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Introduction to State space

Definition of Space

The state of the system at time t0

is the minimum information neededto uniquely specify the system response given the input variable over thetime interval [t0,]

State space representation

The state space equations of thesystem is

x = Ax+ Bu

y = Cx+ Du

wherex IRn ; u IRm ; y IRp

A IRnn B IRnm

C IRpn D IRpm

State space analysis

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Example

State space model of a circuit

Applying KVL to the circuit,

Vin = Ri+ Ldi

dt+

1

C

idt

Vout =1

C

idt

Let x1 =idt

x2 =dx1dt

= x1 = iThen,

x1 = x2x2 =

VinL

Rx2L

x1LC

State space analysis

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Example

Continuation of example

In the standard form,

X =

x1x2

X = 0 1 1LC RL

x1x2 + 0

1LVin

y =

1C

0 x1

x2

+ [0]Vin

x = Ax+ Buy = Cx+ Du

where,

A=

0 1

1LC

RL

B=

01L

C=

1C

0

D=[0]

State space analysis

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Time Domain analysis

Response of a system

Total response = Zero input response + Zero state response

Zero-input state response

u(t) 0x(t0) x(t) =?

Zero-input system response

u(t) 0x(t0) y(t) =?

State space analysis

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Time Domain analysis

Response of a system

Total response = Zero input response + Zero state response

Zero-input state response

u(t) 0x(t0) x(t) =?

Zero-input system response

u(t) 0x(t0) y(t) =?

State space analysis

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Time Domain analysis

Zero-state state response

x(t0) 0

u(t) x(t) =?

Zero-state system response

x(t0) 0

u(t) y(t) =?

State space analysis

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Time Domain analysis

Zero-state state response

x(t0) 0

u(t) x(t) =?

Zero-state system response

x(t0) 0

u(t) y(t) =?

State space analysis

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Time Domain analysis

System response

System response = Zero input system response + Zero state systemresponse

where,Zero input response= eAtX(0)

Zero state response=t

0 eA(t)Bu()d

Impulse response,

h(t)= CeAtBu(t)= k1 (1 Amplitude)

The total response isy(t) = eAtX(0) +

t0eA(t)Bu()d

State space analysis

E l 1

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Example-1

A mechanical system

Md2x

dt2 + B

dx

dt

= f(t)

Let x1 = x ; x2 =dxdt

x1 = x2x2 =

f(t)M

BMx2

x1x

2 =

0 1

0 BM +

01

M u(t)

y =

1 0x1

x2

+ (0)u(t)

Let M=0.5 kg ; B=3 Ns/m

A =

0 10 6

; B =

02

; C =

1 0

; D=(0)

State space analysis

E l 1

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Example-1

Continuation of example..

The solution is

x(t) = eAtx(0) +t

0eA(t)Bu()d

Analysis of system responses

Zero-input state response

x(t) =

1 16

16e

6t

0 e6t

x1(0)x2(0)

x1(t) = x1(0) +x2(0)

6 x2(0)

6e6t

x2(t) = x2(0)e6t

Zero-input system response

y(t) = x1(t) = x1(0) +x2(0)

6 x2(0)

6 e6t

State space analysis

E l 1

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Example-1

Analysis of system responses

Zero-state state responsex(t) =

t0eA(t)Bu()d

Let us consider the impulse input,t

0u()d = 1

x(t) =t

0eA(t)Bd

=t

0

1

1

6

1

6e6(t)

0 e6(t)

02d

=

t0

13 t

13e

6(t)dt

02e6(t)d

x(t) = 1

3

t 1

18

(1 e6t)13 (1 e

6t)

The total output response of the system is

y(t) = x1(0) +x2(0)

6 (1 e6t) + 13 t

118 (1 e

6t)

State space analysis

T f f ti t St t S

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Transfer function to State Space

Example-1

Consider a transfer function, G(s) =5s2+7s+9

s3+8s2+6s+2

Let y(s)u(s) =

y(s)x(s)

x(s)u(s) =

5s2+7s+9s3+8s2+6s+2

Thus,

y(s)

x(s)

= 5s2 + 7s+ 9 (1)

x(s)

u(s)=

1

s3 + 8s2 + 6s+ 2(2)

From eq(2), u(s) = x(s)[s3 + 8s2 + 6s+ 2]

u(t) =d3x

dt3+ 8

d2x

dt2+ 6

dx

dt+ 2x

d3x

dt3= u(t) 8

d2x

dt2 6

dx

dt 2x

State space analysis

T sf f ti t St t S

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Transfer function to State Space

Example-1

Let x1 = x ; x2 = dxdt ; x3 = d2xdt2

x1x2x3

=

0 1 00 0 12 6 8

x1x2x3

+

00

1

u(t)

From eq(1), y(s) = x(s)[5s2 + 7s+ 9]

y(t) = 5d2x

dt2+ 7

dx

dt+ 9x

y(t) = 5x3 + 7x2 + 9x1

y =

9 7 5 x1x2

x3

+ [0]u

State space analysis

Transfer function to State Space

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Transfer function to State Space

Example-2Consider a transfer function, G(s) = 5s

2+7s+9s2+2s+15

Let

x(s)

u(s)

=1

s2

+ 2s+ 15d2x

dt2= 15x 2

dx

dt+ u(t)

Let x1 = x ; x2 =dxdt

x1x2

=

0 115 2

x1x2

+

01

State space analysis

Transfer function to State Space

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Transfer function to State Space

Example-2

y(s)

x(s)= 5s2 + 7s+ 9

y(t) = 5d2x

dt2 + 7dx

dt + 9x

= 5(15x 2dx

dt+ u(t)) + 7

dx

dt+ 9x

= 66x 3dx

dt+ 5u(t)

y(t) =66 3

x1x2

+ 5u(t)

State space analysis

Similarity Transformation

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Similarity Transformation

The state space equations of a system is given as

x = Ax+ Buy = Cx+ Du

Choose any non-singular matrix T such thatz = Tx x = T1z and x = T1z

Substituting in the system equations

z = TAT1z+ TBu

y = CT1z+ Du

It can be written as

z = Azz+ Bzu

y = Czz+ Dzu

where,

Az = TAT1

; Bz = TB ; Cz = CT1

; Dz = DState space analysis

State Space to Transfer function

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State Space to Transfer function

The state space equations of a system is given as

x = Ax+ Bu

y = Cx+ Du

It can be written as

sx(s) = Ax(s) + Bu(s)

y(s) = Cx(s) + Du(s)

Substituting for x(s) in y(s), we get

y(s) = C[sI A]1Bu(s) + Du(s)y(s)

u(s)= C[sI A]1B+ D

State space analysis

State Space to Transfer function

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State Space to Transfer function

State Space to Transfer function is unique

y(s)

u(s)= Cz(sI Az)

1Bz + Dz

= CT1(sI TAT1)1TB+ D

= CT1[TsIT1 TAT1]1TB+ D

= CT1T[sI A]1T1TB+ D

y(s)

u(s)

= C[sI A]1B+ D

State space analysis

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