From 22da0c3b7a1db17524dc459443a94dbbc8c67d33 Mon Sep 17 00:00:00 2001 From: a2nr Date: Thu, 12 Dec 2019 11:08:24 +0700 Subject: [PATCH] Tambah subbab tentang LQR dan revisi minor. Next step adalah tambah sub bab didalem "todo" --- BAB2/bab2.tex | 6 +- BAB3/bab3.tex | 47 +-- BAB3/img/Structur_pen.dia | Bin 1539 -> 1874 bytes BAB3/img/Structur_pen.dia~ | Bin 1539 -> 1880 bytes BAB3/img/structur.tex | 36 +- BAB4/Gref.csv | 52 --- BAB4/Kst.matrix | 4 - BAB4/Nst.matrix | 3 - BAB4/bab4.tex | 350 +++++------------- BAB4/data_response.csv | 202 ++++++++++ BAB4/img/statefeedback.tex | 2 +- BAB4/poleGref.csv | 3 - ...[Asurveyofmulti-agentformationcontrol].pdf | Bin 0 -> 772677 bytes ...ler Distance-based Formation Tracking].pdf | Bin 0 -> 545225 bytes uithesis.sty | 36 +- 15 files changed, 390 insertions(+), 351 deletions(-) delete mode 100644 BAB4/Gref.csv delete mode 100644 BAB4/Kst.matrix delete mode 100644 BAB4/Nst.matrix create mode 100644 BAB4/data_response.csv delete mode 100644 BAB4/poleGref.csv create mode 100644 OTHER/CITESRC/_[2014][Kwang-Kyo][Asurveyofmulti-agentformationcontrol].pdf create mode 100644 OTHER/CITESRC/_[2015][Rozenheck][A PI Controler Distance-based Formation Tracking].pdf diff --git a/BAB2/bab2.tex b/BAB2/bab2.tex index 3f37b12..5bb3e52 100644 --- a/BAB2/bab2.tex +++ b/BAB2/bab2.tex @@ -116,7 +116,7 @@ bernilai kecil, dan dalam persamaan~\eqref{eq:dyn_motor} nilai induktansi dapat Penjabaran dinamika robot bisa diubah dalam bentuk \textit{state-space} \begin{align} - \dot{x}(t) & = A_rx(t) + B_ru(t) + Ksgn(x(t)) \label{eq:ss1} \\ + \dot{x}(t) & = A_rx(t) + B_ru(t) + ksgn(x(t)) \label{eq:ss1} \\ y(t) & = Cx(t) \label{eq:ss2} \end{align} dimana vektor \textit{state} adalah $x(t) = \begin{bmatrix} x_p & y_p & \theta & \dot{x}_r & \dot{y}_r & \dot{\theta}_r \end{bmatrix}^T$, dan @@ -140,9 +140,7 @@ sistem robot \frac{l.K_t}{R_a.r}.\frac{-1}{M} & \frac{l.K_t}{R_a.r}.\frac{\cos(60^\circ)}{M} & \frac{l.K_t}{R_a.r}.\frac{\cos(60^\circ)}{M} \\ \frac{l.K_t}{R_a.r}.\frac{b}{I} & \frac{l.K_t}{R_a.r}.\frac{b}{I} & \frac{l.K_t}{R_a.r}.\frac{b}{I} \\ \end{bmatrix}, - %% \end{align*} - %% \begin{align*} - K = \begin{bmatrix} + k = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & \\ 0 & 0 & 0 & 0 & 0 & 0 & \\ 0 & 0 & 0 & 0 & 0 & 0 & \\ diff --git a/BAB3/bab3.tex b/BAB3/bab3.tex index a0efa47..685c9c4 100644 --- a/BAB3/bab3.tex +++ b/BAB3/bab3.tex @@ -12,9 +12,8 @@ bidan kendali multi-robot, khususnya dalam kendali formasi. \caption{Kerangka Penelitian} \label{fig:kerangka_pen} \end{figure} -\todo{ - Pada Gambar diberi kotak yang didalamnya menunjjukan fokus penelitian lalu diberi legend -} + + \section{Definisi Permasalahan Kendali Formasi} Kendali formasi adalah kendali multi-agent untuk mencapai suatu formasi yang diinginkan. @@ -26,33 +25,27 @@ Pembagian kategori tersebut berdasarkan kemampuan sensor yang digunakan dan penggunaan komunikasi dalam metodenya. Dari ketiga kategori tersebut, kendali formati berbasis jarak sangat dibutuhkan pembahasan mengenai penerapan metode tersebut pada agent yang nyata. +\textit{Simple model, Model real,} dan \textit{Real} dapat dikatakan sebuah tahap pengemabangan. +\kutip{OH2015424} menyatakan bahwa mayoritas dari hasil penelitian yang menggunakan pendekatan ini (\textit{distance-based}) berfokus pada model agent dengan integrator-tunggal di suatu bidang datar. +Gagasan agent \textit{simple model} tersebut memiliki manfaat ketika menginvestigasi karakteristik kendali secara mendasar, model agent yang lebih relistik (\textit{Model real}) perlu untuk dipelajari lebih lanjut untuk menambah kepraktisan pada metode kendali multi-agent berdasarkan jarak. +Dengan bertambahnya kepraktisan pada metode tersebut diharapkan dapat diterapkan dalam agent secara \textit{Real}. Pada penelitian oleh \kutip{Rozenheck2015}, kendali formasi berdasarkan jarak dikendalikan menggunakan kendali PI dan menghasilkan pergerakan yang baik. -Dalam penelitian ini akan difokuskan pada kendali formasi berbasis jarak -dengan mengembangkan kendali PI yang telah dilakukan sebelumnya beserta menggunakan -model nyata. -\todo{ - Diperlukan penjelasan mengenai simple model itu bagaiman, - contohnya pada penjelasan jurnal kebanyakan analisis menggunakan model orde satu yang sangat - sederhana - lalu menggunakan model real untuk mendesain kendali - dan model real adalah penerapan dari model real. -} - +Dapat diperhatikan pada persamaan~\eqref{eq:modelorde1} bahwa peneliti menggunakan \textit{Simple model} untuk mengembangkan kendali multi-robotnya. +Maka, penelitian ini akan difokuskan pada kendali formasi berbasis jarak +kendali PI yang telah dilakukan sebelumnya dengan menggunakan model nyata. \section{Permasalah dan Solusi} -Pada kerangka kendali-PI pada persamaan~\eqref{eq:ss-formasi}, state yang digunakan membutuhkan -koordinat relatif dari tetangganya. Akan tetapi pada batasan penelitian ini, sensor yang digunakan -hanya memberikan jarak terhadap tetangganya. Secara pendekatan, digunakan koordinat polar dan diubah -ke koordinat kartesian. Akan tetapi koordinat polar membutuhkan sudut antara agent dan tetangganya. +Dapat diperhatikan pada persamaan~\eqref{eq:ss-formasi}, state yang digunakan membutuhkan koordinat relatif dari tetangganya. +Akan tetapi pada batasan penelitian ini, sensor yang digunakan hanya memberikan jarak terhadap tetangganya. +Sedangkan koordinat relatif tersebut berupa kartesian. +Koordinat kartesian dapat diubah ke bentuk koordinat polar, atau sebaliknya. +Akan tetapi koordinat polar membutuhkan sudut antara agent dan tetangganya. Oleh karena itu dibutuhkan algoritma khusus untuk menutup permasalahan tersebut. - -Untuk mengembangkan algoritma tersebut, dapat menggunakan hukum \textit{cosinus} segitiga -untuk menentukan sudutnya. -Algoritma \textit{cosinus} tersebut hanya berlaku apabila tetangga tidak melakukan pergerakan dan -akan dijalankan algoritma tersebut ketika inisialisasi. -Ketika tetangga melakukan pergerakan, tetangga mengirimkan informasi percepatan koordinatnya pada agent. -Kegunaannya adalah sebagai referensi perubahan koordinat terhadap tetangga. -Sehingga harapanya adalah kerangka kendali-PI dapat digunakan menggunakan sensor yang hanya mendeteksi jarak saja. - +Untuk mengembangkan algoritma tersebut, dapat menggunakan hukum \textit{cosinus} segitiga untuk menentukan sudutnya. +Dengan memanfaatkan komunikasi antar robot, maka robot dapat mengirimkan informasi state kecepatan kepada tetangganya. +Sehingga informasi tersebut dapat digunakan untuk memantau koordinat relatif terhadap tetangganya. +Akan tetapi state kecepatan tersebut membutuhkan nilai inisialisasi. +Nilai inisialisasi ini akan diperoleh menggunakan algoritma \textit{cosinus}. +Sehingga, harapanya adalah kendali multi-robot menggunakan kendali PI dapat digunakan sesuai batasan penelitian. diff --git a/BAB3/img/Structur_pen.dia b/BAB3/img/Structur_pen.dia index 8dbc126a11954fe9734918cc3f65652e245a485c..986e9b56c6f96ae53acf8a878e34348f58bb32f6 100644 GIT binary patch literal 1874 zcmV-Y2d(%YiwFP!000021MOVPa@#f(-TNyz%2uK9AP6vyooUkXbdgRoO}fcO1BtN2 zh9Wd1mgw~yilR!pS189SRC8DYH zgJ^a&`0LxvTQ>M`eevE8_&fb?7W0LrpNNw5)74kG@;G~jc^(|g5Tu2dXMR!UeAUx=&0gin9WV!4X^L0+mTHl*I-nb8Ex5l#`coW#3UB{N4;sFFVAaYdMV2j+!gC{@9WAyC4ZBp{S~T5UG=ye>{!(a_?m9y?(sq zXvnSA%7f1Y5#)Gzn5_a|BzuP0c9|MW^P<>jU#pkP9ojJYt&&)05FvjQabf=BTBfa{ zgA2y7_*4aVPoFm0L{9z|Q&l*76UzJPoX5&~YkiatgY_gtPeB6zYVZfxZmqI^+L|W9 zo#tA~^_ZeIf`|>74G{L6a!c&;NFB^%QbvijBRrag;wKVxH4?Qws*x^{V`G=$A@pnq zK-dgfgkmwC$~fAcG@RPW3@YQ8T}PkzMY{-y;{gURBbZ^Aw7{B+U^Z9vU=Z3EQRrgm zIv5~ASTnGRjC~RBa}8+M@(XGQA#@>X2A9m`{kUQRyI-A;LD8}-#)*0iMHxUut3_@d zH#5EDzP~N{J5ANF+m(3O>F+WDmdeyMayIqdDp#5%*UXWh5th1Hii}hhPy^fwXS2^CEFCQ%<>gPlOB>=OpIFQ*;oHmz2%4gxlH2^m6cGlobz zxNXUOW?cqAJ0Lxwoka^#-KXo=fd`^g3ToY?V6;}i=8_0oOClRW)HH+-l|qlCAP%`i z*rr41=>K-AfNgv_MEQvAI^M>`+%A>R_O`VM8RdIr^7Lzv=!lt$er4)2S3zX`F7>F; zzqlS1JDT-LuHrz%uT`8CWn}9G$<~+DsE%tRS3SyAx+iE>T73(a(^jw$=~}Rmvlc82`Vg#EJ{iF>f^~+i z!DYy&B4r{YT79mus1dDGi`G|7{yvX=ty$lL*>^ts97JM6x?mw)B1~lz=>X7in(Rp5(_gH) zj_*Ptl+1Sft@U5826Sg2R6GldXYuy4(A`a3E7j zLkzu)gqYJXLo;pBu^u}6d8oq0>QWwbGKQfIc~J-3bXl%RRi?U$?eoU=rts|#ZrDk1 z@86kYr#MQY-5(^m`e$*F#B8tggCuHp<;8%pIn| z45-&7rD-IbBT+)VOC9hK>zP}MOqwDezACHpvyzDbD=OBwNjh1VW$ys&YY(u)7Y_P7 zneQhQx3saeuP2TSsc-52U&SP$^%qn0_`wX2bo_Y8WnG5U8i_E-vi*6=h6via-3#x+m)7^r~%ufMZRF!Y$_C_)it6uYp8jnwLa`WkjpK?ggjbuV4cs??k% zPefepzs};>*g>?9xxg9;L8Rvb>(8DG9P_eS^Rn66!Dz_otmKq`&Rz&o%09_5LJ^%m! 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+\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\draw (25.388559\du,20.807882\du)--(27.083174\du,20.807882\du); +} +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (27.380475\du,20.926802\du){: Fokus Penelitian}; +% setfont left to latex +\definecolor{dialinecolor}{rgb}{0.000000, 0.000000, 0.000000} +\pgfsetstrokecolor{dialinecolor} +\node[anchor=west] at (29.342661\du,20.807882\du){}; \end{tikzpicture} diff --git a/BAB4/Gref.csv b/BAB4/Gref.csv deleted file mode 100644 index 2986465..0000000 --- a/BAB4/Gref.csv +++ /dev/null @@ -1,52 +0,0 @@ -column 1,column 2 -0,0 -0.007519014058975133,0.014 -0.02787288907893497,0.028 -0.05813661124868522,0.04200000000000001 -0.09584103764508942,0.05600000000000001 -0.1389152410439689,0.07000000000000001 -0.1856349017744423,0.08400000000000002 -0.2345762549446597,0.09800000000000002 -0.2845751216558404,0.112 -0.334690575780543,0.126 -0.3841728226304695,0.14 -0.4324348916534503,0.154 -0.4790277715864288,0.168 -0.5236186427736041,0.182 -0.5659718872567027,0.196 -0.6059325824613133,0.21 -0.6434122086135965,0.224 -0.6783763232591336,0.238 -0.7108339783034885,0.2520000000000001 -0.7408286757766605,0.266 -0.7684306779992418,0.28 -0.7937305059825104,0.294 -0.8168334767356632,0.3080000000000001 -0.8378551457061627,0.3220000000000001 -0.8569175348824327,0.3360000000000001 -0.8741460401907847,0.35 -0.8896669237767264,0.364 -0.9036053076360092,0.3780000000000001 -0.9160835949173277,0.3920000000000001 -0.9272202541224777,0.4060000000000001 -0.937128909447261,0.42 -0.9459176877030213,0.4340000000000001 -0.9536887786983711,0.4480000000000001 -0.9605381717052835,0.4620000000000001 -0.966555535742483,0.4760000000000001 -0.9718242159382878,0.49 -0.9764213222379007,0.5040000000000001 -0.980417890246481,0.5180000000000001 -0.9838790970957602,0.532 -0.9868645179317038,0.546 -0.9894284109837901,0.5600000000000001 -0.9916200212296733,0.5740000000000001 -0.9934838944460908,0.5880000000000001 -0.9950601949686966,0.6020000000000001 -0.9963850217981394,0.6160000000000001 -0.9974907188126078,0.6300000000000001 -0.9984061758012728,0.6440000000000001 -0.9991571178393106,0.6580000000000001 -0.9997663812021337,0.6720000000000002 -1.000254174580784,0.6860000000000001 -1.000638324827062,0.7000000000000001 \ No newline at end of file diff --git a/BAB4/Kst.matrix b/BAB4/Kst.matrix deleted file mode 100644 index 54368d1..0000000 --- a/BAB4/Kst.matrix +++ /dev/null @@ -1,4 +0,0 @@ -k1,k2,k3,k4,k5,k6 --0,-82.81000000000003,6.900833333333332,-0,-9.670003622750881,1.028166847804211 -71.71556368738935,41.40500000000003,6.900833333333331,8.386015797373604,4.835001811375442,1.02816684780421 --71.71556368738936,41.40500000000002,6.900833333333334,-8.386015797373604,4.835001811375441,1.028166847804211 diff --git a/BAB4/Nst.matrix b/BAB4/Nst.matrix deleted file mode 100644 index b8aec05..0000000 --- a/BAB4/Nst.matrix +++ /dev/null @@ -1,3 +0,0 @@ --0,-82.81000000000003,6.900833333333335,-0,-0,-0 -71.71556368738936,41.40500000000002,6.900833333333334,-0,-0,-0 --71.71556368738936,41.40500000000002,6.900833333333334,-0,-0,-0 diff --git a/BAB4/bab4.tex b/BAB4/bab4.tex index 3699784..7ed31e8 100644 --- a/BAB4/bab4.tex +++ b/BAB4/bab4.tex @@ -24,7 +24,6 @@ teria yang diinginkan. \caption{State-feedback Sistem} \label{fig:state-feedback} \end{figure} -\todo{ Ganti notasi K untuk friction ke $k$} Pada persamaan~\eqref{eq:ss1} diketahui bahwa state memiliki dimensi $6 \times 1$. Dimensi tersebut tidak menunjukan sistem memiliki orde 6. Apabila diperhatikan orde dari sistem adalah orde 2. Dengan membaginya kedalam 3 persamaan state-space @@ -35,17 +34,17 @@ akan lebih mudah dalam analisis parameter kendalinya. Berikut adalah persamaan \begin{bmatrix}0 & A_{14} \\ 0 & A_{44} \end{bmatrix} \begin{bmatrix}{x}_p \\ \dot{x}_r\end{bmatrix} + \begin{bmatrix}0 & 0& 0 \\ B_{11} & B_{12} & B_{13} \end{bmatrix} - \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + K_{44}sgn(\dot{x}_r) \label{eq:ssx} \\ + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + k_{44}sgn(\dot{x}_r) \label{eq:ssx} \\ \begin{bmatrix}\dot{y}_p \\ \ddot{y}_r \end{bmatrix} & = \begin{bmatrix}0 & A_{25} \\ 0 & A_{55} \end{bmatrix} \begin{bmatrix}{y}_p \\ \dot{y}_r\end{bmatrix} + \begin{bmatrix}0 & 0& 0 \\ B_{21} & B_{22} & B_{23} \end{bmatrix} - \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + K_{55}sgn(\dot{y}_r)\label{eq:ssy} \\ + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + k_{55}sgn(\dot{y}_r)\label{eq:ssy} \\ \begin{bmatrix}\dot{\theta}_p \\ \ddot{\theta}_r\end{bmatrix} & = \begin{bmatrix}0 & A_{34} \\ 0 & A_{66} \end{bmatrix} \begin{bmatrix}{\theta}_p \\ \dot{\theta}_r\end{bmatrix} + \begin{bmatrix}0 & 0& 0 \\ B_{31} & B_{32} & B_{33} \end{bmatrix} - \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + K_{66}sgn(\dot{\theta}_r) \label{eq:ssthe} + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + k_{66}sgn(\dot{\theta}_r) \label{eq:ssthe} \end{align} State feedback membutuhkan kembalian nilai state dari sistem dan mengka- @@ -72,172 +71,59 @@ sistem robot juga observable. Karena pengukuran pada setiap \textit{state} dapat maka \textit{observer} tidak dibutuhkan dalam desain kendali robot. \subsubsection{Desain Kendali} -Kriteria didefinisi menggunakan analisis sistem orde dua pada domain waktu. -Berikut adalah transfer fungsi tertutup dari sistem orde dua (\kutip{Richard2010}). -\begin{align} - Y(s) = \frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} R(s) - \label{eq:Gref} -\end{align} -Dengan input $R(s) = 1/s$ sebagai unit step, maka didapat persamaan keluaran sistem dalam domain waktu. -\begin{align} - y(t) = 1 - \frac{1}{\beta}e^{-\zeta\omega_n t}sin(\omega_n\beta t + \theta) -\end{align} -dimana $\beta = \sqrt{1-\zeta^2}$, $\theta = \cos^{-1}\zeta$, dan $0<\zeta<1$. -Dari persamaan domain waktu tersebut didapat 4 kriteria dalam sistem, -yaitu \textit{satling time} ($T_s$), Prosentase \textit{overshoot} ($P.O$), -\textit{peek time} ($T_p$), dan \textit{Transient Time} ($T_{r1}$). -Dari keempat kriteria tersebut, merupakan fungsi dari $\zeta$ dan $\omega_n$. -Berikut adalah rumus dari keempat kriteria tersebut. -\begin{itemize} - \item $T_s$ adalah rumus pendekatan untuk mengetahui waktu respon sistem - mulai dari kondisi inisial sampai sistem mencapai 2\% dari set poin yang terakhir. - \begin{align} - T_s = \frac{4}{\zeta \omega_n} - \end{align} - \item $P.O$ adalah prosentase \textit{overshoot} dari sistem. - Arti dari \textit{overshoot} pada sistem adalah respon sistem yang melebihi dari nilai set poin. - \begin{align} - P.O. = 100e^{-\zeta \pi/\sqrt{1-\zeta^2}} - \end{align} - \item $T_p$ adalah waktu dimana respon sistem mengalami \textit{overshoot} dimulai dari kondisi inisial. - \begin{align} - T_p = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}} - \end{align} - \item $T_{r1}$ adalah pendekatan nilai waktu respon sistem dimulai dari respon 10\% menuju ke 90\% set poin. - Akurasi dari rumus ini adalah $0.3 \leq \zeta \leq 0.8$. - \begin{align} - T_{r1} = \frac{2.16\zeta + 0.6}{\omega_n} - \end{align} -\end{itemize} -Apabila menggunakan $\zeta = 0.9$ dan $\omega_n = 9.1$, akan diperoleh kriteria $T_s= 0.48$, $T_p=14.75$, $P.O.=0.15$, dan $T_{r1}=0.27$. Grafik step response dapat diperhatiakn pada gamabar~\ref{fig:stepResGref}. -\begin{figure} - \begin{center} - \begin{subfigure}[t]{.4\textwidth} - \begin{tikzpicture} - %%https://www.latex-tutorial.com/tutorials/pgfplots/ - \begin{axis}[ - width=\linewidth, % Scale the plot to \linewidth - grid=major, % Display a grid - grid style={dashed,gray!30}, % Set the style - xlabel=$s$, % Set the labels - ylabel=Response, - no marks - ] - \addplot - table[x=column 2,y=column 1,col sep=comma] {BAB4/Gref.csv}; - \end{axis} - \end{tikzpicture} - \caption{} - \label{fig:stepResGref} - \end{subfigure} - \begin{subfigure}[t]{.4\textwidth} - \begin{tikzpicture} - \begin{axis}[ - width=\linewidth, % Scale the plot to \linewidth - grid=major, % Display a grid - grid style={dashed,gray!30}, % Set the style - xlabel=Re, % Set the labels - ylabel=Im, - legend pos=south east, - only marks - ] - \addplot - table[x=real,y=imag,col sep=comma] {BAB4/poleGref.csv}; - \end{axis} - \end{tikzpicture} - \caption{} - \label{fig:poleSystem} - \end{subfigure} - \caption{(a) Step Respon dari persamaan~\eqref{eq:Gref} dengan parameter $\zeta = 0.9$ dan $\omega_n = 9.1$ \\ - (b) Pole sistem adalah -8.19+3.96j dan -8.19+3.96j } - \end{center} -\end{figure} - -Untuk mendesain parameter K pada \textit{state feedback}, -diasumsikan bahwa \textit{state} pada sistem dapat diperoleh dari sensor, $x(t)$ untuk semua $t$. -Persamaan rumus masukan ke sistem menjadi -\begin{align} - u(t) = -K_s x(t) -\end{align} -sehingga persamaan \textit{state space} menjadi berikut. -\begin{align} - \dot{x}(t) = (A-BK_s)x(t) -\end{align} -Dapat diperhatikan bahwa $(A-BK_s)$ merupakan matriks karakteristik dari sistem. -Sehingga dengan mengatur besaran $K_s$ dapat menjadikan sistem sesuai dengan kriterianya. -\begin{align} - \det[\lambda I-(A-BK_s)]=0 - \label{eq:eigen} -\end{align} -Perhatikan persamaan~\eqref{eq:ssx}, matriks pada persamaan tersebut akan diterapkan pada -persamaan~\eqref{eq:eigen}. +Berdasarkan \kutip{Richard2010}, bahwa kendali optimal berdasarkan indeks kinerja sistem. Indeks tersebut adalah hasil dari meminimalisasi pada integral kuadrat error atau \textit{integration square error} (ISE). +Kendali optimal dilakukan oleh komputer untuk mengkalkulasi minimal indeks tersebut. +Apabila sebuah sistem \textit{state space} \begin{align*} - 0 & =\det \Big[ - \begin{bmatrix}\lambda & 0\\ 0 & \lambda \end{bmatrix} - - \Big( - \begin{bmatrix}0 & A_{14} \\ 0 & A_{44} \end{bmatrix} - - \begin{bmatrix}0 & 0& 0 \\ B_{11} & B_{12} & B_{13} \end{bmatrix} - \begin{bmatrix}k_{11} & k_{12} \\ k_{21} & k_{22} \\ k_{31} & k_{32} \end{bmatrix} - \Big) - \Big] \\ - 0 & = \det \Big[ - \begin{bmatrix}\lambda & 0\\ 0 & \lambda \end{bmatrix} - - \Big( - \begin{bmatrix}0 & A_{14} \\ -B_{11}k_{11}-B_{12}k_{21}-B_{13}k_{31} & - A_{44}- B_{11}k_{12}-B_{12}k_{22}-B_{13}k_{32}\end{bmatrix} - \Big) - \Big] \\ - 0 & = \det \Big[ - \begin{bmatrix}\lambda & 0\\ 0 & \lambda \end{bmatrix} - - \begin{bmatrix}0 & A_{14} \\ Z_{21}& Z_{22}\end{bmatrix} - \Big] - = \det \Big[ \begin{bmatrix}\lambda & -A_{14}\\ -Z_{21} & \lambda-Z_{22} \end{bmatrix}\Big] + \dot{x} & = Ax + Bu \\ + u & = -K_sx \end{align*} -Hasil dari diterminan sebagai berikut. +maka indeks kinerja \begin{align} - 0 & = \lambda^2 -Z_{22} \lambda + Z_{21}A_{14} + J & = \int_0^{\infty} (x^T Q x + u^TRu ) dt \end{align} -Dimana -\begin{align*} - Z_{21} &= B_{11}k_{11} +B_{12}k_{21} +B_{13}k_{31} \\ - Z_{22} &= A_{44}- B_{11}k_{12}-B_{12}k_{22}-B_{13}k_{32} +dimana $Q$ adalah matriks diagonal $n \times n$, $R$ adalah matriks diagonal $m \times m$ dan keduanya adalah matriks pembobot terhadap state sistem dan input. +Ketika indeks terminimalisasi, maka +\begin{align} + K_s = R^{-1}B^TP +\end{align} +dengan matriks P $n \times n$ ditentukan dari solusi persamaan \textit{Riccati}. +\begin{align} + A^TP+PA-PBR^{-1}B^TP+Q=0 +\end{align} +% here is it KST from octave +% 0.00000 -21.08185 3.33333 0.00000 -3.74993 0.99738 +% 18.25742 10.54093 3.33333 3.25168 1.87496 0.99738 +% -18.25742 10.54093 3.33333 -3.25168 1.87496 0.99738 +Kalkulasi konstanta $K_s$ akan dikalkulasi menggunakan persamaan-\eqref{eq:ssx}~\eqref{eq:ssthe}. +Sehingga konstanta $K_s$ akan terbagi dalam sub matriks $K_{sx}, K_{sy},$ dan $K_{s\theta}$. +Berikut adalah hasil kalkulasi. +\begin{align*} + K_{s}^{x} & = \begin{bmatrix} + 0.00000 & 0.00000 \\ + 18.25742 & 3.25168 \\ + -18.25742 & -3.25168 \\ + \end{bmatrix} = \begin{bmatrix} K_{s}^{x1} & K_{s}^{x2} \end{bmatrix} \\ + K_{s}^{y} & = \begin{bmatrix} + -21.08185 & -3.74993 \\ + 10.54093 & 1.87496 \\ + 10.54093 & 1.87496 \\ + \end{bmatrix} = \begin{bmatrix} K_{s}^{y1} & K_{s}^{y2} \end{bmatrix} \\ + K_{s}^{\theta} & = \begin{bmatrix} + 3.33333 & 0.99738 \\ + 3.33333 & 0.99738 \\ + 3.33333 & 0.99738 \\ + \end{bmatrix} = \begin{bmatrix} K_{s}^{\theta 1} & K_{s}^{\theta 2} \end{bmatrix} \end{align*} -Dengan asumsi persamaan orde dua mengunakan parameter $\zeta~=~0.9$ dan $\omega_n~=~9.1$ -(hasil analisa pada gambar~\ref{fig:stepResGref}) sebagai berikut. +Apabila diintegrasi terhadap persamaan~\eqref{eq:ss1} terhadap diagram~\ref{fig:state-feedback} \begin{align} - \triangle \lambda &= \lambda^2+2\zeta\omega_n \lambda +\omega_n^2 \\ - &= \lambda^2+16.38 \lambda + 82.81 -\end{align} -Sehingga akan diperoleh besaran $Z_{21}$ dan $Z_{22}$ - - -% Untuk mendapatkan konstanta $K_s$, digunakan aplikasi Matlab/Octave dengan fungsi yang bernama -% $place()$. -% Fungsi tersebut akan mengatur nilai $K_s$ dengan kriteria pole(perhatikan pada gambar \ref{fig:poleSystem}) yang diinginkan. -% Berikut adalah hasil kalkulasi dari fungsi $place()$ menggunakan matriks pada persamaan~\eqref{eq:ssx},\eqref{eq:ssy}, dan \eqref{eq:ssthe}. -\begin{align} - K_s^{x} & = - \begin{bmatrix} - -0.00000 & -0.00000 \\ - 71.71556 & 8.38602 \\ - -71.71556 & -8.38602 - \end{bmatrix}; - K_s^{y}= - \begin{bmatrix} - -82.81000 & -9.67000 & \\ - 41.40500 & 4.83500 & \\ - 41.40500 & 4.83500 & - \end{bmatrix}; \nonumber \\ - K_s^{\theta} & = - \begin{bmatrix} - 6.90083 & 1.02817 \\ - 6.90083 & 1.02817 \\ - 6.90083 & 1.02817 + K_s = \begin{bmatrix} + K_{s}^{x1} & K_{s}^{y1} & K_{s}^{\theta 1} & + K_{s}^{x2} & K_{s}^{y2} & K_{s}^{\theta 2} \end{bmatrix} \end{align} - Setelah mendapatkan konstanta $K_s$, sistem sudah dalam keadaan stabil. Akan tetapi sistem tidak mencapai set poin yang diinginkan. Maka permasalahan tersebut dapat diselesaikan dengan \textit{input refrence}. @@ -262,24 +148,58 @@ Sehingga dapat diperoleh persamaan $N$. \end{align} Berikut adalah hasil kalkulasi dari rumus $N$ menggunakan matriks pada persamaan~\eqref{eq:ssx},\eqref{eq:ssy}, dan \eqref{eq:ssthe}. +% 0.00000 -21.08185 3.33333 0.00000 0.00000 0.00000 +% 18.25742 10.54093 3.33333 0.00000 0.00000 0.00000 +% -18.25742 10.54093 3.33333 0.00000 0.00000 0.00000 +\begin{align*} + N^x & = \begin{bmatrix} + 0.00000 & 0.00000 \\ + 18.25742 & 0.00000 \\ + -18.25742 & 0.00000 + \end{bmatrix} = \begin{bmatrix}N^{x1} & N^{x2}\end{bmatrix}\\ + N^y & = \begin{bmatrix} + -21.08185& 0.00000 \\ + 10.54093 & 0.00000 \\ + 10.54093 & 0.00000 + \end{bmatrix} = \begin{bmatrix}N^{y1} & N^{y2}\end{bmatrix}\\ + N^\theta & = \begin{bmatrix} + 3.33333 & 0.00000 \\ + 3.33333 & 0.00000 \\ + 3.33333 & 0.00000 + \end{bmatrix} = \begin{bmatrix}N^{\theta 1} & N^{\theta 2}\end{bmatrix}\\ +\end{align*} +Apabila diintegrasi terhadap persamaan~\eqref{eq:ss1} terhadap diagram~\ref{fig:state-feedback} \begin{align} - N^x & = \begin{bmatrix} - -0.00000 & -0.00000 \\ - 71.71556 & -0.00000 \\ - -71.71556 & -0.00000 - \end{bmatrix}; - N^y = \begin{bmatrix} - -82.81000 & -0.00000 \\ - 41.40500 & -0.00000 \\ - 41.40500 & -0.00000 - \end{bmatrix} \nonumber \\ - N^\theta & = \begin{bmatrix} - 6.90083 & -0.00000 \\ - 6.90083 & -0.00000 \\ - 6.90083 & -0.00000 - \end{bmatrix} + N = \begin{bmatrix} + N^{x1} & N^{y1} & N^{\theta 1} & + N^{x2} & N^{y2} & N^{\theta 2} + \end{bmatrix} \end{align} +\begin{figure} + \begin{center} + \begin{subfigure}[t]{.3\linewidth} + \dataGraph{t}{x}{t}{x}{BAB4/data_response.csv} + \end{subfigure} + \begin{subfigure}[t]{.3\linewidth} + \dataGraph{t}{y}{t}{y}{BAB4/data_response.csv} + \end{subfigure} + \begin{subfigure}[t]{.3\linewidth} + \dataGraph{t}{theta}{t}{$\theta$}{BAB4/data_response.csv} + \end{subfigure} + \begin{subfigure}[t]{.3\linewidth} + \dataGraph{t}{dx}{t}{$\dot{x}$}{BAB4/data_response.csv} + \end{subfigure} + \begin{subfigure}[t]{.3\linewidth} + \dataGraph{t}{dy}{t}{$\dot{y}$}{BAB4/data_response.csv} + \end{subfigure} + \begin{subfigure}[t]{.3\linewidth} + \dataGraph{t}{dtheta}{t}{$\dot{\theta}$}{BAB4/data_response.csv} + \end{subfigure} + \captionsetup{singlelinecheck=off} + \caption[.]{\centering Grafik state terhadap t dengan state refrensi $ r = \begin{bmatrix}6 & -3 & -90 &0 &0 &0\end{bmatrix} $ } + \end{center} +\end{figure} \subsection{Kendali Formasi Multi Robot} Pada sub bab~\ref{subbab:KendaliFormasi} dijabarkan bagaimana kendali formasi menggunakan @@ -387,86 +307,6 @@ Sedangkan komunikasi dengan PC akan mempresentasikan aktuator dan sensor untuk s kendali. PC akan merekam setiap keluaran dari model dan masukan dari setiap prangkat kendali sebagai tampilan pergerakan robotnya. -\subsection{Kestabilan Model} -\todo{ - Untuk proposal, kestabilan model tidak perlu dimasukkan - masukkan kestabilan ini pada laporan thesis saja. -} -Pada persamaan~\eqref{eq:disstab} apabila model dikalkulasi akan bergantung dengan besarnya \textit{step size}, $h$. -Oleh karena itu, setelah persamaan~\eqref{eq:ss1}-\eqref{eq:ss2} dilakukan parameterisasi harus dilakukan penentuan \textit{step size} agar model tersebut stabil dalam mensimulasikan modelnya. -Penentuan \textit{step size} harus berdasarkan kriteria kestabilan pada gamabar~\ref{fig:explicit_euler}. - -Apabila didefinisi ulang \textit{state} pada persamaan~\eqref{eq:ss1}-\eqref{eq:ss2} dengan -$x(t) = \begin{bmatrix}\dot{x}_r & \dot{y}_r & \dot{\theta}_r \end{bmatrix}^T$, -maka akan lebih mudah untuk menghitung kestabilan dari matriks $A \in \mathbb{R}^{3 \times 3}$. -Dengan menggunakan parameter dari penelitian oleh \kutip{CORREIA20127}, maka akan diperoleh matriks $A, B, K,$ dan $C$. -\begin{align*} - A & = \begin{bmatrix} - -6.69666 & 0.00000 & 0.00000 \\ - 0.00000 & -6.71000 & 0.00000 \\ - 0.00000 & 0.00000 & -4.04200 \\ - \end{bmatrix} ; \quad - B = \begin{bmatrix} - 0.00000 & 0.57735 & -0.57735 \\ - -0.66667 & 0.33333 & 0.33333 \\ - 4.00000 & 4.00000 & 4.00000 \\ - \end{bmatrix} ; \\ - K & = \begin{bmatrix} - -1.46667 & 0.00000 & 0.00000 \\ - 0.00000 & -1.00000 & 0.00000 \\ - 0.00000 & 0.00000 & -0.06600 \\ - \end{bmatrix}; \quad - C = \begin{bmatrix} - 1 & 0 & 0 \\ - 0 & 1 & 0 \\ - 0 & 0 & 1 - \end{bmatrix}. -\end{align*} - -Dengan menggunakan pendekatan pada persamaan~\eqref{eq:desdotode1} untuk persamaan~\eqref{eq:ss1} maka diperoleh bentuk diskretnya -\begin{align} - x[k+1] & = (I + Ah)x[k] + Bhu[k] + Khsgn(x[k]) \\ -\end{align} -Pengali $sgn(.)$ bersifat penambah dari sistem, maka dalam penentuan kestabilan ini akan dianggap penambah dari matriks sistem. -\begin{align} - x[k+1] & = (I + (A+K)h)x[k] + Bhu[k] \\ -\end{align} -Kriteria kestabilan akan bergantung dari hasil penentuan $h$ pada $I+(A+K)h~=~\Lambda$. -Untuk semua nilai $\lambda$ pada matriks $\Lambda$ harus memenuhi kriteris $\lambda \leq 1$. -Dimungkinkan akan mengalami kebingungan ketika menentukan besar $h$, -akan tetapi nantinya persamaan ini akan diterapkan dan diselesaikan oleh komputer. -Alangkah baiknya apabila diidentifikasi terlebih dahulu konsumsi waktu yang dibutuhkan untuk menyelesaikan -satu iterasi dari persamaan tersebut. -Setelah dilakukan identifikasi, waktu yang dibutuhkan untuk satu kali iterasi berkisar $0.001$ ms (Pembulatan). -Sehingga penentuan \textit{step size} sebesar $0.1$ ms sangat dimungkinkan, dengan pertimbangan -sisa dari waktu yang digunakan kalkulasi dapat digunakan untuk waktu \textit{idle} dan menjalankan program yang lain. Berikut adalah matriks $\Lambda$ setelah dikalkulasi menggunakan $h=0.1$ -\begin{align*} - \Lambda = \begin{bmatrix} - 0.18367 & 0.00000 & 0.00000 \\ - 0.00000 & 0.22900 & 0.00000 \\ - 0.00000 & 0.00000 & 0.58920 \\ - \end{bmatrix}. -\end{align*} -Terbukti bahwa semua nilai item didalam matriks kurang dari sama dengan satu. -Sehingga menggunakan algoritma \textit{Expilicit Euler} sudah cukup untuk menjalankan model robot \textit{omni 3-wheel} sebagai model \textit{holonomic} yang akan digunakan untuk percobaan kendali multi robot. -Hasil plot dari simulasi model dapat dilihat pada gambar~\ref{fig:sim_model}. -\begin{figure} - \centering - \begin{subfigure}[t]{.6\textwidth} - \includegraphics[scale=.4]{BAB3/img/speedRobot_-6_3_3.png} - \caption{} - \end{subfigure} - \begin{subfigure}[t]{.6\textwidth} - \includegraphics[scale=.4]{BAB3/img/speedRobot_0_6_-6.png} - \caption{} - \end{subfigure} - \begin{subfigure}[t]{.6\textwidth} - \includegraphics[scale=.4]{BAB3/img/speedRobot_6_6_6.png} - \caption{} - \end{subfigure} - \caption{(a)$w_1=-6; w_2=3; w_3=3$. (b) $w_1=0; w_2=6; w_3=-6$ (c) $w_1=6; w_2=6; w_3=6$} - \label{fig:sim_model} -\end{figure} \todo{ Tambahkan subsection mengenai \begin{itemize} diff --git a/BAB4/data_response.csv b/BAB4/data_response.csv new file mode 100644 index 0000000..c6b1e57 --- /dev/null +++ b/BAB4/data_response.csv @@ -0,0 +1,202 @@ +t,x,y,theta,dx,dy,dtheta,u1,u2,u3 +0,0,0,0,0,0,0,-6,-6,-6 +0.1,0,0,0,0,0,-7.200000000000001,-6,-6,-6 +0.2,0,0,-0.7200000000000002,0,0,-11.48316156502838,-6,-6,-6 +0.3,0,0,-1.868316156502838,0,0,-14.03507015648193,-6,-6,-6 +0.4,0,0,-3.271823172151031,0,0,-15.55549784996569,-6,-6,-6 +0.5,0,0,-4.827372957147601,0,0,-16.46136900023117,-6,-6,-6 +0.6,0,0,-6.473509857170717,0,0,-17.00108722846407,-6,-6,-6 +0.7,0,0,-8.173618580017125,0,0,-17.32265146616112,-6,-6,-6 +0.7999999999999999,0,0,-9.905883726633236,0,0,-17.51423950887784,-6,-6,-6 +0.8999999999999999,0,0,-11.65730767752102,0,0,-17.62838770637301,-6,-6,-6 +0.9999999999999999,0,0,-13.42014644815832,0,0,-17.69639722725246,-6,-6,-6 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\newcommand{\equ}{Persamaan} % % -%%% Coloring the comment as blue in algorthm +%%% Merubah comment menjadi biru pada perintah algorthm \newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} \SetCommentSty{mycommfont} \SetKwInput{KwInput}{Masukan} % Set the Input

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