Introduction
Why Biology?
Given some dynamical systems, objectives and constraints, control theory can be used to provide mathematical descriptions of the optimal operation and behavior of the system. Of course, we can also combine data-driven techniques and control theory to create new quantitative models, as is done in Chapter VII.
Why Control Theory?
We characterize how the optimal site size changes as a function of system size and horizon length; the results are summarized in Figure 5.3. Wiesel, βReceptive fields of single neurons in the striate cortex of the cat,β The Journal of Physiology, vol.
Technical Setup
Notation and Abbreviations
For example, (π΄)π, π denotes the element in the πth row, πth column of π΄. π΄)π:,:indicates the rows fromπ΄beginning of theπthrow. Where appropriate, we can also overload subscript notation to denote block partitions of a matrix; the meaning will be clear from context.
System Level Synthesis
For a given system and predictable horizon length, Algorithm 5.2 will return the optimal site sizeπβ the smallest value ofπ that achieves optimal global performance. Daoutidis, βThe role of community structures in sparse feedback control,β in IEEE American Control Conference, 2018, p.
Closed-Loop vs. Controller Specifications
Introduction
In the standard SLS framework, the closed-loop responsesΞ¦π₯ andΞ¦π’ are used directly to implement the controller. Thus, any constraints applied to the controller are also imposed on the closed-loop response.
Implementation Matrices
Then we say that (R, M) are implementation matrices for (Ξ¦Λπ₯, Ξ¦Λπ’), and (Ξ¦Λπ₯, Ξ¦Λπ’) are the implemented closed-loop responses of controller matricesRandM. We note that we can also see (R,M) as implementation matrices for the controllerK= Ξ¦π’Ξ¦β1π₯, since there is a one-to-one mapping between controller and closed-loop responses.
Stability
Due to the structure of βπ, the internal dynamic matrix π΄π§ can be computed in parallel using only local information. Here ππ ππ₯ is a predetermined maximum number of iterations, and ππ ππ₯ is a predetermined threshold for the transient state; since β₯π΄π.
Approximate Implementations
Due to the dependence ofβπonRandM, this expression is not convex with respect to the implementation matrices. The first term in the objective is a heuristic for how close the implemented closed-loop responses are to the desired closed-loop responses.
Closed-Loop vs. Controller Constraints
π(π)]π, π =0 βπ < π(π, π) (3.20) As an example for a system with subsystems arranged in chain configuration, with one state and input per subsystem and π(π, π) proportional to distance between subsystems, the delay constraints result in band diagonal π (π) and π(π), with wider bands for higher values ββof π. Thus, delay constraints are only appropriate for implementation matrices and should not be enforced on closed-loop responses.
Simulations
To apply the two-step procedure, we first synthesize a centralized FIR controller with a horizon of 20 time steps using SLS. Interestingly, this lower order controller performs almost as well as the full order controller with only 0.1% performance degradation.
Conclusions and Future Work
In step 1, each subsystem performs operations on matrix[π»]π, which has a size of approximately ππ₯(π +1) by ππ - the complexity will vary depending on the underlying implementations of the pseudoinverse and matrix manipulations, but will generally scale according to π( (ππ)2ππ₯π), orπ(π3π2π). Each pair of adjacent subsystems has a 40% chance of being connected by an edge; the expected number of edges is 0.8ΓπΓ (πβ1).
Distributed Structured Robust Control
Introduction
In this chapter, we exploit the SLS parameterization to provide distributed synthesis methods for structured robust control. To the best of our knowledge, this is the first distributed and localized algorithm for structured robust control.
System Level Synthesis for Linear Control Problems
We note that full control (i.e. full activation, sparse sensitivity) is twice the state response (i.e. full sensitivity, sparse activation). 4.6) The output feedback reachability constraint resembles a combination of state feedback reachability constraints and full control (2.4) and (4.4).
Separability and Computation
Optimization problem (4.1) is a partially separable optimization when all objectives and constraints are either row or column separable, but the overall problem is not fully separable. Partially separable problems thus also enjoy computational complexity that scales with local neighborhood size instead of global system size [3], [15].
Robust Stability
The partially separable problem (4.9) can be decomposed into row and column subfunctionals and under constraints. Although the convergence properties differ, the step Ξ¦ (4.13) can be computed with a complexity that scales with the local size of the neighborhood instead of the global size of the system for all non-Hβ cases.
Simulations
The controller with maximum stability margin for π is found in 18 iterations for both algorithms; for L1, in 30 iterations for Algorithm 4.1 and 35 iterations for Algorithm 4.2; for Lβ, forL1, in 25 iterations for Algorithm 4.1 and 35 iterations for Algorithm 4.2. The results are shown in Figure 4.2, where the cost and robust stability margin are normalized against the LQR solution.
Conclusions and Future Work
With internal feedback, task execution remains close to the centralized optimal (i.e., the case where local controllers can communicate freely without delay). We also experimented with adjusting the forecast horizon so that it is smaller than the muscle lag, with catastrophic consequences; the model mostly just produced noise.
Efficient Distributed Model Predictive Control
Introduction
In this chapter, we address this problem by providing a rigorous characterization of the impact of local communication constraints on performance. In other words, the performance of the system is unchanged by the introduction of communication restrictions.
Localized MPC
To our knowledge, these are the first results of this kind regarding local communication constraints; our findings are useful for both theorists and practitioners. In (5.7), Ξ¨π₯ and Ξ¨π’ represent not only the closed loop of the system, but also the communication structure of the system.
Global Performance of Localized MPC
The localized trajectory set YL(π₯0) denotes the set of trajectories available from the state π₯0 under the dynamics (2.1) and the locality constraint Ξ¨β Las. In our example, πβπ(πΌ βπΉβ πΉ) also has a degree of 2; by Theorem 5.1 , the local trajectory set is equal to the trajectory set , and by Theorem 5.1 , the localized MPC problem achieves global optimal performance.
Algorithmic Implementation of Optimal Locality Selection
First, we notice that the rank of πβπ (= 2) is low compared to the number of non-zero columns (= 12), especially when π₯0 is dense. Given the system and forecast horizon, experts should first determine the optimal locality sizeπusing Algorithm 5.2, and then run the corresponding online algorithm from [27].
Simulations
The optimal local size also increases as a function of system size β but only if we do not have 100% activation. From the previous section, we found that at 100% activation, the optimal site size is always 1.
Efficient Two-Layer MPC
This two-layer controller allows us to achieve MPC-like behavior, but at a fraction of the computational cost. We compared the performance of the distributed two-layer controller with the performance of centralized MPC and centralized LQR (i.e., offline controller).
Conclusions and Future Work
Opposing internal feedback is in the opposite direction of the single-loop model (e.g., from V2 to V1); these signals flow from action to perception. The remaining internal feedback is inherent to the Kalman filter, as described in the previous section and shown in Figure 6.7.
Internal Feedback in Primate Cortex
Introduction
We can divide intrinsic feedback into two broad categories: feedback between brain areas and lateral interactions within or between areas. Intrinsic lateral feedback consists of recurrent connections within and between areas (for example, from V2 to V2, or from MT to IT).
Task Model and Performance
Letπ₯, The dynamics of the time evolution of the tracking error follows from the linear equation of motion (2.1).
Sensing and Actuation Delay
Internal feedback's controller contains sensor delays, and uses internal feedback to compensate for the delays. Counter-directional internal feedback (pink) carries information back to sensation, to compensate for the drive delay.
Functional Localization
Local controllers communicate with each other via internal lateral feedback (pink), with some delay. End) Local controller circuit 1. This analysis shows that when motor function is localized to specialized parts of the motor cortex that control separate parts of the body, cross-talk through internal feedback between local controllers is essential.
Communication Constraints and Distributed Sensorimotor Circuits . 88
Contralateral internal feedback is indicated by the solid color arrows, and lateral internal feedback is indicated by dashed color arrows. Right) Performance (log cost) of the two-layer controller with internal feedback as delays of both layers differ.
Discussion and Interpretation of Results
The counter-directional internal feedback then solves this problem with a loop between the visual cortex and the motor cortex. Reinforcement learning controlled by circuits in the basal ganglia may also benefit from the internal feedback pathways in the cortex.
Conclusions and Future Work
Hakeem, et al., "The von economo neurons in frontoinsular and anterior cingulate cortex in great apes and humans," Brain Structure and Function, vol. Holmes, βThe effects of feedback on stability and maneuverability of a phase-reduced model for cockroach locomotion,β Biological Cybernetics , vol.
A Layered Model of Drosophila Locomotion
Introduction
The ultimate goal of the study of legged locomotion is to produce analyzes and models that unify physiology, physical dynamics and behavior and reveal the design principles behind locomotion. In short, existing approaches typically lack at least one of the following: physiological considerations, physical dynamics, or realistic behavior.
End-to-End Learning and Control Model
At regular intervals, the controller receives a time series of desired state trajectories from the trajectory generator layer. The trajectory generator periodically receives proprioceptive information from the controller about the true state (joint angles and angular velocities) of the leg.
Realistic Model-Generated Walking
The dotted line shows the sample mean differences between different periods of the real walk - we see that the mean difference between the model and the data is comparable to the sample differences in the data itself. Peaks in coupling indicate synchronization: a single peak at zero on the horizontal axis indicates that the two coupling phases are paired to match;.
Dynamic Perturbations and Motor Delay
We see that in the real data the joints show a mixture of strong and weak coupling, which is largely replicated by the model. Including delays in the model allows us to explore different sensor and motor delays and how this affects walking.
Discussion and Interpretation of Results
However, the assumption of perfectly predictable, interference-free gait is unrealistic; therefore, we consider how motor delays affect gait in the presence of slip-like perturbations. During model development, we found that following a reference trajectory requires compensatory prediction for good model performance in the presence of motor delays.
Conclusions and Future Work
Matni, "Distributed and Localized Closed-Loop Model Predictive Control via System-Level Synthesis," in IEEE Conference on Decision and Control, 2020, p. , p.
Common abbreviations
Comparison of LQR costs
Separability of Ξ¦ step of D- Ξ¦ iteration
Scalability of D step of D- Ξ¦ iteration
LQR costs corresponding to Figure 5.7
LQR costs averaged over 30 simulations
Joints included for leg models
