A System Level Approach to Optimal Controller Design for Large-Scale Distributed Systems

Second, we defined a class of system-level planning problems, which we called localized optimal control problems, that are scalable to arbitrarily large systems. Finally, we provide a system-level approach that supports the co-design of an optimal controller and its sensing and actuation architecture.

LIST OF TABLES

INTRODUCTION

• Motivation and Challenges
• Key Ideas
• Theoretical Contributions
• Organization of the Dissertation
• Mathematical Notations

First, we generalize the objectives of the constrained optimal control problem (1.1) from a system norm to arbitrary SLO (1.2a). The set of all SLS problems is a superset of the set of all constrained optimal control problems.

Figure 1.1: Optimal Feedback Control Problem

PROBLEM STATEMENT AND MAIN RESULTS

• System Model
• Youla Parameterization
• Distributed Optimal Control and Quadratic Invariance
• Beyond Quadratic Invariance
• Summary of Main Results

From [81], the centralized optimal control problem (2.6) can be reformulated in terms of the Youla parameter axis. To determine the traceability (convexity) of the restricted optimal control problem (2.8), we first reformulate (2.8) in terms of the Youla parameter Q.

Figure 2.1: Interconnection of the plant P and controller K .

APPENDIX

SYSTEM LEVEL PARAMETERIZATION OF STABILIZING CONTROLLERS

State Feedback

• Necessity
• Sufficiency
• Summary and corollary

Given any system response {R,M} lying in the affine subspace described by (3.4), the state feedback controllerK= MR−1, with structure shown in Figure 3.1, stabilizes the plant internally. For a continuous time state feedback system with state space realization (3.1), the following is true:. a) The affine subspace defined by h.

Figure 3.1: The proposed state feedback controller structure, with R ˜ = I − zR and M˜ = zM .

Output Feedback for Strictly Proper Systems

• Necessity
• Sufficiency
• Alternative controller implementations
• Summary

For any system response {R,M,N,L} lying in the affine subspace defined by (3.15), the controller K = L− MR−1N (with structure shown in Figure 3.2) stabilizes the plant internally. For the controller implementation and structure shown in Figure 3.3a, the closed-loop transfer matrices from disturbances to the internal variables are given by .

Figure 3.2: The proposed output feedback controller structure, with R ˜ + = z R ˜ = z ( I − zR) , M˜ = zM , and N˜ = − zN .

SYSTEM LEVEL SYNTHESIS PROBLEMS

General Formulation

More generally, as long as the intersection of SLC (4.1c) and SLP (4.1b) is convex and the constraint SLO (4.1a) on constraint (4.1b) - (4.1c) is convex, the resulting SLS problem is a convex optimization problem.

Convex System Level Constraints

• Constraints on the Youla Parameter
• Quadratically Invariant Subspace Constraints
• System Performance Constraints
• Controller Robustness Constraints
• Controller Architecture Constraints
• Positivity Constraints and Desired System Response
• FIR Constraints
• Subspace and Sparsity Constraints
• Intersections of SLCs and Spatiotemporal Constraints

The set of constrained internal stabilizing controllers described by (4.4) can be equivalently expressed as K=L−MR−1N, where the system response {R,M,N,L} lies in the set. Note that the support of the RandMi system response elements is defined by the support of A and A2, respectively. Another major advantage of SLCs is that multiple such constraints can be imposed on the system response simultaneously.

Remember that the controllability and the observability of a system is determined by the existence of an FIR system response.

Convex System Level Synthesis Problems

• Distributed Optimal Control
• Localized LQG Control
• Regularization for Design and SLS
• Computational Complexity and Non-convex Optimization

In particular, all decentralized optimal control problems that can be formulated as convex optimization problems in the Youla domain are special cases of the SLS problem (4.1). It can be recovered as a special case of the SLS problem (4.1) by choosing the SLO to be of the form (4.10) (with the k · k system norm chosen as the H2 norm) and choosing the constraint set to be a spatial SLC -time. Thus to integrate RFD with the system-level approach, it suffices to add a suitable regularizer, as mentioned in section 4.2.5 and described in [40, 70], to the objective function of the SLS problem (4.1).

A final advantage of the SLS problem (4.1) is that it is transparent in determining the computational complexity of the optimization problem.

Figure 4.1: Space time diagram for a single disturbance striking the chain described in Example 3.

LOCALIZED LINEAR QUADRATIC REGULATOR

Problem Statement

• Interconnected System Model
• Localized LQR as a SLS Problem

Note that the graph topology is encoded in the matrix reduction model of the global system A. Traditionally, the solution to the stochastic LQR problem (5.4) is obtained by solving a discrete-time algebraic Riccati equation (DARE). The notion of d-localized SLC Ld is defined based on the interaction graph G of the interconnected system.

In this sense, localizability can be considered a stricter notion of a system's controllability.

Figure 5.2: Illustration of the 2-incoming and 2-outgoing sets of subsystem 5.

Localized Linear Quadratic Regulator

• Localized Synthesis
• Localized Implementation
• Localized Re-synthesis
• Summary of LLQR

When non-zero entries of the state vector x[k] are contained in the set Outj(d), we know by definition that non-zero entries of Ax[k] will be contained in the set Outj(d+1). Therefore, the entries of the state vector x[k+1]= Ax[k]+B2u[k] at time (k + 1) will automatically be contained in the set Outj(d+ 1). The complexity of solving (5.12) is determined by the dimension of vectorx(j,d) and u(j,d), which is equal to the cardinality of the set Outj(d +1).

In this case, the complexity of solving (5.12) is independent of the size of the global network, i.e., with computational complexity O(1).

Figure 5.3: The localized region for w 1

State Feedback Localizability

• LLQR with Spatiotemporal Constraints
• Constraint Setup Procedure
• LLQR with Actuator Regularization

For the LLQR problem with a spatiotemporal constraint (5.15), we propose a serial procedure that successively designs the localized subspace Ld, the FIR subspace FT and the activation scheme B2, which guarantee the (d,T)locality of the system. Since the topology of the communication network mimics the topology of the installation, measurements of the state deviationxi[t] are performed. Since the initial perturbation was random, this shows that the delay condition of the lemma is sufficient to guarantee the feasibility of ad-localized constraints.

Since the communication network has the same topology as the physical network, this means equivalently that the kth spectral component of the system responseR[k]j is localized, i.e. the sparse pattern of R[k] is covered by the support of Aj, withA = supp(A) ∪supp(I).

Figure 5.4: Information delay and physical delay

Nearly Localizable Systems

Then the controller implementation(5.23) stabilizes the system(A,B2) internally if and only if(I+∆RM)−1 is stable. Therefore, the stability of I∆ =(I+∆RM)−1 is a necessary and sufficient condition for the controller implementation (5.23) to stabilize the system internally. The robustness of the controller implementation (5.23) allows us to use approximate solution to implement the localized controller, even when the system is not exactly (d,T) localizable.

Specifically, we assume that the localized controller is designed based on the system model (A,B2) but applied to another system model (A+∆A,B2+∆B).

Table 5.1: Closed Loop Maps With Non-localizability

Adaptive Constraint Update with Performance Guarantee

• Lower Bound of Centralized LQR Cost We first assume that the cost matrix
• Adaptive Constraint Update Algorithm

Therefore, a lower bound of the objective function in (5.28) can be calculated by setting xi = 0 andui = 0 for all xi andui. If the size of the localized region is significantly smaller than the size of the global network, (5.34) can be solved efficiently. To calculate a lower bound of the optimal cost, we introduce additional control actions at the limit states x2 and x4.

Relax the constraint by increasing the lengthTj of the FIR constraint or the size of the localized region Outj(d+1);.

Figure 5.5: Localized region for the initial condition x [ 1 ] = e 1 in a large network Example 7

Localized Distributed Kalman Filter

• Motivation of LDKF
• Traditional Kalman Filter
• LDKF Formulation
• Localized Estimator Implementation

To design a state estimation algorithm for large-scale systems, the aforementioned limitations of the traditional Kalman filter must be overcome. This motivates our development of the LDKF architecture, in which the estimator can be implemented and designed in a localized and scalable manner. We now use the system response to analyze the dynamics of the Kalman filter estimation error.

We can see optimization problem (5.52) as an alternative formulation to the delayed form of the Kalman filter problem.

Simulation Results

• Chain Model
• Power System Model

We solve the LLQR problem (5.7) and study the effects of choosing localized areas d with different sizes and the length of the FIR horizon T. The result in Figure 5.8 holds for a large number of parameters (κ, α, γ) of the plant. We choose (d,T) = (7.20) for the previous example and study the trade-off between communication delay tc and the normalized H2 cost (i.e., the square root of the LQR cost).

We now vary the size of the mesh network and compare the computation time required to synthesize a centralized, distributed, and localized LQR optimal controller.

Figure 5.8: Performance vs. FIR horizon T and localized region d . The cost is normalized with respect to the optimal centralized H 2 cost.

LOCALIZED LINEAR QUADRATIC GAUSSIAN

• Problem Statement
• Interconnected System Model
• Localized LQG as a SLS Problem
• Localized Controller Implementation
• Localized Controller Synthesis
• ADMM Algorithm
• Analytic Solution
• Convergence and Stopping Criteria
• Simulation Results
• Power System Model
• LLQG
• Large-Scale Example

Compared to the state feedback problem (5.1), the output feedback problem (6.1) has only a noisy and partial measurement of the state (6.1b). By solving the transposition of the LLQG problem, the method can also be extended to the case where. We now make a series of observations that motivate the use of the ADMM algorithm to solve the LLQG problem (6.8).

A less restrictive condition for the convergence of the ADMM algorithm will be discussed in Appendix 7.B of Chapter 7.

Figure 6.1: An Example of Interconnected System i is assumed to be given by

SYSTEM LEVEL SYNTHESIS FOR LARGE-SCALE SYSTEMS

Problem Setup

• Mathematical Notation
• System Model
• Localized System Level Synthesis
• Localized Implementation
• Problem Statement: Separability

For an LTI system with dynamics given by (7.1), we define a system response {R,M,N,L} to satisfy the maps. Here L can impose arbitrary sparse restriction - in particular, it need not be a d-localized SLC as described in Chapters 5 and 6. The constraint X includes any other convex constraint imposed by the system - in particular, the setX can be the combination of all kinds of convex SLCs introduced in Section 4.2, which includes the communication delay SLC under consideration.

Our approach is to impose a sparsity constraint on the system response (R, M, N, L) via a localized L constraint.

Column/Row-wise Separable Problems

• Column-wise Separable Problems
• Row-wise Separable Problems
• Summary

The system-level objective g(Φ) in (7.8a) is said to be column-wise separable with respect to the column-wise partition {c1,. We note that the objective functionality of the LLQR problem (5.7) is column-wise separable with respect to arbitrary column-wise partitioning. The system-level constraint S in (7.8c) is said to be column-wise separable with respect to the column-wise partition {c1,.

Specifically, we first use the column-wise separability of the problem to decompose the SLS problem into parallel subproblems.

Convex Localized Separable System Level Synthesis Problems

• Partially Separable Problems
• Examples of CLS-SLS Problems
• Analytic Solution and Acceleration

We propose an algorithm to solve a convex, localized and columnwise separable SLS problem in a localized and scalable manner. The localized constraint L and the FIR constraint FT are partially separable with respect to arbitrary row- and column-wise partition. The H2 norm regularizer in (7.23a) is columnwise separable with respect to arbitrary columnwise division.

The sensor norm defined in (7.24) is column separable with respect to any column partition.

Simulation Results

• Localized H 2 with Sensor Actuator Regularization

Recall from (7.2) that the number of rows of the transfer matrix Φ(:,cj) is given by (nx+nu). Note that the optimization problem (7.21) may not have a unique optimal point, so the optimization variables Φk and Ψk need not converge. If we further assume that SLO g(·) is strongly convex with respect to Φ, then problem (7.21) has a unique optimal solution Φ∗.

If the ADMM algorithm does not converge, then we know that the system (A,B2,C2) is not feasible given the given SLC.

Figure 7.1: The upward-pointing triangles represent the subsystems in which the PMU is removed

CONCLUSIONS AND FUTURE WORKS

Summary

Potential Applications and Future Works

• Smart Grid
• Transportation Systems
• Software-Defined Networking
• Layered Control Architecture

Shock waves are generated in transport systems due to a discontinuity in the density profile caused by large disturbances. In [44], the authors study the performance tradeoffs between myopic (fully decentralized), coordinated (distributed), and centralized SDN architectures on the admission control problems. It is worthwhile to see if the CLS-SLS framework can be applied to the same control problem but with larger scale.

It should be noted that the SLS problems and the traditional optimal control problems primarily focus on the design of the feedback control laws for the purposes of disturbance suppression and reference tracking.

BIBLIOGRAPHY

Large-Scale Kalman Filter Solutions for the Electrophysiological Source Localization Problem – A MEG Case Study”.