This is non linear control system questions. Please give me answers.

1. Question One [15 marks total] a) An engineer considers the stability of an aircraft which has malfunctioned. The aircraft dynamics are nonlinear but the engineer obtains a linear model from these and examines the aircraft behaviour around the equilibrium point where the linear model was obtained. The engineer finds that the linear model is stable. Should we trust this assertion of stability? Justify your answer. [3 marks] b) A simple recurrent neural network is modelled using the state-space equations = o(aj|x1 +421x2 +bu) 32 o(a21x1 +422x2 + b2u) where x(t) are the network states and u(t) the input. The activation function o(.) = tanh() (hyper- bolic tangent), and aij and b; are constants. i) Prove that an equilibrium point for the neural network system is x = x2 =u=0. Marks will be awarded for clarity of working and your reasoning. [2 marks] i) Linearise the neural network around the equilibrium point identified in part ii) above, and give expressions for the linearised system's A and B matrices. Express in as simple a form as possible. You must show your working clearly. [7 marks] iii) A software engineer tasked with implementing the neural network system, decides, for simplic- ity, to change the activation function from o(.) = tanh(.), to o(.) = sign(.). Would this cause problems with linear model derivation? Explain your answer. Suggest an alternative activation function, easy to implement and less complex than tanh(.), which would not cause linearisation problems. [3 marks] 2 Question Two [15 marks total] a) Nonlinear systems arising from circuit and mechanical system analysis can often be expressed as a state-space system consisting of a linear part, plus a nonlinear perturbation. One such system is given by the following state-space equations: : = 12 { xi 2 = -2x 3x2 +h(x)x) where h(x) is some nonlinear function to be defined shortly. i) Express this system in the form *=Ax+8(x) and give expressions for the matrix A and the nonlinear function g(x). ii) Determine whether a positive definite solution exists to the Lyapunov equation [2 marks] A'P+ PA = -1 and if it does, calculate P. [6 marks] iii) The matrix P mentioned in part ii) is to be used as the basis of a Lyapunov function for the nonlinear system E. Assume that h(x) = -- 1. Using the Lyapunov function V(x) = x'Pr + 2x1 show that the nonlinear system I is globally asymptotically stable. Explain why a quadratic Lyapunov function would fail to prove stability. [7 marks] 3. Question Three [15 marks total] u a) The dynamics of a washing machine can, approximately, be captured by the state-space equations: il = -ax+ Bx2-pa 8 12 + px212 where a, B. 7, 8 and are all positive constants. The control signal u(t) is applied to provide active vibration control to the system. 1) Derive an NDI control law which linearises the second state-equation. [1 mark] Prove that the NDI control law derived in part i) can be combined with a state-feedback control law v= -k X1 - k2x2 to ensure that the washing machine dynamics are globally asympotically stable for all positive values of ki and k2. You may wish to use the Lyapunov function V(x) = [(*+n2) no [7 marks] iii) Assume that y = 0. Derive an expression for the closed-loop A-matrix of the washing machine system with the NDI control law and state-feedback applied. By appropriate choice of k, and ka, where might one place the closed-loop poles of this system? If, in addition, B = 0, does the answer change? [5 marks] iv) The dynamics of the second state-equation are more accurately modelled as 8 22 4+ ( x2) & +221- where the term g(x.x2) is a "small" but unknown function. How would this affect the suc of the NDI control law derived above? [2 marks] SS 1. Question One [15 marks total] a) An engineer considers the stability of an aircraft which has malfunctioned. The aircraft dynamics are nonlinear but the engineer obtains a linear model from these and examines the aircraft behaviour around the equilibrium point where the linear model was obtained. The engineer finds that the linear model is stable. Should we trust this assertion of stability? Justify your answer. [3 marks] b) A simple recurrent neural network is modelled using the state-space equations = o(aj|x1 +421x2 +bu) 32 o(a21x1 +422x2 + b2u) where x(t) are the network states and u(t) the input. The activation function o(.) = tanh() (hyper- bolic tangent), and aij and b; are constants. i) Prove that an equilibrium point for the neural network system is x = x2 =u=0. Marks will be awarded for clarity of working and your reasoning. [2 marks] i) Linearise the neural network around the equilibrium point identified in part ii) above, and give expressions for the linearised system's A and B matrices. Express in as simple a form as possible. You must show your working clearly. [7 marks] iii) A software engineer tasked with implementing the neural network system, decides, for simplic- ity, to change the activation function from o(.) = tanh(.), to o(.) = sign(.). Would this cause problems with linear model derivation? Explain your answer. Suggest an alternative activation function, easy to implement and less complex than tanh(.), which would not cause linearisation problems. [3 marks] 2 Question Two [15 marks total] a) Nonlinear systems arising from circuit and mechanical system analysis can often be expressed as a state-space system consisting of a linear part, plus a nonlinear perturbation. One such system is given by the following state-space equations: : = 12 { xi 2 = -2x 3x2 +h(x)x) where h(x) is some nonlinear function to be defined shortly. i) Express this system in the form *=Ax+8(x) and give expressions for the matrix A and the nonlinear function g(x). ii) Determine whether a positive definite solution exists to the Lyapunov equation [2 marks] A'P+ PA = -1 and if it does, calculate P. [6 marks] iii) The matrix P mentioned in part ii) is to be used as the basis of a Lyapunov function for the nonlinear system E. Assume that h(x) = -- 1. Using the Lyapunov function V(x) = x'Pr + 2x1 show that the nonlinear system I is globally asymptotically stable. Explain why a quadratic Lyapunov function would fail to prove stability. [7 marks] 3. Question Three [15 marks total] u a) The dynamics of a washing machine can, approximately, be captured by the state-space equations: il = -ax+ Bx2-pa 8 12 + px212 where a, B. 7, 8 and are all positive constants. The control signal u(t) is applied to provide active vibration control to the system. 1) Derive an NDI control law which linearises the second state-equation. [1 mark] Prove that the NDI control law derived in part i) can be combined with a state-feedback control law v= -k X1 - k2x2 to ensure that the washing machine dynamics are globally asympotically stable for all positive values of ki and k2. You may wish to use the Lyapunov function V(x) = [(*+n2) no [7 marks] iii) Assume that y = 0. Derive an expression for the closed-loop A-matrix of the washing machine system with the NDI control law and state-feedback applied. By appropriate choice of k, and ka, where might one place the closed-loop poles of this system? If, in addition, B = 0, does the answer change? [5 marks] iv) The dynamics of the second state-equation are more accurately modelled as 8 22 4+ ( x2) & +221- where the term g(x.x2) is a "small" but unknown function. How would this affect the suc of the NDI control law derived above? [2 marks] SS