CN105974933A - Self-balance manned electric single-wheel vehicle control method - Google Patents

Abstract

The invention discloses a self-balance manned electric single-wheel vehicle attitude balance control method. The method includes that the attitude information about inclination angle and inclination angle speed of a single-wheel self-balance vehicle body is obtained through measurement by an accelerometer and a gyroscope arranged on the electric single-wheel vehicle, and then on the basis of a system model, an attitude balance control method based on an LQR (Linear Quadratic Regulator) is designed to obtain the optical control quantity of an attitude balance controller through calculation, and at the end, the optical control quantity of the attitude balance controller is taken as the torque current given, a motor is driven by means of a vector control method to move based on the given torque to maintain the attitude balance of the vehicle body. According to the attitude balance control method based on an LQR (Linear Quadratic Regulator), the attitude balance controller based on the LQR is good in rapidity and stability, has the advantage of energy saving, and can realize the attitude balance control of the single-wheel self-balance vehicle.

Description

A kind of control method of Self-balance manned electric unicycle

Technical field

The invention belongs to field of vehicles, autonomic balance before and after relating in a kind of traveling, left and right relies on bicyclist artificial Control the control method of the Self-balance manned electric unicycle of balance.

Background technology

Purely mechanic wheelbarrow of riding is to need through ad hoc learning and train a kind of activity that just can complete.When riding, Must simultaneously maintain fore-and-aft direction and left and right directions balance, and ride and need people to exert oneself, gait of march also compares Low.Electric unicycle is by motor-driven device of riding, with similar, it is only necessary to the balance about grasp is the most permissible Ride smoothly.Self-balance manned electric unicycle balances before and after controlling car body by reversible pendulum system principle, bicyclist's handle Foot is individually placed on the folding step of wheel both sides, and body forward is inclined, and in order to keep balance, motor is so quick that to rotate forward, Whole car body the most just moves forward.

Self-balance manned electric unicycle is the energy-conserving and environment-protective of a new generation, walking-replacing tool easily, by the internal limit of program It is 18km/h the most at a high speed that system travels, it is ensured that driving safety.The electric unicycle bodily form is small and exquisite, lightweight, is convenient for carrying, permissible It is placed directly in the boot of automobile, mentions family or office.

The patent of invention of Application No. 201410014863.4 proposes Self-balancing electronic wheelbarrow simple and compact for structure, But it is not set forth in concrete control program.The patent of invention of Application No. 201210217335.X propose a kind of based on The Self-balance manned monocycle of inertia balance wheel, it is achieved that left-right balance controls, but the existence of inertia balance wheel, control program Complex it is not suitable for the most popular electric single-wheel bassinet structure yet.

Summary of the invention

Present invention aims to the deficiencies in the prior art, propose the control of a kind of Self-balance manned electric unicycle Method.

It is an object of the invention to be achieved through the following technical solutions: the control of a kind of Self-balance manned electric unicycle Method, the method comprises the steps:

(1) gather the accelerometer being placed on electric unicycle and the data of gyroscope, obtain electric unicycle xyz tri-axle Accekeration and magnitude of angular velocity；Gather the Hall element signal on permagnetic synchronous motor and threephase stator electric current i_a、i_b、i_c；

(2) accekeration and the magnitude of angular velocity of electric unicycle xyz tri-axle are carried out data by Kalman filtering algorithm Merge and filter noise signal, obtaining tilt angle theta and the inclination angle speed omega of electric unicycle；

(3) Hall element signal is calculated, obtain actual speed n and the rotor position angle θ of permagnetic synchronous motor；

(4) threephase stator electric current is carried out successively Clarke and Park conversion, obtains the actual torque of permagnetic synchronous motor Electric current i_qWith exciting current i_d；

(5) take actual speed n, tilt angle theta, inclination angle speed omega are quantity of state, the state equation of system of setting up of deriving:On the basis of the system model set up, use posture balancing control strategy based on LQR, calculate Excellent controlled quentity controlled variable U (t), U (t)=-R^-1B^TPX (t), wherein A is sytem matrix, and B is input matrix, and C is output matrix, and R is for controlling The weight matrix of amount u (t), P is through the calculated positive definite matrix of Riccati equation, and X is state matrix, wherein

(6) give according to the torque current of permagnetic synchronous motorExciting current givesActual torque electric current i_qWith encourage Magnetoelectricity stream i_d, calculate the d shaft voltage u applied to permagnetic synchronous motor_dWith q shaft voltage u_q；Described torque current is given as step (5) optimum control amount u (t) that in, posture balancing controller calculates, i.e.Described exciting current gives

(7) according to d shaft voltage u_dWith q shaft voltage u_q, utilize Park inverse transformation to calculate the u under α β coordinate system_αAnd u_β；

(8)u_αAnd u_βGenerated the control signal of three-phase inverter by SVPWM (space vector pulse width modulation) technology, pass through Inverter applies three-phase voltage, permagnetic synchronous motor output torque T to permagnetic synchronous motor_m, thus keep electric unicycle Posture balancing.

Further, step (5) described optimum control amount U (t) obtains by the following method:

(5.1) respectively wheel, car body are carried out force analysis to be derived by the mathematical model of system.

Car body is along the x-axis direction from initial displacement point n₀Move to n_p, list the equation of motion of car body:

In formula, n₀For axletree central horizontal displacement, and there is n₀=vt；n_pFor vehicle body barycenter horizontal displacement；a_pFor vehicle body matter The vertical displacement of the heart.

The motion of wheel can be decomposed into before and after translation and around the rotation of axle, can list the equation of motion:

In formula, H_fFrictional ground force suffered by wheel, VN is ground holding power suffered by wheel, H and V is respectively as wheel And the level between vehicle body, vertical direction interaction force, T_mElectromagnetic torque for motor output.

Vehicle body is that vehicle frame becomes a rigid body with people's Approximate Equivalent.Vehicle body barycenter is l to the distance of rotation shaft of wheel.Its motion Can be decomposed into and move horizontally and around the rotation of axletree.

Set up the equation of motion of vehicle body:

In formula, T is the moment of torsion that wheel is applied to vehicle body, and has T=T_m。

N in formula (2-1)_p、a_p、n₀、a₀It is all the function of time t, to n_p、a_pSecond dervative is asked to obtain t respectively:

Formula (2-7) is substituted into formula (2-6) and solves interaction force H and V obtained between vehicle body and wheel:

Formula (2-8) is substituted into respectively formula (2-3) and formula (2-6) eliminates interaction force and can obtain:

To formula (2-8), takeFor state variable, u=T_mInput is controlled for system Amount,Export for system, obtain the mathematical model of this system:

In formula,

a₁=m+m_p+J/r²

a₂=m_pl

a₃=J_p+m_pl²

WhenTime, haveSystem (2-9) approximate linearization is processed, To system linearity state equation:

(5.3) given LQR control optimal performance index functional:

In formula, Y_rT () is system idea output；Q is the weight matrix of quantity of state X；R be input quantity u weights square Battle array.

(5.4) according to LQR control theory, the linear state equations of coupling system, obtain optimum control amount U (t)=-R^-1B (t) PX (t) so that quadratic optimal performance index J takes minima.

Further, described step 5.4 particularly as follows:

(5.4.1) performance indications J to be made are minimum, first construct Hamiltonian function:

(5.4.2) when input signal is unfettered, to H derivation and to make its reciprocal value be 0

Solve and obtain optimum control amount:

U (t)=R^-1B^Tλ(t) (3-20)

In formula,

λ (t)=-P (t) X (t) (3-21)

(5.4.3) as t → ∞, P (t) levels off to constant value matrix, andThus obtain RICATTI equation:

PA+A^TP+PBR^-1BP+Q=0 (3-22)

Unique symmetric positive definite matrix P is solved according to Riccati equation.

(5.4.4) obtain system optimal to control to input:

U (t)=-R^-1B^TPX(t). (3-23)

In formula (3-23), U (t) is that the LQR quadratic optimal performance index J that enables to through being derived by minimizes value System optimal controlled quentity controlled variable.

The beneficial effects of the present invention is: combine the application of native system, this LQR control problem can be described as: uses not Big control energy, make the pitch tilt angle stabilization of single wheel Self-Balancing vehicle near equilibrium point, travel speed can follow the tracks of to The change of definite value, and make tracking error the least.The present invention, by using posture balancing control strategy based on LQR, takes into account The control energy consumption of single wheel Self-Balancing vehicle and control performance, use the fewest energy to obtain preferable control performance.By using Vector control strategy drives electric machine rotation, effectively raises dynamic responding speed and the control accuracy of motor, reduces torque The body oscillating that causes of pulsation and noise.

Accompanying drawing explanation

Fig. 1 is the structural representation of Self-balance manned electric unicycle；

Fig. 2 is the moving process side view of Self-balance manned electric unicycle

Fig. 3 is wheel and the force analysis figure of vehicle body of Self-balance manned electric unicycle.

Fig. 4 is the control system block diagram of Self-balance manned electric unicycle

Fig. 5 is the workflow diagram of Self-balance manned electric unicycle

Detailed description of the invention

The invention will be further described below in conjunction with the accompanying drawings.

The character definition that the present invention uses is as follows:

As the common knowledge of this area, the electric unicycle of the present invention, including auxiliary wheel 1, wheel 2, permanent magnet synchronous electric Machine 3, pedal 4, shell 5, controller 6 and set of cells 7, wherein, described controller 6 and set of cells 7 lay respectively at the left and right of shell 4 In two grooves, then fixing with cover plate integral piece with silica gel protected pad, accelerometer and gyroscope 8 are positioned on controller 6, as Shown in Fig. 1, the electric unicycle for different frame for movements can apply control method of the present invention to realize self-balancing and load People rides function.The control method of the present invention is as follows:

(1) gather the accelerometer being placed on electric unicycle and the data of gyroscope, obtain electric unicycle xyz tri-axle Accekeration and magnitude of angular velocity；Gather the Hall element signal on permagnetic synchronous motor and threephase stator electric current i_a、i_b、i_c；

(2) accekeration and the magnitude of angular velocity of electric unicycle xyz tri-axle are carried out data by Kalman filtering algorithm Merge and filter noise signal, obtaining tilt angle theta and the inclination angle speed omega of electric unicycle；

(3) Hall element signal is calculated, obtain actual speed n and the rotor position angle θ of permagnetic synchronous motor；

(4) threephase stator electric current is carried out successively Clarke and Park conversion, obtains the actual torque of permagnetic synchronous motor Electric current i_qWith exciting current i_d；

(5) take actual speed n, tilt angle theta, inclination angle speed omega are quantity of state, the state equation of system of setting up of deriving:On the basis of the system model set up, use posture balancing control strategy based on LQR, calculate Excellent controlled quentity controlled variable U (t), U (t)=-R^-1B^TPX (t), wherein A is sytem matrix, and B is input matrix, and C is output matrix, and R is for controlling The weight matrix of amount u (t), P is through the calculated positive definite matrix of Riccati equation, and X is state matrix, wherein

Specific as follows:

(5.1) respectively wheel, car body are carried out force analysis to be derived by the mathematical model of system.

As it is shown in figure 1, car body is along the x-axis direction from initial displacement point n₀Move to n_p, list the equation of motion of car body:

In formula, n₀For axletree central horizontal displacement, and there is n₀=vt；n_pFor vehicle body barycenter horizontal displacement；a_pFor vehicle body matter The vertical displacement of the heart.

As shown in Fig. 3 (a), the motion of wheel can be decomposed into before and after translation and around the rotation of axle, motion can be listed Equation:

In formula, H_fFrictional ground force suffered by wheel, V_NGround holding power suffered by wheel, H and V is respectively as wheel And the level between vehicle body, vertical direction interaction force, T_mElectromagnetic torque for motor output.

Shown in vehicle body force analysis figure such as Fig. 3 (b).Vehicle body is that vehicle frame becomes a rigid body with people's Approximate Equivalent.Vehicle body matter The heart is l to the distance of rotation shaft of wheel.Its motion can be decomposed into and move horizontally and around the rotation of axletree.

Set up the equation of motion of vehicle body:

In formula, T is the moment of torsion that wheel is applied to vehicle body, and has T=T_m。

N in formula (2-1)_p、a_p、n₀、a₀It is all the function of time t, to n_p、a_pSecond dervative is asked to obtain t respectively:

Formula (2-7) is substituted into formula (2-6) and solves interaction force H and V obtained between vehicle body and wheel:

Formula (2-8) is substituted into respectively formula (2-3) and formula (2-6) eliminates interaction force and can obtain:

(5.2) to formula (2-8), takeFor state variable, u=T_mFor system control Input quantity,Export for system, obtain the mathematical model of this system:

In formula,

a₁=m+m_p+J/r²

a₂=m_pl

a₃=J_p+m_pl²

Following several conclusions can be obtained by analyzing the system state equation being made up of formula (2-10) and formula (2-11).The One, this system is a nonlinear system, in non-linear relation between output variable and input, between output variable and input quantity Also in non-linear relation, between each state variable, there is coupling.Second, this system is a under-actuated systems, controls input quantity Only one of which, controlling target has travel speed speed, luffing angle and angular velocity multiple.3rd, this system be a parameter not Determine system, people's height l of single wheel Self-Balancing vehicle of riding, body weight m_pCan be variant because of individual different etc. parameter, thus lead Cause the uncertain of control object parameter.These characteristics of single wheel Self-Balancing vehicle, to controlling to bring difficulty, need during design controller Consider these factors

Knowable to the system mathematic model set up above, single wheel Self-Balancing vehicle system is high-order, a nonlinear system, gives The inconvenience of the analytic band of system performance.

(2) system model is carried out linearization process, obtain system linearity state equation

When single wheel Self-Balancing vehicle is properly functioning, luffing angle only changes in little scope near equilibrium point, therefore can be flat Near weighing apparatus point, system model approximate linearization is processed, thus obtain system linearity state equation.

WhenTime, haveSystem (2-9) approximate linearization is processed, To system linearity state equation:

(5.3) given LQR control optimal performance index functional:

In formula, Y_rT () is system idea output；Q is the weight matrix of quantity of state X；R be input quantity u weights square Battle array.

Q and R chooses the control performance that can affect system, and in matrix Q, element is to the size of element in R, indicates design Person controls the attention degree of both energy to tracking error and system.In Q and R matrix in real system, the value of each element needs Determine through experiment, quantity of state weight matrix Q be listed below and control several the principles that energy weight matrix R chooses:

1) generally choosing Q and R is diagonal matrix, in actual application, the most first R value is fixed, and then changes each weights in Q The numerical value of element, to meet the specific control performance requirement of system requirements

2) selection of Q and R is mutually restriction, interactional, if requiring that the control error of system is little, then the most inevitable The consumption of energization；In like manner, in order to reduce energy expenditure, would have to sacrifice the required precision of control performance.

3) increase of a certain element in weighting matrices Q, the dynamic response process of its corresponding x (t) is improved, quickly Property significantly improves；Meanwhile, the steady-state error of other quantity of states can increase, and dynamic response process is deteriorated.

Native system lays particular emphasis on the balance control performance of system, and controls speed real-time less demanding.Therefore take and bow Elevation angle degree weight element q₂More than speed controlling weights element q₁Value, luffing angle stablized to obtain, quickly control System.During real system debugging, first organize parameter according to mentioned above principle test more, eventually find a best parameter of control performance.

(5.4) according to LQR control theory, the linear state equations of coupling system, obtain optimum input quantity U (t)=-R^-1B (t) PX (t) so that quadratic optimal performance index J takes minima.Specific as follows:

(5.4.1) performance indications J to be made are minimum, first construct Hamiltonian function:

(5.4.2) when input signal is unfettered, to H derivation and to make its reciprocal value be 0

Solve and obtain optimum control amount:

U (t)=R^-1B^Tλ(t) (3-20)

In formula,

λ (t)=-P (t) X (t) (3-21)

(5.4.3) as t → ∞, P (t) levels off to constant value matrix, andThus obtain RICATTI equation:

PA+A^TP+PBR^-1BP+Q=0 (3-22)

Unique symmetric positive definite matrix P is solved according to Riccati equation.

(5.4.4) obtain system optimal to control to input:

U (t)=-R^-1B^TPX(t). (3-23)

In formula (3-23), U (t) is that the LQR quadratic optimal performance index J that enables to through being derived by minimizes value System optimal controlled quentity controlled variable.

(6) give according to the torque current of permagnetic synchronous motorExciting current givesActual torque electric current i_qAnd excitation Electric current i_d, calculate the d shaft voltage u applied to permagnetic synchronous motor_dWith q shaft voltage u_q；Described torque current is given as step (5) optimum control amount u (t) that in, posture balancing controller calculates, i.e.Described exciting current gives

Particularly as follows: described step (6) particularly as follows:

Torque current gives to use linear PI method to determine, formula is as follows:

Wherein,For torque current ring scale parameter,For torque current ring integral parameter,Give for torque current Value, i_qFor actual torque current value, e_iqGive for torque currentWith actual torque electric current i_qDifference.

Same, use linear PI algorithm to determine U_d, formula is as follows:

Wherein,For torque current ring scale parameter,For torque current ring integral parameter,Give for exciting current Value, i_dFor actual exciting current value, e_idGive for exciting currentWith actual exciting current i_dDifference.

(7) according to d shaft voltage u_dWith q shaft voltage u_q, utilize Park inverse transformation to calculate the u under α β coordinate system_αAnd u_β；

(8)u_αAnd u_βGenerated the control signal of three-phase inverter by SVPWM (space vector pulse width modulation) technology, pass through Inverter applies three-phase voltage, permagnetic synchronous motor output torque T to permagnetic synchronous motor_m, thus keep electric unicycle Posture balancing.

Claims (3)

1. the control method of a Self-balance manned electric unicycle, it is characterised in that the method comprises the steps:

(1) gather the accelerometer being placed on electric unicycle and the data of gyroscope, obtain adding of electric unicycle xyz tri-axle Velocity amplitude and magnitude of angular velocity；Gather the Hall element signal on permagnetic synchronous motor and threephase stator electric current i_a、i_b、i_c；

(2) accekeration and the magnitude of angular velocity of electric unicycle xyz tri-axle are carried out data fusion by Kalman filtering algorithm And filter noise signal, obtain tilt angle theta and the inclination angle speed omega of electric unicycle；

(3) Hall element signal is calculated, obtain actual speed n and the rotor position angle θ of permagnetic synchronous motor；

(4) threephase stator electric current is carried out successively Clarke and Park conversion, obtains the actual torque electric current i of permagnetic synchronous motor_q With exciting current i_d；

(5) take actual speed n, tilt angle theta, inclination angle speed omega are quantity of state, the state equation of system of setting up of deriving:On the basis of the system model set up, use posture balancing control strategy based on LQR, calculate Excellent controlled quentity controlled variable U (t), U (t)=-R^-1B^TPX (t), wherein A is sytem matrix, and B is input matrix, and C is output matrix, and R is for controlling The weight matrix of amount u (t), P is through the calculated positive definite matrix of Riccati equation, and X is state matrix, wherein

(6) give according to the torque current of permagnetic synchronous motorExciting current givesActual torque electric current i_qAnd exciting current i_d, calculate the d shaft voltage u applied to permagnetic synchronous motor_dWith q shaft voltage u_q；Described torque current is given as in step (5) Optimum control amount u (t) that posture balancing controller calculates, i.e.Described exciting current gives

(7) according to d shaft voltage u_dWith q shaft voltage u_q, utilize Park inverse transformation to calculate the u under α β coordinate system_αAnd u_β；

(8)u_αAnd u_βThe control signal of three-phase inverter is generated, through inversion by SVPWM (space vector pulse width modulation) technology Device applies three-phase voltage, permagnetic synchronous motor output torque T to permagnetic synchronous motor_m, thus keep the attitude of electric unicycle Balance.

Method the most according to claim 1, it is characterised in that step (5) described optimum control amount U (t) is by with lower section Method obtains:

(5.1) respectively wheel, car body are carried out force analysis to be derived by the mathematical model of system.

Car body is along the x-axis direction from initial displacement point n₀Move to n_p, list the equation of motion of car body:

In formula, n₀For axletree central horizontal displacement, and there is n₀=vt；n_pFor vehicle body barycenter horizontal displacement；a_pFor vehicle body barycenter Vertical displacement.

The motion of wheel can be decomposed into before and after translation and around the rotation of axle, can list the equation of motion:

$\left\{\begin{array}{c}m\stackrel{\·}{v}=H_f-H\\ J\stackrel{\·}{\mathrm{\ω}}=T_m-H_{f}r\\ v=r\mathrm{\ω}\\ \stackrel{\·}{v}=r\stackrel{\·}{\mathrm{\ω}}\end{array}\right.---(2-2)$

In formula, H_fFrictional ground force suffered by wheel, V_NGround holding power suffered by wheel, H and V is respectively as wheel and car Level between body, vertical direction interaction force, T_mElectromagnetic torque for motor output.

Vehicle body is that vehicle frame becomes a rigid body with people's Approximate Equivalent.Vehicle body barycenter is l to the distance of rotation shaft of wheel.Its motion is permissible It is decomposed into and moves horizontally and around the rotation of axletree.

Set up the equation of motion of vehicle body:

In formula, T is the moment of torsion that wheel is applied to vehicle body, and has T=T_m。

N in formula (2-1)_p、a_p、n₀、a₀It is all the function of time t, to n_p、a_pSecond dervative is asked to obtain t respectively:

Formula (2-7) is substituted into formula (2-6) and solves interaction force H and V obtained between vehicle body and wheel:

Formula (2-8) is substituted into respectively formula (2-3) and formula (2-6) eliminates interaction force and can obtain:

To formula (2-8), takeFor state variable, u=T_mInput quantity is controlled for system,Export for system, obtain the mathematical model of this system:

$\stackrel{\·}{x}=f\left(x\right)+g\left(x\right)u---(2-9)$

$y=\[\begin{array}{ccc}1& 1& 1\end{array}\]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]---(2-10)$

In formula,

$f\left(x\right)=\left[\begin{array}{c}-\frac{a_2^{2}g/2}{a_1{a}_3-a_2^2{\cos }^2{x}_1}sin2x_1+\frac{a_2{a}_3\sin x_1}{a_1{a}_3-a_2^2{\cos }^2{x}_1}{x}_2^2\\ x_3\\ \frac{a_1{a}_{2}g}{a_1{a}_3-a_2^2{\cos }^2{x}_1}\sin x_1-\frac{a_2^2\sin 2x_1/2}{a_1{a}_3-a_2^2{\cos }^2{x}_1}{x}_2^2\end{array}\right]$

$g\left(x\right)=\left[\begin{array}{c}\frac{a_2\cos x_1+a_3/r}{a_1{a}_3-a_2^2{\cos }^2{x}_1}\\ 0\\ -\frac{a_1+a_2\cos x_1/r}{a_1{a}_3-a_2^2{\cos }^2{x}_1}\end{array}\right]$

a₁=m+m_p+J/r²

a₂=m_pl

a₃=J_p+m_pl²

WhenTime, haveSystem (2-9) approximate linearization is processed, is The linear state equations of system:

$\begin{array}{c}\stackrel{\·}{X}=AX+Bu\\ =\left[\begin{array}{ccc}0& -\frac{{\left(m_{p}l\right)}^{2}g}{(m+m_p+J/r^2)(J_p+m_p{l}^2)-{\left(m_{p}l\right)}^2}& 0\\ 0& 0& 1\\ 0& \frac{(m+m_p+J/r^2)m_p\lg }{(m+m_p+J/r^2)(J_p+m_p{l}^2)-{\left(m_{p}l\right)}^2}& 0\end{array}\right]X+\left[\begin{array}{c}\frac{m_{p}l+(J_p+m_p{l}^2)/r}{(m+m_p+J/r^2)(J_p+m_p{l}^2)-{\left(m_{p}l\right)}^2}\\ 0\\ -\frac{m+m_p+j/r^2+m_{p}l/r}{(m+m_p+J/r^2)(J_p+m_p{l}^2)-{\left(m_{p}l\right)}^2}\end{array}\right]u\end{array}---(2-11)$

(5.3) given LQR control optimal performance index functional:

$J=\frac{1}{2}{\∫}_0^{\mathrm{\∞}}{\[Y_r\left(t\right)-Y\left(t\right)\]}^{T}Q\[Y_r\left(t\right)-Y\left(t\right)\]dt+\frac{1}{2}{\∫}_0^{\mathrm{\∞}}{U}^T\left(t\right)RU\left(t\right)dt---(3-25)$

In formula, Y_rT () is system idea output；Q is the weight matrix of quantity of state X；R be input quantity u weight matrix.

(5.4) according to LQR control theory, the linear state equations of coupling system, obtain optimum control amount U (t)=-R^-1B(t)PX (t) so that quadratic optimal performance index J takes minima.

Method the most according to claim 2, it is characterised in that described step 5.4 particularly as follows:

(5.4.1) performance indications J to be made are minimum, first construct Hamiltonian function:

$H=-\frac{1}{2}{X}^T\left(t\right){\mathrm{QX}}^T\left(t\right)-\frac{1}{2}{U}^T\left(t\right)RU\left(t\right)+{\mathrm{\λ}}^T\left(t\right)\left(AX\right(t)+BU(t\left)\right)---(3-18)$

(5.4.2) when input signal is unfettered, to H derivation and to make its reciprocal value be 0

$\frac{\∂H}{\∂U}=-RU\left(t\right)+B^T\mathrm{\λ}\left(t\right)=0---(3-19)$

Solve and obtain optimum control amount:

U (t)=R^-1B^Tλ(t) (3-20)

In formula,

λ (t)=-P (t) X (t) (3-21)

(5.4.3) as t → ∞, P (t) levels off to constant value matrix, andThus obtain RICATTI equation:

PA+A^TP+PBR^-1BP+Q=0 (3-22)

Unique symmetric positive definite matrix P is solved according to Riccati equation.

(5.4.4) obtain system optimal to control to input:

U (t)=-R^-1B^TPX(t). (3-23)

In formula (3-23), U (t) is that LQR quadratic optimal performance index J minimizes value is through enabling to of being derived by System optimum control amount.