6 chapter 6 state space modeling

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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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đà State Space Modelin c3 ° 3 We have the following state equation and output equation: po = Ax(t) + Bu(t) y(t) = Cx(t) Where: A414 42 +; địn] D4) A412 422 ++ QA2n b>

A=| | | B=|'` C=[cy co Cyl

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

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đà State Space Modelin

+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

„1 ¬

X2 = ky — by) + u

The output equation is y = x,

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

Frictionless

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State Space Modeling ` The differential equation for the network in Figure É L M,- Tự D “+ Kxi - Kx› =0 dt- dt -Kxi + Mạ— “+ Kx = f(t) d*x, dv, d^ˆx; _ du;

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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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:

a" y(t) d”~*y(t) dy(t)

đo gen T17 nề TE nt

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BK

Write the state equations describing the following system:

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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

d™y(t) d™ *y(t) dy(t)

Ao qẹn + 11-1 Fee FAn-4 dt + agy(t)

d"~1u(t) d~^u(t) du(t)

= 0m TÚI eat “++ Du —a dt + bn_1u{(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

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BK

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c2 State space equation in controllable canonical form : 3

Consider a system described by the differential equation d" y(t) d™~*y(t) dy(t) dọ dtn + a4 a re oO dt + agy(t) d™ u(t d™~-u(t du(t) = bọ dựm + 1 dtm-1 + -+ bm-1 + b,,u(t)

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

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c2 Convert from state equations to transfer functions - 3

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c3 Convert from state equations to transfer functions - 9 Solution: Step |: Find the associated differential equation C(s) | 24 R(s) 98 +95? +2654 24 Cross-multiplying yields (s°+9s° +26s + 24)C(s)= 24R(s)

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Convert from state equations to transfer functions

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Solution: We obtain

Which can be written as

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Convert from state equations to transfer functions

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