M-Level Control for Long-Horizon Digital-to-Analog Conversion
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Solution Overview
Problem
Existing methods for computing binary, ternary, or M-level input signals to control analog physical systems or perform digital-to-analog conversion face challenges with computational complexity that grows exponentially with the planning horizon, particularly for long horizons, and often fail to efficiently handle level switches in discrete input signals.
Innovation Solution
A method using sparse Bayesian learning and variational representations of non-Gaussian distributions, combined with Kalman-type recursions, to compute M-level input signals that can handle arbitrarily long planning horizons while minimizing level switches, represented by a combination of parameterized Gaussian distributions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Duration of action of moving object
If standard MPC methods or digital ΔΣ modulators are used to compute binary or M-level input signals, then the control problem can be solved for short planning horizons, but the computational complexity grows exponentially with the planning horizon length
Solution Approach 1:
The patent transforms the discrete M-level control problem into a continuous optimization problem by parameterizing the discrete input signals as continuous variables during the optimization process. The M-level constraints are enforced through a cost function that penalizes deviations from discrete levels, allowing the use of efficient continuous optimization algorithms while maintaining the discrete nature of the final control signals. This parameter transformation enables linear computational complexity with respect to the planning horizon.
Solution Approach 2:
The patent replaces traditional mechanical search methods (such as exhaustive search or dynamic programming) with a probabilistic optimization approach based on sparse Bayesian learning. By formulating the control problem as a probabilistic inference task and using variational Bayes methods, the patent achieves efficient optimization with linear computational complexity, substituting the exponential-time mechanical search with a scalable probabilistic computation.
2Duration of action of moving object
If existing MPC methods are used to handle long planning horizons, then the ability to control systems over extended periods is improved, but the methods become unsatisfactory due to computational intractability
Solution Approach 1:
The patent reformulates the discrete control problem as a continuous optimization problem with a carefully designed cost function. By parameterizing discrete M-level inputs as continuous variables and using a cost function that includes terms for tracking performance, control effort, and deviation from discrete levels, the patent enables the use of efficient continuous optimization algorithms. This parameter transformation makes long-horizon control computationally tractable with linear complexity.
Solution Approach 2:
The patent introduces an intermediate continuous optimization problem as a mediator between the original discrete control problem and the final discrete control solution. The continuous relaxation allows efficient computation, and the solution is then projected back to discrete M-level signals through a rounding or projection step. This intermediate continuous formulation acts as a bridge that enables long-horizon control while maintaining computational efficiency.
3Device complexity
If digital ΔΣ modulators are used for digital-to-analog conversion, then the conversion can be performed with simple single-step processing, but the planning horizon is limited to a single time step
Solution Approach 1:
The patent segments the control problem into a sequence of time steps and formulates it as a global optimization problem over the entire planning horizon rather than processing each time step independently. By considering all time steps simultaneously in a unified optimization framework, the patent achieves multi-step planning while maintaining processing efficiency through the continuous relaxation approach. The segmented time steps are optimized jointly rather than sequentially.
Solution Approach 2:
The patent extends the useful action from single-step modulation to continuous multi-step planning by formulating the control problem as a continuous optimization over the entire planning horizon. The continuous relaxation allows the optimization to proceed smoothly across all time steps, maintaining continuity of the control action. The final discrete control signals are obtained by projecting the continuous solution back to the discrete M-level space, preserving the continuity benefit while satisfying discrete constraints.
Data Source
AI summary
A method is disclosed for steering a physical analog system (e.g., an electric motor) using a discrete-level (e.g., binary) control signal. The discrete-level control signal is computed by an iterative scheme that can handle a long planning horizon. A preference for infrequent level switches can be taken into account. The quality of the fit to the target trajectory can be expressed not only by the quadratic error, but also by other norms. The method can be used also for digital-to-analog conversion.