Teljes Matlab script kiegészítő függvényekkel.
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,B,C,D] = deal(sys.a, sys.b, sys.c, sys.d);
tol = 1e-10;
prec = -log10(tol);
A0 = round(A0,prec);
B = round(B,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;
B0 = B;
B1 = B*0;
% Indices = [3];
% B1(Indices) = rand(size(Indices));
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,B,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 = sdpvar(n_x,n_y,'full');
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*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*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*C-C'*N'
];
$$ M_2 = \spmqty{ { \wt A_c({\color{red} \rho})^T} { P} + { P} { \wt A_c({\color{red} \rho})} & { P} \wt B_c({\red \rho}) - { \wt C^T({\red \rho})} & { \wt C^T({\red \rho})} \\ \wt B_c^T({\red \rho}) { P}- { \wt C({\red \rho})} & 0 & 0 \\ { \wt C({\red \rho})} & 0 & -W^{-1} } $$
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 = value(N);
L = S\N;
P = value(P);
Co = value(Co);
C1 = value(C1);
D1 = value(D1);
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 70
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 : 70
Cones : 0
Scalar variables : 0
Matrix variables : 3
Integer variables : 0
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 70
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 : 2485 after factor : 2485
Factor - dense dim. : 0 flops : 4.63e+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 0.00
1 5.1e-01 2.2e-01 2.3e+00 1.00e+00 8.653260980e-01 0.000000000e+00 2.2e-01 0.01
2 1.7e-01 7.3e-02 1.4e+00 9.99e-01 2.928667199e-01 0.000000000e+00 7.3e-02 0.01
3 6.5e-02 2.7e-02 8.2e-01 9.96e-01 1.092859917e-01 0.000000000e+00 2.7e-02 0.01
4 2.6e-02 1.1e-02 5.2e-01 9.96e-01 4.359699912e-02 0.000000000e+00 1.1e-02 0.01
5 6.5e-03 2.7e-03 2.6e-01 9.93e-01 1.106114712e-02 0.000000000e+00 2.7e-03 0.01
6 1.2e-03 4.9e-04 1.1e-01 9.83e-01 1.983690263e-03 0.000000000e+00 4.9e-04 0.01
7 3.7e-04 1.6e-04 6.0e-02 9.66e-01 6.497328863e-04 0.000000000e+00 1.6e-04 0.01
8 1.0e-04 4.3e-05 3.1e-02 9.72e-01 1.792048672e-04 0.000000000e+00 4.3e-05 0.02
9 3.7e-05 1.5e-05 1.8e-02 9.61e-01 6.549183771e-05 0.000000000e+00 1.5e-05 0.02
10 1.1e-05 4.5e-06 9.3e-03 9.37e-01 1.963054462e-05 0.000000000e+00 4.5e-06 0.02
11 3.8e-06 1.6e-06 5.1e-03 8.75e-01 7.332408972e-06 0.000000000e+00 1.6e-06 0.02
12 1.0e-06 4.4e-07 2.2e-03 7.86e-01 2.163809803e-06 0.000000000e+00 4.4e-07 0.02
13 4.0e-07 1.7e-07 1.1e-03 5.87e-01 9.458365393e-07 0.000000000e+00 1.7e-07 0.02
14 1.1e-07 4.5e-08 3.7e-04 4.42e-01 2.401016369e-07 0.000000000e+00 4.5e-08 0.02
15 4.2e-08 1.8e-08 1.6e-04 2.27e-01 4.545620214e-08 0.000000000e+00 1.8e-08 0.02
16 1.1e-08 4.6e-09 4.6e-05 1.82e-01 -9.180751529e-08 0.000000000e+00 4.6e-09 0.02
17 3.7e-09 1.6e-09 1.5e-05 4.56e-02 -1.688411463e-07 0.000000000e+00 1.6e-09 0.03
18 9.9e-10 4.2e-10 4.6e-06 1.05e-01 -1.624759075e-07 0.000000000e+00 4.2e-10 0.03
19 3.3e-10 1.4e-10 1.3e-06 -4.15e-02 -2.373727832e-07 0.000000000e+00 1.4e-10 0.03
20 7.8e-11 3.3e-11 3.7e-07 8.04e-02 -1.847140321e-07 0.000000000e+00 3.3e-11 0.03
21 2.5e-11 1.1e-11 1.1e-07 -4.23e-02 -2.418477399e-07 0.000000000e+00 1.1e-11 0.03
22 5.9e-12 2.5e-12 2.8e-08 4.44e-02 -2.014976492e-07 0.000000000e+00 2.5e-12 0.03
23 2.3e-12 9.6e-13 9.9e-09 -2.33e-02 -2.318494891e-07 0.000000000e+00 9.6e-13 0.03
24 5.6e-13 2.4e-13 2.5e-09 1.39e-02 -2.143400718e-07 0.000000000e+00 2.4e-13 0.04
25 1.9e-13 8.3e-14 8.5e-10 2.36e-02 -2.133640503e-07 0.000000000e+00 7.8e-14 0.04
26 8.7e-14 4.0e-14 3.9e-10 -2.92e-02 -2.242798185e-07 0.000000000e+00 3.7e-14 0.04
27 2.2e-14 1.1e-14 1.0e-10 4.02e-02 -2.048706677e-07 0.000000000e+00 9.4e-15 0.04
28 1.5e-14 4.5e-15 3.2e-11 -5.15e-02 -2.423431259e-07 0.000000000e+00 3.2e-15 0.04
29 5.7e-15 3.8e-15 8.6e-12 3.29e-02 -1.991483253e-07 0.000000000e+00 7.8e-16 0.04
30 1.1e-14 2.9e-15 3.3e-12 7.07e-03 -2.168408055e-07 0.000000000e+00 3.0e-16 0.04
31 1.9e-14 4.0e-15 8.0e-13 -5.50e-02 -2.151847642e-07 0.000000000e+00 7.7e-17 0.04
32 1.5e-14 5.1e-15 3.2e-13 5.05e-02 -2.044214007e-07 0.000000000e+00 2.9e-17 0.04
33 1.4e-14 3.6e-15 7.7e-14 -2.12e-03 -2.112498488e-07 0.000000000e+00 7.1e-18 0.05
34 1.6e-14 5.8e-15 3.5e-14 1.94e-02 -2.020719376e-07 0.000000000e+00 3.1e-18 0.05
35 1.7e-14 5.3e-15 8.7e-15 1.05e-02 -2.073297244e-07 0.000000000e+00 7.9e-19 0.05
36 1.6e-14 6.7e-15 2.7e-15 1.44e-02 -2.079595344e-07 0.000000000e+00 2.5e-19 0.05
37 4.8e-15 5.2e-15 8.1e-16 -8.84e-03 -2.086795478e-07 0.000000000e+00 7.4e-20 0.05
38 1.9e-15 4.2e-15 3.2e-16 3.66e-02 -2.085400886e-07 0.000000000e+00 2.9e-20 0.05
39 4.9e-16 3.3e-15 8.2e-17 -2.18e-02 -2.122872776e-07 0.000000000e+00 7.6e-21 0.06
40 2.0e-16 4.9e-15 3.4e-17 2.60e-02 -2.094344730e-07 0.000000000e+00 3.1e-21 0.06
41 4.8e-17 7.6e-15 7.9e-18 -2.70e-02 -2.157591766e-07 0.000000000e+00 7.5e-22 0.06
42 2.2e-17 5.8e-15 3.7e-18 8.82e-03 -2.153702299e-07 0.000000000e+00 3.5e-22 0.06
43 5.2e-18 8.1e-15 8.5e-19 -4.88e-02 -2.191924415e-07 0.000000000e+00 8.5e-23 0.07
44 2.6e-18 4.9e-15 4.3e-19 5.61e-02 -2.166711047e-07 0.000000000e+00 4.2e-23 0.07
45 6.2e-19 8.6e-15 1.0e-19 -3.17e-02 -2.216121230e-07 0.000000000e+00 1.0e-23 0.07
46 2.5e-19 9.2e-15 4.1e-20 4.33e-02 -2.139068091e-07 0.000000000e+00 4.1e-24 0.07
47 5.0e-19 5.8e-15 9.5e-21 -1.19e-02 -2.222423383e-07 0.000000000e+00 9.8e-25 0.07
48 2.6e-17 5.8e-15 8.8e-21 2.28e-02 -2.220289699e-07 0.000000000e+00 9.0e-25 0.08
49 3.7e-17 5.1e-15 8.1e-21 1.67e-02 -2.219344403e-07 0.000000000e+00 8.3e-25 0.08
50 3.9e-17 5.3e-15 8.0e-21 1.60e-02 -2.219318921e-07 0.000000000e+00 8.2e-25 0.08
51 3.9e-17 5.8e-15 8.0e-21 1.44e-02 -2.219313216e-07 0.000000000e+00 8.2e-25 0.09
52 3.9e-17 5.8e-15 8.0e-21 1.45e-02 -2.219313216e-07 0.000000000e+00 8.2e-25 0.09
Optimizer terminated. Time: 0.10
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -2.2193132163e-07 nrm: 3e+09 Viol. con: 3e-07 barvar: 1e-07
Dual. obj: 0.0000000000e+00 nrm: 3e+09 Viol. con: 0e+00 barvar: 3e-06
Optimizer summary
Optimizer - time: 0.10
Interior-point - iterations : 53 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: 0.1801
solvertime: 0.1077
info: 'Successfully solved (MOSEK)'
problem: 0
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint| Primal residual| Dual residual|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Matrix inequality| -3.2315e-07| 4.5921e-17|
| #2| Matrix inequality| -1.7472e-07| 1.3704e-16|
| #3| Matrix inequality| 2.4208e-05| 5.8803e-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,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} -0.991 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -0.0188 & 0.978 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0.0217 & -0.0604 & -0.837 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0.112 & 0.0919 & 0.26 & 0.446 & 0 & 0 & 0 & 0 & 0 & 0 \\ -0.0141 & -0.0395 & 0.0778 & 0.1 & 0.974 & 0 & 0 & 0 & 0 & 0 \\ -0.0544 & -0.134 & 0.229 & 0.469 & -0.191 & 0.531 & 0 & 0 & 0 & 0 \\ 0.0208 & 0.0393 & -0.0398 & -0.233 & 0.0622 & 0.469 & -0.826 & 0 & 0 & 0 \\ 0.0272 & 0.0157 & 0.0886 & -0.465 & 0.0502 & 0.475 & 0.428 & 0.563 & 0 & 0 \\ -0.0073 & 0.0609 & -0.281 & 0.417 & 0.0433 & 0.126 & 0.0189 & 0.344 & -0.648 & 0 \\ -0.0074 & 0.0118 & -0.0877 & 0.199 & 0.0003 & -0.0667 & -0.0835 & 0.335 & 0.415 & -0.799 \\ 0.0146 & 0.0684 & -0.189 & 0.0186 & 0.0794 & 0.49 & 0.355 & -0.637 & 0.081 & -0.233 \\ -0.0045 & 0.0458 & -0.206 & 0.295 & 0.0339 & 0.11 & 0.0287 & 0.215 & 0.633 & 0.554 \\ \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 * -2.0433 -0.9166 -0.7192 -0.5903 -0.4123 -0.0797 -0.0000 -0.0000 -0.0000 -0.0000 ans = 1.0e+09 * -2.3565 -0.8375 -0.6957 -0.5840 -0.4172 -0.0936 -0.0000 -0.0000 -0.0000 -0.0000