## LPV systems, robustness and gain-scheduling

Linear Parameter Varying (LPV) systems are a class of linear systems whose internal description depends on some time-varying parameters as $\dot{x}(t)=A(\rho(t))x(t)+B(\rho(t))u(t)$ where $$\rho$$ is the vector of time-varying parameters, $$x$$ is the state of the system and $$u$$ is the control input.

Tools for their analysis are the same as the ones used in robust analysis and robust control. In this regard, the considered LPV systems usually take the form of polytopic LPV systems, generic parameter dependent systems and systems in Linear Fractional Representation. LPV systems can model a wide range of real world processes where time-varying parameters are involved. As an example, a vehicle trajectory and its response to a change in the angle of the steering wheel can be described as a parameter-dependent system with car speed as parameter. This follows from the fact that the behavior of the car highly depends on its speed.

The actual power of the LPV framework is only revealed when control enters the picture. In this case, and under the assumption that the parameters can be measured or estimated in real-time, the parameters can be used to adapt the controller expression according to the values taken by the parameters. This gives rise to the concept of LPV-based gain-scheduled controllers. For instance, gain-scheduled state-feedback controllers take the form $u(t)=K(\rho(t))x(t).$ This framework then offers an elegant way for deriving (conservative) nonlinear controllers when the parameters depend on the state of the system. It is also possible to compute, in a single design, different controllers optimized according to different performance measures, and accordingly scheduling between them. In our car example, we may design for instance two controllers: one optimizing road holding, the other one optimizing comfort. Scheduling between them can be then performed according to the driver's choice or using automatic heuristics.

I am personally interested in the development of new theoretical results for the analysis and control of LPV systems. I have been recently interested in deriving theoretical results that capture more information about the delay trajectories. Indeed, it turns out that the considered Lyapunov functions implies some condition on the trajectories of the parameters: parameter-independent Lyapunov functions correspond to arbitrarily fast parameter variations whereas parameter-dependent ones are associated with bounded rates of variations for the parameter trajectories. This has recently motivated me to consider LPV systems with piecewise constant and piecewise differentiable parameters that can be analyzed using recent hybrid systems methods.

### References:

[1] J. S. Shamma, "Gain scheduling: potential hazards and possible remedies", IEEE Control Systems Magazine, Vol. 12(3), 2002.
[2] P. Apkarian, P. Gahinet and G. Becker, "Self-scheduled $$\mathscr{H}_\infty$$ control of linear parameter-varying systems: A design example", Automatica, Vol. 31(9), pp. 1251-1261, 1995.
[3] P. Akarian and P. Gahinet, A convex characterization of gain-scheduled $$\mathcal{H}_\infty$$ controllers, IEEE Transactions on Automatic Control, Vol. 40(5), pp. 853-864, 1995.
[4] C. W. Scherer, "LPV control and full-block multipliers", Automatica, Vol. 37(3), pp. 361-375, 2001.
[5] J. Mohammadpour and C. W. Scherer (editors), "Control of linear parameter varying systems with applications", Springer, 2012.
[6] C. Briat, "LPV & Time-Delay Systems - Stability, Observation, Filtering and Control", Springer, 2014.
[7] A. P. White, G. Zhu and J. Choi, "Linear Parameter-Varying Control for Engineering Applications", Springer, 2013.
[8] C. Briat, "Stability analysis and control of LPV systems with piecewise constant parameters", Systems & Control Letters, Vol. 82, pp. 10-17, 2015.

## Time-delay systems

Delays are ubiquitous in nature as they arise a wide range of fields such as biology, ecology, economics, particle physics and communication networks. Time-delay systems (or hereditary systems) are a class of dynamical systems whose state evolution depends on past state values. For instance, discrete-delay systems with single delay can be generally expressed as \begin{equation*} \dot{x}(t)=f(x(t),x(t-h)) \end{equation*} where $$h>0$$ is the delay and $$x(t)$$ is the state value of the system at time $$t$$. They can be seen as a generalization of usual dynamical systems whose future evolution only depends on the current value of the state; e.g. $$\dot{x}(t)=f(x(t))$$. In general, delays are detrimental to stability and will more likely destabilize systems and deteriorate performance. In some other problems, however, delays can improve stability properties or induce desirable oscillations (e.g. limit cycles in biological systems).

The analysis of delay-systems is slightly more complicated than for usual dynamical systems, mainly due to their infinite dimensional nature. Many techniques have been developed both in frequency- and time-domains. When delays are time-varying, time-domains approaches are preferable through the use of the Lyapunov-Krasovskii and Lyapunov-Razumikhin Theorems, and robust analysis methods (such as IQCs, well-posedness, etc.). When delays are state-dependent (as in communication networks), the problem is much more complicated and still open since very few general results have been obtained.

Controlling time-delay systems is also a challenging problem and many approaches have been developed over the past years. For input-delay systems, state-predictors can be considered to predict future values of the state in order to use them in the control law. When the delay acts on the state of the system, controllers involving "delay components" can be considered, either by using delayed signals (controllers involving memory) or by having the delay scheduling the controller (delay-scheduled controllers) in an LPV-way (see section above). In any case, however, the knowledge of the delay is critical for implementation. When the delay is time-varying or uncertain, it is difficult to know the exact value of the delay in real time and controllers must be made resilient with respect to delay uncertainty (memory-resilient controllers).

I am personally interested in the development of new theoretical results for the analysis and control of time-delay systems. The application of the theory of time-delay systems to real world processes is part of my research interests, notably the application to communication and biological networks.

### References:

[1] S.-I. Niculescu, "Delay effects on stability - A robust control approach", Springer, 2001.
[2] K. Gu, V. L. Kharitonov and J. Chen, "Stability of time-delay systems", Birkhauser, 2003.
[3] C. Briat, "LPV & Time-Delay Systems - Stability, Observation, Filtering and Control", Springer, 2014.
[4] C. Briat, O. Sename and J.-F. Lafay, "Delay-scheduled state-feedback design for time-delay systems with time-varying delays - A LPV approach", Systems & Control Letters, Vol. 58(9), pp. 664-671, 2009;
[5] C. Briat, O. Sename and J.-F. Lafay, "Memory Resilient Gain-scheduled State-Feedback Control of Uncertain LTI/LPV Time-Delay Systems with Time-Varying Delays", Systems & Control Letters, Vol. 59(8), pp. 451-459, 2010.

## Hybrid dynamical systems: impulsive, switched and sampled-data systems

Looped-functionals are a particular class of functionals allowing to express a discrete-time stability criteria in an alternative way. The term 'looped' comes from the fact that the looped-functionals satisfy a boundary condition, looping both sides of the functional together. These functionals are particularly useful for several reasons. The first one is the use of a discrete-time stability criterion, which is a much weaker condition than a continuous-time condition. Demanding a continuous decrease of a function is indeed a much stronger requirement than asking for a pointwise decrease of a sequence of points extracted from the same function. This hence allows us to relax the constraint on the strict decrease of the continuous-time Lyapunov function. This is particularly interesting when dealing with hybrid systems, such as impulsive or switched systems, where jumps in the Lyapunov function level can occur. Linear impulsive systems are described as $\begin{array}{rcl} \dot{x}(t)&=&Ax(t),\ t\ne t_k\\ x(t_k)&=&Jx(t_k^-) \end{array}$ where $$x$$ is the state and $$\{t_k\}_{k\in\mathbb{N}}$$ is a sequence of increasing impulse instants satisfying $$t_k\to\infty$$ as $$k\to\infty$$. The notation $$x(t_k^-)$$ denotes the left-limit of $$x(s)$$ at $$s=t_k$$. On the other hand, linear switched systems are given by defined as $\dot{x}(t)=A_{\sigma(t)}x(t)$ where $$x$$ is the state and the function $$\sigma(t)\in\{1,\ldots,N\}$$ defines which mode is active at each time $$t$$.

The second one is that the obtained stability conditions depend on the system matrices in a convex way. In this regard, it is not necessary to consider the discrete-time system embedded in the hybrid system. This latter fact becomes particularly interesting when time-varying and nonlinear systems are considered since no closed-form expression for the embedded discrete-time system exists. The convexity of the conditions also permits an easy extension to uncertain systems.

Looped-functionals can be seen as a unifying paradigm for dealing with hybrid systems, such as switched and impulsive systems, but also sampled-data systems, periodic systems, LPV systems, and possibly many others. This paradigm is quite recent and a lot of things remain to be discovered, notably applications to delay systems, extensions to systems with inputs, looped-functionals for control, etc.

Clock-dependent Lyapunov functions are a particular class of Lyapunov functions that depend on a clock measuring the time elapsed since a particular event such as the last jump of the state for impulsive systems or the last change of mode for switched systems. The obtained criteria are similar to the ones obtained using looped-functionals, with the striking difference that they are structurally more suitable for control design. They also involve a much lower number of decision variables, which makes them more scalable as the size of the considered system increases.

### References:

[1] A. Seuret, "A novel stability analysis of linear systems under asynchronous samplings", Automatica, Vol. 48(1), pp. 177-182, 2012.
[2] C. Briat and A. Seuret, "A looped-functional approach for robust stability analysis of linear impulsive systems", Systems & Control Letters, Vol. 61(10), pp. 980-988, 2012.
[3] C. Briat and A. Seuret, "Convex dwell-time characterizations for uncertain linear impulsive systems", IEEE Transactions on Automatic Control, Vol. 57(12), pp. 3241-3246, 2012.
[4] C. Briat and A. Seuret, "Affine minimal and mode-dependent dwell-time characterization for uncertain switched linear systems", IEEE Transactions on Automatic Control, Vol. 58(5), pp. 1304-1310, 2013.
[5] C. Briat, "Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints", Automatica, Vol. 49(11), pp. 3449-3457, 2013;
[6] C. Briat, "Convex lifted conditions for robust $$\ell_2$$-stability analysis and $$\ell_2$$-stabilization of linear discrete-time switched systems with minimum dwell-time constraint", Automatica, Vol. 50(3), pp. 976-983, 2014;
[7] C. Briat, "Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints", Automatica, Vol. 74, pp. 279-287, 2016;

## Positive systems and their applications

### Positive systems

Positive systems [1] are a class of systems having their state confined in the nonnegative orthant and which map nonnegative inputs into nonnegative outputs. Since many physical systems naturally involves positive variables, it is hence natural to (try to) represent them as positive dynamical systems. Interesting examples are biological systems [2,3,4], epidemiological systems [2,5], communication networks [6,7,8], etc. Quite surprisingly, linear positive systems of the form $\begin{array}{rcl} \dot{x}(t)&=&Ax(t),\ A\ \text{Metzler}\\ x(0)&=&x_0\ge0 \end{array}$ have very interesting properties. Their stability can be exactly characterized by sum-separable Lyapunov functions of the form $\begin{array}{rcl} V_\ell&=&\sum_{i=1}^nv_ix_i, v_i>0\\ V_q&=&\sum_{i=1}^nd_ix_i^2, d_i>0\\ &=&x^TDx,\ D=\text{diag}_{i=1}^nd_i. \end{array}$ The former one leads to stability conditions taking the form of a linear programming problem [9] whereas the latter one leads to a semidefinite programming problem [10] (although simpler than usual ones for general linear systems due to the diagonal structure of the Lyapunov matrix $$D$$). The use of such Lyapunov functions allows for the design of structured state-feedback in a nonconservative way [11] (a problem for which certain instances are known to be NP-hard). It has also been recently shown that the computation of the $$L_1$$- and the $$L_\infty$$-gains of linear positive can be exactly cast as a linear program as well [12]. This sharply contrasts with the poor tractability of the $$L_\infty$$-gain for general linear systems. Exact robustness results for uncertain linear positive systems in LFT form have also been obtained [12,13,14] and structural results have been obtained in [15].

### Linear positive systems with delays

Linear positive with delays [16] take the form $\begin{array}{rcl} \dot{x}(t)&=&Ax(t)+A_hx(t-h)\\ x(s)&=&\phi(s)\ge0,\ s\in[-h,0] \end{array}$ where $$A$$ is Metzler and $$A_h$$ is nonnegative. As for undelayed systems, they have very interesting properties such that the above system is stable for any $$h\ge0$$ if and only if $$A+A_h$$ is Hurwitz stable; i.e. the system with zero-delay is stable [12,16]. In other words, the worst-case delay value is $$h=0$$ which is rather unusual. Results for time-varying delays have also been obtained and coincide with the constant-delay stability conditions suggesting that the worst-case time-varying delay is actually the constant-delay, which goes against intuition as time-varying delays usually tend to be destabilizing [12,17,18,19].

### Interval observers

An interesting application of positive systems is in the design of interval-observers [20,21,22]. The goal of such observers is not to estimate the state as closely as possible but instead estimate an upper-bound and lower-bound on the value of the state over time. These observers are hence able to deal with the presence of persistent disturbances that may drive the estimation error away from zero. A state-feedback controller can then be designed using a weighted sum of these bounds; e.g. the mean value.

### References:

[1] L. Farina and S. Rinaldi, "Positive Linear Systems: Theory and Applications", John Wiley & Sons, 2000.
[2] J. D. Murray, "Mathematical Biology Part I. An Introduction", Springer-Verlag, 2002.
[3] U. Müller-Herold, "General mass-action kinetics. Positiveness of concentrations as structural property of Horn's equation", Chemical Physics Letters, Vol. 33(3), pp. 467-470, 1975.
[4] C. Briat and M. Khammash, "Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback", 51st IEEE Conference on Decision and Control, 2012.
[5] C. Briat and E. I. Verriest, "A New Delay-SIR Model for Pulse Vaccination", Biomedical Signal Processing and Control, Vol. 4, pp. 272-277, 2009.
[6] S. H. Low, F. Paganini and J. C. Doyle, "Internet congestion control", IEEE Control Systems Magazine, Vol. 22(1), pp. 28-43, 2002.
[7] R. Shorten, F. Wirth and D. Leith, "A positive systems model of TCP-like congestion control: asymptotic results", IEEE/ACM Transactions on Networking, Vol. 14(3), pp. 616-629, 2006.
[8] C. Briat, E. A. Yavuz, H. Hjalmarsson, K. H. Johansson, U. T. Jönsson, G. Karlsson and H. Sandberg, "The conservation of information, towards an axiomatized modular modeling approach to congestion control", IEEE/ACM Transactions on Networking, Vol. 23(3), pp. 851-865, 2015.
[9] W. M. Haddad and V. Chellaboina, "Stability and dissipativity theory for nonnegative dynamic systems: a unified analysis framework for biological and physiological systems", Nonlinear Analysis: Real World Applications, Vol.6(1), pp. 35-65, 2005.
[10] R. Shorten, O. Mason and D. Leith, "An alternative proof of the Barker, Berman, Plemmons result on diagonal stability and extensions", Linear Algebra and Its Applications, Vol.430, pp. 34-40, 2009.
[11] M. Ait Rami and F. Tadeo, "Controller synthesis for positive linear systems with bounded controls", IEEE Transactions on Circuits and Systems -- II. Express Briefs, Vol. 54(2), pp. 151-155, 2007.
[12] C. Briat, "Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints: $$L_1$$- and $$L_\infty$$-gains characterization", International Journal of Robust and Nonlinear Control, Vol. 23(17), pp. 1932--1954, 2013.
[13] Y. Ebihara, D. Peaucelle and D. Arzelier, "$$L_1$$ Gain Analysis of Linear Positive Systems and Its Application", 50th IEEE Conference on Decision and Control, 2011.
[14] M. Colombino and R. S. Smith, "Convex characterization of robust stability analysis and control synthesis for positive linear systems", 53rd IEEE Conference on Decision and Control, 2014.
[15] C. Briat and M. Khammash, "Sign properties of Metzler matrices with applications", 2016.
[16] W. M. Haddad and V. Chellaboina, "Stability theory for nonnegative and compartmental dynamical systems with time delay", Systems & Control Letters, Vol. 51(5), pp. 355-361, 2004.
[17] M. Ait Rami, "Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays", In "Positive systems - Proceedings of the 3rd Multidisciplinary International Symposium on Positive Systems: Theory and Applications, pp. 205-216, 2009.
[18] J. Shen and J. Lam, " $$\ell_\infty$$/$${L}_\infty$$-Gain Analysis for Positive Linear Systems with Unbounded Time-Varying Delays", IEEE Transactions on Automatic Control, Vol. 60(3), pp. 857-862, 2015.
[19] J. Zhu and J. Chen, "Stability of systems with time-varying delays: An $$\mathscr{L}_1$$ small-gain perspective", Automatica, Vol. 52, pp. 260-265, 2015.
[20] J. L. Gouzé, A. Rapaport and M. Z. Hadj-Sadok, "Interval observers for uncertain biological systems", Ecological modelling, Vol. 133, pp. 45-56, 2000.
[21] F. Mazenc and O. Bernard, "Interval observers for linear time-invariant systems with disturbances", Automatica, Vol. 47, pp. 140-147, 2011.
[22] C. Briat and M. Khammash, "Interval peak-to-peak observers for continuous- and discrete-time systems with persistent inputs and delays", Automatica, Vol. 74, pp. 206-213, 2016.

## Congestion control modeling and analysis in communication networks

### What is congestion ?

Congestion in communication networks like Internet is an important phenomenon responsible of large communication delays and data loss that deteriorate the overall network efficiency. To improve this, congestion control algorithms have been implemented in transmission protocols like TCP.

Congestion control is truly a control problem, in the control theory sense: the system is the network, the controlled output is congestion, the controllers are protocol congestion avoidance/control algorithms and the control input is the user sending rate, the rate at which the user is sending packets. This sending rate is calculated from a congestion windows, which is the number of packets in flight we want to maintain, and from a congestion measure which represents the level of congestion over the route. Usual congestion measures are data loss and delays. While the first one is easy to use, it tends to provoke congestion to detect it. Delay-based protocols are however much smoother and can prevent congestion before it occurs, but are more difficult to consider due to the difficulty of separating the Round Trip Time (RTT) into a sum of queuing delays (varying) and propagation delays (constant).

Besides stability, several other performance criteria may be considered when designing congestion control protocols. Efficiency means the complete use of the available bandwidth of the network. Fairness corresponds to the property that the available bandwidth is shared among users in an equitable way (in a certain sense). Cross-traffic adaptation is a property referring to the fact that unregulated/external traffic is allowed to pass and does not destabilize the congestion control mechanism.

Users usually ignore almost everything about the network they are using: they ignore the number of servers (queues) that are used, they ignore the capacities of the links, they also ignore the path they are using (and thus the corresponding delays) as well as the number of users that are using the very same path, as well as the congestion protocols they are implementing. Hence, the users face the problem of controlling a very complex process with very little information. All these limitations make the design of efficient congestion control protocols rather difficult.

### Why modeling congestion in networks ?

The first reason is mathematical rigor in the modeling: is it possible to provide a mathematical model for congestion in networks based on simple concepts/axioms, similarly as for electrical networks ? Since networks consist of interconnections of different elements, the model should be modular, scalable and share the same topology with the network itself. Therefore, the model is expected to have a suitable structure adapted to the network modeling problem. It turns out that such a model exists and can be derived from the simple concept of conservation of information, which is a conservation law. Using this concept, local models for users, queues and transmission channels can be easily characterized.

The second reason is analysis: by using models for networks and protocols, we can theoretically analyze performance of protocols, such as stability, fairness, efficiency, cross-traffic adaptation, speed of convergence. Analysis is not possible using the NS-2 simulator, which is essentially a computational simulator emulating a network in the behavioral sense. Mathematical models are much more insightful from a theoretical perspective.

The third reason is protocol design: based on the model, it may be possible to design a new generation of protocols relying on mathematical properties, such as the recent FAST-TCP protocol. The fourth reason is simulator design: even though if simulators, like NS-2, already exists, the development of an equation-based simulator is an interesting problem on its own. This would lead to the development of new algorithms for integrating functional differential equations involving state-dependent delays, known to be quite difficult to solve numerically.

### References:

[1] R.Srikant, "The Mathematics of Internet Congestion Control", Springer, 2004.
[2] S. H. Low, F. Paganini and J. C. Doyle, "Internet Congestion Control", IEEE Control Systems, Vol. 22(1), pp.28-43, 2002.
[3] C. Briat, E. A. Yavuz and G. Karlsson, "A conservation-law-based modular fluid-flow model for network congestion modeling", 31st {IEEE} International Conference on Computer Communications (INFOCOM), pp. 2050-2058, 2012.
[4] C. Briat, E. A. Yavuz and G. Karlsson, "The conservation of information, towards an axiomatized modular modeling approach to congestion control", IEEE Transactions on Networking (in press), Vol. 23(3), 2015.

## Deterministic and Stochastic Reaction Networks

Reactions network theory is a very powerful and broadly applicable paradigm that can be used to represent population dynamics, epidemiological dynamics, chemical and biological reaction networks, social networks, etc.

Deterministic models consider the evolution of continuous quantities such as concentrations. These models are well-suited when the homogeneous mixing property is strongly satisfied and the species populations are large (so that the definition of concentration is well-defined and makes perfect sense). These assumptions are generally satisfied in some fields. For instance, this is the case in chemistry since species involved there are in Avogadro numbers. Deterministic models are very common in epidemiology, the so-called SIR-models, or even population models described, for instance, by Lotka-Volterra equations. Analysis of these models include stability of equilibrium points, persistence and permanence of trajectories, boundedness of trajectories. Control and observation of these processes are also interesting and important problems.

Stochastic models, on the other hand, keep track of the actual count of individuals in the reaction network and the state takes integer values. Such models are better able to characterize processes where population of certain species can be small and for which stochastic effects cannot be neglected. This is, for instance, the case in systems biology where randomness in the reactions is the source of intrinsic noise. Under some assumptions, the process is a contnuous-time Markov process that can be exactly described by the forward Kolmogorov-equation (also called Chemical Master Equation). Solving the master equation is, in general, not possible since it consists of a countable infinite set of ordinary differential equations. When the network only involves monomolecular (or monoindividual) reactions, the dynamics of the moments of the probability distribution admit an explicit and closed-form that is finite-dimensional. When some reactions are however bimolecular, the moments do not admit any closed-form anymore and the moment closure problem arises. The analysis of stochastic reaction networks mostly involves open problems, that are the stochastic analogs of the ones in the deterministic setting. Furthermore, not only the actual trajectories of the stochastic process are important, the moments are also critical from a practical viewpoint. Analysis tools are then the first step towards the development of control theory for stochastic reaction networks, a very interesting and mostly open field that will likely be playing a key role in the future of systems and synthetic biology.