Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 387 (2) Robust reliability based design of optimal controller Firstly, if Theorem 3.3 is used, by solving a optimization problem corresponding to (64) with 1 * = α , the gain matrices as follows for deriving the controller are obtained [] 2536.35211.138512.20 1 − − −= G K , [ ] 3799.41299.132143.21 2 − − = G K . The norm of the gain matrices are respectively 0635.25 1 = G K and 3303.25 2 = G K . So, there exist relations 1639.326 1 = L K G1 0135.13 K= , 2767.488 2 = L K G2 2764.19 K= . To examine the effect of the controllers, the initial values of the states of the Lorenz system are taken as [ ] T 101010)0( −−=x , the control input is activated at t=3.89s, all as that of Lee, Park, and Chen (2001), the simulated state trajectories of the controlled Lorenz system without uncertainty are shown in Fig. 2. In which, on the left- and right-hand sides are results of the controller of Lee, Park, and Chen (2001) and of the controller obtained in this paper respectively. Simulations of the corresponding control inputs are shown in Fig. 3, in which, the dash-dot line and the solid line represent respectively the input of the controller of Lee, Park, and Chen (2001) and of the controller in the paper. 3.9 3.95 4 0 2000 4000 6000 Time (sec) 4 4.5 5 5.5 -100 0 100 200 300 400 500 Time ( sec ) Fig. 3. Control input of the two controllers (dash-dot line and solid line represent respectively the result of Lee, Park, and Chen (2001) and the result of the paper) The simulated state trajectories and phase trajectory of the controlled Lorenz system are shown respectively in Figs. 4 and 5, in which, all the uncertain parameters are generated randomly within the allowable ranges. Robust Control, Theory and Applications 388 Fig. 4. Ten-times simulated state trajectories of the controlled chaotic Lorenz system with parametric uncertainties (all uncertain parameters are generated randomly within the allowable ranges, and on the left- and right-hand sides are respectively the results of controllers in Lee, Park, and Chen (2001) and in the paper) 0 10 20 30 40 50 -20 0 20 -20 0 20 40 x 3 (t) x 1 (t) end x 2 (t) Fig. 5. Ten-times simulated phase trajectories of the parametric uncertain Lorenz system controlled by the presented method (all parameters are generated randomly within their allowable ranges) Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method 389 It can be seen that the controller obtained by the presented method is effective, and the control effect has no evident difference with that of the controller in Lee, Park, and Chen (2001), but the control input of it is much lower. This shows that the presented method is much less conservative. Taking 3= α , which means that the allowable variation of all the uncertain parameters are within 90% of their nominal values, by applying Theorem 3.3 and solving a corresponding optimization problem of (64) with 3 * = α , the gain matrices for deriving the fuzzy controller obtained by the presented method become [ ] [ ] 12 -54.0211 32.5959 6.5886 , -50.0340 30.6071 10.4215 GG =−− =−KK. Obviously, the input of the controller in this case is also much lower than that of the controller obtained by Lee, Park, and Chen (2001). Secondly, when Theorem 3.4 is used, by solving two optimization problems corresponding to (69) with 1 * = α and 3 * = α respectively, the gain matrices for deriving the controller are found to be [ ] [ ] [][] * 12 * 12 20.8198 13.5543 3.2560 , 21.1621 13.1451 4.3928 ( 1). -54.0517 32.6216 6.6078 , -50.0276 30.6484 10.4362 ( 3) GG GG α α =−−− =−− = =−− =− = KK KK Note that the results based on Theorem 3.4 are in agreement, approximately, with those based on Theorem 3.3. 5. Conclusion In this chapter, stability of parametric uncertain nonlinear systems was studied from a new point of view. A robust reliability procedure was presented to deal with bounded parametric uncertainties involved in fuzzy control of nonlinear systems. In the method, the T-S fuzzy model was adopted for fuzzy modeling of nonlinear systems, and the parallel- distributed compensation (PDC) approach was used to control design. The stabilizing controller design of uncertain nonlinear systems were carried out by solving a set of linear matrix inequalities (LMIs) subjected to robust reliability for feasible solutions, or by solving a robust reliability based optimization problem to obtain optimal controller. In the optimal controller design, both the robustness with respect to uncertainties and control cost can be taken into account simultaneously. Formulations used for analysis and synthesis are within the framework of LMIs and thus can be carried out conveniently. It is demonstrated, via numerical simulations of control of a simple mechanical system and of the chaotic Lorenz system, that the presented method is much less conservative and is effective and feasible. Moreover, the bounds of uncertain parameters are not required strictly in the presented method. So, it is applicable for both the cases that the bounds of uncertain parameters are known and unknown. 6. References Ben-Haim, Y. (1996). Robust Reliability in the Mechanical Sciences, Berlin: Spring-Verlag Breitung, K.; Casciati, F. & Faravelli, L. (1998). Reliability based stability analysis for actively controlled structures. Engineering Structures, Vol. 20, No. 3, 211–215 Robust Control, Theory and Applications 390 Chen, B.; Liu, X. & Tong, S. (2006). Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay. Fuzzy Sets and Systems, Vol. 157, 2224–2240 Crespo, L. G. & Kenny, S. P. (2005). Reliability-based control design for uncertain systems. Journal of Guidance, Control, and Dynamics, Vol. 28, No. 4, 649-658 Feng, G.; Cao, S. G.; Kees, N. W. & Chak, C. K. (1997). Design of fuzzy control systems with guaranteed stability. Fuzzy Sets and Systems, Vol. 85, 1–10 Guo, S. X. (2010). Robust reliability as a measure of stability of controlled dynamic systems with bounded uncertain parameters. Journal of Vibration and Control, Vol. 16, No. 9, 1351-1368 Guo, S. X. (2007). Robust reliability method for optimal guaranteed cost control of parametric uncertain systems. Proceedings of IEEE International Conference on Control and Automation , 2925-2928, Guangzhou, China Hong, S. K. & Langari, R. (2000). An LMI-based H ∞ fuzzy control system design with TS framework. Information Sciences, Vol. 123, 163-179 Lam, H. K. & Leung, F. H. F. (2007). Fuzzy controller with stability and performance rules for nonlinear systems. Fuzzy Sets and Systems,Vol. 158, 147–163 Lee, H. J.; Park, J. B. & Chen, G. (2001). Robust fuzzy control of nonlinear systems with parametric uncertainties. IEEE Transactions on Fuzzy Systems, Vol. 9, 369–379 Park, J.; Kim, J. & Park, D. (2001). LMI-based design of stabilizing fuzzy controllers for nonlinear systems described by Takagi-Sugeno fuzzy model. Fuzzy Sets and Systems , Vol. 122, 73–82 Spencer, B. F.; Sain, M. K.; Kantor, J. C. & Montemagno, C. (1992). Probabilistic stability measures for controlled structures subject to real parameter uncertainties. Smart Materials and Structures , Vol. 1, 294–305 Spencer, B. F.; Sain, M. K.; Won C. H.; et al. (1994). Reliability-based measures of structural control robustness. Structural Safety, Vol. 15, No. 2, 111–129 Tanaka, K.; Ikeda, T. & Wang, H. O. (1996). Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H ∞ control theory, and linear matrix inequalities. IEEE Transactions on Fuzzy Systems, Vol. 4, No. 1, 1–13 Tanaka, K. & Sugeno, M. (1992). Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems, Vol. 45, 135–156 Teixeira, M. C. M. & Zak, S. H. (1999). Stabilizing controller design for uncertain nonlinear systems using fuzzy models. IEEE Transactions on Fuzzy Systems, Vol. 7, 133–142 Tuan, H. D. & Apkarian, P. (1999). Relaxation of parameterized LMIs with control applications. International Journal of Nonlinear Robust Control, Vol. 9, 59-84 Tuan, H. D.; Apkarian, P. & Narikiyo, T. (2001). Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Transactions on Fuzzy Systems, Vol. 9, 324–333 Venini, P. & Mariani, C. (1999). Reliability as a measure of active control effectiveness. Computers and Structures, Vol. 73, 465-473 Wu, H. N. & Cai, K. Y. (2006). H 2 guaranteed cost fuzzy control design for discrete-time nonlinear systems with parameter uncertainty. Automatica, Vol. 42, 1183–1188 Xiu, Z. H. & Ren, G. (2005). Stability analysis and systematic design of Takagi-Sugeno fuzzy control systems. Fuzzy Sets and Systems, Vol. 151, 119–138 Yoneyama, J. (2006). Robust H ∞ control analysis and synthesis for Takagi-Sugeno general uncertain fuzzy systems. Fuzzy Sets and Systems, Vol. 157, 2205–2223 Yoneyama, J. (2007). Robust stability and stabilization for uncertain Takagi-Sugeno fuzzy time-delay systems. Fuzzy Sets and Systems, Vol. 158, 115–134 This chapter presents for SISO (Single Input Single Output) LTI (Linear Time Invariant) systems, a detailed description of this robust control technique and two real experiences where QFT has successfully applied at the University of Almería (Spain). It starts with a QFT description from a theoretical point of view, afterwards section 3. 1 is devoted to present two well-known software tools for QFT design, and after that two real applications in agricultural spraying tasks and solar energy are presented. Finally, the chapter ends with some conclusions. 2. Synthesis of SISO LTI uncertain feedback control systems using QFT QFT is a methodology to design robust controllers based on frequency domain (Horowitz, 1993; Yaniv, 1999). This technique allows designing robust controllers which fulfil some quantitative specifications. The Nichols plane is the key tool for this technique and is used to achieve a robust design over the specified region of plant uncertainty. The aim is to design a compensator C (s) and a prefilter F(s ) (if it is necessary), as shown in Figure 1, so that performance and stability specifications are achieved for the family of plants ℘( s) describing aplantP (s). Here, the notation ˆ a is used to represent the Laplace transform for a time domain signal a (t). Fig. 1. Two degrees of freedom feedback system. The QFT technique uses the information of the plant uncertainty in a quantitative way, imposing robust tracking, robust stability, and robust attenuation specifications (among others). The 2DoF compensator {F, C}, from now onwards the s argument will be omitted when necessary for clarity, must be designed in such a way that the plant behaviour variations due to the uncertainties are inside of some specific tolerance margins in closed-loop. Here, the family ℘( s) is represented by the following equation ℘( s)=  P (s)=k ∏ n i =1 (s + z i ) ∏ m z =1 (s 2 + 2ξ z ω 0z + ω 2 0z ) s N ∏ a r=1 (s + p r ) ∏ b t=1 (s 2 + 2ξ t ω 0t + ω 2 0t ) :(1) k ∈ [k min , k max ], z i ∈ [z i,min , z i,max ], p r ∈ [p r,min , p r,ma x ], ξ z ∈ [ξ z,min , ξ z,max ], ω 0z ∈ [ω 0z ,min , ω 0z ,max ], ξ t ∈ [ξ t,min , ξ t,max ], ω 0t ∈ [ω 0t,min , ω 0t,max ], n + m < a + b + N  A typical QFT design involves the following steps: 392 Robust Control, Theory and Applications 1. Problem specification. The plant model with uncertainty is identified, and a set of working frequencies is selected based on the system bandwidth, Ω ={ω 1 , ω 2 , ,ω k }. The specifications (stability, tracking, input disturbances, output disturbances, noise, and control effort) for each frequency are defined, and the nominal plant P 0 is selected. 2. Templates. The quantitative information of the uncertainties is represented by a set of points on the Nichols plane. This set of points is called template and it defines a graphical representation of the uncertainty at each design frequency ω. An example is shown in Figure 2, where templates of a second-order system given by P (s)=k/s( s + a),with k ∈ [1, 10] and a ∈ [1, 10] are displayed for the following set of frequencies Ω = { 0.5, 1, 2, 4, 8, 15, 30, 60, 90, 120, 180} rad/s. 3. Bounds. The specifications settled at the first step are translated, for each frequency ω in Ω set, into prohibited zones on the Nichols plane for the loop transfer function L 0 (jω)= C(jω)P 0 (jω). These zones are defined by limits that are known as bounds.Thereexistso many bounds for each frequency as specifications are considered. So, all these bounds for each frequency are grouped showing an unique prohibited boundary.Figure3showsan example for stability and tracking specifications. Fig. 2. QFT Template example. 4. Loop shaping. This phase consists in designing the C controller in such a way that the nominal loop transfer function L 0 (jω)=C(jω)P 0 (jω) fulfils the bounds calculated in the previous phase. Figure 3 shows the design of L 0 where the bounds are fulfilled at each design frequency. 5. Prefilter. The prefilter F is designed so that the closed-loop transfer function from reference to output follows the robust tracking specifications, that is, the closed-loop system variations must be inside of a desired tolerance range, as Figure 4 shows. 393 A Frequency Domain Quantitative Technique for Robust Control System Design Fig. 3. QFT Bound and Loop Shaping example. Fig. 4. QFT Prefilter example. 394 Robust Control, Theory and Applications 6. Validation. This step is devoted to verify that the closed-loop control system fulfils, for the whole family of plants, and for all frequencies in the bandwith of the system, all the specifications given in the first step. Otherwise, new frequencies are added to the set Ω,so that the design is repeated until such specifications are reached. The closed-loop specifications for system in Figure 1 are typically defined in time domain and/or in the frequency domain. The time domain specifications define the desired outputs for determined inputs, and the frequency domain specifications define in terms of frequency the desired characteristics for the system output for those inputs. In the following, these types of specifications are described and the specifications translation problem from time domain to frequency domain is considered. 2.1 Time domain specifications Typically, the closed-loop specifications for system in Figure 1 are defined in terms of the system inputs and outputs. Both of them must be delimited, so that the system operates in a predetermined region. For example: 1. In a regulation problem, the aim is to achieve a plant output close to zero (or nearby a determined operation point). For this case, the time domain specifications could define allowed operation regions as shown in Figures 5a and 5b , supposing that the aim is to achieve a plant output close to zero. 2. In a reference tracking problem, the plant output must follow the reference input with determined time domain characteristics. In Figure 5c a typical specified region is shown, in which the system output must stay. The unit step response is a very common characterization, due to it combines a fast signal (an infinite change in velocity at t = 0 + ) with a slow signal (it remains in a constant value after transitory). The classical specifications such as rise time, settling time and maximum overshoot, are special cases of examples in Figure 5. All these cases can be also defined in frequency domain. 2.2 Frequency domain specifications The closed-loop specifications for system in Figure 1 are typically defined in terms of inequalities on the closed-loop transfer functions for the system, as shown in Equations (2)-(7). 1. Disturbance rejection at the plant output:     ˆ c ˆ d o     =     1 1 + P(jω)C(jω)     ≤ δ po (ω) ∀ω > 0, ∀P ∈ ℘ (2) 2. Disturbance rejection at the plant input:     ˆ c ˆ d i     =     P (jω) 1 + P(jω)C(jω)     ≤ δ pi (ω) ∀ω > 0, ∀P ∈ ℘ (3) 3. Stability:     ˆ c ˆ rF     =     P (jω) C(jω) 1 + P(jω)C(jω)     ≤ λ ∀ω > 0, ∀P ∈ ℘ (4) 4. References Tracking: B l (ω) ≤     ˆ c ˆ r     =     F (jω)P(jω)C(jω) 1 + P(jω)C(jω)     ≤ B u (ω) ∀ω > 0, ∀P ∈ ℘ (5) 395 A Frequency Domain Quantitative Technique for Robust Control System Design 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time (s) c Allowed operation region (a) Regulation problem 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time (s) c Allowed operation region (b) Regulation problem for other initial conditions 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 time (s) c Allowed operation region (c) Tracking problem Fig. 5. Specifications examples in time domain. 396 Robust Control, Theory and Applications [...]... computer-aided control design using quantitative feedback theory: The problem of vertical movement stabilization on a high-speed ferry International Journal of Control, 78:813–825, 2005a J M Díaz, S Dormido, and J Aranda SISO-QFTIT An interactive software tool for the design of robust controllers using the QFT methodology UNED, http://ctb.dia.uned.es/asig/qftit/, 2005b 422 Robust Control, Theory and Applications. .. Berenguel, and L Valenzuela A survey on control schemes for distributed solar collector fields part ii: advances control approaches Solar Energy, 81:1252–1272, 2007b M.C Cirre, J.C Moreno, M Berenguel, and J.L Guzmán Robust control of solar plants with distributed collectors In IFAC International Symposium on Dynamics and Control of Process Systems, DYCOPS, Leuven, Belgium, 2010 J M Díaz, S Dormido, and J Aranda... Templates and bounds for the example described in Eq (14) (a) Loop shaping (b) Validation Fig 15 SISO-QFTIT Loop shaping and validation for the example described in Eq (14) 4 Practical applications This section presents two industrial projects where the QFT technique has been successfully used The first one is focused on the pressure control of a mobile robot which was design 412 Robust Control, Theory and Applications. .. locus of L (s) when the PID is introduced 420 Robust Control, Theory and Applications Fig 26 Tracking specifications (dashed-dotted) and magnitude Bode diagram of some closed loop transfer functions In order to prove the fulfillment of the tracking and stability specifications of the control structure, experiments were performed under several operating points and under different conditions of disturbances... Borghesani, Y Chait, and O Yaniv The QFT Frequency Domain Control Design Toolbox Terasoft, Inc., http://www.terasoft.com/qft/QFTManual.pdf, 2003 E.F Camacho, M Berenguel, and F.R Rubio Advanced Control of Solar Plants (1st edn) Springer, London, 1997 E.F Camacho, F.R Rubio, M Berenguel, and L Valenzuela A survey on control schemes for distributed solar collector fields part i: modeling and basic control approaches... rise time and settling time 400 Robust Control, Theory and Applications 1.6 λ=0.3 1.4 λ=0.5 1.2 λ=0.7 1 λ=1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 ωn·time 6 7 8 9 10 Fig 7 Third-order model with a zero for μ = 5 and ξ = 1 There exist other techniques to translate specifications from time domain to frequency domain, such as model-based techniques, where based on the structures of the plant and the controller,... free software interactive tool for robust control design using the QFT methodology (Díaz et al., 2005a;b) The main advantages of SISO-QFTIT compared to other existing tools are its easiness of use and its interactive nature In the tool described in the previous section, a combination between code and graphical interfaces must be used, where 410 Robust Control, Theory and Applications some interactive features... 27(3):361–386, 1978 K R Krishnan and A Cruickshanks Frequency domain design of feedback systems for specified insensitivity of time-domain response to parameter variations International Journal of Control, 25 (4):609–620, 1977 M Morari and E Zafiriou Robust Process Control Prentice Hall, 1989 J C Moreno Robust control techniques for systems with input constrains, (in Spanish, Control Robusto de Sistemas con Restricciones... 2003 J C Moreno, A Baños, and M Berenguel A synthesis theory for uncertain linear systems with saturation In Proceedings of the 4th IFAC Symposium on Robust Control Design, Milan, Italy, 2003 J C Moreno, A Baños, and M Berenguel Improvements on the computation of boundaries in qft International Journal of Robust and Nonlinear Control, 16(12):575–597, May 2006 J C Moreno, A Baños, and M Berenguel A qft... set Ω, and the corresponding template and boundary are 3 A system is conditionally stable if a gain reduction of the open-loop transfer function L drives the closed-loop poles to the right half plane 406 Robust Control, Theory and Applications computed for that frequency ω p Then, the function L0 is reshaped, so that the new restriction is satisfied Afterwards, the precompensator F is reshaped, and finally . the controlled Lorenz system without uncertainty are shown in Fig. 2. In which, on the left- and right-hand sides are results of the controller of Lee, Park, and Chen (2001) and of the controller. parameters are generated randomly within the allowable ranges, and on the left- and right-hand sides are respectively the results of controllers in Lee, Park, and Chen (2001) and in the paper) . Reliability based stability analysis for actively controlled structures. Engineering Structures, Vol. 20, No. 3, 211 215 Robust Control, Theory and Applications 390 Chen, B.; Liu, X. &