Take deliverables for Robust MPC from yanzhen

According to table 12

  - generate_disturbances
  - simulate_uncertain
  - compute_tube_contorller
  - compute_minRPI
  - compute_tightening
  - MPC_TUBE
  - MPC_TUBE/eval (contained in MPC_TUBE.m)
  - MPC_TUBE_script (and its output MPC_TUBE_params.mat)

10 files changed, 446 insertions, 2 deletions

diff --git a/templates/MPC_TUBE.m b/templates/MPC_TUBE.m
index bd62044..9a630c1 100644
--- a/templates/MPC_TUBE.m
+++ b/templates/MPC_TUBE.m
@@ -15,8 +15,48 @@ classdef MPC_TUBE
     methods
         function obj = MPC_TUBE(Q,R,N,H_N,h_N,H_tube,h_tube,K_tube,params)
             obj.K_tube = K_tube;
-
+    
             % YOUR CODE HERE
+            nx = params.model.nx;
+            nu = params.model.nu;
+
+            A = params.model.A;
+            B = params.model.B;
+
+            [~,P,~] = dlqr(A,B,Q,R);
+
+            V = sdpvar(repmat(nu,1,N),ones(1,N),'full');
+            Z = sdpvar(repmat(nx,1,N+1),ones(1,N+1),'full');
+
+            H_x = params.constraints.StateMatrix;
+            h_x = params.constraints.StateRHS;
+            H_u = params.constraints.InputMatrix;
+            h_u = params.constraints.InputRHS;
+
+            P_x_tightened = Polyhedron('A',H_x,'b',h_x).minus(Polyhedron('A',H_tube,'b',h_tube));
+            H_z = P_x_tightened.A;
+            h_z = P_x_tightened.b;
+
+            P_u_tightened = Polyhedron('A',H_u,'b',h_u).minus(K_tube*Polyhedron('A',H_tube,'b',h_tube));
+            H_v = P_u_tightened.A;
+            h_v = P_u_tightened.b;
+
+            X0 = sdpvar(nx,1,'full');
+            constraints = [H_tube*(X0-Z{1}) <= h_tube];
+            objective = 0;
+            for k = 1:N
+                constraints = [ ...
+                    constraints, ...
+                    Z{k+1} == A*Z{k} + B*V{k} , ...
+%                     H_z * Z{k} <= h_z, ...
+%                     H_v * V{k} <= h_v ...
+                    H_x * Z{k} <= h_x, ...
+                    H_u * V{k} <= h_u ...
+                ];
+                objective = objective + Z{k}'*Q*Z{k} + V{k}'*R*V{k};
+            end
+            constraints = [constraints, H_N * Z{end} <= h_N];
+            objective = objective + Z{end}'*P*Z{end};
 
             opts = sdpsettings('verbose',1,'solver','quadprog');
             obj.yalmip_optimizer = optimizer(constraints,objective,opts,X0,{V{1} Z{1} objective});
@@ -28,7 +68,10 @@ classdef MPC_TUBE
             [optimizer_out,errorcode] = obj.yalmip_optimizer(x);
             solvetime = toc;
             % YOUR CODE HERE
-            
+            [v0, z0, objective] = optimizer_out{:};
+            u = obj.K_tube*(x-z0)+v0;
+
+
             feasible = true;
             if (errorcode ~= 0)
                 feasible = false;
diff --git a/templates/MPC_TUBE_params.mat b/templates/MPC_TUBE_params.mat
new file mode 100644
index 0000000..9db7ef4
--- /dev/null
+++ b/templates/MPC_TUBE_params.mat
diff --git a/templates/MPC_TUBE_script.m b/templates/MPC_TUBE_script.m
new file mode 100644
index 0000000..a021cd0
--- /dev/null
+++ b/templates/MPC_TUBE_script.m
@@ -0,0 +1,15 @@
+p = [0.05, 0.1];
+params = generate_params();
+params = generate_params_z(params);
+K_tube = compute_tube_controller(p,params);
+[H_tube, h_tube, n_iter] = compute_minRPI(K_tube, params);
+params = compute_tightening(K_tube, H_tube, h_tube, params);
+
+Q = diag([300,0.1]);R = 1;N = 50;
+[H_N, h_N] = lqr_maxPI(Q, R, params);
+
+% MPC_TUBE_obj = MPC_TUBE(Q,R,N,H_N,h_N,H_tube,h_tube,K_tube,params);
+matfile = struct("p", p, "K_tube", K_tube, "H_tube", H_tube, "h_tube", h_tube,...
+                "H_N", H_N, "h_N", h_N, "params_z_tube", params);
+save("MPC_TUBE_params.mat", "-struct", "matfile");
+
diff --git a/templates/chebycenter.m b/templates/chebycenter.m
new file mode 100755
index 0000000..cbdce1c
--- /dev/null
+++ b/templates/chebycenter.m
@@ -0,0 +1,20 @@
+function [c,r] = chebycenter(A,b)
+%CHEBYCENTER Compute Chebyshev center of polytope Ax <= b.
+%  The Chebyshev center of a polytope is the center of the largest
+%  hypersphere enclosed by the polytope. 
+%  Requires optimization toolbox.
+
+[n,p] = size(A);
+an = sqrt(sum(A.^2,2));
+A1 = zeros(n,p+1);
+A1(:,1:p) = A;
+A1(:,p+1) = an;
+f = zeros(p+1,1);
+f(p+1) = -1;
+
+options = optimset;
+options = optimset(options,'Display', 'off');
+c = linprog(f,A1,b,[],[],[],[],[],options);
+r = c(p+1);
+c = c(1:p);
+end
diff --git a/templates/compute_minRPI.m b/templates/compute_minRPI.m
index 0b4ffdb..19df517 100644
--- a/templates/compute_minRPI.m
+++ b/templates/compute_minRPI.m
@@ -8,4 +8,32 @@
 
 function [H_tube,h_tube,n_iter] = compute_minRPI(K_tube,params)
     % YOUR CODE HERE
+    A = params.model.A+params.model.B*K_tube;
+    % A = params.model.A;
+%     display(A)
+    nx = params.model.nx;
+    omega = Polyhedron('A',[eye(nx);-eye(nx)],'b',zeros(1,2*nx)');
+    % display(omega.A)
+    % display(omega.b)
+    n_iter = 0;
+    Hw = params.constraints.DisturbanceMatrix;
+    hw = params.constraints.DisturbanceRHS;
+    W = Polyhedron('A',Hw,'b',hw); 
+    while true 
+        omega_last = copy(omega);
+        % display(n_iter)
+        % A_last = omega.A;
+        % b_last = omega.b;
+        display(n_iter);
+        display(omega.b)
+        omega = omega.plus(A^(n_iter)*W);
+        
+        if eq(omega.minHRep(),omega_last.minHRep())
+        % if eq(omega,omega_last)
+            break
+        end
+        n_iter = n_iter+1;
+    end
+    H_tube = omega.A;
+    h_tube = omega.b;
 end
\ No newline at end of file
diff --git a/templates/compute_tightening.m b/templates/compute_tightening.m
index 8919d12..7095f9c 100644
--- a/templates/compute_tightening.m
+++ b/templates/compute_tightening.m
@@ -8,4 +8,22 @@
 
 function params = compute_tightening(K_tube,H_tube,h_tube,params)  
 	% YOUR CODE HERE
+	Hx = params.constraints.StateMatrix;
+	hx = params.constraints.StateRHS;
+	
+	Hu = params.constraints.InputMatrix;
+	hu = params.constraints.InputRHS;
+	
+	Pxz = Polyhedron('A',Hx,'b',hx);
+	Puz = Polyhedron('A',Hu,'b',hu);
+	% Ptube = Polyhedron('A',H_tube,'b',h_tube);
+	
+	Px_tube = Pxz.minus(Polyhedron('A',H_tube,'b',h_tube));
+	params.constraints.StateMatrix = Px_tube.A;
+	params.constraints.StateRHS = Px_tube.b;
+    
+    Pu_tube = Puz.minus(K_tube*Polyhedron('A',H_tube,'b',h_tube))
+% 	Pu_tube = Puz.minus(Polyhedron('A',H_tube*K_tube,'b',h_tube));
+	params.constraints.InputMatrix = Pu_tube.A;
+	params.constraints.InputRHS = Pu_tube.b;
 end
\ No newline at end of file
diff --git a/templates/compute_tube_controller.m b/templates/compute_tube_controller.m
index adca95d..2c567f0 100644
--- a/templates/compute_tube_controller.m
+++ b/templates/compute_tube_controller.m
@@ -8,4 +8,9 @@
 
 function K_tube = compute_tube_controller(p,params)
 	% YOUR CODE HERE
+	Az = params.model.A;
+	Bz  =params.model.B;
+	% Bz = [params.model.B,eye(params.model.nx)];
+	K_tube = -place(Az,Bz,p);
+	% K_tube = K(1:params.model.nu,:);
 end
\ No newline at end of file
diff --git a/templates/cprnd.m b/templates/cprnd.m
new file mode 100755
index 0000000..4183ac3
--- /dev/null
+++ b/templates/cprnd.m
@@ -0,0 +1,296 @@
+function [X S] = cprnd(N,A,b,options)
+%CPRND Draw from the uniform distribution over a convex polytope.
+%   X = cprnd(N,A,b) Returns an N-by-P matrix of random vectors drawn
+%   from the uniform distribution over the interior of the polytope
+%   defined by Ax <= b. A is a M-by-P matrix of constraint equation
+%   coefficients. b is a M-by-1 vector of constraint equation
+%   constants.
+%    
+%   cprnd(N,A,b,options) allows various options to be specified.
+
+%   'method'    Specifies the algorithm. One of the strings 'gibbs',
+%               'hitandrun' (the default), and 'achr'. The default
+%               algorithm 'hitandrun' is a vanilla hit-and-run sampler
+%               [1]. 'gibbs' specifies a Gibbs sampler, 'achr' is the
+%               Adaptive Centering Hit-and-Run algorithm of [2]. 
+%
+%   'x0'        Vector x0 is a starting point which should be interior 
+%               to the polytope. If x0 is not supplied CPRND uses the
+%               Chebyshev center as the initial point.
+%    
+%   'isotropic' Perform an adaptive istropic transformation of the 
+%               polytope. The values 0, 1 and 2 respectively turn
+%               off the transformation, construct it during a runup
+%               period only, or continuously update the
+%               tranformation throughout sample production. The
+%               transformation makes sense only for the Gibbs and
+%               Hit-and-run samplers (ACHR is invariant under
+%               linear transformations).
+%   
+%   'discard'   Number of initial samples (post-runup) to discard.
+%
+%   'runup'     When the method is gibbs or hitandrun and the
+%               isotropic transformation is used, this is the
+%               number of initial iterations of the algorithm in
+%               the untransformed space. That is a sample of size
+%               runup is generated and its covariance used as the
+%               basis of a transformation. 
+%               When the method is achr runup is the number of
+%               conventional hitandrun iterations. See [2].
+%
+%   'ortho'     Zero/one flag. If turned on direction vectors for
+%               the hitandrun algorithm are generated in random
+%               orthonormal blocks rather than one by
+%               one. Experimental and of dubious utility.
+%   
+%   'quasi' Allows the user to specify a quasirandom number generator
+%               (such as 'halton' or 'sobol').  Experimental and of
+%               dubious utility.
+%
+%      
+
+%   By default CPRND employs the hit-and-run sampler which may
+%   exhibit marked serial correlation and very long convergence times.
+%
+% References
+% [1] Kroese, D.P. and Taimre, T. and Botev, Z.I., "Handbook of Monte
+% Carlo Methods" (2011), pp. 240-244.
+% [2] Kaufman, David E. and Smith, Robert L., "Direction Choice for
+% Accelerated Convergence in Hit-and-Run Sampling", Op. Res. 46,
+% pp. 84-95.
+%
+% Copyright (c) 2011-2012 Tim J. Benham, School of Mathematics and Physics,
+%                University of Queensland.
+
+    function y = stdize(z)
+        y = z/norm(z);
+    end
+
+    p = size(A,2);                         % dimension
+    m = size(A,1);                         % num constraint ineqs
+    x0 = [];
+    runup = [];                              % runup necessary to method
+    discard = [];                            % num initial pts discarded
+    quasi = 0;
+    method = 'achr';
+    orthogonal = 0;
+    isotropic = [];
+    
+    % gendir generates a random unit (direction) vector.
+    gendir = @() stdize(randn(p,1));
+
+    % Alternative function ogendir forces directions to come in
+    % orthogonal bunches.
+    Ucache = {};
+    function u = ogendir()
+        if length(Ucache) == 0
+            u = stdize(randn(p,1));
+            m = null(u');               % orthonormal basis for nullspace
+            Ucache = mat2cell(m',ones(1,p-1));
+        else
+            u = Ucache{end}';
+            Ucache(end) = [];
+        end
+    end
+    
+    % Check input arguments
+    
+    if m < p+1                             
+        % obv a prob here
+        error('cprnd:obvprob',['At least ',num2str(p+1),' inequalities ' ...
+                            'required']);
+    end
+
+    if nargin == 4
+        if isstruct(options)
+
+            if isfield(options,'method')
+                method = lower(options.method);
+                switch method
+                  case 'gibbs'
+                  case 'hitandrun'
+                  case 'achr'
+                  otherwise
+                    error('cprnd:badopt',...
+                          ['The method option takes only the ' ...
+                           'values "gibbs", "hitandrun", and "ACHR".']);
+                end
+            end
+            
+            if isfield(options,'isotropic')
+                % Controls application of isotropic transformation,
+                % which seems to help a lot.
+                isotropic = options.isotropic;
+            end
+
+            if isfield(options,'discard')
+                % num. of samples to discard
+                discard = options.discard;
+            end
+
+            if isfield(options,'quasi')
+                % Use quasi random numbers, which doesn't seem to
+                % help much.
+                quasi = options.quasi;
+                if quasi && ~ischar(quasi), quasi='halton'; end
+                if ~strcmp(quasi,'none')
+                    qstr = qrandstream(quasi,p,'Skip',1);
+                    gendir = @() stdize(norminv(qrand(qstr,1),0,1)');
+                end
+            end
+
+            if isfield(options,'x0')
+                % initial interior point
+                x0 = options.x0;
+            end
+
+            if isfield(options,'runup')
+                % number of points to generate before first output point
+                runup = options.runup;
+            end
+            
+            if isfield(options,'ortho')
+                % Generate direction vectors in orthogonal
+                % groups. Seems to help a little.
+                orthogonal = options.ortho;
+            end
+            
+        else
+            x0 = options;                   % legacy support
+        end
+    end
+
+    % Default and apply options
+    
+    if isempty(isotropic)
+        if ~strcmp(method,'achr')
+            isotropic = 2;
+        else
+            isotropic = 0;
+        end
+    end
+    
+    if orthogonal
+        gendir = @() ogendir();
+    end
+    
+    % Choose a starting point x0 if user did not provide one.
+    if isempty(x0)
+        x0 = chebycenter(A,b); % prob. if p==1?
+    end
+    
+    % Default the runup to something arbitrary.
+    if isempty(runup)
+        if strcmp(method,'achr')
+            runup = 10*(p+1);
+        elseif isotropic > 0
+            runup = 10*p*(p+1);
+        else 
+            runup = 0;
+        end
+    end
+
+    % Default the discard to something arbitrary
+    if isempty(discard)
+        if strcmp(method,'achr')
+            discard = 25*(p+1);
+        else
+            discard = runup;
+        end
+    end
+
+    X = zeros(N+runup+discard,p);
+
+    n = 0;                                  % num generated so far
+    x = x0;
+
+    % Initialize variables for keeping track of sample mean, covariance
+    % and isotropic transform.
+    M = zeros(p,1);                         % Incremental mean.
+    S2 = zeros(p,p);                        % Incremental sum of
+                                            % outer prodcts.
+    S = eye(p); T = eye(p); W = A;
+    
+    while n < N+runup+discard
+        y = x;
+        
+        % compute approximate stochastic transformation
+        if isotropic>0
+            if n == runup || (isotropic > 1 && n > runup)
+                T = chol(S,'lower');
+                W = A*T;
+            end
+            y = T\y;
+        end
+        
+        switch method
+            
+          case 'gibbs'
+            % choose p new components
+            for i = 1:p
+                % Find points where the line with the (p-1) components x_i
+                % fixed intersects the bounding polytope.
+                e = circshift(eye(p,1),i-1);
+                z = W*e;
+                c = (b - W*y)./z;
+                tmin = max(c(z<0)); tmax = min(c(z>0));
+                % choose a point on that line segment
+                y = y + (tmin+(tmax-tmin)*rand)*e;
+            end
+            
+          case 'hitandrun'
+            % choose a direction
+            u = gendir();
+            % determine intersections of x + ut with the polytope
+            z = W*u;
+            c = (b - W*y)./z;
+            tmin = max(c(z<0)); tmax = min(c(z>0));
+            % choose a point on that line segment
+            y = y + (tmin+(tmax-tmin)*rand)*u;
+            
+          case 'achr'
+            % test whether in runup or not
+            if n < runup
+                % same as hitandrun
+                u = gendir();
+            else
+                % choose a previous point at random
+                v = X(randi(n),:)';
+                % line sampling direction is from v to sample mean
+                u = (v-M)/norm(v-M);
+            end
+            % proceed as in hit and run
+            z = A*u;
+            c = (b - A*y)./z;
+            tmin = max(c(z<0)); tmax = min(c(z>0));
+            % Choose a random point on that line segment
+            y = y + (tmin+(tmax-tmin)*rand)*u;
+        end
+        
+        if isotropic>0
+            x = T*y;
+        else
+            x = y;
+        end
+        
+        X(n+1,:)=x';
+        n = n + 1;
+        
+        % Incremental mean and covariance updates
+        delta0 = x - M;       % delta new point wrt old mean
+        M = M + delta0/n;     % sample mean
+        delta1 = x - M;       % delta new point wrt new mean
+        if n > 1
+            S2 = S2 + (n-1)/(n*n)*delta0*delta0'...
+                 + delta1*delta1';
+            S0 = S;
+            S = S2/(n-1);           % sample covariance
+        else 
+            S = eye(p);
+        end
+        
+    end
+        
+    X = X((discard+runup+1):(N+discard+runup),:); 
+
+end
diff --git a/templates/generate_disturbances.m b/templates/generate_disturbances.m
index 570a4c9..f08cdb5 100644
--- a/templates/generate_disturbances.m
+++ b/templates/generate_disturbances.m
@@ -8,4 +8,10 @@
 
 function Wt = generate_disturbances(params)
     % YOUR CODE HERE
+    % params_z = generate_params_z(params);
+    Hw = params.constraints.DisturbanceMatrix;
+    hw = params.constraints.DisturbanceRHS;
+    % Pw = Polyhedron('A', Hw, 'b', hw);
+    N = params.model.HorizonLength;
+    Wt = cprnd(N,Hw,hw)';
 end
\ No newline at end of file
diff --git a/templates/simulate_uncertain.m b/templates/simulate_uncertain.m
index dc9df26..f74f7af 100644
--- a/templates/simulate_uncertain.m
+++ b/templates/simulate_uncertain.m
@@ -8,4 +8,17 @@
 
 function [Xt,Ut,ctrl_info] = simulate_uncertain(x0, ctrl, Wt, params)
 	% YOUR CODE HERE
+	Xt = [x0];
+    Ut = [];
+    x = x0
+    ctrl_info = [];
+    % obj = LQR();
+    for i = 1:params.model.HorizonLength
+        [u, ctrl_info_] = ctrl.eval(x);
+        Ut = [Ut u];
+        ctrl_info = [ctrl_info ctrl_info_];
+        x = params.model.A*x + params.model.B*u + Wt(:,i);
+        Xt = [Xt x];
+    end
+    % plot_trajectory(Xt, Ut, ctrl_info, params);
 end
\ No newline at end of file