Riassunti di Sistemi di Controllo

28/09/20

Studying a physical process it's necessary to find mathematical laws which can help understanding the behavior and the characteristics of the problem. In this way, you can also modify changing some parameters of the process, the behavior in the way you want.

The variables of a physical process are:

• inputs
• states
• outputs, which are the variables on which we will design the control system

The controller is the system which acts on the inputs variables to obtain a determined state or something else.

Inputs can be changed by the controller in two ways:

• independently, by outputs, which are available thanks to measurement or observation
• dependently of reference variable (for example if you want to maintain a certain pressure in a tank)

Inputs don't depend on the process, are independent and given, practically.

If P you have to know the independent variables to know its behavior and how they characterize P.

States are variables which completely describe the physical process (represented by x(t)). For example position and velocity of a vehicle which is moving.

28/09/20

Studying a physical process it's necessary to find mathematical laws which can help understanding the behavior and the characteristics of the problem. In this way, you can also modify changing some parameters of the process, the behavior in the way you want.

The variables of a physical process are:

• inputs
• states
• outputs, which are the variables on which we will design the control system

The controller is the system which acts on the inputs variables to obtain a determined state or something else.

Inputs can be changed by the controller in two ways:

• independently, by outputs, which are available thanks to measurement or observation
• dependently, of reference variable (for example if you want to maintain a certain pressure in a tank)

[C] INPUTS —> [physical process] P —> OUTPUTS STATES

Inputs don’t depend on the process, are independent and given practically.

To know P, you have to know the independent variables to know its behavior and how they characterize P.

States are variables which completely describe the physical process (associated by x_i(t)). For example, position and velocity of a vehicle which is moving.

Imagine we have a car moving with determinate mass, position and velocity, its dynamic is governed by the Newton's law: \( M \cdot \ddot{s} = F \; \) and \( \dot{s} = \ddot{s} \cdot t \).

So if i use variable \( s \) i have a mathematical model made by a second order differential equation so i need to know two border conditions.

\(\Rightarrow \) \( \ddot{s}=F/M \quad s(0), \dot{s}(0)\)

On the other way, we can choose two variables \( s=x_1 \) and \( \dot{s}=x_2 \), so i can state

\(\Rightarrow \) \( x_1, \dot{x_1}=x_2 \Rightarrow \) with this eq. too i have a first order differential system and so even a first order model.

• \( x_1 = x_2 \)
• \( \dot{x_2} = F/M \), \( F= u \) = control variable

\(\Rightarrow \) the general form is \( \dot{x} = Ax + Bu \quad \) where in our case \( A=\begin{pmatrix} 0 & 1 \\ 0 & 1/m \end{pmatrix}\), \( B=\begin{pmatrix} 0 \\ 1/m \end{pmatrix} \) \(x=\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\)

Imagine we want to control the velocity x₂, so we have y=x₂, which in general is:

\(\dot{x} = Ax + Bu

y = Cx + Du

\(\begin{pmatrix} 0 & 1 \end{pmatrix}\) \(\begin{pmatrix} 0 & 0 \end{pmatrix}\)

output, so the controlled variable. -> but is even possible that you shall know a variable but you can control just another

If we have the same situation just seen, but added with the friction force, we need to approximate this force, and so we do with the model

\(\dot{x} = x_2\)

\(\dot{x_2} = F/M - kx_2 \Rightarrow F=0 \; \dot{x_2} = -kx_2 \) which as an exponential tending to zero x₂ => x₂->0 and the mass tend to stop exponentially

So know we have a more realistic model, but it has also the approximation of the friction force. This means that we have to well model the physical processes,

However once you really understand the physical process you can change the mathematical model by using different states which for example

can be \(\hat{z} = Tx\), where \(T\) is a squared matrix and even invertible

which has the same dimension of x

in this way is possible to simplify the A and B matrices and so the model

=> \(\hat{\dot{z}} = T\hat{\dot{x}} = T(Ax + Bu) = TAx + TBu = TAT^{-1}\hat{z} + TBu \) and so we have another linear differential problem

=> \(\hat{\dot{z}} = \tilde{A}\hat{z} + \tilde{B}\hat{u}\) \(\tilde{A} = TAT^{-1} \; \tilde{B} = TB\)

=> the same things can be done with the control variables, for example \(v = Gu\), with \(|G| \neq 0\). By the way, we have infinite equivalent models describing a process.

Till now we have seen just lin