Lutz presents Hollywood. Lutz Radio Theatre brings you Tyrone Power and Annabella in the Rage of Manhattan. Ladies and gentlemen, your producer, Mr. Cecil B. DeMille. Greetings from Hollywood, ladies and gentlemen. To the American theatre in every generation, a benign fortune gives at least one Tyrone Power. That's practically an established tradition, established by a century of the theatre and three Tyrone Powers. I saw the second Tyrone Power many times on the stage and screen both. In fact, Mrs. DeMille and I were in the same stock company with him in Denver 35 years ago. But the first Tyrone Power was playing in New York a little more than 100 years ago. Perhaps I'd better reassure you that I did not see him. Tonight, Tyrone Power III stars for the first time in the Lutz Radio Theatre. The play is The Rage of Manhattan. And that title refers to our co-star of the evening, the lovely and charming Annabella, who has captured the screen audiences of two continents. In The Rage of Manhattan, she plays one of those very dangerous characters, a young lady in search of a husband. The young lady happens to be stranded in a strange country, but that's no handicap at all because she also happens to be beautiful. But there's a young man who's determined that she will not find a husband. And of course, that's Tyrone Power, who comes to us this week from 20th Century Fox, where everybody is very happy over his recent hit in the squash-buckling Mark of Zorro. It's a joy to present two stars like Tyrone Power and Annabella in our play tonight. And you know, it's rather staggering, even to an old time I like myself, to realize that literally millions and millions of you are listening to this play.  How many millions of cakes every week. And we intend to keep on deserving your faith by giving you the best possible in plays and the best possible in soap, which of course is luxe-toilet soap. And now try our prescription for your enjoyment tonight, as we raise the curtain on the first act of The Rage of Manhattan, starring Tyrone Power as Jim Trevor and Annabella as Nicole. Manhattan skyscraper, 83 floors above the street, are the offices of the Tower Model Agency. It's a busy morning and the reception room is lined with most of the city's famous glamour girls. Those young ladies of beauty and charm who smile so provokingly from the pages of our leading magazines. In the doorway stands a pretty but obviously inexperienced young girl. She glances timidly about the room and then summoning up all her courage approaches the secretary. Good morning, please. What? I said good morning, please. You're welcome. What do you want? How can I have a job here to model? Are you registered? Registered. Oh, yes, I am registered. Did you get a call from Mr. Wright's office? Well, I... If you didn't, there's nothing for you. Sorry. But I did. He called me. Mr. Wright. Well, then go in, please. That door over there. Thank you very much. Hello. Hello, Miss Ingram. Get me Mr. James Trevor's address, please, right away. Yes, sir. Hello. Hello. Did I call you? Oh, yes, sir. What for? I don't know to be a model. Say, you've got an accent. Yes, I can't help it, but it couldn't show any picture, no? Well, no, that's right. Sit down. Now, let's see. This job is... Oh, here's that address, Mr. Wright. James Trevor. Okay, thank you. Leave it right there. Now, young lady, this man is a first-class photographer. You pose with drapes. Pardon? Yes, like on windows. Oh, drapes. And what else? Nothing else. You mean you want me to pose and have my picture taken just with a few drapes? Who do you think I am? You don't want to do it? Me? No, I never do it. Okay, excuse me. You, that girl over there. Yes, sir? I have a job for you to pose and drape. How much? $2.75 an hour. Well... Wait out here and I'll get you the address. Mr. Wright. Are you still here? Mr. Wright, I changed my mind. I will take the job. Sorry, I just gave it away. Oh, but I needed job so bad. Nothing else today, Miss? Come back next week. Yes, sir. Thank you, sir. Oh, where in heaven's name is that address? Miss Ingram. Yes, sir? I thought I asked you for James Traver's address. Well, I gave it to you, sir. Oh, where is it? It's on your desk. It is not on my desk. But I put it there. Well, it's not there now. You didn't give it to that girl, did you? What? What are you talking about? Well, that girl, it was just in there. She had a slip of paper in her hand when she came out. Miss Ingram, since when has Mr. Traver been a photographer? I know he has. Then why would I give the girl his address? But she might have thought he was a photographer. Miss Ingram, I don't wish to argue. Mr. Traver is our biggest client, and he's waiting for those proofs. Will you please get his address so I can send them to him? Yes, sir, I will. The Tower Agency is sending us the proofs this afternoon, so you'll have a chance to look over the whole setup. Sounds fine to me, Mr. Traver. Good. And if there's anything further you want to talk about, I'll be in my office till lunchtime. Goodbye. Goodbye. How do you do? What the... how did you get in here? You are Mr. James Traver, no? I am Mr. James Traver, yes. But I'd still like to know... Where can I change my clothes, please? Huh? My clothes. I'm ready to change my clothes. Oh, are you? Well, you don't want me in this dress. The man said so. I'll take it right off. Oh, no, wait a minute. Wait a minute. Would you mind telling me just what this is all about, please? About? Yes. What's all this for? Oh, for $2.75 an hour. $2.75? Is that too much? Oh, no, no. Go right ahead. It's worth that to see what happens next.  Let me see. Yeah, that's me, all right. Oh, then where are the draps? The what? The draps. You have the draps. Well, I don't know. I had the measles. What are the draps like? Listen, where is your camera? Cam... oh, my camera. Now, that's a long, sad story. I lost it.  Now, that's where you've got me. I am a photographer, no. What are you? I'm a model from Towers. A model from Towers? Now, that's very interesting. I think I'll just give them a ring. Oh, no, no, please. Don't call them up. I... I'm not supposed to be here. Uh-huh. I am from Towers, but they didn't send me. They said about the draps and I come here. All of which sounds very, very fishy. Oh, Sanders. Yes, sir? Ask Mr. Humphrey and Mr. Charles to step in here. Yes, sir. I think we ought to have witnesses to this. Something's phony. Oh, no. Nothing is funny. I... I just wanted to make a little money. That's all. Yes, that's what I thought. Uh, come in, gentlemen. What's the matter, Mr. Trevor? This girl forced her way into my office. She wants to know where she can change her clothes. Doesn't this look a little haywire to you? Uh-huh. It's a good thing you sent for us, Mr. Trevor. It certainly is haywire. What does this mean, haywire? I'm a model for $2.75 an hour. Oh, it's undoubtedly a bargain, but you see, we can't use it. Oh, vous êtes des gouttants. Vous êtes tous affreux.  You pigs!  Well, gentlemen, undoubtedly, there's a moral in this somewhere. So will you please go back to your offices and think about it? If you find out what it is, let me know. I run from the office and I'm so mad. And then I come back here and still have no job. Come on, drink your tea. You'll feel better in a minute. But I must have a job. The lonely dissay she will lock me out if I don't pay her tomorrow. The old war horse. But don't you worry. I still have a couple of bucks you can have. You? Oh, no, I can't. What do you mean? You gotta have a place to sleep, such as it is. But why are you always so good to me? Well, I suppose you gotta be good to something and my dog died the week before you moved in here. I must get a job tomorrow. Listen, Nicole, I've been telling you for months there's only one job for a woman and that's marriage. Who'd marry me? And there's nothing the matter with you that a nice rich husband couldn't cure. Did you marry a rich husband? Me? I married a hoofer. A hoofer? Dancer. Oh. Great guy, heaven rest his soul. When I married him, I turned down a millionaire. Every time I think of it, I want to cut my throat. But you love the hoofer. Yes, my luck. We used to go to Ryzenwebbers every night after the show. I could have had anybody there from a millionaire down to the bus boy. There was a funny kid, that bus boy. Say, he's at the Grand Savoy right now. He's the head waiter. He'll give you a job. Oh, you think so? I know so, Michael. Do anything I tell him and we're going to tell him tonight. Tonight? Gloria, I do not believe it. How are you, Mike? It is not you, but it is you. We're looking so good. Come into the kitchen where we can talk. Who is this? Your daughter? How do you do? No, Mike, she's just a friend. Nicole, this is Mike. How do you do? It is an honor. Mike, I know you're busy, but this young lady wants a job. A job? But I couldn't put anybody on right now. What about season? Yeah, so I see. What do they do in the good season? Hang from the chandelier? And when it gets better, I can't put her on either because I won't be here myself. I am going to open my own restaurant. You are? When? Very soon. I have found just the place. But not the price? No. I have $3,000, but I need money to renovate. That's great. I'm sure you'll get the rest of it. And when I open my own business, I may be able to put the girl on. I'll address you for Mrs. Bitten. One moment, I will fix it myself. I'll be right back, Gloria. Excuse me, please. Sure. Well... That's too bad, isn't it? It would have been so nice to work here. Mm-hmm. But look at those girls in there in the dining room. That's where you belong, not in the kitchen. If I only looked like you do and know what I know now, I'd be in there brushing caviar off a mink coat this minute. You think I could? Could be like those girls? Sure you could. All you need is clothes, a decent place to live, a chance to meet the right people, and... Yeah. You know, about $3,000 to land you a swell guy. $3,000? Yeah. I think I got an idea. Mike, come here. Yeah. Listen, Mike, you need some more money for that restaurant, don't you? Another $3,000, that is all. You want to be my partner. I'd like to talk about it a little bit. Wonderful, wonderful. We will meet later tonight and we will talk, eh? No, I'll talk. You'll listen. We'll not do it, no. But I tell you, we can rent the clothes. Give her a big build-up. We'll move right into this hotel so you can keep your eye on the restaurant. Oh, it is crazy. It is insane. Look at the work, Mike. Look at Nicole. Imagine what she'll look like dressed up.  And when she marries a millionaire, you'll get your $3,000 back, plus another $3,000 for your restaurant. Think of it. Your own restaurant might live better of it's grills. No, it is too much of a gambler. I am not the gambler. I am... Do you really think it would work? This way, please. Your sweetman, Moselle, the best in the hotel. Oh, thank you. It looks wonderful. How many rooms, manager? Eight, madam, and three baths. Yes, I think we'll be comfortable here. If there's anything Madame requires in the way of special service, we will be only too happy to arrange for it. Good morning, Madame. Will you have lunch and served in your room? Yes. Thank you, Mike. I mean, thank you, my good man. If you need a maid to help you unpack, just form the desk clerk. I think my niece and I can manage. Thank you. Well, remember, we are all at your service at all times for all things. I thought he was going to stay for breakfast. Isn't it wonderful? Gloria, Mike, look at my coat. It's like a dream. It costs $30 a day to rent that coat. That's not a dream. It's a nightmare. Let's get on with it. Let's get unpacked. Mike, I'm worried to pick this suite. Because Bill Duncan lives across the hall. Who's Bill Duncan? He has $10 million. $10 million? Well, we'll dedicate this little blue number to him then.  Oh, it's so wonderful. So beautiful. I can't wait.  Yes. And while she's growing up, it's costing me $60 a day for this suite. That's a nickel a minute. And a penny and a half every time you draw your breath. So hold it. Don't make jokes at these prices, please. Listen, if you lose a cent on this deal, I'll give you my right eye. And I'll take it. Stop worrying. Tell me about this Duncan man. He's got $10 million. Isn't that enough? Maybe. How does Nicole meet him? That is up to you, not me, but make it soon. Yeah. Tomorrow morning, I think, let's see. Nicole meets him in the hall by the elevator. She thinks she's an old friend. A natural mistake. She speaks to him. He speaks to him. You all right now, Mr. Duncan? Thank you. Charlie, Charlie. Attendez-moi. Attendez-moi. It's Nicole. I beg your pardon. Oh, Charlie, qu'est-ce que vous faites ici? Y'a un temps. Faut que je ne vous ai pas vu. Quelle chance de vous rencontrer. Comment allez-vous? I'm sorry, but I think there's some mistake. Charlie, why don't you speak to me in French? Well, principally because I don't speak it. Charlie, are you sick? I don't think so. Are you not Charlie Dumare that I met in Paris four years ago? I wasn't in Paris four years ago. Oh, you are not Charlie. I'm so sorry. My name is Bill Duncan, but please don't be sorry. Oh, but I am. I've made such a big mistake. Excuse me, please. Oh, no, wait. Do you live here, in the hotel? Yes, I live in Suite 1460. Oh, then I'll see you again. Please. Excuse me. Oh, but wait. Wait. Listen, couldn't we just... Nicole, it worked. You think so? I know so, darling. It'll be the same old story in the same old settings. Lunch, one day, dinner, the next, races, hockey games, nightclubs. Oh, I like that. It doesn't mean a thing. But if he invites you to the symphony and sits you in the family box, then, my dear, that's love. Do you like it, Nicole? Oh, it's wonderful. Hey, Bill. Bill, what? Who's calling me? It's Bill. It's me over here. Jim! Why, you all? Why, who is it? An old pal of mine. I haven't seen him since New Year's. Bill, I'll see you outside. Okay. By any means. Nicole, will you excuse me? I'll be right back. Of course. Nicole, what's going on? Gloria, that man Bill just went out to see. It's Mr. James Draper. Who's he? Oh, Gloria. He's the man I told you about. The man in the office. Oh, my goodness.  I got troubles on my own. How've you been? Great. It's good to see you, Jim. Who are you with? A customer. Well, get rid of him and join us. I wish I could, but he has a big contract in his pocket. Oh, forget the contract. Come on. I can't, Bill. It's business. Who are you with? A girl. Jim, she's really something. Two diplomats tried to blow their brains out last year because she wouldn't marry them. Sounds interesting. Interesting. She's marvelous. Daughter of a baron. Her Aunt Gloria told me so. Oh, she has an aunt, huh? All nice girls have aunts. Well, never mind the aunt. When do I meet the girl? Uh, wait. You still like Brunettes? Sure. That's fine. Nicole's a blonde. You can meet her. I can't wait. Oh, you won't have to. There she is. Nicole! Nicole! What's she going that way for? Nicole, here! Wait, Nicole! Here? Nicole, I want you to meet an old friend of mine. Jim Trevor. How do you do? Well, how do you do? I'm charmed. I was afraid of that. Yes, too bad you can't make it for supper with us, Jim. Don't believe a word he says, ma'am Zell. I wouldn't miss it for the world. Too near the music, Nicole? Oh, no. This is wonderful. I think so, too. Tell me, how long have you been in New York, ma'am Zell? In New York? Why? I, uh, just a week. A week? I should have said that you'd been here longer than that. Oh, she fooled everybody, Mr. Trevor. I'm sure she does. That reminds me. I don't know why, but the funniest thing happened to me about three weeks ago in my office. A young lady called and got herself into my office. Would you, young people, like the doll? Oh, yes, yes. It would be a great pleasure if I may. Oh, but, uh, you don't mind, do you, Bill? Uh, well, no, I guess not. Thanks. Now, ma'am Zell, suppose you tell me all about it. About it? About what, Mr. Trevor? I think you know everything. The grand Savoy, the big play for Bill, the clothes, all this stuff. Stuff? Now, please, don't stall. I know there's monkey business going on, but what I'm sure of is that my friend Bill isn't going to be the monkey. Monkey? I don't understand monkey. I'm sorry. I'm sorry, too, but you're going to understand it. Come on outside while we can talk. Again, I am sorry, but I have nothing to talk about with you. No? Then suppose we go back to the table and I'll tell Bill how you were in my office three weeks ago posing as a model. Oh, we will go outside. Good. This way. All right, then I'll do the talking. The whole act is to impress Bill that you're a nice girl from a nice family. You want to marry him, then divorce him in a few months and live happily ever after on the alimony. Right? That's a very dishonest, cheap idea. Shall I go on? Yes, I can took it. Fine. I can dish it out, too.  Tell him the truth, because if you don't, I will. You don't know anything about me. I know enough. Three weeks ago, you were in my office with some other kind of a racket. When he knows about that little episode, you're finy. Who tells him? You or me? I will tell him. When? Tomorrow. Oh, no. You'll tell him right now while I'm here. I started this and I want to be in at the finish. All right. You see the finish. They're coming now. All right. I tried to do a good job. Hello. Remember us? Hey, Jim, I just loaned to Nicole for a dance. I didn't give it to you for keeps. Oh, Bill, please forgive me. But we have been talking and shall I tell him now, Mr. Travo? Sure. Go ahead. Tell me what? His story. Mr. Travo wants me to tell you a story. Nicole, maybe you better tell it to auntie first. Oh, no. I can't tell it to everybody. He wants me to tell you that I once went to his office. Is that right, Mr. Travo? That's right. And he was taking photograph. And I was to be the model to pose with draps for a calendar. That's it. He said it will be a big joke. Don't you think it's funny, Bill? Very funny. Excruciatingly funny. You don't see why it's so funny. I'm not a model, and I never pose for one picture. And he's not a photographer. Everybody knows this. But anyhow, Mr. Travo says, if I tell you this joke, it takes me out to dinner tomorrow night. Oh, so that's it. What are you trying to do? Oh, that's what you said. No, if I tell Bill this story, you and I, we can go to dinner tomorrow all by ourselves. Why, you little... I beg your pardon, Mr. Travo. Bill, I never said that. She's... It's all right, Jim. I get it. But I tell you this... I'll forget it. It's okay, I tell you. Now, wait a minute. Come on, Nicole. I'll take you home. Thank you. I'm so sorry, Mr. Travo, but you see, I could not lie to Bill. He's my good friend. I could not make him the monkey. Good night. After a short intermission, Mr. DeMille and our stars Tyron Power and Annabella will return in act two of The Rage of Manhattan. Now, here's Libby Collins, our Hollywood news reporter and super sleuther. Where have you been sleuthing today, Libby? Well, I've been over at Paramount, but it doesn't take much sleuthing to find out what they think of a certain young star over there. Over at Paramount? Mm-hmm.  Is she blonde or brunette? Both. What? I said, she's both blonde and brunette. Her studio claims she's the only blonde brunette in Hollywood. You see, she has brunette coloring, but casting directors say she has a distinctly blonde personality. She's very vivacious and she's, oh, but you know Patricia Morrison as well as I do. Patricia Morrison? Why, of course, I should have guessed. Well, I know just three things about her. I know she's a fine little actress because I saw her in Rangers of Fortune. I know she's a striking beauty. And I know she's a luxe girl, too. I was just getting around to that. As a matter of fact, Patricia's used luxe toilets over a long time. She says her daily active leather facials help her skin stay just right. Nice and smooth and soft and camera clear, as she calls it. Well, that's fine to be. And now before I tell our listeners exactly how to take a luxe soap active leather facial, is there anything more you want to tell us about Patricia Morrison? Well, I thought maybe you'd like to know that in addition to her acting ability, she has another talent. She's an excellent fashion designer and fashion designs in a spare time. Sort of a hobby, you know. But she hasn't much spare time. She's very busy now making another Paramount picture the roundup. Oh, yes. I'm looking forward to seeing you, Libby. And now about the luxe soap active leather facial that Hollywood stars and starlets use as a daily complexion care. Here's all you do. Just pat the gentle luxe soap leather lightly in, rinse with warm water, a dash of cool, and then pat your skin dry. Active leather has removed every trace of dust, dirt, and stale cosmetics. Now your skin feels wonderfully smooth and soft. Take Hollywood's tip. Try active leather facials for 30 days. See if you don't agree they're the best care you ever found for your skin. Now, our producer, Mr. DeMille, act two of the Rage of Manhattan starring Tyrone Power as Jim Trevor and Annabella as Nicole. It's later the same evening. Determined to protect his friend from the designing Nicole, Jim Trevor has settled himself comfortably in Bill Duncan's apartment. He's still there reading a newspaper when Bill returns. Hello. Hello. Hello. What are you doing here? Just reading the newspaper.  That was a nice trick you pulled tonight, Jim. That's the reason I'm here. Bill, there's something phony about that, young midi. She's not what you think she is at all. Oh, get a new line, Jim. No, I'm serious. Yeah, you were serious in Bermuda, too. Remember, Sonia? Same story. But not the same girl. And then I pulled it on you in Miami. Yeah, what was her name? Uh, Eloise. But Bill, this time let's be sensible. My idea, exactly. Oh, I don't blame you for trying, but it won't work. All right, be smart. But that girl's a phony, and I'm going to prove it to you if it takes me a month. Well, you don't mind if I get a little sleep in the meantime, do you?  Not tonight. I'm too sleepy. Good night, Jim. Okay. But I'm not through yet. Good night, sucker. This Mike with a fur coat. Wait a minute. Where is she? Where is she? What's the matter? Look, her fur coat, she left it downstairs. Does she think fur coats grow on trees? I'm so sorry, Mike. I forgot it. Forgot it. She forgot it. Mike, please calm yourself. We could be very proud of our little girl. She saved all of us tonight. You were wonderful, Nicole. Yes, wonderful. Well, don't cry about it. What's the matter with you? That's the first time anyone ever said anything like that to me before. Said what? Mr. Trevor. He said I am cheap. Well, you paid him back for that. He's got some nerve. Cheap. I figured out just now what it cost tonight. $87.50 for rentals. Cheap. What does he want you to wear? Radium? He meant I was dishonest. All women are dishonest. If they weren't, there'd be just two classes of people in the world. Old maids and bachelors.  Of course. It is so simple. When I put parsley on a steak, does anybody ever eat the parsley? No, it is for sure. What you are doing, Nicole, is just putting parsley on a steak. You are the steak and the clothes and all the other stuff is just the parsley. Exactly. Oh, please, stop talking about it. Let me alone. No, Luke, I am a partner in this. Leave her alone. You're the silent partner. Leave her alone. Somebody tells us she's cheap and right away she gets unreasonable, but can I get unreasonable? No. Now, Mike, everything's going to be all right. Sure, it's going to be great. I started out to get a restaurant. I'll be lucky if I wind up with a sandwich.  Hello? Oh, excuse me. Why, yes, just a moment. Because it's him, that travel guy. I'll speak to him. Hello? Hello? What do you want? I was thinking about that dinner you said I wanted to take. I don't have dinner with you. How are you? It's seven sharp. I won't come. I don't like you. I'm sorry. Yes, sir. I want everything to be just right tonight. It will be, sir. I assure you. Thank you, Wrigley. Mr. Trevor, sir, the young lady you're expecting for dinner. Her name is Miss Day Cotillon, isn't it? Nicole de Cotillon, yes. Oh, sir. I saw this item in the evening paper. The Madden Manhattan column, sir. Oh, let me see. It might explain things in the event that she doesn't come, sir. The Duncan clan is gathering tonight at the Grand Savoy to meet young Bill's new heartbeat, Nicole de Cotillon. If they put the stamp of approval on her, as your correspondent did when he saw her in a night spot recently, there will soon be a formal announcement of their engagement. But you don't have to worry about that, Wrigley. She'll be here all right. It was her command invitation. Yes, sir. But if I might be permitted a suggestion, sir. Well? It's rather a dangerous business, sir, interfering between a man and a maid. Yes, Wrigley. I know I'm in trouble without your telling me. There was a certain affair along those lines in my own family. As a matter of fact, it's the way my second cousin lost his right eye. Oh, you don't say. I know you and Mr. Duncan are great friends, sir, but it was my second cousin's brother who knocked out my second cousin's eye, sir. Well, I'll have to take that chance, Wrigley. Yes, sir. I hope you forgive my mention of the incident affecting one-eyed Wrigley, sir. Oh, there she is. Let her in. Good evening, miss. I think Mr. Tiber expects me. Indeed I do. Hello. Hello. So nice of you to come. Well, and you told me to. But some little girls don't always do what they're told. Where's Bill this evening? He's with his mother and father and some friends. They are waiting for me. Oh, yes. Yes, the reception. I think that is a word. So you see, I'm in a hurry. Tell me what you want me for, and then I will go, please. Dinner is served, sir. There's your answer. Dinner. I don't want to eat dinner. Now, look.  Now, sit down. I have a lot of nasty things to say to you, and all I want you to do is eat and listen. I'm waiting. Why am I here? You may pour the wine, Wrigley. Yes, sir. Well, this is pleasant, isn't it? You know, you look very nice by candlelight. Some people might even think you were beautiful. I wonder how many men have told you that? One too many. But I never said you were. You said it. I didn't, but it's true. You dress with judgment. Your manners are, well, fair. You have the sort of eyes that some men like. Of course you drink your wine too fast, but I'm sure no one notices the little flaws when they look in the general direction of perfection. You said it, Mr. Traver. Your English is rather limited, but if you never opened your mouth, if you never did anything but just sit there like that, you'd still be perfect, except that the tip of your nose is shiny. Will you please explain what I am doing here? All right. I thought there might be a chance for us to find a nice, friendly way to call off hostilities. Pardon? End our little battle. Provided, of course, that there is some way I can make you give up this idea about Bill. I'm sorry. Now, you're not going to tell me that you're madly in love with him. I'm going to tell you nothing.  I don't think so. Oh, yes, you are. Because you know if you told me you were in love with Bill, I wouldn't believe you. But by refusing to answer, you get credit for being honest. Bill is very lucky. To get you? No. To have a friend like you who fights so hard for him. Come on now. Won't you be regular and drop this whole thing? But to do this, I mean money. About $3,000. I thought so. $3,000, eh? Oh, but... Do you hear that, Wrigley? Do you want me to hear it, sir? I do. I did. Good. And remember it when I tell Mr. Duncan tonight. Oh, no, no. You can't do this. I didn't want you to give it to me. This is a trick. It isn't fair. I'm sorry, ma'am Zell, but the age of chivalry is dead. When I come back, we'll go to the funeral. Where are you going? To see Bill. Like to come with me? You, you. I'll see him myself. Come back here. Wrigley. Yes, sir. And get mine, too. We're going to a reception. Mr. Trevor, I don't like this, sir.  You just have to tell the truth. That's all. But I can't seem to forget about my second cousin, sir. Jim. Hello, Bill. Well, this is a surprise. It's always a surprise when somebody that's not invited shows up, isn't it? Oh, I'm sorry, Jim. But I didn't think you'd want to come. I, I think I've met this gentleman, but I, I can't quite place you. You remember good old Wrigley? Oh, yes. How do you do, Mr. Wrigley? How do you do, sir? My butler. Yeah. Oh, your butler. Very nice idea. Make yourself comfortable, Wrigley. I'm very uncomfortable, sir. Go ahead, Wrigley. Speak your little piece, and we'll get out of here. Well, sir. Well, well, you see... Speak up. Speak up. Well, it was a... Mr. Trevor, I can't do it, sir. All right, then. I can. A few minutes ago, Bill, your fiancé was up in my apartment trying to hook me for $3,000. You asked for it, Jim. For what? This. Ah! Sorry, Jim, but I had to do it.  Maybe you'd better leave. Sure. Good night, all. I'm terribly sorry. I'll bet you are. Oh, please come back. I'll explain to everybody what you meant. No thanks. Oh, but I'm so sorry, and I want to help you. Yeah? Get in the car. What? Come on, get in. Out! What are you doing? Let me out. I have to go back there. I don't want to go with you. Let me out, I tell you. Take me back. Take you back when I get good and ready.  You are a... You are a... A beast is the word. Beast?  Well, I don't stay in the car with a beast. If you don't like it here, jump out. We're only doing 40. Oh, you want me to hurt myself? I won't do that. Oh, please. Just as a favor to me. Oh, look out where you are driving. Look out! Hang on. This is when you don't behave. We almost hit that truck. Are you all right? Yes, I... You... I can't talk. Good. It's the best news I've had yet. Will you tell me now where you are going to take me? Keep still and you'll soon find out. I don't care because when I get there, I get back. Sure. How? I hitch hitch. You what? Hitch hitch. Oh, you mean hitchhike. Oh, it's what I said. Now, the only thing that goes by the place we're going to is the milk truck. And that's at four o'clock in the morning. You've got a lot of practice waiting for it. By the way, I suppose you know the accepted hitch hitch gesture is made with the thumb. Like this. Ouch!  Sure. This is nothing. Just wait. I make a lot of trouble for you. Is that so? Well, I can took it. Here's where we get out. What place is this? My little gray home in the mountains. Get out. I won't. All right. Don't. What is that? Don't be afraid. That's the caretaker. Used to be a hog-calling champion. What is a hog?  I could say something, but I don't. Hello, pups. It's me, Jimmy. Jimmy, let me have a look at you. Gosh, but you're looking fine. I feel great. Say, you ain't been here in so long. I was beginning to commenced to forget what you look like. What are you doing up in this lake of the woods, son? It's a long story, pups. Well, it ought to be worth hearing. Later, pups. Sure. I'll open the house and light the fire so it'll be nice and warm for you.  I'm going to sit down and strike. No. Then I've got to get you a coat. I don't want a coat.  Then what will you do with a buddy? I'll show you. Now, here. You're going in here. You're going to like it. Oh, don't. Put me down. Put me down. Stop kicking. Stop it. Put me down. Oh, sure. There you are. Well, well, well. Married, did you mean? Pups, I'd like to. What did you say? Carrying the bride across the threshold, eh? Well, I always said when you get spliced, you'd pick out a lullapalooza. What is it, lullapalooza? You call me. Pups, I just...  Oh, can't you speak English? Yes, speaking English, but it's also nonsense. I hope you'll be as happy as two jade birds in a cherry tree. And I'm mighty glad to see somebody as pretty as you in the family at last. Pardon? Now, Pops, you've got this all wrong. Let me explain. What is this? He thinks we're married. Oh. Oh. Then why don't we tell him the truth? I have every intention of telling him the truth. I think we should tell him. Now listen, you... I didn't need to be told. I knew it the minute I seen him carry over the doorstep. It's the custom in these parts. Yes, I know it's the custom up here, Pops, but it's also the custom...  Jimmy! You see, Pops, it's a secret, but if you have to know, you have to know. Oh, it's a secret, huh? Well, I won't tell nobody. I'll have something for you to eat and do shakes of a lamb's tail. Well, now you've fixed it. I didn't tell him. He told me, and I said, yes, Jimmy. Stop calling me Jimmy. If you like, I'll call you Mr. Trevor, but I think it sounds funny for me, Bride. You're not a Bride, and you're not going to be one. That's why you're here. And you're going to stay here until you make Bill understand what you are and what you did. And if I stay here and I don't say we are married, I get a bad reputation. And I get a bad reputation if you do. Then take me back. No. I don't care what happens, you're staying here. Okay. I told you, didn't I? I make an awful lot of trouble yet. Just wait. In just a moment, Mr. DeMille and our stars Tyrone Power and Annabella will bring us Act 3 of the Rage of Manhattan. Well, what have we here? Or whom have we here, I should say? Well, you remember us, Mr. Rueick. We're the Lux Toilet Soap Symbols. Why, of course. For a moment, I thought you were just six pretty girls. Oh, very pretty girls. Oh, no, Mr. Rueick. We're symbols. We stand for six Lux Toilet Soap qualities. I'm its mildness. That means Lux Toilet Soap cares gently for even delicate skin. You ought to ask each of us what quality we stand for, Mr. Rueick. Well, you then all dressed up in that lovely white satin dress. What quality do you stand for? Well, you just practically said it yourself, Mr. Rueick. I'm Lux Toilet Soap's whiteness, a quality that makes Lux Toilet Soap very popular with choosy women. Oh, I see. Well, who's next? I ought to be next, Mr. Rueick, because I stand for Lux Soap's purity, and that's a very important quality. Only the finest ingredients are used in making Lux Soap, Mr. Rueick. Yes, that's right. And now with the next one... I stand for Lux Soap's... Just a moment, girls. Please, one at a time. We'll take that tall blonde girl next. Oh, I stand for Active Lather. Lux Soap's Active Lather that does such a thorough job. And now the little dark girl. I stand for Lux Soap's perfume, the nice, delicate fragrance that makes it such a wonderful bath soap, too. And now last, but not a bit least, I assure you. Oh, thank you, Mr. Rueick. I stand for the long-lasting quality of Lux Soap. You know, Mr. Rueick, Lux Soap is as hard milled as the finest French Soaps are. And that means it's economical to buy, because it lasts and lasts. Well, thank you. Thank all of you. Thank you. Good night. It's been a pleasure to have you here, and I assure you the best-looking set of symbols any complexion soap could ask for. Thank you, and good night. Good night, Mr. Rueick. We pause now for station identification. This is the Columbia Broadcasting System. The curtain rises on the third act of the Rage of Manhattan. The hour has passed. In the lonely cabin, Jim and his fair prisoner sit before a cheerful fire. Nicole, gazing into the flames, is secretly amused about something. Jim hears a sound suspiciously like laughter. What's so funny? You might tell me what the joke is, you know. I was just thinking of something. It's very amusing. What is? You wouldn't understand. No. Here's your dinner, folks. Just country ham and country cheese. By the moment we're coming up, I'd have shot a couple of rabbits. This'll do nicely, pops. When you want me to clear up, just ring the bell. Where's the bell? Now, son, you know we ain't got any bell. Just give out with the old suey. Okay, pops, thanks. Say, Miss, aren't you going to eat something? No, thank you. Girls nowadays don't eat any more on a hummingbird. I remember when I was young. That's all, pops. Thanks a lot. Oh, won't be alone, eh? All right, son. Wait, Mr. Pops. Yes, ma'am. Mr. Pops, where is the telephone, please? Telephone? Well, I'll tell you. Go to the head of the stairs out there and turn right. You'll see a little door on the wall about so big. No, about so big. Phone's in there. Oh, thank you, Mr. Pops. But they're connected. Oh, but can't you make it connected? I can if you want. Oh, yes, I do. All right. Take about 10 days. I'll write the company in the morning. 10 days.  Have some cheese. No. I'm very particular. About what you eat? About who I eat with. Charming girl. Who is this man? What man? In the picture up there. A funny-looking one. Oh, the funny-looking one. That's my father. I thought so. Who is the other one? My grandfather. He's funny-looking, too. Yes, he is, isn't he? I never noticed. Why do they all have guns? They were hunting. Oh, hunters. Yes, I come from a long line of hunters. I think you come from a long line of sourpusses.  Thank you very much. Look, if you let me in on the joke, I can laugh with more feeling. It's no joke for you, but it's a big joke for me. I've just been thinking what happens when Bill Dakin comes here. Oh, is he coming here? Oh, sure. He comes right in this door, and he says, you ask for it, Jim, bang, and it hits you right on the nose. And you fall over in that direction. Oh, pardon me. I better move the wood box so you don't cut on your head. You think it'll happen just that way? Why not? It happened that way before. It happens that way again. You like the idea? I think that will be a lot of fun. Thanks. No, I'm going to bed. What? I'm going to bed. I've had a rather tiring day. Good night. Wait, I cannot stay here. You'll find everything you need in that room over there. The drama's in the dresser, slippers in the closet. Now, good night. I want to go back to New York. Go ahead. Write down that road and turn to your left. It's 86 miles, and it'll take you four days on foot. If you don't take me back, I'll scream. Oh! That's not bad. Try it again. Oh! Hear any answer? Hmm. No. Not too bad. I guess 86 miles is too far to yell, even for you. Good night. What is it? Just a minute. It's you. Oh! Were you asleep? No. No, I was doing a fan dance. I'm very sorry. I know. You're afraid in that great big room.  I'm not afraid. Well, I am. Go away. I cannot get the window open in my room. A likely story. It's very simple.  But I have to lift up. Nothing happens. Then go to sleep. But I cannot go to sleep if the window is not up. I need fresh air. All right. All right. I'll get you some fresh air. Come on. This is a terrible house. The fresh air is all on the outside. There is no fresh air on the inside. Which window do you want open? Oh! How easy. Such a strong man. Well, is that all? That is all. Thank you. Good night. Oh, wait. How can I get a glass of water, please? A glass of water? Mm-hmm. Oh, that's practically impossible to get up here in the mountains. Because you see, you have to go all the way over to the table here. And take the carafe between the thumb and the forefinger. Remove the top. Lay it gently aside. Lift the glass. And, uh... Oh. It's empty, isn't it? Yes. That's what I meant. Yes. I'll get you some. Hold on, brother. I'm not first anymore. Good. Anything else you want, will you kindly tell me now or forever after?  No, thank you. I'm going to look out the window if you don't mind. No, no. That all comes with the dinner. Pardon? Nothing at all. Go ahead. Look out the window. Do anything you please. The country is nice. Everything is so quiet. I'm going to like it here while we are waiting for Bill Duncan to come and knock you down. It is so lovely. No, keep quiet. Good night. Ah! What's the matter now? All right. Take it easy now. Oh! Oh, my back! How did you ever do that? You did it, slamming doors. I was looking out the window and it fell on me. I'm helping all over. Get out of here and let me alone. No, it's all right. Just sit down quietly. It hurts me. Look. Is it bleeding? No, it's all right. Just kind of red. The blood is red? No, no. The back. Don't make such a fuss.   Just a cramp where the wind came down on the muscle. Try to straighten up. There. You think it's not broken? It's not even bent.  Up, up. Now, how's that? Well, I think me and my back feel better. Thank you. Now, go to bed. Yes, please. And remember in the future that bad girls who do bad things always get hurt. But I didn't do anything. I know, I know. It was my fault and I'm glad you aren't hurt. Good night. Good night. Oh, Mr. Trever. Yes? I have something to tell you. Yes? You won't have to worry. I shall not marry Bill Duncan. What? No. Good. That's all I wanted to hear you say. I know. I don't want to do anything you don't want me to do. You see, I rather you like me. That's all. Why?  Well, I like you too, I think. You mean that? I never say things I don't mean. Really? Well, just tell me one thing. Yes? How in the world did you find out that I had more money than Bill Duncan? Oh, did I? I don't want to see any more.   I'm sorry. I shouldn't have said that. Do you want to slide on? More. Well, good night then. I hope this will be for the last time. Mr. Strava. Yes? I would like to tell you something. What is it? You think I'm very terrible, don't you? Well, I don't know. What do you think? I think so too. But you see, I was all alone and I had nothing. And Gloria told me that the best thing for a girl is a good marriage. And the head waiter, Mike, he gave her the money to meet a nice rich man. And I met Bill, not so. I see. Well, you know, I may be old-fashioned, but it always seemed to me that a girl should be in love with a man she marries. Oh, yes, but when I met Bill, I didn't know anything about love. And now you do. Well, yes. Are you really in love with Bill? No. But you said that I don't get it. No, I know you don't. It doesn't matter now. Good night, please. Good morning, son. How are you this fine morning? I'm tired and hungry. I'll take care of that.  Fine. We had a rather hectic night. Oh, I slept like a log, didn't hear any of it. I'm a little deaf. Bring lots of coffee, will you, pops? Sure, lots of coffee, ham and eggs and some biscuits. Hurry it up. Ham and eggs all right for you too? Who are you yelling to? Mrs. She's not in that room. She's over here. Come on. Time to get started. She don't answer. Maybe she's mad at you.  Hey, come out. Come out or I'll come in. All right then. I told you to hurry up. Hey. What's the matter, son? She isn't here. She's gone. Say, I didn't see your car outside. You suppose she went somewhere with it? I guess so. Well, what are you looking so glum about? She ain't lost. I'm afraid she is, pops. As far as I'm concerned. Where have you been, Nicole? Yes, where? Where? Where? I'm sorry, Gloria and Mike, please. Where were you? Mr. Traybury, he took me.  Why did you do it? Why? Oh, Mike, I don't know. I just run out. You ran out on me. You ran out on Bill Duncan. You ran me out of a restaurant. You just ran out. Now, Mike, listen. I did listen. I listened to you. I listened to her. I listened to everybody. Now you listen to me, both of you. I have something to say to you. Something to say to Duncan. Something to say to everybody. I... Oh, what is to you? Nicole, what are you going to do? I don't know. Feel bad? No. It was all my idea, and everything's my fault. Oh, no. Yes, it was. No. I couldn't do what I promised I would do. I guess you have to figure on every woman doing that. Falling in love, I mean. You're in love with Trevor. I hate him. I knew it. He's a beast. Try and forget him, honey. Yes. It's going to hurt an awful lot for an awful long time if you don't. I feel sorry for Mike, too. I pay every penny back to him. I get a job. I do any kind of work. Sure. And we'll make a deal to pay him so much a week. Just to keep him from going crazy. Hello, Gloria. I am back. Oh, Nicole, my little chicken. You are crying. Do not cry. Why cry? Hey, what happened to you? It is all over but the shouting. Well, that's life. You take a chance. You draw a blank. So what? It's finally got him. Me? Who knows? I am a gambler, so I lose. So what? Do I lose my life? Do I lose my health? Do I lose my good looks? No. I just lose my money. And after all, what's money? Nothing. Nothing at all. It's a good thing they don't have pockets in the straight jacket. Sit down and cool off, Mike. I better go to see Bill Duncan and tell him. Why see Bill Duncan? That is finished. Bear it. Forget all about Bill Duncan. I just saw him and he understands everything. He is a very understanding man. There. Nicole, do not be sad. Go on a vacation. Go to Bermuda on a nice boat. Meet a nice man and marry him. Go to Bermuda? Yes, I would like that. Any place. Yeah, but it's a long swim. Oh, what does it cost to go to Bermuda? A few pennies, poof. There is a boat at noon. Here is a hundred dollars. Right. There is something awful funny here. What is funny? I just told you. Mr. Duncan is a very understanding man. Wrigley, answer that. Yes, sir. Hello, Wrigley. Where's Mr. Trevor? Why, I... Jim. Hey, Jim. Now, wait a minute, Bill. Before we do anything rash, let's each count ten. One, two, three. Here's the greatest pal a fellow ever had. Shake. Huh?  Everything you suspected is true. Why, if I'd married her, the family would have cut me off without a penny. Then you don't want to marry her? Marry her? I just paid five thousand dollars to that waiter. He put up the money for the whole thing. You're not going to marry her then. No. But it's going to cost you more than that. You kidnap her. She's got you good. They'll probably take you for a hundred thousand at least. You're sure that you don't want to marry her? Oh, stop talking about me marrying her. Don't worry about the money. I'll pay it. You won't have to. Wrigley. Yes, sir? Get me out another suit. Quick. Where are you going? I'm going to marry her myself. Wait! Wrigley, I can't let her marry that girl. I have to stop this. Before you do that, sir, permit me to tell you of an incident in which my second cousin, one-eyed Wrigley, was involved. All aboard. All aboard. I'm sorry. Pardon me. Are you Miss Nicola Codion? Yes. Will you please follow me? Why? Get us your bag. Yes. I'll take it alone. Oh, no, no. Right here, Miss. This is your straight room. This? Oh, but it's too big. Mine is... All right in, Miss.  May I change my shirt here? Please, what? Where are the draps? Oh, stop it. Pardon? Oh, please. Don't make fun of me now. Well, I'm not making fun of you. Then why do you do all this? Well, you wouldn't want me to get married without changing my shirt, would you? Married? Sure. Wrigley's outfixing you. And pretty soon, you're going to be the uncertain Mrs. James Trevor. Oh, but you always think I'm so bad at... I'll tell you a little secret. I'm pretty bad myself. Oh, but... Oh, James. Nicola. Mr. Trevor, the captain will marry you in 15 minutes. Oh. I'll go see if I can hurry it up, sir. I don't know how the rest of you feel about it, but this has been one of the most charming evenings I've ever spent in the Luxorator. For two reasons. And here they are, Tyrone Power and Annabella. Thank you, Mr. DeMille. And thanks too for making our first visit to the Luxorator theater so pleasant. This may be your first visit, but it won't be your last. You performed like a radio veteran, Annabella. Yes, I think she did, too, but then maybe I'm a little prejudiced. You'd better be. You know, Tyrone really is a radio veteran, Mr. DeMille. He was a radio actor before he was in pictures. He seemed on pretty familiar terms with that microphone. Oh, it's great training for an actor, Mr. DeMille, especially a job I once had in Chicago. Oh, yes. Tell Mr. DeMille about that. Well, I was one of those fellows who read the funny papers on a children's program, and that was really a chance for an actor to run the gamut of emotions, even before he had Superman. I'm glad you think it's good training, Tyrone. You see, I read the comics to my grandchildren every week. It's the only chance I ever get to act. Well, it's only fair for Emma Nounoud as much about the profession as you do to get one chance to act himself. Oh, by the way, Mr. DeMille, it's a credit to your knowledge of our profession that you say every now and then on this program that a screen star just has to have a nice complexion. That's why so many of us use lux soap all the time. That's a fine tribute, Annabella. You know, I started using lux soap several years ago. Even before I came to Hollywood, it's certainly fine complexion care, and I wouldn't know what to do without it. Fortunately, fortunately, nobody needs to do without it, Annabella. Lux soap is just as handy as the corner store. What's doing around here next week, Mr. DeMille? Next Monday night, Tyrone, there's something very important going on here. To be exact, Loretta is the director of the film to be exact, Loretta Young, a Brian Dunlevy and an all-star cast in the play Jezebel. A hit on Broadway and later on the screen, Jezebel is the drummer of a woman who makes a great sacrifice, who rises from a trivial selfish life to a climax of heroic idealism, a magnificent story and a great cast, headed by Loretta Young and Brian Dunlevy. Oh, with a cast like that in Jezebel, your audience has a guarantee of an exciting evening, Mr. DeMille. Oh, good night. Good night. Good night.  Our sponsors, the makers of Lux Toilet Soap, join me in inviting you to be with us again next Monday night when the Lux Radio Theatre presents Loretta Young and Brian Dunlevy in Jezebel. This is Cecil B. DeMille saying good night to you from Hollywood. Heard in tonight's play that Kristen Poppin as Bill, Ruth Ricketts as Gloria, Drew Merrill as Mike, J. Arthur Young as Wrigley and Charles Seal as Pops. The picture from which the Rage of Manhattan was adapted was produced by Universal Studios, whose latest release is Backstreet with Charles Boyer and Margaret Sullivan. Our music is directed by Louis Silvers and your announcer has been Melville Roy. This is the Columbia Broadcasting System.


Okay, so today we're going to wrap up the Bellman equation and the main concern that I want to present to you today is the problem that we are going to have a policy that depends on a gain times an estimate of state. That's x hat and in our derivation so far we've been using x as if it was x hat. So when we found the value associated with the optimal policy we showed it to be quadratic in x.  So that's not quite right meaning that our policy depends on both x hat and this value function which we showed to be quadratic in x. So we need to take care of things a little bit with that regard which we'll do that today. What we're going to see is that the value of the policy is going to become a quadratic function when we consider a commands u to be a function of x hat rather than x but that it becomes a quadratic function of both x and the difference between x and x hat. So the value function is going to be dependent on x hat and that's going to be its component of our derivation today. The second thing we're going to see is what to do when we have signal dependent noise. So the signal dependent noise condition is going to make it so that our policy g is going to depend on the signal dependent noises. Basically the larger the noise is going to be the smaller the gains associated with our feedback loop is going to be and it also makes it so that the u that we generate is going to depend on the size of this feedback gain k associated with how we're going to estimate the state of the system. So the main thing that I want to come back to is that in the Bellman equation what we're doing is we are finding for the last time step the optimum action then we're finding the value associated with that policy and if it's a closed form solution that we have then we're going to get a function. If it's a game that we're working on then we're going to get a value for each state. So some nonlinear function of state. Then what happens is that in the step after that we're going to find the optimum action that minimizes the sum of the cost per step plus the value of the state that we're going to go to assuming that from then on we're going to have the optimum action. So the value function is really the optimum way to the minimum cost that we're going to incur given that we find ourselves in a particular state. That's what the value function refers to and what the value function computes is iteratively to give you a value associated with a particular state. How good is this state and of course really value here is like a cost because the smaller the value the better that that state is and why is it better because from that state on if you produce the optimum commands you're going to have the minimum accumulated cost which is that value function. Okay, so do we get a sense of what the Bellman equation is? So as it's applied to linear dynamical systems it's only interesting because it becomes a quadratic function of state. The value function becomes a quadratic function. That's really what that is. So even if it's not quadratic whatever function it is as long as we have a representation of it in state space we can use it. So that's just a mathematically convenient way of representing the value function. In your homework you didn't have a value function that was quadratic it was something else. But nevertheless if you have enough memory space you can just represent it as it is. So that's fine. With regard to today's lecture we're going to add to it the concept that well the value function depends not just on the state but also depends on your estimate of state because you really don't know the state. At any given time x, at any given time k you have an estimate of the state but you don't really have state. So well how good is the value of a state if you have an estimate of state x hat? So we're going to see how to incorporate that into our system. Okay, so let me start. As a reminder we will begin with our cost per step alpha and we write that there we go. Our cost step depends on state x and input u. So our policy at time point p will be to minimize this alpha at time point p for state x of p. So we find the u that minimizes that state and the value of this policy at time point p is just going to be alpha at time point p x of p and pi star of x of p. So we just compute the time per step for whatever this is just a general representation of it for this particular alpha you could have a particular pi but this is just a general way of writing our policy for the final time step. Then our policy for a step behind the final time point is going to be one that minimizes alpha at time point p minus one plus the expected value of the value function at time point p given that we were at p minus one and executed action u of p minus one. And if we find this particular policy then the value of that policy at time point p minus one is going to be equal to alpha of p minus one at x of p minus one and given the policy x of p minus one plus the value plus the expected value of producing that action. So what we saw on Wednesday was that when we produce the optimum policy then what we get is that u at time point k is equal to some gain times x hat of k. And if we place this policy inside the value function then the value function for the optimum policy at x of k is going to be equal to some quadratic function plus some constant. But that was assuming that u is equal to g times x. So in reality u is not equal to g times x. In reality u is equal to g times x hat. And so what that's going to cause is that well when we put this u as a function of x hat into this equation we're going to have to show that the value function is still quadratic function and that we can continue using our procedures. So what we're going to see is that if we allow this to be as a function of x hat then of course what we're going to have to know is that what is x hat? How does x hat depend on things that you know we can measure? Well x hat of k given k is for a dynamical system that we typically have b times so our typical estimate of what x hat is going to be at time point k is our prior estimate for what it was plus the difference between what we observe and what we predicted times this thing that we call the common gain. So if we have for example a system of the following form x of k plus 1 is equal to a x of k plus b u of k plus epsilon u plus epsilon x and y of k is equal to c, let me write it like this, h times x of k plus epsilon s plus epsilon y where I'm going to introduce signal dependent noise in both the action and in the sensors so epsilon u is going to be equal to c1 u1 phi 1 c2 u2 phi 2 as many as u's that we have so u is here u here is a vector u1 u2 is the components of that vector so that's how we're going to introduce signal dependent noise phi is just some scalar random variable with a mean zero variance one epsilon s similarly is going to be another random variable that looks like this. Yeah so remember when we do signal dependent noise what we do is that we say we say that the standard deviation of that signal grows as a function of the signal u standard deviation of u grows as the expected value of u with a slope that's is equal to c. So what I'm writing there is that if I have a random variable that is of the following form so if epsilon u is normally distributed with mean that is u and variance that is c squared times u squared then a standard deviation of epsilon u and this is the mean of epsilon u right so standard deviation is going to grow linearly c times u with a slope c yeah so the way I wrote it here is that epsilon is a random variable and you see that its mean is is is is u times phi which is zero but its variance is u squared times c squared so it has a it's it's it's a random variable with mean zero and a variance that grows as the size of the signal and that variance grows quadratically which means the standard deviation of it grows linearly. So epsilon in this case is a vector that depends on the vector u and the slope of each of the elements of this vector is defined by this variable c and what this means is that the variance of epsilon u so it looks like this so I can write epsilon u as follows some of this matrix c i times the vector u times the random variable phi sub i and what what this means is that c i is this matrix that for example c 1 this is c 1 then it's 0 everywhere else it's all 0 one element is one element is c 1 and then c 2 is 0 c 2 then 0 0 0 everywhere else and so I've written epsilon u which is a vector as a matrix times a vector times a scalar and what that means is that the variance of epsilon u is going to be equal to the sum of c i u u transpose c i transpose it's mean is zero its variance depends on the square of u sure yes yes yes yes yes the individual slopes right so signal dependent noise for a vector you can have independent slope for each component of the vector and there's no covariance between the elements so phi 1 is no has no covariance with phi 2 they're independent and similarly with x so you can have signal dependent noise on the things that you measure and so epsilon s is going to be equal to the sum of this matrix d times x times phi i and so the variance of epsilon s is going to be the sum of d i x x transpose d i transpose what this is saying is that for this system the variance of your observation is going to depend on this which we normally have the variance of epsilon y this is your usual random variable with variance q y sorry q y and q x the usual variances that we have in our state equation and our update equation and our measurement equation but this has a variance that depends on x x of in this case x of k so the bigger the state the bigger the noise is going to be the bigger you the bit bigger the input that you have the bigger your uncertainty is going to be on measuring state on estimating state this variance here okay so our problem remains how to incorporate x hat into the into our into our problem all right so what I have is that my x hat of k given k is a times x hat of k given k minus 1 plus k of k common common gain times y of k minus what's my y hat my y hat is going to be h times x hat of k given k minus 1 right the expected value of the observation equation the expected value of y of k the y hat is just the expected value of h times x because these other terms expected value of y hat is just h times x because the expected value of this is 0 and this is 0 and then my x hat of k plus 1 given k is is going to be according to that equation it's going to be oh I'm sorry there's no a here so my my prior estimate for the next trial is going to be my post my posterior from the previous trial x hat of k given k times a plus b times u of k which is my input that I've given to the system so that's that's how I'm going to move things forward which is going to be equal to a times x hat of k k minus 1 plus a times k of k y of k minus h x hat of k k minus 1 plus b u of k so what I'd wrote is that my x hat at time point k plus 1 is going to be equal to a times my x hat at time point k plus a times k of k y of k minus h x hat of k plus b u of k so what I'm calling my prior estimate this is my prior right that's my estimate on trial k for any trial k my estimate of state is my prior so my estimate of state on trial k plus 1 is my estimate of state on trial k plus my correction with the common gain plus the input that I gave on u of k that's the expected value of my state in time k plus 1 is that clear do you see why see what I did if you didn't if you don't just raise your head I'll go over it again all I've done is use the common gain to estimate state at time point my posterior here's my posterior right is my posterior time point k and I say okay what's my prior on the next step my prior on the next step is my posterior from the previous step plus the input that I gave that becomes my prior in time point k plus 1 so then all I've done is that I've written this equation in terms of this equation which is this so this is my estimate of state at time k plus 1 as a function of estimated state in time point k no no it's the same it's exactly the same you haven't done it no we haven't done it no no we haven't done it the reason why I need to write this is because I'm going to say g times x hat of k so what is x hat of k this is what it is x hat of k is a times x hat of k minus 1 blah blah blah right so why is this important because the value function look at the value function the value function is in terms of x of p given x of p minus 1 and u of p minus 1 right and you see that x hat also depends on the previous time points so we just have a way to write what x hat is all right so let's begin at the last time point and minimize and find our optimum optimum command so time point p so for time point p right so my optimum policy at x of p is one that minimizes that function which is going to be equal to zero right so that the thing that minimizes x of p alpha p is just going to be zero and the value function for that policy at x of p is going to be equal to x of p transpose t of p x of p so what's the value function of x of p given that I'm at x of p minus 1 and I've given command u of p minus 1 why do I need to know this why am I writing this yeah because to go to move to time point p minus 1 I need to know what my value function is for time point p given that I'm at some other state and so what I'm going to do is I'm going to write that out to see what that what that what that shape looks like for the particular dynamics that I have and I have a x of p minus 1 plus b u of p minus 1 plus epsilon u plus epsilon x transpose times t of p times this thing again right so this this is just repeated here that's what that is so what's the expected value of this function so what's the expected value of this well the expected value of this quadratic equation is the expected value of this transpose expected value so let's write that down what's expected value it's going to be a x of p minus 1 plus b u of p minus 1 this expected value is zero this expected value is zero transpose t of p times this plus what I have now is that I have to consider the trace of t of p times the variance of this this variance here variance of epsilon u that's our random variable and I have a variance of epsilon x the other random variable so let's begin with the easy one variance of epsilon x see what is that this is going to be this term here I'm going to write it out here so trace of t of p times the variance of this term this term here let's see what that is so the you're going to have the trace of t of p times variance of this term epsilon x which is qx plus the variance of epsilon u times b which is going to be the sum of b times d sub i sorry c sub i u u of p minus 1 u of p minus 1 transpose c of i transpose b transpose right so this is a scalar variable this is a trace so that's just a scalar and so this is going to be equal to the trace of t of p times q of x plus the trace of t of p times this is it's just going to be a scalar variable which means that I can rewrite this by bringing the u's on the outside just doing enough doing a transpose on the scalar scalar quantity I'm going to get u of p1 transpose c i transpose b transpose t of p times b c i put this times the sum of c i transpose b transpose t of p c i b u of p minus 1 and c is a symmetric matrix anyway so it's transpose doesn't mean anything so okay so I'm going to call this quantity here cx at time point p so then my expected value is going to be ax p minus 1 plus b u of p minus 1 transpose t of p plus u of p minus 1 transpose cx of p u of p minus 1 plus the trace of t of p q of x okay so all we've done so far is introduce the concept of signal dependent noise and when we introduce signal dependent noise in our in our equation then the expected value of the the value function the optimal value function here depends on depends on not just depends on this this c of x which has these noises in it so you see that if you have signal dependent noise then the value function is going to depend on your command u scaled by the noises that influence you we didn't have that before so when we don't have signal dependent noise then the value function doesn't depend on the these commands the noises in these commands now that when we have it the expected value the expected value of this of this function here it has these epsilon u's in it where it depends on it depends on you so the big difference between having signal dependent noise and not having signal dependent noise is that having signal dependent noise makes it so that the value function depends on you okay all right so what this means is that when we put when we put in what the u is if it depends on x x hat then it's going to change our value function so okay so what's our next step so this is the expected value of u all right sorry the expected value of under the optimum policy so what we do is that we say the optimum policy at x of p at x of p minus one is going to be the argument of u of alpha of p minus one plus the expected value of v pi star of x of p given that you are at x of p minus one and you did act action u of p minus one all right so and this is alpha p minus one is going to be x of p minus one transpose t of p minus one x of p minus one plus u of p minus one t l u of p minus one plus this expected value is is over there what's so funny yeah yeah yeah but we're going to simplify it that's what what's nice so we're going to write it as a as a quadratic function of x and u so this is x of p minus one transpose times t of p minus one plus I haven't so if you look at this I have x of p minus one x of p minus one a times t so this is a times t of p times a that's all I have times x of p minus one right yep okay that's what I have all right so now I can write it as a quadratic function of u u of p minus one transpose I get c of x plus I have u times b times t here so b times t of p all right so I've taken care of all the squared u's and x's and now I get one one of the interaction terms two times let's do the u so let's do two times u of p minus one t times bt t of p times a x of p minus one and plus the trace of tp qx so my oh this is sorry this is argument of this right so we're going to minimize this to get the the the policy and it's a quadratic function of x and u quadratic function of x quadratic function of u and this interaction term all right so we minimize this with respect to x with respect to u of p minus one and what do I get I get I get the squared term in u which is two times cx plus b transpose tp times b plus l times u of p minus one so what I'm doing I'm finding derivative of this with respect to d u of p minus one plus two times b transpose t of p a x of p minus one so I'm equal to zero so then I get u of p minus one is equal to minus cx plus b transpose t of p b plus l minus times b transpose t of p a times x of p minus one and this is called my gain g at time point p minus one okay and my problem is that I don't know x all I have is x hat so I found for my time point p minus one my policy and here it is and let's look at it for a second it depends on l which is the motor cost obviously the greater your motor cost the smaller the smaller u you're going to produce and it depends on cx what is cx cx is the noise the signal dependent noise on the on the u's bigger the noise is the smaller your gain going to be so that kind of makes sense yeah yeah yeah at this point I'm just finding the policy right so I'm saying at time point XP show me what's your best policy and that's the one that minimizes this right and so because because my v of my value function I pieced I for XP is is just you know it's just x there is no x hat here yeah it's kind now it's gonna now it's gonna make a difference so because if the time point p didn't make a difference but time point p1 is gonna make a difference because the time point p g was zero so didn't matter so the value function didn't depend on x hat it just depend on on x but the value function under the optimal policy for p minus one is going to depend on x hat because you depends on x hat and time point p you didn't you you was just zero so it didn't matter okay so so then right so the important thing is you is equal to minus g of p minus one times x hat of p minus one all right so now we have to put this back to this function all right so now you're going to get g times x hat g times x hat g times x hat so your value function no longer is going to be quadratic in x it's gonna have x hats in it that's the problem we face well not well no because remember we can't we're not going to replace this with with x hat we're only going to replace this with x hat you depends on x hat x doesn't depend on x hat the actual state is independent of what our estimate is it doesn't matter what we can still compute is the value function for any x and any x hat so whatever the true state is we can find the value function for x and x hat you'll see you'll see yes and and x hat it's a function of both yeah yeah we'll see that this value function so so so the value under the optimal policy is this quantity where you is equal to minus g times x hat right so all I can do right now is to say v under the optimal policy for x of p minus one and x hat of p minus one is equal to x of p minus one transpose times this so I'm calling I'm going to call it w of p minus one x times x of p minus one plus this u which is going to be this term here it's going to be x hat of p minus one transpose times g p minus one transpose times minus because this u is not going to have minus g times x hat minus two times x hat of p minus one transpose g of p minus one transpose b t t p a x of p minus one plus the trace of t of p q x so we have some strangeness going on here so we have x we have x hat and we have x hat times x we can do a little bit of simplification so what is g let's look at g g is this quantity inverse times this right so that's exactly this quantity right so this quantity times g is going to cancel so this term here is going to be plus x hat of p minus one transpose g of p minus one transpose this times g this is going to fall out this is going to remain b t t of p a times x hat of p minus one so this term becomes this okay so that's that's kind of nice because here's what we got we have if I call this z you see that it appears here as well right so I have x hat transpose z x hat minus two x hat transpose z x well that's kind of nice why is that nice because x hat transpose z x that's the first term minus two times x hat z x that's a second term right that's going to be something that I can simplify that's x minus x hat transpose z x minus x hat minus x hat z x sorry x x transpose z x so this value function can be written as a quadratic function of x which is what we've always done plus a new term here which I'm going to call error in estimating x how far is the true x away from my estimate of x so the value function under the optimum policy at time point x of p minus one it's going to depend on both x and x hat and it's going to be x of p minus one transpose times t of p minus one plus a transpose t of p a x of p minus one plus yeah so the last term just gets into here so what we see is that the value function becomes a quadratic function of x and the error in estimating x and basically this matrix we're going to call w x at time point p minus one and this matrix we're going to call w e at time point p minus one so it's a quadratic function so in 2005 Todorov his work was based on showing that if you have signal dependent noise in the state update equation like we have now and if you incorporate that into the Bellman equation then what you end up with is a representation of the value function that is quadratic equation but in terms of x and the error in estimation of x so remember how we proceeded last time what we said is that if we can write the value function in terms of a quadratic equation then when we apply the optimum commands and you know we see that the value function is quadratic the next step is going to be just a formula that basically says if we apply this we still going to get a quadratic function on the next trial so that the proof in in basically in your in your book and in what Todorov did in 2005 was to show that for whatever k that you have whatever time point k that you have will basically write it down for you so if you have if you have a time point k let's say time point k plus one you have the value function under the optimum policy at x of k plus one and x hat of k plus one equal to x of k plus one wx of k plus one transpose x of k plus one plus e of k plus one transpose w e k plus one e of k plus one plus a constant so if that's the value function and if you were to apply u equal to the time point k you apply u equal to minus g of k times x hat of k then what happens is that the value function at time point k and x hat of k is going to be equal to a once again at quadratic function of x and the error so any time point if you begin you can write the value function under the optimum policy as a quadratic function of x and error in estimation of x for the next time point you can find a minimum of the sum of the alpha plus the value function and it's going to be a linear function of x hat and when you apply that policy the value function associated with that policy is again going to be a quadratic function of x and the error so that's just like before except now instead of having a value function that's only depends on x we also have a value function that depends on the error in estimating estimating this now yeah so that that's basically it we don't really have any anything I think other than the idea that the value function being a function of x and an x hat now x hat of course depends on k right the common gain so technically this becomes a problem because remember that what happens is that when you do estimation you go forward in time you say I have computed all the common gains I need from beginning to the end right because you can do that you can say at my first time point this is going to be my first common gain my second my time point is going to be my second so forth in in this approach the case are going to influence these ease which are going to become part of the value function so what what Todorov showed is that case and the value function they converge after a few iterations so you run this forward in time you get the case you get the value function backwards in time and then you run it again and you run again and it to the to converge and I did it in your book for the condition where you're moving the eye and the head just to check to see and it and it works quite well it converges after four or five runs through the system question on the x state which is running forward so cx depends on the noise but if you have like state dependent like you said that that signal dependent right in x yes so that means that like you need to know where x is in order to compute cx which means you have to run forward iteration of the model to compute x is to get cx so that you can run backwards iteration with the Bellman equation like it seems like that doesn't that seems weird yeah yeah no you're right you're right because because until you get to that state you don't know what the feedback gain is going to be right so the feedback gain is going to depend on not just x hat but the value of the so it's won't it would be a the noises are going to influence the size of the feedback gain because that's going to affect c so this is this is what effects so cx is the size of you and then we're going to have a variable that also depends on the size of x which is going to depend on d which which let's see where where is the scenario equation it's going to depend on it's going to influence k the common gain and and then that's indirectly going to influence e that you have there so so it's an excellent point basically you can see that optimum u so let's see what is where do we do g up there it is right so you see that g depends on cx right but cx depends on you but this this quantity here is ci right and the variance of it depends on the size of the size of the noise so just just to be clear it isn't the actual value of you that matters what matters is how the noise grows with you so c doesn't depend on you right it just depends on it's just a slope of that line is it's not circular all right this is where we are with the bellman equation and signal dependent noise that's as far as control theory as far as optimal control and signal dependent noise is gone and it's like I said it's about now nine years old this this derivation how does that representation of the noise when you're trying to compute like yeah yeah yeah yeah yeah yeah with common estimation and that's all that would just give you the process so how does this representation of the value function in terms of the error right actually help in terms of like like the control theory what's going on because you never have access to the actual underlying x but it seems like it's kind of a little bit so it's important because when we put you into the into this equation when we say when we say the value of the policy is going to be the cost per step plus the expected value of this well this depends on you but you depends on x hat right so this expected value which is what we just did we found was to be a quadratic function of x and e so the importance of this is that it shows us that no matter what step we're at if we have the value function associated with optimal policy and we minimize the Bellman equation we end up with another value function that still is quadratic and so what this means is that in your in your recipe you're going to see that wx of k depends on this it's just going to be some you know transformation of wx we is going to be some transformation of k and g is going to be a function that depends on wx and we okay so basically it just the fact that that transforms quadratic value functions in the quadratic yeah it really allows you to have that same linear form yeah yeah yeah that's right yeah so before when we didn't have x hat the all that we have was on the left side we just had this right and so when we did the policy that was optimum we still got a value function was quadratic in x so the thing was it just consistent with itself the only thing that was new today was that well my you depends on x hat not depends on x but to handle that we have to introduce this concept of an error in state and as long as we do that we get a value function that's quadratic in error in state