Optimal Control Problem (OCP)

The dynamic model is a set of mathematical equations, typically ordinary differential equations (ODEs) or difference equations, that describe how the system's state variables evolve over time in response to control inputs and disturbances. This internal model is the predictive engine of the OCP.

• Continuous-time: dx/dt = f(x(t), u(t), p)
• Discrete-time: x_{k+1} = f(x_k, u_k, p)

Where x is the state, u is the control input, and p are parameters. The fidelity of this model directly determines the accuracy of the controller's predictions.

The cost function (or objective function) J is a scalar quantity that the OCP seeks to minimize. It mathematically encodes the control performance goals over the prediction horizon N.

A common quadratic form for tracking is: J = Σ (x_k - x_ref)^T Q (x_k - x_ref) + (u_k - u_ref)^T R (u_k - u_ref)

• The first term penalizes deviation from a desired state reference trajectory.
• The second term penalizes excessive control effort.
• The matrices Q and R are tuning weights that balance these competing objectives.

Constraints are mathematical inequalities that define the admissible operating region of the physical system. Explicitly handling these is a primary advantage of MPC.

• State Constraints: x_min ≤ x(t) ≤ x_max (e.g., position limits, tank levels).
• Input Constraints: u_min ≤ u(t) ≤ u_max (e.g., actuator saturation, valve limits).
• Mixed Constraints: May involve both states and inputs.

Constraints can be hard (must never be violated) or soft (allowed violation with a penalty via slack variables to ensure the optimization problem remains feasible).

The OCP is solved over a finite optimization horizon, which consists of two key intervals:

• Prediction Horizon (N_p): The future time window over which the system's behavior is predicted using the dynamic model.
• Control Horizon (N_c): The (often shorter) window over which control moves are optimized. Control inputs are typically held constant for steps beyond N_c.

The principle of Receding Horizon Control is applied: only the first control input from the optimized sequence is applied to the plant. The horizon then "shifts" forward one step, and the OCP is re-solved with new state measurements, providing feedback and disturbance rejection.

The initial condition x_0 is the estimated or measured state of the system at the start of the current optimization window. It serves as the starting point for all predictions made by the dynamic model.

In practice, the full state is often not directly measurable. Therefore, a state estimator (like a Kalman Filter or Moving Horizon Estimator (MHE)) is used to reconstruct x_0 from available sensor outputs. The accuracy of this initial condition is critical for prediction quality.

The OCP is a constrained optimization problem that must be solved numerically at each control step. The choice of solver depends on the problem structure:

• Linear/Quadratic MPC: Results in a Quadratic Program (QP). Solved efficiently with active-set or interior-point methods.
• Nonlinear MPC (NMPC): Results in a Nonlinear Program (NLP). Solved using Sequential Quadratic Programming (SQP) or interior-point methods.

Real-Time Optimization (RTO) demands that the solver converges to a solution within the controller's sampling period (often milliseconds). Techniques like warm-starting (using the previous solution as an initial guess) are essential for meeting this timing requirement.

The cost function is the scalar mathematical expression that an Optimal Control Problem (OCP) seeks to minimize. It quantifies the performance objective over the prediction horizon.

• Typical Form: Often a quadratic function penalizing tracking error (deviation from a reference) and control effort.
• Economic MPC: In advanced applications, the cost can directly represent economic objectives like energy consumption or profit.
• Terminal Cost: An additional term evaluated at the final predicted state, used as a stability-enforcing design tool.

Constraints are mathematical inequalities that define the admissible operating region of the system, explicitly enforced within the OCP optimization.

• State Constraints: Limits on system variables like position, velocity, or temperature (e.g., x_min ≤ x(t) ≤ x_max).
• Input Constraints: Physical limits on actuator commands, such as torque saturation or valve position limits.
• Hard vs. Soft: Hard constraints must never be violated. Soft constraints use slack variables to allow minor, penalized violations, ensuring the optimization problem remains feasible.

The dynamic model is the set of differential or difference equations that describes how the system's state evolves over time in response to control inputs. It is the internal predictor within the OCP.

• Forms: Can be linear time-invariant (LTI), nonlinear, or hybrid.
• Fidelity: Model accuracy directly impacts MPC performance. System identification techniques are used to derive models from data.
• Example: For a drone, the model encodes the rigid-body dynamics relating rotor thrusts to position, velocity, and orientation.

Receding Horizon Control is the real-time execution policy of MPC. At each control interval, the OCP is solved over a finite future window, but only the first control input is applied to the physical system.

• Principle: After application, the horizon "recedes" forward by one time step, new state measurements are taken, and the OCP is re-solved with updated initial conditions.
• Feedback: This closed-loop approach provides inherent compensation for disturbances and model mismatch.
• Core Advantage: It allows the controller to continuously adjust its plan based on the latest information.

A Quadratic Programming (QP) solver is the numerical optimization engine at the heart of Linear MPC. It efficiently solves the convex optimization problem generated at each time step.

• Problem Structure: Minimizes a quadratic cost function subject to linear equality and inequality constraints.
• Real-Time Requirement: Must compute a solution within the system's sampling period (e.g., milliseconds).
• Common Algorithms: Active-set methods and interior-point methods are standard. For embedded systems, highly optimized, code-generated solvers like qpOASES or OSQP are used.

Nonlinear MPC is the generalization of MPC where the OCP involves a nonlinear dynamic model and/or a non-quadratic cost function, resulting in a Nonlinear Programming (NLP) problem.

• Application: Essential for systems with significant nonlinear dynamics (e.g., chemical reactors, agile robotics).
• Computational Challenge: The NLP is non-convex and more computationally intensive. Sequential Quadratic Programming (SQP) and Interior-Point methods are common solution approaches.
• Tools: Frameworks like ACADO Toolkit and CasADi are used to model and solve NMPC problems efficiently.