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On Exact Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control Problems
Lida Lamakani, Efstratios N. Pistikopoulos
TL;DR
Continuous-time multiparametric MPC partitions initial states into critical regions, but region boundaries may be curved and have previously been approximated rather than computed. This paper fixes a switching or event time and solves linear equations in the initial state to obtain exact boundaries. On the evaporator example, it computes four hyperplanes and four curved boundaries, with endpoints and neighboring regions identified.
Problem
Continuous-time critical-region boundaries can be curved and had not been computed, with classifiers used to distinguish regions instead.
Source: §I · p. 1Method
Fix a switching or event time so boundary conditions become linear equations in the initial state, then solve them and step the fixed time to trace a curved boundary.
Source: §I · p. 1, What is fixed, and why · p. 3Results
The Newell–Lee evaporator has seven critical regions and eight boundaries—four hyperplanes and four curved—and the paper computes a boundary of each type with endpoints and adjacent regions.
Source: Evaporator · p. 5, §IV · p. 6Takeaways & Limitations
With no unknown switching time a boundary is one linear equation; with one, fixing it reduces each boundary point to two linear equations.
Source: C. Step 3. Count the unknowns, fix one… · p. 5, §IV · p. 6
Abstract
from arXiv · showhide
Continuous-time multiparametric model predictive control solves the optimal control problem once, offline, for every initial state, and divides the initial-state space into critical regions, each with its own control law and sequence of active constraints. A boundary between two regions is either a hyperplane, given by one linear equation, or curved. The curved boundaries have not been computed to date; a classifier has been trained instead to distinguish the regions, so the partition has been known only approximately. In this paper we show that a curved boundary can be computed exactly, without solving any optimal control problem. The method rests on one property. Once the sequence of active constraints is fixed, together with the instants inside it at which a constraint becomes active or inactive (the switching times) and the instant at which the change of the neighboring region first appears (the event time), every condition on a boundary is a linear equation in the initial state, so the computation of a boundary, curved or not, is exact. We fix one of these instants, a switching time or the event time, and solve the linear equations that correspond to it for the corresponding initial state on the boundary; stepping the fixed instant along the horizon yields the whole boundary. A hyperplane is the case in which no instant is left to fix. We illustrate the method on the Newell-Lee evaporator, which has seven critical regions and eight boundaries, four hyperplanes and four curved, computing one boundary of each type in full with the adjacent regions at its ends.
I. INTRODUCTION
Continuous-time multiparametric MPC partitions initial states by their optimal arc sequences, but its region boundaries can be curved and have previously been approximated by classifiers. This paper computes a boundary exactly by fixing a switching or event time and solving linear equations in the initial state.
I. INTRODUCTION
- A critical region groups initial states whose optimal solutions follow the same sequence of arcs, while switching times vary within the region.A switching time marks an arc ending and the next beginning, when a constraint enters or leaves the active set.Source: §I · p. 1
- Continuous-time boundaries can be curved because their conditions depend nonlinearly on switching times; prior work used classifiers instead of computing them.Earlier results established hyperplane boundaries for a single switch and identified boundary type from neighboring labels.Source: §I · p. 1
- Fixing a switching or event time leaves linear equations in the initial state, whose solutions trace an exact boundary without solving an optimal control problem.Stepping the fixed time traces a curved boundary; with no time to fix, one linear equation gives a hyperplane.Source: §I · p. 1
A. Optimal control problem
- The illustrated formulation uses a two-dimensional state and scalar input; state-only constraints, which make the costate jump, are outside its scope.The setup assumes a linear time-invariant system and nonzero input coefficients for the path constraints.Source: A. Optimal control problem · pp. 1–2
B. The evaporator
The Newell–Lee forced-circulation evaporator supplies the example system, with scaled deviations for its level, product composition, and product-flow input. The setup uses a finite initial-state domain and constraints on flow and product solids.
B. The evaporator
- The evaporator example uses separator level and product composition as states, with product flow as the input and other inputs held nominal.The variables are scaled deviations from the nominal steady state reported for the Newell–Lee process.Source: B. The evaporator · p. 2
- The finite-horizon example sets tf = 10 min and uses the initial-state domain X = [−2, 2]².The cost weights are Q = I, R = 1, and P = I.Source: B. The evaporator · p. 2
- The constraints are bands on product flow and solids leaving with the product; the flow band is never reached, and symmetry gives each boundary a mirror image.Consequently, only constraints 3 and 4 appear in the labels, and the analysis follows constraint 3.Source: B. The evaporator · p. 2
C. Optimality conditions
The optimality conditions determine the input from the Hamiltonian and the active constraints, while costates and multipliers track their effects. A candidate labeled solution is optimal when its constraint and multiplier tests hold throughout the horizon.
C. Optimality conditions
- The costate carries the effect of the remaining horizon into the input and obeys a terminal-value differential equation.Its terminal condition is λ(tf) = Px(tf).Source: C. Optimality conditions · p. 2
- The optimal input minimizes the Hamiltonian, and R > 0 makes the minimizer unique with necessary and sufficient conditions.With no active constraint, the input is ufree = −B⊤λ/R; an active constraint fixes the input and yields its multiplier.Source: C. Optimality conditions · p. 3
- A labeled solution passes the check only if inactive constraints stay feasible and active-constraint multipliers remain nonnegative throughout the horizon.The maxima for this check are located using a closed-form bound on second derivatives, not a time grid.Source: C. Optimality conditions · p. 3
III. COMPUTING A BOUNDARY IN FIVE STEPS
Boundary equations combine switching conditions for the lower-arc label with an event condition for the extra arc. Counting the unknown times determines whether to solve directly or fix and step one instant.
III. COMPUTING A BOUNDARY IN FIVE STEPS
- At each switching time, impose g_i = 0 for an entry or μ_i = 0 for an exit; the extra arc contributes its own event condition.For an extra arc inside a free arc, its constraint function also has zero time derivative at the event; for a free extra arc inside a constrained arc, the multiplier and its derivative vanish.Source: §III · p. 3
- Count the switching times of S and include the event time only when it lies inside an arc; this gives the number m of unknown instants.When the event is at the start or end of the horizon, its time is already known.Source: §III · p. 3
- If m = 0, the boundary is a hyperplane; if m = 1, fix that instant, solve the two linear equations for x0, and step it along the horizon.Each step first illustrates the computation on the evaporator and then states the corresponding general result.Source: §III · p. 3
A. Step 1. The solution on the boundary as a formula in x0
For the evaporator’s no-constraint label, the whole horizon is a free arc, with state and costate propagated from the initial point. The resulting trajectory gives a time-dependent linear expression for constraint 3.
A. Step 1. The solution on the boundary as a formula in x0
- For the empty label, the evaporator trajectory spans the horizon as one free arc, with z(t) = e^Mt z(0).Here z comprises the state and costate, and M comes from the model equations and terminal condition.Source: A. Step 1. The solution on the… · p. 3
- The terminal condition yields z(t; x0) = e^Mt W x0, linking the trajectory at each instant to the initial state.This provides the trajectory formula used to evaluate the constraint along the horizon.Source: A. Step 1. The solution on the… · p. 3
- On the free arc, the input is u_free = 0.25(λ1 + λ2), and constraint 3 is g3 = 5x2 + 12.5u − 7.The passage expresses this constraint function as cᵀz − 7, with cᵀ = [0 5 3.125 3.125].Source: A. Step 1. The solution on the… · p. 3
What is fixed, and why:
With a label and its switching times fixed, the state and costate depend affinely on the initial state. Consequently, boundary conditions at any fixed instant are linear equations in x0, even though their coefficients vary with time.
What is fixed, and why:
- For fixed t, each constraint function, multiplier, or derivative is linear in x0, while its coefficients may vary nonlinearly with t.For example, fixing t = 2.5 min gives 2.6230 x0,1 + 6.2224 x0,2 = 7 for the stated constraint function.Source: What is fixed, and why · p. 3
- For fixed switching times, each arc propagates the state and costate through a linear differential equation with constant coefficients.The flow is applied successively across arcs to obtain the trajectory along the horizon.Source: What is fixed, and why · pp. 3–4
- Lemma 1 gives an affine trajectory in x0 for a fixed label and switching times, with coefficients independent of x0.The terminal condition determines the initial costate through a linear solve; strict convexity makes the relevant matrix invertible.Source: What is fixed, and why · p. 4, Figure 4 · p. 4
B. Step 2. The equations that hold on the boundary
The boundary conditions come from the junction conditions for changes in active constraints and from the extra arc’s event. The evaporator example applies them at the lower-label switch and at the event that separates neighboring regions.
B. Step 2. The equations that hold on the boundary
- For the boundary between {3} and the empty label, constraint 3’s exit condition is μ3(t1x) = 0, while the extra free arc’s event condition is μ3(0) = 0.At point B, t1x = 5.44 min; for the neighboring empty-label region at C, the entry and exit times are 1.38 min and 4.55 min.Source: B. Step 2. The equations that hold on… · p. 4
- At an entry from a free arc, g_i(τ) = 0; at an exit into a free arc, μ_i(τ) = 0.If entry and exit coincide, the relevant quantity and its time derivative both vanish.Source: B. Step 2. The equations that hold on… · p. 4
- When a quantity only touches zero at an interior event, both it and its time derivative vanish; direct switches between two constraints are left to the companion paper.The evaporator example has no direct switch from one constraint to another.Source: B. Step 2. The equations that hold on… · p. 4
C. Step 3. Count the unknowns, fix one and solve
Boundary computation reduces to solving linear equations in the initial state after fixing any unknown switching or event time. With no unknown time the boundary is one equation; with one, stepping the fixed time traces it.
C. Step 3. Count the unknowns, fix one and solve
- For the evaporator’s row-3 boundary, fixing t1x = 4 min gives x0 = (−1.0835, 1.6288), and stepping t1x traces the purple curve.The event time is zero throughout, but the boundary curves because t1x enters the costate through the terminal condition.Source: C. Step 3. Count the unknowns, fix one… · p. 4
- For row 4, the event conditions give N(t∗)x0 = r, so x0(t∗) = N(t∗)−1r traces the boundary as t∗ is stepped.At t∗ = 2.5 min, the whole-horizon check is satisfied and g3 touches zero only at that instant.Source: C. Step 3. Count the unknowns, fix one… · p. 5
- With m unknown times besides x0, there are m + 1 boundary equations; when independent, the boundary has dimension n−1.The unknowns are x0, the K−1 switching times, and an event time only when the extra arc lies inside an arc of S.Source: C. Step 3. Count the unknowns, fix one… · p. 5
- For m = 1, fix the switching or event time, solve two linear equations for x0, then step the fixed time across its range.Each candidate point is checked over the full horizon; a failed check indicates that the point lies past a boundary end.Source: C. Step 3. Count the unknowns, fix one… · p. 5
Evaporator:
The evaporator boundaries end when an additional condition is met, and the end-point conditions identify which neighboring regions meet there. The computed examples include a hyperplane endpoint and two curved boundaries beginning at the same point.
Evaporator:
- The hyperplane reaches x0,1 = 2 at (2.0000, −1.4299), and ends at E = (−0.7221, 1.3090), where g3(0) and its slope are zero.At E, the row-4 curve begins at t∗ = 0 and the row-3 curve begins as t1x → 0.Source: Evaporator · p. 5
- The two curved boundaries from E end on different domain sides: row 4 at x0,1 = −2 with t∗ = 4.995 min, and row 3 at x0,2 = 2 with t1x = 7.74 min.A boundary ends at the first additional condition; for m = 1, that condition determines the fixed switching time.Source: Evaporator · p. 5
- In general, a boundary ends when one extra condition holds: it reaches a domain side, another constraint changes status, an endpoint condition is violated, or an arc vanishes.Each case adds one equation, and the check at computed points identifies which occurs first.Source: Evaporator · p. 5, Table III · p. 6
E. Step 5. The regions at an end
- At E, the signs of g3(0) and its slope distinguish three regions: the constraint is inactive, active from the start, or active on an inner arc.Both quantities are zero at E, where the three regions and their boundaries meet.Source: E. Step 5. The regions at an end · pp. 5–6, Figure 6 · p. 6
- More generally, an interior end where another constraint changes status has four adjacent regions; an endpoint value-and-slope event has three.The adjacent labels correspond to the base solution and the insertion of either or both additional arcs in the four-region case.Source: E. Step 5. The regions at an end · p. 6
IV. CONCLUSION
The paper gives an exact procedure for computing continuous-time critical-region boundaries from linear equations in the initial state, without solving an optimal control problem. On the evaporator, it computes hyperplane and curved boundaries, their ends, and the adjacent regions.
IV. CONCLUSION
- Fixing the label and its switching times makes every boundary condition linear in the initial state, enabling exact boundary computation without solving an optimal control problem.The conditions are the switching conditions of S and the event condition of the extra arc.Source: §IV · p. 6
- With no unknown switching time, one equation gives a hyperplane; with one, solving two linear equations while stepping the fixed time gives the exact curved boundary.Boundary ends are found when one more quantity reaches zero, and signs of the end conditions identify adjacent regions.Source: §IV · p. 6
- For the Newell–Lee evaporator, the method yields four hyperplanes in closed form and four curved boundaries from two-by-two linear systems.The procedure for visiting every boundary and cases with general state dimension or several unknown switching times are left to a companion paper.Source: §IV · p. 6