**2. System formulation and problem statement**

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 nonlin‐ ear 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-

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 timeinvariant (LTI) system, the Lungerber observer [19] design with constant correction gain is

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

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

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 contin‐

straightforward and can be implemented on a circuit board with ease.

varying system and time delay problems remain.

2 Robust Control - Theoretical Models and Case Studies

states can be monitored [21–25].

with optimality property.

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

$$\begin{aligned} \dot{x}(t) &= A(t)x(t) + B\_1 u(t) + B\_2 w(t) \\ z(t) &= -Dx(t) & + D\_2 w(t) \\ y(t) &= -Cx(t) \end{aligned} \tag{1}$$

The first equation describes the *plant* with *n*-vector of *state x* and *control input u* ∈ℝ*<sup>m</sup>* and is subject to *exogenous input w* ∈ℝ*<sup>l</sup>* , which include *disturbances* (to be rejected) or *references* (to be tracked). The second equation defines the *regulated outputs* z∈ℝ*<sup>q</sup>* , which, for example, may include *tracking error*, expressed as a linear combination of the plant state *x* and of the exoge‐ nous input *w*. The last part is the *measured outputs y* ∈ℝ*<sup>p</sup>* . The matrices in (1) are assumed to have the following *system property*:

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

$$A(t) = F(t)A,\tag{2}$$

where *A* is the *n* ×*n* constant matrix that extracts from *A*(*t*). The *n* ×*n* matrix *F*(*t*) lumps all timevarying elements associated with plant matrix *A*(*t*), and it is possible to find a vertex set Ψ<sup>1</sup> defined as follows

$$\mathbf{Y\_1} = \mathbf{Co}\{F\_{1\prime}F\_{2\prime}\cdots, F\_{\ell}\}$$

such that *F* (*t*)∈Ψ<sup>1</sup> , which is equivalent to saying that *F*(*t*) is within the convex set Ψ1 for all time *t* ≥ 0.

**(S2)** The matrices *B*1, *B*2, *D*, and *D*2 are all constant matrices, in which *B*1 and *B*2 quantify the range spaces of control input *u* and exogenous input *w*, respectively, and *D*<sup>2</sup> is chosen to be zero matrix that is *D*2 = 0 for computational simplicity.

**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

$$A(t) = \begin{pmatrix} -2 & f(t) \\ 0 & -1 \end{pmatrix} = F(t)A\_J$$

where

$$F(t) = \begin{pmatrix} 1 & -f(t) \\ 0 & 1 \end{pmatrix}, \quad A = \begin{pmatrix} -2 & 0 \\ 0 & -1 \end{pmatrix}.$$

It should be aware that another equally good choice is to use additive type of representation, that is, *A*(*t*) = *A* + *F*<sup>1</sup> (*t*), where *F*<sup>1</sup> (*t*) lumps all time-varying factors. As a matter of fact, multiplicative and additive type of representations are interchangeable. Let *F*<sup>2</sup> (*t*) be such that *F*1 (*t*) = *F*<sup>2</sup> (*t*)*A*. Thus, *A*(*t*)= *F*(*t*)*A*, where *F* (*t*) = *I* + *F*<sup>2</sup> (*t*).

**Remark 2**. A number of examples are found to show the time-varying bound for *F* (*t*), such as aircraft control systems in which constantly weight decreasing due to fuel consumption, the switching operations of a power circuit board for voltage and current regulations, and the hard disk drives with rotational disks induced time-varying dynamic phenomena [6].

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),

$$\begin{aligned} \dot{\mathfrak{X}}(t) &= A\mathfrak{X}(t) + B\_1 u(t) + B\_2 w(t) + L e(t) \\ \mathfrak{Y}(t) &= \mathbb{C} \mathfrak{X}(t) \end{aligned} \tag{3}$$

where

$$e(t) = \operatorname{diag}[\hat{y}(t)]\boldsymbol{\varsigma}(t) - y(t)\_{\prime}$$

such that *F* (*t*)∈Ψ<sup>1</sup> , which is equivalent to saying that *F*(*t*) is within the convex set Ψ1 for all

**(S2)** The matrices *B*1, *B*2, *D*, and *D*2 are all constant matrices, in which *B*1 and *B*2 quantify the range spaces of control input *u* and exogenous input *w*, respectively, and *D*<sup>2</sup> is chosen to be

**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

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

**Remark 2**. A number of examples are found to show the time-varying bound for *F* (*t*), such as aircraft control systems in which constantly weight decreasing due to fuel consumption, the switching operations of a power circuit board for voltage and current regulations, and the hard

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*

(*t*).

multiplicative and additive type of representations are interchangeable. Let *F*<sup>2</sup>

disk drives with rotational disks induced time-varying dynamic phenomena [6].

(*t*) lumps all time-varying factors. As a matter of fact,

(*t*) be such that

(3)

zero matrix that is *D*2 = 0 for computational simplicity.

4 Robust Control - Theoretical Models and Case Studies

(*t*), where *F*<sup>1</sup>

(*t*)*A*. Thus, *A*(*t*)= *F*(*t*)*A*, where *F* (*t*) = *I* + *F*<sup>2</sup>

representing time-varying matrix, an example is set as follows. Let

time *t* ≥ 0.

where

*F*1 (*t*) = *F*<sup>2</sup>

where

that is, *A*(*t*) = *A* + *F*<sup>1</sup>

*dynamics* is developed for system (1),

*x* ^(*t*) is the observed state of *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*, *B*1, and *B*2 as stated in the Section 2. It is also seen that the observer (3) is Luenbergerlike observer because of the use of observer gain L.

The time-varying vector-valued function *ς*(*t*)∈ℝ*<sup>p</sup>* to be determined in the sequel is an additional degree of freedom for driving observer (3) to estimate the plant state *x*(*t*). We should be aware that the intention of *ς*(*t*) is designed and meant to compensate time-varying effects of *F*(*t*) to the system, that is, the effects of the time-varying functions will be adjusted by one such function *ς*(*t*). Therefore, in addition to input *u*(*t*), *e*(*t*) becomes an additional *driving force* to (3) such that *x* ^(*t*) tracks *x*(*t*) is possible. If *<sup>F</sup>* (*t*) = *<sup>I</sup>* and all elements of the vector *ς*(*t*) being equal to 1, then the system (1) with the observer (3) is a typical textbook example of Luenberger observer control system [21].

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

$$
\dot{\mathfrak{X}}(t) = F(t)A\mathfrak{X}(t) + (I - F(t))A\mathfrak{X}(t) + Le(t) \tag{4}
$$

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

$$\dot{\tilde{x}}(t) = (A + LC)\tilde{x}(t) + (I - F(t))Ax(t) + Ldiag[\epsilon(t)]\hat{y}(t) \tag{5}$$

where *x*˜(*t*) = *x* ^(*t*)<sup>−</sup> *<sup>x</sup>*(*t*) and *ε*(*t*) = *ς*(*t*)−1, in which **1** denotes the vector with all elements being equal to 1.

Once the observed state *x* ^ is available, the control input *u* is chosen to be a memoryless system of the form

$$
\mu(t) = K\mathfrak{X}(t),
\tag{6}
$$

where K is the static gain to be designed. The control purpose has twofold: to *achieve closedloop stability* and to *attenuate the influence* of the exogenous input w on the penalty variable z, in the sense of rendering the L2 gain of the corresponding closed-loop system less than a prescribed number γ, in the presence of time-varying plant. The problem of finding controllers achieving these goals can be formally stated in the following terms.

**Observer-based control via measured feedback**. Given a real number *γ* >0 and {*A*(*t*), *B*1, *B*2, *C*, *D*, *D*2} satisfying system properties (S1) and (S2). Find, if possible, two constant matrices *K* and *L* such that

**(O1)** the matrix

$$
\begin{pmatrix} F(t)A + B\_1K & B\_1K \\ (I - F(t))A & A + LC \end{pmatrix} \tag{7}
$$

has all eigenvalues in C<sup>−</sup> ,

**(O2)** the L2-gain of the closed-loop system

$$\begin{aligned} \dot{x}(t) &= (F(t)A + B\_1K)x(t) + B\_1K\xi(t) + B\_2w(t), \quad y(t) = \mathcal{C}x(t) \\ \dot{\mathcal{H}}(t) &= (A + B\_1K)\mathcal{K}(t) + B\_2w(t) + L\mathcal{e}(t), \quad \mathcal{Y}(t) = \mathcal{C}\mathcal{X}(t) \\ \dot{x}(t) &= Dx(t), \quad \text{with} \quad e(t) = diag[\mathcal{Y}(t)]\xi(t) - y(t) \end{aligned} \tag{8}$$

is strictly less than γ, or equivalently, for each input *u*(*t*)= *K x* ^(*t*)∈L<sup>2</sup> 0, *<sup>∞</sup>*), the response *z*(*t*) of (8) from initial state (*x*(0), *x*˜(0)) = (0, 0) is such that the following performance index is satisfied

$$\mathcal{J}\_{\mathfrak{D}} = \int\_0^{\infty} \|z(t)\|^2 dt \le \gamma^2 \int\_0^{\infty} \|w(t)\|^2 dt \tag{9}$$

for some γ> 0 and every *w*(*t*)∈L<sup>2</sup> 0, *∞*).

**Remark 3**. Here, we will be using the notion of quadratic stability with an L2-gain measure which was introduced in [32]. This concept is a generalization of that of quadratic stabilization to handle L2-gain performance constraint to time-varying system attenuation. To this end, the characterizations of robust performance based on quadratic stability will be given in terms of LMIs, where if LMIs can be found, then the computations by finite dimensional convex programming are efficient (see, for example, [32]).

**Figure 1.** Overall control structure.

**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.

**Figure 2.** Observer–error dynamics.

**Observer-based control via measured feedback**. Given a real number *γ* >0 and {*A*(*t*), *B*1, *B*2, *C*, *D*, *D*2} satisfying system properties (S1) and (S2). Find, if possible, two

(8) from initial state (*x*(0), *x*˜(0)) = (0, 0) is such that the following performance index is satisfied

**Remark 3**. Here, we will be using the notion of quadratic stability with an L2-gain measure which was introduced in [32]. This concept is a generalization of that of quadratic stabilization to handle L2-gain performance constraint to time-varying system attenuation. To this end, the characterizations of robust performance based on quadratic stability will be given in terms of LMIs, where if LMIs can be found, then the computations by finite dimensional convex

(7)

(8)

(9)

^(*t*)∈L<sup>2</sup> 0, *<sup>∞</sup>*), the response *z*(*t*) of

constant matrices *K* and *L* such that

6 Robust Control - Theoretical Models and Case Studies

,

is strictly less than γ, or equivalently, for each input *u*(*t*)= *K x*

**(O2)** the L2-gain of the closed-loop system

for some γ> 0 and every *w*(*t*)∈L<sup>2</sup> 0, *∞*).

**Figure 1.** Overall control structure.

programming are efficient (see, for example, [32]).

**(O1)** the matrix

has all eigenvalues in C<sup>−</sup>
