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 $${\left|u-u_0\right|}^2$$. Here, u₀ is the unmodified control action.

The solver applies the following constraints to the optimization problem.

$$\begin{array}{l}{q}_x{f}_x+q_x{g}_{x}u+\gamma h_x^{\beta }\ge 0\\ u_{\min }\le u\le u_{\max }\end{array}$$

Here:

• f_x and g_x are functions defined by the plant dynamics $$\dot{x}=f(x)+g(x)u$$.

• h_x is the control barrier function.

• q_x is the partial derivative of the control barrier function over states x.

• γ is the constraint factor.

• β is the constraint power.

• u_min is a lower bound for the control action.

• u_max 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.

• Open Live Script

Enforce Barrier Certificate Constraints for Adaptive Cruise Control

Enforce barrier certificate constraints for adaptive cruise control to maintain velocity and following distance.

• Open Live Script

Enforce Barrier Certificate Constraints for Collision-Free Robots

Enforce barrier certificate constraints for two robots to reach their target positions in collision-free manner.

• Open Live Script

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.

• Open Live Script

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.

$$\dot{x}=f(x)+g(x)u$$

Connect fx to an N_x-by-1 signal, where N_x is equal to the Number of states parameter.

gx — Input function

scalar | vector | matrix

Input function g(x) in the following plant dynamics equation.

$$\dot{x}=f(x)+g(x)u$$

Connect gx to an N_x-by-N_u signal, where N_x is equal to the Number of states parameter and N_u 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.

$$\{x:h(x)\ge 0\}$$

Connect hx to an N_c-by-1 signal, where N_c 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.

$$q(x)=\frac{\partial h}{\partial x}$$

Connect qx to an N_c-by-N_x signal, where N_c is equal to the Number of barrier certificates parameter and N_x 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. Rerunning the optimization without action bounds did not produce a feasible solution. Rerunning the optimization without constraint bounds did not produce a feasible solution. In this case, the control actions output in u* are zero.

Dependencies

To enable this output port, select the Optimization status parameter.

Parameters

Parameters Tab

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'

Block Tab

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 R2022a

R2023b: 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.