sampled_data_lqr_2009_11_24_01

sampled_data_lqr_2009_11_24_01 - 18 1 Sampled-Data LQR S...

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Unformatted text preview: 18 - 1 Sampled-Data LQR S. Lall, Stanford 2009.11.24.01 18. Sampled-Data LQR • Implementation of digital controllers • Order of discretization • Sampled-data problem formulation • Equivalent discrete-time problem • Solution of discrete-time problem with cross-terms • Comparison of discrete-time and sampled-data controller • Comparison of continuous-time and sampled-data controller • Performance comparisons at different sampling rates • Variation of cost with sampling rate 18 - 2 Sampled-Data LQR S. Lall, Stanford 2009.11.24.01 The Key Points of This Section • Neither continuous-time design nor discrete-time design are optimal • Both ignore essential features of interconnection of physical systems to digital hard- ware • instead, we use sampled-data design • cost function is continuous-time, accounts for intersample behavior • problem is solved by converting it to a discrete-time problem with same cost 18 - 3 Sampled-Data LQR S. Lall, Stanford 2009.11.24.01 Implementation of Digital Controllers state state samples controller output controller control signal u ( t ) x ( t ) x ç = Ax + Bu S H x d ( k ) u d ( k ) • S is the sampler ; continuous signal x ( t ) is sampled to give discrete signal x d ( k ) = x ( kh ) • H is the hold ; discrete signal is held constant on intervals [ kh, ( k + 1) h ) to give continuous signal u ( t ) 18 - 4 Sampled-Data LQR S. Lall, Stanford 2009.11.24.01 Order of discretization so we have a continuous-time physical system , but need to design a discrete-time controller two approaches 1. design using continuous-time methods, then discretize the controller 2. discretize the plant, then design a controller in discrete-time 18 - 5 Sampled-Data LQR S. Lall, Stanford 2009.11.24.01 implementation of digital controllers which is right? neither method 1 does not account for the effects of sample and hold; it assumes • measurements are available at all times, not just sampling instants • and control signal varies smoothly, not piecewise constant...
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sampled_data_lqr_2009_11_24_01 - 18 1 Sampled-Data LQR S...

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