Distributed Nonlinear Optimal Control Using Sequential ... - IEEE Xplore

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

WeA16.2

Distributed Nonlinear Optimal Control using Sequential Convex Programming and Smoothing Techniques Ion Necoara, Carlo Savorgnan, Quoc Tran Dinh, Johan Suykens and Moritz Diehl Abstract— We regard a network of coupled nonlinear dynamical systems that we want to control optimally. The cost function is assumed to be separable and convex. The algorithm we propose to address the numerical solution of this problem is based on two ingredients: first, we exploit the convex problem structure using a sequential convex programming framework that linearizes the nonlinear dynamics in each iteration. Second, we use distributed dual decomposition methods to address the decomposable convex subproblems, that allow efficient parallel implementation. We analyze the convergence of the algorithm towards a local solution.

I. I NTRODUCTION For the synthesis of large-scale networked systems, centralized control is often impractical and computationally too demanding for online implementation. Consider for example an electric distribution network or a transportation network. Designing a centralized controller for these systems is not practically viable. Gathering to one single place all the data which determine the current state of the system and calculating all the optimal values of control inputs as a single problem is clearly a very difficult task. When the control problem is formulated as an optimization problem, a valid alternative to centralized control can be achieved using decomposition techniques. Exploiting its structure, the system is divided into subsystems for which the input values are calculated locally. Only the necessary information is exchanged between the subsystems to reach the common optimization goal. Solving the problem of how distributing effectively the computations among the subsystems, has challenged many researchers in the last decades. Several contributions on this subject appeared for general control problems (see e.g. [1], [2], [3]) and in the model predictive control framework (see e.g. [4], [5], [6], [7], [8], [9]). When the dynamics of the subsystems are coupled, a common approach to distribute the computation is using Jacobi or subgradient type algorithms. However, it is wellknown that these methods have a very slow convergence rate (see e.g. [10]). In [11], [12] it has been shown that for convex problems we can use smoothing techniques to achieve distributed algorithms with a great improvement of the convergence rate compared to the previous methods. Ion Necoara is with Automation and System Engineering Department, University Politehnica Bucharest, Romania. Carlo Savorgnan, Quoc Tran Dinh, Johan Suykens and Moritz Diehl are with OPTEC, ESATSCD, Katholieke Universiteit Leuven, Belgium. {ion.necoara, carlo.savorgnan, quoc.trandinh, moritz.diehl, johan.suykens}@esat.kuleuven.be.

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

Paper contribution. This paper contains two contributions. First, we show that combining sequential convex programming (SCP) and smoothing techniques, we can obtain a distributed algorithm also for nonlinear control problems that generalizes the results from [13] corresponding to the linear case. Second, we show that SCP locally converges to a minimizer and, consequently, also the distributed control algorithm we propose has the same property. The paper is organized as follows. In Section II we state the optimal control problem we shall consider. In Section III we introduce the SCP framework and we state some convergence results for this method. In Section IV we show how the SCP problem can be decomposed in order to obtain a distributed control scheme. II. P ROBLEM FORMULATION We consider discrete-time systems which can be decomposed into M subsystems described by difference equations of the form xit+1 = φi (xjt , ujt ; j ∈ N i ) ∀i = 1, . . . , M.

(1)

The vectors xit ∈ Rnxi , uit ∈ Rnui represent the state and the input of the subsystem1 i at time t. The index set N i contains the index i and all the indices of the subsystems which interact with the subsystem i. Consider for example the networked system in Figure 1 where the arrows indicate Σ2

Σ1 Σ4 Fig. 1.

Σ3

An example of networked systems.

interaction between the subsystems Σ1 , Σ2 , Σ3 , Σ4 . If we consider Σ4 we have N 4 = {3, 4} and therefore x4t+1 = φ4 (x3t , x4t , u3t , u4t ). In the problem formulation we assume that the control and state vectors must satisfy local mixed constraints: (xit , uit ) ∈ Xit xiN

∈

XiN

∀t = 0, . . . , N −1 ∀i = 1, . . . , M

∀i = 1, . . . , M

(2)

1 Throughout the paper we will use the convention that every superscript indicates a subsystem index.

543

WeA16.2 where Xit ⊆ Rnxi × Rnui , XiN ⊆ RnxN are convex compact sets and N is the control horizon. To obtain a general formulation we also consider possibly nonlinear constraints of the form gti (xit , uit ) = 0

∀t ∈ T i

where T i is a given set containing integer values in the interval [0, N ]. E.g., the previous constraint can be used to express a constraint on the final value of the state, in which case T i = {N }. The system performance is expressed via a stage and a final cost, which are composed of individual convex costs for each subsystem i and have the form N −1 X

i

ℓ

(xit , uit )

xil ,uil

s.t.

+

s.t.

ℓi (xit , uit ) +

M X

ℓif (xiN )

(3)

i=1 i=1 t=0 xi0 = x ¯i , xit+1 = φi (xjt , ujt ; j ∈ N i ) gti (xit , uit ) = 0 ∀t ∈ T i , ∀i = 1, . . . ,M (xit , uit ) ∈ Xit ∀t = 0, . . . ,N −1 ∀i = 1, . . . ,M xiN ∈ XiN ∀i = 1, . . . ,M

(3.1) (3.2) (3.3) (3.4)

i

where x ¯ are the values of the initial state. Note that a similar formulation of distributed control but for coupled linear subsystems with decoupled costs was given in [8], [13] in the context of model predictive control. The centralized optimization problem (3) becomes interesting if the computations can be distributed among the subsystems (agents) and the amount of information that the agents must exchange is limited. Now, we show that the optimization problem (3) can be recast as a separable optimization problem but with a particular structure. To this purpose, define the variables h T T h T i i T T T T T T xi = xi0 ui0 . . . xiN−1 uiN−1 xiN , x = x1 . . . xM , the sets

Xi = Xi0 × · · · × XiN , X = X1 × · · · × XM and the functions M N −1 X X f i (xi ). ℓi (xit , uit ) + ℓif (xiN ), f (x) = f i (xi ) = t=0

i=1

With this notation, problem (3) now reads min

x1 ∈X1 ···xM ∈XM

s.t.

M X

min

ℓif (xiN ).

The centralized optimal control problem reads: −1 M N X X

III. S EQUENTIAL C ONVEX P ROGRAMMING The underlying idea of SCP is that we can solve the nonconvex optimization problem (5) by iteratively solving a convex approximation of the original problem. Starting from an initial guess x0 ∈ X, the SCP algorithm calculates a sequence {xk }k≥0 by solving the convex approximation of (5) xk+1 ∈X

t=0

min

where h : Rnx → Rnh is obtained by stacking all the functions hi (xj ; j ∈ N i ). Notice that the non-convexity in problems (4) and (5) is concentrated in the equality constraint h(x) = 0.

f i (xi )

i=1 i j

, (4) i

h (x ; j ∈ N ) = 0 ∀i = 1, . . . , M

where the functions hi (xj ; j ∈ N i ) are obtained by stacking the constraints (3.1)-(3.2) for a given i. In a more compact form we can also write min f (x) x∈X (5) s.t. h(x) = 0,

f (xk+1 ) h(xk ) + ∇h(xk )T (xk+1 − xk ) = 0,

(6)

where ∇h(x) denotes the Jacobian of h. Note that we implicitly assumed that the function h is differentiable. We will show that the solution of this convex subproblem is under mild conditions unique, and we denote the solution map from xk to xk+1 by ΦSCP , i.e. ΦSCP (xk ) = xk+1 . The optimization problem we have to solve in each iteration is convex and has a decomposable structure, and therefore is suitable for decomposition using smoothing techniques. Section IV will illustrate how we can efficiently compute the map ΦSCP (xk ) using such a distributed algorithm. The remainder of this section establishes local convergence properties of the SCP method. A. Notation and Regularity Assumptions We regard a local solution x∗ of (5) and investigate under which circumstances the SCP method converges to this solution. We will make a few technical assumptions regarding representation of the convex set X and regularity of the solution x∗ . Assumption 1: The set X is the intersection of the sublevel sets of convex functions: X = {x|cj (x) ≤ 0, j = 1, . . . .nc }. Assumption 2: Functions f, h, c are twice continuously differentiable. Assumption 3: At the solution x∗ holds linear independence constraint qualification (LICQ). Lemma 1 (First order necessary conditions (FONC)): Under Assumptions 1, 2, 3 exist unique multipliers λ∗ ∈ Rnh and ν∗ ∈ Rnc such that   ∇f (x∗ ) + ∇h(x∗ )T λ∗ + ∇c(x∗ )T ν∗   = 0. h(x∗ ) (7) max(c(x∗ ), −ν∗ ) In the last condition – which is non-smooth and a short formulation of ν∗ ≥ 0, c(x∗ ) ≤ 0, c(x∗ )T ν∗ = 0 – the maximum is taken componentwise, so that it has nc components. Assumption 4: At the solution (x∗ , λ∗ , ν∗ ) holds strict complementarity, i.e. ν∗ − c(x∗ ) > 0. Strict complementarity allows us to reorder the inequalities and their multipliers into two disjoint sets of “active” and “inactive” components, with ca (x∗ ) = 0 and ν∗i = 0:  a   a c (x) ν (8) c(x) = i and ν = i . ν c (x)

544

WeA16.2 In the neighborhood of the solution (x∗ , λ∗ , ν∗ ), the first order necessary conditions for optimality can now compactly be written as the differentiable nonlinear system   ∇f (x∗ ) + ∇h(x∗ )T λ∗ + ∇c(x∗ )T ν∗   h(x∗ )  = 0.  (9)   ca (x∗ ) −ν∗i B. Local convergence of the SCP method It is well known that Newton’s method, when applied to the system (9) and started sufficiently close to the solution (x∗ , λ∗ , ν∗ ), converges quadratically under the additional assumption that the Jacobian of the equation system is invertible, which would follow from second order sufficient conditions for optimality. However, here we are not interested in the Newton iteration, but in the SCP iteration. For this aim we formulate a peculiar variation of the above smooth nonlinear conditions, and start by defining a map F : Rnx +nh +nc +nx → Rnx +nh +nc as follows   ∇f (x) + ∇h(¯ x)λ + ∇c(x)ν  h(¯ x) + ∇h(¯ x)T (x − ¯ x)  . (10) F (x, λ, ν, ¯ x) =  a   c (x) −ν i ¯ enters the nonlinear equation Note that the argument x only via the linearization of h, and that the square system F (x∗ , λ∗ , ν∗ , x∗ ) = 0 would be equivalent to eq. (9), the FONC of the original problem (5). More importantly, the nonlinear residual F helps us to define the relation between the SCP iterates. We denote the Jacobian matrix of F with respect to its first three components by ∂F J(x, λ, ν, ¯ x) = (x, λ, ν, ¯ x). ∂(x, λ, ν) Assumption 5: Matrix J(x∗ , λ∗ , ν∗ , x∗ ) is invertible. It is important to note that the top-left of the matrix Pnc block J(x, λ, ν, ¯ x) is given by ∇2 f (x) + j=1 νi ∇2 ci (x) which is a positive semi-definite matrix. Lemma 2: Under Assumptions 1-5 and when kxk − x∗ k is sufficiently small, the next SCP iterate xk+1 generated by solution of (6) is unique and there exist unique multipliers λk+1 ∈ Rnh and νk+1 ∈ Rnc so so that F (xk+1 , λk+1 , νk+1 , xk ) = 0. (11) Proof: Due to Assumption 5 and the implicit function theorem, for xk sufficiently close to x∗ , the system F (x, λ, ν, xk ) = 0

(12)

in unknowns (x, λ, ν) admits a solution in a neighborhood of (x∗ , λ∗ , ν∗ ). If the distance kxk − x∗ k is sufficiently small, this solution – that we already call (xk+1 , λk+1 , νk+1 ) – still satisfies, due to Assumption 4 and continuity of all involved functions, the strict complementarity condition νk+1 − c(xk+1 ) > 0, and J(xk+1 , λk+1 , νk+1 , xk ) remains invertible. It remains to be shown that this solution is the unique solution of the convex subproblem (6). This follows from the fact that eq. (11) is equivalent to the FONC of the

subproblem, and that invertibility of J implies uniqueness of the solution. Now that we have shown that the SCP iteration is well defined in the neighborhood of the solution, we can analyze contractivity of the map ΦSCP (x). Theorem 1 (Local Convergence of SCP): Under Assumptions 1-5, the SCP iteration mapping ΦSCP (x) is differentiable in a neighborhood of x∗ , and its derivative at x∗ is given by the matrix ∂ΦSCP ∂x (x∗ )

= M∗ :=

Pnh  2 j=1 λ∗,j∇ hj (x∗ )    − I 0 0 J(x∗ , λ∗ , ν∗ , x∗ )−1  0 0

(13)

If the spectral radius of M∗ is smaller than one, ρ(M∗ ) < 1

(14)

then the SCP iteration is locally linear convergent with asymptotic contraction rate ρ(M∗ ). If one eigenvalue of M∗ has a modulus larger than one, the (unregularized) SCP iteration does not converge. Proof: In view of the previous lemma, only the formula (13) needs to be shown, and the rest follows from standard stability results from nonlinear discrete time system theory. To prove the formula, recall that the map ΦSCP (¯ x) is defined via the implicit equation F (ΦSCP (¯x), λ(¯x), ν(¯ x), ¯ x) = 0 which can be differentiated w.r.t. to ¯x to yield  ∂Φ (¯x)  SCP ∂¯ x  + ∂F (·)  (15) 0 = J(·) ∗ ∂¯x ∗ which evaluated  at ¯x = x∗ gives by invertibility of J and projected by I 0 0 the result (13).

C. Discussion While the technical Assumptions 1-5 are in practice not restrictive, the last condition, ρ(M∗ ) < 1, is. It is satisfied, however, whenever the nonlinear equality constraints are only weakly nonlinear, i.e. have only small second derivatives, or when the corresponding multipliers, λ∗ , are small. The latter case occurs, due to the interpretation of multipliers as “shadow prices”, if the optimal objective function value of (6) does only weakly depend on perturbations of the nonlinear constraints h(x) = 0. Summarizing, the SCP method will only converge well if either the nonlinearity of h is small or if the equality constraints are not very costly. Note that the assumption of strict complementarity would not at all be affected by zero equality multipliers, and that small equality multipliers are very favorable for the convergence of the SCP method. If condition (14) is not satisfied, a possible remedy would be to add a quadratic regularization term αkx − xk k22 to the objective function of the convex subproblem (6), which by a more detailed analysis can be shown to render the SCP method convergent. Note that a quadratic regularization is also used in the smoothing technique for decomposed solution of the convex subproblems, but that we do not discuss the details here.

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WeA16.2 IV. A PPLICATION OF SMOOTHING TECHNIQUES TO

We also define the corresponding augmented dual function:

SEPARABLE CONVEX PROBLEMS

d(µ, λ) =

In this section we explore the particular structure of the SCP optimization problem (6). In particular, it can be easily derived that the SCP problem (6) can be written as a separable convex optimization problem of the following form: f∗ =

min

x1 ∈X1 ···xM ∈XM

s.t.

M X

M X

min

xi ∈Xi ,Gi xi =ai

Lµ (x, λ).

PM Note that by adding the smoothing term µ i=1 φXi the objective function in (16) remains separable in xi , i.e. M X d(µ, λ) = i

min

[fi (xi )+µφXi (xi )+hλ, Hi xi −

x ∈Xi ,Gi xi =ai i=1

fi (xi )

i=1

(16)

Hi xi = a, Gi xi = ai ∀i = 1 · · · M,

where fi : Rn → R are convex functions, Xi are closed convex sets, the local equality constraints Gi xi = ai are obtained by linearization of the local equality constraints PM (3.2) and the linear coupling constraints i=1 Hi xi = a are obtained by linearization of the dynamical constraints (3.1) at the point xk . For simplicity of the exposition we define the matrix H = [H1 · · · HM ]. Note that we do not assume strict/strong convexity of any function fi . To state the results in this section we require that constraint qualification (CQ) holds for problem (16): {x ∈ int(X) : Gi xi = ai ∀i, Hx = a} 6= ∅. Let h·, ·i/k · k denote the Euclidian inner product/norm on Rn . By forming the Lagrangian corresponding to the coupling constraints (with the Lagrange multipliers λ), i.e.

a i] M

Denote by xi (µ, λ) the optimal solution of the previous minimization problem in xi , i.e.: xi (µ, λ) = arg

i=1

(18)

min

xi ∈Xi ,Gi xi =ai

[fi (xi ) + µφXi (xi ) + hλ, Hi xi i].

We are interested in the properties of the family of augmented dual functions {d(µ, ·)}µ>0 . It is obvious that lim d(µ, λ) = d0 (λ) ∀λ.

µ→0

A. Proximal Center method In the sequel we briefly describe the proximal center decomposition method whose efficiency estimate improves by one order the bounds on the number of iterations of the classical dual subgradient method (see [11] for more details). In the proximal center method, the functions φXi are chosen to be continuous, nonnegative and strongly convex on Xi with strong parameter σi . Since Xi are compact, we can always choose finite and positive constants DXi such that DXi ≥ max φXi (xi ) ∀i. i i x ∈X

L0 (x, λ) = f (x) + hλ, Hx − ai, we can define the standard dual function d0 (λ) = min{L0 (x, λ) : xi ∈ Xi , Gi xi = ai ∀i}. x

Clearly, the dual function d0 is concave but, in general (e.g. when f is not strictly convex), d0 is not differentiable. Therefore, for maximizing d0 we have to use involved nonsmooth optimization techniques such as subgradient type algorithms with slow convergence rate of order O( ǫ12 ), where ǫ > 0 is the required accuracy of the approximation of the optimal value f ∗ (see e.g. [10]). The CQ condition guarantees that strong duality holds for problem (16) and thus there exists a primal-dual optimal solution (x∗ , λ∗ ). In order to obtain a smooth dual function we need to use smoothing techniques applied to the ordinary Lagrangian L0 (see e.g. [14]). In [11], [12] the authors proposed two dual decomposition methods for (16) in whichPwe add to the M standard Lagrangian a smoothing term µ i=1 φXi , where i each function φXi associated to the set X (usually called prox function) must have certain properties explained below. The two algorithms differ in the choice of the functions φXi . In this case we define the augmented Lagrangian: Lµ (x, λ) =

M X

i

i

[fi (x ) + µφXi (x )] + hλ, Hx − ai.

In the following lemma we show that the dual function d(µ, ·) has the following smoothness properties: Lemma 3: [11] If CQ holds and the functions φXi are continuous, nonnegative and strongly convex on Xi , then the family of dual functions {d(µ, ·)}µ>0 is concave and differentiable at any λ. Moreover, the gradient ∇d(µ, λ) = PM i H x (µ, λ) − a is Lipschitz continuous with Lipschitz i=1 i 2 PM ik constant Dµ = i=1 kH µσi . Furthermore, d(µ, λ) ≥ d0 (λ) ≥ d(µ, λ) − µ

M X

DXi ∀λ.

i=1

We now describe a distributed optimization method for (16), called the proximal center algorithm, that has the nice feature that the coordination between the agents involves the maximization of a smooth convex objective function (i.e. with Lipschitz continuous gradient). Moreover, the resource allocation stage consists in solving in parallel by each subsystem agent a minimization problem with strongly convex objective using only local information. Algorithm PCM 0. input: λ0 and p = 0 1. given λp compute in parallel for all i xip+1 = arg i

(17)

min

2. compute ∇d(µ, λp ) =

i=1

546

fi (xi ) + µφXi (xi ) + hλp , Hi xi i

x ∈Xi ,Gi xi =ai

PM

i=1

Hi xip+1 − a

WeA16.2 3. find Dµ up = arg maxh∇d(µ, λp ), λ − λp i − kλ − λp k2 λ 2 4. find p X Dµ l+1 vp = arg max − kλk2 + h∇d(µ, λl ), λ−λl i λ 2 2

the case when the prox functions φXi are self-concordant barriers: Lemma 4: If Assumption 6 holds and φXi ’s are Ni -selfconcordant barriers associated to Xi , then the family of dual functions {−d(µ, ·)}µ>0 is self-concordant. Moreover, the Hessian of −d(µ, ·) is positive definite and given by:

l=0

p+1 p+3 up

5. set λp+1 = + Remark 1: Note that there is a difference between the outer iterations xk in the SCP method and the inner iterations xip in the distributed scheme PCM. Note that the maximization problems in Steps 3 and 4 of Algorithm PCM can be solved explicitly and thus computationally very efficiently. The main computational effort arises in Step 1. However, in some applications, Step 1 can be performed also very efficiently (see subsection IV-C). The proximal center algorithm can be applied in decomposition since it is highly parallelizable: the subsystems can solve their corresponding local minimization problems in parallel. After p iterations of Algorithm PCM we define: ˆ xi =

p X l=0

− ∇2 d(µ, λ) =

2 p+3 vp .

2(l + 1) ˆ = λp . xi and λ (p + 1)(p + 2) l+1

In the next theorem we show that the solution generated by our distributed Algorithm PCM converges to the solution of the original SCP subproblem (16) and we provide also estimates for the rate of convergence: Theorem 2: [11] Under the P hypothesis of Lemma 3pand taking µ = ǫ/ and p + 1 = i DXi P P 2 ( i kHi k2 /σi )( i DXi ) 1ǫ , then after p iterations X −kλ∗ k · k xi − ak ≤ f (ˆ x) − f ∗ ≤ ǫ Hi ˆ i

and the constraints satisfy X p
k xi − ak ≤ ǫ kλ∗ k + kλ∗ k2 + 2 , Hi ˆ i

∗

where λ is the minimum norm optimal multiplier. Therefore, the efficiency estimate of Algorithm PCM is of order O( 1ǫ ) and thus improves by one order the complexity of the subgradient algorithm, whose efficiency estimate is O( ǫ12 ) (i.e. we obtained a much faster convergence). B. Interior-point Lagrangian method In this section we describe the second decomposition method, called the interior-point Lagrangian algorithm. In this decomposition method P we add to the standard LaM grangian a smoothing term µ i=1 φXi , where each function φXi is a Ni -self-concordant barrier associated to the convex set Xi (see [12] for precise definitions). Assumption 6: (i) Each function fi is convex quadratic. (ii) The matrix [diag(GT1 · · · GTM ) H T ] has full column rank. Note that assumption (ii) is automatically satisfied when LICQ holds (i.e. when Assumption 3 is valid). In the following lemma we derive some of the properties of the family of augmented dual functions {d(µ, ·)}µ>0 in

M X i=1

 Hi Q−1 i (µ, λ)−

−1 T T Q−1 i (µ, λ)Gi Gi Qi (µ, λ)Gi


−1

 T Gi Q−1 i (µ, λ) Hi ,

where Qi (µ, λ) = ∇2 fi (xi (µ, λ)) + µ∇2 φ(xi (µ, λ)). Proof: Since fi are convex quadratic functions, they are also self-concordant functions. Moreover, since φXi ’s are Ni self-concordant barriers, it follows that fi + µφXi are also self-concordant and with positive definite Hessians. P Since d(µ, ·) is basically the Legendre transformation of i fi + µφXi , in view of well-known properties of the Legendre transformation it follows that the augmented dual function is self-concordant and its Hessian is positive definite. The expression for ∇2 d(µ, λ) follows from Theorem 3.1 in [12]. From Lemma 4 we can conclude that the family of augmented dual functions {−d(µ, ·)}µ>0 is self-concordant. This opens the possibility of deriving a dual interior-point based method for solving distributively the convex separable SCP subproblem (16) using Newton directions for updating the multipliers to speed up the convergence rate. Note that in the proximal center method (Algorithm PCM) we use only first-order information (gradient directions) to update the multipliers. Denote the Newton direction associated to function d(µ, ·) at λ as follows:
−1 ∆λ(µ, λ) = − ∇2 d(µ, λ) ∇d(µ, λ). For every µ > 0, we also define the Newton decrement of the function d(µ, ·) at λ as: q
−1 ∇d(µ, λ). δ(µ, λ) = −1/µ∇d(µ, λ)T ∇2 d(µ, λ)

Algorithm IPLM 0. input: (µ0 , λ0 ) satisfying δ(µ0 , λ0 ) ≤ ǫV , p = 0, 0 < τ < 1 and ǫ > 0 1. if µp ≤ ǫ, then stop 2. (outer iteration) let µp+1 = τ µp and go to inner iteration (step 3) 3. (inner iteration) initialize λ = λp , µ = µp+1 and δ = δ(µp+1 , λp ) while δ > ǫV do 3.1 determine a step size α and compute xi (µ, λ), λ+ = λ + α∆λ(µ, λ) 3.2 compute δ + = δ(µ, λ+ ) and update λ = λ+ and δ = δ+ i 4. xp+1 = xi (µ, λ), λp+1 = λ, replace p by p + 1 and go to step 1 In a practical implementation of the algorithm one may choose the parameters τ and ǫV using heuristic considerations although theoretical choices also exist. In [12] it

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WeA16.2 was shown that Algorithm IPLM is globally convergent under Assumption 6 and moreover it has polynomial-time complexity. Roughly speaking, the following convergence rate was derived: Theorem 3: [12] Under the hypothesis of Lemma 4 the following convergence rate holds for the Algorithm IPLM: 0 ≤ f (xp ) − f ∗ ≤ Nφ µp , P where xp = arg minx Lµp (x, λp ) and Nφ = i Ni . Therefore, the complexity of the interior-point Lagrangian method (Algorithm IPLM) is of order O(ln( µǫ0 )). Note however that at each iteration we have to invert matrices of dimension equal to the dimension of the local variables xi or to the number of coupling constraints and thus such an algorithm is not suitable when the dimension of the local variables xi or the number of coupling constraints is large. C. Efficient Solution of the local subproblems in the PCM method In the special case that the stage costs ℓi are convex T T quadratic costs of the form ℓi (xi , ui ) = xi Qi xi +ui Ri ui , we can check that the distributed PCM scheme leads to decomposition in both “space” and “time”, i.e. the separable problem can be decomposed into small subproblems corresponding to the spatial structure of the system–M subsystems, but also to the prediction horizon–N the length of the prediction (see [13] for more details). Note that this is not the case with Jacobi or primal (incremental) subgradient type algorithms. Note also that in this particular case, the solution of each subproblem depends parametrically on the parameter λ which modifies the QP gradient only. For QP active set strategies, this allows to make efficient use of warm-starting techniques that are implemented in many offline QP solvers [15]. As the QP matrices never change, it is even possible to keep internal matrix factorizations, which is e.g. exploited in the online active set strategy presented in [16]. V. C ONCLUSIONS In this paper we proposed a new method to decompose nonlinear optimal control problems in order to obtain a distributed control algorithm. The idea we exploited can be summarized in the following way: as a first step we tackled the nonlinearity of the system dynamics using sequential convex programming (SCP), and then we decompose the problem obtained using smoothing techniques. Besides proposing this method, the main contribution of the paper is theoretical. We proved that under suitable assumption, SCP converges to a local solution of the problem. There are several research directions we shall pursue in the future. First, we need to validate experimentally the effectiveness of the approach. In this regard, the most natural framework where we can apply our method is model predictive control, where similar methods have already been used mostly for linear dynamic systems. Other possible research directions include new decomposition techniques based on

SCP. For example, SCP can be combined with primal methods or we can obtain new algorithms where, instead of first convexifying the problem and then decomposing it, the opposite approach is used. ACKNOWLEDGMENTS Research supported by Research Council KUL: CoE EF/05/006 Optimization in Engineering(OPTEC), IOFSCORES4CHEM, GOA-MANET, several PhD/postdoc and fellow grants; Flemish Government: FWO: PhD/postdoc grants, projects G.0452.04, G.0499.04, G.0211.05, G.0226.06, G.0321.06, G.0302.07, G.0320.08, G.0558.08, G.0557.08, G.0588.09 research communities (ICCoS, ANMMM, MLDM); IWT: PhD Grants, Belgian Federal Science Policy Office: IUAP P6/04; EU: ERNSI; FP7HDMPC, FP7-EMBOCON, Contract Research: AMINAL. Other: Helmholtz-viCERP, COMET-ACCM. R EFERENCES [1] E. Kalvins and R. Murray, “Distributed algorithms for cooperative control,” Pervasive Computing, IEEE, vol. 3, pp. 56–65, 2004. [2] J. Shamma, Cooperative Control of Distributed Multiagent Systems. Wiley, 2008. [3] A. Rantzer, “On Prize Mechanism in Linear Quadratic Team Theory,” in Proc. 46th IEEE Conf. Decision Contr., 2007. [4] B. Krogh, E. Camponogara, D. Jia, and S. Talukdar, “Distributed model predictive control,” IEEE Control Systems Magazine, vol. 22(1), pp. 44–52, 2002. [5] F. Borrelli, T. Keviczky, and G. J. Balas, “Decentralized receding horizon control for large scale dynamically decoupled systems,” Automatica, vol. 42, pp. 2105–2115, 2006. [6] W. Dunbar and R. Murray, “Distributed receding horizon control for multi-vehicle formation stabilization,” Automatica, vol. 42, pp. 549– 558, 2006. [7] A. Richards and J. How, “Robust distributed model predictive control,” International Journal of Control, vol. 80(9), pp. 1517–1531, 2007. [8] A. Venkat, I. Hiskens, J. Rawlings, and S. Wright, “Distributed MPC strategies with application to power system automatic generation control,” IEEE Transactions on Control Sys. Techn., vol. 16, no. 6, pp. 1192–1206, 2008. [9] E. Franco, L. Magni, T. Parisini, M. M. Polycarpou, and D. M. Raimondo, “Cooperative Constrained Control of Distributed Agents With Nonlinear Dynamics and Delayed Information Exchange: A Stabilizing Receding-Horizon Approach,” IEEE Trans. Automat. Contr., vol. 53, pp. 324–338, 2008. [10] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course. Boston: Kluwer, 2004. [11] I. Necoara and J. Suykens, “Application of a smoothing technique to decomposition in convex optimization,” IEEE Transactions on Automatic Control, vol. 53, no. 11, pp. 2674–2679, 2008. [12] ——, “An interior-point Lagrangian decomposition method for separable convex optimization,” Journal of Optimization Theory and Applications, vol. 143, no. 3, Dec. 2009. [13] I. Necoara, D. Doang, and J. Suykens, “Application of the proximal center decomposition method to distributed model predictive control,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 2008. [14] Y. Nesterov, “Smooth minimization of non-smooth functions,” Math. Programming, vol. 103, no. 1, pp. 127–152, 2005. [15] D. Goldfarb and A. Idnani, “A numerically stable dual method for solving strictly convex quadratic programs,” Mathematical Programming, vol. 27, pp. 1–33, 1983. [16] H. Ferreau, H. Bock, and M. Diehl, “An online active set strategy to overcome the limitations of explicit MPC,” International Journal of Robust and Nonlinear Control, vol. 18, no. 8, pp. 816–830, 2008.

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