CN103631135B - Based on the fractional order PI of vector approach λthe parameter tuning method of controller - Google Patents

Abstract

The present invention relates to a kind of fractional order PI based on vector approach ^λthe parameter tuning method of controller, belongs to technical field of fractional order automatic control.The mathematical model transport function of given controlled device system, controlled device is write as controlled device vector form, write out performance index vector form, utilize a updating vector L to go to correct controlled device vector P and reach performance index vector G, updating vector and controller vector form triangle in complex plane, obtain controller transfer function.Advantage is: can reduce calculated amount, and tuning process is easily understood.When controlled device changes, we only need to calculate the modulus value of controlled device, phase angle and angular velocity, and formula below can be applied mechanically, thus decreases the process of equation of repeatedly deriving, and solves many solutions problem that matlab solves middle existence.

Description

Based on the fractional order PI of vector approach λthe parameter tuning method of controller

Technical field

The invention belongs to technical field of fractional order automatic control, that relate generally to is a kind of PI ^λthe parameter tuning method of the fractional order control device of structure.

Background technology

Servo-drive system occupies very important status at automation field, and the feedback control system of to be a kind of controlled volume be displacement, speed, acceleration, can accurately follow certain control procedure.In addition, some servo-drive system working environments are complicated, and system exists larger uncertainty and delayed, and the change of inertia load in the course of the work, are typical nonlinear system.Conventional PID controllers, fuzzy adaptive controller, nerve network controller etc. mostly are at present based on the controller adopted in the servo-drive system of automatic control technology.Conventional PID controllers control method is too simple, the control performance of parameter also has certain limitation, although conventional PID controllers realizes control performance by three parameters, but its differential order is 1 rank, therefore the type of control system is more fixing, what adopt conventional PID controllers to control for new fractional-order system will produce comparatively big error, and control effects does not reach control accuracy requirement sometimes.Therefore, for non-linear, delayed servo-drive system, adopt conventional controller to be often difficult to meet the requirement of system to robustness and lasting accuracy.Due to fractional order PI ^λcontroller is a many adjustable parameter λ, can regulate the power of integral action, thus make fractional order PI ^λcontroller can obtain than the better robustness of integer rank controller and lasting accuracy.

Simultaneously also because fractional order PI ^λcontroller is a many adjustable parameter λ, makes fractional order PI ^λattitude conirol process is complicated, and operand is large.To different controlled devices, parameter tuning equation needs again to derive and calculate, and makes Controller gain variations become loaded down with trivial details and time-consuming.And controller parameter uses matlab to solve, and generally all there is multiple solution, needs to find satisfactory solution, bring much inconvenience to emulation and experiment.

Summary of the invention

The invention provides a kind of fractional order PI based on vector approach ^λthe parameter tuning method of controller, can either have stable precision can improve system robustness and dirigibility again to meet servo-drive system.

The technical scheme that the present invention takes comprises the following steps:

(1), the mathematical model transport function of given controlled device system and given design objective bandwidth omega _cwith need keep stable phase margin φ _m, treat setting controller transport function form wherein, described T is arithmetic number, and s is Laplace operator, K _prepresent scale-up factor to be adjusted, K _irepresent integral coefficient to be adjusted, λ represents integration order to be adjusted;

(2), by controlled device write as controlled device vector form:

$P\left({\mathrm{j\ω}}_c\right)=\frac{1}{\sqrt{{\left({\mathrm{T\ω}}_c\right)}^2+1}}\∠-\arctan {\mathrm{T\ω}}_c,$ Controlled device vectorial angle speed ${\mathrm{\ψ}}_P=-\frac{T}{{\left({\mathrm{T\ω}}_c\right)}^2+1};$

(3), by design objective bandwidth omega _cwith need keep stable phase margin φ _m, write out performance index vector form: G (j ω _c)=1 ∠ φ _m-180 °, angular velocity is 0;

(4), utilize a updating vector L to go to correct controlled device vector P and reach performance index vector G, i.e. LP=G; According to step (two) and step (three), can in the hope of at ω _cplace updating vector L, try to achieve updating vector at ω simultaneously _cthe angular velocity at place ${\mathrm{\ψ}}_L=\frac{T}{{\left({\mathrm{T\ω}}_c\right)}^2+1};$

(5) by controller be mapped in complex plane and obtain being 0 by phase angle and modulus value is K _pvector (K _p∠ 0 °) add that a phase angle is and modulus value is vector and, namely controller vector, utilize controller vector C go proximity correction vector L, i.e. lim|C-L|=0, thus updating vector and controller vector in complex plane, form triangle;

(6) make

Namely with θ=φ _m+ arctanT ω _c-180 °, the leg-of-mutton cosine law is utilized to obtain:

$\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}=\sqrt{K_p^2+A^2-{2K}_{p}A\cos \mathrm{\θ}}---\left(1\right)$

$\cos (\mathrm{\π}-\frac{\mathrm{\π}}{2}\mathrm{\λ})=\frac{K_p-A\cos \mathrm{\θ}}{\sqrt{K_p^2+A^2-{2K}_{p}A\cos \mathrm{\θ}}}---\left(2\right)$

Utilize lim|C-L|=0 angular velocity equal simultaneously, can obtain:

$\frac{{\mathrm{\λK}}_p\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}+1}}\sin (\mathrm{\λ\π}/2)}{{(K_p+\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\cos (\mathrm{\λ\π}/2))}^2+{\left(\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ\π}/2)\right)}^2}={\mathrm{\ψ}}_L---\left(3\right)$

Notice that there is geometric relationship is ${(K_p+\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\cos (\mathrm{\λ\π}/2))}^2+{\left(\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ\π}/2)\right)}^2=A^2$ With

$\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ\π}/2)=-A\sin \mathrm{\θ},$ So can obtain,

$-\frac{{\mathrm{\λK}}_p\sin \mathrm{\θ}}{{\mathrm{A\ω}}_c}={\mathrm{\ψ}}_L---\left(4\right)$

Formula (2) and (4) are utilized to try to achieve K _pand λ, substitute into formula (1) and can K be tried to achieve _i, obtain controller transfer function namely a kind of fractional order PI of robustness is completed ^λattitude conirol.

The invention has the beneficial effects as follows: servo system controller provided by the invention is having two adjustable parameter K _p, K _iprerequisite under, have again an integration order λ, adjustment parameter K _p, K _inumerical value can improve the response speed of servo-drive system, the dynamic perfromance of improvement system, eliminate systematic steady state error, increase an adjustable parameter λ again, the stable state accuracy of servo-drive system can be improved again, three parameter coordination adjustment can increase the stability of system greatly, can also meet the demand of system robustness simultaneously.Therefore the design of the method is more flexible, and control performance is more superior, be have also been obtained good control by the precision of its servo-drive system, simple to operate, easy.With other fractional order control device PI ^λparameter tuning method is compared, and tool of the present invention has the following advantages:

(1) calculated amount can be reduced, and tuning process is easily understood.

(2) when controlled device changes, we only need to calculate the modulus value of controlled device, phase angle and angular velocity, and formula below can be applied mechanically, thus decreases the process of equation of repeatedly deriving.

(3) by formula (2) and (4), solving with matlab is unique solution, thus solves many solutions problem that matlab solves middle existence.

Accompanying drawing explanation

Fig. 1 is controller vector;

Fig. 2 is the triangle that controller vector sum updating vector is formed;

Fig. 3 is the designed open cycle system Bode diagram in embodiment.

Fig. 4 is the step response diagram of whole closed-loop control system in embodiment; Wherein, three curves are that controlled device molecule 1 becomes 0.9,1 and 1.1 step response curves being respectively.

Embodiment

Comprise the following steps:

(1), the mathematical model transport function of given controlled device system and given design objective bandwidth omega _cwith need keep stable phase margin φ _m, treat setting controller transport function form wherein, described T is arithmetic number, and s is Laplace operator, K _prepresent scale-up factor to be adjusted, K _irepresent integral coefficient to be adjusted, λ represents integration order to be adjusted;

(2), by controlled device write as controlled device vector form:

$P\left({\mathrm{j\ω}}_c\right)=\frac{1}{\sqrt{{\left({\mathrm{T\ω}}_c\right)}^2+1}}\∠-\arctan {\mathrm{T\ω}}_c,$ Controlled device vectorial angle speed ${\mathrm{\ψ}}_P=-\frac{T}{{\left({\mathrm{T\ω}}_c\right)}^2+1};$

(3), by design objective bandwidth omega _cwith need keep stable phase margin φ _m, write out performance index vector form: G (j ω _c)=1 ∠ φ _m-180 °, angular velocity is 0;

(4), utilize a updating vector L to go to correct controlled device vector P and reach performance index vector G, i.e. LP=G; According to step (two) and step (three), can in the hope of at ω _cplace updating vector L, try to achieve updating vector at ω simultaneously _cthe angular velocity at place ${\mathrm{\ψ}}_L=\frac{T}{{\left({\mathrm{T\ω}}_c\right)}^2+1};$

(5) by controller be mapped in complex plane and obtain being 0 by phase angle and modulus value is K _pvector (K _p∠ 0 °) add that a phase angle is and modulus value is vector and, namely controller vector, utilize controller vector C go proximity correction vector L, i.e. lim|C-L|=0, thus updating vector and controller vector in complex plane, form triangle;

(6) make

Namely with θ=φ _m+ arctanT ω _c-180 °, the leg-of-mutton cosine law is utilized to obtain:

$\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}=\sqrt{K_p^2+A^2-{2K}_{p}A\cos \mathrm{\θ}}---\left(1\right)$

$\cos (\mathrm{\π}-\frac{\mathrm{\π}}{2}\mathrm{\λ})=\frac{K_p-A\cos \mathrm{\θ}}{\sqrt{K_p^2+A^2-{2K}_{p}A\cos \mathrm{\θ}}}---\left(2\right)$

Utilize lim|C-L|=0 angular velocity equal simultaneously, can obtain:

$\frac{{\mathrm{\λK}}_p\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}+1}}\sin (\mathrm{\λ\π}/2)}{{(K_p+\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\cos (\mathrm{\λ\π}/2))}^2+{\left(\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ\π}/2)\right)}^2}={\mathrm{\ψ}}_L---\left(3\right)$

Notice that there is geometric relationship is ${(K_p+\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\cos (\mathrm{\λ\π}/2))}^2+{\left(\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ\π}/2)\right)}^2=A^2$ With

$\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ\π}/2)=-A\sin \mathrm{\θ},$ So can obtain,

$-\frac{{\mathrm{\λK}}_p\sin \mathrm{\θ}}{{\mathrm{A\ω}}_c}={\mathrm{\ψ}}_L---\left(4\right)$

Formula (2) and (4) are utilized to try to achieve K _pand λ, substitute into formula (1) and can K be tried to achieve _i, obtain controller transfer function namely a kind of fractional order PI λ attitude conirol of robustness is completed.

The present invention is further illustrated below by embody rule example.

Direct current generator servo-drive system is widely used in Stellungsservosteuerung with its excellent performance, and its transport function is $P\left(s\right)=\frac{K_t}{L_a{\mathrm{Js}}^2+(L_{a}B+R_{a}J)s+R_{a}B+K_e{K}_t},$ If ignore armature inductance L _aand viscous damping coefficient B, transport function can be approximated to be wherein T _mrepresent motor electromechanical time constant, K _erepresent back EMF coefficient respectively, K _trepresent electromagnetic torque coefficient, J is the moment of inertia of motor, L _abe armature inductance, B is viscosity friction coefficient; In running example, in select location servomechanism, conventional direct current motor is as control object, and without loss of generality, its mathematical model transport function can be expressed as wherein, T is time constant;

1. the mathematical model transport function of given controlled device system wherein T=0.4, and given design objective bandwidth omega _c=10rad/s and stable phase margin φ need be kept _m=70 °.

2. write out controlled device vector controlled device vectorial angle speed

3. write out performance index vector G (j ω _c)=1 ∠-110 °, angular velocity 0.

4. obtain updating vector updating vector angular velocity

5. by L (j ω _c)=A ∠ θ obtains thus obtain

$\frac{K_i}{{10}^{\mathrm{\λ}}}=\sqrt{K_p^2+17-6.8332K_p}---\left(5\right)$

$\cos \left(\frac{\mathrm{\λ\π}}{2}\right)=\frac{3.4166-K_p}{\sqrt{K_p^2+17-6.8332K_p}}---\left(6\right)$

Can in the hope of K by (6) and (7) _p=2.3743, λ=0.7299 substitutes into (5) can in the hope of K _i=13.5969.Then required PI ^λcontroller is $C\left(s\right)=2.3743+\frac{13.5969}{s^{0.729}}.$

Fig. 3 is the Bode diagram of designed open cycle system; Wherein, as can be seen from the figure system in bandwidth omega _cneighbouring Phase margin keeps constant.

Fig. 4 be the step response diagram of designed control system wherein, three curves be controlled device molecule 1 when becoming 0.9,1 and 1.1 respectively system also can keep stable output overshoot, namely utilize the PI that method listed by the present invention is adjusted out ^λthe fractional order control utensil of structure has extraordinary robust property.

Claims (1)

1. the fractional order PI based on vector approach ^λthe parameter tuning method of controller, is characterized in that comprising the following steps:

(1), the mathematical model transport function of given controlled device system and given design objective bandwidth omega _cwith need keep stable phase margin φ _m, treat setting controller transport function form wherein, described T is arithmetic number, and s is Laplace operator, K _prepresent scale-up factor to be adjusted, K _irepresent integral coefficient to be adjusted, λ represents integration order to be adjusted;

(2), by controlled device write as controlled device vector form:

$P\left({\mathrm{j\ω}}_c\right)=\frac{1}{\sqrt{{\left({\mathrm{T\ω}}_c\right)}^2+1}}\∠-{\mathrm{arctanT\ω}}_c,$ Controlled device vectorial angle speed ${\mathrm{\ψ}}_P=-\frac{T}{{\left({\mathrm{T\ω}}_c\right)}^2+1};$

(3), by design objective bandwidth omega _cwith need keep stable phase margin φ _m, write out performance index vector form: G (j ω _c)=1 ∠ φ _m-180 °, angular velocity is 0;

(4), utilize a updating vector L to go to correct controlled device vector P and reach performance index vector G, i.e. LP=G; According to step (two) and step (three), can in the hope of at ω _cplace updating vector L, try to achieve updating vector at ω simultaneously _cthe angular velocity at place ${\mathrm{\ψ}}_L=\frac{T}{{\left({\mathrm{T\ω}}_c\right)}^2+1};$

(5) by controller be mapped in complex plane and obtain being 0 by phase angle and modulus value is K _pvector add that a phase angle is and modulus value is vector and, wherein K _p∠ 0 °, i.e. controller vector, utilize controller vector C to remove proximity correction vector L, i.e. lim|C-L|=0, thus updating vector and controller vector forms triangle in complex plane;

(6) make

Namely with θ=φ _m+ arctanT ω _c-180 °, the leg-of-mutton cosine law is utilized to obtain:

$\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}=\sqrt{K_p^2+A^2-2K_{p}Acos\mathrm{\θ}}---\left(1\right)$

$cos(\mathrm{\π}-\frac{\mathrm{\π}}{2}\mathrm{\λ})=\frac{K_p-Acos\mathrm{\θ}}{\sqrt{K_p^2+A^2-2K_{p}Acos\mathrm{\θ}}}---\left(2\right)$

Utilize lim|C-L|=0 angular velocity equal simultaneously, can obtain:

$\frac{{\mathrm{\λK}}_p\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}+1}}sin(\mathrm{\λ}\mathrm{\π}/2)}{{(K_p+\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}cos(\mathrm{\λ}\mathrm{\π}/2\left)\right)}^2+{\left(\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}sin\right(\mathrm{\λ}\mathrm{\π}/2\left)\right)}^2}={\mathrm{\ψ}}_L---\left(3\right)$

Notice that there is geometric relationship is ${(K_p+\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}cos(\mathrm{\λ}\mathrm{\π}/2\left)\right)}^2+{\left(\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}sin\right(\mathrm{\λ}\mathrm{\π}/2\left)\right)}^2=A^2$ With $\frac{K_i}{{\mathrm{\ω}}_c^{\mathrm{\λ}}}\sin (\mathrm{\λ}\mathrm{\π}/2)=-Asin\mathrm{\θ},$ So can obtain,

$-\frac{{\mathrm{\λK}}_{p}sin\mathrm{\θ}}{{\mathrm{A\ω}}_c}={\mathrm{\ψ}}_L---\left(4\right)$

Formula (2) and (4) are utilized to try to achieve K _pand λ, substitute into formula (1) and can K be tried to achieve _i, obtain controller transfer function namely a kind of fractional order PI of robustness is completed ^λattitude conirol.