# Comparison and Interpretation Methods for Predictive Control of Mechanics

## Abstract

**:**

## 1. Introduction

A key contribution is the design of predictive controllers that are designed using optimization as the very first step, including a formulation of state feedback for robustness in the same optimization while a subsequent step converts the optimal solution from time-parameterization to state-parameterization allowing proportional-derivative gains to be expressed as exact functions of the optimal solution. Thus, feedback errors are de facto expressed exactly in terms of the solutions to the original optimization problem and errors are, thereby, optimally rejected. This notion permits the reader to use only this predictive, optimal feedback controller by itself and also together with the optimal feedforward. Lastly, comparing the optimal feedforward to the predictive optimal feedback control permits expression of a proposed controller called “2DOF” to imply the twice-invocation of the original optimization problem. This proposed 2DOF topology achieves near-machine precision target tracking errors while using near-minimal costs.

## 2. Materials and Methods

^{2}, also known as least squares). The controllers were designed to both meet the end conditions and minimize the cost function (often opposing goals) from the outset. Designs were iterated to get as close as possible to the end conditions while keeping cost low (a ubiquitous trade-off). In the main body of the manuscript, each controller is designed with descriptions to address the key issues of control in addition to how well they perform in a Monte Carlo analysis.

#### 2.1. Open Loop Optimum Controller

- Write the control Hamiltonian.$$H\left(\lambda ,\theta ,u,t\right)=F\left(\theta ,u,t\right)+{\lambda}^{T}f\left(\theta ,u,t\right)$$
- Implement the Hamiltonian Minimization Condition for the static problem of Equation (8).$$\mathrm{min}H\left(\dots ;u\right)whereu\in R,thatis\frac{\partial H}{\partial u}=0$$This is a constrained minimization problem, so use Equations (9) and (10) where ${\lambda}_{\omega}$ is the Lagrange multiplier associated with the co-state.$$\frac{\partial H\left(\dots ;u\right)}{\partial u}=u+{\lambda}_{\omega}=0$$$$u\left({\lambda}_{\omega}\right)=-{\lambda}_{\omega}$$Confirm optimality by verifying the convexity condition in Equation (11).$$\frac{{\partial}^{2}H\left(\dots ;u\right)}{\partial {u}^{2}}=1>0$$The result: once we find the co-state, we will have optimum control. Notice Karush-Kuhn-Tucker conditions often used with inequality constraints are not necessary here.
- Apply the Adjoint Equations per Equations (12) and (14), which result in Equations (13) and (15), respectively, and are plotted in Figure 1b.$$-{\dot{\lambda}}_{\theta}=\frac{\partial H}{\partial \theta}=0$$$${\lambda}_{\theta}=a$$$$-{\dot{\lambda}}_{\omega}=\frac{\partial H}{\partial \omega}={\lambda}_{\theta}=a$$Integrating (14)$${\lambda}_{\omega}=-\mathrm{at}+b$$
- Rewrite the Hamiltonian Minimization Condition in Equation (8) by substituting Equation (9).$$\frac{\partial H}{\partial u}=u+{\lambda}_{\omega}=0\to u=-{\lambda}_{\omega}$$
- Substituting Equation (15) into Equation (16) produces Equation (17).$$\mathrm{Optimal}\mathrm{control},{u}^{\ast}\equiv u=\mathrm{at}+b$$
- Implement in SIMULINK as displayed in Figure 2 using Equations (3), (18)–(20), which are simulated per Figure 2 and whose results are plotted in Figure 1a.$$u=\ddot{\theta}=\dot{\omega}=at+b$$$$\omega =\dot{\theta}={\displaystyle \int}\ddot{\theta}dt={\displaystyle \int}\left(at+b\right)dt=\frac{a}{2}{t}^{2}+bt+c$$$$\theta ={\displaystyle \int}\dot{\theta}dt={\displaystyle \int}\left(a{t}^{2}+bt+c\right)dt=\frac{a}{6}{t}^{3}+\frac{b}{2}{t}^{2}+ct+d$$

ASIDE: We’ll see in the next section how to implement the more general optimum control and solve for constants a and b as time, t progresses. That controller is referred to as the continuous predictive optimal closed-loop controller.

#### 2.2. Continuous Predictive Closed Loop Optimum Controller

- Recall Equations (18)–(20) where $\theta \left(1\right)=1,\omega \left(1\right)=0$ yields:$$\theta \left\{1\right\}=\frac{a}{6}\left(1\right)+\frac{b}{2}\left(1\right)+c\left(1\right)+d=\frac{a}{6}+\frac{b}{2}+c+d=1$$$$\omega \left(1\right)=\frac{a}{2}\left(1\right)+b\left(1\right)+c=\frac{a}{2}+b+c=0$$
- Set up a matrix equation in the form $\left[A\right]\left\{x\right\}=\left\{B\right\}$ simulated in Figure 3.$$\underset{\left[A\right]}{\underbrace{\left[\begin{array}{cccc}\frac{{t}_{0}^{3}}{6}& \frac{{t}_{0}^{2}}{2}& {t}_{0}& 1\\ \frac{1}{6}& \frac{1}{2}& 1& 1\\ \frac{{t}_{0}^{2}}{2}& {t}_{0}& 1& 1\\ \frac{1}{2}& 1& 1& 0\end{array}\right]}}\underset{\left\{x\right\}}{\underbrace{\left\{\begin{array}{c}a\\ b\\ c\\ d\end{array}\right\}}}=\underset{\left\{B\right\}}{\underbrace{\left\{\begin{array}{c}\theta \left(0\right)\\ 1\\ \omega \left(0\right)\\ 0\end{array}\right\}}}\to \left\{{x}^{\ast}\right\}={\left[A\right]}^{\u2020}\left\{B\right\}\to \{\begin{array}{c}x\left(1\right)\\ x\left(2\right)\\ x\left(3\right)\\ x\left(4\right)\end{array}$$

^{−1}. The value of [A]

^{−1}is an intermediate calculation step to get [a b c d]. Calculating the matrix inverse can be problematic, so avoid the unnecessary pitfalls by not calculating [A]

^{−1}. Instead, use the LU solver (or QR solver, etc.) or the LU inverse.

#### 2.3. Sampled-Data Predictive Optimum Controller

#### 2.4. Proportional Plus Derivative (PD) Controller Derived Foremost from an Optimization Problem

Defining$a=-\left({K}_{\theta}+{K}_{\omega}\omega \right)$and$b={K}_{\theta}+{K}_{\omega}\omega $, the optimum control$u=at+b$as a function of the time may be written as a function of states:$u={K}_{\theta}\left(1-t\right)+{K}_{\omega}\left(1-t\right)\omega $where the K’s are feedback gains that are functions of the θ and ω error.

#### 2.5. Feedforward/Feedback PD Controller

#### 2.6. Two-DOF Controller: Optimal Control Augmented with Feedback Errors Calcuated with Optimal States

## 3. Results

#### 3.1. Monte Carlo Analysis on a Deterministic Plant with Noise

#### 3.2. Open Loop Optimal Controller

#### 3.3. Continuous-Update Optimal Controller

#### 3.4. Sample-Predictive Optimum Controller

#### 3.5. PD Control Derived Foremost from an Optimization Problem

#### 3.6. Feedforward/Feedback PD Controller

#### 3.7. Two-DOF Controller: Optimal Control Augmented with Feedback Errors Calcuated with Optimal States

#### 3.8. Monte Carlo Analysis on a Mismodeled Plant with Noise

## 4. Discussion

Proposed 2DOF control designed foremost as an optimization problem: The open-loop optimal control is used as a feedforward, while the optimal states derived from a time-parameterized optimal control are compared to the feedback signal to generate the error fed to the feedback controller whose gains are a reparameterization of the optimal solution.

## 5. Future Works

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Optimum Open Loop no noise (Figure A1)
- Optimum Open Loop with noise (Figure A2)
- PD Controller no noise (Figure A3)
- PD Controller with noise (Figure A4)
- Continuous Predictive Optimum no noise (Figure A5)
- Continuous Predictive Optimum with noise (Figure A6)
- Feedforward/Feedback PD no noise (Figure A7)
- Feedforward/Feedback PD with noise (Figure A8)
- 2DOF PD Controller no noise (Figure A9)
- 2DOF PD Controller with noise (Figure A10)
- Sampled Predictive Controller without Noise (Figure A11)
- Sampled Predictive Controller with Noise (Figure A12)
- Mis-modeled Plant (Figure A13)
- Error Norms (Figure A14)

#### Appendix A.1. Optimum Open Loop Controller-No Noise

#### Appendix A.2. Optimum Open Loop Controller-with Noise

#### Appendix A.3. PD Controller No Noise

#### Appendix A.4. PD Controller with Noise

**Figure A4.**Proportional-derivative (PD) controller (${K}_{w}=-13.3,{K}_{q}=40$) simulation model with noise.

#### Appendix A.5. Continuouse Predictive Optimum Controller-No Noise

#### Appendix A.6. Continuouse Predictive Optimum Controlle-with Noise

#### Appendix A.7. Feedforward/Feedback PD Controller-No Noise

#### Appendix A.8. Feedforward/Feedback PD Controlle-with Noise

#### Appendix A.9. 2DOF PD Controller-No Noise

#### Appendix A.10. 2DOF PD Controller-with Noise

#### Appendix A.11. Sampled Predictive Controller-without Noise

#### Appendix A.12. Sampled Predictive Controller-with Noise

#### Appendix A.13. Mismodeled Plant

#### Appendix A.14. Error Norms

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**Figure 1.**External states, co-states, and control for optimally controlled double integrator plant with time on the abscissa and values on the ordinate. (

**a**) State trajectories, θ(t) and ω(t) from Equations (18)–(20). (

**b**) Co-state trajectories in shadow space λ_θ (t), λ_ω (t), and control u(t) solutions to Equations (12)–(15), and (17).

**Figure 4.**Shrinking horizon prediction control problem: Time-varying values of integration constants computed online by a continuous predictive closed-loop optimum controller. Dotted lines are no noise cases and results displayed with solid lines are run with noise. The abscissa contains time, while the ordinate displays the values of the constants. a and b in Equation (18).

**Figure 10.**Monte Carlo analysis of a deterministic plan with noise with $\theta $ (radians) on the abscissa and $\omega $ (rad/s) on the ordinate.

Controller | Deviation, ${\mathit{\sigma}}_{\mathit{\theta}}$ | Mean Error, ${\mathit{\mu}}_{\mathit{\theta}}$ | Deviation, ${\mathit{\sigma}}_{\mathit{\omega}}$ | Mean Error, ${\mathit{\mu}}_{\mathit{\omega}}$ | Mean Cost, J |
---|---|---|---|---|---|

Optimal open loop | 0.0328 | 0.0439 | 0.03 | 0.045 | 5.9787 |

Continuous predictive | 0.0015 | 0.002 | 0.0192 | 0.0251 | 6.0117 |

Sampled predictive | 0.0031 | 0.0074 | 0.0249 | 0.0321 | 6.1137 |

Proportional-derivative (PD) | 9.66 × 10^{−4} | 1.41 × 10^{−2} | 3.14 × 10^{−16} | 1.02 × 10^{−15} | 59.6885 |

Feedforward + feedback | 7.86 × 10^{−16} | 9.83 × 10^{−16} | 0.0058 | 0.0087 | 8.4507 |

2DOF | 7.07 × 10^{−16} | 1.02 × 10^{−15} | 1.89 × 10^{−15} | 2.51 × 10^{−15} | 8.619 |

^{1}note highest and lowest values in bold font.

Controller | Deviation, ${\mathit{\sigma}}_{\mathit{\theta}}$ | Mean Error, ${\mathit{\mu}}_{\mathit{\theta}}$ | Deviation, ${\mathit{\sigma}}_{\mathit{\omega}}$ | Mean Error, ${\mathit{\mu}}_{\mathit{\omega}}$ | Mean Cost, J |
---|---|---|---|---|---|

Optimal open loop | 0.034 | 0.044 | 0.0283 | 0.0484 | 6.004 |

Continuous predictive | 0.0015 | 0.0019 | 0.02 | 0.025 | 6.0137 |

Sampled predictive | 0.0029 | 0.007 | 0.0222 | 0.0307 | 6.0183 |

Proportional-derivative (PD) | 9.89 × 10^{−4} | 0.014 | 3.12 × 10^{−16} | 1.02 × 10^{−15} | 59.3244 |

Feedforward + feedback | 7.36 × 10^{−16} | 9.69 × 10^{−16} | 0.0058 | 0.0086 | 8.9637 |

2DOF | 6.42 × 10^{−16} | 9.02 × 10^{−16} | 1.90 × 10^{−15} | 2.56 × 10^{−15} | 7.8284 |

^{1}note highest and lowest values in bold font.

Controller | Benefits | Weakness |
---|---|---|

Optimal open loop | Establishes optimal case | Not realistically implementable |

Continuous predictive * | Good cost and position | High rate error and deviation |

Sampled predictive * | Good cost and position | High rate error and deviation |

Proportional-derivative (PD) * | Best rate control | Worst cost |

Feedforward + feedback | Best position control | Slight rate error and deviation |

2DOF * | All-around good | Slightly higher cost than optimal |

^{1}Large generalizations; * Not the ubiquitous PD topology.

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Sands, T.
Comparison and Interpretation Methods for Predictive Control of Mechanics. *Algorithms* **2019**, *12*, 232.
https://doi.org/10.3390/a12110232

**AMA Style**

Sands T.
Comparison and Interpretation Methods for Predictive Control of Mechanics. *Algorithms*. 2019; 12(11):232.
https://doi.org/10.3390/a12110232

**Chicago/Turabian Style**

Sands, Timothy.
2019. "Comparison and Interpretation Methods for Predictive Control of Mechanics" *Algorithms* 12, no. 11: 232.
https://doi.org/10.3390/a12110232