2. Synthesis of centralized adaptive gain robust controllers

A centralized adaptive gain robust state feedback control scheme for a class of uncertain linear systems is proposed in this section. The adaptive gain robust controller under consideration is composed of a state feedback with a fixed gain matrix and a time-varying adjustable parameter. In this section, we show an LMI-based design method of the adaptive gain robust state feedback controller, and the allowable perturbation region of unknown parameters is discussed.

#### 2.1. Problem statement

Consider the uncertain linear system described by the following state-space representation:

$$\frac{d}{dt}\mathbf{x}(t) = (A + \Delta(t))\mathbf{x}(t) + Bu(t),\tag{1}$$

where xð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> and u tð Þ<sup>∈</sup> <sup>R</sup><sup>m</sup> are the vectors of the state (assumed to be available for feedback) and the control input, respectively. In Eq. (1) the constant matrices A and B mean the nominal values of the system, and ð Þ <sup>A</sup>; <sup>B</sup> is stabilizable pair. Moreover, the matrix <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> represents unknown time-varying parameters which satisfy <sup>Δ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>Δ</sup>ð Þ<sup>t</sup> <sup>≤</sup> <sup>δ</sup><sup>⋆</sup>In , and the elements of <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> are Lebesgue measurable [1, 34]. Namely, the unknown time-varying matrix <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is bounded, and the parameter <sup>δ</sup><sup>⋆</sup> denotes the upper bound of the perturbation region for the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> . Additionally, we suppose that the nominal system which can be obtained by ignoring the unknown parameter Δð Þt in Eq. (1) is given by

$$\frac{d}{dt}\overline{\mathbf{x}}(t) = A\overline{\mathbf{x}}(t) + B\overline{u}(t). \tag{2}$$

In Eq. (2), <sup>x</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> and u tð Þ<sup>∈</sup> <sup>R</sup><sup>m</sup> are the vectors of the state and the control input for the nominal system, respectively.

First of all, we design the state feedback control for the nominal system of Eq. (2) so as to generate the desirable transient behavior in time response for the uncertain linear system of Eq. (1). Namely, the nominal control input is given as

$$
\overline{u}(t) = \mathbb{K}\overline{\mathbf{x}}(t),\tag{3}
$$

and thus the following nominal closed-loop system is obtained:

$$\frac{d}{dt}\overline{\mathbf{x}}(t) = A\_k \overline{\mathbf{x}}(t),\tag{4}$$

where AK is a matrix given by AK¼ <sup>Δ</sup> <sup>A</sup> <sup>þ</sup> BK. Note that the standard LQ control theory for the nominal system of Eq. (2) for designing the fixed feedback gain K ∈ R<sup>m</sup>�<sup>n</sup> is adopted in the existing result [32]. In this section, for the nominal system of Eq. (2), we derive a state feedback controller with pole placement constraints [8]. Note that for simplicity the sector constraints are introduced only in this chapter, and of course, one can adopt some other design constraints or another controller design approach for designing the fixed gain matrix K ∈ R<sup>m</sup>�n. Therefore, we consider the matrix inequality condition:

$$\left(\left(A\_{\mathcal{K}} + aI\_n\right)^T \mathcal{P} + \mathcal{P}(A\_{\mathcal{K}} + aI\_n) + \mathcal{Q} < 0\right) \tag{5}$$

consists of a fixed gain matrix <sup>K</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup>�<sup>n</sup> and <sup>θ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>∈</sup> <sup>R</sup>. Note that, the robust control input of the form of Eq. (9) is called "adaptive gain robust control" in this chapter. Thus, from Eqs. (1)

From the above, the control objective in this section is to design the adaptive gain robust control which achieves satisfactory transient behavior. Namely, the control problem is to derive the adjustable time-varying parameter θð Þ x; t ∈ R such that the closed-loop system of Eq. (10) can achieve the desired transient response. In addition, we evaluate the allowable

In this subsection, we deal with design problems for the adjustable time-varying parameter θð Þ x; t ∈ R so that the satisfactory transient response for the uncertain linear system of Eq. (1) can be achieved. For the proposed adaptive gain robust control, the following theorem gives

Theorem 1: Consider the uncertain linear system of Eq. (1) and the adaptive gain robust control of

For a given design parameter ϑ > 0 and the known upper bound δ<sup>⋆</sup> for the unknown parameter

<sup>⋆</sup> � <sup>γ</sup>

Proof. In order to prove Theorem 1, by using symmetric positive definite matrix P ∈ R<sup>n</sup>�<sup>n</sup> which satisfies the standard Riccati equation of Eq. (4), we introduce the quadratic function

<sup>δ</sup><sup>⋆</sup> In

<sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> if x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWB<sup>T</sup>Pxð Þ<sup>t</sup> <sup>&</sup>lt; <sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> :

1

<sup>2</sup> if x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWB<sup>T</sup>Pxð Þ<sup>t</sup> <sup>≥</sup> <sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> ,

<sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>xð Þ<sup>t</sup> , (13)

CA <sup>&</sup>lt; <sup>0</sup>, (11)

(12)

<sup>K</sup><sup>P</sup> <sup>þ</sup> <sup>P</sup>AK <sup>þ</sup> <sup>γ</sup>P<sup>2</sup> In

dt <sup>x</sup>ðÞ¼ <sup>t</sup> AKxð Þþ <sup>t</sup> <sup>Δ</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þþ <sup>t</sup> <sup>θ</sup>ð Þ <sup>x</sup>; <sup>t</sup> BKxð Þ<sup>t</sup> : (10)

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

http://dx.doi.org/10.5772/intechopen.71733

191

.

and (9), the uncertain closed-loop system can be written as

perturbation region of the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup>

Eq. (9) with the adjustable time-varying parameter θð Þ x; t ∈ R.

AT

0 B@

the adjustable time-varying parameter θð Þ x; t ∈ R is determined as

<sup>δ</sup><sup>⋆</sup> <sup>p</sup> k k <sup>P</sup>xð Þ<sup>t</sup> k k <sup>x</sup>ð Þ<sup>t</sup>

<sup>δ</sup><sup>⋆</sup> <sup>p</sup> k k <sup>P</sup>xð Þ<sup>t</sup> k k <sup>x</sup>ð Þ<sup>t</sup>

Then, the uncertain closed-loop system of Eq. (10) is asymptotically stable.

�

<sup>V</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>Δ</sup>

ffiffiffiffiffi

8 >>>>>>><

>>>>>>>:

ffiffiffiffiffi

W<sup>1</sup>=<sup>2</sup> <sup>B</sup><sup>T</sup>Pxð Þ<sup>t</sup> � � �

<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup>, if the scalar parameter <sup>γ</sup> <sup>&</sup>gt; <sup>0</sup> exists satisfying

an LMI-based design synthesis.

θð Þ¼ x; t

2.2. Synthesis of centralized adaptive gain robust state feedback controllers

d

where P ∈ R<sup>n</sup>�<sup>n</sup> and Q ∈ R<sup>n</sup>�<sup>n</sup> are a symmetric positive definite matrix and a symmetric semipositive definite matrix, respectively, and the matrix Q ∈ R<sup>n</sup>�<sup>n</sup> is selected by designers. If the symmetric positive definite matrix P ∈ R<sup>n</sup>�<sup>n</sup> satisfying the matrix inequality of Eq. (5) exists, then poles for the nominal closed-loop system of Eq. (4) are located into the subspace S<sup>α</sup> ¼ f g sjRef gs ≤ � α in the complex plane. Namely, the nominal closed-loop system of Eq. (4) is asymptotically stable, and the quadratic function <sup>V</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>Δ</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>xð Þ<sup>t</sup> becomes a Lyapunov function for the nominal closed-loop system of Eq. (4), because the time derivative of the quadratic function Vð Þ x; t can be expressed as

$$\begin{split} \frac{d}{dt} \mathcal{V}(\overline{\mathbf{x}}, t) &< -\overline{\mathbf{x}}^T(t)(\mathcal{Q} + 2\alpha \mathcal{P})\overline{\mathbf{x}}(t) \\ &< 0, \quad \forall \, \overline{\mathbf{x}}(t) \neq 0. \end{split} \tag{6}$$

Now, we introduce complementary matrices Y ∈ R<sup>n</sup>�<sup>n</sup> and W ∈ R<sup>m</sup>�<sup>m</sup> which satisfy the relations Y¼ <sup>Δ</sup> P�<sup>1</sup> , <sup>K</sup> ¼ �WBTP, and <sup>W</sup> <sup>¼</sup> <sup>W</sup><sup>T</sup> <sup>&</sup>gt; 0, respectively. Then, some algebraic manipulations gives

$$
\lambda \mathcal{J} \mathcal{A}^T + A \mathcal{J} - B \mathcal{W}^T \mathcal{B}^T - B \mathcal{W} \mathcal{B}^T + 2a \mathcal{J} + \mathcal{J} \mathcal{Q} \mathcal{J} < 0. \tag{7}
$$

Additionally, applying Lemma 1.2 (Schur complement) to Eq. (7), one can easily see that the matrix inequality condition of Eq. (7) is equivalent to

$$
\begin{pmatrix}
\mathcal{I}\mathcal{A}^T + A\mathcal{Y} - B\mathcal{W}^T\mathcal{B}^T - B\mathcal{W}\mathcal{B}^T + 2\alpha\mathcal{Y} & \mathcal{Y} \\
\star & -\mathcal{Q}^{-1}
\end{pmatrix} < 0. \tag{8}
$$

Thus, the control gain matrix <sup>K</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup>�<sup>n</sup> is determined as <sup>K</sup> ¼ �WB<sup>T</sup><sup>P</sup> ¼ �WBTY�<sup>1</sup> .

Now, for the uncertain linear system of Eq. (1), we define the following control input [37]:

$$
\mu(t) \stackrel{\Delta}{=} (1 + \Theta(\mathbf{x}, t)) \mathbf{K} \mathbf{x}(t), \tag{9}
$$

where <sup>θ</sup>ð Þ <sup>x</sup>; <sup>t</sup> : <sup>R</sup><sup>n</sup> � <sup>R</sup> ! <sup>R</sup> is an adjustable time-varying parameter [32] which plays the important role for correcting the effect of uncertainties, that is, the control input u tð Þ<sup>∈</sup> <sup>R</sup><sup>m</sup>

consists of a fixed gain matrix <sup>K</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup>�<sup>n</sup> and <sup>θ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>∈</sup> <sup>R</sup>. Note that, the robust control input of the form of Eq. (9) is called "adaptive gain robust control" in this chapter. Thus, from Eqs. (1) and (9), the uncertain closed-loop system can be written as

$$\frac{d}{dt}\mathbf{x}(t) = A\_k \mathbf{x}(t) + \Delta(t)\mathbf{x}(t) + \mathcal{O}(\mathbf{x}, t)\mathcal{B}\mathbf{K}\mathbf{x}(t). \tag{10}$$

From the above, the control objective in this section is to design the adaptive gain robust control which achieves satisfactory transient behavior. Namely, the control problem is to derive the adjustable time-varying parameter θð Þ x; t ∈ R such that the closed-loop system of Eq. (10) can achieve the desired transient response. In addition, we evaluate the allowable perturbation region of the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> .

#### 2.2. Synthesis of centralized adaptive gain robust state feedback controllers

where AK is a matrix given by AK¼

190 Adaptive Robust Control Systems

we consider the matrix inequality condition:

quadratic function Vð Þ x; t can be expressed as

matrix inequality condition of Eq. (7) is equivalent to

tions Y¼

tions gives

<sup>Δ</sup> P�<sup>1</sup>

is asymptotically stable, and the quadratic function <sup>V</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>Δ</sup>

d

<sup>Δ</sup> <sup>A</sup> <sup>þ</sup> BK. Note that the standard LQ control theory for the

ð Þ AK <sup>þ</sup> <sup>α</sup>In <sup>T</sup><sup>P</sup> <sup>þ</sup> <sup>P</sup>ð Þþ AK <sup>þ</sup> <sup>α</sup>In <sup>Q</sup> <sup>&</sup>lt; <sup>0</sup>, (5)

<sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>xð Þ<sup>t</sup> becomes a Lyapunov

< 0: (8)

.

(6)

nominal system of Eq. (2) for designing the fixed feedback gain K ∈ R<sup>m</sup>�<sup>n</sup> is adopted in the existing result [32]. In this section, for the nominal system of Eq. (2), we derive a state feedback controller with pole placement constraints [8]. Note that for simplicity the sector constraints are introduced only in this chapter, and of course, one can adopt some other design constraints or another controller design approach for designing the fixed gain matrix K ∈ R<sup>m</sup>�n. Therefore,

where P ∈ R<sup>n</sup>�<sup>n</sup> and Q ∈ R<sup>n</sup>�<sup>n</sup> are a symmetric positive definite matrix and a symmetric semipositive definite matrix, respectively, and the matrix Q ∈ R<sup>n</sup>�<sup>n</sup> is selected by designers. If the symmetric positive definite matrix P ∈ R<sup>n</sup>�<sup>n</sup> satisfying the matrix inequality of Eq. (5) exists, then poles for the nominal closed-loop system of Eq. (4) are located into the subspace S<sup>α</sup> ¼ f g sjRef gs ≤ � α in the complex plane. Namely, the nominal closed-loop system of Eq. (4)

function for the nominal closed-loop system of Eq. (4), because the time derivative of the

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>&</sup>lt; �x<sup>T</sup>ð Þ<sup>t</sup> ð Þ <sup>Q</sup> <sup>þ</sup> <sup>2</sup>α<sup>P</sup> <sup>x</sup>ð Þ<sup>t</sup>

Now, we introduce complementary matrices Y ∈ R<sup>n</sup>�<sup>n</sup> and W ∈ R<sup>m</sup>�<sup>m</sup> which satisfy the rela-

Additionally, applying Lemma 1.2 (Schur complement) to Eq. (7), one can easily see that the

<sup>Y</sup>AT <sup>þ</sup> <sup>A</sup><sup>Y</sup> � <sup>B</sup>WTBT � <sup>B</sup>WBT <sup>þ</sup> <sup>2</sup>αY Y

Thus, the control gain matrix <sup>K</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup>�<sup>n</sup> is determined as <sup>K</sup> ¼ �WB<sup>T</sup><sup>P</sup> ¼ �WBTY�<sup>1</sup>

!

Now, for the uncertain linear system of Eq. (1), we define the following control input [37]:

where <sup>θ</sup>ð Þ <sup>x</sup>; <sup>t</sup> : <sup>R</sup><sup>n</sup> � <sup>R</sup> ! <sup>R</sup> is an adjustable time-varying parameter [32] which plays the important role for correcting the effect of uncertainties, that is, the control input u tð Þ<sup>∈</sup> <sup>R</sup><sup>m</sup>

< 0, ∀ xð Þt 6¼ 0:

, <sup>K</sup> ¼ �WBTP, and <sup>W</sup> <sup>¼</sup> <sup>W</sup><sup>T</sup> <sup>&</sup>gt; 0, respectively. Then, some algebraic manipula-

<sup>⋆</sup> �Q�<sup>1</sup>

<sup>Y</sup>A<sup>T</sup> <sup>þ</sup> <sup>A</sup><sup>Y</sup> � <sup>B</sup>WTBT � <sup>B</sup>WB<sup>T</sup> <sup>þ</sup> <sup>2</sup>α<sup>Y</sup> <sup>þ</sup> YQY <sup>&</sup>lt; <sup>0</sup>: (7)

u tð Þ¼<sup>Δ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>θ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>K</sup>xð Þ<sup>t</sup> , (9)

In this subsection, we deal with design problems for the adjustable time-varying parameter θð Þ x; t ∈ R so that the satisfactory transient response for the uncertain linear system of Eq. (1) can be achieved. For the proposed adaptive gain robust control, the following theorem gives an LMI-based design synthesis.

Theorem 1: Consider the uncertain linear system of Eq. (1) and the adaptive gain robust control of Eq. (9) with the adjustable time-varying parameter θð Þ x; t ∈ R.

For a given design parameter ϑ > 0 and the known upper bound δ<sup>⋆</sup> for the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup>, if the scalar parameter <sup>γ</sup> <sup>&</sup>gt; <sup>0</sup> exists satisfying

$$
\begin{pmatrix} A\_K^T \mathcal{P} + \mathcal{P} A\_K + \gamma \mathcal{P}^2 & I\_n \\ \star & -\frac{\mathcal{V}}{\delta^\star} I\_n \end{pmatrix} < 0,
$$

the adjustable time-varying parameter θð Þ x; t ∈ R is determined as

$$\boldsymbol{\Theta}(\mathbf{x},t) = \begin{cases} \frac{\sqrt{\delta^{\mathbf{x}}} \left\lVert \mathcal{P} \mathbf{x}(t) \right\rVert \left\lVert \mathbf{x}(t) \right\rVert \left\lVert \mathbf{x}(t) \right\rVert}{\left\lVert \mathcal{M}^{1/2} \mathbf{B}^{T} \mathcal{P} \mathbf{x}(t) \right\rVert^{2}} & \text{if} \quad \mathbf{x}^{T}(t) \mathcal{P} \mathbf{B} \mathcal{W} \mathbf{B}^{T} \mathcal{P} \mathbf{x}(t) \geq \mathbf{\mathcal{S}} \mathbf{x}^{T}(t) \mathbf{x}(t), \\\\ \frac{\sqrt{\delta^{\mathbf{x}}} \left\lVert \mathbf{P} \mathbf{x}(t) \right\rVert \left\lVert \mathbf{x}(t) \right\rVert}{\left\lVert \mathbf{x}^{T}(t) \mathbf{x}(t) \right\rVert} & \text{if} \quad \mathbf{x}^{T}(t) \mathcal{P} \mathbf{B} \mathcal{W} \mathbf{B}^{T} \mathcal{P} \mathbf{x}(t) < \mathbf{\mathcal{S}} \mathbf{x}^{T}(t) \mathbf{x}(t). \end{cases} \tag{12}$$

Then, the uncertain closed-loop system of Eq. (10) is asymptotically stable.

Proof. In order to prove Theorem 1, by using symmetric positive definite matrix P ∈ R<sup>n</sup>�<sup>n</sup> which satisfies the standard Riccati equation of Eq. (4), we introduce the quadratic function

$$\mathcal{V}(\mathbf{x},t) \stackrel{\Delta}{=} \mathbf{x}^T(t)\mathcal{P}\mathbf{x}(t),\tag{13}$$

as a Lyapunov function candidate. Let xð Þt be the solution of the uncertain closed-loop system of Eq. (10) for t ≥ t0, and then the time derivative of the quadratic function Vð Þ x; t along the trajectory of the uncertain closed-loop system of Eq. (10) can be written as

$$\begin{split} \frac{d}{dt} \mathcal{V}(\mathbf{x}, t) &= \mathbf{x}^T(t) \{ A\_K^T \mathcal{P} + \mathcal{P} A\_K \} \mathbf{x}(t) \\\\ + 2\mathbf{x}^T(t) \mathcal{P} \Delta(t) \mathbf{x}(t) + 2\boldsymbol{\theta}(\mathbf{x}, t) \mathbf{x}^T(t) \mathcal{P} \mathbf{B} \mathbf{K} \mathbf{x}(t). \end{split} \tag{14}$$

we see from Eq. (14) that some algebraic manipulations give

<sup>K</sup>P þ PAK

<sup>þ</sup> <sup>2</sup>θð Þ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BKxð Þ<sup>t</sup>

AT

then the following relation for the quadratic function Vð Þ x; t holds:

d

<sup>x</sup>ð Þþ <sup>t</sup> <sup>γ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup><sup>2</sup>

<sup>K</sup><sup>P</sup> <sup>þ</sup> <sup>P</sup>AK <sup>þ</sup> <sup>γ</sup>P<sup>2</sup> <sup>þ</sup>

xð Þþ t 1 γ

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

δ⋆ γ In <sup>x</sup>ð Þþ <sup>t</sup> <sup>2</sup>θð Þ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BKxð Þ<sup>t</sup>

> δ⋆ γ

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>&</sup>lt; <sup>0</sup>, <sup>∀</sup>xð Þ<sup>t</sup> 6¼ <sup>0</sup>: (23)

> 0, (24)

<sup>K</sup><sup>P</sup> <sup>þ</sup> <sup>P</sup>AK <sup>þ</sup> <sup>γ</sup>P<sup>2</sup> <sup>þ</sup> In <sup>&</sup>lt; <sup>0</sup>, (25)

Let us consider the last term of the right-hand side of Eq. (21). We see from Eq. (12) and the relation <sup>K</sup> ¼ �WBT<sup>P</sup> that the last term of the right-hand side of Eq. (21) is nonpositive. Thus, if

Furthermore, applying Lemma 1.2 (Schur complement) to Eq. (22), we find that the matrix inequality condition of Eq. (22) can be transformed into the LMI of Eq. (11). Namely, the quadratic function Vð Þ x; t of Eq. (13) becomes a Lyapunov function of the uncertain closedloop system of Eq. (10) with the adjustable time-varying parameter of Eq. (12), that is, asymptotical stability of the uncertain closed-loop system of Eq. (10) is ensured. It follows that

From the above, we show an LMI-based design strategy for the proposed adaptive gain robust control. Namely, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Note that the LMI of Eq. (11) defines a convex solution set of γ, and therefore one can easily see that various efficient convex optimization algorithms can be used to test whether the LMI is solvable and to generate particular solution. Furthermore, the LMI of Eq. (11) can also be exploited to design the proposed adaptive gain robust controller with some additional requirements. Thus, in this paper, we consider the allowable region of the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> and introduce the additional con-

> <sup>γ</sup> � <sup>1</sup> ε

(Schur complement), we find that the LMI of Eq. (11) is equivalent to

AT

where ε is a positive constant. From the relation of Eq. (24), we find that the minimization of the parameter ε means the maximization of the upper bound δ<sup>⋆</sup>. Then, by using Lemma 2

<sup>K</sup><sup>P</sup> <sup>þ</sup> <sup>P</sup>AK <sup>þ</sup> <sup>γ</sup>P<sup>2</sup> <sup>þ</sup>

<sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>Δ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>Δ</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup>

http://dx.doi.org/10.5772/intechopen.71733

In < 0, (22)

(21)

193

d

where γ is a positive constant.

the scalar parameter γ exists satisfying

the result of this theorem is true.

straints <sup>γ</sup> <sup>¼</sup> <sup>δ</sup><sup>⋆</sup> and

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup>

<sup>≤</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup>

Firstly, the case of x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWBTPxð Þ<sup>t</sup> <sup>≥</sup> <sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> is considered. In this case, one can see from the relation k k <sup>Δ</sup>ð Þ<sup>t</sup> <sup>≤</sup> ffiffiffiffiffi <sup>δ</sup><sup>⋆</sup> <sup>p</sup> , Eq. (14), and Lemma 1.1 that the following inequality holds:

$$\begin{split} \frac{d}{dt} \mathcal{V}(\mathbf{x}, t) &\leq \mathbf{x}^{T}(t) \Big( A\_{K}^{T} \mathcal{P} + \mathcal{P} A\_{K} \Big) \mathbf{x}(t) + 2\sqrt{\delta^{\star}} \| \mathcal{P} \mathbf{x}(t) \| \| \mathbf{x}(t) \| \\\\ &+ 2\theta(\mathbf{x}, t) \mathbf{x}^{T}(t) \mathcal{P} \mathbf{B} \mathbf{K} \mathbf{x}(t). \end{split} \tag{15}$$

Moreover, since the relation <sup>K</sup> ¼ �WB<sup>T</sup><sup>P</sup> holds, the inequality of Eq. (15) can be rewritten as

$$\begin{split} \frac{d}{dt} \mathcal{V}(\mathbf{x}, t) &\leq \mathbf{x}^{\top}(t) \Big( A\_{K}^{\top} \mathcal{P} + \mathcal{P} A\_{K} \Big) \mathbf{x}(t) + 2\sqrt{\delta^{\star}} \| \mathcal{P} \mathbf{x}(t) \| \| \mathbf{x}(t) \| \\\\ &- 2\theta(\mathbf{x}, t) \mathbf{x}^{\top}(t) \mathcal{P} \mathbf{B} \mathcal{W} \mathbf{B}^{\top} \mathcal{P} \mathbf{x}(t). \end{split} \tag{16}$$

Substituting the adjustable time-varying parameter θð Þ x; t of Eq. (12) into Eq. (16) gives

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) \le \mathbf{x}^T(t) \left(A\_K^T \mathcal{P} + \mathcal{P}A\_K\right) \mathbf{x}(t) + 2\sqrt{\delta^\star} \left\|\mathcal{P}\mathbf{x}(t)\right\| \left\|\mathbf{x}(t)\right\|$$

$$-2\mathbf{x}^T(t)\mathcal{P}\left(\frac{\sqrt{\delta^\star} \left\|\left|\mathcal{P}\mathbf{x}(t)\right| \left\|\left|\mathbf{x}(t)\right|\right|}{\left\|\left|\mathcal{W}^{1/2}B^T\mathbf{x}(t)\right|\right\|^2}\right) \mathcal{B}\mathcal{V}\mathcal{B}^T\mathcal{P}\mathbf{x}(t)\right.\tag{17}$$

$$\le \mathbf{x}^T(t) \left(A\_K^T \mathcal{P} + \mathcal{P}A\_K\right) \mathbf{x}(t).$$

If the solution of the LMI of Eq. (11) exists, then the inequality

$$A\_K^T \mathcal{P} + \mathcal{P} A\_K < 0 \tag{18}$$

is satisfied. Thus, one can see that the following relation holds:

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) < 0, \quad \forall \mathbf{x}(t) \neq 0. \tag{19}$$

Next, we consider the case of x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWBTPxð Þ<sup>t</sup> <sup>&</sup>lt; <sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> . By using the well-known inequality for any vectors α and β with appropriate dimensions and a positive scalar ζ

$$
\zeta \, 2\alpha^T \beta \le \zeta \alpha^T \alpha + \frac{1}{\zeta} \beta^T \beta,\tag{20}
$$

we see from Eq. (14) that some algebraic manipulations give

$$\begin{split} \frac{d}{dt} \mathcal{V}(\mathbf{x}, t) \leq & \mathbf{x}^T(t) \big( A\_K^T \mathcal{P} + \mathcal{P} A\_K \big) \mathbf{x}(t) + \gamma \mathbf{x}^T(t) \mathcal{P}^2 \mathbf{x}(t) + \frac{1}{\mathcal{V}} \mathbf{x}^T(t) \Delta^T(t) \Delta(t) \mathbf{x}(t) \\ \qquad + 2 \Theta(\mathbf{x}, t) \mathbf{x}^T(t) \mathcal{P} \mathbf{B} \mathbf{K} \mathbf{x}(t) \\ \leq & \mathbf{x}^T(t) \Big( A\_K^T \mathcal{P} + \mathcal{P} A\_K + \gamma \mathcal{P}^2 + \frac{\delta^\star}{\mathcal{V}} I\_n \Big) \mathbf{x}(t) + 2 \theta(\mathbf{x}, t) \mathbf{x}^T(t) \mathcal{P} \mathbf{B} \mathbf{K} \mathbf{x}(t) \end{split} \tag{21}$$

where γ is a positive constant.

as a Lyapunov function candidate. Let xð Þt be the solution of the uncertain closed-loop system of Eq. (10) for t ≥ t0, and then the time derivative of the quadratic function Vð Þ x; t along the

<sup>þ</sup>2x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>Δð Þ<sup>t</sup> <sup>x</sup>ð Þþ <sup>t</sup> <sup>2</sup>θð Þ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BKxð Þ<sup>t</sup> :

Firstly, the case of x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWBTPxð Þ<sup>t</sup> <sup>≥</sup> <sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> is considered. In this case, one can see from

Moreover, since the relation <sup>K</sup> ¼ �WB<sup>T</sup><sup>P</sup> holds, the inequality of Eq. (15) can be rewritten as

<sup>K</sup>P þ PAK � �xð Þþ <sup>t</sup> <sup>2</sup>

<sup>K</sup>P þ PAK � �xð Þþ <sup>t</sup> <sup>2</sup>

Substituting the adjustable time-varying parameter θð Þ x; t of Eq. (12) into Eq. (16) gives

� 2

Next, we consider the case of x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWBTPxð Þ<sup>t</sup> <sup>&</sup>lt; <sup>ϑ</sup>x<sup>T</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þ<sup>t</sup> . By using the well-known

1 ζ

inequality for any vectors α and β with appropriate dimensions and a positive scalar ζ

<sup>2</sup>α<sup>T</sup><sup>β</sup> <sup>≤</sup> ζα<sup>T</sup><sup>α</sup> <sup>þ</sup>

<sup>K</sup>P þ PAK � �xð Þþ <sup>t</sup> <sup>2</sup>

<sup>δ</sup><sup>⋆</sup> <sup>p</sup> k k <sup>P</sup>xð Þ<sup>t</sup> k k <sup>x</sup>ð Þ<sup>t</sup>

AT

!

<sup>K</sup>P þ PAK � �xð Þ<sup>t</sup>

<sup>δ</sup><sup>⋆</sup> <sup>p</sup> , Eq. (14), and Lemma 1.1 that the following inequality holds:

ffiffiffiffiffi <sup>δ</sup><sup>⋆</sup> <sup>p</sup>

ffiffiffiffiffi <sup>δ</sup><sup>⋆</sup> <sup>p</sup>

ffiffiffiffiffi <sup>δ</sup><sup>⋆</sup> <sup>p</sup>

<sup>B</sup>WBTPxð Þ<sup>t</sup>

k k Pxð Þt k k xð Þt

k k Pxð Þt k k xð Þt

k k Pxð Þt k k xð Þt

<sup>K</sup>P þ PAK < 0 (18)

β<sup>T</sup>β, (20)

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>&</sup>lt; <sup>0</sup>, <sup>∀</sup>xð Þ<sup>t</sup> 6¼ <sup>0</sup>: (19)

(14)

(15)

(16)

(17)

trajectory of the uncertain closed-loop system of Eq. (10) can be written as

dt <sup>V</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup>

d

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> AT

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> AT

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> AT

�2θð Þ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BWBTPxð Þ<sup>t</sup> :

ffiffiffiffiffi

<sup>K</sup>P þ PAK � �xð Þ<sup>t</sup> :

If the solution of the LMI of Eq. (11) exists, then the inequality

is satisfied. Thus, one can see that the following relation holds:

d

W<sup>1</sup>=<sup>2</sup> BTPxð Þ<sup>t</sup> � � �

<sup>þ</sup>2θð Þ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>BKxð Þ<sup>t</sup> :

the relation k k <sup>Δ</sup>ð Þ<sup>t</sup> <sup>≤</sup> ffiffiffiffiffi

192 Adaptive Robust Control Systems

d

d

d

�2x<sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>

<sup>≤</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup>

Let us consider the last term of the right-hand side of Eq. (21). We see from Eq. (12) and the relation <sup>K</sup> ¼ �WBT<sup>P</sup> that the last term of the right-hand side of Eq. (21) is nonpositive. Thus, if the scalar parameter γ exists satisfying

$$A\_K^T \mathcal{P} + \mathcal{P} A\_K + \gamma \mathcal{P}^2 + \frac{\delta^\star}{\mathcal{Y}} I\_n < 0,\tag{22}$$

then the following relation for the quadratic function Vð Þ x; t holds:

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) < 0, \quad \forall \mathbf{x}(t) \neq 0. \tag{23}$$

Furthermore, applying Lemma 1.2 (Schur complement) to Eq. (22), we find that the matrix inequality condition of Eq. (22) can be transformed into the LMI of Eq. (11). Namely, the quadratic function Vð Þ x; t of Eq. (13) becomes a Lyapunov function of the uncertain closedloop system of Eq. (10) with the adjustable time-varying parameter of Eq. (12), that is, asymptotical stability of the uncertain closed-loop system of Eq. (10) is ensured. It follows that the result of this theorem is true.

From the above, we show an LMI-based design strategy for the proposed adaptive gain robust control. Namely, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Note that the LMI of Eq. (11) defines a convex solution set of γ, and therefore one can easily see that various efficient convex optimization algorithms can be used to test whether the LMI is solvable and to generate particular solution. Furthermore, the LMI of Eq. (11) can also be exploited to design the proposed adaptive gain robust controller with some additional requirements. Thus, in this paper, we consider the allowable region of the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> and introduce the additional constraints <sup>γ</sup> <sup>¼</sup> <sup>δ</sup><sup>⋆</sup> and

$$
\gamma - \frac{1}{\varepsilon} > 0,\tag{24}
$$

where ε is a positive constant. From the relation of Eq. (24), we find that the minimization of the parameter ε means the maximization of the upper bound δ<sup>⋆</sup>. Then, by using Lemma 2 (Schur complement), we find that the LMI of Eq. (11) is equivalent to

$$A\_K^T \mathcal{P} + \mathcal{P} A\_K + \mathcal{Y} \mathcal{P}^2 + I\_n < 0,\tag{25}$$

and the constraint of Eq. (24) can be transformed into

$$
\begin{pmatrix} \gamma & 1.0 \\ \star & \varepsilon \end{pmatrix} > 0. \tag{26}
$$

Remark 3: The proposed adaptive gain robust controller with the adjustable time-varying parameter has some advantages as follows: the proposed controller design approach is very simple, and by selecting the design parameter, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input (see [32]). Besides, the structure of the proposed control system is also simple compared with the existing results for robust controllers with adjustable parameters (e.g., [29, 30]). However, the online adjustment strategy for the design parameter ϑ has not been established, and this problem is one of our future

Remark 4: In this section, firstly the nominal control input is designed by adopting pole placement constraints, and the fixed gain K ∈ R<sup>m</sup>�<sup>n</sup> can be derived by using the solution of the LMI of Eq. (8). Note that the quadratic function Vð Þ x; t is a Lyapunov function for both the uncertain linear system of Eq. (1) and the nominal system of Eq. (2), that is, the Lyapunov function for the uncertain linear system of Eq. (1) and one for the nominal system of Eq. (2) have same level set. Therefore, by selecting the design parameter ϑ > 0, the proposed adaptive gain robust control system can achieve good

On the other hand, if the design problem for a state feedback control u tðÞ¼ Ksxð Þt is considered, the quadratic function <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> is replaced as <sup>V</sup>sð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>sxð Þ<sup>t</sup> where <sup>P</sup><sup>s</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is a Lyapunov matrix. Moreover, P<sup>s</sup> ∈ R<sup>n</sup>�<sup>n</sup> becomes a variable for resultant LMI conditions, and the standard techniques for

In order to demonstrate the efficiency of the proposed control strategy, we have run a simple

� �xðÞþ <sup>t</sup> <sup>Δ</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þþ <sup>t</sup>

Firstly, we design the nominal control input u tðÞ¼ Kxð Þt . By selecting the design parameters α and Q in Eq. (5) such as α ¼ 3:0 and Q ¼ 1:0 � I<sup>2</sup> and solving the LMI of Eq. (8), we obtain the

> <sup>Y</sup> <sup>¼</sup> <sup>1</sup>:<sup>0855</sup> �1:<sup>5356</sup> <sup>⋆</sup> <sup>4</sup>:5318 !,

> > :

<sup>W</sup> <sup>¼</sup> <sup>2</sup>:<sup>1708</sup> � 101

:

� �u tð Þ: (29)

(30)

0:0 1:0

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

http://dx.doi.org/10.5772/intechopen.71733

195

<sup>K</sup> <sup>¼</sup> �1:<sup>3017</sup> � 101 �9:<sup>2008</sup> � �: (31)

Consider the following linear system with unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup>

1:0 4:0 0:0 �1:0

transient performance which is close to the nominal one or avoid the excessive control input.

research subjects.

the quadratic stabilization can also be used.

d dt <sup>x</sup>ðÞ¼ <sup>t</sup>

Thus, the following fixed gain matrix can be computed:

2.3. Illustrative examples

following solution:

example.

From the above, we consider the following constrained optimization problem:

$$\underset{\gamma > 0, \varepsilon}{\text{Minimize}} \left[ \varepsilon \right] \text{ subject to (25) and (26).}\tag{27}$$

If the optimal solution of the constrained optimization problem of Eq. (27) exists, in which are denoted by γ<sup>⋆</sup> and ε<sup>⋆</sup>, the proposed adaptive gain robust controller can be done, and the allowable upper bound of the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is given by

$$
\delta^\star = \gamma^\star.\tag{28}
$$

Consequently, the following theorem for the proposed adaptive gain robust control with guaranteed allowable region of unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is developed.

Theorem 2: Consider the uncertain linear system of Eq. (1) and the adaptive gain robust control of Eq. (8) with the adjustable time-varying parameter θð Þ x; t ∈ R.

If the optimal solution γ<sup>⋆</sup> of the constrained optimization problem of Eq. (27) exists, then the adjustable time-varying parameter θð Þ x; t ∈ R is designed as Eq. (12), and asymptotical stability of the uncertain closed-loop system of Eq. (10) is ensured. Moreover, the upper bound δ<sup>⋆</sup> for the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is given by Eq. (28).

Remark 1: In this section, the uncertain linear dynamical system of Eq. (1) is considered, and the centralized adaptive gain robust controller has been proposed. Although the uncertain linear system of Eq. (1) has uncertainties in the state matrix only, the proposed adaptive gain robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. Namely, by introducing additional actuator dynamics and constituting an augmented system, unknown parameters in the input matrix are embedded in the system matrix of the augmented system [45]. As a result, the proposed controller design procedure can be applied to such case.

Remark 2: In Theorem 1, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Namely, in order to design the proposed robust control system, designers have to solve the LMI of Eq. (11). If the LMI of Eq. (11) is feasible for ∃δ<sup>⋆</sup> > 0, then one can easily see that the LMI of Eq. (11) is always satisfied for the positive scalar ∀δ� < δ<sup>⋆</sup>. Moreover, if a positive scalar γ exists satisfying the LMI of Eq. (11) for ∃δ<sup>þ</sup> > δ<sup>⋆</sup>, then the proposed adaptive gain robust controller can also be designed, and note that the adaptive gain robust controller for δ<sup>⋆</sup> > 0 coincides exactly with the one for δ<sup>þ</sup> > δ<sup>⋆</sup> > 0. Furthermore, one can see from Theorem 2 that the resultant adaptive gain robust controller derived by solving the constrained convex optimization problem of Eq. (27) is same, because the solution of LMI of Eq. (8) or one of the constrained convex optimization problem of Eq. (27) cannot be reflected the resultant controller. Note that in the general controller design strategies for the conventional fixed gain robust control, the solution of the some constraints can be applied to the resultant robust controller. This is a fascinating fact for the proposed controller design strategy.

Remark 3: The proposed adaptive gain robust controller with the adjustable time-varying parameter has some advantages as follows: the proposed controller design approach is very simple, and by selecting the design parameter, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input (see [32]). Besides, the structure of the proposed control system is also simple compared with the existing results for robust controllers with adjustable parameters (e.g., [29, 30]). However, the online adjustment strategy for the design parameter ϑ has not been established, and this problem is one of our future research subjects.

Remark 4: In this section, firstly the nominal control input is designed by adopting pole placement constraints, and the fixed gain K ∈ R<sup>m</sup>�<sup>n</sup> can be derived by using the solution of the LMI of Eq. (8). Note that the quadratic function Vð Þ x; t is a Lyapunov function for both the uncertain linear system of Eq. (1) and the nominal system of Eq. (2), that is, the Lyapunov function for the uncertain linear system of Eq. (1) and one for the nominal system of Eq. (2) have same level set. Therefore, by selecting the design parameter ϑ > 0, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input.

On the other hand, if the design problem for a state feedback control u tðÞ¼ Ksxð Þt is considered, the quadratic function <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> is replaced as <sup>V</sup>sð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>x</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>P</sup>sxð Þ<sup>t</sup> where <sup>P</sup><sup>s</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is a Lyapunov matrix. Moreover, P<sup>s</sup> ∈ R<sup>n</sup>�<sup>n</sup> becomes a variable for resultant LMI conditions, and the standard techniques for the quadratic stabilization can also be used.

#### 2.3. Illustrative examples

and the constraint of Eq. (24) can be transformed into

194 Adaptive Robust Control Systems

γ 1:0 ⋆ ε 

If the optimal solution of the constrained optimization problem of Eq. (27) exists, in which are denoted by γ<sup>⋆</sup> and ε<sup>⋆</sup>, the proposed adaptive gain robust controller can be done, and the

Consequently, the following theorem for the proposed adaptive gain robust control with

Theorem 2: Consider the uncertain linear system of Eq. (1) and the adaptive gain robust control of

If the optimal solution γ<sup>⋆</sup> of the constrained optimization problem of Eq. (27) exists, then the adjustable time-varying parameter θð Þ x; t ∈ R is designed as Eq. (12), and asymptotical stability of the uncertain closed-loop system of Eq. (10) is ensured. Moreover, the upper bound δ<sup>⋆</sup> for the unknown parameter

Remark 1: In this section, the uncertain linear dynamical system of Eq. (1) is considered, and the centralized adaptive gain robust controller has been proposed. Although the uncertain linear system of Eq. (1) has uncertainties in the state matrix only, the proposed adaptive gain robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. Namely, by introducing additional actuator dynamics and constituting an augmented system, unknown parameters in the input matrix are embedded in the system matrix of the augmented system

Remark 2: In Theorem 1, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Namely, in order to design the proposed robust control system, designers have to solve the LMI of Eq. (11). If the LMI of Eq. (11) is feasible for ∃δ<sup>⋆</sup> > 0, then one can easily see that the LMI of Eq. (11) is always satisfied for the positive scalar ∀δ� < δ<sup>⋆</sup>. Moreover, if a positive scalar γ exists satisfying the LMI of Eq. (11) for ∃δ<sup>þ</sup> > δ<sup>⋆</sup>, then the proposed adaptive gain robust controller can also be designed, and note that the adaptive gain robust controller for δ<sup>⋆</sup> > 0 coincides exactly with the one for δ<sup>þ</sup> > δ<sup>⋆</sup> > 0. Furthermore, one can see from Theorem 2 that the resultant adaptive gain robust controller derived by solving the constrained convex optimization problem of Eq. (27) is same, because the solution of LMI of Eq. (8) or one of the constrained convex optimization problem of Eq. (27) cannot be reflected the resultant controller. Note that in the general controller design strategies for the conventional fixed gain robust control, the solution of the some constraints can be applied to the resultant robust controller. This is a fascinating fact for the proposed controller design strategy.

From the above, we consider the following constrained optimization problem:

allowable upper bound of the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is given by

guaranteed allowable region of unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is developed.

[45]. As a result, the proposed controller design procedure can be applied to such case.

Minimize

Eq. (8) with the adjustable time-varying parameter θð Þ x; t ∈ R.

<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is given by Eq. (28).

> 0: (26)

<sup>γ</sup>><sup>0</sup>, <sup>ε</sup> ½ � <sup>ε</sup> subject to 25 ð Þ and 26 ð Þ: (27)

<sup>δ</sup><sup>⋆</sup> <sup>¼</sup> <sup>γ</sup><sup>⋆</sup>: (28)

In order to demonstrate the efficiency of the proposed control strategy, we have run a simple example.

Consider the following linear system with unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> :

$$
\frac{d}{dt}\mathbf{x}(t) = \begin{pmatrix} 1.0 & 4.0 \\ 0.0 & -1.0 \end{pmatrix} \mathbf{x}(t) + \boldsymbol{\Delta}(t)\mathbf{x}(t) + \begin{pmatrix} 0.0 \\ 1.0 \end{pmatrix} \boldsymbol{u}(t). \tag{29}
$$

Firstly, we design the nominal control input u tðÞ¼ Kxð Þt . By selecting the design parameters α and Q in Eq. (5) such as α ¼ 3:0 and Q ¼ 1:0 � I<sup>2</sup> and solving the LMI of Eq. (8), we obtain the following solution:

$$\begin{aligned} \mathcal{V} &= \begin{pmatrix} 1.0855 & -1.5356 \\ \star & 4.5318 \end{pmatrix} \text{\textquotedbl{}} \\\\ \mathcal{W} &= 2.1708 \times 10^1 \text{.} \end{aligned} \tag{30}$$

Thus, the following fixed gain matrix can be computed:

$$K = \begin{pmatrix} -1.3017 \times 10^1 & -9.2008 \end{pmatrix}. \tag{31}$$

Next, we solve the constrained optimization problem of Eq. (27), then the solutions

$$\begin{aligned} \gamma &= 3.1612, \\ \varepsilon &= 3.1633 \times 10^{-1}, \end{aligned} \tag{32}$$

can be derived, and therefore the allowable upper bound of unknown parameter is given as

$$
\delta^\star = 3.1612.\tag{33}
$$

J <sup>e</sup> ¼ Δ ð∞ 0 e

evaluate the transient performance.

to the nominal system.

> -0.2 0 0.2 0.4 0.6 0.8 1 1.2

State

Control input

0 0.5 1 1.5 2

Time *t*

0 0.5 1 1.5 2

Time *t*

Figure 3. Time histories of the states for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup>

and u tð Þ, (b) The time histories of V(x, t) and Vð Þ x; t .

Figure 2. Time histories of the control inputs and the Lyapunov function for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup><sup>2</sup>

Case 1) Nominal Case 2)

Case 1) Case 2) Nominal

where eð Þt is an error vector between the time response and the desired one generated by the nominal system, that is, e tð Þ¼<sup>Δ</sup> x tðÞ� x tð Þ. Namely, <sup>J</sup> <sup>e</sup> of Eq. (34) is a performance index so as to

0





State

(a) (b)

0

0.5

0 0.5 1 1.5 2

Time *t*

. (a) The time histories of u(t)

Case 1)

197

http://dx.doi.org/10.5772/intechopen.71733

Case 2) Nominal

0.2

 0.4 0.6 0.8

Lyapunov function

(a) (b)

1

1.2

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

From Figures 1–4 the proposed adaptive gain robust state feedback controller stabilizes the uncertain linear system of Eq. (29) in spite of uncertainties. Furthermore, we also find that the proposed adaptive gain robust controller achieves the good transient performance close

<sup>T</sup>ð Þ<sup>t</sup> e tð Þdt, (34)

. (a) The time histories of x1(t), (b) The time histories of x2(t).

0 0.5 1 1.5 2

Time *t*

Case 1) Case 2) Nominal

In this example, we consider the following two cases for the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> :

• Case 1) <sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>δ</sup><sup>⋆</sup> � 0:0 0:0 �7:2289 6:8530 ! � <sup>10</sup>�<sup>1</sup> . • Case 2) <sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>δ</sup><sup>⋆</sup> � sin 5ð Þ� :0 � π � t cos 5ð Þ :0 � π � t ⋆ � sin 5ð Þ :0 � π � t � �.

Note that the unknown parameter of Case 1 satisfies the matching condition [45]. In addition, for the design parameter <sup>ϑ</sup>, the numerical simulation for two cases such as <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 and <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> is run. Moreover, the initial values of the uncertain system of Eq. (29) and the nominal system are selected as x 0ð Þ¼ x 0ð Þ¼ð1:0 �2:0Þ <sup>T</sup>. The results of the simulation of this example are shown in Figures 1–4 and Table 1. In these figures, "Case 1)" and "Case 2)" represent the time histories of the state variables x1ð Þt and x2ð Þt and the control input u tð Þ and Lyapunov function Vð Þ x:t for the proposed adaptive gain robust control, and "nominal" means the desired time response and the desired control input and Lyapunov function Vð Þ x; t for the nominal system. In Table 1, J <sup>e</sup> means

Figure 1. Time histories of the states for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 . (a) The time histories of x1(t), (b) The time histories of x2(t).

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems http://dx.doi.org/10.5772/intechopen.71733 197

Next, we solve the constrained optimization problem of Eq. (27), then the solutions

γ ¼ 3:1612, <sup>ε</sup> <sup>¼</sup> <sup>3</sup>:<sup>1633</sup> � <sup>10</sup>�<sup>1</sup>

can be derived, and therefore the allowable upper bound of unknown parameter is given as

In this example, we consider the following two cases for the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup>

� <sup>10</sup>�<sup>1</sup> .

⋆ � sin 5ð Þ :0 � π � t

Note that the unknown parameter of Case 1 satisfies the matching condition [45]. In addition, for the design parameter <sup>ϑ</sup>, the numerical simulation for two cases such as <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 and <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> is run. Moreover, the initial values of the uncertain system of Eq. (29) and the

example are shown in Figures 1–4 and Table 1. In these figures, "Case 1)" and "Case 2)" represent the time histories of the state variables x1ð Þt and x2ð Þt and the control input u tð Þ and Lyapunov function Vð Þ x:t for the proposed adaptive gain robust control, and "nominal" means the desired time response and the desired control input and Lyapunov function Vð Þ x; t for the





State

(a) (b)

0

0.5

sin 5ð Þ� :0 � π � t cos 5ð Þ :0 � π � t

� �

0:0 0:0 �7:2289 6:8530

!

nominal system are selected as x 0ð Þ¼ x 0ð Þ¼ð1:0 �2:0Þ

Case 1) Nominal Case 2)

0 0.5 1 1.5 2

Time *t*

Figure 1. Time histories of the states for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102

nominal system. In Table 1, J <sup>e</sup> means

• Case 1) <sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>δ</sup><sup>⋆</sup> �

196 Adaptive Robust Control Systems

• Case 2) <sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>δ</sup><sup>⋆</sup> �


State

, (32)

<sup>T</sup>. The results of the simulation of this

0 0.5 1 1.5 2

Time *t*

. (a) The time histories of x1(t), (b) The time histories of x2(t).

Case 1) Nominal Case 2)

:

<sup>δ</sup><sup>⋆</sup> <sup>¼</sup> <sup>3</sup>:1612: (33)

.

Figure 2. Time histories of the control inputs and the Lyapunov function for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup><sup>2</sup> . (a) The time histories of u(t) and u tð Þ, (b) The time histories of V(x, t) and Vð Þ x; t .

Figure 3. Time histories of the states for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> . (a) The time histories of x1(t), (b) The time histories of x2(t).

$$\mathcal{J}\_e \stackrel{\Delta}{=} \int\_0^\infty e^T(t)e(t)dt,\tag{34}$$

where eð Þt is an error vector between the time response and the desired one generated by the nominal system, that is, e tð Þ¼<sup>Δ</sup> x tðÞ� x tð Þ. Namely, <sup>J</sup> <sup>e</sup> of Eq. (34) is a performance index so as to evaluate the transient performance.

From Figures 1–4 the proposed adaptive gain robust state feedback controller stabilizes the uncertain linear system of Eq. (29) in spite of uncertainties. Furthermore, we also find that the proposed adaptive gain robust controller achieves the good transient performance close to the nominal system.

presented. Since the proposed adaptive gain robust controller can easily be obtained by solving the constrained convex optimization problem, the proposed design approach is simple. Moreover, by selecting the design parameter, the proposed adaptive gain robust controller can achieve good transient performance and/or avoid excessive control input. Note that there are trade-offs between achieving good transient performance and avoiding

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

http://dx.doi.org/10.5772/intechopen.71733

199

The future research subject is the extension of proposed robust control scheme to such a broad class of systems as linear systems with state delays, uncertain systems with some constraints, and so on. Additionally, we will discuss the online adjustment for the design parameter ϑ and

In this section, on the basis of the result derived in Section 2, an LMI-based design method of decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems is suggested. The design problem of the decentralized adaptive gain robust controller under consideration can also be reduced to the feasibility of LMIs, and the

Consider the uncertain large-scale interconnected system composed of N subsystems described as

N

j ¼ 1 j 6¼ i

where xið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup><sup>i</sup> and uið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup><sup>i</sup> (<sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, <sup>N</sup> ) are the vectors of the state and the control input

AiiðÞ¼ t Aii þ Δiið Þt ,

AijðÞ¼ t Aij þ Δijð Þt :

values of the system, and matrices <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> show unknown parameters

these unknown parameters are Lebesgue measurable [1, 34]. For Eq. (35), the nominal

ijð Þ<sup>t</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup>

, Aij ∈ R<sup>n</sup>i�n<sup>j</sup>

ij Inj

<sup>1</sup> ð Þ<sup>t</sup> ; <sup>⋯</sup>; <sup>x</sup><sup>T</sup>

Aijð Þt xjð Þþ t Biuið Þt , (35)

<sup>N</sup> ð Þ<sup>t</sup> � �<sup>T</sup> is the state of the overall

, and Bi ∈ R<sup>n</sup>i�m<sup>i</sup> denote the nominal

, respectively. Note that the elements of

(36)

3. Synthesis of decentralized adaptive gain robust controllers

dt <sup>x</sup>iðÞ¼ <sup>t</sup> Aiið Þ<sup>t</sup> <sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup>

system. The matrices Aiið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and Aijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> in Eq. (35) are given by

ii Ini and Δ<sup>T</sup>

subsystem, ignoring the unknown parameters, is given by

excessive control input.

3.1. Problem statement

which satisfy Δ<sup>T</sup>

the design problem for output feedback control systems.

allowable perturbation region of uncertainties is also discussed.

d

for the <sup>i</sup>th subsystem, respectively, and xðÞ¼ <sup>t</sup> <sup>x</sup><sup>T</sup>

In Eqs. (35) and (36), the matrices Aii ∈ R<sup>n</sup>i�n<sup>i</sup>

iið Þ<sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup>

Figure 4. Time histories of the control inputs and the Lyapunov function for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> . (a) The time histories of u(t) and u tð Þ, (b) The time histories of V(x, t) and Vð Þ x; t .


Table 1. The performance index Ie.

For Case 1 in this example, one can see from Table 1 that the adaptive gain robust controller for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> is more desirable comparing with one for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 , that is, the error between the time response and the desired one generated by the nominal system ("nominal" in figures) is small. But for the result of Case 2), we find that the robust controller with the parameter <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 achieves more desirable performance. Additionally, one can see from Figures 2(a) and 4(a) that by selecting the design parameter ϑ the proposed adaptive gain robust controller can adjust the magnitude of the control input. In this example, the magnitude of the control input for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup><sup>2</sup> is suppressed comparing with one for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> . However, the online adjustment way of the design parameter ϑ for the purpose of improving transient behavior and avoiding excessive control input cannot to developed, and thus it is an important problem of our research subjects.

Therefore, the effectiveness of the proposed adaptive gain robust controller is shown.

#### 2.4. Summary

In this section, an LMI-based design scheme of the centralized adaptive gain robust state feedback controller for a class of uncertain linear systems has been proposed, and by simple numerical simulations, the effectiveness of the proposed robust control strategy has been presented. Since the proposed adaptive gain robust controller can easily be obtained by solving the constrained convex optimization problem, the proposed design approach is simple. Moreover, by selecting the design parameter, the proposed adaptive gain robust controller can achieve good transient performance and/or avoid excessive control input. Note that there are trade-offs between achieving good transient performance and avoiding excessive control input.

The future research subject is the extension of proposed robust control scheme to such a broad class of systems as linear systems with state delays, uncertain systems with some constraints, and so on. Additionally, we will discuss the online adjustment for the design parameter ϑ and the design problem for output feedback control systems.
