| # Robust Stability of Linear Dynamical Systems SDP > Conic problems | LMIs | Standard Forms | Applications > Back | Robust Stability Motivation Lyapunov stability criterion Robust stability Robust linear control ## Motivation The time-behavior of many dynamical systems can be well modeled using a linear system of the form where is the state, which encapsulates the state of the system at time , contains control variables, and are matrices of appropriate size. (For more details, see here.) When are constant, we say that the system is time-invariant. A continuous-time version of the above takes the form We focus on these continuous-time models here. A crucial problem in dynamical systems is asymptotic stability. Roughly speaking, we would like to know if, irrespective of the initial condition , and when there are no inputs ( for every ), then when . If not, we would like to see how to act on the system with certain inputs that are functions of the state, so that the resulting system becomes stable. ## Lyapunov stability criterion Lyapunov invented (circa 1890) a method that provides a (in general, only sufficient) condition for asymptotic stability. Assume that there is a positive-definite function of the state, that decreases strictly along every trajectory. Then converges to zero as . The physical interpretation of is that it acts as a total energy for the system. For mechanical systems, can be chosen to contain the kinetic and potential energy. When , and with , we have Thus a sufficient condition for stability is on any trajectory. Expressing this condition for and for arbitrary results in the following sufficient condition for asymptotic stability. Lyapunov's sufficient condition for asymptotic stability: there exist such that is positive definite; for every , is negative definite. It turns out that for time-invariant systems ( independent of ), the above is also necessary. ## Robust stability Now assume that is not completely known, say it can take its values arbitrarily in a finite set . The matrices can represent the behavior of the system under different operating conditions, and allow to model the uncertainty about the model's parameters. Based on Lyapunov's condition above, the following condition guarantees stability irrespective of the choice of in . Lyapunov's sufficient condition for robust asymptotic stability: there exist such that is positive definite; for every , is negative definite. The above is a semidefinite program, in matrix variable . (There is no objective function here; this is a feasibility problem.) ## Robust linear control What if a system is not stable? One way to stabilize it is via linear feeback, using this time an input that is a linear function of the state (which we assume is measured, hence available). That is, we set where is a matrix of control parameters (or gains). The closed-loop system becomes Applying the stability condition to the system above leads to a condition in , that should be negative-definite for every . The above matrix is not affine in the variables . However, the condition for a matrix to be negative-definite is equivalent to the fact that is, where is an arbitrary invertible matrix. Applying this to , and letting , we obtain that if there exist with positive-definite, such that for every , is negative-definite, then the system can be stabilized by linear feedback. In the case when the matrix is not completely known, say it can take its values arbitrarily in a finite set , then the condition: there exist with positive-definite, such that: is negative definite, then the system is robustly stabilizable.