LEC 10 - State feedback controller design PDF

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State space model based dynamic systems and control, State space model based dynamic systems and control, State space model based dynamic systems and control...

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Control Systems Design Lecture 10

UFME5W-20-3

Tool 3 (design) --- State feedback controller design

1 Introduction to state feedback controller Consider a general state space description of linear time invariant systems . x( t)  Ax( t)  Bu( t) ( stateequation)

S( A, B, C, D):

(1.1)

y( t )  Cx( t )  Du (t ) (output eqution)

where input, output, and state variable vector are defined respectively

u(t ) u1( t) ...

um (t )

y( t) y1 (t ) ... . x ( t) x1 ( t) ...

yl ( t)

T

T

(1.2)

xn (t )

T

Dynamic matrix An*n: describes the dynamics of the system and control the trajectory of the state vector x(t). Input matrix Bn*m: shows how each control input effects the state variables of the systems. Output matrix Cl*n: transforms the state vector x(t) into the output vector y(t). Transmission matrix Dl*m: indicates the direct (feedforward) effect of control inputs to output vector y(t). These four matrices are usually known as system matrices, and a system defined by these matrices is denoted by S(A, B, C, D). A general block diagram of such system is shown in Fig. 1. D n

n

i

0i

y(t)

()s ( s ai)  is n i

u(t)



+

B

x(t)

+

C

A Fig. 1 General block diagram of a system S(A, B, C, D) For the closed loop control of the system S(A, B, C, D ) with state feedback is shown in Fig. 2 D n

n

i

0i

()s ( s a)i  nsi i

v(t)

u(t) +

H

+

B

y(t) 

x(t)

A Fs Fig. 2 Closed loop control with state feedback which

v( t )  v1( t )

...

vm ( t )

T

(1.3)

1

+ C

is called reference vector and it should be noticed that there are two gain matrices are introduced, Fs is the m*n state feedback gain matrix to specify the poles of the closed loop system and H : is the m*m input feedforward gain matrix to specify the zeros of the closed loop system. The transfer function matrix between output Y(s) and reference V(s) is given by





Y (s ) 1 G( s)   C  DFs   sI  A  BF s  B  D H V ( s) _ 1      C  DFs   sI  Fs  B  D H     _     adj  sI  Fs         C  DFs  _ B  D H     det sI  Fs      

where _ F s  A  BFs

(1.4)

(1.5)

Therefore the controller design can be divided into the assignment of poles and zeros according to some specifications, which also is called pole and zero assignment approach. The derivation of eqn (1.4) is based on the state feedback control law

u( t)  Fs x( t)  Hv( t)

(1.6)

2 Assign closed loop poles Inspection of eqn (1.4) the poles of the closed loop are determined by the denominator (closed loop characteristic polynomial) _ (s )  det  sI  Fs   

(2.1)

Let the desired poles of the closed loop be assigned as follows n

n

i

i0

 ( s)  ( s  a i )   i s n i

(2.2)

To obtain the state feedback matrix Fs let (2.1) equal to eqn (2.2)

_ n det sI  Fs    i s n i  i 0 

(2.3)

Equal the coefficients associated with si on both side of (2.3) to give a set of n linear equations (equation associated with sn can be removed because of 0 = 1)

 1 (Fs )  1   n (Fs )  n

(2.4)

where i (.) is the linear function of Fs . Theorem: If a system S(A, B, C, D ) given in (1.1) is completely state controllable, then the closed loop poles can be arbitrarily assigned through the state feedback law in (1.6). 3 Assign closed loop zeros

2

The selection of the input feedforward gain H is usually carried out in order to ensure that the actual output y(t) is exactly equal to the reference v(t) once steady state conditions are reached, i.e. H is chosen to counteract the inherent steady state gain of the closed loop system. Therefore it is straightforward, with reference to eqn (1.4), and assuming a step reference v(t), to select _    H  C  DFs   sI  Fs    

1 1 1    _    C  DFs   Fs  B  D B  D      s 0

1

(3.1)

4 Example Consider an n = 2 dimensional system

 0 A   2

1  0 B   C  1 0 D 0  3    1

(4.1)

Design a state feedback controller with desired closed loop poles

(s  4)(s  1) s  5s  4 2

(4.2)

and the steady state error between output y(t) and reference v(t) is zero. 5 PI and PD controllers Traditional PID control schemes also can be applied to MIMO systems, here only introduce two typical implementations PI controller and PD controller shown in Fig. 3 and Fig. 4 respectively.

. x (t

u(t)

v(t) +



H



+

B

+

x(t)

y(t) C

A

Fp

Fig. 3 PI control

v(t) +

n

n

i

0i

(s )  (s a )i  is n i

u(t) B

+

x(t)  A

+

Fp H

Fig. 4 PD control

3

d(.) / dt

y(t) C

It should be noticed that the MIMO PID controller design is much more complicated than the state feedback and output feedback approaches, this shows the restriction of PID controller design to MIMO systems. Exercise 10 E1 Matlab function design Write a program (matlab function with name zgain ) to compute the zeros of closed loop system as formulated in eqn (3.1) E2 Controller design Consider an n = 2 dimensional system

 0 A   2

1  0 B   C  1 0 D 0   3  1

(E 1)

Design a state feedback controller with desired closed loop denominator

s 2  2 ns   2 n

(E.2)

where damping ratio  = 0.5 and undamped natural frequency n = 6, and to guarantee the steady state error between the output y(t) and the step reference v(t) is zero. After the controller designed, draw the whole control system block diagram.

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