DESIGN METHOD FOR UNDISTURBED SWITCHING OF LINEAR CONTROLLERS

Abstract

A design method for undisturbed switching of linear controllers is provided for solving the problem of sudden system response change or an unstable control circuit caused by switching of multiple linear controllers, in which the linear controllers include proportional-integral-derivative (PID), linear-quadratic-Gaussian (LQG), linear active disturbance rejection control (LADRC), H∞, model reference adaptive control (MRAC), and open-loop controllers. Firstly, differentials tbr outputs of the controllers are found, then through controller decision, a controller is selected to be connected to a closed-loop control circuit, and a differential term of the controller connected to the closed-loop control circuit is integrated through a common integrator, thereby ensuring smooth controller switching. The design method has a simple structure and good versatility, is operable, and can be easily applied to various actual control systems without requiring parameter adjustment.

Claims

1. A design method for undisturbed switching of linear controllers, comprising the following steps: step 1: establishing a numerical simulation program of a control system; step 2: configuring controllers to allow performance of a closed-loop control system to meet an expected design requirement, wherein the controllers comprise linear controllers and an open-loop controller; and step 3: combining the open-loop controller and the linear controllers into a switching controller by: differentiating outputs of the linear controllers, then based on a decision of the switching controller, selecting one of the linear controllers to be connected to form a closed-loop control circuit, and taking out a differential term of the linear controllers for integration; wherein taking out the differential term of the linear controllers for integration comprises differentiating the linear controllers, and then using a common integrator to ensure smooth switching, resulting in eliminating adverse effects of sudden change or instability caused by controller switching.

2. The design method for undisturbed switching of linear controllers according to claim 1, wherein in step 1, a plant controlled by the switching controller is a general nonlinear model, which is expressed as: $\{\begin{array}{c}\stackrel{.}{x}=f\left(x,u\right)\\ y=g\left(x,u\right)\end{array},$ wherein f is a nonlinear function of a system state; g is a nonlinear function of a system output; y is an output quantity of the plant; and u is a control quantity output by the switching controller.

3. The design method for undisturbed switching of linear controllers according to claim 1, wherein in step 2, the switching controller comprising the open-loop controller, an H∞ controller, and a linear active disturbance rejection control (LADRC) controller; and the design method further comprises the following steps: first, the open-loop controller is configured based on a nonlinear mode, wherein: an open-loop control law is abstracted as an interpolation function interp according to an interpolation table [r.sub.Table;value.sub.Table], and a reference input is set to r, to obtain:
u=interp(r,[r.sub.Table;value.sub.Table]), second, the H∞ controller is configured based on the nonlinear model; and the nonlinear model is linearized to obtain a linear system: $\{\begin{array}{c}\stackrel{.}{x}=\mathrm{Ax}+\mathrm{Bu}\\ u=\mathrm{Cx}+\mathrm{Du}\end{array},$ wherein r, e, u, and y represent a reference input, a tracking error, a control input, and a system output, respectively, C(s) represents the H∞ controller, and G(s) represents a controlled-object model, wherein closed-loop transfer functions for r to e, u, and y separately are: $\{\begin{array}{c}S\left(s\right)=\frac{e\left(s\right)}{r\left(s\right)}={\left(I+G\left(s\right)C\left(s\right)\right)}^{-1}\\ R\left(s\right)=\frac{u\left(s\right)}{r\left(s\right)}=C\left(s\right)S\left(s\right)=C\left(s\right){\left(I+G\left(s\right)C\left(s\right)\right)}^{-1}\\ T\left(s\right)=\frac{y\left(s\right)}{r\left(s\right)}=G\left(s\right)C\left(s\right){\left(I+G\left(s\right)C\left(s\right)\right)}^{-1}=I-S\left(s\right)\end{array},$ wherein that Ws(s) represents a performance weight function, Wr(s) represents a controller output weight function, and Wt(s) represents a robust weight function to ensure stability of the closed-loop control system, wherein conditions to be satisfied are: $\{\begin{array}{c}\stackrel{\_}{\sigma }\left(S\left(j\omega \right)\right)\le \stackrel{\_}{\sigma }\left[{\mathrm{Ws}}^{-1}\left(j\omega \right)\right]\\ \stackrel{\_}{\sigma }\left(R\left(j\omega \right)\right)\le \stackrel{\_}{\sigma }\left[{\mathrm{Wr}}^{-1}\left(j\omega \right)\right]\\ \stackrel{\_}{\sigma }\left(T\left(j\omega \right)\right)\le \stackrel{\_}{\sigma }\left[{\mathrm{Wt}}^{-1}\left(j\omega \right)\right]\end{array},$ an original problem is converted into a standard H∞ control problem, and an original closed-loop control system is augmented to obtain: $\left[\begin{array}{c}\mathrm{Ws}.Math.e\\ \mathrm{Wr}.Math.e\\ \mathrm{Wt}.Math.e\\ e\end{array}\right]=\left[\begin{array}{cc}\mathrm{Ws}& -\mathrm{WsG}\\ 0& \mathrm{Wr}\\ 0& \mathrm{WtG}\\ I& -G\end{array}\right]\left[\begin{array}{c}r\\ u\end{array}\right],$ and according to a solution to the standard H control problem, a general solution is obtained: $\{\begin{array}{c}\stackrel{.}{x}=\mathrm{Ax}+\mathrm{Be}\\ u=\mathrm{Cx}+\mathrm{De}\end{array},$ third, the LADRC controller is configured based on the nonlinear model; wherein a bandwidth of an extended state observer (ESO) is w.sub.o, an influence coefficient of a control quantity on a system state is b.sub.0, a target value estimated by the ESO is z.sub.1, a target-value derivative estimated by the ESO is z.sub.2, and a total system disturbance estimated by the ESO is z.sub.3, an active disturbance rejection control (ADRC) supports decoupling of a multi-variable loop and the multi-variable loop is directly formed by multiple single-variable control loops connected in parallel, and an ESO of the LADRC controller is expressed as: $\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\\ {\stackrel{.}{x}}_3\end{array}\right]=\left[\begin{array}{ccc}-3w_0& 1& 0\\ -3w_o^2& 0& 1\\ -w_o^3& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{cc}0& 3w_0\\ b_0& 3w_o^3\\ 0& w_o^3\end{array}\right]\left[\begin{array}{c}u\\ r\end{array}\right]\\ \left[\begin{array}{c}{z}_1\\ z_2\\ z_3\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}u\\ r\end{array}\right]\end{array},$ and a control law of the LADRC controller is obtained: $\{\begin{array}{c}u=\frac{-z_3}{b_0}+u_0\\ u_0=\mathrm{KP}\left(r-z_1\right)+\mathrm{KD}\left(\stackrel{.}{r}-z_2\right)\end{array}.$

4. The design method for undisturbed switching of linear controllers according to claim 1, wherein in step 3, when ù is a derivative of a control quantity u, and switch is a function for selecting the switching controller, indicating that an i-th controller is currently connected to the closed-loop control circuit, an output quantity u.sub.k of the controller in a k-th running cycle is expressed as: $\begin{array}{c}{u}_k={\int }_0^{t_k}\mathrm{switch}\left(\stackrel{.}{u},i\right)\mathrm{dt}={\int }_0^{t_{k-1}}\mathrm{switch}\left(\stackrel{.}{u},i\right)\mathrm{dt}+{\int }_{t_{k-1}}^{t_k}\mathrm{switch}\left(\stackrel{.}{u},i\right)\mathrm{dt}=\\ u_{k-1}+\delta u_k^i\end{array},$ wherein a current control quantity u.sub.k is obtained by adding a control increment Su.sub.k.sup.i of a current closed-loop circuit controller to a control quantity u.sub.k−1 at a previous moment, resulting in enabling a smooth controller switching without sudden change of control quantities.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] To make the advantages and implementations of the present disclosure clearer, the present disclosure is described in detail below in conjunction with the accompanying drawings and examples. The accompanying drawings are only used to describe the present disclosure, and are not intended to impose any limitation on the present disclosure.

[0025] FIG. 1 is a schematic diagram of a design method for undisturbed switching of linear controllers.

[0026] FIG. 2. is a schematic diagram for H∞ control.

[0027] FIG. 3 is a schematic diagram for LADRC control.

[0028] FIG. 4 is a control effect diagram of a switching controller not including an undisturbed switching design method.

[0029] FIG. 5 is a control effect diagram of a switching controller including an undisturbed switching design method.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0030] The present disclosure is further described below with reference to the accompanying drawings and examples.

[0031] The present disclosure provides a design method for undisturbed switching of linear controllers, and treats an open-loop controller as a linear controller, to solve the problem of sudden system change or even instability caused by linear controller switching in practical projects.

[0032] A design method for undisturbed switching of linear controllers includes the following steps.

[0033] Step 1: Directly establish a numerical simulation program of a control system without considering impact of controller switching. FIG. 1 shows a control framework of the control system, in which a plant is a general nonlinear model. In this embodiment, the nonlinear model is specifically a dual-rotor turbofan engine, which is expressed as:

[00010]$\{\begin{array}{c}\stackrel{.}{x}=f\left(x,u\right)\\ y=g\left(x,u\right)\end{array}$

where f is a nonlinear function of system state; g is a nonlinear function of system output; u is a control quantity output by a switching controller, which may be expressed as u=[WF M,A8].sup.T; and y is an output quantity of the engine, which may be expressed as y=[N.sub.2,π.sub.T].sup.T.

[0034] Step 2: Without considering impact of controller switching, directly design controllers such that performance of the closed-loop control system meets an expected design requirement. The undisturbed switching design method proposed in the present disclosure is for all linear controllers and open-loop controllers, such as PID, LQG, LADRC, H∞, MRAC, and open-loop controllers. The controller in the present disclosure is specifically a switching controller including an open-loop controller, an H∞ controller, and an LADRC controller.

[0035] First, in terms of the open-loop controller, the nonlinear model is designed as follows: an open-loop control law is abstracted as an interpolation function interp according to an interpolation table [r.sub.Table;value.sub.Table], and a reference input is set to r, to obtain:

u=interp(r,[r.sub.Table;value.sub.Table])

[0036] Based on engineering experience, an open-loop control law may be obtained:

[00011]$\left[\begin{array}{c}\mathrm{WFM}\\ A8\end{array}\right]=\mathrm{interp}\left(r,\left[\begin{array}{c}\left[\begin{array}{c}15,20,25,30,35,40,45,50,55\\ 15,20,25,30,35,40,45,50,55\end{array}\right],\\ \left[\begin{array}{c}5000,6000,10000,15000,18000,20000,22000,25000,28000\\ 500,480,450,440,400,380,350,310,300\end{array}\right]\end{array}\right]\right)$

[0037] Second, in terms of the H∞ controller, the nonlinear model is designed as follows: and the nonlinear model is linearized to obtain a linear system:

[00012]$\{\begin{array}{c}\stackrel{.}{x}=\mathrm{Ax}+\mathrm{Bu}\\ y=\mathrm{Cx}+\mathrm{Du}\end{array}$

[0038] The dual-rotor turbofan engine model is linearized, and according to a small deviation model method, a linear model is obtained through system identification at equilibrium points of idle and above states, which is expressed as:

[00013]$\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\end{array}\right]=\left[\begin{array}{cc}-4.348& 7.408\\ 3.6964& -44.6039\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{cc}0.0131& -0.0139\\ 0.0202& 0.0888\end{array}\right]\left[\begin{array}{c}\mathrm{WFM}\\ A8\end{array}\right]\\ \left[\begin{array}{c}{N}_2\\ {\pi }_T\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}\mathrm{WFM}\\ A8\end{array}\right]\end{array}$

[0039] FIG. 2 is a schematic diagram of a closed-loop control system based on H∞. If r, e, u and y represent a reference input, a tracking error, a control input, and a system output, respectively, C(s) represents an H∞ controller, and G(s) represents a controlled-object model, closed-loop transfer functions from r to e, u, and y separately may be obtained:

[00014]$\{\begin{array}{c}\begin{array}{c}S\left(s\right)=\frac{e\left(s\right)}{r\left(s\right)}={\left(I+G\left(s\right)C\left(s\right)\right)}^{-1}\\ R\left(s\right)=\frac{u\left(s\right)}{r\left(s\right)}=C\left(s\right)S\left(s\right)=C\left(s\right){\left(I+G\left(s\right)C\left(s\right)\right)}^{-1}\end{array}\\ T\left(s\right)=\frac{y\left(s\right)}{r\left(s\right)}=G\left(s\right)C\left(s\right){\left(I+G\left(s\right)C\left(s\right)\right)}^{-1}=I-S\left(s\right)\end{array}$

[0040] Assuming that Ws(s) represents a performance weight function, Wr(s) represents a controller output weight function, and Wt(s) represents a robust weight function, to ensure stability of the closed-loop control system, conditions to be satisfied are:

[00015]$\{\begin{array}{c}\begin{array}{c}\stackrel{\_}{\sigma }\left(S\left(j\omega \right)\right)\le \stackrel{\_}{\sigma }\left[{\mathrm{Ws}}^{-1}\left(j\omega \right)\right]\\ \stackrel{\_}{\sigma }\left(R\left(j\omega \right)\right)\le \stackrel{\_}{\sigma }\left[{\mathrm{Wr}}^{-1}\left(j\omega \right)\right]\end{array}\\ \stackrel{\_}{\sigma }\left(T\left(j\omega \right)\right)\le \stackrel{\_}{\sigma }\left[{\mathrm{Wt}}^{-1}\left(j\omega \right)\right]\end{array}$

[0041] An original problem is converted into a standard H∞ control problem, and the original closed-loop control system is augmented to obtain:

[00016]$\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\mathrm{Ws}.Math.e\\ \mathrm{Wr}.Math.e\end{array}\\ \mathrm{Wt}.Math.e\end{array}\\ e\end{array}\right]=\left[\begin{array}{cc}\mathrm{Ws}& -\mathrm{WsG}\\ 0& \mathrm{Wr}\\ 0& \mathrm{WtG}\\ I& -G\end{array}\right]\left[\begin{array}{c}r\\ u\end{array}\right]$

[0042] According to a solution to the standard H∞ control problem, a general solution may be obtained:

[00017]$\{\begin{array}{c}\stackrel{.}{x}=\mathrm{Ax}+\mathrm{Be}\\ u=\mathrm{Cx}+\mathrm{De}\end{array},$

[0043] According to a DGKF solution to the standard H∞ control problem, a feasible solution may be obtained:

[00018]$\{\begin{array}{c}\stackrel{.}{x}=\left[\begin{array}{cccccc}-1e-5& 0& 0& 0& 0& 0\\ 0& -1e-5& 0& 0& 0& 0\\ 0& 0& -0.8882& 0& 0& 0\\ 0& 0& 0& -0.9288& 0& 0\\ 0& 0& 0& 0& -9.189& 0\\ 0& 0& 0& 0& 0& -63.43\end{array}\right]x+\\ \left[\begin{array}{cc}12.05& 2.802\\ -3.693& 16.49\\ 3.988& 1.233\\ -1.096& 4.362\\ -15.48& -2.872\\ -0.63& -10.24\end{array}\right]\left[\begin{array}{c}{N}_{2_{\mathrm{err}}}\\ N_{T_{\mathrm{err}}}\end{array}\right]\\ \left[\begin{array}{c}\mathrm{WFM}\\ A8\end{array}\right]=\left[\begin{array}{cccccc}12.28& -2.367& -3.376& 0.762& -13& -1.825\\ 1.534& 16.7& -0.7573& -4.406& 1.34& -9.903\end{array}\right]x+\\ \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{N}_{2_{\mathrm{err}}}\\ N_{T_{\mathrm{err}}}\end{array}\right]\end{array}$

[0044] Third, the LADRC controller is designed for the nonlinear model. If bandwidth of an ESO is w.sub.o, an influence coefficient of a control quantity on a system state is b.sub.0, a target value estimated by the ESO is z.sub.1, a target-value derivative estimated by the ESO is z.sub.2, and a total system disturbance estimated by the ESO is z.sub.3, because ADRC supports decoupling of a multi-variable loop and the multi-variable loop is directly formed by multiple single-variable control loops connected in parallel, an LADRC closed-loop control system is shown in FIG. 3, and an ESO of the LADRC may be expressed as:

[00019]$\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\\ {\stackrel{.}{x}}_3\end{array}\right]=\left[\begin{array}{ccc}-3w_0& 1& 0\\ -3w_o^2& 0& 1\\ -w_o^3& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{cc}0& 2w_0\\ b_0& 3w_o^3\\ 0& w_o^3\end{array}\right]\left[\begin{array}{c}u\\ r\end{array}\right]\\ \left[\begin{array}{c}{z}_1\\ z_2\\ z_3\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}u\\ r\end{array}\right]\end{array}$

[0045] A control law of the LADRC may be obtained:

[00020]$\{\begin{array}{c}u=\frac{-z_3}{b_0}+u_0\\ u_0=\mathrm{KP}\left(r-z_1\right)+\mathrm{KD}\left(\stackrel{.}{r}-z_1\right)\end{array}$

[0046] After repeated parameter adjustment, specific parameters of LADRC may be obtained:

[00021]$\{\begin{array}{c}{\mathrm{KP}}_{\mathrm{WFM}}=100,{\mathrm{KD}}_{\mathrm{WFM}}=2,w_{0_{\mathrm{WFM}}}=15,b_{0_{\mathrm{WFM}}}=2\\ {\mathrm{KP}}_{A8}=100,{\mathrm{KD}}_{A8}=2,w_{0_{A8}}=15,b_{0_{A8}}=15,b_{0_{A8}}=1\end{array}$

[0047] Step 3: Combine the open-loop controller, the H∞ controller, and the LADRC controller into a switching controller, and design an undisturbed switching method to achieve the four objectives of the present disclosure. The undisturbed switching design method proposed by the present disclosure first finds differentials for outputs of the controllers, then through controller decision, selects a controller to be connected to a closed-loop control circuit, and takes out a differential term of the controller for integration, that is, finding integrals of the controllers, and then using a common integrator to ensure smooth switching, so as to eliminate adverse effects of sudden change and even instability caused by controller switching.

[0048] If ù is a derivative of a control quantity u, and switch is a function for selecting a switching controller, indicating that the i-th controller is currently connected to the closed-loop control circuit, an output quantity u.sub.k of the controller in the k-th running cycle may be expressed as:

[00022]$u_k={\int }_0^{t_k}\mathrm{switch}\left(\stackrel{.}{u},i\right)\mathrm{dt}={\int }_0^{t_{k-1}}\mathrm{switch}\left(\stackrel{.}{u},i\right)\mathrm{dt}+{\int }_{t_{k-1}}^{t_k}\mathrm{switch}\left(\stackrel{.}{u},i\right)\mathrm{dt}=u_{k-1}+\delta u_k^i$

[0049] The current control quantity u.sub.k is obtained by adding a control increment Su.sub.k.sup.i of the current closed-loop circuit controller to a control quantity u.sub.k−1 at a previous moment, such that smooth controller switching can be implemented without sudden change of control quantities.

[0050] After the above three steps, a switching controller composed of the open-loop controller, the H∞ controller, and the LADRC controller is obtained. FIG. 4 shows a control effect of the switching controller not including the undisturbed switching design method. FIG. 5 shows a control effect of the switching controller including the undisturbed switching design method. It can be seen that the design method for undisturbed switching of linear controllers proposed by the present disclosure has significant improvements, and its beneficial effects are as follows:

(1) The present disclosure uses a common integrator for integration after finding differentials of multiple linear controllers, which ensures smooth controller switching without affecting performance of an original controller, effectively solves the problem of sudden change or even unstable control caused by the controller switching, and meets the industry's control performance requirements for controller switching.
(2) The undisturbed switching design method proposed by the present disclosure is versatile and is suitable for all linear controllers in a control system, including undisturbed switching between a single-variable controller and a multi-variable controller, and between an open-loop controller and a closed-loop controller.
(3) The undisturbed switching method proposed by the present disclosure has a simple structure, is operable, and can be easily applied to various actual control systems without adjusting parameters of the existing controllers.

[0051] The examples of the present disclosure are described above in detail, which are merely preferred examples of the present disclosure and cannot be construed as limiting the scope of implementation of the present disclosure. The implementations of the steps may be quantity, and all equivalent changes and improvements made according to the scope of the present disclosure should still fall within the scope of this patent.