Teljes Matlab script kiegészítő függvényekkel.
File: 2018_01_25_K_prelim_L_codesign_v5_rhofuggoBL.m Directory: projects/3_outsel/2017_11_13_lpv_passivity Author: Peter Polcz (ppolcz@gmail.com)
Created on 2018. January 25.
Inherited from
LPV output passivization
File: d2018_01_09_K_prelim_L_codesign_v4.m Directory: projects/3_outsel/2017_11_13_lpv_passivity Author: Peter Polcz (ppolcz@gmail.com)
Created on 2018. January 09. Modified on 2018. January 17. Modified on 2018. January 25.
FIGYELEM, a rendszermodell, nem stabil!
s = tf('s');
% unstable MIMO
H = @(s) [
(s-1)/(s-2)/(s+1) 1/(s+3)/(s-0.1)
(s-7)/(s+1)/(s+5) (s-6)/(s^2+5*s+6)
];
sys = minreal( ss( H(s) ) );
[A0,B0,C,D] = deal(sys.a, sys.b, sys.c, sys.d);
tol = 1e-10;
prec = -log10(tol);
A0 = round(A0,prec);
B0 = round(B0,prec);
C = round(C,prec);
D = round(D,prec);
[POLES,ZEROS] = pzmap(sys)
2 states removed.
POLES =
-5.0000
-1.0000
2.0000
-3.0000
-2.0000
0.1000
ZEROS =
5.8894 + 0.0000i
-4.3164 + 0.0000i
0.7635 + 0.7978i
0.7635 - 0.7978i
\begin{align} {\LARGE(1) \quad}
H(s) =
\left(\begin{array}{cc}
-\frac{s-1}{-s^2+s+2} & \frac{10}{10s^2+29s-3} \\
\frac{s-7}{s^2+6s+5} & \frac{s-6}{s^2+5s+6} \\
\end{array}\right)
\end{align}
Az előző LTI modellt egy kicsit megperturbálom (kezdetben csak az A mátrixot).
$$ \begin{aligned} &\Sigma: \left\{\begin{aligned} &\dot x = A(\rho) x + B(\rho) u,~~~ \rho \in \mathcal P \\ &y = C x \\ \end{aligned}\right. \\ &\begin{aligned} \text{where: } & A(\rho) = A_0 + A_1 \rho \in \mathbb{R}^{n\times n} \\ & B(\rho) = B_0 + B_1 \rho \in \mathbb{R}^{n\times r} \\ & C \in \mathbb{R}^{m\times n} \\ & D = 0_{m\times r} \end{aligned} \end{aligned} $$
rho_lim = [
-1 1
];
A1 = A0;
A1(abs(A0) < 1) = 0;
A1 = A1 .* randn(size(A1))/10;
B1 = [ 2\B0(:,1)+3\B0(:,2) B0(:,1)*0 ]*0;
A_fh = @(rho) A0 + rho*A1;
B_fh = @(rho) B0 + rho*B1;
Matrix $K$ is preliminarily design for the nominal value of $\rho$, than checked whether the closed loop is stable for other $\rho \in \mathcal P$ values. Let $G = I_m$.
G = eye(n_u);
K = place(A0,B0,linspace(-5,-6,n_x));
Free matrix variables. All matrices are assumed to be parameter independent.
C1 = sdpvar(n_yp,n_x,'full');
D1 = sdpvar(n_yp,n_y,'full');
Q = sdpvar(n_x,n_x,'symmetric');
S = sdpvar(n_x,n_x,'symmetric');
N_1 = sdpvar(n_x,n_y,'full');
N_2 = sdpvar(n_x,n_y,'full');
N_fh = @(rho) N_1 + rho*N_2;
P = blkdiag(Q,S);
where $N = S L$.
$$ \begin{align} &\wt A(\rho) = \pmqty{ A(\rho) & 0 \\ 0 & A(\rho) - L C },~ \wt B = \pmqty{B \\ 0} \\ &\wt C = \pmqty{ D_1 C + C_1 & -C_1 }. \end{align} $$
Ao = @(L,rho) [
A_fh(rho) zeros(n_x,n_x)
zeros(n_x,n_x) A_fh(rho)-L(rho)*C
];
Bo = @(rho) [B_fh(rho) ; zeros(n_x,n_u)];
Co = [D1*C+C1 -C1];
Do = zeros(n_yp,n_r);
$$ \begin{aligned} \wt A_c(\rho) = \begin{pmatrix} A(\rho)-B(\rho) K & B(\rho) K \\ 0 & A(\rho) - L C \end{pmatrix},~ \wt B_c(\rho) = \begin{pmatrix} B(\rho) G \\ 0 \end{pmatrix}. \end{aligned} $$
Ac = @(L,rho) [
A_fh(rho)-B_fh(rho)*K B_fh(rho)*K
zeros(n_x,n_x) A_fh(rho)-L(rho)*C
];
Bc = @(rho) Bo(rho)*G;
W = eye(n_yp);
$$ \begin{aligned} &\wt A_c({\color{red} \rho})^T P + P \wt A_c({\color{red} \rho}) = \spmqty{ {\blue Q} A_K({\red \rho}) + A_K^T({\red \rho}) {\blue Q} & {\blue Q} B({\red \rho}) K \\ K^T B^T({\red \rho}) {\blue Q} & {\blue S} A({\red \rho}) + A({\red \rho})^T {\blue S} - {\blue N}({\red \rho}) C - C^T {\blue N^T}({\red \rho}), } \\ &\text{where } A_K({\red \rho}) = A({\red \rho}) - B({\red \rho}) K. \nonumber \end{aligned} $$
AcP_PAc = @(rho) [
Q*(A_fh(rho)-B_fh(rho)*K)+(A_fh(rho)-B_fh(rho)*K)'*Q , Q*B_fh(rho)*K
K'*B_fh(rho)'*Q , S*A_fh(rho)+A_fh(rho)'*S-N_fh(rho)*C-C'*N_fh(rho)'
];
$$ M_2 = \spmqty{ { \wt A_c({\color{red} \rho})^T} { P} + { P} { \wt A_c({\color{red} \rho})} & { P} \wt B_c({\red \rho}) - { \wt C^T({\red \rho})} & { \wt C^T({\red \rho})} \\ \wt B_c^T({\red \rho}) { P}- { \wt C({\red \rho})} & 0 & 0 \\ { \wt C({\red \rho})} & 0 & -W^{-1} } $$
M2 = @(rho) [
AcP_PAc(rho) P*Bc(rho)-Co' Co'
Bc(rho)'*P-Co zeros(n_r,n_r) zeros(n_r,n_yp)
Co zeros(n_yp,n_r) -inv(W)
];
% Constraints
Constraints = [
M2(rho_lim(1)) <= 0
M2(rho_lim(2)) <= 0
P - 0.0001*eye(size(P)) >= 0
]
++++++++++++++++++++++++++++++++++ | ID| Constraint| ++++++++++++++++++++++++++++++++++ | #1| Matrix inequality 16x16| | #2| Matrix inequality 16x16| | #3| Matrix inequality 12x12| ++++++++++++++++++++++++++++++++++
optimize(Constraints)
check(Constraints)
Q = value(Q);
S = value(S);
N_1 = value(N_1);
N_2 = value(N_2);
L_1 = S\N_1;
L_2 = S\N_2;
L_fh = @(rho) L_1 + rho*L_2;
P = value(P);
Co = value(Co);
C1 = value(C1);
D1 = value(D1);
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 82
Cones : 0
Scalar variables : 0
Matrix variables : 3
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 82
Cones : 0
Scalar variables : 0
Matrix variables : 3
Integer variables : 0
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 82
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 3 scalarized : 350
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 3403 after factor : 3403
Factor - dense dim. : 0 flops : 5.86e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.4e+00 1.0e+00 5.0e+00 0.00e+00 3.998800000e+00 0.000000000e+00 1.0e+00 0.01
1 5.2e-01 2.2e-01 2.3e+00 1.00e+00 8.755693332e-01 0.000000000e+00 2.2e-01 0.02
2 1.9e-01 7.9e-02 1.4e+00 9.99e-01 3.166073696e-01 0.000000000e+00 7.9e-02 0.02
3 7.3e-02 3.1e-02 8.7e-01 9.96e-01 1.231920274e-01 0.000000000e+00 3.1e-02 0.02
4 2.7e-02 1.1e-02 5.3e-01 9.96e-01 4.570574136e-02 0.000000000e+00 1.1e-02 0.02
5 5.8e-03 2.5e-03 2.4e-01 9.92e-01 9.875092415e-03 0.000000000e+00 2.5e-03 0.02
6 1.2e-03 5.0e-04 1.1e-01 9.82e-01 2.040980985e-03 0.000000000e+00 5.0e-04 0.02
7 3.8e-04 1.6e-04 6.0e-02 9.67e-01 6.547027256e-04 0.000000000e+00 1.6e-04 0.02
8 1.1e-04 4.7e-05 3.2e-02 9.74e-01 1.972824594e-04 0.000000000e+00 4.7e-05 0.02
9 4.4e-05 1.9e-05 2.0e-02 9.66e-01 7.865094703e-05 0.000000000e+00 1.9e-05 0.02
10 9.3e-06 3.9e-06 8.7e-03 9.46e-01 1.700041537e-05 0.000000000e+00 3.9e-06 0.03
11 2.6e-06 1.1e-06 4.2e-03 8.88e-01 5.056939028e-06 0.000000000e+00 1.1e-06 0.03
12 7.5e-07 3.2e-07 1.8e-03 7.49e-01 1.585240685e-06 0.000000000e+00 3.2e-07 0.03
13 3.1e-07 1.3e-07 8.7e-04 5.06e-01 7.471267729e-07 0.000000000e+00 1.3e-07 0.03
14 9.2e-08 3.9e-08 3.6e-04 4.43e-01 2.384750889e-07 0.000000000e+00 3.9e-08 0.03
15 3.0e-08 1.3e-08 1.1e-04 1.94e-01 -1.371440162e-08 0.000000000e+00 1.3e-08 0.03
16 6.0e-09 2.5e-09 2.9e-05 2.22e-01 -8.385578824e-08 0.000000000e+00 2.5e-09 0.03
17 1.6e-09 6.6e-10 7.1e-06 6.03e-02 -1.577668375e-07 0.000000000e+00 6.6e-10 0.03
18 4.3e-10 1.8e-10 2.0e-06 1.23e-02 -1.889399310e-07 0.000000000e+00 1.8e-10 0.03
19 1.7e-10 7.2e-11 7.2e-07 -4.84e-02 -2.287530519e-07 0.000000000e+00 7.2e-11 0.04
20 4.7e-11 2.0e-11 2.4e-07 1.02e-01 -1.594187513e-07 0.000000000e+00 2.0e-11 0.04
21 1.3e-11 5.4e-12 5.3e-08 -4.32e-02 -2.478096800e-07 0.000000000e+00 5.4e-12 0.04
22 2.8e-12 1.2e-12 1.4e-08 6.76e-02 -1.809508529e-07 0.000000000e+00 1.2e-12 0.04
23 7.8e-13 3.3e-13 3.8e-09 3.36e-02 -1.877296559e-07 0.000000000e+00 3.3e-13 0.04
24 2.3e-13 9.8e-14 1.1e-09 1.21e-02 -1.940468187e-07 0.000000000e+00 9.7e-14 0.04
25 9.2e-14 4.2e-14 3.9e-10 -9.85e-02 -2.503239479e-07 0.000000000e+00 3.9e-14 0.04
26 2.4e-14 1.9e-14 1.3e-10 1.02e-01 -1.599584278e-07 0.000000000e+00 1.0e-14 0.04
27 8.3e-15 8.2e-15 3.7e-11 -1.45e-02 -2.216619486e-07 0.000000000e+00 3.4e-15 0.04
28 9.0e-15 3.6e-15 7.7e-12 -4.67e-02 -2.078673001e-07 0.000000000e+00 7.2e-16 0.04
29 9.6e-15 1.4e-14 2.4e-12 6.01e-02 -1.881283600e-07 0.000000000e+00 2.1e-16 0.05
30 3.5e-14 8.0e-15 7.0e-13 1.78e-02 -1.913183009e-07 0.000000000e+00 6.1e-17 0.05
31 1.3e-14 3.9e-15 2.7e-13 -2.11e-02 -2.022821419e-07 0.000000000e+00 2.4e-17 0.05
32 7.8e-15 4.2e-15 6.4e-14 -4.37e-02 -1.962618795e-07 0.000000000e+00 5.7e-18 0.05
33 3.1e-14 3.9e-15 2.8e-14 5.85e-02 -2.018576626e-07 0.000000000e+00 2.5e-18 0.05
34 8.6e-15 5.5e-15 6.0e-15 -5.72e-02 -2.061561207e-07 0.000000000e+00 5.7e-19 0.05
35 1.8e-14 5.2e-15 2.6e-15 4.81e-02 -2.135388074e-07 0.000000000e+00 2.5e-19 0.05
36 4.1e-15 3.4e-15 5.8e-16 -6.22e-02 -2.161701279e-07 0.000000000e+00 5.8e-20 0.05
37 1.7e-15 4.0e-15 2.4e-16 5.55e-02 -2.215534097e-07 0.000000000e+00 2.4e-20 0.06
38 3.8e-16 6.2e-15 5.4e-17 -5.73e-02 -2.221187378e-07 0.000000000e+00 5.7e-21 0.06
39 1.6e-16 3.5e-15 2.2e-17 5.83e-02 -2.345455988e-07 0.000000000e+00 2.4e-21 0.06
40 3.6e-17 8.9e-15 4.9e-18 -7.84e-02 -2.349898597e-07 0.000000000e+00 5.7e-22 0.06
41 1.6e-17 7.2e-15 2.2e-18 7.95e-02 -2.445369317e-07 0.000000000e+00 2.4e-22 0.06
42 3.5e-18 1.4e-14 4.7e-19 -7.27e-02 -2.471943501e-07 0.000000000e+00 5.8e-23 0.07
43 1.6e-18 4.9e-15 2.1e-19 7.82e-02 -2.580022299e-07 0.000000000e+00 2.5e-23 0.07
44 3.5e-19 6.7e-15 4.5e-20 -6.49e-02 -2.621010202e-07 0.000000000e+00 5.9e-24 0.07
45 1.5e-19 4.9e-15 1.9e-20 6.77e-02 -2.708044697e-07 0.000000000e+00 2.5e-24 0.07
46 3.3e-20 5.8e-15 4.2e-21 -5.55e-02 -2.739479729e-07 0.000000000e+00 6.0e-25 0.07
47 2.5e-17 7.1e-15 3.0e-21 5.69e-02 -2.779917531e-07 0.000000000e+00 4.3e-25 0.08
48 2.5e-17 5.8e-15 3.0e-21 1.59e-02 -2.780344052e-07 0.000000000e+00 4.2e-25 0.08
49 2.5e-17 8.4e-15 3.0e-21 1.54e-02 -2.780370327e-07 0.000000000e+00 4.2e-25 0.08
50 2.5e-17 1.1e-14 3.0e-21 1.55e-02 -2.780396983e-07 0.000000000e+00 4.2e-25 0.08
51 2.5e-17 1.1e-14 3.0e-21 1.61e-02 -2.780396983e-07 0.000000000e+00 4.2e-25 0.09
Optimizer terminated. Time: 0.10
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.7803969833e-07 nrm: 5e+09 Viol. con: 3e-07 barvar: 2e-07
Dual. obj: 0.0000000000e+00 nrm: 6e+09 Viol. con: 0e+00 barvar: 1e-05
Optimizer summary
Optimizer - time: 0.10
Interior-point - iterations : 52 time: 0.09
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
ans =
struct with fields:
yalmiptime: 0.3025
solvertime: 0.1106
info: 'Successfully solved (MOSEK)'
problem: 0
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint| Primal residual| Dual residual|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Matrix inequality| -1.5857e-07| 1.424e-16|
| #2| Matrix inequality| -1.1524e-07| 1.0903e-16|
| #3| Matrix inequality| 2.4627e-05| 3.7307e-17|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[ OK ] M2(0) <= 0 [ OK ] M2(0.1) <= 0 [ OK ] M2(-0.1) <= 0 [ OK ] P > 0 [ OK ] A - LC < 0 [ OK ] norm(D1) > 1e-3 [ OK ] Zeros of the open loop system are stable [ OK ] Zeros of the closed loop system are stable
Written on 2018. January 17.
Ebben a részben $A(\rho)$, $B(\rho)$, $C$, $D$ új értelmet nyer. Ezek adják meg az observerrel kiegészített dinamika mátrixait.
A_fh = @(rho) Ao(L_fh,rho);
B_fh = Bo;
A0 = A_fh(0);
A1 = A_fh(1) - A0;
B0 = B_fh(0);
B1 = B_fh(1) - B0;
C0 = Co;
C1 = C0*0;
D0 = Do;
D1 = Do*0;
A_fh = @(rho) A0 + A1*rho;
B_fh = @(rho) B0 + B1*rho;
C_fh = @(rho) C0 + C1*rho;
D_fh = @(rho) D0 + D1*rho;
A = {A0, A1};
B = {B0, B1};
C = {C0, C1};
D = {D0, D1};
AA = [A{:}];
BB = [B{:}];
Nálam az $E_c$ most az $Im(B)$.
Im_B = IMA([B{1} B{2}]);
Ker_C = INTS(KER(C{1}), KER(C{2}));
We compute $\mathcal V^*$ and $\mathcal R^*$.
[R,V] = CSA([A{1} A{2}], Im_B, Ker_C);
dim_V = size(V,2);
dim_Ker_V = size(V,1) - size(V,2);
assert(rank(R) == 0 && rank([V Ker_C]) == rank(Ker_C), ...
'Strong invertibility condition is not satisfied!');
assert(rank(INTS(V, Im_B)) == 0)
We ensure that $\mathcal R^* = \{0\}$ and that $\mathcal V^* \subseteq \mathrm{Ker}(C)$. Furthermore, we checked the strong invertibility condition $\mathcal V^* \cap \mathrm{Im}\big(B(\rho)\big) = \{0\}$.
\begin{align} {\LARGE(10) \quad} \mathcal V^* = \left(\begin{array}{cccccccccc} -0.992 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -0.0143 & 0.978 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0.028 & -0.0591 & -0.843 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0.105 & 0.0804 & 0.277 & 0.448 & 0 & 0 & 0 & 0 & 0 & 0 \\ -0.0165 & -0.06 & 0.115 & 0.13 & 0.941 & 0 & 0 & 0 & 0 & 0 \\ -0.0426 & -0.121 & 0.184 & 0.444 & -0.264 & 0.595 & 0 & 0 & 0 & 0 \\ 0.017 & 0.0341 & -0.0252 & -0.223 & 0.0832 & 0.376 & -0.878 & 0 & 0 & 0 \\ 0.0274 & 0.0201 & 0.0756 & -0.469 & 0.0794 & 0.467 & 0.356 & 0.611 & 0 & 0 \\ -0.0129 & 0.0599 & -0.268 & 0.442 & 0.0711 & 0.0554 & -0.04 & 0.401 & -0.616 & 0 \\ -0.0091 & 0.0069 & -0.0704 & 0.198 & -0.005 & -0.1 & -0.0926 & 0.323 & 0.451 & -0.786 \\ 0.0129 & 0.0834 & -0.212 & 0.0148 & 0.153 & 0.514 & 0.304 & -0.571 & 0.0364 & -0.256 \\ -0.0068 & 0.0477 & -0.195 & 0.283 & 0.0628 & 0.094 & 0.0093 & 0.19 & 0.645 & 0.563 \\ \end{array}\right) \end{align}1. variáns
Let $T = \begin{pmatrix} {V^*}^\perp & L \end{pmatrix}^T$, where ${V^*}^\perp$ and $L$ are orthonormal bases for ${\mathcal V^*}^\perp \not\perp \mathcal L \subseteq \mathrm{Im}\big(B(\rho)\big)$, respectively.
L = ORTCO(Im_B);
dim_L = size(L,2);
assert(dim_L >= dim_V);
T = [ ORTCO(V) L(:,1:dim_V) ]';
tol = 1e-10;
prec = -log10(tol);
N = numel(A);
tA = cell(1,N);
tB = cell(1,N);
tC = cell(1,N);
tD = cell(1,N);
for i = 1:numel(A)
tA{i} = round(T*A{i}/T, prec);
tB{i} = round(T*B{i}, prec);
tC{i} = round(C{i}/T, prec);
tD{i} = round(D{i}, prec);
end
tA_fh = @(rho) tA{1} + tA{2}*rho;
tB_fh = @(rho) tB{1} + tB{2}*rho;
tC_fh = @(rho) tC{1} + tC{2}*rho;
tD_fh = @(rho) tD{1} + tD{2}*rho;
tP = T' \ P / T;
tP22 = tP(dim_Ker_V+1:end,dim_Ker_V+1:end);
for rho = rho_lim
tA_rho = tA_fh(rho);
tA22 = tA_rho(dim_Ker_V+1:end,dim_Ker_V+1:end);
eig(tP22 * tA22 + tA22' * tP22)
end
ans = 1.0e+09 * -3.1174 -1.1826 -0.1368 -0.9343 -0.7637 -0.8061 -0.0000 -0.0000 -0.0000 -0.0000 ans = 1.0e+09 * -2.7092 -0.1686 -1.2891 -1.0916 -0.7694 -0.8056 -0.0000 -0.0000 -0.0000 -0.0000