Pole Placement Observer Design - nptel

Lecture – 22

Pole Placement Observer Design Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Outline z

Philosophy of observer design

z

Full-order observer

z

Reduced (Minimum) order observer

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Philosophy of Observer Design z

In practice all the state variables are not available for feedback. Possible reasons include:

• Non-Availability of sensors • Expensive sensors • Available sensors are not acceptable (due to high noise, high power consumption etc.) z

z

A state observer estimates the state variables based on the measurements of the output over a period of time. The system should be “observable”. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Full-order Observer Design

Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

State Observer Block Diagram Plant :

X = AX + BU y = CX (single output)

Plant

Let the observed state be X . Let the observer dynamics be X = AX + BU + K e y Error :

(

E X − X

)

State observer

Ref: K. Ogata: Modern Control Engineering, 3rd Ed., Prentice Hall, 1999


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Observer Design: Concepts Error Dynamics: E = X − X + BU + K y) = ( AX + BU ) − ( AX e and substitute y = CX Add and Substract AX + AX − AX + BU − BU − K CX = AX − AX e

= ( A − A ) X + A ( X − X ) + ( B − B )U − K e CX + ( A − A − K C ) X + ( B − B )U ∴ E = AE e

Strategy: 1. Make the error dynamics independent of X 2. Eliminate the effect of U from eror dynamics ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Design: Concepts A = A − K eC B = B

z

This leads to:

z

Error dynamics:

z

Observer dynamics

= ( A− K C) E E = AE e

(

)

X = AX + BU + K e y − CX

Residue


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Observer Design: Full Order z

z

Goal: Obtain gain Ke such that the error dynamics are asymptotically stable with sufficient speed of response.

Necessary and sufficient condition for the existence of Ke : The system should be completely observable!

ÃT =AT – CTKeT. Hence the problem here becomes the same as the pole placement problem! ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Comparison with Pole Placement Design Controller Design

z

Dynamics

Observer Design

z

X = ( A − BK ) X z

Objective

Dynamics = ( A− K C) E E = AE e

z

X ( t ) → 0, as t → ∞

Objective

E ( t ) → 0, as t → ∞

z

Notice that T λ ( A − K e C ) = λ ⎡( A − K e C ) ⎤ ⎣ ⎦

(

= λ AT − C T K eT ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

)

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Observer Design as a Dual Problem Consider the dual problem with input v and output y * Z = AT Z + C T v y* = BT Z Pole placement design for this problem with desired observer roots at μ1 "" μn yields

(

)

sI − AT − C T K o = ( s − μ1 )" ( s − μn )

Now equating observer characteristic equation to the RHS of the above equation we get

K e = K oT ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Design: Method – 1 z

For systems of low order (n ≤ 3)

z

Check Observability

z

Define Ke = [k1 k2 k3]T

z

Substitute this gain in the desired characteristic polynomial equation

sI − ( A − K e C ) = ( s − μ 1 )" ( s − μ n )

z

Solve for the gain elements by equating the like powers on both sides ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Design: Method – 2 Step:1

| sI − A | = s n + a 1 s n − 1 + a 2 s n − 2 + " + a n − 1 s + a n fin d a i 's

Step:2

( s − μ 1 ) " ( s − μ n ) = s n + α 1 s n −1 + α 2 s n − 2 + " + α n find α i ' s

Step:3

Follow a similar approach as in pole placement control design (i.e. Bass-Gura approach) to compute the observer gain. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Design: Method – 2

(

K e = WN T

)

⎡ (α n − a n ) ⎤ ⎢ ⎥ − 1 ⎢ ( α n −1 − a n −1 ) ⎥ ⎢ ⎥ # ⎢ ⎥ ⎢ (α 1 − a1 ) ⎥ ⎣ ⎦

N = ⎡⎣ C T ⎡ a n −1 " a1 ⎢ # $ ⎢ W = ⎢ a1 $ " ⎢ " " ⎣ 1 W here

A T C T """ ( A T ) n -1 C T ⎤⎦ 1⎤ 0 ⎥⎥ #⎥ ⎥ 0⎦


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Observer Design: Method – 3 Ackerman’s Formula −1

⎡ C ⎤ ⎡0 ⎤ ⎢ CA ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ # ⎥ ⎢0 ⎥ K e = φ ( A) ⎢ ⎥ ⎢ ⎥ ⎢ # ⎥ ⎢# ⎥ ⎢CAn − 2 ⎥ ⎢ # ⎥ ⎢ n −1 ⎥ ⎢ ⎥ ⎢⎣ CA ⎥⎦ ⎢⎣1 ⎥⎦ φ ( A) = An + α1 An −1 + """ + α n −1 A + α n I ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Example: Observer Design ⎡0 20.6 ⎤ ⎡0 ⎤ ; B = ⎢ ⎥ ; C = [ 0 1] A=⎢ ⎥ 0 ⎦ ⎣1 ⎣1 ⎦ Assume the desired eigen values of the observer μ1 = −1.8 + 2.4 j; μ 2 = −1.8 − 2.4 j

Step : 1 observability n = 2 ⎡1 0 ⎤ ⎡⎣C ; A C ⎤⎦ = ⎢ ⎥ ⎣0 1 ⎦ Step : 2 Characteristic equation T

T

T

rank = 2

s −20.6 sI − A = = s 2 − 20.6 = s 2 + a1 s + a2 = 0 s −1 ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Example: Observer Design a1 = 0;

Step : 3

a2 = −20.6

Desired Characteristic Equation ( s + 1.8 − 2.4 j )( s + 1.8 + 2.4 j ) = s 2 + 3.6 s + 9 = s 2 + α1 s + α 2 = 0

α1 = 3.6; α 2 = 9

Step : 4

Observer gain ⎡α 2 − a2 ⎤ K e = (WN ) ⎢ = ⎥ ⎣ α1 − a1 ⎦ ⎡ 29.6 ⎤ Ke = ⎢ ⎥ ⎣ 3.6 ⎦ T

−1

⎡1 0 ⎤ ⎡9 + 20.6 ⎤ ⎢0 1 ⎥ ⎢ 3.6 − 0 ⎥ ⎣ ⎦⎣ ⎦


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Separation Principle System dynamics

X = AX + BU y = CX

State feedback control based on observed state is U = − K X State equation X = AX − BKX = ( A − BK ) X + BK X − X

(

hence

)

error E (t ) = X − X X = ( A − BK ) X + BKE

observer error equation E = ( A − K e C ) E ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Separation Principle ⎡ X ⎤ ⎡ A − BK Combined equation: ⎢ ⎥ = ⎢ ⎣ E ⎦ ⎣ 0

BK ⎤ ⎡ X ⎤ A − K e C ⎦⎥ ⎢⎣ E ⎥⎦

Characteristic equation for the Observer-State-Feedback system Hence Observer design and sI − A + BK − BK =0 Pole placement are sI − A + K e C 0 independent of each other! sI − A + BK sI − A + K e C = 0 This is known as “Separation Theorem”. Poles due to Poles due to controller Observer ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Closed Loop System Ref: K. Ogata: Modern Control Engineering, 3rd Ed., Prentice Hall, 1999

Fig: Observed State feedback Control System ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Reduced-order Observer Design

Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Reduced Order Observer z

Some of the state variables may be accurately measured .

z

Suppose X is an n - vector and the output y is an m - vector that can be measured .

• We need to estimate only (n-m)

state

variables.

• The reduced-order observer becomes (n-m)th order observer. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Block diagram:

State feedback control with minimum order observer

Ref : K. Ogata: Modern Control Engineering, 3rd Ed., Prentice Hall, 1999


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State Equation for the Reduced order observer Let m = 1,

X = AX + Bu y = CX

⎡ xa ⎤ ⎡ Aaa A ab ⎤ ⎡ xa ⎤ ⎡ Ba ⎤ + ⎢ ⎥u ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎣ X b ⎦ ⎣ Aba Abb ⎦ ⎣ X b ⎦ ⎣ Bb ⎦ ⎡ xa ⎤ y = [1 0] ⎢ ⎥ ⎣ Xb ⎦ xa = scalar , X b = (n − 1) vector ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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State Equation for the Reduced order observer • The equation for the measured portion of the state, xa = Aaa xa + Aab X b + Ba u xa − Aaa xa − Ba u = Aab X b • The equation for the unmeasured portion of the state , X = A x + A X + B u b

ba a

bb

b

b

• Terms Aba xa and Bbu are "known quantities" ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Full order and Reduced order observer comparison • State/output equation for the full order observer : X = AX + Bu y = CX • State/output equation for the reduced order observer: X = A X + A x + B u b

bb

b

ba a

b

xa − A aa xa − Ba u = Aab X b ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Full order and Reduced order observer comparison Full – Order State Observer Reduced Order State observer

X

X b

A

Abb

Bu

Aba xa + Bbu

y

xa − Aaa xa − Ba u

Aab

C K e (n × 1 matrix)

K e [(n − 1) ×1 matrix]

Fig : List of Necessary Substitutions for Writing the Observer Equation for the Reduced Order State Observer. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Equation • Full order Observer equation : X = ( A − K eC ) X + Bu + K e y • Making substitutions from the table, X b = ( Abb − K e Aab ) X b + Aba xa + Bb u + K e ( xa − A aa xa − Ba u ) i.e. X b − K e xa = ( Abb − K e Aab ) X b + ( Aba − K e Aaa ) y + ( Bb − K e Ba )u = ( A − K A )( X − K y ) bb

e

ab

b

e

+ [ ( Abb − K e Aab ) K e + Aba − K e Aaa ] y + ( Bb − K e Ba )u ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Equation • Define X b − K e y = ( X b − K e xa ) η

(

)

X b − K e y = X b − K e xa η • Then η = ( Abb − K e Aab )η +

[ ( Abb − K e Aab ) K e + Aba − K e Aaa ] y + ( Bb − K e Ba )u

This is reduced order observer. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Observer Error Equation We have: X b = Abb X b + ( Aba xa + Bbu ) X b = ( Abb − K e Aab ) X b + ( Aba xa + Bb u ) + K e Aab X b Subtracting: X b − X b = ( Abb X b − K e Aab X b ) − ( Abb − K e Aab ) X b

(

)

= ( Abb − K e Aab ) X b − X b

E

i.e. E = ( Abb − K e Aab ) E

(

)

where E X b − X b = (η −η ) ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Gain Matrix Computation Necessary Condition The error dynamics can be chosen provided the rank of matrix ⎡ Aab ⎤ ⎢ ⎥ A A ⎢ ab bb ⎥ ⎢ . ⎥ is ⎢ ⎥ ⎢ . ⎥ ⎢ A An-2 ⎥ ⎣ ab bb ⎦

( n − 1) . This is

complete observability condition

Characteristic Equation: sI − Abb + K e Aab = ( s − μ1 )( s − μ2 )........( s − μn -1 ) = s n −1 + αˆ1s n − 2 + .......... + αˆ n -2 s + αˆ n-1 = 0 where μ1 , μ2 ,.....μn-1 are desired eigenvalues of error dynamics ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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The Characteristic Equation ⎡αˆ n −1 − aˆn −1 ⎤ ⎢αˆ − aˆ ⎥ ⎢ n−2 n−2 ⎥ ˆ ˆ T ) −1 ⎥ = (WN K e = Qˆ ⎢ . ⎢ ⎥ . ⎢ ⎥ ⎢ αˆ1 − aˆ1 ⎥ ⎣ ⎦ where T Nˆ = ⎡⎣ Aab | ⎡ aˆn − 2 ⎢ aˆ ⎢ n −3 ⎢. Wˆ = ⎢ ⎢. ⎢ aˆ1 ⎢ ⎣⎢1

⎡αˆ n −1 − aˆn −1 ⎤ ⎢αˆ − aˆ ⎥ ⎢ n−2 n−2 ⎥ ⎢ . ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ αˆ1 − aˆ1 ⎥ ⎣ ⎦

T T T n−2 T Abb Aab | ..... | ( Abb ) Aab ⎤⎦ : (n − 1) × (n − 1) matrix.

aˆn −3 ....... aˆ1 1⎤ aˆn − 4 ....... 1 0 ⎥⎥ . . . ⎥ ⎥: . . . ⎥ 1 0 0 ⎥ ⎥ 0 . . . 0 0 ⎦⎥

(n − 1) × (n − 1) matrix .


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The Characteristic Equation • aˆ1 , aˆ2 ,......aˆn-2 are coefficients in the characteristic equation sI − Abb = s n-1 + aˆ1s n -2 + .... + aˆn-2 s + aˆn-1 = 0. ⎡ Aab ⎤ ⎢ ⎥ ⎢ Aab Abb ⎥ ⎢ . ⎥ ⎢ ⎥ • Ackermann's formula : K e = φ ( Abb ) ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ n −3 ⎢ Aab Abb ⎥ ⎢ ⎥ n−2 ⎢⎣ Aab Abb ⎥⎦ where φ ( Abb ) = Abbn-1 + αˆ1 Abbn-2 + ..... + αˆ n − 2 Abb + αˆ n −1 I ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

−1

⎡0 ⎤ ⎢0 ⎥ ⎢ ⎥ ⎢. ⎥ ⎢ ⎥ ⎢. ⎥ ⎢. ⎥ ⎢ ⎥ ⎢ 0⎥ ⎢ ⎥ ⎣1 ⎦

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Separation Principle • The system characteristic equation can be derived as sI - A + BK sI - Abb + K e Aab = 0

Poles due to pole placement

Poles due to reduced order Observer

• Therefore the pole-placement design and the design of the reduced order observer are independent of each other.


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Example Problem : Consider the system X = AX + Bu y = CX where 1 0⎤ ⎡ 0 ⎡0 ⎤ 0 1 ⎥⎥ , B = ⎢⎢0 ⎥⎥ , C = [1 0 0] A = ⎢⎢ 0 ⎢⎣ −6 − 11 − 6 ⎥⎦ ⎢⎣1 ⎥⎦ Assume that the output y can be accurately measured. Design minimum order observer assuming that the desired eigen values are:

μ1 = − 2 + j 2 3 , μ2 = − 2 − j 2 3 ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Example Characteristic equation: sI − Abb + K e Aab = ( s − μ1 )( s − μ2 ) = ( s + 2 − j 2 3)( s + 2 + j 2 3) = s 2 + 4s + 16 = 0 Ackermann's formula: ⎡ Aab ⎤ K e = φ ( Abb ) ⎢ ⎥ A A ⎣ ab bb ⎦

−1

⎡ 0⎤ ⎢1⎥ ⎣ ⎦ where φ ( Abb ) = Abb2 + αˆ1 Abb + αˆ 2 I = Abb2 + 4 Abb + 16 I ⎡ xa ⎤ ⎡ xa ⎤ ⎢ ⎥ X = ⎢ ⎥ = ⎢ x2 ⎥ , A = ⎣ Xb ⎦ ⎢x ⎥ ⎣ 3⎦

⎡ 0 ⎢ 0 ⎢ ⎢⎣ −6

1 0 − 11

0⎤ ⎡0 ⎤ 1 ⎥⎥ , B = ⎢⎢0 ⎥⎥ ⎢⎣1 ⎥⎦ − 6 ⎥⎦


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Example ⎡ 0⎤ Here Aaa = 0 , Aab = [1 0], Aba = ⎢ ⎥ ⎣ −6 ⎦ 1⎤ ⎡ 0 ⎡0 ⎤ Abb = ⎢ , Ba = 0, Bb = ⎢ ⎥ ⎥ ⎣ −11 − 6 ⎦ ⎣1 ⎦ Hence 2 −1 ⎧⎪ ⎡ 0 ⎫ 1⎤ ⎡ 0 1⎤ ⎡1 0 ⎤ ⎪ ⎡1 0 ⎤ ⎡0 ⎤ Ke = ⎨⎢ + 4⎢ + 16 ⎢ ⎬⎢ ⎥ ⎥ ⎥ ⎥ ⎢1 ⎥ 11 6 11 6 0 1 0 1 − − − − ⎦ ⎣ ⎦ ⎣ ⎦ ⎪⎭ ⎣ ⎦ ⎣ ⎦ ⎪⎩ ⎣ ⎡5 − 2 ⎤ ⎡0 ⎤ ⎡ −2 ⎤ = ⎢ =⎢ ⎥ ⎥ ⎢ ⎥ ⎣ 22 17 ⎦ ⎣1 ⎦ ⎣17 ⎦ ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Example Observer equation: η = ( Abb − K e Aab )η + ⎡⎣( Abb − K e Aab ) K e + Aba − K e Aaa ⎤⎦ y Note: η X − K y = X − K x + (B − K B )u b

e

a

(

b

e

1⎤ ⎡ −2 ⎤ ⎡ 0 ⎡ 2 − ⎢ ⎥ [1 0] = ⎢ Abb − K e Aab = ⎢ ⎥ ⎣ −11 − 6 ⎦ ⎣17 ⎦ ⎣ −28 Substituting various values, ⎡η2 ⎤ ⎡ 2 1⎤ ⎡η2 ⎤ ⎡ 13 ⎤ ⎡0 ⎤ y + ⎢ ⎥u ⎢ ⎥+⎢ ⎢ ⎥=⎢ ⎥ ⎥ ⎣1 ⎦ ⎢⎣η3 ⎥⎦ ⎣ −28 − 6 ⎦ ⎢⎣η3 ⎥⎦ ⎣ −52 ⎦ ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

b

e 1

)

1⎤ − 6 ⎥⎦

37

Example : ⎡η2 ⎤ ⎡ x2 ⎤ ⎢η ⎥ = ⎢ x ⎥ − K e y ⎣ 3⎦ ⎣ 3⎦ ⎡ x2 ⎤ ⎡η2 ⎤ ⎢ x ⎥ = ⎢η ⎥ + K e x1 ⎣ 3⎦ ⎣ 3⎦ If the observed state feedback is used, then ⎡ x1 ⎤ u = − KX = − K ⎢⎢ x2 ⎥⎥ ⎢⎣ x3 ⎥⎦ where K is the state feedback matrix. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Comment z

Reduced order observers are computationally efficient.

z

Reduced order observers may converge faster.

z

Sometimes its advisable to use a fullorder observer even if its possible to design a reduced-order observer. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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References z

K. Ogata: Modern Control Engineering, 3rd Ed., Prentice Hall, 1999.

z

B. Friedland: Control System Design, McGraw Hill, 1986.


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