JP2011008360A - Design method of sliding mode control system and design support device thereof - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a simple design method of a sliding mode control system which includes a state feedback gain obtaining means, and a computing means for computing a hyperplane vector and a proportional transmission gain.SOLUTION: The design method of a sliding mode control system (10) includes an equivalence controller (11) which computes an equivalence control input (u); a proportion attainment controller (13) which computes a control input (u) which is proportional to a value of a switching function (σ); a nonlinear controller (14) which computes a nonlinear control input (u) to a state of control object which is determined, based on the switching function (σ); and an adder (15) which computes a control input (u) by adding three control inputs (u, uand u). The control method includes a step for computing a state feedback gain (F), when controlling a control object by a linear control system; and a design step for designing the equivalence controller (11) and the proportion attainment controller (13), assuming that the control input derived by a state feedback gain (F) is equal to the sum total of the control input (u) and control input (u).

Description

  The present invention relates to a design method and a design support apparatus for a sliding mode control system that controls a multi-inertia resonance system.

  Conventionally, based on a sliding mode control theory (see, for example, Non-Patent Document 1), a position control device that uses a servo motor to track the position of a moving body to a command value is known (see, for example, Patent Document 1). .)

  The position control device described in Patent Document 1 includes a sliding mode controller that receives a position command and a state variable of a control system and outputs a control input, and a control input that the sliding mode controller outputs based on a feedback speed. A switching function in the sliding mode controller is defined using a predetermined hyperplane vector.

JP 2005-293564 A

Kenzo Nonami, Hiroki Ta, “Sliding Mode Control: Design Theory of Nonlinear Robust Control”, Corona (1994)

  However, since Patent Document 1 does not disclose any means for determining the hyperplane vector, the designer cannot apply the sliding mode controller to a specific design based on Patent Document 1.

  In view of the above, an object of the present invention is to provide a simple design method of a sliding mode control system and a design support apparatus thereof.

  In order to achieve the above-described object, a design method for a sliding mode control system according to the present invention includes an equivalent controller that calculates a first control input that is an equivalent control input, and a second control input that is proportional to the value of the switching function. A proportional arrival controller to calculate, a nonlinear controller to calculate a third control input that is determined based on a switching function and is nonlinear with respect to a state to be controlled, and the first control input, the second control input, and the A method for designing a sliding mode control system including an adder that adds a third control input to calculate a final control input, and calculates a state feedback gain when a controlled object is controlled by a linear control system And the equivalent controller and the control input, assuming that the control input derived by the state feedback gain is equal to the sum of the first control input and the second control input. It characterized by having a design step, to design the proportional arrival controller.

In the sliding mode control system design method according to the present invention, the sliding mode control system is a continuous-time control system, and the hyperplane vector is S, the system matrix is A, the unit matrix is I, the state vector is x, When the proportional reach gain is η and the state feedback gain is F _c , the first control input is represented by u ₁ = −SAx, the second control input is represented by u ₂ = −ηSx, and the control target The control input based on the state feedback gain in the case of controlling with a linear control system is represented by u _C = −F _c x. In the design step, the equivalent controller and the proportional arrival controller are represented by u ₁ + u _2. It is preferably designed by deriving the hyperplane vector S and the proportional reach gain η from the equation S (A + ηI) = F _c based on the relationship = _C.

Also, in the sliding mode control system design method according to the present invention, the sliding mode control system is a discrete time control system, wherein the hyperplane vector is S, the system matrix is Φ, the unit matrix is I, the state vector is x, When the number of samples is k, the proportional reach gain is μ, and the state feedback gain is F _d , the first control input is represented by u ₁ [k] = − S (Φ−I) × [k], The second control input is represented by u ₂ [k] = − μSx [k], and the control input by the state feedback gain when the control target is controlled by the linear control system is u _d [k] = − F _d x [k], and in the design step, the equivalent controller and the proportional arrival controller are represented by an equation S {Φ + () based on the relationship u ₁ [k] + u ₂ [k] = u _d [k]. μ-1) I} = F _d and beyond It is preferably designed by deriving the plane vector S and the proportional reach gain μ.

  In the sliding mode control system design method according to the present invention, it is preferable that in the design step, the nonlinear controller is designed using a saturation function.

  Further, a design support apparatus for supporting the design of a sliding mode control system according to the present invention includes an equivalent controller that calculates a first control input that is an equivalent control input using a hyperplane vector, the hyperplane vector, and a proportional transfer gain. A proportional arrival controller that calculates a second control input proportional to the value of the switching function using a non-linear controller that calculates a third control input determined based on the switching function and nonlinear with respect to the state of the controlled object; and A design support apparatus that supports the design of a sliding mode control system including an adder that adds the first control input, the second control input, and the third control input to calculate a final control input. State feedback gain acquisition means for acquiring a state feedback gain when the controlled object is controlled by a linear control system, and the state feedback gain. Control input is derived is characterized by comprising a calculating means for calculating the hyperplane vector and the proportional transmission gain as equal to the sum of said second control input and said first control input.

  As described above, the present invention can provide a simple design method of a sliding mode control system and a design support apparatus thereof.

It is a block diagram which shows the structural example of the sliding mode controller which concerns on this invention. It is a flowchart which shows the flow of the design procedure of the sliding mode controller of a continuous time control system. It is a block diagram which shows the structural example of the sliding mode controller designed according to the design procedure of FIG. It is a flowchart which shows the flow of the design procedure of the sliding mode controller of a discrete time control system. It is a block diagram of the control system of the electric motor with a reduction gear designed according to this invention. It is a figure which shows the simulation result of the time transition of each rotation position of the electric motor and load in the control system of FIG.

  FIG. 1 is a block diagram showing a configuration example of a sliding mode controller according to the present invention. The sliding mode controller 10 includes an equivalent controller 11, a switching function calculator 12, a proportional arrival controller 13, a nonlinear controller 14, and An adder 15 is included.

  In FIG. 1, the control target of the sliding mode controller 10 is a continuous time control system, and the dynamics of the control target is described by a differential equation (state equation) of Formula (1).

ここで、ｎを任意の自然数とすると、ｘは、ｎ次の列ベクトルで表される時間の変数（状態ベクトル）であり、Ａは、ｎ×ｎの行列で表される制御対象のシステム行列であり、Ｂは、ｎ次の列ベクトルで表される制御対象の入力行列であり、ｕは、スカラーで表される時間の変数（制御入力）である。また、この状態方程式は、可制御正準形であるものとする。

Here, when n is an arbitrary natural number, x is a time variable (state vector) represented by an nth-order column vector, and A is a system matrix to be controlled represented by an n × n matrix. B is an input matrix to be controlled represented by an nth-order column vector, and u is a time variable (control input) represented by a scalar. This state equation is assumed to be a controllable canonical form.

The equivalent controller 11 is an element that receives an input of the state vector x and calculates an equivalent control input u ₁ , and the switching function calculator 12 is an element that receives the input of the state vector x and outputs a switching function σ. . In addition, the switching function σ is expressed by Expression (2), the switching hyperplane is expressed by Expression (3), and the equivalent control input u ₁ is expressed by Expression (4).

ここで、Ｓは、ｎ次の行ベクトルで表される超平面ベクトルである。また、切換超平面は、ｎ次元ベクトル空間の超平面であり、ｎ＝２ならば直線、ｎ＝３ならば平面となる。切換関数σが０（ゼロ）であれば、制御対象の状態が切換超平面上にあることが分かり、また、切換関数σが０（ゼロ）でなければ、その符号に基づいて、制御対象の状態が切換超平面によって二分されるｎ次元ベクトル空間の何れの側に存在するかが分かる。

Here, S is a hyperplane vector represented by an nth-order row vector. The switching hyperplane is a hyperplane of the n-dimensional vector space, and is a straight line if n = 2 and a plane if n = 3. If the switching function σ is 0 (zero), it can be seen that the state of the controlled object is on the switching hyperplane, and if the switching function σ is not 0 (zero), based on the sign, It can be seen on which side of the n-dimensional vector space the state is bisected by the switching hyperplane.

The proportional arrival controller 13 is an element that receives the input of the switching function σ and calculates a control input u ₂ that is proportional to the value of the switching function σ. The nonlinear controller 14 switches the state of the controlled object to the switching hyperplane. This is an element for calculating the control input u ₃ for reaching and restraining. The control input u ₂ is expressed by the equation (5), and the control input u ₃ is expressed by the saturation function of the equation (6).

ここで、ηは、比例到達ゲインであり、Ｋは、非線形ゲインであり、φは、境界層を示す定数である。

Here, η is a proportional reach gain, K is a non-linear gain, and φ is a constant indicating the boundary layer.

The adder 15 adds the equivalent control input u ₁ output from the equivalent controller 11, the control input u ₂ output from the proportional arrival controller 13, and the control input u ₃ output from the nonlinear controller 14. This is an element for calculating the control input u.

  Here, a design method of the sliding mode controller 10 of the continuous time control system will be described with reference to FIG. FIG. 2 is a flowchart showing the flow of the design procedure. In the following, it is assumed that the state equation obtained by modeling the controlled object is not a controllable canonical form, but if a controllable canonical state equation is obtained directly from the controlled object, the controllable canonical form is obtained. No conversion procedure is required. For the controllable canonical form and equivalence conversion, known methods (for example, the method described in Tsuneo Yoshikawa, Junichi Imura, “Modern Control Theory”, Shosodo (1994)) may be used.

  First, the designer obtains a controllable canonical state equation by modeling a controlled object, calculates a conversion matrix to a controllable canonical form, and performs equivalence conversion (step S1).

Thereafter, the designer calculates the state feedback gain F _c in the case of controlling the control object in a linear control system (step S2). As a calculation method, for example, a well-known pole placement method or optimal control theory (for example, the method described in Tsuneo Yoshikawa, Junichi Imura, “Modern Control Theory”, Shosodo (1994)) is used. The control input u _c by the state feedback control is expressed by the equation (7).

その後、設計者は、状態フィードバックゲインＦ_ｃによる制御入力ｕ_ｃが、等価制御器１１が出力する等価制御入力ｕ_１と比例到達制御器１３が出力する制御入力ｕ_２との合計に等しいものとして、式（４）、（５）、（７）から式（８）を導き出し、式（８）を解くことによって比例伝達ゲインη及び超平面ベクトルＳを算出する（ステップＳ３）。

Thereafter, the designer, the state feedback gain F _c by the control input u _c is, as equal to the sum of the control input u ₂ of the equivalent control input u ₁ and proportional reaching controller 13 to output the equivalent control unit 11 outputs Equation (8) is derived from Equations (4), (5), and (7), and proportional transmission gain η and hyperplane vector S are calculated by solving Equation (8) (step S3).

その後、設計者は、算出した比例伝達ゲインη及び超平面ベクトルＳを用いて等価制御器１１、切換関数演算器１２及び比例到達制御器１３を設計し（ステップＳ４）、例えば、図３に示すような構成を得る。

Thereafter, the designer designs the equivalent controller 11, the switching function calculator 12, and the proportional arrival controller 13 using the calculated proportional transfer gain η and hyperplane vector S (step S4), for example, as shown in FIG. A configuration like this is obtained.

  Finally, the designer designs the nonlinear controller 14 by adopting control (for example, saturation control using a saturation function) for causing the state of the controlled object to reach the switching hyperplane and constraining it ( Step S5), for example, to obtain a sliding mode controller 10 as shown in FIG.

  With the above configuration, the designer can apply the sliding mode control to the continuous-time control system in which the dynamics to be controlled are described by a differential equation.

  Next, a design method of the sliding mode controller 10 of the discrete time control system will be described with reference to FIG. FIG. 4 is a flowchart showing the flow of the design procedure.

  First, as in the case of the continuous-time control system, the designer models the object to be controlled and obtains the state equation represented by the equation (1) (it does not have to be a controllable canonical form) ( Step S11).

Thereafter, the designer, as in the case of a continuous time control system, to calculate the state feedback gain F _c of the formula (7) (step S12).

  After that, the designer discretizes the state equation of the continuous-time control system represented by Equation (1), calculates the transformation matrix to a controllable canonical form, and performs equivalence transformation, thereby obtaining Equation (9) The state equation of the discrete time control system represented and its transformation matrix are acquired (step S13).

ここで、ｎ、ｋを任意の自然数とすると、ｘは、所定のサンプリング間隔で更新されるｎ次の列ベクトルで表される状態ベクトルであり、Φは、ｎ×ｎの行列で表される制御対象のシステム行列であり、Γは、ｎ次の列ベクトルで表される制御対象の入力行列であり、ｕは、所定のサンプリング間隔で更新されるスカラーで表される制御入力である。また、［］（大括弧）内の値はサンプリングの順番を表す。なお、この状態方程式は、連続時間制御系の状態方程式に基づくことなく、離散時間におけるモデル化を行うことによって直接的に導出されてもよい。

Here, when n and k are arbitrary natural numbers, x is a state vector represented by an nth-order column vector updated at a predetermined sampling interval, and Φ is represented by an n × n matrix. It is a system matrix to be controlled, Γ is an input matrix of the control target expressed by an nth-order column vector, and u is a control input expressed by a scalar updated at a predetermined sampling interval. The value in [] (brackets) represents the sampling order. Note that this state equation may be derived directly by modeling in discrete time without being based on the state equation of the continuous-time control system.

Thereafter, the designer obtains a state feedback gain F _d corresponding to the discrete time control system by discretizing the state feedback gain F _c represented by Expression (7) (step S14). In addition, the designer can apply state design feedback in the discrete time control system to the state equation expressed by Equation (9) without using the state feedback gain F _c of the continuous time control system. it may acquire a gain F _d.

For example, the designer calculates P _d = exp (P _c T _s ) based on the eigenvalue P _c of the matrix A-BF _c , and the state feedback gain so that the eigenvalue of the matrix Φ-ΓF _d becomes P _d. F _d is calculated. Note that T _s is a sampling time.

Thus, the control input u _d by the state feedback control is expressed by the equation (10).

その後、設計者は、状態フィードバックゲインＦ_ｄによる制御入力ｕ_ｄが、等価制御器１１が出力する等価制御入力ｕ_１と比例到達制御器１３が出力する制御入力ｕ_２との合計に等しいものとして、式（１１）を導き出し、式（１１）を解くことによって比例伝達ゲインμ及び超平面ベクトルＳを算出する（ステップＳ１５）。なお、等価制御入力ｕ_１及び制御入力ｕ_２はそれぞれ、式（１２）、式（１３）で表される。

Thereafter, the designer, the state feedback gain F _d by the control input u _d is, as equal to the sum of the control input u ₂ of the equivalent control input u ₁ and proportional reaching controller 13 to output the equivalent control unit 11 outputs The equation (11) is derived, and the proportional transmission gain μ and the hyperplane vector S are calculated by solving the equation (11) (step S15). Note that the equivalent control input u ₁ and the control input u ₂ are expressed by Expression (12) and Expression (13), respectively.

その後、設計者は、算出した比例伝達ゲインμ及び超平面ベクトルＳを用いて等価制御器１１、切換関数演算器１２及び比例到達制御器１３を設計し（ステップＳ１６）、最後に、制御対象の状態を切換超平面に到達させ且つ拘束するための制御（例えば、飽和関数を用いた飽和制御である。）を採用して非線形制御器１４を設計する（ステップＳ１７）。

Thereafter, the designer designs the equivalent controller 11, the switching function calculator 12, and the proportional arrival controller 13 using the calculated proportional transfer gain μ and hyperplane vector S (step S16), and finally, the control target. The nonlinear controller 14 is designed by adopting control for reaching and constraining the state to the switching hyperplane (for example, saturation control using a saturation function) (step S17).

The control input u ₃ for causing the state of the controlled object to reach and restrain the switching hyperplane is expressed by, for example, Expression (14).

以上の構成により、設計者は、制御対象のダイナミクスが差分方程式で記述される離散時間制御系に対してスライディングモード制御を適用することができる。

With the above configuration, the designer can apply the sliding mode control to the discrete time control system in which the dynamics to be controlled are described by the difference equation.

  Next, a procedure for calculating the proportional transfer gain η and the hyperplane vector S by solving the equation (8) in the continuous time control system will be described in detail.

  In the controllable canonical form, the system matrix A and the input matrix B are expressed by the equations (15) and (16), respectively.

ここで、α_１、α_２、・・・、α_ｎ−１は、式（１７）で表される同値のシステムの特性多項式の係数である。

Here, α ₁ , α ₂ ,..., Α _n−1 are coefficients of the characteristic polynomial of the system of the same value expressed by the equation (17).

状態フィードバックゲインＦ_ｃによって状態フィードバック制御が施されたときの特性多項式が式（１８）で表すものであるとすれば、状態フィードバックゲインＦ_ｃは、式（１９）で表されることとなる。

If the characteristic polynomial of when the state feedback control is performed by the state feedback gain F _c is the those represented by the formula (18), the state feedback gain F _c is a fact represented by the formula (19).

式（３）によると、互いに線形従属な超平面ベクトルＳは、同じ切換超平面を規定するので、第ｎ列の値を１とするように超平面ベクトルＳを選ぶと、超平面ベクトルＳは、式（２０）のようになる。

According to equation (3), hyperplane vectors S that are linearly dependent on each other define the same switching hyperplane, and therefore, if the hyperplane vector S is selected so that the value of the n-th column is 1, the hyperplane vector S is (20).

以上より、式（８）は、式（２１）で表すことができ、式（２１）からηの連立方程式（２２）が得られる。

From the above, equation (8) can be expressed by equation (21), and simultaneous equations (22) of η can be obtained from equation (21).

この連立方程式は、第１式がηに関するｎ次方程式となっており、ηが求まると第２式からＳ_１が求まり、第３式からＳ_２が求まるというように順次求まっていく。なお、ηの解は、ｎ次方程式の正の実数解から任意のものを選ぶことができるが、ηの値が大きいと制御系の応答が振動的になり易いことから、好適には、最小の値が選択される。

In the simultaneous equations, the first equation is an n-order equation relating to η, and when η is obtained, S ₁ is obtained from the _second equation, and S ₂ is obtained from the third equation. The solution of η can be selected from positive real solutions of n-order equations. However, since the response of the control system tends to vibrate when the value of η is large, The value of is selected.

  Next, a procedure for calculating the proportional transfer gain μ and the hyperplane vector S by solving the equation (11) in the discrete time control system will be described in detail.

  Expression (11) in the discrete time control system is expressed by Expression (23) when developed in the same manner as Expression (8) in the continuous time control system, and Expression (23) is further expressed by Expression (24). The

更に、式（２４）からμの連立方程式（２５）が得られる。

Further, the simultaneous equation (25) of μ is obtained from the equation (24).

この連立方程式は、第１式が（μ−１）に関するｎ次方程式となっており、（μ−１）が求まると第２式からＳ_１が求まり、第３式からＳ_２が求まるというように順次求まっていく。（μ−１）の解は、ｎ次方程式の実数解のうち、０＜μ＜２を満たし且つμ＝１以外のものを任意に選ぶことができるが、チャタリングが発生しないものは０＜μ＜１であり、好適には、連続時間制御系の場合と同様、正の最小値が選択される。

In this simultaneous equation, the first equation is an n-order equation relating to (μ−1), and when (μ−1) is obtained, S ₁ is obtained from the _second equation, and S ₂ is obtained from the third equation. Will be sought sequentially. The solution of (μ−1) can be arbitrarily selected from real solutions of n-order equations that satisfy 0 <μ <2 and other than μ = 1, but those that do not cause chattering are 0 <μ <1, and preferably the positive minimum value is selected as in the case of the continuous time control system.

  Next, the details of the nonlinear controller 14 using the saturation function will be described.

The nonlinear controller 14 in the continuous time control system employs a saturation function represented by Expression (6), and the nonlinear controller 14 in the discrete time control system employs a saturation function represented by Expression (14). Thus, chattering is prevented from occurring (if the nonlinear controller 14 is used as a switching function without providing a boundary layer and the nonlinear control input u ₃ is expressed as shown in Equation (6) ′, nonlinear control is performed. (The input u ₃ causes chattering that frequently switches between −K and K in a region where the switching function σ is in the vicinity of 0 (zero).)

具体的には、切換関数σの絶対値がφ未満となる領域において、非線形制御入力ｕ_３は、式（２６）（連続時間制御系）又は式（２７）（離散時間制御系）で表されるように線形となり、切換関数σが０（ゼロ）の近傍となる領域におけるチャタリングを防止する。

Specifically, in the region where the absolute value of the switching function σ is less than φ, the nonlinear control input u ₃ is expressed by Equation (26) (continuous time control system) or Equation (27) (discrete time control system). Thus, chattering in a region where the switching function σ is in the vicinity of 0 (zero) is prevented.

以上の構成により、設計者は、線形状態フィードバックゲインに基づいて超平面ベクトル及び比例到達ゲインを算出することで、制御対象のダイナミクスが微分方程式で記述される連続時間制御系、及び、制御対象のダイナミクスが差分方程式で記述される離散時間制御系の双方に対してスライディングモード制御を適用することができ、線形状態フィードバックゲインの設計法が確立している線形制御系を容易にスライディングモード制御系に拡張することができる。

With the above configuration, the designer calculates the hyperplane vector and the proportional reach gain based on the linear state feedback gain, and thereby the continuous-time control system in which the dynamics of the controlled object are described by a differential equation, and the controlled object Sliding mode control can be applied to both discrete-time control systems whose dynamics are described by differential equations, and linear control systems with established linear state feedback gain design methods can be easily converted to sliding mode control systems. Can be extended.

  Each of the state quantities of the state vector x to be fed back may be detected directly from the control target, or may be estimated by an observer. However, the state quantity directly detected from the controlled object and the state quantity estimated by the observer are generally different from the controllable canonical state quantity. Therefore, the state quantity directly detected from the control object or the state quantity estimated by the observer is fed back after being converted into a controllable canonical state quantity.

  Next, the control system 100 of the motor with a reduction gear designed according to the design method of the sliding mode control system according to the present invention will be described with reference to FIG.

The control system 100 is a resonance mechanical system in which an electric motor 101 and a load 104 are coupled by a connecting portion 103, and a target signal generator 105 determines a position θ _m of the electric motor 101 detected by a position detector 102 such as a resolver. It operates to match the target signal theta _r generated.

The position controller 106 outputs a speed command ω _r based on the position deviation (θ _r −θ _m ).

  The speed controller 107 corresponds to the sliding mode controller 10 according to the present invention, receives an input of the controllable canonical state vector x, and outputs a torque command as the control input u.

The torque controller 108 supplies electric power τ _r to the motor 101 so that the torque output from the motor 101 matches the control input u.

The state observer (observer) 109 is based on the control input u output from the speed controller 107 and the position θ _m of the motor 101 detected by the position detector 102, and the estimated twist value (the position θ _{m of the} motor 101 and the load 104). and it outputs the differential value omega _t of shift) of the theta _t and twisted estimate theta _t between the position of the.

The subtractor 110 subtracts the position θ _m of the electric motor 101 from the target signal θ _r in order to derive the position deviation (θ _r −θ _m ).

Subtractor 111 inputs the result of subtracting a speed command omega _r from the differential value omega _t of torsion estimates theta _t the transducer 112.

Converter 112 converts the controllable canonical form a column vector and a value obtained by subtracting a speed command omega _r from the differential value omega _t of torsion estimates theta _t and twisted estimate theta _t by the transformation matrix T, the state The vector x is output.

  The state equation of the control system 100 is expressed by Expression (28), and the state vector x, the system matrix A, the input matrix B, and the output matrix C are respectively expressed by Expression (29), Expression (30), Expression (31), Expression (32)

ここで、ｊ_ｍは、電動機１０１の慣性モーメントであり、ｊ_ａは、合成モーメント＝（１／ｊ_ａ＝１／ｊ_ｍ＋１／ｊ_ｌ）であり、ｊ_ｌは、負荷１０４の慣性モーメントである。また、ｋは、ねじり剛性であり、ｂは、ねじり粘性である。

Here, j _m is the moment of inertia of the motor 101, j _a is the combined moment = (1 / j _a = 1 / j _m + 1 / j _l ), and j _l is the moment of inertia of the load 104. is there. Further, k is torsional rigidity, and b is torsional viscosity.

The inertia moment j _m of the electric motor 101 is set to 0.0001 [kgm ² ] and the inertia moment j _l of the load 104 is set to 0.001 [kgm ² ] (as a result, the composite moment j _a is 0.000091 [kgm ^2]. When the torsional rigidity k is 98.7 [Nm / rad] and the torsional viscosity b is negligible (a value close to 0 [Nm / rad / s]), the control system 100 is designed. A transformation matrix T for transforming the system represented by Expression (28) into a controllable canonical form is represented by Expression (33).

電動機１０１と負荷１０４とから成るこの二慣性系の主共振周波数を５０［Ｈｚ］とし、速度制御帯域を７５［Ｈｚ］とし、且つ、減衰性を適度に調整すると、状態フィードバックゲインＦ_ｃは、式（３４）で表されるものとなる。

When the main resonance frequency of this two-inertia system composed of the electric motor 101 and the load 104 is set to 50 [Hz], the speed control band is set to 75 [Hz], and the attenuation is appropriately adjusted, the state feedback gain _Fc is This is expressed by Expression (34).

この状態フィードバックゲインＦ_ｃと変換行列Ｔとを用いて式（８）を式（３５）とし式（３５）を解くことによって比例到達ゲインη＝１３１０、及び、超平面ベクトルＳ＝［１５０７００ ２０５ １］が得られ、このようにして、制御系１００の設計が実現される。

By using the state feedback gain F _c and the transformation matrix T, the equation (8) is converted into the equation (35) and the equation (35) is solved to obtain the proportional reach gain η = 1310 and the hyperplane vector S = [150700 205 1 In this way, the design of the control system 100 is realized.

図６は、制御系１００における電動機１０１及び負荷１０４のそれぞれの回転位置の時間的推移のシミュレーション結果を示す図であり、縦軸に回転位置、横軸に時間軸をそれぞれ配する。また、電動機１０１の回転位置の推移を一点鎖線で示し、負荷の回転位置の推移を点線で示し、位置指令の推移を実線で示すものとする。

FIG. 6 is a diagram showing a simulation result of temporal transitions of the respective rotational positions of the electric motor 101 and the load 104 in the control system 100. The vertical axis represents the rotational position, and the horizontal axis represents the time axis. In addition, it is assumed that the transition of the rotational position of the electric motor 101 is indicated by a one-dot chain line, the transition of the rotational position of the load is indicated by a dotted line, and the transition of the position command is indicated by a solid line.

6A and 6B show the case where the torsional stiffness k (98.7 [Nm / rad]) and the torsional viscosity b (0.0001 [Nm / rad / s]) as given in the design values are given. 6A shows a case where position control is performed without using the non-linear controller 14, and FIG. 6B shows a case where position control is performed using the non-linear controller 14. Indicates. Note that the nonlinear controller 14 adopts the saturation function expressed by the equation (6), the nonlinear gain K is 1.5, and the constant φ indicating the boundary layer is 8.044 × 10 ⁻⁴ .

  Referring to FIGS. 6A and 6B, it can be seen that the simulation result in FIG. 6B has better settling than the simulation result in FIG.

  FIGS. 6C and 6D are simulation results when a torsional rigidity k (69.1 [Nm / rad]) of 70% of the design value is given (other conditions are shown in FIG. 6). (A) and FIG. 6 (B) are equivalent.) FIG. 6 (C) shows the case where position control is performed without using the nonlinear controller 14, and FIG. 6 (D) shows nonlinear control. The case where position control is performed using the device 14 is shown.

  Referring to FIGS. 6A to 6D, the simulation result in FIG. 6C has a larger residual vibration than the simulation result in FIG. 6A, and the simulation in FIG. As a result, it can be seen that the residual vibration is reduced as compared with the simulation result in FIG.

  Although the preferred embodiments of the present invention have been described in detail above, the present invention is not limited to the above-described embodiments, and various modifications and substitutions can be made to the above-described embodiments without departing from the scope of the present invention. Can be added.

  For example, the sliding mode controller 10 can be applied to both a regulator that controls the output power to be kept constant, and a servo system that controls to follow a target value using the position of an object as a control amount.

  A series of design procedures to acquire linear state feedback gain and calculate hyperplane vector and proportional arrival gain based on the acquired linear state feedback gain is realized as a design support device using hardware such as a microprocessor. It may also be realized as design support software that can be distributed through various computer-readable storage media and telecommunication lines.

  DESCRIPTION OF SYMBOLS 10 ... Sliding mode controller 11 ... Equivalent controller 12 ... Switching function calculator 13 ... Proportional transfer controller 14 ... Nonlinear controller 15 ... Adder 100 ... With reduction gear Motor control system 101 ... Motor 102 ... Position detector 103 ... Connector 104 ... Load 105 ... Target signal generator 106 ... Position controller 107 ... Speed controller 108 ..Torque controller 109 ... State observer 110, 111 ... Subtractor 112 ... Converter

Claims (5)

An equivalent controller for calculating the first control input, which is an equivalent control input, a proportional arrival controller for calculating the second control input proportional to the value of the switching function, and a non-linear with respect to the state of the controlled object determined based on the switching function And a non-linear controller that calculates a third control input, and a sliding device including an adder that calculates the final control input by adding the first control input, the second control input, and the third control input A mode control system design method,
A calculation step for calculating a state feedback gain when the control object is controlled by a linear control system, and a control input derived by the state feedback gain is equal to the sum of the first control input and the second control input A design step of designing the equivalent controller and the proportional reach controller;
Method for designing a sliding mode control system having The sliding mode control system is a continuous time control system,
If the hyperplane vector is S, the system matrix is A, the unit matrix is I, the state vector is x, the proportional gain is η, and the state feedback gain is F _c ,
The first control input is represented by u ₁ = −SAx,
The second control input is represented by u ₂ = −ηSx,
The control input by the state feedback gain when the control target is controlled by a linear control system is represented by u _C = −F _c x,
In the design step, the equivalent controller and the proportional reaching controller derive a hyperplane vector S and a proportional reaching gain η from an equation S (A + ηI) = F _c based on the relationship u ₁ + u ₂ = u _C. Designed by,
The sliding mode control system design method according to claim 1. The sliding mode control system is a discrete time control system,
If the hyperplane vector is S, the system matrix is Φ, the unit matrix is I, the state vector is x, the number of samples is k, the proportional gain is μ, and the state feedback gain is F _d ,
The first control input is represented by u ₁ [k] = − S (Φ−I) × [k],
The second control input is represented by u ₂ [k] = − μSx [k],
The control input by the state feedback gain when the control target is controlled by a linear control system is represented by u _d [k] = − F _d x [k],
In the design step, the equivalent controller and the proportional arrival controller are configured to use the equation S {Φ + (μ−1) I} = F based on the relationship u ₁ [k] + u ₂ [k] = u _d [k]. designed by deriving the hyperplane vector S and the proportional reach gain μ from _d ,
The sliding mode control system design method according to claim 1.   4. The sliding mode control system design method according to claim 1, wherein in the designing step, the nonlinear controller is designed using a saturation function. 5. An equivalent controller that calculates a first control input that is an equivalent control input using a hyperplane vector, and a proportional attainment control that calculates a second control input proportional to the value of the switching function using the hyperplane vector and a proportional transfer gain A non-linear controller that is determined based on a switching function and calculates a non-linear third control input with respect to the state of the control target, and the first control input, the second control input, and the third control input A design support apparatus that supports the design of a sliding mode control system including an adder that adds and calculates a final control input,
State feedback gain acquisition means for acquiring a state feedback gain when controlling a controlled object with a linear control system, and a control input derived by the state feedback gain is the sum of the first control input and the second control input Calculating means for calculating the hyperplane vector and the proportional transfer gain as being equal;
Sliding mode control system design support device comprising:

Images