CN106126940B - Formalized verification method for stability of robot fractional order PID controller - Google Patents
Formalized verification method for stability of robot fractional order PID controller Download PDFInfo
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Abstract
The invention provides a formalization method proved by a high-order logic theorem to verify the stability of a robot fractional order PID control system. Aiming at a fractional order PID controller of a robot, firstly, a high-order logic formalized model of fractional order Laplace transformation and a high-order logic formalized model of the fractional order PID controller are established, on the basis, the high-order logic formalized model of a fractional order closed-loop control system is established, and the stability of the fractional order PID control system is verified by utilizing the formalized model and the theorem. The invention adopts the formalized verification of the high-order logic theorem prover to provide guarantee for the high reliability of the robot fractional order PID control system and provide a solid foundation for the safety of the human-computer interaction robot.
Description
Technical Field
The invention provides high-order logic formalized verification for the stability of a robot fractional order PID control system, and belongs to the field of information safety and mathematics.
Background
As robots are continuously developed from low levels to high levels, the robots have penetrated into various fields of human life. Higher requirements are put on the intelligence level and the reliability of the robot. The inventor also provides a robot controlled by a fractional order PID controller, so that the robot can complete various actions more accurately. In addition to precise control, close verification in designing a robot is also an important direction for reliability research of the robot at present. The traditional simulation verification method cannot guarantee hundred percent accurate analysis of the system. For some safety performance requirements, especially for human-computer interaction robots, inaccurate applications pose a great threat and even harm to life.
Formalization has been successfully applied to the precise analysis of software and hardware as a highly reliable analysis technique. The theorem proving is one of formalized technologies, and takes basic theories such as high-order logics and set theory as tools to realize a formal calculation system for computational modeling and logical reasoning. The theorem proving method can be used for any system which can be represented by a mathematical model, is not limited by the number of states, and is an ideal verification method. The American aviation and space administration adopts a theorem prover PVS (Protopype Verification System) system to establish the specification of control software in a space shuttle and verify a production line used by Rockwell International in the aviation industryProcessor AAMP5 from which four design errors were found, one of which was hard to find with simulation and testing (handsome, dui smart. formal verification of digital hardware [ M]Beijing: beijing university Press, 2001.12). Holger of artificial intelligence research center of Bremen, Germany
In the literature of Design for development and comprehensive-aid Verification (Autonomous Robots, Vol.32, No.3, April 2012, pp.303-331), they Design Safety algorithms that implement the avoidance of collisions between Autonomous vehicles and static obstacles, and use the theoretic proving system Isabella to provide formal proof of the avoidance of collisions for Autonomous mobile Robots.
Disclosure of Invention
The invention aims to verify the stability of a robot fractional Order PID control system by using a theorem prover HOL4 (which is a version of HOL (high Order logic)). Firstly establishing a formal model of fractional order Laplace transform and a formal model of a fractional order PID controller, establishing a formal model of a fractional order closed-loop control system on the basis, and finally verifying the stability of the fractional order control system by utilizing the established fractional order formal model and theorem. Formal verification based on a high-order logic theorem prover provides guarantee for high reliability of a robot fractional order control system and provides a solid foundation for safety of a human-computer interaction robot.
The invention provides high-order logic verification of stability of a robot fractional order PID control system, which comprises the following steps:
(1) and establishing a high-order logic formalized model of Laplace transformation in a high-order logic theorem prover. The laplacian transform is the basic tool for converting fractional order systems between the time domain and the frequency domain. Here, a high-order logical formalized model of the laplace transform is established:
|-！s f t.LAPLACE s f t＝lim(\n.lim(\b.integral(1/2pow n,&b) (\t.f t*exp(-(s*t)))t))
the definition of Laplace transform describes the interval [0 ]+，+∞]An integral of (a) and (b), the integrand being f (t) and e-stThe product of (a). Where lim is the limit and integer is the integral. The laplace transform is a tool for the system to transform in the time domain and the frequency domain. A fractional order system formed by corresponding fractional order controllers introduces fractional order Laplace transformation on the basis of a Laplace transformation high-order logic formalized model.
(2) And establishing a high-order logic formalized model of fractional Laplace transform in a high-order logic theorem prover.
|-FRAC_LAPLACE<＝>！F s f t v.(F s＝LAPLACE s f t)＝＝>(s rpow v *F s＝frac_cal f v 0t t)
The fractional laplacian transform can transform the fractional system model in time domain and frequency domain. Among them, the inclusion condition F s ═ LAPLACE s f t requires that the function f (t) of the fractional LAPLACE transform must satisfy the LAPLACE correlation condition of the ordinary differential equation. In the above transformation, the function before transformation is represented by f (t), the function after transformation is represented by f(s), and the high-order logic definition of fractional calculus is represented by frac _ cal.
(3) And obtaining a high-order logic formalized model of the fractional order PID controller by utilizing the high-order logic formalized model of the fractional order Laplace transform. The mathematical model is as follows:
Gc(s)＝JP+KIs-λ+KDsμ
in a high-order logic theorem prover, establishing and verifying a high-order logic formalization model of a fractional order PID controller:
Theorem：val FRAC_PID_TD_FD
|-0<s/\0<E s/\FRAC_LAPLACE/\(LAPLACE s u t＝U s)/\(LAPLACE s e t＝E s)/\(u t＝u_t Lambda Mu K_P K_I K_D e_t t)＝＝>(U s/E s ＝K_P+K_I*s rpow-Lambda+K_D*s rpow Mu)
here, LAPLACE transform conditions are satisfied with LAPLACE transform input function e (t) and controller output function u (t) using LAPLACE s u t U s and LAPLACE s e t E s, and then the fractional LAPLACE transform of e (t) and u (t) is derived using fractional LAPLACE transform definition FRAC _ LAPLACE. Finally, the transfer function of the fractional order PID controller with output u (t) is derived as K _ P + K _ I _ s rpow-Lambda + K _ D _ s rpow Mu.
(4) And establishing a high-order logic formalized model of the fractional order closed-loop system in a high-order logic theorem prover. The fractional order closed loop control system is composed of a fractional order PID controller and a controlled system, and is also a fractional order system. In the time domain, a fractional order system can be described by an n-term fractional order differential equation, which is as follows:
wherein the content of the first and second substances,
ai(i-0, 1, …, n) is an arbitrary constant, βj(j-0, 1, …, n) is an arbitrary real number, and βn＞βn-1＞…＞β1＞β0Is more than or equal to 0. The high order logic modeling of the fractional order system using the high order logic definition of fractional order calculus, frac _ cal, and the sum function sum, is represented in HOL4 as follows:
|-FRAC_ORDER_SYSTEM<＝>！n p a y u t.0<＝p 0/\(！j.j<n＝＝>p j<p(SUC j))＝＝>(sum(0,SUC n)(\i.a i*frac_cal y(p i)0t t)＝ u t)
the high-order logic formalized model is formalized based on a mathematical model of fractional differential equation. In the formalization, both the constants and the order are subscripted. Wherein a (i) representsiWherein a (i) is variable i type num->The function of real. By p (j) representing betajWherein p (j) is variable j with num->The function of real. Preconditions! j.j<n＝＝>p j<p (sucj) makes this form meet the requirement of increasing the order of the fractional calculus in a fractional order system.
The transfer function of the fractional order system is:
the time domain model and the frequency domain model of the fractional order system are equivalent and can be transformed into each other, and the following theorem verifies the equivalence of the fractional order time domain model and the frequency domain model based on the laplace transform.
Theorem：FRAC_ORDER_SYSTEM_TD_FD
|-0<s/\(！i.0<a i)/\U s<>0/\(LAPLACE s u t＝U s)/\(LAPLACE s y t＝Y s)/\FRAC_LAPLACE/\FRAC_ORDER_SYSTEM＝＝>(Y s/U s＝1/ sum(0,SUC n)(\i.a i*s rpow p i))
The theorem proves that the time domain model of the fractional order system can obtain a corresponding frequency domain model through Laplace transformation. Wherein LAPLACE s u t U s and LAPLACE y t Y s require that the input function u (t) and the output function y (t) must satisfy the LAPLACE transform condition, so that u (t) and y (t) can be derived using the definition FRAC _ LAPLACE of the fractional LAPLACE transform, and that the fractional LAPLACE transform can be performed. In practice, U(s) and sum (0, SUC n) (\\ i.a i.s rpow p i)) are required to satisfy the condition of not being equal to zero, U s < >0 apparently satisfies that U(s) is not equal to zero and sum (0, SUC n) (\ i. a i s rpow p i)) is not equal to zero, and the sum (0, SUC n) (\\ i. a i s rpow p i)) is equal to zero, and is represented by 0< s and! i.0< a i. Under these conditions, a corresponding high-level logical formalized model of the fractional ORDER SYSTEM FRAC _ ORDER _ SYSTEM is available, which can be expressed as 1/sum (0, SUC n) (\ i.a i s rpow p i).
(5) Establishing a high-order logic formal model of the fractional order control system by utilizing the fractional order system and the high-order logic formal model of the fractional order PID controller, wherein the corresponding mathematical model is as follows:
in a high-order logic theorem prover, a high-order logic formalized model of a robot fractional order PID control system is established as follows:
|-！a p n Lambda Mu K_P K_I K_D s.G_s a p n Lambda Mu K_P K_I K_D s＝G_f a p s n*G_c Lambda Mu K_P K_I K_D s/(1+G_f a p s n*G_c Lambda Mu K_P K_IK_D s)
wherein G _ f, G _ c and G _ s respectively represent a controlled object in the robot, a fractional PID controller and a fractional closed-loop system, and G _ s is obtained through G _ f and G _ c.
In the robot fractional order PID control system, the stability of the system is verified by formally verifying the steady state output of the robot fractional order PID control system. For convenient application, theorems are established in a high-order logic theorem prover
Theorem：FRAC_ORDER_CLOSED_LOOP_SYSTEM_TRANSFER_GENERAL
|-！n a s p K_P K_D K_I Lambda Mu.0<s/\(！i.0<a i)/\u_t Lambda Mu K_P K_I K_D e_t t<>0＝＝>(G_s a p n Lambda Mu K_P K_I K_D s＝(K_D *s rpow(Lambda+Mu)+K_P*s rpow Lambda+K_I)/(sum(0,SUC n)(\i. a i*s rpow(p i+Lambda))+K_D*srpow(Lambda+Mu)+K_P*s rpow Lambda+K_I))
Where Lambda, Mu are the order of integration and differentiation, respectively, K _ P, K _ I, K _ D are the proportional gain, the integral coefficient and the derivative coefficient, respectively, and s is a variable of the transfer function. Proof of theorem by s >0 and (| i.0< a I) two preconditions, it follows that sum (0, sucn) (\\ i.a I: _ srpow P I) is not equal to zero, then, depending on the fact that srpow Lambda is always greater than zero, it can be concluded that srpow Lambda sum (0, sucn) (\ i.a I: _ srpow P I) is not equal to zero, and also that the transfer function u _ t Lambda Mu _ P K _ I K _ D e _ t t of the controller is not equal to zero, it can be concluded that (1+1/sum (0, sucn) (\\ i. a I: _ srpow I) (K _ P + K _ I _ srpow-lada + K _ D _ srpow)) is not zero. The equations are not zero, and the condition that denominator is not equal to zero is met, so that the establishment of theorem is deduced according to the correlation theory in the theorem prover. The theorem can be directly applied in the verification, so that the verification process is simplified.
In the robot fractional order PID control system, when the input is unit step response, the steady state output of the system is verified through the high-order logic formalization model and theorem, namely the stability of the robot fractional order PID control system is verified in a man-machine interaction mode.
Drawings
FIG. 1 is a diagram of a system formal verification process in an embodiment of the present invention
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
For a bottom layer system of a five-degree-of-freedom robot, the transfer function is as follows:
the applicant also proposed a designed fractional order PID controller:
Gfc＝34+89/s0.2+28*s0.6
the stability of the robot under control of a fractional order PID controller has been verified by simulation methods. The theorem proposed by the present invention is used here to demonstrate the logical verification of the stability of a system under control of a fractional order PID controller. The time domain model and the frequency domain model of the fractional order system are equivalent and can be transformed into each other by the laplace transform. For the closed-loop control system, whether the steady-state output can be achieved or not is the most basic requirement, and if the system cannot achieve the steady state and the robot cannot be reasonably controlled, the robot can be caused to be mad. Here the steady state output of the system is verified in a formalized way. The formal verification process is shown in fig. 1.
The invention verifies the steady state output of the fractional order control system of the robot when the input is unit step response. The unit step signal means that the signal amount is constantly 0 when t <0 and is constantly 1 when t > 0. Firstly, a high-order logic formalized model of unit step response is established in a high-order logic theorem prover. High-order logical formalized model in HOL4 defining unit step response:
|-！x.unit x＝if 0<x then 1else 0
after the high-order logic formalized model with unit step response is provided, some basic theorems of fractional calculus are needed, namely, first, the fractional differentiation of a constant. The fractional order differential, which defines a constant, is the same as the integer order differential, which is a constant, and the result is zero. The high-order logic formalized model established for the method is as follows:
|-FRAC_CAL_CONST<＝>！v t c.0<v/\(frac_cal(\t.c)v 0t t＝ 0)
and verifying that the fractional order control system can obtain a steady state value according to the established high-order logic formal model. On the basis of a time domain model of a fractional order control system, establishing a formalized model of high-order logic and verifying:
Theorem：POSITION_SERVO_SYSTEM_UNIT
|-？t.0<t/\FRAC_CAL_CONST＝＝>((6.58*frac_cal(\t.unit t)0.8 0t t+7.99*frac_cal(\t.unit t)0.2 0t t+20.915)/(0.00001196 *frac_cal(\t.unit t)3.2 0t t+0.002404*frac_cal(\t.unit t)2.2 0t t+0.7523*frac_cal(\t.unit t)1.2 0t t+6.58*frac_cal(\t. unit t)0.8 0t t+7.99*frac_cal(\t.unit t)0.2 0t t 20.915)＝1)
where frac _ cal is a high-order logical formalization of the definition of the fractional calculus GL, which is defined as follows:
with the rewrite strategy we can get:
the unit step response is equal to the constant at the moment when the unit step response is larger than zero, and the fractional order differential of the unit step response can be obtained according to the fractional order differential of the constant. The unit step response is in a jump state at zero time, and is discontinuous at the zero time, fractional order differentiation cannot be obtained, and the unit step response is constant at the time (t-jh) greater than zero. This moment is present. Since the maximum value of the variable j of the summation function is t/h, jh is less than or equal to t, and since the theorem gives a precondition of 0<t, so (t-jh) is greater than zero. Thus, the function unit (t-jh) for obtaining the fractional calculus is constant 1, and the function can be obtained
Is zero. Finally, the establishment of the theorem is deduced. That is, the fractional order position servo control system can achieve a steady state output.
High-level logic formal verification of the stability of the control system proves that the robot is stable and effective under the designed fractional order PID controller, which is a basic condition. The verification also shows that the theorem proving based on high-order logic can formalize a control system of the verification robot, and provides a solid foundation for the development of the human-computer interaction robot.
Claims (2)
1. The high-order logic verification method for the stability of the robot fractional order PID control system is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: establishing a high-order logic formalized model of Laplace transformation in a high-order logic theorem prover; a high-order logical formalized model of the Laplace transform:
i to | But! s f t. LAPLACE s f t ═ lim (\ n.lim (\ b.integral (1/2pow n, & b) (\ t.f t × exp (- (s × t))) t), where lim is the limit and integral is the integral;
step two: on the basis of a high-order logic formalized model of the Laplace transform, establishing a high-order logic formalized model of the fractional order Laplace transform in a high-order logic theorem prover:
I-FRAC _ LAPLACE < ═! F s F t v. (F s) (LAPLACE s F t) (s rpow v x F s) (frac _ cal F v 0 t t), where the inclusion condition F s (LAPLACE s F t) requires that the function F (t) of the fractional LAPLACE transform must satisfy the LAPLACE correlation condition of the ordinary differential equation, the pre-transform function is denoted by F (t), the post-transform function is denoted by F(s), and the high-order logic definition of the fractional calculus is denoted by frac _ cal;
step three: obtaining a high-order logic formalized model of the fractional order PID controller by utilizing the high-order logic model of the fractional order Laplace transform; a high-order logic formalized model of the fractional order PID controller:
-0< s/\0< E s/\ FRAC _ LAPLACE/\ (LAPLACE s t ═ U s)/\ (LAPLACE e t ═ E s)/\ (u t ═ u _ t Lambda Mu K _ P K _ I K _ D e _ t t) ═ U s/E s ═ K _ P + K _ I ═ s rpow-Lambda + K _ D _ s rpow Mu); wherein e (t) is an input function e (t) of the controller, and u (t) is an output function u (t) of the controller;
step four: establishing a fractional order system in a high-order logic theorem prover; the fractional order system:
-FRAC ORDER SYSTEM | > |! n pa y u t.0< ═ p0/\ (| j.j < n ═ p j < p (SUC j) >) ═ sum (0, SUC n) (\ i.a i ═ frac _ cal y (p i)0 t t) ═ u t, where a (i) is a function with a variable of type i num- > real, and p (j) is a function with a variable of type j num- > real; frac _ cal is a high-order logic definition, and sum is a summation function;
step five: establishing a high-order logic formal model of the robot fractional order PID control system by utilizing the fractional order system and the high-order logic formal model of the fractional order PID controller; the high-order logic formalization model of the robot fractional order PID control system is as follows:
|-！a p n Lambda Mu K_P K_I K_D s.G_s a p n Lambda Mu K_P K_I K_D s＝G_f a p s n*G_c Lambda Mu K_P K_I K_D s/(1+G_f a p s n*G_c Lambda Mu K_P K_I K_D s)
g _ f, G _ c and G _ s respectively represent a fractional order controlled object, a fractional order PID controller and a fractional order closed-loop system, and the G _ s is obtained through the G _ f and the G _ c;
step six: verifying the stability of the robot fractional order PID control system in a man-machine interaction mode; the man-machine interaction mode is as follows: verifying the stability of the fractional order PID control system by using the established high-order logic formalization model and theorem of the fractional order PID control system of the robot; the theorem is as follows: i to | But! n a s P K _ P K _ D K _ I Lambda mu.0< s/\\ (| i.0< a I)/\ u _ t Lambda Mu _ P K _ I K _ D e _ t t < >0 ═ G _ s P n Lambda Mu K _ P K _ I K _ D s ═ K _ D _ s rpow (Lambda + Mu) + K _ P _ s rpow Lambda + K _ I)/(sum (0, sucn) (\\ i.a I _ s rpow (P I + Lambda)) + K _ D _ s rpow (Lambda + Mu) + K _ P _ s rpow Lambda + K _ I)), lambda and Mu are respectively integral and differential orders, K _ P, K _ I and K _ D are respectively proportional gain, an integral coefficient and a differential coefficient, and s is a variable in a high-order logic formal model of the fractional order PID control system of the robot.
2. The high-order logic verification method for robot fractional order PID control system stability according to claim 1, characterized in that: mathematical model of transfer function for fractional order PID controller:
Gfc(s)＝KP+KIs-λ+KDsμ
mathematical model of the transfer function of a fractional order system:
the transfer function mathematical model corresponding to the robot fractional order PID control system is as follows:
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