Separation principle in stochastic control

The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system

with a state process , an output process and a control , where is a vector-valued Wiener process, is a zero-mean Gaussian random vector independent of , , and , , , , are matrix-valued functions which generally are taken to be continuous of bounded variation. Moreover, is nonsingular on some interval . The problem is to design an output feedback law which maps the observed process to the control input in a nonanticipatory manner so as to minimize the functional

where denotes expected value, prime () denotes transpose. and and are continuous matrix functions of bounded variation, is positive semi-definite and is positive definite for all . Under suitable conditions, which need to be properly stated, the optimal policy can be chosen in the form

where is the linear least-squares estimate of the state vector obtained from the Kalman filter

where is the gain of the optimal linear-quadratic regulator obtained by taking and deterministic, and where is the Kalman gain. There is also a non-Gaussian version of this problem (to be discussed below) where the Wiener process is replaced by a more general square-integrable martingale with possible jumps.[1] In this case, the Kalman filter needs to be replaced by a nonlinear filter providing an estimate of the (strict sense) conditional mean

where

is the filtration generated by the output process; i.e., the family of increasing sigma fields representing the data as it is produced.

In the early literature on the separation principle it was common to allow as admissible controls all processes that are adapted to the filtration . This is equivalent to allowing all non-anticipatory Borel functions as feedback laws, which raises the question of existence of a unique solution to the equations of the feedback loop. Moreover, one needs to exclude the possibility that a nonlinear controller extracts more information from the data than what is possible with a linear control law.[2]

Choices of the class of admissible control laws

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Linear-quadratic control problems are often solved by a completion-of-squares argument. In our present context we have

 

in which the first term takes the form[3]

 

where   is the covariance matrix

 

The separation principle would now follow immediately if   were independent of the control. However this needs to be established.

The state equation can be integrated to take the form

 

where   is the state process obtained by setting   and   is the transition matrix function. By linearity,   equals

 

where  . Consequently,

 

but we need to establish that   does not depend on the control. This would be the case if

 

where   is the output process obtained by setting  . This issue was discussed in detail by Lindquist.[2] In fact, since the control process   is in general a nonlinear function of the data and thus non-Gaussian, then so is the output process  . To avoid these problems one might begin by uncoupling the feedback loop and determine an optimal control process in the class of stochastic processes   that are adapted to the family   of sigma fields. This problem, where one optimizes over the class of all control processes adapted to a fixed filtration, is called a stochastic open loop (SOL) problem.[2] It is not uncommon in the literature to assume from the outset that the control is adapted to  ; see, e.g., Section 2.3 in Bensoussan,[4] also van Handel [5] and Willems.[6]

In Lindquist 1973[2] a procedure was proposed for how to embed the class of admissible controls in various SOL classes in a problem-dependent manner, and then construct the corresponding feedback law. The largest class   of admissible feedback laws   consists of the non-anticipatory functions   such that the feedback equation has a unique solution and the corresponding control process   is adapted to  . Next, we give a few examples of specific classes of feedback laws that belong to this general class, as well as some other strategies in the literature to overcome the problems described above.

Linear control laws

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The admissible class   of control laws could be restricted to contain only certain linear ones as in Davis.[7] More generally, the linear class

 

where   is a deterministic function and   is an   kernel, ensures that   is independent of the control.[8][2] In fact, the Gaussian property will then be preserved, and   will be generated by the Kalman filter. Then the error process   is generated by

 

which is clearly independent of the choice of control, and thus so is  .

Lipschitz-continuous control laws

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Wonham proved a separation theorem for controls in the class  , even for a more general cost functional than J(u).[9] However, the proof is far from simple and there are many technical assumptions. For example,   must square and have a determinant bounded away from zero, which is a serious restriction. A later proof by Fleming and Rishel[10] is considerably simpler. They also prove the separation theorem with quadratic cost functional   for a class of Lipschitz continuous feedback laws, namely  , where   is a non-anticipatory function of   which is Lipschitz continuous in this argument. Kushner[11] proposed a more restricted class  , where the modified state process   is given by

 

leading to the identity  .

Imposing delay

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If there is a delay in the processing of the observed data so that, for each  ,   is a function of  , then  ,  , see Example 3 in Georgiou and Lindquist.[1] Consequently,   is independent of the control. Nevertheless, the control policy   must be such that the feedback equations have a unique solution.

Consequently, the problem with possibly control-dependent sigma fields does not occur in the usual discrete-time formulation. However, a procedure used in several textbooks to construct the continuous-time   as the limit of finite difference quotients of the discrete-time  , which does not depend on the control, is circular or a best incomplete; see Remark 4 in Georgiou and Lindquist.[1]

Weak solutions

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An approach introduced by Duncan and Varaiya[12] and Davis and Varaiya,[13] see also Section 2.4 in Bensoussan[4] is based on weak solutions of the stochastic differential equation. Considering such solutions of

 

we can change the probability measure (that depends on  ) via a Girsanov transformation so that

 

becomes a new Wiener process, which (under the new probability measure) can be assumed to be unaffected by the control. The question of how this could be implemented in an engineering system is left open.

Nonlinear filtering solutions

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Although a nonlinear control law will produce a non-Gaussian state process, it can be shown, using nonlinear filtering theory (Chapters 16.1 in Lipster and Shirayev[14] ), that the state process is conditionally Gaussian given the filtration  . This fact can be used to show that   is actually generated by a Kalman filter (see Chapters 11 and 12 in Lipster and Shirayev[14]). However, this requires quite a sophisticated analysis and is restricted to the case where the driving noise   is a Wiener process.

Additional historical perspective can be found in Mitter.[15]

Issues on feedback in linear stochastic systems

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At this point it is suitable to consider a more general class of controlled linear stochastic systems that also covers systems with time delays, namely

 

with   a stochastic vector process which does not depend on the control.[2] The standard stochastic system is then obtained as a special case where  ,   and  . We shall use the short-hand notation

 

for the feedback system, where

 

is a Volterra operator.

In this more general formulation the embedding procedure of Lindquist[2] defines the class   of admissible feedback laws   as the class of non-anticipatory functions   such that the feedback equation   has a unique solution   and   is adapted to  .

In Georgiou and Lindquist[1] a new framework for the separation principle was proposed. This approach considers stochastic systems as well-defined maps between sample paths rather than between stochastic processes and allows us to extend the separation principle to systems driven by martingales with possible jumps. The approach is motivated by engineering thinking where systems and feedback loops process signals, and not stochastic processes per se or transformations of probability measures. Hence the purpose is to create a natural class of admissible control laws that make engineering sense, including those that are nonlinear and discontinuous.

The feedback equation   has a unique strong solution if there exists a non-anticipating function   such that   satisfies the equation with probability one and all other solutions coincide with   with probability one. However, in the sample-wise setting, more is required, namely that such a unique solution exists and that   holds for all  , not just almost all. The resulting feedback loop is deterministically well-posedin the sense that the feedback equations admit a unique solution that causally depends on the input for each input sample path.

In this context, a signal is defined to be a sample path of a stochastic process with possible discontinuities. More precisely, signals will belong to the Skorohod space  , i.e., the space of functions which are continuous on the right and have a left limit at all points (càdlàg functions). In particular, the space   of continuous functions is a proper subspace of  . Hence the response of a typical nonlinear operation that involves thresholding and switching can be modeled as a signal. The same goes for sample paths of counting processes and other martingales. A system is defined to be a measurable non-anticipatory map   sending sample paths to sample paths so that their outputs at any time   is a measurable function of past values of the input and time. For example, stochastic differential equations with Lipschitz coefficients driven by a Wiener process induce maps between corresponding path spaces, see page 127 in Rogers and Williams,[16] and pages 126-128 in Klebaner.[17] Also, under fairly general conditions (see e.g., Chapter V in Protter[18]), stochastic differential equations driven by martingales with sample paths in   have strong solutions who are semi-martingales.

For the time setting  , the feedback system   can be written  , where   can be interpreted as an input.

Definition. A feedback loop   is deterministically well-posed if it has a unique solution   for all inputs   and   is a system.

This implies that the processes   and   define identical filtrations.[1] Consequently, no new information is created by the loop. However, what we need is that   for  . This is ensured by the following lemma (Lemma 8 in Georgiou and Lindquist[1]).

Key Lemma. If the feedback loop   is deterministically well-posed,   is a system, and   is a linear system having a right inverse   that is also a system, then   is a system and   for  .

The condition on   in this lemma is clearly satisfied in the standard linear stochastic system, for which  , and hence  . The remaining conditions are collected in the following definition.

Definition. A feedback law   is deterministically well-posed for the system   if   is a system and the feedback system   deterministically well-posed.

Examples of simple systems that are not deterministically well-posed are given in Remark 12 in Georgiou and Lindquist.[1]

A separation principle for physically realizable control laws

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By only considering feedback laws that are deterministically well-posed, all admissible control laws are physically realizable in the engineering sense that they induce a signal that travels through the feedback loop. The proof of the following theorem can be found in Georgiou and Lindquist 2013.[1]

Separation theorem. Given the linear stochastic system

 

where   is a vector-valued Wiener process,   is a zero-mean Gaussian random vector independent of  , consider the problem of minimizing the quadratic functional J(u) over the class of all deterministically well-posed feedback laws  . Then the unique optimal control law is given by   where   is defined as above and   is given by the Kalman filter. More generally, if   is a square-integrable martingale and   is an arbitrary zero mean random vector,  , where  , is the optimal control law provided it is deterministically well-posed.

In the general non-Gaussian case, which may involve counting processes, the Kalman filter needs to be replaced by a nonlinear filter.

A Separation principle for delay-differential systems

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Stochastic control for time-delay systems were first studied in Lindquist,[19][20][8][2] and Brooks,[21] although Brooks relies on the strong assumption that the observation   is functionally independent of the control  , thus avoiding the key question of feedback.

Consider the delay-differential system[8]

 

where   is now a (square-integrable) Gaussian (vector) martingale, and where   and   are of bounded variation in the first argument and continuous on the right in the second,   is deterministic for  , and  . More precisely,   for  ,   for  , and the total variation of   is bounded by an integrable function in the variable  , and the same holds for  .

We want to determine a control law which minimizes

 

where   is a positive Stieltjes measure. The corresponding deterministic problem obtained by setting   is given by

 

with[8]  .

The following separation principle for the delay system above can be found in Georgiou and Lindquist 2013[1] and generalizes the corresponding result in Lindquist 1973[8]

Theorem. There is a unique feedback law   in the class of deterministically well-posed control laws that minimizes  , and it is given by

 

where   is the deterministic control gain and   is given by the linear (distributed) filter

 

where   is the innovation process

 

and the gain   is as defined in page 120 in Lindquist.[8]

References

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  1. ^ a b c d e f g h i Tryphon T. Georgiou and Anders Lindquist (2013). "The Separation Principle in Stochastic Control, Redux". IEEE Transactions on Automatic Control. 58 (10): 2481–2494. arXiv:1103.3005. doi:10.1109/TAC.2013.2259207. S2CID 12623187..
  2. ^ a b c d e f g h Anders Lindquist (1973). "On Feedback Control of Linear Stochastic Systems". SIAM Journal on Control. 11 (2): 323–343. doi:10.1137/0311025..
  3. ^ Karl Johan Astrom (1970). Introduction to Stochastic Control Theory. Vol. 58. Academic Press. ISBN 978-0-486-44531-1..
  4. ^ a b A. Bensoussan (1992). Stochastic Control of Partially Observable Systems. Cambridge University Press..
  5. ^ Ramon van Handel (2007). Stochastic Calculus, Filtering, and Stochastic Control (PDF). unpublished notes.
  6. ^ Jan C. Willems. (1978). "Recursive filtering". Statistica Neerlandica. 32 (1): 1–39. doi:10.1111/j.1467-9574.1978.tb01382.x..
  7. ^ M.H.A. Davis (1978). Linear Estimation and Stochastic Control. Chapman and Hall..
  8. ^ a b c d e f Anders Lindquist (1973). "Optimal control of linear stochastic systems with applications to time lag systems". Information Sciences. 5: 81–126. doi:10.1016/0020-0255(73)90005-4..
  9. ^ Murray Wonham (1968). "On the separation theorem of stochastic control". SIAM J. Control. 6 (2): 312–326. doi:10.1137/0306023.
  10. ^ W.H. Fleming and R.W. Rishel (1968). Deterministic and Stochastic Optimal Control. Springer-Verlag..
  11. ^ H. Kushner (1971). Introduction to Stochastic Control. Holt, Rinehart and Winston..
  12. ^ Tyrone Duncan and Pravin Varaiya (1971). "On the solutions of a stochastic control system" (PDF). SIAM J. Control. 9 (3): 354–371. doi:10.1137/0309026. hdl:1808/16692..
  13. ^ M.H.A. Davis and P. Varaiya (1972). "Information states for stochastic systems". J. Math. Anal. Applications. 37: 384–402. doi:10.1016/0022-247X(72)90281-8..
  14. ^ a b R.S. Liptser and A.N. Shirayev (1978). Statistics of Random Processes II, Applications. Springer-Verlag..
  15. ^ S. Mitter (1996). "Filtering and stochastic control: A historical perspective". IEEE Control Systems Magazine. 13 (3): 67–76..
  16. ^ Rogers, L. Chris G., and David Williams (2000). Diffusions, Markov processes and martingales: Volume 2, Itô calculus. Cambridge university press.{{cite book}}: CS1 maint: multiple names: authors list (link)
  17. ^ Klebaner, Fima C. (2012). Introduction to Stochastic Calculus with Applications. Imperial College Press – via World Scientific Publishing Company.
  18. ^ Protter, P. E. (2004). Stochastic Integration and Differential Equations. Springer.
  19. ^ Anders Lindquist (1968). "On optimal stochastic control with smoothed information". Information Sciences. 1: 55–85. doi:10.1016/0020-0255(68)90007-8..
  20. ^ Anders Lindquist (1969). "An innovations approach to optimal control of linear stochastic systems with time delay". Information Sciences. 1 (3): 279–295. doi:10.1016/S0020-0255(69)80014-9..
  21. ^ R. Brooks (1972). "Linear Stochastic Control: An extended separation principle". J. Math. Anal. Appl. 38 (3): 569–587. doi:10.1016/0022-247X(72)90069-8..