## Abstract

A class of time-varying systems can be quadratically stabilized with satisfactory performance by a modified time-invariant-based observer. The modified observer driven by the additional adaptation forces with static correction gains is used to estimate the time-varying system states. Under the frame of quadratic stability, the closed-loop systems satisfying induced norm bounded performance criterion are exponentially stabilized while the states are exponentially approaching by the modified observer. This paper deals with the time-varying systems that can be characterized as the multiplicative type of time-invariant and time-varying parts. The time-invariant part is then used to construct the modified observer with additional driving forces, which are ready to adjust time-varying effect coming from the measured outputs feeding into the modified observer. The determination of the adaptation forces can be derived from the minimization of the cost of error dynamics with modified least-squares algorithms. The synthesis of control and observer static correction gains are also demonstrated. The developed systems have been tested in a mass-spring-damper system to illustrate the effectiveness of the design.

### Keywords

- quadratic stablilization
- time-invariant-based observer
- error dynamics
- least-squares algorithm
- adaptation forces
- time-varying parts

## 1. Introduction

The study of optimal control for time-varying systems involves, in general, the solutions of Riccati differential equations (RDEs) and computations of the time-varying correction gains [1–4]. It is noted that the system is typically computer-implemented system, upon which the RDE and correction gains are calculated. The computations, however, induces unavoidable time delay. Although the time-delayed control has been considered, it leads to two disadvantages—complication of control mechanism and bulk of the control board. For some systems, for example, hard disk drives (a typically time-varying system) can only tolerate no delayed or very limited time delay control [5, 6] and use very small compartment. Hence, many literatures have focused on the static gain control of time-varying systems or the systems with time-varying or nonlinear uncertainties [7, 8]. It represents the simplest closed-loop control form but still encounters problems. One should aware that static output control is nonconvex, in which iterative linear matrix inequality approaches are exploited after it is expressed as a bilinear matrix inequality formulation( see [9–12]). As a result, it cannot be easily implemented in controlling the time-varying system and time delay problems remain.

It is a great challenge problem to design a linear continuous time-invariant observer with constant correction gains that regulate linear continuous time-varying plants. Although the vast majority of continuous TV control applications are implemented in digital computers [6, 13, 14], there are still opportunities to implement control with Kalman observer in continuous time (i.e., in analog circuits) [Hug88]. In particular, those control systems requiring fast response ask no or little delay effects. The difficulties for setting up those boards are because the algorithm of the design is too complex to implement in board level design, too expensive which can only be realized in a laboratory, or digital computation time induced unsatisfactory delay. It should be noticed that to realize the Kalman observer involves the computation of Riccati differential equations and inversion of matrices, which cause the obstacles of the board level design. A survey of linear and nonlinear observer design for control systems has been conducted in the literatures [15–18] and references therein. For controlling an linear time-invariant (LTI) system, the Lungerber observer [19] design with constant correction gain is straightforward and can be implemented on a circuit board with ease.

Many practical control systems implement time-invariant controllers with observers in the feedback loop, which can be easily realized not only in the laboratory but also in the industrial merchandize [20]. The advantages of realization for the time-invariant controllers and observers are due to the constant parameters, which can be easily assembled by using resistors and other analog integrated elements in circuits board. The use of observers is also essential in industrial controls due to, in some cases, the states can be either not reachable or expensive to be sensed. Therefore, the use of observers are undoubtedly required to estimate unmeasured states since not merely full-state feedback control can be easily implemented but unmeasured states can be monitored [21–25].

With the aforementioned disadvantages and advantages, the control of time-varying systems is naturally arisen by designing a time-invariant observer-based controller that stabilizes, in particular exponentially, this time-varying plant. It is believed that this is a great challenge problem since we found no literatures tackling this problem. In what follows time-varying system control is first reviewed for laying the foundation of the robust control of the system with optimality property.

The feedback control of linear time-varying system has been extensively studied [1, 6, 7, 26–31]. The key observation of early works for exponential stability of time-varying systems requires that the time-dependent matrix-valued functions be bounded and piecewise continuous satisfying Lyapunov quadratic stability [29, 31]. In this regard, many, but not all, of them can be translated to robust control framework since time-dependent matrices are essentially bounded and are treated as uncertainties [8, 32]. This gives an opportunity for the control system design without solving RDEs, although what we pay for the avoiding solving RDEs is the conservative of control. The conservativeness comes from two reasons—solutions of RDEs are avoided and admit fast varying parameters. This, however, can be reduced by designing parameter-dependent type of criterion or introducing slack variables such that reduces the tightness of dependent variables( see, e.g., [33] and reference therein).

This paper is organized as follows. The following section, Section 2, sets up the time-varying systems to be tackled, time-invariant observer to be built, feedback control problems to be solved, and system properties (assumptions) to be with the systems. Section 3 gives the main results for solving the feedback control problems, in which LMIs characterize the quadratically stability of the closed-loop system with

## 2. System formulation and problem statement

We consider a nonlinear time-varying system described by a set of equations

The first equation describes the *plant* with *n*-vector of *state x* and *control input* *exogenous input* _{,} which include *disturbances* (to be rejected) or *references* (to be tracked). The second equation defines the *regulated outputs* *tracking error*, expressed as a linear combination of the plant state *x* and of the exogenous input *w*. The last part is the *measured outputs* *system property*:

**(S1)** *A*(*t*) denotes the matrix with nonlinear time-varying properties satisfying

where *A* is the *A*(*t*). The *F*(*t*) lumps all time-varying elements associated with plant matrix *A*(*t*), and it is possible to find a vertex set

such that *F*(*t*) is within the convex set

**(S2)** The matrices *D*, and *u* and exogenous input *w*, respectively, and

**Remark 1**. It is highlighted that *F*(*t*) in (*S*1) is not merely to lump all possible time-varying functions but to include the parametric uncertainties. For the parametric uncertainties, it is seen by simply observing that *F*(*t*) can be multiplicative uncertainties shown in [8]. For representing time-varying matrix, an example is set as follows. Let

where

It should be aware that another equally good choice is to use additive type of representation, that is,

**Remark 2**. A number of examples are found to show the time-varying bound for

The control action to (1) is to design an observer-based output feedback control system, which processes the measured outputs *y*(*t*) in order to determine the plant states and generate an appropriate control inputs *u*(*t*) based on the estimated plant states. The following *observer dynamics* is developed for system (1),

where

*x*(*t*) and the gain L is to be designed for the sake of stability. It should be noted that the usage of constant matrix *A* in (3) instead of using time-varying *A*(*t*) is due to the fact that *it is not possible or may be too expensive to build the time-varying plant matrix A*(*t*) *for the time-varying observers* in a real analog circuit board that controls the system. On the contrary, we are able to establish a time-invariant observer with ease for constant system matrix *A*,

The time-varying vector-valued function *x*(*t*). We should be aware that the intention of *F*(*t*) to the system, that is, the effects of the time-varying functions will be adjusted by one such function *u*(*t*), *e*(*t*) becomes an additional *driving force* to (3) such that *x*(*t*) is possible. If

In order to facilitate the closed-loop system, the *error dynamics* can thus be found by manipulating (1) and (3) as follows

or, equivalently, by taking the advantages of polytopic bound of (S1)

where **1** denotes the vector with all elements being equal to 1.

Once the observed state *u* is chosen to be a memoryless system of the form

where K is the static gain to be designed. The control purpose has twofold: to *achieve closed-loop stability* and to *attenuate the influence* of the exogenous input w on the penalty variable z, in the sense of rendering the

**Observer-based control via measured feedback**. Given a real number *K* and *L* such that

**(O1)** the matrix

has all eigenvalues in

**(O2)** the

is strictly less than *z*(*t*) of (8) from initial state

for some

**Remark 3**. Here, we will be using the notion of quadratic stability with an

**Remark 4**. Figure 2 shows the overall feedback control structure of (8) to be designed in the sequel, where the feedback loop, namely *observer–error dynamics*, serves as filtering process with *y*(*t*) and *w*(*t*) as inputs such that proper control inputs *u*(*t*) and additional driving force of (3) *e*(*t*) are produced.

## 3. Analyses and characterizations

Two issues will be addressed in this section. Firstly, the theorem states the sufficiency condition showing that the problem of observer-based control via measured feedback of time-varying system is solvable. Secondly, an identification process based on least-squares algorithms for

### 3.1. LMI characterizations

**Theorem 1**. Consider the time-varying system (1), observer dynamics (3), and error dynamics (4) satisfying system property (S1) and (S2). Then, (T1) implies (T2), where (T1) and (T2) are as follows.

**(T1)** There exist matrices *K*, and *L* and positive scalars γ and β such that

and matrices

The matrices,

**(T2)** (O1) and (O2) hold, that is, the problem of observer-based control via contaminated measured feedback is solvable.

**Proof**: the implication between (T1) and (T2) is shown in the Appendix.

**Remark 5**. It is shown in the Theorem 1 that if the matrix inequality (10) is satisfied and if

### 3.2. Modified least-squares algorithms

Prior to stating the modified least-squares scheme for computing

where

To minimize the cost function

where

In view of (16), the *least-squares estimate* for

where *covariance matrix* and is defined as follows

To assure positive definiteness and thus the invertibility, the covariance matrix will be further polished in the sequel. The covariance matrix plays an important role in the estimation of

To find the least-squares estimator with recursive formulations, which parameters are updated continuously on the basis of available data, we differentiate (17) with respect to time and obtain

where

for *covariance resetting propagation law* is developed. Within each time window, we modify (18) as follows,

and

The scalar

**Lemma 1**. Assuming that (21) and (22) hold. Then,

**Proof**: At the resettings, the covariance matrix *t*, if it exists, on which

Before presenting the theorem for modified least-squares algorithms of

**Lemma 2**. There exists a positive number *k* such that the transition matrix,

**Proof**: The proof is constructive. We first notice that the solution to (19) is given by

where

A constructive method is suggested by letting a differential equation

Let

Without loss of generality, let

At the point of resetting, that is, the point of discontinuity of

It follows from (24) and (25), we conclude that the Lyapunov candidate along the solution

**Theorem 2**. Assuming that the problem of observer-based control via contaminated measured feedback is solvable. If there exist the identifier structure of least-squares algorithm (19) with covariance resetting propagation law (21) and (22), then

**Proof**: To prove the claim is true, we need to show that

where

The boundedness of *F*(*t*) is clearly bounded for all

Therefore,

which indicates that

**Remark 6**. In this section, a modified least-squares algorithm is shown to find the estimated *F*(*t*) produced in the plant (1). Figure 3 depicts the complete structure of observer–error dynamics that has been shown in Figure 2, in which two filters, namely *observer dynamics* and *error dynamics*, and one lest squares algorithm construct the feedback control. The observer dynamics produces the estimated state of plant by filtering the signals *u*(*t*), *w*(*t*), and *e*(*t*). It is worth noting that the signal *e*(*t*) from least-squares algorithm plays an additional drive force to the observer dynamics. The error dynamics is to find the error state

## 4. Control and observer gain synthesis

The synthesis of control and observer gains is addressed in Theorem 1. For the simplicity of expression, the time argument of matrix-valued function *F*(*t*) will be dropped and denoted by *F*. A useful and important Lemma will be stated in advance for clarity:

**Lemma 3** (Elimination Lemma see [32]). Given

if and only if

where

**Lemma 4**. Given a real number

**(Q1)** There exist matrices *K* and *L*, and positive scalars

**(Q2)** There exist matrices *K* and *L*, and positive scalars

**(Q3)** There exist matrices *W* and *Y*, and the positive scalars

**Proof**: to prove

Next, the orthogonal complement of

which

and

It is seen that matrix inequalities (28) and (29) hold if and only if (32) is true. Given (32), (33) is also true. Therefore, by Lemma 3,

To prove *iff* condition for inequality (28),

where *iff* holds is due to Schur complement in that the positive definiteness of

Again, the last *iff* of (35) is due to Schur complement and

**Remark 7**. It is seen that *K* and *L* are solely determined by (34) and (35), respectively. From rigorous point of view, we may not be able to say that the separation principle is completely valid for this case. But, loosely speaking, it fits by small modification.

**Lemma 5**. (Q1) implies (10).

**Proof**: let

Thus, (Q1) implies (10). This completes the proof.

**Theorem 3**. Given a real number

**Proof**: by Lemma 5, (Q1) implies matrix inequality (10). Moreover, by Lemma 4, we have

**Remark 8**. Theorem 3 states that the problems post in observer-based control via contaminated measured feedback, that is, (O1) and (O2), are solvable by proving that (T2) holds.

## 5. Illustrative application

In this application, a simple time-varying mass-damper-spring system is controlled to demonstrate that the time-varying effects appearing in the system matrix can be transferred to a force term in the observer structure. Thus, consider the system shown in Figure 3 without sensor fault.

where time-varying functions are

where

Here, we consider the parameters *F*(*t*) is

By applying linear matrix inequalities (30) and (31) of (Q3) in Lemma 4, the control and observer gain, *K* and *L* can be found by implementing Matlab Robust Control Toolbox. It is also noted that the computation of two matrix inequalities can be separated by justifying

The control input is then computed by

with

The implementation are coded in Matlab using the initial states: *e*(*t*) can actually trace the plant (37). The control input *u*(*t*) to the system is shown in Figure 4(c). The covariance resetting propagation law *e*(*t*) and 2-norm value of the time-varying matrix function *F*(*t*) are depicted in Figure 5(c) and (d). It is seen clearly that the *e*(*t*) shows the same results. Figure 5(c) depicts that the time-varying matrix *F*(*t*) is indeed varying with time.

## 6. Conclusion

This paper has developed the modified time-invariant observer control for a class of time-varying systems. The control scheme is suitable for the time-varying system that can be characterized by the multiplicative type of time-invariant and time-varying parts. The time-invariant observer is constructed directly from the time-invariant part of the system with additional adaptation forces that are prepared to account for time-varying effects coming from the measured output feeding into the modified observer. The derivation of adaptation forces is based on the least squares algorithms in which the minimization of the cost of error dynamics considers as the criteria. It is seen from the illustrative application that the closed-loop systems are showing exponentially stable with system states being asymptotically approached by the modified observer. Finally, the LMI process has been demonstrated for the synthesis of control and observer gains and their implementation on a mass-spring-damper system proves the effectiveness of the design.

## 7. Appendix

It is noted that in this appendix all time arguments of either vector-valued or matrix-valued time functions will be dropped for the simplicity of expression. They can be easily distinguished by their contents.

**Proof**:

with

for all states satisfying (1) and (3) with initial states

The second integrand in (39) is

The right-hand side of equality (41) can be reorganized by using the closed-loop system (8), and thus, the first term is

Completing the square of (42), we have

Similarly, the second term of (41) is

Applying completing the square to (44), we obtain

Substituting (40), (43), and (45) into (39), we have

where definition of Π1(P1) and Π2(P2) are defined as (12) and (13), respectively. Therefore, by eliminating the negative terms from (46), the following inequality is drawn,

Given that

In view of (10) of (T1), we thus find that the inequality (47) is simply

Therefore, the inequality (48) satisfies the performance index (9), which completes the proof (O2).

To prove that (O1) holds, we use the inequality (10) in (T1) and get the equivalent inequality as follows,

where

It is concluded, by a standard Lyapunov stability argument, that