Page - (000095) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

92 12 ModelPredictiveControl Modelpredictivecontrolmaybeenhancedbyadaptivefeedbackthatmodifiesthe parametersor the formfor themodelof internaldynamics.Thegeneral approaches ofmodelpredictivecontrolandadaptivecontrolprovidebenefitsof robustnesswith respect to themodelof internaldynamics. 12.1 TrackingaChaoticReference Figure12.1showstheperformanceofasimplemodelpredictivecontrolsystemwhen tracking a chaotic reference signal. Thefigure caption describes the calculation of the chaotic input signal (blue curve). In this example, themodel predictive control systembeginswithan internalprocessgivenby x¨=u (12.1) forcontrol input,u.Theinputcanbethoughtofasaforcethatalters theacceleration of the systemstate. In thisexample, thecost function is summedoveraseriesofdiscrete timepoints fromthepresentuntil a futurehorizonat timeT , yielding J = T∑ t=0 [ x(t)−r(t)]2+ρu(t)2. Thefirst term is the squared tracking error. The second term is the squared control input signal, weighted by ρ, which is the relative cost of the input signal to the tracking error. The sum includes a series of input controls and tracking deviations over a finite period from the present to the future horizon, T . As time passes, the timewindow (0,T)moves so that t=0 isalways thepresent time. The system has sensors to measure the current locations and time derivatives (velocities) for the system state and the reference signal, given as x0, x˙0, r0, and ρ=1 ρ=10 ρ=100 Time Fig. 12.1 Tracking an environmental reference signal by model predictive control. The blue curvesshowtheenvironmental signal, achaotic formofDuffing’sequation, r¨+0.15r˙−r+r3= 0.3cos(t),with initial conditions r˙ =1.001 and r =−1.The gold curves show the tracking per- formance for different values for the cost,ρ, of control input signals relative to the trackingerror. Ahighercontrol input cost favorsweakercontrol inputs andgreater trackingerror