JP7347790B2 - Output estimation method - Google Patents

Description

The present invention relates to a method for estimating an output in response to an input to a controlled object via a controller K, and particularly relates to a method for estimating an output in response to an input to a controlled object (hereinafter referred to as a "single input/output response"). ) is used to estimate the output for any input to a controlled object connected to a controller.

Control engineering usually takes the approach of identifying an accurate model system of the controlled object, designing a controller that can be expected to have good control performance based on that model, and verifying the control performance through experiments.

However, in many actual controlled objects, it is not possible to obtain rapidly changing responses, open-loop responses, or long-term experiments. In this case, it is difficult to obtain an accurate model through system identification.

For such cases, data-driven control in which the controller is directly adjusted from experimental data without determining a model has been actively studied in recent years (Non-Patent Documents 1 to 4). Among them, Virtual Reference Feedback Tuning (VRFT) and Fictitious Reference Iterative Tuning (FRIT) can optimize the control gain so that the closed-loop system approaches the reference model based on only one experimental data. Therefore, many application examples have been reported.

M. C. Campi, A. Lecchini, S. M. Savaresi, Virtual reference feedback tuning: a direct method for the design of feedback controllers, Automatica, Vol. 38, pp. 1337-1346, 2002. Shotaro Soma, Osamu Kaneko, Takao Fujii, A new approach to controller parameter tuning using data from a single experiment - Proposal of Fictitious Reference Iterative Tuning, Transactions of the Institute of Systems, Control and Information Engineers, Vol. 17, No. 12, pp. 528-536, 2004 Osamu Kaneko, Toru Yamamoto, et al., Special feature on data-driven control: New innovations and new horizons, Measurement and Control, Vol. 52, No. 10, pp. 892-897, 2013 Kenichi Tasaka, Manabu Kano, Morimasa Ogawa, Shiro Masuda, Toru Yamamoto, Direct PID adjustment based on closed-loop data and its application to unstable processes, Transactions of the Institute of Systems, Control and Information Engineers, Vol. 22, No. 4, pp. 137-144, 2009

In VRFT and FRIT, if the reference model is not appropriate, there is a problem that not only the control performance deteriorates but also the control system may become unstable.

The present invention was conceived in view of the above problems, and estimates the time response of a closed-loop system when a certain controller is introduced based on only one experimental data.

More specifically, the output estimation method according to the present invention includes:
A method for estimating an output for an input to a controlled object via a controller K, the method comprising:
obtaining a periodic input/output response (periodic input signal u ₀ and periodic output signal y ₀ ) to the controlled object based on one trial;
The present invention is characterized in that it includes a step of determining the response of the controlled object (virtual output signal y) to the virtual input signal r using equation (1).

Here, r ₁ (jω) follows equation (2). r ₁ (jω), r (jω), u ₀ (jω), and y ₀ (jω) represent signals obtained by Fourier transform of the signals r ₁ , r, u ₀ , and y _{0 ,} respectively. Further, K ⁻¹ (jω) represents 1/K(jω), and K(jω) represents the transfer function of the controller K. F ⁻¹ represents the inverse Fourier transform.

The output estimation method according to the present invention is characterized in that the frequency components of the input/output response of the feedback system are obtained from the Fourier transform of experimental data and the frequency characteristics of the controller, and the frequency components of the input/output response of the feedback system are obtained by inverse Fourier transform to obtain the virtual time response. . In other words, since no transfer function model is used to obtain the virtual output signal (estimated output signal), problems caused by the suitability of the model or ignorance of the structure of the model can be avoided. can.

FIG. 3 is a diagram showing the flow of processing of the output estimation method according to the present invention. FIG. 3 is a diagram showing a procedure for converting an aperiodic input signal u ₀₀ and an aperiodic output signal y ₀₀ into a periodic input signal u ₀ and a periodic output signal y ₀ . FIG. 3(a) is a block diagram showing the relationship between the aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ , and FIG. 3(b) is a block diagram showing the relationship between the virtual input signal r and the virtual output signal y. FIG. 3C is a block diagram showing the flow of signals when r1 is input to the control system. It is a figure which shows the observation result of an aperiodic output signal when a step signal is inputted as an aperiodic input signal with respect to control objects G1 to G9. It is a graph comparing a true time response and a virtual time response to a controlled object. It is a graph comparing the true time response and the virtual time response when the parameters of each controller are optimized. It is a diagram showing the true output response and virtual time response of the feedback system when the one-time input/output responses u ₀₀ and y ₀₀ are closed-loop data and G is unstable. It is a graph comparing a true time response and a virtual time response when parameters are optimized. It is a photograph of a motor used in an example. It is a graph comparing a one-time input/output response to a motor, a true time response (true output signal), and a virtual time response. FIG. 3 is a diagram showing a true time response (true input signal) of a one-time input/output response to a motor.

The output estimation method according to the present invention will be described below with reference to drawings and embodiments. In addition, the following description illustrates one embodiment and one example of the present invention, and the present invention is not limited to the following description. The following description can be modified without departing from the spirit of the present invention.

In addition, an aperiodic input signal u ₀₀ , an aperiodic output signal y ₀₀ , a _periodic input signal u 0 , a periodic output signal y ₀ , a virtual input signal r to the control system, a virtual output signal y to the control system, a virtual output signal y to the controlled object The virtual input signal u is a time function and a continuous signal. However, it can always be treated as discrete processing using known methods. That is, in this specification, expression as a continuous function includes discrete expression. When handling signals discretely thereafter, the number k representing the number of samples may be used in parentheses, such as r(k).

FIG. 1 shows the processing flow of the output estimation method according to the present invention, and then details of each step are shown. In the output estimation method according to the present invention, the periodic output signal y ₀ when the periodic input signal u ₀ is input to the controlled object is obtained by one input/output response (one trial), and the periodic input/output response is used to calculate the periodic output signal y 0. Estimate a virtual output signal y for a virtual input signal r. Here, there are two methods to obtain the periodic input/output response.

Refer to the process P100 in FIG. G(z) indicates a controlled object such as a heating furnace or a motor. z is a advance operator. The first method is to first create a periodic input signal u ₀ and input it to the controlled object G to obtain a periodic output signal y ₀ . This is a direct method. This is the only time the input and output are actually measured.

Next, refer to process P101 and process P102. In the second method, an aperiodic input signal u ₀₀ is input to the controlled object G to obtain an aperiodic output signal y ₀₀ (processing P101). Then, the aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ obtained in process P101 are converted into periodic response signals (process P102). Conversion into periodic response signals can be obtained by successively adding inverted signals to the original signal.

The converted periodic input signal u ₀ and periodic output signal y ₀ are periodic response signals, and may be regarded as the aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ under certain conditions. In this method, the signal input/output in the controlled object G is actually checked only once in the process P101 in which the aperiodic input signal u ₀₀ is input to the controlled object G and the aperiodic output signal y ₀₀ is obtained. In addition, in this specification, the method of obtaining the periodic input/output response (periodic input signal u ₀ , periodic output signal y ₀ ) using the methods of process P101 and process P102 will be mainly explained.

Next, refer to process P103. Here, a controller K is introduced to control the controlled object G(z). Let these discrete time frequency transfer functions be G(jω) and K(jω), respectively. Furthermore, the Fourier-transformed virtual input signal r for the entire control system including the controller K is assumed to be r(jω). The output of the controller K(jω) based on the virtual input signal r(jω) is assumed to be the virtual input signal u(jω) for the controlled object G(jω), and the virtual input signal u(jω) is input to the controlled object G(jω). Let y(jω) be the virtual output signal when this happens. Then, the virtual output signal y of the controlled object G(z) controlled by the controller K(z) is expressed by equation (1).

Here, r ₁ (jω), r (jω), u ₀ (jω), and y ₀ (jω) represent signals obtained by Fourier transform of the signals r ₁ , r, u ₀ , and y _{0 ,} respectively. Further, K ⁻¹ (jω) represents 1/K(jω), and K(jω) represents the transfer function of the controller K. F ⁻¹ represents the inverse Fourier transform.

Further, r ₁ (jω) is expressed as in equation (2). u ₀ (jω) and y ₀ (jω) are obtained by Fourier transforming the periodic input signal u ₀ and the periodic output signal y ₀ . The virtual input signal r is of course a known signal. r ₁ (jω) is expressed by a transfer function of the controller K set by itself, and known information such as a periodic input signal u ₀ (jω) and a periodic output signal y ₀ (jω).

Therefore, equation (1) indicates that the virtual output signal y when the virtual input signal r is input to the control system can be estimated from only known information. The virtual input signal u and virtual output signal y to the controlled object obtained in this way are called a virtual time response.

Note that the virtual input signal r(k) is a target input to the closed loop system (control system) to be evaluated. The discrete Fourier transforms of the periodic input signal u ₀ , periodic output signal y ₀ , and virtual input signal r(k) are respectively converted into the periodic input signal u ₀ (jω), the periodic output signal y ₀ (jω), and the virtual input signal r(jω). shall be. Here, j is an imaginary unit, and ω [rad/s] is an angular frequency given by equation (3) with t _s [s] as the sampling time.

Furthermore, in the output estimation method of the present invention, it is assumed that the periodic input signal u ₀ , the virtual input signal r, and the discrete time controller K satisfy the following assumptions.
A1) u ₀ , r are time series of integer periods of a periodic signal. At this time, _y0 also becomes a periodic signal.
A2) u ₀ includes the same frequency component as the frequency component included in r. That is, if r(jω _a )≠0 at a certain ω _a , then u ₀ (jω _a )≠0. Taking the contraposition, if u ₀ (jω _a )=0 at a certain ω _a , then r(jω _a )=0.
A3) K stabilizes the feedback system.
A4) The final value of the inverse discrete Fourier transform of K ⁻¹ (jω)u ₀ (jω) can be regarded as the steady state. This is satisfied if the zero point of K(s) obtained by inverse Z-transformation or inverse bilinear transformation of K(z) is not on the imaginary axis and the number of data N is sufficiently large.
A5) When the zero point of G(s) obtained by inverse Z transformation or inverse bilinear transformation of G(z) is on the imaginary axis, K(s) does not have that zero point as a pole.
A6) When the aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ are used as step response data, the initial and final values of the aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ can be considered to be in a steady state.

<Processing P101>
Next, detailed explanation will be given starting from process P101. In process P101, an actual aperiodic input signal u ₀₀ is input to the controlled object G(z), and its output is an aperiodic output signal y ₀₀ .

The controlled object G(z) is a linear time-invariant discrete time one-input/output system, and one experimental data is an aperiodic input signal u ₀₀ (k), which is a time series of the input/output of G(z), and an aperiodic Suppose that the output signal is y ₀₀ (k). As already mentioned, z is the advance operator, k (=1, 2, . . . , N) is the number of samples, and N is the total number of data. The aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ may be either a closed loop response or an open loop response.

<Processing P102>
Based on assumption A6, periodic signals u ₀ and _{y 0} that satisfy assumption A1 are created as follows using an aperiodic input signal u ₀₀ and an aperiodic output signal y ₀₀ .

If the open-loop or closed-loop aperiodic input signal u ₀₀ and aperiodic output signal y ₀₀ are made into step responses, they do not satisfy assumption A1, but the waveforms of the aperiodic input signal u ₀₀ and aperiodic output signal y ₀₀ can be copied. However, the waveform obtained by turning the top and bottom upside down and joining them satisfies assumption A1. Specifically, FIG. 2(a) shows a case where the non-periodic input/output responses u ₀₀ and y ₀₀ are step response waveforms. The horizontal axis is time (the number of steps since it is discretized), and the vertical axis is the signal strength. Moreover, FIG. 2(b) shows a waveform obtained by copying and inverting the waveform of FIG. 2(a) and joining them together. Similarly to FIG. 2(a), the horizontal axis is the number of steps, and the vertical axis is the signal strength.

The signal in FIG. 2(b) can be obtained by processing the step response in FIG. 2(a) using equations (4) and (5). The signal in FIG. 2(b) is a periodic input/output response for one period (periodic input signal u ₀ and periodic output signal y ₀ ), and these satisfy assumption A1. Under assumption A6, these periodic input signal u ₀ and periodic output signal y ₀ match the true response of the controlled object G.

By following the above procedure, a one-time input/output response (aperiodic input signal u ₀₀ , aperiodic output signal y ₀₀ ) is converted into a periodic input/output response (periodic input signal u ₀ , periodic output signal y ₀ ). be able to. Further, under a predetermined assumption, the periodic signal matches the true response of the controlled object G.

<Processing P103>
Next, it will be explained that when the controlled object G is controlled by the controller K, the virtual output signal y(k) when the virtual input signal r(k) is input to the controller K can be expressed by equation (1).

A periodic signal is represented by a sum of sine waves. Therefore, based on assumption A1, the periodic input signal u ₀ , periodic output signal y ₀ , and virtual input signal r can be expressed as a sum of sine waves. At this time, the frequency response is a steady characteristic and does not include transient characteristics. When the discrete time frequency transfer function of the controlled object G is G(jω), the periodic output signal y ₀ for the periodic input signal u ₀ is expressed as in equation (6).

Expression (6) is expressed in a block diagram as shown in FIG. 3(a). Next, a controller K that controls the controlled object G is introduced. A block diagram when controller K is introduced is shown in FIG. 3(b). Equation (7) is obtained from FIG. 3(b).

Substituting equation (6) into equation (7), equation (8) is obtained.

Here, by setting the denominator of equation (8) as r ₁ (jω), equation (9) is obtained. Note that if r ₁ (jω) is written explicitly, it is expressed as equation (10). Furthermore, K ⁻¹ (jω) in formula (10) means 1/K(jω).

The right side of equation (9) is the virtual input signal r(jω) to the controller K, the periodic input signal _u0 , the periodic output signal _y0 , and the transfer function of the controller K, all of which are known information. . That is, based on this known information, a virtual output signal y(jω) for a virtual input signal r(jω) to the controller K is determined. Note that equation (2) is the same as equation (10).

Expressions (6) and (10) are expressed as a block diagram as shown in FIG. 3(c). From the figure, r ₁ is the target input of the feedback system when the input/output response of the controlled object G becomes the periodic input signal u ₀ and the periodic output signal y ₀ . Under assumption A4, r ₁ can be considered as a periodic signal.

Further, at this time, a virtual input signal u(jω) to the controlled object G is obtained as shown in equation (11) from FIG. 3(b) and equations (6) and (9).

The virtual output signal y(k) was obtained as the output when the virtual input signal r(k) was input to the controller K, but this was not actually observed. However, it can be estimated as described above from the aperiodic input signal u ₀₀ and the aperiodic output signal y ₀₀ which are one-time input/output responses. Under conditions in which assumptions A1, A3, and A4 are satisfied, virtual time The response (virtual input signal u and virtual output signal y to the controlled object G) can be obtained. Furthermore, instead of using equation (11), u can be obtained by simulating u=(Kr−y) using y.

On the other hand, in this virtual time response, it can be said that if the virtual input signal r(k) is fixed under the controller K, the virtual output signal y(k) can be estimated. Therefore, once the configuration of the controller K is determined, the parameters of the controller K can be optimized by giving certain constraint conditions to the virtual output signal y(k).

For example, controller K is assumed to be a PID controller. The transfer function of this controller K is expressed as in equation (12).

The parameters of this controller K are set as θ=(K _p , K _i , K _d ), and are expressed as controller K(θ). Optimize the controller K(θ) using the following procedure.
1) Set the initial value of the controller K(θ) that stabilizes the system and the allowable value for overshoot.
2) Using overshoot as a constraint and settling time as an evaluation function, solve an optimization problem to search for K(θ) that minimizes the evaluation function while keeping the overshoot below an allowable value. At this time, the overshoot and settling time are determined from the virtual time response of the feedback system when using a controller in which K(θ) is discretized. In addition, in order to satisfy assumption A4, it is also included as a constraint that K ^-1 (θ) is stable and does not have a slow pole.

Note that this method cannot be applied when K(θ) destabilizes the system, and there is a problem in that an error occurs between the virtual time response and the true response. However, when the feedback mechanism is effective, when the stability limit is approached, the oscillation condition is approached and the time response becomes oscillatory.

In this procedure, the initial value of the controller stabilizes the system and overshoot is included as a constraint, so it is thought that optimizing in a direction that reduces overshoot will suppress vibration and move away from the stability limit. It will be done.

Therefore, it is necessary to use an optimization method that searches for local optimal solutions around the initial value, rather than a type that searches for global optimal solutions.

As a solution to such an optimization problem, for example, fmincon of MATLAB can be used. This is a method of searching for the minimum value of a constrained nonlinear multivariable function called an interior-point algorithm, and can find a local optimal solution around an initial value. Of course, other methods may also be used.

<When the one-time input/output responses u ₀₀ and y ₀₀ are open loop data>
(Comparison of true time response and virtual time response)
The virtual time response during PID control was calculated using the output estimation method according to the present invention, and its accuracy was confirmed. The following nine continuous time transfer functions (Equations (13) to (20)) were controlled.

The same PID gain can be obtained for these controlled objects by the limit sensitivity method. A specific transfer function of the controller K is as shown in equation (21).

The non-periodic output signal y ₀₀ when G ₁ (s) to G ₉ (s) are Z-transformed with a sampling time t _s =0.01 sec and the non-periodic input signal u ₀₀ is applied as a step signal for 100 seconds to each of them. Obtained.

However, for _G4 , the final value of the step response does not reach a steady state in a non-orientating system. Therefore, only the non-periodic input signal _u00 of _G4 was input as a pulse input of 1 for up to 10 seconds and 0 thereafter. The obtained aperiodic output signal _y00 is shown in FIG. Referring to FIG. 4, the horizontal axis is the elapsed time, and the vertical axis is the signal intensity of the non-periodic output signal _y00 .

Although G5 converges slowly compared to other controlled objects and is still changing just before 100 seconds, the 100 second time point was considered to be a steady state.

A feedback system using a discretized controller K was obtained by substituting the following equation (22) into equation (21). The true time response when a step function is input as the target value (virtual input signal r) of the controller K, and the virtual time response (virtual output signal y, virtual input signal to the controlled object) obtained by the output estimation method according to the present invention. The input signal u) and its error are shown in FIG. 5, respectively.

Referring to FIG. 5, FIG. 5(a) is the output y of the controlled object, and FIG. 5(b) is the input signal (output signal of the controller K) to the controlled object u. In both figures, the horizontal axis is time (sec) and the vertical axis is signal strength (unitless). From FIGS. 5A and 5B, the values of the true time response and the virtual time response almost overlap, and the difference between the true time response and the virtual time response cannot be discerned. The error between the two was small, less than 0.01.

(Optimization of controller K)
Next, with an overshoot of 20% or less as a constraint and a settling time to within 2% of the steady value as an evaluation function, an optimization problem of parameters to minimize the evaluation function was solved using MATLAB's fmincon. The parameters are θ=(K _p , K _i , K _d ) in equation (12).

The initial values were the coefficients of each term in equation (21). Furthermore, in order to satisfy assumption A4, we added to the constraint conditions that the zero point of K(θ) is stable and the size of its real part is 1 [rad/s] or more.

FIG. 6 shows the true time response and virtual time response of the feedback system when using K(θ) obtained through optimization. FIG. 6(a) shows the output y of the controlled object, and FIG. 6(b) shows the input signal (output signal of the controller K) u to the controlled object. In both figures, the horizontal axis is time (sec) and the vertical axis is signal strength (unitless). From FIGS. 6(a) and 6(b), it can be seen that the values of the true time response and the virtual time response almost overlap, and the difference between the two is indistinguishable. The errors were small: the output y was less than 0.01, and the input u to the controlled object was less than 0.03.

Comparing with Figure 5 before optimization, only _G9 fell into an extreme value close to the initial value and did not change much, but overshoot was suppressed and the settling time became shorter in all other than _G9 .

<When the single input/output responses u ₀₀ , y ₀₀ are closed loop data and G is an unstable non-minimum phase system>
(Comparison of true time response and virtual time response)
As an unstable non-minimum phase system, the transfer function (G ₁₀ ) of equation (23) was used.

A continuous time controller K(s) that combines state feedback and a same-dimensional observer was set using the pole placement method, and G(s) and K(s) were Z-transformed at a sample time of 0.01 sec. The one-time input/output response was a response for 100 seconds when the poles of the closed loop system were set to -10, -10, -20, and -20, and the target value was set as a step function. The true output response and virtual response of each feedback system when the poles of the closed loop system are set to p, p, -20, and -20, and p is set to -1, -5, -10, -25, and -50. The time response is shown in FIG.

In FIG. 7, the horizontal axis is time (sec), and the vertical axis is signal strength (unitless). From FIG. 7, the true output response and virtual time response almost overlapped, and the error was small, less than 10 ^-9 .

(Optimization of controller K)
Next, with the overshoot of 20% or less as a constraint and the settling time to within 10% of the steady-state value as the evaluation function, solve an optimization problem using MATLAB's fmincon to find a controller K(θ) that minimizes the evaluation function. there was. The controller K(θ) was expressed as equation (24). The parameters were given as θ=(θ ₁ , θ ₂ , θ ₃ , θ ₄ ).

In order to satisfy assumption A4, we added to the constraint conditions that the zero point of K(θ) is stable and the size of its real part is 1 [rad/s] or more.

The initial value is K(θ) when a single input/output response (non-periodic input signal u ₀₀ , non-periodic output signal y ₀₀ ) is obtained, and when K(θ) obtained by optimization is used, FIG. 8 shows the true time response and virtual time response of the feedback system. In FIG. 8, the horizontal axis is time (sec), and the vertical axis is signal strength (unitless). From FIG. 8, compared to _y00 , y has suppressed overshoot and the settling time has become shorter. Furthermore, the true output response and the virtual time response almost overlapped, and the error was small, less than 10 ⁻¹⁰ .

<When the controlled object is a motor>
The object to be controlled was a system shown in FIG. 9 in which a DC motor was attached with wheels having a diameter of 68 mm and a weight of 100 g. This DC motor (Bringsmart JGA25-371-201) is equipped with a gear head with a reduction ratio of 1:21.3 and an encoder with 12 counts per rotation of the motor.

The output pulse signal of the encoder is frequency-voltage converted by an FV converter (JRCNJM4151) and quantized by a 10-bit AD converter of a microcomputer Arduino Mega2560. The sample time is 0.01 seconds, and the control input u calculated by the controller is quantized to 8 bits.

The quantized control input u is output to a motor driver (DFROBOT L298P) with a chopping frequency of 980 Hz PWM. This motor has a dead zone due to friction and will not rotate with a small motor input voltage v [V]. To compensate for this, dead band widths h ₁ and h ₂ on the plus side and minus side were measured, and h ₁ =-h ₂ =2.0V was added to the control input u as shown in equation (25).

Since this equation is an approximation of the inverse function of the dead zone, it is considered that the characteristics from the control input u to the control output y become approximately linear due to this compensation. Here, the step response when the target speed r was changed from 11.67 rpm to 20 rpm was evaluated.

A non-periodic input signal u ₀₀ and a non-periodic output signal y ₀₀ , which are one-time input/output response data, were obtained for 10 seconds using a PID controller designed using a step response method. The waveform for the first two seconds is shown in FIG. FIG. 10(a) represents the output signal, and FIG. 10(b) represents the input to the motor (control target). In both figures, the horizontal axis is time (sec), and the vertical axis is output. FIG. 10(a) shows the rotational speed (rpm), and FIG. 10(b) shows the voltage (V). The non-periodic output signal y ₀₀ is represented by "y ₀₀ " in FIG. 10(a), and the non-periodic input signal u ₀₀ is represented by "u ₀₀ " in FIG. 10(b).

The overshoot of 10% or less was used as a constraint, the settling time to within 2% of the steady value was used as an evaluation function, and the evaluation function was minimized using MATLAB's fmincon. Specifically, we solved an optimization problem to find K(θ) in equation (26).

The adjustment parameters θ=(K _p , K _i , K _d ) were set, and the PID parameters at the time when the non-periodic output signal y ₀₀ was obtained were set as the initial values. Since noise exceeding 2% of the steady value is superimposed on the non-periodic output signal _y00 and it will not settle as it is, the virtual output signal y is passed through the moving average filter of equation (27) and then the settling time and overshoot are measured. did.

The true time response of the feedback system when using K(θ) obtained by optimization in Figures 10 and 11 ("True y" in Figure 10(a) and "True u" in Figure 11) and a virtual time response ("Virtual y" in FIG. 10(a) and "Virtual u" in FIG. 10(b)). Note that the true time response is the input voltage u and rotational speed y of the motor when the motor is driven by a controller of K(θ) optimized using the virtual time response.

As can be seen from the figure, overshoot is suppressed and settling time is reduced in the virtual time response and the true response compared to the non-periodic output signal _y00 . However, oscillations mainly at about 5 Hz occurred in the virtual time response, which were not observed in the true output response, and high-frequency oscillations occurred in the true input response. Although there is such an error, control performance can be improved by optimizing controller parameters using the output estimation method according to the present invention, even though it is based on a step response that does not have abundant frequency components. I was able to confirm that it was improved.

<Capture>
In equations (9) and (11), at a certain angular frequency ωa, when r ₁ (jω)=0, division by zero may occur. In that case, it is sufficient to set y(jω)=u(jω)=0. The details are explained below.
Since the target input of the feedback system in FIG. 3(c) is r ₁ (jω)=0, equations (28) and (29) are obtained for y ₀ (jω) and u ₀ (jω).

From equation (28), if [G(jω _a )K(jω _a )] ⁻¹ +1≠0, then y ₀ (jω _a )=0. According to the Nyquist stability determination, G(jω _a )K(jω _a )=−1 only at the stability limit where the stability margin becomes zero. However, this will not happen because the feedback system is more stable than assumption A3. Therefore, we obtain y ₀ (jω _a )=0.

From equation (29), if K ⁻¹ (jω _a )+G(jω _a )≠0, then u ₀ (jω _a )=0. K ^-1 (jω _a ) + G (jω _a ) = 0 when G (jω _a ) K (jω _a ) = -1 or K ^-1 (jω _a ) = G (jω _a ) = 0. Only.

From assumption A3, G(jω _a )K(jω _a )≠−1, and from assumption A5, when G(jω _a )=0, K ⁻¹ (jω _a )≠0, so u ₀ (jω _a ) =0 is obtained. Therefore, according to assumption A2, r(jω _a )=0.

When r(jω _a )=0, if a similar argument is made for FIG. 3(b), y(jω _a )=u(jω _a )=0 is obtained.

The output estimation method according to the present invention is useful in cases where trial signal input to a controlled object is limited.

Claims (3)

A method for estimating an output for an input to a controlled object via a controller K, the method comprising:
obtaining a periodic input/output response (periodic input signal u ₀ and periodic output signal y ₀ ) to the controlled object based on one trial;
An output estimation method comprising the step of determining a response (virtual output signal y) of the controlled object to a virtual input signal r using equation (1).
Here, r ₁ (jω) follows equation (2). r ₁ (jω), r (jω), u ₀ (jω), and y ₀ (jω) represent signals obtained by Fourier transform of the signals r ₁ , r, u ₀ , and y _{0 ,} respectively. Further, K ⁻¹ (jω) represents 1/K(jω), and K(jω) represents the transfer function of the controller K. F ^-1 represents the inverse Fourier transform In the step of obtaining the periodic input/output response, in the one trial, an aperiodic input signal u ₀₀ is input to the controlled object, an aperiodic output signal y ₀₀ is obtained, and the aperiodic signal (aperiodic input signal u ₀₀ , non-periodic output signal y ₀₀ ) is converted into a periodic signal (periodic input signal u ₀ , periodic output signal y ₀ ) by successively adding an inverted signal to the aperiodic signal. The output estimation method according to claim 1. The controlled object is a PID controller, the parameters of the controller to be controlled are θ=(K _p , K _i , K _d ), and the transfer function K(θ) of the controller is set to equation (12) , 3. A controller optimization method for optimizing K(θ) using the virtual output signal and virtual input signal obtained by the output estimation method according to claim 2.