Integrated squared error

Usually the smallest square error ISE (Integral Square Error) is used as a measure of the quality of the control ... 

Jones uses an integrated squared error loss value [L(x) as the performance criterion ... 

Figure 22.4 illustrates a different way to adjust the parameters of the controller. We postulate a reference model which tells us how the controlled process output ideally should respond to the command signal (set point). The model output is compared to the actual process output. The difference (error em) between the two outputs is used through a computer to adjust the parameters of the controller in such a way as to minimize the integral square error ... 

Describe an on-line tuning procedure for PI controllers which uses as criterion the ISE (integral square error). 

In order to improve the performance of time-delay systems, special control algorithms have been developed to provide time-delay compensation. The Smith predictor technique is the best known algorithm a related method is called the analytical predictor. Various investigators have found that based on integral squared error, the performance of the Smith predictor can be as much as 30% better than a conventional controller (Donoghue 1977 Schleck and Hanesian 1978). 

Chapter 8 considers optimal control problems for systems that are either linear or nonlinear in the state variables but are linear in the controls. The solution of this class of problems leads to bang-bang control strategies. The existence of singular or intermediate control must also be investigated. Both time-optimal control and minimum integral square error problems are discussed. 

Define a dynamic objective function DOF (for example, sum of the integral squared error (ISE) of all controlled variables). 

In order to effectively perform dynamic operability analysis of a process, one needs a quantitative measure of dynamic performance. Rise time, settling time, overshoot, and integral square error are well-known and commonly used examples of such measures. For the purpose of this framework, a dynamic operability measure is defined as the shortest time it would take a system to settle to the desired set point after a set-point change and/or a disturbance occurrence. 

Using the integral squared error of the external outputs z(t) for step changes of the individual set-points r(t) of the inputs as the objective function and the weighted sum of all individual criteria, the optimization problem can be written as ... 

Here, the integrated design problem is formulated as a multiobjective optimization problem, where the objective functions to be minimized are a weighted sum of economic terms (Fi) and the Integral Square Error (ISE) ... 

Regarding differential equations (18-23) are the mass balances (mg/l-h) of the aeration tanks (subscripts 1 and 2). They represent the rate of change of oxygen (c/, C2), biomass (x/, x ) and organic substrate (si, s ). Equations (24-29) are the mass balances for the two settlers (subscripts 1 and 2) where three layers of different and increasing biomass concentration (xdi, xbj, xri and xd2, xb2, xr2) are considered. Finally, equations 30 and 31 correspond to the integral term of the controller and to the ISE (Integral Square Error) measure respectively. 

The integral squared error between the unit impulse response of the process and that of the Nth Laguerre model is defined as... 

Equation (2.32) corresponds to the original definition of the Laguerre coefficients in Equations (2.6). It can be shown that the solutions of the coefiicients given by Equation (2.32) minimize the integral squared error in... 

The integrated squared error (ISE) criterion uses the square of the error, thereby penalizing larger errors more than smaller errors. This gives a more conservative response, i.e. faster return to the set point. 

---