Closed-loop eigenvalues

Continue the recursive steps until the solution settles down (when k = 50, or kT = 5 seconds) and hence determine the steady-state value of the feedback matrix K(0) and Riccati matrix P(0). What are the closed-loop eigenvalues ... 

Rlccatl matrix, see equation (9.45) %Closed-loop eigenvalues... 

Note that these eigenvalues are neither the openloop eigenvalues nor the closed-loop eigenvalues of the system They are eigenvalues of a completely different matrix, not the 4 or the 4cl matrices. 

The close loop response depends not only on the closed loop eigenvalues but also on eigenvectors. Intuitive specification of closed loop eigenvalues may be difficult. 

The system uses a full order closed-loop rotor flux observer, by configuring the closed-loop eigenvalues to achieve a smooth handover between current and voltage model, to combines the both advantages at different speed segments effectively, which makes the system suitable for the flux observation in a wide speed range . ... 

The admissible region for closed-loop eigenvalues is denoted as F, a system is called F-stable if all its eigenvalues are located in this region and an imcertain system is called robustly F-stable if all eigenvalues for all operating conditions are contained in F. The definition of F-stability permits arbitrary regions in the complex 5 -plane and does not imderlie any restrictions. It also includes the special cases of the left half-plane for Hurwitz stability and the unit circle for Schur stability. 

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

The roots of equation (8.96) are the closed-loop poles or eigenvalues. 

Estimation error covariance matrix %Closed-loop estimator eigenvalues... 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

The eigenvalues of the system matrix (A - BK) are called the regulator poles. What we want is to find K such that it satisfies how we select all the eigenvalues (or where we put all the closed-loop poles). 

We next return to our assertion that we can choose all our closed-loop poles, or in terms of eigenvalues, ib K- This desired closed-loop characteristic equation is... 

This is the result of full state feedback pole-placement design. If the system is completely state controllable, we can compute the state gain vector K to meet our selection of all the closed-loop poles (eigenvalues) through the coefficients a . 

And we can easily obtain a state-space representation and see that the eigenvalues of the state matrix are identical to the closed-loop poles ... 

The first objective can be reached by using a general property of discretized system namely, if for the system (23) there exists a matrix Kd such that Ad + BdKd has all its eigenvalues inside the unit circle, then the controller u k) = Kdx(k) will also stabilize the continuous system. Thus, the closed-loop stability can be then assured by properly assigning the discrete poles inside the unit circle. 

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 second important bifurcation that is connected with a stability change in a stationary state is the /fop/bifurcation. At a Hopf bifurcation, the real parts of two conjugate complex eigenvalues of J vanish, and as Hopf s theorem ensures, a periodic orbit or limit cycle is bom. A limit cycle is a closed loop in phase space toward which neighboring points (of the kinetic representation) are attracted or from which they are repelled. If all neighboring points are attracted to the limit cycle, it is stable otherwise it is unstable (see Ref. 57). The periodic orbit emerging from a Hopf bifurcation can be stable or unstable and the existence of a Hopf bifurcation cannot be deduced from the mere fact that a system exhibits oscillatory behavior. Still, in a system with a sufficient number of parameters, the presence or absence of a Hopf bifurcation is indicative of the presence or absence of stable oscillations. 

According to stability analysis of linear time invariant system, stability of the closed-loop system x= A- BK)x depends on the eigenvalue of eigenmatrix (A - BK). In other words, the condition that the stabifity is positive is all the eigenvalues of matrix (A - BK) are negative. The switching function of SMC is... 

Every dynamic adaptronic system must be checked for stability in the case of disturbances. For linear elastic adaptronic structures, asymptotic stability as defined in Sect. 5.2.4 is guaranteed if the poles (or eigenvalues) of the closed-loop active system lie in the left complex half-plane, i.e. if they have negative real parts. More stringent stability criteria, such as the generalized Nyquist criterion [7], also consider the zeros of the adaptronic system. 

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