In The Name of Allah The Most Beneficent The Most Merciful

1. In The Name of Allah The Most Beneficent The Most Merciful

2. ECE4545:Control Systems Lecture: Mathematical Modeling of Systems-Time Domain Engr. Ijlal Haider UoL, Lahore

3. Control Talk • Classical Control • Modern Control • Continuous and Discrete • Analog and Digital (Samplers and Reconstructor) • Hybrid Systems Modern control is older!!!

4. Control Talk • Example of Hybrid System: Car Computer Most modern automobiles today have integrated computer systems, that monitor certain aspects of the car, and actually help to control the performance of the car. The speed of the car, and the rotational speed of the transmission are analog values, but a sampler converts them into digital values so the car computer can monitor them. The digital computer will then output control signals to other parts of the car, to alter analog systems such as the engine timing, the suspension, the brakes, and other parts. Because the car has both digital and analog components, it is a hybrid system.

5. Control Talk A Computer Controlled System

6. Control Talk • Feedback (is reactive) • Negative • Subtracts output from setpoint • Positive • Accumulates output with set point • E.g. to create switching behavior, system maintains a given state until threshold. **Instability** • Feedforward (is proactive) • Applied when disturbances can be predicted • Feedback vs. Feedforward: market based economy vs. a planned economy

7. Need of Another Approach Why are we interested to use another approach?? • Transfer Function has limitation: • Dealing with MIMO systems is complex • It cannot give enough information on internal stability • We need more insight into the system

8. Modern Control • The power of modern control has its roots in the fact that the it can represent a MIMO (multi-input multi-output) system as a SISO (single-input single-output) system due to the use of vectors and matrices

9. Concept of State • We identify the internal variables that drives the system dynamics • Input and internal states drives the states, states drives the output

10. Large classes of engineering, biological, social and economic systems may be represented by state-determined system models. System models constructed with the pure and ideal (linear) one-port elements (such as mass, spring and damper elements) are state-determined • Number of state variables “n” depends on independent energy storage elements • The values of the state variables at any time t specify the energy of each energy storage element within the system and therefore the total system energy, and the time derivatives of the state variables determine the rate of change of the system energy

11. Definitions • State Variable • Is the smallest set of a dynamic system such that the knowledge of these variables at t=to together with the knowledge of input for t>=to completely determines the behavior of the system

12. Definitions • State Vector • If n state variable are needed to completely describe the behavior of the system then the n-dimensional vector containing these variables is called the state vector • State Space • The n-dimensional space whose coordinate axis consists of all the states is called state space

13. How to get State Space! • Transformation from the nth order differential equation into n first order differential equations, called State Equations • Output is defined by variable of interest

14. State Equation • In state equations the time derivative of each state variable is expressed in terms of the state variables x1(t), . . . , xn(t) and the system inputs u1(t), . . . , ur(t)

15. For LTI system • In matrix form

16. Output Equation • All state variables may not necessarily be of direct engineering interest • An arbitrary output variable in a system of order n with r inputs may be written:

17. In general form • In matrix form

18. State Space Model • Dimension of matrices • For r number of inputs, m number of outputs and n number of states

19. Properties of States • States are not unique however the number of state variable is unique • It is possible to transform one set of variable to other • State variable of a systems may be formulated in terms of physical, measurable and controllable variables, or in terms of variables that are not directly measurable and controllable

20. Why State Space • Why use state-space approach?? • State variable form is a convenient way to work with complex dynamics; matrix format easy to use on computers • Transfer functions only deal with input/output behavior, while state-space form provides easy access to the internal features and response of the system • State-space approach is great for MIMO (multi-input multi-output) system, which are very hard to work with using transfer functions • State variables can be used for feedback

21. Block Diagram

22. Converting back to TF

23. Thank You!