CN112327636B - Preset performance control method based on preset track - Google Patents
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- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- 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
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
- G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
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; step two, 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
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 preset performance control means that a control algorithm is designed aiming at a given controlled object to enable a closed-loop system to be stable and have preset transient performance and steady-state performance, for example, the tracking error e (t) moves in a given performance envelope, namely, the condition that e (t) epsilon (-delta) is met m ρ(t),δ M ρ (t)), where δ m ＞0,δ M > 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 not difficult to find that these predetermined performance control methods inevitably have the singularity problem of the control law. This problem is exemplified below by a predetermined performance control method based on a non-linear mapping.
In order to ensure that the error e (t) meets the performance constraint e (t) epsilon (-delta) m ρ(t),δ M Rho (t)) or e (t)/rho (t) epsilon (-delta) m ,δ M ) The core step of the method is to construct a mapping phi to constrain the space (-delta) m ,δ M ) 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 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 thatMake e (t) satisfy the constraint e (t)/rho (t) epsilon (-delta) m ,δ M ) But may nevertheless occur
Thus, a singularity problem may occur when calculating ε (t).
Disclosure of Invention
Aiming at solving 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 predetermined performance control method based on a preset track comprises the following steps:
step one, defining control objects and performance requirements:
consider a second order nonlinear system common in engineering as follows:
wherein, the ratio of x,
is a status variable of the system, is>
Represents the derivative of x with respect to time t, t ≧ 0, u is the input variable of the system,
is a continuous function, is->
D is a bounded disturbance signal for a control coefficient, satisfies >>
Wherein->
Is a known constant;
the design control law is required to make variable x track given second-order continuous micro-reference signal x d (t), and a tracking error e (t) = x (t) -x d (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 -ct C > 0 is a given constant; the adjustment time is not more than T f ，T f 0 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) 0 -ρ ∞ )e -ct +ρ ∞ Wherein the constant ρ 0 Satisfy rho 0 ＞|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/ρ 0 ；
(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/ρ 0 ；
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)，
and &>
Is bounded;
(iii)η(0)＝e(0)，
(iv)η(t)＝0，
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:
wherein λ > 0 is a design parameter, z (t) = e (t) - η (t) represents a deviation between an actual error trajectory and an expected error trajectory, and a sliding mode control law is as follows:
wherein sign represents a sign function,
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 design flow chart 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:
wherein, the ratio of x,
is a status variable of the system, is>
Represents the derivative of x with respect to time t, t ≧ 0, u is the input variable of the system,
is a continuous function, based on>
To control systemA number, d is a bounded interfering signal satisfying >>
Wherein +>
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 x d (t), and a tracking error e (t) = x (t) -x d (t) satisfies: the maximum overshoot is not more than sigma, sigma > 0 is a given constant, and the convergence speed of the dynamic process is not slower than e -ct C > 0 is a given constant and the adjustment time is not more than T f ，T f 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 ρ (t) = (ρ) according to design requirements 0 -ρ ∞ )e -ct +ρ ∞ Wherein constant ρ 0 Satisfies rho 0 > | e (0) | > 0. The performance envelope is constructed from the performance function as follows:
(1) 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/ρ 0 ；
(2) If the initial error e (0) < 0, the performance envelope is the region sandwiched by the curve- ρ (t) and the curve δ ρ (t), where δ = | e (0) | σ/ρ 0 ；
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 -ct C > 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)，
and &>
Is bounded;
(iii)η(0)＝e(0)，
(iv)η(t)＝0，
according to the four principles, T =0 and T = T f When 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:
wherein λ > 0 is a design parameter.
The sliding mode control law is designed as follows:
wherein sign represents a sign function of the symbol,
are design parameters.
The Lyapunov function is constructed as:
its derivative with time satisfies:
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).
Because the expected error track eta (t) is strictly contained in the performance envelope, the actual error e (t) also moves strictly in the performance envelope, so that the maximum overshoot is not more than sigma and more than 0, and the convergence speed of the dynamic process is not slower than e -ct C > 0, steady state error not greater than rho ∞ Is greater than 0. And because e (t) ≡ η (t) =0,
therefore, the adjustment time is not more than T f Is 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 test 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:
wherein, the first and the second end of the pipe are connected with each other,
d(t)＝0.1sin10t，x(0)＝1.4，/>
requiring a design control law to make variable x track reference signal x d (t) = cost, and tracking error e (t) = x (t) -x d (t) satisfies: the maximum overshoot is not more than sigma =0.1, and the convergence speed of the dynamic process is not slower than e -ct C =1, the adjustment time is not more than T f =1s, steady state error not greater than ρ ∞ ＝0.01。/>
The performance function is ρ (t) = (ρ) according to design requirements 0 -ρ ∞ )e -ct +ρ ∞ Where ρ is 0 =0.5. Constructing a performance envelope as an area between a curve- δ ρ (t) and the curve ρ (t) according to a performance function, where δ = | e (0) | σ/ρ 0 ＝0.08。
The expected error tracking trajectory is generated by using piecewise cubic polynomial interpolation as follows:
wherein, a 0 ＝e(0)，
Wherein e (0) = x (0) -x d (0)＝0.4，/>
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:
wherein sign represents a sign function,
z (t) = e (t) - η (t), λ > 0, k > 0.1 are design parameters.
Setting the control parameter as lambda =1, k =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 x d (t) = cost, and tracking error e (t) = x (t) -x d (t) satisfies: the maximum overshoot is 0, and the convergence speed of the dynamic process is faster than e -t Adjusting the time T f And 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 (1)
1. A method for controlling predetermined performance based on a predetermined trajectory, the method comprising the steps of:
the method comprises the following steps of firstly, defining control objects and performance requirements, and specifically comprising the following steps:
consider the following van der pol oscillator system with interference:
wherein, the ratio of x,
is a status variable of the system, is>
Represents the derivative of x with respect to time t, t ≧ 0, u is the input variable of the system,
d is a bounded interfering signal, i.e. < >>
Wherein->
Is a known normal number;
requiring a design control law to make variable x track reference signal x d (t), and a tracking error e (t) = x (t) -x d (t) satisfies: maximum overshoot is not more than sigma, and convergence speed of dynamic process is not slower than e -ct C is more than 0, and the adjusting time is not more than T f Steady state error no greater than rho ∞ ；
Step two, constructing a performance function and a performance envelope of the system, and specifically comprising the following steps:
(1) The performance function for constructing a system according to design requirements is ρ (t) = (ρ) 0 -ρ ∞ )e -ct +ρ ∞ Where ρ is 0 ＞|e(0)|＞0；
(2) Constructing a performance envelope from the performance function:
(a) If the initial error e (0) ≧ 0, the performance envelope is the region sandwiched by the curve- δ ρ (t) and the curve ρ (t), where δ = | e (0) | σ/ρ 0 ；
(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/ρ 0 ；
Step three, generating an expected error track in the performance envelope by adopting segmented cubic polynomial interpolation:
one possible way to find the coefficients is: a is 0 ＝e(0)，
Wherein e (0) = x (0) -x d (0)，/>
Step four, designing a sliding mode control law to enable the actual error to move along an expected error track, and accordingly ensuring that the actual error moves within a performance envelope, wherein: the sliding mode control law is designed as follows:
wherein sign represents a sign function,
z (t) = e (t) - η (t) represents the deviation between the actual error trajectory and the expected error trajectory, λ > 0, k > 0.1 are design parameters;
step five, checking the performance of a preset performance control law:
after a group of design parameters is selected, carrying out simulation or experimental verification, and if the performance of the closed-loop system meets the requirements, ending the design; 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.
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| 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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| 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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| 马广富 ; 朱庆华 ; 王鹏宇 ; 郭延宁 ; .基于终端滑模的航天器自适应预设性能姿态跟踪控制.航空学报.2018,(06),321763-1-321763-11. * |
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