State feedback gain matrix

In equation (8.93), r(t) is a vector of desired state variables and K is referred to as the state feedback gain matrix. Equations (8.92) and (8.93) are represented in state variable block diagram form in Figure 8.7. 

In reverse-time, starting with P(A ) = 0 at NT = 20 seconds, compute the state feedback gain matrix K(kT) and Riccati matrix P(kT) using equations (9.29) and (9.30). Aiso in reverse time, use the desired state vector r(/c7 ) to drive the tracking equation (9.53) with the boundary condition s(N) = 0 and hence compute the command vector y kT). 

Ifae state feedback gain matrix K and the observer spdn matrix thus obtained are as folloess ... 

Now suppose a feedback controller is added to the system. The manipulated variables tp will now be set by the feedback controller. To keep things as simple as possible, let us make two assumptions that arc not very good ones, but permit us to illustrate an important point. We assume that the feedback controller matrix consists of just constants (gains). 7 3itd we assume that there are as many manipulated variables m as state variables x. 

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. 

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. 

Essentially, a feedforward gain matrix and state feedback were used in a state space representation to achieve the desired result in the classical decoupling methods. In general, state feedback can be used to place poles as well as to affect the element zeros[4]-[6] of transfer function matrices in MIMO systems. The invariant zeros [4]-[6] of MIMO systems are, however, not affected by state feedback or feedforward gain. In the classical decoupling methods, the invariant zeros are typically cancelled by a number of the new system poles, thus, effectively leading to an overaU reducedr-order system. 

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