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H$$\infty$$ Optimal Stabilization of a Class of Uncertain Impulsive Systems

Impulsive Systems are a special class of hybrid dynamical system which contains many abrupt changes of states at some certain time instances. Many examples can be found in finance, engineering, and other disciplines. This LMI addressed the problem of designing a feedback control law such that the uncertain closed-loop system is asymptotically stable, and the H$$\infty$$ norm bound constrains on disturbance attentuation for all admissible uncertainties are satisfied.

The System
The following system is assumed:

$$ \begin{align} \dot{x}(t) &= A x(t) + C u(t) \,\,(1)\\ \end{align} $$

When the system (1) is time-invariant and involves uncertain disturbances and proportional impulsive effects is can be rewritten as seen below in (2).

The Data
The system can then be described as:

$$

\begin{align} \dot{x}(t)   &= A x(t) + B w(t) + C u(t) \,\,\, &t\ne t_{k}\\ \ \Delta x(t) &= d_{k} x(t) &t=t_{k} \\ \ z(t) &= E x(t)\\ \ x(t) &= 0 &t=t_{0}=0 \\ \end{align}

$$

Where

$$x(t) \in R^{n} $$ is the state.

$$u(t) \in R^{m} $$ is the control input.

$$w(t) \in R^{p} $$ is the disturbance input with limited energy.

$$z(t) \in R^{n} $$ is the controlled output.

The LMI
Using the Lyapunov function, Schur Complement, the time-domain expression of the L2-norm, and Parseval's Theorem.

$$ \begin{align} &\quad \varepsilon > 0 \,\,\, \& \,\,\, P>0 \in R^{nxn} \\

&\begin{bmatrix} A^{T}P + PA + P(\gamma^{-2}BB^{T}-\varepsilon^{-2}CC^{T})P & E^{T} \\ E & -I \\ \end{bmatrix} < 0 \end{align}$$

Conclusion:
The goal of this LMI is to provide a state feedback robust H$$\infty$$ optimal control technique for impulsive dynamical systems with time varying uncertainty. Based on a positive definite solution of a linear matrix inequality, the proposed robust H$$\infty$$ static state feedback control law guarantees both internally asymptotical stability and robust H$$\infty$$ optimal performance.