Source-linked AI summary
On generalized terminal state constraints for model predictive control
Lorenzo Fagiano, Andrew R. Teel
TL;DR
MPC terminal-state constraints can ensure recursive feasibility and constraint satisfaction, but fixed terminal states may restrict feasibility and require longer horizons. This paper develops a generalized terminal-state constraint and a receding-horizon strategy that preserve these guarantees while enlarging feasibility and approaching the performance of an optimally chosen fixed terminal state.
Problem
Fixed terminal-state constraints can be restrictive, requiring longer prediction horizons and greater computational complexity, while economic MPC may lack convergence and stability guarantees.
Method
The paper replaces the fixed terminal state with a generalized terminal state constraint and develops a receding-horizon strategy exploiting it.
Results
Under reasonable or mild assumptions, the generalized approach preserves recursive feasibility and constraint satisfaction, provides a larger feasibility set with lower computational complexity, and achieves performance similar to fixed optimally chosen terminal constraints.
Takeaways & Limitations
The technique applies to tracking and economic MPC for linear and nonlinear systems, with its features illustrated through three examples.
Takeaways & Limitations
In general settings, the generalized approach does not guarantee convergence to the optimal stage-cost value or any near-optimal value, so performance guarantees may be unavailable.
Abstract
from arXiv · showhide
This manuscript contains technical results related to a particular approach for the design of Model Predictive Control (MPC) laws. The approach, named "generalized" terminal state constraint, induces the recursive feasibility of the underlying optimization problem and recursive satisfaction of state and input constraints, and it can be used for both tracking MPC (i.e. when the objective is to track a given steady state) and economic MPC (i.e. when the objective is to minimize a cost function which does not necessarily attains its minimum at a steady state). It is shown that the proposed technique provides, in general, a larger feasibility set with respect to existing approaches, given the same computational complexity. Moreover, a new receding horizon strategy is introduced, exploiting the generalized terminal state constraint. Under mild assumptions, the new strategy is guaranteed to converge in finite time, with arbitrarily good accuracy, to an MPC law with an optimally-chosen terminal state constraint, while still enjoying a larger feasibility set. The features of the new technique are illustrated by three examples.
1 Introduction
MPC handles nonlinear dynamics, multiple inputs and outputs, and hard state and input constraints through finite-horizon optimization. The paper distinguishes tracking MPC, which follows prescribed references, from economic MPC, which directly optimizes a performance objective.
- MPC framework: MPC computes inputs by solving a finite-horizon optimal control problem while accommodating nonlinear dynamics and hard state and input constraints.The finite-horizon objective averages predicted stage costs over N steps.
- Tracking MPC: Tracking MPC uses a stage cost based on deviations from a prescribed set point or reference trajectory.
- Economic MPC: Economic MPC uses a stage cost expressing a performance to optimize rather than deviation from a prescribed set point or trajectory.It can integrate set-point computation and tracking into one predictive controller.
- Economic MPC: Economic MPC is relevant when the economically best performance is not attained at a steady state or when pre-computing a trajectory is undesirable.
2 Notation and problem formulation
The formulation defines constrained finite-horizon MPC problems around an optimal fixed point and applies them recursively through a terminal-state-constrained receding-horizon algorithm. This standard scheme guarantees feasibility and constraint satisfaction, but its fixed terminal constraint can restrict the feasible set and require longer horizons.
- Problem formulation: The system evolves in discrete time as x(t+1)=f(x(t),u(t)) with state constraints X, compact input constraints U, and a finite prediction horizon N.Predicted variables are indexed as y(j|t), and the stage cost is l(x,u).
- Problem formulation: The FHOCP constrains predicted dynamics, inputs, states, the initial condition x(0|t)=x(t), and the terminal state x(N|t)=x_s.
- Problem formulation: The terminal pair (x_s,u_s) is selected from fixed points that minimize the stage cost.The formulation names such a pair an optimal fixed point.
- Feasibility and receding-horizon control: The feasibility set F_s contains states for which the terminal-state-constrained problem admits a solution, and it is assumed nonempty and bounded.The receding-horizon algorithm starts from x(0)∈F_s and repeatedly resolves the FHOCP using the previous solution tail and u_s.
- Recursive guarantees: With no noise or model uncertainty, the terminal-state-constrained algorithm guarantees recursive feasibility and satisfaction of state and input constraints.
- Tracking and economic MPC: For tracking costs, additional assumptions can yield convergence and stability, whereas economic costs generally guarantee feasibility and constraint satisfaction without guaranteeing convergence or stability.Economic performance is instead characterized through an asymptotic time-average cost criterion.
- Motivation: The fixed terminal-state constraint may restrict the feasibility set, often requiring long horizons and higher computational complexity.The generalized terminal-state approach is introduced to reduce this drawback.
3 Generalized terminal state constraint
The generalized terminal state constraint replaces a fixed terminal steady state with a reachable fixed point whose terminal cost is bounded, enlarging feasibility while preserving recursive feasibility and constraint satisfaction. A modified receding-horizon strategy is then introduced to obtain convergence toward the optimal terminal cost under additional assumptions.
- Generalized terminal constraint: The generalized problem adds a terminal input and permits the terminal fixed point to vary with the current state and optimization solution.Its formulation uses N + 1 predicted inputs and a terminal state-input pair satisfying x(N|t) − f(x(N|t), v(N|t)) = 0.
- Computational trade-off: The generalized formulation requires m additional optimization variables, but examples show that a much shorter horizon can deliver similar closed-loop performance to a longer fixed-terminal formulation.The additional variables are identified as the computational price of generalizing the terminal constraint.
- Generalized feasibility set: The generalized feasibility set F contains states from which an admissible sequence reaches a fixed point within N steps.The reachable fixed-point set X(x(t), N) characterizes terminal states satisfying the fixed-point condition after an admissible trajectory.
- Feasibility and invariance: For every initial state in F, the generalized receding-horizon strategy guarantees recursive feasibility and recursive satisfaction of state and input constraints.The proof uses the shifted previous solution together with its terminal input to preserve the fixed-point terminal condition at the next time step.
- Feasibility and invariance: The generalized feasibility set F is positively invariant and at least as large as the fixed-terminal feasibility set Fs.Depending on the system, the generalized formulation can be feasible for more initial states at the same horizon or achieve comparable feasibility with a shorter horizon.
- Performance guarantees: Without further assumptions, the generalized strategy guarantees neither convergence of terminal costs to the optimum nor tracking stability or asymptotic average-cost performance.The terminal cost sequence is nonincreasing under the generalized constraint, but convergence to the optimal value or a nearby value is not guaranteed; a modified strategy is proposed to address this.
4 Guaranteed properties of MPC with generalized terminal state constraint
The generalized terminal constraint supports performance guarantees for tracking and economic MPC under stated regularity and optimization assumptions. Its receding-horizon strategy can bring the terminal stage cost within an arbitrary tolerance of the optimal value in finite time.
- Generalized terminal-cost optimality: Under Assumptions 1–4, the terminal state-input pair can have stage cost within any prescribed ǫ of the minimum among reachable steady states.Proposition 2 establishes a finite β(ǫ) sufficient for this guarantee.
- Economic MPC: For economic MPC, the generalized-terminal scheme achieves asymptotic average performance no worse than the optimal fixed point plus an arbitrarily small tolerance ǫ.The performance statement is given as part of Theorem 2-c.
- Tracking MPC: For tracking MPC, sufficiently large β makes the generalized terminal pair arbitrarily close to the desired steady state as ǫ approaches zero.The result extends convergence and stability properties associated with fixed terminal state constraints.
- Parameter tuning and scope: The sufficient condition on β can be conservative, while a reasonably high β can match the performance of a fixed, optimally chosen terminal constraint over Fs.This matching is reported as an empirical observation from the paper’s examples.
- Finite-time receding-horizon strategy: Algorithm 3 produces terminal stage costs that converge in finite time to within ǫ of the optimal value for every initial state in the generalized feasibility set.Theorem 3 applies under Assumptions 1–5 and 7 with a sufficiently long horizon.
5 Examples
Three examples illustrate the generalized terminal state constraint in linear-quadratic tracking, economic CSTR control, and inverted-pendulum tracking. Across these cases, it enlarges feasibility or achieves comparable performance with shorter horizons, while convergence speed depends on β and sufficient horizon length.
- Linear-quadratic tracking MPC: The generalized terminal constraint enlarges the linear tracking MPC feasibility set for a given horizon, reaching the maximal region with N = 3 versus N ≥7 for a fixed constraint.The generalized set combines N-step null-controllable regions associated with different terminal fixed points.
- Economic MPC for an isothermal CSTR: In the CSTR example, generalized-constraint performance is equivalent to an optimally chosen fixed constraint, while the asymptotic average cost is 21.14.The terminal stage cost rapidly converges to its optimal value; larger β increases convergence speed.
- Tracking MPC of an inverted pendulum system: For the inverted pendulum, N = 60 approaches a limit cycle because the target region is not reachable, whereas N = 141 satisfies the sufficient horizon condition.The N = 60 trajectory has average cost 195.89, compared with 213.33 for the best reachable steady state.
- Tracking MPC of an inverted pendulum system: The inverted-pendulum generalized constraint reaches the target in about 6.6 s with N = 100, while the fixed constraint requires N ≥200 and takes about 11 s.The comparison supports similar swing-up performance with substantially shorter prediction horizons for the generalized constraint.
6 Conclusions
The paper concludes that generalized terminal state constraints can deliver comparable performance to fixed, optimally chosen constraints while offering a larger feasibility set and lower computational complexity.
- The proposed generalized terminal state constraint achieves performance similar to MPC schemes with a fixed, optimally chosen terminal state constraint.
- It provides a larger feasibility set and lower computational complexity than the compared fixed-constraint schemes.
- Three examples demonstrate the technique for tracking and economic MPC with both linear and nonlinear systems.
Appendix
The appendix bounds terms appearing in the proof of equations (17)–(18), including stage-cost differences and a second term bounded by η.
- A term involving l(x(t)) and l(ˆx, ˆu) is shown to be less than −βǫ under β > 0 and the stated cost inequality.
- The proof of equations (17)–(18) begins by considering equation (16).
- Stage-cost differences are bounded using Assumption 3 for each prediction index i ∈ [0, N − 1].
- The second term in equation (16) is bounded by η = αl.