### BibTeX

@MISC{Yubai_,

author = {Kazuhiro Yubai and Akitaka Mizutani and Junji Hirai},

title = {},

year = {}

}

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### Abstract

synthesis. However, since it is difficult to give a physical interpretation to the dual Youla parameter in general, we must select the weighting function for identification and the order of the identified model by trial and error. For implementation aspect, a low-order controller is much preferable, which means that a low-order model of the dual Youla parameter should be identified. However, it is difficult to identify the low-order model of the dual Youla parameter which contains enough information on the actual dual Youla parameter to design the appropriate Youla parameter. Moreover, there may be the cases where an accurate and reasonably low-order model of the dual Youla parameter can not be obtained easily. To avoid these difficulties in system identification of the dual Youla parameter, this article addresses the design method of the Youla parameter by model-free controller synthesis. Model-free controller syntheses have the advantages that the controller is directly synthesized or tuned only from the input/output data collected from the plant, and no plant mathematical model is required for the controller design, which avoids the troublesome model identification of the dual Youla parameter. Moreover, since the order and the controller structure are specified by the designer, we can easily design a low-order Youla parameter by model-free controller syntheses. A number of model-free controller syntheses have been proposed, e.g., the Iterative Feedback Tuning (IFT) 390 Recent Advances in Robust Control -Novel Approaches and Design Methods www.intechopen.com Robust control by the GIMC structure This section gives a brief review of the GIMC (Generalized Internal Model Control) structure and it is a control architecture solving the trade-off between the control performance and the robustness. GIMC structure A linear time-invariant plant P 0 is assumed to have a coprime factorization where RH ∞ denotes the set of all real rational proper stable transfer functions. A nominal controller C 0 stabilizing P 0 is also assumed to have a coprime factorization on RH ∞ as where X and Y satisfy the Bezout identity XN + YD = 1. Then a class of all stabilizing controllers C is parameterized as (3), which is called as Youla parameterization, by introducing the Youla parameter Q ∈RH ∞ where Q is a free parameter and is determined arbitrarily as long as Then, the GIMC structure is constructed as Dual Youla parameterization and robust stability condition For appropriate compensation of plant uncertainties, information on plant uncertainties is essential. In the design of the Youla parameter Q, the following parameterization plays an important role. On the assumption that the nominal plant P 0 factorized as (1) and its deviated version, P, are stabilized by the nominal controller C 0 ,thenP is parameterized by introducing adualY oulaparameterR ∈RH ∞ as follows: This parameterization is called as the dual Youla parameterization, which is a dual version of the Youla parameterization mentioned in the previous subsection. It says that the actual plant P, which is deviated from the nominal plant P 0 , can be represented by the dual Youla parameter R. By substituting (4) to the block-diagram shown by We must design Q so as to meet this stability condition. Direct design of the Youla parameter from experimental data As stated in the previous subsection, the role of Q is to suppress plant variations and disturbances. This article addresses the design problem of Q to approach the closed-loop performance from r to y, denoted by G ry , to the its nominal control performance as an example. This design problem is formulated in frequency domain as a model matching problem; where . it depends on the coprime factors, N, D, X and Y, which makes it difficult to give a physical interpretation for R. As a result, the identification of R requires trial-and-error for the selection of the structure and/or the order of R. As is clear from Review of the Virtual Reference Feedback Tuning (VRFT) The Virtual Reference Feedback Tuning (VRFT) is one of model-free controller design methods to achieve the model matching. The VRFT provides the controller parameters using only the input/output data set so that the actual closed-loop property approaches to its reference model given by the designer. In this subsection, the basic concept and its algorithm are reviewed. The basic concept of the VRFT is depicted in where M is a reference model to be achieved. Now, assume that the output of the feedback system consisting of P and C(θ) parameterized by the parameter vector θ coincides with y 0 (t) when the virtual reference signalr(t) is given as a reference signal. Then, the output of C(θ), denoted byũ(t, θ) is represented as Ifũ(t, θ)=u 0 (t), then the model matching is achieved, i.e., . Since the exact model matching is difficult in practice due to the restricted structural controller, the measurement noise injected to the output etc., we consider the alternative optimization problem:θ L is a prefilter given by the designer. By selection of L = W M M(1 − M),θ would be a good approximation of the exact solution of the model matching problemθ even ifũ(t, θ) = u 0 (t) where ϕ(t)=Lσ(r(t) − y 0 (t)), u L (t)=Lu 0 (t). Direct tuning of Q from experimental data by the VRFT This subsection describes the application of the VRFT to the design of the Youla parameter Q without any model identification of the dual Youla parameter R. The experimental data set used in the controller design, r 0 (t), u 0 (t), y 0 (t) , is collected from the closed-loop system composed of the perturbed plant P and the nominal controller C 0 . Define the Youla parameter Q(z, θ) linearly parameterized with respect to θ as where σ(z) is a discrete-time transfer function vector defined as and θ is a parameter vector of length n defined as Then the model matching problem formulated as (6) can be rewritten with respect to θ as where Under the condition that the dual Youla parameter R is unknown, we will obtain the minimizerθ of J MR (θ) using the closed-loop experimental data set r 0 (t), u 0 (t), y 0 (t) . Firstly, we obtain the input and the output data of R denoted by α(t),andβ(t), respectively. In Although α(t) is an internal signal of the feedback control system, α(t) is an function of the external signal r 0 (t) given by the designer as is clear from If there exists the parameter θ such that α(t)=Xr(t) − Q(θ)β(t), the exact model matching is achieved (G ry = M). According to the concept of the VRFT, the approximated solution of the model matching problem,θ, is obtained by solving the following optimization problem: where Since Q(θ) is linear with respect to the parameter vector θ as defined in where The minimizer of J N VR (θ) is then calculated using the least-squares method aŝ The filter L M is specified by the designer. By selecting L M = W M MYΦ α (ω) −1 ,θ could be a good approximation ofθ in case N → ∞,whereΦ α (ω) is a spectral density function of α(t). Moreover, this design approach needs an inverse system of the reference model, M −1 ,wheñ r(t) is generated. However, by introducing L M , we can avoid overemphasis by derivation in M −1 in the case where the noise corrupted data y 0 (t) is used. Stability constraint on the design of Q by the VRFT The design method of Q based on the VRFT stated in the previous subsection does not explicitly address the stability issue of the resulting closed-loop system. Therefore, we can not evaluate whether the resulting Youla parameter Q(θ) actually stabilizes the closed-loop . Data acquisition of α(t) and β(t). system or not in advance of its implementation. To avoid the instability, the data-based stability constraint should be introduced in the optimization problem (17). As stated in the subsection 2.2, the robust stability condition when the plant perturbs from P 0 to P is described using R and Q as (5). However, The alternative constraint RQ(θ) by introducing the virtual signal ξ(t, θ)=Q(θ)β(t). Assuming that α(t) is a p times repeating signal of a periodic signal with a period T, i.e., α(t) is of length N = pT,t h eH ∞ norm of RQ(θ) denoted by δ(θ) can be estimated via the spectral analysis method as the ratio between the power spectral density function of α(t), denoted by Φ α (ω k ), and the power cross spectral density function between α(t) and ξ(t, θ), denoted by Φ αξ (ω k ) where ,a n dT s is a sampling time. The frequency points ω k must be defined as a sequence with a much narrow interval for a good estimate of δ(θ). A shorter sampling time T s is preferable to estimate δ(θ) in higher frequencies, and a longer period T improves the frequency resolution. Similarly, Φ αξ (ω k , θ) is estimated as a DFT of the cross-correlation between α(t) and ξ(t, θ), Using the p-periods cyclic signal α(t) in the estimate ofR αξ (τ, θ),t h ee f f e c to ft h e measurement noise involved in ξ(t, θ) is averaged and the estimate error in Φ αξ (ω k , θ) is then reduced. Especially, the measurement noise is normalized, the effect on the estimate of Φ αξ (ω k , θ) by the measurement noise is asymptotically reduced to 0. Since Q(θ) is linearly defined with respect to θ,R αξ (τ, θ) andΦ αξ (ω k , θ) are also linear with respect to θ. As a result, the stability constraint of Since this constraint is convex with respect to θ at each frequency point ω k , we can integrate this H ∞ norm constraint into the optimization problem (17) and solve it as a convex optimization problem. Design algorithm This subsection describes the design algorithm of Q(θ) imposing the stability constraint. [step 1] Collect the input/output data set {u 0 (t), y 0 (t)} of length N in the closed-loop manner in the unity feedback control structure shown in [step 2] Calculate α(t) and β(t) using the data set {r 0 (t), u 0 (t), y 0 (t)} as [step 3] Generate the virtual referencer(t) such that y 0 (t)=Mr(t). [step 4] Solve the following convex optimization problem; Design example To verify the effectiveness of the proposed design method, we address a velocity control problem of a belt-driven two-mass system frequently encountered in many industrial processes. Controlled plant The plant to be controlled is depicted as Moreover, the delay time of 14 ms is emulated by the software as the plant perturbation in P, but it is not reflected in P 0 . Due to the delay time, the closed-loop system tends to be destabilized when the gain of the feedback controller is high. This means that if the reference model with the high cut-off frequency is given, the closed-loop system readily destabilized. Experimental condition For the simplicity, the design problem is restricted to the model matching of G ry approaching to its reference model M in the previous section. However, the proposed method readily address the model matching of multiple characteristics. In the practical situations, we must solve the trade-off between several closed-loop properties. In this experimental set-up, we show the design result of the simultaneous optimization problem approaching the tracking performance, G ry , and the noise attenuation performance, G ny to their reference models, M and T, respectively. The evaluation function is defined as To deal with the above multiobjective optimization problem, we redefine ϕ(t) and y L (t) in The reference models for G ry and G ny are given by discretization of M = 50 2 (s + 50) 2 ,a n d with the sampling time T s = 1[ms]. The nominal controller stabilizing P 0 is evaluated from the relation The weighting functions W M and W T are given to improve the tracking performance in low frequencies and the noise attenuation performance in high frequencies as The Youla parameter Q(s, θ) is defined in the continuous-time so that the properness of Q(s, θ) and the relation, Q ∈RH ∞ , are satisfied as The discrete-time Youla parameter Q(z, θ) is defined by discretization of Q(s, θ), i.e., σ(s), with the sampling time T s = 1 [ms]. In order to construct the type-I servo system even if the plant perturbs, the constant term of the numerator of Q(s, θ) is set to 0 such that Q(s, θ)| s=0 = 0 in the continuous-time Experimental result The VRFT can be regarded as the open-loop identification problem of the controller parameter by the least-squares method. We select the pseudo random binary signal (PRBS) as the input for identification of the controller parameter as same as in the general open-loop identification problem, since the identification input should have certain power spectrum in all frequencies. The PRBS is generated through a 12-bit shift register (i.e., T = 2 12 − 1 = 4095 samples), the reference signal r 0 is constructed by repeating this PRBS 10 times (i.e., p = 10, N = 40950). Firstly, we obtain the parameterθ w/o as (26) when the stability constraint is not imposed. Secondly, we obtain the parameterθ w/ as (27) when the stability constraint is imposed. constraint. Therefore, we can predict in advance of implementation that the closed-loop system could be stabilized if the Youla parameter Q(z,θ w/ ) was implemented. Conclusion In this article, the design method of the Youla parameter in the GIMC structure by the typical model-free controller design method, VRFT, is proposed. By the model-free controller design method, we can significantly reduce the effort for identification of R and the design of Q compared with the model-based control design method. We can also specify the order and the structure of Q, which enable us to design a low-order controller readily. Moreover, the stability constraint derived from the small-gain theorem is integrated into the 2-norm based standard optimization problem. As a result, we can guarantee the closed-loop stability by the designed Q in advance of the controller implementation. The effectiveness of the proposed controller design method is confirmed by the experiment on the two-mass system. As a future work, we must tackle the robustness issue. The proposed method guarantees the closed-loop stability only at the specific condition where the input/output data is collected. If the load condition changes, the closed-loop stability is no longer guaranteed in the proposed method. We must improve the proposed method to enhance the robustness for the plant perturbation and/or the plant uncertainties. Morevover, though the controller structure is now restricted to the linearly parameterized one in the proposed method, the fully parameterized controller should be tuned for the higher control performance. References