Mathematical Problems in Engineering

Volume 2017, Article ID 6843614, 11 pages

https://doi.org/10.1155/2017/6843614

## Solving a Class of Nonlinear Inverse Problems Using a Feedback Control Approach

Department of Computer Science, San Diego State University, San Diego, CA 92128-7720, USA

Correspondence should be addressed to Mahmoud Tarokh; ude.usds.liam@hkoratm

Received 28 February 2017; Revised 8 April 2017; Accepted 10 April 2017; Published 28 May 2017

Academic Editor: J.-C. Cortés

Copyright © 2017 Mahmoud Tarokh. 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

Inverse problems have applications in many branches of science and engineering. In this paper we propose a new approach to solving inverse problems which is based on using concepts from feedback control systems to determine the inverse of highly nonlinear, discontinuous, and ill-conditioned input-output relationships. The method uses elements from least squares solutions that are formed within a control loop. The stability and convergence of the inverse solution are established. Several examples demonstrate the applicability of the proposed method.

#### 1. Introduction

Inverse problems form an interdisciplinary filed encompassing science and engineering. Applications of inverse problems are diverse and include robotics, optics, nondestructive detection, geophysics, imaging, acoustics, and civil and mechanical engineering, just to list a few. Briefly and roughly speaking, “forward” problems estimate the effect (output) from the cause (input). In contrast, “inverse” problems require estimating the cause or the parameters from the effect. For example, in robot manipulators the forward kinematics equations are readily available and relate the robot manipulator joints angles to the end effector (hand) position and orientation. However the fundamental problem in robotics is the inverse kinematics; that is, given hand position and orientation, determine the joint angles. The mathematical formulation of inverse problems leads to models that are typically ill-posed, meaning that a solution may not exist, the solution may not be unique, or the solution is sensitive to small changes in data causing instability. In the robot manipulator example, this situation often happens when during the robot motion the joint angles form a configuration in which the so-called manipulator Jacobian matrix is singular.

Inverse problems can be categorized into two groups: linear and nonlinear. Linear inverse problems are usually formulated as where is vector of (possibly noisy) data and the input vector is to be extracted, given that the matrix can be ill-conditioned such that its inverse either does not exist or is meaningless and does not reflect the actual physical problem. In other to overcome these difficulties, one has to use regularization which replaces an ill-posed problem by a neighbouring well-posed one. There are two main related regularization approaches, namely, Tikhonov and truncated singular value decomposition. In the Tikhonov approach a small constant is included in the pseudo-inverse computation of the matrix to limit to magnitude of . In the truncated SVD method, tiny singular values of are removed to prevent the pseudo-inverse of to become excessively large. Variations and improvement of these two basic approaches have been proposed (e.g., [1–3]).

Many real-world inverse problems are nonlinear and unlike the linear ones have not been fully explored due to the complexity of the problem. Many of the existing techniques are based on least squares formulation of the ill-posed nonlinear problem of the form [4–8]. Most damped least squares solutions are based on Levenberg–Marquardt iterative solution which is further explored in [9]. A more recent work describes a two-stage method combining Levenberg–Marquardt regularization of the linearized problems at each stage [10]. When the solution is assumed to have a sparse expansion, an iterative method has been proposed to solve the nonlinear inverse problem [11]. This method shows good results for colour image inpainting applications.

A recent book describes the application of inverse problems to imaging [12]. A survey of nonlinear inverse problems can be found in [13] and references therein. Neural networks have also played a major role in the solution of inverse problems (see, e.g., [14] and references therein). This is due to the fact that neural networks are considered universal approximators. A method using function decomposition and approximation is proposed by the author that is applicable to a wide-class of nonlinear inverse problems [15].

Many approaches to inverse problems propose solutions for specific and often simple cases and thus are not generally applicable to situations and applications other than those that they are developed for. In this paper we present a new approach to the inverse problems that have complex structures with highly nonlinear and discontinuous behaviours as well as being ill-conditioned. We employ concepts from feedback control systems and stability theory to prove the convergence and accuracy of the solutions. Examples are provided to demonstrate the concept.

#### 2. Statement of the Problem

Consider the input vector of dimension and the corresponding output vector of dimension and the forward relationship between and of the form It is noted that (1) also covers cases where the input-output relationship is expressed in the integral form

We assume that the forward relationship of the form (1) or (2) is known. The inverse problem is given ; find . In most applications there is no analytical solution to the inverse problem; that is, cannot be expressed in analytical form in terms of .

There are a number of challenges to the inverse problems. Noise can corrupt the data, the forward and/or the inverse relationship can be discontinuous, and the problem can be ill-conditioned. Furthermore, in some applications the forward relationship can be very complex and the inverse can have multiple solutions for a particular value of . These are especially the case for many practical problems such as robotic inverse kinematics problem. In such an application even the forward equations consist of highly nonlinear trigonometric functions with sever discontinuities due to joint limits and other physical constraints, as well as varying number of the inverse solutions.

We propose to solve the above inverse problem using a feedback control approach and a generalized pseudo-inverse method, as well as optimizing a performance index in some cases. Suppose that the desired output vector is and we want to find an input such that . Suppose further that, for an initial value , for example, randomly chosen, the corresponding output is obtained from (1) as . The problem is then to transfer from to . We propose to do this transition through a smooth reference trajectory parameterized by time . Among other possible functions such as quintic, we propose a cycloid function defined by where is the time to move from the initial value to the desired value . It is noted that the first, second, and higher order time derivatives of (3) are all continuous, smooth, and finite which is an important characteristic of this particular trajectory. Higher values of correspond to slowly varying trajectory and have a favorable effect on the stability of the system, as will be seen later. The cycloid function has a simple form and is parameterized by only three quantities, namely, , , and , each of which has a meaningful interpretation. On the other hand, other smooth functions are described by more parameters whose roles in the overall form of the function are not easily recognizable. For example, the quintic function requires six parameters, and it is not easy to determine these six parameters to achieve the desired transition time, final value, and maximum value of its derivative, all of which are needed later in Sections 3 and 4. Furthermore, a simple step function is undesirable since its first and higher order derivatives are infinite, resulting in instability and large steady-state error for this application, as will be seen in Sections 3 and 4. In addition, a step function has no parameters to adjust the transition time between initial and final values.

Since the problem is nonlinear we use an incremental approach by taking the time derivative of (1) to obtain where is the Jacobian matrix. The partial derivative is computed numerically. When the trajectory passes through the points where the function is discontinuous, some or all elements of become infinity at certain values of . We assume that the desired is not at the points of discontinuity. Consequently at the points where the derivative is infinite, we set the corresponding values of the elements of to a large but finite value to prevent numerical difficulties when the inverse of is found. Since we seek values of for finite , therefore high values of the derivative can only happen during the transition, that is, at times . As a result, even though can be very different from the desired during the transient, it will become arbitrarily close to at steady-state, that is, , as will be shown in Section 3.

Equation (4) is a differential equation relating the incremental change in the input to the incremental change in the output. The generalized inverse solution to (4) is [16] where is the pseudo-inverse of , is the identity matrix, is an arbitrary free vector, and is an arbitrary positive scalar. The free vector and scalar will be used for optimization as will be discussed in Section 4. The second term on the right hand side of (5) is orthogonal to the first term. It is possible to substitute for from (3) into (5) to obtain and then integrate to get whose steady-state value inverse corresponds to the desired. However, integration of (5) is not a viable solution due to (i) integration drift and numerical difficulties and (ii) ill-conditioned matrix *.* This means that the matrix can have singular values close to zero and as a result the inverse or pseudo-inverse of the matrix, that is, , can produce solutions that are grossly in error, especially when noise is present. To overcome (i) we propose to form an error defined byand apply a feedback signal , where is the gain matrix. To remedy (ii) we decompose into two matrices as follows:where is well-conditioned with a desired conditioning number and is a residual ill-conditioned matrix that can be found in a number of different ways, two of which will be described shortly. Thus, (5) is modified to We will prove that the steady-state solution to dynamic equation (8) will be inverse solution for a desired when a suitable feedback matrix is determined. The feedback block diagram representing (2) and (8) is shown in Figure 1.