OUTLINE
SLIDING‐MODE CONTROL
Variable structure systems
Sliding mode control
Motivating example (Khalil)
M. SAMI FADALI
Equivalent Control
PROFESSOR EBME
VARIABLE STRUCTURE SYSTEMS
2
1
UNIVERSITY OF NEVADA, RENO
VARIABLE STRUCTURE CONTROL
• Dynamic systems of the form
where has discontinuities with respect to some arguments • State‐dependent switching feedback control that intentionally changes the structure of the system.
• Occur in problems in physics, control engineering and mathematics.
• Origins: relay control, bang‐bang control.
• Occur naturally in some physical systems, e.g. for some electric motors and power converters.
• Variable structure control system: composed of independent structures and a switching logic.
3
4
• Overall system behavior is unlike any of its structures.
• For such systems, the control law is naturally discontinuous
EXAMPLE: LINEAR PLANT
Assume a=1 and |k|=12
LINEAR STATE FEEDBACK CONTROL LAW:
PLANT:
x1 (t ) x2 (t )
x2 (t )ax2 (t )u (t )
u ( t )   kx 1 ( t )
1, 2  0.5  j 3.4278
x2(t)
3
Discontinuous argument
2
Closed‐loop eigenvalues
1 
Unstable equilibrium point at the origin.
2
CASE 1: k  12  a 4
1
a  a  4k
2
2
1 
a  a  4k
2
-1
2
-0.8
-0.6
x1(t)
0.2
-0.4 -0.2
0.4
0.6
0.8
1
-1
-2
For the eigenvalues have different
properties for different and parameter values.
-3
CASE 2: k  12   a 2 4
 1  3,  2  4
The green dashed line equation can defined as
Saddle point at the origin.
x 2 ( t)
6
5
-4
x 2 ( t)
4
4
s (x)  x 2 (t )  1 x1 (t )
3
3
2
 x 2 (t )  3 x1 (t )
2
1
x 1 (t )
-1
- 0 .8
-0.6
- 0 .4
-0 . 2
0 .2
0 .4
0 .6
0 .8
1
1
-1
x 1 (t )
-1
- 0 .8
-0. 6
- 0 .4
-0 .2
0 .2
0 .4
0 .6
0 .8
-2
1
-3
-1
-4
s(x)=0
-2
Choose the discontinuous control law
-3
-4
s(x) x1  0
s(x) x1  0
8
 12 x1 (t ),
u (t )  
 12 x1 (t ),
7
Green dashed line (eigenvector of stable eigenvalue):
system trajectories converge to the origin.
VSS THEORY: use this structure with a discontinuous to make the system stable.
OVERALL SYSTEM BEHAVIOR
SLIDING MODES
• Unlike any of its structures.
• s(x)  x2  3 x1  0 switching function is chosen from system
trajectories. • In general, the switching function is chosen using the system trajectories. These are known as sliding modes. • Variable Structure System (VSS) can possess new properties not present in any of the structures used. x2(t)
3
New switching
function
• In the example, we have
2
1
Case 1: Unstable Equilibrium x1(t)
-1
-0.5
Case 2: Saddle Point
0.5
-1
1
x2+3x1=0
x2+c1x1=0
 VSS System: Asymptotically Stable
(0<c1<3)
9
-3
SLIDING MODE CONTROL (SMC)
10
-2
SMC TRAJECTORIES
SMC design involves two steps:
A trajectory starting from a non‐zero initial condition, evolves in two phases:
(i) Selection of stable hyperplane(s) in the state/error space on which motion should be restricted, called the switching function, and
a) Reaching mode, in which it reaches the sliding surface, and
11
12
b) Sliding mode, in which the trajectory on reaching the sliding surface, remains there for all times and thus evolves according to the dynamics specified by the sliding surface.
(ii) Synthesis of a control law which makes the selected sliding surface attractive .
PLANT AND SWITCHED CONTROL
Each matrix entry: continuously differentiable w.r.t. Switched (Corrective) Control: I: DESIGN OF SWITCHING FUNCTION Properties of Switching Functions
14
13
Switching Surface: ‐dimensional manifold in determined by constraints.
Importance of Switching Functions
a) Order of switching function is less than order of plant
Example: 2nd order system  1st order switching function
e2
III.
IV.
e2e
e1(0), (0)
c1
II.
2 c3
e1
I.
c2
16
Upper Limit c3 : depends on the physical properties of the system and the technology used.
Lower Limit c1 : depends on the allowable tracking time
Trade off between PERFORMANCE and ROBUSTNESS
1 15
0 e1  e2
e2  ce2  ksign( s )
c1<c2<c3
b) Sliding mode does not depend on plant dynamics and is determined by parameters of the switching function only (in the example on only)
c) Switching function does not depend on the control law.
Switching Function Design
Constant
e
Linear
e
Nonlinear
e
ADVANTAGES
+Easy to obtain the surface parameters
DISADVANTAGES
‐ May not be appopriate for system dynamics, in general.
‐Magnitude of the control signal increases directly proportional to the tracking error.
‐Fewer design options
e
e2
0.5
0.4
0.3
0.2
e(0), (0)
e
e
0.1
-1
-0.8
-0.6
-0.4
-0.2
0
e1
0
0.2
0.4
0.6
0.8
1
-0.1
-0.2
-0.3
-0.4
-0.5
17
ADVANTAGES
+ Appropriate for global dynamic properties of nonlinear systems
+Numerous design options
DISADVANTAGES
‐Difficult to find non‐linear functions
‐Difficult to obtain the surface parameters
e
e
e
e
e(0), (0)
e
e(0), (0)
e
e(0), (0)
e
e(0), (0)
e
Time‐Varying
e
18
Switching Function Design
II: FIND A CONTROL LAW TO REACH Switching Function Design
AND STAY THEREAFTER
Most common choice is LTI Switching Surfaces
Attractive Surface (Sliding Surface): Trajectories outside the surface converge to it. Once on the surface, trajectories remain on it
u(t)=uc(t)+ueq(t)
2,
1
x0
s=0
s2=0
e
3,
CORRECTIVE CONTROL
(compensate the deviations
from the sliding surface
to reach the sliding surface)
2
s1=0
EQUIVALENT CONTROL
(makes the derivative of the sliding surface
equal zero to stay on the sliding surface)
20
x0
19
e
EXISTENCE OF SLIDING MODE
LYAPUNOV FUNCTION APPROACH
• A system with inputs can have switching functions and sliding surfaces.
• The control law design and existence of sliding mode are surveyed in:
Hung et.al., Variable Structure Control. A Survey, IEEE Tran. Industrial Electronics, 40(1), 1993.
• Direct Switching Approach
• Reaching Law Approach
• Lyapunov Function Approach
• Positive definite Lyapunov function
• Derivative CORRECTIVE CONTROL
SWITCHING SURFACE • Corrective control : Use high speed switching to drive the state trajectory to a specified switching surface.
• Choose the input amplitude sufficiently large negative definite for any : to make Robust w.r.t. modeling errors.
• Attractive surface: trajectories outside the surface converge to it and ones starting on the surface stay on it.
24
• Typical choice
23
: Design surface so that the • Local control system has good dynamic behavior on
(e.g. by pole placement for a linear surface).
22
: 21
• Choose for a negative definite trajectories converge to the surface.
CORRECTIVE CONTROL : SI SYSTEM/SCALAR LINEAR SURFACE
EQUIVALENT CONTROL • Motion tangent to the switching surface
Corrective Control: EQUIVALENT CONTROL : LINEAR SYSTEM/LINEAR SURFACE
26
25
• Equivalent dynamics (on the surface)
POLE PLACEMENT DESIGN
Linear Dynamics
• Motion tangent to the switching surface
Similarity transformation Equivalent Control: Sliding Mode Dynamics (equivalent system): order zero eigenvalues
nonsingular
28
27
Constraint: LEMMA: CONTROLLABILITY
is If is controllable then controllable.
Proof: Controllability is invariant under similarity transformation, 29
Proof: Controllability is invariant under similarity transformation, is EQUIVALENT CONTROL
30
If is controllable then controllable.
LEMMA: CONTROLLABILITY
ON SWITCHING SURFACE
Use the transformed system Assign eigenvalues using pole placement to select and select any nonsingular 32
31
Eigenvalues invariant under similarity transformation
33
CHATTERING
>> k=place(A11,A12,‐5)
k =
1.6000
3.2000
>> Ku=[k*A11+A21,k*A12+A22]/T
Ku =
1.2000 ‐2.6000 1.4000
‐9.6000 ‐3.2000 ‐6.2000
>> eig(A‐B*Ku)
ans =
‐0.0000
‐5.0000
0
34
MATLAB
EXAMPLE
EXAMPLE (SLOTINE)
• In theory, the trajectories slide along the switching function.
• In practice, there is high frequency switching.
Model Uncertainty • Occurs in the vicinity of the switching surface due to non‐ideal switching e.g. delays, hysteresis, etc.
36
35
• Called chattering because of the sound made by old mechanical switches.
SLIDING CONDITION
EXAMPLE (KHALIL)
unknown
• Design a state feedback law that stabilizes the origin.
SLIDING MANIFOLD
38
37
Choose LYAPUNOV FUNCTION
• Constrain the system to the surface (manifold)
• Motion on manifold • Stable for for any .
40
39
• Drive the trajectories to the surface and maintain them on it.
SLIDING MODE CONTROL
COMPARISON PRINCIPLE
Assume
satisfies • Decreases till it reaches its minimum (
• The manifold >0
)
is reached in finite time.
REGION OF ATTRACTION
CONSTANT BOUND
Assume 42
41
• Stays there because (invariant set)
, i.e.
(relay)
Relay leads to a finite region of attraction
(invariant set)
44
43
Invariant set REGION OF ATTRACTION
TRAJECTORIES IN Trajectories approach the invariant set
4
3
2
Inside the invariant set, trajectories approach if 1
0
-5
-3
-1
1
3
5
-1
-2
-4
5
0
5
46
45
-3
SPECIFIC SYSTEM: PENDULUM
SIMULATION DIAGRAM
48
• Simulate the system assuming negligible switching delays.
47
• Let NO SWITCHING DELAYS
SWITCHING DELAYS
0
Chattering: high frequency switching.
-0.5
Occurs in the vicinity of the switching surface due to switching delays.
-1
If the relay switches to positive from negative at and from negative to positive at , the system exhibits chattering behavior.
-1.5
-2
0.5
1
1.5
2
CHATTERING (SWITCH 50
0
49
-2.5
REDUCING CHATTERING
0.2
Add a continuous control component
0
Change the sgn(.) function to sat(.): linear control inside a boundary layer of width -0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
0.4
0.6
0.8
1
52
0.2
51
0
With signum function
0.2
SWITCHING DELAY 0.1
EXAMPLE
0
-0.2
-0.4
-0.6
-0.8
Add a continuous control component
-1
0.2
-1.2
Ultimately bounded but does not converge to the origin.
0
Change the sgn(.) function to sat(.): linear control inside a boundary layer of width -0.2
-0.4
-1.4
-1.6
0
0.2
0.4
0.6
0.8
1
-0.6
-0.8
-1
-1.2
CONCLUSION
-1.6
-0.2
0
0.2
0.4
0.6
0.8
1
54
53
-1.4
REFERENCES
• Sliding Mode Control is
DETERMINISTIC (only bounds of variations are
considered)
NONLINEAR (the corrective term is nonlinear)
ROBUST (once on the sliding surface, the system is robust to BOUNDED PARAMETERS VARIATIONS and
BOUNDED DISTURBANCES)
• Switching function design and control law design determine the performance. • Although chattering is a problem, various methods are available for mitigating or eliminating it.
1. R. A. DeCarlo, S. H. Zak, and G. P. Mathews, “Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial,” Proceedings IEEE, Vol. 76, No.3, March 1988.
2. H. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, 2003.
3. J.‐J. Slotine and W. Li, Applied Nonlinear Control, Prentice‐
Hall, Englewood Cliffs, NJ, 1991.
4. Kar‐Keung Young, P. Kokotovic, and V. Utkin, “A singular perturbation analysis of high‐gain feedback systems,” IEEE Trans. Automat. Contr., Vol. 22, No. 6, pp931‐938, 1977.
56
55
5. S. H. Zak, Systems and Control, Oxford Univ. Press, NY, 2003.