Lyapunov Function

A Lyapunov function, denoted as V(x), must be positive definite around the equilibrium point (typically chosen as the origin, x=0). This means:

• V(0) = 0 (zero at the equilibrium).
• V(x) > 0 for all x ≠ 0 in a region around the origin.

This property ensures the function acts like an abstract measure of "energy" or "distance" from the equilibrium. Common examples include quadratic forms like V(x) = xᵀPx, where P is a positive definite matrix.

The time derivative of the Lyapunov function along the trajectories of the system, denoted V̇(x), must be negative definite (or at least negative semi-definite). For a system ẋ = f(x), this derivative is computed as V̇(x) = (∂V/∂x) f(x).

• V̇(0) = 0.
• V̇(x) < 0 for all x ≠ 0 in a region.

This is the critical property: it guarantees that the "energy" V(x) is strictly decreasing over time, forcing the system state x(t) to converge to the origin.

To prove global asymptotic stability (stability from any initial state), a Lyapunov function must be radially unbounded. This means:

• V(x) → ∞ as ||x|| → ∞.

This property ensures that the sub-level sets of V(x) (the regions where V(x) ≤ c) are bounded for all finite values of c. It prevents trajectories from escaping to infinity while the function value decreases. A quadratic Lyapunov function V(x) = xᵀPx is inherently radially unbounded.

In Model Predictive Control (MPC), a Lyapunov function is often employed as a terminal cost and used to define a terminal constraint set. This is a key technique in the dual-mode or quasi-infinite horizon MPC paradigm.

• Terminal Cost: The function V_f(x) is added to the MPC cost function evaluated at the end of the prediction horizon.
• Terminal Set: A constraint requires the terminal state to lie within a region where V_f(x) is a valid Lyapunov function for the system under a local stabilizing controller.

Together, these enforce that the MPC optimization constructs a feasible, stabilizing sequence of controls.

For linear time-invariant (LTI) systems of the form ẋ = Ax, stability analysis is greatly simplified. A quadratic Lyapunov function V(x) = xᵀPx is sought. Its derivative is V̇(x) = xᵀ(AᵀP + PA)x.

Stability is proven if there exists a positive definite matrix P satisfying the Lyapunov equation: AᵀP + PA = -Q where Q is any positive definite matrix. This equation is directly related to the solution of the Algebraic Riccati Equation (ARE) from Linear Quadratic Regulator (LQR) design, where the optimal cost-to-go is itself a Lyapunov function.

A Control Lyapunov Function (CLF) is an extension for systems with control inputs ẋ = f(x, u). It is a Lyapunov function candidate for which, for every state x ≠ 0, there exists a control input u such that V̇(x, u) < 0.

• Existence vs. Synthesis: A CLF proves the system is stabilizable. Finding the actual control law u(x) is a separate step.
• Integration with MPC: CLF constraints can be added to the MPC optimization problem to guarantee stability, often forming a safety filter or being used in conjunction with Control Barrier Functions (CBFs) for provably safe control.

A Control Barrier Function (CBF) is a mathematical construct used to enforce safety constraints by defining a forward-invariant safe set. It certifies that if a system starts within this safe set, it will remain there for all future time under an appropriate control law. In practice, CBFs are often used as safety filters alongside performance-oriented controllers like MPC.

• Key Mechanism: Derives a constraint on the control input that guarantees the time derivative of the barrier function is non-positive on the boundary of the safe set.
• Relation to Lyapunov: While a Lyapunov function certifies stability (convergence to an equilibrium), a CBF certifies safety (remaining within a set). They can be combined in a CLF-CBF framework for stable and safe control.

Stability is the fundamental property that a system's state will remain bounded or converge to a desired equilibrium under specified conditions.

• Nominal Stability: Guarantees convergence when the system model used for controller design is perfectly accurate, with no disturbances. Lyapunov's direct method is the primary tool for proving nominal stability.
• Robust Stability: Ensures the system remains stable despite model uncertainties, disturbances, or noise. This is a more stringent requirement. In MPC, robust stability is often addressed via techniques like tube-based MPC or constraint tightening, which use the Lyapunov function analysis as a foundation but account for bounded errors.
• Lyapunov's Role: A Lyapunov function decreasing over time is the certificate for nominal stability. For robust stability, one might use a Lyapunov function for the error system or a robust control Lyapunov function (RCLF).

In Model Predictive Control (MPC), the terminal cost and terminal constraint are design elements added to the finite-horizon optimal control problem to guarantee closed-loop stability, using Lyapunov theory.

• Terminal Cost (P): An additional term, often a quadratic form x(t+N)^T P x(t+N), added to the MPC cost function. Matrix P is typically chosen as the solution to the Algebraic Riccati Equation (ARE) for the linearized system, making the terminal cost a Lyapunov function for the system under a local stabilizing controller.
• Terminal Set (X_f): A constraint requiring the predicted state at the end of the horizon to lie within a positively invariant set. This set is often defined as a sub-level set of the terminal Lyapunov function.
• Stability Proof: This combination ensures the optimal cost of the MPC problem acts as a Lyapunov function for the closed-loop system, proving stability.

The Algebraic Riccati Equation (ARE) is a foundational matrix equation in optimal control theory. For a linear system with quadratic cost, the solution to the ARE provides the optimal state feedback gain for the Linear Quadratic Regulator (LQR), which minimizes an infinite-horizon cost.

• Direct Link to Lyapunov: The optimal cost-to-go for the LQR problem, V(x) = x^T P x, where P solves the ARE, is a quadratic Lyapunov function for the closed-loop system. Its time derivative equals the negative of the stage cost.
• Role in MPC: The matrix P from the ARE is the standard choice for the terminal cost in Linear MPC. This elegantly connects the infinite-horizon optimal solution (LQR) to the finite-horizon MPC formulation, providing a ready-made Lyapunov function to ensure stability.

An Optimal Control Problem (OCP) is the core mathematical formulation solved (repeatedly) by Model Predictive Control. It defines the objective and rules for controlling a dynamic system.

• Components: An OCP is defined by: 1) A dynamic model (e.g., x_{k+1} = f(x_k, u_k)), 2) A cost function to minimize (e.g., tracking error + control effort), 3) Constraints on states and inputs, 4) An initial condition and a time horizon.
• Lyapunov Connection: The value function of an OCP—the optimal cost achieved from a given initial state—can often serve as a Lyapunov function for the closed-loop system. This is the theoretical bedrock for proving MPC stability. Designing the OCP with a terminal cost derived from a Lyapunov function ensures this property holds.

Receding Horizon Control is the real-time implementation strategy of MPC, describing the closed-loop execution loop. It is not an algorithm itself but the operational principle.

• The Loop: At each time step t:
  1. Measure/estimate the current system state.
  2. Solve an Optimal Control Problem (OCP) over a finite future horizon [t, t+N].
  3. Apply only the first control input from the optimized sequence to the system.
  4. At time t+1, shift the horizon forward and repeat.
• Stability Challenge: Solving a finite-horizon OCP at each step does not inherently guarantee infinite-horizon stability. This is where Lyapunov theory intervenes. By designing the OCP with a terminal cost/constraint based on a Lyapunov function, the receding-horizon application of the control law is provably stable.