Mathematical Problems in Engineering

Volume 2018, Article ID 5217427, 10 pages

https://doi.org/10.1155/2018/5217427

## Time Optimal Control Laws for Bilinear Systems

Advanced System Laboratory (LSA), Tunisia Polytechnic School (EPT), University of Carthage, BP 743, 2078 La Marsa, Tunisia

Correspondence should be addressed to Salim Bichiou; moc.liamg@uoihcib.milas

Received 27 July 2017; Revised 8 February 2018; Accepted 5 March 2018; Published 8 April 2018

Academic Editor: J.-C. Cortés

Copyright © 2018 Salim Bichiou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this paper is to determine the feedforward and state feedback suboptimal time control for a subset of bilinear systems, namely, the control sequence and reaching time. This paper proposes a method that uses Block pulse functions as an orthogonal base. The bilinear system is projected along that base. The mathematical integration is transformed into a product of matrices. An algebraic system of equations is obtained. This system together with specified constraints is treated as an optimization problem. The parameters to determine are the final time, the control sequence, and the states trajectories. The obtained results via the newly proposed method are compared to known analytical solutions.

#### 1. Introduction

Most engineering applications are aimed at solving complex mathematical models. This usually comes with a computational burden and is of a major concern. Researchers are therefore striving to reduce that burden. This is particularly true for minimum time optimal control problems. Using Pontryagin maximum principle, the solution to this problem is known to be Bang-Bang, that is, control values switches between lower and upper boundaries. This type of control is required in some types of systems such as the thermostat switching between the on- and off-position. Time optimal control problems’ aim is driving systems from an initial state to a desired final state in minimum time while satisfying given constraints. To this day, time optimal control problem still attracts interest among researchers [1–3].

Most engineering systems are interpreted as models. These models often feature nonlinear components which are challenging during the resolution process. Thus, simplified nonlinear representations such as bilinear models have gained momentum. Bilinear models have been introduced since the 1960s and are approximate representations for a wide range of systems. The bilinear structure can be used to describe a nonlinear system while maintaining a linear structure: the bilinearity expresses a double linearity with respect to the state vector and [4]. A detailed review of bilinear systems can be found in [5]. This type of representation has been used extensively by researchers during the previous decade for a variety of fields: engineering, biology, and economics [6, 7].

Solving the minimum time control problem analytically means finding switching times through the resolution of the Hamiltonian equation. This is a computationally intensive task. Some researchers tried to solve it analytically through the determination of switching surfaces [8]. Due to mathematical difficulty, numerical algorithms have been developed and introduced such as GPOPS-II, which is based on orthogonal collocation at Legendre–Gauss or Legendre–Gauss–Radau points [9]. Other tools are based on the control parametrization enhancing technique [10] and the optimal switching (TOS) algorithm [11].

In this paper, we use orthogonal functions to solve the minimum time control problem. In fact, this mathematical tool was widely used in the past and still is in different ways such as solution determination for optimal control problems [12–14]. Using this technique, a transposition of state space equation on an orthogonal base of functions and transformation of bilinear nonlinearity and integration operation into a matrical product is made. These approximations bypass all the mathematical difficulties associated with nonlinearities.

Generally, two types of orthogonal base of functions are considered in the literature: piecewise orthogonal functions such as Walsh [15], Haar wavelets [16], and Block pulse functions [17] and orthogonal polynomials such as Legendre [18], Chebyshev [19], and Lagerre polynomials [20]. In [21] authors used Chebyshev polynomials and expanded linear systems in that base to determine the time optimal control input. In this paper piecewise orthogonal functions, namely, Block pulse functions, are used to capture the Bang-Bang control nature associated with bilinear systems.

The determination of minimum time control problem is generally done in open loop which can easily be altered by perturbation. Thus, many suboptimal structures and formulations were introduced for linear systems [22].

The main contributions in this work are determining an open loop time optimal control solution for bilinear systems through orthogonal functions and numerical resolution; determining a state suboptimal time feedback controller (see Table 2) using a new formulation based on Kronecker product for that class of systems.

The remainder of the paper is organised as follows. In the second section, formulation of time optimal control is presented. In the third section, orthogonal functions and their properties are introduced. The optimization model for an open loop minimum time control is presented in the fourth section. In the fifth section, the minimum time state feedback control is formulated and a solution is proposed. Finally, concluding remarks and some open problems are presented.

#### 2. Time-Optimal Constrained Feedforward Control Problem

Consider the bilinear systems described by the following state space form:where is the state vector, is the input control vector, and , are square matrices, is an matrix and and , , are the column of the matrix , and is the Kronecker product [24].

We assume that the input variables satisfy the instantaneous constraintThe target state being the originTo minimize the final time, the cost function is taken as [25]:where .

Applying the Pontryagin Maximum Principle (PMP) [26], we define the Hamiltonian [27] equation for system (1)The canonical Hamilton equations are given byMaximizing the Hamiltonian equation, one obtains the following control law:Thus, the obtained control is Bang-Bang which is a feedforward controller that switches abruptly between two states.

The main difficulty related to this control law formulation is the determination of the switching times . are the times when the quantity changes sign. It is still until today a difficult task to solve this problem. Even for linear systems only reduced-order systems as second-order systems have been treated in the literature [28]. To overcome this difficulty for the class of bilinear systems, an original approach, based on the formulation of the studied system and the associated optimization problem in an orthogonal functions base, is proposed and developed in the following sections. In the next section, the main properties of the orthogonal functions, which will be used to derive our main results, are recalled.

#### 3. Orthogonal Functions-Block Pulse Functions

Orthogonal functions and their operational matrices have been used for modeling dynamic systems [29] and identification [15]. They have been largely used in quadratic optimal control for different dynamic systems [30, 31]. To our knowledge, until today, there are no works dealing with the problem of minimum time control for bilinear systems using orthogonal functions.

##### 3.1. General Idea

Let be a set of orthogonal polynomials or piecewise functions. Any analytical function absolutely integrable on the time interval can be expanded by the following infinite series:where the coefficients are evaluated by the following scalar product:For numerical purposes, a truncation of (8) until a convenient number of elementary functions is considered.where is the orthogonal base and is the coefficient vector.

Integrating (10), one obtainswhere is the operational matrix of integration depending on the considered orthogonal base [17].

As a result, the differential equations describing dynamic processes can be reduced into algebraic relations allowing important simplifications in problem synthesis.

The considered piecewise orthogonal functions and precisely the Block pulse functions are described in the following section.

##### 3.2. Properties of Block Pulse Functions

Block pulse functions constitute a complete set of orthogonal functions. They are defined as follows [32]:A function can be approximated by relation (10), where is a vector of Block pulse functions and the coefficients of the vector are given by the following formula:Many interesting properties and tools of the approximation of an analytic function by a series of Block pulse functions have been defined in literature, as the operational matrix of integration, the operational matrix of product, the operational matrix of delay, and the operational matrix of derivative [17].

###### 3.2.1. Block Pulse Operational Matrix of Integration

The integration of an analytic function using the operational matrix of integration for Block pulse functions is given by [17]

###### 3.2.2. Block Pulse Operational Matrix of Product

The product of orthogonal base vectors can be approximated through operational matrix of product denoted . It is defined by [33]: , , one haswith .

Using (15), one may obtain

#### 4. Synthesis of an Open Loop Minimum Time Control Using Orthogonal Functions

In this work we have chosen to use the Block pulse orthogonal functions due to the fact that solutions for minimum time problems are of type Bang-Bang which has the same structure as Block pulse functions.

##### 4.1. Studied System Transposition on Orthogonal Base

Consider bilinear system (1). To make use of the orthogonal functions properties and mainly the operational matrix of integration one needs to know the operating time interval . For simplicity, in the rest of the work and without loss of generality, let . We introduce here the following time variable change:where denotes the final time to be minimized.

This change of variable allows a transformation of the time domain from to . Then system state becomesNotice that the latter variable change leads to a constant time interval for the used series. The final time becomes an additional unknown variable.

Consequently, we deduceThe original state equation of system (1) is now equivalent toThe development of the state and input vectors on the considered base of Block pulse functions with the new time variable can be written asIntegrating equation (20) leads toIntroducing coefficients of and and operational matrix of integration one obtainsKnowing that [33]one can writewhere is the transposition of the initial state over orthogonal functions. It depends on the chosen set of functions.

##### 4.2. Optimization Problem Formulation Using Orthogonal Functions

To find the transition time from the initial to the target position, we need to solve the following nonlinear problem.

* OP1: Original Optimization Problem*

The problem transposition over Block pulse base leads to the following nonlinear optimization problem named “orthogonal functions optimization problem.” This algorithm determines the time optimal input for given bilinear system.

*OP2: Orthogonal Functions Optimization Problem*is subject to the following constraints:(i)Initial condition is(ii)Equality constraint is denotes the transposition of the final sate over the orthogonal functions base.(iii)Inequality constraint is(iv)Nonlinear equality constraint isTo solve this optimization problem, an interior point method such as the one implemented in the function “fmincon” under “Optimization” Toolbox of Matlab environment is used.

##### 4.3. Simulation Results

In this subsection, two examples of bilinear systems are presented to evaluate the effectiveness of our developed approach.

###### 4.3.1. Example 1: First-Order Bilinear System

Consider the following bilinear system:The system needs to be shifted from to the origin; the input must be within interval .

To validate the proposed approach, a comparative study between the theoretical method and the developed Block pulse functions approach is presented.

Applying both methods on the studied system, we get the optimal minimum time value s with for and so without switching time. From Table 1, it is clear that the obtained result thought the BPF algorithm meets the analytic one. Figure 1(a) shows that the control law with BPF approach has the same control sequence as the analytical method. The state trajectory of the system provided by optimal control input is given in Figure 1(b).