# Additive-State-Decomposition-Based Tracking Control for TORA Benchmark

###### Abstract

In this paper, a new control scheme, called additive-state-decomposition-based tracking control, is proposed to solve the tracking (rejection) problem for rotational position of the TORA (a nonlinear nonminimum phase system). By the additive state decomposition, the tracking (rejection) task for the considered nonlinear system is decomposed into two independent subtasks: a tracking (rejection) subtask for a linear time invariant (LTI) system, leaving a stabilization subtask for a derived nonlinear system. By the decomposition, the proposed tracking control scheme avoids solving regulation equations and can tackle the tracking (rejection) problem in the presence of any external signal (except for the frequencies at ) generated by a marginally stable autonomous LTI system. To demonstrate the effectiveness, numerical simulation is given.

TORA, RTAC, Nonminimum phase, Additive state decomposition.

## I Introduction

The tracking (rejection) problem for a nonlinear benchmark system called translational oscillator with a rotational actuator (TORA) and also known as rotational-translational actuator (RTAC) has received a considerable amount of attention these years [1]-[8]. Some results were presented concerning the tracking (rejection) problem for general external signals [2],[3]. However, the proposed control methods cannot achieve asymptotic disturbance rejection. Taking this into account, the nonlinear output regulation theory was applied to track (reject) external signals generated by an autonomous system. In this case, asymptotic disturbance rejection can be achieved. By using different measurement, the tracking (rejection) problem for translational displacement of the TORA were investigated [4]-[6]. Readers can refer to [6] for details. Based on the same benchmark system, some other work was also presented concerning the tracking (rejection) problem for rotational position by nonlinear output regulation theory [7],[8]. For the two types of tracking (rejection) problems, regulator equations have to be solved and then the resulting solutions will be further used in the controller design. However, the difficulty of constructing and solving regulator equations will increase as the complexity of external signals increases. Moreover, it may fail to design a controller if regulator equations have no solutions. These are our major motivation.

In this paper, the tracking (rejection) problem for rotational position of the
TORA as [7],[8] is revisited by a new control
scheme called additive-state-decomposition-based tracking control,
which is based on the additive state decomposition^{1}^{1}1In this
paper we have replaced the term “additive
decomposition” in [9] with the more
descriptive term “additive state
decomposition”.. The proposed additive state decomposition
is a new decomposition manner different from the lower-order subsystem
decomposition methods. Concretely, taking the system for example, it is decomposed into two subsystems: and , where and respectively. The lower-order subsystem decomposition satisfies

By contrast, the proposed additive state decomposition satisfies

In our opinion, lower-order subsystem decomposition aims to reduce the complexity of the system itself, while the additive state decomposition emphasizes the reduction of the complexity of tasks for the system.

By following the philosophy above, the original tracking (rejection) task is ‘additively’ decomposed into two independent subtasks, namely the tracking (rejection) subtask for a linear time invariant (LTI) system and the stabilization subtask for a derived nonlinear system. Since tracking (rejection) subtask only needs to be achieved on an LTI system, the complexity of external signals can be handled easier by the transfer function method. It is proved that the designed controller can tackle the tracking (rejection) problem for rotational position of the TORA in the presence of any external signal (except for the frequency at ) generated by a marginally stable autonomous LTI system.

This paper is organized as follows. In Section 2, the problem is formulated and the additive state decomposition is recalled briefly first. In Section 3, an observer is proposed to compensate for nonlinearity; then the resulting system is ‘additively’ decomposed into two subsystems; sequently, controllers are designed for them. In Section 4, numerical simulation is given. Section 5 concludes this paper.

## Ii Nonlinear Benchmark Problem and Additive State Decomposition

### Ii-a Nonlinear Benchmark Problem

As shown in Fig.1, the TORA system consists of a cart attached to a wall with a spring. The cart is affected by a disturbance force . An unbalanced point mass rotates around the axis in the center of the cart, which is actuated by a control torque The translational displacement of the cart is denoted by and the rotational position of the unbalanced point mass is denoted by

For simplicity, after normalization and transformation, the TORA system is described by the following state-space representation [1]:

(1a) | ||||

(1b) | ||||

(1c) | ||||

(1d) | ||||

where , , is the unknown dimensionless disturbance, is the dimensionless control torque. In this paper, the tracking (rejection) problem for rotational position of the TORA as [7],[8] is revisited. Concretely, for system (1), it is to design a controller such that the output as meanwhile keeping the other states bounded, where is a known constant. Obviously, this is a nonlinear nonminimum phase tracking problem, or say a nonlinear weakly minimum phase tracking problem. For system (1), the following assumptions are imposed. | ||||

Assumption 1. The state can be obtained. | ||||

Assumption 2. The disturbance is generated by an autonomous LTI system |

(2) |

where , are constant matrix, and the pair is observable.

Remark 1. If all eigenvalues of have zero real part, then, in suitable coordinates, the matrix can always be written to be a skew-symmetric matrix. The matrix in previous literature on the output regulation problem is often chosen in a simple form where is a positive real [4]-[8]. In such a case, is in the form as sin and the solution to the regulator equation is easier to obtain. However, this is a difficulty when is complicated.

### Ii-B Additive State Decomposition

In order to make the paper self-contained, the additive state decomposition [9] is recalled here briefly. Consider the following ‘original’ system:

(3) |

where . We first bring in a ‘primary’ system having the same dimension as (3), according to:

(4) |

where . From the original system (3) and the primary system (4) we derive the following ‘secondary’ system:

(5) |

where is given by the primary system (4). Define a new variable as follows:

(6) |

Then the secondary system (5) can be further written as follows:

(7) |

From the definition (6), we have

(8) |

Remark 2. By the additive state decomposition, the system (3) is decomposed into two subsystems with the same dimension as the original system. In this sense our decomposition is “additive”. In addition, this decomposition is with respect to state. So, we call it “additive state decomposition”.

As a special case of (3), a class of differential dynamic systems is considered as follows:

(9) |

where and Two systems, denoted by the primary system and (derived) secondary system respectively, are defined as follows:

(10) |

and

(11) |

where and . The secondary system (11) is determined by the original system (9) and the primary system (10). From the definition, we have

(12) |

## Iii Additive-State-Decomposition-Based Tracking Control

In this section, in order to decrease nonlinearity, an observer is proposed to compensate for the nonlinear term After the compensation, the resulting nonlinear nonminimum phase tracking system is decomposed into two systems by the additive state decomposition: an LTI system including all external signals as the primary system, leaving the secondary system with a zero equilibrium point. Therefore, the tracking problem for the original system is correspondingly decomposed into two subproblems by the additive state decomposition: a tracking problem for the LTI ‘primary’ system and a stabilization problem for the secondary system. Obviously, the two subproblems are easier than the original one. Therefore, the original tracking problem is simplified. The structure of the closed-loop system is shown in Fig.2.

### Iii-a Nonlinearity Compensation

First, in order to estimate the term an observer is designed, which is stated in Theorem 1.

Theorem 1. Under Assumptions 1-2, for system (1), let the observer be designed as follows

(13) | ||||

where Then where

Proof. See Appendix A.

### Iii-B Additive State Decomposition of Original System

Introduce a zero term into the system (14), where , and Then the system (14) becomes

(15) |

where

(16) |

The additive state decomposition is ready to apply to the system (15), for which the primary system is chosen to be an LTI system including all external signals as follows

(17) |

where Then, according to the rule (11), the secondary system is derived from the original system (15) and the primary system (17) as follows

(18) |

where According to (12), we have

(19) |

Remark 3. The pair is uncontrollable, while the pair is controllable. Therefore, there always exists a vector such that is a stable matrix.

Remark 4. If and then is a zero equilibrium point of the secondary system (18).

So far, the nonlinear nonminimum phase tracking system (15) is decomposed into two systems by the additive state decomposition, where the external signal is shifted to (17) and the nonlinear term is shifted to (18). The strategy here is to assign the tracking (rejection) task to the primary system (17) and stabilization task to the secondary system (18). More concretely, in (17) design to track , and design to stabilize (18). If so, by the relationship (19), can track In the following, controllers and are designed separately.

### Iii-C Tracking Controller Design for Primary System

Before proceeding further, we have the following preliminary result.

Consider the following linear system

(20) |

where is a marginally stable matrix, and

Lemma 1. Suppose i) is bounded on and ii) every element of are bounded on and can be generated by with appropriate initial values, where iii) the parameters in (20) satisfy

(21) |

Then in (20) meanwhile keeping and bounded.

Proof. See Appendix B.

Define a filtered tracking error to be

(22) |

where and . Let us consider the tracking problem for the primary system (17). With Lemma 1 in hand, the design of is stated in Theorem 2.

Theorem 2. For the primary system (17), let the controller be designed as follows

(23) |

where diag and satisfy

(24) |

Then and meanwhile keeping and bounded.

Proof. Incorporating the controller (23) into the primary system (17) results in

where the definition (22) is utilized. Moreover, every element of andcan be generated by an autonomous system in the form with appropriate initial values, where By Lemma 1, if (24) holds, then meanwhile keeping and bounded. It is easy to see from (22) that both and can be viewed as outputs of a stable system with as input. This means that and are bounded if is bounded. In addition, and

In most of cases, the controller parameters and in (23) can be always found. This is shown in the following proposition.

Proposition 1. For any without eigenvalues the parameters

(25) |

can always make where

Proof. See Appendix C.

Remark 5. Proposition 1 in fact implies that, in the presence of any external signal (except for the frequencies at ), the controller (23) with parameters (25) can always make and meanwhile keeping and bounded. In other words, the disturbance like cannot be dealt with, which is consistent with [7]. If the external signal contains the component with frequencies at then such a frequency component can be chosen not to compensate for, i.e., in (23) will not contain eigenvalues .

### Iii-D Stabilized Controller Design for Secondary System

So far, we have designed the tracking controller for the primary system (17). In this section, we are going to design the stabilized controller for the secondary system (18). It can be rewritten as

(26) |

where Our constructive procedure has been inspired by the design in [3]. We will start the controller design procedure from the marginally stable -subsystem.

Step 1. Consider the -subsystem of (26) with viewed as the virtual control input. Differentiating the quadratic function results in

(27) |

Guided by the state-feedback design [10], we introduce the following “Certainty Equivalence” (CE) based virtual controller

(28) |

Then

(29) |

where

(30) |

In order to ensure the parameter is chosen to satisfy Since is a constant, always exists. The term CE is used here because in (28) makes in (27) negative semidefinite as

Step 2. We will apply backstepping to the -subsystem and design a nonlinear controller to drive to the origin. By the definition (28), atan Then the time derivative of the new variable is

(31) |

where Define a new variable as follows

(32) |

Then (31) becomes

By the definition (32), the time derivative of the new variable is

where

Design for the secondary system (26) as follows

(33) |

Then the -subsystem becomes

(34) |

It is easy to see that and as

We are now ready to state the theorem for the secondary system.

Theorem 3. Suppose and Let the controller for the secondary system (26) be designed as (33), where . Then meanwhile keeping bounded.

Proof. See Appendix D.

### Iii-E Controller Synthesis for Original System

It should be noticed that the controller design above is based on the condition that and are known as priori. A problem arises that the states and cannot be measured directly except for . By taking this into account, the following observer is proposed to estimate the states and , which is stated in Theorem 4.

Theorem 4. Let the observer be designed as follows

(35) |

where is stable. Then and

Proof. Since we have Consequently, (35) can be rewritten as

(36) |

Subtracting (18) from (36) results in

(37) |

where Then Furthermore, with the aid of the relationship we have

Remark 6. Unlike traditional observers, the proposed observer can estimate the states of the primary system and the secondary system directly rather than asymptotically or exponentially. This can be explained that, although the initial value is unknown, the initial value of either the primary system or the secondary system can be specified exactly, leaving an unknown initial value for the other system. The measurement and parameters may be inaccurate. In this case, it is expected that small uncertainties lead to close to (or close to ). From (37), a stable matrix can ensure a small in the presence of small uncertainties.

Theorem 5. Suppose that the conditions of Theorems 1-4 hold. Let the controller in the system (1) be designed as follows

(38) |

where is given by (13), and are given by (35), is defined in (23), and is defined in (33). Then meanwhile keeping and bounded.

Proof. Note that the original system (1), the primary system (17) and the secondary system (18) have the relationship: and With the controller (38), for the primary system (17), meanwhile keeping and boundedby Theorem 2. On the other hand, for the secondary system (18), we have meanwhile keeping bounded on by Theorem 3. In addition, Theorem 4 ensures that and Therefore, meanwhile keeping and bounded.

## Iv Numerical Simulation

In the simulation, set and the initial value in (1). The unknown dimensionless disturbance is generated by an autonomous LTI system (2) with the parameters as follows

The objective here is to design a controller such that the output as meanwhile keeping the other states bounded.

The parameters of the observer (13) are chosen as . In (16), the parameters of are chosen as and Then Re Since matrix does not possess the eigenvalues the parameters of the tracking controller (23) of the primary system can be chosen according to Proposition 1 that and These make in (24) satisfies . The parameter of the stabilized controller (33) is chosen as

The TORA system (1) is driven by the controller (38) with the parameters above. The evolutions of all states of (1) are shown in Fig.3. As shown, the proposed controller drives the output as , meanwhile keeping the other states bounded.

Unlike the output regulation theory, the proposed method does not require the regulator equations. If the disturbance consists of more frequency components, i.e., is more complicated, the designed controller above does not need to be changed except for the corresponding and . This demonstrates the effectiveness of the proposed control method. For example, we consider that the unknown dimensionless disturbance is generated by an autonomous LTI system (2) with the parameters as follows

The controller in the first simulation is still applied to this case except for replacing and (the dimension is changed correspondingly). Driven by the new controller, the evolutions of all states of (1) are shown in Fig.4. As shown, the proposed controller drives the output as meanwhile keeping the other states bounded.

## V Conclusions

In this paper, the tracking (rejection) problem for rotational position of the TORA was discussed. Our main contribution lies in the presentation of a new decomposition scheme, named additive state decomposition, which not only simplifies the controller design but also increases flexibility of the controller design. By the additive state decomposition, the considered system was decomposed into two subsystems in charge of two independent subtasks respectively: an LTI system in charge of a tracking (rejection) subtask, leaving a nonlinear system in charge of a stabilization subtask. Based on the decomposition, the subcontrollers corresponding to two subsystems were designed separately, which increased the flexibility of design. The tracking (rejection) controller was designed by the transfer function method, while the stabilized controller was designed by the backstepping method. This numerical simulation has shown that the designed controller can achieve the objective, moreover, can be changed flexibly according to the model of external signals.

## Vi Appendix

### Vi-a Proof of Theorem 1

The disturbance is generated by an autonomous LTI system (2) with an initial value It can also be generated by the following system

(39) |

with the initial value Subtracting (1d) and (39) from (13) results in

(40) |

where and Design a Lyapunov function as follows

Taking the derivative of along (40) results in

By Assumption 2, Then the derivative of becomes

Since from the inequality above, it can be concluded by LaSalle’s invariance principle [12] that and

### Vi-B Proof of Lemma 1

Before proving Lemma 1, we need the following preliminary result.

Lemma 2. If the pair is controllable, then there exists a such that