\r\n\tThis book will be a self-contained collection of scholarly papers targeting an audience of practicing researchers, academics, PhD students and other scientists. The contents of the book will be written by multiple authors and edited by experts in the field. The area of interest and scope of the project can be described with (but are not limited to) the following keywords: Alcoholism, Depression, Addiction, Blackouts, Relapse, Binge Drinking, Genetic basis, Neurological Aspects, Treatment, Organ Damage.
\r\n\r\n\tAuthors are not limited in terms of topic, but encouraged to present a chapter proposal that best suits their current research efforts. Later, when all chapter proposals are collected, the editor will provide a more specific direction of the book.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:null,priceUsd:null,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"cb00568f155a16350f11d29aabfc4ba9",bookSignature:"Associate Prof. Palash Mandal",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8315.jpg",keywords:"Alcoholism, Depression, Addiction, Blackouts, Relapse, Binge Drinking, Genetic basis, Neurological Aspects, Treatment, Organ Damage",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 3rd 2018",dateEndSecondStepPublish:"October 24th 2018",dateEndThirdStepPublish:"December 23rd 2018",dateEndFourthStepPublish:"March 13th 2019",dateEndFifthStepPublish:"May 12th 2019",remainingDaysToSecondStep:"2 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"217215",title:"Dr.",name:"Palash",middleName:null,surname:"Mandal",slug:"palash-mandal",fullName:"Palash Mandal",profilePictureURL:"https://mts.intechopen.com/storage/users/217215/images/system/217215.jpeg",biography:null,institutionString:"Charusat University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"3",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"16",title:"Medicine",slug:"medicine"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"270934",firstName:"Ivan",lastName:"Butkovic",middleName:null,title:"Mr.",imageUrl:"//cdnintech.com/web/frontend/www/assets/author.svg",email:"ivan.b@intechopen.com",biography:null}},relatedBooks:[{type:"book",id:"6550",title:"Cohort Studies in Health Sciences",subtitle:null,isOpenForSubmission:!1,hash:"01df5aba4fff1a84b37a2fdafa809660",slug:"cohort-studies-in-health-sciences",bookSignature:"R. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"18235",title:"Stem Cell Therapy for Patients with Chronic Liver Disease",doi:"10.5772/20286",slug:"stem-cell-therapy-for-patients-with-chronic-liver-disease",body:'\n\t\tAdult stem cells are a population of immature cells that have the capability of self renewal to provide the human body with a constant source of cells for maintaining healthy tissues or replacing those that are damaged. Our present day scientific developments alongside clinical experiences have collectively consolidated our understanding of the signals that mediate stem cell lineage commitment and differentiation. From this, the concept of using adult stem cells to repopulate chronically diseased organs is now a feasible option for regenerative medicine.
\n\t\t\tWhilst the use of human embryonic stem cells in the clinical setting has been overwhelmingly marred by its oncogenic potential as well as ethical concerns, the rate of progress in understanding the complex developmental plasticity of adult stem cells, in particular those derived from the bone marrow, has successfully allowed their use for translational research with unattached ethical issues. The pioneering use of bone marrow stem cells in the 1960s as a viable treatment option for leukaemia, myeloma and lymphoma has come a long way since then. Today we are able to use these stem cells to give rise to bone and cartilage [1] and to repair both cardiac and liver function [2, 3]. Since a detailed review for each of these exciting developments is too broad for this chapter, we will mainly focus on the use of bone marrow derived stem cell for the treatment of chronic liver disease.
\n\t\tLiver cirrhosis is the end stage of chronic liver disease and is associated with many serious systemic complications resulting from both liver failure and portal hypertension. This condition has a poor prognosis and is difficult to treat. In the UK alone as many as one in ten people have some form of liver disease where many die prematurely as a result [4]. Liver disease is currently the fifth most common cause of mortality in the UK for both men and women and whilst the mortality rates for coronary heart disease, cancer, respiratory and cerebro-vascular diseases are falling, the death rates from all types of liver disease in England and Wales over the last 25 years showed over 150% increase in men and 100% in women [5]. This significant upward trend clearly indicates a need to plan ahead for the health service. Worldwide, the common causes of liver fibrosis and cirrhosis include hepatitis B, hepatitis C and alcohol consumption. Other causes include immune mediated damage, genetic abnormalities, and non-alcoholic steatohepatitis which are mostly associated with diabetes and metabolic syndrome [6-8].
\n\t\tLiver cirrhosis or fibrosis is a common progressive pathological lesion of chronic liver disease that occurs in response to various liver-damaging factors. Cirrhosis is defined as the histological development of regenerative nodules surrounded by fibrous bands which leads to portal hypertension; the development of hepatocellular carcinoma and end-stage liver disease. Over the past two decades the main cellular and molecular mechanisms of cirrhotic initiation and progression have been clearly defined. Various causative factors including hepatitis virus infection; ischemia; parasite infection; abnormal copper or iron load, all result in chronic inflammation which initially leads towards an excessive synthesis of the extracellular matrix (ECM). Activated hepatic stellate cells; injured or regenerated hepatocytes; Kupffer cells; sinusoidal cells and natural killer (NK) cells all produce cytokines and immunoreactive factors which exert various biological effects in an autocrine and paracrine manner to initiate fibrotic growth [9-12].
\n\t\t\tThe detailed understanding of the natural history and pathophysiology of cirrhosis has resulted in better management of its complications to improve the quality of life of patients. Whilst pharmacological treatments can halt progression to decompensated cirrhosis, orthotopic liver transplantation (OLT) still remains the only highly successful and curative option for end stage cirrhosis. Survival data from the United Network of Organ Sharing (UNOS) study in 2004 indicated a survival rate of only 61% at 8 year post transplantation [13]. OLT cannot be sustained as the only option for advanced liver failure. The limited availability of organs for a constantly growing list of patients requiring transplant in addition to issues of compatibility and comorbid factors means that not everyone is eligible for transplantation. There is therefore is an unmet need to find a suitable alternative.
\n\t\tHepatocyte transplantation has been identified as a promising alternative to OLT for certain liver-based metabolic disorders and acute liver failure. With the advantage of preserving the native liver, hepatocyte transplantation is less invasive and can be performed repeatedly to allow a higher chance for improved recovery in patients with acute liver failure [14-16]. The mechanisms by which intraportally injected donor hepatocytes engraft into the host liver parenchyma involves cell migration within the liver sinusoids [17]. These cells become trapped causing portal hypertension and ischemic–reperfusion [18, 19]. This consequently initiates clearance by the innate immune system where the associated release of cytokines induces vascular permeability thus allowing surviving hepatocytes to translocate through the sinusoidal fenestrations and integrate into the liver parenchyma [20]. Although the initial engraftment success of hepatocytes after transplantation is low, repeated hepatocyte transplantation in animal models have shown a sufficient increase in the number of engrafted cells to allow recovery from some metabolic defects [21]. Isolation of hepatocytes is carried out from donor liver that are unused or not suitable for whole organ transplantation. Cells harvested from a single donor have the advantage of being used for multiple recipients in addition to being cryopreserved for use in emergency treatment of acute liver failure or for repeat treatment of liver-based metabolic disorders [22-24]. To date hepatocyte transplantation has only been used to address single gene defects and some metabolic disorders where the risks associated with OLT are not justified or are aimed at avoiding or postponing liver transplantation. The most successful clinical outcomes of OLT have been in patients with metabolic disorders including those with urea cycle defects [23, 24]; Crigler-Najjar syndrome type 1 (CN1) [25, 26]; glycogen storage disease type 1 [27] and Factor VII deficiency [28]. Although it was initially thought that transplantation of hepatocytes would be less immunogenic than whole organ transplantation, data from animal models demonstrated that the innate immune system clears a significant amount of hepatocyte irrespective of whether syngeneic or allogeneic cells are used [18]. As a result most transplant facilities use the same immunosupressive treatment as for liver transplantation [29]. Since the clinical outcome following hepatocyte transplantation is still highly variable due to a multitude of factors including fibrotic damage to the liver which limits the proper engraftment of cells; the availability of the cells required to provided sufficient liver function over repeated infusion per patient, and often the very marginal quality of the cells isolated from donor liver, a new and readily available cell source must be identified. This strong demand has fuelled the interest in haematopoietic stem cells for treating acute liver failure.
\n\t\tIt had previously been assumed that adult stem cells were lineage restricted, however the culmination of research over the past decade has now fully confirmed the remarkable developmental plasticity of human adult stem cells. The ability of adult bone marrow derived myogenic progenitors to participate in the regeneration of damaged skeletal muscle [30]; ischemic myocardium [31-33]; neurogenesis [34, 35] and the conversion of adult neural stem cells back into hematopoietic cells [36], redefined the biology of development. It was now accepted that the original three germs layers were not necessarily vital for transdifferentiation to occur [37]. This implied that adult stem cells could exhibit similar pluripotency as embryonic stem cells.
\n\t\tThe demonstration that adult bone marrow stem cells could differentiate towards a hepatic lineage was made almost a decade ago. Two independent research groups showed that the adult rat bone marrow contains a subpopulation of cells (about 3%) co-expressing the haematopoietic stem cell markers (CD34+, c-Kit, Thy-1); α-fetoprotein (AFP) and c-met. When they cultured this crude extract of bone marrow with hepatocyte growth factor (HGF) and epidermal growth factor (EGF), they were able to detect the expression of albumin, a marker of fully differentiated hepatocyte [38, 39]. Over the years further refinement of the in vitro culture conditions led to the differentiation of haematopoietic cells that expressed more mature liver specific transcription factors including human Hepatocyte nuclear factor-1α (HNF-1α); cytokeratin (CK8); CK19 and AFP [40, 41]. Studies on human adult bone marrow similarly confirmed their capability of differentiating into liver like cells [42, 43] where numerous cytokines and growth factors were shown to be important for driving this differentiation under in vitro conditions [44-46]. More interestingly was the discovery that exposing rat haematopoietic stem cells to injured liver tissues induced the expression of functional hepatocyte factors (CK18, albumin and transferrin) [47]. This suggested that the liver micro-environment alone was sufficient in providing the appropriate cues for inducing stem cell conversion towards the hepatic lineage. Several human post-mortem studies confirmed this observation where the presence of bone marrow derived cells were found in the liver. Theise et al., (2000) investigated archival autopsy and biopsy liver specimens from recipients of sex-mismatched therapeutic bone marrow transplantation and OLT [48]. By immunohistochemical staining for CK8, CK18 and CK19, they identified hepatocytes and cholangiocytes of bone marrow origin as well as the detection of the Y-chromosome by fluorescent in situ hybridization (FISH). Using double staining analysis, they found a large number of engrafted hepatocytes (4%-43%) and cholangiocytes (4%-38%) - that were bone-marrow derived- replenishing the hepatic parenchyma. Despite a general variability of this observation amongst parallel studies [49, 50], the ultimate conclusion was that bone marrow derived haematopoietic stem cells were able to successfully engraft into the liver and gradually adopt a phenotype very similar to the hepatic lineage.
\n\t\tToday, the most widely studied adult stem cells are derived from the hematopoietic lineage. Hematopoietic stem cells (HSCs) form part of the bone marrow compartment which constitutes a heterogenous population of mesenchymal stem cells; committed progenitor cells and non-circulating stromal cells. The identification of HSCs from the bone marrow compartment is largely based on the expression of cell surface markers. Detecting the cluster of differentiation, CD34 is generally used in human studies as a surrogate marker for progenitor stem cells although there also exists a subpopulation of more primitive HSCs which do not express CD34 [51, 52]. Despite the numerous studies that have shown bone marrow stem cells can give rise to hepatocytes, their use as a therapeutic agent is still in its infancy. Many studies are still at a pilot stage requiring randomisation and controls, however those reported show interesting results that require conformation. The first clinical study was performed by a German group, am Esch et al., (2005), where three patients were infused with autologous CD133+ cells subsequent to portal vein embolisation of right liver segments [3]. A computerised tomography (CT) scan for volume analysis of the left lateral segments showed a 2.5 fold increase in growth rate of the three patients when compared to the control group without bone marrow stem cell administration. Despite the small number of patients and the lack of adequately sized randomised control group, this data showcased the promising potential of bone marrow derived stem cells in enhancing liver regeneration. A year later, a parallel study was reported by Gordon et al., (2006) which included a phase I clinical trial on five patients with liver insufficiency [53]. These patients were given granulocyte colony-stimulating factor (G-CSF) to mobilise their stem cells for collection by leukapheresis followed by purification of CD34+ expressing cells which were then injected into either the portal vein or hepatic artery. Three of the five patients showed improvement in serum bilirubin whereas four out of the five patients displayed significant increase in serum albumin. Clinically, the procedure was well tolerated with no observed procedure-related complications and the data concluded that there was a marked contribution of the stem cells towards regeneration of the damaged liver [53]. In another independent study, Terai et al., (2006) enrolled nine patients with liver cirrhosis where they were injected with autologously derived bone marrow stem cells enriched for the expression of CD34+, CD45+ and c-kit+. These patients were followed for 24 weeks where they showed a significant improvement in serum albumin levels in addition to significantly improved Child-Pugh scores [54]. A year later, Rajkumar et al., (2007) reported another small scale trial where 22 patients with chronic liver disease (Child-Pugh scores B-C) were enrolled [55]. 200ml to 300ml of bone marrow aspirate were subjected to density gradient fractionation for isolation of CD34+ cells. These were then administered intravenously through the median cubital vein. Liver function tests; ultrasound and CT scans were performed before, 4 weeks and 8 weeks following cell infusion. 32% of the patients showed a drop in bilirubin levels; 67% of the patients showed an increase in serum albumin levels and 73% showed reduction in ascites. No patients from this study showed severe adverse effects following transfusion and the overall quality of life index was significantly improved in the majority (82%) [55]. In 2007, Mohamadnejad et al., [56, 57] performed two clinical studies in patients with decompensated liver cirrhosis. In the first study they treated four patients each with 31.73 x 106 of cultured autologous mesenchymal stem cells, infused through a peripheral vein. The Phase I study demonstrated no side effects and the quality of life of all four patients improved by the end of follow-up. Furthermore, the model for end-stage liver disease scores of two patients (patient 1 & patient 4) showed marked amelioration in their Child Pugh scores by the end of follow-up. In the second study, they treated four patients with autologous CD34+ cells isolated from the bone marrow, which was slowly infused through the hepatic artery of the patients. In two patients they observed mild albumin improvement, however, the health of two patients further deteriorated where one died of liver failure a few days after the transplantation. This trial was prematurely stopped due to the severity of the side effects where it was concluded that infusion of CD34+ stem cells through the hepatic artery was not safe in decompensated cirrhosis, but suggested that it may be beneficial to use alternative routes to transplant CD34+ cells. Having corroborated the use of bone marrow stem cells as a regenerative therapy and demonstrated their safety in patients with liver insufficiency, Pai et al.,(2008) conducted a prospective clinical efficacy study of expanded adult CD34+ stem cells infused into the hepatic artery in nine patients with alcoholic liver cirrhosis (ALC) to determine whether clinical benefit was conferred [58]. The primary end point was to assess the safety of infusing autologous stem cells into the hepatic artery of these patients; the secondary end point was to assess the improvement in liver function through serological and biochemical analysis; and to determine whether there were any symptomatic improvements. Following CD34+ stem cell infusion, CT scans showed normal enhancement in the liver parenchyma, with no evidence of focal liver lesions; additionally, duplex Doppler ultrasound scans showed normal flow in the portal veins and hepatic artery. To corroborate the safety and efficacy of improving liver function in patients with liver cirrhosis, Levicar et al., (2008) then reported the results from a long-term follow-up (12-18 months) of patients with chronic liver disease injected with CD34+ enriched stem cells [59]. During this time the patients were monitored for side effects, toxicity and changes in clinical, haematological and biochemical parameters. All the patients tolerated the treatment protocol without any complications or side effects related to the procedure. Four patients showed an initial improvement in serum bilirubin level, which was maintained for up to 6 months; whilst only a marginal increase in serum bilirubin was observed in three of the patients at 12 months. Only one patient showed an increase in serum bilirubin over slightly longer period of 18 months post-infusion. CT scans and serum AFP monitoring did not show any lesions or tumor formation in the patients. This successful study provided the basis for subsequent trials where more recently Li Nan et al., (2010) reported the clinical outcome of autologous CD34+ infusion in patients with hepatitis induced liver failure [60]. Twenty seven patients with Child- Pugh C cirrhosis were enrolled of which 22 were positive for Hepatitis B and 5 were positive for Hepatitis C. 50ml to 120ml of bone marrow aspirate were subjected to a Percoll gradient for the isolation of myeloid stem cells. 20 of the patients received cell infusion via the hepatic portal vein whilst 7 patients were infused via the hepatic artery using the Seldinger percutaneous technique. The patients were closely monitored for 3 months following transplantation where it was initially noticed that liver function tests decreased for 3 days. The authors attributed this with the contrast media used to perform arteriography. Since all the patients had Child- Pugh C cirrhosis, the contrast media may have aggravated liver injury. Despite this, the patients did subsequently show an improvement in liver function by 1 week of transplantation. The most significant recovery was seen at 3 months post procedure where total bilirubin and albumin levels were highest. Overall, an improved clinical outcome was observed in the majority of the patients where jaundice was resolved and ascites improved 3 months after therapy. From this cohort of patients, two died, one due to peritoneal cavity infection and liver failure, and the second from a massive hemorrhage of the upper alimentary tract. The latest clinical study on patients with end stage liver disease has been reported by Nikeghbalian et al., (2011). Six patients were intraportally injected with autologous bone marrow-derived CD133+ cells. This arm of patients were compared with patients subjected short term infusion (6months) and long term infusion (12 months) of mononuclear cells. Liver function test was positive at 24 months of follow up in all the patients enrolled in this trial. Since there were no differences between the groups receiving MNC or CD133+ cell, this recent study has highlighted the versatility of bone marrow-derived progenitor mononuclear stem cells irrespective of the subpopulation of their cluster of differentiation markers. Furthermore this study also confirms their safe use to circumvent the need for liver transplantation in end stage liver disease.
\n\t\t\tThe culmination of clinical trials thus far mentioned have shown a trend towards decreased serum fibrosis markers and improvements in bilirubin, albumin and Child-Pugh scores following stem cell infusion. The past decade has marked an important progress for regenerative liver therapy; however the accumulated data is still in its infancy. It is still not clear whether the route of infusion is important or whether mobilization of myeloid progenitor stem cells with G-CSF without leukapheresis is sufficient. Until more is learnt about the mechanisms by which HSCs contribute to hepatocyte regeneration, and the mechanism of clinical benefit in the recipient patients; controversy will inevitably shadow this form of treatment. Some authors have proposed that the observed conversion of infused stem cell to hepatocytes is simply a byproduct of cell fusion with no true induction of transdifferentiation [59, 60]. Others have suggested that instead of adopting a new lineage, the stem cells indirectly serve a regenerative capability by stimulating activation of tissue specific stem cells and by inducing the release of vascular endothelial growth factor (VEGF) thereby increasing the blood supply to the cells and aiding in repair of the damaged tissue [61, 62]. There are reports that also suggest HSCs may act in a regenerative capacity simply by inducing the expression of the B-Cell leukemia/lymphoma-2 gene (Bcl-2) and interleuking-6 (IL-6) in a paracrine manner thus suppressing apoptosis and inflammation of the surrounding tissue [62-64]. In addition to these uncertainties, the homing mechanism of infused adult stem cells to the liver also remains unclear. There are suggestions that chemokines similar to stroma derived factor 1 may be involved, however, this remains to be validated [65]. Although the application of stem cell therapy will not be prevented despite an incomplete understanding of the mechanisms involved, a clearer picture will ultimately enable better tailoring of stem cell therapy to the disease in question.
\n\t\tFor cell replacement therapy is still too early to demonstrate its long-term effectiveness or its improvement in survival rate and in the quality of life. This will undoubtedly have to be evaluated by more randomised control studies in the future. A standardised protocol will need to be established for the most efficient route of delivering infused HSCs; for the efficient long term culture of HSCs; for the optimal cell density and repeated transplantation required to re-establish liver function.
\n\t\tAutologous bone marrow infusion for patients with chronic liver disease has been shown to improve liver function parameters in contrast to observations accompanied by abstinence from alcohol. The degree of effectiveness of this therapy will likely to vary among different patients. Although the mechanisms underlining adult stem cell plasticity is still far from being fully characterised, the general enthusiasm regarding its potential clinical implication is heightened by the numerous clinical trials currently underway. In hepatology, the data presented here provides hope that human adult stem cells could eventually be used in tissue replacement protocols for the treatment of inherited and acquired end-stage liver diseases.
\n\t\tThe Kalman filter and the Kalman-Bucy filter [1, 2] solved the problem of optimal estimation of stochastic and deterministic linear systems. Since then, there is a continuing research on estimation of nonlinear systems.
There are many different approaches for the state reconstruction, estimation, and filtering of nonlinear systems, for a recent review, see [3, 4] and the references within. The space in this chapter is too short to cover them. These approaches can be classified roughly into two types: the stochastic approach and the statistical-deterministic-likelihood approach.
The stochastic approach is based on the Itȏ calculus and computation of the conditional probabilities by the Kolmogorov’s forward/Fokker-Plank equation or Zakai’s equation that are difficult to solve and usually need numerical solution, e.g., see a numerical approach to the filtering problem for a class of nonlinear time-varying systems [5]. The innovations approach to the nonlinear estimation in a white noise is presented in [6]. However, explicit result for a specific nonlinear system is difficult to arrive at. Thus, when a closed-form estimator is sought, the stochastic approach leads, in general, to suboptimal and approximate solutions. The exceptions are [7, 8], where some restricted cases for which closed-form solutions of the optimal filtering equations of continuous systems are presented. Moreover, it was shown that generally the stochastic approach leads to infinite dimensional solution of the optimal estimator [9]. Different classes of nonlinear systems for which there is a closed-form explicit solution are presented in [10, 11] for the nonlinear problem of estimating the parameters of linear system with unknown coefficients. These belong to the specific class of nonlinear systems for which a general solution is presented in [12], Chapter 10.
The Kalman filter [1, 2] was obtained as well by solving the dual of the linear quadratic control problem criterion [13, 14, 15] by calculus of variations within the framework using the statistical-deterministic-likelihood approach. The dual of the LQ criterion is the least squares (LS) criterion also called the mean squares error (MSE) criterion, or joint maximum likelihood (JML) criterion [15, 16, 17], or just maximum likelihood (ML). The statistical-deterministic-likelihood approach has been used to derive filters of linear systems [13, 15]. For linear system, this approach leads to the structure of the Kalman and Kalman-Bucy filters. This shows that the Kalman and Kalman-Bucy filters are not only optimal estimators on the average but also optimal estimators for a single sample. Within the likelihood approach [18], the noises are white and the criterion is the likelihood functional [15]. The deterministic variational approach has been applied in [18] to nonlinear system. Within the statistical-deterministic-likelihood approach [13, 19], the input disturbance and output measurement error are considered as disturbances with unknown statistics ([20], p. 361). This approach is based on the calculus of variations [13] and has been widely used for numerical implicit computations of estimates and smoothers for nonlinear dynamic systems [21].
Thus, the statistical-deterministic-likelihood approach is most tempting for application in developing filters of nonlinear systems [18]. Mortensen [18] derives the general structure of the optimal recursive estimator’s state propagation equation derived from the likelihood approach point of view. This solution has the structure of the state propagation equation of the extended Kalman filter (EKF) thus justifying its usage beyond the heuristic of usage as the first-order Taylor series expansion. However, Mortensen [18] does not derive the respective equation of the gain. Moreover, Mortensen [18] states that the computation of the gain “…suffers from the same kind moment problem or closure problem as does the minimum variance nonlinear filtering.” This means that the derived estimation error gain is not feasible. The solution in this chapter shows that the statistical-deterministic-likelihood approach based on the calculus of variations leads to a solution that is not plagued with the closure problem.
The most popular estimation filter of nonlinear systems is the EKF. The EKF uses the Jacobian fx of the system’s differential equations function
An additional estimation filter of nonlinear systems that has been developed in the recent years with success is the state-dependent differential/difference Riccati equation (SDRE/SDDRE)-based filter of nonlinear system [22, 23, 24, 25]. This has been enabled by the introduction of the state-dependent coefficient (SDC) form [23, 24] approach to filtering. The SDC form represents the nonlinear equation in the quasilinear form
Since Mortensen’s derivation [18], no progress has been made [4, 34, 35] in explicitly solving the optimal nonlinear filtering problem till [16, 17, 36, 37, 38] for continuous nonlinear systems and [39] for discrete nonlinear systems.
This chapter combines: (i) the LS criterion based on the statistical-deterministic-likelihood approach to estimation; (ii) the SDC form representation of the nonlinear system; and (iii) the calculus of variations; for derivation of a recursive filter in the form of a differential equation as the filter-estimator for nonlinear systems with nonlinear dynamics and nonlinear measurement.
This chapter is based on the preliminary publication [16]. The results for nonlinear time-varying system are presented in [17], for system with input in [37] and for the H∞ criterion in [38].
The presented approach leads to an optimal, exact, explicit, closed-form, and recursive solution, where state propagation equation is as derived in [18] (and is that as of the EKF). This filter is called here the recursive nonlinear least squares (RNLS) filter. The optimal gain is computed via the solution of a nonsymmetric differential matrix Riccati equation (NDMRE) that uses the respective Jacobians and the SDC form representation.
The importance and novelty of the result in this chapter are:
An optimal, exact, explicit, closed-form, and recursive solution to the estimation of nonlinear time-varying systems based on the quadratic least-squares criterion is presented.
The fact that the optimal filter of nonlinear systems can be derived by calculus of variations is highlighted.
The optimal filter can be taught to students that are familiar with calculus of variations before mastering stochastic calculus.
The RNLS-based filter, the EKF, and the SDDRE-based filter were compared on a common basis in [36, 40].
In the chapter, derivation of the result is presented. The performances of the RNLS-based filter are demonstrated with the Van der Pol differential equation driven by a band-limited noise, and the nonlinear measurement is noise corrupted.
A general nonlinear system is dealt with. Let the reality be represented by:
where ζ(t) is the real state (unknown and of unknown dimension), y(t) is the measured output, υ(t) is the measurement noise, ω(t) is the system driving noise, and the functions φ and η represent the reality. The functions φ and η that describe the real system cannot be either precisely represented or are unknown precisely up to the last detail (e.g., the output measurement function may include some measurement noise or themselves exhibit random uncertain behavior). For the design of the observer, we use the representation model given by:
where x(t) ∈
The problem: Derive a recursive estimator (in form of a differential equation) for the state of the model, x(t), from the output measurements.
The continuous least square criterion is used [13, 14, 15] in the evaluation of the optimal estimator of linear systems. The covariance constraint and the minimum model error concepts [21] rationalize this approach as well.
The continuous least squares criterion is the dual of the LQ criterion for the control problem.
The objective is ([15], Eq. 5.24)
where Q is an a priori estimate of the driving force errors, w(t), Q ∈
We wish to minimize the continuous least squares error objective (3) with respect to w(τ), to ≤ τ ≤ t subject to the model (2) in order to find an estimate of x(τ). That is, we are looking for the representation-realization of the difference between the reality and the model, w(t), that best fits, the observations. In other words and roughly speaking, “we want to pass the solution to Eq. (3) as closely as possible, through the observations.” The presented approach also constitutes the statistical methods approach to filtering ([15], Section 5.3).
The problem above is solvable by a batch solution [21] that will minimize the objective (3). Here, we look for a recursive solution in the form of differential equations.
Throughout the chapter, it is assumed that all functions satisfy the necessary boundedness, smoothness, and differentiability conditions for existence of solution.
In this chapter, we deal with a specific structure of the model of the nonlinear system (2). It is assumed that:
I. Eq. (2) is partitioned as (with a slight abuse of notation):
II. At the origin, we have
Then, by defining the state-dependent coefficient form (SDC) [23] as:
The dynamic equations of the system (4) are written as
where F ∈
In this section, the main result is derived for the specific structure of the nonlinear system (7), i.e., nonlinear dynamics, f(x(t)), nonlinear measurement, m(x(t)), that are represented in the SDC form given in Eq. (6), and the quadratic criterion
that is minimized with respect to, w(t), subject to Eq. (7). Calculus of variations is applied in derivation of the main result for nonlinear systems (7). The Hamiltonian is
where λ(t) is the costate.
The necessary conditions for optimality ([15], Example 7.11) are
This gives
This leads to the nonlinear two-point boundary value problem (TPBVP) for
The system’s dynamic equation (Eq. (4)) in the SDC form is Eq. (7). Thus, the optimal solution is given by the TPBVP.
The usage of the SDC form converts the nonlinear TPBVP (Eq. (12)) to a time-varying TPBVP (Eq. (13)) thus enables a causal solution. This is as up to the current time, as the solution propagates forward in time,
By setting
The solution of the nonhomogeneous time-varying TPBVP (Eqs. (13) and (14)) is hinted by the necessary condition
The solution in the form of differential equations, the continuous recursive nonlinear least squares (RNLS) filter, is given by:
where the filter’s gain is
and
where
Notice:
The first term of the right-hand side of Eq. (19) includes the SDC form and the second term includes the Jacobian and same is in the last term. The SDC and the Jacobian are equal for linear systems only.
The solution requires the solution of the nonsymmetric differential matrix Riccati equation (Eq. (19)) and the solution, P, is nonsymmetric.
The solution of the nonsymmetric Riccati matrix equation depends on the estimated state
Notice that the state propagation Eq. (19) has exactly the same structure as derived by Mortensen [18] and used by the EKF. The solution of Eqs. (18) and (19) gives explicitly the filter’s gain.
In [18], it is claimed that computation of the filter’s optimal gain, P, (Eqs. (18) and (19)) suffers from “…the moment or closure problem…”. In this chapter, it is shown that the filter’s optimal gain is solved completely and explicitly by the NDMRE (Eq. (19)).
In order to enable better understanding of Eq. (17–19), the following presents Eq. (17–19) by suppressing the explicit and implicit dependence on time.2 The optimal filter is
or
One can clearly see that for linear system, Eq. (20–22) gets the structure of the Kalman filter as then
The deterministic stability of the RNLS estimator-filter along the filter’s trajectories Eq. (20–22) is considered. Recall that optimality does not guarantee stability. The stability of the RNLS filter is connected to the stability of the NDMRE equation. Let us consider the system/observer:
where the explicit and implicit time dependency is suppressed as in the previous section. Eqs. (24–26) are actually the deterministic SDDRE-based observer of Eq. (7) whose stability is treated in [31, 32, 33]. The matrix Riccati equation (Eq. (26)) is symmetric.
First, existing result on the stability of optimal estimators of system Eqs. (24–26) as a linear time-varying system is cited. The following result is valid for linear time-invariant and time-variant systems. [31, 32, 41] Consider the symmetric Riccati equation (Eq. (26)) where
A Lyapunov function for the autonomous system Eqs. (24–26) (i.e. y = 0) is
For which
where
Next, the NDMRE equation is considered. It is dealt with in [42, 43, 44, 45, 46]. The only reference that is directly addressing the stability issue of an NDMRE is [42] (Chapter 9). The Riccati equation related to the time-invariant control problem is dealt with in [42] (Theorem 9.1.23 and Remark 9.1.24). Although not explicitly stated, these results apply as well to time-varying systems. Motivated by this theorem and remark, translated by duality to the estimation problem, the following conjecture is formulated. Consider the nonsymmetric differential Riccati matrix equation.
where
This conjecture is supported by [42] (Theorem 9.1.23 and Remark 9.1.24). The requirement of detectability (observability) and stabilizability (controllability) is not explicitly required in [42] (supposedly they appear implicitly). This conjecture means that the filter given by Eqs. (20–22) is stable. An issue under research is (loosely): in addition to the conditions in Conjecture 1, the boundedness conditions of all matrices and variables (the output and system driving noise and measurement noise) are sufficient conditions for this stability, as for the SDDRE-based filter [31, 32, 33]?
Notice that for the symmetric case, this well-known result for linear system results in Theorem 1.
The stability of the RNLS filter is investigated via Lyapunov analysis. As the solution of the nonsymmetric Riccati equation in Eq. (19) is eventually not symmetric, the following symmetric Lyapunov function is dealt with here:
The derivative of the Lyapunov function is [47]
For linear system,
The first terms in Eq. (31) are potentially nonnegative definite
The second term in Eq. (31) is negative (nonpositive) definite
The next two terms in Eq. (31) are indefinite and can be negative
The last two terms in Eq. (31)
are indefinite.
The discussion above hints that for small nonsymmetry, for sure, the NDMRE stabilizes the RNLS filter. The stability of the RNLS filter is summarized in the following conjecture. Further results are beyond the scope of this chapter. If The nonlinearities are such that The observability and controllability conditions are satisfied along the filter trajectories, then the RNLS filter is asymptotically stable.
Remark: Simulation results show/hint that as long as the incremental observability matrices
and the incremental controllability matrices
along the estimator’s trajectory of the RNLS filter are nonsingular, i.e.,
then: (i) the estimation errors of the filter for the deterministic case, i.e., w(t) = 0 and v(t) = 0, converge to zero; and (ii) for the case with bounded disturbance and bounded measurement noise, the estimation errors are bounded, i.e., do not diverge.
This section demonstrates the performance of the RNLS-based estimator on a generalized nonlinear time-varying Van der Pol differential equation driven by band-limited noise and noise-corrupted nonlinear measurement. The state is
That can be put in matrix form as:
The noisy measurement is
Then, we have
The SDC form system matrix is selected as:
and the Jacobian is
The observability matrices are
and controllability matrices are
The observability and controllability matrices have full rank for all bounded trajectories.
The system and the RNLS estimator were implemented in SIIMULINK® with the following parameters:
Ts = 0.1 msec | Sampling interval |
μ = 1 | Mass |
c = 0.01 | Damping coefficient |
k = 0.1 | Spring stiffness |
R = 1e-5 [1/Hz] | Spectral density of the measurement noise—v |
Q = 1e0 [(1/sec2)2/Hz] | Spectral density of the system driving noise—w |
Po = [0.001 0; 0 0.001] | Initial condition of the P matrix |
x(to) = [2 0]T | Initial conditions of the state |
The measurement noise and system driving noises are white in 100 [rad/sec] bandwidth.
The following figures present the performances of the RNLS filter. Figure 1 presents the measured output—y and the estimated output versus time. Figure 2 presents the real (true) position—ξ and the estimated position—
The measured output—y and the estimated output versus time.
The real position—x and the estimated position state—x̂ versus time.
The real velocity—ẋ and the estimated velocity—ẋ̂, versus time.
Filter’s gains, K1 gain of the position state, K2 gain of the velocity state, versus time.
The terms of the solution of the Riccati equation—P matrix, versus time.
Figure 6 presents the phase plane plot of the velocity estimation error versus the position estimation errors. One can see that following the initial transient, the estimation errors concentrate around the origin.
Phase plane plot of velocity versus position estimation errors.
The mean least square error criterion has been used to derive the optimal estimator for continuous nonlinear systems with nonlinear dynamics and nonlinear measurement. The solution is exact, explicit, in closed form, and in recursive form. Simulation example demonstrates the performance.
This research was partially supported by the Prof. Pazy Research Foundation.
IntechOpen publishes different types of publications
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