## 1. Introduction

This primary purpose of this research is concerned with adaptive tracking control of a nonlinear system [6, 9]. Particularly, time-varying control approach has been designed for tracking of the system with application to a nonlinear dynamic model [1]. Furthermore, the time-varying system is further complicated by parametric uncertainty or disturbances such as external forces, continuous or discrete noise where the parameters are unknown. Over the past several years, trajectory tracking issue as a high-level control of a nonlinear system has been received a wide attention from control community. Hence, the discussion here is principally devoted to model-based adaptive trajectory tracking control algorithm of linear time-varying (LTV) systems in the presence of uncertainty [4, 5].

A system undergoing slow time variation in comparison to its time constants can usually be considered to be linear time invariant (LTI) and thus, slow time-variation is often ignored in dealing with systems in practice. An example of this is the aging and wearing of electronic and mechanical components, which happens on a scale of years, and thus does not result in any behavior qualitatively different from that observed in a time invariant system on a day-to-day basis. There are many well developed techniques for dealing with the response of linear time invariant systems such as Laplace and Fourier transforms, but not applicable to linear time varying or nonlinear systems, nor feasible to implement for complicated real-world systems. In addition, time-varying system may be difficult to satisfy global controllability or to show whether the time-varying system is even stable or not, due to difficulties in computing or finding solution. Unlike LTI systems, linear time varying systems may behave more like nonlinear systems [1, 2, 3]. In general all systems are time-varying in principle and a large number of systems arising in practice are time-varying. Time variation is a result of system parameters changing as a function of time [5], such as aerodynamic coefficients of aircrafts, hydrodynamic terms in marine vessels, circuit parameters in electronic circuits, and mechanical parameters in machinery. Thus, we characterize systems as time-varying if the parameter variation is happening on time scales close to that of the system dynamics. Time variation also occurs as a result of linearizing a nonlinear system about a family of operating points and/or about a time-varying trajectory for developing control system. However, due to the desire to achieve better accuracy and quality in a wide range of applications [11], there have been increasing interests to include the effects of time-variation [12] while designing controllers or observers at the time analyzing and/or applying to such systems.

In this work, tracking error system is formulated based on its model-reference system which has a reference input and the nonlinear dynamic model of the inverted pendulum. We found a solution of the tracking nonlinear system after developing its linear time varying systems. For the development of subsequent control approach, the error system is linearized about given desired trajectory using a perturbation approach and produced a linear time-varying tracking error equation [3] with system matrices, *A(t)* and/or *B(t)*. At this time the controllability of this time-varying system only shows that the system is stable in an instant time or about a trajectory which can be locally controllable or stabilized. Then, a novelty of this research is that a controllability grammian matrix is found to be a necessary and sufficient conditions of the global controllability and the inverse of the grammian matrix exists, which is nonsingular, and is used for the designing the control input of the closed-loop system. In this research, a complicated solution of state transition matrix is obtained based on Taylor series expansion, categorized into feasible forms based on the system and the shape of matrix. The control input of the tracking system is designed from the state transition matrix and the grammian matrix, which makes the system globally controllable, and the control input of the actual system is redesigned via the tracking controller while compensating for the uncertainty as disturbances, which also yields the system globally stable. This chapter consists of as follows: a time-varying system is briefly described relative to a time-invariant system and a non-homogeneous system is introduced for linear time-varying system for the development of the solution which is state transition matrix in Section II followed by Introduction. Then a cart-pole nonlinear dynamic model where the system parameters are unknown is developed for the application of a proposed control algorithm and expressed into a state space form. For the trajectory tracking control, error signals are formulated from desired model-based reference system. Based on the analysis of the developed time-varying error system, the solution of the system, state transition matrix, is derived in a series form and then a special form of the matrix is obtained for the second-order error differential equation, which is used for the grammian matrix and the closed-loop controller. The control system is also developed to reject disturbances via a projection-based adaptive control approach and update laws for the parameter update in Section III. Numerical simulation results with analyses demonstrate the validity of the proposed system. This approach can be extended to other nonlinear time-varying dynamic systems such as aerial-, marine, or ground vehicles.

## 2. Linear time-varying system

A linear time-invariant system (LTI) is described as

where the equilibrium point is at the origin and if det(*A*)≠0, the fixed point is isolated and the stability of the origin depends on the location of the eigenvalues of the matrix

Another LTI state equations is given by

where

where this is a convolution control solution and the state transition matrix

### 2.1. Homogeneous system

A time-varying system is described as

where*n*-dimensional linear vector space, *n* linearly independent initial condition. According to the linearly independent solutions, a system is defined as

where

where

### 2.2. Nonhomogeneous system

A linear time-varying system (LTV) is described as

where*n* is the number of control inputs. Note that in case the control input is underactuated, then *n-m* is the underactuation, or the number of underactuated inputs. For the controllability of time-varying systems given in (8), the state transition matrix (or known as fundamental solution matrix) is the overall solution and used to perform the function of integrating factor where the solution is derived from a linear independence on the columns of a matrix that was a function of

### 2.3. Solution of the state transition matrix

The system is controllable if the controllability gramian (or grammian) matrix *i.e.*, invertible for the necessary and sufficient condition

where the rows of the matrix product

where

where the solution of linear time-varying system

The expression (11) yields by factoring

where (9) was used and this implies the control input,

## 3. Application to nonlinear inverted pendulum system

### 3.1. Dynamic model

A continuous nonlinear time-varying system is given as a combined model based on the inverted pendulum [1] expressed by the second-order differential equation by

where

where

where

where the first term

### 3.2. Error formulation

Then, the error equation can be derived from the subtraction between the desired and the actual system as

and subtracting

Let the error,

which results in

where

where the parameter

where

### 3.3. Solution of the linear time-varying system

The solution of linear time-varying error system for (22) is given by

However, it is difficult to find the state transition matrix of (22) since the system has a function of time in the

where

where

Thus, the state transition matrix,

where it is identified that

where this gramian matrix is positive definite and nonsingular, whose inverse exists and satisfies the sufficient and necessary condition of the controllability due to the time-varying system. Applying (27) to (24) for solving (26) yields

where

where

where the first term,

the second term,

### 3.4. Adaptation laws for parameter update

Substituting (32) for

where

in which

Then, the adaptation law is designed as

Hence, the final error system utilized (37) results in

The following is assumed to define the upper and lower bounds of each unknown parameters

where

where

It is straightforward to make a conclusion that the above adaptive control approach is applied to (36) and then the parenthesis term in (36) will be going to zero, resulting in (38) if both are perfectly canceled, which yields globally stable tracking result.

## 4. Numerical results

The initial condition of inverted pendulum angles is given as

where

## 5. Conclusion

A tracking control of a model-based linear time-varying system is developed in application to the nonlinear inverted pendulum model. A novelty of this paper is that not only found a gramian matrix which is difficult to find or compute but also utilized to the linear time-varying tracking controller which satisfies the necessary and sufficient of the global stability of the system. Another is that the linear time-varying system is further complicated by parametric uncertainty where the combined parameters are unknown. The suggested adaptive control approach and update laws are applied for estimating the parameters while preserving the system to be stable and converging the tracking error close to zero. Numerical simulation results are demonstrated the validity of the proposed system.

### Acknowledgement

This research is supported by Office of Naval Research (ONR, N00014-09-1-1195), which we gratefully acknowledge.