Term Paper on State Space Analysis

7/30/2019 Term Paper on State Space Analysis

1/22

1

ASSIGNMENT ON

CONTROL SYSTEMS

State Space Analysis of Control Systems

DEPARTMENT OF

ELECTRONICS & COMMUNICATION ENGINEERING

INDIAN SCHOOL OF MINES, DHANBAD

DHANBAD-826004

SUBMITTED TO -

DR. S. K. RAGHUWANSHI

SUBMITTED BY:

Madhuri Suthar (ADM. NO: 2010JE1034)

B-TECH 3rd

YEAR,

ECE DEPT., ISM DHANBAD


2/22

2

ACKNOWLEDGEMENT:

It gives me immense pleasure in presenting my term paper. I would like to take this

opportunity to express my deepest gratitude to the people, who have contributed their

valuable time for helping me to successfully complete this training.

In recent years, the concept of automatic control system has achieved a very importantposition in advancement of modern science. Classical approaches to modern approach for

Automatic control systems have played a very important role in advancement and

improvement of engineering skills.

With great pleasure and acknowledgement I extend my deep gratitude to Dr. Sanjeev

Kumar Raghuwanshi for providing me the in-depth knowledge about the subject on Control

systems.

It is my profound to express my deep sense of gratitude towards Mr. Santosh Kumar for his

precious guidance, constructive encouragement and support.

I would also like to thank my college who has directly or indirectly helped me for providing

this opportunity to nurture my educational skills.

The facilities provided by the library section in data collection have played a pivotal role in

completion of the project. It is my obligation to acknowledge this help. My thanks are also

due to the computer section and other support provided by the administrative section.

At last, by concluding I would like to acknowledge that it would not have been possible for

me to complete the paper without the above mentioned cooperative assistance.

Madhuri Suthar


3/22

3

CONTENTS

INTRODUCTION -4

STATE SPACE MODEL -5-10

CONTROLLABILITY AND OBSERVABILITY -11,12

METHODS OF STATE SPACE EQUATION FROM TRANSFER FUNCTION OF A

CONTROL SYSTEM -13,15

STATE SPACE REPRESENTATION FROM TRANSFER FUNCTION OF A

ELECTRICAL NETWORK -16,17


FEEDBACK CONTROL SYSTEM -18,19


FEEDBACK CONTROL SYSTEM -20

NON-LINEAR SYSTEMS -21

REFERENCES -22


4/22

4

INTRODUCTION

Control theory is an interdisciplinary branch of engineering and mathematics that deals with

the behavior of dynamical systems with inputs. The external input of a system is called

thereference. When one or more output variables of a system need to follow a certainreference over time, acontrollermanipulates the inputs to a system to obtain the desired

effect on the output of the system.

The usual objective of a control theory is to calculate solutions for the proper corrective

action from the controller that result in system stability, that is, the system will hold the set

point and not oscillate around it.

The inputs and outputs of a continuous control system are generally related by differential

equations. If these are linear with constant coefficients, a transfer function relating the input

and output can be obtained by taking their Laplace transform.

If the differential equations are nonlinear and have a known solution, it may be possible tolinearism the nonlinear differential equations at that solution. If the resulting linear

differential equations have constant coefficients one can take their Laplace transform to

obtain a transfer function.

Thetransfer functionis also known as the system function or network function. The transfer

function is a mathematical representation, in terms of spatial or temporal frequency, of the

relation between the input and output of a linear time-invariant solution of the nonlinear

differential equations describing the system.

Why control?

Control is a key enabling technology underpinning:

enhance product quality

waste minimization

environmental protection

greater throughput for a given installed capacity

greater yield

deferring costly plant upgrades

higher safety margins

The "control design" process involves.

Plant study and modelling

Determination of sensors and actuators (measured and controlled outputs, controlinputs).

Performance specifications

Control design (many methods)

Simulation tests

Implementation, tests and validation.
http://en.wikipedia.org/wiki/Referencehttp://en.wikipedia.org/wiki/Referencehttp://en.wikipedia.org/wiki/Referencehttp://en.wikipedia.org/wiki/Controller_(control_theory)http://en.wikipedia.org/wiki/Controller_(control_theory)http://en.wikipedia.org/wiki/Controller_(control_theory)http://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Controller_(control_theory)http://en.wikipedia.org/wiki/Reference


5/22

5

State Space Representation

The classical control theory and methods (such as root locus) that we have been using are

based on a simple input-output description of the plant, usually expressed as a transfer

function. These methods do not use any knowledge of the interior structure of the plant, and

limit us to single-input single-output (SISO) systems, and as we have seen allows onlylimited control of the closed-loop behavior when feedback control is used.

Modern control theory solves many of the limitations by using a much richer description of

the plant dynamics.

A state space representation is a mathematical model of a physical system as a set of input,

output and state variables related by first-order differential equations. To abstract from the

number of inputs, outputs and states, the variables are expressed as vectors. Additionally, if

the dynamical system is linear and time invariant, the differential and algebraic equations

may be written in matrix form. The state space representation (also known as the "time-

domain approach") provides a convenient and compact way to model and analyze systems

with multiple inputs and outputs.

With inputs and outputs, we would otherwise have to write down Laplacetransformsto encode all the information about a system. Unlike the frequency domain

approach, the use of the state space representation is not limited to systems with linear

components and zero initial conditions. "State space" refers to the space whose axes are the

state variables.

The state of the system can be represented as a vector within that space.

Why state space equations?

Dynamical systems where physical equations can be derived: electrical engineering,

mechanical engineering, aerospace engineering, microsystems, process plants.

include physical parameters: easy to use when parameters are changed fordesign

State variables have physical meaning.

Allow for including non-linearity (state constraints )

Easy to extend to Multi-Input Multi-Output (MIMO) systems

Advanced control design methods are based on state space equations (reliablenumerical optimisation tools).

The internal state variables are the smallest possible subset of system variables that canrepresent the entire state of the system at any given time. The minimum number of state

variables required to represent a given system is usually equal to the order of the system'sdefining differential equation. If the system is represented in transfer function form, the

minimum number of state variables is equal to the order of the transfer function's denominator

after it has been reduced to a proper fraction. It is important to understand that converting a

state space realization to a transfer function form may lose some internal information about

the system, and may provide a description of a system which is stable, when the state-space

realization is unstable at certain points.

In electric circuits, the number of state variables is often, though not always, the same as the

number of energy storage elements in the circuit such as capacitors and inductors. The statevariables defined must be linearly independent; no state variable can be written as a linear

combination of the other state variables or the system will not be able to be solved.
http://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/wiki/Laplace_transform


6/22

6

State

The state of a dynamical system is a minimal set of variablesx1(t),x2(t),x3(t) ......xn(t) such

that the knowledge of these variables at t=t0 (initial condition), together with the knowledge

of inputs u1(t), u2(t), u3(t)...... um(t) for tt0, completely determines the dynamic behavior of

the system for t> t0.

This definition asserts that the dynamic behavior of a state-determined system is completely

characterized by the response of the set ofn variablesxi(t), where the number n is defined

to be the orderof the system.

State-Variables

The variables x1(t), x2(t), x3(t) ...... xn(t) such that the knowledge of these variables

at t = t0 (initial condition), together with the knowledge of inputs u1(t), u2(t), u3(t)......um(t)

for t t0, completely determines the behavior of the system for t t0; are called state-

variables. In other words, the variables that determine the state of a dynamical system, are

called state-variables.

Large classes of engineering, biological, social and economic systems may be represented by

state-determined system models. System models constructed with the pure and ideal (linear)

one-port elements (such as mass, spring and damper elements) are state-determinedsystem

models. For such systems the number of state variables, n, is equal to the number of

independentenergy storage elements in the system. The values of the state variables at anytime tspecify the energy of each energy storage element within the system and therefore the

total system energy and the time derivatives of the state variables determine the rate of

change of the system energy. Furthermore, the values of the system state variables at any

time tprovide sufficient information to determine the values of all other variables in the

system at that time.

There is no unique set of state variables that describe any given system; many different setsof variables may be selected to yield a complete system description. However, for a given

system the order n is unique, and is independent of the particular set of state variables chosen.

State variable descriptions of systems may be formulated in terms of physical and measurable

variables, or in terms of variables that are not directly measurable. It is possible to

mathematically transform one set of state variables to another; the important point is that anyset of state variables must provide a complete description of the system. In this note we

concentrate on a particular set of state variables that are based on energy storage variables in

physical systems.


7/22

7

State space models of continuous-time linear systems.

The state space model of a continuous-time dynamic system can be derived either from the

system model given in the time domain by a differential equation or from its transfer function

representation.

The State Space Model and Differential Equations

Consider a general -order model of a dynamic system represented by an

-orderdifferential equation At this point we assume that all initial conditions for the above differential equation, i.e. are

Equal to zero.In order to derive a systematic procedure that transforms a differential equation of order to astate space form representing a system of first-order differential equations, we first start witha simplified version, namely we study the case when no derivatives with respect to the input

are present Introduce the following change of variables

which after taking derivatives leads to


8/22

8

The state space representation of above equation is then given by

[

] [

] [

] [

]

With the corresponding output equation obtained as [

]

The above two equation define state space form which is known in the literature as thephasevariable canonical form.

In order to extend this technique to the general case, which includes derivatives with respect

to the input, we form an auxiliary differential equation having the form as For which the change of variables is applicable

And then apply the superposition principle. Since is the response of above equation, thenby the superposition property the response is given by

This produce the state space equations in the form already shown above. The output equation

can be obtained by eliminating i.e.


9/22

9

This leads to the output equation

It is interesting to point out that for which is almost alwaysthe case, the outputequation also has an easy-to-remember form given by

Thus, in summary, for a given dynamic system modeled by differential equation, one is able

to write immediately its state space form, just by identifying coefficients And, and using them to form the corresponding entries in matrices.

The most general state-space representation of a linear system with inputs, outputsand state variables is written in the following form:

Where: is called the "state vector", ;is called the "output vector", ; is called the "input (or control) vector", is the "state matrix", , is the "input matrix", ,

is the "output matrix",

,

is the "feedthrough (or feedforward) matrix" (in cases where the system modeldoes not have a direct feedthrough, is the zero matrix), , .

In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can

depend on time); however, in the common LTI case, matrices will be time invariant. The time

variable . can be continuous (e.g. ) or discrete (e.g. ). In the latter case, the timevariable .

is usually used instead of

.


10/22

10

State Equation Based Modeling Procedure

The complete system model for a linear time-invariant system consists of (i) a set ofn state

equations, defined in terms of the matrices A and B, and (ii) a set of output equations that

relate any output variables of interest to the state variables and inputs, and expressed in terms

of the C and D matrices. The task of modeling the system is to derive the elements of the

matrices, and to write the system model in the form:

x = Ax + Buy = Cx + Du

The matrices A and B are properties of the system and are determined by the system structure

and elements. The output equation matrices C and D are determined by the particular choice

of output variables.

The overall modeling procedure developed in this chapter is based on the following steps:

1. Determination of the system order n and selection of a set of state variables from the linear

graph system representation.

2. Generation of a set of state equations and the system A and B matrices using a well defined

methodology. This step is also based on the linear graph system description.

3. Determination of a suitable set of output equations and derivation of the appropriate C and

D matrices.

Hybrid systems allow for time domains that have both continuous and discrete parts.

Depending on the assumptions taken, the state-space model representation can assume the

following forms:

System type State-space model

Continuous time-invariant

Continuous time-variant

y

Explicit discrete time-invariant

Explicit discrete time-variant


11/22

11

Laplace domain of

continuous time-invariant

Z-domain of

discrete time-invariant

Block Diagram Representation of Linear Systems

Described by State EquationsThe matrix-based state equations express the derivatives of the state-variables explicitly in

terms of the states themselves and the inputs. In this form, the state vector is expressed as the

direct result of vector integration. The block diagram representation is shown in Fig. below.

This general block diagram shows the matrix operations from input to output in terms of the

A, B, C, D matrices, but does not show the path of individual variables.

In state-determined systems, the state variables may always be taken as the outputs of

integrator blocks. A system of order n has n integrators in its block diagram. The derivatives

of the state variables are the inputs to the integrator blocks, and each state equation expresses

a derivative as a sum of weighted state variables and inputs. A detailed block diagram

representing a system of order n may be constructed directly from the state and outputequations as follows:

Step 1: Draw n integrator (S1) blocks, and assign a state variable to the output of each

block.

Step 2: At the input to each block (which represents the derivative of its state variable) draw

a summing element.

Step 3: Use the state equations to connect the state variables and inputs to the summing

elements through scaling operator blocks.

Step 4: Expand the output equations and sum the state variables and inputs through a set of

scaling operators to form the components of the output

.


12/22

12

Example: Continuous-time LTI case

Stability and natural response characteristics of a continuous-time LTI system (i.e., linear with

matrices that are constant with respect to time) can be studied from the eigenvalues of the matrix A.

The stability of a time-invariant state-space model can be determined by looking at the

system's transfer function in factored form. It will then look something like this:

The denominator of the transfer function is equal to the characteristic polynomial found by

taking the determinant of, | |.The roots of this polynomial (the eigenvalues) are the system transfer function's poles (i.e.,the singularities where the transfer function's magnitude is unbounded). These poles can be

used to analyze whether the system is asymptotically stable or marginally stable.

An alternative approach to determining stability, which does not involve calculating

eigenvalues, is to analyze the system's Lyapunov stability.

The zeros found in the numerator of can similarly be used to determine whether the system

is minimum phase.

The system may still be inputoutput stable (see BIBO stable) even though it is not

internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if

those singularities in the transfer function are removable).

Controllability and Observability

Controllability and Observability are main issues in the analysis of a system before deciding

the best control strategy to be applied, or whether it is even possible to control or stabilize the

system. Controllability is related to the possibility of forcing the system into a particular state

by using an appropriate control signal. If a state is not controllable, then no signal will ever

be able to control the state. If a state is not controllable, but its dynamics are stable, then the

state is termed Stabilizable. Observability instead is related to the possibility of "observing",

through output measurements, the state of a system. If a state is not observable, the controller

will never be able to determine the behavior of an unobservable state and hence cannot use itto stabilize the system. However, similar to the stabilizability condition above, if a state

cannot be observed it might still be detectable.

From a geometrical point of view, looking at the states of each variable of the system to be

controlled, every "bad" state of these variables must be controllable and observable to ensure

a good behavior in the closed-loop system. That is, if one of the eigenvalues of the system is

not both controllable and observable, this part of the dynamics will remain untouched in the

closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will

be present in the closed-loop system which therefore will be unstable. Unobservable poles are

not present in the transfer function realization of a state-space representation, which is why

sometimes the latter is preferred in dynamical systems analysis.

Solutions to problems of uncontrollable or unobservable system include adding actuators and

sensors


13/22

13

Controllability

State controllability condition implies that it is possibleby admissible inputsto steer the

states from any initial value to any final value within some finite time window. A continuous

time-invariant linear state-space model is controllable if and only if

Rank [ Where rankis the number of linearly independent rows in a matrix.Observability

Observability is a measure for how well internal states of a system can be inferred by

knowledge of its external outputs. The Observability and controllability of a system are

mathematical duals (i.e., as controllability provides that an input is available that brings any

initial state to any desired final state, Observability provides that knowing an output

trajectory provides enough information to predict the initial state of the system).

A continuous time-invariant linear state-space model is observable if and only if

Rank

State Space Representation from Transfer function of a Control System

The "transfer function" of a continuous time-invariant linear state-space model can be derived

in the following way:

First, taking the Laplace transform of Yields. Next, we simplify for X(s) giving

( and thus Substituting for X(s) in the output equation

Giving ()
http://en.wikipedia.org/wiki/Iffhttp://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Iff


14/22

14

Because the transfer function G(s) is defined as the ratio of the output to the input of a

system, we take and substitute the previous expression for Y(s) with respect to U(s), giving

Clearly G(s) must have by dimensionality, and thus has a total of elements. So for

every input there are transfer functions with one for each output. This is why the state-

space representation can easily be the preferred choice for multiple-input, multiple-output

(MIMO) systems.

Canonical realizationsAny given transfer function which is strictly proper can easily be transferred into state-space

by the following approach (this example is for a 4-dimensional, single-input, single-output

system):

Given a transfer function, expand it to reveal all coefficients in both the numerator and

denominator. This should result in the following form:

The coefficients can now be inserted directly into the state-space model by the following

approach:

This state-space realization is called controllable canonical form because the resulting

model is guaranteed to be controllable (i.e., because the control enters a chain of integrators,

it has the ability to move every state).

The transfer function coefficients can also be used to construct another type of canonical

form


15/22

15

This state-space realization is called observable canonical form because the resulting model

is guaranteed to be observable (i.e., because the output exists from a chain of integrators,

every state has an effect on the output).

Proper transfer functions

Transfer functions which are only proper (and not strictly proper) can also be realized quite

easily. The trick here is to separate the transfer function into two parts: a strictly proper part

and a constant. The strictly proper transfer function can then be transformed into a canonical state space

realization using techniques shown above. The state space realization of the constant is

trivially. Together we then get a state space realization with matricesA,B and Cdetermined

by the strictly proper part, and matrixD determined by the constant.

Here is an example to clear things up a bit: which yields the following controllable realization * + *+

Notice how the output also depends directly on the input. This is due to the constant inthe transfer function.


16/22

16

State Space Representation from Transfer function of a Electrical network

The state equations, written in the form of Eq. (16), are a set ofn simultaneous operational

expressions. The common methods of solving linear algebraic equations, for example

Gaussian elimination, Cramers rule, the matrix inverse, elimination and substitution, may be

directly applied to linear operational equations such as Eq. (16).

For low-order single-input single-output systems the transformation to a classical formulationmay be performed in the following steps:

1. Take the Laplace transform of the state equations.

2. Reorganize each state equation so that all terms in the state variables are on the left-hand

side.

3. Treat the state equations as a set of simultaneous algebraic equations and solve for those

state variables required to generate the output variable.

4. Substitute for the state variables in the output equation.

5. Write the output equation in operational form and identify the transfer function.

6. Use the transfer function to write a single differential equation between the output variable

and the system input. This method is illustrated in the following example.

ExampleUse the Laplace transform method to derive a single differential equation for the

capacitor voltage in the series R-L-C electric circuit shown in figure.

Figure : A series RLC circuit.

Solution: The linear graph method of state equation generation selects the capacitor voltage and the inductor current as state variables, and generates the following pair ofstate equations:

*+ The required output equation is:

Step 1: In Laplace transform form the state equations are:


17/22

17

Step 2: Reorganize the state equations:

Step 3: In this case we have two simultaneous operational equations in the state variables and

. The output equation requires only

. If Eq.

Is multiplied by [s +R/L], and Eq. is multiplied by 1/C, and the equations added, is eliminated:

Step 4: The output equation is

Operate on both sides of Eq.

and write in quotient form:

* + Step 5: The transfer function

is:

* + Step 6: The differential equation relating to is:


18/22

18

State Space Representation from Transfer function of a Feedback Control

System

Typical state space model with feedback

A common method for feedback is to multiply the output by a matrix Kand setting this as theinput to the system: Since the values ofKare unrestricted the values can easily be negated for negative feedback.

The presence of a negative sign (the common notation) is merely a notational one and its

absence has no impact on the end results becomes

Solving the output equation for andSubstituting in the state equation results in

The advantage of this is that the Eigen values ofA can be controlled by

setting Kappropriately through Eigen decomposition of This assumes that the closed-loop system is controllable or that the unstable Eigen values

ofA can be made stable through appropriate choice ofK.


19/22

19

Example

For a strictly proper systemD equals zero. Another fairly common situation is when all states

are outputs, i.e.y =x, which yields C=I, the Identity matrix. This would then result in the

simpler equations

This reduces the necessary Eigen decomposition to just .Feedback with set point (reference) input

In addition to feedback, an input,, can be added such that . Becomes

Solving the output equation for and substituting in the state equation results in

One fairly common simplification to this system is removingD, which reduces the equationsto common simplification to this system is removingD, which reduces the equations to


20/22

20


PHYSICAL EXAMPLE

Moving object example

A classical linear system is that of one-dimensional movement of an object. The Newton's

laws of motion for an object moving horizontally on a plane and attached to a wall with a

spring Where Is position; is velocity; is acceleration Is an applied force

Is the viscous friction coefficient

Is the spring constant. Is the mass of the object.The state equation would then become

Where

Represents the position of the object. Is the velocity of the object.

Is the acceleration of the object.the output is the position of the objectThe Controllability test is the

Which has full rank for all and .The Observability test is then

* + * +


21/22

21

This also has full rank. Therefore, this system is both controllable and observable.

Nonlinear systems

The more general form of a state space model can be written as t

wo functions.

( ) ( )The first is the state equation and the latter is the output equation. If the function is alinear combination of states and inputs then the equations can be written in matrix notation

like above. The argument to the functions can be dropped if the system is unforced (i.e.,it has no inputs).

Pendulum example

A classic nonlinear system is a simple unforced pendulum

where

is the angle of the pendulum with respect to the direction of gravity is the mass of the pendulum (pendulum rod's mass is assumed to be zero) is the gravitational acceleration is coefficient of friction at the pivot point

is the radius of the pendulum (to the center of gravity of the mass )

The state equations are then where

is the angle of the pendulum

is the rotational velocity of the pendulum

is the rotational acceleration of the pendulumInstead, the state equation can be written in the general form

( ) The equilibrium/stationary points of a system are when

and so the equilibrium

points of a pendulum are those that satisfy ( ) for integers n.


22/22

22

REFERENCES

Books:

Linear Control Systems :By- B. S. Manke

Control Engineering :By M. N. Bandyopadhyay

Modern control system: By Dr. V. Sakarnarayanan

Websites:

www.wikipedia.com

www.ece.rutgers.edu

http://reference.wolfram.com

http://www.samson.de

Date post:	04-Apr-2018
Category:	Documents
Upload:	madhuri-suthar
View:	218 times
Download:	0 times

Term Paper on State Space Analysis

Documents