CN112327636A - Preset performance control method based on preset track - Google Patents

Preset performance control method based on preset track Download PDF

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CN112327636A
CN112327636A CN202011363531.9A CN202011363531A CN112327636A CN 112327636 A CN112327636 A CN 112327636A CN 202011363531 A CN202011363531 A CN 202011363531A CN 112327636 A CN112327636 A CN 112327636A
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侯明哲
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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Abstract

The invention discloses a preset performance control method based on a preset track, which comprises the following steps: firstly, defining a control object and a performance requirement; constructing a performance function and a performance envelope of the system; thirdly, generating an expected error track in the performance envelope; designing a sliding mode control law to enable the actual error to move along an expected error track, so that the actual error is guaranteed to move within a performance envelope; and step five, checking the performance of the preset performance control law. The invention avoids the common singularity problem of the existing predetermined performance control method and provides a novel predetermined performance control method based on the preset track. Because the singularity problem of the control law brings potential risks to actual engineering, the method can ensure that higher control reliability is obtained.

Description

Preset performance control method based on preset track
Technical Field
The invention belongs to the technical field of automatic control, relates to a nonlinear control method, and particularly relates to a preset performance control method based on a preset track.
Background
The predetermined performance control means that a control algorithm is designed for a given controlled object to enable a closed-loop system to be stable and have preset transient performance and steady-state performance, for example, a tracking error e (t) moves in a given performance envelope, namely, the condition that e (t) epsilon (-delta) is metmρ(t),δMρ (t)), where δm＞0,δM> 0 is two constants and ρ (t) > 0 is the given performance function. The existing predetermined performance control methods include three major categories of funnel control, predetermined performance control based on nonlinear mapping, and predetermined performance control based on barrier Lyapunov function. However, through careful analysis, it is readily apparent that these predetermined performance control methods all inevitably present a control law odds with oneThe problem of the opposite sex. This problem is exemplified below by a predetermined performance control method based on a non-linear mapping.
To ensure that the error e (t) satisfies the performance constraint e (t) epsilon (-delta)mρ(t),δMRho (t)) or e (t)/rho (t) epsilon (-delta)m,δM) The core step of the method is to construct a mapping phi to constrain the space (-delta)m,δM) Mapping to an unconstrained space (— ∞, + ∞), and then designing the control law so that ε (t) — Φ (e (t)/ρ (t)) is bounded, thereby ensuring that e (t) satisfies the predetermined performance constraint. Usually, the mapping Φ is taken in logarithmic form, so that there is ∈ (t) ═ ln ((1+ e (t)/ρ (t))/(1-e (t)/ρ (t))). ε (t) is used in the designed control law. In a real non-ideal situation (e.g. a sensor failure), the measured error signal becomes e (t) + Δ (t), where Δ (t) is an interference signal. Therefore, ∈ (t) ═ ln ((1+ (e (t) + Δ (t))/ρ (t))/(1- (e (t) + Δ (t)))/ρ (t)). It can be seen that even if e (t) itself satisfies the constraint e (t)/ρ (t) e (- δ)m,δM) But may nevertheless occur
Figure BDA0002804730110000028
Thus, a singularity problem may occur when calculating ε (t).
Disclosure of Invention
Aiming at overcoming the problem of control law singularity existing in the existing predetermined performance control method, the invention provides a predetermined performance control method based on a preset track aiming at a common second-order nonlinear system in engineering. The method can essentially avoid the singularity problem of the predetermined performance control law, thereby providing theoretical reference and technical support for relevant engineering practice.
The purpose of the invention is realized by the following technical scheme:
a preset track-based predetermined performance control method comprises the following steps:
step one, defining control objects and performance requirements:
consider a second order nonlinear system common in engineering as follows:
Figure BDA0002804730110000021
wherein, the ratio of x,
Figure BDA0002804730110000022
is a state variable of the system and is,
Figure BDA0002804730110000023
represents the derivative of x with respect to time t, t ≧ 0, u is the input variable of the system,
Figure BDA0002804730110000024
is a function of the continuity of the function,
Figure BDA0002804730110000025
d is a bounded interference signal satisfying the control coefficient
Figure BDA0002804730110000026
Wherein
Figure BDA0002804730110000027
Is a known constant;
the control law is required to be designed to make variable x track given second-order continuous micro-reference signal xd(t), and the tracking error e (t) ═ x (t) -xd(t) satisfies: the maximum overshoot is not more than sigma, and sigma is more than 0 and is a given constant; dynamic process convergence speed is not slower than e-ctC > 0 is a given constant; the adjustment time is not more than Tf，T f0 is a given constant; steady state error not greater than rho∞，ρ ∞0 is a given constant;
step two, constructing a performance function and a performance envelope of the system:
(1) the performance function for constructing a system according to design requirements is ρ (t) ═ ρ (ρ)0-ρ∞)e-ct+ρ∞Wherein the constant ρ0Satisfy rho0＞|e(0)|＞0；
(2) The performance envelope of the system is constructed from the performance function as follows:
(a) if the initial error e: (0) And more than or equal to 0, the performance envelope is a region between a curve-delta rho (t) and the curve rho (t), wherein delta is | e (0) | sigma/rho0；
(b) If the initial error e (0) < 0, the performance envelope is the region bounded by the curve- ρ (t) and the curve δ ρ (t), where δ ═ e (0) | σ/ρ0；
Step three, generating an expected error track eta (t) in the performance envelope, wherein t is more than or equal to 0:
within the performance envelope, the expected error trajectory for a second order system is constructed according to the following conditions:
(i) η (t) is a second order continuously differentiable function;
(ii)η(t)，
Figure BDA0002804730110000031
and
Figure BDA0002804730110000032
is bounded;
(iii)η(0)＝e(0)，
Figure BDA0002804730110000033
(iv)η(t)＝0，
Figure BDA0002804730110000034
step four, designing a control law to enable the actual error to move along the expected error track, so that the actual error is guaranteed to move in the performance envelope:
the control law of a second-order system can be designed by adopting a sliding mode control method, wherein a sliding mode variable is defined as:
Figure BDA0002804730110000035
wherein λ > 0 is a design parameter, z (t) ═ e (t) - η (t) represents the deviation between the actual error trajectory and the expected error trajectory, and the sliding mode control law is as follows:
Figure BDA0002804730110000036
wherein sign represents a sign function,
Figure BDA0002804730110000037
Figure BDA0002804730110000038
is a design parameter;
step five, checking the performance of a preset performance control law:
selecting a group of design parameters lambda and k, and then carrying out simulation or experimental verification, wherein if the performance of the closed-loop system meets the requirements, the design is finished; otherwise, the design parameters need to be adjusted, and the performance inspection is carried out again until the performance of the closed-loop system meets the requirements.
Compared with the prior art, the invention has the following advantages:
the invention avoids the common singularity problem of the existing predetermined performance control method and provides a novel predetermined performance control method based on the preset track. Because the singularity problem of the control law brings potential risks to actual engineering, the method can ensure that higher control reliability is obtained.
Drawings
FIG. 1 is a flow chart illustrating the design of a predetermined performance control method based on a predetermined trajectory according to the present invention;
fig. 2 is a tracking error variation curve.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings, but not limited thereto, and any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.
The invention provides a preset performance control method based on a preset track, which comprises the following design steps as shown in figure 1:
the first step is as follows: explicit control objects and performance requirements.
Consider a second order nonlinear system common in engineering as follows:
Figure BDA0002804730110000041
wherein, the ratio of x,
Figure BDA0002804730110000042
is a state variable of the system and is,
Figure BDA0002804730110000043
represents the derivative of x with respect to time t, t ≧ 0, u is the input variable of the system,
Figure BDA0002804730110000044
is a function of the continuity of the function,
Figure BDA0002804730110000045
d is a bounded interference signal satisfying the control coefficient
Figure BDA0002804730110000046
Wherein
Figure BDA0002804730110000047
Is a known constant.
The control design requirements are as follows: designing a control law to make variable x track given second-order continuous micro-reference signal xd(t), and the tracking error e (t) ═ x (t) -xd(t) satisfies: the maximum overshoot is not more than sigma, sigma is more than 0 and is a given constant, and the convergence speed of the dynamic process is not slower than e-ctC > 0 is a given constant and the adjustment time is not more than Tf，Tf> 0 is a given constant, and the steady state error is not more than rho∞，ρ∞> 0 is a given constant.
The second step is that: and constructing a performance function and a performance envelope of the system.
The performance function is made to be rho (t) ═ rho (rho) according to the design requirements0-ρ∞)e-ct+ρ∞Wherein the constant ρ0Satisfy rho0> | e (0) | > 0. The performance envelope is constructed from the performance function as follows:
(1) if the initial error e: (0) And more than or equal to 0, the performance envelope is a region sandwiched by a curve-delta rho (t) and the curve rho (t), wherein delta is | e (0) | sigma/rho0；
(2) If the initial error e (0) < 0, the performance envelope is the region bounded by the curve- ρ (t) and the curve δ ρ (t), where δ ═ e (0) | σ/ρ0；
Obviously, as long as the tracking error e (t) moves within the performance envelope described above, it can be guaranteed that it satisfies: maximum overshoot is not more than sigma > 0, and convergence speed of dynamic process is not slower than e-ctC > 0, steady state error not greater than rho∞＞0。
The third step: expected error trajectories are generated within the performance envelope.
Within the designed performance envelope, an expected error trajectory eta (t) of a second-order system can be generated according to the following conditions, wherein t is more than or equal to 0:
(i) η (t) is a second order continuously differentiable function;
(ii)η(t)，
Figure BDA0002804730110000051
and
Figure BDA0002804730110000052
is bounded;
(iii)η(0)＝e(0)，
Figure BDA0002804730110000053
(iv)η(t)＝0，
Figure BDA0002804730110000054
according to the four principles, T is equal to 0 and T is equal to TfWhen the interpolation node is regarded as an interpolation node, an expected error track can be constructed by using various interpolation methods, and then whether the error track is strictly contained in the performance envelope is checked, and if the error track exceeds the performance envelope, proper adjustment is needed.
The fourth step: the control law is designed so that the actual error moves along the expected error trajectory, thereby ensuring that the actual error moves within the performance envelope.
The control law of the second-order system can be designed by adopting a sliding mode control method. Defining the deviation between the actual error trajectory and the expected error trajectory as:
z(t)＝e(t)-η(t) (2)；
further defining the sliding mode variables as:
Figure BDA0002804730110000061
wherein λ > 0 is a design parameter.
The sliding mode control law is designed as follows:
Figure BDA0002804730110000062
wherein sign represents a sign function,
Figure BDA0002804730110000063
Figure BDA0002804730110000064
are design parameters.
The Lyapunov function is constructed as:
Figure BDA0002804730110000065
its derivative with time satisfies:
Figure BDA0002804730110000066
since z (0) ═ e (0) — η (0) ═ 0, s (0) ═ 0, and further V (0) ═ 0. Thus, V (t) ≡ 0 and thus s (t) ≡ 0 can be obtained. From the definition of the sliding mode variables and the expected error trajectory, z (t) ≡ 0, i.e. e (t) ≡ η (t).
Since the desired error trajectory η (t) is strictly contained within the performance envelope, the actual error e (t) also moves strictly within the performance envelope, so that the maximum overshoot is no greater thanSigma is more than 0, and the convergence speed of the dynamic process is not slower than e-ctC > 0, steady state error not greater than rho∞Is greater than 0. And because e (t) ≡ η (t) ═ 0,
Figure BDA0002804730110000071
therefore, the adjustment time is not more than TfIs greater than 0. In summary, the designed control law can ensure that the closed loop system meets the predetermined performance.
The fifth step: the performance of a predetermined performance control law is checked.
To verify the performance of the designed control law, it can be applied to a practical second-order nonlinear system. Selecting a group of design parameters lambda and k, and then carrying out simulation or experimental verification, wherein if the performance of the closed-loop system meets the requirements, the design is finished; otherwise, the design parameters need to be adjusted, and the performance inspection is carried out again until the performance of the closed-loop system meets the requirements.
Example (b):
the design in the solution according to the invention is further illustrated here by way of a description of a certain representative embodiment.
Consider the following van der pol oscillator system with interference:
Figure BDA0002804730110000072
wherein,
Figure BDA0002804730110000073
d(t)＝0.1sin10t，x(0)＝1.4，
Figure BDA0002804730110000074
requiring a design control law to make variable x track reference signal xd(t) cost, and tracking error e (t) x (t) -xd(t) satisfies: maximum overshoot is not more than 0.1, and convergence speed of dynamic process is not slower than e-ctC is 1, the adjusting time is not more than Tf1s, steady state error no greater than ρ∞＝0.01。
According to the design requirement, make a performance function ofρ(t)＝(ρ0-ρ∞)e-ct+ρ∞Where ρ is00.5. Constructing a performance envelope from the performance function as the region bounded by the curve- δ ρ (t) and the curve ρ (t), where δ ═ e (0) | σ/ρ0＝0.08。
The expected error tracking trajectory is generated by using piecewise cubic polynomial interpolation as follows:
Figure BDA0002804730110000081
wherein, a0＝e(0)，
Figure BDA0002804730110000082
Wherein e (0) ═ x (0) -xd(0)＝0.4，
Figure BDA0002804730110000083
It is easy to verify that the expected error trajectory is always moving within the performance envelope.
The sliding mode control law is designed as follows:
Figure BDA0002804730110000084
wherein sign represents a sign function,
Figure BDA0002804730110000085
z (t) ═ e (t) - η (t), λ > 0, and k > 0.1 are design parameters.
Setting the control parameters as lambda being 1 and k being 0.2, and verifying the performance of the closed-loop system through simulation. The variation curve of the tracking error is shown in fig. 2. As can be seen from fig. 2, the control law designed in this embodiment can make the variable x track the reference signal xd(t) cost, and tracking error e (t) x (t) -xd(t) satisfies: maximum overshoot is 0, and the dynamic process convergence speed is faster than e-tAdjusting the time TfAnd the steady-state error is 0 for 1s, so that the design requirement is completely met.
The simulation result shows the correctness and the effectiveness of the control method provided by the invention.

Claims (5)

1. A method for controlling predetermined performance based on a predetermined trajectory, the method comprising the steps of:
firstly, defining a control object and a performance requirement;
constructing a performance function and a performance envelope of the system;
thirdly, generating an expected error track in the performance envelope;
designing a control law to enable the actual error to move along the expected error track, so that the actual error is guaranteed to move in the performance envelope;
step five, checking the performance of a preset performance control law:
carrying out simulation or experimental verification after selecting a group of design parameters, and finishing the design if the performance of the closed-loop system meets the requirements; otherwise, the design parameters need to be adjusted, and the performance inspection is carried out again until the performance of the closed-loop system meets the requirements.
2. The method according to claim 1, wherein the specific steps of the first step are as follows:
consider a second order nonlinear system as follows:
Figure FDA0002804730100000011
wherein, the ratio of x,
Figure FDA0002804730100000012
is a state variable of the system and is,
Figure FDA0002804730100000013
represents the derivative of x with respect to time t, t ≧ 0, u is the input variable of the system,
Figure FDA0002804730100000014
is a function of the continuity of the function,
Figure FDA0002804730100000015
to control the coefficient, d is a bounded interfering signal,
Figure FDA0002804730100000016
wherein
Figure FDA0002804730100000017
Is a known constant;
the control law is required to be designed to make variable x track given second-order continuous micro-reference signal xd(t), and the tracking error e (t) ═ x (t) -xd(t) satisfies: the maximum overshoot is not more than sigma, and sigma is more than 0 and is a given constant; dynamic process convergence speed is not slower than e-ctC > 0 is a given constant; the adjustment time is not more than Tf，Tf0 is a given constant; steady state error not greater than rho∞，ρ∞> 0 is a given constant.
3. The method according to claim 1, wherein the second step comprises the following steps:
(1) the performance function for constructing a system according to design requirements is ρ (t) ═ ρ (ρ)0-ρ∞)e-ct+ρ∞Wherein the constant ρ0Satisfy rho0＞|e(0)|＞0；
(2) The performance envelope of the system is constructed from the performance function as follows:
(a) if the initial error e (0) ≧ 0, the performance envelope is the region bounded by the curve- δ ρ (t) and the curve ρ (t), where δ ═ e (0) | σ/ρ0；
(b) If the initial error e (0) < 0, the performance envelope is the region bounded by the curve- ρ (t) and the curve δ ρ (t), where δ ═ e (0) | σ/ρ0。
4. The preset trajectory-based predetermined performance control method according to claim 1, wherein in the third step, the expected error trajectory η (t), t ≧ 0 satisfies:
(i) η (t) is a second order continuously differentiable function;
(ii)η(t)，
Figure FDA0002804730100000021
and
Figure FDA0002804730100000022
is bounded;
(iii)η(0)＝e(0)，
Figure FDA0002804730100000023
(iv)η(t)＝0，
Figure FDA0002804730100000024
5. the preset trajectory-based predetermined performance control method according to claim 1, wherein in the fourth step, the control law is a sliding mode control law, and the calculation formula is as follows:
Figure FDA0002804730100000025
wherein sign represents a sign function,
Figure FDA0002804730100000026
Figure FDA0002804730100000027
for design parameters, the sliding mode variables are defined as:
Figure FDA0002804730100000028
where λ > 0 is a design parameter and z (t) ═ e (t) - η (t) represents the deviation between the actual error trajectory and the desired error trajectory.
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Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111596545A (en) * 2020-04-27 2020-08-28 江苏建筑职业技术学院 Self-adaptive fault-tolerant preset performance control method for multi-input multi-output mechanical system
CN111650943A (en) * 2020-06-19 2020-09-11 哈尔滨理工大学 Track tracking preset performance control method for speed-limited still water dynamic positioning ship

Patent Citations (2)

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Publication number Priority date Publication date Assignee Title
CN111596545A (en) * 2020-04-27 2020-08-28 江苏建筑职业技术学院 Self-adaptive fault-tolerant preset performance control method for multi-input multi-output mechanical system
CN111650943A (en) * 2020-06-19 2020-09-11 哈尔滨理工大学 Track tracking preset performance control method for speed-limited still water dynamic positioning ship

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Title
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