I System variables: states, inputs, outputs, & measurements. I Linear independence. I State space representation

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1 EE C28 / ME C34 Feedback Control Systems Lecture Chapter 3 Modeling in the Time Domain Lecture abstract Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley Topics covered in this presentation I System variables: states, inputs, outputs, & measurements I Linear independence I State space representation I Conversion between systems in time-, frequency-domain, TF, & state space representations September 0, 203 Bayen (EECS, UCB) Feedback Control Systems September 0, 203 / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Chapter outline 3 Introduction 35 Converting a transfer function to state space 36 Converting from state space to a transfer function 3 Introduction 35 Converting a transfer function to state space 36 Converting from state space to a transfer function Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 SS representation, [, p 9] Size of system states, inputs & outputs, [, p 22] Procedure System variables: Select a subset of all possible system variables as states and determine inputs & outputs 2 State di erential equations: Write n simultaneous, first-order DEs of the states in terms of the states and inputs for an nth-order system 3 Initial conditions: If we know the initial conditions of all the states at t 0 as well as the inputs for t t 0, we can solve the simultaneous DEs for the states for t t 0 4 Output-state relation equations: Write linear relations of the outputs in terms of the states and inputs for t t 0 5 State space (SS) representation: The state and output equations represent a viable representation of the system I States: Typically the minimum number of states required to describe a system equals the order of the system DE We can define more states than the minimum set; however, within this minimal set the states must be linearly independent (defined later) I Inputs & outputs: Single-input, single-output (SISO) systems are a unique case of general multiple-input, multiple-output (MIMO) systems The output and input of a SISO system are represented by scalar quantities The outputs and inputs of a MIMO system are represented by vector quantities Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29

2 Motivational example, [, p 20] Motivational example, [, p 20] A quick example to introduce the terminology and concept before we generalize the definition of SS representation System variables I States I Current through the RLC loop, i(t) I Capacitor charge, q(t) I Input I Voltage, v(t) I Output I Inductor voltage, v L(t) Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 2 State di erential equations I Kirchho s voltage law L di(t) + Ri(t)+ Z i(t) = v(t) C I I Charge definition i(t) = dq(t) Two simultaneous, first-order DEs dq(t) = i(t) di(t) = LC q(t) R L i(t)+ L v(t) Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Motivational example, [, p 20] Motivational example, [, p 20] 3 Initial conditions I Assume we know the initial conditions of the states at t 0 and the input for t t 0 Figure: RLC system 4 Output-state relation equations v L (t) = C q(t) Ri(t)+v(t) Figure: RLC system Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Motivational example, [, p 20] 5 SS representation apple q(t) x = i(t) ; u = v(t) apple 0 A = ; B = LC R L apple 0 L C = C R ; D = Figure: RLC system 3 Introduction 35 Converting a transfer function to state space 36 Converting from state space to a transfer function Bayen (EECS, UCB) Feedback Control Systems September 0, 203 / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29

3 Definitions, [, p 23] Definitions, [, p 23] I Linear combination: A linear combination of n variables, x i,fori = to n, is given by the following sum, S where each K i is a constant S = K n x n + K n x n + + K x I Linear independence: None of the variables can be written as a linear combination of the others Variables x i,fori =to n, aresaiobe linearly independent if their linear combination, S, equals zero only if every K i =0and no x i =0for all t>0 I System variable: Any variable that responds to an input or initial condition in a system I State: The state variables are a non-unique set of linearly independent system variables such that the values of the members of the set at time t 0 along with known inputs completely determine the value of all system variables for all t>t 0 I State vector: A vector whose elements are the states I State space: The n-dimensional space whose axes are the states A trajectory can be thought of as being mapped out by the state vector, x(t), for a range of t Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Equations, [, p 23] Equations, [, p 23] I State equation: A set of n simultaneous, first-order DEs that expresses the time derivatives of the n states of a system as linear combinations of the states and inputs x u a, a,n b, b,m 6 x = 4 5 ; u = 4 5 ; A = 4 5 ; B = x n u m a n, a n,n b n, b n,m I Output equation: An equation that expresses the measured output variables of a system as linear combinations of the states and inputs y c, c,n d, d,m 6 y = 4 5 ; C = 4 5 ; D = y p c p, c p,n d p, d p,m Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Variables & their dimensions, [, p 23] ẋ 2 R n x 2 R n u 2 R m y 2 R p A 2 R n n B 2 R m C 2 R p n D 2 R p m time derivative of state vector state vector control input vector measured output vector system matrix input matrix output matrix feedforward matrix 3 Introduction 35 Converting a transfer function to state space 36 Converting from state space to a transfer function Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29

4 Selecting the states, [, p 24] Selecting the states, [, p 24] State requirements If I The states must be linearly independent I A minimum number of states must be selected and must be su cient to describe completely the state of the system Typically the number required equals the sum of the orders of a set of DEs describing the system I too few states are selected or I a minimum number of states are selected and are linearly dependent, it may be impossible to completely express state and output equations as linear combinations of the states and inputs Notes concerning adding states to the minimal set of linear independent states I Linear independent states: These additional linear independent states are also decoupled, ie, they are not required in order to solve for any of the other linearly independent states or any other dependent system variable I Linear dependent states: The dimension of the system matrix is increased unnecessarily, adding di culty to the solution of the state vector [, Ch 4] and hindering the designer s ability to use state space methods for design [, Ch 2] Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Representing an electrical system, [, p 26] Representing a translational mechanical system, [, p 30] Example (RLC system) I Problem: Find a state-space representation in vector-matrix form if the states are the capacitor voltage, v C,anhe inductor current, i L,anhe input is the applied voltage, v, and the output is the resistor current, i R Figure: Electrical system Example (translational inertia-spring-damper system) I Problem: Find the state equations in vector-matrix form if the states are the positions, x and x 2,anheinputisthe applied force, f Figure: Translational mechanical system Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / Converting a TF to state space 35 Converting a TF to state space Phase-variable representation, [, p 32] 3 Introduction 35 Converting a transfer function to state space 36 Converting from state space to a transfer function Select a set of state variables, called phase variables, whereeach subsequent state variable is defined to be the derivative of the previous state variable d n y n + a d n y n n + + a dy + a 0y = b 0 u x = y ẋ = x 2 x 2 = dy ẋ 2 = x 3 x n = dn 2 y x n = dn d n 2 t y d n t ẋ n = x n ẋ n = a 0 x a x 2 a n x n + b 0 u Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29

5 35 Converting a TF to state space Phase-variable representation, [, p 32] 35 Converting a TF to state space Phase-variable representation, [, p 34] 2 ẋ = a 0 a a 2 a n y = 0 0 x x + u b 0 Example (arbitrary system) I Problem: Find the state-space representation in vector-matrix form for the transfer function from R(s) to C(s) Figure: a TF; b equivalent block diagram showing phase variables Note: y(t) =c(t) Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / Converting from state space to a TF 36 Converting from state space to a TF Converting from SS to a TF, [, p 39] State and output equations 3 Introduction 35 Converting a transfer function to state space 36 Converting from state space to a transfer function Laplace transform assuming zero initial conditions sx(s) =AX(s)+BU(s) Y (s) =CX(s)+DU(s) Transfer function matrix T (s) = Y (s) U(s) = C(sI A) B + D Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29 Bibliography 36 Converting from state space to a TF Norman S Nise Control Systems Engineering, 20 Bayen (EECS, UCB) Feedback Control Systems September 0, / 29

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