Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

Intl J Robust & Nonlinear

2018

Summary In this paper, several equivalent stability conditions for switched linear systems with dwell time are presented. Both continuous‐time and discrete‐time cases are considered. For the continuous‐time case, the conditions that are convex in system matrices are presented in terms of infinite‐dimensional linear matrix inequalities (LMIs), which are not numerically testable. Then, by adopting the sum of square (SOS) and piecewise linear approach, computable conditions are formulated in terms of SOS program and LMIs. Compared to the literature, less conservative results can be obtained through solving these conditions for the same polynomial degree or discretized order. For the discrete‐time case, the stability conditions, which are convex in system matrices, are numerically testable. The convexity comes at the price of increment of computational complexity. Furthermore, by adopting the convexification approach, sufficient stability conditions of switched linear systems with polytopic uncertainties are derived, both for continuous‐time and discrete‐time cases. At last, several examples are given to demonstrate the correctness and advantages of our results.

“…Definition (See the work of Liu and Zhao) Let

x^{false{m false}} \in R^{d false(n, m false)}

be a vector, which contains all monomials of degree m in

x \in R^{n}

. The matrix

A_{i m} \in R^{d false(n, m false) \times d false(n, m false)}

i \in scriptI

, which satisfies,

\forall x \in R^{n}

, and

\forall τ \in N^{+}

{((), A_{i}^{τ} x)}^{false{m false}} {= 𝒜}_{i m}^{τ} x^{false{m false}},

is called extended matrix of

A_{i} \in R^{n \times n}

with respect to x { m } and can be calculated by

𝒜_{i m} = {((), K_{0}^{T} K_{0})}^{- 1} K_{0}^{T} A_{i}^{\otimes m} K_{0},

where K 0 satisfies that x ⊗ m = K 0 x { m } .…”

Section: Nominal Stability For Discrete‐time Switched Linear Systemsmentioning

confidence: 99%

“…Given scalars

m, τ^{*} \in N^{+}

, if det( A i ) ≠ 0 holds,

\forall i \in scriptI

, then the following statements are equivalent. (a) (See the work of Liu and Zhao) There exist N matrices

P_{i} \in R^{d false(n, m false) \times d false(n, m false)}

, P i > 0,

\frac{N false(N + 1 false)}{2}

vectors

α_{i}, α_{i j} \in R^{d_{p a r} false(n, m false)}

such that,

\forall false(i \times j false) \in false(scriptI \times scriptI false)

, i ≠ j , the following conditions hold:

{((), 𝒜_{i m}^{τ^{*}})}^{T} P_{j} 𝒜_{i m}^{τ^{*}} - P_{i} + L_{m} false(α_{i j} false) < 0,

𝒜_{i m}^{T} P_{i} 𝒜_{i m} - P_{i} + L_{m} false(α_{i} false) < 0 .

(b) There exist N matrices

P_{i} \in R^{d false(n, m false) \times d false(n, m false)}

, P i > 0,

\frac{N false(N + 1 false)}{2}

vectors

α_{i}, α_{i j} \in R^{d_{p a r} false(n, m false)}

such that,

\forall

…”

Section: Nominal Stability For Discrete‐time Switched Linear Systemsmentioning

confidence: 99%

“…Proof of (a) ⟹ Exponential stability of system . See the proof of theorem 1 in the work of Liu and Zhao …”

Section: Nominal Stability For Discrete‐time Switched Linear Systemsmentioning

confidence: 99%

“…Example Consider the following discrete‐time switched linear system appearing in the work of Liu and Zhao:

A_{1} = [\begin{matrix} a & 0.8 \\ 0.2 & - 0.9 \end{matrix}], A_{2} = [\begin{matrix} - 0.8 & - 0.1 \\ - 0.7 & b \end{matrix}],

where a and b are known system parameters. Table records the

τ_{\min}^{*}

computed by solving the conditions in statement (b) and (d) of Theorem .…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Equivalence of several stability conditions for switched linear systems with dwell time

Intl J Robust & Nonlinear

2018

Adaptive Control & Signal

Multi‐model predictive control of converter inlet temperature in the process of acid production with flue gas

Liu,

Li,

Wang

et al. 2024

SummaryThe smelting of non‐ferrous metals produces substantial quantities of sulfur dioxide (SO)‐laden flue gas, which is seriously harmful to environment and humans. To improve the conversion ratio of SO and minimize environmental pollution, controlling converter inlet temperature during acid production has proven to be an efficient approach. However, unsteadiness of smelting procedure leads to frequent changes in the concentration of SO, which affects the catalytic conversion of SO and the production of sulfuric acid. To regulate converter inlet temperature, a proposed method of multi‐model predictive control is introduced. First, working conditions are divided and characterized according to the range of SO concentration. Then, the mathematical model is established for each working condition and the model predictive controller is designed. Finally, an effective switching mechanism is established to realize smooth switching under different working conditions and closed‐loop control of the whole system. Through simulation validation, compared with traditional single‐model predictive controllers and multi‐model PID controllers, the proposed approach demonstrates improved transient performance and steady‐state performance. Simulation outcomes clearly highlight the superiority of the proposed algorithm.

Improved Controller Design for Uncertain Positive Systems and its Extension to Uncertain Positive Switched Systems

Zhao

Asian Journal of Control

et al. 2017

Self Cite

This paper is concerned with the controller design of uncertain positive systems. First, we decompose the feedback gain matrix K m×n into m × n non-negative components and m × n non-positive components. For the non-negative components, each component contains only one positive element and the other elements are zero. Similarly, each non-positive component contains only one negative element and the other ones are zero. Then, a simple and effective controller design approach of uncertain positive systems is proposed by incorporating the decomposed feedback gain matrix into the resulting closed-loop systems and further applied to uncertain positive switched systems. It is shown that the designed controller is less conservative compared with those in the literature. Finally, a numerical example is provided to verify the validity of the proposed design.