A Nonlinear Robust Aerodynamic Control Systems, an Application to Missile Control System Abdulhmeed Mohamed Elhassan June 2012 Thanks to Dr. Elhassan Bashier Elaàgab Objectives Designing a nonlinear controller by scheduling linear H∞ controller designs at four constant operating conditions bounding the operating range. That is ,the linear designs based on linearization of the missile model at four distinct operating conditions and application of an H∞ software tool (MATLAB) to calculate four respective linear dynamic controllers Specifications Missile Model Linearization in Modern Control Feedback Linearization Method State-Dependent Riccati Equation Method Quickest Descent Method Recursive Back-stepping Method Gain Scheduling method Feedback Linearization Method transformation of the nonlinear system into an equivalent linear system through a change of variables and a suitable control input Advantages Disadvantages a precise knowledge of the system model is required in order to synthesize the nonlinear controller, the system zero dynamics must be stable, the system states must be measurable. Simple State-Dependent Riccati Equation Method Advantages avoiding intensive interaction calculation Disadvantages is computationally demanding, requiring the solution of a 7*7 algebraic Riccati equation at each sample Recursive Backstepping Method Advantages imposes the desired properties of stability by fixing the functions initially, then by calculating the other functions in a recursive way Disadvantages GAIN SCHEDULING Advantages. 1. 2. 3. 4. 5. Employs powerful linear design tools on difficult nonlinear problems. Most performance specifications are in linear terms, involving a mixture of time-domain and frequencydomain specifications. Carried out using the physical variables . (nonlinear control approaches involve coordinate transformations). Gain scheduling enables a controller to respond rapidly to changing operating conditions (which themselves must vary ‘slowly’ in the LPV or QuasiLPV approach; The computational burden of linearization scheduling approaches is often much less than for other nonlinear design approaches. Quasi-LPV approaches offer guaranteed stability and performance properties. GAIN SCHEDULING Disadvantages Quasi-LPV approaches are computationally intensive. Gain scheduling often involves several ad hoc steps, beginning with problem formulation. This can be suitable in simple situations, but increasingly troublesome as more complicated controllers are designed. Linearization gain scheduling stability can be assured only locally and in a `slow-variation setting, and typically there are no performance guarantees. GAIN SCHEDULING Two methods Classical one (y-y1)/(x-x1) = (y2-y1)/(x2-x1) GAIN SCHEDULING Quasi-LPV the plant dynamics are rewritten to hide nonlinearities as time-varying parameters that are then used as scheduling variables. Controller Steps in Designing Gain Scheduled compute a linear parameter-varying model for the plant. use linear design controller techniques for the LPV plant model implementing family of linear controllers such that the controller coefficients (gains) are varied (scheduled) according to the current value of the scheduling variables. performance assessment. Controller Design parameter range: A1min =.5 A2min =0 A1max =4; A2max =106; parameter range and rate of variation of time-varying pv = pvec('box',[A1min A1max ; A2min A2max ]) affine model: • pdP = psys(pv,[s0 s1 s2]) • s0 = ltisys([0 1;0 0],[0;1],[-1 0;0 1],[0;0]) • s1 = ltisys([-1 0;0 0],[0;0],zeros(2),[0;0],0) % A1_al • s2 = ltisys([0 0;-1 0],[0;0],zeros(2),[0;0],0) % A2_al s0 , s1 , and s2 are given system matrices form the plant interconnection and append the shaping filters For loop-shaping purposes, we must form the augmented plant [pdP,r] = sconnect('r','e=rGP1;K','K:e;G(2)','G:K',pdG); Paug = smult(pdP,sdiag(w1,w2,eye(2))) Filters • Using Magshape GUI in matlab and the command • LPF W1(s) = 2.01/ (s + 0.201) • HPF W2(s) = (9.678s3 + 0.029s2)/ (s3 + 1.206e4s2 + 1.136e7s + 1.066e10) perform the gain-scheduled controller • [gopt,pdK] = hinfgs(Paug,r) simulate the step response of the gainscheduled system • spiral trajectory • A1α(t) = 2.25 + 1.70 e–4t cos(100 t) • A2α(t) = 50 + 49 e–4t sin(100 t) • function p = spiral(t) p = [2.25 + 1.70*exp(-4*t).*cos(100*t) ; ... 50 + 49*exp(-4*t).*sin(100*t)]; plot the closed-loop step response [t,x,y]=pdsimul(pCL,'spiralt',0.5) where pcl is the polytopic representation of the closed-loop system plot(t,1-y(:,1)) RESULTS Step response information Settling Time: 0.3343 Rise Time: 0.0805 Settling Min : 0.9132 Settling Max: 1.0949 Overshoot: 11.4086% Undershoot: 0 Peak: 1.0949 Peak Time: 0.1840