# Preview Control for MIMO Discrete-Time System with Parameter Uncertainty

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**A1**: The uncertain matrices are given by

**A2**: Let $r(k)={\left[\begin{array}{cccc}{r}_{1}(k)& {r}_{2}(k)& \cdots & {r}_{q}(k)\end{array}\right]}^{T}\in {R}^{q}$ be the reference signal. Assume that the component reference signal ${r}_{i}(k)$ $(i=1,2,\cdots q)$ is available from current time $k$ to $k+{\mathrm{h}}_{i}$. The future values are assumed not to change beyond $k+{\mathrm{h}}_{i}$, namely,

**Remark**

**1.**

- (i)
- The output tracks the reference signal without steady-state error, that is,$$\underset{k\to \infty}{\mathrm{lim}}{e}_{i}(k)=0$$
- (ii)
- The closed-loop system is robustly stable and exhibits acceptable transient responses for all $\theta \in \Theta $.

## 3. Derivation of AES

**Remark**

**2.**

## 4. PC Design

**Lemma**

**1.**

**Proof.**

#### 4.1. State Feedback PC

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

#### 4.2. Static Output Feedback PC

**Lemma**

**2.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Corollary**

**3.**

**Corollary**

**4.**

**Remark**

**3.**

**Remark**

**4.**

## 5. Numerical Example

#### Output Feedback Case

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zheng, D.Z. Linear System Theory; Tsinghua University Press: Beijing, China, 2012. [Google Scholar]
- Tan, K.K.; Zhao, S.; Xu, J.X. Online automatic tuning of a proportional integral derivative controller based on an iterative learning control approach. IET Control Theory A
**2007**, 1, 90–96. [Google Scholar] [CrossRef] - Li, Y.; Zhao, S.; He, W.; Lu, R. Adaptive finite-time tracking control of full state constrained nonlinear systems with dead-zone. Automatica
**2019**, 100, 99–107. [Google Scholar] [CrossRef] - Wang, Y.; Wang, R.; Xie, X.; Zhang, H. Observer-based H∞ fuzzy control for modified repetitive control. Neurocomputing
**2018**, 286, 141–149. [Google Scholar] [CrossRef] - Birla, N.; Swarup, A. Optimal preview control: A review. Optim. Control Appl. Methods
**2015**, 36, 241–268. [Google Scholar] [CrossRef] - Zhen, Z.Y. Research development in preview control theory and applications. Acta Autom. Sin.
**2016**, 42, 172–188. [Google Scholar] - Sheridan, T.B. Three models of preview control. IEEE Trans. Hum. Factors Electron.
**1996**, 7, 91–102. [Google Scholar] [CrossRef] - Bender, E.K. Optimum linear preview control with application to vehicle suspension. J. Basic Eng.
**1968**, 90, 213–221. [Google Scholar] [CrossRef] - Tomizuka, M. Optimal continuous finite preview problem. IEEE Trans. Autom. Control
**1975**, 20, 362–365. [Google Scholar] [CrossRef] - Tomizuka, M. Optimal discrete finite preview problem (Why and how is future information important?). J. Dyn. Syst. ASME
**1975**, 97, 319–325. [Google Scholar] [CrossRef] - Katayama, T.; Ohki, T.; Inoue, T.; Kato, T. Design of an optimal controller for a discrete-time system subject to previewable demand. Int. J. Control
**1985**, 41, 677–699. [Google Scholar] [CrossRef] - Katayama, T.; Hirono, T. Design of an optimal servomechanism with preview action and its dual problem. Int. J. Control
**1987**, 45, 407–420. [Google Scholar] [CrossRef] - Tsuchiya, T.; Egami, T. Digital Preview and Predictive Control; Beijing Science and Technology Press: Beijing, China, 1994. [Google Scholar]
- Wu, J.; Liao, F.; Tomizuka, M. Optimal preview control for a linear continuous-time stochastic control system in finite-time horizon. Int. J. Syst. Sci.
**2017**, 48, 129–137. [Google Scholar] [CrossRef] - Wang, D.; Liao, F.; Tomizuka, M. Adaptive preview control for piecewise discrete-time systems using multiple models. Appl. Math. Model
**2016**, 40, 9932–9946. [Google Scholar] [CrossRef] - Lu, Y.; Liao, F.; Deng, J.; Pattinson, C. Cooperative optimal preview tracking for linear descriptor multi-agent systems. J. Frankl. Inst.
**2019**, 356, 908–934. [Google Scholar] [CrossRef] [Green Version] - Lu, Y.; Liao, F.; Deng, J.; Liu, H. Cooperative global optimal preview tracking control of linear multi-agent systems: An internal model approach. Int. J. Syst. Sci.
**2017**, 48, 2451–2462. [Google Scholar] [CrossRef] [Green Version] - Running, K.D.; Martins, N.C. Optimal preview control of Markovian jump linear systems. IEEE Trans. Autom. Control
**2009**, 54, 2260–2266. [Google Scholar] [CrossRef] - Liao, F.; Wang, Y.; Lu, Y.; Deng, J. Optimal preview control for a class of linear continuous-time large-scale systems. Trans. Inst. Meas. Control
**2018**, 40, 4004–4013. [Google Scholar] [CrossRef] [Green Version] - Bidyadhar, S.; Ogeti, P.S. Optimal preview stator voltage-oriented control of DFIG WECS. IET Gener. Transm. Distrib.
**2018**, 12, 1004–1013. [Google Scholar] - Kojima, A. H∞ controller design for preview and delayed systems. IEEE Trans. Autom. Control
**2015**, 60, 404–419. [Google Scholar] [CrossRef] - Gershon, E.; Shaked, U. H∞ preview tracking control of retarded state- multiplicative stochastic systems. Int. J. Robust Nonlinear Control
**2014**, 24, 2119–2135. [Google Scholar] [CrossRef] - Kristalny, M.; Mirkin, L. On the H2 two-sided model matching problem with preview. IEEE Trans. Autom. Control
**2012**, 57, 207–212. [Google Scholar] [CrossRef] - Hamada, Y. Preview feedforward compensation: LMI synthesis and flight simulation. IFAC-Pap.
**2016**, 49, 397–402. [Google Scholar] [CrossRef] - Li, L.; Liao, F. Robust preview control for a class of uncertain discrete-time systems with time-varying delay. ISA Trans.
**2018**, 73, 11–21. [Google Scholar] [CrossRef] [PubMed] - Li, L.; Yuan, Y. Output feedback preview control for polytopic uncertain discrete-time systems with time-varying delay. Int. J. Robust Nonlinear Control
**2019**, 29, 2619–2638. [Google Scholar] [CrossRef] - Shao, Y.-F.; Gao, S.-J. Robust preview control of uncertain discrete systems based on an internal model approach. Sci. Technol. Vis.
**2018**, 1–3. [Google Scholar] [CrossRef] - Lan, Y.; Xia, J. Observer based design of preview repetitive control for linear discrete systems. Comput. Integr. Manuf. Syst.
**2019**, 1–18. [Google Scholar] - Lan, Y.; Xia, J.; Shi, Y. Robust guaranteed-cost preview repetitive control for polytopic uncertain discrete-time systems. Algorithms
**2019**, 12, 20. [Google Scholar] [CrossRef] [Green Version] - Han, K.; Feng, J.; Li, Y.; Li, S. Reduced-order simultaneous state and fault estimator based fault tolerant preview control for discrete-time linear time-invariant systems. IET Control Theory A
**2018**, 12, 1601–1610. [Google Scholar] [CrossRef] - Han, K.; Feng, J. Data-driven robust fault tolerant linear quadratic preview control of discrete-time linear systems with completely unknown dynamics. Int. J. Control
**2019**, 1–11. [Google Scholar] [CrossRef] - Yu, X.; Liao, F. Preview tracking control for a class of discrete-time Lipschitz non-linear time-delay systems. IMA J. Math. Control Inf.
**2019**, 36, 849–867. [Google Scholar] [CrossRef] - Lio, W.H.; Jones, B.L.; Rossiter, J.A. Preview predictive control layer design based upon known wind turbine blade-pitch controllers: MPC layer design based upon known blade-pitch controllers. Wind Energy
**2017**, 20, 1207–1226. [Google Scholar] [CrossRef] [Green Version] - Zhao, Y.; Cai, Y.; Song, Q. Energy control of plug-in hybrid electric vehicles using model predictive control with route preview. IEEE/CAA J. Autom. Sin.
**2018**, 1–8. [Google Scholar] [CrossRef] [Green Version] - Al Khudir, K.; Halvorsen, G.; Lanari, L.; De Luca, A. Stable torque optimization for redundant robots using a short preview. IEEE Robot. Autom. Lett.
**2019**, 4, 2046–2053. [Google Scholar] [CrossRef] [Green Version] - Pak, H.A.; Shieh, R. On mimo optimal preview tracking control for known trajectory models. Optim. Control Appl. Methods
**1991**, 12, 119–130. [Google Scholar] [CrossRef] - Li, L.; Liao, F. Parameter-dependent preview control with robust tracking performance. IET Control Theory Appl.
**2017**, 11, 38–46. [Google Scholar] [CrossRef] - Chen, Y.; Fei, S.; Li, Y. Stabilization of neutral time-delay systems with actuator saturation via auxiliary time-delay feedback. Automatic
**2015**, 52, 242–247. [Google Scholar] [CrossRef] - He, Y.; Wu, M.; She, J.-H. Improved bounded-real-lemma representation and H∞ control of systems with polytopic uncertainties. IEEE Trans. Circuits Syst. II Express Briefs
**2005**, 52, 380–383. [Google Scholar] - Chang, X.H.; Zhang, L.; Park, H.P. Robust static output feedback H∞ control for uncertain fuzzy systems. Fuzzy Sets Syst.
**2015**, 273, 87–104. [Google Scholar] [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, L.; Liao, F.
Preview Control for MIMO Discrete-Time System with Parameter Uncertainty. *Mathematics* **2020**, *8*, 756.
https://doi.org/10.3390/math8050756

**AMA Style**

Li L, Liao F.
Preview Control for MIMO Discrete-Time System with Parameter Uncertainty. *Mathematics*. 2020; 8(5):756.
https://doi.org/10.3390/math8050756

**Chicago/Turabian Style**

Li, Li, and Fucheng Liao.
2020. "Preview Control for MIMO Discrete-Time System with Parameter Uncertainty" *Mathematics* 8, no. 5: 756.
https://doi.org/10.3390/math8050756