Introduction to Signals and Systems Lectures #3 - State ...guilldrion/Files/SYST0002-2017-18... ·...

Introduction to Signals and Systems Lectures #3 - State-space Representation of Systems

Guillaume Drion Academic year 2017-2018

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Outline

Modeling in state-space - time invariance - linearity

Block diagram and state-space representation (discrete and continuous)

Solutions of state-space equations: transition matrix and matrix exponential

Response of N-dimensional systems

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Systems modeling

Modeling and analysis of systems: open loop. “Observing and analyzing the environment”

Can be used to understand/analyze the behavior of a dynamical system. A “good” model can predict the future evolution of a system.

How can we use systems modeling to predict the future? What is a “good model” or a “good system” to model?

SYSTEMInput Output

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In “state-space”, dynamical systems are modeled using difference equations (discrete domain) or differential equations (continuous domain).

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, are variables of the systems (the model describes their evolution). A system can have many variables (dimension).x[n]

is a parameter of the system. A system can have many parameters.a

x =dx

dt

(where )x[n+ 1] = ax[n]

! x[n] = a

n

x = ax

! x(t) = e

at

Discrete domain Continuous domain

x(t)


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, are variables of the systems (the model describes their evolution). A system can have many variables (dimension).x[n]


x =dx

dt


! x[n] = a

n

x = ax

! x(t) = e

at


x(t)

What defines a parameter?

Time-variance vs time-invariance

Example: think about the evolution of a stock market.

Can we accurately predict the evolution of a stock market using a model?

where the evolution depends on the market growth rate, i.e the market growth rate is a parameter of the system.

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We all know that the market growth rate evolves over time, so the prediction that we will make at a time “t” using the rate at this time “t” will not accurately describe the evolution of the stock market! No good prediction!


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We all know that the market growth rate evolves over time, so the prediction that we will make at a time “t” using the rate at this time “t” will not accurately describe the evolution of the stock market! No good prediction!

Systems whose parameters evolve over time are called time-variant. In this course, we will study Time-Invariant (TI) systems.


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Time-Invariant (TI) systems

In a time-invariant system, the input/output relationship does not depend on time:

i.e. a system whose parameters do not vary over time!

if then

Examples: is time-invariant. is time-variant.

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In this course, we focus in time-invariant systems.

State-space representation in continuous-time: general form

General form of a state-space representation in continuous-time:

is the input is the output, are the states (i.e. energy storage/memory).

!! State-space representations are not unique !! But all representations describe the same system, i.e. with the same (and unique) inputs, outputs and transfer function.

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u(t)y(t)x(t)

State-space representation in continuous-time: identifying states

Method #1

Identify inputs and outputs.

Identify independent means of energy storage.

Identify state variables.

Write model (using laws of physics, etc.).

Method #2




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State-space representation in continuous-time: identifying states

Method #1


Identify independent means of energy storage.



Method #2




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Modeling TI systems in state-space: driven pendulum

u

θ

m

l

mg

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Driven pendulum: modelingu

θ

m

l

mg with moment of inertia.

which gives .

What do I do with a ? vs

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Driven pendulum: statesu

θ

m

l

mg with moment of inertia.

which gives .

What do I do with a ? vs

States: . If I want to write the system in the form , the second state should be (the energy is stored in the position and speed).

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Driven pendulum: state-space modelu

θ

m

l

mg

with and .

This gives the state-space representation

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Driven pendulum: model analysisu

θ

m

l

mg

with and .


(i) Is the system linear? No!

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Nonlinear vs linear models

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A dynamical system is represented by its update function and its output function

For linear systems, it writeswhere , and are vectors,

, , and are matrices (dimensionality of the system).

x = f(x, u)

y = g(x, u)

x = Ax+Bu

y = Cx+Du

BA

x u y

DC

x[n+ 1] = Ax[n] +Bu[n]

y[n] = Cx[n] +Du[n]

x[n+ 1] = f(x[n], u[n])

y[n] = g(x[n], u[n])


θ

m

l

mg

with and .


(i) Is the system linear? No! (ii) Find equilibrium points , i.e. points such that .

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(x1, x2) x1 = x2 = 0


θ

m

l

mg

with and .


(i) Is the system linear? No! (ii) Find equilibrium points , i.e. points such that . (ii) If the system is nonlinear, linearize around the equilibrium point(s). You will end up with one linear, time-invariant (LTI) system for each equilibrium points!

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(x1, x2) x1 = x2 = 0

Driven pendulum: linearizationu

θ

m

l

mgIdea behind linearizing a system around a specific point:

x

f(x)

+ See tutorials.

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x = f(x, u)

y = g(x, u)

x = Ax+Bu

y = Cx+Du

Modeling LTI system: a RC circuit

R

CV

i

vC(t)vR(t)

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RC circuit: method #1


Identify energy storage.


Write model.

R

CV

i

vC(t)vR(t)

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RC circuit: method #1 - inputs and outputs




Write model.

Inputs: active sources voltage source:

Outputs: or

R

CV

i

vC(t)vR(t)

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RC circuit: method #1 - energy storage




Write model.

A capacitor accumulates charges

R

CV

i

vC(t)vR(t)

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RC circuit: method #1 - state variables




Write model.


State variable:

R

CV

i

vC(t)vR(t)

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RC circuit: method #1 - model




Write model.

R

CV

i

vC(t)vR(t)

Input: . Output: or . State:

Kirchhoff:

Which gives

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RC circuit: method #2


Write model


R

CV

i

vC(t)vR(t)

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RC circuit: method #2 - model


Write model


Input: . Output: or .

Kirchhoff:

State variable:

R

CV

i

vC(t)vR(t)

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RC circuit: method #2 - state variable


Write model


Input: . Output: or .

Kirchhoff:

State variable:

R

CV

i

vC(t)vR(t)

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RC circuit: method #2 - state-space model


Write model


Input: . Output: or . State:

Kirchhoff:

Which gives

R

CV

i

vC(t)vR(t)

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RC circuit: two different, but equivalent, state-space models


The accumulation of charges creates a voltage gradient . They both represent the same energy storage.

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RLC circuit: inputs, outputs, states, equations

R

LV

i

vL(t)vR(t)

C

vC(t)




Write model.

Input: . Output: or or .

Energy storage/variable. Capacitor: . Inductor: .

Kirchhoff:

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RLC circuit: state-space model

R

LV

i

vL(t)vR(t)

C

vC(t)




Write model.

Kirchhoff: with and .

State-space model:

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RLC circuit: state-space model

R

LV

i

vL(t)vR(t)

C

vC(t)




Write model.

Kirchhoff:

You could also use (try it at home) or or

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Outline





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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)


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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


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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!


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What kind of LTI systems admit a state-space representation?

Continuous time:

Discrete time:

yes if

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


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y goes back in u for the next “update”. effect of the past value of y!


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Input-output approach using the block diagram: If , we have


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Let’s define a state in which we store the history of the system (here, ):


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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 the book)

Block diagram of a N-dimensional system in continuous time

52


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54


55


56

States are outputs of integrators:

State-space representation of a N-dimensional system in continuous time

57

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

58

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

59

and


60

and


61

and


62

and


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and

State-space of a general N-dimensional system in continuous time

64

and

The update function is the same: states are outputs of integrators:

x1 = ⌫, x2 = ⌫

(1), x3 = ⌫

(3), . . . , xn = ⌫

N�1


65

and

The output function is more complex:

x1 = ⌫, x2 = ⌫

(1), x3 = ⌫

(3), . . . , xn = ⌫

N�1


66

and

It gives the following state-space representation:

x1 = ⌫, x2 = ⌫

(1), x3 = ⌫

(3), . . . , xn = ⌫

N�1

State-space representation of LTI systems in discrete time

67

We will derive a general structure for the state-space representation of LTI systems in discrete time.

Let’s start with a first order system (i.e. one state):

This system can be done with one delay, multipliers, and one adder (or summer).

We will start by building a block diagram of the system using delays, multipliers and adders.

Block diagram of a first order system in discrete time

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69



70



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States are outputs of delays:

A general N-dimensional system in discrete time

72

Let’s now consider the general case

We have to consider an intermediate signal where

We can rewrite the first equation:

and

State-space of a general N-dimensional system in discrete time

73

States are output of delays (and )

and

x1[n] = ⌫[n�N ], x2[n] = ⌫[n�N + 1], · · · , xN [n] = ⌫[n� 1]


74


and

x1[n] = ⌫[n�N ], x2[n] = ⌫[n�N + 1], · · · , xN [n] = ⌫[n� 1]


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and

x1[n] = ⌫[n�N ], x2[n] = ⌫[n�N + 1], · · · , xN [n] = ⌫[n� 1]

Block diagram of a general N-dimensional system in discrete time

76

and

Solutions of state-space equations: discrete case

A. Update function:

77

Outline





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A. Update function:

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A. Update function:

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A. Update function:

81


A. Update function:

is the transition matrix of the system :

If you know the solution at any time , the transition matrix gives you the solution at any future time .

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Using we have


B. Output function:

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Using we have


B. Output function:

zero inputzero state

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Solutions of state-space equations: continuous case

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!

85

Solutions of state-space equations: the matrix exponential

The system has the solutionwhere is the matrix exponential and is the transition matrix:

Reminder: in the discrete case, we had

86

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 textbook).

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).

89

Outline





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91


x =dx

dt


! x[n] = a

n

x = ax

! x(t) = e

at


a > 0a < 0


92


x =dx

dt


! x[n] = a

n

x = ax

! x(t) = e

at


Systems with more than one variable:

x[n+ 1] = Ax[n]

! x[n] =?

x = Ax

! x(t) =?


is a square matrix of parameters.A

Response of N-dimensional continuous systems

93


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

94

The complex exponential

95

Special case: , giving .

Response of N-dimensional continuous systems

96



A


xi(t) = e

�it = e

(�i+j!i)t


x = Ax

! x(t) =?

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

i.e. negative feedback

97

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).


a < 0x = ax+ bu

AA

x = Ax+Bu

R(eig(A)) < 0

i.e. negative feedback

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99

STABLE UNSTABLE

Response of N-dimensional discrete systems

100



A



x[n+ 1] = Ax[n]

! x[n] =?

xi[n] = �

ni = ⇢

ni e

j!in

The discrete time complex exponential

101

u[n] = zn, z = ⇢ej!

⇢ < 1 ⇢ = 1 ⇢ > 1


102

u[n] = zn, z = ⇢ej!

⇢ < 1 ⇢ = 1 ⇢ > 1

STABLE UNSTABLE

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