Abstract Generated abstract
This paper studies stability in probability for automatic control systems with variable structure whose dynamics include a random perturbation term. Using a positive definite quadratic form as a Lyapunov-type function and interpreting solutions of the discontinuous deterministic system in the sense of Filippov, it derives an upper exponential estimate for the form under explicit inequalities on the switching control coefficients and the perturbation matrix. The main theorem gives sufficient conditions for asymptotic stability in probability in the large: the magnitude of the random process must satisfy a law of large numbers, and a bound involving its expected value must be negative. The result also indicates that, for some processes such as Gaussian processes, the estimate can be averaged to bound the expectation of the quadratic form.
Full Text
Reports of the Academy of Sciences of the USSR
1968. Volume 183, No. 4
UDC 62-501.32
CYBERNETICS AND CONTROL THEORY
I. A. GANIN
SUFFICIENT CONDITIONS FOR STABILITY IN PROBABILITY OF CONTROL SYSTEMS WITH VARIABLE STRUCTURE
(Presented by Academician B. N. Petrov on 19 IV 1968)
- We consider automatic control systems with variable structure (see \((^1,{}^2)\), etc.), whose motion is described by the differential equation
\[ dx/dt = F(x,u,t)+B(x,t)\xi(t). \tag{1} \]
Here \(x\) and \(F\) are vectors in \(E_n\); \(\xi(t)\) is a random process with values in \(E_k\); \(B(x,t)\) is an \(n\times k\) matrix; \(B(0,t)\equiv 0\). The components of the vector \(F\) have the form
\[ F_i=x_{i+1}\quad (i=1,\ldots,n-2), \]
\[ F_{n-1}=x_n-\sum_{i=1}^{n-1}c_i x_i, \tag{2} \]
\[ F_n=(c_{n-1}-a_n)x_n-\sum_{i=1}^{n-1}x_i\,[c_i(c_{n-1}-a_n)-c_{i-1}+a_i]-u; \]
\(c_i\) are constant positive coefficients, \(a_i\) are piecewise-continuous bounded functions of time (object parameters).
The control function is formed as a linear combination
\[ u=\sum_{i=1}^{s}\psi_i x_i+\rho x_n,\qquad 1\le s\le n-1, \tag{3} \]
where
\[ \psi_i= \begin{cases} \omega_i, & \text{for } x_i x_n>0,\\ \lambda_i, & \text{for } x_i x_n<0; \end{cases} \]
\(\rho\) is a nonnegative constant.
The solution of the equation
\[ dx/dt=F(x,u,t) \]
will be understood in the sense of A. F. Filippov \((^3)\).
- Introduce a positive definite quadratic form with constant coefficients
\[ A(x)=\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}a_{ij}x_i x_j+a_{nn}x_n^2\quad (a_{ij}=a_{ji}). \tag{4} \]
We also denote
\[ \|B\|=\left(\sum_{i=1}^{n}\sum_{j=1}^{k}b_{ij}^2\right)^{0.5}. \]
Lemma 1. Let \(x^0(t,x_0,t_0)\) be a solution of equation (1). Suppose, furthermore, that the inequalities are valid
\[ \begin{gathered} \omega_i > \sup_t [c_{i-1}-a_i-c_i(c_{n-1}-a_n)],\\ \lambda_i < \inf_t [c_{i-1}-a_i-c_i(c_{n-1}-a_n)],\\ \|B\| \leqslant \beta \sqrt{A(x)} . \end{gathered} \tag{5} \]
Then for the form (4) the following upper estimate holds:
\[ A(x^0,t) \leqslant A(x^0,0)\exp\left\{Lt\left(\frac{\bar\mu}{L}+\frac{1}{t}\int_0^t |\xi(u)|\,du\right)\right\}, \tag{6} \]
Here \(L>0\) and \(\bar\mu\) are certain constants.
Definition. The trivial solution of equation (1) is called asymptotically stable in probability in the large if, for any \(\varepsilon>0,\delta>0\), there exists \(r\) such that
\[ P\{|x(t,x_0,t_0)|>\varepsilon\}<\delta \quad \text{for } t\geqslant t_0,\ |x_0|<r \]
and, moreover, for any \(x_0,\varepsilon>0\),
\[ P\{|x(t,x_0,t_0)|>\varepsilon\}\to 0 \quad \text{as } t\to\infty . \]
Theorem. Suppose that the conditions of the formulated Lemma 1 are satisfied. Then the trivial solution of the variable-structure system (1) is asymptotically stable in probability in the large if the process \(|\xi(t)|\) satisfies the law of large numbers and
\[ \bar\mu/L+\sup_t \mathbf{M}|\xi(t)|<0 . \tag{7} \]
Here \(\mathbf{M}\) is the expectation operator.
The proof is based on the results of (4).
For a certain class of random processes, in particular for Gaussian processes, by averaging (6) one can obtain an estimate for the expectation of the form \(A(x)\).
The author expresses his gratitude to Academician B. N. Petrov for his attention to the present work.
Moscow Aviation Institute
named after S. Ordzhonikidze
Received
17 IV 1968
REFERENCES
- S. V. Emel’yanov, V. I. Utkin, DAN, 152, No. 2 (1964).
- B. N. Petrov, S. V. Emel’yanov, M. B. Grineko, DAN, 169, No. 1 (1966).
- A. F. Filippov, Mat. sborn., 51(93), No. 1 (1960).
- R. Z. Khas’minskii, PMM, 30, issue 5 (1966).