## On the design of a piecewise-constant feedback control for the linear regulator problem |

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Page 4

This approach is the basis for this thesis. Constraining the structure of the

feedback gains to be

becomes an optimization problem over the parameters which describe the

feedback control; in particular, the constant gain matrices and the set of times

denoting the discontinuities in control. In general, the optimum set of parameters

cannot be determined in closed form and must be computed by some nonlinear

search technique.

This approach is the basis for this thesis. Constraining the structure of the

feedback gains to be

**piecewise constant**, the linear regulator problem effectivelybecomes an optimization problem over the parameters which describe the

feedback control; in particular, the constant gain matrices and the set of times

denoting the discontinuities in control. In general, the optimum set of parameters

cannot be determined in closed form and must be computed by some nonlinear

search technique.

Page 66

S. USE OF GAIN MATRICES FOR DESIRED TRANSFER FUNCTION

CHARACTERISTICS -- AN EXAMPLE As previously mentioned, one advantage

of constraining the feedback control to have

optimizing over the switch times is that the constant gain matrices can be used to

satisfy further constraints. In this section, an example is given where an extra

constraint is placed on the characteristic values (natural modes) of a second

order system.

S. USE OF GAIN MATRICES FOR DESIRED TRANSFER FUNCTION

CHARACTERISTICS -- AN EXAMPLE As previously mentioned, one advantage

of constraining the feedback control to have

**piecewise**-**constant**gains and thenoptimizing over the switch times is that the constant gain matrices can be used to

satisfy further constraints. In this section, an example is given where an extra

constraint is placed on the characteristic values (natural modes) of a second

order system.

Page 69

CONCLUSION It has been shown that by restricting the structure of the feedback

gain matrix for the linear regulator problem to be

optimality can be traded for ease of implementation, and the loss of optimality can

.be reduced by increasing the number of constant-gain intervals. Three

algorithms based on truncated Taylor expansions were developed and their

performance compared for determining an optimal set of switch times given the

gain matrices ...

CONCLUSION It has been shown that by restricting the structure of the feedback

gain matrix for the linear regulator problem to be

**piecewise**-**constant**, a degree ofoptimality can be traded for ease of implementation, and the loss of optimality can

.be reduced by increasing the number of constant-gain intervals. Three

algorithms based on truncated Taylor expansions were developed and their

performance compared for determining an optimal set of switch times given the

gain matrices ...

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### Contents

Surface Modeling Algorithms | 5 |

Structure of the Suboptimal Feedback Control | 13 |

Structure of the State Transition Matrix | 19 |

8 other sections not shown

### Common terms and phrases

accuracy approximation bound building blocks computed Consider constant gain matrices constraint algorithm contraction mapping cost index cubically convergent define a convergent denote determine eigenvalues equation example feedback control feedback gain matrix finite Gauss-Legendre quadrature gradient and Hessian Hess Hessian matrix Hessian p.d. Hessian implementation integral interpolation iterations to solution K!WK linear regulator problem matrix norm method modified Hessian nonsingular null space number of evaluations number of iterations number of switches OSV algorithm p.d. Hessian p.d. partial derivatives performance index piecewise-constant point quadrature polynomial positive definite positive definite matrix positive semidefinite Proof Q+K'WK QR algorithm quadratic R(tQ rate of convergence respect Riccati feedback gain Riccati solution satisfy scalar spectral norm structure suboptimal gain sufficient condition surface model symmetric Table Taylor expansion tensor tf;T Theorem A-2 thesis third-variation algorithm tion TjTi TjTk Tk Tj transition matrix U(KR variation algorithm vector