This paper presents an optimal control approach to robust control of linear and non-linear systems. The approach translates the robust control problems into optimal control problems of minimizing cost functionals. Because of the uncertainty bound that is reflected in the cost functionals, the solution of the optimal control problems are solutions of the robust control problems. The optimal control approach allows us to solve some non-linear robust control problems that cannot be easily solved otherwise.   Keywords: Robust Control, Optimal Control, Cost Functional, Dynamic Equation, Riccati equation, Uncertainty

OKE, M.O.

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  69 Optimal Control Approach to Robust Control of Linear and Non-linear Systems M.O. OKE Department of Mathematical Sciences, Ekiti State University, P.M.B. 5363, Ado-Ekiti, Nigeria E-mail: femioke91@gmail.com Abstract This paper presents an optimal control approach to robust control of linear and non-linear systems. The approach translates the robust control problems into optimal control problems of minimizing cost functionals. Because of the uncertainty bound that is reflected in the cost functionals, the solution of the optimal control problems are solutions of the robust control problems. The optimal control approach allows us to solve some non-linear robust control problems that cannot be easily solved otherwise.  Keywords: Robust Control, Optimal Control, Cost Functional, Dynamic Equation, Riccati equation, Uncertainty.  1. Introduction Robust control is concerned with modeling uncertainties. Nearly all models in engineering and applied sciences have errors and uncertainties. A controller is said to be robust if it works under uncertainties and performs even when the actual system deviates from its ideal model on which the controller’s design is based. Robust control design has a very good tolerance to modeling errors and uncertainties. There are several approaches available for the design of robust control. These include parametric, fuzzy control, optimal control and /  approaches. In the optimal control approach, we do not attempt to solve the robust control problems directly; rather we translate the robust control problems into optimal control problems. The solutions of the optimal control problems are now taking as the solutions of the robust control problems because of the uncertainty bound that is reflected in the cost functionals. To use the optimal control approach, the matching condition must be satisfied. We applied the optimal control approach to both linear and non-linear systems. Optimal control is a particular control technique in which the control signal optimizes a certain cost index. After stabilizing a system, the next thing is to optimize the system performance.  A lot of research work had been conducted on robust control systems. Lin et al. (1992) considered the compensation for uncertainty in the robust control of non-linear systems while Kuo and Wang (1991) worked on the robust position control of robotics manipulator in cartesian coordinates. Kehinde (2000) applied the fuzzy control to robust control problems while Mohammed and Hamson (2008) looked the Kharitorov and Hankel norms approach to some robust control problems. Daniel and Kuo (2010) studied the application of fuzzy logic to the robust control of linear systems while Landau et al. (1996) worked on the robust control of a 360 flexible arm by using the combined pole placement/sensitivity function shaping method. Lin and Olbrot (1996) used a linear quadratic regulator approach for robust control of linear systems with uncertainty parameters while Lin and Zhang (1993) concentrated on the Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  70 robust control of non-linear systems without matching condition. None of these researchers considered the optimal control approach to the robust control of linear and non-linear systems, hence the need for this research paper.   2. Materials and Methods The optimal control problem to be considered in this paper is of the form   which minimizes the cost functional   where  is the initial time,  is the final time,  is the current state and  characterizes the cost objectives.  Applying the principle of optimality to the problems in (1) and (2) above and considering a time  in between and , then the cost functional  becomes   which gives   where   is the state at  and . Let  denote the minimal cost functional. Then, by the principle of optimality we have   Approximating and using the Taylor’s series expansion, we get   Now since  and  are independent of . (6) can therefore be written as   which implies that    For a time-invariant system with an infinite horizon (i.e. , we obtain, the Hamilton-Jacobi-Bellman equation   For a linear quadratic time invariant regulator problem, we have Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  71   which minimizes the cost functional   where  is a symmetric, positive semi-definite matrix and  is a symmetric, positive definite matrix. To find the solution to the linear quadratic regulator problem, we assume the minimal cost to be quadratic in the form   where  is a symmetric, positive semi-definite matrix function of t. By the Hamilton-Jacobi-Bellman equation, the optimal control  satisfies (9) above or   To find this minimum, we set the derivative of  with respect to u to be equal to zero to get   Therefore the optimal control is given by   Now  can be therefore calculated as   which now gives     Simplifying this we have   That is   By the Hamilton-Jacobi-Bellman equation, we have   In other words  satisfies   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  72 (21) is the Riccati equation.  If the cost functional is quadratic with infinite horizon, then we have our optimal control problem as   which minimizes the cost functional   The optimal control of this linear quadratic regulator problem with an infinite horizon will only exist if the system is stabilizable and the Riccati equation will further reduce to the following algebraic equation    3. Robust Control of Linear Systems A system (network) is said to be linear if the principle of superposition and the principle of proportionality holds. By the superposition principle, if the excitation of the system is given by , then the response would be given by . That is, the responses produced by the simultaneous application of two different forcing functions are the sum of the two individual responses. By the proportionality principle, if the excitation is given by , where  is a constant of proportionality, then the response would be . That is, the constant of proportionality ( ) is preserved by the linear network, Franklin (1966). Let us consider the robust control of linear time invariant systems of the form   where  and  are state and control variables respectively. Matrices A and B are used to model the uncertainty in the system dynamics. The system to be controlled is therefore given by   where  is an  matrix representing the uncertainty in the system. Our goal is to design a state feedback to stabilize the system for all possible values of  within a specified bound. The solution of this robust problem depends on whether the uncertainty satisfies a matching condition which requires that the uncertainty is within the range of . If the uncertainty satisfies the matching condition, then the solution to the robust control problem exists and can be obtained by solving the associated linear quadratic regulator problem. The associated linear quadratic regulator problem is obtained by just including the bounds on the uncertainty in the cost functional.  If the uncertainty does not satisfy the matching condition, then we first decompose the uncertainty into the matched and unmatched parts. We then use an augmented control to deal with the unmatched uncertainty, Lin (1997) and Lin et al. (1992).  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  73 For a matching condition to be satisfied, the uncertainty in A can be written as   For an m   matrix  which is bounded, where  is a nominal value of ,  the system dynamics can therefore be written as   where  To solve the robust control problem of stabilizing the system under uncertainty, we find a control law  such that the closed-loop system    is asymptotically stable for all , Mohammed and Hamson (2008). For the auxiliary system  we find a feedback control law  that minimizes the cost functional   where   is an upper bound on the uncertainty . To solve the linear quadratic regulator problem, we just solve the algebraic Riccati equation given by    to get   For an unmatched uncertainty, we consider a system   The system dynamics can therefore be written as   where  We now find a control law  such that the closed-loop system   is asymptotically stable. In order to solve this robust control problem, we first decompose the uncertainty  into the sum of matched and unmatched components. This can be done by using pseudo-inverse of  given by , Lin and Olbrot (1996). Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  74 Let    Then  is the matched component and   is the unmatched component. If the matching condition is satisfied, then the unmatched part will be equal to zero. Now let    So that   we also have     where F and H are upper bounds on the uncertainties and  is a design parameter. We now solve the robust control problem indirectly by translating it into a linear quadratic regulator problem. For the auxiliary system given by    we will find feedback control laws  and  that minimizes the cost functional    where  and  are design parameters. In this linear quadratic regulator problem,   is an augmented control that is used to deal with the unmatched uncertainty. If the matching condition is satisfied, then the design parameters are taken to be and  and the solution of the linear quadratic regulator problem is given by    where S is the unique positive definite solution to the following algebraic Riccati equation    4. Robust Control of Non-Linear Systems Robust control design is more complex for non-linear systems. In fact many approaches to robust control problems are applicable only to linear systems. But the optimal control approach is applicable to both linear and non-linear systems. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  75 Let us consider a non-linear system   where  is an  positive semi-definite matrix representing the uncertainty in the input matrix.  and  are non-linear (matrix) function of    and  are used to model the uncertainty in the system dynamics. Since the uncertainty is in the range of , the matching condition is satisfied. For the robust control problem, we find a feedback control law  such that the closed loop system given by   is globally asymptotically stable for all uncertainties  satisfying , Lin and Zhang (1993). For the nominal system    we find a feedback control law  that minimizes the following cost functional     If the uncertainty is not in the range of , then we consider the following non-linear system   where   models the uncertainty in the system dynamics and it does not satisfy the matching condition.  is a matrix introduced to make  flexible. We now find a feedback control law  such that the closed loop system given by    is globally asymptotically stable for all uncertainties . For the auxiliary system given by    we find a feedback control law   that minimizes the following cost functional   where  , , and  are design parameters.  and  are non-negative functions such that   and    Conclusion We have used the optimal control approach to represent the robust control of linear and non-linear systems in this paper. We accomplished this by translating the robust control problems into optimal control problems of minimizing cost functionals. The uncertainties of the robust control systems are reflected in the performance index of the optimal control problems. This approach allows us to solve some non-linear robust control problems that cannot be easily solved by the use of other methods. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.3, 2015  76  References Kehinde, A.O. (2000), “Application of Fuzzy Control to Robust Control”, REBEV Journal of Engineering, 5(2), 68 – 82. Mohammed, J.I. and Hamson, O.R. (2008), “Kharitorov and Hankel Norms Approach to Robust Control”, DAVE Journal of Scientific Applications, 6(3), 112 – 120. Franklin, F.K. (1966),  “Network Analysis and Synthesis”, John Wiley & Sons Inc. Daniel, A.O. and Kuo, F. (2010), “Application of Fuzzy Logic to the Robust Control of Linear Systems”, Journal of Science and Engineering Applications, 5(4), 88 – 94. Lin, F. (1997), “Robust Control Design: An Optimal Control Approach”, AFI Press.  Lin, F., Brandt, R.D. and Sun, J. (1992), “Robust Control of Non-linear Systems: Compensating for uncertainty”, International Journal of Control, 56, 1453 – 1459. Kuo, C.Y. and Wang, S.P.T. (1991), “Robust Position Control of Robotics Manipulator in Cartesian Coordinates”, IEEE Transactions on Robot Automation, 7, 653 – 659. Landau, I.D., Langer, J., Rey, D. and Barnier, J. (1996), “Robust Control of a 360 Flexible Arm using the Combined Pole Placement/Sensitivity Function Shaping Method”, IEEE Transactions on Control System Technology, 4, 369 – 383 Lin, F. and Olbrot, A.W. (1996), “An LQR Approach to Robust Control of Linear Systems with Uncertainty Parameters”, Proceedings of the 35th IEEE Conference Decision Control, 35, 4158 – 4163.  Lin, F. and Zhang, W. (1993), “Robust Control of Nonlinear Systems without Matching Condition”, Proceedings of the 32nd IEEE Conference Decision Control, 32, 2572 – 2577.  The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   Prospective authors of journals can find the submission instruction on the following page: http://www.iiste.org/journals/  All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself.  Paper version of the journals is also available upon request of readers and authors.   MORE RESOURCES Book publication information: http://www.iiste.org/book/ Academic conference: http://www.iiste.org/conference/upcoming-conferences-call-for-paper/   IISTE Knowledge Sharing Partners EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar   

Optimal Control Approach to Robust Control of Linear and Non-linear Systems

https://www.iiste.org/Journals/index.php/MTM/article/download/20422/21114

Optimal Control Approach to Robust Control of Linear and Non-linear Systems

Abstract

Similar works

Full text

Available Versions

International Institute for Science, Technology and Education (IISTE): E-Journals