Full State Feedback for State Space Approach

State Space Equations

• Using Cramer’s rule it can be shown that the characteristic equation of the system is :

0]det[ AsI

• Roots (for s) of the resulting polynomial will be the poles of the system.

• These values for s in the above equation are the eigenvalues of [A].

Full-State Feedback

Full-State Feedback

BKAA

Kxu

CL

2

1

x

xx

𝑠2 − 3𝑠 + 2=0

We have examined linear state space models in a little more depth for the SISO case. Many of the ideas will carry over to the MIMO case.

• similarity transformations & equivalent state representations,

• state space model properties: – controllability, reachability, and stabilizability,

– observability, reconstructability, and detectability,

• special (canonical) model formats.

Linear Continuous-Time State Space Models

A continuous-time linear time-invariant state space model takes the form

where x Rn is the state vector, u Rm is the control signal, y Rp is the output, x0 Rn is the state vector at time t = t0 and A, B, C, and D are matrices of appropriate dimensions.

State Space Characteristics

• Controllability

– Can a system be controlled, fully?

• Each state requires control.

• Observability

– Are all states observable ?

• Must be observed to be used as feedback

– Sensors may be needed to measure states

– Models may be constructed to estimate states that cannot be measured (Model based control).

Controllability • You are sitting in your car on an infinite, flat plane and facing

north. The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line. If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north). The lack of steering case would be analogous to when the rank of is 1 (the two distances you drove are on the same line).

• Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of is 2.

Controllable Canonical Form We can then choose, as state variables, xi(t) = vi(t), which lead to the following state space model for the system.

The above model has a special form. Any completely controllable system can be expressed in this way.

Controllability of State Space

제 14강 16

• Controllability

A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t0) to any other desired location x(t) in a finite time t0tT.

u x Ax B

rank[ ]c nP

2 1Controllability Matrix [ ]n

c

P B AB A B A B

nn

nm Controllable

0c P

nn

Example

제 14강 17

• Controllability

Example 1: Controllability of a system

0 1 2

0 1 0 0

0 0 1 0 u

1

y= 1 0 0 0 u

a a a

x x

x

2 1

2

2

2 2 1

0 0 1

[ ] 0 1

1 ( )

n

c a

a a a

P B AB A B A B 1c P

0 1 2

2

2

2

2 2 1

0 1 0 0

0 0 1 , 0

1

0 0

1 ,

( )

a a a

a

a a a

A B

AB A B

Det. Not 0

Rank is full,

Controllable

Ackermann’s Formula & Full State Feedback

Observability of State Space

제 14강 20

• Observability

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).

u

y=

x Ax B

Cx

rank[ ]o nP

1Observability Matrix [ ]n T

o

P C CA CA

Controllable

0o P

1n

n1

nn

Example 2

제 14강 21

Example 2: Observability of a system

0 1 2

0 1 0 0

0 0 1 0 u

1

y= 1 0 0 0 u

a a a

x x

x

1

1 0 0

[ ] 0 1 0

0 0 1

n T

o

P C CA CA 1o P

0 1 2

2

0 1 0

0 0 1 ,

1 0 0

0 1 0 ,

0 0 1

a a a

A

C

CA

CA

Observable

• Observability

Example 3

Example 3: Controllability and Observability of a two-state system

2 0 1u

1 1 1

y= 1 1

x x

x

0c o P P

1 2 1 2, ,

1 2 1 2

1 11 1 , 1 1 ,

1 1

c

T

o

B AB P B AB

C CA P C CA

Not Controllable and Not Observable

• Observability

1 2

1 2 1 2 1

1 2

2 ( )

y x x

x x x x x u u

x x

Exercise

Is the following system completely state controllable and completely

observable?

Matlab Commands • obsv

– obsv (A,C) returns the observability matrix [C; CA; C(A^2) ... C(A^n)]

– If rank is full then it is observable

– rank(obsv(A,C) )

• ctrb

– ctrb(A,B) returns the controllability matrix [B AB (A^2)B ...]

– If rank is full then it is controllable

– rank(ctrb(A,B) )

More Matlab • K = place(A,B,p)

– computes a feedback gain matrix K that achieves the desired closed-loop pole locations p, assuming all the inputs of the plant are control inputs. The length of p must match the row size of A.

• K= acker (A,B,p)

– uses Ackermann's formula to calculate a gain vector k such that the state feedback places the closed-loop poles at the locations p. Limited to single input systems.

Matlab Commands

A=[0 1 0; 0 0 1; -6 -11 -6]

B=[0 0 1]'

C=[20 9 1]

D=[0]

Mo=obsv(A,C)

rank(Mc)

Mc=ctrb(A,B)

rank(Mo)

Mo =

20 9 1

-6 9 3

-18 -39 -9

ans =

3

Mc =

0 0 1

0 1 -6

1 -6 25

ans =

3

Thus, the answer is YES!

A = 0 1 0

0 0 1

-6 -11 -6

B =

0

0

1

C = 20 9 1

D = 0

Test for Controllability

Theorem : Consider the state space model

(i) The set of all controllable states is the range space of the controllability matrix c[A, B], where

(ii) The model is completely controllable if and only if where c[A, B] has full row rank.

Example

Consider the state space model

The controllability matrix is given by

Clearly, rank c[A, B] = 2; thus, the system is completely

controllable.

Example

For

The controllability matrix is given by:

Rank c[A, B] = 1 < 2; thus, the system is not completely controllable.

We see that controllability is a black and white issue: a model either is completely controllable or it is not. Clearly, to know that something is uncontrollable is a valuable piece of information. However, to know that something is controllable really tells us nothing about the degree of controllability, i.e., about the difficulty that might be involved in achieving a certain objective.