| linsys(A, B, U, G, V, C, W, D)
|
function ls = linsys(A, B, U, G, V, C, W, D)
%
% LINSYS - constructor for linear system object.
%
%
% Description:
% ------------
%
% Continuous-time linear system:
% dx/dt = A(t) x(t) + B(t) u(t) + G(t) v(t)
% y(t) = C(t) x(t) + w(t)
%
% Discrete-time linear system:
% x[k+1] = A[k] x[k] + B[k] u[k] + G[k] v[k]
% y[k] = C[k] x[k] + w[k]
%
% x - state, vector in R^n.
% u - control, vector in R^k.
% v - disturbance, vector in R^l.
% w - noise, vector in R^m.
% y - output, vector in R^m.
% A in R^(nxn), B in R^(nxk), G in R^(nxl) and C in R^(mxn).
%
% S = LINSYS(A, B, U) Defines linear system
% dx/dt = A(t) x(t) + B(t) u(t)
% y(t) = x(t)
% where U defines ellipsoidal control bounds
% U = E(p(t), P(t)). If p(t) and P(t) are
% constant, parameter U should have type ellipsoid,
% or be a fixed vector, otherwise U should be
% a structure:
% U.center - symbolic vector p(t),
% U.shape - symolic matrix P(t).
% If U is empty, it means there are no bounds
% on control.
% S = LINSYS(A, B, U, G, V) Defines linear system
% dx/dt = A(t) x(t) + B(t) u(t) + G(t) v(t)
% y(t) = x(t),
% where V defines ellipsoidal disturbance bounds
% V = E(q(t), Q(t)). If q(t) and Q(t) are
% constant, parameter V should have type ellipsoid,
% or be a fixed vector, otherwise V should be
% a structure:
% V.center - symbolic vector q(t),
% V.shape - symolic matrix Q(t).
% If G and/or V is empty, it means there is no
% disturbance.
% S = LINSYS(A, B, U, G, V, C) Defines linear system
% dx/dt = A(t) x(t) + B(t) u(t) + G(t) v(t)
% y(t) = C(t) x(t).
% To consider the system without disturbance,
% make G = [], V = [], Or, make G zero matrix.
% S = LINSYS(A, B, U, G, V, C, W) Defines linear system
% dx/dt = A(t) x(t) + B(t) u(t) + G(t) v(t)
% y(t) = C(t) x(t) + W(t),
% where W defines ellipsoidal noise bounds
% W = E(r(t), R(t)). If r(t) and R(t) are
% constant, parameter W should have type ellipsoid,
% or be a fixed vector, otherwise W should be
% a structure:
% W.center - symbolic vector r(t),
% W.shape - symolic matrix R(t).
% S = LINSYS(A, B, U, G, V, C, W, D) If flag D is set to 'd', then linear system
% is considered discrete-time:
% x[k+1] = A[k] x[k] + B[k] u[k] + G[k] v[k]
% y[k] = C[k] x[k] + W[k].
% To define discrete-time system without
% disturbance and/or noise, set G = [], V = [],
% W = [].
%
% In case one or more of the matrices A, B, G and C depend on time t,
% they should be in symbolic form. For example, to represent matrix
% _ _
% A(t) = | 1 -t |
% |_ 0 1 _|,
% type A = {'1' '-t'; '0' '1'}.
% In discrete-time case, for
% _ _
% A[k] = | 1 -k |
% |_ 0 1 _|
% type A = {'1' '-k'; '0' '1'}.
%
%
% Output:
% -------
%
% S - structure that describes linear system.
% Following fields of structure S can be accessed directly:
% S.A - matrix A,
% S.B - matrix B,
% S.G - matrix G,
% S.C - matrix C,
% S.control - control bounds U,
% S.disturbance - disturbance bounds V,
% S.noise - noise bounds W.
%
%
% See also:
% ---------
%
% LINSYS/DIMENSION, ISEMPTY, ISDISCRETE, ISLTI, HASDISTURBANCE, HASNOISE.
%
%
% Author:
% -------
%
% Alex Kurzhanskiy <akurzhan@eecs.berkeley.edu>
%
global ellOptions;
if ~isstruct(ellOptions)
evalin('base', 'ellipsoids_init;');
end
if nargin == 0
ls.A = [];
ls.B = [];
ls.control = [];
ls.G = [];
ls.disturbance = [];
ls.C = [];
ls.noise = [];
ls.lti = 0;
ls.dt = 0;
ls.constantbounds = [0 0 0];
ls = class(ls, 'linsys');
return;
end
lti = 1;
[m, n] = size(A);
if m ~= n
error('LINSYS: A must be square matrix.');
end
if iscell(A)
lti = 0;
elseif ~(isa(A, 'double'))
error('LINSYS: matrix A must be of type ''cell'' or ''double''.');
end
ls.A = A;
[k, l] = size(B);
if k ~= n
error('LINSYS: dimensions of A and B do not match.');
end
if iscell(B)
lti = 0;
elseif ~(isa(B, 'double'))
error('LINSYS: matrix B must be of type ''cell'' or ''double''.');
end
ls.B = B;
cbu = 1;
if nargin > 2
if isempty(U)
; % leave as is
elseif isa(U, 'ellipsoid')
U = U(1, 1);
[d, r] = dimension(U);
if d ~= l
error('LINSYS: dimensions of control bounds U and matrix B do not match.');
end
if (d > r) & (ellOptions.verbose > 0)
fprintf('LINSYS: Warning! Control bounds U represented by degenerate ellipsoid.\n');
end
elseif isa(U, 'double') | iscell(U)
[k, m] = size(U);
if m > 1
error('LINSYS: control U must be an ellipsoid or a vector.')
elseif k ~= l
error('LINSYS: dimensions of control vector U and matrix B do not match.');
end
elseif isstruct(U) & isfield(U, 'center') & isfield(U, 'shape')
cbu = 0;
U = U(1, 1);
msg = l_check_ell_struct(U, l);
if ~(isempty(msg))
error(sprintf('LINSYS: control bounds U: %s.', msg));
end
else
error('LINSYS: control U must be an ellipsoid or a vector.')
end
else
U = [];
end
ls.control = U;
if nargin > 3
if isempty(G)
; % leave as is
else
[k, l] = size(G);
if k ~= n
error('LINSYS: dimensions of A and G do not match.');
end
if iscell(G)
lti = 0;
elseif ~(isa(G, 'double'))
error('LINSYS: matrix G must be of type ''cell'' or ''double''.');
end
end
else
G = [];
end
cbv = 1;
if nargin > 4
if isempty(G) | isempty(V)
G = [];
V = [];
elseif isa(V, 'ellipsoid')
V = V(1, 1);
[d, r] = dimension(V);
if d ~= l
error('LINSYS: dimensions of disturbance bounds V and matrix G do not match.');
end
elseif isa(V, 'double') | iscell(V)
[k, m] = size(V);
if m > 1
error('LINSYS: disturbance V must be an ellipsoid or a vector.')
elseif k ~= l
error('LINSYS: dimensions of disturbance vector V and matrix G do not match.');
end
elseif isstruct(V) & isfield(V, 'center') & isfield(V, 'shape')
cbv = 0;
V = V(1, 1);
msg = l_check_ell_struct(V, l);
if ~(isempty(msg))
error(sprintf('LINSYS: disturbance bounds V: %s.', msg));
end
else
error('LINSYS: disturbance V must be an ellipsoid or a vector.')
end
else
V = [];
end
ls.G = G;
ls.disturbance = V;
if nargin > 5
if isempty(C)
C = eye(n);
else
[k, l] = size(C);
if l ~= n
error('LINSYS: dimensions of A and C do not match.');
end
if iscell(C)
lti = 0;
elseif ~(isa(C, 'double'))
error('LINSYS: matrix C must be of type ''cell'' or ''double''.');
end
end
else
C = eye(n);
end
ls.C = C;
cbw = 1;
if nargin > 6
if isempty(W)
; % leave as is
elseif isa(W, 'ellipsoid')
W = W(1, 1);
[d, r] = dimension(W);
if d ~= k
error('LINSYS: dimensions of noise bounds W and matrix C do not match.');
end
elseif isa(W, 'double') | iscell(W)
[l, m] = size(W);
if m > 1
error('LINSYS: noise W must be an ellipsoid or a vector.')
elseif k ~= l
error('LINSYS: dimensions of noise vector W and matrix C do not match.');
end
elseif isstruct(W) & isfield(W, 'center') & isfield(W, 'shape')
cbw = 0;
W = W(1, 1);
msg = l_check_ell_struct(W, k);
if ~(isempty(msg))
error(sprintf('LINSYS: noise bounds W: %s.', msg));
end
else
error('LINSYS: noise W must be an ellipsoid or a vector.')
end
else
W = [];
end
ls.noise = W;
ls.lti = lti;
ls.dt = 0;
if (nargin > 7) & ischar(D) & (D == 'd')
ls.dt = 1;
end
ls.constantbounds = [cbu cbv cbw];
ls = class(ls, 'linsys');
return;
%%%%%%%%
function msg = l_check_ell_struct(E, N)
%
% L_CHECK_ELL_STRUCT - checks if given structure E represents an ellipsoid
% of dimension N.
%
global ellOptions;
msg = '';
q = E.center;
Q = E.shape;
[k, l] = size(q);
[m, n] = size(Q);
if m ~= n
msg = 'shape matrix must be symmetric, positive definite';
return;
elseif n ~= N
msg = sprintf('shape matrix must be of dimension %dx%d', N, N);
return;
elseif l > 1
msg = sprintf('center must be a vector of dimension %d', N);
return;
elseif k ~= N
msg = sprintf('center must be a vector of dimension %d', N);
return;
end
if ~(iscell(q)) & ~(iscell(Q))
msg = 'for constant ellipsoids us ellipsoid object';
return;
end
if ~(iscell(q))
if ~(isa(q, 'double'))
msg = 'center must be of type ''cell'' or ''double''';
return;
end
end
if ~(iscell(Q))
if ~(isa(Q, 'double'))
msg = 'shape matrix must be of type ''cell'' or ''double''';
return;
end
if (Q ~= Q') | (min(eig(Q)) < 0)
msg = 'shape matrix must be symmetric, positive definite';
return;
end
else
if ellOptions.verbose > 0
fprintf('LINSYS: Warning! Cannot check if symbolic matrix is positive definite.\n');
end
for i = 1:n
for j = i:n
if min(Q{i, j} == Q{j, i}) < 1
msg = 'shape matrix must be symmetric, positive definite';
return;
end
end
end
end
return;
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