Robust control of seismic structures employing active suspension elements (Q2725702)

The authors comment that passive as well as active controls have been tried to deal with vibrations of structures caused by seismic phenomena. From a purely theoretical view, active controls are always preferred, but their cost is prohibitive. The authors follow the ideas of Leitmann, published in 1994, for using semi-active controls. Let \(x(t)\) denote the \(n\)-dimensional state of a plant modelled by the equation: \(dx/dt=Ax +b(x,z,u) +e(x,z,t)\), where \(A\) is an \(n\times n\) constant matrix, \(z\) denotes a disturbance, \(u\) is an external control, while \(b(x,z,u)\) is the actual control function, which is linear in \(u\), and a continuous function of \(x\) and \(z\). The feedback \((x,z) \to u\) is denoted by \(p(x,z)\). The norm \(\|x\|_P\) is defined by: \(\|x\|_P^2= \langle x,P(x) \rangle\), where \(P\) is an \(n\times n\) constant positive definite matrix. The authors study stability through the use of Lyapunov functions. The obvious choice for such a function \(V\) is the norm \(\|x\|_P\). Thus they could look for a control input \(u\) that minimizes \(\|x\|_P\) at some future time. But they choose to minimize the derivative of their Lyapunov function along the path of the system, arguing that the smaller the derivative is, the stronger the tendency shall be to pull the system towards the origin, which is of course true. Historically this is consistent with the steepest descent idea of Weierstrass. The reviewer comments that Pontryagin showed that one can do better by using his ideas of following the optimum of Pontryagin's Hamiltonian (commonly resulting in the bang-bang principle) if one wishes to reduce some form of energy to the smallest value in the shortest time. Here the authors proceed to choose another positive definite matrix \(Q\), solving the Lyapunov algebraic equation \(Q=-(PA+A^TP)\) for \(P\). As is well known, each Lyapunov function determines a region of stability. Using a similar idea, the authors determine a region in which the functional based on the derivative of the Lyapunov function is less than a negative quantity including the minimal and maximal eigenvalues. Thus they determine the ball of ultimate boundedness of \(x\), with \(\|x\|_P\geq r\), where \(r\) is of the form: \(c\lambda_{\max} (P)/ \lambda_{\min} (Q)\), while the ball of ultimate boundedness is given by \(\rho=r \{\lambda_{\max} (P)/ \lambda_{\min} (P)\}\), so that any response \(x(t)\) that enters this ball at time \(t=t^0\) remains in it for all times greater than \(t^0\). Finally, the authors offer a structural model on which they test both a ``strictly Lyapunov'' controller and also a fuzzy Lyapunov control.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00020].