# Output Feedback Tracking Control for a Class of Uncertain Systems subject to Unmodeled Dynamics and Delay at Input

## Abstract

Besides parametric uncertainties and disturbances, the unmodeled dynamics and time delay at the input are often present in practical systems, which cannot be ignored in some cases. This paper aims to solve output feedback tracking control problem for a class of nonlinear uncertain systems subject to unmodeled high-frequency gains and time delay at the input. By the additive state decomposition, the uncertain system is transformed to an uncertainty-free system, where the uncertainties, disturbance and effect of unmodeled dynamics plus time delay are lumped into a new disturbance at the output. Sequently, additive state decomposition is used to decompose the transformed system, which simplifies the tracking controller design. To demonstrate the effectiveness, the proposed control scheme is applied to three benchmark examples.

Additive state decomposition, tracking, input delay, unmodeled dynamics, output feedback, nonlinear systems.

## 1 Introduction

Tracking control of an uncertain system is a challenging problem. Most of research mainly focuses on systems subject to parametric uncertainties and additive disturbances [1]-[3]. Also, some research focuses on systems subject to uncertainties at the input, such as backlash, dead zone or other nonlinearities [4]-[5]. It is well known that unmodeled dynamics and time delay at the input are also often present in practical systems. For example, the unmodeled dynamics and time delay at the input often exist in flight control systems [6]-[8]. These uncertainties at the input may produce a significant degradation in the tracking performance or even cause instability if not dealt with properly. In the literature, there are some academic examples to demonstrate that uncertainties at the input cannot be ignored in some cases. For example, in [9], the authors constructed a simple example, later known as Rohrs’ example, to show that conventional adaptive control algorithms lose their robustness in the presence of unmodeled dynamics. Also, some control algorithms may lose their robustness in the presence of input delay, see for example the repetitive control example considered in [10]. Therefore, it is important to explicitly consider unmodeled dynamics and time delay at the input in the controller design.

In this paper, the output feedback tracking control problem is investigated for a class of single-input single-output (SISO) nonlinear systems subject to mismatching parametric uncertainty, mismatching additive disturbances, unmodeled high-frequency gains and time delay at the input. Before introducing our main idea, some accepted control methods in the literature to handle uncertainties are briefly reviewed. A nature way is to estimate all of the unknown parameters, then compensate for them. In [11], the tracking problem for a linear system subject to unknown parameters and the unknown input delay was considered, where both the parameters and input delay were estimated by the proposed method. However, this method cannot handle unparameterized uncertainties such as unmodeled high-frequency gains. The second way is to design adaptive control with robustness against unmodeled dynamics and time delay at the input. In [12], the Rohrs’ example and the two-cart example, which are tracking problems for uncertain linear systems subject to unmodeled dynamics and time delay at the input respectively, were revisited by the adaptive control. In [13], the authors analyzed that their proposed method is robust against time delay at the input. The third way is to convert a tracking problem to a stabilization problem by the idea of internal model principle [14], if disturbances or desired trajectories are limited to a special case. In [15], the problem of set point output tracking of an uncertain linear system with multiple delays in both the state and control vectors was considered. There also exist other methods to handle uncertainties. However, some of them such as high-gain feedback cannot be applied to the considered system directly as they rely on rapid changing control signal to attenuate uncertainties and disturbance. After passing unmodeled high-frequency gains or time delay at the input, the rapid changing control signal will be distorted a lot which will affect the feedback and then may destabilize the system. This explains why high-gain feedback is often avoided in practice.

Compared with these existing literature, the problem studied in this
paper is more general since not only the uncertainties at the input
but also the output feedback and mismatching are considered. For
output feedback, the state needs to be estimated which is difficult
mainly due to the uncertainties and disturbances in the state
equation. Even if parameters and disturbance can be estimated, it is
also difficult to compensate for mismatching uncertain parameters
and disturbance directly. To tackle these difficulties, two new
mechanisms are adopted in this paper. First, the input is redefined
to make it smooth and bounded to handle uncertainties at input. As a
consequence, the effect of unmodeled high-frequency gains and time
delay at the input is always bounded. And then, to handle estimate
and mismatching problem, the input-redefinition system is
transformed to an uncertainty-free system, which is proved to be
input-output equivalent with the aid of the *additive state
decomposition*^{1}*
*[16]. All mismatching uncertainties,
mismatching disturbance and effect of unmodeled dynamics plus time
delay are lumped into a new disturbance at the output. An observer
is then designed for the transformed system to estimate the new
state and the new disturbance. Next, the transformed system is
‘additively’ decomposed into two independent subsystems in charge of
corresponding subtasks, namely the tracking (including rejection)
subtask and the input-realization subtask. Then one can design
controller for each subtask respectively, and finally combines them
to achieve the original control task. Three benchmark examples are
given to demonstrate the effectiveness of the proposed control
scheme.

The additive state decomposition is a decomposition scheme also proposed in our previous work [17], where the additive state decomposition is used to transform output feedback tracking control for systems with measurable nonlinearities and unknown disturbances and then to decompose it into three simpler problems. This hence makes a challenging control problem tractable. In this paper, a different control problem is investigated by using additive state decomposition. Correspondingly, the transform and decomposition are different. The major contributions of this paper are: i) a tracking control scheme proposed to handle mismatching parametric uncertainty, mismatching additive disturbances, unmodeled high-frequency gains and time delay at the input; ii) a model transform proposed to lump various uncertainties together; iii) additive state decomposition in the controller design, especially in how to handle saturation term.

This paper is organized as follows. In Section II, the problem formulation is given and the additive state decomposition is introduced briefly first. In Section III, input is redefined and the input-redefinition system is transformed to an uncertainty-free system in sense of input-output equivalence. Sequently, controller design is given in Section IV. In Section V, two-cart example is revisited by the proposed control scheme. Section VI concludes this paper.

## 2 Problem Formulation and Additive State Decomposition

### 2.1 Problem Formulation

Consider a class of SISO nonlinear systems as follows:

(1) |

Here and are constant vectors, belongs to a given compact set is the state vector, is the output, is a bounded disturbance vector, and is the control subject to an unmodeled high-frequency gain and a time delay as follows:

(2) |

where is an unknown stable proper transfer function with representing the unmodeled high-frequency gain at the input and is the input delay. It is assumed that only is available from measurement. The desired trajectory is known a priori, . In the following, for convenience, we will drop the notation except when necessary for clarity.

For system (1), the following assumptions are made.

Assumption 1. The function satisfies and is bounded when is bounded on . Moreover, for given there exist positive definite matrices and such that

(3) |

where

Definition 1 [18]. The gain of a stable proper SISO system is defined d where is the impulse response of .

Assumption 2. There exists a known stable proper transfer function with such that , where are positive real.

Under Assumptions 1-2, the objective here is to design a tracking controller such that with a good tracking accuracy, i.e., is ultimately bounded by a small value.

Remark 1. From Assumption 1, since by the Taylor expansion. Consequently, the system is exponentially stable by (3). In practice, many systems are stable themselves or they can be stabilized by output feedback control. The following three benchmark systems all satisfy Assumption 1.

Example 1 (Rohrs’ Example). Consider the Rohrs’ example system as follows [9]:

(4) |

The nominal system is assumed to be here. In this case, the system (4) can be formulated into (1) as

(5) |

where the parameter is assumed unknown and . It is easy to see that Assumption 1 is satisfied. Choose Then Assumption 2 is satisfied with and

Example 2 (Nonlinear). Consider a simple nonlinear system as follows [19]:

(6) |

where the parameter , the input delay and are assumed unknown. The system (6) can be formulated into (1) with and It is easy to verify Therefore, Assumption 1 is satisfied. Let Then Assumption 2 is satisfied with and

Remark 2. The Rohrs’ example system in Example 1 is proposed to demonstrate that conventional adaptive control algorithms developed at that time lose their robustness in the presence of unmodeled dynamics [9]. For the tracking problem in Example 2, there exist robustness issues by using exact feedback linearization [19]. Compared with the system in [19], the input delay is added in (6) to make system worse. The two benchmark examples tell us that the uncertainties either on the system parameters or at the input cannot be ignored in practice when design a tracking controller, even if the original systems are stable. This is also the initial motivation of this paper.

### 2.2 Additive State Decomposition

In order to make the paper self-contained, additive state decomposition [16] is introduced briefly here. Consider the following ‘original’ system:

(7) |

where . We first bring in a ‘primary’ system having the same dimension as (7), according to:

(8) |

where . From the original system (7) and the primary system (8) we derive the following ‘secondary’ system:

(9) |

where is given by the primary system (8). Define a new variable as follows:

(10) |

Then the secondary system (9) can be further written as follows:

(11) |

From the definition (10), we have

(12) |

Remark 3. By the additive state
decomposition, the system (7) 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 (7), a class of differential dynamic systems is considered as follows:

(13) |

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

(14) |

and

(15) |

where and . The secondary system (15) is determined by the original system (13) and the primary system (14). From the definition, we have

(16) |

## 3 Input Redefinition and Model Transformation

Since is the unmodeled high-frequency gain and is the input delay, the control signal should be smooth (low-frequency signal) so that it will maintain its original form as far as possible after passing . Otherwise, the control signal will be distorted a lot. This explains why high-gain feedback in practice is often avoided. For such a purpose, the input is redefined to make control signal smooth and bounded first. This makes the effect of under control, i.e., the effect will be predicted and bounded.

### 3.1 Input Redefinition

Redefine the input as follows:

where is the redefined control input and is a saturation function defined as sign. Then is written as

(17) |

where represents the effect of the unmodeled high-frequency gain and the time delay. The function can be further written as

(18) |

From the definition of , we have In this paper denotes the inverse Laplace transform. By Assumption 2, is bounded as follows:

(19) |

where The input redefinition makes bounded not matter what the redefined control input is. Therefore, the redefined control input can be designed freely. According to input redefinition above, the controller (2) is rewritten as

(20) |

Here can be written in the form of state equation as follows

(21) |

where the vectors and matrices are compatibly dimensioned depending on Substituting (20) into the system (1) results in

(22) |

where The system (22) with the
redefined controller (21) is called as the
*input-redefinition system *here.

### 3.2 Model Transformation

The unknown parameter and the unknown disturbances are not appear in “matching” positions for the control input, i.e., and do not appear like Therefore, in a general system except for one dimensional system, the unknown uncertainties cannot be often compensated for directly. Even if and satisfy the “matching condition”, it is also difficult to compensate for since the state is unknown. To tackle this difficulty, we first transform the input-redefinition system (22) to an uncertainty-free system, which is proved to be input-output equivalent with the aid of the additive state decomposition as stated in Theorem 1. Before proving the theorem, the following lemma is needed.

Lemma 1. Consider the following system

(23) |

where is bounded. Under Assumption 1, the solutions of (23) satisfy

(24) |

where is a class function [20, p.144] and .

Proof. By the Taylor expansion, the function can be written as

where Then the system (23) can be rewritten as

(25) |

Choose Lyapunov function By Assumption 1, the derivative of along (25) satisfies

By Theorem 4.19 [20, p.176], we can conclude this proof.

With Lemma 1 in hand, we have

Theorem 1. Under Assumption 1, there always exists an estimate of namely such that the system (22) is input-output equivalent to the following system:

(26) |

Here and satisfy

(27) |

where is a class function, and

Proof. In the following, additive state decomposition is utilized to decompose the system (22) first. Consider the system (22) as the original system and choose the primary system as follows:

(28) |

Then the secondary system is determined by the original system (22) and the primary system (28) with the rule (15) that

(29) |

According to (16), we have and Consequently, we can get an uncertainty-free system as follows

where and are the same to those in (22). Let and We can conclude that the system (22) is input-output equivalent to (26). Next, we will prove that (27) is satisfied. The system (29) can be rewritten as

(30) |

where . Then, by Lemma 1, we have

For the uncertainty-free transformed system (26), we design an observer to estimate and , which is stated in Theorem 2.

Theorem 2. Under Assumption 1, an observer is designed to estimate state and in (26) as follows

(31) |

Then and

Proof. Subtracting (31) from (26) results in

where and Then . This implies that Consequently, by the relation in (26), we have

Remark 4. By (21), the control signal is always bounded. Therefore, by Lemma 1, the state is always bounded. Consequently, by (27), is always bounded as well. It is interesting to note that the new state and disturbance in the transformed system (26) can be observed directly rather than asymptotically or exponentially. This will facilitate the analysis and design later.

## 4 Controller Design

In this section, the transformed system (26) is ‘additively’ decomposed into two independent subsystems in charge of corresponding subtasks. Then one can design controller for each subtask respectively, and finally combines them to achieve the original control task.

### 4.1 Additive State Decomposition of Transformed System

Currently, based on the new transformed system (26), the objective is to design a tracking controller such that with a good tracking accuracy, i.e., is ultimately bounded by a small value. While, is realized by (21). According to this fact, the transformed system (26) is ‘additively’ decomposed into two independent subsystems in charge of corresponding subtasks, namely the tracking (including rejection) subtask and the input-realization subtask. This is shown in Fig.1.

Consider the transformed system (26) as the original system. According to the principle above, we choose the primary system as follows:

(34) |

Then the secondary system is determined by the original system (26) and the primary system (34) with the rule (15), and we can obtain that

(35) |

According to (16), we have

(36) |

The strategy here is to assign the tracking (including rejection) subtask to the primary system (34) and the input-realization subtask to the secondary system (35). It is clear from (34)-(36) that if the controller drives in (34) and drives in (35), then as . The benefit brought by the additive state decomposition is that the controller will not affect the tracking and rejection performance since the primary system (34) is independent of the secondary system (35). Since the states and are unknown except for addition of them, namely , an observer is proposed to estimate and

Remark 5. Although the proposed additive state decomposition gives clear how to decompose a system, it still leaves a freedom to choose the primary system. By the additive state decomposition, the transformed system (26) can be also decomposed into a primary system

(37) |

and the derived secondary system

(38) |

where is an arbitrary constant matrix. Therefore, there is an infinite number of decompositions. The principle here is to derive the secondary system with an equilibrium point close to zero as far as possible. If so, the problem for the secondary system is only a stabilization problem, which is easier compared with a tracking problem. In (35), is an equilibrium point of whereas in (38), is not an equilibrium point of This is why we choose the primary system as (34) not (37). From the mention above, a good additive state decomposition often depends on a concrete problem.

Theorem 3. Under Assumption 1, suppose that an observer is designed to estimate state and in (34)-(35) as follows:

Then and | ||||

Proof. Similar to the proof of Theorem 2. | ||||

So far, we have transformed the original system to an uncertainty-free system, in which the new state and the new disturbance can be estimated directly. And then, decompose the transformed system into two independent subsystems in charge of corresponding subtasks. In the following, we are going to investigate the controller design with respect to the two decomposed subtasks respectively. |

(40) |

such that
^{2}

Remark 6 (on Problem 1). Since Problem 1 can be also considered to design such that Here, the difference between and should be clarified. The reference is often known a priori, i.e., is known at the time where Moreover, its derivative is often given or can be obtained by analytic methods. Whereas, the new disturbance only can be obtained at the time whose derivative only can be obtained by numerical methods. By recalling (27), the new disturbance depends on the disturbance the parameters and the effect of unmodeled high-frequency gain namely the state and initial value One way of reducing the complexity is to design an observer to estimate and makes as As a result, the new disturbance finally depends on and as In practice, low frequency band is often dominant in the reference signal and disturbance. Therefore, from a practical point of view, we can also modify the tracking target, namely For example, let pass a low-pass filter to obtain its major component. If the major component of belongs to a fixed family of functions of time, Problem 1 can also be considered as an output regulation problem [21].

### 4.3 Problem for Input-Realization Subtask

As shown in Fig.1, the input realization subtask aims to make Let us investigate the secondary system (35). By Lemma 1, we have

(41) |

This implies that as , where It is noticed that only can be realized by (21). Therefore, problem for input-realization subtask can be stated as follows:

Problem 2. Given a signal design a controller for (21) such that as

This is also a tracking problem but with a saturation constraint. Here we give a solution to the Problem 2. The main difficult is how to handle the saturation in (21). Here, additive state decomposition will be used again. Taking (21) as the original system, we choose the primary system as follows

(42) |

Then the secondary system is determined by the original system (21) and the primary system (42) with the rule (15), and we can obtain that

(43) |

According to (16), we have and The benefit brought by the additive state decomposition is that the controller saturation will not affect the primary system (42). Moreover, the controller can be designed only based on the primary system (42), where the controller uses the state not So, the strategy here is to design in (42) to drive as and neglect the secondary system (43). Since is bounded, the state of the secondary system (43) will be bounded as well. If as then as Consequently, as For (42), the transfer function from to is If is designed to be minimum phase, an easy way is to design to be

(44) |

The design will make the signal close to the idea one, meanwhile maintaining the signal smooth as far as possible. By recalling (18), it will make the effect of the unmodeled high-frequency gain and the time delay smaller.

### 4.4 Controller Integration

With the solutions of the two problems in hand, we can state

Theorem 4. Under Assumptions 1-2, suppose i) Problems 1-2 are solved; ii) the controller for system (1) (or (26)) is designed as

Observer:

(45) |

Controller:

(46) |

Then the output of system (1) (or (26)) satisfies that as meanwhile keeping all states bounded. In particular, if then the output in system (1) (or (26)) satisfies that as

Proof. It is easy to follow the proof in Theorems 2-3 that the observer (45) will make

(47) |

Suppose that Problem 1 is solved. By (40) and (47), the controller can drive as in (34). Suppose that Problem 2 is solved. By (47), the controller can drive as in (35). Further by (41), we have Since we have

Example 5 (Rohrs’ Example, Example 3 Continued). According to (34), the primary system of linear system (32) can be rewritten as follows:

Design Then the system above becomes where Therefore, as According to (44), is designed as Here and are approximated by and respectively. Suppose and given and sin respectively. Driven by the resulting controller (46), the simulation result is shown in Fig.2.