Page - (000096) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

12.1 TrackingaChaoticReference 93 r˙0. From thosemeasurements, and from the projected set of inputs, u(t), over the interval t=0,1,...,T , the systemcanproject thevaluesof x(t)andr(t)and thus estimate the trackingerrors and the total cost,J . Inparticular, x(t)= x0+ x˙0t+ t−1∑ α=0 (t−α)u(α) r(t)= r0+ r˙0t. Here, u(α) is interpreted as an impulse that acts at the continuous time offset,α, relative to the current time.Because the systemhasdynamics x¨=u, an impulse at α causes an instantaneous increase in velocity by u(α), which then acts to change the futurepredictedvalueat time t by (t−α)u(α). These predicted values for x and r allowcalculation of the sequence of control inputsu(t)over the interval t=0,1,...,T thatminimize thecost,J . The plots in Fig.12.1 show the system trajectory that reflects theminimization ofJ .At each time step, the systemcalculates the sequenceu tominimizeJ , then applies u(0) as the control input. The remaining u values for t=1,2,...,T are ignored. In the next time step, the same procedure gives the newcontrol input for that timeperiod, and the future inputs for the optimal trajectory are again ignored. Theprocesscontinues for each timeperiodas the futurehorizon recedes. 12.2 QuickCalculationHeuristics Thesolutionfor the inputu(0) typicallydependsonthefull sequenceof inputsover t=0,1,...,T . In somecases, a relatively simpleexplicit solution foru(0)canbe obtained that requiresonly thecurrentmeasured inputs for x0, x˙0,r0, and r˙0. If asystemappliesonly thefirst input,u(0), before recalculating in thenext time step, then that systemonlyneeds theexplicit solution foru(0) toupdate thecontrol inputs ineach timestep. Forexample,withT =2, theexact solution for theabovecase is u(0)= Δ+ρ [ 3Δ+2(r˙0− x˙0 )] 1+6ρ+ρ2 , withΔ= (r0+ r˙0 )−(x0+ x˙0 ) .WithlargerT , thesolutionhasmoretermsaspow- ers ofρ, but nonetheless remains a relatively simple ratio of polynomials inρ that couldbeapproximatedbyaquickheuristic calculation.