### Abstract

In this brief, we present a set of techniques for finding a cost function to the time-invariant linear quadratic regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient linear matrix inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. In addition, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrix K from the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.

Original language | English |
---|---|

Article number | 6880317 |

Pages (from-to) | 770-777 |

Number of pages | 8 |

Journal | IEEE Transactions on Control Systems Technology |

Volume | 23 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 Mar 1 |

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### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Control Systems Technology*,

*23*(2), 770-777. [6880317]. https://doi.org/10.1109/TCST.2014.2343935

}

*IEEE Transactions on Control Systems Technology*, vol. 23, no. 2, 6880317, pp. 770-777. https://doi.org/10.1109/TCST.2014.2343935

**Solutions to the inverse LQR problem with application to biological systems analysis.** / Priess, M. Cody; Conway, Richard; Choi, Jongeun; Popovich, John M.; Radcliffe, Clark.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Solutions to the inverse LQR problem with application to biological systems analysis

AU - Priess, M. Cody

AU - Conway, Richard

AU - Choi, Jongeun

AU - Popovich, John M.

AU - Radcliffe, Clark

PY - 2015/3/1

Y1 - 2015/3/1

N2 - In this brief, we present a set of techniques for finding a cost function to the time-invariant linear quadratic regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient linear matrix inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. In addition, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrix K from the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.

AB - In this brief, we present a set of techniques for finding a cost function to the time-invariant linear quadratic regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient linear matrix inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. In addition, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrix K from the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.

UR - http://www.scopus.com/inward/record.url?scp=85027957809&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027957809&partnerID=8YFLogxK

U2 - 10.1109/TCST.2014.2343935

DO - 10.1109/TCST.2014.2343935

M3 - Article

AN - SCOPUS:85027957809

VL - 23

SP - 770

EP - 777

JO - IEEE Transactions on Control Systems Technology

JF - IEEE Transactions on Control Systems Technology

SN - 1063-6536

IS - 2

M1 - 6880317

ER -