Barrier Certificate Enforcement
Modify control actions to satisfy barrier certificate constraints and action bounds
Since R2022a
Libraries:
Simulink Control Design /
Constraint Control
Description
The Barrier Certificate Enforcement block computes the modified control actions that are closest to specified control actions subject to barrier certificate constraints and action bounds.
The block uses a quadratic programming (QP) solver to find the control action u that minimizes the function . Here, u0 is the unmodified control action.
The solver applies the following constraints to the optimization problem.
Here:
fx and gx are functions defined by the plant dynamics .
hx is the control barrier function.
qx is the partial derivative of the control barrier function over states x.
γ is the constraint factor.
β is the constraint power.
umin is a lower bound for the control action.
umax is an upper bound for the control action.
The Barrier Certificate Enforcement block requires Optimization Toolbox™ software.
For more information on barrier certificate enforcement, see Barrier Certificate Enforcement for Control Design.
Examples
Enforce Barrier Certificate Constraints for PID Controllers
Apply barrier certificate constraints to a PID control application using the Barrier Certificate Enforcement block.
Enforce Barrier Certificate Constraints for Adaptive Cruise Control
Enforce barrier certificate constraints for adaptive cruise control to maintain velocity and following distance.
Enforce Barrier Certificate Constraints for Collision-Free Robots
Enforce barrier certificate constraints for two robots to reach their target positions in collision-free manner.
Enforce Barrier Certificate Constraints for Collision-Free Multi-Robot System
Enforce barrier certificate constraints for three robots to reach their target positions in collision-free manner.
Ports
Input
u0 — Control actions
scalar | vector
Unmodified control actions, specified as a scalar or a vector.
If the Number of actions parameter is
1
, connect u0 to a scalar
signal. Otherwise, connect u0 to a vector signal
with length equal to Number of actions.
fx — State function
scalar | vector
State function f(x) in the following plant dynamics equation.
Connect fx to an Nx-by-1 signal, where Nx is equal to the Number of states parameter.
gx — Input function
scalar | vector | matrix
Input function g(x) in the following plant dynamics equation.
Connect gx to an Nx-by-Nu signal, where Nx is equal to the Number of states parameter and Nu is equal to the Number of actions parameter.
hx — Control barrier function
scalar | vector
Control barrier function, defined as the following safety set for plant states.
Connect hx to an Nc-by-1 signal, where Nc is equal to the Number of barrier certificates parameter.
qx — Partial derivative of control barrier function
scalar | vector | matrix
Partial derivative of the control barrier function over plant states.
Connect qx to an Nc-by-Nx signal, where Nc is equal to the Number of barrier certificates parameter and Nx is equal to the Number of states parameter.
umax — Action signal upper bounds
scalar | vector
To specify run-time upper bounds to the action signals, enable this input port. If this port is disabled, the block does not apply any upper bounds to the control actions.
If the Number of actions parameter is
1
, connect umax to a scalar
signal. Otherwise, connect umax to a vector signal
with length equal to Number of actions.
Dependencies
To enable this input port, select the Use external source for upper bound parameter.
umin — Action signal lower bounds
scalar | vector
To specify run-time lower bounds to the action signals, enable this input port. If this port is disabled, the block does not apply any lower bounds to the control actions.
If the Number of actions parameter is
1
, connect umin to a scalar
signal. Otherwise, connect umin to a vector signal
with length equal to Number of actions.
Dependencies
To enable this input port, select the Use external source for lower bound parameter.
Output
u* — Modified control action
scalar | vector
Modified control action returned by the QP solver.
If the solver finds a solution before reaching the maximum number of iterations, u* outputs this optimal solution.
If the solver reaches the maximum number of iterations, optimization stops and u* outputs a suboptimal solution.
If the initial optimization problem is infeasible, the returned control action depends on the whether the block is configured to ignore constraint or action bounds. For more information, see the exitflag parameter.
If the Number of actions parameter is
1
, u* outputs a scalar
signal. Otherwise, u* outputs a vector signal with
length equal to Number of actions.
exitflag — Optimization status
1
| 0
| negative integer
Optimization status of the QP solver. The following table shows the possible status values.
Exit Flag | Description |
---|---|
1 | The solver converged to an optimal solution with all constraints and bounds active. In this case, u* outputs the optimal control actions. |
0 | The solver reached the maximum number of iterations. The control actions output in u* might be suboptimal. |
negative integer | The initial optimization problem was infeasible and one of the following scenarios applies.
In this case, the control actions output in u* are zero. |
Dependencies
To enable this output port, select the Optimization status parameter.
Parameters
Number of states — Number of plant states
1
(default) | positive integer
Specify the number of states in your plant.
Programmatic Use
Block Parameter:
nx |
Type: character vector |
Default:
'1' |
Number of actions — Number of control actions
1
(default) | positive integer
Specify the number of actions to apply bounds to and optimize.
Programmatic Use
Block Parameter:
nu |
Type: character vector |
Default:
'1' |
Number of barrier certificates — Number of barrier certificate constraints
1
(default) | positive integer
Specify the number of barrier certificate constraints to enforce.
Programmatic Use
Block Parameter:
nc |
Type: character vector |
Default:
'1' |
Constraint factor — Constraint factor
10
(default) | positive scalar | vector
Specify the constraint factor γ in the barrier certificate constraint.
If the Number of barrier certificates parameter is
1
, specify Constraint factor as
a finite positive scalar. Otherwise, you can specify Constraint
factor as either a finite positive scalar value or a column
vector of positive scalars with length equal to Number of barrier
certificates.
Programmatic Use
Block Parameter:
gamma |
Type: character vector |
Default:
'10' |
Constraint power — Constraint power
1
(default) | positive odd integer | vector
Specify the constraint power β in the barrier certificate constraint.
If the Number of barrier certificates parameter is
1
, specify Constraint power as a
positive odd integer. Otherwise, you can specify Constraint
power as either a positive odd integer or a column vector of
positive odd integers with length equal to Number of barrier
certificates.
Programmatic Use
Block Parameter:
beta |
Type: character vector |
Default:
'1' |
Use external source for upper bound — Add upper action bound input port
off
(default) | on
Select this parameter to add the umax input port for external upper action bounds.
Programmatic Use
Block Parameter:
external_umax |
Type: character vector |
Values:
'off' |'on' |
Default:
'off' |
Use external source for lower bound — Add lower action bound input port
off (default) | on
Select this parameter to add the umin input port for external lower action bounds.
Programmatic Use
Block Parameter:
external_umin |
Type: character vector |
Values:
'off' |'on' |
Default:
'off' |
Sample time — Optimization sample time
0.1
(default) | positive scalar
Specify the sample time for running the optimization.
Programmatic Use
Block Parameter:
Ts |
Type: character vector |
Default:
'0.1' |
Maximum iterations — Maximum optimization iterations
200
(default) | positive integer
Specify the maximum number of optimization iterations.
Programmatic Use
Block Parameter:
maxiter |
Type: character vector |
Default:
'200' |
Constraint tolerance — Tolerance for constraint violations
1e-6
(default) | nonnegative scalar
Specify a tolerance value for constraint violations.
Programmatic Use
Block Parameter:
tol |
Type: character vector |
Default:
'1e-6' |
Optimization status — Add exit flag output port
off
(default) | on
Select this parameter to add the exitflag output port for the optimization status of the QP solver.
Programmatic Use
Block Parameter:
exitflag |
Type: character vector |
Values:
'off' |'on' |
Default:
'off' |
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
The Barrier Certificate Enforcement block supports code generation for double-precision signals only.
Version History
Introduced in R2022aR2023b: Library location changed
The Barrier Certificate Enforcement block is now in the Simulink Control Design/Constraint Control sublibrary.
In R2023b, when you open a model saved in a previous release, the library links for Barrier Certificate Enforcement blocks update automatically.
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