Design of state variable FS2017b mk

State Variable Design Using Control Design Software s i tes.google.com/site/ncpdhbkhn 13... State Variable Design Using Control Design Software... State Variable Design Using Control D

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Nguyễn Công Phương

CONTROL SYSTEM DESIGN

The Design

of State Variable Feedback Systems

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I Introduction

II Mathematical Models of Systems

III State Variable Models

IV Feedback Control System Characteristics

V The Performance of Feedback Control Systems

VI The Stability of Linear Feedback Systems

VII The Root Locus Method

VIII.Frequency Response Methods

IX Stability in the Frequency Domain

X The Design of Feedback Control Systems

XI The Design of State Variable Feedback Systems

XII Robust Control Systems

XIII.Digital Control Systems

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The Design

1 Introduction

2 Controllability & Observability

3 Full-State Feedback Control Design

4 Observer Design

5 Integrated Full-State Feedback & Observer

6 Reference Inputs

7 Optimal Control Systems

8 Internal Model Design

9 State Variable Design Using Control Design

Software

s i tes.google.com/site/ncpdhbkhn 3

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Three steps for state variable design:

1 Use a full-state feedback control law.

2 Construct an observer to estimate the states that

are not directly sensed and available as outputs.

3 Connect appropriately the observer to the

full-state feedback control law.

Compensator

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The Design

1 Introduction

2 Controllability & Observability

4 Observer Design

6 Reference Inputs

Software

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Controllability & Observability

(1)

exists an unconstrained control u(t) that can

transfer any initial state x(t 0 ) to any other desired

location x(t) in a finite time, t 0 ≤ t ≤ T.

we can determine whether the system is controllable

by examining the algebraic condition:

A is n×n & B is n×1.

is the number of inputs.

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(2)

• A system is completely controllable if there

exists an unconstrained control u(t) that can

transfer any initial state x(t 0 ) to any other

desired location x(t) in a finite time, t 0 ≤ t ≤ T.

• For a single-input, single-output system, the

controllability matrix P c is:

• If the determinant of P c is nonzero, the system

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(5)

• A system is completely observable if and only if there exists a finite

time T such that the initial state x(0) can be determined from the

observation history y(t) given the control u(t), t 0 ≤ t ≤ T.

• For the SISO system ,C is 1×n & x is n×1,

we define the observability matrix

• This system is completely observable when the determinant of P o is nonzero.

u y

CA

⋮

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The Design

1 Introduction

3 Full-State Feedback Control Design

4 Observer Design

6 Reference Inputs

Software

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Full-State Feedback Control

Determining the gain matrix K is the objective

of the full-state feedback design procedure

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170.8 79.1 9.4

k k k

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Step Response

Time (seconds)

Ex 1

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The Design

1 Introduction

4 Observer Design

6 Reference Inputs

Software

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ˆ = ˆ + u + ( y − ˆ )

ˆ ( ) t = ( ) t − ( ) t

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22 59

L L

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Observer Design (4)

T n

Compensator

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16 100

β β

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The Design

1 Introduction

4 Observer Design

5 Integrated Full-State Feedback & Observer

6 Reference Inputs

Software

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Integrated Full-State Feedback &

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Observer (3)

1 Determine K such that det(λI – (A – BK)) = 0 has roots in

the left half-plane and place the poles appropriately to

meet the control system design specifications The ability

to place the poles arbitrarily in the complex plane is

guaranteed if the system is completely controllable.

2 Determine L such that det(λI – (A – LC)) = 0 has roots in

the left half-plane and place the poles to achieve

acceptable observer performance The ability to place the observer poles arbitrarily in the complex plane is

guaranteed if the system is completely observable.

3 Connect the observer to the full-state feedback law using

λ λ

ˆ

u t = − Kx t

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,

0.825kg

0 0 100 0 1.2621 8.085kg

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CA CA

P

CA

⋮ → det( Po) = ≠ 1 0

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Observer (3)

λ λ

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λ λ

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λ λ

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0.1237 1.2621 16.8 0.1237 1.2621 100 113.05 11.1079 84

11.1079 25

K K

2.2507 7.5622 169.2065 14.0523

K K K K

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Compensator

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Observer (3)

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λ λ λ

λ λ

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64 2546.2 51911 760300

L L L L

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64 2546.2 51911 760300

L L L L

Compensator

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The Design

1 Introduction

4 Observer Design

6 Reference Inputs

Software

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ˆ = ˆ + u + y + r

x ɺ Ax B ɶ L ɶ M

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Reference Inputs (2)

Two methods:

1 Select M and N so that the estimation error e(t) is

independent of the reference input r(t),

2 Select M and N so that the tracking error y(t) – r(t)

is used as an input to the compensator.

+

r

uɶ

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2 Select M and N so that the tracking error y(t) – r(t) is

used as an input to the compensator.

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The Design

1 Introduction

4 Observer Design

6 Reference Inputs

7 Optimal Control Systems

Software

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Systems that are adjusted to provide a minimum performance

index are often called optimal control systems.

0f ( , , )

t

J = ∫ g x u t dt

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Optimal Control Systems (2)

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→ x Px = − x x

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0

f

t T

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The design steps:

1 Determine the matrix P that satisfies Equation

(2), where H is known,

2 Minimize J by determining the minimum of

Equation (1) by adjusting one or more

unspecified system parameters.

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Ex 1

1 2

2 2

1 1

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k J

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Ex 1

2 12

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0 2

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The Design

1 Introduction

4 Observer Design

6 Reference Inputs

8 Internal Model Design

Software

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The Design

1 Introduction

4 Observer Design

6 Reference Inputs

9 State Variable Design Using Control Design

Software

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State Variable Design Using Control Design Software (1)

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State Variable Design Using Control Design Software (2)

Ex 2

1,22

[ 1 j 1 j ]

= − + − −

P

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