Linear quadratic problem

Reformulating the necessaiy conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem ana let the ac tive set be selected by the algorithm used to solve the linear quadratic program. 

Atlians, M. (1971) The Role and Use of the Stochastic Linear-Quadratic-Gaussian Problem in Control System Design, IEEE Trans, on Automatic Control AC-16, 6, pp. 529-551. 

Burch, P.R.J., Problems With the Linear-Quadratic Dose-Response Relationship, Health Physics 44 411-413 (1983)... 

Problem 4.1 is nonlinear if one or more of the functions/, gv...,gm are nonlinear. It is unconstrained if there are no constraint functions g, and no bounds on the jc,., and it is bound-constrained if only the xt are bounded. In linearly constrained problems all constraint functions g, are linear, and the objective/is nonlinear. There are special NLP algorithms and software for unconstrained and bound-constrained problems, and we describe these in Chapters 6 and 8. Methods and software for solving constrained NLPs use many ideas from the unconstrained case. Most modem software can handle nonlinear constraints, and is especially efficient on linearly constrained problems. A linearly constrained problem with a quadratic objective is called a quadratic program (QP). Special methods exist for solving QPs, and these iare often faster than general purpose optimization procedures. 

Illustration 3.2.4 Consider the following convex quadratic problem subject to a linear equality constraint ... 

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004 Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

Edgar, T. F. Vermeychuk, J. G. and Lapidus, L., "The Linear-quadratic Control Problem A Review of Theory and... 

Such a design procedure is clearly a far cry from the linear quadratic Gaussian techniques in which robustness is obtained in an indirect manner by inventing measurement noise and introducing stochastic processes into an essentially deterministic problem. Nevertheless, the two approaches have amazing mathematical parallels [16]. 

In the sixties and seventies, in contrast to literature references to the constrained on-line optimization performed by MFC, which were only sporadic, there was an already vast and growing literature on a related problem, the linear-quadratic regulator (LQR) either in deterministic or stochastic settings. Simply stated, the LQR problem is... 

The boundary conditions are applied in the finite element method in a different way than in the finite difference method, and then the linear algebra problem is solved to give the approximation of the solution. The solution is known at the grid points, which are the points between elements, and a form of the solution is known in between, either linear or quadratic in position as described here. (FEMLAB has available even higher order approximations.) The result is still an approximation to the solution of the differential equation, and the mesh must be refined and the procedure repeated until no further changes are noted in the approximation. 

LQG-Benchmark The achievable performance of a linear system characterized by quadratic costs and Gaussian noise can be estimated by solving the linear quadratic Gaussian (LQG) problem. The solution can be plotted as a trade-off curve that displays the minimal achievable variance of the controlled variable versus the variance of the manipulated variable [115] which is used as a CPM benchmark. Operation close to optimal performance is indicated by an operating point near this trade-off curve. For multivariable control systems, H2 norms are plotted. The LQG objective function and the corresponding H2 norms are [115]... 

Quadratic programming problems are an important class of linearly constrained problems having the following form ... 

Kuhn-Tucker conditions and reduces the quadratic programming problem to what is referred to as a linear complementarity problem. 

Many problems in engineering mathematics lead to the construction of models that can be used to describe physical systems. Because of the power of technology, a model may be derived from a system of a few equations that may be linear, quadratic, exponential, or trigonometric—or a system of many equations of even greater complexity. In engineering, such equations include ordinary differential equations, differential algebraic equations, and partial differential equations. 

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