不确定动态系统自适应控制策略详解

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本章深入探讨了不确定动态系统自适应增益鲁棒控制策略，这是在复杂工程系统中提高控制性能和鲁棒性的重要方法。作者Shunya Nagai、Hidetoshi Oya、Tsuyoshi Matsuki和Yoshikatsu Hoshi在他们的专著中，针对一类不确定线性系统的集中式自适应控制器设计进行了详尽的研究。首先，他们展示了如何通过将设计问题转化为一个有约束的凸优化问题，来实现对未知参数的自适应增益控制。这种技术允许设计者在系统存在不确定性的情况下，确定控制器的增益范围，从而确保系统的稳定性和性能。章节中详细阐述了如何计算允许的参数扰动区域，这对于实际应用中的系统稳定性和抗干扰能力至关重要。进一步，作者将集中式的控制器设计扩展到了更复杂的不确定大型互联系统。他们提出了基于矩阵不等式（Linear Matrix Inequalities, LMIs）的方法，这是一种创新的分散式（decentralized）自适应增益鲁棒控制器设计策略。这种分布式架构允许在大型系统中进行局部控制器的设计和调整，提高了响应速度和资源利用率，同时保持整体系统的稳定性。关键词包括“自适应”，“不确定动态系统”，“鲁棒控制”，“集中式控制”，“分布式控制”，“矩阵不等式”，以及“系统稳定性”。通过本章的学习，读者能够理解如何在面对系统不确定性时，利用自适应增益策略来构建稳健且高效的控制系统，这对于工业自动化、航空航天、电力系统等多个领域的工程师具有很高的实用价值。

where A

is a matrix given by A

A þ BK. Note that the standard LQ control theory for the

nominal system of Eq. (2) for designing the fixed feedback gain K ∈ R

mn

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

mn

. Therefore,

we consider the matrix inequality condition:

þ αI

ðÞ

P þ P A

þ αI

ðÞþQ < 0, (5)

where P ∈ R

nn

and Q ∈ R

nn

are a symmetric positive definite matrix and a symmetric semi-

positive definite matrix, respectively, and the matrix Q ∈ R

nn

is selected by designers. If the

symmetric positive definite matrix P ∈ R

nn

satisfying the matrix inequality of Eq. (5) exists,

then poles for the nominal closed-loop system of Eq. (4) are located into the subspace

¼ sjR

≤  α

in the complex plane. Namely, the nominal closed-loop system of Eq. (4)

is asymptotically stable, and the quadratic function V

x; tðÞ¼

tðÞPx tðÞbecomes a Lyapunov

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

quadratic function V

x; tðÞcan be expressed as

x; tðÞ< x

tðÞQ þ 2αPðÞx tðÞ

< 0, ∀

x tðÞ6¼ 0:

(6)

Now, we introduce complementary matrices Y ∈ R

nn

and W ∈ R

mm

which satisfy the rela-

tions Y¼

1

, K ¼WB

P, and W ¼ W

> 0, respectively. Then, some algebraic manipula-

tions gives

þ AY  BW

 BWB

þ 2αY þ YQY < 0: (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

þ AY  BW

 BWB

þ 2αYY

⋆ Q

1

< 0: (8)

Thus, the control gain matrix K ∈ R

mn

is determined as K ¼WB

P ¼WB

1

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

utðÞ¼

1 þ θ x; tðÞðÞKx tðÞ, (9)

where θ x ; tðÞ: R

 R ! R is an adjustable time-varying parameter [32] which plays the

important role for correcting the effect of uncertainties, that is, the control input utðÞ∈ R

Adaptive Robust Control Systems190

consists of a fixed gain matrix K ∈ R

mn

and θ x; tðÞ∈ R. 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

x tðÞ¼A

x tðÞþΔ tðÞx tðÞþθ x; tðÞBKx tðÞ: (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 θ x; tðÞ∈ 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 Δ tðÞ∈ R

nn

2.2. Synthesis of centralized adaptive gain robust state feedback controllers

In this subsection, we deal with design problems for the adjustable time-varying parameter

θ x; tðÞ∈ 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 θ x; tðÞ∈ R.

For a given design parameter ϑ > 0 and the known upper bound δ

⋆

for the unknown parameter

Δ tðÞ∈ R

nn

, if the scalar parameter γ > 0 exists satisfying

P þ PA

þ γP

⋆ 

⋆

< 0, (11)

the adjustable time-varying parameter θ x; tðÞ∈ R is determined as

θ x; tðÞ¼

ﬃﬃﬃﬃﬃ

⋆

Px tðÞ

x tðÞ

1=2

Px tðÞ



if x

tðÞPBWB

Px tðÞ≥ ϑx

tðÞx tðÞ,

ﬃﬃﬃﬃﬃ

⋆

Px tðÞ

x tðÞ

ϑx

tðÞx tðÞ

if x

tðÞPBWB

Px tðÞ< ϑx

tðÞx tðÞ:

(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

nn

which satisfies the standard Riccati equation of Eq. (4), we introduce the quadratic function

V x; tðÞ¼

tðÞPx tðÞ, (13)

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

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

191

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