**Significance Statement**

Dr. Ambrose Adegbege at The College of New Jersey and Zachary E. Nelson, now a PhD student at The Johns Hopkins University, provided solutions to problems of embedded constrained control that occur in feedback interconnected systems with multivariable algebraic loops and where the feedback gain is not assumed to be symmetric. The work which was published in the journal *Systems & Control Letters*, implemented the projected Gauss-Seidel algorithm to solve problems with symmetric or asymmetric feedback gains. The authors considered an interconnected system containing a linear time invariant subsystem and a multivariable algebraic loop subsystem with nonlinear functions.

The efficiency of the projected Gauss-Seidel algorithm was tested at first with an input-constrained linear quadratic regulator problem with a simple system. After adding certain control parameters, they found that the symmetric multivariable algebraic loop could be simply solved.

When they compared the Gauss-Seidel algorithm with other algorithm methods such as the gradient projection method and accelerated Nesterov method, the projected Gauss-Seidel method solved the problem of the input-constrained linear quadratic regulator that follows the embedded constrained control faster when viewed from the results of the closed-loop response and average computational time per quadratic program for the samples. The projected Gauss-Seidel method also had the least computation time per iteration.

The second test involved an anti-windup problem, which conforms to an asymmetric algebraic loop. The projected Gauss-Seidel algorithm had a convergence rate of real numbers less than 2. When compared with the Lemke algorithm, time complexity for different input regions was lower, indicating a lower computational burden for the projected Gauss-Seidel algorithm method.

This study was able to show that the projected Gauss-Seidel algorithm would definitely perform well for a fast and approximate solution in control applications.

**Reference**

Adegbege, A.A., Nelson, Z.E. **A Gauss–Seidel Type Solver for the Fast Computation of Input-Constrained Control Systems**, Systems & Control Letters 97 (2016) 132–138.

Department of Electrical and Computer Engineering, The College of New Jersey, Ewing, NJ 08628-0718, USA.

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