MIMO Systems with Degraded Actuators and Sensors

This paper presents a reliable control design technique for linear, time-invariant, multi-input multi-output (MIMO) systems with degraded actuators and sensors. The degradation defined in this paper ranges from normal operational conditions to complete failure of actuators and sensors. We derive linear matrix inequality (LMI) conditions ensuring robust stability of the system using static state feedback. The potential of the proposed technique has been demonstrated by an example of three coupled inverted pendulums. Keywords—linear matrix inequality; static state feedback; MIMO; degradation.


I. Introduction
used the same methodology as [3] except that the second stabilizing controller is of adaptive controller which again ends up with high order.
In this paper, we design a static state feedback control system to tolerate the degradation ranging from normal operational condition to complete failure of actuator and sensor. The degradation is modeled as a multiplicative uncertainty at the plant input or output.
We accomplish this via diagonal weighting and norm @ IJTSRD | Available Online @ www.ijtsrd.com | Volume -1 | Issue -5 Page:91 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 bounded matrices. To simplify the derivation we will first develop and prove LMI conditions which tolerate sensor or actuator degradation independently. Our approach is extended to the simultaneous degradation actuators and sensors in the system.

Problem Formulation
The purpose of this section is to define the framework on which our approach of reliable control systems is based. The degradation for actuators and sensors will be considered separately to simplify the derivation.

A. Actuator case:
Actuator case represents the degradation of actuators and is shown as follows, where xR n denotes the state of the system, uk R m is the output of the control gain shown in Figure 1. The control signal u(t) can be readily computed as where the I is identity matrix and u = diag (u 1 .... u r ... u m ), u r  R. Since |u r (t)|  1, i.e. -1  u r (t)  1, the degradation is modeled by a priori weighting u as follows.
(1) u r = 1 indicates the rth actuator may fail fully.
(2) u r = 0 indicates there is no possibility that the rth actuator fails partially or fully.
The degraded sensor signal xs(t) can be readily represented as where the possibility of full or partial failure of sensors is defined in the similar way as in the actuator case using variables s and s in place of u and u, respectively. x

III. Controller Synthesis
In this section we present the main results. Lyapunov stability theory is used to establish stability criteria. Although there is no trivial method to establish a Lyapunov function, by experience, quadratic Lyapunov functions, i.e. V() =  T L, have been proven to be efficient and easily implemented [7]. We consider static state feedback, i.e. uk(t) = Kx(t) for the actuator case, u(t) = Kxs(t) for sensor case and uk(t) = Kxs(t) for combination of actuator and sensor case.
If there exist matrices Q, Y, and M satisfying for a given weighting matrix u, then the following statements are equivalent, (1) The closed-loop system (3.1) is asymptotically stable.
Proof. See Appendix A for proof and notation. If there exist matrices P, W, S and  satisfying for a given weighting matrix s, then the following statements are equivalent.
(1) The closed-loop system (3.3) is asymptotically stable. If there exist matrices P, W, Su, Sx, Ss, , and  satisfying where      u T u , for a given weighting matrix u and s, then the following statements are equivalent, (1) The closed-loop system (3.6) is asymptotically stable.
Proof. See Appendix B for proof.

IV. Numerical Example
Consider the system in Figure 3 consisting of three coupled inverted pendulums of point masses mi, and length li. The pendulums interact via three springs and three dampers of stiffness kij and damping bij; i, j = 1,2,3, and ij. The distances from attached point of springs and dampers to the platform baseline are ai.
The system data are shown in Table 1. Using this system, we will demonstrate several examples for actuator and sensor degradation. The system dynamics for pendulums are written in the general form, We will first demonstrate actuator degradation. The weighting is chosen u=diag(1, 0.1, 0.28) which represents the possibility of degradation for each actuator. For instance, actuator for m1 may fail during its operation, actuators for m2 and m3 are subject to 10% and 28% variation of its nominal operation signal.

Figure 3 Three Inverted Pendulums Systems
We will demonstrate nominal operation of actuators, i.e. u = 0, at time t  0.5. Then, we will show the case where the actuator for m1 is subject to fail and the actuator for m2 and m3 are partial failure. Detail conditions are shown in Table 2. Table 2 time International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 0.5  t < 2.5 failure 0.9*S 2 0.72*S 3 2.5  t  6 2*S 1 1.1*S 2 1.28*S 3 where Si = the nominal operation control signal of actuator associated with each mass. The computed gain followed by Theorem 1 is   variations. We will demonstrate the extreme case, which is shown in  Figure 6 and 7. The sudden jumps in Figure   7 show the instantaneous switching at time t = 0.5 sec.
The following results are observed from simulation.
The system will oscillate if we fail three velocity sensors. Since the damping control signals given by velocity sensor are zero, the occurrence of oscillation is not unexpected.

V. Conclusions
We ensure the stability of a MIMO system by establishing LMI conditions when the system is subject to degradation of actuators and sensors. The theory developed in Section 3 is demonstrated by three inverted coupled pendulums, which show the system can be stabilized.

VI. Acknowledgment
This work was supported by the Center for Automation and Intelligent Research and DARPA/ARO grant.

Appendix A
Proof.
We consider the quadratic Lyapunov function V(x)= x T Lx. The consideration of perturbation will be incorporated in the derivative of Lyapunov function for stabilization of overall system. We have

Appendix B
Proof.
We need the following lemma used in the proof. (3)