CN110361967A - The construction method of sliding mode observer - Google Patents
The construction method of sliding mode observer Download PDFInfo
- Publication number
- CN110361967A CN110361967A CN201910420812.4A CN201910420812A CN110361967A CN 110361967 A CN110361967 A CN 110361967A CN 201910420812 A CN201910420812 A CN 201910420812A CN 110361967 A CN110361967 A CN 110361967A
- Authority
- CN
- China
- Prior art keywords
- sliding mode
- mode observer
- sliding
- state
- state variable
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- 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
Landscapes
- Engineering & Computer Science (AREA)
- Health & Medical Sciences (AREA)
- Artificial Intelligence (AREA)
- Computer Vision & Pattern Recognition (AREA)
- Evolutionary Computation (AREA)
- Medical Informatics (AREA)
- Software Systems (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Automation & Control Theory (AREA)
- Electric Propulsion And Braking For Vehicles (AREA)
- Feedback Control In General (AREA)
Abstract
The invention discloses a kind of construction methods of sliding mode observer, include the following steps, step 1, vehicle drive shaft is reduced to the system being made of spring and damping, the state space equation of sliding mode observer is established to the system, step 2, the output estimation error feedback term in utilization index Reaching Law and exponential function building sliding mode observer state space equation；Step 3, based on output estimation error feedback term building sliding mode observer equation and structure；The sliding mode observer that the present invention constructs can the torque estimation result in time to vehicle correct so that drive system torque estimated value approach actual value, alleviate chattering phenomenon, reduce phase delay, observed result is more accurate.
Description
Technical field
The invention belongs to sliding formwork control technical fields, more particularly to a kind of construction method of sliding mode observer.
Background technique
When calculating drive shaft torque using line integral method in the prior art, wheel speed sensor, motor rotation will receive
Become error influence, signals of rotational speed sensor noise, external interference etc. also due to integral action in torque estimation result by
It gradually adds up, estimated value and actual value is finally made to generate large error, when Assumption torque initial value and actual value have deviation, evaluated error
May be very big, application difficult in practice.
Luenberger observer has preferable observation effect, but vehicle for the system for being mainly shown as linear characteristic
It is typical non-thread since there are the non-linear factors such as Coulomb damping, air drag and backlash in power drive system
Property system, while more simplification has been carried out in transmission system modeling process, there is likely to be unknown disturbances in real system,
Zero can not be leveled off to using the evaluated error that Systems with Linear Observation device exports drive system, and the system mode of estimation is difficult to converge to
System virtual condition.
Therefore, it is necessary to be fed back using practical can measure, torque estimation result is corrected, enable estimated value with
With actual value, and the influence of external interference is resisted, that is, needs to design a sliding mode observer and driving shaft torque is observed.
Summary of the invention
The purpose of the present invention is to provide a kind of construction methods of sliding mode observer, and building process is simple, utilization index letter
Number realizes that under nonlinear state, the torque estimated value of drive system and the deviation of actual value reduce, and eliminates drive system shape
State approaches buffeting when sliding-mode surface, reduces phase delay, so that the sliding mode observer observed result accuracy of building is higher.
The technical scheme adopted by the invention is that the construction method of sliding mode observer specifically includes the following steps:
Step 1, vehicle drive shaft is reduced to the system being made of spring and damping, the state for establishing sliding mode observer is empty
Between equation, the general type of sliding mode observer such as formula (1):
If x is state variable,Y is output quantity,For state estimation error,It is defeated
Evaluated error out, v are the feedback term of output estimation error, GnFor gain matrix,
For motor speed,To export revolving speed, TsFor output shaft torque, CtFor retarder damping, CvFor wheel damping, CsFor
Drive shaft damping, JvIt is equivalent to the sum of the equivalent inertia of wheel and wheel inertia, J for car bodypFor the rotor inertia of motor B, irFor
Final driver ratio, ksFor drive shaft rigidity, i is final driver ratio, TpFor driving torque, TvFor vehicle load torque；
Step 2, using exponentially approaching rule and exponential function, the output estimation error feedback term of sliding mode observer is set；
Step 3, sliding mode observer equation and structure are established.
Further, in step 2 output estimation error feedback term are as follows:
V=- ε sigmoid (s)-qs ε > 0, q > 0
Wherein ε is sliding formwork gain, and q is exponential approach gain, and s is drive system state at a distance from sliding-mode surface,It is any
Positive integer.
Further, the equation for the sliding mode observer that step 3 is established are as follows:
WhereinIt is state variable x1Measured value,It is state variable x2Measured value,For state variable x3Measurement
Value, s1Be state variable be x1When drive system state at a distance from sliding-mode surface, s2Be state variable be x2When drive system state
At a distance from sliding-mode surface, q1It is state variable x1Exponential approach gain, q2It is state variable x2Exponential approach gain, ε1It is shape
State variable x1Sliding formwork gain, ε2It is state variable x2Sliding formwork gain, L1、L2For feedback gain；
The state error of sliding mode observerEquation are as follows:
Further, sliding mode observer sliding formwork gain selection should meet the following conditions:
Wherein e3It is state variable x3State estimation error.
The beneficial effects of the present invention are: the process that the present invention constructs sliding mode observer is simple, index letter is used in building
Number eliminates the buffeting generated when drive system state approach sliding-mode surface, when so that the sliding mode observer of building being used to detect, detection
Order of accuarcy is high.
Detailed description of the invention
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below
There is attached drawing needed in technical description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this
Some embodiments of invention for those of ordinary skill in the art without creative efforts, can be with
It obtains other drawings based on these drawings.
Fig. 1 is dynamical system simplified model figure.
Fig. 2 is torque sliding mode observer structure chart.
Fig. 3 is that the present invention cuts down result figure to the torsional oscillation of linear model.
Fig. 4 is that the present invention cuts down result figure to the torsional oscillation of nonlinear model.
Specific embodiment
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation description, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, it is obtained by those of ordinary skill in the art without making creative efforts every other
Embodiment shall fall within the protection scope of the present invention.
The construction method of sliding mode observer, specifically includes the following steps:
Step 1: vehicle drive shaft being reduced to the system being made of spring and damping, structure is as shown in Figure 1, establish sliding formwork
The state space equation of observer, shown in the general type of sliding mode observer such as formula (1):
X is state variable in formula (1),Y is output quantity, ifFor state estimation error,For output estimation error, v is the feedback term of output estimation error, GnFor gain matrix,
WhereinFor motor speed,To export revolving speed, TsFor output shaft torque, CtFor retarder damping, CvFor wheel resistance
Buddhist nun, CsFor drive shaft damping, JvIt is equivalent to the sum of the equivalent inertia of wheel and wheel inertia, J for car bodypIt is used for the rotor of motor B
Amount, irFor final driver ratio, ksFor drive shaft rigidity, i is final driver ratio, TpFor driving torque, TvIt is negative for vehicle
Set torque；
Step 2: using exponentially approaching rule and exponential function, the output estimation error feedback term v of sliding mode observer is set；
Feedback term of the utilization index Reaching Law as output estimation error, so that torque estimated value rapidly converges to reality
Value, shown in exponentially approaching rule such as formula (2):
V=- ε sgn (s)-qs ε > 0, q > 0 (2)
Wherein ε is sliding formwork gain, and q is exponential approach gain, and s is drive system state at a distance from sliding-mode surface；Index becomes
Nearly rule converges to any point in state space on sliding-mode surface with exponential form, and convergence rate increases with exponential approach gain q
And accelerate, but the excessive state variable that will lead to of exponential approach gain has biggish chatter on sliding-mode surface；Drive system state
Far from sliding-mode surface, i.e., when s absolute value is larger, drive system state can fast approach sliding-mode surface, so that s=0；Drive system state
Close to sliding-mode surface, i.e., when s absolute value is smaller, the speed of drive system state approach sliding-mode surface is gradually reduced, and switches item-ε sgn
(s) it can guarantee that drive system reaches sliding-mode surface in finite time；
The symbolization function sgn (s) in the feedback term of output estimation error, so that drive system state reaches sliding formwork
With certain speed when face, sliding-mode surface can be crossed under effect of inertia, then control action is reversed, subtracts drive system state
Speed reversely approaches sliding-mode surface, repeatedly, causes to buffet, and the present invention eliminates drive system using exponential function sigmoid (s)
State approaches the buffeting generated when sliding-mode surface, shown in the calculating of exponential function such as formula (3):
WhereinFor any positive integer；
With the sgn (s) in formula (3) replacement formula (2), the feedback term such as formula (4) of output error in sliding mode observer
It is shown:
V=- ε sigmoid (s)-qs ε > 0, q > 0 (4)
Step 3, sliding mode observer equation and structure are established；
According to the state space equation of sliding mode observer, sliding-mode surface function S is set are as follows:
Gain matrix GnForm beWherein L ∈ R1 ×2, I is unit matrix, and L is the coefficient matrix of 1*2, and R indicates that L is real number matrix；
Then shown in the expression formula of sliding mode observer such as formula (5):
Wherein JpFor the rotor inertia of motor B, JvIt is equivalent to the sum of the equivalent inertia of wheel and wheel inertia, T for car bodyv
For vehicle load torque, CtFor retarder damping, CvFor wheel damping, CsFor drive shaft damping, ksFor drive shaft rigidity, TPIt is
Force system driving torque,It is state variable x1Measured value,It is state variable x2Measured value,For state variable x3's
Measured value, s1Be state variable be x1When drive system state at a distance from sliding-mode surface, s2Be state variable be x2When drive system
State is at a distance from sliding-mode surface, q1It is state variable x1Exponential approach gain, q2It is state variable x2Exponential approach gain, ε1
It is state variable x1Sliding formwork gain, ε2It is state variable x2Sliding formwork gain, L1、L2For feedback gain；
The error of sliding mode observerEquation are as follows:
Shown in the error such as formula (7) for the state variable and virtual condition that sliding mode observer measures:
e1It is state variable x1State estimation error, e2It is state variable x2State estimation error, e3It is state variable
x3State estimation error；
Utilize the stability of Lyapunov Theory of Stability verifying sliding mode observer, the Lyapunov function of sliding mode observer
As shown in formula (8):
Wherein S is the matrix that drive system state is formed with sliding-mode surface distance s；
It substitutes into formula (7) and can obtain to formula (8) derivation, and by derivation result:
By formula (9) it is found that being the stability for guaranteeing sliding mode observer, the sliding formwork gain of sliding mode observer is answered when choosing
Meet following condition:
1)e1> 0 and e2> 0,
2)e1> 0 and e2< 0,
3)e1< 0 and e2> 0,
4)e1< 0 and e2< 0,
JPFor the rotor inertia of motor B, ε is sliding formwork gain, and i is transmission ratio, and e is state estimation error；
Condition is chosen to the synovial membrane gain of sliding mode observer and carries out computational short cut, is derived as guaranteeing the progressive of sliding mode observer
The sliding formwork gain of stability, sliding mode observer should meet following condition:
Utilization index Reaching Law and exponential function in the sliding mode observer that the present invention constructs construct the anti-of output estimation error
Item is presented, so that drive system state can slow down in time when close to sliding-mode surface, alleviate drive system trembling near sliding-mode surface
Phenomenon of shaking reduces phase delay, so that the observed result of sliding mode observer so that the speed of estimated value approach actual value is faster
It is more accurate.
Embodiment
Transmission system simplified model is built in Matlab/Simulink, and the delay of power source is considered in modeling process
Characteristic and power source, the frictional resistance of actuated element etc. imitate sliding mode observer observation effect based on this simplified model
True verifying, as a result as shown in Figure 3 and Figure 4, from the figure 3, it may be seen that sliding mode observer has good observing capacity in a linear system,
The observed result of sliding mode observer can be good at tracking the virtual condition of vehicle, reach good observation effect, sliding formwork in Fig. 4
Observer is still able to maintain preferable observed result in nonlinear model, exists to stablize between observed result and actual torque and miss
Difference, sliding mode observer is when being observed driving shaft torque, directly by the torque command of power drive system as system
Torque output is inputted in sliding mode observer and is calculated, due to the presence of frictional resistance moment, so that the reality output of power source turns
There are steady state difference value between square and torque command, being reflected in is stable state observation error on sliding mode observer observed result, still
Error can guarantee within 5% substantially.
Each embodiment in this specification is all made of relevant mode and describes, same and similar portion between each embodiment
Dividing may refer to each other, and each embodiment focuses on the differences from other embodiments.Especially for system reality
For applying example, since it is substantially similar to the method embodiment, so being described relatively simple, related place is referring to embodiment of the method
Part explanation.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the scope of the present invention.It is all
Any modification, equivalent replacement, improvement and so within the spirit and principles in the present invention, are all contained in protection scope of the present invention
It is interior.
Claims (4)
1. the construction method of sliding mode observer, which is characterized in that specifically includes the following steps:
Step 1, vehicle drive shaft is reduced to the system being made of spring and damping, establishes the state space side of sliding mode observer
Journey, the general type of sliding mode observer such as formula (1):
If x is state variable,Y is output quantity,For state estimation error,For output estimation
Error, v are the feedback term of output estimation error, GnFor gain matrix,
For
Motor speed,To export revolving speed, TsFor output shaft torque, CtFor retarder damping, CvFor wheel damping, CsFor drive shaft resistance
Buddhist nun, JvIt is equivalent to the sum of the equivalent inertia of wheel and wheel inertia, J for car bodypFor the rotor inertia of motor B, irFor main reducing gear
Transmission ratio, ksFor drive shaft rigidity, i is final driver ratio, TpFor driving torque, TvFor vehicle load torque；
Step 2, using exponentially approaching rule and exponential function, the output estimation error feedback term of sliding mode observer is set；
Step 3, sliding mode observer equation and structure are established.
2. the construction method of sliding mode observer according to claim 1, which is characterized in that output estimation in the step 2
The feedback term of error are as follows:
V=- ε sigmoid (s)-qs ε > 0, q > 0
Wherein ε is sliding formwork gain, and q is exponential approach gain, and s is drive system state at a distance from sliding-mode surface,It is arbitrarily just whole
Number.
3. the construction method of sliding mode observer according to claim 1, which is characterized in that the sliding formwork that the step 3 is established
The equation of observer are as follows:
WhereinIt is state variable x1Measured value,It is state variable x2Measured value,For state variable x3Measured value, s1
Be state variable be x1When drive system state at a distance from sliding-mode surface, s2Be state variable be x2When drive system state and cunning
The distance of die face, q1It is state variable x1Exponential approach gain, q2It is state variable x2Exponential approach gain, ε1It is that state becomes
Measure x1Sliding formwork gain, ε2It is state variable x2Sliding formwork gain, L1、L2For feedback gain；
The state error of sliding mode observerEquation are as follows:
4. the construction method of sliding mode observer according to claim 1, which is characterized in that the sliding mode observer sliding formwork increases
Benefit, which is chosen, to meet the following conditions:
Wherein e3It is state variable x3State estimation error.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910420812.4A CN110361967B (en) | 2019-05-20 | 2019-05-20 | Construction method of sliding-mode observer |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910420812.4A CN110361967B (en) | 2019-05-20 | 2019-05-20 | Construction method of sliding-mode observer |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN110361967A true CN110361967A (en) | 2019-10-22 |
| CN110361967B CN110361967B (en) | 2022-01-11 |
Family
ID=68215234
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201910420812.4A Active CN110361967B (en) | 2019-05-20 | 2019-05-20 | Construction method of sliding-mode observer |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN110361967B (en) |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112526880A (en) * | 2020-11-25 | 2021-03-19 | 哈尔滨工业大学 | Real-time estimation method for road surface height in vehicle driving process |
| CN113824114A (en) * | 2021-11-22 | 2021-12-21 | 广东电网有限责任公司惠州供电局 | Power distribution network state estimation method, device, system and medium based on sliding mode observation |
| CN114785216A (en) * | 2022-05-12 | 2022-07-22 | 合肥工业大学 | Design method of self-adaptive gain sliding-mode observer |
Citations (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2013021453A1 (en) * | 2011-08-09 | 2013-02-14 | トヨタ自動車株式会社 | Sliding mode controller, and internal combustion engine system control apparatus |
| CN103647490A (en) * | 2013-09-27 | 2014-03-19 | 天津大学 | Permanent magnet motor sliding mode control strategy |
| CN104300863A (en) * | 2014-10-21 | 2015-01-21 | 天津大学 | Self-adaption sliding mode control method for speed regulation of variable-load permanent magnet synchronous motor |
| CN106541855A (en) * | 2016-10-27 | 2017-03-29 | 同济大学 | A kind of electric automobile starting shake Active Control Method based on POLE PLACEMENT USING |
| CN106777677A (en) * | 2016-12-14 | 2017-05-31 | 华南理工大学 | A kind of modeling method suitable for passenger car difference operating mode power train Analysis of Torsional Vibration |
| CN110361965A (en) * | 2019-05-20 | 2019-10-22 | 北京理工大学 | The construction method of linear Luenberger observer |
-
2019
- 2019-05-20 CN CN201910420812.4A patent/CN110361967B/en active Active
Patent Citations (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2013021453A1 (en) * | 2011-08-09 | 2013-02-14 | トヨタ自動車株式会社 | Sliding mode controller, and internal combustion engine system control apparatus |
| CN103647490A (en) * | 2013-09-27 | 2014-03-19 | 天津大学 | Permanent magnet motor sliding mode control strategy |
| CN104300863A (en) * | 2014-10-21 | 2015-01-21 | 天津大学 | Self-adaption sliding mode control method for speed regulation of variable-load permanent magnet synchronous motor |
| CN106541855A (en) * | 2016-10-27 | 2017-03-29 | 同济大学 | A kind of electric automobile starting shake Active Control Method based on POLE PLACEMENT USING |
| CN106777677A (en) * | 2016-12-14 | 2017-05-31 | 华南理工大学 | A kind of modeling method suitable for passenger car difference operating mode power train Analysis of Torsional Vibration |
| CN110361965A (en) * | 2019-05-20 | 2019-10-22 | 北京理工大学 | The construction method of linear Luenberger observer |
Non-Patent Citations (5)
| Title |
|---|
| KAI LIAO，YAN XU: "A Robust Load Frequency Control Scheme for Power Systems Based on Second-Order Sliding Mode and Extended Disturbance Observer", 《IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS》 * |
| KAI LIAO: "A robust damping controller for DFIG based on variable-gain sliding mode", 《ELECTRICAL POWER AND ENERGY SYSTEMS》 * |
| LIANGZHANG: "Fixed-time extended state observer based non-singular fast terminal sliding mode control for a VTVL reusable launch vehicle", 《AEROSPACE SCIENCE AND TECHNOLOGY》 * |
| 张冬旭: "可控滚转二维弹道修正机构的研究", 《中国博士学位论文全文数据库》 * |
| 陈云微，赵强: "车辆半主动座椅悬架滑模变结构控制", 《哈尔滨工程大学学报》 * |
Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112526880A (en) * | 2020-11-25 | 2021-03-19 | 哈尔滨工业大学 | Real-time estimation method for road surface height in vehicle driving process |
| CN112526880B (en) * | 2020-11-25 | 2022-06-10 | 哈尔滨工业大学 | Real-time estimation method for road surface height in vehicle driving process |
| CN113824114A (en) * | 2021-11-22 | 2021-12-21 | 广东电网有限责任公司惠州供电局 | Power distribution network state estimation method, device, system and medium based on sliding mode observation |
| CN113824114B (en) * | 2021-11-22 | 2022-03-18 | 广东电网有限责任公司惠州供电局 | Power distribution network state estimation method, device, system and medium based on sliding mode observation |
| CN114785216A (en) * | 2022-05-12 | 2022-07-22 | 合肥工业大学 | Design method of self-adaptive gain sliding-mode observer |
| CN114785216B (en) * | 2022-05-12 | 2024-03-26 | 合肥工业大学 | Design method of self-adaptive gain sliding mode observer |
Also Published As
| Publication number | Publication date |
|---|---|
| CN110361967B (en) | 2022-01-11 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN110361967A (en) | The construction method of sliding mode observer | |
| CN107561935B (en) | Motor position servo system friction compensation control method based on multilayer neural network | |
| CN106444799B (en) | Quadrotor drone control method based on Fuzzy Extension state observer and adaptive sliding mode | |
| Lagerberg et al. | Backlash estimation with application to automotive powertrains | |
| CN111546346B (en) | Flexible joint disturbance observation method, moment control method and equipment | |
| CN104635492A (en) | Parametric adaptive feed-forward control method of guide head stabilizing platform | |
| CN110083062A (en) | A kind of optic central extract composite control method based on velocity disturbance observer and Fuzzy-PID | |
| CN111176118A (en) | Turntable servo system identification method based on robust adaptive algorithm | |
| CN103713520A (en) | Gyroscope stabilized platform adaptive composite control method | |
| CN113093538A (en) | Non-zero and game neural-optimal control method of modular robot system | |
| CN112965383B (en) | Self-adaptive neural network optimal timing synchronization control method of unidirectional coupling fractional order self-sustaining electromechanical seismograph system | |
| CN116079741B (en) | Self-adaptive control method for motor-driven single-link mechanical arm | |
| Rajan et al. | Robust control methods for swing-up and stabilization of a rotary inverted pendulum | |
| Linfeng et al. | Robust chattering-free sliding mode control of electronic throttle systems in drive-by-wire vehicles | |
| Lamraoui et al. | Dynamic velocity tracking control of differential-drive mobile robot based on LADRC | |
| CN112737456B (en) | Traction asynchronous motor control method | |
| CN113872477A (en) | Anti-disturbance composite controller design of low-speed high-torque direct-drive motor | |
| CN103219936B (en) | Permanent magnet synchronous motor feed-forward decoupling method based on neural network online learning | |
| CN109309468B (en) | Linear motor speed regulating system and method | |
| Pan et al. | Dual-objective torque control of electric vehicles drivetrain with robust global finite-time stability guarantee | |
| Chen et al. | Robust indirect adaptive control of electromechanical servo systems with uncertain time-varying parameters | |
| Reichhartinger et al. | Robust cascaded automatic cruise control of electric vehicles | |
| CN114257130B (en) | Predictive speed synchronous control method suitable for double permanent magnet motor system and based on quadratic form cost function | |
| Hidalgo et al. | Steady-State Sensitivities of Marine Propulsion Control Techniques | |
| Leng et al. | Design of low speed controller for traveling wave ultrasonic motor |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |