Observer-based sliding mode controller design for output tracking

In this section, using reduced-order observer-based sliding mode controller design to simulates three cases, there are output tracking that tracking trajectory yd

( )

t is constant with matched disturbance, output track that tracking trajectory yd

( )

t is constant with mismatching disturbance, and output track that tracking trajectory yd

( )

t is not constant with mismatching disturbance.

In case 1, use the matched disturbance of LTI system, and output tracks control ^y

( )

^{t to}

( )

d t

y . The system and the observer design same with Section 4.1 case 2, hence, it is completed which can successfully estimate x t3

( )

as given in (4-20), and the observer rewrites as

( ) ( ) [ ]

¹

^{( )} _{( )}

2

10 9.2588 3.6821 t

t t

The control goal is to fulfill tracking control for the output

( )

^{t =}

[

^{1 2 0}

] ( )

^t

y x (4-37)

Assume the tracking error e as

( )

t =

( )

t − d

( )

t

e y y (4-38)

whereyd

( )

t is output tracking control trajectory, and design yd

( )

t =¹⁰ is constant. Hence, the system can be reconstructed as

( ) ( )

The first step in the sliding mode controller design is to choose an appropriate sliding function, such that the system trajectory can trace the control goal in the sliding mode. Let the sliding

function be

Because the eigenvalues of A are 0, -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 −4 −6.67}

]

^{, and CB I=} , hence, the design C as

[

^0.4694 1.0692 1.4694 0.6575

]

= − − −

and ^z

( )

^{0 =0}. Figure 4.28 shows the MATLAB simulink connection diagram, that contains system, reduced-order observer, sliding mode controller, and disturbance. Figure 4.29 shows the observer state error, that at 0.9773s convergence to approach zero, and the convergence speed from (16.91, 0s) to (0.0007658, 1s), conform the ^eig

( )

^{F = −10}. Figure 4.30 shows the sliding surface, that at 0.725s into the sliding layer, and then it affect the control input that have disjunctive parts in this time shows in Figure 4.32. Figure 4.31 shows the sliding surface bound into s ≤ε . Figure 4.33 shows the disturbance, that the range between -1.2 and 1.25.

Figure 4.34 shows the output tracking control ^y

( )

^{t to} yd

( )

t , because this is matched disturbance system and yd

( )

t is constant, then the control output can complete track to

( )

¹⁰

d t =

y , that introduced in Section 3.3. Figure 4.35 shows the control output error.

y2

Figure 4.28 The MATLAB slimulink connection diagram

Figure 4.29 The observer state error

Figure 4.30 The sliding surface ^s

( )

^t

Sliding Mode Control

Reduced Order Observer

Figure 4.31 The sliding surface ^s

( )

^{t bound in s ≤ε}

Figure 4.32 The control input ^u

( )

^t

Figure 4.33 The disturbance ^d

( )

^t

t:0.725s u:-3.795

t:0.9733s u:-0.9378 t:0.725s s:0.01

t:0.9733s s:0.005563

Figure 4.34 The output tracking control ^y

( )

^{t to}yd

( )

t

Figure 4.35 The output tracking control error

t:0.725s Y:2.857 t:0.9733s Y:6.934

t:0.725s eY:-7.143 t:0.9733s

eY:-3.066

In Case 2 and Case 3, use the mismatching disturbance of LTI system, and output tracking control ^y

( )

^{t to} yd

( )

t , and then will consider two condition of yd

( )

t =¹⁰ is constant in case 2 and yd

( )

t ⁼cos t

( )

is not constant in case 3. The system and the observer design same with Section 4.2, hence, it is completed that can successfully estimate x t3

( )

as given in (4-32), and the observer rewrites as

( ) ( ) [ ]

¹

^{( )} _{( )}

2

10 -3.1212 2.4084 t

t t

The control goal is to fulfill tracking control for the output

( )

^{t =}

[

^{1 2 0}

] ( )

^t

y x (4-44)

Assume the tracking error e as

( )

t =

( )

t − d

( )

t

e y y (4-45)

Hence, the system (3-32) can be reconstructed as

( ) ( )

The first step in the sliding mode controller design is to choose an appropriate sliding function, such that the system trajectory can trace to control goal in the sliding mode. Let the sliding function be

Because the eigenvalues of A are 0, -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 −2 −3}

]

^,

[

^{−1 − +4 2i − −4 2i}

]

^and

[

^{−2 −6.67 −12}

]

respectively, and CB I= , hence, the design C as

[ ]

[ ]

[ ]

= 0.0046 0.4114 1.0046 0.1479

= 0.3219 0.7977 1.3219 0.4929

= 0.3681 1.9060 1.3681 3.9451

− − − −

In case2, Figure 4.36 to Figure 4.43 are simulation results with initial condition

( )

^{0 =}

[

^{10 7 5}

]

^T

x and ^z

( )

^{0 =0}. Figure 4.36 shows the MATLAB simulink connection diagram that contains system, reduced-order observer, sliding mode controller, and disturbance. Figure 4.37 shows the observer state error, because constant the eigenvalues of F, hence, the state error at 0.8338s convergence to zero, and the convergence speed from (4.181, 0s) to (0.0001898, 1s), conform the ^eig

( )

^{F = −10}. Figure 4.38 shows the sliding surface, that at 0.2092s, 0.2086s, and 0.1964s into the sliding layer respectively, and then it effect the control input that have disjunctive parts in these time shows in Figure 4.40 and the design eigenvalues of C distant from origin in left phase plane, then the input u must be higher gain.

Figure 4.39 shows the sliding surface bound into s ≤ε . Figure 4.41 shows the disturbance, that the range between -1.12 and 1.32. Figure 4.42 shows the output tracking control ^y

( )

^{t to}

( )

d t

y , because the system affected mismatching disturbance and constant yd

( )

t , then the control output can not complete track toyd

( )

t =¹⁰, conform with introduced Section 3.3.

Figure 4.35 shows the control output error, and the design eigenvalues of C distant from origin in left phase plane more approach yd

( )

t =¹⁰.

y2

MAT LAB Fun_Bu_d x3

x2

Figure 4.36 The MATLAB slimulink connection diagram

Figure 4.37 The observer state error

Figure 4.38 The sliding surface ^s

( )

^t

Sliding Mode Controller Reduced Order Observer

System

Figure 4.39 The sliding surface ^s

( )

^{t bound in s ≤ε}

Figure 4.40 The control input ^u

( )

^t

Figure 4.41 The disturbance ^d

( )

^t

t:0.2086s u:-1.375 t:0.1964s u:-16.15

t:0.2092s u:-15.67

t:0.8338s u:-4.097 t:0.2086s

s:-0.01 t:0.1964s s:-0.01

t:0.2092s s:0.01

Figure 4.42 The output tracking control ^y

( )

^{t to} yd

( )

t

Figure 4.43 The output tracking control error

t:0.2086s Y:1.379

t:0.1964s Y:-3.751

t:0.2092s Y:3.015

t:0.8338s Y:0.5354

t:0.2086s eY:-8.621 t:0.1964s eY:-13.75

t:0.8338s eY:-9.465 t:0.2092s eY:-6.985

In case 3, Figure 4.44 to Figure 4.50 are simulation results with initial condition

( )

^{0 =}

[

^{10 7 5}

]

^T

x and ^z

( )

^{0 =0}. Figure 4.44 the observer state error, because constant the eigenvalues of F, hence, the state error at 0.8338s convergence to zero, and the convergence speed from (4.181, 0s) to (0.0001898, 1s), conform the ^eig

( )

^{F = −10}. Figure 4.45 shows the sliding surface, that at 0.1643s, 0.2883s, and 0.5745s into the sliding layer respectively, and then it effect the control input have disjunctive parts in these time shows in Figure 4.47, and the design eigenvalues of C distant from origin in left phase plane, then the input u must be higher gain. Figure 4.46 shows the sliding surface bound into s ≤ε . Figure 4.48 shows the disturbance, that the range between -1 and 1.25. Figure 4.49 shows the output tracking control

( )

^t

y to yd

( )

t , because the system affected mismatching disturbance and not constant

( )

d t

y , then the control output can not complete track toyd

( )

t =cos t

( )

, conform with introduced Section 3.3. Figure 4.50 shows the control output error, and the design eigenvalues of C distant from origin in left phase plane more approach yd

( )

t =cos t

( )

.

Figure 4.44 The observer state error

Figure 4.45 The sliding surface ^s

( )

^t

Figure 4.46 The sliding surface ^s

( )

^{t bound in s ≤ε}

t:0.8338s ex3:0.001

t:1s

ex3:0.0001898 t:0s

ex3:4.181

t:0.2883s s:-0.01

t:0.5745s s:-0.01 t:0.1643s

s:0.01

t:0.2883s

s:-0.01 t:0.5745s s:-0.01 t:0.1643s s:0.01

Figure 4.47 The control input ^u

( )

^t

Figure 4.48 The disturbance ^d

( )

^t

Figure 4.49 The tracking control output ^y

( )

^{t to} yd

( )

t

t:0.2883s u:-3.407 ^t:0.5745s_u:-5.303

t:0.1643s u:-15.77

t:0.8338s u:-3.036

t:0.2883s Y:-3.181

t:0.5745s Y:-7.679 t:0.1643s Y:5.41

t:0.8338s Y:-3.01

Figure 4.50 The tracking control output error

t:0.2883s eY:-4.14

t:0.5745s eY:-8.518 t:0.1643s eY:4.423

t:0.8338s eY:-3.682

Chapter 5

在文檔中 具降階估測器之可變結構控制器設計 (頁 47-61)