Sliding Mode Controller (Reaching Law)
System controller in sliding mode with selectable sliding plane hit rate function.
Description
Block Sliding Mode Controller (Reaching Law) implements the system regulator in the sliding mode with the selection of the function of the speed of hitting the sliding plane for nonlinear systems described by the differential equation . The possibility of selecting the function of speed of hitting the slip plane allows to effectively control the process of movement of trajectory points to the slip plane.
The block can operate in the following modes:
-
control mode
,
-
tracking mode
.
The formulas use the following notations:
-
- coefficient matrix;
-
- sliding (switching) plane;
-
- the function of hit rate on the sliding plane.
This block allows you to set the parameters of the sliding plane and the sliding plane hit rate function. These features provide the possibility to adjust the control law for a particular system for a specific control objective under changing external conditions.
Ports
Input
#
x
—
system states
vector
Details
States of the system. Defined as a vector, the length of the vector should be equal to the number of states.
| Data types |
|
| Complex numbers support |
No |
#
f(x)
—
system dynamics
`vector
Details
Function , describing the dynamics of the system. It is defined as a vector, the length of the vector should be equal to the number of states.
This function can be obtained from the differential equation .
| Data types |
|
| Complex numbers support |
No |
#
g(x)
—
input influences
vector
Details
A function describing the input influences. Defined as a matrix of size , where is the number of states, is the number of input influences.
This function can be obtained from the differential equation .
| Data types |
|
| Complex numbers support |
No |
#
x.ref
—
specified trajectories
vector
Details
The trajectories along which the system moves in tracking mode. They are specified as a vector, the length of the vector should be equal to the number of states.
In control mode, a vector of zero elements should be supplied to this port.
| Data types |
|
| Complex numbers support |
No |
#
x.ref.dot
—
derivatives of given trajectories
vector
Details
The derivatives of the trajectories along which the system moves in tracking mode. They are defined as a vector, the length of the vector should be equal to the number of states.
In the control mode, a vector of zero elements should be supplied to this port.
| Data types |
|
| Complex numbers support |
No |
Output
#
u
—
control actions
`vector'
Details
Computed control actions in the form of a vector whose length is equal to the number of input actions.
| Data types |
|
| Complex numbers support |
No |
#
s(x)
—
coordinates
vector
Details
The coordinates of the system in sliding mode as a vector whose length is equal to the number of input actions.
| Data types |
|
| Complex numbers support |
No |
Parameters
Switching Function
#
Switching Function —
mode of operation
Regulation Mode | Tracking Mode
Details
The operating mode of the unit:
-
Regulation Mode- regulation mode; use this mode if you want the system to reach a steady state at a given point, with all the states of the system equal to zero. -
Tracking Mode- tracking mode; use this mode if you want the system to follow a given trajectory .
| Values |
|
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
Sliding Surface Parameter
# Sliding Coefficients Matrix (C) — coefficient matrix
Details
Coefficient matrix defining a sliding hyperplane. Defined as a matrix by , where is the number of states, is the number of input actions.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
Reaching Law
#
Reaching Law —
function of sliding plane hit rate
Constant Rate | Exponential | Power Rate
Details
Function , which determines how fast the trajectory point hits the slip plane (switching). The following functions are available:
-
Constant Rate- linear function, .This function ensures uniform movement of the trajectory point.
-
Exponential- polynomial function, .This function provides faster tractor point hitting speed than linear function, in case the tractor point deviation from the sliding plane is large.
-
Power Rate- step function, .This function provides a fast trajectory point hit rate when the trajectory point deviation from the slip plane is large, but the hit rate decreases as the trajectory point approaches the slip plane. This allows to reduce high-frequency oscillations of the control action and increase convergence.
The following notations are used in the formulas:
-
- coefficient of reaching -y of the control action, determined by the parameters Reaching Rate (Eta);
-
- boundary layer of -th control action, defined by parameters Select Boundary Layer:;
-
- is the coefficient scaling the control effect of -th variable;
-
- degree exponent, determined by the parameters Power Rate Exponent (Alpha).
| Values |
|
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
Boundary Layer
#
Select Boundary Layer: —
boundary layer
Sign | Relay | Hyperbolic Tangent | Saturation
Details
A function describing the boundary layer. The following functions are available:
-
Sign- function sgn, .When this function is selected, switches between -1 and 1 are made with a gap.
-
Relay- hysteresis, . -
Hyperbolic Tangent- hyperbolic tangent, . -
Saturation- saturation, .This function provides reduction of high-frequency oscillations of the control action by smooth switching between
-1and1inside the boundary layer.
In the formulas is the boundary layer width defined by the parameters Phi.
| Values |
|
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
Reaching Law Parameter
# Reaching Rate (Eta) — achievement rate
Details
A coefficient characterising the rate of approach of a trajectory point to the sliding (switching) plane. It is set as a positive scalar or a vector of positive numbers, the length of the vector should be equal to the number of control actions.
Setting a large value of the coefficient leads to faster reaching of the sliding plane, but also increases the energy consumption required for control realisation.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
# Control Gain (K) — control coefficient
Details
The coefficient of the proportional part of the polynomial function of the speed of hitting the sliding (switching) plane. It is set as a positive scalar or a vector of positive numbers, the length of the vector should be equal to the number of control actions.
This coefficient determines the energy consumption required to realise the control. When it increases, the cost of control implementation increases proportionally to the deviation of the trajectory point from the slip plane.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
# Power Rate Exponent (Alpha) — degree index
Details
Degree exponent in the stepped function of the rate of hitting the sliding (switching) plane. It is specified as a positive scalar from 0 to 1 or a vector of positive numbers from 0 to 1, the length of the vector should be equal to the number of control actions.
This parameter determines the smoothness of the process of approaching the sliding plane. The lower the value of the parameters, the smoother the process, and also, most likely, the lower the high-frequency oscillations of the control action.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
Boundary Layer Parameter
# Phi — boundary layer width
Details
Boundary layer width. It is set as a positive scalar or vector of positive numbers, the length of the vector should be equal to the number of control actions.
The wider the boundary layer, the lower the high-frequency oscillations of the control action, but the higher the static error. On the contrary, the narrower the boundary layer, the smaller the static error and the higher the high-frequency oscillations of the control action.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |