6 chapter 6 state space modeling

Xn| State equations: A set of n simultaneous, first-order differential equations with n variables, where the n variables to be solved are the state variables Output equation: The algeb

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p SYSTEM DYNAMICS & CONTROL

TP.HCM

CHAPTER 6 STATE SPACE MODELING

Dr Vo Tuong Quan HCMUT - 2011

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State Space Modeling

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State: The state of a dynamic system is the smallest set of variables

State Variables: The state variables of a dynamic system are the variables making up the smallest set of variables that determine the state of the dynamic

system

State Vector: If n state variables are needed to completely describe the behavior

of a given system, then these n state variables can be considered the n

components of a vector x

x= [x, xX Xn|

State equations: A set of n simultaneous, first-order differential equations with

n variables, where the n variables to be solved are the state variables

Output equation: The algebraic equation that expresses the output variables of a system as linear combinations of the state variables and the inputs

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Any An2 + Ann by!

Depending on how we chose the state variables, a system can be described by many different state equations

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fy State Space Modeling

Consider the mechanical system shown in Figure We assume that the system is linear The external force u(t) 1s the input to the system, and the displacement y(t) of the mass is the output The displacement y(t) is measured from the equilibrium position in the absence of the external force This system is a single-input, single-output system

Find the block diagram of the mechanical system and state space equation

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+2 From the diagram the system equation 1s ° 3

my+by+ky=u

This system is of second order This means that the system involves two

integrators Define state variables x,(t) and x2(t) as

x(t) = y(t), x2(t) = y(t)

We obtain: X14 = X2

X2 = ky — by) + u

The output equation is y = x,

Vector matrix form can be written as

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Find the state equations for the translational mechanical system shown in Figure

Frictionless

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Now let a2 =— and — > =—

Select x1,X2,X3 and vz as state variable

Add — = = v, and ax 7, = V2 to complete the set of state equations

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The state-space representation of the translational mechanical system 1s

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BK

ca Method for establishing state equations from differential equations

Case#1: the differential equation do not involve the input derivatives

The differential equation describing the system dynamics 1s:

+ Aany(t) = bou(t)

Xn (t) = Xn-1(0)

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BK

Write the state equations describing the following system:

2y(t) + Sy(t) + 6y(t) + 10y(t) = ult)

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Bk State Space Modeling

Case #2: the differential equation involve the input derivatives

Consider a system described by the differential equation

Ao qẹn + 11-1 Fee FAn-4 dt + agy(t)

Define the state variables: The first state is the system output and The i““ state

(i = 2 n) is chosen to be the first derivative of the (i — 1)" state minus a

quantity proportional to the input:

x(t) = y(t)

X2(t) = x1(t) — Byr(t) x3(t) = x2(t) — Bor (t)

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BK

2y(t) + Sy(t) + 6y(t) + 10y(t) = 10u(t) + 20u(t)

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—15

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đà State Space Modelin

c2 State space equation in controllable canonical form : 3

Consider a system described by the differential equation

Or equivalently by the transfer function:

bos™ + bys™ 1 + + Dy_1s + bm

G(s) =

(s) đoS” + a+s~† + - + an_1S + dạn

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đà 2 State space equation in controllable canonical form State Space Modelin : 3

2y(t) + y(t) + Sy(t) + 4y(t) = ù(t) + 3u(t)

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Establish the state equations describing the system below:

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Define the state variables as in the block diagram:

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=> xX3(t) = —x,(t) + r(t) Combining all of them we ha the state equations:

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c2 Convert from state equations to transfer functions - 3

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ca Convert from state equations to transfer functions

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Solution to the state equation: x(t) = Ax(t) + Bu(t) is:

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Example 2: Find the state-space representation in phase-variable form for the transfer function shown in Figure below:

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Select the state variables: Xj = Cc

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Convert from state equations to transfer functions

Example 3: Give the system defined follow equation, find the transfer

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Example 4: Find the state-space equation and output equation for the system defined by

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Example 5: Find the state-space model of the system shown in Figure below:

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Solution: We obtain

Which can be written as

X; (5) 10 Xs) s+5

X3(5) _ 1

U(s)— X3(s) - s

X:(s) _ 1 Xi(s) s + Y(s) = X,(s)

sX,(s) = —5X,(s) + 10X,(s) sX,(s) = —X,(s) + U(s)

sX3(s) = X,(s) — X3(s)

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By taking the inverse Laplace transforms of the previous four equations, we

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