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Reports of the Academy of Sciences of the USSR
1962. Volume 145, No. 4
CYBERNETICS AND CONTROL THEORY
S. V. EMEL’YANOV and M. A. BERMANT
ON THE PROBLEM OF CONSTRUCTING HIGH-QUALITY AUTOMATIC CONTROL SYSTEMS FOR PLANTS WITH VARYING PARAMETERS
(Presented by Academician B. N. Petrov on 14 III 1962)
In a number of cases it becomes necessary to construct automatic control systems for plants whose parameters vary within fairly wide limits. In this paper an attempt is made to solve this problem by using certain properties of automatic control systems with variable structure (¹). The possibility is considered of constructing, by means of a certain fixed piecewise-linear control law, such automatic control systems in which the quality of the transient processes, when the plant parameters vary within wide limits, changes only slightly (within admissible limits).
Let, in a domain \(G\) of the \(n\)-dimensional space \(x_1,\ldots,x_n\), the motion of a dynamical system be described by the system of differential equations
\[ \frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x}), \tag{1} \]
where \(\mathbf{x}=(x_1,\ldots,x_n)\); \(\mathbf{f}=(f_1,\ldots,f_n)\); \(f_i=x_{i+1}\) \((i=1,2,\ldots,n-1)\);
\[ f_n=\sum_{i=1}^{n}\psi_i(\mathbf{x})x_i; \]
\[ \psi_i(\mathbf{x})= \begin{cases} a_i, & \text{when } \displaystyle\sum_{j=1}^{n}c_jx_j>0,\\[6pt] b_i, & \text{when } \displaystyle\sum_{j=1}^{n}c_jx_j<0 \end{cases} \qquad (i=1,2,\ldots,n); \]
\(a_i,b_i,c_j\) are constant quantities.
Let the hyperplane \(S\), defined by the equation \(\sum_{j=1}^{n}c_jx_j=0\), divide the domain \(G\) into the domains \(G^+\left(\sum_{j=1}^{n}c_jx_j>0\right)\) and \(G^-\left(\sum_{j=1}^{n}c_jx_j<0\right)\), in which the vector function \(\mathbf{f}(\mathbf{x})\) of system (1) is bounded, and for any constant value of time \(t\), as \(S\) is approached from \(G^+\) and \(G^-\), there exist its limiting values \(\mathbf{f}^+\) and \(\mathbf{f}^-\). Suppose that, as the solution \(\mathbf{x}(t)\) approaches some domain \(U\subset S\), the vector functions \(\mathbf{f}^+\) and \(\mathbf{f}^-\) are directed toward the hyperplane \(S\) \((f_N^+(\mathbf{x})\leq 0,\)
\[ {}^* \text{ In the case } \sum_{j=1}^{n}c_jx_j=0 \]
\[ \psi_i(\mathbf{x})=a_i \quad \text{when } \sum_{j=1}^{n}c_jx_j\to +0,\qquad \psi_i(\mathbf{x})=b_i \quad \text{when } \sum_{j=1}^{n}c_jx_j\to -0. \]
\(f_N^+(0) \leqslant 0,\ f_N^- - f_N^+ > 0\), where \(f_N^+\) and \(f_N^-\) are the projections of the vectors \(\mathbf f^+\) and \(\mathbf f^-\) onto the normal to the hyperplane \(S\), directed from \(G^-\) to \(G^+\)). Then, when \(\mathbf x(t)\) enters \(U\), the so-called “sliding mode” arises, and the solution of system (1) does not depend on \(a_i\) and \(b_i\).
Indeed, in this case, as shown in the work of A. F. Filippov (2), in the domain \(U\) there exists a solution \(\mathbf x(t)\) of system (1), and the vector \(d\mathbf x/dt=\mathbf f^0(\mathbf x)\) lies in the hyperplane \(S\) and is determined by the values of the vector-functions \(\mathbf f^+\) and \(\mathbf f^-\).
From the condition that \(\mathbf f^0 \in S\), there follows the linear dependence of the components of the vector \(\mathbf f^0\)
\[ \sum_{j=1}^{n} c_j f_j^0 = 0, \tag{2} \]
where \(f_j^0\) is the \(j\)-th component of the vector \(\mathbf f^0\), whence
\[ f_n^0 = \frac{1}{c_n}\sum_{j=1}^{n-1} c_j f_j^0 . \tag{3} \]
Consequently, the solution of system (1) for \(\mathbf x(t)\in U\) coincides with the solution of the linear system of differential equations
\[ \frac{d\mathbf x}{dt}=\mathbf f^0(\mathbf x), \tag{4} \]
where \(\mathbf x=(x_1,\ldots,x_n)\); \(\mathbf f^0=(f_1^0,\ldots,f_n^0)\); \(f_j^0=x_{j+1}\) \((j=1,2,\ldots,n-1)\);
\[ f_n^0=\frac{1}{c_n}\sum_{j=1}^{n-1} c_j x_j;\quad c_j \]
are constant quantities.
It is evident that the solution of system (4) does not depend on \(a_i\) and \(b_i\). Let the parameters of the controlled plant that determine the coefficients \(a_i\) and \(b_i\) of the system of equations (1) vary in such a way that, during the transient process, \(a_i\) and \(b_i\) may be regarded as constant. We shall construct the automatic control system in such a way that, when these parameters vary within sufficiently wide limits: 1) the domain \(U\) exists, includes the origin, and the solution of the system of differential equations (4) satisfies the prescribed requirements for the quality of the control process (the settling time and the maximum dynamic error must not exceed previously assigned values); 2) for any initial conditions the solution of the system of equations (1) enters the domain \(U\), and the coordinate \(x_1\) changes sign no more than once.
Then the solution of the original system (1) will depend on the parameters of the plant only up to the moment when \(\mathbf x(t)\) enters the domain \(U\), where the solution depends only on the coefficients \(c_j\).
Consequently, the influence of changes in the plant parameters on the quality of the control process is substantially weakened. Physically, this effect can be explained by means of analogies between relay systems operating in a sliding mode and systems having an infinite gain, considered by Ya. Z. Tsypkin (3). Indeed, for \(\mathbf x(t)\in U\),
\[ \sum_{j=1}^{n} c_j x_j \]
is the input quantity of the element implementing the function \(\psi_i(\mathbf x)\), and is close to zero, whereas its output quantity \(x(t)\) in the nonstationary mode is different from zero. Consequently, in the sliding mode this element may be regarded as an amplifying link with an infinite gain.
Thus, the solution of the problem reduces to choosing a control law satisfying the requirements stated above, i.e., to completing the definition of the function \(\psi_i(\mathbf x)\).
Example. Let the following system (Fig. 1) be described by the system of differential equations
Fig. 1
\[ \frac{d x_1}{d t}=x_2, \qquad \frac{d x_2}{d t}=\psi(x_1,x_2)x_1, \tag{5} \]
where
\[ \psi(x_1,x_2)= \begin{cases} a_1, & \text{for } \Phi(x_1,x_2)>0,\\ b_1, & \text{for } \Phi(x_1,x_2)<0^*; \end{cases} \]
\[ \Phi(x_1,x_2)=x_1(c_1x_1+c_2x_2); \qquad a_1=+\omega_0^2; \qquad b_1=-\omega_0^2; \qquad \omega_0^2=\frac{K_sK_a}{T_s}; \qquad K_a, \]
\(K_s, T_s, c_1, c_2\) are constants.
Let the plant parameter \(K_a\) vary within the limits from \(K_{a\min}\) to \(K_{a\max}\to\infty\). It is required to choose such values of \(K_s/T_s\) and \(c_1/c_2\) that, when \(K_a\)
Fig. 2. \(a\)—\(\omega_0=c_1/c_2\); \(b\)—\(\omega_0'\); \(v\)—\(\omega_0''\); \(\omega_0<\omega_0'<\omega_0''\)
varies within the specified limits, the transient processes in the tracking system proceed without overshoot and the tracking time \(t_{\text{track}}\) does not exceed the prescribed value.
On the basis of an analysis of the phase portrait of the system (Fig. 2), taking into account that the application of a step-like control action \(g(t)\) is equivalent in this case to nonzero initial conditions in \(x_1\), Fig. 3 gives plots of the dependence of \(t_{\text{track}}\) on \(\omega_0\). By tracking time is meant the time over which the dynamic error \(x_1\) decreases to a value equal to \(5\%\) of \(x_{1\max}\). The dashed curve in Fig. 3 corresponds to the dependence of \(t_{\text{track}}\) on \(\omega_0\) for \(c_1/c_2=\omega_0\). The solid line represents the dependence of \(t_{\text{track}}\) on \(\omega_0\) for a constant controller setting \(c_1/c_2=\mathrm{const}\) and variation of \(\omega_0\) from \(\omega_0=c_1/c_2\) to \(\omega_0\to\infty\).
\[ {}^*\ \text{In the case } \Phi(x_1,x_2)=0 \]
\[ \psi(x_1,x_2)=a_1 \quad \text{for } \Phi(x_1,x_2)\to +0, \]
\[ \psi(x_1,x_2)=b_1 \quad \text{for } \Phi(x_1,x_2)\to -0. \]
The dependence of \(t_{\text{track}}\) on \(\omega_0\) for \(c_1/c_2=\mathrm{const}\) in Fig. 3 was obtained under the assumption that \(\omega_0\) changes due to a change in \(K_a\) within the limits from \(K_{a\min}\) to \(K_{a\max}\).
Let \(K_a\) vary within the limits from 1 to 1000. It is required to construct a system in which the transient processes occur without overshoot and \(t_{\text{track}}\leqslant 2\) sec.
Fig. 3
From the graph in Fig. 3, for the given value \(t_{\text{track}}=2\) sec, we determine the value
\[
c_1/c_2=\omega_0=K_{a\min}K_s/T_s=1.6,
\]
corresponding to the value \(K_{a\min}=1\). Hence we find the required value \(K_s/T_s=1.6\). When \(K_a\) changes from 1 to 1000, the transient process proceeds without overshoot, and \(t_{\text{track}}\) decreases (Fig. 3).
Institute
of Automation and Telemechanics
Received
12 III 1962
REFERENCES
- S. V. Emel’yanov, Avtomatika i telemekh., 20, No. 7 (1959).
- A. F. Filippov, Matem. sbornik, 51, No. 1 (1960).
- Ya. Z. Tsypkin, Tr. Vsesoyuzn. zaochn. energetich. inst., issue 6 (1956).