Optimal feedback matrix

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

The feedback matrix can be calculated based on optimal control theory like linear quadratic regulator, LQG. Optimal control methods are based on the concept of minimizing a cost criterion. The criterion of cost, represented by J, has a common format ... 

Photoswitchable enzymes could have an important role in controlling biochemical transformations in bioreactors. Various biotechnological processes generate an inhibitor, or alter the environmental conditions (pH, for example) of the reaction medium. Photochemical activation of enzymes that adjust environmental conditions or deplete the inhibitor to a low concentration may maintain the bioreactor at optimal performance. More specifically, integration of the photoswitchable biocataly-tic matrix with a sensory electrode might yield a feedback mechanism in which the sensor element triggers the light-induced activation/deactivation of the photosensitive biocatalyst. 

The LQP is the only general optimal control problem for which there exists an analytical representation for the optimal control in closed-loop or feedback form. For the LQP, the optimal controller gain matrix K becomes a constant matrix for tf>°°. K is independent of the initial conditions, so it can be used for any initial condition displacement, except those which, due to model nonlinearities, invalidate the computed state matrices. 

The term in braces is the process transfer function. Inasmuch as x has three components and c only two, the system is underdetermined—i.e. many combinations of x s will yield the same c s, at least in the steady-state. In this paper a method of selecting the x vector which is optimal in the steady-state is described. The method is shown schematically in Figure 1. The input vector x is determined by an uncoupling matrix Gm whose inputs come from the feedback controller Gc. A performance function is attached to the x vector. The elements of Gm are computed so as to minimize (or maximize) this function in the steady-state. 

The weight on the error, d, is a. P x P diagonal matrix. Similarly, 1 is the weight on the input (suppression move) and is an M x M diagonal matrix. The optimization problem is solved at every sampling time when a new prediction is updated by recent feedback measurements. Here, is a matrix of the set... 

Gluck et al. (1996) adapted optimal control theory (OCT) to the damper placement problem. OCT is used to minimise the performance objective by optimising the location of linear passive devices. Since passive dampers cannot provide feedback in terms of optimal control gains, three approaches (response spectrum approach, single mode approach, and truncation approach) are proposed to remove the off-diagonal state interactions within the gain matrix and allow approximation of floor damping coefficients. Combination of these methods with OCT and passive devices achieves an equivalent effect compared to active control. 

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