Robust Control of Missile

A Nonlinear Robust Aerodynamic Control Systems, an Application to Missile Control System

Abdulhmeed Mohamed ElhassanJune 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  fourrespective linear dynamic controllers

SpecificationsSpecifications

Missile ModelMissile Model

Linearization in Modern ControlLinearization in Modern ControlFeedback Linearization MethodFeedback Linearization Method

StateState--Dependent Dependent RiccatiRiccati Equation MethodEquation Method

Quickest Descent MethodQuickest Descent Method

Recursive BackRecursive Back--stepping Methodstepping Method

Gain Scheduling methodGain Scheduling method

Feedback Linearization MethodFeedback Linearization Methodtransformation of the nonlinear system into an equivalent lineartransformation of the nonlinear system into an equivalent linear system system

through a change of variables and a suitable control inputthrough a change of variables and a suitable control input

AdvantagesAdvantages

SimpleSimple

DisadvantagesDisadvantagesa precise knowledge of the a precise knowledge of the system model is required in system model is required in order to synthesize the order to synthesize the nonlinear controller,nonlinear controller,the system zero dynamics the system zero dynamics must be stable,must be stable,the system states must be the system states must be measurable.measurable.

StateState--Dependent Dependent RiccatiRiccati Equation MethodEquation Method

AdvantagesAdvantagesavoiding intensive avoiding intensive interaction interaction calculationcalculation

DisadvantagesDisadvantagesis computationally is computationally demanding, requiring demanding, requiring the solution of a 7*7 the solution of a 7*7 algebraic algebraic RiccatiRiccatiequation at each equation at each samplesample

Recursive Recursive BacksteppingBackstepping MethodMethod

AdvantagesAdvantagesimposes the desired imposes the desired properties of stability properties of stability by fixing the by fixing the functions initially, functions initially, then by calculating then by calculating the other functions the other functions in a recursive wayin a recursive way

DisadvantagesDisadvantages

GAIN SCHEDULINGGAIN SCHEDULING Advantages.Advantages.

1.1. Employs powerful linear design tools on difficult Employs powerful linear design tools on difficult nonlinear problems.nonlinear problems.

2.2. Most performance specifications are in linear terms, Most performance specifications are in linear terms, involving a mixture of timeinvolving a mixture of time--domain and frequencydomain and frequency--domain specifications.domain specifications.

3.3. Carried out using the physical variables . (nonlinear Carried out using the physical variables . (nonlinear control approaches involve coordinate control approaches involve coordinate transformations).transformations).

4.4. Gain scheduling enables a controller to respond Gain scheduling enables a controller to respond rapidly to changing operating conditions (which rapidly to changing operating conditions (which themselves must vary themselves must vary ‘‘slowlyslowly’’ in the LPV or Quasiin the LPV or Quasi--LPV approach; The computational burden of LPV approach; The computational burden of linearization scheduling approaches is often much linearization scheduling approaches is often much less than for other nonlinear design approaches.less than for other nonlinear design approaches.

5.5. QuasiQuasi--LPV approaches offer guaranteed stability and LPV approaches offer guaranteed stability and performance properties.performance properties.

GAIN SCHEDULINGGAIN SCHEDULING DisadvantagesDisadvantages

QuasiQuasi--LPV approaches are computationally LPV approaches are computationally intensive.intensive.

Gain scheduling often involves several ad hoc Gain scheduling often involves several ad hoc steps, beginning with problem formulation. This steps, beginning with problem formulation. This can be suitable in simple situations, but can be suitable in simple situations, but increasingly troublesome as more complicated increasingly troublesome as more complicated controllers are designed.controllers are designed.

Linearization gain scheduling stability can be Linearization gain scheduling stability can be assured only locally and in a `slowassured only locally and in a `slow--variation variation setting, and typically there are no performance setting, and typically there are no performance guarantees.guarantees.

GAIN SCHEDULINGGAIN SCHEDULING

Two methodsTwo methodsClassical oneClassical one

(y-y1)/(x-x1) = (y2-y1)/(x2-x1)

GAIN SCHEDULINGGAIN SCHEDULING

QuasiQuasi--LPVLPV

the plant dynamics are rewritten to hide nonlinearities as time-varying parameters that are then used as scheduling variables.

Steps in Designing Gain ScheduledSteps in Designing Gain Scheduled ControllerController

compute a linear parametercompute a linear parameter--varying varying model for the plant.model for the plant.use linear design controller techniques use linear design controller techniques for the LPV plant modelfor the LPV plant modelimplementing family of linear controllers implementing family of linear controllers such that the controller coefficients such that the controller coefficients (gains) are varied (scheduled) according (gains) are varied (scheduled) according to the current value of the scheduling to the current value of the scheduling variables.variables.performance assessment.performance assessment.

Controller DesignController Design

parameter range:parameter range:

A1min =.5 A1max =4; A2min =0 A2max =106;

parameter range and rate of parameter range and rate of variation of timevariation of time--varying varying

pv = pvec('box',[A1min

A1max ; A2min A2max ])

affine model: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_als0 , s1 , and s2 are given system matrices

form the plant interconnection and form the plant interconnection and append the shaping filtersappend the shaping filters

[pdP,r] = sconnect('r','e=r-GP1;K','K:e;G(2)','G:K',pdG);

Paug = smult(pdP,sdiag(w1,w2,eye(2)))

For loop-shaping purposes, we must form the augmented plant

FiltersFilters

• 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 gainperform the gain--scheduled scheduled controllercontroller

• [gopt,pdK] = hinfgs(Paug,r)

simulate the step response of the gainsimulate the step response of the gain--scheduled systemscheduled 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 closedplot the closed--loop step responseloop 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))

RESULTSStep response information

Settling Time:      0.3343Rise Time:            0.0805Settling Min :      0.9132Settling Max:       1.0949Overshoot:     11.4086%Undershoot:                 0Peak:                    1.0949

Peak Time:           0.1840