OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/pages/Modified Minimum Gain Lemma

LMIs in Control/pages/Modified Minimum Gain Lemma
Modified Minimum Gain Lemma

## The System

Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.
With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $G$ corresponds to the following matrices $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.
Essentially the same concept as the Minimum Gain Lemma; however, this modification ensure that the system being optimized is also Lyapunov stable. This Essentially means that the variable $Q$ must ensure that the system is stable to this degree, otherwise the system itself is infeasible.

## The Data

A system of the plant $G$ will be required with the following matrices: $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.

## The LMI: Modified Minimum Gain Lemma

The following two LMI's are equivalent, with the same variable:
Suppose there exists $Q\in \mathbb {S} ^{n}$, and $\upsilon \in \mathbb {R} _{\geq 0}$, where $Q>0$ such that:
       ${\begin{aligned}{\begin{bmatrix}AQ+QA^{T}&B-QC^{T}D\\(B-QC^{T}D)^{T}&\upsilon ^{2}I-D^{T}D\end{bmatrix}}&\leq 0\end{aligned}}$
Or (Obtained using Schur compliment):
       ${\begin{aligned}{\begin{bmatrix}AQ+QA^{T}&B-QC^{T}D&0\\(B-QC^{T}D)^{T}&-D^{T}D&\upsilon I\\0&\upsilon I&-I\end{bmatrix}}&\leq 0\end{aligned}}$

## Conclusion:

Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $\upsilon$ obtained from the optimization. This system will also prove that the plant is Lyapunov stable.

## Implementation

* Minimum Gain Lemma Example

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* The minimum gain lemma - A paper about Minimum gain lemma by Bridgeman and Forbes
# LMIs in Control/pages/NI-Lemma

LMIs in Control/pages/NI-Lemma Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

## The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$
    ${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}$

## The Data

$A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},D\in \mathbb {S} ^{m}$

## The LMI: LMI for Negative Imaginary Lemma

According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions
* There exists a $P$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$$^n$,where $P\geq 0$, such that,
    ${\begin{bmatrix}A^{T}P+PA&PB-A^{T}C^{T}\\B^{T}P-CA&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$
* There exists a $Q$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$$^n$,where $Q\geq 0$, such that,
    ${\begin{bmatrix}QA^{T}+AQ&B-QA^{T}C^{T}\\B^{T}-CAQ&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$

## Conclusion

The system is strictly negative if det($A$) $\neq$ 0 and either of the above LMIs are feasible with resulting $P>0$ or $Q>0$

## Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/ygovada

## Related LMIs

Positive Real Lemma

## External Links

* - LMI in Control Systems Analysis, Design and Applications
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* Xiong, Junlin & Petersen, Ian & Lanzon, Alexander. (2010). A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems.
# LMIs in Control/pages/Mu Analysis

Mu Synthesis. The technique of $\mu$ synthesis extends the methods of $H\infty$ synthesis to design a robust controller for an uncertain plant. You can perform $\mu$ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. $\mu$ analysis is an extremely powerful multivariable technique which has been applied to many problems in the almost every industry including Aerospace, process industry etc.

## The System:

Consider the continuous-time generalized LTI plant with minimal states-space realization
    ${\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\\\end{aligned}}$
where it is assumed that $D$ is Invertible.

## The Data

The matrices needed as inputs are only, $A$ and $D$.

## The LMI: $\mu$- Analysis

The inequality ${\overline {\sigma }}(DAD^{-1})<\gamma$ holds if and only if there exist $X\in \mathbb {S} ^{n}$ and $\gamma \in \mathbb {R} _{>0}$, where $X>0$, satisfying:
       ${\begin{aligned}A^{T}XA-\gamma ^{2}X<0\end{aligned}}$

## Conclusion:

The inequality ${\overline {\sigma }}(DAD^{-1})<\gamma$ holds for $D=X^{1/2},$ where X satisfies the above Inequality.

## Implementation

* https://github.com/Ricky-10/coding107/blob/master/Mu%20Analysis - $\mu$-Analysis

## External links

* -MATLAB $\mu$-synthesis Implementation
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
# LMIs in Control/pages/Mixed H2 Hinf optimal state feedback control

LMIs in Control/pages/Mixed H2 Hinf optimal state feedback control
Mixed $H_{\infty }/H_{2}$ Optimal State Feedback Control

## The Optimization Problem

This Optimization problem involves the same process used on the full-feedback control design; however, instead of optimizing the full output-feedback design of the Optimal output-feedback control design. This is done by defining the 9-matrix plant as such: $A\in \mathbb {R} ^{m\times m}$, $B_{1}\in \mathbb {R} ^{m\times n}$, $B_{2}\in \mathbb {R} ^{m\times p}$, $C_{1}\in \mathbb {R} ^{n\times m}$, $D_{11}\in \mathbb {R} ^{n\times n}$, $D_{12}\in \mathbb {R} ^{n\times p}$, $C_{2}=I$, $D_{21}=0$, and $D_{22}=0$. Using this type of optimization allows for stacking of optimization LMIs in order to achieve the controller synthesis for both $H_{\infty }$ and $H_{2}$.

## The Data

The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $A\in \mathbb {R} ^{m\times m}$, $B_{1}\in \mathbb {R} ^{m\times n}$, $B_{2}\in \mathbb {R} ^{m\times p}$, $C_{1}\in \mathbb {R} ^{n\times m}$, $D_{11}\in \mathbb {R} ^{n\times n}$, $D_{12}\in \mathbb {R} ^{n\times p}$, $C_{2}=I$, $D_{21}=0$, and $D_{22}=0$.

## The LMI: Mixed $H_{\infty }/H_{2}$ Optimal State Feedback Control

There exists the scalars $\gamma _{1}$, $\gamma _{2}$, along with the matrices $X>0$, $W$ and $Z$ where:
       ${\begin{aligned}||S(P,K(0,0,0,F))||_{H_{\infty }}^{2}+||S(P,K(0,0,0,F))||_{H_{2}}^{2}\leq \gamma _{1}+\gamma _{2}\\\\{\begin{bmatrix}XA^{T}+Z^{T}B_{2}^{T}+AX+B_{2}Z&B_{1}&XC_{1}^{T}+Z^{T}D_{12}^{T}\\B1_{1}^{T}&-I&D_{11}^{T}\\C_{1}X+D_{12}Z&D_{11}&-\gamma _{1}I\end{bmatrix}}&<0\\\\\\traceW<\gamma _{2}\\\\AX+XA^{T}+B_{2}Z+Z'B_{2}'+B_{1}B_{1}^{T}<0\\\\{\begin{bmatrix}X&(C_{1}X+D_{12}Z)^{T}\\C_{1}X+D_{12}Z&W\\\end{bmatrix}}&>0\\\end{aligned}}$
Where $F=ZX^{-1}$ is the controller matrix.

## Conclusion:

The results from this LMI give a controller that is a mixed optimization of both an $H_{\infty }$, and $H_{2}$ optimization.

## Implementation

* Hinf/H2 Mixed state-feedback optimization example

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# LMIs in Control/pages/Modified Exterior Conic Sector Lemma
## The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

## The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

## The Data

The matrices The matrices $A,B,C$ and $D$

## LMI : Modified Exterior Conic Sector Lemma

The system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c (i.e. ${\mathcal {G}}\in$excone$_r$(c)), where $r\in {\mathcal {R}}_{>0}$ and $\in {\mathcal {R}}$, under either of the following sufficient conditions.
    1. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that
       ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq 0.$
    Proof. The term $-C^{T}C$ in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.
Therefore,
       ${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq {\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}$
    2. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that
       ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}(D-CI)&0\\(PB-C^{T}(D-CI))^{T}&-(D-cI)^{T}(D-cI)&rI\\0&(rI)^{T}&-I\end{bmatrix}}\leq 0.$
    Proof. Applying the Schur complement lemma to the $r^{2}I$ terms in (1) gives (2).

## Conclusion:

If there exist a positive definite $P$ matrix satisfying above LMIs then the system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c.

## Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

## Related LMIs

KYP Lemma
State Space Stability Exterior Conic Sector Lemma

## References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.
# Pinyin/Say

See also
# Pinyin/Schistosomiasis

Schistosomiasis, or bilharzia, is a parasitic disease caused by trematode flatworms of the genus Schistosoma. Larval forms of the parasites, which are released by freshwater snails, penetrate the skin of people in the water.
In the body, the larvae develop into adult schistosomes, which live in the blood vessels. The females release eggs, some of which are passed out of the body in the urine or faeces. Others are trapped in body tissues, causing an immune reaction.
In urinary schistosomiasis, there is progressive damage to the bladder, ureters and kidneys. In intestinal schistosomiasis, there is progressive enlargement of the liver and spleen, intestinal damage, and hypertension of the abdominal blood vessels.
Control of schistosomiasis is based on drug treatment, snail control, improved sanitation and health education.
Xuèxīchóngbìng (yòuchēng liètǐxīchóngbìng) shì yóu liètǐxīchóngshǔ xuèxīchóng yǐnqǐ de yīzhǒng jìshēngchóngbìng. Zài shuǐzhōng, dànshuǐluó zhōng shìfàng chū de wěiyòu qīnrù réntǐ pífū, zàochéng gǎnrǎn.
Zài réntǐ nèi, wěiyòu fāzhǎn wéi chéngchóng. Chéngchóng jìshēng zài xuèguǎn zhōng, cíchóng zài xuèguǎn zhōng chǎnluǎn. Yīxiē chóngluǎn suí niàoyè huò fènbiàn páichū tǐwài. Qítā chóngluǎn zé kùn zài réntǐ zǔzhī nèi, yǐnqǐ miǎnyì fǎnyìng.
Niàodào xuèxīchóngbìng huànzhě zhújiàn chūxiàn pángguāng, shūniàoguǎn hé shènzàng shòusǔn zhèngzhuàng. Chángxuèxīchóngbìng huànzhě zé huì zhújiàn chūxiàn gānzàng hé pízàng zhǒngdà, chángwèi shòusǔn yǐjí fùqiāng xuèguǎn gāoyā děng zhèngzhuàng.
Kòngzhì xuèxīchóngbìng de jīchǔ shì, kāizhǎn yàowù zhìliáo, miè luó, gǎishàn huánjìng wèishēng, bìngqiě jìnxíng wèishēng jiàoyù.
# Pinyin/School

Search Pinyin school
# Pinyin/Saudi Arabia

Search Pinyin Saudi
Saudi Arabia is a country in Southwest Asia, on the Arab Peninsula.
Saudi Arabia shih Yahzhou xinarnbuh Arab Bahndaao d guorjia.
# Pinyin/Sauerkraut

See also
Sauerkraut (directly translated from German) is shredded cabbage that has been fermented by various lactic acid bacteria. It has a long shelf-life and a distinctive sour flavor, both of which result from the lactic acid that forms when the bacteria ferment the sugars in the cabbage. It is therefore not to be confused with coleslaw, which receives its acidic taste from vinegar.
sauerkraut (cóng Déwén zhíjiē fānyì) shì jīngguò gèzhǒng rǔsuānjūn fāxiào d yēcài-sī. Yóuyú xìjūn fāxiào yēcài de tángfèn, chǎnshēng rǔsuān, shǐ sauerkraut jùyǒu cháng de bǎozhìqī hé dútè de suānwèi. Yīncǐ, sauerkraut yǔ suānwèi láizì cù de coleslaw bùtóng.

## Health Effects
### Benefits

It is a source of vitamins C, B, and K; the fermentation process makes sauerkraut more nutritious than the original cabbage. It is also low in calories and high in calcium and magnesium, and it is a very good source of dietary fiber, folate, iron, potassium, copper and manganese.
Tā shì wéishēngsù C, B hé K de láiyuán; fāxiào guòchéng shǐ sauerkraut bǐ yuánlái de yēcài gèng yǒu yíngyǎng. Tā rèliàng dī, Gài hé Měi hánliàng gāo, bìngqiě fùhán shànshí-xiānwéi, yèsuān, Tiě, Jiǎ, Tóng hé Měng.

### Disadvantages

Excessive consumption of sauerkraut may lead to bloating and flatulence due to the trisaccharide raffinose, which the human small intestine cannot break down.
Rén de xiǎocháng bùnéng fēnjiě sāntáng-miánzǐtáng, guòliàng shíyòng sauerkraut huì dǎozhì fùzhàng hé wèichángzhàngqì.
# Pinyin/School Bus

Search Pinyin school
Search Pinyin bus
    See also: Pinyin/Bus
# Linux Certified LC2520S
## CPU

* CPU:

## Memory

* Memory:
* Memory Type:

## Video Graphics Display

* Video Graphics Chipset: Nvidia
* Screen Size:
* Maximum Screen Resolution:

## Audio

* Audio Chipset:

## Pointing Device

* Pointing Device Type: Touchpad
# Thinkpad R61
## CPU

* CPU:

## Memory

* Memory:
* Memory Type:

## Video Graphics Display

* Video Graphics Chipset: Nvidia Quadro NVS 140M (???MB)
* Screen Size:
* Maximum Screen Resolution:

## Audio

* Audio Chipset:

## Pointing Device

* Pointing Device Type: Touchpad
# Thinkpad T61p 14W
## CPU

* CPU:

## Memory

* Memory:
* Memory Type:

## Video Graphics Display

* Video Graphics Chipset: Nvidia Quadro FX 570M (???MB)
* Screen Size:
* Maximum Screen Resolution:

## Audio

* Audio Chipset:

## Pointing Device

* Pointing Device Type: Touchpad
# Laptop Computer Models/Linux Certified

    LC2430S
    LC2520S
# Thinkpad T61
## CPU

* CPU: Core 2 Duo

## Memory

* Memory:
* Memory Type:

## Video Graphics Display

* Video Graphics Chipset: Nvidia Quadro NVS 140M (???MB)
* Screen Size: 14.1"
* Maximum Screen Resolution: 1440x900

## Audio

* Audio Chipset:

## Pointing Device

* Pointing Device Type: Touchpad
# Linux Certified LC2430S
## CPU

* CPU:

## Memory

* Memory:
* Memory Type:

## Video Graphics Display

* Video Graphics Chipset: Nvidia
* Screen Size:
* Maximum Screen Resolution:

## Audio

* Audio Chipset:

## Pointing Device

* Pointing Device Type: Touchpad
# Thinkpad T61p 15W
## CPU

* CPU:

## Memory

* Memory:
* Memory Type:

## Video Graphics Display

* Video Graphics Chipset: Nvidia Quadro FX 570M (???MB)
* Screen Size:
* Maximum Screen Resolution:

## Audio

* Audio Chipset:

## Pointing Device

* Pointing Device Type: Touchpad
# Practical Electronics/Decreasing Sin Wave Oscillator

With an RLC series
    $L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt+IR=0$
    ${\frac {d^{2}I}{dt}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$
    $s=-(\alpha \pm {\sqrt {\alpha ^{2}-\beta ^{2}}})t=-(\alpha \pm \lambda )t$
    $I=e^{-}\alpha t(e^{\lambda }t+e^{-}\lambda t)$
    $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}$
    $I=e^{-}\alpha t$
    $I=e^{-}\alpha t(e^{\lambda }t+e^{-}\lambda t)$
    $I=ACos\lambda t$
    $A={\frac {e^{-}\alpha t}{2}}$
    $I=A(e^{j}\lambda t+e^{-}j\lambda t)$
    $I=ASin\lambda t$
    $A={\frac {e^{-}\alpha t}{2j}}$
The response of an LC series is a sinusoidal wave, or LC series can be used to produce decreasing sine wave oscillator.
# Practical Electronics/Diodes/Diode applications
## Electronics switch

    Switch ON when $v<v_{d}$
    Switch OFF when $v>v_{d}$

## Electronics rectifier

    When diode conducts DC voltage , current will flow in the same direction as the arrow
# Practical Electronics/Diodes/Diode operation characteristics
## P-N Junction
### Forward Bias

The rectifying action of semiconductor diode is due to the electrical effects that occur at the junction between the n-type semiconductor and p-type semiconductor, at a region known as p-n junction (gray areas in images shown below)
As seen in the images below: If the positive terminal of the power source is connected to the p-type semiconductor and the negative terminal of the power source is connected to the n-type semiconductor, the electron flows across the junction from n-type material to p-type material, and the holes flow in reverse. This will cause the junction area to become narrower allow easy movement of both holes and electrons.
These phenomena are known as forward bias. The p-n junction will be narrower and will allow more electrons to fill into holes and the holes will move to capture electrons.

### Reverse Bias

If the power source is installed in reverse, the positive charge in n-type and negative charge in p-type will oppose electron movement causing potential barrier.
The p-n junction will widen, both electrons and holes will be moving away from the junction and causing no current that can flow across the junction. These phenomena are known as reverse bias.
Hence, it acted more or less like a one-way valve for electricity flow.

## IV curve

The graph shown is a current-voltage characteristic of p-n silicone diode.
From IV curve
| :$v<v_{d}$ . $i=0$ | Diode does not conduct current |
| :$v=v_{d}$ . $i=1mA$ | Current starts to flow |
| :$v>v_{d}$ . $i=e^{\frac {v}{v_{d}}}mA$ | Diode conducts current exponentially |

## Conducting voltage of diode

Conducting voltage of diode represents diode biasing voltage to conduct current . Conducting voltage of diode has a symbol $v_{d}$ Silicon (Si) diode is 0.7 V Germanium (Ge) diode is 0.3 V

## Polarities & Biased Voltage

When Diode is connected with a voltage source in such that the polarities of the diode is in opposite with the polarities of the voltage source Diode is said to be Forward Biased and when voltage is greater than Forward biased voltage current will flow and increase exponentially with time in the direction of Pointing Arrow
When Diode is connected with a voltage source in such that the polarities of the diode is in the same with the polarities of the voltage source Diode is said to be Backward Biased Diode does not conduct current
# Practical Electronics/Diodes

Diodes are semiconductor devices that allow electricity to flow through them in only one direction. Diode is made from joining one negative-typed semiconductor with one positive-typed semiconductor
    $o--[N|P]--o$
Circuit's symbol of diode
* Diode operation characteristics
* Types of diode
* Diode applications
# Practical Electronics/Finding component information/Capacitors
## Ceramic Numbering Code

Generally, they use an alphanumeric marking system. The number part is the same as for SMT resistors, except that the value represented is in pF. They may also be written out directly, for instance, 2n2 = 2.2 nF. For capacitors the tolerance code is:
    Tolerance CodeTolerance A± 0.05 pF B± 0.1 pF C± 0.25 pF D± 0.5 pF E± 0.5% F± 1% G± 2% H± 3% J± 5% K± 10% L*± 15% M± 20% N*± 30% P*−0% +100% S*−20% +50% W*−0% +200% X*−20% +40% Z−20% +80%
letters with * are not used consistently.
how similar is this to resistors?
For example, .047K = .047 µF 10%.
The dielectric material is sometimes designated by another code: (note: the materials in the table below are not ceramic capacitors as implied, but plastic and paper)
    MarkingMaterial KTPolyester Film / Foil MKTMetallized Polyester (PETP) KCPolycarbonate Film / Foil MKCMetallized Polycarbonate KSPolystyrene Film / Foil KPSfoil/pps? MPSMetallized pps KPPolypropylene Film / Foil MKPMetallized Polypropylene MPMetallized paper
Please note that the table above is valid most of the time but not always because it is not a real standard. To be real sure you may check the data sheet from the manufacturer in mind.
The cap color can also indicate the dielectric:
| Black | Brown | Red | Orange | Yellow | Green | Blue | Violet | Red/Orange | Orange/Orange | Yellow/Orange | Green/Orange | Blue/Orange | Red/Violet |
| C0G/NP0 | N030 | N080 | N150 | N220 | N330 | N470 | N750 | N1000 | N1500 | N2200 | N3300 | N4700 | P100 |
more info at http://my.execpc.com/~endlr/markings.html

#### Axial ceramic capacitors

These often have a yellow body, and use the same color band system as resistors, except that the value indicated is in pF.

### Multilayer Ceramic capacitors

These usually use the same alphanumeric system.