Chapter 3: Fourier Representation of Signals and LTI Systems
Introduction to Signals and Systems Lectures #3 - State-space Representation … · 2020. 10....
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Introduction to Signals and Systems Lectures #3 - State-space Representation - LTI systems Guillaume Drion Academic year 2020-2021 1
Transcript of Introduction to Signals and Systems Lectures #3 - State-space Representation … · 2020. 10....
Introduction to Signals and Systems Lectures #3 - State-space Representation - LTI systems
Guillaume Drion Academic year 2020-2021
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Outline
Response of N-dimensional closed LTI systems
Open LTI systems
Block diagram and state-space representation
Solutions of state-space equations: transition matrix and matrix exponential
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Response of N-dimensional continuous systems
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Systems with more than two variables:
x = Ax
! x(t) =?
is a square matrix of parameters. The response will be a sum of exponentials whose coefficients are the eigenvalues of the matrix .A
A
Eigenvalues can be real or complex conjugate pairs. In general:
xi(t) = e
�it = e
(�i+j!i)t
where is the imaginary unit.j
The complex exponential
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The complex exponential
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Special case: , giving .
How to assess stability of a general linear system (1D): System is stable if
How to assess stability of a general linear system (N-D): Multiple state-interactions: what are the feedbacks?N-feedback directions (eigenvectors of matrix ), each having its own amplitude (eigenvalues of ). The system is stable if (all feedbacks negative).
Condition for stability of continuous linear systems
a < 0x = ax+ bu
AA
x = Ax+Bu
R(eig(A)) < 0
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Condition for stability of continuous linear systems
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STABLE UNSTABLE
Outline
Response of N-dimensional closed LTI systems
Open LTI systems
Block diagram and state-space representation
Solutions of state-space equations: transition matrix and matrix exponential
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Outline
Response of N-dimensional closed LTI systems
Open LTI systems
Block diagram and state-space representation
Solutions of state-space equations: transition matrix and matrix exponential
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State-space representation of LTI systems
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A LTI system can be represented as followswhere is the state vector whose dimension gives the dimension of the system.
In a single-input/single-output (SISO) system, we have (ex: 4 states)
State-space representation of LTI systems
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A LTI system can be represented as followswhere is the state vector whose dimension gives the dimension of the system.
In a multiple-inputs/multiple-outputs (MIMO) system (m inputs, p outputs, n states):
Update function Output function
States Inputs States Inputs
State-space representation of LTI systems
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Some LTI systems cannot be represented with a finite state-space!
Example: delay
At t=t0, we need to know all the values of the input between t0-T and t0.
In continuous time, it represents an infinite number of values to be stored, i.e. an infinite number of states.
In discrete time, a delay can be represented by a finite number of states!
State-space representation of LTI systems
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What kind of LTI systems admit a state-space representation?
yes if
State-space representation of LTI systems in continuous time
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We will derive a general structure for the state-space representation of LTI systems in continuous time.
Let’s start with a first order system (i.e. one state):
This system can be done with one integrator (not a differentiator!!!), multipliers, and one adder (or summer).
We will start by building a block diagram of the system using integrators, multipliers and adders.
Block diagram of a first order system in continuous time
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Block diagram of the first order system
Block diagram of a first order system in continuous time
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Block diagram of the first order system
Block diagram of a first order system in continuous time
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Block diagram of the first order system
Block diagram of a first order system in continuous time
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Block diagram of the first order system
y goes back in u for the next “update”. effect of the past value of y!
Block diagram of a first order system in continuous time
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Block diagram of the first order system
Input-output approach using the block diagram: If , we have
Block diagram of a first order system in continuous time
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Block diagram of the first order system
Let’s define a state in which we store the history of the system (here, ):
Block diagram of a first order system in continuous time
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Block diagram of the first order system
Now, we have ,
The past of the system is stored in the integrator. States are therefore the outputs of the integrators.
The past of the system (or energy) is stored in the integrators
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t
y
t0 t0+t1
At t=t0+t1, the integral contains the history of the past values.
State-space representation/block diagram in continuous time
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Block diagram: representation of the system using integrators.
State-space: representation of the system using states, which are the outputs of the integrators!
Example: RLC circuitthe current in the inductor and the voltage across the capacitor are outputs of integrators!
A N-dimensional system in continuous time
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Let’s first consider the case
Which can be written(incorrect in TXB)
Block diagram of a N-dimensional system in continuous time
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Block diagram of a N-dimensional system in continuous time
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Block diagram of a N-dimensional system in continuous time
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Block diagram of a N-dimensional system in continuous time
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Block diagram of a N-dimensional system in continuous time
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States are outputs of integrators:
State-space representation of a N-dimensional system in continuous time
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States are outputs of integrators:
8>>>>>>>>><
>>>>>>>>>:
x1 = x2
x2 = x3
x3 = x4...xn�1 = xn
xn = �a0x1 � a1x2 � . . .� aN�1xn + b0u
y = x1
A general N-dimensional system in continuous time
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Let’s now consider the general case
We have to consider an intermediate signal where
Justification:
and
P (D)y = Q(D)u ( P (D)⌫ = u and y = Q(D)⌫
Block diagram of a general N-dimensional system in continuous time
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and
Block diagram of a general N-dimensional system in continuous time
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and
Block diagram of a general N-dimensional system in continuous time
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and
Block diagram of a general N-dimensional system in continuous time
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and
Block diagram of a general N-dimensional system in continuous time
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and
State-space of a general N-dimensional system in continuous time
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and
The update function is the same: states are outputs of integrators:
x1 = ⌫, x2 = ⌫
(1), x3 = ⌫
(3), . . . , xn = ⌫
N�1
State-space of a general N-dimensional system in continuous time
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and
The output function is more complex:
x1 = ⌫, x2 = ⌫
(1), x3 = ⌫
(3), . . . , xn = ⌫
N�1
State-space of a general N-dimensional system in continuous time
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and
It gives the following state-space representation:
x1 = ⌫, x2 = ⌫
(1), x3 = ⌫
(3), . . . , xn = ⌫
N�1
Outline
Response of N-dimensional closed LTI systems
Open LTI systems
Block diagram and state-space representation
Solutions of state-space equations: transition matrix and matrix exponential
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Solutions of state-space equations
A. Update function: .
Let’s start with the autonomous system (zero-input).
If (scalar, i.e. one dimensional case), we have , which gives the solution (zero-input response):
We want to extend this solution to the general, N-dimensional case!
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Solutions of state-space equations: the matrix exponential
The system has the solutionwhere is the matrix exponential and is the transition matrix:
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Solutions of state-space equations
Now, we will try to find the complete solution of the system
We can define such that
The solution of for any input is therefore
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Solutions of state-space equations: zero-input and zero-state responses
The solution of for any input is
We can use the change of variable , which gives
zero input zero state
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Computing the matrix exponential: the Jordan form
The solution of the system involves A only if A is the matrix exponential, i.e. it has to respect the properties described earlier.
It means that to find the solution of the state-space equations, the matrix A has to be into a specific form, called the Jordan form (see TXB).
This implies that we have to change the variables of the state-space representation, i.e. make a state-space transformation (same to reach the canonical form).
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State-space approach: limitations
zero input zero state
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can be high dimensional! (curse of dimensionnality)
The input needs to be integrated, but it can be quite complex. Other approach?
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