# 0.3 Trạng thái của hệ thống  (Page 3/4)

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Trong đó X(t) là ma trận cột biểu diễn các biến số trạng thái gọi là các véctơ trạng thái.

R(t) là ma trận cột, biểu diễn input gọi là các véctơ input.

$\begin{array}{}{x}_{1}t\\ {x}_{2}t\\ \text{}⋮\\ {x}_{n}t\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\left[\right]\end{array}\\ Xt=\\ \end{array}$ $\begin{array}{}{r}_{1}t\\ {r}_{2}t\\ \text{}⋮\\ {r}_{p}t\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\left[\right]\end{array}\\ Rt=\\ \end{array}$ (4.19)

A là ma trận vuông n x n :

$\begin{array}{}{a}_{\text{11}}{a}_{1n}\cdots {a}_{1n}\\ {a}_{\text{21}}{a}_{\text{22}}\cdots {a}_{2n}\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ {a}_{\mathrm{n1}}{a}_{\mathrm{n2}}\cdots {a}_{\text{nn}}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\left[\right]\end{array}\\ A=\\ \end{array}$ (4.20)

B là ma trận n x p (vì có p input r )

$\begin{array}{}{b}_{\text{11}}{b}_{\text{12}}\cdots \cdots {b}_{1p}\\ {b}_{\text{21}}{b}_{\text{22}}\cdots \cdots {b}_{2p}\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ {b}_{\mathrm{n1}}{b}_{\mathrm{n2}}\cdots \cdots {b}_{\text{np}}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\left[\right]\end{array}\\ B=\\ \end{array}$ (4.21)

Tương tự như vậy, q phương trình trong (4.2) cũng có thêû được trình bày bằng một ma trận duy nhất

$Ct=g\left[Xt+Rt\right]=\text{DX}t+\text{ER}t$ (4.22)

Trong đó D là ma trận q x n và E là ma trận q x p.

Thí dụ, các phương trình trạng thái của phương trình (4.11) được viết dưới dạng ma trận:

$\begin{array}{}\text{}n\mathrm{x1}\text{n x n n x 1 n x 1}\\ \stackrel{˙}{{x}_{1}}t\\ \stackrel{˙}{{x}_{2}}t\\ \text{}⋮\\ ⋮\\ ⋮\\ \stackrel{˙}{{x}_{n}}t\\ \\ \mathrm{righ}\\ \\ \\ \\ \left[\right]=\\ \\ 01000\cdots \cdots 0\\ 00100\cdots \cdots 0\\ 00010\cdots \cdots 0\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ 00000\cdots \cdots 1\\ n-1\cdots \cdots \cdots \cdots \cdots \cdots \text{}-{a}_{1}\\ -{a}_{n}-{a}_{}\mathrm{righ}\\ \\ \\ \\ \left[\right]\\ \\ {x}_{1}t\\ {x}_{1}t\\ ⋮\\ ⋮\\ ⋮\\ {x}_{n}t\\ \\ \mathrm{righ}\\ \\ \\ \\ +\\ 0\\ 0\\ ⋮\\ ⋮\\ ⋮\\ 1\\ \\ \mathrm{righ}\\ \\ \\ \\ \\ \begin{array}{}\begin{array}{}\left[\right]\left[\right]\left[\right]\left[\right]\left[\right]\end{array}\\ \end{array}\\ \end{array}$ (4.23)

Khi so sánh phương trình (4.23) với phương trình (4.18), các ma trận A và B sẽ được đồng nhất dễ dàng. Trường hợp này, phương trình output (4.22) là một phương trình vô hướng.

$D=\left[100\cdots 0\right]$ (4.24)

Và E = 0 (ma trận không ( 4.25 )

Tương tự các ma trận A, B,C,D đối với phương trình (4.13) sẽ là

$\begin{array}{}A=\\ 01000\cdots \cdots 0\\ 00100\cdots \cdots 0\\ 00010\cdots \cdots 0\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ 00000\cdots \cdots 1\\ n-1\cdots \cdots \cdots \cdots \cdots \cdots \text{}-{a}_{1}\\ -{a}_{n}-{a}_{}\mathrm{righ}\\ \\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\left[\right]\left[\right]\left[\right]\end{array}\\ \end{array}$ (4.26)

$\begin{array}{}B=\\ {h}_{1}\\ {h}_{2}\\ ⋮\\ {h}_{n}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\left[\right]\end{array}\\ \end{array}$ (4.27)

$D=\left[100\cdots 0\right]$ (4.28 )

$E=\left[{b}_{0}\right]$ (4.29)

## Vài thí dụ.

Thí dụ 4.1:

Xem một hệ thống tuyến tính, có hàm chuyển cho bởi:

$GS=\frac{CS}{RS}=\frac{5}{{S}^{3}+{8S}^{2}+9S+2}$ (4.30)

Phương trình vi phân tương ứng diển tả hệ thống là:

$\frac{{d}^{3}c}{{\text{dt}}^{3}}+8\frac{{d}^{2}c}{{\text{dt}}^{2}}+9\frac{\text{dc}}{\text{dt}}+2c=5r$ (4.31)

Các biến số trạng thái được định nghĩa:

$\begin{array}{}\begin{array}{}{x}_{1}t=ct\\ \stackrel{˙}{{x}_{1}}t={x}_{2}t\\ \stackrel{˙}{{x}_{2}}t={x}_{3}t\\ \stackrel{˙}{{x}_{3}}t=-{2x}_{1}-{9x}_{2}-{8x}_{3}+5r\end{array}\\ \end{array}$ (4.32)

Do đó hệ thống có thể được diễn tả bằng ma trận:

$\stackrel{˙}{X}=\text{AX}+\text{BR}$ (4.33)

và C = DX + ER (4.34)

Với

$\begin{array}{}010\\ 001\\ -2-9-8\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\end{array}\\ A=\\ \end{array}$ ; $\begin{array}{}B=\\ 000\\ 000\\ 005\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\end{array}\\ \end{array}$

$\begin{array}{}R=\\ 0\\ 0\\ r\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\end{array}\\ \end{array}$ ; $\begin{array}{}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\end{array}\\ X=\\ \end{array}$ ; $\begin{array}{}\stackrel{˙}{X}=\\ \stackrel{˙}{{x}_{1}}\\ {\stackrel{˙}{x}}_{2}\\ {\stackrel{˙}{x}}_{3}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\left[\right]\end{array}\\ \end{array}$

$D=\left[100\right]$ ; E = 0

Thí dụ 4.2:

Xem một hệ thống điều khiển như H.4.2. Hàm chuyển vòng kín của hệ là:

Hình 4.2

$\frac{CS}{RS}=\frac{2}{{S}^{2}+S+2}$ (4.35)

Phương trình vi phân tương ứng

$\frac{{d}^{2}c}{{\text{dt}}^{2}}+\frac{\text{dc}}{\text{dt}}+2c=2r$ (4.36)

Các biến trạng thái:

${x}_{1}=c$

$\stackrel{˙}{{x}_{1}}={x}_{2}$ (4.37)

$\begin{array}{}\stackrel{˙}{{x}_{2}}=-2{x}_{1}-{x}_{2}+2r\\ \end{array}$

Vậy hệ thống có thể diển tả bằng hệ thống véctơ:

$\stackrel{˙}{X}=\text{AX}+Br$ (4.38)

C = DX+Er

Trong đó :

$\begin{array}{}01\\ -2-1\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\end{array}\\ A=\\ \end{array}$ ; $\begin{array}{}0\\ 2\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\end{array}\\ B=\\ \end{array}$ ; $\begin{array}{}{x}_{1}\\ {x}_{2}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\end{array}\\ X=\\ \end{array}$ ; $\begin{array}{}\stackrel{˙}{{x}_{1}}\\ {\stackrel{˙}{x}}_{2}\\ \\ \mathrm{righ}\\ \\ \\ \\ \begin{array}{}\left[\right]\end{array}\\ \stackrel{˙}{X}=\\ \end{array}$ ;

$D=\left[10\right]$

Thí dụ 4.3 :

Xem một mạch RLC như H. 4.3

Trạng thái của hệ có thể mô tả bởi tập hợp các biến trạng thái

x1 = vc(t) ( 4.39)

x2 = iL(t) ( 4.40)

Đối với mạch RLC thụ động, số các biến số trạng thái cần thiết thì bằng với số các bộ phận tích trữ năng lượng độc lập. Các định luật Kirchhoff cho:

${i}_{c}=c\frac{{\text{dv}}_{c}}{\text{dt}}=r\left(t\right)-{i}_{L}$ (4.41)

$L\frac{{\text{di}}_{L}}{\text{dt}}=-{\text{Ri}}_{L}+{v}_{C}$ (4.42)

Output của hệ : v0 = RiL (4.43)

 Viết lại(4.41) và (4.42) như là tập hợp các phương trình vi phân cấp 1:

$\stackrel{}{{x}_{1}}=\frac{{\text{dv}}_{c}}{\text{dt}}=-\frac{1}{C}{x}_{2}+\frac{1}{C}r\left(t\right)$ (4.44)

$\stackrel{}{{x}_{2}}=\frac{1}{L}{x}_{1}-\frac{R}{L}{x}_{2}$ (4.45)

Tín hiệu ra c(t) = v0 = Rx2 (4.46)

Dùng các phương trình (4.44), (4.45), (4.46) và các điều kiện đầu của mạch x1(t0), x2(t0) ta có thể xác định trạng thái tương lai của mạch và tín hiệu ra của nó.

Dưới dạng véctơ, trạng thái của hệ được trình bày:

$\stackrel{}{X}=\text{AX}+\text{Br}$

$\stackrel{}{C}=\text{DX}+\text{Er}$

Trong đó:

$A=\mid \begin{array}{cc}0& -\frac{1}{C}\\ \frac{1}{L}& -\frac{R}{L}\end{array}\mid$ ; $B=\left[\begin{array}{c}\frac{1}{C}\\ 0\end{array}\right]$ ; $D=\left[\begin{array}{cc}0& \end{array}R\right]$

$X=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]$ ;  $\stackrel{\text{.}}{X}=\left[\begin{array}{c}{\stackrel{\text{.}}{x}}_{1}\\ {\stackrel{\text{.}}{x}}_{2}\end{array}\right]$ ;  E=0

Lưu ý là các biến trạng thái của hệ thống không phải là duy nhất. Tùy theo cách chọn lựa, có thể có những tập hợp khác của các biến trạng thái.

## Đồ hình trạng thái .

Đồ hình truyền tín hiệu mà ta đã nói ở chương 3 chỉ áp dụng cho các phương trình đại số. Ở đây, ta sẽ đưa vào các phương pháp đồ hình trạng thái, như là một sự mở rộng cho đồ hình truyền tín hiệu để mô tả các phương trình trạng thái ,và các phương trình vi phân. Ý nghĩa quan trọng của đồ hình trạng thái là nó tạo được một sự liên hệ kín giữa phương trình trạng thái, sự mô phỏng trên máy tính và hàm chuyển.

What is your reason for learning Psychology? I'd like to know what all of you are thinking. ('~')
To understand my self, and help my self to cope with problems, then if there's a change I would like to help others.
Hindun
I could only encourage you to continue doing so. ('~')
J.
Thank you. 😉
Hindun
I'm diagnosed with ADD, Autism level 1 and Bipolar 2 so.... Knowledge about what's going on inside of my head may help me how to find the best ways to handle it. sort of. :P
Nathalie
That's morbidly interesting. In fact, I myself have never been professionally diagnosed with any psychological disorders. It's scary, but at the same time curious. At least you're educating yourself, so more power to you! ^_^
J.
the mind is a muscle, train it and work it out to condition and shape it to what ever you need. Dont believe you are diseased just because they have labels. think and be positive about who you can be with personal development. your on track im sure👌
Rey
I'ts my job!😊
Diferent
Well said @Rey Ordonez. Also, this is only my opinion, but the way I see a Psychological condition is like a unique window; it gives you a different angle of how the world could look like. Taken with a grain of salt, of course.
J.
I just can't understand why some people think that it's a bad thing to get a diagnose. For me, it's amazing because I can get the help that I need like therapist, medication etc.
Nathalie
your right, its not bad at all, what is bad is the negative assumption that comes from the idea of being sick. Its a necessary step to take towards self recognition. i havnt been diagnosed but from what i have studied i definitely have PTSD, and depression. which i have taught myself to cope with.
Rey
@Nathalie Bodn...because some people will be scared rather than nurturing positive thinking when they got diagnosed and knew they have some diseases. In fact knowing early your problem will help you to overcome it earlier before it getting worse.
Hindun
It has gotten a bad reputation, unfortunately. It's seen as something which makes you less human. A negative stigma, if you will. However, it's nice knowing that there are those, for example you and I, who know it's otherwise the complete opposite.
J.
Yes... some people are like "nope I'm sick so I can't do that, I can't do better etc." My reaction: wtf don't you want to improve yourself? 😅
Nathalie
@Nathalia getting a diagnosis in psychology is a good thing! those who do not visit a psychologist are afraid of stereotypes! Those who are well informed in psychology realize that we all have discorder!😊
Diferent
@J. Grimace...in some countries it becomes more open, and people will see it's a true disease, and not less human creatures. Only those who have conventional thought will be assumed that people with kind of disease is different from them.
Hindun
@Hindun Zhang Quite. But in my country it isn't. But we're still trying to spread the awareness, conventionally.
J.
I got my ADD diagnosed when I was 23. The Bipolar at 24. And the Autism this year, at the age of 28. I've learnt to deal with them really good in some ways, that's why I got them so late.
Nathalie
Sorry for writing so slow (and bad) 😂 I'm from Sweden and the keyboard and autocorrect are not my friends when I want to write in English 😂
Nathalie
because it deals with society Pasay
kitwala
It's fine, there aren't an stringent english professors around, ha! Besides, so far you've been engaging in sharing a part of your life with us today. Just a side note, I'm currently learning Swedish, so yeah. Tack på förhand!
J.
nathalia what type of autism if it is not secret?
Diferent
1
Nathalie
Nathalie
What if I'm just...I don't know..like without feelings?
why do you say that
Olivia
my dad died 3week ago... I wasn't feel sad or something... I was first at my school...but I wasn't proud of myself or happy... I don't have feelings...
Iza
because she is looking for an outlet, she says she is without feelings which is disconnection to others, which is a feeling itself. seems like she is having a hard time connecting with others emotionally due to mistrust, that probably comes from dishonesty from others and a defensive mechanism.
Rey
sorry to hear abt that. okay tell me was u close to your dad?
mahliqa
I know someone in a similar situation. How can i make this person happy?It s my boyfriend
Stefana
Do you think you have gone through so much pain before that your brain is trying to numb you to emotions? Do you watch unhealthy videos such as gore? If you do your brain is so used to sadistic content that you cant feel emotions and I'd recommend that you would stay away from anything unhealthy for
CW
I was somehow close to my dad...like..yeah... I was closer to him than to my mom
Iza
Stay away from it for a month or so and see how you feel.
CW
you know the thing is u are not accepting the fact that ur dad is not there anymore. that does not mean that u dont have feeling its just u dont want to believe it. see ur childhood pictures with him some video's of yours with him. the u wll know that u are not feeling less. any question u can ask m
mahliqa
lmao...but I'm like that like when I was a child...when I was 5 I wasn't "happy"
Iza
y u wasnt happy
mahliqa
see i need to know everything abt your childhood. is there anything in your childhood that something bad happened? i am here to help u so u have to tell me everything so i can reach to the core of the problem
mahliqa
Iza
@mahliqa I can understand ur trying for Psychotherapy But kindly go and read , practice well. just a chat doesn't make clients to open up their problem also this is not proved anywhere yet. if you really concern about their welfare and wanna help them. ask them to meet a Psychologist near to her
Dear Iza Pingu kindly get an appointment to ur nearest Psychologist for a long term betterment. they will made u to deal and make u better in all possible ways. Good Luck!
yall dont understand, obviously she is seeking attention, she is not seeking an answer. i come from a lot of mistrust and pain but i know we have very strong minds so one can overcome anything. its all about control of your emotions, since you say you have none then you must create these moments.
Rey
if you don't have feelings then your attitude dulls and may lead to some kind of disorder and people around think you as careless person. To feel joy some situation practice showing all the feeling whether good/bad in a Normal way without harm just for 21-30 days & control all ur Negative Emotions.
Jaswanth
Mr.rey how come u say that she is attention seeking without diagnose her ? have u created any rapport with her ? did u investigate her ? never judge ur client. kindly go and study what is counseling and what is the role of Psychologist.
@Jaswanth nothing will make this to disorders. for a disorders there are lots of things behind the reason. in this case the patient is mild Depressed. lose of hope, feels nothing, feeling hope less all these are symptoms of depression.
@pradeep u plz mind ur own bussiness. i am a psychologist. i know what to do and what not to do. and honestly saying u need a psychologist go get an appointment.
mahliqa
guys kindly don't provide any thing for clients just having an Psychology degree. practice before u deal with clients. your misguidence will make the clients worst. Also understand Psychologist never give solution they help the clients to find out the solution by themselves.
not having feelings after a traumatic event is a way for the mind to cope with what has happened. if this is the case use it to your advantage and go over the trauma with a fine tooth comb. how does it effect you, what can you learn from it, what areas in your life feel removed due to this event.
Selphira
the kore you go over it the more it because a common experience the more sensory adaptation will occur when you have emotions associated with it. and it will affect you less. seek out friends and talk about it. if you are comfortable with that. 😊
Selphira
If psyche means soul and ology means study of. how does psychology mean the scientific study of the mind?
The term *PSYCHOLOGY* was coined when mind and soul were not considered two different things. That may be because the concept was not clear. I guess that is the reason of psychology being the scientific study of mind, even though psyche means soul.
Bellatrix
yes. i think so u need to show love for the teamm.thay respect
can losing someone in a traumatic way cause someone to develop things like anxiety and panic attacks?
it can but not always. depends upon the coping capabilities of the person.
Bonito
say this person has never dealt with such grief before and they don't know how to cope
Chloe
PTSD
Maria
maybe but things related to this event don't trigger them
Chloe
a neurotransmitter is a chemical messenger that carries signals between neurons/nerve cells. These chemical messengers influence both physical and psychological functionings. Some important neurotransmitters are acetylcholine, Norepinephrine, Seratonin, GABA and Dopamine
aha
Ihrar
what is a neurotransmitter
they're simply chemical messengers which go from one neuron to another
Chloe
it is a chemical which uses in the communication of neurons may be excitatory or inhibitory
natnael
Neuro transmitter,it is a biological or chemical substance that carries information from the nervous systems expertialy from the Brian to all parts of the body of both humans and animals.
Tanam
what are the limitations of case study in psychology?
Ethics
Maria
sample selection biasness, cannot generalized to wider population, difficult to establish cause effect relationship, observational biasness, limited data collection etc
prachi
what is the educational psychology
Educational psychology is the application of psychological theories and principles in school or classroom.
Tanam
it is a branch of modern psychology concerned with the learning process and psychological problems of teachers and students
Chika
how does limbic system works?
I am doing a counselling course in person centred but I like to help and advise what should I do extra ,?
During the time you are being taught textbook psychology you should round your education with real world experience by volunteering at institutions you have an interest in making a possible career of after you graduate.
Carey
applied psychology is a branch of psychology
Kaneez
how does psychology is different from applied psychology
applied psychology is a kind of psychology which tried to show the application of psychology in your practical life though there is no such differences between the content but it's more practical than basic psychology which includes lots of distribution within the discipline & theoretical in nature
deepashree
could anyone explain what course are available for applied psychology I'm highly interested and use psychology to help me understand and process like through my experiences and would love to develop a deeper understand and knowledge bases so I'm able to extract the wisdom?
Michael
applied psychology is in its a course . there are 2 types of courses 1. B.A. /M.A.psychology 2 . B.A./ M.A. applied psychology
deepashree
if you want to be an clinical psycologist you can also opt for B.A. or M.A in clinical psycology
Raksha
psychology is the study of behavior and mental functioning of both human and animals.On the other hand applied psychology is the use of psychological theories and principles to solve psychological problems.
Tanam
reasons why we conduct personality assessment
we conduct it for knowing one person's attitude and identifying the faults from it easily
Jaswanth
for knowing our strengths and weaknesses
Kris
personality assessment can provide us with a way to categorize different charateristics or traits that we may not be aware of.Also, this categorization will help us learn how others might react to something in their environment.
Tanam
These assignments can be used for self reflection and understanding for job placement and for learning how to better interact with others in a team or work group.
Tanam
well explained Tanam Ibrahim
Kris
thank you all
Tanam
what is applied psychology then ?
applied psychology is psychology for a narrow audience basic psychology is broad, in comparrison to applied psychology. Basic psychology wants to discover people's motive to every day environmental wellness. Applied psychology wants to discover what motivates you to buy an eco friendly product.
Briana
Applied psychology refers to applying the concepts and theories to understand why people do what they do and perhaps change the behaviour in cases where it is abnormal.
Maria
does industrial psychology come under this ? wt abt social media and product designing psychology ....does it come under this applied psychology
Varshaa
Industrial psychology basically deals with issues in the workplace and focuses more on increasing productivity and the well-being of the employees.
Maria
Social media and product designing is not a sub field in psychology. It is a topic tho.
Maria
wt abt career in those Feilds..of social media and product design
Varshaa
Social psychologists and product designers usually takes a behavioural approach. If it's directly in the field of psychology I'd suggest social aspect. Psychology is within every field. What are you currently studying?
Maria
I am studying BA Hons applied psychology
Varshaa
Product designing falls under consumer psychology which is a branch of Social psychology through which conscious and unconscious influence prompts customers to buy or not buy the products.
Maria
oh kay cool..soo wt abt u ?...Ur are studying?
Varshaa
I'm studying Psychology. In the future I'll specialize in Clinical Psychology.
Maria
oh great !...
Varshaa
Do you guys have to obtain a phD before the license is granted?
Maria
hi guys
TECHNICAL
not exactly ...we need to have minimum MA I guess....
Varshaa
am not quite sure abt that ......
Varshaa
Okay.
Maria
What is Neuro psychology?
Michael
it aim to study the relationship between nervous system and cognitive behaviour. it culminate to identify pathway of neuron functioning followed in course of psychological phenomena. it is widely used to identify biological differences in an individual psychology.
prachi
its true
kitwala
how to get license of professional psychologists in India
Basavaraj
you have to do MPhil to practise as a psychologist in India
Himshikha
Can you name some good colleges in India, where I can do MPhil in Clinical Psychology.
Khursheede
The best is NIMHANS Bangalore
Himshikha
Nimhans is great my junior have doing der P.G
helo
parsa
what is mean by cognitive science
Umesh
Cognitive science is the study of the mind and what is does, including many scientific disciplines that touch on the subject. In this lesson, we will briefly explore the definition of cognitive science, and review its history.
Alicia
you probably got on here the same way I did you download the app
Richard
how to learn this cognitive science and where are the colleges?
Umesh
not exactly what I expected
Richard
the app is pretty decent though
Richard
I'm going to lee community college and this app is actually the text book they have us reading... you can find it for free in kindle but the app helps because it's interactive and has built in vocab cards
Taylor
umesh, what are you talking about? do you know english well?
Rey
This is the app they have students reading 🤔 The Caribbean really do have it harder and do much more in a year.
Maria
not the app itself, this app is made from the actual text book.... It's basically exactly the same
Taylor
I've seen the content.
Maria
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