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learn:courses:real-analog-chapter-9:start [2017/01/11 20:23] – Marthalearn:courses:real-analog-chapter-9:start [2022/10/04 11:38] (current) – external edit 127.0.0.1
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 ====== Real Analog: Chapter 9 ====== ====== Real Analog: Chapter 9 ======
 +[[{}/learn/courses/real-analog-chapter-8/start|Back to Chapter 8]]
 +-->Chapter 9 Materials#  
 +  * Lecture Material:
 +    * {{ :learn:courses:real-analog-chapter-9:lecture24.ppt |Lecture 24 PowerPoint Slides}}: Introduction to state-variable modeling, simulating system response using MATLAB
 +    * [[http://www.youtube.com/watch?v=2cQMO3NZz4Y&list=PLDEC730F6A8CDE318&index=27&feature=plpp_video| Lecture 24 Video]]
 +  * Chapter 9 Video:
 +    * [[http://www.youtube.com/watch?v=vl-lemzB1w8&list=PL170A01159D42313D&index=19&feature=plpp_video| Lab 9 Video 1]]: State Variable Models: The step response of the state variables of a series RLC circuit are measured. The measurements are compared to the simulated response obtained by using MATLAB.
 +  * {{ :learn:courses:real-analog-chapter-9:real-analog-chapter-9.pdf | Chapter 9 Complete PDF}}
 +    * {{ :learn:courses:real-analog-chapter-9:lab9p3p1.pdf |Lab 9.3.1}}
 +      * {{ :learn:courses:real-analog-chapter-9:lab9p3p1_worksheet.docx |Worksheet 9.3.1}}
 +    * {{ :learn:courses:real-analog-chapter-9:lab9p3p2.pdf |Lab 9.3.2}}
 +      * {{ :learn:courses:real-analog-chapter-9:lab9p3p2_worksheet.docx |Worksheet 9.3.2}}
 +  * {{ :learn:courses:real-analog-chapter-9:realanalog-exercisesolutions-chapter9.pdf |Exercise Solutions}}: Chapter 9 exercise solutions
 +  * {{ :learn:courses:real-analog-chapter-9:homework9.docx |Homework}}: Chapter 9 homework problems
 +
 +<--
 +
 ====== 9. Introduction and Chapter Objectives ====== ====== 9. Introduction and Chapter Objectives ======
 In our analysis approach of dynamic systems so far, we have defined variables which describe the energy in the circuit energy storage elements (voltages across capacitors and currents through inductors). We then used KVL and KCL to write differential equations describing the circuit, in terms of those variables. The resulting equations were then combined into a single differential equation governing the parameter in which we were interested (the //input-output equation// for the system); this equation was solved to determine the response of the circuit. This approach is useful for low-order systems, such as the first or second order systems we have examined so far, but it becomes cumbersome for higher-order systems. An alternate modeling approach, //state variable// (or //state space//) modeling, has a number of advantages over the approach we have been using to date, especially for higher-order systems. In our analysis approach of dynamic systems so far, we have defined variables which describe the energy in the circuit energy storage elements (voltages across capacitors and currents through inductors). We then used KVL and KCL to write differential equations describing the circuit, in terms of those variables. The resulting equations were then combined into a single differential equation governing the parameter in which we were interested (the //input-output equation// for the system); this equation was solved to determine the response of the circuit. This approach is useful for low-order systems, such as the first or second order systems we have examined so far, but it becomes cumbersome for higher-order systems. An alternate modeling approach, //state variable// (or //state space//) modeling, has a number of advantages over the approach we have been using to date, especially for higher-order systems.
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 \cdot \\ \cdot \\
  
-y_P=c_{P1}x_1+c_{P2}x_2+ \dots + C_{PN}x_N+d_Pu \\+y_P=c_{P1}x_1+c_{P2}x_2+ \dots + c_{PN}x_N+d_Pu \\
  
 (Eq. 9.5)$$ (Eq. 9.5)$$
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 $$ $$
-> c = [1 0; 0 1]; +> c = [1 0; 0 1]; \\ 
-> d = [0; 0]; +> d = [0; 0]; \\ 
-> sys = ss(A,b,c,d); +> sys = ss(A,b,c,d); \\ 
-> [y,t] = \text{step}(sys);+> [y,t] = \text{step}(sys); \\
 > y = 2*y; > y = 2*y;
 $$ $$
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 $$ $$
-> \text{plot}(y(1,:),y(:,2)) +> \text{plot}(y(1,:),y(:,2)) \\ 
-> \text{grid} +> \text{grid} \\ 
-> \text{xlabel}('X_1(t)') +> \text{xlabel}('X_1(t)') \\ 
-> \text{ylabel}('X_2(t)')+> \text{ylabel}('X_2(t)') \\
 > \text{title('State trajectory for Example 1')} > \text{title('State trajectory for Example 1')}
 $$ $$
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 {{ :learn:courses:real-analog-chapter-9:chapter9l.png |Example image 2.}} {{ :learn:courses:real-analog-chapter-9:chapter9l.png |Example image 2.}}
 +
 +
 +[[{}/learn/courses/real-analog-chapter-8/start|Back to Chapter 8]]
 +[[{}/learn/courses/real-analog-chapter-10/start|Go to Chapter 10]]