State-Space...1. The inductor current q1 and the capacitor voltage q2 as the state variables. 2. 3....

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State-Space

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh (with material from Dr. Paul Pounds)

Lecture 22

[email protected]://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

May 22, 2013

Today in Linear Systems…

ELEC 3004: Systems 22 May 2013 - 2

Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control17-MayApplications in Industry

12 22-MayState-Space24-MayControllability & Observability

1329-MayInformation Theory/Communications & Review31-MaySummary and Course Review

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Today:• State-Space

• Compensator Design

Friday:• Controllability

• Observability


Goals for the Week

• More general mathematical model – MIMO, time-varying, nonlinear

• Matrix notation (think LAPACK MATLAB)• Good for discrete systems• More design tools!

Or more aptly…

Welcome to

State-Space!( It be stated -- Hallelujah ! )

22 May 2013 -ELEC 3004: Systems 4

3

Affairs of state• Introductory brain-teaser:

– If you have a dynamic system model with history (ie. integration) how do you represent the instantaneous state of the plant?

Eg. how would you setup a simulation of a step response, mid-step?

t = 0t

start


Introduction to state-space• Linear systems can be written as networks of simple dynamic

elements:

2

7 1224

13

7

1

12

2

u y


4

Introduction to state-space• We can identify the nodes in the system

– These nodes contain the integrated time-history values of the system response

– We call them “states”

7

1

12

2

u yx1 x2


Linear system equations• We can represent the dynamic relationship between the states

with a linear system:

7 12 0 0

2 0


5

State-space representation• We can write linear systems in matrix form:

7 121 0

10

1 2 0

Or, more generally:

“State-space equations”


1. Choose all independent capacitor voltages and inductor currents to be the state variables.

2. Choose a set of loop currents; express the state variables and their first derivatives in terms of these loop currents.

3. Write loop equations, and eliminate all variables other than state variables (and their first derivatives) from the equations derived in steps 2 and 3.


A Procedure for Determining State Equationsin Electrical Circuits

6

1. The inductor current q1 and the capacitor voltage q2 as the state variables.

2.

3.


A Quick Example

•

•

•

•


Another Example

7


Another Example

• N1(t)=N1(0)exp(-λ1t)

•

•

•


Another Example

8

State-space representation• State-space matrices are not necessarily a unique

representation of a system– There are two common forms

• Control canonical form– Each node – each entry in x – represents a state of the system

(each order of s maps to a state)

• Modal form– Diagonals of the state matrix A are the poles (“modes”) of the

transfer function


Control canonical form

• CCF matrix representations have the following structure:

⋯1 0 0 0 00 1⋮ ⋱ ⋮

1 0 00 0 ⋯ 0 1 0

Pretty diagonal!


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State variable transformation• Important note!

– The states of a control canonical form system are not the same as the modal states

– They represent the same dynamics, and give the same output, but the vector values are different!

• However we can convert between them:– Consider state representations, x and q where

x = Tq

T is a “transformation matrix”


State variable transformation• Two homologous representations:

and

We can write:

Therefore, and

Similarly, and


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Controllability matrix

• To convert an arbitrary state representation in F, G, H and J to control canonical form A, B, C and D, the “controllability matrix”

⋯must be nonsingular.

Why is it called the “controllability” matrix?


Controllability matrix

• If you can write it in CCF, then the system equations must be linearly independent.

• Transformation by any nonsingular matrix preserves the controllability of the system.

• Thus, a nonsingular controllability matrix means x can be driven to any value.


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Why is this “Kind of awesome”?• The controllability of a system depends on the particular set of

states you chose

• You can’t tell just from a transfer function whether all the states of x are controllable

• The poles of the system are the Eigenvalues of F, ( ).


State evolution• Consider the system matrix relation:

The time solution of this system is:

If you didn’t know, the matrix exponential is:12!

13!

⋯


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Stability• We can solve for the natural response to initial conditions :

∴

Clearly, a system will be stable provided eig 0


Characteristic polynomial

• From this, we can see or, I 0

which is true only when det I 0Aka. the characteristic equation!

• We can reconstruct the CP in s by writing:det I 0


13

Great, so how about control?• Given , if we know and , we can design a

controller such thateig 0

• In fact, if we have full measurement and control of the states of , we can position the poles of the system in arbitrary locations!

(Of course, that never happens in reality.)


Example: PID control• Consider a system parameterised by three states:

– , ,– where and

=1

12

0 1 0 0

is the output state of the system; is the value of the integral;

is the velocity.


14

• We can choose to move the eigenvalues of the system as desired:

det1

12

All of these eigenvalues must be positive.

It’s straightforward to see how adding derivative gain can stabilise the system.


Just scratching the surface

• There is a lot of stuff to state-space control

• One lecture (or even two) can’t possibly cover it all in depth

Go play with Matlab and check it out!


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Discretisation FTW!• We can use the time-domain representation to produce

difference equations!

Notice is not based on a discrete ZOH input, but rather an integrated time-series.We can structure this by using the form:

,


Discretisation FTW!• Put this in the form of a new variable:

Then:

Let’s rename and


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Discrete state matricesSo,

1

Again, 1 is shorthand for

Note that we can also write as:

where

2! 3!⋯


Simplifying calculation• We can also use to calculate

– Note that:

Γ1 !

itself can be evaluated with the series:

≅2 3

⋯1


17

State-space z-transformWe can apply the z-transform to our system:

which yields the transfer function:


State-space control design

• Design for discrete state-space systems is just like the continuous case.– Apply linear state-variable feedback:

such that detwhere is the desired control characteristic equation

Predictably, this requires the system controllability matrix ⋯ to be full-rank.


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2nd Order System Response


• Response of a 2nd order system to increasing levels of damping:


2nd Order System Response

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Damping and natural frequency

[Adapted from Franklin, Powell and Emami-Naeini]-1.0 -0.8 -0.6 -0.4 0-0.2 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

Re(z)

Img(z)

where 1

0.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9

235

710

910

25

1

25

0

310

5

10

20


• Poles inside the unit circleare stable

• Poles outside the unit circleunstable

• Poles on the unit circleare oscillatory

• Real poles at 0 < z < 1give exponential response

• Higher frequency ofoscillation for larger

• Lower apparent dampingfor larer and r


Pole positions in the z-plane

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Characterizing the step response:


2nd Order System Specifications

• Rise time (10% 90%):

• Overshoot:

• Settling time (to 1%):

• Steady state error to unit step: ess

• Phase margin:

Characterizing the step response:


2nd Order System Specifications

• Rise time (10% 90%) & Overshoot: tr, Mp ζ, ω0 : Locations of dominant poles

• Settling time (to 1%): ts radius of poles:

• Steady state error to unit step: ess final value theorem

21

Design a controller for a system with:• A continuous transfer function:• A discrete ZOH sampler • Sampling time (Ts): Ts= 1s• Controller:

The closed loop system is required to have:• Mp < 16%• ts < 10 s• ess < 1


Ex: System Specifications Control Design [1/4]



22





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Approximation Methods


Euler’s method*• Dynamic systems can be approximated† by recognising that:

≅1

T

x(tk)

x(tk+1)

*Also known as the forward rectangle rule†Just an approximation – more on this later

• As → 0, approximation error approaches 0


24

Euler’s method• Euler approximation can produce a system z-transform directly• Use the substitution:

1


Tustin’s method• Tustin uses a trapezoidal integration approximation (compare

Euler’s rectangles)• Integral between two samples treated as a straight line:

1Taking the derivative, then z-transform yields:

which can be substituted into continuous models

1

x(tk)

x(tk+1)


25

Matched pole-zero• If , why can’t we just make a direct substitution and go

home?

• Kind of!– Still an approximation– Produces quasi-causal system (hard to compute)– Fortunately, also very easy to calculate.


Matched pole-zero• The process:

1. Replace continuous poles and zeros with discrete equivalents:

2. Scale the discrete system DC gain to match the continuous system DC gain

3. If the order of the denominator is higher than the enumerator, multiply the numerator by 1 until they are of equal order*

* This introduces an averaging effect like Tustin’s method


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Modified matched pole-zero• We’re prefer it if we didn’t require instant calculations to

produce timely outputs

• Modify step 2 to leave the dynamic order of the numerator one less than the denominator– Can work with slower sample times, and at higher frequencies


Approximation comparison• We’ve been making a lot of approximations

– Just how goods are these approximations?– As you might expect, it depends on how closely T matches the

bandwidth of the system– Also varies by order of the approximation

Let’s consider the system

sampled at 10 Hz


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Approximation comparison

• .

. .

•. .

. .

•. . .

. .

•.

. .

• .

. .

*FOH: First Order Hold ‘triangle approximation’


Approximation comparison

All approximations give good results for /4


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∴ Effect of ZOH delay• Recall the intrinsic time-delay associated with ZOH? What

about that?• This is the discrete domain; we can model the delay exactly:

1

This can be thought of a as a step input, followed by an immediate negative step one sample time later


LTID Stability


29


LTID Stability


Characteristic roots location and the corresponding characteristic modes [1/2]

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Characteristic roots location and the corresponding characteristic modes [2/2]


S-Plane to z-Plane [1/2]

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S-Plane to z-Plane [2/2]


Relationship with s-plane poles and z-plane transforms

32

• Final Exam:Saturday, June 15 at 9:30 AM (sorry!)

• Problem Set 2 is due this Friday!


Announcements:

!

• AKA: I can see that. Yes, I can control that!


Next Time in Linear Systems ….1

27-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control17-MayApplications in Industry

1222-MayState-Space

24-MayControllability & Observability13

29-MayInformation Theory/Communications & Review31-MaySummary and Course Review

State-Space...1. The inductor current q1 and the capacitor voltage q2 as the state variables. 2. 3....

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Transcript of State-Space...1. The inductor current q1 and the capacitor voltage q2 as the state variables. 2. 3....