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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. The LINEAR QUADRATIC REGULATOR (LQR) is a well-known design technique that provides practical feedback gains. (Continued on next page) Full-State Feedback 93 Full-State Feedback For the derivation of the LINEAR QUADRATIC REGULATOR , we assume the plant to be written in state-space form x = Ax + Bu, and that all of the n states x are available for the controller.

above equations. 19.4 Gradient Method Solution for the General Case Numerical solutions to the general problem are iterative, and the simplest approach is the gradient method. It is outlined as follows: 1. For a given xo, pick a control history u(t). 2. Propogate x˙ = f(x, u, t) forward in time to create a state trajectory. 3.

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