1 00:00:00,880 --> 00:00:06,220 In this video, we are going to look at linear dynamics systems and the state based representation of 2 00:00:06,220 --> 00:00:13,030 the systems, a system can be considered a collection of interrelated entities or in our case, differential 3 00:00:13,030 --> 00:00:15,400 equations that can be considered as a whole. 4 00:00:16,760 --> 00:00:22,100 If the different processes that make up the systems change of time, then it is considered as a dynamic 5 00:00:22,100 --> 00:00:22,540 system. 6 00:00:24,170 --> 00:00:29,210 The differential equations that make up the system are called the state equations of the dynamic system. 7 00:00:29,480 --> 00:00:34,130 The state variables of the system are dependent variables of those state equations. 8 00:00:35,570 --> 00:00:41,870 So if we consider the system of time varying first order differential equations as shown on slide here, 9 00:00:42,680 --> 00:00:45,230 and we'll be using the Newton's dot notation. 10 00:00:45,230 --> 00:00:50,300 So the dot notation just means that if the variable has a dot over the top, it means we just taking 11 00:00:50,300 --> 00:00:52,950 the derivative of that variable with respect to time. 12 00:00:53,540 --> 00:00:56,670 So I can say these here are first order differential equations. 13 00:00:56,730 --> 00:01:03,170 They're a function of time of the different states x different inputs. 14 00:01:03,170 --> 00:01:10,220 You and they calculate the state rates or the rate of change of the variable or state. 15 00:01:11,210 --> 00:01:18,140 So on the slide here we have instead equations and each of these equations are going to be a time varying. 16 00:01:18,150 --> 00:01:22,660 So you can see that they're a function of time and we have nine states. 17 00:01:22,660 --> 00:01:27,740 So x1 x2 all the way up to extend all the different states of the equations. 18 00:01:27,740 --> 00:01:33,620 And you can see for every state we have a corresponding state right equation that calculates the derivative 19 00:01:33,620 --> 00:01:34,430 of that state. 20 00:01:35,420 --> 00:01:37,790 We also have an input. 21 00:01:37,790 --> 00:01:41,250 So you can see we have you want you to own up to um. 22 00:01:41,810 --> 00:01:48,530 So each of these functions here are a function of the the state's inputs and time. 23 00:01:48,530 --> 00:01:50,800 And you can see that all these are correlated with each other. 24 00:01:50,810 --> 00:01:55,760 So each of these functions can be a function of the other states themselves. 25 00:01:56,210 --> 00:02:01,700 So this is why it's a system of interrelated entities, because each function can be a function of the 26 00:02:01,700 --> 00:02:02,360 different states. 27 00:02:03,020 --> 00:02:09,620 So how all these evolve time is a function of how all the other states and inputs evolve with time. 28 00:02:10,910 --> 00:02:15,710 So the system of equations in the last slide can be completely written in this form here. 29 00:02:16,220 --> 00:02:21,950 So instead of writing out all the different functions individually, what we do is we change the representations 30 00:02:22,070 --> 00:02:24,450 so that we write everything out in vector form. 31 00:02:24,830 --> 00:02:30,390 So here we have a vector of the states and the vector of the inputs to get the vector of the derivatives. 32 00:02:30,800 --> 00:02:38,030 So here exactly is a vector of the different states you have here is a vector of the different inputs 33 00:02:38,420 --> 00:02:44,510 and then X of T here is going to be the derivative of the state vector with respect to time. 34 00:02:45,230 --> 00:02:47,510 So this is normally called the state vector. 35 00:02:48,200 --> 00:02:50,120 This is normally called the input vector. 36 00:02:51,230 --> 00:02:53,590 This derivative here is usually called the state, right? 37 00:02:53,640 --> 00:02:58,250 So it's a derivative of the states with respect to time and if function. 38 00:02:58,280 --> 00:02:59,860 Here are the state equations. 39 00:03:00,950 --> 00:03:05,960 Many of the processes that occur in the world can be considered as linear or nonlinear differential 40 00:03:05,960 --> 00:03:06,620 equations. 41 00:03:07,130 --> 00:03:12,170 We can describe these processes in state's best form, which then allows us to use different mathematical 42 00:03:12,170 --> 00:03:16,860 tools to extract useful information and perform various analysis on the system. 43 00:03:17,540 --> 00:03:22,880 So using state's best form, we've come up with a standard mathematical tool that allows us to analyze 44 00:03:22,880 --> 00:03:26,990 systems in general so we can write something in the state's best form. 45 00:03:26,990 --> 00:03:30,850 Then we can use these tools that we've developed for other systems on the system. 46 00:03:31,920 --> 00:03:37,500 If we know the state of the system for the current time and all the current and future inputs to that 47 00:03:37,500 --> 00:03:43,080 system, then we can predict values of the future states and the future outputs of that system and a 48 00:03:43,080 --> 00:03:43,610 lot more. 49 00:03:44,550 --> 00:03:49,860 So if we consider the solar system as a dynamic system, then we can use the same process on this. 50 00:03:50,910 --> 00:03:56,940 Isaac Newton's first use of calculus was to describe the differential equations of the orbit of objects 51 00:03:56,940 --> 00:03:57,970 in the solar system. 52 00:03:58,590 --> 00:04:04,230 So if we consider the solar system as a dynamic system, if you know the position of the planets for 53 00:04:04,230 --> 00:04:09,570 a given point of time, then you can predict where the planets will be at any point in the future or 54 00:04:09,570 --> 00:04:12,270 in the past by using these states based analysis. 55 00:04:13,820 --> 00:04:18,980 So this is the power of using state space allows us to use different mathematical tools and allows us 56 00:04:18,980 --> 00:04:24,800 to completely predict or estimate different values in the future or the past of that system.