A Continuous-Time Model of the Temperature in a Room

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Из курса от партнера University of California, Santa Cruz

Cyber-Physical Systems: Modeling and Simulation

6 оценки

University of California, Santa Cruz

Cyber-Physical Systems: Modeling and Simulation

6 оценки

Cyber-physical systems (CPS for short) combine digital and analog devices, interfaces, networks, computer systems, and the like, with the natural and man-made physical world. The inherent interconnected and heterogeneous combination of behaviors in these systems makes their analysis and design an exciting and challenging task. CPS: Modeling and Simulation provides you with an introduction to modeling and simulation of cyber-physical systems. The main focus is on models of physical process, finite state machines, computation, converters between physical and cyber variables, and digital networks. The instructor of this course is Ricardo Sanfelice (https://hybrid.soe.ucsc.edu), Associate Professor in the Department of Computer Engineering at the University of California Santa Cruz.

Из урока

Basic Modeling Concepts: Discrete-time and Continuous-Time Systems

Welcome to the Course5:35

Introduction5:39

Modeling Cyber-Physical Systems5:47

Discrete-Time Systems Concepts (Part 1)9:18

Discrete-Time Systems Concepts (Part 2)8:35

A Discrete-Time Model of a Ground Vehicle8:35

Simulation of a Discrete-Time Model of a Ground Vehicle16:54

Continuous-Time Concepts (Part 1)6:40

A Continuous-Time Model of a Ground Vehicle8:34

Simulation of a Continuous-Time Model of a Ground Vehicle20:27

Continuous-Time Concepts (Part 2)8:13

A Continuous-Time Model of a Linear Time-Invariant System8:39

A Continuous-Time Model of the Temperature in a Room9:09

Simulation of the Temperature in a Room13:22

Познакомьтесь с преподавателями

Ricardo Sanfelice
Associate Professor
Department of Computer Engineering

[MUSIC]

Now if we want to consider a concrete example where these linear type of

form appears naturally when we model the system,

module of some constant that might lead us to a an assigned system actually.

But, let's consider the model of a temperature.

In a room.

So given a room.

We denote its temperature.

As T. This can be any real number.

And the device.

That.

Can change T.

Is given by a heater.

A similar analysis could be done for the case of a cooler.

The idea for us will be to determine how this temperature changes over time.

So that automatically leads to a derivative of temperature with respect to

time, which will basically boil down to a differential equation.

A simple model of such a variation of temperature is given

by the following linear affine differential equation.

The constant a.

Is the decay rate.

Of a temperature.

It actually characterizes objects, walls, and

all sort of things present in the room.

And will give us a rough approximation of how rapidly that temperature changes.

This parameter here,

Tr will correspond to the information of temperature outside a room.

And how that affects the room that we are interested in, its temperature.

Since we're considering a heating type of scenario,

we would potentially consider this signal, u,

as either zero, or off whenever the heater is off, or

one, or on whenever the heater is on.

So whenever it's off, we will say that u = 0.

And whenever it's on, we say that u = 1.

Therefore, this is somehow the effect or

the amount of temperature change that the heater can introduce into the room.

This is what is called the heater capacity.

So, we have a temperature model.

This model can be written as a physical model,

given by z dot = Fp,

(z,u) with the following assignments.

Z is the state in our model, therefore will equal to T,

u is equal to the input itself that is also labeled as u.

And the function is given by this entire expression that we have right here.

Which notice that because we are using z,

these Ts actually should be z.

Coming back to the model in the previous video,

this -a will correspond to the ap matrix, in this case a scalar.

This T Delta will correspond to the BP.

And as I said, we have an additional constant that generates what is called

an affine linear system.

So if someone gives you an initial state, in this case initial temperature and

the values of these constants, then we can predict forward in

time what the values of the temperature will be for a given input profile, okay?

So, when the input is equal to zero.

Then.

The model we obtain.

T dot minus aT plus this constant here.

I would like to talk about this particular model, because it is simple to analyze.

And it will allow us to essentially figure out where the temperature will go to.

So consider for a second, the fact that we actually reach

a particular value of the temperature that is not infinity.

And we call that value of the temperature, the steady state temperature.

So, in a steady state.

The temperature.

Is constant.

And equal to.

A temperature that I'm going to label as Tss, ss for steady state.

Then.

Since it's constant, what we actually obtain is that

the derivative of that state is equal to zero.

So therefore, we satisfy the following equation.

From that equation, we can actually solve for the value of the steady state.

So this told us that the temperature as a function of time will

converge to this particular value if there is a steady state.

It turns out that because we said that the temperature is actually

governed with a decay rate that is positive, we can always guarantee that

this steady state exists and is actually equal to this value.

And moreover, we can breakdown from this expression using

the general formula from the previous video.

The expression of T, temperature, as a function of time.

[MUSIC]

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