How fast can the MPC solution be evaluated?

Assuming a processor speed of 1 Gflops/s, the explicit multiscale control law for this four-dimensional example can be evaluated in 119 ns or 2.5 µs with the search tree and perfect hash data structures respectively, and even faster if implemented on a parallel processor.

What is the way to prove stability for a linear system?

By the properties of barycentric coordinates (6), (7), and (8) and because ∑v∈extr(R) J(û(v))λv is defined on the convex hull conv({(v, J(û(v))) : v ∈ extr(R)}), for all x0 ∈ RJ∗(x0) + ǫl(x0, 0) − ∑v∈extr(R)J(û(v))fv(x0) ≥ err(R)Therefore, if the worst case error between the optimal cost function and the convex hull of the approximate cost function is less than ǫl(x0, 0) (i.e. err(R) ≥ 0), then the entire region spanned by the interpolation by barycentric coordinates satisfies the Lyapunov criteria of Theorem 1.Corollary 2: The optimal cost function J∗(x) defined over the set R∗ := {x ∈ R : R ∈ Rδ, err(R) ≥ 0} is a Lyapunov function over R∗. Since the system must also be invariant or feasible for all time, a Lyapunov function alone is insufficient to prove stability for a constrained system.

What is the transformation between basis’ and the hierarchical basis?

A~φ,where the matrix A is square and invertible (i.e. there also exists an inverse transformation from the hierarchical basis to the nodal basis).

What is the simplest way to reduce the hierarchical detail?

Common thresholding, |wk,i| ≥ δ, simply reduces to taking the absolute value of the hierarchical detail greater than the threshold, and discarding the weight wk,i if it is lower than the threshold.

What is the simplest way to store the hypercubic regions of guaranteed stability?

It is necessary to store the hypercubic regions of guaranteed stability Rs, guaranteed infeasibility Rf (i.e. no feasible point exists within any hypercube in Rf ), and regions at level lbdry which intersect the boundary of the feasible set Rb, all of which are initialized as the null set.

what are the future considerations for reducing the conservative nature of the verification procedure?

Future considerations include reducing the conservative nature of the verification procedure and extending the results to systems with nonlinear dynamics.

What is the cost function for the approximate control law?

The authors now show that the cost function (14) evaluated for the approximate control law (17) over R ∈ Rδ can be upper bounded by an approximate cost defined by the interpolation by barycentric coordinates of the approximate cost at theextremal points.

What is the approximate control law defined on R R?

By Corollary 1, the authors see that the approximate control law defined on R ∈ Rδ, where Rδ is defined in (12), can be expressed as an interpolation by barycentric coordinatesû(x) = ∑v∈extr(R)û(v)fv(x), if x ∈ R. (17)The following Lemma states that the resulting interpolation (i.e. the approximate control law) defined across R ∈

What is the approximate cost function and control law?

Note that as lmax → ∞, lbdry → ∞, and ǫ → 0, the approximate cost function and approximate control law converge to the true solutions.

Open AccessJournal Article10.1109/TAC.2011.2146990

A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

Sean Summers, +3 more

- 25 Apr 2011

- IEEE Transactions on Automatic Control

- Vol. 56, Iss: 11, pp 2530-2541

TL;DR: An algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second-order interpolets that is shown to be everywhere feasible and continuous.

Abstract: A model predictive control law is given by the solution to a parametric optimization problem that can be pre-computed offline and provides an explicit map from state to control input. In this paper, an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second-order interpolets. The resulting interpolation is shown to be everywhere feasible and continuous. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a grid hierarchy, convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. Two examples are provided; A two-dimensional example with an evaluation speed of 31 ns and a four-dimensional example with an evaluation speed of 119 ns.

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Most frequently asked questions

1. What is the approach to approximate the optimal control law?

The approach approximates the optimal control law directly, and because of the basis functions used to build the function approximation, can provide guaranteed stability, feasibility, and bounds on the performance.

2. Why is it necessary to create a smart data structure for the hierarchical details?

In the current approach, because of the adaptive nature of Algorithm 2, it is necessary to create a smart data structure for the hierarchical details.

3. How many Monte Carlo simulations were completed?

10000 Monte Carlo simulations were completed with the initial states drawn randomly from the feasible region according to a uniform distribution.

4. What is the influential benefit of the approximate control law in this case?

the most influential benefit of the explicit approximate control law in this case is the online evaluation speed that can be achieved.

Fig. 4. An example of ad = 2 hierarchical function approximation inΩ2 over levelsk ∈ {1, 2, 3}. The active hierarchical indices are given by the corresponding centers (dots) on the grids. The hypercubic regions in which the basis functions form an interpolation bybarycentric coordinates are marked gray.

Fig. 6. (a) Sample four-dimensional system trajectory with the optimal online receding horizon control (red dashed line) and the multiscale approximate receding horizon control (blue solid line). (b) Error trajectory for the sample four-dimensional system trajectory given by the optimal solution minus the approximate solution (Note the difference in scale compared to Figure 6(a)). (c) Sample control trajectory for the four-dimensional system with the optimal online receding horizon control value (red dashed line) andthe multiscale approximate receding horizon control value (blue solid line).

Fig. 5. The approximate control law and optimal control law areshown in (a) and (b) respectively. Notice the sparsity of therequired hierarchical details as shown in (c). In (d), the (inner) set denoting the feasible and stable region with bounded performance and the (outer) set denoting the regions which intersect with the boundary are given with respect to the the solid linerepresenting the boundary of the optimal feasible set. In (e), w see the resulting hierarchical tree structure if speed is chosen over storage space. In (f) we illustrate optimality (ǫ) versus approximation complexity.

Fig. 1. Example of a piecewise linear approximation of the functio u(x) defined in (4) with a maximum resolution of levell = 4 given in the nodal basis 1(a) and the hierarchical basis 1(b). 1(a) The piecewise linear approximationu4(x) (solid line) with corresponding nodal basis functions (dashed lines). 1(b) The piecewise linear approximationu4(x) (solid line) with corresponding hierarchical basis functions (dashed lines). In both 1(a) and 1(b), the piecewise linear function (solid line) is the result of the summation of the basis functions (dashed lines).

Fig. 3. Grid points for subspacesW 2 0 (circles),W 2 1 (x’s), andW 2 2 (dots).

Citations

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- 27 Jun 2018

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- 01 Sep 2021

- IEEE Transactions on Control Systems and...

TL;DR: A novel framework for approximating the MPC policy for linear parameter-varying systems using supervised learning that enables the deployment of MPC on resource-constrained systems and demonstrates a speedup of up to up to $62\times on a desktop computer and on an automotive-grade electronic control unit, while maintaining high control performance.

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An EKF-Based Fast Tube MPC Scheme for Moving Target Tracking of a Redundant Underwater Vehicle-Manipulator System

Yong Dai, +3 more

- 23 Sep 2019

- IEEE-ASME Transactions on Mechatronics

TL;DR: A robust fast tube model predictive controller with an extended Kalman filter (EKF) target observer is used to estimate the moving target trajectory and overcome the sensory measurement ( system) noises, which can eliminate the assumption that most prior existing research works need to know the tracking reference trajectory precisely.

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Ankush Chakrabarty, +5 more

- 01 Jan 2017

- IEEE Transactions on Automatic Control

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References

•Book

Introduction to Algorithms

Thomas H. Cormen, +2 more

- 01 Jan 1990

TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.

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•Journal Article•10.1109/34.192463

A theory for multiresolution signal decomposition: the wavelet representation

Stéphane Mallat

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- IEEE Transactions on Pattern Analysis an...

TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.

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Journal Article•10.1007/BF02837777

Introduction to algorithms: 4. Turtle graphics

R. K. Shyamasundar

- 01 Sep 1996

- Resonance

TL;DR: In this article, a language similar to logo is used to draw geometric pictures using this language and programs are developed to draw geometrical pictures using it, which is similar to the one we use in this paper.

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•Journal Article•10.1063/1.4823127

Ten Lectures on Wavelets

Ingrid Daubechies, +1 more

- 01 Nov 1992

- Computers in Physics

TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.

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•Journal Article•10.1002/CPA.3160410705

Orthonormal bases of compactly supported wavelets

Ingrid Daubechies

- 01 Oct 1988

- Communications on Pure and Applied Mathe...

TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.

...read moreread less

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A Multiresolution Approximation Method for Fast Explicit Model Predictive Control

Chat with Paper

AI Agents for this Paper

Most frequently asked questions

1. What is the approach to approximate the optimal control law?

2. Why is it necessary to create a smart data structure for the hierarchical details?

3. How many Monte Carlo simulations were completed?

4. What is the influential benefit of the approximate control law in this case?

5. How fast can the MPC solution be evaluated?

6. What is the way to prove stability for a linear system?

7. What is the transformation between basis’ and the hierarchical basis?

8. What is the simplest way to reduce the hierarchical detail?

9. What is the simplest way to store the hypercubic regions of guaranteed stability?

10. what are the future considerations for reducing the conservative nature of the verification procedure?

11. What is the cost function for the approximate control law?

12. What is the approximate control law defined on R R?

13. What is the approximate cost function and control law?

Figures

Citations

Predictive Control for Linear and Hybrid Systems

Approximating Explicit Model Predictive Control Using Constrained Neural Networks

Near-Optimal Rapid MPC Using Neural Networks: A Primal-Dual Policy Learning Framework

An EKF-Based Fast Tube MPC Scheme for Moving Target Tracking of a Redundant Underwater Vehicle-Manipulator System

Support Vector Machine Informed Explicit Nonlinear Model Predictive Control Using Low-Discrepancy Sequences

References

Introduction to Algorithms

A theory for multiresolution signal decomposition: the wavelet representation

Introduction to algorithms: 4. Turtle graphics

Ten Lectures on Wavelets

Orthonormal bases of compactly supported wavelets

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