Control Lyapunov Function

R2026b

Modify control actions to satisfy Lyapunov stability constraints and optional barrier certificate constraints, subject to control bounds

Since R2026b

expand all in page

Libraries:
Simulink Control Design / Constraint Control

Description

The Control Lyapunov Function block computes the modified control actions that are closest to specified control actions subject to Lyapunov stability constraints, optional barrier certificate constraints, and control bounds.

The block uses a quadratic programming (QP) solver to find the control action u and the decision variable δ that minimize the cost function $\frac{1}{2} (| u - u_{0} |^{2} + p δ^{2})$ . Here, u₀ is the unmodified control action. p is the relaxation penalty weight and δ is the relaxation variable.

The solver applies the following constraints to the optimization problem.

$\begin{array}{l} L_{f} V (x) + L_{g} V (x) u + c V (x) \leq δ \\ L_{f} h (x) + L_{g} h (x) u + γ h^{β} (x) \geq 0 \\ u_{\min} \leq u \leq u_{\max} \\ δ \geq 0 \end{array}$

Here:

V(x) is the Lyapunov function.
L_fV(x) is the partial derivative of the Lyapunov function V(x) over the plant states multiplied by the state function f(x).
L_gV(x) is the partial derivative of the Lyapunov function V(x) over the plant states multiplied by the input function g(x).
c is the CLF rate.
h(x) is the control barrier function.
L_fh(x) is the partial derivative of the control barrier function h(x) over the plant states multiplied by the state function f(x).
L_gh(x) is the partial derivative of the control barrier function h(x) over the plant states multiplied by the input function g(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.

Additionally,

f(x) and g(x) are functions defined by the plant dynamics $\dot{x} = f (x) + g (x) u$ .
L_fV(x)+L_gV(x)u+cV(x)≤δ is the CLF constraint.
L_fh(x)+L_gh(x)u+γh^β(x)≥0 is the barrier certificate constraint.

The Control Lyapunov Function block requires Optimization Toolbox™ software.

Examples

Enforce Lyapunov Stability for Adaptive Cruise Control

Enforce exponential convergence of adaptive cruise control system using the Control Lyapunov Function block.

Open Live Script

Ports

Input

expand all

u₀ — Control actions
scalar | vector

Unmodified control actions, specified as a scalar or a vector.

If the Number of control inputs parameter is 1, connect u₀ to a scalar signal. Otherwise, connect u₀ to a vector signal with length equal to Number of control inputs.

V — Lyapunov function
scalar

Lyapunov function V(x), specified as a scalar.

V(x) must be positive definite, that is, V(0)=0 and V(x)>0 for all x≠0.

L_fV — Lie derivative of V along a vector field f
scalar

Lie derivative of V(x) along a vector field f(x), denoted as L_fV(x). It is the partial derivative of the Lyapunov function V(x) over the plant states multiplied by the state function f(x).

$L_{f} V (x) = \frac{\partial V (x)}{\partial x} f (x)$

The state function f(x) is defined by the plant dynamics.

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

L_gV — Lie derivative of V along a vector field g
scalar | vector

Lie derivative of V(x) along a vector field g(x), denoted as L_gV(x). It is the partial derivative of the Lyapunov function V(x) over the plant states multiplied by the input function g(x).

$L_{g} V (x) = \frac{\partial V (x)}{\partial x} g (x)$

The input function g(x) is defined by the plant dynamics.

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

L_gV must be nonzero for the control Lyapunov function to be effective.

Connect L_gV to a 1-by-N_u signal, where N_u is equal to the Number of control inputs parameter.

h — Control barrier function
scalar | vector

Control barrier function, defined as the following safety set for plant states.

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

Connect h to an N_c-by-1 signal, where N_c is equal to the Number of barrier certificates parameter.

Dependencies

To enable this input port, specify the Number of barrier certificates parameter as a positive integer.

L_fh — Lie derivative of h along a vector field f
scalar | vector

Lie derivative of h(x) along a vector field f(x), denoted as L_fh(x). It is the partial derivative of the control barrier function h(x) over the plant states multiplied by the state function f(x).

$L_{f} h (x) = \frac{\partial h (x)}{\partial x} f (x)$

The state function f(x) is defined by the plant dynamics.

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

Connect L_fh to an N_c-by-1 signal, where N_c is equal to the Number of barrier certificates parameter.

Dependencies

To enable this input port, specify the Number of barrier certificates parameter as a positive integer.

L_gh — Lie derivative of h along a vector field g
scalar | vector | matrix

Lie derivative of h(x) along a vector field g(x), denoted as L_gh(x). It is the partial derivative of the control barrier function h(x) over the plant states multiplied by the input function g(x).

$L_{g} h (x) = \frac{\partial h (x)}{\partial x} g (x)$

The input function g(x) is defined by the plant dynamics.

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

expand all

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 is the same as the solution obtained by using quadprog (Optimization Toolbox) with its Algorithm option set to "active-set".

If the Number of control inputs parameter is 1, u* outputs a scalar signal. Otherwise, u* outputs a vector signal with length equal to Number of control inputs.

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 or unbounded. The control actions output is the same as the solution obtained by using `quadprog` (Optimization Toolbox) with its `Algorithm` option set to `"active-set"`.

Dependencies

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

δ — Relaxation variable
nonnegative scalar

Relaxation variable returned by the QP solver. If δ=0, the solver fully enforces the CLF constraint. If δ>0, the solver relaxes the CLF constraint.

Dependencies

To enable this output port, select the Relaxation variable (δ) parameter.

Parameters

expand all

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Parameters Tab

CLF rate (c) — Exponential decay rate
`0.1` (default) | positive scalar

Specify the exponential decay rate.

When the relaxation variable δ=0, the CLF constraint enforces $\frac{d V}{d t} < = - c V$ , where V is the Lyapunov function and c is the exponential decay rate or the CLF rate.

Programmatic Use

Block Parameter: c

Type: character vector

Default: '0.1'

Relaxation penalty weight (p) — Penalty weight on the square of the relaxation variable δ
`10` (default) | positive scalar

Specify the penalty on δ² in the cost function.

If you specify a small value of p, the QP solver relaxes the CLF constraint. If p is too small, the solver ignores the CLF constraint and the modified control action will be approximately equal to the unmodified control action.

If you specify a large value of p, the QP solver strongly enforces the CLF constraint. If p is too large, the modified control action will be aggressive.

Programmatic Use

Block Parameter: p

Type: character vector

Default: '10'

Number of barrier certificates — Number of barrier certificate constraints
`0` (default) | positive integer

Specify the number of barrier certificate constraints to enforce.

Programmatic Use

Block Parameter: nh

Type: character vector

Default: '0'

CBF gain (γ) — 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 CBF gain (γ) as a finite positive scalar. Otherwise, you can specify CBF gain (γ) 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'

Dependencies

To specify CBF gain (γ), specify the Number of barrier certificates parameter as a positive integer.

CBF exponent (β) — 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 CBF exponent (β) as a positive odd integer. Otherwise, you can specify CBF exponent (β) 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'

Dependencies

To specify CBF exponent (β), specify the Number of barrier certificates parameter as a positive integer.

Number of control inputs — Number of control actions
`1` (default) | positive integer

Specify the number of control actions to apply bounds to and optimize.

Programmatic Use

Block Parameter: nu

Type: character vector

Default: '1'

Use external source for upper bound (umax) — Add upper control bound input port
`off` (default) | `on`

Select this parameter to add the umax input port for external upper control bounds.

Programmatic Use

Block Parameter: external_umax

Type: boolean

Values: 'off'|'on'

Default: 'off'

Use external source for lower bound (umin) — Add lower control bound input port
off (default) | on

Select this parameter to add the umin input port for external lower control bounds.

Programmatic Use

Block Parameter: external_umin

Type: boolean

Values: 'off'|'on'

Default: 'off'

Block Tab

Data type — Data type
`double` (default) | `single` | `<data type expression>`

Specify the output data type.

The Data Type Assistant helps you set data attributes. To use the Data Type Assistant, click .

You can specify the following data types.

Built in — single or double data types.
Expression — Use a MATLAB^® expression that specifies the type.

Relaxation variable (δ) — Add relaxation variable output port
`off` (default) | `on`

Select this parameter to add the δ output port for the value of the relaxation variable.

Programmatic Use

Block Parameter: output_delta

Type: boolean

Values: 'off'|'on'

Default: 'off'

Extended Capabilities

expand all

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2026b

Control Lyapunov Function

Description

Examples

Enforce Lyapunov Stability for Adaptive Cruise Control

Ports

Input

u0 — Control actions scalar | vector

V — Lyapunov function scalar

LfV — Lie derivative of V along a vector field f scalar

LgV — Lie derivative of V along a vector field g scalar | vector

h — Control barrier function scalar | vector

Dependencies

Lfh — Lie derivative of h along a vector field f scalar | vector

Dependencies

Lgh — Lie derivative of h along a vector field g scalar | vector | matrix

Dependencies

umax — Action signal upper bounds scalar | vector

Dependencies

umin — Action signal lower bounds scalar | vector

Dependencies

Output

u* — Modified control action scalar | vector

exitflag — Optimization status 1 | 0 | negative integer

Dependencies

δ — Relaxation variable nonnegative scalar

Dependencies

Parameters

Parameters Tab

CLF rate (c) — Exponential decay rate 0.1 (default) | positive scalar

Programmatic Use

Relaxation penalty weight (p) — Penalty weight on the square of the relaxation variable δ 10 (default) | positive scalar

Programmatic Use

Number of barrier certificates — Number of barrier certificate constraints 0 (default) | positive integer

Programmatic Use

CBF gain (γ) — Constraint factor 10 (default) | positive scalar | vector

Programmatic Use

Dependencies

CBF exponent (β) — Constraint power 1 (default) | positive odd integer | vector

Programmatic Use

Dependencies

Number of control inputs — Number of control actions 1 (default) | positive integer

Programmatic Use

Use external source for upper bound (umax) — Add upper control bound input port off (default) | on

Programmatic Use

Use external source for lower bound (umin) — Add lower control bound input port off (default) | on

Programmatic Use

Block Tab

Data type — Data type double (default) | single | <data type expression>

Programmatic Use

Sample time (seconds) — Optimization sample time 0.1 (default) | positive scalar

Programmatic Use

Maximum iterations — Maximum optimization iterations 200 (default) | positive integer

Programmatic Use

Constraint tolerance — Tolerance for constraint violations 1e-6 (default) | nonnegative scalar

Programmatic Use

Optimization status (exitflag) — Add exit flag output port off (default) | on

Programmatic Use

Relaxation variable (δ) — Add relaxation variable output port off (default) | on

Programmatic Use

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

See Also

u₀ — Control actions
scalar | vector

V — Lyapunov function
scalar

L_fV — Lie derivative of V along a vector field f
scalar

L_gV — Lie derivative of V along a vector field g
scalar | vector

h — Control barrier function
scalar | vector

L_fh — Lie derivative of h along a vector field f
scalar | vector

L_gh — Lie derivative of h along a vector field g
scalar | vector | matrix

u_max — Action signal upper bounds
scalar | vector

u_min — Action signal lower bounds
scalar | vector

u* — Modified control action
scalar | vector

exitflag — Optimization status
`1` | `0` | negative integer

δ — Relaxation variable
nonnegative scalar

CLF rate (c) — Exponential decay rate
`0.1` (default) | positive scalar

Relaxation penalty weight (p) — Penalty weight on the square of the relaxation variable δ
`10` (default) | positive scalar

Number of barrier certificates — Number of barrier certificate constraints
`0` (default) | positive integer

CBF gain (γ) — Constraint factor
`10` (default) | positive scalar | vector

CBF exponent (β) — Constraint power
`1` (default) | positive odd integer | vector

Number of control inputs — Number of control actions
`1` (default) | positive integer

Use external source for upper bound (umax) — Add upper control bound input port
`off` (default) | `on`

Use external source for lower bound (umin) — Add lower control bound input port
off (default) | on

Data type — Data type
`double` (default) | `single` | `<data type expression>`

Sample time (seconds) — Optimization sample time
`0.1` (default) | positive scalar

Maximum iterations — Maximum optimization iterations
`200` (default) | positive integer

Constraint tolerance — Tolerance for constraint violations
`1e-6` (default) | nonnegative scalar

Optimization status (exitflag) — Add exit flag output port
`off` (default) | `on`

Relaxation variable (δ) — Add relaxation variable output port
`off` (default) | `on`

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.