cross sectional area, 191 cm2,
x1 is the liquid level in cm, x2 is the liquid temperature,
°C, To is the temperature of the feed 18°C, Cp
is the specific heat, = 4.2 J-1 K-1, ς is the density
of the liquid (water).
Extensive numerical simulations are carried out on the proposed algorithm as
detailed below. A standard Runge-Kutta Gill algorithm is used for the numerical
integration of the set of ordinary differential equations. Prior to decoupling,
liquid level Y1 depended on u1 (flow rate) and liquid
temperature Y2 depended on both u1 and u2 (heater
input). After the application of decoupling algorithm Y1 depends
only on u1 and Y2 developed here depends only on u2
as can be observed from Eq. 9 and 10.
DEVELOPMENT OF HIRSCHORNS CONTROL LAW
In order to calculate a control law that induces linear input/ output behavior
of a MIMO system, a Decoupling and Linearization (Hirschorns) algorithm
(Kravaris and Soroush, 1990) was developed. It helps
to find a differential operator such that, when applied to the outputs, it will
provide a set of algebraic expressions in x and u that gives solution to u.
The use of this control law does not require any structured constraints to be imposed on the closed loop system dynamics. Therefore, the control designer has the flexibility to adjust the parameters βik for fast closed-loop response and desirable level of coupling.
From Kravaris and Soroush (1990) if:
F1(x) = constant, 1 = 0,..., k*-1, then the system Eq. 1 is input/output linearizable. [mx1] matrice βik, i = 0,.., m, k = 0,..., ri-1.
mx(m-ς(0)), mx(m-ς(1)),..., mx(m-ς(k*-1)) matrices γ0, γ1,
, γk*-1 and an m×m invertible matrix Γ.
Applying the linearising algorithm, the decoupling and linearization control
law obtained as given by (Kravaris and Soroush, 1990)
is reproduced below:
Applying the procedure and data as given in (Akkari
et al., 2009) and substituting u1 and u2 in the
Eq. 3 and 4 the state equation obtained
is both decoupled and linearised forms. The resulting Eq. 9
and 10 are in decoupled form:
The advantage of using Hirschorns algorithm is that the control law is less complex. In addition to that it also offers more dynamic feed flow rate of liquid u1 and heat input rate u2. The simulation results show that Hirschorns algorithm has better effect.
IMPLEMENTATION OF KALMAN OBSERVER
Kalman observer is a recursive predictive filter that is based on the use of
state space techniques and recursive algorithms, i.e., only the estimated state
from the previous time step and the current measurement are needed to compute
the estimate of the current state. The Kalman filter operates by propagating
the mean and covariance of the state through time. The notation
represents the estimate of the state vector X at time n given observations
The state of the filter is represented by two variables (Tiano
et al., 2007):
a posteriori state estimate at time k. The given observation is up to and
including at time k
||Pk|k, a posteriori error covariance matrix which
measure the estimated accuracy of the state
||The Kalman filter has two distinct phases, prediction and
In a typical situation, first prediction phase provides an estimate of the current state which holds until the present scheduled observation. In the correction phase this observed information is used to update the estimate produced by the prediction phase. These two phases alternatively produce new estimate of the states. However, if the observation is not possible for some cases, the estimate can made by multiple prediction phases, skipping observation phase. Consider a linear time invariant discrete system given by the following equation:
where, F is the state transition matrix, B is the control input matrix, Wk is the process noise with zero mean multivariate normal distribution having covariance Qk. H is the observation matrix, Vk+1 is the observation noise which is zero mean Gaussian white noise having covariance Rk. Uk is the control input.
Prediction (time update) equations: Predicted state estimate:
Predicted estimate covariance:
Correction (measurement update) equations: Innovation or measurement residual:
Innovation (or residual) covariance:
Optimal Kalman gain:
Updated (a posteriori) state estimate:
Updated (a posteriori) estimate covariance:
The results that are obtained using the Kalman observer technique are explained in the simulation results section.
IMPLEMENTATION OF UNKNOWN INPUT OBSERVER
Observer is capable of estimating the states with unknown inputs. The unknown
inputs generally could be a combination or any of the unmeasurable or unmeasured
disturbances, unknown control action or unmodelled system dynamics. This observer
is very useful when we are dealing with problem of instrument fault detection.
It can be implemented as reduced order observer or full order observer (Wang
and Gao, 2003).
Consider a continuous linear time invariant steady space model of the system:
where, xεRnx1 is the state vector, u is the input vector, y is the sensor output, A is the system coefficient matrix, B is the input coefficient matrix, C is the output coefficient matrix, dεRqx1 is the unknown input vector and EεRnxq is the unknown input distribution matrix.
The structure of the UIO is described as:
is the estimated state vector and TεRnxn, KεRnxn
and HεRnxn are matrices satisfying requirements.
The error vector is given by:
Using Eq. 21, error vector is obtained:
Using Eq. 23, derivative of the vector is:
The following relations also hold true:
K1 = (AT, CT, p); The observer gain matrix:
H = L*inverse C*L)T*(C*L))*(C*L)T; F, G, H, K1, K2 are the coefficients matrices with appropriate dimensions. I is the identity matrix. p is the desired closed loop poles. The desired observer response can be achieved by assigning suitable poles through the design of K1 and K2. The results that are obtained when the UIO observer technique is implemented, is explained in the simulation results section.
RESULTS AND DISCUSSION
Utilizing the model given by Kravaris and Chung (1987),
decoupling and linearization algorithm was designed. At first the simulation
was carried out without decoupling. Figure 2 shows the output
response of the level and temperature when the set point of the level and temperature
were changed from 1 to 60 cm and 1 to 30°C, respectively. When sudden disturbance
was introduced at 200 sec in level, it affected the temperature process due
Simulation was carried out after applying Hirschorns algorithm with external
PI controller (Nejati et al., 2012) as shown
in Fig. 3. The sudden disturbance introduced at 200 sec in
level does not affect the temperature process. It can be seen from the Fig.
2 that under the influence of the controller, ISE is improved.
Figure 4 illustrates the level tracking error between plant and Kalman observer. Kalman Observer was designed for the level process of the state space model. Actual level output of y and observer level output were obtained directly from simulation model of the plant and state estimation error was calculated. The level error varied between -2 to +2 cm when the setpoint of the level is changed from 1 to 60 cm. So, 6.66% error occurred as mentioned in Table 1.
Figure 5 illustrates the temperature tracking errors for
plant and Kalman observer. Kalman observer was designed for the temperature
process of the state space model. Actual temperature output of y and observer
were obtained directly from the simulation model of the plant and state estimation
error was calculated.
||Out put response for the step change in the level without
decoupling with PI controllers
||Out put response for the step change in the level with decoupling
with PI-SPW controllers
||Estimated error of the plant using Kalman observer (KO) for
||Estimated error of the plant using Kalman observer (KO) for
||Estimated error of the plant using UI observer (UIO) for level
|| Comparative performance of observers
The temperature error varied between -2 to +2°C when the setpoint of the
level is changed from 1 to 30°C. So 13.3% error occurred as mentioned in
Figure 6 illustrates the level tracking errors for plant
and Unknown Input Observer. Unknown Input Observer was designed for the level
process of the state space model. Actual level output of y and observer level
output were obtained directly from the simulation model of the plant and state
estimation error was calculated. The level error varied between +0.02 to -0.02
cm when the setpoint of the level is changed from 1 to 60 cm.
||Estimated error of the plant using UI observer (UIO) for temperature
So the observer is designed in such a way that the observer output follows
the system output and only 0.06% error occurred.
Figure 7 illustrates the temperature tracking errors for
plant and Unknown Input Observer. Unknown Input Observer was designed for the
temperature parameter of the state space model. Actual temperature output of
y and observer temperature output were obtained directly from simulation model
of the plant and the state estimation error was calculated. The temperature
error varied between +0.02 to -0.02 cm when the setpoint of the level is changed
from 1 to 30°C. so the UI observer is designed in such a way that the observer
output follows the system output and only 0.13% error occurred as mentioned
in Table 1. Table 1 show that UIO gives
The decoupling linearization algorithm was applied to a nonlinear MIMO interacting thermal process. The simulation results had shown that even if the processes are non-linear and interactive a satisfactory control performance could be obtained. The Kalman and Unknown Input Observer (UIO) was designed to estimate the system parameters like level and temperature for a non interacting linear MIMO system. Results of these simulations are presented in Table 1. Performance of a UIO was found to be better. The obtained outputs for UIO give less error when compared to Kalman observer.