Linear quadratic performance

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 ... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

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... 

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

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... 

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... 

The proposed strategies for stabilization of gas-lifted oil wells are offline methods which are unable to track online dynamic changes of the system. However, system parameters such as flow rate of injected gas and also noise characteristic are not constant with respect to time. An adaptive Linear Quadratic Gaussian (LQG) approach is presented in this paper in which the state estimation is performed using an Adaptive Unscented Kalman Filter (AUKF) to deal with unknown time-varying noise statistics. State-feedback gain is adaptively calculated based on Linear Quadratic Regulator (LQR). Finally, the proposed control scheme is evaluated on a simulation case study. 

Various performance indices have been suggested [54, 53, 149, 20, 148] and several approaches have been proposed for estimating the performance index for SISO systems, including the normalized performance index approach [53], the three estimator approach [175[, and the filtering and correlation analysis (FCOR) approach [115[. A model free approach for linear quadratic CPM from closed-loop experiments that uses spectrum analysis of the input and output data has been suggested [136]. Implementation of SISO loop based CPM tools for refinery-wide control loop performance assessment has been reported [294]. 

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]... 

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator... 

In the optimization block, the control input applied to the smart structural system is obtained by minimizing a generalized linear quadratic (LQ) performance index with weights on the control moves. The performance index is given by... 

Chapter 12 considers the combination of optimal control with state and parameter estimation. The separation principle is developed, which states that the design of a control problem with measurement and model uncertainty can be treated by first performing a Kalman filter estimate of the states and then developing the optimal control law based upon the estimated states. For linear regulator problems, the problem is known as the linear quadratic Gaussian (LQG) problem. The inclusion of model parameter identification results in adaptive control algorithms. 

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