Multivariable Model Reference Adaptive Control Further Study

Both schemes, the direct model reference adaptive control using Kp = LDU actorizationfor multivariable discrete-time plants and immersion and invariance adaptive control of linearmultivariable systems, are considered in the paper which is composed of the following threeparts:1. The design and analysis of the direct model reference adaptive control using Kp =LDU factorization for multivariable discrete-time plants.Consider the following multivariable discrete-time system described byy{t) = G{z)u{t), t = {0,1,2…},where u(t),y(t)∈R^m are the input and output, respectively, G(z)∈R^m×m. The referencemodel is chosen asy_m(t)=W_m(z)r(t),where W_m(z)∈R^m×m, r∈R^m is the reference input which is assumed to be bounded. Theobjective of MRAC is to find an output feedback control signal u(t) for the plant such that allthe signals in the closed-loop plant are bounded and the tracking error e(t)(?)y(t)-y_m(t)â†'0as tâ†'∞. To design and analyze the MRAC scheme, the assumptions of the system andthe reference model are needed in the second chapter.In this part, For a class of discrete-time multivariable systems, by reproving the keytechnical lemma and the discrete-time (?)_p and (?)_2δ norm relationship between inputs andoutputs, the stability and convergence properties of the discrete-time MRAC scheme areanalyzed rigorously in a systematic fashion as in the continuous-time case, which is propitiousto establish a unified MARC schemes system between the continuous-time systems and thediscrete-time systems.2. The design and analysis of the robust model reference adaptive control using Kp =LDU factorization for multivariable discrete-time plants.Consider the following multivariable discrete-time system with the unmodeled dynamicsdescribed by(I+μ₁â–³₁(z))y(t))=G(z)(I+μ₂â–³₂(z))u(t), t ={1,2,…},where u(t), y(t)∈R^m are the input and output, respectively, G(z)∈R^m×m, and the parame-ters of G(z) are considered to be unknown.â–³₁(z),â–³₂(z)∈R^m×m are unknown multiplicative uncertainties,μ₁,μ₂>0 are parameter indicting the magnitude ofâ–³₁(z),â–³₂(z), Control ob-jective is to design u(t) such that all the signals in the closed-loop plant are all bounded andthe output y tracks the following reference model output y_m as close as possible.y_m(t)=W_m(z)r(t),where W_m(z)∈R^m×m, r∈R^m is the reference input which is assumed to be bounded. Tomeet the control objective, to design and analyze the MRAC scheme, the assumptions ofthe system, the reference model and the unmodel dynamic are needed in the third chapter.In this part, for general plants with unmodeled dynamics and relative degree greater one.using a new parametric models established by the high frequency gain matrix factorizationK_p=LDU, the problem of robust model reference adaptive control is considered under theweaker assumptions.3. The design and analysis of immersion and invariance model reference adaptive controlof linear multivariable systems.Consider the following multivariable system described byy=G(s)u,where u,y∈R^m are the input and output, respectively, G(s)∈R^m×m, The symbol s is usedto denote the Differential Operators, i.e, s(x) =(?), and the parameters of G(s) are consideredto be unknown. Control objective is to design u such that all the signals in the closed-loopplant are all bounded and the output y tracks the following reference model output y_m asclose as possible.y_m =W_m(s)r,where W_m(s)∈R^m×m, r∈R^m is the reference input which is assumed to be bounded. Tomeet the control objective, the assumptions of the system, the reference model are neededin the forth chapter.In this part, for multivariable linear continuous-time systems, model ference adaptivecontrol based on system immersion and manifold invariance, a new method presented lately,is considered in this paper. it is proved that all signals of the closed-loop plant bounded andthe tracking error converges to zero asymptotically.