Numerical Solution of Non Linear Ware Equations Applying ...
Numerical Solution of Nonlinear Optimal Control Problems
Using Nonlinear Programming
K.P.BADAKHSHAN, A.V. KAMYAD, M.M.FARD
Engineering, Mathematics, Engineering Department
Ferdowsi University of Mashhad
IRAN
Abstract: In solving nonlinear control (NOC) problems and especially nonlinear optimal control problems, classical methods usually are not efficiency. In this paper we submit a new approach for solving this class of problems by using nonlinear programming. First, we put the problem to a new problem in form calculus of variations. Then we discretize the new problem and solve it by using nonlinear packages such as Lingo, Matlab. Then we obtain the optimal control and states which they are the exact solution of the original problem (nonlinear optimal control). Also we can obtain the nonlinear optimal control problems as a combination of power functions and periodic functions. Then, we bringing some different examples that efficiency of the proposed method has been confirmed.
Key words and phrases: nonlinear optimal control, nonlinear programming, approximation, discretization.
1 Introduction
Many methods are proposed to control of nonlinear systems, such as feedback linearization [1], Robust control [2], adaptive control [3] and other people apply combinational numerical algorithms, as genetic algorithm, Neural network, and fuzzy theory. Some of them are able to solve a few nonlinear systems but are not a general algorithm for all systems and have some difficulty, such as instability, non robust with respect to system disturbance, non causal, expensive, solution time explication and many other difficulties [4,5]. Kamyad and his coworker converted partial differential equations (PDEs) to a classical optimal control problem by using artificial control then solved this problem by measure theory [6] and solved nonlinear PDEs as a control problem by using measure theory [7-11]. He was applied measure theory in PDEs and time-varying system in [12].
In this paper we continue our algorithm [13] that worked on linear systems. This algorithm introduces an approximate solution of the NOC based on optimization. In this approach, we define an equivalent minimization problem to the NOC. Then by discretizing the new problem, we reach to a nonlinear programming (NLP) problem that in fact the analytic solution of the new problem is the solution of the original problem. But the solution of the new problem is an approximate solution of the original problem. Moreover error of this obtained approximate solution is controllable. Finally our algorithm is proposed to solve different control problems such as optimal control problem; especially control of nonlinear systems. This approach may be applied to all systems as linear, nonlinear and time varying systems. In case of linear time varying systems, main problem is approximated to a linear programming (LP) problem, which is solved with large variants of powerful software such as LINDO and so on, which for nonlinear systems it is changed to a nonlinear programming (NLP) problem, and can be solved by nonlinear programming packages such as LINGO. Also we applied our algorithm to some example of NOC problems and the result confirms the efficiency of the algorithm.
2 Nonlinear Optimal Control (NOC) Problem:
1.1 Definition 1:
We focused on following nonlinear time varying system:
|(1) |[pic] |
| |[pic] |
| |[pic] |
Where [pic] is a continues nonlinear time varying function, [pic], [pic], [pic], [pic] and [pic] [pic] are compact subset and must be chosen so that the system reach from initial state [pic]to final state [pic]. Also [pic] is the state function and [pic] is the control function, and [pic][pic] are given initial and final state in [pic]respectively, that may be fix or free.
1.2 Definition 2:
We may use the above formulation to obtain a solution for the NOC problem as follows:
|[pic][pic] |(2) |
|[pic][pic] | |
|[pic] | |
|[pic] | |
Where [pic] is a continues function. Particularly in optimal control problems[pic] can be an energy or fuel function as below:
| |[pic] |
| |[pic] |
In general, [pic] also may be a multi purpose or multi objective functional, for example minimization of fuel dissipation or maximization of benefit is a common multi objective function.
3 A Calculus Variation Problem Equivalent to NOC Problem:
Consider nonlinear system (1) at first, we define following functional that is called the error functional. Let:
|[pic] |(3) |
where [pic] is the absolute value function and other parameter introduced in definition.1. We define the following problem in calculus of variations:
|[pic][pic] |(4) |
|[pic][pic] | |
|[pic] | |
And for NOC problem (2) we define
|[pic][pic] |(5) |
|[pic][pic] | |
|[pic] | |
|[pic] | |
Where [pic] and other parameters are as same as calculus variable problem (4). In the following, we assume original NOC problem is (5) thus for reach to problem (4) this is sufficient to change [pic] from equation (3).
To solve a NOC problem using Euler-Lagrange method may cause many difficulties. So, numerical solution used to obtain an approximate solution for NOC problem. Here we convert the problem to a nonlinear programming problem and finally we obtain an approximate solution for the original problem by using the solution of the NLP problem.
Now, the following theorem that is a key lemma is demonstrated.
3.1 Lemma.1
If [pic]is a nonlinear continuous function on [pic] and non-negative [pic] , then it may be written:
|[pic] | |
Proof:
[pic]Let assume [pic] and [pic] for some[pic] and let [pic], since [pic] is a continues function so [pic] for some neighborhood of [pic]. Therefore [pic] this is contradiction to our assumption.
[pic] Where [pic] on [pic] then obviously [pic], thus proves the lemma.(
3.2 Theorem.1
Necessary and sufficient condition for [pic] be a solution of NOC problem (1) is [pic] in minimization problem (4).
proof:
[pic]If [pic] is the solution of NOC problem, the [pic] on [pic] so by definition we have [pic].
[pic] If [pic] and [pic]be a function where [pic], then we have [pic], [pic]. Because the absolute value function [pic] is continuous, and also functions [pic] and [pic] are continuous. Then [pic] is a nonnegative continues function. Therefore from assumption [pic], we conclude [pic] every where on [pic] , or:
|[pic] [pic] | |
Therefore, theorem.2 is proved.(
Note: Without loss of generality, we may assume [pic] and [pic].
4 Discretization
We partition interval [pic] to [pic] parts and also [pic] to [pic] equal subinterval (cells), where [pic] and [pic] be arbitrary fixed positive integer, then [pic] and [pic]. Let:
|[pic] | |
|[pic] |[pic] |
|[pic] | |
|[pic] | |
|[pic] |[pic] |
|[pic] | |
|[pic] |[pic] |
and also assume:
|[pic] | | |
Where [pic]are [pic] unknown control parameters. For the first derivative we have:
|[pic][pic][pic] | |
Thus, we obtain discretized problem (6) in the form:
|[pic] |(7) |
|[pic] | |
|[pic][pic] | |
| [pic] , [pic] | |
4.1 Remark:
As we know, an approximate value of integral [pic] is [pic], where [pic] is any point such that [pic]. So, applying above remark, the minimization problem (5) is formed as
|[pic] |(8) |
|[pic] | |
|[pic][pic] | |
| [pic] , [pic] | |
In general the problem (8) is a NLP problem and we may obtain its solution by many packages (Lingo, Matlab, Gino, etc). Finally by obtaining the solution of problem (8), we recognize the value of unknown admissible pair ([pic],[pic]) state and control function at [pic] point. We can construct the optimal solution for NOC problem by two piecewise functions ([pic],[pic]). Also we may fit two curves by those functions and introduce two time variant continues functions for optimal state and control. Theorem 3 will show existence of the optimal solution for the NOC problem (1).
4.2 Theorem.2
If ([pic],[pic]) be the pair constructed solution of the above NLP problem and optimal objective function be zero
|[pic] | |
then ([pic], [pic]) is a solution for problem (1).
proof:
By assumption ([pic], [pic]) is a pair piecewise continues function with the corresponding objective zero, so by theorem.1 we can write ([pic], [pic]) is a solution of problem (1).(
4.3 Remark:
If we want to control the total error of the problem (8) which is discretized of the original problem [6], it is sufficient to add the following restriction to problem (8)
|[pic] | |
Where [pic] is a known acceptable error. Then we have new minimization problem:
|[pic][pic] |(9) |
| | |
|[pic][pic] | |
|[pic], [pic] | |
|[pic] | |
Note: When we are going to solve problem (9), it is possible that we have no solution for the problem (infeasible problem). In this case we should to increase the numbers [pic] and [pic].
4.4 LP formulation for Linear wave Problem
If [pic] is a linear function, we can transfer the new obtained nonlinear programming problem (8) or (9) to a linear programming problem [18]. So, we may decompose the value of [pic] by a difference of two non negative values [pic] and [pic], i.e. [pic] as follows:
[pic]
where [pic] and [pic]. Considering the determination of absolute value in the definition norm function (3), such that:
|[pic] | |
at the end, the minimization problem (8) is formed as below LP problem:
|[pic] |(10) |
|[pic] | |
|[pic][pic] | |
| [pic] | |
| [pic] | |
It can be solved by many powerful packages, such as Lindo and Matlab.
4.5 Theorem.3
Supposing the nonlinear system [pic] that [pic] and [pic], [pic] then assume:
|[pic] |(11) |
to obtain the answer, we expect [pic]. In this case if we discretize the problem and [pic] is the problem’s solution by sampling [pic] we have:
|[pic][pic] | |
Where [pic] is the number of subinterval of partition [pic]. So,
|[pic] | |
proof:
Let functional [pic], [pic]. So, [pic] and we can define the following function:
|[pic] | |
Considering the given discretization method that explained in the previous part, it may be defined:
|[pic] | |
thus, by minimization [pic] and solving the NLP problem, the optimal solution of NLP problem is [pic] and [pic].
Therefore, because [pic] then:
|[pic] | |
with indicate [pic] in the equation (11), it’s enough to specify
|[pic][pic] | |
So, as [pic] ,[pic] that [pic]may be taken such small that [pic], Thus,
|[pic] | |
Therefore, the theorem is proved.(
4.6 Degree of Desirability:
In general it may be written that solution of the NOC equation (1) by Theorem.4 about optimization may be solved by applying our method. In this case, if a suboptimal approximated solution for the NOC equation (1) is obtained by solving NLP problem (9) with desirability solution may be defined as follow. If ([pic], [pic]) be a solution obtained by (9), then let:
[pic]
so, the desirability of the solution is:
|[pic] | |
Note: Usually we may obtain also the exact solution for the original problem, it sufficient to show [pic], where we assume the control function be piecewise constant function.
5 Simulation:
In this section we use our algorithm for some nonlinear systems:
5.1 Example 1:
Find a suitable control for minimization of following nonlinear functional:
| |[pic] |
| |[pic][pic] |
| |[pic], [pic], [pic] |
| |[pic], [pic] |
For this system we choose partitioning [0, l] to 10 equal subintervals. So [pic], [pic], … , [pic]. The formulation of corresponding NLP problem is as the following:
|[pic] |
|[pic] |
|[pic], [pic] |
|[pic] [pic] |
|[pic] [pic] |
|[pic] |
|[pic] |
|[pic] free |
The optimal objective function value for this NLP is zero. The approximation of trajectory function obtained from the solution of the above NLP is shown in fig.1 (dotted line). Another approach to show the result is curve fitting by Fourier series. We fit state function in fig. 1 (bold line) with following Fourier series:
[pic]
where [pic], [pic], [pic], [pic], [pic], [pic], [pic] and [pic].
Then the approximation of control function is shown in fig.2 (dotted line) and fitting a curve by Fourier series is scheme in fig. 2 (bold line).
|[pic] |
|Fig.1: Approximation solution of the state function with fitting the |
|curve |
|[pic] |
|Fig.2: the control function with fitting the curve |
5.2 Example 2:
Obtain a control [pic] and simultaneously state of the following nonlinear system:
| |[pic] |
| |[pic][pic] |
| |[pic] |
| |[pic] |
Obtained objective function value is 0.1468870E-06
5.3 Example 3:
Find suitable control for the following nonlinear system subject to minimize this functional.
| |[pic] |
| |[pic] |
| |[pic] |
| |[pic] |
| |[pic], [pic] |
| |[pic], [pic] |
The optimal objective function value for this NLP is 0.01348171 with LINGO software, and given result in reference [14] is 0.024, which show our algorithm advantage. The sate functions (dotted line) and fitting curves by polynomial are scheme in fig. 3 (bold line). The approximation solution of the control function is obtained from the above NLP (dotted line) with fitting the curve by following polynomial (bold line) is shown in fig.4.
[pic]
where [pic], [pic], [pic], [pic], [pic], [pic] and [pic].
|[pic] |
|Fig.3: Approximation solution of the state functions with fitting |
|curves |
|[pic] |
|Fig.4: the control function with fitting the curve |
6- conclusion:
Discretize the classic control problem to a NLP or LP to obtain an approximate solution of the original problem, is the main goal of this paper. Because the classic methods for the solution of nonlinear systems and in easier case for linear time varying systems are caused a linear or nonlinear set of equations. Common methods as nonlinear system linearization cause non-predictable and non-controllable errors. All found solutions are local optimum, which are tailor expansion around Equilibrium point, and are not global solutions. However in non convex search space, classic theory is inefficient. In this paper, our algorithm was applied to NOC systems. By using the algorithm, in addition, analyze system controllability in general form, can be enable the users to restrict actuating signal. This is an important result to make more applied control problems. Although, this approach removes all difficulties that contact in optimal control problem solution. Simulations confirmed the approach ability to solve NOC problems, which is compared with [14] and better solution was presented.
References:
[1] Jean-Jacques E. Slotine, Weiping Li, Applied Nonlinear Control, Printice-Hall, 1991
[2] B. Fernandez R. and J. K. Hedrick, “Control of multivariable non-linear systems by the sliding model”, Int. J. Control, vol. 46, no. 3, pp. 1019-1040, 1987.
[3] D. Leith and W. Leithead, “Survey of gain-scheduling analysis and design,” Int. J. Control, vol. 73, no. 11, pp. 1001–1025, 2000.
[4] H.K. Khalil, Nonlinear Systems, Prentice Hall, 2nd edition, 1996.
[5] P. Kokotovi and M. Arcak, Constructive nonlinear control: A historical perspective” (survey paper). Automatica, 37(5):637–662, May 2001.
[6]. M. Gachpazan and A. V. Kamyad, “Solving of second order nonlinear PDE problems by using artificial controls with controlled error”, J. Appl. Math. & Computing, Vol.15, no.1-2, 2004, pp.173-184.
[7]. A. V. Kamyad, A. H. Borzabadi, “Strong controllability and optimal of the heat equation with a thermal source”, to appear in The Korean Journal of Computational Applied Mathematics.
[8]. A. V. Kamyad, J. E. Rubio, and D. A. Wilson, “An optimal control problem for the multidimensional diffusion equation with a generalized control variable”, J. Opt. Theory and Apl., Vol. 75, No.1, 1992, pp.101-132.
[9]. A. V. Kamyad, J.E. Rubio, and D. A. Wilson, “The optimal control of the multidimensional diffusion equation”, J. Opt. Theory and Apl., Vol. 70, 1991, pp.191-209.
[10]. S. A. Alavi, A. V. Kamyad, and M. H. Farahi, “The optimal control of an inhomogeneous wave problem with internal control and their numerical solution”, Bulletin of the Iranian Mathematical society, Vol. 23, No.2, 1997, pp.9-36.
[11]. S. A. Alavi, A. V. Kamyad, and M. Gachpazan, “Solving of nonlinear ordinary differential equations as a control problem by using measure theory” , Scientia Iranica, Vol. 7, No.1, 2000, pp.1-7.
[12]. A. H. Borzabadi, A. V. Kamyad, M.H. Farahi, “Optimal control of the heat equation in an inhomogeneous body”, J. Appl. Math. & Computing, Vol.15, no.1-2, 2004, pp.127-146.
[13]. A. V. Kamyad, H.H. Mehne, “A linear programming approach to the controllability of time-varying system”, IUST- International J. of eng. Science, vol.14, no,4, 2003, pp.143-151.
[14] J. E. Rubio, control and optimization the linear treatment of nonlinear problems,Manchester university press, 1985.
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