CN111610712B - Bounded PID control algorithm for SISO and MIMO systems - Google Patents
Bounded PID control algorithm for SISO and MIMO systems Download PDFInfo
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Abstract
The application discloses an bounded PID control algorithm for SISO and MIMO systems, comprising two core parts: PID control unit with integral feedback in single or multiple paths, and single-output or multi-output bounded control unitThe method comprises the steps of carrying out a first treatment on the surface of the And the single-output or multi-output bounded control unit controls a time-varying control parameter k 0 And feeding back to a single-path or multi-path PID control unit with integral feedback. The stability and the reliability of the closed-loop control system are enhanced, and the bounded requirement of final output of the PID control algorithm is met; the method solves the requirement that the output of a controller in a SISO system is bounded or an actuator is bounded, and simultaneously solves the phenomenon of integral saturation or integral runaway of the traditional PID control algorithm; the method further solves the requirement that the output of the controller is bounded or the executor is bounded in the MIMO system, and simultaneously solves the limitation of the traditional PID control algorithm when the problem of the MIMO system is bounded.
Description
Technical Field
The invention belongs to the field of control and automation, and particularly relates to a bounded PID control algorithm for a SISO (single input single output) and MIMO (multiple input multiple output) system.
Background
PID control algorithms originate in the 20 th to 30 th century. PID is: pro-port, integral, differential abbreviations. The PID control algorithm is a control algorithm that is very widely used in an automatic control system, and for example, the PID control algorithm can be used for adjusting a control target in an industrial control system such as a mechanical system, an electrical system, a hydraulic system, or various composite systems. The device has the advantages of simple structure, good stability, wide application range, reliable work, convenient adjustment and the like.
Although PID control algorithms have many advantages and wide application, they also have certain drawbacks. For example, in most control systems, the actuator or actuator has a boundary requirement that is related to the physical characteristics of the actuator or to the physical characteristics or safety characteristics of the controlled object. Once the actuator has a borderline requirement, it often causes integral saturation (integral windup) or loss of integral control phenomena of the PID control algorithm. The main reason for this is that if the input error of the PID varies greatly, the integrator cannot respond to the PID controller output in time because the greatly varying error results in a large cumulative amount and the actuator has reached the control boundary. This accumulated error often results in the PID not being able to accurately and reliably control the control object, and even in instability of the overall control system.
Although there are some methods to deal with the saturation problem of PID control, such as Anti-saturation (Anti-windup) method, intelligent integration management, or integration accumulation amount upper and lower limit design, etc., this approach has more or less drawbacks in system stability, complexity, or control response. And none of these methods is well suited to PID control algorithms for MIMO systems.
Disclosure of Invention
In order to solve the technical defects in the existing PID algorithm, the invention provides a bounded PID control algorithm for SISO (single input single output) and MIMO (multiple input multiple output) systems, which is used for avoiding the traditional phenomenon of integral saturation or integral runaway. The bounded PID control algorithm can be applied to SISO systems and can be expanded to MIMO systems.
An bounded PID control algorithm for SISO and MIMO systems comprising: a single-path or multi-path PID control unit with integral feedback and a single-output or multi-output bounded control unit;
the PID control unit with integral feedback and the single-output bounded control unit are used for a SISO system, and the PID control unit with integral feedback and the multi-output bounded control unit are used for a MIMO system;
the bounded PID control algorithm for SISO and MIMO system makes the difference between the single-path or multi-path reference signal and the single-path or multi-path system output to obtain single-path or multi-path control error, and then inputs the control error to the single-path or multi-path PID control unit with integral feedback; the output of the single-path or multi-path PID control unit with integral feedback is connected with the single-output or multi-output bounded control unit; the final output of the single-output or multi-output bounded control unit is the final output of the bounded PID control algorithm; and the single-output or multi-output bounded control unit controls a time-varying control parameter k 0 And feeding back to a single-path or multi-path PID control unit with integral feedback.
Wherein the bounded problem of SISO system is u E (u min ,u max ) Problem, u is control algorithmFinal output of method or control input of actuator (actuator), u max And u min The final output of the control algorithm or the upper (maximum) and lower (minimum) limits of the actuator control input, respectively. The limiting problem of the MIMIIO system is that
Problem u i For the final output of the multiplex control algorithm or for the control input of the multiplex actuator (actuator), u max And > 0 is the upper bound of the sum of orthogonal vectors output by the multipath control algorithm.
An bounded PID control algorithm for SISO systems comprises two core parts: a single-path PID control unit with integral feedback and a single-output bounded control unit;
to solve u E (u) min ,u max ) The problem of bounded and integral saturation is solved, the bounded PID control algorithm of the SISO system makes the difference between the single-path reference signal and the single-path system output to obtain a single-path control error, and then the control error is input into the PID control unit of single-path band integral feedback; the output of the PID control unit with integral feedback is connected with the single-output bounded control unit; the final output of the single-output bounded control unit is the final output of the bounded PID control algorithm for a SISO system, and the single-output bounded control unit controls the parameter k in a time-varying manner 0 And feeding back to a PID control unit with single-path integral feedback.
The PID control unit for single-path integral feedback comprises: the system comprises a proportion unit, a differentiation unit and an integration unit with feedback;
the PID control unit with single-path integral feedback receives a single-path control error obtained by differencing a single-path reference signal and a single-path system output, the single-path control error is respectively input into a proportional unit, a differential unit and an integral unit with feedback of the PID control unit with single-path integral feedback, and then the outputs of the proportional unit, the differential unit and the integral unit with feedback are added to obtain the output of the PID control unit with single-path integral feedback.
The proportion unit is as follows: k (K) P E, wherein K P E is an error obtained by the difference between the reference signal and the system output;
the differentiating unit is as follows:
wherein K is D Is a differential coefficient +.>
Is the differentiation of the error;
the integrating unit with feedback is as follows: k (k) 0 K I Jedt, where k 0 K for time-varying control parameters fed back from said bounded control unit 0 ∈(0,1]，K I As an integral coefficient, +.Edt is the integral of the error;
the PID control unit of the single-path integral feedback is expressed as:
wherein u is m The output of the PID control unit with integral feedback is an intermediate control variable, and can be input into the single-output bounded control unit.
The single-output bounded control unit is used for receiving the output of the PID control unit fed back by single-path integration, then outputting the final output of the bounded PID control algorithm for the SISO system, and controlling the parameter k in a time-varying way 0 Feeding back to a PID control unit with integral feedback in a single path;
the single output bounded control unit is represented as:
where u is both the output of the single-output bounded control unit and the final output of the entire bounded PID control algorithm for the SISO system. u (u) m The PID control unit is derived from the single-path integral feedback; u (u) max And u min And controlling the maximum value and the minimum value of the output for the bounded PID control algorithm. k is a constant control parameter which can be adjusted, and k is more than 0, k 1 Is also a constant control parameter which can be adjusted, and k 1 ＞0，k 0 Is a time-varying control parameter, is further fed back to a PID control unit with single-path integral feedback, and k 0 ∈(0,1]。
Give u and k 0 The relation is:
the equation is an elliptic equation. Wherein u and k can be demonstrated by the above formulas (2) and (3) of the bounded control unit and the lyapunov stability analysis method 0 The formula of the relation is correct, and the Lyapunov analysis process is detailed in the specific embodiment.
By the elliptic equation, the final control output of the bounded PID control algorithm for the SISO system can be easily obtained to meet u E (u min ,u max ) Thereby guaranteeing the output of the bounded PID control algorithm. Then through the selection of reasonable initial points, the time-varying control parameter k 0 Will be guaranteed at k 0 ∈(0,1]Within the range. And it can be noted that if the final control output of the bounded PID control algorithm for a SISO system has approached a maximum or minimum, i.e., u→u min Or u.fwdarw.u max Time-varying control parameter k 0 Will approach 0, i.e. k 0 And 0. Then pass through k 0 Feedback of integral term of PID control unit of the single-path band integral feedback makes integral term of PID control unit of the single-path band integral feedback approach 0, namely k 0 K I ∫EdtAnd 0, thereby naturally avoiding the phenomenon of integral saturation or integral runaway.
The bounded PID control algorithm for the SISO system can be further expanded into a MIMO system. The bounded PID control algorithm for MIMO systems also includes two core parts: a multi-path PID control unit with integral feedback and a multi-output bounded control unit;
to solve the problems that
The method comprises the steps of solving the problems of bounded and integral saturation, according to a bounded PID control algorithm of a MIMO system, performing difference between a multipath reference signal and multipath system output to obtain multipath control errors, and then respectively inputting the multipath control errors into a PID control unit with integral feedback; the outputs of the multipath PID control units with integral feedback are respectively connected with a unified multi-output bounded control unit; the multi-output bounded control unit has multiple outputs, namely the final output of the bounded PID control algorithm for the MIMO system, and the multi-output bounded control unit outputs a time-varying control parameter k 0 Respectively feeding back to the PID control units with integral feedback.
Wherein each PID unit with integral feedback can be expressed as:
where i=1, 2, …, n, n is the total number of ways of the bounded PID control algorithm for the MIMO system, which is related to the total number of ways of the MIMO system itself, u mi Is the intermediate control variable of the ith path, K Pi 、K Di 、K Ii Respectively a proportional control parameter, an integral control parameter and an integral control parameter E i ＝R i -Y i Is the i-th error signal. R is R i For the ith reference signal, Y i For the ith system output, k 0 K, a time-varying control parameter fed back from the bounded control unit 0 ∈(0,1]。
The multi-output bounded control unit includes a multi-path control output, represented as:
wherein u is i The multiplexed output of the bounded control unit is also the final output of the bounded PID control algorithm for the entire MIMO system. u (u) mi The PID control unit is derived from the multipath integrated feedback; u (u) max > 0 is the output u of the multi-path control algorithm i The upper bound of the sum of orthogonal vectors. Similar to the bounded PID control algorithm for SISO systems, k > 0 is an adjustable constant control parameter, k i The value of > 0 is also an adjustable constant value control parameter corresponding to each path, k 0 ∈(0,1]Further feedback from the multi-output bounded control unit to the PID unit of each of the integrated feedback paths is provided for a time-varying control parameter.
By the design of the bounded control unit and the Lyapunov stability analysis method, u can be obtained i And k is equal to 0 The relation is:
the demonstration process is detailed in the detailed description. Thus can be easily obtained
Thereby ensuring the output of the bounded PID control algorithm in the MIMO system. And it can be noted that if the final control output of the bounded PID control algorithm has already approached an extreme value, i.e.>
Time-varying control parameter k 0 Will approach 0, i.e. k 0 -0; through k 0 Integral term of PID control unit with integral feedback to the multiple pathsFeedback, make the integral term of the PID control unit with integral feedback also approach 0, namely k 0 K Ii ∫E i dt→0; thus naturally avoiding the phenomenon of integral saturation or integral runaway.
The beneficial effect that this application reached:
the invention provides an bounded PID control algorithm for a SISO and MIMO system, which can be applied to the SISO system and can be expanded to the MIMO system. Compared with the traditional PID control algorithm, the invention realizes the output limitation of the final controller through a single-output or multi-output limitation unit and the limitation of the final controller through the parameter k 0 The parameter feedback is carried out on the single-path or multi-path PID control unit with integral feedback, thereby naturally avoiding the phenomenon of integral saturation or integral runaway of the traditional PID algorithm and enhancing the stability and reliability of the closed-loop control system. The invention realizes the final output of the PID control algorithm, and solves the requirement that the controller output is limited or the actuator is limited in the SISO system, namely u min ≤u≤u max The problem is solved, and the phenomenon of integral saturation or integral runaway of the traditional PID algorithm is solved. And further solves the finite requirement of final output of PID control algorithm in MIMO system, namely
The problem is solved, and the phenomenon of integral saturation or integral runaway of the traditional PID algorithm is solved. />
Drawings
FIG. 1 is a block diagram of an bounded PID control algorithm for SISO and MIMO systems according to an embodiment of the present invention;
FIG. 2 is a block diagram of an bounded PID control algorithm for a SISO system according to an embodiment of the invention;
FIG. 3 is a block diagram of an bounded PID control algorithm for a MIMO system according to an embodiment of the invention;
FIG. 4 is a block diagram of a conventional PID control algorithm for a SISO system;
FIG. 5 is a block diagram of a control system based on an bounded PID controller;
FIG. 6 shows the practice of the present inventionIn the bounded PID control algorithm for SISO system of the embodiment, u and k are as follows 0 A relationship;
FIG. 7 is a block diagram of an interconnection system for stability demonstration in accordance with an embodiment of the present invention;
FIG. 8 is an example of an embodiment of the present invention in a boost DC/DC converter system;
FIG. 9 shows the control results of the bounded PID control algorithm based on the SISO system in the boost DC-DC converter system according to the embodiment of the invention, wherein (a) is the output voltage control result of the algorithm of the invention, (b) is the inductance current control result of the algorithm of the invention, and (c) is the control input curve of the algorithm of the invention;
fig. 10 shows the control result of the conventional PID control algorithm combined with the saturation unit in the boost dc-dc converter system, wherein (a) is the output voltage control result of the conventional method, (b) is the inductor current control result of the conventional method, and (c) is the control input curve of the conventional method.
Detailed Description
The present application is further described below with reference to the accompanying drawings. The following examples are only for more clearly illustrating the technical solutions of the present invention and are not intended to limit the scope of protection of the present application.
Aiming at the limiting requirement of an actuator or an actuating mechanism of a control system, a limiting PID control algorithm is designed to be used for a SISO (single input single output) system, so that the phenomenon of integral saturation or integral runaway of the traditional PID algorithm is avoided, and the stability and reliability of the system are enhanced. And the bounded PID control algorithm can also be extended to MIMO (multiple input multiple output) systems.
Fig. 5 is a block diagram of a control system based on a bounded PID controller, i.e. comprising a control system for a SISO system, and also comprising a control system for a MIMO system.
FIG. 2 is a bounded PID control algorithm for a SISO system according to the invention. Compared with the conventional PID control algorithm for the SISO system in FIG. 4, the invention provides a bounded PID control algorithm for avoiding the phenomenon of integral saturation or integral runaway of the conventional PID algorithm, which comprises two core parts: a single-path PID control unit with integral feedback and a single-output bounded control unit;
the bounded PID control algorithm for SISO and MIMO systems, as shown in FIG. 1, comprises: a single-path or multi-path PID control unit with integral feedback and a single-output or multi-output bounded control unit;
the PID control unit with integral feedback and the single-output bounded control unit are used for a SISO system, and the PID control unit with integral feedback and the multi-output bounded control unit are used for a MIMO system;
the bounded PID control algorithm for SISO and MIMO system makes the difference between the single-path or multi-path reference signal and the single-path or multi-path system output to obtain single-path or multi-path control error, and then inputs the control error to the single-path or multi-path PID control unit with integral feedback; the output of the single-path or multi-path PID control unit with integral feedback is connected with the single-output or multi-output bounded control unit; the final output of the single-output or multi-output bounded control unit is the final output of the bounded PID control algorithm; and the single-output or multi-output bounded control unit controls a time-varying control parameter k 0 And feeding back to a single-path or multi-path PID control unit with integral feedback.
The bounded PID control algorithm for the SISO system performs difference between a single-path reference signal and a single-path system output to obtain a single-path control error, and then inputs the control error to a PID control unit of single-path band integral feedback; the output of the PID control unit with integral feedback is connected with the single-output bounded control unit; the final output of the single-output bounded control unit is the final output of the bounded PID control algorithm for a SISO system, and the single-output bounded control unit controls the parameter k in a time-varying manner 0 And feeding back to a PID control unit with single-path integral feedback.
The PID control unit for single-path integral feedback comprises: the system comprises a proportion unit, a differentiation unit and an integration unit with feedback;
the PID control unit with single-path integral feedback receives a single-path control error obtained by differencing a single-path reference signal and a single-path system output, the control error is respectively input into a proportional unit, a differential unit and an integral unit with feedback of the PID control unit with single-path integral feedback, and then the outputs of the proportional unit, the differential unit and the integral unit with feedback are added to obtain the output of the PID control unit with single-path integral feedback.
The proportion unit is as follows: k (K) P E, wherein K P E is an error obtained by the difference between the reference signal and the system output;
the differentiating unit is as follows:
wherein K is D Is a differential coefficient +.>
Is the differentiation of the error;
the integrating unit with feedback is as follows: k (k) 0 K I Jedt, where k 0 K for time-varying control parameters fed back from said bounded control unit 0 ∈(0,1]，K I As an integral coefficient, +.Edt is the integral of the error;
the PID control unit of the single-path integral feedback is expressed as:
wherein u is m The output of the PID control unit with integral feedback is an intermediate control variable, and can be input into the single-output bounded control unit.
In the PID control unit of the single-path integral feedback, the parameter k is passed through 0 And the parameter of the integral unit with feedback is fed back, so that the phenomenon of integral saturation or integral runaway of the traditional PID algorithm is naturally avoided.
The bounded control unit is used for realizing the bounded requirement of the SISO system controller output so as to solve the problem that the single controller output is bounded orThe finite requirements of the actuator, i.e. u min ≤u≤u max Problems. And will change the control parameter k 0 And feeding back to a PID control unit with single-path integral feedback.
The single output bounded control unit is represented as:
where u is both the output of the single-output bounded control unit and the final output of the entire bounded PID control algorithm for the SISO system. u (u) max And u min And controlling the maximum value and the minimum value of the output for the bounded PID control algorithm. k is a constant control parameter which can be adjusted, and k is more than 0, k 1 Is also a constant control parameter which can be adjusted, and k 1 ＞0，k 0 Is a time-varying control parameter, is further fed back to a PID control unit with single-path integral feedback, and k 0 ∈(0,1]。
Consider the following lyapunov function:
the derivative of the Lyapunov function is obtained:
substituting the bounded algorithm for SISO systems, namely formulas (2) and (3), into the available:
solving the formula (10) to obtain:
thus by proper choice of V s Initial value V of (t) s (0) Such as making
And k 0 (0) =1, further by the formula (11), thereby obtaining
Once V is s (t) =1, readily available u and k 0 The relation is:
this equation is an elliptic equation, as shown in fig. 6. As can be seen from FIG. 6, the final control unit output u is ultimately controlled to u ε (u min ,u max ) Is a fixed range of (c). And if the control unit output u reaches a minimum or maximum value, k 0 Will approach 0, so that in the PID control unit with integral feedback, the integral term k 0 K I The Edt also stops increasing, thereby naturally avoiding the phenomenon of integral saturation or integral runaway of the traditional PID control algorithm.
FIG. 3 is a finite PID control algorithm for a MIMO system, which is developed from the finite PID control algorithm for a SISO system of FIG. 2, to solve the problem
Bounded problems and integral saturation phenomena.
The bounded PID control algorithm for a MIMO system comprises n channels for a system of n inputs and n outputs. It still contains two core parts: a multi-path PID control unit with integral feedback and a multi-output bounded control unit.
Multipath PID control unit with integral feedback: the PID control unit with integral feedback of each path is the same as the bounded PID control algorithm for the SISO system, and is realized by the parameter k 0 And the PID control unit with integral feedback is subjected to parameter feedback, so that the traditional phenomenon of integral saturation or integral runaway is naturally avoided.
A multi-output bounded control unit: implementing the bounded requirement of the multi-path controller output to solve the simultaneous bounded requirement of the square sum of the multi-path controller output or the multi-path actuator, i.e
Problems. And the bounded control unit generates a time-varying control parameter k 0 And feeding back to each PID control unit with integral feedback. The bounded PID control algorithm for the MIMO system comprises a multi-path control error obtained by taking difference between a multi-path reference signal and multi-path system output, and then respectively inputting the multi-path control error into a multi-path PID control unit with integral feedback; the outputs of the multipath PID control units with integral feedback are respectively connected with a unified multi-output bounded control unit; the multi-output bounded control unit has multiple outputs, namely the final output of the bounded PID control algorithm, and the multi-output bounded control unit controls the parameter k in a time-varying manner 0 Respectively feeding back to the PID control units with integral feedback.
Wherein each PID unit with integral feedback is expressed as:
where i=1, 2, …, n, n is the total number of ways of the bounded PID control algorithm for the MIMO system, which is related to the total number of ways of the MIMO system itself, u mi Is the intermediate control variable of the ith path, K Pi 、K Di 、K Ii Respectively a proportional control parameter, an integral control parameter and an integral control parameter E i ＝R i -Y i Is the i-th error signal. R is R i For the ith reference signal, Y i For the ith system output, k 0 K, a time-varying control parameter fed back from the bounded control unit 0 ∈(0,1]。
In each PID control unit with integral feedback, the parameter k is used for 0 And the parameter feedback of the PID control unit with integral feedback is carried out, so that the integral saturation or integral runaway phenomenon of the traditional PID algorithm is naturally avoided.
The multi-output limiting control unit is used for meeting the limiting requirements of a plurality of actuators or actuating mechanisms of the control system, namely
Problem, and will time-varying control parameter k 0 And the feedback is fed into a multi-path PID control unit with integral feedback and is used for avoiding the phenomenon of integral saturation or integral runaway of the traditional PID algorithm.
The multi-output bounded control unit includes a multi-way control output, represented as:
wherein u is i The multiplexing output of the bounded control unit, also the final output of the entire bounded PID control algorithm for MIMO systems, u mi The PID control unit is derived from the multipath integrated feedback; u (u) max > 0 is the output u of the multi-path control algorithm i An upper bound of the sum of orthogonal vectors; k is a constant control parameter which can be adjusted, and k is more than 0, k i Is also an adjustable constant value control parameter corresponding to each path, and k i ＞0，k 0 Is a time-varying control parameter, and k 0 ∈(0,1]And the PID control unit with integral feedback is further fed back to each path from the multi-output bounded control unit.
Consider the following lyapunov function:
the derivative of the Lyapunov function is obtained:
substituting a bounded algorithm for a MIMO system, equation (6), into the above-described similar reduction calculation yields:
the equation can be obtained by solving the above equation:
thus by proper choice of V m Initial value V of (t) m (0) For example, let u i (0) =0 and k 0 (0) =1, further by equation (16), thereby obtaining
Once V is m (t) =1, u can be obtained i And k is equal to 0 The relation is:
from this equation, the final control output of the bounded PID control algorithm is obtained
Thereby ensuring the bounded PID control algorithmOutput limitations in MIMO systems. And it can be noted that if the final control output of the bounded PID control algorithm has approached an extreme value, i.e
Time-varying control parameter k 0 Will approach 0, i.e. k 0 -0; through k 0 Feedback of the integral term of the multi-path PID control unit with integral feedback causes the integral term of the multi-path PID control unit with integral feedback to approach 0, namely k 0 K Ii ∫E i dt→0; thus naturally avoiding the phenomenon of integral saturation or integral runaway.
Closed loop system stability based on bounded PID control algorithm is demonstrated as follows:
consider the following interconnection system, as shown in fig. 7. Wherein the system Σ 1 Is a controlled object and meets the requirement of bounded input and output stability, and the system sigma 2 Is a bounded PID control algorithm. The interconnection system can be applied to SISO systems and MIMO systems. In SISO systems, Y, u, R are scalar quantities and d is the disturbance of the controlled object. In a MIMO system, Y, u, R are vectors, i.e., y= [ Y ] 1 ,Y 2 ,…Y n ]，u＝[u 1 ,u 2 ,…u n ]，R＝[R 1 ,R 2 ,…R n ]。
Consider the system Σ 1 Satisfying the stability of bounded input and output, there is a KL function beta 1 () Two K ∞ Function gamma u () And gamma d () And a positive fixed value K 1 ，K 2 And K 3 The method comprises the steps of carrying out a first treatment on the surface of the And satisfies Y (0) | < K for an arbitrary initial state Y (0) 1 To any input u (t) satisfy sup t≥0 ||u(t)||＜K 2 And to any external disturbance d (t) satisfy sup t≥0 ||d(t)||＜K 3 The method comprises the steps of carrying out a first treatment on the surface of the For all t.gtoreq.0, there is
Wherein alpha is 1 And 0 is a fixed value.
System sigma 2 Is a bounded PID control algorithm. For the bounded PID control algorithm for SISO systems, there are:
||u(t)||≤max(||u min ||,||u max ||) (19)
for the bounded PID control algorithm for MIMO systems, there are:
||u(t)||≤u max (20)
therefore consider the system Σ 2 If there is a KL function beta 2 () Two K ∞ Function gamma R () And gamma Y () And a positive fixed value K 4 ，K 5 And K 6 The method comprises the steps of carrying out a first treatment on the surface of the And satisfies u (0) < K for any initial state u (0) 4 To any input Y (t) satisfy sup t≥0 ||Y(t)||＜K 5 And R (t) is satisfied for any external reference t≥0 ||R(t)||＜K 6 The method comprises the steps of carrying out a first treatment on the surface of the For all t.gtoreq.0, there is
Wherein alpha is 2 Max is max (||u) min ||,||u max I) or u max Respectively correspond to a SISO system and a MIMO system.
To sum up the equation (18) and the equation (21), two K can be easily found ∞ Function ρ 1 And ρ 2 And a non-negative real number s l The following equation is satisfied:
wherein I is d Is a unit function, and is a function compound operator. Based on the small gain theorem, the closed-loop control system based on fig. 7 satisfies the bounded input-output stabilization, corresponding to the bounded external disturbance d (t) and the external reference R (t).
The application of the bounded PID control algorithm based on the SISO system in a boost DC-DC conversion system is exemplified as follows:
the boost DC-DC converter system is shown in FIG. 8, with the DC input V on the left side in The right side is the direct current output V out The power electronic device (MOSFET or IGBT) is used for controlling the energy storage and discharge of the inductor, and the unidirectional effect of the diode and the energy storage of the capacitor are further combined, so that the voltage boost conversion is realized. The control input u of the system is the average working state of the power device, and the system output is the system output voltage V out . Due to the limitation of the boost DC-DC conversion system, the control input u of the system needs to be limited within a certain working range.
In order to further verify the effectiveness, stability and reliability of the bounded PID control algorithm proposed by the present invention, a conventional PID control algorithm in combination with a saturation unit is also used for comparison in the boost dc-dc conversion system.
The direct current output V of the boost direct current-direct current conversion system out The target of (2) is 100V, and the control parameters of the system are as follows:
P＝0.05；
I＝1；
k＝1000；
k1＝200；
0≤u≤0.75.
DC input V in The initial value of (1) is 50V, at t=1s, the dc input V in The variation was 30V.
The control results based on two different control algorithms are shown in fig. 9 and 10 respectively, wherein fig. 9 (a) is the algorithm output voltage control result of the present invention, (b) is the algorithm inductor current control result of the present invention, and (c) is the algorithm control input curve of the present invention; fig. 10 (a) shows the output voltage control result of the conventional method, (b) shows the inductor current control result of the conventional method, and (c) shows the control input curve of the conventional method. Fig. 9 is a diagram corresponding to the bounded PID control algorithm based on SISO system, and fig. 10 is a diagram corresponding to the conventional PID control algorithm combined with a saturation unit. By comparison, before t=1s, both control algorithms can achieve the control purpose wellTarget, make direct current output voltage V out Stabilize to 100V. And the inductor current and the control input u can also remain stable. When at t=1s, the DC input V in The SISO system based bounded PID control algorithm is still well controlled and stabilizes the output voltage at 30V. Although the output voltage and input current contain a small spike, the system can settle quickly. Due to the DC input voltage V in Is increased to a state near the upper limit of 0.75. Whereas after t=1s, the conventional PID control algorithm cannot control and stabilize the output voltage well in combination with the saturation unit, and there is a wide range of jitter in the output voltage, inductor current and control input signal, which is caused by the integral saturation or integral runaway phenomenon of the conventional PID control algorithm. Therefore, by comparison, the bounded PID control algorithm provided by the invention can ensure the output bounded requirement of the controller or the bounded requirement of the executor, can avoid the phenomenon of integral saturation or integral runaway of the traditional PID control algorithm, and can enhance the stability and reliability of the system.
While the applicant has described and illustrated the embodiments of the present invention in detail with reference to the drawings, it should be understood by those skilled in the art that the above embodiments are only preferred embodiments of the present invention, and the detailed description is only for the purpose of helping the reader to better understand the spirit of the present invention, and not to limit the scope of the present invention, but any improvements or modifications based on the spirit of the present invention should fall within the scope of the present invention.
Claims (9)
1. A bounded PID control algorithm for a SISO and MIMO system, comprising: a single-path or multi-path PID control unit with integral feedback, and a single-output or multi-output bounded control unit;
the PID control unit with integral feedback and the single-output bounded control unit are used for a SISO system, and the PID control unit with integral feedback and the multi-output bounded control unit are used for a MIMO system;
the bounded PID control algorithm for SISO and MIMO system makes the difference between the single-path or multi-path reference signal and the single-path or multi-path system output to obtain single-path or multi-path control error, and then inputs the control error to the single-path or multi-path PID control unit with integral feedback; the output of the single-path or multi-path PID control unit with integral feedback is connected with the single-output or multi-output bounded control unit; the final output of the single-output or multi-output bounded control unit is the final output of the bounded PID control algorithm; and the single-output or multi-output bounded control unit controls a time-varying control parameter k 0 Feedback to a single-path or multi-path PID control unit with integral feedback;
the single-output bounded control unit is used for receiving the output of the PID control unit fed back by single-path integration, then outputting the final output of the bounded PID control algorithm for the SISO system, and controlling the parameter k in a time-varying way 0 Feeding back to a PID control unit with integral feedback in a single path;
the single output bounded control unit is represented as:
wherein u is the output of the single-output bounded control unit, and is the final output of the entire bounded PID control algorithm for the SISO system; u (u) m The PID control unit is derived from the single-path integral feedback; u (u) max And u min Controlling the maximum value and the minimum value of the output for the bounded PID control algorithm; k is a constant control parameter which can be adjusted, and k is more than 0, k 1 Is also a constant control parameter which can be adjusted, and k 1 ＞0，k 0 Is a time-varying control parameter and is further fed back toIn the PID control unit with integral feedback in one path, and k 0 ∈(0,1]；
For the output u of the single output bounded control unit, give u and k 0 The relation is:
the equation is an elliptic equation by which the final control output u e (u) of the bounded PID control algorithm is obtained min ,u max ) Thereby ensuring the output of the bounded PID control algorithm;
the following conclusions are drawn from the elliptic equation: if the final control output of the bounded PID control algorithm for a SISO system has approached a maximum or minimum, i.e., u→u min Or u.fwdarw.u max Then time-varying control parameter k 0 Will approach 0, i.e. k 0 -0; through k 0 Feedback of the integral term of the PID control unit with integral feedback causes the integral term of the PID control unit with integral feedback to approach 0, namely k 0 K I ∫Edt→0。
2. The bounded PID control algorithm for a SISO and MIMO system according to claim 1,
the method is characterized in that:
in the SISO system, a single-path control error is subjected to PID control unit and single-output bounded control unit fed back by single-path band integration to obtain final controller output;
a bounded problem for SISO systems is u.epsilon (u) min ,u max ) The problem, u, is the final output of the control algorithm or the control input of the actuator, u max And u min The upper and lower bounds of the final output of the control algorithm or the actuator control input, respectively.
3. The bounded PID control algorithm for a SISO and MIMO system according to claim 2,
the method is characterized in that:
the PID control unit of the single-path integral feedback comprises: the system comprises a proportion unit, a differentiation unit and an integration unit with feedback;
the PID control unit with single-path integral feedback receives a control error obtained by differencing a reference signal and system output, the control error is respectively input into a proportional unit, a differential unit and an integral unit with feedback of the PID control unit with single-path integral feedback, and then the outputs of the proportional unit, the differential unit and the integral unit with feedback are added to obtain the output of the PID control unit with single-path integral feedback.
4. The bounded PID control algorithm for a SISO and MIMO system according to claim 3,
the method is characterized in that:
in the PID control unit of the single-path integral feedback:
the proportion unit is as follows: k (K) P E, wherein K P E is an error obtained by performing difference between a reference signal and system output;
the differentiating unit is as follows:
wherein K is D Is a differential coefficient +.>
Is the differentiation of the error;
the integrating unit with feedback is as follows: k (k) 0 K I Jedt, where k 0 K, which is a time-varying control parameter fed back from the single-output bounded control unit 0 ∈(0,1]，K I As an integral coefficient, +.Edt is the integral of the error;
the PID control unit of the single-path integral feedback is expressed as:
wherein u is m The output of the PID control unit with integral feedback is an intermediate control variable, and can be input into the single-output bounded control unit.
5. The bounded PID control algorithm for a SISO and MIMO system according to claim 1,
the method is characterized in that:
in the MIMO system, the multipath control error is output by a multipath PID control unit with integral feedback and a multiple-output bounded control unit to obtain a final controller;
the bounded problem for the MIMO system is that
Problem u i For the final output of the multiplex control algorithm or for the control input of the multiplex actuator, u max The upper bound of the sum of the orthogonal vectors is output for the multipath control algorithm.
6. The bounded PID control algorithm for a SISO and MIMO system according to claim 5,
the method is characterized in that:
in the multi-path PID control unit with integral feedback, each PID control unit with integral feedback is expressed as:
where i=1, 2, …, n, n is the total number of paths of the bounded PID control algorithm for MIMO system, u mi Is the intermediate control variable of the ith path, K Pi 、K Di 、K Ii Respectively a proportional control parameter, an integral control parameter and an integral control parameter E i ＝R i -Y i For the ith error signal, R i For the ith reference signal, Y i For the ith systemGo out, k 0 K, a time-varying control parameter fed back from the bounded control unit 0 ∈(0,1]。
7. The bounded PID control algorithm for a SISO and MIMO system according to claim 5,
the method is characterized in that:
the multi-output bounded control unit, expressed as:
wherein i=1, 2, …, n, n is the total number of paths of the bounded PID control algorithm for MIMO system; u (u) i The multipath output of the bounded control unit is also the final output of the bounded PID control algorithm for the whole MIMO system; u (u) mi The PID control unit is derived from the multipath integrated feedback; u (u) max > 0 is the output u of the multi-path control algorithm i An upper bound of the sum of orthogonal vectors; k is a constant control parameter which can be adjusted, and k is more than 0; k (k) i Is also an adjustable constant value control parameter corresponding to each path, and k i ＞0；k 0 Is a time-varying control parameter, and k 0 ∈(0,1]And the PID control unit with integral feedback is further fed back to each path from the multi-output bounded control unit.
8. The bounded PID control algorithm for a SISO and MIMO system according to claim 7,
the method is characterized in that:
multiplexed output u for the bounded control unit i Give u i And k is equal to 0 The relation is:
from this equation, the final control output of the bounded PID control algorithm is obtained
Thereby ensuring the output of the bounded PID control algorithm in the MIMO system.
9. The bounded PID control algorithm for SISO and MIMO systems as claimed in claim 6 or 7, characterized in that:
the integration unit with feedback and u for the multiple paths of the bounded PID control algorithm of the MIMO system i And k is equal to 0 Is the relation of:
the bounded PID control algorithm solves the problem of MIMO systems
The problem, and once the final control output of the bounded PID control algorithm for the MIMO system has approached an extremum, namely
Time-varying control parameter k 0 Will approach 0, i.e. k 0 -0; through k 0 Feedback of the integral term of the multi-path PID control unit with integral feedback causes the integral term of the multi-path PID control unit with integral feedback to approach 0, namely k 0 K Ii ∫E i dt→0。/>
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