【Applied Algebra】Algebraic control method for AT89C51 microcontroller

【Applied Algebra】Algebraic control method for AT89C51 microcontroller (Talk for WJLeuang)

In this discussion we will introduce an efficient control algorithm for embedded devices based on computational exchange algebra.

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problem background

Considering an automatic control system, the microcontroller is a central decision-making module that receives system state feedback and then gives control signals; its architecture can be visualized as follows:

Then the question is, how to make the MCU receive as little feedback as possible - the control sequence pair sample set { u ( t ) , v ( t ) } \{u(t),v(t)\}{ u(t),After v ( t ) } , it can continuously feed back$v\left(t\right)$ gives the correct control signal$u(t)$;

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Use ${\mathbb{F}}_2$The polynomial ring simulates predicate logic

We use predicate logic to descriptively model the system. For example, predicate logic can be used to fuzzily describe the control rules. The temperature is very high (conveyor) $\wedge Excessive$ flow (primary circuit pipeline)$\to$ Open (No. 2 check valve), such empirical rules can be established on the basis of accurate data signals, making the system more robust (the membership function is used to model in fuzzy logic control, the degree of nonlinearity is not enough, Now we try to simulate it using a polynomial ring);

Consider such a homomorphic map

$$r:\mathcal{L}\to {\mathbb{F}}_2[x_1...x_n]$$

Next we can verify that for $\forall p_i,p_j\in L$ , have

$$p(p_i\diamond p_j)=p\left(p_i\right)\oslash p\left(p_j\right)=f_i\oslash f_j$$

where $\diamond$ is a logical operation connection symbol,$\oslash$ is the corresponding operation on the ring;

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Solution ideas

Theorem.1 (Ideal Elimination Therom $I$ at$k[x_1...x_n]$ ’s elimination ideal$I_l$The corresponding Groebner basis can be obtained as follows:

$$G_l=G\cap k[x_{l+1}...x_n]$$

The ideal representation of the predicate logic corresponding to the feedback signal is: $I_p=<f_1...f_m>\subset {\mathbb{F}}_2[x_1...x_n]$ ;
Let the ideal expression of the predicate logic corresponding to the associated information be:$I_r=<r_1...r_d>\subset {\mathbb{F}}_2[z_1...z_h]$ ;
Let the ideal representation of the predicate logic corresponding to the feedback signal be:$I_q=<g_1...g_k>\subset F_2[z_1...z_h,x_1...x_n]$;

Then according to the above theorem, the solution process is to find the satisfying $G_p=G_q\cap {\mathbb{F}}_2[x_1...x_n]$ of algebraic varieties;

Summarize

The following evaluations and reflections on the current work:

• It currently appears to be feasible;
• How to limit the degree rise problem in the process of polynomial calculation;
• How symbolic computation can be efficiently implemented in embedded devices to outperform current control schemes;