A Parallel Riccati Factorization Algorithm with Applications to Model Predictive Control

Model Predictive Control (MPC) is increasing in popularity in industry as more efficient algorithms for solving the related optimization problem are developed. The main computational bottle-neck in on-line MPC is often the computation of the search step direction, i.e. the Newton step, which is often done using generic sparsity exploiting algorithms or Riccati recursions. However, as parallel hardware is becoming increasingly popular the demand for efficient parallel algorithms for solving the Newton step is increasing. In this paper a tailored, non-iterative parallel algorithm for computing the Riccati factorization is presented. The algorithm exploits the special structure in the MPC problem, and when sufficiently many processing units are available, the complexity of the algorithm scales logarithmically in the prediction horizon. Computing the Newton step is the main computational bottle-neck in many MPC algorithms and the algorithm can significantly reduce the computation cost for popular state-of-the-art MPC algorithms

Reduced Memory Footprint in Multiparametric Quadratic Programming by Exploiting Low Rank Structure

In multiparametric programming an optimization problem which is dependent on a parameter vector is solved parametrically. In control, multiparametric quadratic programming (mp-QP) problems have become increasingly important since the optimization problem arising in Model Predictive Control (MPC) can be cast as an mp-QP problem, which is referred to as explicit MPC. One of the main limitations with mp-QP and explicit MPC is the amount of memory required to store the parametric solution and the critical regions. In this paper, a method for exploiting low rank structure in the parametric solution of an mp-QP problem in order to reduce the required memory is introduced. The method is based on ideas similar to what is done to exploit low rank modifications in generic QP solvers, but is here applied to mp-QP problems to save memory. The proposed method has been evaluated experimentally, and for some examples of relevant problems the relative memory reduction is an order of magnitude compared to storing the full parametric solution and critical regions

Low-Rank Modifications of Riccati Factorizations for Model Predictive Control

In Model Predictive Control (MPC) the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem at each sample in the control loop. The main computational effort is often spent on computing the search directions, which in MPC corresponds to solving unconstrained finite-time optimal control (UFTOC) problems. This is commonly performed using Riccati recursions or generic sparsity exploiting algorithms. In this work the focus is efficient search direction computations for active-set (AS) type methods. The system of equations to be solved at each AS iteration is changed only by a low-rank modification of the previous one, and exploiting this structured change is important for the performance of AS type solvers. In this paper, theory for how to exploit these low-rank changes by modifying the Riccati factorization between AS iterations in a structured way is presented. A numerical evaluation of the proposed algorithm shows that the computation time can be significantly reduced by modifying, instead of re-computing, the Riccati factorization. This speed-up can be important for AS type solvers used for linear, nonlinear and hybrid MPC

Discrete adipose-derived stem cell subpopulations may display differential functionality after in vitro expansion despite convergence to a common phenotype distribution

BACKGROUND: Complex immunophenotypic repertoires defining discrete adipose-derived stem cell (ASC) subpopulations may hold a key toward identifying predictors of clinical utility. To this end, we sorted out of the freshly established ASCs four subpopulations (SPs) according to a specific pattern of co-expression of six surface markers, the CD34, CD73, CD90, CD105, CD146, and CD271, using polychromatic flow cytometry. METHOD: Using flow cytometry-associated cell sorting and analysis, gating parameters were set to select for a CD73(+)CD90(+)CD105(+) phenotype plus one of the four following combinations, CD34(−)CD146(−)CD271(−) (SP1), CD34(−)CD146(+)CD271(−) (SP2), CD34(+)CD146(+)CD271(−) (SP3), and CD34(−)CD146(+)CD271(+) (SP4). The SPs were expanded 700- to 1000-fold, and their surface repertoire, trilineage differentiation, and clonogenic potential, and the capacity to support wound healing were assayed. RESULTS: Upon culturing, the co-expression of major epitopes, the CD73, CD90, and CD105 was maintained, while regarding the minor markers, all SPs reverted to resemble the pre-sorted population with CD34(−)CD146(−)CD271(−) and CD34(−)CD146(+)CD271(−) representing the most prevalent combinations, followed by less frequent CD34(+)CD146(−)CD271(−) and CD34(+)CD146(+)CD271(−) variants. There was no difference in the efficiency of adipo-, osteo-, or chondrogenesis by cytochemistry and real-time RT-PCR or the CFU capacity between the individual SPs, however, the SP2(CD73+90+105+34-146+271-) outperformed others in terms of wound healing. CONCLUSIONS: Our study shows that ASCs upon culturing inherently maintain a stable distribution of immunophenotype variants, which may potentially disguise specific functional properties of particular downstream lines. Furthermore, the outlined approach suggests a paradigm whereby discrete subpopulations could be identified to provide for a therapeutically most relevant cell product. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s13287-016-0435-8) contains supplementary material, which is available to authorized users

On Structure Exploiting Numerical Algorithms for Model Predictive Control

One of the most common advanced control strategies used in industry today is Model Predictive Control (MPC), and some reasons for its success are that it can handle multivariable systems and constraints on states and control inputs in a structured way. At each time-step in the MPC control loop the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem on-line. There exist several optimization methods to solve the CFTOC problem, where two common types are interior-point (IP) methods and active-set (AS) methods. In both these types of methods, the main computational effort is known to be the computation of the search directions, which boils down to solving a sequence of Newton-system-like equations. These systems of equations correspond to unconstrained finite-time optimal control (UFTOC) problems. Hence, high-performance IP and AS methods for CFTOC problems rely on efficient algorithms for solving the UFTOC problems. The solution to a UFTOC problem is computed by solving the corresponding Karush-Kuhn-Tucker (KKT) system, which is often done using generic sparsity exploiting algorithms or Riccati recursions. When an AS method is used to compute the solution to the CFTOC problem, the system of equations that is solved to obtain the solution to a UFTOC problem is only changed by a low-rank modification of the system of equations in the previous iteration. This structured change is often exploited in AS methods to improve performance in terms of computation time. Traditionally, this has not been possible to exploit when Riccati recursions are used to solve the UFTOC problems, but in this thesis, an algorithm for performing low-rank modifications of the Riccati recursion is presented. In recent years, parallel hardware has become more commonly available, and the use of parallel algorithms for solving the CFTOC problem and the underlying UFTOC problem has increased. Some existing parallel algorithms for computing the solution to this type of problems obtain the solution iteratively, and these methods may require many iterations to converge. Some other parallel algorithms compute the solution directly (non-iteratively) by solving parts of the system of equations in parallel, followed by a serial solution of a dense system of equations without the sparse structure of the MPC problem. In this thesis, two parallel algorithms that compute the solution directly (non-iteratively) in parallel are presented. These algorithms can be used in both IP and AS methods, and they exploit the sparse structure of the MPC problem such that no dense system of equations needs to be solved serially. Furthermore, one of the proposed parallel algorithms exploits the special structure of the MPC problem even in the parallel computations, which improves performance in terms of computation time even more. By using these algorithms, it is possible to obtain logarithmic complexity growth in the prediction horizon length

Modellering och Reglering av Friction Stir Welding i 5 cm tjocka Kopparkapslar

Friction stir welding has become a popular forging technique used in many applications. The Swedish Nuclear Fuel and Waste Management Company (SKB) evaluates this method to seal the 5 cm thick copper canisters that will contain the spent nuclear fuel. To produce repetitive, high quality welds, the process must be controlled, and today a cascade controller is used to keep the desired stir zone temperature. In this thesis, the control system is extended to also include a plunge depth controller. Two different approaches are evaluated; the first attempt is a decentralized solution where the cascaded temperature controller is kept, and the second approach uses a non-linear model predictive controller for both depth and temperature. Suitable models have been derived and used to design the controllers; a simpler model for the decentralized control and a more extensive, full model used in the non-linear model predictive controller that relates all the important process variables. The two controller designs are compared according to important performance measures, and the achieved increase in performance with the more complex non-linear model predictive controller is evaluated. The non-linear model predictive controller has not been implemented on the real process. Hence, simulations of the closed loop systems using the full model have been used to compare and evaluate the control strategies. The decentralized controller has been implemented on the real system. Two welds have been made using plunge depth control with excellent experimental results, confirming that the decentralized controller design proposed in this thesis can be successfully used. Even though the controller manages to regulate the plunge depth with satisfying performance, simulations indicate that the non-linear model predictive controller achieves even better closed loop performance. This controller manages to compensate for the cross-connections between the process variables, and the resulting closed loop system is almost decoupled. Further research will reveal which control design that will finally be used.''Friction stir welding'' har blivit en populär svetsmetod inom många olika tillämpningar. På Svensk Kärnbränslehantering AB (SKB) undersöks möjligheten att använda metoden för att försegla de 5 cm tjocka kopparkapslarna som kommer innehålla det använda kärnbränslet. För att kunna producera repeterbara svetsar utav hög kvalité krävs det att processen regleras. Idag löses detta med en temperaturregulator som reglerar svetszonens temperatur. I detta examensarbete utökas styrsystemet med en regulator för svetsdjupet. Två olika lösningar har utvärderats; först en decentraliserad lösning där temperatur-regulatorn behålls och sedan en lösning med en olinjär modellprediktiv reglering (MPC) som reglerar både djup och temperatur. Passande modeller har tagits fram och har använts för att designa regulatorerna; en enklare modell för den decentraliserade regulatorn och en utökad, komplett modell som används i den olinjära MPC:n och som beskriver alla viktiga variabler i processen. Viktiga prestandamått har jämförts för de båda regulatorstrukturerna och även prestandaökningen med den olinjära MPC:n har utvärderats. Då denna regulator inte har implementerats på den verkliga processen har simuleringar av den kompletta modellen använts för att jämföra och utvärdera regulatorstrukturerna. Den decentraliserade regulatorn har implementerats och testats på processen. Två svetsar har gjorts och de har givit utmärkta resultat, vilket visar att regulatorstrukturen som presenteras i rapporten fungerar bra för reglering av svetsdjupet. Trots att den implementerade regulatorn klarar av att reglera svetsdjupet med godkänt resultat, så visar simuleringar att den olinjära MPC:n ger ännu bättre reglerprestanda. Denna regulator kompenserar för korskopplingar i systemet och resulterar i ett slutet system som är nästan helt frikopplat. Ytterligare forskning kommer avgöra vilken av strategierna som kommer att användas i slutprodukten

Structure-Exploiting Numerical Algorithms for Optimal Control

Numerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (MPC), but can also be useful for advanced estimation techniques, such as moving horizon estimation (MHE). In MPC, the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem on-line, and in MHE the estimated states are obtained by solving an optimization problem that often can be formulated as a CFTOC problem. Common types of optimization methods for solving CFTOC problems are interior-point (IP) methods, sequential quadratic programming (SQP) methods and active-set (AS) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (UFTOC) problems. Hence, high-performing second-order methods for CFTOC problems rely on efficient numerical algorithms for solving UFTOC problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a CFTOC problem is computed using an AS type method, the aforementioned system of equations is only changed by a low-rank modification between two AS iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the UFTOC problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in IP, SQP and AS methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the UFTOC problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear MPC problems, an alternative approach to solving the CFTOC problem on-line is to use multiparametric quadratic programming (mp-QP), where the corresponding CFTOC problem can be solved explicitly off-line. This is referred to as explicit MPC. One of the main limitations with mp-QP is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-QP and explicit MPC more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the QP problem in the parametric solution in order to reduce the memory footprint of general mp-QP solutions, and in particular, of explicit MPC solutions. The algorithm can be used directly in mp-QP solvers, or as a post-processing step to an existing solution.Numeriska algoritmer för att effektivt lösa optimala styrningsproblem är en viktig komponent i avancerade regler- och estimeringsstrategier som exempelvis modellprediktiv reglering (eng. model predictive control (MPC)) och glidande horisont estimering (eng. moving horizon estimation (MHE)). MPC är en reglerstrategi som kan användas för att styra system med flera styrsignaler och/eller utsignaler samt ta hänsyn till exempelvis begränsningar i styrdon. Den grundläggande principen för MPC och MHE är att styrsignalen och de estimerade variablerna kan beräknas genom att lösa ett optimalt styrningsproblem. Detta optimeringsproblem måste lösas inom en kort tidsram varje gång som en styrsignal ska beräknas eller som variabler ska estimeras, och således är det viktigt att det finns effektiva algoritmer för att lösa denna typ av problem. Två vanliga sådana är inrepunkts-metoder (eng. interior-point (IP)) och aktivmängd-metoder (eng. active-set (AS)), där optimeringsproblemet löses genom att lösa ett antal enklare delproblem. Ett av huvudfokusen i denna avhandling är att beräkna lösningen till dessa delproblem på ett tidseffektivt sätt genom att utnyttja strukturen i delproblemen. Lösningen till ett delproblem beräknas genom att lösa ett linjärt ekvationssystem. Detta ekvationssystem kan man exempelvis lösa med generella metoder eller med så kallade Riccatirekursioner som utnyttjar strukturen i problemet. När man använder en AS-metod för att lösa MPC-problemet så görs endast små strukturerade ändringar av ekvationssystemet mellan varje delproblem, vilket inte har utnyttjats tidigare tillsammans med Riccatirekursionen. I denna avhandling presenteras ett sätt att utnyttja detta genom att bara göra små förändringar av Riccatirekursionen för att minska beräkningstiden för att lösa delproblemet. Idag har behovet av  parallella algoritmer för att lösa MPC och MHE problem ökat. Att algoritmerna är parallella innebär att beräkningar kan ske på olika delar av problemet samtidigt med syftet att minska den totala verkliga beräkningstiden för att lösa optimeringsproblemet. I denna avhandling presenteras parallella algoritmer som kan användas i både IP- och AS-metoder. Algoritmerna beräknar lösningen till delproblemen parallellt med ett förutbestämt antal steg, till skillnad från många andra parallella algoritmer där ett okänt (ofta stort) antal steg krävs. De parallella algoritmerna utnyttjar problemstrukturen för att lösa delproblemen effektivt, och en av dem har utvärderats på parallell hårdvara. Linjära MPC problem kan också lösas genom att utnyttja teori från multiparametrisk kvadratisk programmering (eng. multiparametric quadratic programming (mp-QP)) där den optimala lösningen beräknas i förhand och lagras i en tabell, vilket benämns explicit MPC. I detta fall behöver inte MPC problemet lösas varje gång en styrsignal beräknas, utan istället kan den förberäknade optimala styrsignalen slås upp. En nackdel med mp-QP är att det krävs mycket plats i minnet för att spara lösningen. I denna avhandling presenteras en strukturutnyttjande algoritm som kan minska behovet av minne för att spara lösningen, vilket kan öka det praktiska användningsområdet för mp-QP och explicit MPC

Modellering och Reglering av Friction Stir Welding i 5 cm tjocka Kopparkapslar

Friction stir welding has become a popular forging technique used in many applications. The Swedish Nuclear Fuel and Waste Management Company (SKB) evaluates this method to seal the 5 cm thick copper canisters that will contain the spent nuclear fuel. To produce repetitive, high quality welds, the process must be controlled, and today a cascade controller is used to keep the desired stir zone temperature. In this thesis, the control system is extended to also include a plunge depth controller. Two different approaches are evaluated; the first attempt is a decentralized solution where the cascaded temperature controller is kept, and the second approach uses a non-linear model predictive controller for both depth and temperature. Suitable models have been derived and used to design the controllers; a simpler model for the decentralized control and a more extensive, full model used in the non-linear model predictive controller that relates all the important process variables. The two controller designs are compared according to important performance measures, and the achieved increase in performance with the more complex non-linear model predictive controller is evaluated. The non-linear model predictive controller has not been implemented on the real process. Hence, simulations of the closed loop systems using the full model have been used to compare and evaluate the control strategies. The decentralized controller has been implemented on the real system. Two welds have been made using plunge depth control with excellent experimental results, confirming that the decentralized controller design proposed in this thesis can be successfully used. Even though the controller manages to regulate the plunge depth with satisfying performance, simulations indicate that the non-linear model predictive controller achieves even better closed loop performance. This controller manages to compensate for the cross-connections between the process variables, and the resulting closed loop system is almost decoupled. Further research will reveal which control design that will finally be used.''Friction stir welding'' har blivit en populär svetsmetod inom många olika tillämpningar. På Svensk Kärnbränslehantering AB (SKB) undersöks möjligheten att använda metoden för att försegla de 5 cm tjocka kopparkapslarna som kommer innehålla det använda kärnbränslet. För att kunna producera repeterbara svetsar utav hög kvalité krävs det att processen regleras. Idag löses detta med en temperaturregulator som reglerar svetszonens temperatur. I detta examensarbete utökas styrsystemet med en regulator för svetsdjupet. Två olika lösningar har utvärderats; först en decentraliserad lösning där temperatur-regulatorn behålls och sedan en lösning med en olinjär modellprediktiv reglering (MPC) som reglerar både djup och temperatur. Passande modeller har tagits fram och har använts för att designa regulatorerna; en enklare modell för den decentraliserade regulatorn och en utökad, komplett modell som används i den olinjära MPC:n och som beskriver alla viktiga variabler i processen. Viktiga prestandamått har jämförts för de båda regulatorstrukturerna och även prestandaökningen med den olinjära MPC:n har utvärderats. Då denna regulator inte har implementerats på den verkliga processen har simuleringar av den kompletta modellen använts för att jämföra och utvärdera regulatorstrukturerna. Den decentraliserade regulatorn har implementerats och testats på processen. Två svetsar har gjorts och de har givit utmärkta resultat, vilket visar att regulatorstrukturen som presenteras i rapporten fungerar bra för reglering av svetsdjupet. Trots att den implementerade regulatorn klarar av att reglera svetsdjupet med godkänt resultat, så visar simuleringar att den olinjära MPC:n ger ännu bättre reglerprestanda. Denna regulator kompenserar för korskopplingar i systemet och resulterar i ett slutet system som är nästan helt frikopplat. Ytterligare forskning kommer avgöra vilken av strategierna som kommer att användas i slutprodukten