This dissertation provides a distributed implementation of moving horizon control with guaranteed convergence and performance comparable to a centralized implementation. This dissertation provides a distributed implementation of moving horizon control with guaranteed convergence and performance comparable to a centralized implementation.
Literature Review
Receding Horizon Control
Receding horizon control is also easy to describe and understand compared to other control approaches. Recent work on distributed receding horizon control includes Jia and Krogh [33], Motee and Sayyar-Rodsaru [54], and Acar [1].
Multi-Vehicle Coordinated Control
The multi-vehicle formation stabilization problem explored in this thesis is used as a meeting point for the theory. Ultimately, the distributed receding horizon control developed here will be applied to the Caltech Multi-Vehicle Wireless Testbed [13], where the individual vehicles have hovercraft dynamics and communicate using wireless ethernet.
Decentralized Optimal Control and Distributed Optimization . 9
Although the centralized solution of the original problem is not recovered, numerical experiments show that the distributed implementation performs comparably (in terms of closed system performance) to the centralized implementation of the original problem. The distributed implementation is first generated by decomposing the single optimal control problem into local optimal control problems.
Receding Horizon Control
An admissible control is any piecewise, right-continuous function u(·) : [0, T] → U, for any T ≥ 0, such that given an initial state z(0) ∈ Z, the control generates the state trajectory curve . i) the function f : Rn×Rm → Rn is twice continuously differentiable, satisfies 0 =f(zc,0) and f linearized around (zc,0) is stabilizable;. ii) the system (2.1) has a unique, absolutely continuous solutionz(·;z(0)) for any initial state z(0) ∈ Z and any admissible control;. iii) the set U ⊂ Rm is compact, convex and contains the origin in its interior, and the set Z ⊆Rn is convex, connected and contains zc in its interior;. iv) full state measurement is available and the computation time is negligible compared to the development of the closed-loop dynamics. The notation above shows the implicit dependence of the optimal open-loop controlu∗(·;z(t)) on the initial state z(t) through the optimal.
Implementation Issues
Computational Delay
It is also required that δ be sufficiently small as the authors consider quantization errors in the numerical implementation of the law of the backward horizon control. In the absence of uncertainty, the stability proofs still follow as before, since the predicted state and actual state coincide.
Robustness to Uncertainty
An interesting paper that considers non-trivial computation times in implementing receding horizon control on an air traffic control experiment, the same experiment described in the next chapter, is [51]. It is well known that if a Lyapunov function is used to prove asymptotic stability of a closed-loop system, then the feedback exhibits inherent robustness properties [40], especially in the form of multiplicative uncertainty on the input.
Relaxing Optimality
Regarding the conditions on the constraint sets in Assumption 2.1, we note that convexity of both U and Z is relevant to guarantee that the closed-loop system will have nominal robustness properties [26]. In the absence of uncertainty, concatenating the rest of this control with the terminal controller K defined in Assumption 2.2 leads to subsequent feasibility.
Summary
This chapter is concerned with the application of descending horizon control to a high-performance flight control experiment shown in Figure 3.1. More recent work on the application of drag horizon control to the Caltech ducted fan was performed by Milam et al.
Flight Control Experiment
Hardware
The ducted fan has three degrees of freedom: the rod that holds the ducted fan can act on a cylinder 2 m high and 4.7 m in diameter, which allows horizontal and vertical movements. Optical encoders mounted on the duct fan, the gear and the bottom of the stand measure three degrees of freedom.
Software
Model of the Ducted Fan
The parameters are the mass m, the moment of inertia J, g is the acceleration due to gravity, and r is the distance from the force f2 to the center of mass, i.e., the origin of the x-y frame. The configuration variables x and y represent, respectively, horizontal and vertical inertial translations of the fan while θ is the rotation of the fan about the boom axis.
Application of Receding Horizon Control
Receding Horizon Control Formulation
This amount of time will be described in detail in the next section, which describes how the timing of the control of the receding horizon is carried out. The reference statezref is used to explore the response of the closed-loop system to step changes in the value of the objective state.
Timing Setup
The logic for this is that if the inputs on the real hardware are saturated, there is no control left for the theta controller in the inner loop and the system can become unstable. Note also that the same discontinuity is present for the corresponding open-loop optimal state trajectory, again with a probability of greater discontinuity for longer computation times.
LQR Controller Design
Results
Comparison of Response for Different Horizon Times
The first comparison is between different receding horizon control laws where the time horizon is varied to be or 6.0 seconds. There is also an initial transient increase in average computation time that is larger for shorter horizon times.
LQR vs. Receding Horizon Control
Referring to Figure 3.6, the average calculation profile for the 1 second trailing horizon control with 11 breakpoints looks like the 3 second horizon control with 16 breakpoints.
Summary
As discussed in the introductory Chapter 1, we are interested in the application of backward horizon control to multiagent systems. Finally, the computational and communication costs of implementing a centralized law on the descending horizon can be prohibitive in many multiagent applications.
Outline of the Chapter
A distributed implementation should not only enable the autonomy of individual agents, but also improve the usability of a corresponding centralized implementation. A chapter outline is now presented to help understand the main ideas behind distributed implementation.
Structure of Centralized Problem
The coupling between agents is assumed to occur in the cost function of the state. We can now define the distributed integrated costs that will be included in each agent's local optimal control problem.
Distributed Optimal Control Problems
The collaboration is achieved by minimizing the distributed integrated cost functions given in Definition 4.1, as included in the optimal control problems defined below. In the centralized implementation, initialization requires finding a feasible solution to the optimization problem.
Distributed Implementation Algorithm
The reason for the test is its use in the proof of Proposition 4.1 in Section 4.6. We note that the asymptotic stability result in the next section guarantees that only in the limit astk→ ∞ does viz(tk)→zc.
Stability Analysis
Finally, from the properties of the feedback in Assumptionb 4.3 and the notation in equation (4.6), we have. To mitigate these problems, a dual-mode version of the distributed backward horizon control law is formulated in the next chapter.
Summary
Finally, Section 5.4 discusses extensions of the theory in detail, outlining some of the future work to be explored. A more general discussion of extensions of the theory, in terms of its relevance in other disciplines and possible applications, is provided later in Chapter 7.
Interpretation of the Distributed Receding Horizon Control Law
Comparison with Centralized Implementations
This implementation is not scalable, in terms of computational or communication cost. Recall that only the applied part of the optimal control trajectories is communicated to all agents in the centralized implementation with a single computational node.
Effect of Compatibility Constraint on Closed-Loop Performance 82
In this sense, the compatibility constraint does imply that the transient response will be slower, especially for smaller update times. The experiments also indicate that, even if the compatibility constraints are removed, the distributed implementation is slower, if only slightly, than the centralized implementations.
Alternative Formulations
Dual-Mode Distributed Receding Horizon Control
Therefore, the dual-mode distributed receding horizon controller results in asymptotic stability with the region of attraction ZΣ. The dual-mode distributed receding horizon control law in Theorem 5.1 requires that all agents have the following information available: the horizon time T, all parameters in the computation forδmaxin Assumption 5.1, and the parameters of the distributed consensus algorithm satisfying the conditions in Lemma 5.2.
Alternative Exchanged Inter-Agent Information
We note that for the controller of Theorem 4.1, another consensus algorithm could be included for the distributed computation of kz(tk)−zck2Q at each receding horizon update. As a final note, we note that the computations required to integrate each neighbor's model to generate the necessary hypothesized trajectories are negligible compared to the computations required to find the optimal control for each agent.
Alternative Compatibility Constraint
Extensions of the Theory
A General Coupling Cost Function
In the theory from the previous sections, neighboring agents are coupled by links in the integrated cost function. The decomposition can now be performed in terms of the unique cost functions Lzi constructed in the proof of Proposition 5.1.
Inter-Agent Coupling Constraints
In the next section, we explore the implications of linking state constraints on the implementation of distributed receding horizon control. In the ith local optimal control problem, for each i, the collision avoidance constraints are included as.
Locally Synchronous Timing
The timing is locally synchronous, as each agent only needs to know when it receives the last putative control trajectory from each neighbor relative to the time of the next update. For the rest of the time interval, i.e. [τk+1,j, τk,i+T], agent i has an assumed state trajectory for neighbor j, which deviates at most by δ2κ from the actual state trajectory, according to equation (5.2).
Summary
Moreover, the closed-loop performance of the distributed implementation is comparable to that of the centralized implementation. For the distributed implementation, coordination only takes place between vehicles connected in the local optimal control problems, i.e. vehicles that are neighbors.
Formation Stabilization Objective
Likewise, the tail of the edge ei, denoted t(ei), is the first element of the corresponding ordered pair, and the head of the vector h(ei) is the second element. The vector ˆd= (.., dij, ..,−qd) has the order of vectorsdij according to the definition of F.
Optimal Control Problems
Centralized Receding Horizon Control
In the next section, the optimal control problem associated with the multi-vehicle formation stabilization objective is defined for centralized and distributed towing horizon control. The centralized optimal control problem, the descending horizon control law, and the sufficient conditions for stabilization are presented in Chapter 2.
Distributed Receding Horizon Control
In the stability proof, the main premise is that the sum of the distributed integrated costs is equal to the centralized cost multiplied by γ. What is required is that the distributed integrated costs are summed as the centralized cost, multiplied by a factor (γ) greater than one.
Numerical Experiments
Centralized Implementation
At time 12.0, the snapshot shows the formation reconfiguring to the change in direction of the reference trajectory that occurred at time 10.0. The figure shows greater discontinuity during the transient phase of the closed-loop response, with the largest discontinuity occurring at time 0.0 and at time 10.0, when the reference trajectory changed course.
Distributed Implementation
The closed-loop fingertip shaping response is shown in Figure 6.4 and the time history of the falling horizon control law for vehicle 3 is shown in Figure 6.5. The resulting response looks exactly like the results with κ = +∞ for all update times, i.e., the formation response in Figure 6.4 and the control law history for vehicle 3 in Figure 6.5.
Alternative Description of Formations
The following edge and face deviation variables (also known as shape variable [58]) associated with the edges and faces of the triangulated graph G are defined, respectively, as . In the single optimal control problem, the integrated cost is defined as the sum of the above formation cost plus a tracking cost and collision avoidance cost, defined in [56].
Summary
The design of terminal costs and constraints can still be based on the linearization techniques presented in Chapter 4. In the next chapter, connections between the theory of Chapter 4 and other fields, as well as other possible places for the theory, are identified for the future . research.
Relevant Areas of Research
- Parallel and Distributed Optimization
- Optimal Control and Neighboring Extremals
- Multiagent Systems in Computer Science
- Other Areas
The receding horizon distributed control algorithm and, at a lower level, the algorithms used to solve each optimization problem should be characterized in the same way. A connection between receding horizon control and rollout policies was recently initiated in the work of Chang [7, 8].
Potential Future Applications
Mobile Sensor Networks
To be practically feasible, algorithms for coverage problems must address all these issues together, which is really challenging. To handle the non-convexity of collision avoidance, the initialization procedure must be designed to work on top of each optimization problem.
Control of Networks
If any local optimization problem can be solved efficiently, the distributed implementation presented here is especially useful for mobile sensor network problems, due to its generality and ability to handle reconfigurations. It would not be too difficult to formulate a discrete-time version of the distributed implementation in this thesis and compare it with the results in [2].
Summary
To date, the predominant number of successful examples of descending horizon control in practice arise in the field of process control, where the time scales of the dynamics are slow enough to allow the required online optimization calculations. The implementation is compared qualitatively, in terms of computational and communication cost, with two centralized implementations of descending horizon control.