1 st Tutorial on EG4321/EG7040 Nonlinear Control

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1 1 st Tutorial on EG4321/EG7040 Nonlinear Control Introduction to State-Space Concepts Dr Angeliki Lekka 1 1 Control Systems Research Group Department of Engineering, University of Leicester February 9, 2017 Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

2 Structure of the Tutorial 1 General Info 2 Revision on Classical Control Concepts 3 Revision of State-Space Concepts Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

3 General Info Personal Background 2006: BSc in Electronics (final year Control project) 2008: MSc in Embedded Systems & Control 2015: PhD in Nonlinear & Anti-windup Control 2016: PostDoc in Adaptive & Nonlinear Control Dr Angeliki Lekka Introduction to State-Space February 9, / 13

4 General Info Purpose of the Tutorials Revision of concepts introduced/mentioned in lectures Examples solving Informal setting/opportunity to interrupt and ask questions Dr Angeliki Lekka Introduction to State-Space February 9, / 13

5 Revision on Classical Control Concepts Eigenvalues Why do we care about the eigenvalues of a system? Their values determine whether a system is stable or not Different system output response wrt the location of its eigenvalues How can we find the eigenvalues of a system? 1 roots of the denominator polynomial of a transfer function 2 the characteristic polynomial of a matrix A must be zero, eg det(λi A) = 0 How can stability of a system be determined wrt to its eigenvalues? 1 roots of the denominator of a transfer function located on the open LHP 2 each eigenvalue of a matrix should have a negative real part Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

6 Revision on Classical Control Concepts System Response wrt to Eigenvalue Location Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

7 Revision on Classical Control Concepts Finding the Poles of a Transfer Function Example Suppose a linear system is described by the following transfer function: G(s) = 1 s 2 + 5s + 6 We need to derive the roots of the denominator polynomial: (1) s 2 + 5s + 6 = 0 (2) One way is to use the quadratic equation s 1,2 = β± β 2 4αγ 2α So, s 1 = 2 and s 2 = 3 and the system described by Equation 1 is stable Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

8 Revision of State-Space Concepts State Space - Notation A system written in state-space form: where and x = [x 1, x 2,, x n ], u = [u 1, u 2,, u m ], y = [y 1, y 2,, y p ], A R n n : state matrix B R n m : input matrix C R p n : output matrix ẋ = Ax + Bu y = Cx + Du x R n : system state vector u R m : control input vector y R p : system output vector D R p m : direct transition (feed-through) matrix Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

9 Revision of State-Space Concepts State Space - Notation Advantages of this representation include: It is very compact - even large systems can be represented by two simple equations All systems are represented by the same notation - very easy to develop general techniques to solve them Computers can easily simulate 1 st -order equations - easy implementation in Matlab/Simulink Study of MIMO systems - not possible with TF Study/analysis of nonlinear systems Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

10 Revision of State-Space Concepts Finding the Eigenvalues of a Matrix Example Suppose the dynamics of a linear system are described by the following state matrix A: [ 2 10 ] 0 3 To find the eigenvalues of matrix A, we need to solve the characteristic equation det(λi A) = 0 Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13 (3) [ ] det(λi A) = λ ( 2) 10 (4) 0 λ ( 3) = (λ + 2)(λ + 3) 0 10 = 0 So, λ 1 = 2 and λ 2 = 3 and the system described by matrix A is stable

11 Revision of State-Space Concepts Choice of State Variables How can we choose the state variables in a system? There is not a unique way of choosing them! We need a first order differential equation for each state variable The number of states must be integer For example, choose as state variables quantities determining energy in a system: mechanical system: choose extension of springs (potential energy) & velocity of masses (kinetic energy) electrical system: choose voltage across capacitors (electrical energy) & current through inductors (magnetic energy) Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

12 Revision of State-Space Concepts Choice of State Variables - An Example From the Equation of Motion: Mẍ + Bẋ + Kx = F a (t) (5) Choosing the states as x 1 := x and x 2 = ẋ, yields: and ẋ 1 = ẋ (6) ẋ 2 = ẍ (7) = 1 M ( Bẋ Kx + F a(t)) = 1 M ( Bx 2 Kx 1 + F a (t)) Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

13 Revision of State-Space Concepts Choice of State Variables - An Example (continued ) and if we want to express this in state space form: [ ] [ ] [ ] [ ] ẋ1 0 1 x1 0 = ẋ 2 K M B + 1 F M x a (t) (8) 2 M }{{}}{{} A B y = [ 0 1 ] [ ] x1 + [ 0 ] F }{{} x a (t) (9) 2 }{{} C D Dr Angeliki Lekka (al385@leacuk) Introduction to State-Space February 9, / 13

3 rd Tutorial on EG4321/EG7040 Nonlinear Control

3 rd Tutorial on EG4321/EG7040 Nonlinear Control Lyapunov Stability Dr Angeliki Lekka 1 1 Control Systems Research Group Department of Engineering, University of Leicester arch 9, 2017 Dr Angeliki Lekka