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Vehicle.asv
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Vehicle.asv
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classdef Vehicle < handle
%FACILITY Summary of this class goes here
% Detailed explanation goes here
properties
% Indices
platoonIndex % k
vehicleIndex % i
% Parameters
vehicleParameters
noiseMean
noiseStd
% Passivity Indices
nu
rho
% Local Controller Gains
localControllerGains1 = []
localControllerGains2 = []
% Global ControllerGains
controllerGains1 = []
controllerGains2 = []
% Data to be distributed (in decentralized schemes)
dataToBeDistributed
controllerGainsCollection
% States
desiredSeparation %From the leader
desiredSeparations %From all others
states % x_ik
noise % v_ik
controlInput
errors
outputs
% state history
stateHistory = []
% error history
errorHistory = []
% Predefined controls
plannedControls = [] % matrix of paris [t_i,u_i]
% GeometricProperties (for plotting)
inNeighbors = []
outNeighbors = []
% graphicHandles
graphics = []
R_i
end
methods
function obj = Vehicle(k,i,parameters,states,desiredSeparation,noiseMean,noiseStd)
% Constructor
obj.platoonIndex = k;
obj.vehicleIndex = i;
obj.vehicleParameters = parameters; %[mass,length,height1,height2]
obj.states = states; % states of the i^{th} vehicle
obj.desiredSeparation = desiredSeparation; % need to track this signal (desired position,velocity and 0 acceleration for i^{th} vehicle)
% External disturbances represented by random noise
obj.noiseMean = noiseMean;
obj.noiseStd = noiseStd;
obj.noise = noiseMean + noiseStd*randn(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Initial controller values
obj.errors = zeros(3,1);
obj.controlInput = zeros(1);
obj.outputs = zeros(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
obj.inNeighbors = [];
obj.outNeighbors = [];
end
% This function is used to draw a "car" shape object
function outputArg = drawVehicle(obj,figNum)
figure(figNum); hold on;
if ~isempty(obj.graphics)
delete(obj.graphics);
end
length = obj.vehicleParameters(2);
height1 = obj.vehicleParameters(2)/8;
height2 = obj.vehicleParameters(2)/8;
radius = obj.vehicleParameters(2)/16;
pos = obj.states(1);
% Draw the car body
% poly1 = polyshape([pos, (pos-length), (pos-length), pos],[0.5, 0.5, 0.5+height1, 0.5+height1]+8*(obj.platoonIndex-1)*[1, 1, 1, 1]);
poly1 = polyshape([pos, (pos-length), (pos-length), pos],[0.5, 0.5, 0.5+height1, 0.5+height1]);
obj.graphics(1) = plot(poly1,'FaceColor','r');
% poly2 = polyshape([0.5*(pos+0.5*(pos+(pos-length))), 0.5*((pos-length)+0.5*(pos+(pos-length))),...
% 0.5*((pos-length)+0.5*(pos+(pos-length))), 0.5*(pos+0.5*(pos+(pos-length)))],...
% [0.5+height1, 0.5+height1, (0.5+height1)+height2, (0.5+height1)+height2]+8*(obj.platoonIndex-1)*[1, 1, 1, 1]);
poly2 = polyshape([0.5*(pos+0.5*(pos+(pos-length))), 0.5*((pos-length)+0.5*(pos+(pos-length))),...
0.5*((pos-length)+0.5*(pos+(pos-length))), 0.5*(pos+0.5*(pos+(pos-length)))],...
[0.5+height1, 0.5+height1, (0.5+height1)+height2, (0.5+height1)+height2]);
obj.graphics(2) = plot(poly2,'FaceColor','r');
% Draw two wheels
% viscircles([0.5*((pos-length)+0.5*(pos+(pos-length))), 0.3+8*(obj.platoonIndex-1)], 0.3,'Color','k');
% viscircles([0.5*(pos+0.5*(pos+(pos-length))), 0.3+8*(obj.platoonIndex-1)], 0.3,'Color','k');
obj.graphics(3) = viscircles([0.5*((pos-length)+0.5*(pos+(pos-length))), 0.3], radius,'Color','k');
obj.graphics(4) = viscircles([0.5*(pos+0.5*(pos+(pos-length))), 0.3], radius,'Color','k');
% Vehicle number
obj.graphics(5) = text(pos,0.2,num2str(obj.vehicleIndex));
end
% This function is used to plot the real time states (i.e., position) of the
% i^{th} vehicle at the position before each vehicle
% function outputArg = drawState(obj,figNum)
% figure(figNum); hold on;
%
% if ~isempty(obj.graphics)
% delete(obj.graphics)
% end
%
% end
function outputArg = redrawVehicle(obj,figNum)
figure(figNum); hold on;
if ~isempty(obj.graphics)
delete(obj.graphics);
length = obj.vehicleParameters(2);
height1 = obj.vehicleParameters(2)/8;
height2 = obj.vehicleParameters(2)/8;
radius = obj.vehicleParameters(2)/16;
pos = obj.states(1);
% Draw the car body
poly1 = polyshape([pos, (pos-length), (pos-length), pos],[0.5, 0.5, 0.5+height1, 0.5+height1]);
obj.graphics(1) = plot(poly1,'FaceColor','r');
poly2 = polyshape([0.5*(pos+0.5*(pos+(pos-length))), 0.5*((pos-length)+0.5*(pos+(pos-length))),...
0.5*((pos-length)+0.5*(pos+(pos-length))), 0.5*(pos+0.5*(pos+(pos-length)))],...
[0.5+height1, 0.5+height1, (0.5+height1)+height2, (0.5+height1)+height2]);
obj.graphics(2) = plot(poly2,'FaceColor','r');
% Draw two wheels
obj.graphics(3) = viscircles([0.5*((pos-length)+0.5*(pos+(pos-length))), 0.3], radius,'Color','k');
obj.graphics(4) = viscircles([0.5*(pos+0.5*(pos+(pos-length))), 0.3], radius,'Color','k');
% Vehicle number
obj.graphics(5) = text(pos,0,num2str(obj.vehicleIndex));
end
end
function outputArg = generateNoise(obj)
if obj.vehicleIndex==1
w = 0; % Leader is not affected by the noise.
else
w = obj.noiseMean + obj.noiseStd.*randn(3,1);
end
obj.noise = w;
%%%% Some old code
% stepSize = 1;
% if w < obj.noise - stepSize
% w = obj.noise - stepSize;
% elseif w > obj.noise + stepSize
% w = obj.noise + stepSize;
% end
end
function outputArg = computePlatooningErrors1(obj,leaderStates,neighborInformation)
locationError = 0;
velocityError = 0;
for jInd = 1:1:length(obj.inNeighbors)
j = obj.inNeighbors(jInd);
k_ijBar = obj.controllerGains1{j};
d_ij = obj.desiredSeparations(j);
X_j = neighborInformation{j};
locationError_j = k_ijBar*(obj.states(1)-X_j(1)-d_ij);
locationError = locationError + locationError_j;
velocityError_j = k_ijBar*(obj.states(2)-X_j(2));
velocityError = velocityError + velocityError_j;
end
accelerationError = obj.states(3)-leaderStates(3); %a_i-a_0
newErrors = [locationError;velocityError;accelerationError];
obj.errors = newErrors;
obj.errorHistory = [obj.errorHistory, newErrors];
end
function outputArg = computePlatooningErrors2(obj,leaderStates)
separationFromLeader = obj.desiredSeparation;
newErrors = obj.states - leaderStates + [separationFromLeader;0;0];
obj.errors = newErrors;
obj.errorHistory = [obj.errorHistory, newErrors];
end
function outputArg = computeControlInputs1(obj,t)
if obj.vehicleIndex==1 % Leader's control (from planned)
if obj.plannedControls(1,1)==t
obj.controlInput = obj.plannedControls(1,2);
obj.plannedControls = obj.plannedControls(2:end,:); % Delete the executed planned control
else
obj.controlInput = 0;
end
else % Followers control (based on errors) under Error-Dynamics - I
i = obj.vehicleIndex;
L_ii = obj.controllerGains1{i} + obj.localControllerGains1;
e_i = obj.errors;
obj.controlInput = L_ii*e_i;
end
end
function outputArg = computeControlInputs2(obj,t,neighborInformation)
if obj.vehicleIndex==1 % Leader's control (from planned)
if obj.plannedControls(1,1)==t
obj.controlInput = obj.plannedControls(1,2);
obj.plannedControls = obj.plannedControls(2:end,:); % Delete the executed planned control
else
obj.controlInput = 0;
end
else % Followers control (based on errors) under Error-Dynamics - II
i = obj.vehicleIndex;
L_ii = obj.controllerGains2{i} + obj.localControllerGains2;
e_i = obj.errors;
controlInput = L_ii*e_i;
for jInd = 1:1:length(obj.inNeighbors)
j = obj.inNeighbors(jInd);
if j~=1
L_ij = obj.controllerGains2{j};
e_j = neighborInformation{j};
controlInput = controlInput + L_ij*(e_i - e_j);
end
end
obj.controlInput = controlInput;
end
end
% Update the state values of the system dynamics
function vehicleError = update(obj,t,dt)
vehicleError = obj.errors;
A = [0 1 0; 0 0 1; 0 0 0];
B = [0 0 1]';
updateValue = A*obj.states + B*obj.controlInput + obj.noise;
newStates = obj.states + dt*(updateValue);
obj.states = newStates;
% Collect all the state points at each step
obj.stateHistory = [obj.stateHistory, newStates];
end
function outputArg = loadPassivityIndices(obj,nu,rho)
obj.nu = nu;
obj.rho = rho;
end
function status = synthesizeLocalControllers(obj,errorDynamicsType,nuBar,rhoBar)
% Here we will synthesize the local controllers for local error
% dynamics to optimize the passivity properties
% Error Dynamics Type
% When nu = 0, both methods seems to lead to rho = -1/2 (this will not do)
% When rho = 0, second methods lead to nu = -1/2
if errorDynamicsType == 1
A = [0,1,0;0,0,0;0,0,0]; % For error dynamics type 1
else
A = [0,1,0;0,0,1;0,0,0]; % For error dynamics type 2
end
B = [0;0;1];
I = eye(3);
O = zeros(3);
rhoBarBar = 1/rhoBar;
nuHat = -0.01; % nuBar <= nu <= nuHat < 0
rhoHat = 0.01; % 0 <= rhoHat <= rho <= rhoBar ( or 0 <= rhoBarBar <= rhoBar <= rhoHatBar )
rhoHatBar = 1/rhoHat;
% Set up the LMI problem
solverOptions = sdpsettings('solver','mosek','verbose',0);
P = sdpvar(3,3,'symmetric');
K = sdpvar(1,3,'full');
rhoTilde = sdpvar(1,1,'full'); %Representing: 1/rho
nu = sdpvar(1,1,'full');
% Basic Constraints
con1 = P >= 0;
%%con2 = trace(P) == 1; % In this setting this is not required actually
% Approach 4 with rho = prespecified, nu < 0 and nu is free to maximize
DMat = [rhoTilde*I];
MMat = [P, O];
ThetaMat = [-A*P-P*A'-B*K-K'*B', -I+0.5*P; -I+0.5*P, -nu*I];
W = [DMat, MMat; MMat', ThetaMat];
con3 = W >= 0;
% Some modesty constraints on resulting nu and rho from the local design
con4 = nu >= nuBar; % -8.1
con5 = rhoTilde >= rhoBarBar; % 1/4.1
con6 = nu <= nuHat;
con7 = rhoTilde <= rhoHatBar;
% To help the global design (\gammaSqBar=10, p_i = 1/N)
gammaSqBar = 10;
con8 = nu >= -gammaSqBar/(1/5) ; % nu >= -50
con9 = rhoTilde <= 4*gammaSqBar/(1/5); % rhoBar >= 200
% Total Cost and Constraints
cons = [con1,con3,con4,con5,con6,con7,con8,con9];
%costFun = 0*(-nu + rhoBar); % For stabilizing, set coefficient to 0
%costFun = 0.0000001*(-nu + rhoBar); % Otherwise set to 0.0000001.
costFun = 0.0000001*(- nu + rhoTilde);
% Solution
sol = optimize(cons,costFun,solverOptions);
status = sol.problem == 0; % sol.info;
PVal = value(P)
KVal = value(K)
LVal = KVal/PVal
nuVal = value(nu)
rhoVal = 1/value(rhoTilde)
disp('TestVal = -nu,rhoTilde/4')
val = [-value(nu), value(rhoTilde)/4]
% Updating the information
obj.nu = nuVal;
obj.rho = rhoVal;
obj.localControllerGains1 = LVal; % Here we need \bar{k}_{i0}^{Local} = 1
obj.localControllerGains2 = LVal;
if status == 1
disp(['Synthesis Success at Vehicle ',num2str(obj.vehicleIndex),'.'])
else
disp(['Synthesis Failed at Vehicle ',num2str(obj.vehicleIndex),'.'])
end
end
%% Parametrized Approach to Synthesize Local Controllers (while also assisting synthesis of global controllers)
function [status,PVal,KVal,LVal,nuVal,rhoVal,gammaSqVal] = synthesizeLocalControllersParameterized(obj,errorDynamicsType,pVal)
% Here we will synthesize the local controllers for local error
% dynamics to optimize the passivity properties
% This is the true local controller synthesis for a given p_i value
if errorDynamicsType == 1
A = [0,1,0;0,0,0;0,0,0]; % For error dynamics type 1
else
A = [0,1,0;0,0,1;0,0,0]; % For error dynamics type 2
end
B = [0;0;1];
I = eye(3);
O = zeros(3);
% Set up the LMI problem
solverOptions = sdpsettings('solver','mosek','verbose',0);
P = sdpvar(3,3,'symmetric');
K = sdpvar(1,3,'full');
nu = sdpvar(1,1,'full');
rhoTilde = sdpvar(1,1,'full'); % Representing: 1/rho
gammaSq = sdpvar(1,1,'full');
% For: nuBar < nu < nuHat < 0
nuBar = -gammaSq/pVal;
% For: 0 < rhoHat1,rhoHat2 < rho < rhoBar
% For: 0 < rhoTildeHat < rhoTilde < rhoTildeBar1,rhoTildeBar2
rhoTildeBar1 = 4*gammaSq/pVal;
rhoTildeBar2 = pVal;
% Basic Constraints
con1 = P >= 0;
% Approach 4 with rho = prespecified, nu < 0 and nu is free to maximize
DMat = [rhoTilde*I];
MMat = [P, O];
ThetaMat = [-A*P-P*A'-B*K-K'*B', -I+0.5*P; -I+0.5*P, -nu*I];
W = [DMat, MMat; MMat', ThetaMat];
con2 = W >= 0;
%%Constraints on resulting nu and rho from the local design
% nuBar < nu < nuHat < 0
con3 = nu >= nuBar; % Helps global design
% 0 < rhoTildeHat < rhoTilde < rhoTildeBar1,rhoTildeBar2
con4 = rhoTilde <= rhoTildeBar1; % Helps global design
con5 = rhoTilde <= rhoTildeBar2; % Helps global design
% Total Cost and Constraints
cons = [con1,con2,con3,con4,con5];
%costFun = 0*(-nu + rhoBar); % For stabilizing, set coefficient to 0
%costFun = 0.0000001*(-nu + rhoBar); % Otherwise set to 0.0000001.
costFun = 0*gammaSq;
% Solution
sol = optimize(cons,costFun,solverOptions);
status = sol.problem == 0; % sol.info;
PVal = value(P) % This is the P_i value obtained from solving the local control synthesis
KVal = value(K) % This is \tilde{L}_{ii}
LVal = KVal/PVal % KVal is L and LVal is K in our paper. This LVal is the local controller gain \bar{L}_{ii}
nuVal = value(nu);
rhoVal = 1/value(rhoTilde);
gammaSqVal = value(gammaSq);
% Updating the information
obj.nu = nuVal;
obj.rho = rhoVal;
obj.localControllerGains1 = LVal; % Here we need \bar{k}_{i0}^{Local} = 1
obj.localControllerGains2 = LVal;
% if status == 1
% disp(['Synthesis Success at Vehicle ',num2str(obj.vehicleIndex),' with gammaSq=',num2str(value(gammaSqVal)),'.'])
% else
% disp(['Synthesis Failed at Vehicle ',num2str(obj.vehicleIndex),'.'])
% end
end
%% Decentralized Stabilizing Controller Synthesis (Error Dynamics I)
function [isStabilizable,K_ii,K_ijVals,K_jiVals] = stabilizingControllerSynthesis1(obj, previousSubsystems, subsystems, nuBar, rhoBar)
i = length(previousSubsystems)+1;
iInd = obj.vehicleIndex-1;
disp(['Stabilizing at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
status = obj.synthesizeLocalControllers(1,nuBar,rhoBar);
% Seting up the LMI problem
solverOptions = sdpsettings('solver','mosek','verbose',0);
isSoft = 1; % Whether to use a soft or hard graph constraint
normType = 2; % Type of the norm to be used in the LMI
I_n = eye(3);
O_n = zeros(3);
nu_i = obj.nu;
rho_i = obj.rho;
X_i_11 = -nu_i*I_n;
X_i_12 = 0.5*I_n;
% X_i_21 = X_i_12';
X_i_22 = -rho_i*I_n;
X_ii_12 = X_i_11\X_i_12;
X_ii_21 = X_ii_12';
obj.dataToBeDistributed.X = X_ii_12;
null_ii = [1,1,1; 1,1,0; 0,0,0];
% adj_ii = [0,0,0; 0,0,1; 1,1,1];
cost_ii = 1*[0,0,0; 0,0,1; 1,1,1];
if isempty(previousSubsystems)
% The subsystem only need to test W_ii > 0
p_i = sdpvar(1,1);
Q_ii = sdpvar(3,3,'full');
con1 = p_i >= 0;
Theta_ii = - X_ii_21*Q_ii - Q_ii'*X_ii_12 - p_i*X_i_22;
W_ii = [p_i*X_i_11, Q_ii; Q_ii', Theta_ii];
con2 = W_ii >= 0;
con3 = Q_ii.*(null_ii==1)==O_n;
costFun = norm(Q_ii.*cost_ii,normType);
sol = optimize([con1,con2,con3],costFun,solverOptions);
isStabilizable = sol.problem==0;
p_iVal = value(p_i);
Q_iiVal = value(Q_ii);
W_iiVal = value(W_ii);
tildeW_i = W_iiVal; % Note that, here, \tilde{W}_ii = W_ii = \tilde{W}_i. This also needs to be stored
if abs(det(tildeW_i))<0.000000001
disp("Error: det(tildeW_ii) low");
isStabilizable = 0;
end
K_ii = (p_iVal*X_i_11)\Q_iiVal;
obj.controllerGainsCollection.decenStabCont1{iInd} = K_ii;
K_jiVals = [];
K_ijVals = [];
obj.dataToBeDistributed.P = p_iVal;
obj.dataToBeDistributed.tildeW = tildeW_i; % Storing
disp(['Data saved ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
if ~isStabilizable
disp(['Not stabilizable at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
end
else
% This subsystem has to talk with all the previosSubsystems
% tildeW_ii > 0 iff [M_i, W_i'; W_i, W_ii] > 0 is required where
% M_i =
% inv((scriptD_i*scriptA_i^T)^{-1}*(scriptD_i)*(scriptD_i*scriptA_i^T)^{-1}') = scriptA_i*scriptD_i*scriptA_i'
% M_i term
blockSize = 3;
scriptA_i = [];
scriptD_i = [];
for j = 1:1:length(previousSubsystems)
jInd = previousSubsystems(j);
% Getting stored info from jInd to create \mathcal{A}_i and \mathcal{D}_i matrices (their j-th columns)
tildeW_j = subsystems(jInd).dataToBeDistributed.tildeW;
Z = zeros(blockSize, blockSize*(i-1-j)); % (i-1)-j blocks of blockSize X blockSize zero matrices
z = zeros(blockSize, blockSize*(j-1));
if j==1
tildeW_jj = tildeW_j;
scriptA_i = [tildeW_jj, Z]; % The first row of \mathcal{A}_i.
scriptD_i = [inv(tildeW_jj), Z]; % The first row of \mathcal{D}_i.
else
tildeW_jj = tildeW_j(:,blockSize*(j-1)+1:blockSize*j); % last blockSizeXblockSize block in the row block vector
tildeW_j = tildeW_j(:,1:blockSize*(j-1)); % first (j-1) blocks in the row block vector
scriptA_i = [scriptA_i; [tildeW_j, tildeW_jj, Z]]; % The j-th row of \mathcal{A}_i.
scriptD_i = [scriptD_i; [z, inv(tildeW_jj), Z]]; % The j-th row of \mathcal{D}_i.
end
end
disp(['Data at ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
scriptA_i
scriptD_i
M1_i = inv(scriptD_i*scriptA_i');
% M_i = inv(M1_i*scriptD_i*M1_i') % THis fills (i-1)x(i-1) blocks in the LMI
M_i = scriptA_i*scriptD_i*scriptA_i';
if issymmetric(scriptD_i) & issymmetric(scriptA_i) & ~issymmetric(M_i)
tf = norm(M_i-M_i.',inf);
disp(['Symmetry Error !!! Magnitude:',num2str(tf)]);
% M_i
M_i = 0.5*(M_i + M_i');
end
% W_ii and W_i terms
p_i = sdpvar(1,1);
Q_ii = sdpvar(3,3,'full');
Theta_ii = - X_ii_21*Q_ii - Q_ii'*X_ii_12 - p_i*X_i_22;
W_ii = [p_i*X_i_11, Q_ii; Q_ii', Theta_ii];
W_i = [];
for j = 1:1:length(previousSubsystems)
jInd = previousSubsystems(j);
null_ij{j} = [1,1,1; 1,1,0; 1,1,1];
null_ji{j} = [1,1,1; 1,1,0; 1,1,1];
if any(obj.inNeighbors==jInd) % iInd <----- jInd
adj_ij{j} = [0,0,0; 0,0,1; 0,0,0];
cost_ij{j} = 0.01*[0,0,0; 0,0,1; 0,0,0];
else
adj_ij{j} = [0,0,0; 0,0,0; 0,0,0];
cost_ij{j} = 1*[0,0,0; 0,0,1; 0,0,0];
end
if any(obj.outNeighbors==jInd) % iInd -----> jInd
adj_ji{j} = [0,0,0; 0,0,1; 0,0,0];
cost_ji{j} = 0.01*[0,0,0; 0,0,1; 0,0,0];
else
adj_ji{j} = [0,0,0; 0,0,0; 0,0,0];
cost_ji{j} = 1*[0,0,0; 0,0,1; 0,0,0];
end
Q_ij{j} = sdpvar(3,3,'full');
Q_ji{j} = sdpvar(3,3,'full');
X_jj_12 = subsystems(jInd).dataToBeDistributed.X;
Theta_ij = - X_ii_21*Q_ij{j} - Q_ji{j}'*X_jj_12;
W_ij = [O_n, Q_ij{j}; Q_ji{j}', Theta_ij];
W_i = [W_i, W_ij];
end
con1 = p_i >= 0;
con2 = [M_i, W_i';W_i, W_ii] >= 0;
con3 = [];
con4 = [];
costFun = 0;
for j = 1:1:length(previousSubsystems)
con3_ij = Q_ij{j}.*(null_ij{j}==1)==O_n;
con3_ji = Q_ji{j}.*(null_ji{j}==1)==O_n;
con3 = [con3, con3_ij, con3_ji];
con4_ij = Q_ij{j}.*(adj_ij{j}==0)==O_n;
con4_ji = Q_ji{j}.*(adj_ji{j}==0)==O_n;
con4 = [con4, con4_ij, con4_ji];
costFun = costFun + norm(Q_ij{j}.*cost_ij{j},normType) + norm(Q_ji{j}.*cost_ji{j},normType);
end
con3_ii = Q_ii.*(null_ii==1)==O_n;
con3 = [con3, con3_ii];
costFun = costFun + norm(Q_ii.*cost_ii,normType);
if isSoft
cons = [con1,con2,con3];
else
cons = [con1,con2,con3,con4];
end
sol = optimize(cons,costFun,solverOptions);
isStabilizable = sol.problem==0;
p_iVal = value(p_i);
Q_iiVal = value(Q_ii);
W_iVal = value(W_i);
W_iiVal = value(W_ii);
K_ii = (p_iVal*X_i_11)\Q_iiVal;
obj.controllerGainsCollection.decenStabCont1{iInd} = K_ii;
for j = 1:1:length(previousSubsystems)
jInd = previousSubsystems(j);
Q_ijVal = value(Q_ij{j});
Q_jiVal = value(Q_ji{j});
p_jVal = subsystems(jInd).dataToBeDistributed.P;
X_j_11 = -subsystems(jInd).nu*I_n;
K_ij = (p_iVal*X_i_11)\Q_ijVal;
K_ijVals{jInd} = K_ij;
obj.controllerGainsCollection.decenStabCont1{jInd} = K_ij;
K_ji = (p_jVal*X_j_11)\Q_jiVal
K_jiVals{jInd} = K_ji; % these values will be loaded outside the function
end
% Need to compute \tilede{W}_i and \tilde{W}_{ii} for storage
tildeW_i = W_iVal*M1_i;
tildeW_ii = W_iiVal - tildeW_i*scriptD_i*tildeW_i'; % Note that here, \tilde{W}_ii, W_ii, \tilde{W}_i are different.
tildeW_i = [tildeW_i, tildeW_ii];
if abs(det(tildeW_ii))<0.000000001
disp("Error: det(tildeW_ii) low");
isStabilizable = 0;
end
obj.dataToBeDistributed.P = p_iVal; % Storing
obj.dataToBeDistributed.tildeW = tildeW_i; % Storing
disp(['Data saved ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
if ~isStabilizable
disp(['LMI is not feasible at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
end
end
end
%% Decentralized Robust Controller Synthesis (Error Dynamics I)
function [isRobustStabilizable,K_ii,K_ijVals,K_jiVals] = robustControllerSynthesis1(obj, previousSubsystems, subsystems, nuBar, rhoBar, gammaSqBar)
i = length(previousSubsystems)+1;
iInd = obj.vehicleIndex-1;
disp(['Stabilizing at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
status = obj.synthesizeLocalControllers(1,nuBar,rhoBar);
% Seting up the LMI problem
solverOptions = sdpsettings('solver','mosek','verbose',0);
isSoft = 1; % Whether to use a soft or hard graph constraint
normType = 2; % Type of the norm to be used in the LMI
I_n = eye(3);
O_n = zeros(3);
nu_i = obj.nu;
rho_i = obj.rho;
X_i_11 = -nu_i*I_n;
X_i_12 = 0.5*I_n;
% X_i_21 = X_i_12';
X_i_22 = -rho_i*I_n;
X_ii_12 = X_i_11\X_i_12;
X_ii_21 = X_ii_12';
obj.dataToBeDistributed.X = X_ii_12;
null_ii = [1,1,1; 1,1,0; 0,0,0];
% adj_ii = [0,0,0; 0,0,1; 1,1,1];
cost_ii = 1*[0,0,0; 0,0,1; 1,1,1];
if isempty(previousSubsystems)
% The subsystem only need to test W_ii > 0
p_i = sdpvar(1,1);
Q_ii = sdpvar(3,3,'full');
gammaSq_i = sdpvar(1,1);
con1 = p_i >= 0;
con2 = gammaSq_i >= 0;
con3 = gammaSq_i <= gammaSqBar;
% DMat = [X_p_11, O; O, I];
% MMat = [Q, X_p_11; I, O];
% ThetaMat = [-X_21*Q-Q'*X_12-X_p_22, -X_p_21; -X_p_12, gammaSq*I];
% con4 = [DMat, MMat; MMat', ThetaMat] >= 0; % The real one
DMat_ii = [p_i*X_i_11, O_n; O_n, I_n];
MMat_ii = [Q_ii, p_i*X_i_11; I_n, O_n];
TMat_ii = [-X_ii_21*Q_ii-Q_ii'*X_ii_12-p_i*X_i_22, -p_i*X_i_21; -p_i*X_i_12, gammaSq_i*I_n];
W_ii = [DMat_ii, MMat_ii; MMat_ii', TMat_ii];
con4 = W_ii >= 0;
con5 = Q_ii.*(null_ii==1)==O_n;
costFun = 1*norm(Q_ii.*cost_ii,normType) + 1*gammaSq;
sol = optimize([con1,con2,con3,con4,con5],costFun,solverOptions);
isRobustStabilizable = sol.problem==0;
p_iVal = value(p_i);
Q_iiVal = value(Q_ii);
gammaSq_iVal = value(gammaSq_i);
W_iiVal = value(W_ii);
tildeW_i = W_iiVal; % Note that, here, \tilde{W}_ii = W_ii = \tilde{W}_i. This also needs to be stored
if abs(det(tildeW_i))<0.000000001
disp("Error: det(tildeW_ii) low");
isRobustStabilizable = 0;
end
K_ii = (p_iVal*X_i_11)\Q_iiVal;
obj.controllerGainsCollection.decenRobustCont1{iInd} = K_ii;
K_jiVals = [];
K_ijVals = [];
obj.dataToBeDistributed.P = p_iVal;
obj.dataToBeDistributed.gammaSq = gammaSq_iVal;
obj.dataToBeDistributed.tildeW = tildeW_i; % Storing
disp(['Data saved ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
if ~isRobustStabilizable
disp(['Not stabilizable at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
end
else
% This subsystem has to talk with all the previosSubsystems
% tildeW_ii > 0 iff [M_i, W_i'; W_i, W_ii] > 0 is required where
% M_i =
% inv((scriptD_i*scriptA_i^T)^{-1}*(scriptD_i)*(scriptD_i*scriptA_i^T)^{-1}') = scriptA_i*scriptD_i*scriptA_i'
% M_i term
blockSize = 3;
scriptA_i = [];
scriptD_i = [];
for j = 1:1:length(previousSubsystems)
jInd = previousSubsystems(j);
% Getting stored info from jInd to create \mathcal{A}_i and \mathcal{D}_i matrices (their j-th columns)
tildeW_j = subsystems(jInd).dataToBeDistributed.tildeW;
Z = zeros(blockSize, blockSize*(i-1-j)); % (i-1)-j blocks of blockSize X blockSize zero matrices
z = zeros(blockSize, blockSize*(j-1));
if j==1
tildeW_jj = tildeW_j;
scriptA_i = [tildeW_jj, Z]; % The first row of \mathcal{A}_i.
scriptD_i = [inv(tildeW_jj), Z]; % The first row of \mathcal{D}_i.
else
tildeW_jj = tildeW_j(:,blockSize*(j-1)+1:blockSize*j); % last blockSizeXblockSize block in the row block vector
tildeW_j = tildeW_j(:,1:blockSize*(j-1)); % first (j-1) blocks in the row block vector
scriptA_i = [scriptA_i; [tildeW_j, tildeW_jj, Z]]; % The j-th row of \mathcal{A}_i.
scriptD_i = [scriptD_i; [z, inv(tildeW_jj), Z]]; % The j-th row of \mathcal{D}_i.
end
end
disp(['Data at ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
scriptA_i
scriptD_i
M1_i = inv(scriptD_i*scriptA_i');
% M_i = inv(M1_i*scriptD_i*M1_i') % THis fills (i-1)x(i-1) blocks in the LMI
M_i = scriptA_i*scriptD_i*scriptA_i';
if issymmetric(scriptD_i) & issymmetric(scriptA_i) & ~issymmetric(M_i)
tf = norm(M_i-M_i.',inf);
disp(['Symmetry Error !!! Magnitude:',num2str(tf)]);
% M_i
M_i = 0.5*(M_i + M_i');
end
% W_ii and W_i terms
p_i = sdpvar(1,1);
Q_ii = sdpvar(3,3,'full');
gammaSq_i = sdpvar(1,1);
con1 = p_i >= 0;
con2 = gammaSq_i >= 0;
con3 = gammaSq_i <= gammaSqBar;
DMat_ii = [p_i*X_i_11, O_n; O_n, I_n];
MMat_ii = [Q_ii, p_i*X_i_11; I_n, O_n];
TMat_ii = [-X_ii_21*Q_ii-Q_ii'*X_ii_12-p_i*X_i_22, -p_i*X_i_21; -p_i*X_i_12, gammaSq_i*I_n];
W_ii = [DMat_ii, MMat_ii; MMat_ii', TMat_ii];
W_i = [];
for j = 1:1:length(previousSubsystems)
jInd = previousSubsystems(j);
null_ij{j} = [1,1,1; 1,1,0; 1,1,1];
null_ji{j} = [1,1,1; 1,1,0; 1,1,1];
if any(obj.inNeighbors==jInd) % iInd <----- jInd
adj_ij{j} = [0,0,0; 0,0,1; 0,0,0];
cost_ij{j} = 0.01*[0,0,0; 0,0,1; 0,0,0];
else
adj_ij{j} = [0,0,0; 0,0,0; 0,0,0];
cost_ij{j} = 1*[0,0,0; 0,0,1; 0,0,0];
end
if any(obj.outNeighbors==jInd) % iInd -----> jInd
adj_ji{j} = [0,0,0; 0,0,1; 0,0,0];
cost_ji{j} = 0.01*[0,0,0; 0,0,1; 0,0,0];
else
adj_ji{j} = [0,0,0; 0,0,0; 0,0,0];
cost_ji{j} = 1*[0,0,0; 0,0,1; 0,0,0];
end
Q_ij{j} = sdpvar(3,3,'full');
Q_ji{j} = sdpvar(3,3,'full');
X_jj_12 = subsystems(jInd).dataToBeDistributed.X;
DMat_ij = [O_n, O_n; O_n, O_n];
MMat_ij = [Q_ij{j}, O_n; O_n, O_n];
MMat_ji = [Q_ji{j}, O_n; O_n, O_n];
TMat_ij = [-X_ii_21*Q_ij{j}-Q_ji{j}'*X_jj_12, O_n; O_n, O_n];
W_ij = [DMat_ij, MMat_ij; MMat_ji', TMat_ij];
W_i = [W_i, W_ij];
end
con4 = [M_i, W_i';W_i, W_ii] >= 0;
con5 = [];
con6 = [];
costFun = 0;
for j = 1:1:length(previousSubsystems)
con5_ij = Q_ij{j}.*(null_ij{j}==1)==O_n;
con5_ji = Q_ji{j}.*(null_ji{j}==1)==O_n;
con5 = [con5, con5_ij, con5_ji];
con6_ij = Q_ij{j}.*(adj_ij{j}==0)==O_n;
con6_ji = Q_ji{j}.*(adj_ji{j}==0)==O_n;
con6 = [con6, con6_ij, con6_ji];
costFun = costFun + norm(Q_ij{j}.*cost_ij{j},normType) + norm(Q_ji{j}.*cost_ji{j},normType);
end
con5_ii = Q_ii.*(null_ii==1)==O_n;
con5 = [con5, con5_ii];
costFun = costFun + norm(Q_ii.*cost_ii,normType);
if isSoft
cons = [con1,con2,con3,con4,con5];
else
cons = [con1,con2,con3,con4,con5,con6];
end
sol = optimize(cons,costFun,solverOptions);
isRobustStabilizable = sol.problem==0;
p_iVal = value(p_i);
Q_iiVal = value(Q_ii);
gammaSq_iVal = value(gammaSq_i);
W_iVal = value(W_i);
W_iiVal = value(W_ii);
K_ii = (p_iVal*X_i_11)\Q_iiVal;
obj.controllerGainsCollection.decenStabCont1{iInd} = K_ii;
for j = 1:1:length(previousSubsystems)
jInd = previousSubsystems(j);
Q_ijVal = value(Q_ij{j});
Q_jiVal = value(Q_ji{j});
p_jVal = subsystems(jInd).dataToBeDistributed.P;
X_j_11 = -subsystems(jInd).nu*I_n;
K_ij = (p_iVal*X_i_11)\Q_ijVal;
K_ijVals{jInd} = K_ij;
obj.controllerGainsCollection.decenStabCont1{jInd} = K_ij;
K_ji = (p_jVal*X_j_11)\Q_jiVal;
K_jiVals{jInd} = K_ji; % these values will be loaded outside the function
end
% Need to compute \tilede{W}_i and \tilde{W}_{ii} for storage
tildeW_i = W_iVal*M1_i;
tildeW_ii = W_iiVal - tildeW_i*scriptD_i*tildeW_i'; % Note that here, \tilde{W}_ii, W_ii, \tilde{W}_i are different.
tildeW_i = [tildeW_i, tildeW_ii];
if abs(det(tildeW_ii))<0.000000001
disp("Error: det(tildeW_ii) low");
isRobustStabilizable = 0;
end
obj.dataToBeDistributed.P = p_iVal;
obj.dataToBeDistributed.gammaSq = gammaSq_iVal;
obj.dataToBeDistributed.tildeW = tildeW_i; % Storing
disp(['Data saved ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
if ~isRobustStabilizable
disp(['LMI is not feasible at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
end
end
%%%% Revised upto this
end
%% Decentralized Stabilizing Controller Synthesis (Error Dynamics II)
function [isStabilizable,K_ii,K_ijVals,K_jiVals] = stabilizingControllerSynthesis2(obj, previousSubsystems, subsystems, nuBar, rhoBar)
%%%% Revised upto this
i = length(previousSubsystems)+1;
iInd = obj.vehicleIndex-1;
disp(['Stabilizing at: ',num2str(iInd),' after ',num2str(previousSubsystems),'.']);
status = obj.synthesizeLocalControllers(2,nuBar,rhoBar);
% Seting up the LMI problem
solverOptions = sdpsettings('solver','mosek','verbose',0);
isSoft = 1; % Whether to use a soft or hard graph constraint
normType = 2; % Type of the norm to be used in the LMI
I_n = eye(3);
O_n = zeros(3);