OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Stability Analysis/KYP Lemma With Feedthrough
## Definition

Consider a square, continuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\longrightarrow {\mathcal {L}}_{2e}$, with minimal state-space realization ($A$, $B$, $C$, $D$), where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{m\times n}$, and $D\in \mathbb {R} ^{m\times m}$.

## LMI 1.1

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalent necessary and sufficient conditions.
* There exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$.
* There exists $Q\in \mathbb {S} ^{n}$, where $Q>0$, such that ${\begin{bmatrix}AQ+QA^{T}&B-QC^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$
This is a special case of the KYP Lemma for QSR dissipative systems with $Q=0$, $S={\frac {1}{2}}\cdot 1$, and $R=0$.

## LMIs 1.2

The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalent necessary and sufficient conditions.
* There exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$.
* There exists $Q\in \mathbb {S} ^{n}$, where $Q>0$, such that ${\begin{bmatrix}AQ+QA^{T}&B-QC^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$
This is a special case of the KYP Lemma for QSR dissipative systems with $Q=\epsilon 1$, $S={\frac {1}{2}}\cdot 1$, and $R=0$, where $\epsilon \in \mathbb {R} _{>0}$.

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
# LMIs in Control/Stability Analysis/LMI for Schur Stabilization

LMIs in Control/Stability Analysis/LMI for Schur Stabilization

## The System

We consider the following system:
${\begin{aligned}x(k+1)=Ax(k)+Bu(k)\end{aligned}}$
where ${\begin{aligned}x\in \mathbb {R} ^{n}{\text{and}}u\in \mathbb {R} ^{r}\end{aligned}}$, are the state vector and the input vector, respectively. Moreover, the state feedback control law is defined as follows:
${\begin{aligned}u(k)=Kx(k)\end{aligned}}$
Thus, the closed-loop system is given by:
${\begin{aligned}x(k+1)=(A+BK)x(k)\end{aligned}}$

## The Data

    ${\begin{aligned}{\text{Given matrices}}\quad A\in \mathbb {R} ^{n\times n}{\text{,}}\quad B\in \mathbb {R} ^{n\times r}\quad {\text{, and the scalar}}\quad 0<\gamma \leq 1.\end{aligned}}$

## The Optimization Problem

Find a matrix ${\begin{aligned}K\in \mathbb {R} ^{r\times n}\end{aligned}}$ such that,
${\begin{aligned}||A+BK||_{2}<\gamma \end{aligned}}$
According to the definition of the spectral norms of matrices, this condition becomes equivalent to:
${\begin{aligned}(A+BK)^{T}(A+BK)<\gamma ^{2}I\end{aligned}}$
One can use the Lemma 1.2 in [1] page 14, the aforementioned inequality can be converted into:
${\begin{aligned}{\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

## The LMI: LMI for Schur stabilization

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{min}}\;\quad \gamma :&\\{\text{s.t.}}\quad {\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

This problem is a special case of Intensive Disk Region Design (page 230 in [1]). This problem may not have a solution even when the system is stabilizable. In other words, once there exists a solution, the solution is robust in the sense that when there are parameter perturbations, the closed-loop system's eigenvalues are not easy to go outside of a circle region within the unit circle [1].

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability

## Related LMIs

LMI for Hurwitz stability

## External Links

A list of references documenting and validating the LMI.
* - LMI in Control Systems Analysis, Design and Applications
# LMIs in Control/Stability Analysis/Hurwitz Stability

LMIs in Control/Stability Analysis/Hurwitz Stability
This is a set of LMI conditions for determining Hurwitz Stability of continuous time systems.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

## The Data

* The matrices $A,B,C,D$ are system matrices of appropriate dimensions.
* $x\in \mathbb {R} ^{n}$, $y\in \mathbb {R} ^{m}$ and $u\in \mathbb {R} ^{r}$ are state vector, output vector and input vector respectively.

## The Optimization Problem

Find a symmetric postive definite matrix $X$, where $X\in \mathbb {R} ^{n}$. Thus $X>0$ and $K=ZX^{-1}$ where $Z\in \mathbb {R} ^{r}$.

## The LMI: The Lyapunov Inequality

Matrix pair $(A,B)$, is Hurwitz stabilizable if and only if there exist a symmetric positive definite matrix $X$ and a matrix $Z$ satisfying $AX+XA^{T}+BZ+Z^{T}B^{T}<0$
Proof : Matrix pair $(A,B)$, is Hurwitz stabilizable if and only if
$rank[sI-AB]=n$, $\forall s\in \lambda (A)$ and $Re(s)\geq 0$
This is the definition of Hurwitz Stability. Now, using this definition we can prove the above LMI if we find matrix $X>0$ and matrix $Z$ and thus by substituting $Z=KX$ in the above LMI we get,
$(A+BK)X+X(A+BK)^{T}<0$, which brings us to the Lyapunov Stability Theory.

## Conclusion:

Thus by proving the above conditions we prove that the matrix pair $(A,B)$ is Hurwitz Stabilizable. At the same time, we also prove that the $rank[sI-AB]=n$ i.e. it is full rank and the real part of $s$ is $>0$.

## Implementation

Please find the MATLAB implementation at this link below https://github.com/omiksave/LMI

## Related LMIs

Links to other closely-related LMIs
* Schur Stability
* Quadratic Hurwitz Stability
* Quadratic Schur Stability
* Quadratic D-Stability

## External Links

A list of references documenting and validating the LMI.
* 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.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Return to Main Page:

* https://en.wikibooks.org/wiki/LMIs_in_Control
# LMIs in Control/Stability Analysis/LMI for the Continuous-Time D-Stability Controller

LMIs in Control/pages/Continuous Time D-Stability Controller
# LMIs in Control/Stability Analysis/Discrete Time/Transient Bound for Discrete-Time Non-Autonomous LTI Systems
## The System

For a discrete-time LTI system with a state-space representation of:
$x_{k+1}=A_{d}x_{k}+B_{d}u_{k},$
where $A_{d}\in \mathbb {R} ^{nxn}$, and $B_{d}\in \mathbb {R} ^{nxm}$,
the transient bound can be analyzed with the LMI below.

## The Data

$A_{d}\in \mathbb {R} ^{nxn}$, and $B_{d}\in \mathbb {R} ^{nxm}$

## The LMI:

The Euclidean norm of the state satisfies:
$\lVert x_{k}\rVert _{2}^{2}\leq \gamma ^{2}(\lVert x_{0}\rVert _{2}^{2}+\lVert u\rVert _{2k}^{2}),\forall k\in \mathbb {Z} _{\geq 0}$
if there exists some $P\in \mathbb {S} ^{n}$ and $\gamma \in \mathbb {R} _{>0}$, where P > 0 such that:
* $P-\gamma 1\leq 0,$
* ${\begin{bmatrix}P&1\\*&\gamma 1\end{bmatrix}}\geq 0,$
* ${\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}\\*&B_{d}^{T}B_{d}-\gamma 1\end{bmatrix}}\leq 0.$
if x$_0$ = 0 and u is a unit-energy input ($\lVert u\rVert _{2k}\leq 1,\forall k\in \mathbb {Z} _{\geq 0}$), then the above LMIs ensure that $\lVert x_{k}\rVert _{2}\leq \gamma ,\forall k\in \mathbb {Z} _{\geq 0}$

## Conclusion

By using this LMI the transient state bound can be analyzed for a given non-autonomous LTI system.

## Implementation

The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.

## Related LMIs

* Output Energy Bound for Discrete-Time Autonomous LTI Systems

## External Links

* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
# LMIs in Control/Stability Analysis/Kharitonov-Bernstein-Haddad
## Kharitonov-Bernstein-Haddad

Consider the set of matrices A = { ${\begin{aligned}A={\begin{bmatrix}0_{(n-1)\times 1}&1_{(n-1)\times (n-1)}\\-a_{0}&-a_{n-1}\\\end{bmatrix}}\qquad |\qquad {\underline {a}}_{j}\leq {\underline {a}}_{j}\leq {\overline {a}}_{j},\qquad j=1,2,3,4,...,n-1\end{aligned}}$ }
Every matrix in the set A is Hurwitz if and only if there exist $P_{i}\in {\text{S}}^{n}$, i = 1, 2, 3, 4, where P$_i$ > 0, i = 1, 2, 3, 4, such that
    ${\begin{aligned}\qquad P_{i}A_{i}+A_{i}^{T}P_{i}<0\qquad i=1,2,3,4,\end{aligned}}$
where
${\begin{aligned}\qquad A_{i}={\begin{bmatrix}[0_{(n-1)\times 1}&1_{(n-1)\times (n-1)}]\\\qquad a_{i}\\\end{bmatrix}}\end{aligned}},\qquad i=1,2,3,4,$
${\begin{aligned}\qquad a_{1}=-{\begin{bmatrix}{\underline {a}}_{0}&{\underline {a}}_{1}&{\overline {a}}_{2}&{\overline {a}}_{2}&...&{\underline {a}}_{n-4}&{\underline {a}}_{n-3}&{\overline {a}}_{n-2}&{\overline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$
${\begin{aligned}\qquad a_{2}=-{\begin{bmatrix}{\underline {a}}_{0}&{\overline {a}}_{1}&{\overline {a}}_{2}&{\underline {a}}_{2}&...&{\underline {a}}_{n-4}&{\overline {a}}_{n-3}&{\overline {a}}_{n-2}&{\underline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$
${\begin{aligned}\qquad a_{3}=-{\begin{bmatrix}{\overline {a}}_{0}&{\underline {a}}_{1}&{\underline {a}}_{2}&{\overline {a}}_{2}&...&{\overline {a}}_{n-4}&{\underline {a}}_{n-3}&{\underline {a}}_{n-2}&{\overline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$
${\begin{aligned}\qquad a_{4}=-{\begin{bmatrix}{\overline {a}}_{0}&{\overline {a}}_{1}&{\underline {a}}_{2}&{\underline {a}}_{2}&...&{\overline {a}}_{n-4}&{\overline {a}}_{n-3}&{\underline {a}}_{n-2}&{\underline {a}}_{n-1}\end{bmatrix}},\end{aligned}}$
Equivalently, every matrix in the set A is Hurwitz if and only if there exist $Q_{i}\in {\text{S}}^{n}$, i = 1, 2, 3, 4, where Q$_i$ > 0, i = 1, 2, 3, 4, such that
    ${\begin{aligned}\qquad A_{i}Q_{i}+Q_{i}A_{i}^{T}<0\qquad i=1,2,3,4,\end{aligned}}$

## 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.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Stability Analysis/LMI for the Continuous-Time D-Stability Observer

LMIs in Control/pages/Continuous Time D-Stability Observer
# LMIs in Control/Stability Analysis/LMI for the Controllability Grammian

LMIs in Control/Stability Analysis/LMI for the Controllability Grammian
This is a Template for Creating An LMI Page

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}$

## The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z}$.

## The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;&P>0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}}),T=diag(\tau _{1}):\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## External Links

A list of references documenting and validating the LMI.
* 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.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# Medal of Honor: Frontline/Q&A

The following are some of the questions I've been asked.
Q: I can't climb ladders!
    A: To climb a ladder (using MOH Classic) face upwards and press R3 up wards.
Q: Why am I not able to reload the M1 Garand?
    A: The M1 Garand was not designed in real life to be reloaded in the middle of a magazine, and EA wanted to keep the game real, so they made it just like that, it is advisable to just shoot out the remaining bullets if you need to reload.
Q: "Hi, I am trying to beat under the radar [the level], when I have to destroy the two radar towers and what not [???], my only problem is, I cannot find the way into the second tower." [he continues on to say that the is at the bunkers and can't progress.]
    A: Ok, this is a fairly common mistake, at the bunkers there is a wall dividing the two areas, you have obviously not noticed this because the two bunkers are noticeable from the first area. To get to the second, head back to the top of area two after you get the health pack and move on to the side of the second bunker to find a door that brings you inside.
Q: I'm constantly running out of ammo for all my good guns, and then am left with a stupid pistol to kill the next 20 Germans, how can I retain more ammo?
    A: This is a common mistake, my best advice is, when up against small groups of Germans, use your pistol, only use the better guns in conflict zones, so keep alternating your weapon, otherwise you will easily never have enough ammo, and of course never pass a dead body without taking his ammo.
# Medal of Honor: Frontline/Glitches

The following are some of the games flaws, mistakes and glitches. Some are minor - others quite humorous!

### Missing Covers

On the D-Day mission the soldiers don't have covers for there guns, whereas in the real event they were equipped with them, these covers were needed to make sure the guns didn't jam in the water.

### BAR problem

The BAR gun gives a bit of a kick in the game, however in real life this does not occur at all.

### Camera Angle

Now this has never happened to me before, but several people have informed me of it. It is when you press an unspecified combination of buttons including look up, and your view is stuck on either up or down. I do not know of an unlocking combination of buttons, so I can merely advise you to restart that particular level.

### Walk through Walls

In the level Arnhem Knights if certain events happen at the right time and you press certain buttons during this, walls may become only visual and you may walk through them, this can happen anywhere between after you talk to the English Commander and when you go around the first heap of rubble. It has happened to me twice and both times I was able to restore proper gameplay by collecting ammo or heading past the first bundle of rubble. It is purely a small glitches that went un-noticed in testing, it can actually be fun to have this glitch activated. The main problem is that you cannot go up a set of steps while like this, this is because you are fixed at ground level and during the glitch the material value of a set of steps is taken away leaving only the visual effect.

### Fall through floor (Removed)

This glitch was rumored to have occurred in the second mission of D-Day. EA realized that if invincibility was left on after the gun bunker had been bombed, the player would fall through the floor. For this reason, invincibility doesn't work inside the bunker when the plane takes its bombing run.
GERMAN BAR in the beginning of the D-day level, one of the Germans has a BAR.
# Medal of Honor: Frontline/Secrets
### Bread

You can gain 5% health by eating bread, you can find it on tables in several levels, like Operation Repunzel, or Yard by Yard, to indicate you have eaten bread, a message will come up and say, "The Bread is Delicious!"

### Poor Chef!

During the course of the game you will encounter the Chef several times, in the first encounter he appears perfectly healthy, but gradually he starts looking worse, with tears in his cloths and he will become bruised!

### Panzerknacker Tank Eater

You can see this hidden secret on the Golden Lion level (Needle in a Haystack.) Somewhere in the pub at the end, you can see it through one of the doors.

### Wierd

In Nijmegen Bridge (Several Bridges too Far) if you turn around and shoot the other American soldier with a sniper rifle in the face, it will show the damage without killing him, aim at his face and he will continue smiling even if you blow off his lips!
# Medal of Honor: Frontline/Medals and Stars

There are 9 different medals to collect. To earn a medal, you must get gold stars in all of a campaign's individual levels. The medals are as follows going by each campaign.
* D-Day (prologue): Army Distinguished Service Medal
* Campaign 1: Army Commendation Medal
* Campaign 2: Soldier's Medal
* Campaign 3: Dutch WWII Commemorative Cross
* Campaign 4: Legion of Merit
* Campaign 5: Distinguished Flying Service Cross, Silver Star, EA Los Angeles Medal of Valor, Bronze Star, Distinguished Service Cross.
The star requirements for each level are as follows:
* Bronze Star Requirements - Complete the level
* Silver Star Requirements - Complete the level & 95% of enemies neutralized.
* Gold Star Requirements - Complete the level, 95% of enemies neutralized & 75% health left.
For a rough guide to getting gold stars, see the following section: Getting a Gold Star - Level by Level.
# Medal of Honor: Frontline/Rest in Series

The Medal of Honor Series at the time of the initial writing of this book contained the following titles:
* Medal of Honor (PlayStation)
* Medal of Honor: Underground (PlayStation, Gameboy Advance)
* Medal of Honor: Frontline (PlayStation 2, GameCube, Xbox)
* Medal of Honor: Rising Sun (PlayStation 2, GameCube, Xbox)
* Medal of Honor: Allied Assault (PC)
    Medal of Honor: Allied Assault - Breakthrough (Expansion *pack) (PC)
* Medal of Honor: Allied Assault - Spearhead (Expansion Pack) (PC)
* Medal of Honor: Allied Assault - War Chest (Expansion Pack) (PC)
* Medal of Honor: Pacific Assault (PC)
* Medal of Honor: Infiltrator (Gameboy Advance)
Many games in the Medal of Honor series were released after Medal of Honor: Frontline. For more information, read the Wikipedia article on Medal of Honor.
# Microprocessor Design/Control and Datapath

Most processors and other complicated hardware circuits are typically divided into two components: a datapath and a control unit. The datapath contains all the hardware necessary to perform all the necessary operations. In many cases, these hardware modules are parallel to one another, and the final result is determined by multiplexing all the partial results.
The control unit determines the operation of the datapath, by activating switches and passing control signals to the various multiplexers. In this way, the control unit can specify how the data flows through the datapath.
The width of the data path ...
For good code density, you want the ALU datapath width to be at least as wide as the address bus width. Then every time you need to increment an address, you can do it in a single instruction, rather than requiring multiple instructions to manipulate an address one piece at a time.
After a person has designed the data path, that person finds all the control signal inputs to that datapath -- all the control signals that are needed to specify how data flows through that datapath.
* Each general-purpose register needs at least one control signal to control whether it maintains the current value or loads a new value from elsewhere.
* The ALU needs some control signals to tell it whether to add, subtract, etc.
* The program counter section needs control signals to tell it whether the program counter gets reloaded with an incremented version of the previous value, or with some completely different branch value.
* etc.
Once we know what control signals we need to generate, we need to design an Microprocessor Design/Instruction Decoder to generate those signals.

## References

* ↑ Engineering Education "Today in History" PDP-11 minicomputer introduced by Gordon Bell 2009; referring to "What we learned from the PDP-11" by Gordon Bell and Bill Strecker, 1975. Early version of the PDP-11 had a 16-bit address space. ... also quoted in the book "Electronics"
* ↑ Hollingsworth, Sachs, and Smith. "The Fairchild Clipper: Instruction set architecture and processor implementation". Section 9.4: "Address Space Size". 1987.
* ↑ "It seems that the 16-bit ISA hits somehow the "sweet spot" for the best code density, perhaps because the addresses are also 16-bit wide and are handled in a single instruction. In contrast, 8-bitters need multiple instructions to handle 16-bit addresses." -- "Insects of the computer world" by Miro Samek 2009.
* ↑ "it just really sucks if the largest datum you can manipulate is smaller than your address size. This means that the accumulator needs to be the same size as the PC -- 16-bits." -- Allen "Opcode considerations"
# Microprocessor Design/Control Unit

The control unit reads the opcode and instruction bits from the machine code instruction, and creates a series of control codes to activate and operate the various components to perform the desired task.

## Simple Control Unit

In its most simple form, a control unit can take the form of a lookup table. The machine word opcode is used as the index into the table, and the various control signals are output to the respective destinations.

## Complex Control Unit

A more complex version of a control unit is implemented as a finite state machine (FSM). Multi-cycle, pipelined, and other advanced processor designs may require an FSM-based control unit.

## Further reading

* Electronics/Control Unit
* Digital Circuits/Finite State Machines
* IB/Group 4/Computer Science/Computer Organisation
# Microprocessor Design/Compiler

With an assembler written, it is typically a good idea (although not always) to write a high-level language compiler for your new processor. Typically the high-level language in these situations is C because of its small number of built in constructions, and the close relationship that C shares with the underlying assembly language.
Compilers help to speed up the development process, so that more complicated software can be written without the tedium of writing large assembly language programs. Another benefit to this is that there are a number of pre-existing tools for use with higher-level languages, such as simulators and debuggers that can increase the efficiency of your software team.

## Further reading

* Compiler Construction discusses how to write a compiler from scratch
# Microprocessor Design/Exceptions

Exceptions, are situations where the processor needs to stop executing the current code because of an error. In these cases, the processor typically begins running an exception handling routine to resolve the error, and then returns to the normal program flow. For instance, if the ALU attempts to divide by zero, or if an addition causes overflow, an exception might be triggered. The processor needs to stop operation and fix the error before the program can be resumed.
Some common examples of exceptions are arithmetic overflow or underflow, division by zero, or attempting to access a memory location that does not exist.
# Microprocessor Design/FPGA

Field-Programmable Gate Arrays (FPGA) are programmable logic elements. FPGAs can be designed using a hardware description language (HDL) such as Verilog or VHDL, and that design can be mapped to a hardware design by the HDL synthesizer. FPGAs are the successors of their previous similar components, PLAs and PALs, used at the first steps of the programmable logic era.
FPGAs are quick to design, and because they are reprogrammable, troubleshooting is quick and easy.
FPGAs are useful for designing microcontrollers, which is why we have discussed HDL implementations of various components in the text of this book. In this chapter we will will discuss the implementation of a microcontroller in HDL, and some of the consequences of that implementation.
Dozens of FPGA CPU designs are available for download and tinkering. An appendix to this book, Microprocessor Design/Resources, lists details on how to get them.