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Determination of robust multivariable I-controllers by means of experiments and simulation. (English) Zbl 0569.93045
Consider the linear, I/O stable plant $$(1a,b)\quad \dot x=Ax+Bu,\quad y=Cx+Du$$ where $$x(t)\in R^ n$$, $$y(t)\in R^ r$$ and $$u(t)\in R^ m$$ are state, output and input vectors respectively. Here A, B, C and D are assumed to be unknown. By applying $$(2a,b,c,d)\quad e=y-w,\quad \dot z=e,\quad u=Kz,\quad K=\alpha \tilde K$$ with the command input w(t) and control error e(t) to system (1), a closed-loop system (1), (2) is constructed. When it is possible to make experiments with plant (1), determine $$\tilde K$$ and tune $$\alpha$$ such that the closed-loop system is stable and attains asymptotically regulation for all step commands and step disturbances. This is called the problem of tuning regulation. Let H(t) be the step response matrix of (1), $$K_ s=\lim_{t\to \infty}H(t),$$ and $$K^+_ s=K'\!_ s(K_ sK'\!_ s)^{-1}.$$ The design methods obtained so far require only knowledge of $$K_ s$$ and use the matrix in the form $$\tilde K=K^+_ s.$$ The tuning parameter $$\alpha$$ is chosen by means of further experiments. Exact knowledge of $$K_ s$$ is, however, required, and there is no algorithm to calculate an initial value of $$\alpha$$.
This paper gives two tuning algorithms. Algorithm I requires not exact knowledge of $$K_ s$$ but its measurement value of its error bound.
On the basis of information about the dynamical behaviour, Algorithm II gives an upper bound $${\bar \alpha}$$ for $$\alpha$$ such that the closed- loop system is stable.
Reviewer: M.Kono

##### MSC:
 93C35 Multivariable systems, multidimensional control systems 93B35 Sensitivity (robustness) 93C40 Adaptive control/observation systems 93B40 Computational methods in systems theory (MSC2010) 93D20 Asymptotic stability in control theory 93C05 Linear systems in control theory 93B50 Synthesis problems 68U20 Simulation (MSC2010)
##### Keywords:
experiments; tuning regulation; tuning algorithms