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linear algebra in control systems 4,597 views Share Like Download ...

Ganesh Bhat, ASIC DvDs Engineer Follow Control system… Device or a set of devices that mange, command, direct or regulate the behavior of other device or syste... Published on Mar 9, 2012

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The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. The whole control system is based on the feedback, and its type. An example… Negative feed back control system

Time- varying system: missiles with varying fuel levels or lift in an aircraft flying through varying altitude.MIMO- multiple input and multiple output:

linear algebra in control systems 1. 1. Aalok P Bhat(1rv09ec002) Ganesh V Bhat(1rv09ec027)Sujith Chandra P S(1rv09ec105) 2. 2. Control system… Device or a set of devices that mange, command, direct or regulate the behavior of other device or system. Control system can be thought of as having four functions; Measure, Compare, Compute, and Correct These functions are completed by five elements: Detector, Transducer, Transmitter, Controller, and Final Control Element. Practically a control system is implemented in embedded system using a microcontroller or a PLD’s 3. 3. Control theory… It is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. Usual objective of control theory is to calculate solutions for the proper corrective action from the controller that result in system stability. Transfer function (also known as the system function or network function) is a mathematical tool to define the relation between input and the output. 4. 4. Classification… Logical or sequential controls usually implemented using logic gates(combinational circuit) Feedback or linear controls Usually implemented using combinational circuit and flip flops(sequential circuit) Fuzzy control attempts to combine some of the design simplicity of logic with the utility of linear control. Some devices or systems are inherently not controllable. 5. 5. An example… Negative feed back control system 6. 6. Mathematical approach…y(s)=p(s)u(s)u(s)=c(s)e(s)e(s)=r(s)-f(s)y(s) C-controller P-plant F-sensor * All the transfer function(system) are assumed to by linearly time invariant 7. 7. Analysis and design of a feedbackcontrol system…Two approaches are available for the analysis and design of feedback control systems. Classical or frequency-domain approach Algebraic approach of converting a single input single output system’s differential equation into transfer function by transforming the system equation into the frequency domain equivalent called transfer function. State-space approach Linear algebraic approach of representing a multiple input multiple output system into a mathematical model 8. 8. Classical approach… It is the frequency domain approach where the mathematical tools like laplace transforms are applied Steps followed in a classical approach: System Mathematical approach…y(s)=p(s)u(s)u(s)=c(s)e(s)e(s)=r(s)-f(s)y(s) C-controller P-... equation (in time domain) is transformed into a frequency domain transfer function. Computation, simplification and analysis is done in frequency domain. The obtained result is transformed back to time domain using inverse transforms. 9. 9. Advantages: they rapidly provide stability and transient response informationDisadvantage: The primary disadvantage of the classical approach is its limited applicability: It can be applied only to linear, time-invariant systems or systems that can be approximated as such. 10. 10. State-space approach… System functions are represented in the form of matrices instead of a single system equation. It is a unified method for modeling, analyzing, and designing a wide range of systems. It can be used to represent nonlinear systems. It can handle, conveniently, systems with nonzero initial conditions. The modeling of timevarying system is easy with the help of state-space approach. 11. 11. STATE SPACE REPRESENTATION 12. 12. • In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations• To abstract from the number of inputs, outputs and states, the variables are expressed as vectors, and the differential and algebraic equations are written in matrix form 13. 13. • When the complexity of the equation is more, it is very difficult to work in time domain• So we decompose the higher-order differential equations into multiple first-order equations, and we solve them using state variables method• State space refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space 14. 14. STATE SPACE• The state-space is the vector space that consists of all the possible internal states of the system• In a state space system, the internal state of the system is explicitly accounted for by an equation known as the state equation• The system output is given in terms of a combination of the current system state, and the current system input, through the output equation• These two equations form a system of equations known collectively as state-space equations 15. 15. STATE• The state of a system is an explicit account of the values of the internal system componentsSTATE VARIABLES• Input variables : We need to define all the inputs to the system, and we need to arrange them into a vector, denoted by u(t)• Output variables : Output variables should be independent of oneanother, and only dependent on a linear combination of the inputvector and the state vector and it is denoted by y(t) 16. 16. • State Variables• State variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time•The state variables represent values from inside the system, that can change over time• In an electric circuit, for instance, the node voltages or the mesh currents can be state variables• State variable is denoted by x(t) 17. 17. State-Space Equations• In a state-space system representation, we have a system of two equations • equation for determining the state of the system x(t) = g[t0,t,x(t),x(0),u(t)] • Analysis and design of a feedbackcontrol system…Two approaches are available for the analysis and design of feedback contr... equation for determining the output of the system y(t) = h[t,x(t),u(t)] • x‘(t) = A(t)x(t) + Bu(t) • y(t) = C(t)x(t) + Du(t) 18. 18. Mattrices:A,B,C,D• Matrix A is the system matrix, and relates how the current state affects the state change x‘• Matrix B is the control matrix, and determines how the system input affects the state change• Matrix C is the output matrix, and determines the relationshipbetween the system state and the system output•Matrix D is the feed-forward matrix, and shows how the systeminput to affects the system output directly 19. 19. State-Space Basis TheoremAny system that can be described by a finite number ofnth order differential equations or nth order differenceequations, or any system that can be approximated bythem, can be described using state-space equations. Thegeneral solutions to the state-space equations, therefore, aresolutions to all such sets of equations 20. 20. Representing Systems ByState Space Approach Select a particular subset of all possible system variables and call it as state variables. For an nth-order system, write n simultaneous, first- order differential equations in terms of the state variables. We call this system of simultaneous differential equations state equations. 21. 21. We algebraically combine the state variables with the systems input to get output equation. State equations and output equations combined form a state-space representation. 22. 22. Representation of an ElectricNetwork Step 1:Identify variables in the system. Ic,Vc,Il,Vl Step 2:Select the state variables by writing the derivative equation for all energy storage elements 23. 23. Step 3:Represent other variables as linear combination of state vectors and input. Step 4:Obtain State equations. 24. 24. Step 5:Obtain output equations and represent in matrix form. 25. 25. RepresentingDifferential/difference equation inState Space Model. Consider the differential equation We choose output y and it’s n-1 derivatives as state variable.(Phase Variables). 26. 26. Representing Transfer Function toState Space we first convert the transfer function to a differential equation Then we represent the differential equation in state space in phase variable form. 27. 27. Stability of System A linear state space model is asymptotically stable if all real parts of eigenvalues of A are negative. Correspondingly, a time-discrete linear state space model is asymptotically stable if all the eigenvalues of A have a modulus smaller than one. 28. 28. Controllability state controllability condition implies that it is possible – by admissible inputs – to steer the states from any initial value to any final value within some finite time window. A continuous time-invariant linear state-space model is controllable if and only if 29. 29. Observability Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. A continuous time-invariant linear statespace model is observable if and only if Classical approach… It is the frequency domain approach where the mathematical tools like laplace transforms are applied...

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Representation of an ElectricNetwork Step 1:Identify variables in the system. Ic,Vc,Il,Vl Step 2:Select the state ...

Step 3:Represent other variables as linear combination of state vectors and input. Step 4:Obtain State equations.

Step 5:Obtain output equations and represent in matrix form.

RepresentingDifferential/difference equation inState Space Model. Consider the differential equation We choose output y ...

Representing Transfer Function toState Space we first convert the transfer function to a differential equation Then we ...

Stability of System A linear state space model is asymptotically stable if all real parts of eigenvalues of A are negati...

Controllability state controllability condition implies that it is possible – by admissible inputs – to steer the states...

Observability Observability is a measure for how well internal states of a system can be inferred by knowledge of its e...

linear algebra in control systems

linear algebra in control systems

linear algebra in control systems

linear algebra in control systems

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