UDC 517.93

MATHEMATICS

Academician B. N. PETROV, R. A. NELEPIN, B. M. SHAMRIKOV

CONSTRUCTION OF SEPARATRIX SURFACES AND REGIONS OF STABILITY IN THE PHASE SPACES OF NONLINEAR SYSTEMS

1. Consider the system

\[ \dot{\eta}=A\eta+b f(\sigma), \qquad \sigma=c'\eta, \tag{1} \]

where \(\eta\) is a column of \(n\) variables \(\eta_k\); \(A=(a_{ki})\) is an \(n\times n\) matrix; \(b,c\) are columns of \(n\) constants \(b_k,c_k\) (the prime denotes transposition); \(f(\sigma)\) is a scalar function; \(\dot{}=d/dt\); \(t\) is the independent variable (time).

By means of the transformation

\[ \eta=Kx \tag{2} \]

we pass from system (1) to the system

\[ \dot{x}=\Lambda x+e f(\sigma), \qquad \sigma=\gamma'x. \tag{3} \]

Here \(x\) is a column of \(n\) variables \(x_i\); \(\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)\); \(e=(1,\ldots,1)\) is an \(n\)-dimensional column; \(\gamma'\) is a row of \(n\) quantities \(\gamma_i\);

\[ \gamma_i=-[D'(\lambda_i)]^{-1}\sum_{k=1}^{n} c_k N_k(\lambda_i); \qquad N_i(\lambda_k)=\sum_{\alpha=1}^{n} b_\alpha D_{\alpha i}(\lambda_k), \quad i=1,\ldots,n; \tag{4} \]

\(\lambda_i\) are the roots of the equation \(D(\lambda)\equiv \det[a_{ki}-\delta_{ki}\lambda]=0\); \(D'(\lambda)=dD(\lambda)/d\lambda\); \(D_{\alpha i}(\lambda_k)\) is the algebraic cofactor of the element of row \(\alpha\), column \(i\), of the determinant \(D(\lambda_k)\). The matrix \(K\) is nonsingular under the conditions indicated in \((^1)\).

Let \(a_{ki}, b_k\) be prescribed numbers, and \(c_k\) parameters. In the \(n\)-dimensional space \(C_n\) of parameters \(c_1,\ldots,c_n\), define the revealing sections \(G_2^{(s,r)}\)—planes of dimension 2—by the equations

\[ \sum_{k=1}^{n} c_k N_k(\lambda_i)=\delta_{is}A_s+\delta_{ir}A_r, \qquad i=1,\ldots,n, \tag{5} \]

where \(\delta_{ij}\) is the Kronecker symbol, \(A_s,A_r\) are real (if \(\lambda_s,\lambda_r\) are real) or complex conjugate (if \(\lambda_s,\lambda_r\) are complex conjugate), and otherwise arbitrary constants. Under the conditions of the section (5) we have \(\gamma_i=0\) \((i\ne s,r)\), and equations (3) for \(x_s,x_r,\sigma\) form an independent subsystem \(T\), from which we find \(\sigma(t)\); after this, equations (3) for \(x_i\) \((i\ne s,r)\) are integrated as linear inhomogeneous equations.

2. Consider the space \(X\) of the variables \(x_k\); since a pair of equations with complex \(x_i,x_{i+1}=\bar{x}_i \pm j\bar{x}_{i+1}\) corresponds to a pair of equations with real \(\bar{x}_i,\bar{x}_{i+1}\), by the space \(X\) we shall mean the Euclidean space of the variables \(x_1,\ldots,\bar{x}_i,\bar{x}_{i+1},\ldots,x_n\).

Under the conditions of the section \(G_2^{(s,r)}\), the phase portrait on the plane with Cartesian coordinates \(x_s,x_r\) of the subsystem \(T\) may be regarded as the projection of the phase portrait in the \(n\)-dimensional space \(X\) of the complete system (3) onto the plane \(x_s,x_r\) of this space. To a limit cycle \(\Gamma\) in the plane \(x_s,x_r\) there corresponds a separatrix surface \(S_\Gamma\) of dimension \(n-1\) in the space \(X\), representing a cylindrical surface parallel to the axes \(x_i\) \((i\ne s,r)\) and forming, in its intersection with the plane \(x_s,x_r\), a closed-

curve \(\Gamma\). Analogously, the separatrix surfaces \(S_c, S_{\mathrm{o.p}}\), generated by separatrices and rest segments of the plane \(x_s, x_r\), are defined.

Let there be an equation of a limit cycle in the plane \(x_s, x_r\)

\[ S(x_s,x_r)=0. \tag{6} \]

Then the equation of the separatrix surface \(S_\Gamma\) in the space of the variables \(\eta_1,\ldots,\eta_n\) will be

\[ S\left[(\det K)^{-1}\sum_{i=1}^{n}K_{is}\eta_i;\,(\det K)^{-1}\sum_{i=1}^{n}K_{ir}\eta_i\right]=0, \tag{7} \]

where \(K_{ij}\) is the algebraic cofactor of the element in row \(i\), column \(j\) of the determinant of the matrix \(K\).

1. Let an unstable limit cycle \(\Gamma\) bound the region of asymptotic stability of the equilibrium state of the subsystem \(T\) in the plane \(x_s,x_r\), and let \(\operatorname{Re}\lambda_i<0\) \((i\ne s,r)\); then surface (7) in the section \(G_2^{(s,r)}\) serves as the exact boundary of the domain of attraction of the equilibrium state in the \(n\)-dimensional phase space of system (1).

Construct the Lyapunov function

\[ V(x_1,\ldots,x_n)=\sum_{\substack{i=1\\ i\ne s,r}}^{n}h_i x_i^2+h_r\xi_r^2+h_s\xi_s^2, \tag{8} \]

where \(h_r\xi_r^2+h_s\xi_s^2=\mathrm{const}\) is the equation of an ellipse approximating the cycle \(\Gamma\) and determined by the method of harmonic linearization; \(\xi_s,\xi_r\) are linear functions of \(x_s,x_r\), determined by reducing the equation of the ellipse to its principal axes; \(h_i\) are positive constants.

Using Sylvester’s inequalities, we determine the values of the parameters \(c_k\) in \(G_2^{(s,r)}\) for which \(\dot V<0\) in a neighborhood of the origin.

Consider the surfaces

\[ F(x_1,\ldots,x_n)\equiv \dot V(x_1,\ldots,x_n)=0; \tag{9} \]

\[ \Phi(x_1,\ldots,x_n)\equiv \sum_{\substack{i=1\\ i\ne s,r}}^{n}h_i x_i^2+h_r\xi_r^2+h_s\xi_s^2-V_i=0, \tag{10} \]

where \(V_i\) is some number. Define the domain of attraction of the equilibrium state \(Q_i\) as the largest domain bounded by surface (10) and wholly inscribed in the domain bounded by surface (9). In particular, for a smooth function \(f(\sigma)\), the boundary of the domain \(Q_i\) is described by equation (10) for \(V_i=V_i^*\), where \(V_i^*\) is the smallest of the numbers \(V_i\) determined by solving the system

\[ \left(\frac{\partial \Phi}{\partial x_i}\right)_* \left(\frac{\partial F}{\partial x_i}\right)_*^{-1} =\mathrm{const},\qquad i=1,\ldots,n; \tag{11} \]

\[ F(x_1^*,\ldots,x_n^*)=\Phi(x_1^*,\ldots,x_n^*)=0. \]

Under the conditions of the section \(G_2^{(s,r)}\), system (11) has a solution as \(h_i\to0\) \((i\ne s,r)\) and determines the domain of attraction of the equilibrium state as the interior of an \(n\)-dimensional cylinder.

1. Consider a neighborhood of the section \(G_2^{(s,r)}\), where \(\gamma_i=\varepsilon_i\le\varepsilon\) \((i\ne s,r)\), \(\varepsilon>0\) is a small number. By an orthogonal transformation of variables

\[ x=Lz \tag{12} \]

we reduce system (3) to the form

\[ \dot z=Dz+g f(\sigma),\qquad \sigma=\beta' z. \tag{13} \]

Here \(L=(l_{ij})\), \(D=(d_{ij})\) are \(n\times n\) matrices; \(g\) is an \(n\)-dimensional column; \(\beta'=(0,\ldots,\beta_s,\ldots,\beta_r,\ldots,0)\) is an \(n\)-dimensional row; \(\beta_s\beta_r^{-1}=\gamma_s\gamma_r^{-1}\). Elements

the matrices \(L'\) satisfy the relations

\[ \sum_{j=1}^{n} l_{jk}\gamma_j = 0, \qquad k = 1,\ldots, s-1,\ s+1,\ldots,\ r-1,\ r+1,\ldots,\ n; \]

\[ \sum_{j=1}^{n} l_{ij}l_{jk}=\delta_{jk}, \qquad j,k=1,\ldots,n, \tag{14} \]

and are determined up to \(\frac{1}{2}(n^2-3n+2)\) arbitrary constants. These constants can be chosen so that the numbers \(|d_{ij}|\) \((i\ne j)\) are sufficiently small, and from the conditions \(\varepsilon_i\to 0\) \((i\ne s,r)\) it follows that \(d_{ij}\to 0\) \((i\ne j)\). By the method of harmonic linearization we determine a cycle \(\Gamma\) in the space \(Z\) of the variables \(z_1,\ldots,z_n\) and consider the ellipse generated by it in the Cartesian coordinates \(z_s,z_r\). After this we determine the function \(V\) by formula (8), in which we replace \(x_i\) by \(z_i\), and repeat the exposition of item 3. The solution of system (11) will now determine the surface bounding a certain elongated body. The choice of \(h_i\) \((i\ne s,r)\) makes it possible to construct a family of regions \(Q_i\); the desired region will be their union. Transformations (12) and (2) make it possible to recalculate this region into the space of the variables \(\eta_1,\ldots,\eta_n\) of the original system (1).

The method considered for constructing regions of stability finds application in the design of control systems for moving objects \((^{2})\).

REFERENCES

1. R. A. Nelepin, Exact Analytical Methods in the Theory of Nonlinear Automatic Systems, 1967.
2. B. N. Petrov, B. M. Shamrikov, in the book: Exact Methods for the Study of Nonlinear Automatic Control Systems, 1970 (in press).