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 . 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 , 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 , there have been increasing interests to include the effects of time-variation  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  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 which is not a function of time. The solution of (1) with the initial state is given by
Another LTI state equations is given by
where are time-invariant. It is known that the solution of the equation (3) using an integrating factor yields can be time-varying or time-invariant. The solution of (3) with the initial state is given by
where this is a convolution control solution and the state transition matrix. The solutions (2) and (4) make clear the importance of the matrix exponential and its eigenvalues. However, these techniques are not strictly valid for time-varying systems.
2.1. Homogeneous system
A time-varying system is described as
where, and the matrix is not a constant as a function of time; it is nonautonomous [6, 7]. The general solution of the (5) in n-dimensional linear vector space, is unique for the space on in case is smooth where is a basis set of n linearly independent initial condition. According to the linearly independent solutions, a system is defined as
where is a matrix which has the linearly independence solutions which shows. The general solution of (6) is given as
where is the matrix of the initial value of state, is called the state transition matrix as known as a fundamental solution matrix associated with, having a form of exponential function.
2.2. Nonhomogeneous system
A linear time-varying system (LTV) is described as
where, , in which can be input configuration matrix and is the control input where n is the number of control inputs. Note that in case the control input is underactuated, then and where 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 and.
2.3. Solution of the state transition matrix
The system is controllable if the controllability gramian (or grammian) matrix below is nonsingular, i.e., invertible for the necessary and sufficient condition
where the rows of the matrix product are linearly independent in an interval. In order to prove the invertible exists, the control input of the system can be designed based on the gramian matrix as
where is a control input gain matrix and is the inverse of (9). Convolution integral solution to determine the state at the end of the interval, , yields
where the solution of linear time-varying system is given by
The expression (11) yields by factoring of the left side as
where (9) was used and this implies the control input, in (10), drives the system to reach the zero state, in which should be the identity matrix. Now the system is controllable and shows that the controllability gramian is invertible.
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  expressed by the second-order differential equation by
where is the angle of the pole of the inverted pendulum which is subjected to the external force, is the gravitational force, and is the combined parameter term given by
where is an unknown lumped parameter, in which is the inertia of the pole, is the mass of the pole, and is the length of the pole. The system can be expressed into a state space model in order to analyze as
where is the actual control input of the inverted pendulum to be designed later. Let be the desired model-based reference system as follows
where the first term in the right side of the second row equation is added because the response of the second equation can be divergent due to the positive reference input, in which is a positive constant.
3.2. Error formulation
Then, the error equation can be derived from the subtraction between the desired and the actual system as
and subtracting from and substituting the second equation of (16) yields
Let the error, , assumed to be small. Then, produces
which results in where is positive constant. Substituting this into the parenthesis term for in (18) and using the sum of sines yields
where and. Hence, (18) yields
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 and coupled. However, the fact that (17) and (21) can be considered as decoupled between two equations helps to find the state transition matrix,. The solution of the first equation in (22), i.e., (17), yields
where and the equation can be easily decoupled from the (25).
Thus, the state transition matrix, , yields
where it is identified that is a scalar form,. In this case the gramian matrix is defined by utilizing (27) as
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. Then the open-loop control input for the second equation of (22) using a controllability gramian term is
where the first term, , is designed for subsequent control development as
3.4. Adaptation laws for parameter update
Substituting (32) for into and rearranging yields
in which is constant gain value. Rearranging yields
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 is the estimated constant parameters, , are unknown lower and upper bounds of the estimated parameters as shown in system parameters, respectively, which will be set to the amount of percentage of their true values. vector is designed to update using a projection-based algorithm as
where is the projection operator  and each parameter is adaptively updated using adaptation laws for online estimation of unknown parameter as follows:
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 as shown in Figure 1 where each actual angle of the pendulum track the desired angle, starting from its initial value. Note that the initial angular rate, , is zero. In Figure 2, the actual angular rate tracks the desired angular rate of the inverted pendulum. In Figure 3, their tracking angle errors are shown in the top plot and the error rates are shown in the bottom plot, where the errors and error rates are close to zero and thus the tracking system works well. The control inputs are in Figure 4; the control input shown in plot (a) is the designed control input in (32), which is used for the control input of the system dynamic model given in (15), the control input shown in plot (b) is the control solution given in (31) of the tracking error dynamics in (34), which enables the global stability, and finally the plot in (c) is proposed controller of this research, i.e., the closed-loop adaptive tracking control input designed in (35). Figure 5 is the estimate of the time varying parameter, of, in which the simulation parameters such as mass (), length of the pole (), and inertia of the pole () are combined together and the values used in simulation are as follows: and . The percentage of the upper and lower bounds given in (39) is set to 100%. The nominal value of 1/is 0.612. Thus, the upper bound is 1.232 and lower bound is zero as shown in Figure 5 and the time-varying parameter estimate is varying within the bounds. The error dynamics, , developed in the main body of this chapter are shown in Figure 6; the plot (a) is the second equation of (22) with the control input given in (31), the plot (b) is the output of the error dynamics given in (34) with the control input (35), and the plot in (c) is the final error dynamics given in (38). In Figure 7, those velocity errors with regard to the dynamics are shown. The reference velocity, error control gain constants, gain value, and control input gain matrix are
where is identity matrix.
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.
This research is supported by Office of Naval Research (ONR, N00014-09-1-1195), which we gratefully acknowledge.