Optimal induced \(l^1\)-norm state feedback control
The authors discuss first the advantages of \(\ell^2\), \(\ell^\infty\), and \(\ell^1\) norms in closed loop feedback systems synthesis. Amplification in \(\ell^2\) generally refers to amplification of some form of energy, the \(\ell^\infty\) norm amplifies some component of the system, while the \(\ell^1\) norm distributes the effects of amplification among the components. Frequently it is used to take care of the worst possible case. It is known that closed loop robustness is easily handled within the \(\ell^1\) framework. Because of the duality between \(\ell^\infty\), and \(\ell^1\) norms, in linear systems with linear controls, one can manipulate the control of a system and the one of the transposed (adjoint) system. However, in nonlinear systems these are distinct, separate problems. The following (standard) linear discrete time system is considered: \[ x(k+ 1)= Ax(k)+ B_1w(k)+ B_2u(k),\tag{1} \] \[ z(k)= C_1x(k)+ D_{11}w(k)+ D_{12}u(k),\quad y(k)= x(k), \] where \(x\in\mathbb{R}^n\), \(w\in \mathbb{R}^{n_1}\), \(u\in \mathbb{R}^{n_2}\), \(z\in\mathbb{R}^n\), and \(B_1\) is a full rank vector. Substituting a given control law \(u= u(x)\) the closed loop system reduces to \[ x(k+ 1)= f[x(k)]+ B_1w(k),\quad z(k)= g[x(k)]+ D_{11}w(k),\tag{2} \] with \(f(x)= Ax+ B_2u(k)\), and \(g(x)= C_1x+ D_{12}u(x)\). The transfer function \(G\) maps the input \(w\) into the output \(z\). A definition of the ``analysis problem'' as interpreted by the reviewer consists in finding the solution (the state of the system: \(x(k)\)) at all times \(k= 0,1,2,\dots\) when all the other quantities (and in particular \(G\)) are known, and the transfer function \(G\) does not exceed some a priori given \(\ell^1\) norm: \(\|G\|_{\ell^1}= \sup\|z\|_1/\|w\|_1\), if \(\|w\|_1\neq 0\). The authors' definition differs from that one, which is certainly admissible, since it is consistent. The synthesis (or control) problem consists of finding a control law (it is not necessarily unique) \(u= u(x)\) such that the transfer function \(G\) is internally stable and its norm does not exceed some specified value \(\gamma\). The authors establish global asymptotic stability for the analysis problem. The proof consists of producing a gauge function \(V\), which turns out to be a Lyapunov function, plus application of the triangular inequality in the \(\ell^1\) norm. This argument also proves the so-called `generalized bounded real inequality': \(V[x(k+ 1)]- V[x(k)]+ \|z(k)\|_1- \gamma\|w(k)\|_1\leq 0\) for all \(k\), \(w(k)\) and \(x(k)\). A proof follows showing that if the system is stabilizable, then there exists a function \(V\) as stated above. The authors use the existence of a \(\lambda\)-contractive set and linearity in the proof. Then, they prove the existence of maximal \(\lambda\)-contractive set, and establish a bound \(\gamma\). The main theorem establishes the existence of a feedback law consistent with the required \(\ell^1\) peformance. Finally, a numerical algorithm is given for improving this performance within a polyhedral set determined by an induced closed-loop Lyapunov function. It reduces in the present setting the iterative procedure to linear programming. This paper contains much useful information, which will compete successfully in many cases with the presently more popular \({\mathcal H}^\infty\) approach. However, on occasions the style of writing could be improved by offering additional information. For example: ``Given \(G\), establishing that \(\|G\|_{i1}\) does not exceed a given bound \(\gamma\) is called\dots'' etc.\dots was at first a little confusing. The reviewer also drew his own (rather messy) diagram of the feedback loop. Had the authors included such a diagram, it would have helped the readers to clarify immediately the physical interpretation of all of their equations, and of subsequent inequalities.
scientific article; zbMATH DE number 1325752