Abstract
We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered.
Highlights
Consider a control system defined by an ordinary differential equation with time-varying coefficients of n-th order x (n) + p1 (t) x (n−1) + . . . + pn (t) x = u, (1)where x ∈ R is the state variable, u ∈ R is the control input, t ∈ R+ := [0, +∞)
We suppose that the functions pi (t) are measurable but exact values of these functions at time moments t are unknown, we know only that the functions are bounded on R+ and lower and upper bounds are known: αi ≤ pi (t) ≤ β i, t ∈ R+, i = 1, n
In this work, using Theorem 1, we prove results on exponential stabilization with any pregiven decay rate by linear stationary static state or output feedback for a control system defined by a linear time-varying differential equation of the n-th order with uncertain coefficients
Summary
(12)) are negative necessarily due to positivity of σi , ωi , i = 1, n It follows from the proof of Theorem 1 [6] that every solution x (t) of (9) along with its derivatives up to (n − 1)-th order has the form O(e−νn t ) as t → +∞, where −νn < 0 is the largest of the roots of polynomials (11), (12). In contrast to [34], which uses the Second Lyapunov Method (Method of Lyapunov Function), we apply, in some sense, the First Lyapunov Method (which uses the roots of characteristic polynomial) and non-oscillation theory We extend these stabilization results to systems with static output feedback control. In this work, using Theorem 1, we prove results on exponential stabilization with any pregiven decay rate by linear stationary static state or output feedback for a control system defined by a linear time-varying differential equation of the n-th order with uncertain coefficients
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