The property that every control system should posses is stability, which
translates into safety in real-life applications. A central tool in systems
theory for synthesizing control laws that achieve stability are control
Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting
closed-loop state trajectory is contained within a cone with a fixed,
predefined shape, and which is centered at and converges to a desired
converging point. However, such a requirement often proves to be
overconservative, which is why most of the real-time controllers do not have a
stability guarantee. Recently, a novel idea that improves the design of CLFs in
terms of flexibility was proposed. The focus of this new approach is on the
design of optimization problems that allow certain parameters that define a
cone associated with a standard CLF to be decision variables. In this way
non-monotonicity of the CLF is explicitly linked with a decision variable that
can be optimized on-line. Conservativeness is significantly reduced compared to
classical CLFs, which makes \emph{flexible CLFs} more suitable for
stabilization of constrained discrete-time nonlinear systems and real-time
control. The purpose of this overview is to highlight the potential of flexible
CLFs for real-time control of fast mechatronic systems, with sampling periods
below one millisecond, which are widely employed in aerospace and automotive
applications.Comment: 2 figure

Lazar, M.

Electronic Proceedings in Theoretical Computer Science

English

arXiv

The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable for stabilization of constrained discrete-time nonlinear systems and real-time control. The purpose of this overview is to highlight the potential of flexible CLFs for real-time control of fast mechatronic systems, with sampling periods below one millisecond, which are widely employed in aerospace and automotive applications

Mircea Lazar

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M. Bujorianu and M. Fisher (Eds.):Workshop on Formal Methods for Aerospace (FMA)EPTCS 20, 2010, pp. 76–79, doi:10.4204/EPTCS.20.8c© M. LazarThis work is licensed under theCreative Commons Attribution License.Flexible Lyapunov Functionsand Applications to Fast Mechatronic SystemsMircea LazarDept. of Electrical Eng., Eindhoven Univ. of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlandsm.lazar@tue.nlThe property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stabilityare control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loopstate trajectory is contained within a cone with a fixed, predefined shape, and which is centered at andconverges to a desired converging point. However, such a requirement often proves to be overcon-servative, which is why most of the real-time controllers do not have a stability guarantee. Recently,a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of thisnew approach is on the design of optimization problems that allow certain parameters that define acone associated with a standard CLF to be decision variables. In this way non-monotonicity of theCLF is explicitly linked with a decision variable that can be optimized on-line. Conservativenessis significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable forstabilization of constrained discrete-time nonlinear systems and real-time control. The purpose ofthis overview is to highlight the potential of flexible CLFs for real-time control of fast mechatronicsystems, with sampling periods below one millisecond, which are widely employed in aerospace andautomotive applications.1 Introductory OverviewOne of the interesting problems in nonlinear control systems is the synthesis of control laws that achievestability [9, 17]. Control Lyapunov functions (CLFs) [2, 16] represent a powerful tool for providing asolution to this problem. The classical approach is based on the off-line design of an explicit feedbacklaw that renders the derivative of the CLF negative. An alternative to this approach is to construct anoptimization problem to be solved on-line, such that any of its feasible solutions renders the derivative ofa candidate CLF negative. This method can be traced back to the early results presented in [14], followedby the more recent articles [13,15], where synthesis of CLFs is performed in a receding horizon fashion.All the above works mainly deal with the continuous-time case, while conditions under which theseresults can be extended to sampled-data nonlinear systems using their approximate discrete-time modelscan be found in [5]. An important article on control Lyapunov functions for discrete-time systems is [8].Therein, classical continuous-time results regarding existence of CLFs are reproduced for the discrete-time case. A significant relaxation in the off-line design of CLFs for discrete-time systems was presentedin [4], where parameter dependent quadratic CLFs are introduced. Also, interesting approaches to theoff-line construction of Lyapunov functions for stability analysis were recently presented in [11], [7]and [1].Despite the popularity of CLFs within systems theory there is still a significant gap in the applicationof CLFs in real-time control in general, and control of fast systems (i.e. systems with a very small sam-pling interval) in particular. The main reason for this is conservativeness of the sufficient conditions forarXiv:1003.0634v1  [cs.OH]  2 Mar 2010M. Lazar 77Figure 1: A graphical illustration of classical CLFs (ρ ∈ [0,1), c ∈ R>0).Figure 2: A graphical illustration of flexible CLFs (c ∈ R>0).Lyapunov asymptotic stability which are employed by most off-line and on-line methods for constructingCLFs.To illustrate this consider the graphical depiction in Figure 1. Classically, a CLF enforces that theresulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, whichis centered at and converges to a desired converging point. This cone is obtained by characterizingthe evolution of the state via its state-space position at each discrete-time instant, with respect to acorresponding sublevel set of the Lyapunov function. Typical examples of relevant classes of systemsfor which classical CLFs are overconservative are linear and nonlinear chains of integrators with boundedinputs and state constraints [12] and discontinuous nonlinear and hybrid systems [3]. Furthermore, inmany real-life control problems classical CLFs prove to be overconservative. For example, considerthe control of a simple electric circuit, such as the Buck-Boost DC-DC converter. At start-up, to drivethe output voltage to the reference very fast, the inductor current must rise and stay far away (e.g.,5[A]) from its corresponding steady-state value (e.g., 0.01[A]) for quite some time. Another typicaland very relevant real-life example is control of position and speed in mechatronic devices, such aselectromagnetic actuators. For a given position reference, the speed must increase very fast at start-upand then return to its steady state value, which is equal to zero. In both cases enforcing a classical CLF78 Flexible Lyapunov Functionsdesign is obviously conservative.Motivated by such examples, recently, in [10], a methodology that reduces the conservatism of CLFdesign for discrete-time nonlinear systems was proposed. Rather than searching for a global CLF (i.e. onthe whole admissible state-space), therein the focus was on relaxing CLF-type conditions for a predeter-mined local CLF through on-line optimization problems, as it is graphically illustrated in Figure 2. Thegoal of this overview is to highlight the potential of flexible CLFs for real-time control of fast mecha-tronic systems, with sampling periods below one millisecond, which are widely used in aerospace andautomotive applications. This includes control of electro-magnetic actuators [6] and a real-time applica-tion to the control of DC-DC power converters.Acknowledgements This research is supported by the Veni grant “Flexible Lyapunov Functions forReal-time Control”, grant number 10230, awarded by STW (Dutch Science Foundation) and NWO (TheNetherlands Organization for Scientific Research).References[1] A. A. Ahmadi & P. A. Parrilo (2008): Non-monotonic Lyapunov functions for stability of discrete time non-linear and switched systems. In: 47th IEEE Conference on Decision and Control, Cancun, Mexico, pp.614–621.[2] Z. Artstein (1983): Stabilization with relaxed controls. Nonlinear Analysis 7, pp. 1163–1173.[3] M. S. Branicky, V. S. Borkar & S. K. Mitter (1998): A unified framework for hybrid control: model andoptimal control theory. IEEE Transactions on Automatic Control 43(1), pp. 31–45.[4] J. Daafouz, P. Riedinger & C. Iung (2002): Stability analysis and control synthesis for switched systems: Aswitched Lyapunov function approach. IEEE Transactions on Automatic Control 47, pp. 1883–1887.[5] L. Grüne & D. Nesić (2003): Optimization based stabilization of sampled-data nonlinear systems via theirapproximate discrete-time models. SIAM Journal of Control and Optimization 42(1), pp. 98–122.[6] R. M. Hermans, M. Lazar, S. Di Cairano & I. V. Kolmanovsky (2009): Low complexity model predictivecontrol of electromagnetic actuators with a stability guarantee. In: 28th American Control Conference, St.Louis, Missouri.[7] H. Ito (2008): A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability ofinterconnected systems. Automatica 44(9), pp. 2340–2346.[8] C. M. Kellett & A. R. Teel (2004): Discrete-time asymptotic controllability implies smooth control-Lyapunovfunction. Systems & Control Letters 52, pp. 349–359.[9] P. Kokotović & M. Arcak (2001): Constructive nonlinear control: a historical perspective. Automatica37(5), pp. 637–662.[10] M. Lazar (2009): Flexible Control Lyapunov Functions. In: 28th American Control Conference, St. Louis,USA, pp. 102–107.[11] M. Malisoff & F. Mazenc (2008): Construction of strict Lyapunov functions for discrete time and hybridtime-varying systems. Nonnlinear Analysis: Hybrid Systems 2, pp. 394–407.[12] F. Mazenc (1997): Stabilization of feedforward systems approximated by a non-linear chain of integrators.Systems & Control Letters 32, pp. 223–229.[13] P. Mhaskar, N. H. El-Farra & P. D. Christofides (2006): Stabilization of nonlinear systems with state andcontrol constraints using Lyapunov-based predictive control. Systems & Control Letters 55, pp. 650–659.[14] E. Polak & D. Q. Mayne (1981): Design of nonlinear feedback controllers. IEEE Transactions on AutomaticControl AC-26(3), pp. 730–733.M. Lazar 79[15] J. A. Primbs, V. Nevistić & J. C. Doyle (2000): A receding horizon generalization of pointwise min-normcontrollers. IEEE Transactions on Automatic Control 45(5), pp. 898–909.[16] E. D Sontag (1983): A Lyapunov-like characterization of asymptotic controllability. SIAM Journal of Controland Optimization 21, pp. 462–471.[17] E. D. Sontag (1999): Stability and stabilization: Discontinuities and the effect of disturbances. In: NonlinearAnalysis, Differential Equations, and Control, Clarke, F.H., Stern, R.J. (eds.), Kluwer, Dordrecht, pp. 551–598.

Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems

The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable for stabilization of constrained discrete-time nonlinear systems and real-time control. The purpose of this overview is to highlight the potential of flexible CLFs for real-time control of fast mechatronic systems, with sampling periods below one millisecond, which are widely employed in aerospace and automotive applications.</p

Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems

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