Abstract Generated abstract
The paper introduces the concept of double invariance in automatic control systems, defined as the case where conditions ensuring invariance of a controlled coordinate with respect to specified external actions also make the system operator independent of plant parameters. Starting from a general system of differential equations with variable coefficients, the authors reduce the invariance problem to a single operator equation and relate double invariance to zero sensitivity to parameter variations. They show that searchless self-adjusting systems with a model possess this property when plant parameters change arbitrarily and then remain constant, deriving the corresponding conditions and demonstrating that the resulting operator no longer contains the plant parameters. The paper also notes extensions to actuator and controller coefficients, and describes a partial form of double invariance for systems using real differentiating circuits.
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CYBERNETICS AND CONTROL THEORY
Academician B. N. PETROV, V. Yu. RUTKOVSKII
DOUBLE INVARIANCE OF AUTOMATIC CONTROL SYSTEMS
Let an automatic control system be described by the equations
\[ \begin{aligned} a_{11}(D,t)x_1 + a_{12}(D,t)x_2 + \ldots + a_{1m}(D,t)x_m &= a_{f_1}(D,t)f_1,\\ a_{21}(D,t)x_1 + a_{22}(D,t)x_2 + \ldots + a_{2m}(D,t)x_m &= a_{f_2}(D,t)f_2,\\ &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots\\ a_{m1}(D,t)x_1 + a_{m2}(D,t)x_2 + \ldots + a_{mm}(D,t)x_m &= a_{fm}(D,t)f_m, \end{aligned} \tag{1} \]
where \(x_1, x_2, \ldots, x_m\) are the generalized coordinates of the system; \(f_1, f_2, \ldots, f_m\) are disturbing and control actions, some of which may consist of several identical terms, and we assume that \(f_1, f_2, \ldots, f_m\) are analytic functions bounded in modulus, holomorphic at \(t=t_0\); \(a_{ij}(D,t)\), \(a_{fi}(D,t)\), \(i,j=1,2,\ldots,m\), are second-order operators in \(D=d/dt\) with variable coefficients. Some of them contain variable parameters of the plant.
Without reducing generality, suppose that we are interested in the invariance of the coordinate \(x_1\) with respect to certain \(M\) actions among \(f_1, f_2, \ldots, f_m\). By transformations that do not violate the equivalence of system (1) and the new form of its notation \((^1)\), we reduce (1) to a single equation of the form
\[ A(D,t)x_1=\sum_{i=1}^{M} A_{f_i}(D,t)f_i, \tag{2} \]
where, in multichannel systems, \(M\le m\) (in single-channel systems \(M\le m-2\) \((^2)\)). For \(M<m\), the numbering of \(f_i\) in (2) and (1) may be different, i.e., it is not necessarily only the first \(M\) of the \(f_1, f_2, \ldots, f_m\) actions in (1) that are considered.
We shall call an automatic control system a system possessing double invariance if, for it, the conditions of invariance with respect to the actions \(f_i(t)\) are simultaneously conditions of invariance of the operator \(A(D,t)\) with respect to the plant parameters, i.e., if from the conditions of invariance of the system with respect to \(f_i(t)\) it follows, as a consequence, that the operator \(A(D,t)\) is independent of the plant parameters.
Double invariance obviously ensures zero sensitivity of the system with respect to the plant parameters. In the general case one may assert that the problem of insensitivity is the problem of invariance with respect to the system parameters.
Let us show that self-adjusting searchless systems with a model \((^3,^4)\), in the regime when the plant parameters, having changed in an arbitrary manner, then remain constant, possess double invariance. Indeed, let the equations of the system be \((^4)\):
\[ \sum_{\alpha=0}^{k} a_\alpha^{*}(t)\varphi^{(i)} = -\,b^{*}(t)\mu \quad\text{— plant;} \]
\[ \sum_{\gamma=0}^{r} c_\gamma \mu^{(\gamma)} = \sigma,\quad r+k=n, \quad\text{— actuator;} \]
\[ \sigma = k_b\left(\sum_{i=0}^{n-1} k_i \varphi^{(i)} - k_g o g\right) \quad\text{— control law;} \]
\[ \begin{gathered} k_i=\bar{k}_i+k_{iu}z_i,\\ z_i=\int_0^t\left(\varphi^{(i)}-\varphi_{\mathrm{M}}^{(i)}\right)\Phi\left(\varphi_{\mathrm{M}}^{(i)}\right)\operatorname{sign}\varphi^{(i)}\,dt,\\ \Phi\left(\varphi_{\mathrm{M}}^{(i)}\right)= \begin{cases} 1, & \text{for } |\varphi_{\mathrm{M}}^{(i)}|>\Delta_i,\\ 0, & \text{for } |\varphi_{\mathrm{M}}^{(i)}|\leq \Delta_i, \end{cases}\\ \sum_{\xi=0}^{n} d_\xi \varphi_{\mathrm{M}}^{(\xi)}=g \end{gathered} \qquad \left. \begin{array}{l} \\ \\ \\ \\ \\ \end{array} \right\} \text{— law of variation of the adjustable coefficients,} \tag{3} \]
where \(\varphi\) is the controlled coordinate; \(\varphi_{\mathrm{M}}\) is the model coordinate; \(\mu\) is the coordinate of the regulating element; \(a_\alpha^*(t)\), \(b^*(t)\) are the coefficients of the plant; \(c_\gamma,\ \bar{k}_i,\ k_{iu},\ \Delta_i,\ d_\xi,\ k_{g0}\) are constant coefficients; \(k_b\) is the overall controller gain; \(g\) is the control action.
System (3), for \(z_i=\tilde{z}_i+\Delta z_i,\ \tilde{z}_i=z_{i\,\mathrm{st}}=\mathrm{const},\ |\varphi_{\mathrm{M}}^{(i)}|>\Delta_i,\ \Delta z_i=D^i\Delta z_0 \operatorname{sign}D^i(\varepsilon+\varphi_{\mathrm{M}})\operatorname{sign}(\varepsilon+\varphi_{\mathrm{M}}),\ i=1,2,\ldots,n-1,\ \varepsilon=\varphi-\varphi_{\mathrm{M}}\), reduces to a single equation
\[ \left\{D^{n+1}+\sum_{i=0}^{n-1}(a_i+\varkappa\bar{k}_i+\varkappa k_{iu}\tilde{z}_i)D^{i+1} +\varkappa\sum_{i=0}^{n-1}k_{iu}(D^i\varphi_{\mathrm{M}})\operatorname{sign}D^i\varphi_{\mathrm{M}}\cdot D^i\right\}\Delta z_0= \]
\[ =-\left\{(1-k_{g0}\varkappa d_n)D^n+ \sum_{i=0}^{n-1}(a_i+\varkappa\bar{k}_i+\varkappa k_{iu}\tilde{z}_i-\varkappa k_{g0}d_i)D^i\right\}\varphi_{\mathrm{M}}\operatorname{sign}\varphi_{\mathrm{M}}, \tag{4} \]
where \(\varkappa=k_b b^*/c_r a_k^*=\mathrm{const}\) (see (4)), and \(a_i\) are the coefficients of the polynomial
\[ \sum_{i=0}^{n}a_iD^i=\frac{1}{c_r a_k^*}\sum_{\alpha=0}^{k}a_\alpha^*D^\alpha\sum_{\gamma=0}^{r}c_\gamma D^\gamma. \tag{5} \]
When the invariance conditions \(\varepsilon=\varphi-\varphi_{\mathrm{M}}\) with respect to \(g\) are satisfied (conditions (10) in (4)),
\[ d_n=\frac{1}{\varkappa k_{g0}},\qquad \tilde{z}_i=\frac{\varkappa k_{g0}d_i-\varkappa\bar{k}_i-a_i}{\varkappa k_{iu}}, \tag{6} \]
equation (4) can be written in the form
\[ \left(D^{n+1}+\varkappa k_{g0}d_{n-1}D^n+ (\varkappa k_{g0}d_{n-2}+\varkappa k_{n-1,u}|\varphi_{\mathrm{M}}^{(n-1)}|)D^{n-1}+\ldots\right. \]
\[ \left.\ldots+(\varkappa k_{g0}d_0+\varkappa k_{1u}|\varphi_{\mathrm{M}}|)D+\varkappa k_{0u}|\varphi_{\mathrm{M}}|\right)\Delta z_0=0. \tag{7} \]
It follows from this that, when (6) is satisfied, the operator \(A(D,t)\) of order \((n+1)\) of equation (4) does not depend on the plant parameters \(a_\alpha^*\), which is what was required to be proved.
Let us note that the systems considered above in the indicated mode are also invariant with respect to the coefficients of the actuating mechanism \(c_\gamma\) and the constant coefficients \(\bar{k}_i,\ i=0,1,\ldots,n=1\).
It can be shown that, for real differentiating circuits, double invariance of searchless self-adjusting systems with a model is fulfilled only with respect to the coefficients of the operator of order \(2n\), \(A(D,t)\), at \(D^0,D,D^2,\ldots,D^n\). The coefficients before \(D^{n+1},D^{n+2},\ldots,D^{2n}\) include \(a_\alpha\), but multiplied by the intrinsic time constants of the differentiating circuits, which are small quantities. On the right-hand side of (7) there appear terms with \(\varphi_{\mathrm{M}}^{(2n-1)},\ldots,\varphi_{\mathrm{M}}^{(n+1)}\), whose coefficients are likewise small quantities. In this case one may speak of partial double invariance of system (5).
Institute of Automation and Telemechanics
(Technical Cybernetics)
Received
25 XII 1964
CITED LITERATURE
- N. N. Simonov, Theory of Invariance in Automatic Control Systems, Nauka, 1964.
- V. S. Kulebakin, O. I. Larichev, Izv. AN SSSR, Energetics and Automation, No. 5 (1961).
- I. N. Krutova, V. Yu. Rutkovskii, Izv. AN SSSR, Technical Cybernetics, No. 1, 2 (1964).
- B. N. Petrov, V. Yu. Rutkovskii, DAN, 161, No. 3 (1965).
- B. A. Ryabov, Automation and Telemechanics, No. 4 (1939).