By Feng Lin
Comprehensive and obtainable consultant to the 3 major methods to strong keep watch over layout and its functions
optimum keep an eye on is a mathematical box that's all for regulate regulations that may be deduced utilizing optimization algorithms. The optimum keep an eye on method of strong keep watch over layout differs from traditional direct techniques to powerful keep an eye on which are typically mentioned by way of to start with translating the strong keep watch over challenge into its optimum keep an eye on counterpart, after which fixing the optimum keep an eye on challenge.
powerful keep an eye on layout: An optimum regulate process bargains an entire presentation of this method of powerful keep an eye on layout, offering smooth keep watch over conception in an concise demeanour. the opposite significant methods to strong keep an eye on layout, the H_infinite method and the Kharitonov process, also are coated and defined within the least difficult phrases attainable, with a purpose to offer an entire review of the realm. It contains up to date learn, and provides either theoretical and useful functions that come with versatile buildings, robotics, and car and plane keep watch over.
strong regulate layout: An optimum keep an eye on procedure may be of curiosity to these desiring an introductory textbook on powerful keep an eye on thought, layout and functions in addition to graduate and postgraduate scholars desirous about platforms and keep watch over study. Practitioners also will locate the purposes awarded precious while fixing sensible difficulties within the engineering box.
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Additional resources for Robust Control Design: An Optimal Control Approach (RSP)
We need to find some effective ways to compute eAt . One such way is to use the Laplace transform. It is not difficult to see that the Laplace transform of eAt , denoted by L eAt , is L eAt = sI − A −1 Hence, eAt is the inverse Laplace transform of Is − A −1 . eAt = L−1 sI − A −1 Let us show the computation in the following example. 1 Let 1 0 0 2 A= then s 0 1 0 s−1 0 − = 0 s 0 2 0 s−2 ⎡ ⎤ 1 −1 0 s−1 0 ⎢ ⎥ sI − A −1 = = ⎣ s−1 1 ⎦ 0 s−2 0 s−2 sI − A = The inverse Laplace Transform can be calculated as eAt = L−1 sI − A −1 = Let us consider another example.
5 Consider a linear time-varying system t 0 x˙ 1 = x˙ 2 0 1 x1 x2 We can rewrite the state equation as x˙ 1 = tx1 and x˙ 2 = x2 . We know the solution to x˙ 2 = x2 : x2 t = et−t x2 t . To solve x˙ 1 = tx1 , we separate the variables: x˙ 1 = tx1 dx1 = tx1 dt dx1 ⇒ = tdt x1 ⇒ ⇒ ln x1 x1 t x1 t = 1 2 t 2 t t ⇒ ln x1 t − ln x1 t = ⇒ ln ⇒ 1 2 1 2 t − t 2 2 1 x1 t 1 = t2 − t 2 x1 t 2 2 x1 t 1 2 1 2 = e2t −2t x1 t 1 2 1 2 −2t ⇒ x1 t = e 2 t x1 t Combine two equations: x1 t x1 t = 1 2 1 2 −2t e2t 0 0 et−t x1 t x1 t 24 FUNDAMENTALS OF CONTROL THEORY In other words, the state transition matrix is given by 1 2 1 2 −2t e2t t t = 0 et−t 0 We may not always be able to find the analytical expression of the state transition matrix of a system.
3 Let us calculate the response of the following system with the initial condi1 tion x 0 = and the input u t = 1. 2 2e−t − e−2t −2e−t + 2e−2t eAt = e−t − e−2t −e−t + 2e−2t Hence x t =eA t x0 + =eA t x0 + = eA t− Bu t 0 d 0 t eA Bd 0 2e−t − e−2t −2e−t + 2e−2t + = t e−t − e−2t −e−t + 2e−2t 2e− − e−2 −2e− + 2e−2 e−t + −e−t e− − e−2 −e− + 2e−2 e− − e−2 −e− + 2e−2 t 0 0 d 1 d t 1 e−t −e− + e−2 = 2 −t + −e e− − e−2 = 1 −1 0 1 1 − −e−t + e−2t − + 2 2 0 e−t − e−2t e−t −e−t 1 1 −2t + e = 2 2 −e−2t The output response is y= 1 2 1 1 −2t + e 2 2−2t −e = 1 1 −2t − e 2 2 Another way to find the response of a linear time-invariant system is to first find its transfer function (matrix).