Equilibrium Endpoint Constraint

A simple way of constructing stabilizing terminal constraints consists of explicitly including the desired reference solution in the optimization constraints. We intro-duce this variant for the case of a constant referencex^ref=x_∗and the corresponding Algorithm 3.10 and discuss the general case of a time varying reference at the end of this section, cf. Problem (5.14). Within Algorithm 3.10 we use the optimization problem (OCPN,e) withX0= {x_∗},F ≡0 andωk=1 fork=0, . . . , N−1. This means that we specialize (OCPN,e) to

minimize JN

x0, u(·) :=

N−1 k=0

xu(k, x0), u(k) with respect to u(·)∈U^N_X0(x0) withX0= {x_∗} subject to x_u(0, x0)=x₀, x_u(k+1, x0)=f

x_u(k, x₀), u(k) .

(5.5)

5.2 Equilibrium Endpoint Constraint 89 Recall from Definition 3.9 that each trajectoryx_u(·, x₀)withu(·)∈U^N_X0(x₀) sat-isfiesx_u(N, x₀)∈X0, i.e., x_u(N, x₀)=x_∗. Thus, in (5.5) we only optimize over trajectories satisfying this equilibrium endpoint constraint. Note that (5.5) is only well defined ifx₀is an element of the feasible setXNfrom Definition 3.9.

The idea behind the equilibrium endpoint constraintx_u(N, x₀)=x_∗is intuitive:

since we want our closed-loop system to converge tox_∗ we simply add this re-quirement as a constraint to the optimal control problem. And since “convergence”

is difficult to formalize for the finite horizon predictions it appears reasonable to require the predictions to end exactly at the desired equilibrium.

For the analysis of this problem we will use the following assumptions.

Assumption 5.1

(i) The pointx_∗∈Xis an equilibrium for an admissible control valueu_∗, i.e., there exists a control valueu_∗∈U(x_∗)withf (x_∗, u_∗)=x_∗.

(ii) The running cost:X×U→R⁺₀ satisfies(x_∗, u_∗)=0 foru_∗from (i).

Observe that Assumption5.1(i) is nothing else but a viability assumption for X0= {x_∗}, cf. Assumption 3.3. In order to show that Inequality (5.4) is indeed re-versed for Problem (5.5) satisfying Assumption5.1, we first need an auxiliary re-sult for the feasible setsXN and the corresponding admissible control sequences U^N_X0(x0)from Definition 3.9. For the specific constraintxu(N, x0)=x_∗in (5.5), the following lemma states that each admissible control sequence on the horizonN−1 can be extended to an admissible control sequence on the horizonN.

Lemma 5.2 If Assumption5.1(i) holds for the terminal constraint set X0= {x_∗}, then for eachN≥2 the following properties hold.

(i) For eachx₀∈XN−1and eachu_N−1(·)∈U^N_X0⁻¹(x₀)the control sequence u_N(k):=u_N−1(k), k=0, . . . , N−2, u_N(N−1):=u_∗ (5.6) satisfiesu_N∈U^N_X0(x₀).

(ii) The inclusionXN−1⊆XNholds.

Proof (i) The idea of the proof is simple: since the trajectory related tou_N−1(·)∈ U^N_X0⁻¹(x₀)ends up inx_∗, the trajectory corresponding to the prolonged control se-quenceu_Nfrom (5.6) satisfiesx_uN(N, x0)=x_∗.

In order to verify u_N ∈U^N_X0(x0)we need to show thatx_uN(k, x0)∈Xfork= 0, . . . , N,u_N(k)∈U(x_uN(k, x0))fork=0, . . . , N−1 andx_uN(N, x0)=x_∗.

FromuN−1∈U^N_X0⁻¹(x0)and Lemma 3.12 we obtain xu_N−1(k, x0)∈XN−1−k⊆X, uN−1(k)∈U

xu_N−1(k, x0)

, k=0, . . . , N−2 (5.7) and

xu_N−1(N−1, x0)=x_∗∈X0⊆X (5.8)

90 5 Stability and Suboptimality Using Stabilizing Constraints and from (5.8) and the definition ofu_Nwe get

x_uN(k, x0)=x_uN−1(k, x0), k=0, . . . , N−1, x_uN(N, x0)=x_∗. (5.9) Hence (5.7) and (5.8) are also valid forxu_N. In addition, we getuN(N−1)=u_∗∈ U(x_∗)=U(xu_N(N−1, x0))andxu_N(N, x0)=f (xu_N(N−1, x0), uN(N−1))= f (x_∗, u_∗)=x_∗. Thus,uN∈U^N_X0(x0).

(ii) Letx0∈XN−1. Then there existsuN−1∈U^N−_X0 ¹(x0)and by (i) we can con-clude that there exists u_N ∈U^N_X0(x₀). Thus, U^N_X0(x₀)= ∅, from which x₀∈XN

follows.

Observe that we did not need to impose viability of the constraint setXin this proof. In fact, we implicitly used that under Assumption5.1(i) the setXNis forward invariant for all admissible control sequencesU^N_X0(x). We explicitly formulate a consequence of this property for the NMPC closed-loop system in the following lemma.

Lemma 5.3 Under Assumption5.1(i) for eachN∈Nthe NMPC-feedback lawμ_N obtained from Algorithm 3.10 with (OCPN,e)=(5.5) renders the set XN forward invariant, i.e.,f (x, μ_N(x))∈XN for allx∈XN.

Proof Follows immediately from Corollary 3.13 and Lemma5.2(ii).

This lemma shows that if a statex is feasible, i.e., contained in the feasible set XNthen its closed-loop successor statef (x, μ_N(x))is again feasible. Thus,XNis recursively feasible in the sense defined after Theorem 3.5. Using Lemma5.2it is now easy to establish that Inequality (5.4) is reversed for (5.5).

Lemma 5.4 If Assumptions 5.1(i) and (ii) hold, then for each N ≥2 and each x0∈XN−1the optimal value functions of Problem (5.5) satisfy

V_N(x0)≤VN−1(x0). (5.10) Proof We first show that for eachu_N−1∈U^N_X0⁻¹(x₀) the control sequenceu_N ∈ U^N_X0(x0)from (5.6) satisfies

JN(x0, uN)≤JN−1(x0, uN−1). (5.11) To this end, recall from the proof of Lemma5.2that the trajectoriesx_uN(·, x₀)and x_uN−1(·, x₀)satisfy

x_uN(k, x0)=x_uN−1(k, x0), k=0, . . . , N−1, x_uN(N, x0)=x_∗. Together with (5.6) this yields

J_N(x₀, u_N)=

N−1 k=0

x_uN(k, x₀), u_N(k)

5.2 Equilibrium Endpoint Constraint 91

This shows (5.11). In fact, we even proved “=” but we will only need “≤” for proving (5.10). In order to prove (5.10), letu^k_N−1∈U^N−_X0 ¹(x0),k∈N, be a sequence

Note that for Problem (5.5) in general (5.4) does no longer hold, because the terminal constraint is more restrictive for smaller horizon than for larger ones. Thus, with the terminal constraint we do not get (5.10) on top of (5.4). Rather, we replaced (5.4) by (5.10).

Lemma5.4in conjunction with (5.3) enables us to conclude that the optimal value functionV_N satisfies Inequality (5.1). This will be used in the proof of our following first stability theorem for an NMPC scheme in which we simply assume (5.2). Sufficient conditions for these inequalities will be discussed after the theorem.

Theorem 5.5 Consider the NMPC Algorithm 3.10 with (OCPN,e)=(5.5) and opti-mization horizonN∈N. Let Assumptions5.1(i) and (ii) hold and assume that (5.2) holds for suitableα1, α2, α3∈K∞. Then the nominal NMPC closed-loop system (3.5) with NMPC-feedback lawμNis asymptotically stable onXN.

In addition, forJ_∞(x, μ_N)from Definition 4.10 the inequality J_∞(x, μ_N)≤V_N(x)

holds for eachx∈XN.

Proof Combining Equality (3.20) from Theorem 3.17 with Inequality (5.10) from Lemma5.4withx0=f (x, μN(x)), for eachx∈XNwe obtain

92 5 Stability and Suboptimality Using Stabilizing Constraints Thus, the assumptions of Theorem 4.11 are satisfied with V =V_N, μ=μ_N, S(n)=XN (which is forward invariant by Lemma5.3) and α=1 which yields

the assertion.

The fact that each predicted trajectoryx_u in (5.5) satisfiesx_u(N, x₀)=x_∗does by no means imply that the NMPC closed-loop trajectory satisfiesxμ_N(N )=x_∗. The following example illustrates this fact.

Example 5.6 Consider again Example 2.1, i.e., x⁺=x+u=:f (x, u)

withX=X=U=U=Randx_∗=0. We use the running cost(x, u)=x²+u² and the terminal constraintsX0= {x_∗}.

Observing that everyu(·)∈U¹_X0(x)must satisfyf (x, u(0))=0 we getu(0)=

−x. Hence, (3.19) yields V1(x)= inf

u∈U¹_X

0(x)

x, u(0)

=x²+(−x)²=2x² andμ₁(x)= −x. Now, using (3.19) for (5.5) withN=2 we get

μ2(x)=argmin

u∈R (x, u)+V1

f (x, u)

=argmin

u∈R x²+u²+2(x+u)²

= −2 3x,

which is easily computed by setting the first derivative w.r.t.uof the term in braces to 0, observing that the second derivative is strictly positive. Thus, the NMPC closed loop forN=2 becomes

x⁺=f

x, μ₂(x)

=x−μ₂(x)=x−2 3x=1

3x with solutions

x_μ2(n, x₀)= 1 3ⁿx₀.

Hence, the closed-loop solution asymptotically converges to x_∗ =0 but never reaches 0 in finite time.

In Theorem5.5we have made the assumption thatV_N satisfies the inequalities in (5.2). In terms of the problem dataf andthis is an implicit condition which may be difficult to check. For this reason, in the following proposition we give a sufficient condition onf andfor these inequalities to hold true.

Proposition 5.7 Let V_N denote the optimal value function of Problem (5.5) for optimization horizonN∈N.

5.2 Equilibrium Endpoint Constraint 93 (i) Assume there exists a functionα₃∈K_∞such that the inequality

(x, u)≥α₃

|x|x_∗

holds for allx∈Xand allu∈U. Then

V_N(x)≥α₃

|x|x_∗

holds for allx∈XN.

(ii) Assume thatf andare continuous inX×U,U is compact and there exists a ballBν(x_∗)⊂X,ν >0, and a functionα˜2∈K∞with the following property:

For eachx∈Bν(x_∗)∩Xthere isu_x∈U(x)withf (x, u_x)=x_∗and (x, ux)≤ ˜α2

|x|x_∗

. (5.12)

Then there existsα2∈K∞such that VN(x)≤α2

|x|x_∗

(5.13)

holds for allx∈XN.

Proof (i) is immediate from the definition ofV_N.

In order to prove (ii), first observe that forx∈Bν(x_∗)∩Xthe existence ofu_x withf (x, u_x)=x_∗immediately impliesx∈X1and

V₁(x)= inf

u∈U¹_X0(x)

x, u(0)

≤(x, u_x)≤ ˜α₂

|x|x_∗

,

because the control sequenceu(·)∈U¹defined byu(0)=u_xlies inU¹_X0(x)since by assumptionf (x, u_x)=x_∗. Now by Lemma5.4the inequality

V_N(x)≤ ˜α₂

|x|x_∗

follows for eachN∈Nand eachx∈Bν(x_∗)∩X.

Forx∈XN outside this ball consider an arbitrary closed ballBr(x_∗)forr >0 and an arbitraryN∈N. Sincef andare assumed to be continuous, the functional JN:X×U^N→R⁺₀ is continuous, too, and sinceU is assumed to be compact the setBr(x_∗)×U^N is also compact (both continuity and compactness hold with the usual product topology onX×U^N). Thus the value

ˆ

α2(r):=max J_N(x, u)x∈Br(x_∗), u∈U^N

exists and is finite and the resulting functionαˆ2is continuous and monotone increas-ing inr. By this definition, for eachx∈XN we get

VN(x)≤ ˆα2

|x|x_∗

. Now define a functionα₂:R⁺₀ →R⁺₀ by

α2(r):=r+

α˜₂(ν)+ ˆα2(r), r≥ν,

˜

α₂(r)+rαˆ₂(ν)/ν, r∈ [0, ν).

This function is continuous since both expressions are equal forr=ν, equal to 0 atr=0 and strictly increasing and unbounded due to the addition ofr. Hence,

94 5 Stability and Suboptimality Using Stabilizing Constraints α₂∈K_∞. Furthermore, it satisfiesα₂(r)≥ ˜α₂(r)for r∈ [0, ν]andα₂(r)≥ ˆα₂(r) forr≥ν. Thus, it is the desired upper bound forV_N. The specific upper boundα2constructed in this proof depends on N and will in general grow unboundedly asN → ∞. However, since by Inequality (5.10) we know that the functionsV_N are decreasing inN we can actually conclude the ex-istence of an upper bound inK_∞which is independent ofN. However, since we did not (and do not want to) assume continuity of theVN the construction of this K_∞-function is somewhat technically involved which is why we skip the details.

If we want to deduce local asymptotic controllability from the asymptotic stabil-ity onXN we need thatXN contains a ballBν(x_∗)around the equilibriumx_∗, cf.

the comment after Definition 2.14. The following example shows that this does not necessarily mean thatXkfork≤N−1 must contain such a ball, too.

Example 5.8 Consider the system x⁺=f (x, u)withx ∈X=X=R²,u∈U= [0,1] ⊂U=R,x_∗=0 and

f (x, u)=

x1(1−u) xu

.

We use the NMPC Algorithm 3.10 with (OCPN,e)=(5.5). Forx=0 the system is controllable toX0= {0}in one step if and only ifx1=0. ThusX1= {x∈R²|x1= 0}which obviously does not contain a neighborhood ofx_∗=0.

On the other hand, using the control sequenceu(0)=1,u(1)=0 for each initial valuex∈R²one obtains

xu(0, x)=x, xu(1, x)=

0,x

, xu(2, x)=0, which impliesX2=R=Xand thusXN=R=Xfor allN≥2.

Furthermore, using the running cost(x, u)= x²we obtain the upper bound V2(x)≤ x²+0,x²=2x²

and the lower boundx²≤V_N(x)for allN≥2. Hence, Theorem5.5implies that the NMPC-feedback lawμNstabilizes the system onXN=R²for eachN≥2.

The method described in this section is easily extended to time varying reference trajectoriesx^ref(·)replacing (5.5) by

minimize J_N

n, x₀, u(·) :=

N−1 k=0

n+k, x_u(k, x₀), u(k) with respect to u(·)∈U^N_X0(n, x0) withX0(n)= x^ref(n) subject to x_u(0, x0)=x0, x_u(k+1, x0)=f

x_u(k, x0), u(k)

(5.14)

and choosing (OCPⁿ_N,e) = (5.14) in Algorithm 3.11. Since x^ref(·)is known, the constraintx_u(n+N, x0)=x^ref(n+N )is as easy to implement as its time invariant counterpart in (5.5). All proofs in this section are easily extended to the time varying case by including the appropriate time instants in,J_N,V_Nandμ_N.

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