Global Performance for Localized MPC - Global Performance Guarantees

Chapter IV: Global Performance Guarantees

4.3 Global Performance for Localized MPC

In this section, we analyze the effect of locality constraints𝚽 ∈ L on the optimal performance of the MPC problem (2.7). We are especially interested in scenarios where localized MPC achievesoptimal global performance, i.e., 𝑓^∗ = 𝑓^∗

L, where 𝑓^∗ and 𝑓^∗

L are the solutions to the global MPC problem and localized MPC problem, respectively, for some state𝑥₀.

First, we must analyze the space of available trajectories from state𝑥₀for both global and localized MPC. We denote an available trajectoryy:=

x_1:^⊺_𝑇 u^⊺ i_⊺

Definition 6. Trajectory set Y (𝑥₀) denotes the set of available trajectories from state𝑥₀under dynamics(2.1):

Y (𝑥₀) :={y:∃𝚽s.t. 𝑍_{𝐴 𝐵}𝚽=𝐼 ,ˆ y=𝚽₂𝑥₀}.

Localized trajectory setY_L(𝑥₀) denotes the set of available trajectories from state 𝑥₀under dynamics(2.1)and locality constraint𝚽∈ L𝑑:

Y_L(𝑥₀) :={y:∃𝚽s.t. 𝑍_{𝐴 𝐵}𝚽=𝐼 ,ˆ 𝚽∈ L𝑑, y=𝚽₂𝑥₀}.

Lemma 7. (Optimal global performance) Given an initial condition𝑥₀, if the local communication constraint set L𝑑 is chosen such that Y (𝑥₀) = Y_L(𝑥₀), then the localized MPC problem will attain optimal global performance.

Proof. Global MPC problem (4.1) can be written as miny

𝑓(y)

s.t. y∈ Y (𝑥(𝜏)) ∩ P.

Similarly, the localized MPC problem can also be written in this form by replacing Y (𝑥(𝜏)) with Y_L(𝑥(𝜏)). Thus, if Y (𝑥(𝜏)) = Y_L(𝑥(𝜏)), the two problems are

equivalent and will have the same optimal values. □

Remark: This is a sufficient but not necessary condition for optimal global performance. Even if this condition is not satisfied, i.e., Y_L(𝑥₀) ⊂ Y (𝑥₀), the optimal global trajectory may be contained withinY_L(𝑥₀). However, this is dependent on objective 𝑓. Our analysis focuses on stricter conditions which guarantee optimal global performance foranyobjective function.

We now explore cases in whichY (𝑥₀) =Y_L(𝑥₀), i.e., the localized MPC problem attains optimal global performance. Localized trajectory setY_L(𝑥₀) is shaped by the dynamics and locality constraints:

𝑍_{𝐴 𝐵}𝚽=𝐼ˆ (4.2a)

𝚽 ∈ L. (4.2b)

To obtain a closed-form solution forY_L(𝑥₀), we will parameterize these constraints.

Two equivalent formulations are available. The dynamics-first formulation parameterizes constraint (4.2a), then (4.2b), and the locality-formulation parameterizes the constraints in the opposite order. The dynamics-first formulation clearly shows how local communication constraints affect the trajectory space; the locality-first formulation is less clear in this regard, but can be implemented in code with lower computational complexity than the dynamics-first formulation. We now derive each formulation.

Dynamics-first formulation

We first parameterize (4.2a), which gives a closed-form expression for trajectory set Y (𝑥₀).

Lemma 8. (Image space representation of trajectory set) The trajectory set from state𝑥₀is described by:

Y (𝑥₀) ={y:y=𝑍_𝑝𝑥₀+𝑍_ℎ𝑋 𝜆, 𝜆∈R^𝑁^Φ}, where 𝑍_𝑝 := (𝑍^†

𝐴 𝐵)𝑁_𝑥+1:,:𝐼ˆ and 𝑍_ℎ := (𝐼 − 𝑍^†

𝐴 𝐵𝑍_{𝐴 𝐵})𝑁_𝑥+1:,:; and the size of the trajectory set is

dim(Y (𝑥₀)) =rank(𝑍_ℎ𝑋). If𝑥₀has at least one nonzero value, thendim(Y (𝑥₀))= 𝑁_𝑢𝑇 .

Proof. We start by noticing that for any𝚽𝑥satisfying constraint (4.1d),𝚽₁=𝐼. This is due to the structure of𝑍_{𝐴 𝐵}. Also, constraint (4.1d) is always feasible, with solution space of dimension𝑁_𝑢𝑇. To see this, notice that rank(𝑍_{𝐴 𝐵}) =rankh

𝑍_{𝐴 𝐵} 𝐼ˆ i

always holds, since𝑍_{𝐴 𝐵}has full row rank due to the identity blocks on its diagonal; apply the Rouché-Capelli theorem [78] to get the desired result. Hence, we can parameterize the space of solutions𝚽as:

𝚽=𝑍^†

𝐴 𝐵

𝐼 + (𝐼−𝑍^†

𝐴 𝐵

𝑍_{𝐴 𝐵})Λ (4.3)

whereΛ is a free variable with the same dimensions as𝚽. Since 𝚽₁ = 𝐼 always holds, we can omit the first 𝑁_𝑥 rows of (4.3). Hence,

𝚽₂= 𝑍_𝑝+𝑍_ℎΛ. (4.4)

Combining (4.4) and the definition ofy, we have

y=𝚽₂𝑥₀= 𝑍_𝑝𝑥₀+𝑍_ℎΛ𝑥₀. (4.5) Making use of augmented state𝑋, rewrite this as

y=𝚽₂𝑥₀= 𝑍_𝑝𝑥₀+𝑍_ℎ𝑋

⃗⃗⃗⃗

Λ. (4.6)

This gives the desired expression forY (𝑥₀)and its size.

Notice that if 𝑥₀ has at least one nonzero value, then rank(𝑍_ℎ𝑋) = rank(𝑍_ℎ) due to the structure of 𝑋. All that is left is to show rank(𝑍_ℎ) = 𝑁_𝑢𝑇. First, note that rank(𝑍_{𝐴 𝐵}) = 𝑁_𝑥(𝑇 +1) due to the identity blocks on the diagonal of 𝑍_{𝐴 𝐵}. It follows that rank(𝑍^†

𝐴 𝐵) =rank(𝑍^†

𝐴 𝐵

𝑍_{𝐴 𝐵}) = 𝑁_𝑥(𝑇 +1). Thus, rank(𝐼 −𝑍^†

𝐴 𝐵

𝑍_{𝐴 𝐵}) = 𝑁_Φ−𝑁_𝑥(𝑇 +1) = 𝑁_𝑢𝑇. Recall that𝑍_ℎis simply𝐼−𝑍^†

𝐴 𝐵

𝑍_{𝐴 𝐵}with the first𝑁_𝑥rows removed; this does not result in decreased rank, since all these rows are zero (recall that𝚽₁is always equal to 𝐼). Thus, rank(𝑍_ℎ) =rank(𝐼− 𝑍^†

𝐴 𝐵

𝑍_{𝐴 𝐵})= 𝑁_𝑢𝑇. □

To write the closed form of localized trajectory setY_L(𝑥₀), we require some definitions:

Definition 7. Constrained vector indices 𝔏denote the set of indices of𝚽₂that are constrained to be zero by the locality constraint(4.2b), i.e.

(⃗⃗⃗⃗⃗⃗⃗⃗

𝚽₂)_𝔏=0⇔𝚽 ∈ L. Let𝑁_𝔏be the cardinality of𝔏.

We now parameterize (4.2b) and combine this with Lemma 8, which gives a closed- form expression for localized trajectory setY_L(𝑥₀).

Lemma 9. (Image space representation of localized trajectory set) Assume there exists some 𝚽 that satisfies constraints (4.2a) and (4.2b). Then, the localized trajectory set from state𝑥₀is described by the following:

Y_L(𝑥₀) ={y: y=𝑍_𝑝𝑥₀+𝑍_ℎ𝑋 𝐹^†𝑔+𝑍_ℎ𝑋(𝐼 −𝐹^†𝐹)𝜇, 𝜇∈R^𝑁^𝔏}, where𝐹 := (𝑍^blk

ℎ )_𝔏,_:and𝑔 :=−(⃗⃗⃗⃗⃗⃗⃗

𝑍_𝑝)_𝔏; and the size of the localized trajectory set is dim(YL(𝑥₀)) =rank(𝑍_ℎ𝑋(𝐼−𝐹^†𝐹)).

Proof. Using the augmented matrix of𝑍_ℎ, we can write the vectorization of (4.4)

as ⃗⃗⃗⃗⃗⃗⃗⃗

𝚽₂ =

⃗⃗⃗⃗⃗⃗⃗

𝑍_𝑝+𝑍^blk

ℎ

⃗⃗⃗⃗

Λ (4.7)

where ⃗⃗⃗⃗

Λ is a free variable. Incorporate locality constraint (4.2b) using the constrained vector indices:

(⃗⃗⃗⃗⃗⃗⃗

𝑍_𝑝+𝑍^blk

ℎ

⃗⃗⃗⃗

Λ)_𝔏 =0. (4.8)

This is equivalent to(⃗⃗⃗⃗⃗⃗⃗

𝑍_𝑝)_𝔏+ (𝑍^blk

ℎ )_𝔏,:

⃗⃗⃗⃗

Λ =0, or𝐹

⃗⃗⃗⃗

Λ =𝑔. We can parameterize this constraint as

⃗⃗⃗⃗

Λ = 𝐹^†𝑔+ (𝐼−𝐹^†𝐹)𝜇 (4.9)

where𝜇is a free variable. Plugging this into (4.6) gives the desired expression for Y_L(𝑥₀)and its size. We remark that there is no need to consider𝚽₁ =𝐼 in relation to the locality constraints, since the diagonal sparsity pattern of the identity matrix (which corresponds to self-communication, i.e., node𝑖 ‘communicating’ to itself)

satisfies any local communication constraint. □

Theorem 2. (Optimal global performance) If 𝑥₀ has at least one nonzero value, then localized MPC attains optimal global performance if:

1. there exists some𝚽that satisfies constraints(4.2a)and(4.2b), and 2. rank(𝑍_ℎ𝑋(𝐼 −𝐹^†𝐹))= 𝑁_𝑢𝑇.

Proof. By definition,Y_L(𝑥₀) ⊆ Y (𝑥₀). Equality is achieved if and only if the two sets are of equal size. Applying Lemmas 8 and 9 shows that the conditions of the theorem are necessary and sufficient forY_L(𝑥₀)andY (𝑥₀)to be equal. Then, apply

Proposition 7 for the desired result. □

This formulation provides intuition on how the inclusion of locality constraints affects the trajectory set–this is made clear in numerical examples in Section 4.3.

However, checking the conditions of Theorem 2 is computationally expensive. In particular, we must assemble and compute the rank of matrix𝑍_ℎ𝑋(𝐼 −𝐹^†𝐹). The complexity of this operation is dependent on the size of the matrix, which increases as𝚽becomes more sparse (as enforced by locality constraints). This is a problem, since it is generally preferable to use very sparse𝚽, as previously mentioned–this would correspond to an extremely large matrix 𝑍_ℎ𝑋(𝐼 − 𝐹^†𝐹), which is time- consuming to compute with. Ideally, sparser𝚽should instead correspond tolower complexity; this is the motivation for the next formulation.

Locality-first formulation

We first parameterize (4.2b), then (4.2a). This directly gives a closed-form expression for localized trajectory set Y_L(𝑥₀). We introduce some definitions that will prove helpful in the derivation of this formulation.

Definition 8. Support vector indices 𝔐 denote the set of indices of

⃗⃗⃗⃗⃗

𝚽 such that (⃗⃗⃗⃗⃗

𝚽)_𝔐 ≠ 0is compatible with locality constraint(4.2b). Let𝑁_𝔐 be the cardinality of𝔐.

Remark: this is complementary to Definition 7. Instead of looking at which indices are constrained to be zero, we now look at which indices are allowed to be nonzero.

A subtlety is that this definition considers the entirety of𝚽, while Definition 7 omits the first𝑁_𝑥rows of𝚽.

Lemma 10. (Image space representation of localized trajectory set) Assume there exists some 𝚽 that satisfies constraints (4.2a) and (4.2b). Then, the localized trajectory set from state𝑥₀is described by the following:

Y_L(𝑥₀) ={y: y=(𝑋₂)_:_,𝔐𝐻^†𝑘 + (𝑋₂)_:_,𝔐(𝐼 −𝐻^†𝐻)𝛾 , 𝛾 ∈R^𝑁^𝔐}, where 𝑋₂ := (𝑋)𝑁_𝑥+1:,:, 𝐻 = (𝑍^blk

𝐴 𝐵)_:_,𝔐 and 𝑘 :=

⃗⃗

𝐼ˆ; and the size of the localized trajectory set is

dim(Y_L(𝑥₀)) =rank( (𝑋₂)_:,𝔐(𝐼−𝐻^†𝐻)). Proof. Future trajectoryycan be written as

y=𝑋₂

⃗⃗⃗⃗⃗

𝚽= (𝑋₂)_:,𝔐(⃗⃗⃗⃗⃗

𝚽)_𝔐 (4.10)

where the first equality arises from the definitions ofyand𝑋, and the second equality arises from the fact that zeros in ⃗⃗⃗⃗⃗

𝚽 do not contribute to y; thus, we only need to consider nonzero values(⃗⃗⃗⃗⃗

𝚽)_𝔐.

Using the augmented matrix of𝑍_{𝐴 𝐵}, constraint (4.2a) can be rewritten as𝑍^blk

𝐴 𝐵

⃗⃗⃗⃗⃗

𝚽=𝑘. Nonzero values (⃗⃗⃗⃗⃗

𝚽)_𝔐must obey

𝐻(⃗⃗⃗⃗⃗

𝚽)_𝔐 =𝑘 . (4.11)

Constraint (4.11) is feasible exactly when constraints (4.2a) and (4.2b) are feasible.

By assumption, solutions exist, so we can parameterize the solution space as (

⃗⃗⃗⃗⃗

𝚽)_𝔐 =𝐻^†𝑘+ (𝐼−𝐻^†𝐻)𝛾 (4.12) where𝛾is a free variable. Substituting (4.12) into (4.10) gives the desired expression

forY_L(𝑥₀)and its size. □

Theorem 3. (Optimal global performance) If 𝑥₀ has at least one nonzero value, then localized MPC attains optimal global performance if:

1. there exists some𝚽that satisfies constraints(4.2a)and(4.2b), and 2. rank( (𝑋₂)_:,𝔐(𝐼−𝐻^†𝐻)) =𝑁_𝑢𝑇.

Proof. Similar to Theorem 2; instead of applying Lemma 9, apply Lemma 10. □

To check the conditions of Theorem 3, we must assemble and compute the rank of matrix(𝑋₂)_:,𝔐(𝐼−𝐻^†𝐻). The complexity of this operation is dependent on the size of this matrix, which decreases as𝚽becomes more sparse (as enforced by locality constraints). This is beneficial, since it is preferable to use very sparse 𝚽, which corresponds to a small matrix(𝑋₂)_:_,𝔐(𝐼−𝐻^†𝐻)that is easy to compute with. This is in contrast with the previous formulation, in which sparser 𝚽 corresponded to increased complexity. However, from a theoretical standpoint, this formulation does not provide any intuition on the relationship between the trajectory setY (𝑥₀) and the localized trajectory setY_L(𝑥₀).

For completeness, we now use the locality-first formulation to provide a closed-form expression of Y (𝑥₀). The resulting expression is equivalent to–though decidedly more convoluted than–the expression in Lemma 8:

Lemma 11. (Image space representation of trajectory set) The trajectory set from state𝑥₀is described by:

Y (𝑥₀)= {y: y= 𝑋₂𝑍^blk

𝐴 𝐵𝑘+𝑋₂(𝐼−𝑍^blk†

𝐴 𝐵 𝑍^blk

𝐴 𝐵)𝛾 , 𝛾 ∈R^𝑁^Φ}. Proof. In the absence of locality constraints,𝔐 includes all indices of ⃗⃗⃗⃗⃗

𝚽since all entries are allowed to be nonzero. Here,𝑁_𝔐 =𝑁_Φ, (𝑋₂)_:_,𝔐 = 𝑋₂, (⃗⃗⃗⃗⃗

𝚽)_:_,𝔐 = ⃗⃗⃗⃗⃗

𝚽, and 𝐻 = 𝑍^blk

𝐴 𝐵. Substitute these into the expression in Lemma 10 to obtain the desired

result. □

Remark: By definition,𝑋₂𝑍^blk

𝐴 𝐵

𝑘 =𝑍_𝑝𝑥₀and𝑋₂(𝐼−𝑍^blk†

𝐴 𝐵

𝑍^blk

𝐴 𝐵) =𝑍_ℎ𝑋. Substituting these quantities into Lemma 11 recovers Lemma 8.

Numerical example

To provide some intuition on the results from the previous subsections, we present a simple numerical example. We work with a system of three nodes in a chain interconnection, as shown in Fig. 4.1.

𝑢₁ 𝑢₂

plant actuation

𝑥₁ 𝑥₂ 𝑥₃

Figure 4.1: Example system with three nodes, two of which are actuated.

The system matrices are:

𝐴=







1 2 0 3 4 5 0 6 7





 , 𝐵 =





 1 0 0 0 0 1







. (4.13)

We set initial state𝑥₀= h

1 1 1 i⊤

, and choose a predictive horizon size of𝑇 =1.

Here,𝑁_Φ =𝑁_𝑥(𝑇 +1) +𝑁_𝑢𝑇 =8. We choose locality constraintsL such that each node may only communicate with its immediate neighbors; node 1 with node 2, node 2 with both nodes 1 and 3, and node 3 with node 2. Then, locality constraint (4.2b) is equivalent to

𝚽=







∗ ∗ 0

∗ ∗ ∗ 0 ∗ ∗

∗ ∗ 0

∗ ∗ ∗ 0 ∗ ∗

∗ ∗ 0 0 ∗ ∗







(4.14)

where∗indicate values that are allowed to be nonzero. The support vector indices are𝔐 = {1,2,4,5,7,9−16,18,19,21,22,24}, and the constrained vector indices are𝔏 ={3,5,11,14}(recall that indices in𝔏do not include the first𝑁_𝑥rows of𝚽).

We confirm that there exists some𝚽that satisfies both dynamics constraints (4.2a) and locality constraints (4.2b) by checking that constraint (4.11) is feasible.

We start with dynamics-first formulation. In our case, 𝑍_ℎ=

0₅×3 𝑐₁ 0₅×1 𝑐₂ 𝑐₁ 𝑐₂ i

(4.15) where𝑐₁and𝑐₂are defined as

𝑐₁:= 1 2





 1 0 0 1 0







, 𝑐₂:= 1 2





 0 0 1 0 1







(4.16)

𝑍_ℎhas a rank of 2. Then,𝑍_ℎ𝑋 = h

𝑍_ℎ 𝑍_ℎ 𝑍_ℎ i

, also has a rank of 2. Per Lemma 8, this is the size of the trajectory setY (𝑥₀), which is exactly equal to𝑁_𝑢𝑇, as expected.

𝑍_ℎ𝑋 =

0_5×3 𝑐₁ 0_5×1 𝑐₂ 𝑐₁ 𝑐₂ 0_5×3 𝑐₁ 0_5×1 𝑐₂ 𝑐₁ 𝑐₂ 0_5×3 𝑐₁ 0_5×1 𝑐₂ 𝑐₁ 𝑐₂

(4.17) 𝑍_ℎ𝑋(𝐼 −𝐹^†𝐹)=

0_5×3 𝑐₁ 0_5×2 𝑐₁ 0_5×4 𝑐₁ 0_5×1 𝑐₂ 𝑐₁ 𝑐₂ 0_5×5 𝑐₂ 0_5×1 𝑐₂

(4.18) (𝑋₂)_:,𝔐(𝐼−𝐻^†𝐻) =

0₅×2 𝑐₁ 0₅×1 𝑐₁ 0₅×3 𝑐₁ 0₅×1 𝑐₂ 𝑐₁ 𝑐₂ 0_5×3 𝑐₂ 𝑐₂

(4.19) We write out𝑍_ℎ𝑋 and𝑍_ℎ𝑋(𝐼−𝐹^†𝐹)in full in equations (4.17) and (4.18). Boxed columns represent columns zeroed out as a result of locality constraints, i.e., if we replace the boxed columns in 𝑍_ℎ𝑋 with zeros, we obtain 𝑍_ℎ𝑋(𝐼 − 𝐹^†𝐹). In our example, 𝑍_ℎ𝑋(𝐼−𝐹^†𝐹)also has a rank of 2; by Theorem 2, the local trajectory set is equal to the trajectory set, and by Theorem 2, the localized MPC problem attains optimal global performance.

Two observations are in order. First, we notice that the rank of 𝑍_ℎ𝑋 (= 2) is low compared to the number of nonzero columns (= 12), especially when 𝑥₀ is dense. Additionally, the structure of𝑍_ℎ𝑋 is highly repetitive; the only two linearly independent columns are𝑐₁and𝑐₂, and each appears 6 times in𝑍_ℎ𝑋. Furthermore, the specific values of𝑥₀do not affect the rank of these matrices–only the placement of nonzeros and zeros in 𝑥₀ matters. Second, notice that post-multiplying 𝑍_ℎ𝑋 by (𝐼 −𝐹^†𝐹) effectively zeros out columns of 𝑍_ℎ𝑋. However, due to the repetitive structure of 𝑍_ℎ𝑋, this does not result in decreased rank for𝑍_ℎ𝑋(𝐼−𝐹^†𝐹). In fact, it is difficult to find a feasible locality constraint that results in decreased rank. This observation is corroborated by simulations in Section 4.5, in which we find that locality constraints that are feasible also typically preserve global performance. For more complex systems and larger predictive horizons, post-multiplication of 𝑍_ℎ𝑋 by (𝐼 − 𝐹^†𝐹) no longer cleanly corresponds to zeroing out columns, but similar intuition applies.

We now apply the locality-first formulation. To check the conditions of Theorem 3, we must construct the matrix (𝑋₂)_:,𝔐(𝐼−𝐻^†𝐻)and check its rank. This matrix is written out in equation (4.19). As expected, the rank of this matrix is also equal to two. Additionally, notice that(𝑋₂)_:,𝔐(𝐼−𝐻^†𝐻)contains the same nonzero columns as𝑍_ℎ𝑋(𝐼−𝐹^†𝐹): 𝑐₁and𝑐₂are each repeated four times, in slightly different orders.

This is unsurprising, as the two formulations are equivalent.

Dalam dokumen Distributed and Localized Model Predictive Control (Halaman 69-78)