Quadratic performance index

The performance index for MPC applications is usually a linear or quadratic function of the predic ted errors and calculated future control moves. For example, the following quadratic performance index has been widely used ... 

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

A quadratic performance index for this example is therefore... 

The Linear Quadratic Regulator (LQR) provides an optimal control law for a linear system with a quadratic performance index. 

Using Eq. (4.1.13), the estimation problem is cast in discrete form as the determination of the coefficients c = [c1 . .., cns] that minimize the quadratic performance index ... 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

Several multivariable controllers have been proposed during the last few decades. The optimal control research of the 1960s used variational methods to produce multivariable controllers that rninirnized some quadratic performance index. The method is called linear quadratic (LQ). The mathematics are elegant but very few chemical engmeering industrial applications grew out of this work. Our systems are too high-order and nonlinear for successful application of LQ methods. 

The gain factors c, i=l,...,4 are obtained by minimizing a time-invariant quadratic performance index by the solution of the steady state matrix RICCATI-equation. 

Prett and Garcia (1988) pose the validation problem as a discrete time linear optimal control problem under uncertainty. The uncertainty is defined by simple bounds, giving a polyhedral set of uncertain parameters V. For this problem, certain forms of uncertainty, e.g., in gains only, together with a quadratic performance index can be shown to satisfy the convexity requirements for the worst-case parameters to lie at vertices of V. This allows the algorithm of Gross-mann et al, based on examination only of vertices of V, to be applied (see Section II.A.l). The mathematical formulation is... 

Each day the panel members can now be ranked according to their best performance by adding the square root of the quadratic performance index to the standard deviation. The best panel member is the one with the lowest result. 

---