full state feedbak control: full-state variable feedback control is considered to achive the desired...

FULL STATE FEEDBAK CONTROL:

Full-state variable feedback control is considered to achive the desired pole locations of the closed-loop system.

The first step in the state variable design process requires us to assume that all the states are available for feedback-that is, we have access to the complete state, x(t), for all t. The system input u(t), is given by,

xKu Determining the gain matrix K is the objective of the full-state feedback design procedure. The beauty of the state variable design process is that the problem naturally seperates into a full-state feedback component and an observer design component. These two design procedures can occur independently, and in fact, the seperation principle provides the proof that this approach is optimal. The stability of the closed-loop system is guaranteed if the full-state feedback control law stabilizes the system and the observer is stable.

Dorf and Bishop, Modern Control Systems

The full-state feedback block diagram is illustrated in Figure 1.

System Model

uBxAx

Kontrol Law

-K

xu

Full-state feedback

Figure 1. Full-state feedback block diagram (with no reference input)

uBxAx With the system defined by the state variable model

and the control feedback given by

xKu we find the closed-loop system to be

xKBAxKBxAuBxAx The characteristic equation associated with above equation is

0KBAIdet If all the roots of the characteristic equation lie in the left-half plain, then the closed loop is stable. In other words, for any initial condition x(t0), it follows that


tas0)t(xe)t(x 0tBKA

Given the pair (A,B), we can always determine K to place all the system closed loop poles in the left half-plane if and only if the system is completely controllable-that is, if and only if the controllability matrix PC is full rank (for a SISO system, full rank implies that PC is invertible).

The addition of a reference input can be considered as

)t(rN)t(xK)t(u

System Model

uBxAx

Kontrol Law

-K

x

uN

r(t)

Where r(t) is the reference input. When r(t)=0 for all t>t0, the control design problem is known as the regulator problem. That is, we desire to compute K so that all initial conditions are driven to zero in a desirable fashion (as determined by the design specifications).

When using this state variable feedback, the roots of the characteristic equation are placed where the transient performance meets the desired response.

Example:

Consider the third-order system with the differential equation

uy2dt

dy3

dt

yd5

dt

yd2

2

3

3


uy2dt

dy3

dt

yd5

dt

yd2

2

3

3

We can select the state variables as x1=y, x2=dy/dt, x3=d2y/dt2. (Phase variables)

ux2x3x5dt

ydx

xdt

ydx

xdt

dyx

1233

3

3

32

2

2

21

uBxAu

1

0

0

x

x

x

532

100

010

x

x

x

3

2

1

3

2

1

x001y

1xy and

If the state variable matrix is

321 kkkK

xKu and

then the closed-loop system is

xBKAxKBxAx The state feedback matrix is

321 k5k3k2

100

010

BKA


and the characteristic equation is

0k2k3k5BKAIdet 122

33

clc;clear

syms k1,k2,k3

mtrx=[0,1,0;0,0,1;(-2-k1),(-3-k2),(-5-k3)];

det(mtrx)

If we seek a rapid response with a low overshoot, we choose a desired characteristic equation such that

n2nn

2 2 We choose ζ=0.8 for minimal overshoot and ωn to meet the settling time requirement.

0152.0eov

eov2

2

8.01/8.0

1/

If we want a settling time (with a 2% criterion) equal to 1 second, then

s/rad618.0

44t n

nns

If we choose ωn=6 rad/s, the desired characteristic equation is

8.1721.824.148.4366.9 232

Comparing two characteristic equations yields

8.172k2

1.82k3

4.14k5

1

2

3

Therefore, we require that k1=170.8, k2=79.1 and k3=9.4.

4.91.798.170K

ACKERMANN’S FORMULA:

For a single-input, single output system, Ackermann’s formula is useful for determining the state variable feedback matrix

n21 kkkK where

xKu

n1n

1nq

Given the characteristic equation

The state feedback gain matrix is

AqP100K 1C

where PC is the controllability matrix.

where

IAAAAq n1n1n

1n

Example:

Consider the system

2s

1)s(G

)s(U

)s(Y

and determine the feedback gain to place the closed-loop poles at s=-1±i. Therefore, we require that

>>poly([-1+i -1-1i]) 22)(q 2

and α1= α2=2. With x1=y and x2=dy/dt, the matrix equation for the system G(s) is

u1

0x

00

10x


)t(udt

yd2

2

)t(udt

ydx

xdt

dyx

2

2

2

21

A B

The controllability matrix is

01

10ABBPC

Aq01

1010K

1

Thus we obtain

where

01

10

01

10

1

1P 1C

and

20

22

10

012

00

102

00

10Aq

2

Then we have

2222

2010

20

22

01

1010K

k1 k2

Note that computing the gain matrix K using Ackermann’s formula requires the use of . We see that complete controllability is essential because only then we can guarantee that the controllabilty matrix PC has full rank and hence that exists.

1

CP

1

CP

With Matlab

>>A=[0 1;0 0];

>>B=[0;1];

>>K=acker(A,B,[-1+i,-1-i])

>>K=place(A,B,[-1+i,-1-i])

K =

2 2

Result:


OBSERVER DESIGN:

In the full-state feedback design procedure discussed in the previous section, it was assumed that all the states were available for feedback at all times. This is a good assumption for the control law design process. However, generally speaking, only a subset of the states are readily measurable and available for feedback. Having all the states available for feedback implies that these states are measured with a sensor or sensor combinations. The cost and complexity of the control system increase as the number of required sensor increases. So, even in situations where extra sensors are available, it may not be cost-effective to employ these extra sensors, if indeed, the control system design goals can be accomplished without them. Fortunately, if the system is completely observable with a given set of outputs, then it is possible to determine (or estimate) the states that are not directly measured (or observed).

According to Luenberger, the full-state observer for the system

xCy

uBxAx

is given by

where denotes the estimate of the state x. The matrix L is the observer gain matrix and is to be determined as part of the observer design procedure. The observer is depicted in Figure 2. The observer has two inputs, u and y, and one output, .

x̂

x̂


x̂CyLuBx̂Ax̂

x̂Cyy~ +

-

Observer

u

y~LuBx̂Ax̂

C

x̂

y

Figure 2. Observer.


The goal of the observer is to provide an estimate so that as t ∞. Remember that we do not know x(t0) precisely; therefore we must provide an initial estimate to the observer. Define the observer estimation error as

x̂ xx̂

)t(x̂ 0

)t(x̂)t(x)t(e

The observer design should produce an observer with the property that e(t) 0 as t ∞. One of the main results of systems theory is that if the system is completely observable, we can always find L so that the tracking error is asymptotically stable, as desired.

Taking the time-derivative of the estimation error in the error equation yields

x̂xe and using the system model and the observer, we obtain

x̂CyLBux̂AuBxAe

or

)t(eCLA)t(e

We can guarantee that e(t)0 as t∞ for any initial tracking error e(t0) if the characteristic equation

0CLAIdet has all its roots in the left half-plane. Therefore, the observer design process reduces to finding the matrix L such that the roots of the characteristic equation lie in the left half-plane. This can always be accomplished if the system is completely observable; that is, if the observability matrix, PO, has full rank.

Example:

Consider the second-order system

x01y

u1

0x

41

32x

In this example, we can only directly observe the state y=x1. The observer will provide estimates of the second state, x2.

The observer design begins by checking the system observability to verify that an observer can be constructed to guarantee the stability of the estimation error. From the system model, we find that

01Cand41

32A

The corresponding observability matrix

32

01

AC

CPO

Since det PO=3, the system is completely observable. Suppose that the desired charactersictic equation is given by

2nn

2d 2

We can select ζ=0.8 and ωn=10, resulting in an expected settling time of less than 0.5 second. Computing the actual characteristic equation yields

1L32L46LLCAIdet 2112

where

2

1

L

LL

Equating the coefficients yields the two equations

1001L32L4

166L

21

1

which, when solved, produces

59

22

L

LL

2

1


1x̂y59

22u

1

0x̂

41

32x̂

The observer is thus given by

)t(e460

320)t(e

)t(e04591

03222)t(e

)t(e0159

22

41

32)t(e


)t(eCLA)t(e

)t(e460

320)t(e

0 0.2 0.4 0.6 0.8 1-5

-4

-3

-2

-1

0

1

Time (seconds)

e1

e2

The response of the estimation error to an initial error of

2

1e

is shown in the figure.

clc;clearA=[-20 3;-60 4];B=[0;0];C=[0 0];D=[0];sys=ss(A,B,C,D) %state-space modelx0=[1 -2]; %initial conditionst=[0:0.01:1];u=0*t; %zero input[y,T,x]=lsim(sys,u,t,x0);plot(T,x(:,1),T,x(:,2))xlabel('Time (seconds)')

Matlab Code

Ackermann’s formula can also be employed to place the roots of the observer characteristic equation at the desired locations. Consider the observer gain matrix

n

2

1

L

L

L

L

and the desired observer characteristic equation

n1n1n

1np


The β’s are selected to meet given performance specifications for the observer. The observer gain matrix is then computed via

T1O 10PApL

where PO is the observability matrix

IAAAAp n1n1n

1n

Example:

Consider the second-order system given in previous example. The desired characteristic equation is given

2nn

2 2p where ζ=0.8 and ωn=10, hence, β1=16 and β2=100. Computing p(A) yields

17722

66133

10

01100

41

3216

41

32Ap

2

32

01

AC

CPO

We have the observability matrix

which implies that

3132

01P 1

O

Using Ackermann’s formula yields the observer gain matrix

59

22

1

0

3132

01

17722

6613310PApL T1

O

COMPENSATOR DESIGN: INTEGRATED FULL-STATE FEEDBACK AND OBSERVER

The state variable compensator is constructed by appropriately connecting the full-state feedback conrol law to the observer. The compensator is shown in Figure 3.

Figure 3. State variable compensator employing full-state feedback in series with a full-state observer.

x̂Cyy~ +

-

Observer

u

y~LuBx̂Ax̂

C

x̂

y

System model

uBxAx

Conrol Law

-K

xC

Compensator

Our strategy was to design the state feedback control law as u(t)=-Kx(t), where we assumed that we had access to the complete state x(t). Then we designed an observer to provide an estimate of the state . It seems reasonable that we can employ the state estimate in the feedback control law in place of x(t). In other words, we can consider the feedback law

)t(x̂

)t(x̂K)t(u

The feedback gain matrix K was designed to guarantee stability of the closed-loop system; that is, the roots of the charactersitic equation

0BKAIdet

are in the left-half plane. Under the assumption that the complete state x(t) is available for feedback, the feedback control law (with properly designed gain matrix K) leads to the desired results that x(t)0 as t ∞ for any initial condition x(t0). We need to verify that, when using the feedback control law for observed states, we retain the stability of the closed-loop system.


Consider the observer again

Substituting the feedback law and rearranging terms in the observer yields the compensator system

x̂Ku

yLx̂LCBKAx̂

x̂CyLuBx̂Ax̂

)t(x̂K)t(u

Notice that the system in the equation has the form of a state variable model with input y and output u as illustrated in Figure 3.

Computing the estimation error using the compensator yields

x̂CLLyBux̂ABuAxx̂xe or

eLCAe Dorf and Bishop, Modern Control Systems

This is the same result as we obtained for the estimation error. The estimation error does not depend on the input as seen in the error equation, where input terms cancel.

Recall that the underlying system model is given by

Cxy

BuAxx

Substituting the feedback law

)t(x̂K)t(u into the system model

x̂KBxABuAxx and with

exx̂ we obtain eKBxKBAx

We can write the equations in matrix form

e

x

CLA0

KBKBA

e

x

Recall that our goal is to verify that, with , we retain stability of the closed-loop system and the observer. The characteristic equation associated with the matrix equation

)t(x̂K)t(u

LCAIdetBKAIdet

So if the roots of det[λI-(A-BK)]=0 lie in the left half-plane, and if the roots of det [λI-(A-LC)]=0 lie in the left half-plane, then the overall system is stable. Therefore, employing the strategy of using the state estimates for the feedback is in fact a good strategy.

The fact that the full-state feedback law and the observer can be designed independently is an illustration of the seperation principle.


The design procedure is summarized as follows

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 poles arbitrarily in the complex plane is guaranteed if the system is completely observable.

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

)t(x̂K)t(u


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