Terminal Cost and Terminal Constraint

The core challenge in finite-horizon MPC is that optimizing over a short window can lead to myopic control actions, potentially driving the system to a suboptimal or unstable state beyond the horizon. This is known as the infinite-horizon approximation problem. Terminal elements are the primary theoretical mechanism to ensure that the finite-horizon optimization yields a policy with desirable infinite-horizon properties, namely asymptotic stability.

A terminal cost is an additional term, $V_f(x_N)$, added to the standard MPC cost function and evaluated at the final predicted state, $x_N$, at the end of the prediction horizon. Its role is to approximate the cost-to-go from that state to infinity.

• Common Design: Often chosen as the value function of the associated infinite-horizon Linear Quadratic Regulator (LQR), solving the Algebraic Riccati Equation (ARE).
• Effect: It penalizes the optimizer for leaving the system in a state from which future control would be expensive, effectively 'looking beyond' the finite horizon and steering the solution toward long-term optimality.

A terminal constraint requires the final predicted state, $x_N$, to lie within a pre-defined terminal region or set, $\mathbb{X}_f$. This set is designed to be a positively invariant and control-invariant set under a local stabilizing controller (e.g., the LQR gain).

• Purpose: It ensures the system reaches a 'safe harbor' by the end of the horizon, from which a simple backup controller can maintain stability and constraint satisfaction indefinitely.
• Typical Form: Often an ellipsoidal or polyhedral set where the local controller is known to be effective and safe.

Terminal cost and constraint work together to construct a Lyapunov function for the closed-loop MPC system, which is the standard tool for proving stability.

1. The combined MPC cost function (including terminal cost) serves as a candidate Lyapunov function.
2. The terminal constraint ensures that from the terminal state, a feasible (and stabilizing) control action exists for the next step.
3. The optimization guarantees that the cost at the next time step is non-increasing. This monotonic decrease proves asymptotic stability.

This is formalized in Mayne's Stability Theorem for MPC.

Implementing terminal elements involves engineering trade-offs:

• Conservatism vs. Performance: A small, restrictive terminal set ($\mathbb{X}_f$) guarantees stability but can severely limit the region of attraction (the set of initial states the controller can handle). A larger set improves performance but is harder to design.
• Computational Complexity: Enforcing a terminal constraint adds to the online optimization problem's complexity. For nonlinear systems, defining a suitable invariant set is non-trivial.
• Zero-Terminal-State: A common simplification is setting $\mathbb{X}_f = {0}$ (the origin) and $V_f = 0$. This guarantees stability but often requires a very long prediction horizon to be feasible, increasing computation.

Modern MPC research has developed methods to reduce conservatism:

• Quasi-Infinite Horizon MPC: Uses a terminal cost from the LQR over an infinite horizon but only enforces a terminal constraint for a finite period, balancing guarantee and feasibility.
• Constraint Tuning: The terminal set can be computed offline via algorithms that find the maximal control-invariant set for the local controller.
• Stability without Terminal Constraints: Techniques like using a sufficiently long prediction horizon (horizon tuning) or a contractive constraint can sometimes ensure stability without an explicit terminal set, at the cost of less rigorous guarantees.

A Lyapunov function is a scalar, energy-like function used to prove the stability of a dynamical system's equilibrium point. In MPC stability proofs, the terminal cost is often designed as a Lyapunov function for the system under a local controller. This guarantees that the predicted state at the horizon's end is moving toward the desired setpoint, ensuring the overall MPC law is stabilizing.

• Role in MPC: Provides the theoretical foundation for proving that an MPC controller will drive the system to its target.
• Terminal Cost Design: A common approach is to set the terminal cost equal to a Lyapunov function for the system linearized around the setpoint.
• Decrease Condition: Stability is proven by showing this function's value decreases along the system's trajectory.

The Optimal Control Problem (OCP) is the core mathematical formulation solved at each step of MPC. It defines the optimization over a prediction horizon.

• Components: An OCP consists of a dynamic model (state equations), a cost function to minimize, and constraints on states and control inputs.
• Terminal Elements: The terminal cost and terminal constraint are specific components added to the standard OCP formulation to enforce stability guarantees.
• Solving: The MPC algorithm solves a discretized version of this OCP in real-time to generate the optimal control sequence.

Receding Horizon Control is the fundamental operating principle of MPC. Only the first control input from the optimized sequence is applied to the system. The horizon then "recedes" forward one step, and the OCP is re-solved with a new initial state measurement.

• Connection to Terminal Elements: The terminal cost and constraint work within this receding horizon framework. They ensure that each finite-horizon optimization, when applied in a receding horizon fashion, results in infinite-horizon stability.
• Practical Impact: This principle is what makes MPC feedback-based and adaptive to disturbances.

Constraint handling refers to the methods by which MPC enforces limits on system states and control inputs. Terminal constraints are a specific, powerful form of constraint.

• Hard vs. Soft Constraints: Hard constraints (like a terminal constraint) must never be violated. Soft constraints can be breached at a quantified penalty, often using slack variables to ensure the optimization problem remains feasible.
• Terminal Constraint Set: This is a hard constraint requiring the final predicted state to lie within a pre-computed invariant set (often a positively invariant set).
• Feasibility: A primary challenge is ensuring the OCP with a terminal constraint has a feasible solution at every time step.

Stability guarantees that the closed-loop system converges to a desired setpoint or region. Terminal cost and constraint are primary tools to achieve nominal stability (perfect model).

• Nominal Stability: Guaranteed if the terminal cost is a local Lyapunov function and the terminal set is positively invariant under a local controller. The combined design ensures the optimal cost function acts as a Lyapunov function for the closed-loop MPC system.
• Robust Stability: Extends guarantees to systems with model error or disturbances. Techniques like tube-based MPC or constraint tightening are used, which may also involve robust versions of terminal ingredients.
• Verification: Stability is a theoretical property proven during controller design, not just an observed outcome.

The Algebraic Riccati Equation (ARE) is a key matrix equation in linear optimal control. Its solution is central to designing the terminal cost for Linear Quadratic Regulator (LQR)-based MPC.

• LQR Connection: For an unconstrained linear system with a quadratic cost, the infinite-horizon optimal controller is a constant state-feedback gain found by solving the ARE.
• Terminal Cost Design: In linear MPC, a common and effective choice for the terminal cost is the quadratic form defined by the solution matrix P of the ARE. This makes the finite-horizon MPC cost approximate the infinite-horizon LQR cost.
• Terminal Set: The associated LQR controller often defines the local controller used to compute the terminal constraint set.