CYBERNETICS AND CONTROL THEORY

Academician B. N. PETROV, V. Yu. RUTKOVSKII

ON THE INVARIANCE OF MODEL-BASED NON-SEARCH SELF-ADJUSTING SYSTEMS

Model-based non-search self-adjusting systems (^1) ensure stability and high quality of control processes when the parameters of the plant vary at a rate comparable with the “rate” of transient processes in the system. We shall show that in these systems, in certain regimes, the error between the output coordinates of the system and of the model is invariant with respect to the control action.

Consider a system described by the equations:

\[ \sum_{\alpha=0}^{k} a_{\alpha}^{*}(t)\varphi^{(\alpha)}=-b^{*}(t)\mu \quad \text{— plant;} \]

\[ \sum_{\gamma=0}^{r} c_{\gamma}\mu^{(\gamma)}=\sigma \quad \text{— controller;} \]

\[ \sigma=k_b\left(\sum_{i=0}^{n-1} k_i\varphi^{(i)}-k_g g\right) \quad \text{— control law;} \]

\[ k_i=\bar{k}_i+k_{iu}z_i,\quad z_i=\int_{0}^{t}\left(\varphi^{(i)}-\varphi_M^{(i)}\right) \Phi\!\left(\varphi_M^{(i)}\right)\operatorname{sign}\varphi^{(i)}\,dt \quad \]

\[ \text{— law of variation of the adjustable coefficients;} \tag{1} \]

\[ \Phi\!\left(\varphi_M^{(i)}\right)= \begin{cases} 1, & \text{if } |\varphi_M^{(i)}|>\Delta_i,\\ 0, & \text{if } |\varphi_M^{(i)}|\leqslant \Delta_i, \end{cases} \quad \text{— nonlinear function;} \]

\[ \sum_{\xi=0}^{n} d_{\xi}\varphi_M^{(\xi)}=g \quad \text{— model,} \]

where \(r+k=n\); \(\varphi\) is the controlled coordinate of the plant; \(\mu\) is the coordinate of the regulating element; \(\sigma\) is the control law; \(\varphi_M\) is the coordinate of the model; \(a_{\alpha}^{*}(t)\), \(b^{*}(t)\) are variable coefficients of the plant; \(k_b\) is the overall coefficient of the controller; \(c_{\gamma}\), \(\bar{k}_i\), \(k_{iu}\), \(d_{\xi}\), \(\Delta_i\) are constant coefficients; \(g\) is the control action (an analytic function bounded in modulus, holomorphic at \(t=t_0\)).

We shall assume that \(a_{\alpha}^{*}(t)\) and \(b^{*}(t)\) varied in an arbitrary manner and then, beginning from some \(t=T\), remain constant. Then the equations of the perturbed motion of the system, upon linearization of the terms \(\varphi^{(i)}z_i\) and for \(|\varphi_M^{(i)}|>\Delta_i\), will be

\[ \operatorname{sign}(\varepsilon+\varphi_M) \sum_{\alpha=0}^{k} a_{\alpha}^{*}D^{\alpha+1}\Delta z_0 +b^{*}\mu = -\sum_{\alpha=0}^{k} a_{\alpha}^{*}D^{\alpha}\varphi_M, \]

\[ \sum_{\gamma=0}^{r} c_{\gamma}D^{\gamma}\mu-\sigma=0, \]

\[ k_b\operatorname{sign}(\varepsilon+\varphi_M) \left[ \sum_{i=0}^{n-1}(\bar{k}_i+k_{iu}\tilde{z}_i)D^{i+1} + \sum_{i=0}^{n-1}k_{iu}(D^{i}\varphi_M)\operatorname{sign}D^{i}(\varepsilon+\varphi_M)D^{i} \right]\Delta z_0- \]

\[ -\sigma=\left[k_{g0}k_b d_nD^n+k_b\sum_{i=0}^{n-1}(\bar k_i+k_{iu}\tilde z_i-k_{g0}d_i)D^i\right]\varphi_{\mathrm m}, \tag{2} \]

where \(\varepsilon=\varphi-\varphi'_{\mathrm m}\), \(z_i=\tilde z_i+\Delta z_i\), \(\tilde z_i=\mathrm{const}\), \(i=0,1,\ldots,n-1\),

\[ \Delta z_i=D^i\Delta z_0\,\operatorname{sign}D^i(\varepsilon+\varphi_{\mathrm m})\,\operatorname{sign}(\varepsilon+\varphi_{\mathrm m}),\quad i=1,2,\ldots,n-1. \]

System (2) is nonlinear with variable coefficients. Suppose that the control action \(g\) is such that, for

\[ |z_i(t)-\tilde z_i|<\bar z_i,\quad t>\bar T,\quad \bar T>T \tag{3} \]

the inequalities are satisfied

\[ |\varphi_{\mathrm m}-\varphi_{\mathrm m_0}|<\delta,\quad |\varphi_{\mathrm m}^{(i)}|<\delta,\quad i=1,2,\ldots,n-1, \tag{4} \]

where \(\bar z_i,\delta\) are small positive quantities, \(\varphi_{\mathrm m_0}=\mathrm{const}\ne0\), \(|\varepsilon|<\varphi_{\mathrm m}\).

Then, neglecting in the coefficients of \(D^i\Delta z_0\), \(i=1,2,\ldots,n-1\), in the last equation (2) the terms containing the factor \(\delta\), we obtain the linear system (since \(\varphi_{\mathrm m_0}\) does not change sign) with constant coefficients \((t>\bar T)\)

\[ \operatorname{sign}\varphi_{\mathrm m_0}\sum_{\alpha=0}^{k}a_\alpha^*D^{\alpha+1}\Delta z_0+b^*\mu =-\sum_{\alpha=0}^{k}a_\alpha^*D^\alpha\varphi_{\mathrm m}, \]

\[ \sum_{\gamma=0}^{r}c_\gamma D^\gamma\mu-\sigma=0,\quad r+k=n, \]

\[ k_b\operatorname{sign}\varphi_{\mathrm m_0} \left[\sum_{i=0}^{n-1}(\bar k_i+k_{iu}\tilde z_i)D^{i+1} +k_{0u}\varphi_{\mathrm m_0}\operatorname{sign}\varphi_{\mathrm m_0}\right]\Delta z_0-\sigma = \]

\[ =\left[k_{g0}k_b d_nD^n-k_b\sum_{i=0}^{n-1}(\bar k_i+k_{iu}\tilde z_i-k_{g0}d_i)D^i\right]\varphi_{\mathrm m}. \tag{5} \]

Represent (5) in the form of a matrix equation

\[ Ax=F, \tag{6} \]

where \(A=\|a_{ij}\|\), \(i,j=1,2,3\), is a polynomial or \(D\)-matrix of order 3; \(x,F\) are column matrices, with

\[ x_1=\Delta z_0,\quad x_2=\mu,\quad x_3=\sigma, \]

\[ F_1=-\sum_{\alpha=0}^{k}a_\alpha^*D^\alpha\varphi_{\mathrm m},\quad F_2=0, \]

\[ F_3=\left[k_{g0}k_b d_nD^n-k_b\sum_{i=0}^{n-1}(\bar k_i+k_{iu}\tilde z_i-k_{g0}d_i)D^i\right]\varphi_{\mathrm m}, \]

\[ a_{11}=\operatorname{sign}\varphi_{\mathrm m_0}\sum_{\alpha=0}^{k}a_\alpha^*D^{\alpha+1},\quad a_{12}=b^*,\quad a_{13}=0, \]

\[ a_{21}=0,\quad a_{22}=\sum_{\gamma=0}^{r}c_\gamma D^\gamma,\quad a_{23}=-1, \]

\[ a_{31}=k_b\operatorname{sign}\varphi_{\mathrm m_0} \left[\sum_{i=0}^{n-1}(\bar k_i+k_{iu}\tilde z_i)D^{i+1} +k_{0u}\varphi_{\mathrm m_0}\operatorname{sign}\varphi_{\mathrm m_0}\right], \]

\[ a_{32}=0,\quad a_{33}=-1. \]

We reduce (6) to Hermite’s canonical form (²). As the canonicalizing matrix \(\lambda\) we choose

\[ \lambda=\|\lambda_{ij}\|= \left\| \begin{array}{ccc} 0 & 0 & 1\\[2mm] \dfrac{1}{b^*} & 0 & 0\\[3mm] \dfrac{1}{c_r a_k^*}\displaystyle\sum_{\gamma=0}^{r} c_\gamma D^\gamma & -\dfrac{b^*}{c_r a_k^*} & \dfrac{b^*}{c_r a_k^*} \end{array} \right\|. \tag{7} \]

Multiplying the augmented matrix of system (6) by \(\lambda\) on the left, we find the augmented matrix of Hermite’s canonical form and the corresponding system

\[ \begin{gathered} \lambda_{31}a_{31}x_1+0x_2+\lambda_{13}a_{33}x_3=\lambda_{13}F_3,\\ \lambda_{21}a_{11}x_1+\lambda_{21}a_{12}x_2+0x_3=\lambda_{21}F_1,\\ (\lambda_{31}a_{11}+\lambda_{33}a_{31})x_1+0x_2+0x_3 =\lambda_{31}F_1+\lambda_{33}F_3 . \end{gathered} \tag{8} \]

A necessary and sufficient condition for absolute invariance of \(x_1\), and consequently also of \(\varepsilon=\varphi-\varphi_{\mathrm{m}}\), with respect to \(g\) or \(\varphi_{\mathrm{m}}\), is the condition

\[ \lambda_{31}F_1+\lambda_{33}F_3=0. \tag{9} \]

Expanding (9), we obtain

\[ d_n=\frac{1}{\varkappa k_{g0}},\qquad \widetilde z_i=\frac{\varkappa k_{g0}-\varkappa k_i-a_i}{\varkappa k_{iu}}, \tag{10} \]

where \(\varkappa=k_b b^*/c_r a_k^*\); \(a_i\) are the coefficients of the polynomial

\[ \sum_{i=0}^{n} a_iD^i = \frac{1}{c_r a_k^*} \sum_{\alpha=0}^{k} a_\alpha^*D^\alpha\cdot \sum_{\gamma=0}^{r} c_\gamma D^\gamma, \qquad a_n=1. \tag{11} \]

It is easy to show that, for \(\varkappa=\mathrm{const}\), in the steady state in the regime under consideration \(z_{i\,\mathrm{st}}=\widetilde z_i\). Hence the error \(\varepsilon=\varphi-\varphi_{\mathrm{m}}\) in a searchless self-adjusting system with a model is absolutely invariant with respect to the control action \(g(t)\) in the regime when \(a_\alpha^*(t)\), \(b^*(t)\), after changing in an arbitrary manner, then remain constant. The condition \(\varkappa=\mathrm{const}\) is achieved by means of a special self-adjustment loop for the overall gain coefficient of the regulator \(k_b\), which can be constructed on the basis of monitoring the amplitude-frequency characteristic of the closed system at one point.

Since \(z_i\) attains the values \(z_{i\,\mathrm{st}}\) as \(t\to\infty\), absolute invariance can be attained only at \(t=\infty\). However, this difficulty is easily avoided if it is assumed that the transient processes end when \(|\varepsilon|<\bar\varepsilon\) and \(|z_i-z_{i\,\mathrm{st}}|<\bar z_i\), where \(\bar\varepsilon,\bar z_i\) are small quantities. The indicated formulation of the invariance problem is, in essence, “invariance up to \(\varepsilon\).”

The necessary realizability condition (³) for the relations (10) is satisfied, and the system will be coarse for bounded \(|\varphi_{\mathrm{m}}|\).

In conclusion, we note that the invariance conditions (10) retain their form also for an arbitrary control action \(g(t)\).

Thus, in studying and selecting the structure of searchless self-adjusting systems with a model, a new approach can be applied, connected with the problem of invariance in the theory of automatic control.

Institute of Automation and Telemechanics
(Technical Cybernetics)

Received
25 XII 1964

REFERENCES

¹ I. N. Krutova, V. Yu. Rutkovskii, Izv. Akad. Nauk SSSR, Technical Cybernetics, No. 1, 2 (1964).
² N. N. Luzin, Automation and Telemechanics, No. 5 (1940).
³ B. N. Petrov, Proc. I International Congress of IFAC, 1, Publishing House of the USSR Academy of Sciences, 1961.