PaperlayerView exact revision on arXiv

Source-linked AI summary

A nonlinear model predictive control framework using reference generic terminal ingredients -- extended version

Johannes Köhler, Matthias A. Müller, Frank Allgöwer

arXiv:1909.12765v2eess.SYcs.SYmath.OC

TL;DR

The paper tackles the difficulty of obtaining terminal ingredients for nonlinear MPC across generic reference trajectories and changing operating conditions. It computes parameterized terminal ingredients offline and reports stability and feasibility guarantees, alongside improved benchmark performance over schemes without terminal constraints or with terminal equality constraints.

Abstract

from arXiv · show

In this paper, we present a quasi infinite horizon nonlinear model predictive control (MPC) scheme for tracking of generic reference trajectories. This scheme is applicable to nonlinear systems, which are locally incrementally stabilizable. For such systems, we provide a reference generic offline procedure to compute an incrementally stabilizing feedback with a continuously parameterized quadratic quasi infinite horizon terminal cost. As a result we get a nonlinear reference tracking MPC scheme with a valid terminal cost for general reachable reference trajectories without increasing the online computational complexity. As a corollary, the terminal cost can also be used to design nonlinear MPC schemes that reliably operate under online changing conditions, including unreachable reference signals. The practicality of this approach is demonstrated with a benchmark example. This paper is an extended version of the accepted paper [1], and contains additional details regarding \textit{robust} trajectory tracking (App.~B), continuous-time dynamics (App.~C), output tracking stage costs (App.~D) and the connection to incremental system properties (App.~A).

I. INTRODUCTION

The paper addresses the difficulty of designing terminal ingredients for nonlinear MPC across changing references, where existing approaches are often restricted to linear systems, setpoints, or specific trajectories. It proposes reference-generic terminal ingredients and uses them to extend nonlinear tracking MPC to changing operating conditions.

I. INTRODUCTION

  • Existing terminal-ingredient procedures often target a specific reference, while many broader MPC approaches remain limited to linear systems or setpoint stabilization.Such reference-specific designs can require repeated offline computations when the setpoint or trajectory changes.Source: §I · p. 1, Related work · p. 2

Related work

  • Prior approaches include setpoint-specific designs,, trajectory-specific designs,, and methods with practical limitations for general nonlinear systems,,.The cited methods face issues including difficult pseudo-linearization, increased online demand, complex partitioning, or nondifferentiable terminal costs.Source: Related work · pp. 1–2

Contribution

  • The proposed procedure computes parameterized terminal costs for setpoint or trajectory stabilization and extends tracking MPC to nonlinear dynamics and optimal periodic operation.The design requires local incremental stabilizability of the nonlinear dynamics.Source: Contribution · p. 2
  • The framework extends existing tracking schemes to exogenous output signals and periodic operation, and the paper also outlines robust tracking, continuous-time dynamics, and output-cost extensions.The appendix also discusses connections between generic terminal ingredients and incremental system properties.Source: Contribution · p. 2, Outline · p. 2, C. Terminal cost and terminal set · p. 2

B. Setup

  • The tracking setup assumes reachable references that remain inside the constraints and follow the system dynamics; unreachable references can instead be handled using an artificial reference.The tracking error is the difference between the state and reference state, and the goal includes constraint satisfaction.Source: B. Setup · p. 2, C. Terminal cost and terminal set · p. 2

C. Terminal cost and terminal set

  • The terminal ingredients use a reference-dependent quadratic cost and terminal set with a local feedback controller, and suitable ingredients are the paper’s main design contribution.The stated design conditions apply to references in Zr and their permitted successors.Source: C. Terminal cost and terminal set · p. 3

D. Preliminary results

The tracking MPC scheme applies the first optimized input at each step and uses terminal ingredients to establish recursive feasibility and uniform exponential stability. Its terminal-set assumption and the alternative without terminal ingredients mark important scope and horizon trade-offs.

D. Preliminary results

  • A terminal equality constraint can satisfy the terminal assumption, but may cause numerical problems and reduce MPC performance and robustness.Some tracking schemes also require a non-vanishing terminal set for exponential stability.Source: D. Preliminary results · p. 3
  • Applying the first input of the optimized trajectory yields a feasible shifted candidate and a value-function decrease by the stage cost.The decrease supports the stability argument for the closed loop.Source: D. Preliminary results · p. 3
  • If Assumptions 1 and 2 hold and the problem is initially feasible, the MPC problem remains recursively feasible and the tracking error is uniformly exponentially stable.The result follows from standard Lyapunov arguments using the value function.Source: D. Preliminary results · p. 3
  • Without terminal ingredients, similar closed-loop properties require a sufficiently long prediction horizon, while unreachable references can still be handled.With terminal ingredients, a short horizon is a core advantage, but unreachable references are generally not supported by terminal constraints.Source: D. Preliminary results · p. 3

III. REFERENCE GENERIC OFFLINE COMPUTATIONS

The offline procedure designs reference-parameterized terminal ingredients for nonlinear tracking MPC. Linearization-based conditions and suitable terminal-set bounds ensure the ingredients satisfy the required properties across the considered references.

III. REFERENCE GENERIC OFFLINE COMPUTATIONS

  • The offline procedure computes reference-parameterized terminal ingredients for nonlinear reference-tracking MPC, with extensions described for continuous-time dynamics and output-tracking stage costs.It also discusses existing procedures for setpoint tracking and summarizes the overall design in an algorithm.Source: §III · pp. 3–4

A. Sufficient conditions based on the linearization

  • If continuous matrices Kf(r) and positive definite Pf(r) satisfy the linearization-based matrix inequality, then a sufficiently small terminal-set size α satisfies Assumption 2.The sufficient conditions require twice continuous differentiability of f.Source: A. Sufficient conditions based on the… · p. 4
  • The linearization-based construction bounds terminal-set size by α1 for decrease and α2 for constraint satisfaction; any α ≤ min{α1, α2} satisfies Assumption 2.The result extends, to arbitrary dynamic references.Source: A. Sufficient conditions based on the… · p. 4

B. Quasi-LPV based procedure

The quasi-LPV procedure turns linearization-based terminal-ingredient conditions into tractable offline designs. It offers convexification and gridding approaches, trading computational scaling against conservatism.

B. Quasi-LPV based procedure

  • The quasi-LPV parameterization makes the optimization variables finite, but the original constraints still range over all references and successors.Sum-of-squares optimization is not pursued because its computational complexity grows exponentially in n d for high-order polynomial approximations.Source: B. Quasi-LPV based procedure · p. 5

1) Convexify:

  • Convexification encloses reference-dependent parameters in polytopic sets and reduces verification to finitely many LMI checks at vertices.Multi-concavity along the constraint-set edges makes vertex checks sufficient; the construction adds conservatism.Source: 1) Convexify · p. 5

2) Gridding:

  • Gridding checks the conditions at sampled state-input pairs; continuity means a sufficiently fine grid typically supports satisfaction over the full constraint set.The approach can be computationally challenging for high-dimensional systems, while convexification typically scales better but adds conservatism.Source: 2) Gridding · pp. 5–6

B. Quasi-LPV based procedure

  • The design casts offline terminal-ingredient computation as gain-scheduling synthesis for quasi-LPV systems, drawing on established offline LMI procedures.The underlying optimization is convex but initially infinite-dimensional, requiring finite parameterization and sufficient finite constraints.Source: B. Quasi-LPV based procedure · p. 5, 2) Gridding · p. 6

C. Non-conservative terminal set size α

The section gives less conservative ways to choose the terminal-set size by separately checking constraint satisfaction and local stability. The local-stability computation is more involved, but the offline procedure is performed only once, even when operating conditions change.

1) Constraint satisfaction - α2:

  • For constraint satisfaction, α2 can be found by gridding the constraint set and solving a linear program at each point, then taking the minimum.When Pf and Kf are constant, this reduces to one small-scale linear program.Source: 1) Constraint satisfaction - α2 · p. 6

2) Local Stability - α1:

  • The local-stability constant α1 is harder to determine; the described sampling procedure decreases a candidate α1 until all considered combinations satisfy the stability condition.The passage contrasts this with a non-convex optimization check for setpoint stabilization and notes that solver-based alternatives can get stuck in local minima.Source: 2) Local Stability - α1 · p. 6

C. Non-conservative terminal set size α

  • The terminal-set size is chosen as α = min{α1, α2}, using the local-stability and constraint-satisfaction bounds, respectively.The procedure summary allows either the algorithms or the corresponding equations to compute each bound.Source: 2) Local Stability - α1 · pp. 6–7

2) Local Stability - α1:

  • The offline terminal-ingredient procedure is more involved than a setpoint-specific computation, but it is performed once without repeat computations for changing operating conditions.The paper notes that numerical examples detail its applicability to nonlinear systems and offline computational effort.Source: 2) Local Stability - α1 · p. 7

D. Setpoint tracking

Setpoint tracking is a special case of the reference-tracking derivation, with simplified offline computation and continuously parameterized terminal ingredients. The paper also applies these ingredients to nonlinear MPC under changing output references, establishing recursive feasibility and conditional stability for periodic signals.

D. Setpoint tracking

  • Setpoint tracking is a special case of the reference-tracking derivation, and both gridding and convex computation simplify in that case.Gridding can be restricted to the typically low-dimensional steady-state manifold; in the convex approach, θ+ = θ and only the 2p vertices of Θ are considered.Source: D. Setpoint tracking · p. 7

IV. NONLINEAR MPC SUBJECT TO CHANGING OPERATION CONDITIONS

  • Changing or unreachable exogenous signals motivate an extension that combines reference-generic terminal ingredients with artificial-reference optimization for nonlinear MPC.The section presents a scheme that exponentially stabilizes the periodic trajectory that best tracks an exogenous output signal; extending economic MPC to periodic artificial trajectories is left for future research.Source: §IV · p. 7

A. Nonlinear periodic tracking MPC subject to changing exogenous output references

  • For periodic exogenous output references, the proposed nonlinear MPC scheme is recursively feasible independently of the signal; under stated conditions, the optimal reachable periodic trajectory is uniformly exponentially stable.The conditions include a convexity and continuity condition on feasible periodic orbits and the output function.Source: A. Nonlinear periodic tracking MPC… · p. 8

V. NUMERICAL EXAMPLES

  • The numerical examples illustrate applicability to nonlinear systems through CSTR periodic tracking and car trajectory-tracking tasks, including robust tracking and an evasive maneuver.The examples compare discrete-time and continuous-time formulations; additional periodic output tracking examples with a nonlinear ball-and-plate system are cited in.Source: §V · p. 8

A. Periodic reference tracking - CSTR

The examples apply reference-generic terminal ingredients to periodic CSTR operation and robust car-trajectory tracking, with the offline computation performed once rather than for each reference. In the evasive maneuver test, the proposed scheme outperforms the alternatives at a short prediction horizon, while the examples also expose conservatism and implementation constraints.

Offline computations:

  • For the CSTR, gridding gives the smallest, least conservative terminal cost, while a more detailed convex parameter description only marginally improves performance and raises offline computation demand.The continuous-time formulation is more efficient to compute, but its terminal ingredients require parameters chosen to remain stabilizing for the discrete-time implementation.Source: Offline computations · pp. 8–9
  • For the CSTR, the reference-generic offline computation is performed once without explicit knowledge of the reference trajectory or its period, unlike trajectory-specific methods,.The example tracks reachable periodic references corresponding to economically beneficial periodic operation.Source: System model · p. 8, Offline computations · p. 9

B. Automated driving - robust reference tracking

  • In the automated-driving example, the convex approach is infeasible because its conservative hyperbox includes linearized dynamics that are not stabilizable.For the considered parameters, the continuous-time terminal cost is valid for a zero-order-hold implementation at h = 2 ms, but this is not generally true.Source: B. Automated driving - robust… · p. 9

Robust trajectory tracking - Evasive maneuver test:

  • With ε = 2.5·10^-4, the robust tracking MPC scheme guarantees uniform practical exponential stability and robust constraint satisfaction using tightening based on contraction rate ρ = 0.9995.The simulation considers additive disturbances with prediction horizon N = 10. Robust recursive feasibility can be ensured when the terminal set is robust positively invariant for ∥w(t)∥ ≤ 1.82·10^-5.Source: Robust trajectory tracking - Evasive… · p. 10
  • In the evasive maneuver test, UC and TEC have closed-loop tracking stage costs 10 and 3.000 times larger, respectively, than the proposed scheme with terminal cost (QINF).Source: Robust trajectory tracking - Evasive… · p. 10, Figure 3 · p. 10
  • UC and TEC have tracking stage costs 10 and 3.000 times larger than QINF; at short horizons, UC shows growing position error and TEC deadbeat-like input oscillations.Similar performance to QINF is reported at N = 10 after increasing the prediction horizon to N = 23 and N = 59, respectively; online computational demand increases by 100% and 300%, respectively.Source: Robust trajectory tracking - Evasive… · pp. 10–11
  • State-only and input-only tracking criteria favor different alternatives: TEC has 13% of QINF's state tracking error, while UC has 48% of QINF's input tracking error.The evasive maneuver uses two consecutive lane changes, and its terminal set is restricted by input and path constraints.Source: Robust trajectory tracking - Evasive… · p. 10

VI. CONCLUSION

The paper presents an offline procedure for parameterized terminal ingredients for nonlinear reference-tracking MPC. Its generic computation supports operation under changing conditions, while large-scale distributed systems remain future work.

VI. CONCLUSION

  • The offline terminal-ingredient computation is performed independently of the setpoint or trajectory to be stabilized.The procedure parameterizes the terminal ingredients and treats the reference trajectory as a parameter in a local quasi-LPV approximation.Source: §VI · p. 11
  • Extending the procedure to large-scale nonlinear distributed systems using a separable formulation is left for future work.Source: §VI · p. 11

APPENDIX

The appendices connect reference-generic terminal ingredients to incremental stabilizability and extend the approach to robust tracking. They also state conditions under which reference trajectories can be incrementally stabilized.

APPENDIX

  • Appendix A relates incremental system properties to the terminal ingredients, while Appendices B–D extend the approach to robust tracking, continuous-time dynamics, and output-tracking stage costs.Appendix B uses constraint tightening for robust constraint satisfaction under additive disturbances.Source: APPENDIX · p. 13

A. (Local) Incremental exponential stabilizability

  • Local incremental exponential stabilizability requires a feedback and contraction bound that keeps trajectories near the reference converging exponentially, subject to the stated local-error condition.The definition considers reference trajectories satisfying a dynamic inclusion and an initial error no greater than c.Source: A. (Local) Incremental exponential… · p. 13
  • Under Lemma 1 conditions, the tracking feedback kf(x, r) = ur + Kf(r)(x − xr) locally incrementally exponentially stabilizes references satisfying Assumption 1.The terminal cost acts as a local incremental Lyapunov function based on linearization properties.Source: A. (Local) Incremental exponential… · p. 13
  • With no constraints, uniformly bounded Pf and Kf(r) = K(xr) yield exponential incremental stabilization for references satisfying Assumption 1, with the stated geometric error bound.The bound is ∥x(t) − xr(t)∥ ≤ Mρ^t∥x(0) − xr(0)∥.Source: A. (Local) Incremental exponential… · p. 13

B. Robust reference tracking

3) Constraint tightening:

The robust design tightens constraints and uses terminal ingredients that account for disturbances. Its MPC optimization applies tightened state and input constraints.

3) Constraint tightening:

  • Constraint tightening replaces each reference constraint Ljr ≤ 1 with Ljr ≤ 1 − ϵj,k, and requires a disturbance bound so the tightened set remains non-empty.The method describes the tightening as scalar operations.Source: 3) Constraint tightening · p. 14

4) Terminal ingredients:

  • Robust terminal ingredients use a quadratic tracking cost and feedback, with a terminal set sized by αw and conditions that include an RPI requirement.Compared with the nominal case, the terminal set is smaller because of tightened constraints.Source: 4) Terminal ingredients · p. 14

5) Robust tracking MPC:

6) Theoretical guarantees:

Under the stated assumptions and disturbance bound, the robust tracking MPC is recursively feasible and practically exponentially stabilizes the tracking error. The approach has hard limits on disturbance size.

6) Theoretical guarantees:

  • With Assumptions 1, 3, and 4, the stated disturbance bound, and initial feasibility, the MPC remains recursively feasible and the tracking error is uniformly practically exponentially stable.Source: 6) Theoretical guarantees · p. 15
  • The considered disturbance size is hard-bounded by both the constraint set and local incremental stabilizability conditions.Source: 6) Theoretical guarantees · p. 15
  • The approach can also use a general nonlinear state- and input-dependent disturbance characterization to reduce conservatism.Source: 6) Theoretical guarantees · p. 15

C. Continuous-time dynamics

The continuous-time appendix extends the reference-generic terminal-cost construction to differentiable dynamics and references. It gives sufficient conditions and implementation options for computing the terminal ingredients.

C. Continuous-time dynamics

  • The continuous-time formulation assumes differentiable reference signals whose derivatives lie in an admissible set, with bounded input-reference derivatives.For non-differentiable references, the terminal cost can instead be parameterized by Pf(xr), independent of ur.Source: C. Continuous-time dynamics · p. 15
  • Continuous-time terminal ingredients use a quadratic tracking cost and local feedback, with continuously differentiable Pf and conditions required over admissible references and their derivatives.Source: C. Continuous-time dynamics · p. 15
  • A matrix inequality based on linearization provides sufficient conditions for the continuous-time terminal ingredients when α is sufficiently small.The derivation uses a first-order Taylor approximation and bounds higher-order terms locally.Source: C. Continuous-time dynamics · p. 15
  • The appendix gives LMI conditions and notes that continuous-time terminal costs may also satisfy discrete-time conditions for sufficiently small sampling time.This can make the offline optimization easier to formulate and faster to solve; validity with zero-order-hold inputs can be checked using Algorithm 1.Source: C. Continuous-time dynamics · pp. 15–16

D. Output tracking stage cost

The section extends the terminal-cost derivation to output tracking costs and gives sufficient conditions for feedback and terminal-cost design. Output tracking may stabilize a set or manifold rather than a particular reference unless suitable incremental detectability conditions hold.

D. Output tracking stage cost

  • The output tracking stage cost uses a nonlinear output h and continuous positive definite weighting S, and applies to regulation, trajectory tracking, path following, or manifold stabilization.For linear h(x, u) = [Q1/2x; R1/2u] and S = I, the conditions recover those of Lemma 1 with stage cost (3).Source: D. Output tracking stage cost · pp. 16–17
  • For twice continuously differentiable dynamics and output, the stated matrix inequality conditions ensure that continuous Pf and Kf satisfy Assumption 2 for sufficiently small α.The resulting conditions support LMI-based computation of Pf and Kf.Source: D. Output tracking stage cost · pp. 16, 18
  • Exact output tracking can admit multiple solutions, so stability may concern a corresponding set or manifold rather than the specific reference trajectory.Suitable incremental detectability conditions on h can recover stability of the specific reference.Source: D. Output tracking stage cost · p. 17
Loading 1909.12765v2…