Identification of Periodic Regimes in a Dynamic System

Abstract—

For a dynamic system given by first-order ordinary differential equations, the problem of identification of periodic regimes is investigated. This problem is the establishment the periodicity of an arbitrary solution via the periodicity of the observed value of solution. The conditions under which the problem of identification of periodic regimes is solvable are found. Formulated and proven theorems supplement the well-known results on the observability of dynamic systems.

APPENDIX

Let us verify the validity of the following lemma.

Lemma 1. For an arbitrary vector u ∈ ${{\mathbb{R}}^{n}}$ the identity Ce^tAu ≡ 0, t ∈ (t₁, t₂) is equivalent to

$$Cu = 0,\quad CAu = 0,\quad ...,\quad C{{A}^{{n - 1}}}u = 0.$$

(A.1)

Proof of Lemma 1. Let the identity Ce^tAu ≡ 0, t ∈ (t₁, t₂) holds. Let’s check that Ce^tAu ≡ 0, t ∈ R. To do this, it suffices to show that for any ${v}$ ∈ ${{\mathbb{R}}^{m}}$ the function φ(t) = 〈Ce^tAu, ${v}$〉 is identically equal to zero on $\mathbb{R}$.

Let’s find the derivatives of the function φ(t): φ^(k)(t) = 〈CA^ke^tAu, ${v}$〉, k = 1, 2, …. Next, we use the fact that according to the Hamilton–Cayley theorem [10, p. 93] matrix A satisfies its characteristic equation

$${{A}^{n}} + {{q}_{1}}{{A}^{{n - 1}}} + ... + {{q}_{{n - 1}}}A + {{q}_{n}}E = O,$$

where

$${{\lambda }^{n}} + {{q}_{1}}{{\lambda }^{{n - 1}}} + ... + {{q}_{{n - 1}}}\lambda + {{q}_{n}} \equiv \det (\lambda E - A).$$

From here it follows that function φ(t) satisfies the linear homogeneous differential equation

$${{y}^{{(n)}}}(t) + {{q}_{1}}{{y}^{{(n - 1)}}}(t) + ... + {{q}_{{n - 1}}}y'(t) + {{q}_{n}}y(t) = 0,\quad t \in {{R}^{n}}.$$

For this equation, only the zero solution can vanish identically on some interval. Since according to the condition φ(t) ≡ 0, t ∈ (t₁, t₂), so φ(t) ≡ 0, t ∈ R. Therefore, the identity Ce^tAu ≡ 0, t ∈ R holds. Differentiating this identity k times and setting k = 0, 1, …, n – 1, t = 0, we obtain equalities (A.1).

Conversely, if equality (A.1) holds, then from the Hamilton–Cayley theorem follows that CA^ku = 0 for any integer k ≥ 0. Hence, by the definition of the matrix exponent, we derive Ce^tAu ≡ 0, t ∈ $\mathbb{R}$. The lemma is proven.

Proof of Theorem 1. Let condition (4) holds and x(t) be a solution of the system of Eq. (2) satisfying the conditions

$$Cx(t + \omega ) = Cx(t),\quad t \in ( - \infty , + \infty ).$$

(A.2)

We solve the system of Eqs. (2) with respect to x(t), assuming that the vector-function f(t, Cx(t)) is given:

$$x(t) = {{e}^{{tA}}}\left( {x(0) + \int\limits_0^t {{{e}^{{ - sA}}}f(s,Cx(s))ds} } \right).$$

(A.3)

Given this equality, condition (A.2) takes the following form:

$$C{{e}^{{tA}}}\left( {\left( {{{e}^{{\omega A}}} - E} \right)x(0) + \int\limits_0^{t + \omega } {{{e}^{{(\omega - s)A}}}f(s,Cx(s))ds - \int\limits_0^t {{{e}^{{ - sA}}}f(s,Cx(s))ds} } } \right) = 0.$$

It is easy to verify that

$$\begin{gathered} \frac{d}{{dt}}\left( {\int\limits_0^{t + \omega } {{{e}^{{(\omega - s)A}}}} f(s,Cx(s))ds - \int\limits_0^t {{{e}^{{ - sA}}}f(s,Cx(s))ds} } \right) \\ = f(t + \omega ,Cx(t + \omega )) - f(t,Cx(t)) = 0,\quad t \in ( - \infty , + \infty ). \\ \end{gathered} $$

Consequently

$$\int\limits_0^{t + \omega } {{{e}^{{(\omega - s)A}}}f(s,Cx(s))ds - \int\limits_0^t {{{e}^{{ - sA}}}f(s,Cx(s))ds \equiv \int\limits_0^\omega {{{e}^{{(\omega - s)A}}}} f(s,Cx(s))ds,} } $$

and we obtain the equality

$$C{{e}^{{tA}}}\left( {\left( {{{e}^{{\omega A}}} - E} \right)x(0) + \int\limits_0^\omega {{{e}^{{(\omega - s)A}}}} f(s,Cx(s))ds} \right) = 0,\quad t \in ( - \infty , + \infty ).$$

From here by virtue of the Lemma we obtain:

$$B\left( {\left( {{{e}^{{\omega A}}} - E} \right)x(0) + \int\limits_0^\omega {{{e}^{{(\omega - s)A}}}f(s,Cx(s))ds} } \right) = 0.$$

(A.4)

Thus, for the solution x(t) of the system (2) from (A.2) follows (A.4) and

$$f(t + \omega ,Cx(t + \omega )) = f(t,Cx(t)),\quad t \in ( - \infty , + \infty ).$$

(A.5)

The converse is also true, if (A.4) and (A.5) hold for the solution x(t) of the system of Eqs. (2), then (A.2) holds.

Since rank(B) = n, therefore (A.4) is possible if only under the condition

$$\left( {{{e}^{{\omega A}}} - E} \right)x(0) + \int\limits_0^\omega {{{e}^{{(\omega - s)A}}}f(s,Cx(s))ds = 0.} $$

(A.6)

From (A.3) and (A.6) it follows ω-periodicity of x(t).

Theorem 1 is proven.

Proof of Theorem 2. Above it was shown that for the solution x(t) of the system of Eqs. (2), condi- tion (A.2) is equivalent to the conditions (A.4) and (A.5). Assuming f(s, Cx(s)) ≡ g(s) in these conditions, it follows that the system of Eqs. (6) has a unique solution with ω-periodic observed value Cx(t) if and only if the system of algebraic equations

$$B\left( {\left( {{{e}^{{\omega A}}} - E} \right)x(0) + \int\limits_0^\omega {{{e}^{{(\omega - s)A}}}g(s)ds} } \right) = 0$$

has a unique solution with unknown x(0) ∈ ${{\mathbb{R}}^{n}}$. But this is possible only under the condition

$${\text{rank}}\left( {B\left( {{{e}^{{\omega A}}} - E} \right)} \right) = n.$$

This condition, according to the definition and the general properties of the rank of a matrix, is equivalent to the conditions (4) and (7).

Theorem 2 is proven.

Proof of Theorem 3. Suppose the inequality (10) doesn’t hold. Then there is an infinite sequence of vector-functions z_j(t) ∈ C¹([0, 1]; ${{\mathbb{R}}^{n}}$), j = 1, 2, … such that

$$\mathop {\max }\limits_{0 \leqslant t \leqslant 1} {\text{|}}{{z}_{j}}(t){\text{|}} > j\left( {\mathop {\max }\limits_{0 \leqslant t \leqslant 1} \left| {\frac{{d{{z}_{j}}(t)}}{{dt}} - A{{z}_{j}}(t)} \right| + \mathop {\max }\limits_{0 \leqslant t \leqslant 1} {\text{|}}C{{z}_{j}}(t){\text{|}}} \right),\quad j = 1,2,....$$

Consider the vector-functions

$${{{v}}_{j}}(t) = r_{j}^{{ - 1}}{{z}_{j}}(t),\quad t \in [0,\,\,1],\quad j = 1,\,\,2,\,\,...,$$

where r_j is the maximum of the function |z_j(t)| on the interval [0, 1]. For these vector-functions we have:

$$1 = \mathop {\max }\limits_{0 \leqslant t \leqslant 1} {\text{|}}{{{v}}_{j}}(t){\text{|}} > j\left( {\mathop {\max }\limits_{0 \leqslant t \leqslant 1} {\text{|}}{v}_{j}^{'}(t) - A{{{v}}_{j}}(t){\text{|}} + \mathop {\max }\limits_{0 \leqslant t \leqslant 1} {\text{|}}C{{{v}}_{j}}(t){\text{|}}} \right),\quad j = 1,2,....$$

Passing to the limit along a uniformly convergent subsequence of the vector-functions ${{{v}}_{{{{j}_{1}}}}}$(t), ${{{v}}_{{{{j}_{2}}}}}$(t), …, as a limit we obtain the function ${v}$(t) ∈ C¹ ([0, 1]; ${{\mathbb{R}}^{n}}$) such that

$$\mathop {\max }\limits_{0 \leqslant t \leqslant 1} {\text{|}}{v}(t){\text{|}} = 1,\quad {v}{\kern 1pt} '(t) - A{v}(t) \equiv 0,\quad C{v}(t) \equiv 0.$$

From here it follows that

$${v}(t) \equiv {{e}^{{tA}}}{v}(0),\quad {v}(0) \ne 0,\quad C{{e}^{{tA}}}{v}(0) \equiv 0.$$

By virtue of the Lemma 1 from the last identity it follows that the system of Eqs. (A.1) has a non-zero solution, which contradicts the condition rank(B) = n. The inequality (10) is proven.

Let LM < 1 and x(t) be an arbitrary solution of the system of Eqs. (9). Substituting x(t + a + ω) – x(t + a) in(10) instead of z(t), we get

$$\begin{gathered} \mathop {\max }\limits_{a \leqslant t \leqslant a + 1} {\text{|}}x(t + \omega ) - x(t){\text{|}} \\ \leqslant M\left( {\mathop {\max }\limits_{a \leqslant t \leqslant a + 1} {\text{|}}G(t,x(t + \omega )) - G(t,x(t)){\text{|}} + \mathop {\max }\limits_{a \leqslant t \leqslant a + 1} {\text{|}}Cx(t + \omega ) - Cx(t){\text{|}}} \right). \\ \end{gathered} $$

Further, by using the Lipschitz condition

$$\mathop {\max }\limits_{a \leqslant t \leqslant a + 1} {\text{|}}G(t,x(t + \omega )) - G(t,x(t)){\text{|}} \leqslant \mathop {L\max }\limits_{\,\,\,\,\,\,a \leqslant t \leqslant a + 1} {\text{|}}x(t + \omega ) - x(t){\text{|}}{\text{,}}$$

we obtain the inequality (11).

Theorem 3 is proven.