Observer-based sliding mode controller design for matched disturbance

for controller stability, the output tracking with matched disturbance system and tracking trajectory is constant, the output tracking with mismatching disturbance and tracking trajectory is constant, and the output tracking with mismatching disturbance system and tracking trajectory is not constant.

4.1 Observer-based sliding mode controller design for matched disturbance

In this section, using reduced-order observer-based sliding mode controller design to simulates two case, there are matched disturbance system that dimension of measurable equal dimension of disturbance, and matched disturbance system that dimension of measurable greater than dimension of disturbance.

In Case 1, consider a LTI system (3-1) suffering from the matched disturbance, the system state be decomposed into two parts

( )

1

( )

2

( )

T T T

t = ⎣⎡ t t ⎤⎦

x x x , where x1

( )

t is measurable and x2

( )

t is not obtainable. Then the system reconstruct as:

( ) ( ) ( )

The matched disturbance is assumed as

( )

0.5 sin 2

( )

cos 0.1sin

( )

1

4

d t = ^⎛⎜⎝ πt + ^⎛⎜⎝πt^⎞⎟⎠^⎞⎟⎠+ t x (4-2)

Apparently, the upper bound of the matched disturbance is obtained as

( ) ( )

⁺¹

d t ≤ x t (4-3)

The system dimension of measurable part and the dimension of disturbance represent as

(

^{11 12}

) ( [

^{1 1}

] ) ^{( )}

^{1 1}

rank rank p rank rank⎛⎡ ⎤1 ⎞ q

= − = = = ⎜⎢ ⎥⎟= =

⎝⎣ ⎦⎠

A A B (4-4)

The first step is to design the reduced order observer, and the observer (2-3) represent as

( )

t =

( )

t + 1

( )

t +

( )

t

And the reduced order observer should be designed under two conditions of (2-9) and (2-10).

Because the dimension p= , hence, the matrix L can be chosen as (2-13) by q

1 2 1⁻ = −1

L = B B (4-7)

where L is not chosen by Γ . By substituting (4-7) into (4-6) and the observer (4-5) rewrite as

( )

t = −2

( )

t +3 1

( )

t

z z x (4-8)

In this case, the eigenvalue of F is stable, then it is completed estimate x as given in (4-8) ₂ successfully. The second step is using this information to design the sliding mode controller, Let the sliding function be

( )

^{t = ˆ}

( )

^t

s Cx (4-9)

Because the eigenvalues of A are -0.2679 and -3.7321, then the matrix C can be determined by the transformation matrix method and via the pole-assignement method to appoint the eigenvalue is -4, and CB I= , hence, the design C as

z . Figure 4.1 shows the MATLAB simulink connection diagram, that contains system, reduced-order observer, sliding mode controller, and disturbance. Figure 4.2 shows the observer state error, that at 4.87s convergence to approach of zero, that after into sliding layer,

and the convergence speed from (17, 0s) to (2.3, 1s), conform the ^eig

( )

^{F = −2}. Figure 4.3 shows the sliding surface, that at 3.115s into the sliding layer, and then it effect the control input u that has a disjunctive part that from -0.9769 to 0.08283 at this time shows in Figure 4.5, it possible two reason that the observer does not completely estimated, and sliding surface must bound in sliding layer, hence, must has higher gain instant to conform two possible reasons. Figure 4.4 shows the sliding surface bound into s ≤ε after complete estimate, Figure 4.6 shows the disturbance, that the range between -1 and 1.35. Figure 4.7 and 4.8 illustrate the system state variable all converge to x=0.

Figure 4.1 The MATLAB slimulink connection diagram

Figure 4.2 The observer state error

Figure 4.3 The sliding surface ^s

( )

^t

Disturbanc

Sliding Mode Controller Reduced Order Observer System

y

x_h2

x_d2 x_d1 x2 x1

u M AT LAB Fun_S_u

1 M AT LAB Fun_w_h

x1 x2 fcny

Em bedded M AT LAB Fun_Gx d M AT LAB Fun_Bu_d

x2 M AT LAB Fun_Ax 0.5si n(2pi t)

ex2:0.03348 t:4.87s ex2:0.001 t:1s

ex2:2.3 t:0s

ex2:17

Figure 4.4 The sliding surface ^s

( )

^{t bound in s ≤ε}

Figure 4.5 The control input ^u

( )

^t

Figure 4.6 The disturbance ^d

( )

^t

t:4.87s u:0.7718

t:3.115s u:-0.9769 t:3.151s

u:0.08283 t:3.115s s:0.01

t:4.87s s:-0.007645

Figure 4.7 The system state x1

( )

t

Figure 4.8 The system state x2

( )

t and observer state xˆ2

( )

t

t:3.115s x2:-0.1065

t:4.87s x2:-0.006052 t:3.115s

x1:0.009881 t:4.87s

x1:-0.007645

In case 2, consider a LTI system (3-1) suffering from the matched disturbance, the

The matched disturbance is assumed as

( )

0.5 sin 2

( )

cos 0.1sin

( )

1

4

d t = ^⎛⎜⎝ πt + ^⎛⎜⎝πt^⎞⎟⎠^⎞⎟⎠+ t x (4-13)

Apparently, the upper bound of the matched disturbance is obtained as

( ) ( )

⁺¹

d t ≤ x t (4-14)

The system dimension of measurable part and the dimension of disturbance represent as

( )

where the dimension p> . The first step is to design the reduced order observer, and the q observer (2-3) represent as

( )

t =

( )

t + 1

( )

t +

( )

t

And the reduced order observer should be designed under two conditions of (2-9) and (2-10).

Because the dimension p> , hence, the matrix L can be chosen as (2-12) by q

where L can be chosen by Γ . The matrix L can affect the observer estimate speed, that

determine for eigenvalues of F. By substituting (4-18) into (4-16) and design the eigenvalues of F are -1, -5, and -10 respectively, then the L represent separately as

[ ]

By substituting (4-19) into (4-17), then the observer (4-16) rewrite as

( ) ( ) [ ] ^{( )} _{( )}

Therefore, it is completed which can successfully estimate x as given in (4-20). The second ₃ step is using this information to design the sliding mode controller, let the sliding function be

( )

^{t = ˆ}

( )

^t

s Cx (4-21)

Because the eigenvalues of A are -0.2275±1.3067i and -6.67, then the matrix C can be determined by the transformation matrix method and via the pole-assignment method to appoint the eigenvalues are -1 and -4, andCB I= , hence, the design C as

[ ]

= 0.1323 -0.2659 0.8677

C (4-22)

0.4180 0.1796 -2.5438 1 ,

t t sgn

and ^z

( )

^{0 =5}. Figure 4.9 shows the MATLAB simulink connection diagram, that contains system, reduced-order observer, sliding mode controller, and disturbance. Figure 4.10 shows the observer state error, that at 8.453s, 1.771s and 0.9199s convergence to approach of zero respectively, therefore, Figure 4.11 shows the convergence speed from (4.746, 0s) to (1.744, 1s), (7.042, 0s) to (0.04736, 1s), (9.911, 0s) to (0.0004488, 1s), conform the^eig

( )

^{F = −1,}

( )

⁵

eig F = − , ^eig

( )

^{F = −10} respectively, and the eigenvalues of F distant from origin in

left phase plane, the observer accurate estimate soon. Figure 4.12 shows the sliding surface, that at 0.3244s, 0.191s, and 0.2413s into the sliding layer respectively, and then it lead to the control input have disjunctive parts shows in Figure 4.14 at these time, and the sliding surface guarantee bound into sliding layer after accurate estimate show in Figure 4.13. Figure 4.15 shows the disturbance, that the range between -1 and 1.3. Figure 4.16, 4.17 and 4.18 illustrate the system state variable all converge to x=0.

Figure 4.9 The MATLAB slimulink connection diagram

Figure 4.10 The observer state error

Figure 4.11 The observer state error of convergence speed

y2 M AT LAB Fun_z_d

x1 M AT LAB Fun_Gx d

MAT LAB Fun_Bu_d x3

x2

Sliding Mode Controller Reduced Order Observer

Figure 4.12 The sliding surface ^s

( )

^t

Figure 4.13 The sliding surface ^s

( )

^{t bound in s ≤ε}

Figure 4.14 The control input ^u

( )

^t

t:0.191s s:-0.01

t:0.3244s s:-0.01 t:0.2413s

s:0.01

t:1.771s

u:-0.3943 t:8.453s

u:-0.6072 t:0.9199s

u:-3.152 t:0.191s

u:2.855

t:0.2413s u:-16.39

t:0.3244s u:-1.577 t:0.191s

s:-0.01

t:0.3244s s:-0.01 t:0.2413s s:0.01

Figure 4.15 The disturbance ^d

( )

^t

Figure 4.16 The system state x1

( )

t

Figure 4.17 The system state x2

( )

t

t:1.771s x2:-0.004996

t:8.453s x2:-0.0311 t:0.9199s

x2:-0.006323 t:0.191s x2:-0.4076 t:0.2413s

x2:-0.5392

t:0.3244s x2:-4.26

t:1.771s x1:0.8727

t:8.453s x1:-0.0398 t:0.9199s

x1:3.407 t:0.191s

x1:10.22

t:0.2413s x1:8.285

t:0.3244s x1:9.889

Figure 4.18 The system state x3

( )

t and observer state xˆ3

( )

t

t:1.771s x3:-0.1355

t:8.453s x3:0.004553 t:0.9199s

x3:-0.5194 t:0.191s

x3:1.014

t:0.2413s x3:-0.5299

t:0.3244s x3:0.605

4.2 Observer-based sliding mode controller design for

在文檔中 具降階估測器之可變結構控制器設計 (頁 28-41)