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19 LINEAR QUADRATIC REGULATOR - MIT OpenCourseWare

19 LINEAR QUADRATIC REGULATOR Introduction The simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead of transfer functions. The notion of optimality is closely tied to MIMO control system design. Optimal controllers, , controllers that are the best possible, according to some figure of merit, turn out to generate only stabilizing controllers for MIMO plants. In this sense, optimal control solutions provide an automated design procedure we have only to decide what figure of merit to use.

Consider a scalar, continuous, positive definite function of the state V(x). If this function is always decreasing, except at the origin, then the origin is asymptotically stable. There is an intuitive feel to the Liapunov function; it is like the energy of a physical system. If it can

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