Event-Triggered Embedded Control

This section discusses the design of event-triggering schemes for embedded control systems. The main idea is to first design a continuous-time controller that guarantees some stability concept such as input-to-state stability or L2-stability. We then develop an event-triggering threshold such that the sporadically triggered control system preservers this underlying stability concept. This approach is sometimes called the emulation-based method [1]

Sampled-Data System Model: Let's first consider how a sampled-data system might be configured. The following figure shows a block diagram of the system under study. The plant (G) has two types of inputs. There is an external "uncontrolled" disturbace, w, and a control input, u. The plant's state, x, satisfies the inhomogeneous differential equation

for t>0. The output of the plant is the system state, x.

Rather than working directly with the continuous-time state, the controller works with a sampled version of the state. In particular, we introduce a sampler (S) system that is characterized by a monotone increasing sequence of sampling instants , {r[j]}. The time r[j] denotes the jth consecutive sampling instant. The output of the sampler is therefore a sequence of sampled states , , in which A state feedback controller, k, maps the sampled state onto a control vector, . The resulting sequence of controls is then transformed into a continuous-time signal through a zero-order hold (H).

ISS Event-triggersUnder the emulation-based approach for developing sampled-data systems, we assume that the controller enforces a specified stability concept. In particular, let's confine our attention to input-to-state stability and let's consider the "continuously" sampled closed-loop system

where e and w are L-infinity disturbances. We assume that the controller, k, leaves this closed loop system input-to-state stable with respect to the inputs. The ISS assumption implies the existence of an ISS Lyapunov function, V with class K functions such that

Now consider the sampled-data version of this system. Define the gap function as . The controller uses the sampled state, rather than the true state, so the system's state equation must satisfy the above differential equation where e is now the gap defined above. Under the ISS assumption, we see that if we can restrict the gap so that

and a is between zero and one, then inserting this relation into the original dissipative inequality for Vdot implies that

which is sufficient to assure the input-to-state stability of the sampled-data system.

The constraint given above can be viewed as a state-dependent thresholding condition. In particular, we know that at the beginning of a sampling interval, that the gap is zero. After that, the gap will grow. When the gap equals the state dependent threshold, we again sample the state, thereby forcing the gap to zer again. In this way the condition is always satisfied and we can assure the sampled-data system is ISS. [2]

L2-event triggers State dependent event-triggers can be determined that assure the L2-stability of the closed-loop system [3][4]. In this case we narrow our attention to systems that are affine in the external disturbance and control. In particular, we assume that the system state satisfies the following differential equation

where the control is generated by

where gamma is a constant greater than zero and V is a storage function satisfying the Hamilton Jacobi inequality (HJI). For this particular control, one can show that the induced L2 gain of the system from the input w to [x,u] is less than gamma. The derivation of the L2-event trigger starts by introducing a sequence of release or sampling times, . This is a sequence of times when the system state is sampled and used by the controller.

for t between release times r[j] and r[j+1]. We introduce the gap between the current state and the previously sampled state, and assume that k is Lipschitz with constant L.

We now examine the directional derivative of the storage function V,

where beta is between 0 and 1 and the HJI inequality and a completing the square argument were used to obtain the bounding inequalities. Note that the last inequality will be a dissipative inequality provided we can guarantee that the last three terms are collectively negative definite. If we can guarantee this, then the entire sampled data system will satisfy a dissipative inequality ensuring the L2 stability of the overall sampled-data system. In particular, the inequality ensuring this property is the L2-event triggering condition

Example: We now provide a simple example illustrating the behavior of the ISS and L2 event-triggers. We consider a process model without process noise of the form

The release times are selected when the following ISS event-trigger is used , and we consider three different types of systems shown below.

The following plots show the system response for the linear, superlinear and sublinear choices of f. The top plot shows the gap and the threshold as a function of time. The bottom plot shows the intersample period generated by this system.

For the linear f, we see that the choice of event-trigger and system yield periodic sampling of the system state. For the superlinear case, the intersample time gets longer as the system state approaches the equilibrium point of the unforced system. For the sublinear case, the intersample time gets shorter and the sampling time becomes infinite at a finite time into the simulation. This case, therefore exhibits zeno-type sampling behavior. This occurs becasue the sublinear system function is not Lipschitz at 0. In fact it can be shown that the Lipschitz continuity of f is essential for assuring that intersample times remain bounded below by a non-zero constant. The results for the L2 event trigger are also shown below. This was done for the superlinear case and here we see that as the system approaches its equilibrium, the intersample time becomes periodic in nature.

1. D. Nesic and A.R. Teel, Input-output stability properties of networked control systems, IEEE Transactions on Automatic Control, 49(10):1650-1667, 2004.
2. P. Tabuada. Event-triggered real-time scheduling of stabilizing control tasks. IEEE transactions on Automatic Control, 52(9):1680-1685, Sept. 2007.
3. M. Lemmon, T. Chantem, X.S. Hu, and M. Zyskowski, On self-triggered full information h-infinity controllers. Hybrid Systems: computation and control, Pisa, Italy, July 2007.
4. X. Wang and M.D. Lemmon, Self-triggered feedback control systems with finite-gain L2 stability. IEEE Transactions on Automatic Control, 54(3):452-467, March 2009.