Beda Teorema dan Lemma??

Teorema adalah pernyataan matematis yang telah dibuktikan secara logis sesuai dengan kaidah matematika dengan menggunakan asumsi - asumsi matematis yang telah diketahui

Lemma adalah suatu pernyataan matematis di dalam pembuktian suatu teorema. Dengan kata lain, lemma adalah "teorema kecil" yang digunakan untuk membuktikan suatu teorema. Lemma adalah alat bantu untuk membuktikan suatu teorema

Pseudo Inverse Matrix

Matriks yang tidak persegi membutuhkan suatu metode khusus untuk menginverskan, yang biasa disebut pseudo inverse matriks.

Properti 1
Suppose that A is mn real matrix
If m<n, then the inverse of A^TA does not exist
If mn and if the inverse of A^TA exists
A⁺=(A^TA)^-1A^T satisfies the definition of pseudoinverse
Here, A⁺A=I holds. I is identity matrix.
A: mn, A^T: nm, A⁺: nm, A^TA: nn, I: nn
The rank of A and A⁺ is n

Properti 2
Suppose that A is mn real matrix
If m>n, then the inverse of AA^T does not exist
If mn and if the inverse of AA^T exists
A⁺=A^T(AA^T)^-1 satisfies the definition of pseudoinverse
Here, AA⁺=I holds. I is identity matrix.
A: mn, A^T: nm, A⁺: nm, AA^T: mm, I: mm
The rank of A and A⁺ is m

Pseudoinverse dengan Teknik Singular Decomposition Value (SVD)
Suppose A is mn matrix. If m<n, attach the row of 0 and make the size m=n, a priori. Here, mn.
A=UWV^T is supposed to be the result of SVD
Assume that the left-upper part of W has larger number, and the right-lower part of W has smaller number
If the component of W is less than a threshold, set it to be 0, and define such matrix as W'
When W'=diag(w₁,w₂,...,w_k,0,0,...,0), we define
W''=diag(1/w₁,1/w₂,...,1/w_k,0,0,...,0)
Define A⁺=VW''U^T
A⁺A is nn matrix. Left-upper kk is identity matrix. Otherwise 0.
AA⁺ is mm matrix
If the rank of A is n, A⁺ satisfies the above property 1
Even if the rank of A is less than n, A⁺ satisfies the definition of pseudoinverse
Suppose that we want to solve Ax=b. Calculate x=VW''U^Tb
If the rank of A is n, then x is the value where the error is minimum
If the rank of A is less than n, then x is the solution where the norm ||x|| is minimum

Definition of Moore - Penrose generalized matrix inverse
Given mn real matrix A, nm matrix pseudoinverse A⁺ is defined as follows
AA⁺A=A
A⁺AA⁺=A⁺
(AA⁺)^T=AA⁺
(A⁺A)^T=A⁺A


Reference : Diktat Mata Kuliah Teknik Numerik Sistem Linier, ITS Surabaya

Model Predictive Control (MPC) - Preface

Model Predictive Control (MPC), variously known as rolling-horizon control, receding-horizon control, etc, is a powerful technique employed in diverse engineering applications, and valued for its inherent ability to handle constraints while minimizing some cost (or maximizing some reward). The underlying idea of MPC is to approximate an infinite horizon constrained optimization problem by a finite horizon one, and then an optimal control law is calculated at every time step and applied in a rolling-horizon fashion. In most applications uncertainty in the model is involved, either due to some external disturbances or due to imprecise modeling of the controlled process. Uncertainty is commonly addressed in literature by formulating a robust MPC problem, assuming that the uncertainty is bounded and adopting a worst-case approach. Although there has been some major advances in this field, this approach is often too pessimistic.

But what happens if the uncertainty is not bounded? Or when it is not uniformly distributed? A less conservative approach would clearly be to take these possibilities into account, identifying appropriate distributions of the uncertainties and formulating a stochastic MPC problem instead. In this context, there are several points one should take into consideration for the problem formulation:  

• Probabilistic Constraints:
In the context of unbounded stochastic disturbances, satisfying all constraints with probability one is impossible, as at any time the system might encounter a very big disturbance that forces it to violate its constraints. It is important thus, to substitute the hard constraints with soft, probabilistic ones, ensuring that they are respected with a desired probability

• Expected Value Constraints:
  Another alternative for dealing with constraints is to make sure that they are respected on an average for the optimization problem considered. Depending on the nature of the problem such expected value constraints may be more meaningful than the probabilistic ones.
  
  
• Cost (Reward):  
  The simplest cost (reward) is perhaps an expected cost (reward), formulated as the expected value of the sum of discounted cost-per-stage functions. Depending on the application, alternative and more difficult formulations may be to consider long-run expected average cost (reward), or long-run pathwise cost (reward).

Inverter 3 Fasa

Inverter merupakan suatu peralatan yang dapat digunakan untuk mengkonversikan sumber daya 3 phasa menjadi tegangan DC yang kemudian dikonversikan lagi menjadi sumber daya 3 phasa dengan frekuensi yang sesuai. Cara ini bisa dipakai karena diketahui bahwa kecepatan sinkron motor induksi berbanding lurus dengan frekuensi sumber dayanya.

Salah satu cara yang efektif untuk menghasilkan tegangan dengan frekuensi yang bisa diatur yaitu dengan jalan membangkitkannya sendiri. Untuk itu diperlukan suatu sumber daya DC. Sumber daya ini diperoleh dari sumber daya PLN yang disearahkan dengan penyearah. Selanjutnya sumber daya ini ditapis dengan filter DC untuk mendapatkan sumber daya DC yang lebih rata. Kemudian dengan melalui suatu rangkaian switch (disebut sebagai jembatan inverter) yang bisa dikendalikan sedemikian rupa, sumber daya itu bisa diubah menjadi sumber daya 3 phasa pada ujung beban. Dengan cara mengontrol waktu pensaklaran dari switch-switch tersebut dengan menggunakan sinyal PWM (Pulse Width Modulation). Prinsip kerja inverter ditunjukkan oleh Gambar 1.

Gambar 1. Prinsip Kerja Inverter 

Pada inverter 3 phasa, dikenal yang namanya enam langkah (six-steps) pembangkitan tegangan 3-phasa. Disebut demikian karena akan dijumpai enam langkah (tingkatan) pada tiap tegangan phasenya yang didapat dari enam macam kombinasi on-off dari keenam saklar inverter yang dilakukan secara berulang pada tiap periodenya.

 

 

Gambar 2. Enam Langkah Pembangkitan Tegangan Tiga Fasa pada Inverter

 Gambar 3. Inverter Tiga Fasa