STOCHASTIC STABILITY FOR MODEL-BASED NETWORKED CONTROL SYSTEMS

Transcription

1 Luis Montestruque, Panos J.Antsalis, Stochastic Stability for Model-Based etwored Control Systems, Proceedings of the 3 American Control Conference, pp , Denver, Colorado, June 4-6, 3. SOCHASIC SABILIY FOR MODEL-BASED EWORKED COROL SYSEMS Luis A. Montestruque, Panos J. Antsalis Abstract In this paper we study the stochastic stability properties of certain networed control system. Specifically we study the stability of the Model-Based etwor Control System introduced in [9] under time-varying communication. he Model-Based etwor Control System uses nowledge of the plant to reduce the number of pacet exchanges and thus the networ traffic. Stability conditions for constant data pacet exchange rates are analysed in [9,, ]. Here we concentrate on the stochastic stability of the networed system when the pacet exchange times are time varying and have some nown statistical properties. Conditions are derived for Almost Sure and Mean Square stability for independent, identically distributed update times. Mean Square stability is also studied for Marov chain driven update times. University of otre Dame, otre Dame, I 46556, U.S.A. Department of Electrical Engineering fax: ( {lmontest, pantsal}@nd.edu of tass in the transmitter microprocessor. his allows the tas of periodically transmitting the plant information to the controller/actuator to be executed at times t that can vary according to certain probability distribution function. his translates into an update time h( that can acquire a certain value according to a probability distribution function. Most wor on networed control systems assumes deterministic communication rates [4, 5] or time-varying rates without considering the stochastic behavior of these rates [8, ]. Little wor has concentrated in characterizing stability or performance on a networed control system under time-varying, stochastic communication. Introduction he use of data networs in control applications is rapidly increasing. Advantages of using networs in a control system include low cost installation, maintenance, and reconfigurability. In spite of the benefits, a networed control system also has some drawbacs inherent to data networs. Data networs are discrete systems that move information from one point to another at discrete times. heir impact on the system is that the controller will not have a continuous flow of information from the plant; in addition the information might be delayed. In [9] a model-based networed control system was introduced. he model-based networed control system architecture is shown in Figure. his control architecture has as main obective the reduction of the data pacets transmitted over the networ by a networed control system. o achieve this, the model-based networed control system uses nowledge of the plant. A model of the plant is used to predict the plant s state vector dynamics and generate the appropriate control signal in between pacet updates. In this way the amount of bandwidth necessitated by each control system using the networ is minimized. he pacets transmitted by the sensor contain the measured value of the plant state and are used to update the plant model on the actuator/controller node. hese pacets are transmitted at times t. We define the update times as the times between transmissions or model updates: h(=t + -t. In [9] we made the assumption that the update times h( are constant. his might not always be the case in applications. he transmission times of data pacets from the plant output to the controller/actuator might not be completely periodic due to networ contention and the usual nondeterministic nature of the transmitter tas execution scheduler. Soft real time constraints provide a way to enforce the execution /3/$7. 3 IEEE 49 Figure. Model-Based etwored Control System We will present different stability criteria that can be applied when the update times h( vary with time. he first case is when the statistical properties of h are unnown. he actual update times observed will itter on the range [h min, h max ]. his criterion may be used to provide a first cut on the stability properties of a system perhaps for comparison purposes. his is the strongest and most conservative stability criterion. his stability type is based on the classical Lyapunov function approach and thus will be referred to as Lyapunov Stability. ext we will present two stochastic stability criteria for systems in which the update times are independent, identically distributed random variables with probability distribution F(h. he first and strongest criterion is called Almost Sure Asymptotic Stability or Probability- Asymptotic Stability. his is the stochastic stability criterion that mostly resembles deterministic stability. he second and weaer type of stability is referred to as Mean Square Asymptotic Stability or Quadratic Asymptotic Stability. Finally a sufficient condition for Mean Square Stability for the model-based networed control system with update times following a finite state Marov chain is presented. Denver, Colorado June 4-6, 3

2 Luis Montestruque, Panos J.Antsalis, Stochastic Stability for Model-Based etwored Control Systems, Proceedings of the 3 American Control Conference, pp , Denver, Colorado, June 4-6, 3. We focus our analysis on the Model-Based etwored Control System developed in [9], specifically the case where the plant is continuous and the states are available (full state feedbac. Other networed systems (e.g. output feedbac, discrete plants, etc studied in [,] can be analyzed in a similar fashion. Model-Based etwored Control System Response he dynamics of the system shown in Figure are given by: ( A BK BK xt + xt ( xt ( xt ( ;, et ( = A BK Aˆ BK et ( = + et ( ( t [ t, t, with t t = h( + + Where A and B are the matrices of actual plant state-space representation, Aˆ and B ˆ are the matrices of plant model statespace representation, A = A Aˆ and B = BBˆ represent the modeling error matrices, and et ( = xt ( xt ˆ( represents the error between the plant state and the plant model state. Define A + BK BK now z( t = [ x( t e( t ], and Λ= A BK Aˆ so + BK that Equation ( can be rewritten as z =Λzfor t [ t, t +. Proposition # he system described by Equation ( with initial conditions [ ] z( t = x( t = z, has the following response: Λ( tt z( t = e M( z; t [ t, t+, t+ t = h(, ( I Λh( I and = e. he proof is similar to the one presented in [9] for constant update times and will be omitted here. We observe in Equation ( that the response is given by a product of matrices that share the same structure but are in general different. Stability cannot be guaranteed even if all matrices in the product are stable or equivalently if they have their eigenvalues inside the unit circle. ext the stability criteria mentioned above are considered. 3 Lyapunov Stability his is the strongest and most conservative stability criterion. It is based on the well-nown Lyapunov second method for determining the stability of a system. We will not consider the statistical properties of h(. his criterion is not stochastic but provides a first approach to stability for cases in which the only thing that is nown about the update times is that they are time varying and are contained within some interval. heorem # he system described by Equation ( is Lyapunov Stable for h [ h, h ] if there exists a symmetric positive definite min max matrix X such that Q = X MXM is positive definite for all I h I h [ hmin, hmax ], where M = e Λ. oted that the output norm can be bounded by e M z e M z Λ( tt Λ( tt ( ( σ ( Λ hmax e M( z ( t t hat is, since e Λ has a finite growth rate that will cease at most at h max. hen convergence of the product of matrices M( to zero ensures the stability of the system. It is clear that the range for h, that is the interval [ hmin, h max ], will vary with the choice of X. Another observation is that the interval obtained this way will always be contained in the set of constant update times where the system is stable. hat is, an update time contained in the interval [ hmin, h max ] will always be a stable constant update time. 4 Almost Sure Asymptotic Stability ow we assign to the update times some stochastic properties. We will assume that the update times h( are independent, identically distributed random variables. We now characterize the notion of stochastic stability. We will use the definition of almost sure asymptotic Lyapunov stability [6] that is the one that provides a stability criterion based on the sample path. herefore, this stability definition resembles more the deterministic stability definition [7], and is of practical importance. It ensures that the system is asymptotically stable. he only weaness might be that the decaying rate may vary. But an expected decaying rate can still be specified. Since the stability condition has been relaxed, we expect to see less conservative results than those obtained using the Lyapunov stability previously considered. We now define Almost Sure or Probability- Asymptotic stability. Definition # he equilibrium z = f t, z with initial condition z( t = z is almost sure (or with probability- asymptotically stable at large (or globally if for any β > and z = f t, z satisfies > the solution of ( Whenever z z = of a system described by ( lim P sup z( t, z, t > = δ t δ < β. his definition is similar to the one presented for deterministic systems. We will examine the conditions under which a full state feedbac continuous networed system is stable. Let h=h( be independent identically distributed random sequence with probability distribution function F(h. We now present the (3 (4 4 Denver, Colorado June 4-6, 3

3 Luis Montestruque, Panos J.Antsalis, Stochastic Stability for Model-Based etwored Control Systems, Proceedings of the 3 American Control Conference, pp , Denver, Colorado, June 4-6, 3. condition under which the system described by Equation ( is asymptotically stable with probability. heorem # he system described by Equation (, with update times h( independent identically distributed random variable with probability distribution F(h is globally almost sure (or with probability- asymptotically stable around the solution x ( h z = = if = E ( e σ Λ / < and the e expected value of the maximum singular value of the test matrix M, E M = E[ σ M ], is strictly less than one, I Λ I where M = e h. We will represent the system by using a technique similar to lifting [] to obtain a discrete linear time invariant system. he system will be represented by ξ =Ωξ, with ξ L and ξ = zt ( + t, t [, h. (5 + e L e stands for the extended L. It is easy to obtain from ( Here the operator Ω as: I ( Ω ν ( t = e Λ δτ ( h( ντ ( dτ h ( t (6 ow we can restructure the definition on almost sure stability or probability- stability given in Definition # to better fit the new system representation. We will say that the system represented by (5 is almost sure stable or stable with probability- if for any β > and > the solution of ξ + =Ω ξ satisfies whenever z lim P sup ξ ( t, z > = δ,[, t ] δ < β. It is obvious that the norm / i,[, h ( ] h ( given by ξ = ξ (,[, h ( ] τ dτ. We will omit the subscript from now on. ow lets assume that the supremum of the norm braceted is achieved at ~ δ, that is sup ξ = ξ. So now we can use Chebyshev bound for > δ positive random variables [3] to bound the probability in our definition: P { sup ξ } P{ ξ } δ E ξ > = > Using ( and basic norm properties, we proceed to bound the expectation on the right hand side (7 is (8 E h ( ξ ( τ dτ / / ( h Λτ E e M( z dτ / h ( σ( Λ τ E ( e dτ M( z = E e d E M / h ( σ( Λ τ ( τ ( z he last equation follows from the independency of the update times h(. Analyzing the first term on the last equality we see that is bounded for the trival case where Λ=. When Λ, the integral can be solved, and can be showed to be equal to / ( h ( E ( e σ Λ σ ( Λ which by assumption is bounded. he second term can also be bounded by using the independency property of the update times h(. E M( E M( = E M ( (9 ( We can now evaluate the limit over the expression obtained. { ξ } E ξ lim P sup > lim δ δ δ lim δ σ ( Λ h ( ( ( / E e E M z σ ( Λ It is obvious that the right hand side of the expression will be identically zero (note that ~ δ if the averaged maximum singular value E[ σ M ] = E[ M ] <. ( It is important to note that if this condition is applied directly over the test matrix M, we may end up with very conservative values. o correct this problem we can apply a similarity transformation over the test matrix M so to obtain a less conservative conclusion. An observation over the condition on the matrix is that it ensures that the probability distribution function for the update times F(h assigns smaller occurrence probabilities to increasingly long destabilizing update times. hat is F(h decays rapidly. In particular we observe that can always be bounded if there exists h m such that F(h= for h larger than h m. We can also bound the expression inside the expectation to obtain ( h ( / ( h E σ Λ σ Λ e < E e and formulate the following corollary. 4 Denver, Colorado June 4-6, 3

4 Luis Montestruque, Panos J.Antsalis, Stochastic Stability for Model-Based etwored Control Systems, Proceedings of the 3 American Control Conference, pp , Denver, Colorado, June 4-6, 3. Corollary # he system described by Equation (, with update times h( that are independent identically distributed random variable with probability distribution F(h is globally almost sure (or with probability- asymptotically stable around the solution ( h z = [ x e] = [ ] if = E e σ Λ < and the expected value of the maximum singular value of the test matrix M, E[ M ] = E[ σ M ], is strictly less than one, I Λ I where M = e h. Figure. Average Maximum Singular Value for σ [ M ] for h~u(.5,h max as a function of h max. Example M = E We use an unstable plant, the double integrator where: A =, B =. For our plant model we chose: ˆ ˆ ˆ ˆ x ˆ = Axˆ + Bu, A=, B =. Our feedbac law is K =. We have seen in [9,, ] given by u=kx with [ ] that the maximum constant update time h for which this system is stable is second. We now assume a uniform probability distribution function U(.5,h max. he plot of averaged maximum singular value of a similarity transformation of the original test matrix M is shown in Figure. he similarity transformation used here was one that diagonalizes the matrix M for h=. We see that the maximum value for h max is around.3 seconds (maximum constant update time for stability is h= second. So we see that, for example, a system with uniformly distributed update time from.5 to. seconds is stable, while a system with a constant update time of second is unstable. 5 Mean Square Asymptotic Stability We now present a weaer type of stability, namely Mean Square Asymptotic Stability that is defined next. Definition # he equilibrium z = f t, z with initial condition z( t = z is mean square stable asymptotically stable at large (or globally if the solution of z = f ( t, z satisfies z = of a system described by ( lim E z( t, z, t = t ( A system that is mean square stable as defined previously will have the expectation of system states to be driven to zero with time. It is clear that this does not prevent the system states to be zero the whole time. For example, a system can have non-zero values at times that are separated in time by periods that increase with time can still satisfy the mean square stability condition since its average is converging to zero. So it is clear that this definition of stability is not as strong as probability- stability, but it is still attractive since many optimal control problems use the squared norm in their calculations. So we will present the conditions under which the networed control system described in ( is mean square stable and how to these relate to the ones for probability- stability. heorem #3 he system described by Equation (, with update times h( independent identically distributed random variable with probability distribution F(h is globally mean square asymptotically stable around the solution z = [ ] if ( h K = E ( e σ Λ < and the maximum singular value of the expected value of M M, EM M = σ E M M, is I Λ I strictly less than one, where M = e h. Lets start by evaluating the expectation of the squared norm of the output of the system described by (. ( t t E Λ e z Λ( tt Λ( tt = Ez ( e e z Λ( tt Λ( tt E σ ( e e z z σ ( Λ h ( + E( e z M ( z (3 4 Denver, Colorado June 4-6, 3

5 Luis Montestruque, Panos J.Antsalis, Stochastic Stability for Model-Based etwored Control Systems, Proceedings of the 3 American Control Conference, pp , Denver, Colorado, June 4-6, 3. ow that the expectation is all in terms of the update times, we can use the independently identically distributed property of the update times and the assumption that K is bounded: σ ( Λ h ( + E( e z z = K z E M( M( z = K z E EM M z K σ ( EM M z E z ( We can repeat the last 3 steps recursively to finally obtain. ( t t E Λ e z K( σ ( EM M z z ( So now it is easy to see that if EM M = σ E M M < then the limit of the expectation as time approaches infinity is zero, which concludes the proof. he first thing we note is the similarity between the conditions given by heorems # and #3. For the first and strongest one we require the expectation of the maximum singular value of the test matrix to be less than one. While for the second weaer stability it is required to have the maximum singular value of the expectation of M M to be less than one. 6 Mean Square Asymptotic Stability for Marovian Update imes Sometimes it is useful to represent the dynamics of the update times as driven by a Marov chain. A good example of this is when the networ is susceptible to experience traffic congestion or has queues for message forwarding. We now present a stability criterion for the model-based control system in which the update times h( are driven by a finite state Marov chain. Assume that the update times can tae a value from a finite set: { } h h h h (4 (,,..., Lets represent the Marov chain process by { ω } with state space {,,..., } and transition probability matrix Γ and x matrix with elements p. he transition probability matrix is defined as p Ρ { ω+ = ω= i}. So now we can represent the update times more appropriately as h ( = h ω. We now present a sufficient condition for the mean square stability of the system under marovian umps. heorem #4 he system described by Equation (, with update times h ( = h ω driven by a finite state Marov chain{ ω } with state space {,,..., } and transition probability matrix Γ with elements p is globally mean square asymptotically stable around the solution z = [ x e] = [ ] if there exists positive definite matrices P(, P(,, P( such that p ( H( i P( H( i P( i < =,..., = I i with H( i = e Λh. We now that since the Marov chain has a finite number of states, and thus that the update times are bounded, we can analyze the system s stability by sampling it at a certain time between each update time. We will evaluate the response of the system described by ( at times t : Lets define ς ( z( t I z t e z t Λh ( + + = ( (5 Λh I = and H( ω = e ω. ow we can represent the sampled networed control system as: ( ς ( + = H ω ς ( (6 o ensure mean square stability we will mae use of a Lyapunov function of quadratic form and analyze the expected value of its difference between two consecutive samples. We will use the following Lyapunov function: ( ( V ς(, ω = ς( P ω ς( (7 ow we can compute the expected value of the difference: [ ς, ] E V ( ς(, ω+ V ( ς(, ω ς( ς, ω i E ( P( ( (, i + P( i Eς H( ω P( ω H( ω ς ω i ς P( i ς E V i = + = = = ς + ω ς + ς = ς ω = ς ς = + = ( ( ( ( ( = p ς H i P H i ς ς P i ς ( ( ( ( ( = ς p H i P H i P i ς (8 From this last equality is it obvious that to ensure mean square stability we need to have: 43 Denver, Colorado June 4-6, 3

6 Luis Montestruque, Panos J.Antsalis, Stochastic Stability for Model-Based etwored Control Systems, Proceedings of the 3 American Control Conference, pp , Denver, Colorado, June 4-6, 3. ( ( ( ( ( p H i P H i P i < (9 = his type of stability criteria depend on our ability to find appropriate P(i matrices. Several other results in ump system stability [, 3] use a similar procedure that can be extended to obtain other conditions on stability of networed control systems. ote that most of the results available in the literature deal with similar but not identical type of systems. he relationship with Almost Sure stability can be studied by closely analyzing the implications of such simplification. 7 Conclusions In this paper four types of stability criteria for model-based networed control systems were presented. he first one ensures deterministic stability for an update time that can vary within an interval. he second stability criterion, called Almost Sure (or Probability- Stability, is a true stochastic stability criterion that ensures deterministic-lie stability of a system. It assumes that the update times are independent identically distributed random variables. A weaer type of stochastic stability, Mean Square Stability, is also presented. It assumes independent identically distributed update times. Both stochastic stability criterion have similar structures and can be equivalent depending on the nature of the probability distribution function and the structure of the test matrix, as is the case for example when F(h= for h>h m and the test matrix is diagonal. Finally, the case where the update time is governed by a Marov chain is studied and a sufficient condition for Mean Square Stability is derived. he stability types presented here only represent some of the different stochastic stability types available on the literature. We consider that the insight provided by the stability analysis studied here can be used to determine conditions for other stability types. Proceedings Of he 4 th IEEE Conference On Decision And Control, December, pp [6] F. Kozin, A Survey of Stability of Stochastic Systems, Automatica Vol5 pp. 95-, 968. [7] M. Marion, Jump Linear Systems in Automatic Control Marcel Deer, ew Yor 99. [8]A.S. Matveev, A.V. Savin, Optimal Control of etwored Systems via Asynchronous Communication Channels with Irregular Delays, 4 th IEEE Conference on Decision and Control, Orlando Florida, December, pp [9] L.A. Montestruque, P.J. Antsalis, Model-Based etwored Control Systems: ecessary and Sufficient Conditions for Stability, th Mediterranean Conference On Control And Automation, July. [] L.A. Montestruque, P.J. Antsalis, State And Output Feedbac Control In Model-Based etwored Control Systems, 4st IEEE Conference on Decision and Control, December. [] L.A. Montestruque and P.J. Antsalis, Model-Based etwored Control System - Stability, ISIS echnical Report ISIS--, January, [] G. Walsh, H. Ye, and L. Bushnell, Stability Analysis of etwored Control Systems, Proceedings of American Control Conference, June 999. [3] S. Woods, Probability, Random Processes, and Estimation heory for Engineers, nd edition, Prentice Hall, Upper Saddle River 994. Acnowledgements he partial support of the DARPA/IO-ES Program (AF- F is gratefully acnowledged. References [] O.L.V. Costa, M.D. Fragoso, Stability Results for Discrete- ime Linear Systems with Marovian umping Parameters, Journal of Mathematical Analysis and Applications, Vol 79, pp , 993 []. Chen, B. Francis, Optimal Sampled-Data Control Systems, nd edition, Springer, London 996. [3]Y. Fang, A ew General Sufficient Condition for Almost Sure Stability of Jump Linear Systems, IEEE ransactions on Automatic Control, Vol 4, o3, pp , March 997. [4] D. Hristu-Varsaelis, Feedbac Control Systems as Users of a Shared etwor: Communication Sequences that Guarantee Stability, Proceedings of the 4 th IEEE Conference on Decision and Control, December, pp [5] H. Ishii and B. Francis, Stabilizing a Linear System by Switching Control and Output Feedbac with Dwell ime, 44 Denver, Colorado June 4-6, 3