State Variables Outline • State variables. - PowerPoint Presentation

Slide1

State VariablesSlide2

Outline

• State variables.

• State-space representation.

• Linear state-space equations.• Nonlinear state-space equations.• Linearization of state-space equations.

2Slide3

Input-output Description

The description is valid for

a) time-varying systems:

ai

,

c

j , explicit functions of time.b) multi-input-multi-output (MIMO) systems: l input-outputdifferential equations, l = # of outputs.c) nonlinear systems: differential equations includenonlinear terms.

3Slide4

State Variables

To solve the differential equation we need

(1) The system input

u(t

)

for the period of interest.

(2) A set of constant initial conditions.• Minimal set of initial conditions: incompleteknowledge of the set prevents complete solutionbut additional initial conditions are not needed toobtain the solution.• Initial conditions provide a summary of theHistory of the system up to the initial time.

4Slide5

Definitions

System State

:

minimal set of numbers {xi(

t

),

i = 1,2,...,n}, needed together with the inputu(t), t ∈ [t0,

tf) to uniquely determine thebehavior of the system in the interval [t0,tf].

n

= order of the system.

State Variables

:

As

t

increases, the state ofthe system evolves and each of thenumbers xi(t) becomes a time variable.State Vector: vector of state variables

5Slide6

Notation

Column vector

bolded

Row vector bolded and transposed xT

.

6Slide7

Definitions

State Space

:

n-dimensional vector space where {

x

i

(t), i = 1,2,...,n} represent the coordinate axes State plane: state space for a 2nd order system

Phase plane: special case where the state variables are proportional to the derivatives of the output.Phase variables: state variables in phase plane. State

trajectories

: Curves in state

space

State portrait

: plot of state trajectories in the plane

(

phase portrait for the phase plane).7Slide8

Example 7.1

• State for equation of motion of a point

mass

m driven by a force f• y

= displacement of the point mass.

2

⇒ system is second order

8Slide9

Example 7.1 State Equations

State

variables

State

vector

2

Phase Variables: 2nd = derivative of the first.Two first order differential equations1. First equation: from definitions of state variables.2. Second equation: from equation of motion.

9Slide10

Solution of State Equations

Solve the 1st order differential equations then substitute in

y

= x1

2 differential equations + algebraic expression are

equivalent to the 2nd order differential equation.

Feedback Control Law 2nd order underdamped system u /m = −3x −

9x1. Solution depends only on initial conditions.2. Obtain phase portrait using MATLAB command lsim,3. Time is an implicit parameter.4. Arrows indicate the direction of increasing time.5. Choice of state variables is not unique.

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Phase Portrait

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State Equations

Set of first order equations governing the state

variables obtained

from the input-output differential equation and the definitions of the state variables.• In general,

n

state equations for a

nth order system.• The form of the state equations depends on the nature of the system (equations are time-varying for time-varying systems, nonlinear for nonlinear systems, etc.)• State equations for linear time-invariant systems can also be obtained from their transfer functions.

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Output Equation

• Algebraic equation expressing the output

in terms of the state variables.

• Multi-output systems: a scalar output equation is needed to define each output.• Substitute from solution of state equation

to obtain output.

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State-space Representation

• Representation for the system

described by

a differential equation in terms of state and output equations.

•

Linear Systems:

More convenient to writestate (output) equations as a single matrix equation14Slide15

Example 7.2

The state-space equations for the system of Ex.

7.1

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General Form for Linear Systems

16Slide17

State Space in MATLAB

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Linear Vs. Nonlinear State-Space

Example 7.3

: The following are examples

of state-space equations for linear systems a) 3rd order 2-input-2-output (MIMO) LTI

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Example 7.3 (b)

2nd order 2-output-1-input (SIMO)

linear time-varying

19

1. Zero direct

D,

constant

B and C.2. Time-varying system: A has entries that are functions of t.Slide20

Example 7.4: Nonlinear System

Obtain a state-space representation

for the

s-D.O.F. robotic manipulator from the equations of motion with output q.

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Solution

order 2

s

(need 2 s initial conditions to solve completely. State Variables

21Slide22

Example 7.5

Write the state-space equations for the

2- D.O.F

. anthropomorphic manipulator.22Slide23

Equations of Motion

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Solution

24Slide25

Nonlinear State-space Equations

f

(.) (

n×1) and

g

(.) (

l ×1) = vectors of functions satisfying mathematical conditions to guarantee the existence anduniqueness of solution.affine linear in the control: often encountered in practice(includes equations of robotic manipulators)

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Linearization of State Equations

• Approximate nonlinear state equations

by linear

state equations for small ranges of the control and state variables.

• The linear equations are based on

the first

order approximation.26

x0 constant, Δx

=

x - x

0 =

perturbation

x

0

.Approximation Error of order Δ2xAcceptable for small perturbations.Slide27

Function of

n

Variables

27Slide28

Nonlinear State-space Equations

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Perturbations

Abt

’ Equilibrium

(x0, u0)

29Slide30

Output Equation

30Slide31

Linearized State-Space

Equations

31Slide32

Jacobians

(drop "

Δ"s)32Slide33

Example 7.6

Motion of nonlinear spring-mass-damper.

y

= displacement f

=

applied force

m = mass of 1 Kgb(y) = nonlinear damper constantk(y) = nonlinear spring force.Find the equilibrium position corresponding

to a force f0 in terms of the spring force,then linearize the equation of motion aboutthis equilibrium.33Slide34

Solution

Equilibrium of the system with a force

f

0 (set all the time derivatives equal to zero and solve for

y

) Equilibrium

is at zero velocity and the position y0.34Slide35

Linearize about the equilibrium

• Entries of state matrix: constants whose

values depend on the equilibrium.

• Originally linear terms do not change withlinearization.

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