Optimization-Based Robust Control
- 44 Downloads
This entry describes the basic setup of linear robust control and the difficulties typically encountered when designing optimization algorithms to cope with robust stability and performance specifications.
KeywordsLinear systems Optimization Robust control
Linear Robust Control
Robust control allows dealing with uncertainty affecting a dynamical system and its environment. In this section, we assume that we have a mathematical model of the dynamical system without uncertainty (the so-called nominal system) jointly with a mathematical model of the uncertainty. We restrict ourselves to linear systems: if the dynamical system we want to control has some nonlinear components (e.g., input saturation), they must be embedded in the uncertainty model. Similarly, we assume that the control system is relatively small scale (low number of states): higher-order dynamics (e.g., highly oscillatory but low energy components) are embedded in the uncertainty model. Finally, for conciseness, we focus exclusively on continuous-time systems, even though most of the techniques described in this section can be transposed readily to discrete-time systems.
Unstructured uncertainty, also called norm-bounded uncertainty, where
Structured uncertainty, also called polytopic uncertainty, where
We can find more complicated uncertainty models (e.g., combinations of the two above: see Zhou et al. 1996), but to keep the developments elementary, they are not discussed here.
Nonconvex Nonsmooth Robust Optimization
Nonconvexity: The stability conditions are typically nonconvex in K.
Nondifferentiability: The performance criterion to be optimized is typically a non-differentiable function of K.
Robustness: Stability and performance should be ensured for every possible instance of the uncertainty.
There exist various approaches to handling nonconvexity. One possibility consists of building convex inner approximations of the stability region in the parameter space. The approximations can be polytopes, balls, ellipsoids, or more complicated convex objects described by linear matrix inequalities (LMI). The resulting stability conditions are convex, but surely conservative, in the sense that the conditions are only sufficient for stability and not necessary. Another approach to handling nonconvexity consists of formulating the stability conditions algebraically (e.g., via the Routh-Hurwitz stability criterion or its symmetric version by Hermite) and using converging hierarchies of LMI relaxations to solve the resulting nonconvex polynomial optimization problem: see, e.g., Henrion and Lasserre (2004) and Chesi (2010).
The third difficulty for optimization-based robust control is the uncertainty. As explained above, optimization of a performance criterion with respect to controller parameters is already a potentially difficult problem for a nominal system (i.e., when the uncertainty parameter is equal to zero). This becomes even more difficult when this optimization must be carried out for all possible instances of the uncertainty δ in Δ. This is where the above assumption that the uncertainty set Δ has a simple description proves useful. If the uncertainty δ is unstructured and not time varying, then it can be handled with the complex stability radius (Ackermann, 1993), the pseudospectral abscissa (Trefethen and Embree, 2005), or via an H∞ norm constraint (Zhou et al., 1996). If the uncertainty δ is structured, then we can try to optimize a performance criterion at every vertex in the polytopic description (which is a relaxation of the problem of stabilizing the whole polytope). An example is the problem of simultaneous stabilization, where a controller K must be found such that the maximum spectral abscissa of several matrices Ai(K), i = 1, …, N is negative (Blondel, 1994). Finally, if the uncertainty δ is time varying, then performance and stability guarantees can still be achieved with the help of Lyapunov certificates or potentially conservative convex LMI conditions: see, e.g., Boyd et al. (1994) and Scherer et al. (1997).
Algorithms for nonconvex nonsmooth optimization have been developed and interfaced for linear robust multiobjective control in the public domain Matlab package HIFOO released in 2006 (Burke et al., 2006a) and updated in 2009 (Gumussoy et al., 2009) and based on the theory described in Burke et al. (2006b). In 2011, The MathWorks released HINFSTRUCT, a commercial implementation of these techniques based on the theory described in Apkarian and Noll (2006).
- Burke JV, Henrion D, Lewis AS, Overton ML (2006a) HIFOO – a Matlab package for fixed-order controller design and H-infinity optimization. In: Proceedings of the IFAC symposium robust control design, ToulouseGoogle Scholar
- Gumussoy S, Henrion D, Millstone M, Overton ML (2009) Multiobjective robust control with HIFOO 2.0. In: Proceedings of the IFAC symposium on robust control design (ROCOND 2009), HaifaGoogle Scholar