# Linear State-Space Control Systems

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1 Linear State-Space Control Systems Prof. Kamran Iqbal College of Engineering and Information Technology University of Arkansas at Little Rock

2 Course Overview State space models of linear systems Solution to State equations Controllability and observability Stability, dynamic response Controller design via pole placement Controllers for disturbance and tracking systems Observer based compensator design Linear quadratic optimal control Kalman filters, stochastic control Linear matrix inequalities in control design Course assessment

3 Learning Objectives Formulate and solve state-variable models of linear systems Apply analytical methods of controllability, observability, and stability to system models Controller synthesis via pole placement Observer based compensator design Formulate and solve the optimal control problem Design optimal observers and Kalman filters LMI based controller design

4 Resources Core Text: Bernard Friedland, Control System Design: An Introduction to State- Space Methods, Dover Publications, ISBN: References: Professor Raymond Kwong s notes Professor Jongeun Choi s notes Professor Perry Li s notes Astrom and Murray, Feedback Systems, An Introduction for Scientists and Engineers, Princeton University Press, 2012,

5 Course Schedule Session Topic 1. State space models of linear systems 2. Solution to State equations, canonical forms 3. Controllability and observability 4. Stability and dynamic response 5. Controller design via pole placement 6. Controllers for disturbance and tracking systems 7. Observer based compensator design 8. Linear quadratic optimal control 9. Kalman filters and stochastic control 10. LM in control design

6 State-Space Models of Linear Systems

7 State-Variable Models State variables Energy variables, e.g., velocity (KE), position (PE) Alternate variables, momentum (KE) Flow and across variables, e.g., current, voltage Dynamic Equations Based on physical principles Ordinary differential equations Partial differential equations State-variable equations First order differential equation

8 Transfer Function Models Describe input-output relation Restricted to LTI systems Can be of lower order than actual system Example: Let x + 3x + 2x = u, y = x + x H s = 1 s+2

9 Example: dc Motor Electrical subsystem e v = L di dt + Ri τ = k i i Mechanical subsystem J dω dt = τ v = k ω ω Assume L = 0 k i = k ω = k

10 DC Motor Motor equation J dω dt = τ = k i(e k ω ω)/r Or dω = k2 k ω + e dt JR JR Let K2 JR = α, K JR = β Then dω dt = αω + βe State variables: θ, ω d dt θ ω = α θ ω + 0 β e

11 DC Motor State-space model Let x = θ ω Then dx dt Let y = θ = x = Ax + B u y = 1 0 θ ω = Cx Transfer function model θ s = β e s s s+α, ω s e s = β s+α

12 Example: Inverted Pendulum on Cart Let x cart displacement θ pendulum displacement f applied force Dynamic equations M + m x + ml cos θ θ mlθ 2 sin θ = f ml cos θ x + ml 2 θ mgl sin θ = 0 Linearization (θ 0, sin θ θ, cos θ 1) M + m x + mlθ = f mlx + ml 2 θ mglθ = 0

13 Inverted Pendulum on Cart State variables: x, θ, x, θ State equations: d dt x θ x θ = m g M M+m Ml g 0 0 x θ x θ M 1 Ml f Output variables: [x, θ] Output equations: y θ = y θ y θ

14 Inverted Pendulum on Motor-Driven Cart Let x cart displacement θ pendulum displacement r wheel radius Then, f = τ r, ω = x r f = k2 Rr 2 x + k Rr e Dynamic equations M + m x + mlθ + k2 Rr 2 x = k Rr e x + lθ gθ = 0

15 Inverted Pendulum on Motor-Driven Cart Solve for accelerations x θ = 1 Ml l 1 ml M + m State variables: x, θ, x, θ State equations: d dt x θ x θ = k2 Rr 2 x + k Rr e gθ m M g k2 MRr M+m Ml Or x = Ax + Bu g k 2 MRr 2 l 0 x θ x θ k MRr k MRrl e

16 Example: Two-Axis Gyro Rigid body dynamics (true in an inertial frame): dp = f, dh = τ dt dt Euler s equations for a spinning body: J x ω xb + J z J y ω yb ω zb = τ xb J y ω yb + J x J z ω xb ω zb = τ yb J z ω zb + J y J x ω xb ω yb = τ zb

17 Two-Axis Gyro Assume that z-axis is the spin axis and ω z is constant; let H z = J z ω z, (angular momentum); H = H z J x = J y = J d (diametrical moment of inertia) Dynamic Equations: 1 J d J z gyro constant ω xb ω yb + 1 J d 0 H H 0 ω xb ω yb = 1 J d Gyro equations including the spring and damping terms: τ x τ y ω xb ω + 1 B H yb J d H B ω xb ω yb 1 J d B 0 0 B ω xe ω ye + 1 J d K D K Q K Q K D δ x δ y = 1 J d τ x τ y

18 Two-Axis Gyro Angular displacements (gyro pick off) are: δx = ω xb ω xe δy = ω yb ω ye Define x = δ x, δ y, ω xb, ω yb, u = τ x τ y, x 0 = ω xe ω ye Let b 1 = B J d, b 2 = H J d, c 1 = K D J d, c 2 = K Q J d, β = 1 J d, A 1 = c 1 c 2 c 2 c 1, A 2 = b 1 b 2 b 2 b 1 Then x = 0 I A 1 A 2 x + I b 1 I x βi u Or x = Ax + Bu + Ex 0

19 Two-Axis Gyro The characteristic equation of the gyroscope is: si A = s 2 + b 1 s + c 1 2 b 2 s + c 2 2 The precession and nutation frequencies are given as: s = α p + ω p, α p = b 1c 1 b 2 c 2 b 2 1 +b2, ω p = b 2c 1 b 1 c 2 2 b 2 1 +b2 2 s = α n + ω n, α n = α p b 1, ω n = ω p + b 2 The transfer function of a free gyro is given as: δ x δ y = H(s) ω xe ω ye ; H s = s 2 +b 1 s+c 1 b 2 s+c 2 b 2 s+c 2 s 2 +b 1 s+c 1 s 2 +b 1 s+c 1 2 b 2 s+c 2 2

20 Two-Axis Gyro An ideal gyro is one with zero damping and stiffness Then H s = b 2 s b 2 s s 2 s 2 +b 1 s+c 2 1 b 2 s+c 2 2 s 2 Assume a step input ω xe = 1, ω ye = 0 δ x t = 1 b cos b 2 t δ y t = 1 b 2 t 1 b 2 sin b 2 t

21 Example: Aerodynamics Define α angle of attack, β side slip angle φ roll angle, θ pitch angle, ψ yaw angle p roll rate, q pitch rate, r yaw rate L rolling moment, M pitching moment, N yawing moment X longitudinal force, Y lateral force, Z vertical force V aircraft speed, Δu change in speed δ E elevator deflection δ A aileron deflection δ R rudder deflection

22 Aerodynamics Longitudinal motion: Let x = Δu, α, q, q ; u = δ E Then, Δu = X u Δu + X α α gθ + X E δ E α = Z u V Δu + Z α V α + q + Z E V δ E q = M u Δu + M α α + M q q + M E δ E θ = q

23 Constant-Altitude Autopilot The simplified dynamics of an aircraft at constant speed are described as: α = Z α V α + q + Z E V δ E q = M α α + M q q + M E δ E θ = q Define Δh = (h h 0 )/V Then Δh = γ = θ α

24 Aerodynamics Lateral motion: Let x = β, p, r, φ, ψ ; u = δ A, δ R Then, β = Y β V β + Y p V p + Y r V 1 r + g V φ + Y A V δ A + Y R V δ R p = L β β + L p p + L r r + L A δ A + L R δ R r = N β β + N p p + N r r + N A δ A + N R δ R φ = p ψ = r

25 Missile Dynamics Define V missile velocity α N normal acceleration θ pitch angle γ flight path angle Assume that X u 0, Z u 0, M α 0 Then α q = Z α V 1 α M α M q + q α N = Z α α + Z δ δ Zδ V M δ δ

26 Missile Guidance Define Then λ line-of-sight angle z projected miss distance V missile speed V T target speed T t = T time to go λ z = 0 1 VT λ z + 0 T a N i.e., the state equations are time varying

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