**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. *k*1 and *c*1 are linear spring and damping constant, respectively. *k*<sup>2</sup> (*t*), *c*<sup>2</sup> (*t*), and *c*<sup>3</sup> (*t*) are time-varying spring and viscous damping coefficients. The system is described by the following equation of motion

$$\begin{aligned} 0 &= -k\_2(t)y(t) + c\_3(t)\dot{y}(t) \\ m\ddot{y}(t) &= -k\_1y(t) - (c\_1 + c\_2(t))\dot{y}(t) + u(t), \end{aligned} \tag{36}$$

where time-varying functions are *c*<sup>2</sup> (*t*)=*c*2(sin*ω*2*t*), *k*<sup>2</sup> (*t*)= *<sup>k</sup>*<sup>1</sup> *<sup>m</sup> <sup>e</sup>* <sup>−</sup>*bt*(cos*ω*1*t*), *c*<sup>3</sup> (*t*)= <sup>−</sup> *<sup>c</sup>*<sup>1</sup> *<sup>m</sup> <sup>e</sup>* <sup>−</sup>*bt*(cos*ω*1*t*), and the constants are *k*1 = *k* and *c*<sup>1</sup> =*c*2 = *c*. Define *x*<sup>1</sup> (*t*)= *y*(*t*) and *x*<sup>2</sup> (*t*)= *y*˙(*t*), the state space representation of (36) is

$$\begin{aligned} \dot{x}(t) &= F(t)Ax(t) + B\_1 u(t) \\ y(t) &= \mathbb{C}x(t) \end{aligned} \tag{37}$$

where

$$\begin{aligned} \mathbf{x}(t) &= \begin{pmatrix} \mathbf{x}\_1(t) \\ \mathbf{x}\_2(t) \end{pmatrix}, \; F(t) = \begin{pmatrix} 1 & e^{-bt}\cos\omega\_1 t \\ -\frac{c\_2}{m}\sin\omega\_2 t & 1 \end{pmatrix}, \\\ A &= \begin{pmatrix} 0 & 1 \\ -\frac{k\_1}{m} & -\frac{c\_1}{m} \end{pmatrix}, \; B\_1 = \begin{pmatrix} 0 \\ \frac{1}{m} \end{pmatrix}, \; \text{and } \mathbf{C} = \begin{pmatrix} 1 & 0 \end{pmatrix}. \end{aligned}$$

Here, we consider the parameters *m* = 1, *c* = 1, *k* = 1, *b* = 0, *ω*1 = 1, and *ω*<sup>2</sup> =10 (rad / sec). Thus, the set of vertices of polytope Ψ1 associated with time-varying matrix *F*(*t*) is

$$\mathbf{Co} \left\{ \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \right\}.$$

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 *δ* = 0.05. We thus find

$$K = \begin{pmatrix} -9.0322 & -10.2123 \\ \end{pmatrix}, \ L = \begin{pmatrix} -7.0950 \\ -2.8926 \\ \end{pmatrix}.$$

The control input is then computed by *u* = *K x* ^, where the observed state *<sup>x</sup>* ^ is from

**Remark 8**. Theorem 3 states that the problems post in observer-based control via contaminated

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

are time-varying spring and viscous damping coefficients. The system is described by the

(*t*)=*c*2(sin*ω*2*t*), *k*<sup>2</sup>

Here, we consider the parameters *m* = 1, *c* = 1, *k* = 1, *b* = 0, *ω*1 = 1, and *ω*<sup>2</sup> =10 (rad / sec). Thus,

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

the set of vertices of polytope Ψ1 associated with time-varying matrix *F*(*t*) is

(*t*)= *<sup>k</sup>*<sup>1</sup>

(*t*)= *y*(*t*) and *x*<sup>2</sup>

*<sup>m</sup> <sup>e</sup>* <sup>−</sup>*bt*(cos*ω*1*t*), *c*<sup>3</sup>

(*t*), *c*<sup>2</sup>

(*t*)= <sup>−</sup> *<sup>c</sup>*<sup>1</sup>

(*t*)= *y*˙(*t*), the state space

(*t*), and *c*<sup>3</sup>

*<sup>m</sup> <sup>e</sup>* <sup>−</sup>*bt*(cos*ω*1*t*),

(*t*)

(36)

(37)

measured feedback, that is, (O1) and (O2), are solvable by proving that (T2) holds.

sensor fault. *k*1 and *c*1 are linear spring and damping constant, respectively. *k*<sup>2</sup>

**5. Illustrative application**

16 Robust Control - Theoretical Models and Case Studies

following equation of motion

representation of (36) is

where

where time-varying functions are *c*<sup>2</sup>

and the constants are *k*1 = *k* and *c*<sup>1</sup> =*c*2 = *c*. Define *x*<sup>1</sup>

$$\begin{aligned} \dot{\mathfrak{X}}(t) &= (A + B\_1 K)\mathfrak{X}(t) + Le(t) \\ \dot{\mathfrak{y}}(t) &= \mathbf{C}\mathfrak{X}(t) \end{aligned} \tag{38}$$

with *e*(*t*)=*diag y* ^(*t*) *<sup>ς</sup>*(*t*)<sup>−</sup> *<sup>y</sup>*(*t*), in which the time-varying vector-valued function *ς*(*t*) is estimat‐ ed via the set of recursive formulations (19)–(22).

**Figure 4.** (a) shows plant state (solid line) *x*1 and observer state (dash-dot line) *x* ^ 1. (b) is the plant state (solid line) *x*<sup>2</sup> and observer state (dash-dot line) *x* ^ 2. (c) gives the control input *u*(*t*).

The implementation are coded in Matlab using the initial states: *x*<sup>1</sup> (0)=0.5, *x*<sup>2</sup> (0)= −0.6, *x* ^ 1 (0)=0.0, *x* ^ 2 (0)=0.1, *ς* ^(0)=0.1, Γ(0)=*k*<sup>0</sup> =2, and *k*<sup>1</sup> =0.3. The simulation results are depicted in **Figures 4** and **5**. **Figure 4(a)** and **(b)** shows that the observer states *x* ^ cohere with the plant states *x*. It is, therefore, seen that the observer (38) being driven by time-varying term *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 Γ(*t*) and the estimated *ς*(*t*), that is, *ς* ^(*t*) are shown in **Figure 5(a)** and **(b)**. The observer driving force *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 Γ(*t*) and *ς* ^(*t*) are adjusted to accommodate the time-varying effects that driving the observer dynamics as the closed-loop system is approaching equilibrium point. The driving force to the observer dynamics *e*(*t*) shows the same results. **Figure 5(c)** depicts that the time-varying matrix *F*(*t*) is indeed varying with time.

**Figure 5.** (a) is the values of Γ(*t*). (b) demonstrates the least-squares estimated results of *ς* ^(*t*). (c) is the driving force of the observer dynamics e(t). (d) computes the 2-norm value of time-varying matrix function *F* (*t*) with *b* = 0.
