State Space Control - Philadelphia University. State Space...References •Advanced Control...

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H O C H S C H U L E B O C H U M

G E R M A N Y

M A Y 1 9 - 2 1 , 2 0 1 5

State Space Control

State Space Description

Transfer functions concentrates on the input-output relationship only. But, it hides the details of the inner workings.

To get a better insight into the system’s behavior, variables ‘states’ are introduced.

State Space Description

State variables describe the complete dynamic behavior of a system

State variables change as a function of time and form a trajectory in dimensional space (referred to state-space)

Block Diagram Example

Properties of States

Memory.

The state summarizes the past.

Dynamics.

The effect of the input is directly connected to the derivative (the change) in the state vector.

Not unique.

The state representation is not unique.

Ordinary Differential Equations

The state of a system is a collection of variables that summarize the past of a system for the purpose of predicting the future

A system can be represented by the differential equation

x –state variable, u – input, y – output

f and h are functions

Linear Systems

where A, B, C and D are constant matrices. Such a system is said to be linear and time-invariant, or LTI for short.

Matrix A is called the dynamics (or system) matrix

Matrix B is called the control (or input-gain) matrix

Matrix C is called the sensor (or output-gain) matrix

Matrix D is called the direct term.

State Space Matrices

The system matrix captures the internal structure of the system and determines many fundamental properties.

The input-gain and output-gain matrices can be modified by adding, modifying or deleting some actuators (to control) or sensors (to measure) from the process.

State-Vector Differential Equation

Example: Spring-Mass with Damping

Example: Circuit

Two Mass Example

System Response using MATLAB

>> k1=1; k2=1; c=0.2; m1=5; m2=2;

>> A=[0 1 0 0;-(k1+k2)/m1 -c/m1 k2/m1 0;0 0 0 1;k2/m2 0 -k2/m2 0];

>> sys=ss(A,B,C,D);

>> B=[0 1/m1 0 0]'; >> C=[1 0 0 0;0 0 1 0]; >> D=0;

>> step(sys);

0

0.5

1

1.5

2

To:

Out(

1)

0 50 100 150 200 250 300 350 400 450 500-1

0

1

2

3T

o:

Out(

2)

Step Response

Time (seconds)

Am

plit

ude

Alternative Problem

Derive the state-space equations using three states 𝑥1 = 𝑦1, 𝑥2 = 𝑦 1, 𝑥3 = 𝑦2 and two outputs y1 and y2

State-Space and Transfer Functions

Direct Canonical Form

State Space and Transfer Functions

Example

Controllability

A System is controllable if a control vector u(t) exists that will transfer the system from any initial state x(t0) to some final state x(t)

Controllability Matrix MATLAB Command M=ctrb(A,B)

If Full Rank Controllable

Observability

A system is observable if the system states x(t) can be exactly determined from the measured output y(t)

Observability Matrix MATLAB Command N=obsv(A,C)

If Full Rank Observable

Controllability Flow

2

1

2

1

2

1

01

)(0

1

10

12

x

xy

tux

x

x

x

1s 1s 1

1 2

u y1x2x

s

x )0(2

s

x )0(1

1 1x2x

1

controllable

uncontrollable

Observability Flow

2

1

2

1

2

1

01

)(1

3

10

02

x

xy

tux

x

x

x

1s 1s 1

1 2

u y1x2x

s

x )0(2

s

x )0(1

1 1x2x

3 observable

unobservable

MIMO Example

>> M=ctrb(A,B) M = 0 0 1 3 3 9 1 3 3 9 7 21 0 1 1 6 11 42

>> r=rank(M) r = 3

Practical Example: Orbiting Satellite

Reference: Mauricio de Oliveira

State-Feedback Control

AND

Regulator Design via Pole Placement

MATLAB Command K=place(A,B,P)

Example

Motor State-Space Model

Motor Control Example

Motor Control Example

Motor Control Example

In general

If desired poles at -2, -3, -4

MATLAB k=place(A,B,[-2; -3 ;-4])

k = [6.0 4.5 2.0]

Full-State Observer

Example

Closed-Loop Control with Observer

AND

Reduced State-Observer

A full-order state observer estimates all state variables

In practice, some states are already measured. Then, we use a reduced-state observer.

Consider the case with three states: x1, x2, and x3

Assume x1 is measured. Then, need to estimate x2 and x3 only

Example

Desired Char. Eq. for the Controller

Desired Char. Eq. Observer

Controller-Observer

Controller-Observer

References

• Advanced Control Engineering (Chapter 8: State Space Methods for Control System Design) by Roland Burns 2001

• Modern Control Engineering (Chapters 9 and 10 Control System Analysis and Design in State Space) by Ogata 5th edition 2010

• Modern Control Engineering (Chapter 10: State Space Design Methods) by Paraskevopoulos 2002

• Feedback Systems: An Introduction to Scientists and Engineers (Chapter 8: System Models) by Astrom and Muray 2009