Page - (000078) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

72 9 StateFeedback 2 4 6 8 10 -0.4 -0.2 0.2 0.4 0.6 0.8 2 4 6 8 10 -0.4 -0.2 0.2 0.4 0.6 0.8 2 4 6 8 10 -0.5 0.5 1.0 2 4 6 8 10 -0.5 0.5 1.0 (a) (b) (c) (d) Fig. 9.2 Response to impulse perturbations of systems with state feedback, as in Fig.9.1. a Responseofthestate-spacesysteminEq.9.2.Curvesshowx1 = y forcostratioρ= (0.25,1,4) in blue,gold,andgreen, respectively. In thiscase, the impulseperturbationenters thesystemthrough u inEq.9.2, affecting x˙2.bModified state-spacemodel that has two inputs, one each into x˙1 and x˙2, associatedwith the statematrixB= I2. Impulse perturbation comes into x˙2 as in the original model. In this case, there are two control inputs for feedback via the gainmatrix,K. The opti- mizationusesboth inputs,allowingthefeedback tocontroleachstateseparately.Thatextensionof control directly to all states allows the feedback system to bring the state responses back to zero more quickly than in the original systemwith only one state feedback. c and dResponse of the second state, x2. Systems for eachpanelmatch to the correspondingpanels above.Note ind that the second input for feedbackdrives the state to zeromore quickly than in c,whichhas only one input Methods thatminimize quadratic costs orH2 norms can produce systemswith poor stability margins. To obtain guaranteed stability margins, one canminimize costs subject toaconstrainton theminimumstabilitymargin. 9.2 TrackingExample Consider the trackingexample fromthepreviouschapter.That examplebeganwith theprocess inEq.4.1as P(s)= 1 (s+a)(s+b) = 1 s2+αs+β,